Fluid dynamics

Methods of Experimental Physics VOLUME 18 FLUID DYNAMICS PART 6

METHODS OF EXPERIMENTAL PHYSICS: L. Marton and C. Marton, Editors-in-Chief

Volume 18

Fluid Dynamics PART B

Edited by R. J. EMRICH Department of Physics Lehigh University Bethlehem, Pennsylvania

1981

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L i b r a r y of Conpress C a t a l o g i n g i n P u b l i c a t i o n Data Main e n t r y under t i t l e : F l u i d dynamics. (Methods o f experimental p h y s i c s ; v . 18)

1. F l u i d dynamic measurements. 2. F l u i d dynamics. I . Emrich, Raymond J a y , Date 11. S e r i e s . TA357.FL83 620.1 'U64 80-27897 1SBP.l 0-12-475956-4 ( V . 188)

PRINT1 D IN 1 H T U N I T I D STATFS OF AMtRICA XI828184

9 8 7 6 5 4 . 1 2 1

CONTENTS CONTRIBUTORS ..............................................

ix

......................... TO VOLUME18, PARTA . . .................... CONTRIBUTORS LIST OF VOLUMES I N TREATISE.. ............................

xi

OF VOLUME18, PARTA . CONTENTS

...

xiii

xv

3. Measurement of Density by Beam Absorption and Scattering

3.0. Introduction ........................................ by R. J. EMRICH

405

......................

405

3. I . Beam Attenuation Densitometry by R. J. EMRICH

3.2. Analysis of Raman and Rayleigh Scattered Radiation. by MARSHALL LAPPA N D C. MURRAY PENNEY

...

3.2.1. Introduction .................................. 3.2.2. Rayleigh Scattering ............................ 3.2.3. Raman Scattering. ............................. 3.3. Measurement of Density by Analysis of Electron Beam Excited Radiation ................................... by E. P. MUNTZ 3.3.1. High Probability Transitions for Excitation and Emission: Selection Rules ...................... 3.3.2. Equations Connecting Fluorescent Intensity to Gas Density.. ................................. 3.3.3. Density Measurements ......................... 3.3.4. Beam Generation, Spreading, and Plasma Effects.. 3.3.5. Flow Field Studies. ............................

408 408 414 418 434

438 441 450 451 453

4. Measurement of Temperature

4.1. Probe Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by W. PAULTHOMPSON V

457

vi

CONTENTS

4.1.1. Definitions of Flow Temperatures . . . . . . . . . . . . . . . 457 4.1.2. Temperature Sensors .......................... 460 4.2. Measurement of Temperature by Radiation Analysis 4.2.0. Introduction

. . . . 463

..................................

463

by N . A . GENERALOV 4.2.1. Emitted and Absorbed Radiation . . . . . . . . . . . . . . . . 465 by N.A.GENERALOV 4.2.2. Temperature Measurement by Analysis of Scattered Light ................................ 487 LAPPA N D c . MURRAY PENNEY by MARSHALL 4.2.3. Measurement of Temperature by Analysis of Electron Beam Excited Radiation . . . . . . . . . . . . . . . 489 by E . P . MUNTZ 5. Measurement of Pressure by R . I . SOLOUKHIN. C . W . CURTIS.A N D R . J . EMRICH 5.1. Introduction

.......................................

499

5.2. Gages for Measuring Constant and Slowly Varying

Pressures ..........................................

505

. . . . . . . . . . . . . . . 515 Time-Dependent Pressure Measurements: Preview . . . . . 525 Gage Characterization .............................. 527 Sensors............................................ 534 Pressure-Time Recording ........................... 552

5.3. Pressure Measurement in Moving Fluid 5.4. 5.5. 5.6. 5.7.

5.8. Dynamic Calibration

................................

555

5.9. Diaphragm Gages: Strain by Bending and Stretching . . . 559 5.10. Fast Response Gages: Compressional Strain . . . . . . . . . . .

576

6. Measurement of Composition by JOHNE . DOVE

........................................ Analysis of Sampled Fluids ...........................

6.1. Introduction

611

6.2.

616

CONTENTS

vii

6.3. Analysis of Radiation Absorbed by in Situ Fluid . . . . . . . . 630 6.4. Analysis of Radiation Emitted by in Situ Fluid . . . . . . . . . .

637

6.5. Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

645

7. Heat Transfer Gages by W . PAULTHOMPSON

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........... Instrumented Models ................................ Thin Membrane Calorimeters ......................... Thick Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thin Film Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiation Heat Transfer Gages ........................

663

7.2. One-Dimensional Heat Conduction Relations

665

7.3.

670

7.4. 7 .5 . 7.6. 7.7.

672 676 679 683

8. Light Sources and Recording Methods by M . HUGENSCHMIDT A N D K . VOLLRATH

8.1. Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

687

8.2. Recording Methods ..................................

725

9. Apparatus

9.0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by R . J . EMRICH

755

9.1. Wind Tunnels and Free Flight Facilities . . . . . . . . . . . . . . . . 756 by DANIELBERSHADER 9.1.1. 9.1.2. 9.1.3. 9.1.4. 9.1.5. 9.1.6. 9.1.7.

Overview of Wind Tunnel Systems . . . . . . . . . . . . . . Classification of Wind Tunnels . . . . . . . . . . . . . . . . . . Low Speed Tunnels ............................ Transonic and Supersonic Tunnels . . . . . . . . . . . . . . Free Flight Apparatus .......................... Hypersonic Experimentation and Facilities . . . . . . . Low Density Wind Tunnels .....................

757 758 764 771 779 781 784

...

CONTENTS

Vlll

9.2. Shock Tubes and Tunnels ............................ by DANIELBERSHADER 9.2.1. 9.2.2. 9.2.3. 9.2.4. 9.2.5.

Basic Flow Regimes ........................... Production of Strong Shock Waves . . . . . . . . . . . . . . Aero- and Thermodynamic Testing Apparatus . . . . Studies of Chemical Kinetics . . . . . . . . . . . . . . . . . . . Further Uses of Shock Tubes . . . . . . . . . . . . . . . . . . .

9.3. Low Reynolds Number Flows by DANIELBERSHADER

.......................

785 785 788 791 792 795 796

9.3.1. Features of Highly Viscous Flows . . . . . . . . . . . . . . . 798 9.3.2. Measurements with Viscous Flows . . . . . . . . . . . . . . 798 9.3.3. Measurements with Non-Newtonian Fluids . . . . . . . 800 9.4. Apparatus for Rotating Geophysical Fluid Dynamic Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by ALANJ . FALLER 9.4.1. 9.4.2. 9.4.3. 9.4.4.

Flow in Rotating Systems ...................... Rotating Apparatus ............................ Basic Methods of Generating Fluid Circulation . . . Special Observational Methods. . . . . . . . . . . . . . . . . .

801 802 806 813 817

10. Dimensional Analysis and Model Testing Principles by MAURICE HOLT

10.1. Mathematical Foundations of Dimensional Analysis . . . . 822

. . . . . . . . . . . . . . . . 828 829 10.3. Applications in Fluid Dynamics .................... 843 10.4. Model Testing Principles ............................ 10.2. Geometrical and Dynamical Similarity

AUTHORINDEX.............................................

849

SUBJECTINDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

861

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors' contributions begin.

DANIEL BERSHADER, Department of Aeronautics and Astronautics, Stanford University, Stanford, California 94305 (756)

C . W . CURTIS," Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015 (499) JOHN E. DOVE,Department of Chemistry, University of Toronto, Toronto, Canada M5S 1Al (611)

R. J. EMRICH, Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015 (405, 499, 755) ALANJ. FALLER, Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 (801) N . A. GENERALOV, Institute of Problems of Mechanics, U S S R Academy of Sciences, Moscow A-40, U S S R (463) MAURICEHOLT,Department of Mechanical Engineering, University of California, Berkeley, California 94720 (821)

M . HUGENSCHMIDT, Deutsch-Franzosisches Forschungsinstitut, SaintLouis, 7858 Weil a m Rhein, Federal Republic of Germany (687) MARSHALLLAPP,General Electric Research and Development Center, Schenectady, New York 12301 (408, 487) E. P. MUNTZ,Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90007 (434, 489) C. MURRAY PENNEY, General Electric Research and Development Center, Schenectady, New York 12301 (408, 487) R. I . SOLOUKHIN, Institute of Heat and Mass Transfer, Byelorussian Academy of Sciences, Minsk 220728, U S S R (499)

W . PAULTHOMPSON, Advanced Systems Technology Division, The Aerospace Corporation, Los Angeles, California 90009 (457, 663)

K.

VOLLRATH,Deutsch-Franzosisches Forschungsinstitut, SaintLouis, 7858 Weil a m Rhein, Federal Republic of Germany (687)

* Professor Emeritus. ix

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CONTENTS OF VOLUME 18, PART A 1. Measurement of Velocity

1 . 1 . Tracer Methods* by E. F. C. SOMERSCALES 1.2. Probe Methods for Velocity Measurement 1.2.1. Introduction by R. J. EMRICH 1.2.2. Velocity Measurement by Pitot Probe by R. J. EMRICH 1.2.3. Propeller and Vane Anemometers by R. J. EMRICH 1.2.4. Hot-wire and Hot-Film Anemometers by RON F. BLACKWELDER 1.2.5. Velocity Measurement by Other Probes by R. J. EMRICH 1.2.6. Flowmeters by R. J. EMRICH 1.3. Measurement of Velocity by Analysis of Doppler Shift of Characteristic Radiation by R. J. EMRICH 2. Density Sensitive Flow Visualization by W. MERZKIRCH

2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Introduction Refractive Behavior of Fluids Visualization by Means of Light Deflection Interferometry Evaluation Procedures Radiation Emission

AUTHORINDEX-SUBJECTINDEX

* Section 1.1.4.5, “Fabry-Perot Spectrometer,” is by A. N . Papyrin and R . I. Soloukhin xi

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CONTRIBUTORS TO VOLUME 18, PART A RON F. BLACKWELDER, Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90007 R. J. EMRICH, Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015

W. MERZKIRCH, Institut fiir Thermo- und Fluiddynnmik, Ruhr-Universitat Bochum, 4630 Bochum, Federal Republic of Germany A . N . PAPYRIN, Institute of Theoretical and Applied Mechanics, U S S R Academy of Sciences, Siberian Division, Novosibirsk 630090, U S S R R. I. SOLOUKHIN, Institute of Heat and Mass Transfer, Byelorussian Academy of Sciences, Min.\X 220728, USSR E. F. C. SOMERSCALES, Department of Mechanical Engineering, RensJelaer Polytechnic Institute, Troy, New York 12181

...

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ETHODS OF EXPERlMENTAL PHYSICS Editors-in- Chief L. Marton C. Marton

Volume 1. Classical Methods Edited by lmmanuel Estermann Volume 2. Electronic Methods. Second Edition (in two parts) Edited by E. Bleuler and R. 0. Haxby Volume 3. Molecular Physics, Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics-Part A: Atomic Sources and Detectors, Part B: Free Atoms Edited by Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Edited by Luke C. L. Yuan and Chien-Shiung Wu Volume 6. Solid State Physics (in two parts) Edited by K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics-Atomic two parts) Edited by Benjamin Bederson and Wade L. Fite

Interactions (in

Volume 8. Problems and Solutions for Students Edited by L. Marton and W. F. Hornyak Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation By L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman Volume 12. Astrophysics-Part A: Optical and Infrared Edited by N. Carleton Part 9: Radio Telescopes, Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams xv

xvi

METHODS OF EXPERIMENTAL PHYSICS

Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers (in three parts) Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics Edited by Peter D. Edmonds Volume 20. Biophysics (in preparation) Edited by Harold Lecar and Gerald Ehrenstein

3. MEASUREMENT OF DENSITY BY BEAM ABSORPTION AND SCATTERING

3.0. Introduction * The fields of plasma physics, astrophysics, and all of chemistry use absorption and scattering of optical radiation as primary diagnostic methods. Volumes 9 and 12 of this series present a great deal of material relevant to the measurement of the existence of matter by beam absorption and scattering, and Part 6 of this volume furnishes guidance to the experimentalist seeking to learn of methods for measurement of the composition of a fluid in motion. Usually consideration of the interaction of the different species making up the fluid with the beam of radiation is necessary in order to interpret measurements to yield the total density. This becomes less true as the radiation considered becomes “harder,” i.e., for x rays, gamma rays, mesons, and neutrinos. For measurements of density of fluids in motion, the chief characteristic of interest is short-time response. The achievement of microsecond, nanosecond, and even picosecond responses usually involve intense sources emitting for short periods of time or short-time recording periods. These methods are described in Part 8 of this volume. In this part, the qualitative aspects of the density sensitivity of matter to beams of radiation are outlined, and some of the nomenclature is introduced. The analysis of electron beam excited radiation has resulted in a method used successfully for steady rarefied gas flows, which is unique to the subject of this volume.

3.1. Beam Attenuation Densitometry Radiation may be directed at a sample of matter from a source of such small extent that it may be considered a point source; it may radiate in all directions into a solid angle of 47r, or have a quite directional radiation pattern. Or a source may be a broad source, each little area of which is

* Chapters 3.0 and 3.1 are by

R. J. Emrich. 405

METHODS OF EXPERIMENTAL PHYSICS, VOL. 18B

Copyright @ 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-475956-4

406

3.

MEASUREMENT OF DENSITY

radiating in all directions (more usually thought of as being into solid angles of 27r because other parts of the source get in the way of backward radiation), or each part may have its own directional radiation pattern. In some cases, the broad source may be thought of as many sources all radiating only in a single common direction, in which cases one speaks of a beam ofradiation. Since there can be no radiation into zero solid angle absolutely all in a single direction, a beam always involves a range of directions, and different parts of the source may or may not be emitting equally intensely in a given direction; the degree to which they do is measured by the spatiaf coherence of the beam. In addition to the intensity and direction of the components of radiation making up the beam, there are various frequencies in any one direction, and for each of them there can be various phases. The specification of the content of any beam is therefore rather complex. When a beam is passing through a sample of matter, its interaction with the matter depends on the density and the constitution of the matter, and somewhat on the temperature, as well as on the content of the beam. The types of interaction are roughly classified as absorption, scattering and reradiation. Negative absorption, or amplification, is an important type of interaction which leads to laser sources. Assuming that there is no amplification, the most gross method of employing radiation for the measurement of the density of matter deals with the removal of radiation from a beam by absorption and scattering. The measurement of simply “what is left” after a beam has traversed a given thickness of matter is called beam attenuation densitometry . Under the overwhelmingly large set of circumstances, the intensity 1 left in the beam after traversing a thickness dx of material, where lois the incident intensity, can be described as’

p is called the linear attenuation coefficient, p m the mass attenuation coefficient, and p, the atomic attenuation coefficient; p is the mass density of the material, N is the Avogadro number (= 6.02 X lV3 mol-l), and A

is the atomic or molecular weight of the material. The radiation leaving the beam is either absorbed or scattered. If the beam is monochromatic and narrow (so that scattered radiation does not reenter the beam), Eq. (3.1.1) can be integrated for homogeneous material to yield

(3.1.2)

’ R. D. Evans, Gamma rays. I n “American Institute of Physics Handbook” (D. E. Gray, ed.), 3rd ed., pp. 8-190 to 8-218. McGraw-Hill, New York, 1957.

3.1.

BEAM ATTENUATION DENSITOMETRY

407

x is the distance traversed from the place where the intensity was Zo. Chemists call this exponential relation Eq. (3.1.2) the Lambert-Beer relation. It is valid for a monochromatic beam of spatially coherent radiation of very small cross section. p is strongly dependent on the frequency of the radiation and on the density and molecular weight of the absorbing material. The study of processes that lead to attenuation shows that there are several competing processes, called photoelectric effect, Rayleigh scattering, Compton effect, pair production, photodisintegration of the nucleus and meson production, for example. The probabilities of the many competing processes occurring are expressed in terms of cross sections for the processes, and extensive tables are available to predict the total attenuation. When the radiation is not monochromatic, but contains many frequencies, the character of the radiation changes with depth of penetration. For example, radiation produced by bremsstrahlung from electrons of 0.1 MeV or less (x rays) typically has the lower frequencies removed and consists of the “harder” high frequency components after traversing a few centimeters of water or aluminum. All frequency components are attenuated, of course, but the total intensity does not follow the simple exponential relation Eq. (3.1.2). A variety of radiation detectors is used, ranging from photographic film, ionization chambers, and photomultipliers to scintillating crystals in combination with photomultipliers. The efficiency of the detector varies with the radiation frequency, and the overall combination of the radiation source, the attenuation, and the detector finally determines the accuracy with which a density value is measured. Imaging is often desirable, so that inhomogeneities in a material are detected. This procedure is highly developed in medical radiography and in metallurgical flaw detection. The radiation, typically x rays, is emitted from a small area by concentrating a fine electron beam on a copper or tungsten target. The radiation is emitted with about equal intensity in different directions over a large solid angle, showers the specimen to be studied, and falls on photographic film, whose sensitivity is increased by being in contact with an intensifying screen-a luminescent material effectively converting x rays to visible radiation to which the film is more sensitive. If the specimen is fairly thick, radiation scattered out of one line may reach the detector at a different place; this may seriously affect the quality of the image produced. Since sources of penetrating radiation are seldom monochromatic, and since the exponentiallike attenuation of the radiation puts great stress on detectors in their need to serve over many orders of magnitude, the method of attenuation densitometry turns out to be used only under spe-

3. MEASUREMENT

408

OF DENSITY

cia1 conditions where other methods fail. Achievement of reliability and high accuracy requires careful attention to the problem of scattered radiation reentering the beam. Interaction of x rays and gamma rays with the matter being studied can change the state of the matter, as is illustrated by radiation damage to living tissue. Nevertheless flash radiography has played an important role in diagnosing explosive events, and measurements in shocked solids have given information on high pressure properties up to 20-30 GPa.2 Information on the techniques employed is given in book^^,^ and conference proceeding^.^ An x-ray source consisting of a linear accelerator for electrons impacting a tungsten target with energies as high as 30 MeV and emitting with a duration of the order of 100 ns is employed for measuring shock angles, arrival times of detonation waves loading solids, and densities in both solids and high-explosive gas products .6 Shadowgraphs of laser-imploded microballoons have been made’ with x rays from a separate laser-produced plasma of duration 100 ps.

3.2. Analysis of Raman and Rayleigh Scattered Radiation”? 3.2.1.Introduction During the past decade, light scattering has developed into a powerful approach for measurements in fluid This scattering arises when a beam of light, passing through a medium, interacts with it and consequently part of the beam is diverted or “scattered.” A fraction of this diverted light can be collected and analyzed to determine thermodynamic

* T. Neal, J. Appl. Phys. 46,2521-2527

(1975). K. Vollrath and G. Thomer, “Kurzzeitphysik.” Springer-Verlag, Berlin and New York, 1967. F. Jamet and G. Thomer, “Flash Radiography.” Elsevier, Amsterdam, 1976. “Proceedings of the 4th Symposium (International) on Detonation.” US Govt. Printing Office, Washington, D.C., 1965; 5th, 1970; 6th, 1976. Available from Clearinghouse for Scientific and Technical Information, Springfield, Virginia. D. Venable, ed., “PHERMEX: A Pulsed High-energy Radiographic Machine Emitting X-rays,” Rep. LA-3241 Los Alamos Sci. Lab., Los Alamos, New Mexico, 1967. Available from Clearinghouse for Scientific and Technical Information, Springfield, Virginia. ’ M. H. Key, C. L. S. Lewis, J. G. Lunney, A. Moore, T. A. Hall, and R. G. Evans, Phys. Rev. Lett. 41, 1467 (1978).

‘

* Chapter 3.2 is by

Marshall Lapp and C. Murray Penney. t The work described in this chapter is treated in greater detail in a forthcoming book tentatively titled “Light Scattering for Fluid Dynamics,” to be published by Academic Press.

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

409

INCIDENT LASER BEAM

FIG.1. Conceptual drawing of Raman or Rayleigh light scattering configuration.

fluid properties from the intensity and its spectral distribution, using an arrangement such as that shown schematically in Fig. 1. Since the scattering processes considered here are generally weak the intense optical probe beam provided by a laser source is usually required for practical utilization in diagnostic applications. A primary reason why light scattering is employed for gas measurements is that it can provide precise space- and time-resolved data for temperature, density, and composition. For example, in the configuration shown in Fig. 1, spatial resolution is determined by the dimensions of the incident beam and the field of view defined by the lens and slit, while time resolution can be achieved by sampling a continuous signal, if it is sufficiently strong, or by using a pulsed laser scattering source. The detailed M. Lapp, C. M. Penney, and J . A. Asher, Application of light-scattering techniques for measurements of density, temperature, and velocity in gasdynamics. Aerospace Research Laboratories, Wright-Patterson Air Force Base, Report No. ARL 73-0045, 1973. Available from U.S. National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161. M. Lapp and C. M. Penney, eds., “Laser Raman Gas Diagnostics.” Plenum, New York, 1974. R. Goulard, ed., “Combustion Measurements.” Academic Press, New York, 1976. B. T. Zinn, ed., “Experimental Diagnostics in Gas Phase Combustion Systems.” Progr. Astronuut. Aeronaut. 53 (1977). S. Lederman, Prog. Energy Combust. Sri. 3, 1 (1977); Phys. Fluids 22, 1065 (1979). M. Lapp and C. M. Penney, in “Advances in Infrared and Raman Spectroscopy” (R. J. H. Clark and R. E. Hester, eds.), Vol. 3 . Heyden and Son Ltd., London, 1977, Chapt. 6. L. A. Kennedy, ed., “Turbulent Combustion.” Progr. Astronaut. Aeronaut. 58 (1977). 8 A . C. Eckbreth, P. A. Bonczyk, and J. F. Verdieck, Appl. Spertrmr. Rev. 13 (I), 15 (1978); published with revisions in Prog. Energy Combust. Sci. 5 , 253 (1979). M. Lapp, in “Laser Probes for Combustion Chemistry” (D. R. Crosley, ed.). Amer. Chem. SOC.Symp. Series, Vol. 134, Chapt. 17, Washington, D.C., 1980. A . C. Eckbreth, Symp. (Int.)Cornbust. [Proc.],18th. Combustion Institute, Pittsburgh (to appear).

’

@

410

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MEASUREMENT OF DENSITY

information that can be obtained from these techniques depends in part on the processes that contribute to the observed light. Among these scattering and “scatteringlike” processes are Rayleigh scattering, Raman scattering, and fluorescence from gas molecules. Additionally, if the gas system contains particulates (aggregates of molecules), as most do, then there can be relatively strong scattering from these. (Scattering from particles is often called Mie scattering, although this name applies more directly to scattering from spherical particles.) In some cases, fluorescence or Raman scattering can be observed from the particulates as we11.11J2 In this brief chapter, we concentrate on the scattering from molecules in the gas phase; i.e., Rayleigh and Raman scattering. Our goal is to make clear the general processes involved in obtaining fluid dynamic information from these types of scattering. General descriptions of the various techniques are given in Refs. 1 - 10 and in references quoted there, and background material appears in previous volumes of the Methods of Experimental Physics. l 3 However, since diagnostic scattering measurements have evolved into a relatively large and fast-moving field, we will cite representative examples of recent works that illustrate key points and some of the forefront directions in this field. Absorption and emission spectroscopy are often more familiar to workers in fluid dynamics than is light scattering spectroscopy, perhaps because the equipment necessary for widespread application of scattering techniques has become available more recently. Therefore, in order to make clear the relationship between these approaches and thereby to increase an appreciation of the useful characteristics of light scattering, it is helpful to consider some of the key features that distinguish them. We have already noted that scattering provides direct threedimensional resolution, while absorption and emission are intrinsically line of sight measurements. Although inversion and tomographic techniques can be used in some cases to convert line of sight data to threedimensional information, direct access to this information is often advantageous. Many of the basic features of Rayleigh and Raman scattering can be understood in terms of a classical model, in which a beam of incident radiation passes through a cloud of molecules. The electric field of the radiaD . 3 . Wang, M. Kerker, and H. W. Chew, Appl. Opr. 19, 2315 (1980). R . E. Benner, J . F. Owen, and R. K . Chang,J. Phys. Chem. 84, 1602 (1980). l 3 B . P. Stoicheff, in “Methods of Experimental Physics-Molecular Physics” (D. Williams, ed.), Vol. 3, Chapter 2.3, Academic Press, New York, 1%2; D. H . Rank and T. A. Wiggins, in “Methods of Experimental Physics-Molecular Physics” (D. Williams, ed.), 2nd ed., Vol. 3, Part A , Chapter 3, Academic Press, New York, 1974. l]

3.2.

ANALYSIS OF R A M A N A N D RAYLEIGH SCATTERED RADIATION

41 1

tion E ( t ) distorts the electron cloud of each molecule, creating oscillating dipoles. The relationship between the induced dipole p ( t ) and the incident field is given by the molecular polarizability a ; thus At) =

aE(t),

(3.2.1)

where E(t) = Eo cos wot

(3.2.2)

is the incident field at light frequency wo. (In general and E are vector quantities which are not necessarily parallel and a in its most general form is a tensor. However, the points essential here are contained in the simpler scalar form.) The oscillating dipoles create a secondary radiating field of light at the same frequency as the incident light (neglecting small Doppler shifts from molecular motion). This unshifted frequency field is the Rayleigh scattering. Rotation and vibration of molecules modulate the polarizability at the rotational and vibrational frequencies, because in some orientations and shapes the molecules are more polarizable than in others. This modulation has the same effect as modulation of radio signals; it creates sidebands shifted from the primary frequency by molecular rotational and vibrational frequencies. If we represent the polarizability modulation by a

= a0

+

a1

cos w t ,

(3.2.3)

where w is a vibrational or rotational frequency, then p ( t ) = aoEo

cos wot

+ ( ~ , E o / ~ ) [ c o+ sw() ~t +~ C O S ( W ~- w ) t ] .

(3.2.4)

Here, the first term on the right-hand side, unshifted in frequency, is conventionally called Rayleigh scattering, while the last two terms, displaced symmetrically above and below wo by frequency shifts w, are the sidebands of Raman scattering. Thus, the Rayleigh signal occurs at the same frequency as that of the incident light beam, while the Raman signals occur at different frequencies separated from the incident light frequency by constant shifts which are characteristic of the particular molecule. (See Fig. 2 for a schematic of these effects, illustrated for the case of N2.) If the incident beam is tuned to another frequency, the Rayleigh and Raman lines shift accordingly. Conversely, optical emission and absorption occur at fixed frequencies which cannot be shifted to a spectral region of greater convenience, or to avoid an interfering spectral feature. Scattering also displays other properties which contrast with those of fluorescence, a process which can be viewed as an absorption of light by a

412

3.

MEASUREMENT OF DENSITY LOG

2400 2200

400

0

400 2200 2400

RAMAN S H I F T (ern‘') WAVELENGTH, A-FIG. 2. Raman and Rayleigh scattering from N2at ambient (300 K ) and elevated (1500 K ) temperatures for an exciting laser line in the midvisible spectral region (- 500 nm).6 The unshifted lines representing the intensity of Rayleigh scattering (i.e., at the same wavelength as the incident laser beam) are flanked by rotational Raman scattering represented here by wings showing the peak intensities of the rotational lines. Similar weak wings surround the Stokes and anti-Stokes vibrational Q-branch Raman bands (i.e., the “spikes” shown at Raman shifts of about 2331 cm-I). Note that a log ordinate scale is used in the main plot while a linear scale is used for the encircled inset diagrams showing the vibrational Qbranches at two temperatures. (These were calculated as they would appear if viewed with a spectrometer slit function of triangular shape and 6 cm-’ full width at half maximum.)

molecule followed, after a short delay, by reemission. Fluorescence quenching occurs because the finite delay allows time for collisions which can shift the excited molecule into other internal or translational modes before it can reemit the absorbed light energy. On the other hand, Rayleigh and Raman scattering are effectively instantaneous processes, when excited in the spectral regions we consider (i.e., away from resonance). Because of this instantaneous nature, scattering intensities are generally free of quenching; they are proportional to the number density of scattering molecules and independent of background pressure up to pressures large for fluid dynamic studies, which permits pressure to be determined by simultaneously obtained scattering data for temperature and total gas density. (The shapes of scattering signatures, which are important for temperature analyses and temperature-dependent effects upon density measurements, can be sensitive to pressure under limiting conditions. Thus, Rayleigh profiles can change from Gaussian to more complicated shapes at moderate pressures, as is discussed in Section 3.2.2.1, while Raman profiles are not affected by

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

413

pressure from the point of view of practical diagnostics up to high pressures, as is discussed in Section 3.2.3.1.) in the types of measurements of interest here, the quenching insensitivity of scattering is desirable because it allows the determination of some of the important fluid properties at a point without requiring knowledge of all the remaining properties involved in quenching corrections. Nonresonance scattering cross sections vary moderately with light frequency; in most cases they are closely proportional to the fourth power of the scattered light frequency. However, if scattering is observed as the incident laser source is tuned into resonance, a regime termed “resonance scattering” is entered. The scattering intensity is then increased, sometimes remarkably, and as the frequency approaches coincidence with the resonance level, fluorescence ultimately results. The potential increased strength of the Rayleigh and Raman effects near resonance has not been found to have widespread application for fluid dynamic experimentation, however, since significant enhancement is obtained only in the vicinity of extremely strong absorption and few molecules of interest in fluid mechanics possess such features in spectral regions convenient for scattering. The classical model just discussed gives a good qualitative explanation for the frequency shifts of Raman scattering, but the quantum nature of molecular levels must be taken into account to fully explain the observed scattering amplitudes and the selection rules. These selection rules govern which of the possible molecular transitions actually result in Raman signals. Many of the Raman shifts are relatively large; i.e., they are significant fractions of light frequencies. Consequently, many Raman bands are substantially shifted in color from that of the incident light beam. Since rotational and vibrational frequencies are different for different molecular species (with, of course, occasional coincidental spectral overlaps), and since the intensity of a given molecular band is generally proportional to the number density of its corresponding molecular species, spectral analysis of the Raman scattering can be used in gas mixtures composed of simple molecules (e.g., diatomics, H 2 0 , C 0 2 , CH,, etc.) to determine the concentrations of the various gases, and the resulting gas density. In gas mixtures containing several similar, more complicated molecules (e.g., higher hydrocarbons) it is often difficult to determine the individual concentrations of these molecules because of the overlap and similarity of their Raman bands.14 However, even in such cases the density of particular bonds, such as the C-H bond, can be determined because they D. A. Stephenson, J . Quanf. Specfrosc. Radiaf. Transfer 14, 1291 (1974).

414

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MEASUREMENT OF DENSITY

scatter into a characteristic band which has approximately the same intensity per bond in different molecules.6 This type of determination is not easily obtained in absorption or emission spectroscopy. Furthermore, although not apparent from the classical model, the ratios of intensities of different bands from the same molecular species can yield temperature ; this fact arises from the basic quantum nature of molecular energy levels and the scattering process. It is important also that the selection rules for Raman scattering differ from those for electric dipole transitions. Thus, Raman scattering can probe important homonuclear molecules, such as N2 and 0 2 ,while no infrared spectra exist for such species because of their lack of a permanent dipole moment. Since these common fluid mechanic molecules have no visible resonance absorption spectrum, their Raman response is of even greater importance. For molecules which do have electric dipole moments (such as, for example, CO), Raman scattering moves molecular detection, in effect, from the spectrally crowded infrared, or the spectrally more difficult ultraviolet, into the visible. On the one hand, the Rayleigh effect is less specific than the Raman effect, because Rayleigh scattering from different molecules cannot be spectrally separated easily, and the overall Rayleigh scattering cannot be distinguished readily from scattering by particulates and walls. On the other hand, Rayleigh scattering is typically stronger than Raman scattering by three orders of magnitude, and this strength contributes an important advantage in particular cases; for example, it is useful where a continuous time record of fast fluctuations is required. 3.2.2. Rayleigh Scattering 3.2.2.1. Composition and Temperature Measurements from Rayleigh Scattering Line Shapes. In this process, light is scattered from a molecule without exchanging any energy with the internal states of the molecule. Only the very small amount of energy necessary to conserve momentum between the light photon and the scattering molecule is lost or gained by the photon. If the incident beam is monochromatic, and the gas behaves as a single component ideal gas, this exchange produces a Doppler-broadened line with a Gaussian profile. The line shape is a function of composition (in a multicomponent mixture) and of temperature; in principle, this information can be extracted from Rayleigh line-shape measurements. However, in practice this approach has been found useful only in a few cases, largely because of the following reasons:

(i) The linewidth is extremely narrow, and thus precise measurements of its shape require a highly monochromatic source and exceptional

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

415

detector resolution. For example, for 90” scattering of blue light from N z , the linewidth for monochromatic excitation is about 0.001 nm. The necessity for high resolution generally reduces the available source power and fraction of scattered light that can be utilized, while increasing the delicacy of experimental requirements. (ii) Background scattering from particles, lenses, walls, etc., is centered at the same wavelength as the Rayleigh scattering. This scattering produces a strong, narrower line overlapping the Rayleigh line and distorting its shape. Consequently, careful line shape measurements involve minimization of background, and high contrast, in addition to high resolution spectral analysis. (iii) The line shape for a pure gas is not truly Gaussian in many practical cases.15 At very small scattering angles and/or for very short molecular collision mean free paths, much of the Rayleigh scattering can be viewed as coming from thermally excited sound waves, and corresponds to a hydrodynamic explanation. This phenomenon causes the Rayleigh line to break up into two peaks (conventionally called Brillouin peaks) surrounding a central component. Each Brillouin peak is shifted from the central peak by an amount approximately equal to the Doppler shift produced by the sound wave velocity. Conversely, a purely Gaussian shape is obtained only in a single component mixture, under conditions in which the mean free path between molecular collisions is much longer than the density fluctuation (i.e., sound) wavelength 1, that would produce the Brillouin peaks:

I,

=

X0/2 sin(B/2),

(3.2.5)

where Xo is the light wavelength and 8 is the scattering angle. This “collisionless” regime corresponds to scattering from each test gas molecule independently. Many practical cases are in between these extremes (i.e., in the “kinetic” regime, for which the density fluctuation wavelength and the mean free path are comparable); thus, 90” scattering from N2near STP is approximately Gaussian in shape, but shows vestigial Brillouin peaks which increase its width to a value about 15% in excess of the ideal gas ~ a 1 u e . l (See ~ Fig. 3.) However, the decreased density (about 6 normal) in an Hz-air flame causes 90” Rayleigh scattering from this test gas to closely follow a Gaussian contour.15 (See Fig. 4.) (iv) In unknown gas mixtures, the line-shape dependence on composition and temperature is difficult to separate in order to determine these characteristics individually. For example, within the ideal gas approximation, the line shape in a gas mixture is an overlay of Gaussian peaks, Is R. Cattolica, F. Robben, and L. Talbot, P r o p . Astronuuf. Aeronaut. 53 575 (1977); R. W. Pitz, R. Cattolica, F. Robben, and L. Talbot, Combust. Flume 27, 313 (1976).

3.

416

MEASUREMENT OF DENSITY

10

8

6

4

2

0

20

0

80

60

40

0

FREQUENCY CHANNELS

FIG.3. Rayleigh scattering at 90" from Na at ambient condition^.'^ The solid theoretical curve is a Gaussian shape. The experimental data can be fit well with a trimodal model calculation, which includes Gaussian wings Brillouin-shifted slightly from the central contours.

whose linewidths are inversely proportional to the square root of the various molecular masses and directly proportional to the square root of temperature.

For these reasons, even in gases of known composition, temperature measurements from Rayleigh line shapes require careful experimental techniques and analysis. These requirements can be compounded when

1

0

~ 0

"

"

20

"

"

"

' 40

~

~

60

"

'

~

80

'

'

100

FREQUENCY CHANNELS

FIG.4. Rayleigh scattering at 90" from postflame gases of a premixed hydrogen-air flame at an equivalence ratio of 0.85, with an adiabatic flame temperature of 2200 K.15 The experimental data agree to within experimental accuracy with theory, fitting a Gaussian profile corresponding to 2100 K.

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

417

the composition vanes in an unknown way because of the line-shape dependence on both composition and temperature. Thus, there have been relatively few successful measurements of temperature from Rayleigh scattering line shapes, and no such measurements of composition of which we are aware.

3.2.2.2.Density, Concentration, and Temperature from Rayleigh Scattering Intensities. The basic equation used in relating integrated Rayleigh scattering intensities to density is S = const x Ncreff= const x

Npj,

(3.2.6)

J

where S is the Rayleigh scattering intensity, N is the overall molecule number density, reff is the effective cross section (defined by Eq. (3.2.6)], N j is the concentration of thejth species, cr, is the corresponding Rayleigh scattering cross section, and the sum extends over all species. This expression, which is independent of the Rayleigh line shape, can be used to determine density, and, under some circumstances, composition and temperature as well, for gas mixtures under restrictive conditions apparent from the nature of this elastic scattering. Thus, if one type of molecule scatters much more strongly than others, its concentration can be determined from the scattering intensity. On the other hand, if the gas composition is known, if each type of molecule has nearly the same scattering cross section, or if each of the various observed mixtures (in, say, a reacting flow) has nearly the same effective Rayleigh cross section, then the molecular density N can be determined from Rayleigh scattering. The probability of a nearly constant effective Rayleigh cross section in a reactive flow is favored by the tendency for atoms to make approximately the same contribution to Rayleigh scattering in different molecules.16 In recent work,17 this naturally favorable situation was improved by adjusting the ratio of methanol and hydrogen feeding a turbulent diffusion flame so that the effective Rayleigh cross section of the fuel, combustion products and background air have the same effective Rayleigh cross section to within about 3%. The resultant density data were used t o determine the temperature fluctuations in this constant pressure flame. As has been previously mentioned, these methods are complicated by the difficulty of separating Rayleigh scdtering from the background light scattered from particles, optical elements, and cell walls. However, the latter two background sources can be minimized by careful experimental technique, and their remnants, if constant, can be determined separately M. Born and E. Wolf, "Principles of Optics," p. 89. Pergamon, New York, 1953. R . W. Dibble and R . E. Hollenbach, Symp. (Int.) Combust. [Proc.],18th. Combustion Institute, Pittsburgh (to appear). l6

418

3.

MEASUREMENT OF DENSITY

and subtracted. If the particle background is due primarily to a relatively low density of particles moving through the focused beam, it will appear in the form of isolated short pulses on the (typically) more slowly varying Rayleigh scattering signal; these pulses can be deleted from the density record in subsequent analysis. Thus, ways have been found to utilize Rayleigh scattering in particular cases where the data can be separated from interferences and interpreted clearly. A major motivation of this work is precise time and space resolution in continuous measurements of density, temperature, or concentration fluctuations. For example, if a continuous laser is used to measure density in consecutive time segments of length 7 , the average number of scattered photons detected per segment is

n = Kpcrlr,

(3.2.7)

where K is a proportionality constant containing optical parameters, detector efficiency, etc., and p is the gas density, 1 is the length segment along the incident light beam from which scattered light is detected (determining spatial resolution in one direction), and r is the scattering cross section. Since light scattering in the regions we consider is an essentially random process, the best possible measurement accuracy is characterized by a relative standard deviation h p l p = h n / n = nP1l2.

(3.2.8)

Suppose that the measurement accuracy required in a particular experiment is characterized by a standard deviation of 3%. Then the required average number of photons detected per time resolution segment is approximately 1000, and the product of length resolution and time resolution is given by IT = 1000/Kpa from Eq. (3.2.7), indicating that a large cross section is desirable for precise resolution. Using readily available equipment (in particular, an argon laser supplying 1 W of incident light power at 488 nm), time resolution on the order of 100 /.LS can be obtained along with space resolution of 1 mm from Rayleigh scattering in gases near STP. 3.2.3. Raman Scattering 3.2.3.1. Applications to Fluid Dynamic Methods. The resolution product derived from Eq. (3.2.7) for continuous time measurements from Raman scattering is typically about 1000 times larger than that for Rayleigh scattering because of the smaller cross sections. (Under some circumstances, and with precautions, this resolution product can be made smaller through use of special multipass cells, which pass the incident

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

419

laser beam many times through the test zone.I8) Despite the weakness of the Raman effect, however, precisely resolved information can be obtained from Raman scattering using pulsed lasers which provide light energy on the order of 0.1 to 1 J in microsecond or shorter pulses. (Less energy is required in the UV or for high gas densities, but much more is required in the IR because scattering cross sections and detector efficiencies are smaller in the IR. The cross-section frequency dependence in proportion to the fourth power of scattered light frequency will be apparent subsequently from inspection of Eq. (3.2.9) for vibrational Raman scattering radiant flux.) This approach, which has been called pulsed Raman spectroscopy, allows utilization of several important advantages of Raman scattering. Previously we have described some of these favorable characteristics for use in fluid dynamic probes. These characteristics are summarized below: Specijicity and Relative Lack of Interference. The signals obtained from many different types of relatively simple molecules are spectrally separated. These separations are usually sufficiently large to allow noninterfering quantitative measurements so that concentrations can be obtained independently. The densities of particular structural groups, such as a C-H bond, can often be obtained in mixtures of more complicated m01ecules.~ Well-Determined and Independent Response. The signal intensity is directly proportional to the molecular number density of probed molecules, and is independent of all other molecular densities. Ability to Determine Pressure from Temperature and Density Data. Pressure does not affect the integrated intensity of Raman scattering for most conditions of interest for fluid dynamic studies. Thus, pressure can be obtained for gases from the equation of state combined with temperature and total density data obtained by Raman scattering measurements of all major constituents. (Increasing pressure does broaden and shift slightly the individual Raman rotational or rotational-vibrational lines in a given band, but the experimental resolution required for diagnostic studies is insufficient to resolve such effects. At many atmospheres pressure, vibrational Raman scattering contours can display significant collisional band narrowing and shifting.lg) Nonperturhing Nature. Over a wide range of experimental conditions, no significant perturbations of the test gas occur which are caused by the laser probe beam. Remote In Situ Capability. Probes do not require hardware specifiR. A. Hill and D. L. Hartley, Appl. Opt. 13, 186 (1974). Is

A. D. May, J. C. Stryland, and G . Varghese, Can. J . Phys. 48, 2331 (1970)

420

3.

MEASUREMENT OF DENSITY

cally at the measurement zone and, with reasonable optical access, can be placed in relatively remote locations. Three-Dimensional Spatial Resolution. Volumes considerably less than a 1-mm cube can be probed. Time Resolution. Precise time resolution can be obtained by using pulsed lasers, in which case the resolution is limited only by the laser pulse width, which can be in the nanosecond range or less, if necessary. Generally, microsecond time scales are considered broadly useful for fluid dynamic studies, and can be easily achieved by commercially available types of lasers. Range resolution (LIDAR techniques) can be obtained in remote experiments by measuring the time dependence of the observed scattering. Accessibility of Information. Density information from vibrational Raman scattering data is very largely independent of temperature below temperatures where substantial vibrational excitation occurs. At high temperatures, there is a moderate temperature correction for density which can be calculated directly from the light scattering information and fundamental molecular Cupubility to Probe Systems not in Equilibrium. Thermal equilibrium is not necessary to obtain meaningful data concerning population distributions. In fact, within thermal nonequilibrium systems, the degree of excitation of the major molecular level structure (such as the significantly populated lower vibrational levels) can be determined in many cases.” Simultuneous Multiplicity of Duta. Data can be obtained from many species simultaneously. Wide Sensitivity. As noted previously, all molecules have some form of Raman spectrum, as opposed to near infrared absorption or emission, which is absent for homonuclear diatomic molecules, such as N2 and 02. These advantages must be balanced against the single major disadvantage of Raman scattering, which is its weakness. Thus, in air at STP, only about of an incident light beam is scattered per centimeter along the beam into all Raman lines and all directions. This characteristic places substantial demands upon laser sources and optical detection apparatus. For the most part, it restricts Raman scattering measurements to major and intermediate species (say, near or above one part per thousand relative to STP) and to low or moderate levels of background light. Furthermore, when precise space and time resolution are needed, the resultant incident laser beam intensity required to produce a useful signal M . Lapp, C. M . Penney, and L. M. Goldman, Opr, Comm. 9, 195 (1973). L. Y . Nelson, A. W. Saunders, A. B. Harvey, and G . 0. Neely, J . Chem. Phys. 55, 5127 (1971); A . B. Harvey, in “Laser Raman Gas Diagnostics” (M.Lapp and C. M. Penney, eds.), p. 150, Plenum, New York, 1974. zo 21

3.2.

A N A L Y S I S OF RAMAN A N D RAYLEIGH SCATTERED R A D I A T I O N

421

can, under some circumstances, be great enough to sensibly perturb the measured system. For example, in flame measurements carried out in our l a b o r a t ~ r y ,we ~ ~ obtain * ~ ~ spatial resolution of 0.3 X 0.3 X 0.7 mm and time resolution of 2 p s (the duration of the laser pulse). For length resolution of 0.7 mm along the incident beam, and high temperature (-2000 K) combustion products, about 1 J of incident light energy is required for incident light in the blue-green region to obtain accurate (- 3 to 7% standard deviation) Raman measurements of temperature and major species concentrations. The resultant laser beam intensity, roughly 7 x lo8 W/cm2, does not significantly perturb room air or clean flame gases, but it is sufficient to vaporize small carbon particles in sooting regions of a flame, preventing the acquisition of useful Raman data.8 However, stronger nonlinear Raman processes discussed subsequently can provide a solution to this problem. 3.2.3.2. Temperature Effects. Although this chapter is focused on density measurements, the temperature dependence of Raman scattering will be discussed briefly because: (i) temperature and dehsity information are available simultaneously from the observed scattering, (ii) at high temperatures, where significant vibrational excitation occurs, the density information from integrated vibrational Raman scattering contours must be corrected for a small to moderate dependence on and (iii) for many practical experiments, the use of fixed spectrometer slits or filter bandpasses in combination with the temperature-sensitive shape of the Raman scattering spectral contour results in obtaining only portions of the entire vibrational Raman contour; this signal fraction can have a strong temperature dependence. A direct way to illustrate these temperature effects is to consider, as an example, the temperature-sensitive Nz vibrational Raman contour .24 Near and below room temperature, pure vibrational scattering (such as from N,) appears as a narrow band at a large displacement to the red of the exciting line. (See insert in Fig. 2.) This band is often described as a “Stokes Q-branch,” where the symbol Q is the spectroscopic designation for a transition corresponding to no change of rotational quantum number 22 M. Lapp and C. M. Penney, “Proc. Dynamic Flow Conf. 1978 on Dynamic Measurements in Unsteady Flows, p. 665. P. 0. Box 121, DK-2740 Skovlunde, Denmark, 1979. 23 M. C. Drake, M. Lapp, C. M . Penney, S. Warshaw, and B. Gerhold, Symp. ( I n r . ) Cumbusr. [Proc.], l8th Combustion Institute, Pittsburgh (to appear). M. Lapp, in “Laser Raman Gas Diagnostics” (M. Lapp and C. M. Penney, eds.), p. 107. Plenum, New York, 1974.

422

3.

MEASUREMENT OF DENSITY

(AJ = 0). For this vibrational scattering, “Stokes” denotes a transition in which the molecule gains one vibrational quantum of energy (Av = 1) from the light beam during the scattering process. The term “band” rather than “line” is used here because the vibrational transition associated with different values of the rotational quantum number appears at slightly different energy shifts, producing an overall spectral envelope encompassing a number of closely spaced lines, but appearing under moderate spectral resolution to resemble simply a broad line for many molecules of interest here (with the notable exception of H2,which has widely spaced rotational contributions to the vibrational Q-branch). As temperature increases, the vibrational band broadens because higher rotational and vibrational levels are populated significantly, and thus more individual rotational-vibrational lines begin to contribute significantly to its shape. This can be seen in Fig. 5b, where the increased number of contributing vibration-rotation lines is shown to cause a substantial growth in the v = 1 + v = 2 peak as the temperature is increased from 1300 to 1700 K . (Note, however, that the spectral contours shown in Fig. 5b are all normalized at the v = 0 -+ v = 1 peak, in order to facilitate curve-fitting temperature determination procedures to be discussed subsequently. On an absolute scale, the v = 0 .+ v = 1 peak decreases as the temperature is increased.) The spectral shifts of Raman scattering occur because the Raman effect involves an exchange of energy between the scattered photons and the internal energy states (rotation, vibration, and in some cases, electronic) of the molecule. This quantum interpretation explains a key characteristic of the Raman effect; namely, that at low temperatures, so-called “antiStokes” vibrational Raman lines (the term “anti-Stokes’’ refers to scattering signals shifted to shorter wavelengths than that of the incident light beam) are observed to be significantly weaker for most common molecules than corresponding “Stokes” Raman lines (scattering signals shifted to longer wavelengths). This characteristic occurs because of the decreased populations of the higher initial vibrational levels which lead to the anti-Stokes lines. At higher temperatures, the population of elevated levels is greater, and the anti-Stokes signal becomes stronger relative to the Stokes signal. This temperature effect, illustrated in Fig. 2, and subsequently explained further in Fig. 6, gives rise to a widely used Raman temperature measurement technique, based directly upon the ratio of Stokes to anti-Stokes signal i n t e n ~ i t i e s . ~ ~ ~ * * ~ * * - ~ ~ Another useful spectral characteristic associated with increased vibrational excitation is that Stokes ( v = 1 + 2 , 2 -+ 3, . . . ) and anti-Stokes (v = 2 + 1, 3 + 2, . . . ) “hot bands” appear. These bands are slightly shifted from the (v = 0 + 1 or 1 .--, 0) bands because vibrational

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

423

(a) Band Area Method for Temperature Measurement From Stokes Vibrational Q-Branch Contours

3

Scattered Flux for Upper state Band = G Scattered Flux lor Ground State Band

G Corresponds to v = 0 U, Corresponds to Y = 1 etc

--

v = 1 v = 2

~

,v,

+

,) e-hCwevlkT

>

:

-.E. c

0

d

549

(b) Contour Fit Method

for T Meas.

550 Wavelength (nm)

551

(c) Band Peak Intensity

Method for T Meas.

A 550 Wavslenpth (nm)

Temperature (K)

FIG. 5. Saw-tooth spectral shape for vibrational Raman scattering contour, illustrated here for nitrogen. (a) Contributions of individual vibrational bands at elevated (ca. 1700 K) temperature. (b) Composite vibrational profiles calculated at three temperatures, and normalized at their peaks, compared to typical experimental data.z' The data were acquired with a commercial grating double monochromator with 300-pm entrance and exit slits, producing a triangular-shaped spectral slit function of 0.16-nm full width at half-maximum. The best fit temperature of 1546 K agrees well with corresponding thermocouple measurements. (c) Signal ratio for indicated bandpasses, showing approximately linear variation with temperature from 1200-2400 K. The bandpasses X and Y utilized for the intensity ratio in the ordinate of this plot are specified in part (b).

anharmonicity causes a slight change in the separation between vibrational states with increasing vibrational quantum number. (See Figs. 2,5, and 6.) Thus, we have characteristic spectral shapes for each molecule, which present opportunities for temperature determination by techniques such as contour fitting, ratios of intensities transmitted through wellchosen specific passbands, or spectral shift of band peaks. Particularly, the first of these methods has been used in fluid flow and combustion studies .24

3.

424

MEASUREMENT OF DENSITY

PROBABILITY DENSITY FUNCTIONS (HISTOGRAMS) FOR TEMPERATURE AT POSITIONS:

IN I

SCATTERED WAVELENGTH

INTENSITY

TEMPERATURE

FIG.6. Schematic of turbulent combustor geometry and optical data acquisition system for vibrational Raman scattering temperature measurements using Stokes/anti-Stokes ratios.z2 Also shown are the expected Raman contours viewed by each of the photomultiplier detectors, the temperature calibration curve, and several representative probability distribution functions of temperature at different flame radial positions.

The type of Raman measurement technique chosen for a particular experimental goal will vary with the goal; thus, in the case of temperature data using vibrational Raman scattering, for instance, we can determine probability density functions (pdfs), average values and higher moments, time dependence, or spatial gradient data through choice of various possible laser sources (from cw to p ~ l s e d ) . ~For ~ , example, ~~ average tem23 M. Lapp and R. M. C. So, AGARD Con$ Proc. N o . 281, Tesfing a n d Measurement Techniques in Heat Transfer and Combustion, Published by Advisory Group for Aeronautical Research and Development, NATO, 1980; available from NTIS.

3.2.

ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

425

perature measurements for a steady flame, calculated from data such as that shown in Fig. 5b, can be obtained conveniently using a I W blue or green line from an argon ion laser and a commercial grating scanning monochromator. The data shown in Fig. 5b corresponded to moderatewidth slits on a $-m double monochromator adjusted to provide a bandpass of about 0.16 nm; these data were acquired during the scan at 0.02 nm intervals, at a rate of one point every 10 s. On the other hand, a pulsed dye laser ( 1 J/pulse, 1 pulse/s, 2 ps pulse duration, 0.2 nm spectral pulse width) was utilized with a commercial 2-m grating single monochromator (combined with a polychromator at its exit plane, and augmented by interference filters in front of the polychromator slits in order to increase stray light rejection), to acquire time-resolved data from a turbulent flame. An experimental schematic used for this application is shown in Fig. 6. In this case, we have used the vibrational Raman Stokes/anti-Stokes ratio for NZ to determine statistically significant values of temperature from each laser shot. Relatively wide monochromator exit slit widths (i.e., polychromator slit widths) of 3 to 5 mm (corresponding to 3- to 5-nm spectral widths) were used to give flattopped bandpasses tailored to transmit most of the contours of the species to be detected. (Here, 3-nm bandpasses were used for the N z Stokes and anti-Stokes signals, 4 nm for the H 2 0 Stokes signal, and 5 nm for the Hz Stokes signal.) A typical temperature standard deviation for these datazz was roughly 7% (more recent dataz3yield 4%). Results for pdf’s illustrating the intermittent entrainment of ambient air into the flame boundary22 are shown in Fig. 7. 3.2.3.3. Vibrational Contour Analysis. Considerations about temperature effects for vibrational Raman scattering, and the various methods for determining gas temperature utilizing vibrational Raman techniques, are based upon the expression for the intensity S(v, J ) for a rotational line contribution to this Stokes Q-branch (AJ = 0) of the fundamental band series ( u + 1 t u),1,26-30 where the notation (upper state, lower zB 27

M. Lapp, L . M. Goldman, and C . M. Penney, Science 175, 1 1 12 (1972). G. Placzek, in “Handbuch der Radiologie,” (G. Marx, ed.), Vol. 6, Part 2, p. 205.

Akademische Verlagsgesellschaft, Leipzig, 1934; English trunsl. by A. Werbin, U.C.R.L. Translation No. 526(L), Lawrence Radiation Laboratory, 1959. A. Weber, in “The Raman Effect,” Vol. 2: “Applications” (A. Anderson, ed.), Sect. V1I.B of Chapt. 9. Dekker, New York, 1973. 2e G . Herzberg, “Molecular Spectra and Molecular Structure,” VoI. 1: “Spectra of Diatomic Molecules,” 2nd ed., Chapt. 3, Sect. 2(e) and 2(f). Van Nostrand-Reinhold, New York, 1950. L. A. Woodward, in “Raman Spectroscopy” (H. A . Szymanski, ed.), p. 35. Plenum, New York, 1967.

426

3.

o,201

MEASUREMENT OF DENSITY

0.30

r = 14.5 mm

0.10

0

0.10 I

ta

I 1

A I

r = 13mm

o

0.30

k 4

%!

m

0.20

B

a 0.10 0

mm

___)

*0

FIG.7. Probability density functions (histograms) of temperature for H,-air turbulent diffusion flame for various radial positions r , at an axial position 134 mm downstream of the fuel linetip ( x / d = 50).22 These data were found using pulsed vibrational laser Raman spectroscopy, from the Stokes/anti-Stokes intensity ratio for nitrogen. The measurement positions are drawn schematically at the right-hand side of the figure. Shaded parts of these pdf curves, which increase near the flame boundary, correspond largely to scattering from ambient temperature air.

state) is used. Neglecting depolarization effects (most vibrational Raman scattering is strongly polarized), we have ~ ( uJ,) = const x

'( 2 J

-+ ')('

+

Q rot Q vib

')04Rco exp

[

hc

- =G(u,

J ) ] , (3.2.9)

where the proportionality constant is determined by calibration experiments, and where u is the vibrational quantum number, J the rotational quantum number, k Boltzmann's constant, h Planck's constant, c the speed of light, T the temperature, r] a factor t o account for the effect of

3.2.

A N A L Y S I S OF R A M A N A N D RAYLEIGH SCATTERED R A D I A TI O N 427

nuclear spin of the molecule, oRthe wave number of the Raman fundamental line ( u + 1, J t v , J), C, a factor to account for the magnitude of the scattering cross section, Qrotthe rotational partition function, Qvib the vibrational partition function, and G(u, J ) the term value for the initial molecular level. The term value, which represents the energy level structure of the molecule, can contain terms which account for anharmonic behavior as well as vibration-rotation interactions. These corrections to simple harmonic motion are easily included, and can be especially significant for light molecules. On the other hand, the factor ( v + l ) results from using the harmonic oscillator approximation to determine the transition moment for the probability of the Raman scattering event,27*28-30 a good approximation for the purposes of the spectral contour calculations under consideration here. The ( u + 1) factor is replaced by the factor u for calculation of the anti-Stokes contour spectral intensity, i.e., S ( u , J) for the fundamental series (v + v - 1) anti-Stokes Q-branch. The spectral contour for N2,computed from Eq. (3.2.9), i.e., a calculation of S ( v , J) transformed into a plot of S(o) versus w (since every vibration-rotation transition can be associated with a spectral wave number position), displays a “saw-tooth’’ shape characteristic of diatomic molecules, and possesses temperature sensitivity through the exponential term as well as through Qviband Qrot. (See Fig. 5.) This representation of Eq. (3.2.9), i.e., S ( w ) versus o,gives the basic formulation for computing temperature. Calculation of this shape for any particular experimental situation must also include convolution of the Raman intensity S ( w ) with the spectrometer spectral slit function g (or filter bandpass), in order to account for Thus, we have, as an expression the actual instrumental for the slit-convoluted intensity S c ( w ’ ) , S C ( 0 ‘ )

S ( o ) g o ( o ’ , 0) d o ,

=

(3.2.10)

slit

where g o ( w ‘ , w ) is the experimentally determined response of the spectrometer at wave number w, when it is set at wave number w ’ , for the specific spectrometer configuration and settings denoted by the subscript “0”, and where the integral is to be taken over all wave numbers o contained within the slit profile. A common approximate shape for g o ( w ’ , w), using equal entrance and exit slit widths, is triangular. In practice, slit functions are measured for each spectral arrangement used. 31 S. S. Penner, “Quantitative Molecular Spectroscopy and Gas Emissivities,” Chapt. 5. Addison-Wesley, Reading, Massachusetts, 1959.

428

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MEASUREMENT OF DENSITY

The integral of the spectral contour of S(W)is proportional to Qvibfor diatomics, in the harmonic oscillator and rigid rotor appro~imation,~' which is equal to 1 at low to moderate temperatures, but is greater than I at elevated temperatures. Here, Qvib

= [l - exp(-hc~,/kT)]-~,

(3.2.11)

where W, is the vibrational constant. Thus, measurements of the gas concentration, which are proportional to the integral of s ( ~ )must , incorporate this temperature correction at temperatures where appreciable vibrational excitation occurs. If the experimental spectrometer slit width or filter bandpass does not encompass the full spectral profile width, then only a part of the contour represented by S(W)will be observed. (See Fig. 5 . ) The temperature dependence of that contour must then be known, in order to relate the density to the observed part of the spectrum by numerical evaluation. For some molecules and temperature ranges, a part of the contour can be isolated which gives density information nearly independent of temperature,8,32-34i.e., some parts of the isolated contour portion increase in intensity with temperature, while others decrease commensurably. This is an obviously useful characteristic, but it requires careful determination of spectral bandpass and corrections must still be applied if the temperature fluctuates outside the independent range. 3.2.3.4. Density and Concentration Measurements from Vibrational Raman Scattering. Concentrations of the major molecular species can be determined by measuring the intensity of light scattered into vibrational Raman bands. (See also Section 6.4.5.) If the composition remains constant, or if its variation is known, then the total density can be obtained from the measured concentration of any constituent. If the composition is unknown, density can be obtained by summing the individual measured concentrations. Filters and/or a spectrometer are commonly used to separate the various Raman bands which are monitored for concentration measurements. For example, in an experiment evolved from that described at the end of the last section, we have monitored major species in an H,-air flame (N,, H, , H20) simultaneously, using a spectrometer and a bank of four photomultipliers, each of which observes the Stokes vibrational Raman scattering from one of these species. In this configuration, a fifth 32 J. L. Bribes, R . Gaufres, R. Monan, M. Lapp, and C . M. Penney, Appl. Phys. Lett. 28, 336 (1976). 33 D. A. Leonard and P. M. Rubins, ASME Paper No. 75-GT-83 (1975). '' R. E. Setchell, AIAA Paper No. 76-28 (1976); J. R . Smith in "Laser Probes for Combustion Chemistry" (D. R. Crosley, ed.), Amer. Chem. SOC. symp. Series, Vol. 134, Chapt. 22, Washington, D.C., 1980.

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ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED R A D I A T I O N

429

photomultiplier is used to monitor the N2 anti-Stokes spectrum in order to provide a simultaneous temperature measurement .23 An exciting alternative to this approach which has been demonstrated both for unreactive flows and for involves replacing the photomultiplier bank with an optical multichannel detector. Such detectors can provide several thousand independent channels arranged in a twodimensional array. This array provides a convenient package to monitor many species simultaneously at a number of points along the path of the incident light beam. Multichannel detectors with sufficient sensitivity, low noise, and wide dynamic range for a variety of fluid dynamic applications have been developed at several laboratories and are now commercially available with increasingly satisfactory specifications. A Raman scattering system must be calibrated in order to determine the relationship between light scattering signals and the corresponding species concentrations. This calibration can be calculated from the basic optical quantities -light scattering cross sections, spectrometer transmission, and detector sensitivity as functions of wavelength. Vibrational Raman cross sections relative to N 2 are well known for many species of importance to fluid dynamics and the absolute cross section for N2 has been accurately determined from measurements performed by a number of research groups .37 The remaining characteristics can be measured to similar or better precision. However, in practice, calibration is often obtained by comparison to a known standard, such as room air or a prepared mixture. In our flame work, we use this simpler approach with room air, a hydrogen cell, and a standard flame as reference targets. For additional caibration purposes, a useful source can be provided by Rayleigh scattering from a clean (particle-free) sample of gas of accurately determined density and of well-known index of r e f r a c t i ~ n .This ~ ~ calibration is based upon the fact that Rayleigh scattering intensities can be calculated from the index of refraction and the density, but extreme care must be taken to prevent interference from any entrapped particulates. 3.2.3.5. Measurements from Rotational Raman Scattering. Rotational Raman scattering provides a useful measure of temperature in pure P. C. Black and R. K . Chang, AIAA J . 16, 295 (1978). L. R. Sochet, M. Lucquin, M. Bridoux, M. Crunelle-Cras, F. Grase, and M. Delhaye, Combust. Flume 36, 109 (1979). 37 H. W. Schrotter, in “Advances in Infrared and Raman Spectroscopy” (R. J. H . Clark and R. E. Hester, eds.) Vol. 8, p. 1 . Heyden, London, 1981; H. W. Schrotter and H. W. Klockner, in “Raman Spectroscopy of Gases and Liquids” (A. Weber, ed.), Topics in Current Physics, Vol. 1 l , Chapt. 4, Springer-Verlag, Berlin, 1979; H. Inaba, in “Laser Monitoring of the Atmosphere” (E. D. Hinkley, ed.), Topics in Applied Physics, Vol. 14, Chapt. 5, Springer-Verlag, Berlin, 1976. a s C .M. Penney, R. L. St. Peters, and M. Lapp, J . Opt. SOC. Amer. 64, 712 (1974). 35

98

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MEASUREMENT OF DENSITY

molecular gases, in mixtures of known composition, and in chemically reactive flows. We have shown as an example in Fig. 2 the variation of the “envelope” of this scattering with temperature for the rotational line manifold for N,. Again, the ratio of scattering intensities in selected spectral bands can display sensitive temperature dependence; this type of measurement has been shown to provide high quality atmospheric meas u r e ~ , ~ as ~ ’ well ~ O as data for fluids41and combustion s t u d i e ~ . ~ ’ - ~ ~ Rotational Raman scattering can be used advantageously to measure density of pure gases; in some experimental configurations, this can be accomplished with high s e n ~ i t i v i t y .Data ~ ~ can also be obtained in nonreactive gas m i x t ~ r e s * and ~ * ~in ~ chemically reactive although this latter measurement requires particular care because the rotational lines are typically numerous, closely spaced (with the exception of H2 and other very light molecules), and present with intensities that can vary substantially because of large temperature excursions. 3.2.3.6. Other Diagnostic Characteristics. One additional property of Raman scattering that is of potentially strong value for fluid dynamic studies is its sensitivity to thermal nonequilibrium conditions for test gases. This property becomes evident upon examination of the fact that the scattering signal is proportional to the population of the initial energy levels, and, therefore, if the molecular species is out of rotational or vibrational equilibrium, the Raman signature will be correspondingly affected. This desirable feature permits one to determine “population” temperatures of internal modes, a valuable attribute for the study of many types of gas flows. Our illustrations for Raman diagnostics here have all been for N2;other molecules of major fluid flow interest (O,, CO, CO, , HzO, . . . ) have, to varying degrees, similar spectra. The diatomics are all treated in a reaJ . Cooney and M. Pina, Appl. Opt. 15, 602 (1976). J . A. Salzman and T. A. Coney, NASA TN D-7679 (1974). J . Smith and W. H. Giedt, f n r . J . Heat Mass Transfer 20, 899 (1977). M. C . Drake and G . M. Rosenblatt, in “Characterization of High Temperature Vapors and Gases” ( J . W. Hastie, ed.), Vol. I , p. 609. National Bureau of Standards Special Publication 56111, 1979. 43 M . C. Drake, L. H. Grabner, and J . W . Hastie, in “Characterization of High Temperature Vapors and Gases” (J. W. Hastie, ed.), Vol. 2, p. 1105. Nat. Bur. Stand. Spec. Publ. 56 112, 1979. 44 W. D. Williams, H. M. Powell, R . L. McGuire, L. L. Price, J . H. Jones, D. P. Weaver, and J . W. L. Lewis, P r o p . Astronaut. Aeronauf. 58, 273 (1977). l5 J . J . Barrett, in “Laser Raman Gas Diagnostics” (M. Lapp and C. M . Penney, eds.), p. 63. Plenum, New York, 1974. M. C. Drake, G . J . Rosasco, R. Schneggenberger, and R. L. Nolen, Jr., J . A p p l . P h y s . 50, 7894 (1979). 40

‘‘

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ANALYSIS OF RAMAN A N D RAYLEIGH SCATTERED RADIATION

431

sonably similar fashion. The polyatomics have different features, but the spectroscopic details do not alter the basic measurement principles. Finally, we note that Raman scattering instrumentation is highly compatible with several complementary light scattering techniques. For example, the same laser can be used advantageously to pump a narrow-band tunable dye laser for observation of minor species such as OH, SOz, NO2, CH, CN, and NH, by fluorescence; often the same windows, lenses, mirrors, spectrometer, and detector systems will suffice for fluorescence measurements. Continuous wave (CW) lasers employed for time-averaged Raman measurements are also excellent for laser velocimetry. (See Section 1.1.4.) In situations where Raman signals are difficult to observe (e.g., very bright flames, or for time-resolved measurements in sooty flames) much of the same equipment can be used to observe strong nonlinear processes such as coherent anti-Stokes Raman scattering (CARS), to be discussed in the next section. Thus, it is possible to assemble a facility which enables one to obtain a powerful set of complementary optical measurements. 3.2.3.7. CARS Measurements of Temperature and Density. CARS is one of several non-linear optical processes that have received considerable attention for gas composition and temperature measurements during the last few yearss.10.47-58since its initial application to flame measurements .47 Although CARS requires significant additional equipment so-

‘’ P. R. Regnier and J. P. E. Taran, Appl. Phys. Lett. 23,240 (1973); P. R. Regnier and J. P. E . Taran, in “Laser Raman Gas Diagnostics” (M. Lapp and C. M. Penney, eds.) p. 87, Plenum, New York, 1974. “ S . A. J. Druet and J. P. E. Taran, in “Chemical and Biochemical Applications of Lasers” (C. B. Moore, ed.), p. 187. Academic Press, 1979. B. Attal, M. Pealat, and J. P. Taran,AGARD Conf. Proc. No. 281, Testing and Measurement Techniques in Heat Transfer Combustion, Chapt. 17. Published by Advisory Group for Aeronautical Research and Development, NATO, 1980; available from NTIS. J. W. Nibler, W. M. Shaub, J. R. McDonald, and A. B. Harvey, in “Vibrational Spectra and Structure: A Series of Advances,” Vol. 6 (J. R. Durig, ed.), Chapt. 3. Elsevier, Arnsterdam, 1977. 51 A. C. Eckbreth, R. J. Hall, and J. A. Shirley, AGARD Conf. Proc. Testing and Measurement Techniques in Heat Trunsfer and Combustion, Chapt. 18. Published by Advisory Group for Aeronautical Research and Development, NATO, 1980; available from NTIS. 52 L . P. Goss, J. W. Fleming, and A. B. Harvey, Opt. Lett. 5 , 345 (1980). 53 A. C. Eckbreth, Appl. Phys. Lett. 32, 421 (1978). s4 A Compaan and S. Chandra, Opt. Lett. 4, 170 (1979). 55 W. B. Roh, P. W. Schreiber, and J. P. E . Taran, Appl. Phys. Lett. 29, 174 (1976). 56 A. C. Eckbreth, Combust. FIame 39, 133 (1980). 57 G. L. Switzer, W. M. Roquemore, R. B. Bradley, P. W. Schreiber, and W. B. Roh, Appl. Opt. 18,2343 (1979); G . L. Switzer, L. P. Goss, W. M. Roquemore, R. B. Bradley, P. W. Schreiber, W. B. Roh, AIAA Paper No. 80-0353 (1980). s8 B. Attal, M. Pealat and J. P. E. Taran, AIAA Paper No. 80-0282 (1980). @ ‘

43 2

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MEASUREMENT OF DENSITY

phistication beyond that used for spontaneous Raman scattering, it appears to have developed into a reliable technique for time-resolved temperature measurements in gases which are highly luminous and/or carry a high density of particles, such as a sooting flame. In a typical CARS configuration, a high power laser pulse is propagated through the gas region to be observed. Before this “pump” beam passes through the gas, part of it is diverted to excite a second laser, whose output is adjusted to be at the Stokes Raman wavelength produced by the first laser in the species to be observed. This Stokes beam is combined with the pump beam so that both propagate essentially colinearly, overlapping in the measurement region. As a result of a nonlinear optical interaction between the observed molecules and the two beams, a third beam is generated at the corresponding anti-Stokes wavelength. This beam is well collimated, propagating in a direction almost coincident with the two original beams. The anti-Stokes beam can be isolated by a prism, optical stops, or filters. In a variant of CARS called BOXCARS,53 which provides improved spatial resolution, the pump beam is split into two intersecting beams. The Stokes beam is directed through the intersection region at a slight angle to one of the pump beams, generating the anti-Stokes beam at a slight angle to the other pump beam. Another promising variation of the conventional CARS geometry is based upon use of three incident laser frequencies in a counterpropagating a ~ r a n g e m e n t . ~ ~ A major advantage of the CARS technique over ordinary Raman techniques is that much lower pulsed laser energies are required to generate useful signals from major species. (The total energy in the CARS pulsed laser beams is typically only about 1 to 10% of that required for a comparable Raman measurement.) Thus, signals can be obtained from the sooting region of a flame with less danger that the incident laser beams will significantly perturb the measured region. Furthermore, the tight collimation observed in the anti-Stokes beam allows much stronger discrimination against natural or laser-induced luminosity. One successful method for CARS temperature measurements involves broadening the spectral extent of the Stokes beam so that several hot bands can be observed simultaneously, using a multichannel optical det e ~ t o r . The ~ ~ resulting anti-Stokes spectrum is more complicated than the ordinary Raman spectrum but it displays a similar temperature sensitivity, allowing temperature to be calculated from the ratio of observed bands. Concentration measurements can be based on the intensity of the CARS signal, which is proportional to the square of the density of the observed species. However, this intensity also depends on the Raman line-

3.2.

ANALYSIS OF R A M A N A N D R A Y L E I G H SCATTERED RADIATION

433

shape and the degree of overlap of the pump and Stokes beams. Since the former depends in turn on the measurement zone temperature, composition, and gas pressure, whereas the latter can be affected by turbulence, concentration measurements from CARS intensities in fluctuating environments are somewhat subject to uncertainty. However, new work may help to relax this difficulty by allowing concentration to be determined from CARS bandshapes (which are sensitive to concentration because of interaction with the electronic susceptibility background);51 another method uses this nonresonant susceptibility signal as an in situ referen~e.~~ In addition to the equipment sophistication and concentration measurement difficulties mentioned above, there are several other potential problem areas for CARS diagnostic applications:

(i) A double-ended (or multiple-ended) optical configuration is generally required, involving both entrance and opposing observation windows. (ii) It is difficult to monitnr two or more species with well-separated Raman bands simultaneously, since each species needs its own Stokes beam. (iii) Since the CARS signal is proportional to the square of the concentration, sensitivity decreases rapidly with decreasing concentration. Consequently, minor species (those with molar fractions smaller than 0.1%) are generally difficult to determine with CARS. (iv) CARS spectra depend in a fairly complicated way (relative to ordinary Raman scattering) on both molecular constants and the local environment. These spectra must be calculated or measured as a function of environmental factors in order to interpret measurements. Despite these difficulties, impressive results have been obtained in CARS applications to date. For example, consistent data have been obtained in independent measurement series within combustors typical of gas turbine^.^^-^^ Furthermore, potential ways to overcome the remaining difficulties are being investigated. Thus, the dependence of CARS and stimulated Raman scattering signals on experimental parameters has been established well enough to allow detailed comparison of measurement capabilities between these techniques and the spontaneous Raman scattering methods discussed earlier.6o These developments indicate that CARS and related optical techniques are likely to be of high value for probing difficult experimental environments. 58

R. L. Farrow, R. E. Mitchell, L. A. Rahn, and P. L. Mattern, A I A A Paper No. 81-0182

( 1981 1. Bo L. A. Rahn, P. L. Mattern, and R. L. Farrow, Symp. ( I n t . ) Combust. [Proc.], 18th. Combustion Institute, Pittsburgh (to appear).

3. MEASUREMENT

434

OF DENSITY

3.3. Measurement of Density by Analysis of Electron Beam Excited Radiation* The electron beam fluorescence (EBF) technique has been widely used for the measurement of specie concentrations and temperatures in low density gas flows for about fifteen years. It has been most useful at densities of atoms or molecules below n = 1OI6 ~ m - although ~, there have been several recent investigations reaching, in one case, as high as n = 10l8~ m - ~ . In the EBF technique a narrow (= 1 mm diameter) collimated beam of energetic (10- 100 keV) electrons is passed through the gas flow of interest. The beam electrons have inelastic collisions with a small proportion of the gas atoms or molecules, causing electronic excitations, ionizations, and dissociations. Those atoms or molecules that are raised to an unstable electronic configuration subsequently spontaneously emit fluorescent radiation while decaying to a lower energy level. If the gas density is reasonably low, say below that equivalent to a pressure of 500 Pa at room temperature ( n S 5 x 10l6 cm-9, an electron beam with an energy around 50 keV will not be significantly attenuated over a distance of 10 cm. Generally, the beam is visible as a thin filament of fluorescence. With suitable optics, the intensity and spectral distribution of the fluorescence can be measured at any chosen “point” along the beam. The size of the “point” is determined roughly by the diameter of the fluorescent filament and the length of the segment selected for observation. The sketch in Fig. 1 illustrates a typical EBF experimental arrangement. With suitable interpretation, it is possible to deduce directly from the intensity and spectral distribution of the fluorescence the state of the gas at the point of measurement. Among the properties that can be measured (although not necessarily in all gases) are specie number density, rotational temperature, vibrational temperature, and translational temperature. It is also possible to measure flow velocity and to provide flow visualization. In addition, nonequilibrium population distributions have been measured. Since the beam electrons do not significantly disturb the motions of the gas particles, the technique provides an almost ideal nonperturbing probe. The principal difficulties with the technique are associated with the reliability of the interpretation of the fluorescence in terms of the properties of the gas. Although the beam induced excitation-emission situation is

* Chapter 3.3 is by

E. P. Muntz.

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

435

FIG.1 . Outline of arrangement for observation of EBF in a gas flow.

about as simple a version of this type of phenomena as can be imagined, there are nevertheless many complicating factors which can lead to substantial errors without correct analysis. Compared to laser scattering measurements of similar gas flow properties (see Chapter 3.2), the E B F technique has the advantage that the collision cross sections of the gas particles are greater for electrons than photons, thus leading to more intense fluorescence. This is important at very low gas densities where the intensity can be extremely weak. It is also important in higher density experiments designed to study density and temperature fluctuations in turbulent flows, when a short sampling time is required. The application of electron excitation to the measurement of gas density at low levels was initiated in 1955 by a suggestion of Schumacher and Griin in a German patent application. This was followed, in 1958, by a report of a preliminary investigation of the technique by Schumacher and Gadamer.' It was not until 1962, however, that Gadamer* reported a study, using air, which verified a predictable and understandable relationship between fluorescent intensity and gas density. This relationship had the form of a typical quenching curve such as shown in Fig. 5 and described by Eq. (3.3.14) of Section 3.3.2.3. Starting from very low densities, the fluorescent intensity at first increases approximately linearly with density. With a further increase in density radiationless quenching collisions become relatively more important with the result that the intensity becomes less sensitive to density change. Finally, the intensity apB . W. Schumacher and E. 0 . Gadamer, Cun. J . Phys. 36, 659 (1958).

* E. 0 . Gadamer, Ulniv. Toronto Insr. Aerophys. Rep. 83 (1962).

436

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MEASUREMENT OF DENSITY

proaches an asymptotic limit and becomes totally insensitive to density change. This characteristic limits the usefulness of the method to low densities. Development of a general relationship basic to density measurement, of which Eq. (3.3.14) is a useful simplification, is contained in Sections 3.3.2.1-3.3.2.3. Slightly before electron beam excitation was proposed as a means of measuring gas density, Griin3 in 1954 had looked qualitatively at the changing rotational structure of CO in free jets, suggesting that this might provide a way of indicating temperature levels. This possibility was not investigated quantitatively, however, until the work of M u n t ~published ,~ in 1962. Muntz showed, by observation of the first negative bands of nitrogen (N;) produced by an electron beam, that the intensity distribution of the fine structure was predictable as a function of the rotational and vibrational temperatures of the molecules prior to their excitation. This established EBF as a temperature measuring technique, but numerous later studies5-28led t o refinements. The use of EBF for measuring temperatures is described in Section 4.2.3. A. E. Griin, Z. Nuturforsch., Teil A 9, 833 (1954).

‘ E. P. Muntz, Phys. Fluids 5 (I), 80 (1962).

E. P. Muntz and D. J. Marsden, in “Rarefied Gas Dynamics” (J. A. Laurmann, ed.), Vol. 2, p. 495. Academic Press, New York, 1963. E. P. Muntz and S. J. Abel, Hypervelvcity Tech. Symp., Jrd, Denver, 1964. E. P. Muntz, S. J. Abel, and B. L. Maguire,/EEE Trans. Arrosp., Suppl. p. 210 (1965). * S. L . Petrie. Arronaur. Rrs. Lob. ARL-65-122(1965). D. I. Sebacher and R. J . Duckett, NASA Tkch. R e p . R-114(1964). lo F. Robben and L. Talbot, Phys. Fluids 9 (4), 644 (1966). I I E. P. Muntz, in “Rarefied Gas Dynamics” (J. H . d e Leeuw, ed.). Vol. 2, p. 128. Academic Press, New York, 1966. D. J. Marsden, in “Rarefied Gas Dynamics” ( J . H. de Leeuw, ed.), Vol. 2, p. 566. Academic Press, New York, 1966. I3 P. V. Marrone, Phys. Fluids 10, 521 (1967). R. S. Hickman, U . S . C . A . E . 104 Sept. (1966). W. W. Hunter, Jr., I S A Prepr. 16 (12-4-66) (1966). IRH. Ashkenas, Phys. Fluids 10, 2509 (1967). ” B. L. Maguire, in “Rarefied Gas Dynamics,” (L. Trilling and H. Wachman, eds.), Vol. 2, p. 1761. Academic Press, New York, 1969. I’ R. B. Smith, in ”Rarefied Gas Dynamics” (L. Trilling and H. Wachman, eds.), p. 1749. Academic Press, New York, 1969. la S. L. Petrie and A. A. Boiarski, in “Rarefied Gas Dynamics” (L. Trilling and H. Wachman, eds.), Vol. 2. p. 1551. Academic Press, New York, 1969. *O D. C. Lillicrap and J . K. Harvey, A / A A J . 7 ( 5 ) . 980 (1969). D. C. Lillicrap and L. P. Lee, NASA Tech. Nore D-6576(1971). ** F. Shelby and R. A. Hill, Phys. Fluids 14 (I I ) , 2543 (1971). 23 W. C. Ho and G. Schweiger, Phys. Fluids 15 (8), 1447 (1972). 14 A. E. Kassem and R. S. Hickman, AIAA J . 13 (6), 770 (1975).

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ANALYSIS OF E L E C T R O N BEAM E X C I T E D R A D I A T I O N

437

Flow visualization has been accomplished both by using short lifetime fluorescence emanating directly from the exciting electron and by employing afterglow emissions produced by upstream electron excitation.34-36 This is discussed in Chapter 2.6, Section 2.6.1 of Part 2 of this volume. There have been a number of rather special applications of the EBF technique, such as in studies of time varying turbulent flows at relatively high densities ,37-43 in investigations of high enthalpy flows where species such as NO and 0 are e x p e ~ t e dand ~ ~in ' ~high ~ altitude experiments using rockets carrying EBF a p p a r a t ~ s . ~ ~ , ~ ~ There are two major reviews of the EBF technique. The first by M ~ n t is z ~current ~ through 1968. The second by Butefisch and Vennemann4eis current through 1973. An excellent minireview of the use of the technique at high densities has also been published by Smith and Dris~011.4' Chapter 3.3 deals primarily with EBF as a means of measuring the den-

W. D. Williams,AEDC TR 68-265 (1969). S . L. Petrie, S . S. Lazdinis, and A . A. Boiarski, AIAA Pup. No. 69-329 (1969). 27 S. S. Lazdinis and S. L. Petrie, AIAA Pup. No. 72-683 (1972). 28 C. Dankert and K. A. Biitefisch, in "Rarefied Gas Dynamics" (M. Becker and M. Fiebig, eds.), Vol. 1, p. 820. DFVLR Press, Porz-Wahn, 1974. pa D. Rothe, AIAA J. 3 (lo), 1945 (1965). B. L . Maguire, E. P. Muntz, and J. R . Mallin, IEEE Trans. Aerosp. Electron. Syst. aes-3 (2), 321 (1967). 31 R . S. Hickman, Bufl. APS 11,612 (1966). K. A . Biitefisch, Dtsch. Luft- Raumfuhrt, Forschungsber. DLR-FB-69-63 (1969). 53 H. F. Lee and S. L . Petrie, J. Aircr. 10 (4), 239, April (1973). 34 A. E. Griin, E. Schopper, and B. Schumacher, 2. Angew. Phys. 6 (3, 198 (1954). 35 D. I. Sebacher, J. Chem. Phys. 44 ( l l ) , 4131 (1966). 36 L. M. Weinstein, R . D. Wagner, Jr., and S . L. Ocheltree, A f A A J. 6, 7 (1973). 37 A . G. Boyer and E. P. Muntz, AGARD Conf. Proc. No. 19 (1967). 38 J. E. Wallace, AIAA J . 7 (4), 757 (1969). B. L. Maguire, E. P. Muntz, and K. M. Thomas, AIAA Pap. No. 72-118 (1972). W. N. Harvey and W. W. Hunter, Jr., NASA Tech. Note D-7981 (1975). 'I J. A . Smith and J. F. Driscoll, J. FIuid Mech. 72, 2 (1975). " A . D. McRonald, Ph.D. Thesis, University of Southern California, Los Angeles (1975). S. S. Lazdinis, AIAA J . 14 (2), 133 (1976). S. L. Petrie and J. J. Komar, AFFDL TR 74-8 (1974). S. L. Petrie, A . A. Boiarski, and S. S. Lazdinis, A f A A Pup. No. 71-271 (1971). 4~ J. H. d e Leeuw and W. E. R. Davies, Cun. J . Phys. 50 (19), 1044 (1972). A . A. Haasz, J . H. de Leeuw, and W. E. R . Davies, J . Geophys. Res. 81 (13), 2383 (1976). E. P. Muntz, AGARDOgruph 132, (1969). 4s K. A. Biitefisch and D. Vennemann, Prog. Aerosp. Sci. 15, 217 (1974). 2s z6

43 8

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MEASUREMENT OF DENSITY

sity of atoms or molecules of a single species, both in a pure gas and in a mixture with other species. However, a number of the topics considered -such as the mechanisms by which fluorescence is excited and quenched, electron beam generation, beam spreading and the effects of gas flow-are also pertinent to other applications of the EBF technique. (See Section 4.2.3 for temperature measurement.) 3.3.1. High Probability Transitions for Excitation and Emission: Selection Rules

When an atom or molecule is struck by an electron with an energy much greater than the ionization potential of the molecule, any of many products may result. The molecule may be left in a normal or excited energy state of its neutral, singly ionized, or multiply ionized spectrum. Or, it may be dissociated, with any of the many possible ramifications. From an excited energy state the molecule may spontaneously decay to a lower state with the emission of fluorescent radiation or it may be removed from the excited state by a competing process-such as a quenching collision with another molecule. Analysis of electron beam fluorescence begins by considering radiation of a particular frequency, which is tantamount to considering a particular pair of energy levels as excited and final states. A third energy state on which analysis focuses attention is the initial state from which the molecule is excited-the ground state or at least a low lying level. These three states determine an excitation-emission path transition from initial to excited state and from excited to final state with fluorescent emission. Although the number of possible excitation-emission paths is large, most are unimportant because of low transition probabilities. Some of the factors on which transition probabilities depend are considered in this subsection. Two criteria basic to the choice of an excitation-emission path are that the resulting fluorescence be intense and that there be a short delay time between excitation and emission. As remarked previously, intensity is important because the fluorescence is at best weak, requiring sensitive measuring equipment, particularly if the observation time is limited. A short delay time is needed because during an extended delay the active molecules are blown downstream and the observed intensity at a particular position has the undesirable feature, for density measurements, of being dependent on flow velocity. To satisfy these criteria the transition probabilities for both emission and excitation should be high. As far as emission is concerned, the transition should not disobey any important optical selection rule. In this case the probability of emission will be greater than about 10' s-l.

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

26 r

'S

'P

'D

3s

'F n

439

'p n

4-T 3-'.lj*/

/ /

2-

2-

FIG.2. Energy level diagrams showing states important for excitation and emission of EBF in helium. Observed relative intensities of the lines in visible spectrum excited by an 18-kV electron beam. 3'P + 2IS, 100; 4'P 2'P. 1 .

+

2 9 , 15; 4lD + 2lP, 10; 5ID --* 2'P, 5; 4 5 +

With regard to excitation, the transition should be produced mainly by direct electron impact, rather than by some multiple step process. For direct excitation the collision cross section (probability of causing transition) will depend on the process involved (transition to excited state of neutral molecule, ionized molecule, dissociated fragment, etc.) as well as the electron energy and quantum properties of the initial and final states. Consider first excitation to states of an unionized molecule. Helium, which is often used in EBF studies, provides an illustration for this case. Figure 2 shows several types of excitation-emission path and Figure 3 contains excitation curves for three specific paths.50 For each type the excitation is from the ground state, l'S, to an excited state from which a representative fluorescent transition originates. The first type (A), with a fluorescent transition such as from 3lP to 2lS, has an excitation transition from 1'sto 3IP. In this case the excitation transition is characteristic of a type in which the optical selection rules are obeyed (AS = 0, AL = 1). The second type (B), for a typical fluorescent transition from 43S to 23P, has an excitation transition from 1's to 43S. Here the excitation transition is of a type in which the selection rules for both multiplicity and orbitai angular momentum are broken (AS# 0, AL # 1). The third type (C), with fluorescent transition from 4lD to 2lP, has an excitation transition from 1's to 4'D. This is an intermediate type in which the selection rule 50 B. L. Maguire, in "Rarefied Gas Dynamics" (C. L. Brundin, ed.), Vol. 2, p. 1497. Academic Press, New York, 1967.

440

3.

MEASUREMENT OF DENSITY

for multiplicity is not broken, but the one for orbital angular momentum is broken (AS = 0, AL # I). For low energy electrons, comparison of cross sections for the three types is complicated by the presence of maxima in the excitation curves. But for electrons with energies greater than those at which the maxima occur, say 3 keV, Fig. 3 indicates that the cross section is greatest for type A (optical selection rules obeyed), least for type B (important selection rules broken) and intermediate for type C (only rule for orbital angular momentum broken). Furthermore, these differences in cross section become relatively greater as the electron energy is increased further, since a cross section of type A varies as E;l In E,, type B as EL2 and type C as E;’, where E, is electron energy. This guide, that the cross section is largest when the optical selection rules are obeyed, holds for primary electrons of an EBF probe, since their energy must be high (> 10 keV) to avoid excessive beam spreading and attenuation. Also, collisions by primary electrons are usually the principal cause of excitation, but secondary electrons which are produced by ionizing collisions of the primaries and have low energies (a few electron volts) may contribute, and for some transitions the excitation is almost exclusively by secondaries. (See Section 3.3.2.1). A case in point is shown in Fig. 4a for the neutral nitrogen molecule. For the transition

z (C) w 2

,

,

3 5 60 400 3000 EXCITATION VOLTAGE

FIG.3. Excitation cross sections of types of excitation-emission paths in helium. Type A: 5016 8, (2lS-3’P); type B: 4713 8, (2”P-4”S);type C: 4922 A (2IP-4lD).

3.3.

I

I1 z 10

W

r

441

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

n

~

I

I

EXCITATION ---*

2nd POSITIVE

I

VEGARD KAPLAN

(0)

v,"

-

I

N,X'Z,f

-

(b)

FIG.4. Energy level diagram showing states important for excitation of emission of EBF in nitrogen. (a) Neutral molecule, N,; (b) neutral-to-ionized molecule transition, Nz, N:.

indicated, XlZ; to CW,,the optical rule that there should be no change in multiplicity is broken, but for low energy secondary electrons the cross section can, as illustrated for helium in Fig. 3, be very large compared to the cross section for excitation of the same transition by the much higher energy primaries. Transition to an excited state of an ion is a more complicated process, since not only is the electron configuration changed but in addition at least one electron is ejected. The usual optical selection rules can be applied providing one accounts for the loss of spin associated with the ejected e l e ~ t r o n . For ~ the nitrogen molecule, which like helium is often used in EBF investigations, a suitable fluorescent transition is from an excited state of the ion. Fig. 4b shows a commonly used excitation-emission path. Although a large cross section for electron excitation and a high probability of spontaneous emission are prerequisites for a suitable excitation-emission path, they are only two of many factors affecting the intensity of the fluorescence. Other factors are considered in the following subsections. 3.3.2. Equations Connecting Fluorescent Intensity to Gas Density The fluorescence excited by a beam of electrons in a gas has two properties that are easy to measure: the spectral composition of the emission and the intensity of prominent lines. The spectrum gives an indication of the gas species that are present, although not invariably because of possible dissociative excitations. The intensity of suitably chosen lines or

442

3.

MEASUREMENT OF DENSITY

bands is directly related to the excited state population which in turn is related to the number density of atoms or molecules through which the electron beam passes. The emission intensity also depends on the beam current and electron energy. In Section 3.3.2 the relationship between fluorescent emission intensity and gas specie concentration is developed. Consider the following situation. Assume that spectrally resolved observations are made of the fluorescence excited by a homogeneous beam of electrons. Also, assume that a particular spectral characteristic is selected that originates from an electronic state j. If the populating and depopulating processes for the state j are in equilibrium, there is a steady j state number density n , . The emission intensity in a transition from j to a lower state k is ljk = h c v j k A j k n j , (3.3.1) where Vjk is the wave number and AJkis the Einstein spontaneous emission coefficient. Since for a given transition, nj is the only quantity that can vary (leaving out possibilities of the presence of strong magnetic and electric fields) in Eq. (3.3.1), ljkis a measure of n , . Consequently, if n, can be predicted as a function of the ground state number density n, the density can be measured. Notice it might also be practicable to measure an absorption using the state j as the lower state with the local concentration of absorbers nj. 3.3.2.1. Mechanisms for Populating the Excited State. The several populating mechanisms for the state j are: primary electron collisions, secondary electron collisions, photon absorption, collision with excited particles, cascade population, transport of excitation along the beam by secondary processes, convection due to gas flow, and diffusion of excited species. These are now considered in more detail. 3.3.2.1.1. PRIMARY ELECTRONS. For a gas of number density nB the primary excitation rate due to the energetic electrons in the electron beam is

(3.3.2)

where n, and v, are the number density and velocity of the primary electrons and Qoj(u,) is the cross section for excitation from the ground state * excito j. Values for Qoj(v,) can be found in the review by M ~ n t z . ~No tation from other electronic states created by the beam electrons to the j state, is considered. This is in accord with the usual situation that the number densities in these states are low and their excitation cross sections are generally smaller than Qoj. Kassem and H i ~ k r n a n however, ,~~ have pointed out that this may not always be the case when the beamcreated electronic state happens to be associated with an ion.

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

443

The consideration of transitions only from the ground state may not be sufficient for excitation in high temperature gases; to allow for such a case, a sum over all states in the gas flow under investigation, i.e., QZ,, would replace Q o , . Since, for temperatures high enough to make this consideration important, there are other problems, such as selfluminosity of the gas, the excitation cross section is taken to be Qo,. Note that this argument applies only to the case of electronic transitions. 3.3.2.1.2. SECONDARY ELECTRONS.The secondary electrons' role in exciting certain transitions that are observed in electron beam fluorescence investigations is extremely important. The production rate of secondary electrons is (3.3.3) J

where (&)k is the total ionization cross section (including multiple ionizations) for the kth gas species. Ck is over all species of number density nk leading to ionization. Of the secondaries that are produced, only those with energy sufficient to excite the jth state need be considered. The number of secondaries with sufficient energy is qes

jui

f ( u s ) dus

(3.3.4)

wheref(u,) is the secondary velocity distribution function and u,* is such that irnv;' = E j . At this point, the description of the secondaries' action becomes complicated, even assuming that the total ionization and original distribution function of the secondaries are both known. To obtain the secondaries distribution nesf(u,) at some point in or around the electron beam, a complicated electron diffusion equation must be solved including all the inelastic cross sections that can lead to depletion of the secondary electron population. If the number of secondaries ejected as a function of angle from the beam direction is uniform, as is reasonable, and secondary electron self-collisions are neglected, the problem can be simplified. For a velocity us corresponding to energy E, the total inelastic cross section is Q ~ ( V , )Say . the average energy loss per inelastic collision is AE, which corresponds to a Avs. The mean free path of the secondary until it drops below a velocity u,* is, if its initial velocity is v i , (3.3.5)

The number of excitations along the path of a secondary electron can be

3.

444

MEASUREMENT OF DENSITY

found. The excitation rate of the secondaries psjbecomes for u: > v,* and n, constant for all A,

3

ngneve9'.

(3.3.6)

The ratio of secondary to primary excitation, using Eq. (3.3.6) becomes,

(3.3.7) For pure nitrogen (refer to Fig. 4 for excitation diagram) enough information is available to estimate that psj/pp,has a value for 50 keV primary electrons of about for the allowed excitation transitions to the u' = 0 state of N$B2C. On the other hand, for excitation of the N2C311state the excitation would all be due to the secondaries since a change in multiplicity is involved. In this situation only relatively low energy electrons (secondaries) will have a significant effect and thus (oSj % ppj for this state. So far, the effects of geometry have been ignored with Eq. (3.3.6) implying observation of all secondary excited emissions. Some feeling for this may be obtained by estimating A and comparing it to the beam or flow field dimensions. For nitrogen a typical cross section QITmay be about 5 x 10-le cm2 for 50-eV electrons which leads to a secondary mean free path somewhat less than 1 mm at 100 Pa pressure and room temperature. Thus, at 10 Pa the secondary electrons would fill a significant volume of a typical flow field since they can be considered to be ejected with an approximately uniform distribution for angles measured from the beam direction. Results given by Camacsl indicate that the secondaries have a range of about 0.5 mm at an N2pressure around 270 Pa. which is consistent with the present estimate based on QIT= 5 X cm2. The effect of the secondary path length compared to the dimension of the region from which light is accepted by an optical system is critical. If the emission is sampled from a very small volume using dimensions much smaller than a secondary mean free path the excitation rate that is observed (rpsj, say) will be proportional to the number of secondaries of sufficient energy (thus proportional to &{(QIT(De))knk} J&f(u!) duz and the number density ng). For a pure gas, & reduces to one term with Itk = n, so pqjis proportional to ni and by analogy to Eq. (3.3.6), Y is proportional to n, . On the other hand, if all the secondaries are observed (a short sec51

M.Camac, A l A A Pup. No. 68-722 (1968).

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

445

ondary free path), the effective excitation rate for a pure gas will be, from Eq. (3.3.6) cpsl ng (thus Y = const) since the denominator in Eq. (3.3.6) contains n, in this case. Thus when secondary excited emission is observed, the range of the secondaries compared to the observation volume must be considered very carefully in the design of an experiment. There is one other complication which appears in gas mixtures and was first recognized by Grun and S c h ~ p p e under r ~ ~ somewhat different but similar circumstances to those encountered in application of the fluorescent probe. The observation that led to identifying the phenomena was made when a beam of fast particles was passed into argon at 10 to 15kPa. A small percentage of N2 (3 percent) was added and the light output from the mixture increased by a factor of ten. The increase was ascribed to the large number of secondaries that are present exciting the C311 state of the nitrogen. The Nz second positive system (C3n upper state) in emission produces the large increase. A related effect was also demonstrated by Grun and Schopper5*for secondary excited emission from mixture of 8.7 percent N2in A at 80 kPa when a small amount of O2 was added. In this case the emission was reduced significantly, at least in part due to the interception of secondaries by the oxygen. All the observations corresponded to a situation where the secondary range was small and an excitation equation like Eq. (3.3.6) would apply. In the case of the few percent nitrogen in argon the &{(&(Ve))knk} would be large, whereas, for the oxygen addition the term in the denominator of Eq. (3.3.6) apparently becomes large. The possible interchange of energy and thus excitation, with the secondary electrons acting as transfer agent between different gas species, is the phenomenon that must be emphasized. The whole process has not been investigated in any detail for applications of the fluorescent probing technique. The possible consequences of this type of interchange should be considered in any EBF probe application. There are unfortunately very few quantitative data that can be used for estimates of the exchange effect. 3.3.2.1.3. PHOTONABSORPTION. Here there are two possibilities: (i) the absorption of electron beam generated resonant photons by surrounding unexcited molecules, and (ii) the absorption of beam fluorescence by molecules which have a metastable lower state as the photons and metastables diffuse away from the excitation zone. The second situation is not too important because of relatively low metastable populations. However, R ~ t h believes e ~ ~ he has observed this effect in argon.

-

A . E. Griin and E. Schopper,Z . Naturforsch. Teil A 9, 134 (1954). D. E. Rothe, Phvs. Fluids 9 (9), 1943 (1966).

446

3.

MEASUREMENT OF DENSITY

The first situation is encountered when the excited state j can decay to the ground state by emission of resonant photons. The behavior of the absorption is completely analogous to secondary electron excitation except that it is very sharply tuned to the resonant wave number. Energy exchange in mixtures by this means is extremely unlikely. Intensity of emission excited by resonant photons will have a density sensitivity similar to the secondary electrons. For observation on a scale much smaller than a resonant photon mean free path, resonant photon excitation will vary as n:, whereas for dimensions much greater than a mean free path it varies as n,. The resonant photon excitation rate will be called (Prphj. 3 . 3 . 2 . 1 . 4 . COLLISIONS W I T H EXCITED PARTICLES. Certain states j of a molecule or atom can be excited by collisional exchange with excited particles in some other state r. If there is a steady population density n, of state r particles, the collisional population rate to j will be proportional to ngnr. If n, is produced by primary excitation in the beam and it is assumed that the r, j collision is the only r state depopulator (thus, r is a metastable state), an expression for n, can be found.4x The population rate for j due to collision is48 (3.3.8)

This is only true if there are no convection or diffusion effects and all the activating collisions occur within the scale of the observation of the j state. There are clearly endless opportunities for complicating the picture. It is perhaps only necessary to keep the possible complications in mind, so that they can be avoided. Collisionaly excited emission is primarily useful to produce afterglow for flow ~ i s u a l i z a t i o n . ~ ~ - ~ ~ 3 . 3 . 2 . 1 . 5 . CASCADEPOPULATION. Population of the state j can occur by cascade from higher electronic states. Since the emission that is observed when applying the fluorescence technique usually corresponds to a strong excitation transition the contribution from higher states is generally small. Also, if it is at all significant it will likely be a result of cascade from states excited by direct excitation processes. Thus, the population of j in this manner will behave in essentially the same manner as if the state were excited directly from the ground state but by a slightly slower than expected process. 3.3.2.1.6. TRANSPORT OF EXCITATION ALONG A BEAMBY SECONDARY PROCESSES. If only excitation by secondaries or resonant photons is considered, density gradients in the probed gas in the direction of the electron beam can be studied with a resolution of only the order of the secondary or photon mean free path. High density close to a lower density region

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

447

could cause a high flux of, say, secondary electrons to excite an enhanced emission in the region of the lower density. 3.3.2.1.7. CONVECTION A N D DIFFUSION. Even for fast excitationemission processes, which may take only lo-* sec, there will be flow velocities at which the excited j molecules drift downstream a significant distance before emitting. Thus, depending on the location of observation, convection of the j state population could result in a populating or depopulating mechanism. Diffusion of excited particles could also, at very high temperatures, be significant in either populating or depopulating the j state in a particular region. For applications of the fluorescent probe these problems have been avoided by adjusting conditions or choosing the frequency of the radiation studied to preclude effects of either diffusion or convection. 3.3.2.2. Mechanisms for Depopulating the Excited State. Depopulating mechanisms operate simultaneously with the exciting mechanisms just discussed. These are spontaneous transitions other than the emission of interest, spontaneous and stimulated transition to state k producing the emission that is being studied at wavelength hlk, quenching collisions, and convection and diffusion which have already been discussed and need not be discussed again. 3.3.2.2.1. SPONTANEOUS TRANSITIONS OTHER THAN THE EMISSION OF INTEREST. This is simply the sum of all the spontaneous transition probabilities from the state j , apart from the transition probability to state k(Al,k). The sum will be designated by A T . 3.3.2.2.2. TRANSITION OF INTEREST. Spontaneous transitions from j to k with probability Alk are those providing the observed emission. Typically this will be an allowed electric dipole transition with a probability of around 108/sec. Stimulated emission is negligible at the low level of excited molecule densities involved. 3.3.2.2.3. QUENCHING COLLISIONS. Competing with the spontaneous emission processes, particularly at high densities, are quenching collisions. For the purposes of the fluorescent probe, a quenching collision is one that removes a molecule in the j state by transfer of all or part of its excitation energy in a radiationless collision with an ambient gas atom or molecule. Radiationless here means that no emission corresponding to a j + k transition is observed. In general only quenching by ground state species need be considered, as excited species will usually be present in such small concentrations that they have no significance in quenching phenomena. The quenching collision rate can be written as

40= C { W Q d T ) { 2 W m i + m 3 / ~ m i m 3 ~ ’ ~ 3 ) , 1

(3.3.9)

3.

448

MEASUREMENT OF DENSITY

where the quenching is summed over all the 1 ambient species in the gas of mass mg . Here the subscript g refers to the specie that is under observation and which has the state j. For a pure gas the sum reduces to one term in which nz = n, and ml = m y . 3.3.2.3. Relationship between Fluorescent Intensity and Gas Density in the Steady State. With all the population and depopulation mechanisms operating, a balance equation for the state j can be written. In particular, the number density nj at a location inside a relatively large electron beam will be found. Using the expressions developed before for the populating mechanisms, the population for the j state is p1 =

(Ppj

+ PSI +

VrPhj

+

(Prj

+

Pcj

+

(PdJ 9

(3.3.10)

where pCjand cpdj are the convection and diffusion contributions. The depopulation rate is given by Aj

= n d J k + nfiT

+ AJO +

+

Aid?

(3.3.1 I )

where Ajd and Aj,. are the depopulating diffusion and convection terms. For a steady state t o exist

v3 = 4-

(3.3.12)

For illustration, one set of conditions frequently encountered in fluorescence probing will be considered. If secondary electron, resonant photon, or collisional excitations are important they will have mean free paths much smaller than the scale of observation. Also, if flow velocity changes in the flow direction are not large on the scale of a characteristic emission time, or collisional excitation time, the convective populating and depopulating terms will vanish. The diffusion term will also be neglected with no error if there are no major changes in density or temperature along or across the beam over a characteristic distance corresponding to the emission time at local thermal velocities, or if the scale of observation in the beam direction is larger than this characteristic distance. With these conditions, Eq. (3.3.1) [with Eq. (3.3.12) used to determine nJ]becomes for the fairly common case of a pure gas with no collisional excitation term

Y is defined in Eq. (6).

3.3.

ANALYSIS OF ELECTRON B E A M EXCITED RADIATION

i

449

/

/

,’ //

--

n,*

FIG. 5. Characteristic variation of EBF emission intensity with gas density.

n!?

For a constant beam current density n,v, and constant beam energy (u, = const),

(3.3.1 4) where K is a constant. The term {(Alk + AT)/2Q,,VgJkT is equivalent to p’ used by G ~ i i and n~~ Camac’sS1k7 is its reciprocal. Also, = (4kT/!JmK)”2. The characteristic shape of the emission intensity curve is shown in Fig. 5. At low densities the emission 1jk is essentially directly proportional to ng . As n, increases the term 2ngQg,vg,/(Ajk + AT) becomes significant and eventually much greater than unity. The emission saturates at a value

vKg

(3.3.15)

If the saturation level and the initial linear rise (Ijk= Kn,)are extrapolated to intersect, the intersection occurs at np“ =

(All,

+ AT)/~Q,,~~K

(3.3.16)

where np* can be considered a characteristic quenching density. Note that n,*kT = Griin’s p ’ . Two examples of experimental work are shown in Fig. 6 drawn from data obtained by C a m a ~ .The ~ ~ curve for the N; IN is excited by primary electrons (Y<< Qoj)and has the form of Eq. (3.3.14). The N22P (second positive) system is excited by secondary electrons (Y>> Qoj). As discussed earlier 9has a characteristic nKvariation until the secondary mean free path becomes less than a characteristic dimension of the observation A. E. Griin. Can. J . Phys. 36, 858 (1958).

450

3.

MEASUREMENT OF DENSITY

n,( MOL ECU LES/cm') 1

I

I

I-

OPTICAL VIEWING

I

n,( MOLECULES/cm')

FIG.6. Calibration curves for EBF in Nitrogen. (a) First negative system demonstrating low density linearity and high density quenching. (b) Second positive system demonstrating low pressure n i variation due to secondary electrons. [Data from Carna~.~']

volume. Thus from Eq. (3.3.13) Ijk- n: at low densities for the second positive system. The quenching cross section of the neutral Nz is also much smaller than for the ion. 3.3.3. Density Measurements

The majority of EBF studies for the measurement of density have taken place in the approximately linear portion of the response curve shown in

3.3.

ANALYSIS OF ELECTRON REAM E X C I T E D RADIATION

451

Fig. 5. In the linear region, the accuracy of a specie concentration measurement can approach & l percent if great care is taken in the calibration and choice of the spectral emission feature used for the measurement. Under less ideal circumstances f 5 percent might be more typical. As an example, at 50 keV the excitation cross section for the (0,O) band of nitrogen's first negative system is about 0.45 X lo-'" cm2. For an electron beam current of 1 mA/mm2 and ng = l0l5 cmP3a cubic millimeter will emit about 3 x 10" quanta per second. Allowing for an f / l O optical system leaves 2 x lo8 quanta per second. There is, consequently, generally no limitation on accuracy due to statistical uncertainties. The principal source of error will be in detector stability, calibration and so forth. If there is any significant quenching, the measurement becomes much more difficult since the quenching is not only specie dependent but also temperature dependent. For example, Smith and DriscolF5 report on an apparent significant quenching effect on their calibration curves of intensity versus number density at high helium densities, due to only a few hundred parts per million impurity. The effects of quenching have been studied by Harvey and HunterJO (nitrogen), L i I 1 i c t - a ~(nitrogen ~~ and helium), Hunter and LeinhardP' (helium), Hilliard et (helium), and M c R ~ n a l d(air). ~~ The specific problems associated with high density measurements have been briefly but very well reviewed by Smith and D r i s ~ o l l .There ~ ~ is also a discussion of quenching phenomena in Muntz's A general conclusion of Smith and D r i s ~ o lis l ~that ~ where quenching is important, calibration should be done in a rapidly flowing gas, rather than in gas at rest or in a slow flow, of the correct composition and temperature. Each gas or gas mixture has its own peculiarities and the reader is referred to the review by M ~ n t for z ~ detail ~ on the emission features from various gases. Since the time of that review Petrie and his collaborator^^^*^^ have added new or more complete information on NO and 02.Information on helium is available from works surveyed by Smith and D r i ~ c o l l . ~ ~ 3.3.4. Beam Generation, Spreading, and Plasma Effects

The electron beams used in applications of the EBF technique are generated by conventional electron guns. These guns are isolated from the flow under investigation by one or several small orifices of about 1 mm diameter drilled in metal diaphragms which must have a high thermal con55 J . A . Smith and J . F. Driscoll, A G A R D Symp. Non-intrusive F l o n ~ Instrum.. 1976. No. 193, p. 10-1. 56 D. C. Lillicrap, N A S A Tech. Memo. X-2842(1973). 57 W. W. Hunter and T. E. Leinhardt, J . Chem. Phys. 58, (3). 941 (1973). 58 M. E. Hilliard, S. L. Ocheltree, and R . W. Storey, N A S A TND-6005 (1970).

3.

452

MEASUREMENT OF DENSITY

I

ELECTRON ENERGY, keV

m

CUNNINGHAM -FISHER RESULTS FOR ONE HALF OF RMS WIDTH

/

/

I

o-'

lo2'

lozz

loz3

GAS T H I C K N E S S , n l (rn-')

FIG.7. Electron beam spreading in air.58*80

ductivity and a relatively high melting point. Copper is frequently used for this purpose. The vacuum system is designed to have pumping capacity sufficient to remove the gas that passes through the orifices connecting the gun chamber to the gas flow under study, and thus maintain the gun chamber at density levels that can be tolerated by the cathodes without breakdown. Oxide-coated, heated-filament, and plasma-source cathodes have been used for electron generation. Most of these mechanical details * very high current requireare referred to in the review by M ~ n t z . ~For is of interest. ments the duoplasmatron source used by Petrie et Once it has entered the gas flow under investigation the electron beam will spread due to collisions. Clearly a great amount of spreading cannot be tolerated. In all applications it is essential to know by measurement the beam current that is being observed by the optical system. Extreme beam spreading makes it impossible to satisfy this requirement. A few results are available for spreading and have been reviewed by M ~ n t z . ~ ~ M c R ~ n a l dand ~ ~ Smith and D r i s ~ o l ldiscuss ~~*~~ the problem in detail. Some available experimental results for beam spreading as a function of gas target thickness are shown here in Fig. 7 from Cunningham and FisheF and Center.6o The angle 8 is measured from the beam direction with the point of electron injection the origin. Some information on this is also available from the work of Bogdan and McCaa.*l Js

J . W . Cunningham and C. H . Fisher, Arnold E n g . D e v . Cent. TR-66-211(1967). R. E. Center, Phys. F1uid.s 13 ( I ) , 79 (1970). L. bogdan and D. J. McCaa, Cornell Aeronuut. Lab., Tech. R e p . AG-2079-4-1 (1970).

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

453

In addition to the spreading of the electron beam there is a secondary electron effect that appears to be important. The plasma created by the electron beam has been shown to have a relatively high concentration of low energy (1 eV) electrons and ions in the vicinity of the beam. The presence of these low energy electrons has been used to explain certain features of the EBF temperature measurements that are described in Section 4.2.3. The beam created plasma has been analyzed including collisions by Harbour, Bienkowski, and Smith,62although again only in a stationary gas. These plasma effects do not appear to be important for density measurements. 3.3.5. Flow Field Studies The implementation of electron beam measurements in flow about immersed bodies presents a number of items requiring consideration. Beam spreading will frequently present difficulties. It is avoided, in many supersonic flows at least, by requiring the beam to transit only a portion of the flow field. An installation used by Muntz and Softleys3in their shock tunnel studies of near wakes is shown in Fig. 8. Notice that a long evacuated drift tube is used to bring the beam to a point near the measurement area before exposing it to the ambient gas. In hypersonic flow the disturbances associated with the drift tube (and beam receiver) will propagate downstream at small angles and thus be carried away from the measurement area. Another matter that must be considered for flow field studies is an interaction of the beam generated plasma with the models. In addition, reflection of scattered beam electrons from models can have an influence on measurements in certain circumstances. The questions associated with flow field probing are discussed by Butefisch and Vennemann40 to which the reader is referred for details. Flow field investigations have gone on in a number of countries with a representative proportion of the work appearing in the proceedings of the Rarefied Gas Dynamics Symposia.64 Studies carried out in the U.S.S.R. have been described by Bochkarjov et a/.65and most recently by Rebrov .66 P. J. Harbour, G . K. Bienkowski, and R. B. Smith, Phys. Nuids 11 (4). 800 (1968). E. P. Muntz and E. Softley, AIAA J . 4, 961 (1966). Ed. 64 R. G. Sharafutdinov, in “Rarefied Gas Dynamics” (D. Dini, ed.), Vol. I , p. 563. Tech. Sci., Pisa, 1971. A . A. Bochkarjov, A . K . Rebrov, S. P. Chekmayov, and R. G.Sharafutdinov, in ”Rarefied Gas Dynamics” (D. Dini, ed.), Vol. 1 , p. 589. Ed. Tech. Sci., Pisa, 1971. A. K . Rebrov, in “Rarefied Gas Dynamics” ( J . L. Potter, ed.), Part 11, p. 811. AIAA Press, New York, 1977. 6p

63

SHOCK TUNNEL

T~~~~~~~~~~~~~~~~ GENE R AT0 R

AERO-EKTAR LENS

AM S P L I ~ T E R

FRONT SURFACE

R

PACKAGE

B E A M SPLllTER NONABSORBING REFLECTS 113 TRANSMITS 213

\PHOTOMULTIPLIERS

ROTATIONAL TEMPERATURE PACKAGE

FIG.8. EBF apparatus used for aerodynamic studies in a shock tunnel.63

3.3.

ANALYSIS OF ELECTRON BEAM EXCITED RADIATION

455

Flow visualization is an important use of the EBF technique. Reference to the literature was made in the introduction of this article and more detail is given in Part 2 of this volume. ACKNOWLEDGMENTS The writing of this article was made possible through partial support of the United States Air Force Office of Scientific Research, N o . AFOSR 77-3142.

This Page Intentionally Left Blank

4. MEASUREMENT OF TEMPERATURE

4.1. Probe Methods* 4.1.1. Definitions of Flow Temperatures

There are many temperatures which characterize a fluid flow. They arise because in the process of inserting a probe into a moving fluid we divert the flow causing compression or expansion, and we introduce sources of heat transfer to and from the flow. Thus the interaction of the probe with the fluid may in itself determine the particular “temperature” measured. The conversion of directed motion kinetic energy into internal energy, and the conduction, viscous and radiation transport losses require careful definition and calibration of each physical configuration. The conventional, intuitive definition of temperature is the static temperature T , , measured by a thermometer which is moving with the fluid. It is the “true” temperature since the moving thermometer and the gas are relatively “at rest.” Static temperature represents the random translational component of the molecular motion (not including collective motion at velocity V). The static temperature can also be determined by various spectroscopic techniques covered in .Chapter 4.2. (Note that temperature as defined here is an equilibrium property; in many nonequilibrium flows the molecular translational, vibrational, and rotational motions may all be characterized by different Boltzmann distributions, hence different temperatures which can be determined spectroscopically). In the usual laboratory situation, however, the probe must remain at rest and the gas or liquid flows past it, so that the static temperature cannot be determined directly by probes, but must be inferred from other measurements. Since the fluid is at rest at the surface of any probe or bounding wall (neglecting slip and free molecular effects), and since at some previous time it was moving, kinetic energy must have been converted to thermal energy. We define a total temperature or stagnation temperature To for the moving fluid, which is that temperature attained by the fluid when brought adiabatically to rest. The enthalpy h of a fluid in steady stream-

* Chapter 4.1 is by W. Paul Thompson. 457 M E T H O D S OF E X P E R I M E N T A L PHYSICS, VOL. 18B

Copyright 0 1981 by Academic Press, Inc All rights of reproduction in any form reserved.

ISBN 0-1’2-475956-4

458

4. MEASUREMENT OF

TEMPERATURE

line adiabatic motion is conserved, such that’.‘ h

+ V/2

= =

const ho the total enthalpy.

(4.1.1)

For an ideal gas with constant specific heat C , then, ho = C,T,

+ V/2.

(4.1.2)

The total temperature and static temperature are related by To

=

T,

+

vL/2C,

(4.1.3)

when velocity is known (cf. Part 1). For supersonic flow, this can be written in terms of Mach number M , as

3= 1+y-lM2, 2 T8.m

=

1

+ 0.2=

(4.1.4)

for air, where y = 1.4. The subscript 03 denotes the free stream, outside any viscous or thermal boundary layer. If the flow in a wind tunnel, for example, is adiabatic and nondissipative, then the total temperature can be measured in the essentially zero-velocity gas in an upstream settling chamber or reservoir. It can be as low as 300 K or less for a subsonic or low-density tunnel, and as high as 10,000 K for the reservoir gas behind the reflected shock in a hypersonic shock tunnel. (Cf. Chapters 9.2 and 9.3.)

For air at 300 K with a C, of order 35 J/kg K, the contribution of the velocity term to total temperature becomes of order 10 percent at V = 45 m/s; whereas for water with C, = 4.18 x lo3 J/kg K, the difference becomes 10 percent at V = 500 m/s, a velocity seldom reached in any realistic liquid flow experiment. Thus a simple thermometer inserted in a gas flow will read the static temperature only at very low subsonic speeds; while for liquids which are essentially incompressible and have high conductivity and specific heat, a glass thermometer or thermocouple probe may read static temperature directly with small error. When a real fluid is actually brought to rest at the surface of a model wall or at the stagnation point of a probe, conduction, radiation and viscous dissipation processes enter, and all of the total temperature or total enthalpy is not “recovered” by the measuring system. The udicihatic wall temperature Tad is defined as the steady state temperature achieved on an insulated wall (no heat conduction or radiation H . W. Liepmann and A. Roshko, “Elements of Gasdynamics.” Wiley, New York, 1957. A. M. Kuethe and .I.D. Schetzer, “Foundations of Aerodynamics,” 2nd ed. Wiley, New York, 1959.

4.1.

PROBE METHODS

45 9

loss). It is frequently determined by inserting an insulated model (e.g., a sharp flat plate) into a continuous flow facility, and measuring its final temperature. In short duration flows, it must be calculated from knowledge of T, and Toand the flow geometry. The value of Taddepends on the nature of the wall boundary layer, and on the Prandtl number of the fluid (cf. Chapter 10.3), Pr = C p p / K = v / a ,

(4.1.5)

which can be interpreted as the ratio of viscous energy degradation to the thermal conduction; or as the ratio of viscous diffusivity to thermal diffusivity. In the particular case where Pr = 1 , the heating due to the one is balanced by the cooling due to the other, and no net energy is lost in bringing the gas to rest, so that Tad= To. The value of Pr is of order 1 for gases, less than 1 for many liquids and molten metals, and as large as lo3 for some viscous oils. In general we can define a recovery factor r for wall temperature T, which can be shown to depend on the boundary layer stru~ture~,~ r = T~ Tad

- Tm

-

Tm

= ~r

for Couette flow

(4.1.6)

= Pr1/2for a laminar flat plate boundary layer =

Pr1I3for a turbulent flat plate boundary layer.

Similarly, a recovery factor can be defined for stagnation temperature T , actually read by a probe, r =

T, - T , TO - Tm

(4.1.7)

representing the fraction of totaI temperature “recovered” by the probe. The practical importance of knowing To or Tadis that they are the reference temperatures for stagnation point or wall convective heat transfer. As is discussed further in Part 7, and in Eckert and Drake3 convective heat transfer rate is characterized for practical purposes by a surface heat transfer coefficient H ; (4.1.8) 4 = H(Tre, - Tw). For a given experimental configuration, H combines all the geometric and gasflow parameters. It is customary in the heat transfer literature to express relationships in terms of nondimensional parameters which are shown by experience to scale the key physical quantities. (See Chapter E. R. G. Eckert and R. M . Drake, “Heat and Mass Transfer.” McGraw-Hill, New York, 1959. R . W. Ladenburg, B. Lewis, R . N . Pease, and H. S . Taylor, eds., “High Speed Aerodynamics and Jet Propulsion,” Vol. 9. Princeton Univ. Press, Princeton, New Jersey, 1954.

460

4.

MEASUREMENT OF TEMPERATURE

10.3). Thus for example, the Nusselt number

NU = H x / K

(4.1.9)

scales the surface heat transfer coefficient with gas conductivity K and some characteristic dimension x . Frequently the heat transfer, e.g., for a flat plate or a cylinder, can be expressed as N u =f(Pr, Re)

(4.1.10)

in simple power law forms. 4.1.2. Temperature Sensors

There are many possible methods for sensing temperature. For continuous flows, especially in liquids, a liquid-in-glass thermometer may suffice. Thermocouples are favored in practice for their stability and well-known calibration over wide temperature ranges. Thermistors offer about 10-fold sensitivity improvement over thermocouples, but tend to be nonlinear and are limited to fairly low temperatures (400-500 K). Resistance thermometers-fine wires, or thin metal films on insulated substrates-also offer good sensitivity, but are somewhat less stable over long periods than thermocouples. They are, however, extremely useful in short-duration flows where microsecond response time is required. The various techniques are reviewed in detail in recent texts,5 as are detailed practical cautions on thermocouple application.6 Copper/constantan or iron/constantan couples are favored at low temperatures because of their greater sensitivity. Platinum/Pt- 10 percent rhodium couples can be used up to 2000 K , close to their melting point, and are more resistant to chemical attack by the flow. For pipe flow of a liquid or gas (Fig. l), a thermometer may be inserted through the wall, or a temperature sensor placed at the bottom of a probe well of insertion length I , radius R , and conductivity K . If H is the heat transfer coefficient to the thermometer or sensor well, T,, the probe reading, T, the pipe wall temperature, and T, the desired fluid temperature, then conduction losses can be minimized by properly choosing insertion length.g 7p -

Tm

=

T , - Tm cosh ml ’

where m

=

(2H/KR)’I2.

(4.1.11)

For point measurements of gas temperature, e.g., in mapping boundary a R. P. Benedict, “Fundamentals of Temperature, Pressure and Flow Measurements.” Wiley, New York, 1969. R. J. Moffat, I S A Prepr. ASIT 74206, 1 1 1-124 (1974).

4.1.

46 1

PROBE METHODS

layers in continuous flow tunnels, a hotwire anemometer or hot film (cf. Section 1.2.4) can be used to measure gas temperature, by adjusting the heating current until there is no net heat transfer to the wire, in which case Twire= Tad. The wire here is used as a resistance thermometer. An excellent small probe can be made by suspending a fine thermocouple bead between needle mounts.’ Pt/Pt- 10 percent Rh couples made from 20- to 40-pm wire with a 50-pm-diameter welded bead at the center, are welded to 0.25-mm nickel alloy supports. If the spacing between bead and support is 1.5 mm or greater, then conduction losses are less than 1 percent at the maximum wire temperature of 2100 K. The bead temperature is determined by the balance of convective heat input and radiative loss from the hot wire to the colder radiating surroundings. The adiabatic wall temperature Tad is then found from: H(Tad - Twire)= U E T & ~.,

(4.1.12)

Over a wide range of wire bead Reynolds numbers convective heat transfer coefficient H is given by7

< Re < lo4),the

Nu = H R / K

=

0.42 Pro.2+ 0.57

Re0.5

(4.1.13)

where Reynolds number is based on wire radius R and freestream gas properties. A much more rugged probe also useful for steady flows was constructed from a small sharp platinum cone facing the flow and mounted on an insulating ceramic sting.8 The cone temperature was measured by an embedded thermocouple, and the sting was instrumented to measure thermal gradient (conduction loss). By minimizing radiation loss from the low emissivity Pt,and correcting for the measured conduction loss, a recovery factor close to the ideal r = Pr1’2was achieved. The small cone angle and attendant weak shock led to small errors, and gave a measured Tad equal to that calculated for a flat plate. The probe was useful at stagna-

’ D . Bradley and K. J . Matthews, 1. Mech. E n g . Sci. 10, 299 (1968).

* J. E. Danberg, in “Advances in Hypervelocity Techniques” (A. M. Krill, ed.), p. 693. Plenum, New York, 1962.

4.

462

MEASUREMENT OF TEMPERATURE

-5mm

PLATINUM CONE

-ir

19 mm----

1 IRON CONSTANTAN IHtHMOCOUPLES

FIG.

\

\

2. Equilibrium temperature probe. [From Ref. 8.1

tion pressures up to 3.5 MPa (35 atm) at To of 550 K, and to M = 6.7 (see Fig. 2). The classic total temperature probe is a thermocouple placed in the stagnation region of a well-designed diffuser, and shrouded to reduce radiation loss from the thermocouple junction to the cold tunnel walls. Several proven designs are a ~ a i l a b l e .In ~ general it is impossible to bring the flow perfectly adiabatically to rest, and to account for all conduction and radiation losses. The design must also take into account the thermal capacity and time response of the probe elements. Comprehensive tables and charts for the solution of many sensor thermal problems may be found in Ref. 5 . In practice all probes are extensively calibrated in order to establish empirical recovery factors. When pains are taken to surround the thermocouple with low-emissivity radiation shields, and to carefully vent the probe to bring the flow to rest without shocks or further compression (Fig. 3), recovery factors of 0.95 to 0.99 are achievable up to M = 5 . Most simple probes are insensitive to yaw angles up to 10 deg. Specially constructed probes have been instrumented with electrically heated base mounts and radiation shields, so that radiation and conduction losses could be measured d i r e ~ t l y .A~ recovery factor of 1 .O, with an error of 0.5 K at To = 430 K has been achieved after correction for measured losses. In high-enthalpy flows characteristic of arc heated jets (15,000 K), water cooling is required to maintain reasonable probe temperatures. A typical unitlo gives an accuracy of 3 percent in stagnation enthalpy. The mass flow of coolant and sampled gas, and the temperature rise of the coolant must be known in order to correct for spurious heat transfer. In shock tube and shock tunnel flows where stagnation temperatures reach 6- 12 kK in milliseconds or less, the gas stagnation temperature is often inferred from a stagnation point heat transfer measurement, using fast-response film calorimeters (cf. Chapters 7.4 to 7.6). Calorimetry has @

lo

R. D. Wood, J . Aerosp. Sci. 27, 556 (1960). J . Grey, P. F. Jacobs, and M . P. Sherman, Rev. Sci. Instrum. 33, 738 (1962).

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS PT CUAIED SILICA\ THERMOCOUPLE BEAD 7

,-0.8 mm VENT HOLE

\ /

r

463

STEEL HOLDER

+ -19cm-d FIG. 3. Shrouded thermocuple total temperature probe. [From Ref. 4.1

also been applied using cooled or uncooled slug calorimeters in a transient mode, to survey arcjet uniformity and total enthalpy (cf. Chapter 7.4). Established theoretical models11.12backed by experimental verification13”*permit reliable measurements even in highly dissociated and ionized flows.

4.2. Measurement of Temperature by Radiation Analysis 4.2.0. Introduction*

Optical methods are unique among methods of temperature measurement of heated gases, plasmas, and flames. Optical methods are practically inertialess and do not disturb the phenomenon under investigation. Such methods, with the exception of that of scattering laser radiation by electrons, atoms, or molecules, give the temperature averaged over an optical path; this constitutes their chief disadvantage. Apart from fluid dynamics, optical methods are widely used in physical chemistry, and in many practical There is no method J. A. Fay and F. R. Riddell, J. Aerosp. Sci. 25, 73 (1958). J. A. Fay and N. Kemp, AIAA J. 1, 2741 (1963). l3 P. H. Rose and W. I. Stark, J . Aerosp. Sci. 25, 86 (1958). “ P. H. Rose and J. 0. Stankevics, AIAA J. 1 , 2752 (1963). l1 l2

B. Lewis and G. von Elbe, “Combustion, Flames and Explosions of Gases.” Academic Press, New York, 1951. A. G. Gaydon and H . G. Wolfhard, “Flames, their Structure, Radiation and Temperature,” 4th ed. Chapman & Hall, London, 1979. C. M. Herzfeld, ed., “Temperature, its Measurement and Control in Science and Industry.” Van Nostrand-Reinhold, New York, 1955. S. S. Penner, in “Temperature, its Measurement and Control in Science and Industry” (C. M. Herzfeld, ed.), p. 561. Van Nostrand-Reinhold, New York, 1962.

‘

* Sections 4.2.0 and 4.2.1 are by N. A. Generalov.

464

4. MEASUREMENT

OF TEMPERATURE

which could be considered universal, but of several methods, each having advantages for some experimental conditions, one may be chosen the simplest and most reliable technique for the conditions. The means of detecting the radiation, absorption, or scattering of light from gases must be considered fundamental to the optical method. Radiation, absorption, and scattering are the outer manifestation of the physical processes dependent on temperature; i.e., bremsstrahlung radiation, excitation. deactivation, line broadening, etc. The measured parameter is related to temperature by means of the Planck law or the Boltzmann or Maxwell distributions. It is customary to attribute temperatures to a medium according to the degrees of freedom where quasi states of equilibrium may exist: translational temperature Tt , rotational temperature T, , vibrational temperature T, , electron temperature T, , and thermodynamic (equilibrium) temperature T ; this reflects the fact that for separate degrees of freedom Boltzmann or Maxwell distributions can exist with their characteristic temperatures. If there is no distribution for a degree of freedom, one cannot attribute such a temperature. For example, there is no single vibrational temperature in the molecular gases behind the shock front when the rates of dissociation and vibrational relaxation are close to each other; in this case there is a translational temperature characterizing the upper levels and vibrational temperature characterizing the lower. For such conditions temperature measurements become more complicated, although Raman scattering techniques show promise for such molecular nonequilibrium conditions.2 ~

~.

S. S. Penner, “Quantitative Molecular Spectroscopy and Gas Emissivities.” Addison-Wesley, Reading, Massachusetts, 1959. E. V. Stupochenko, S. A. Losev, and A. 1. Osipov, “Relaxation in Shock Waves.” Springer-Verlag. Berlin and New York, 1967[ 19651. ‘ A . G. Gaydon and I . R. Hurle, “The Shock Tube in High-Temperature Chemical Physics.” Van Nostrand-Reinhold. New York, 1963. R. I. Soloukhin, “Shock Waves and Detonations in Gases.” Mono Book Corp., Baltimore, Maryland, 1966. Yu. E. Nesterikhin and R. I. Soloukhin, “Methods of High-speed Measurements in Gas Dynamics and Plasma Physics,” AD 682 067. NTIS, Springfield, Virginia, 1968. lo F. S. Faizullov, 7 r . Fiz. l n s i . , Akad. Nouk SSSR 18, 105 (1962); in Proc. P . N. tebedev Phys. I n s / . [ A ( w l . Sci. USSR]. ** K . Vullrath and G.Thomer, “High-speed Physics. Springer-Verlag, Berlin and New York, 1967. ’’ W. Lochte-Holtgreven, “Plasma Diagnostics.” North-Holland Publ., Amsterdam, 1968. l 3 R. I. Soloukhin, Shork Tubes, Proc. I t i t . Shock Tube Symp., 71h. Toronlo, 1969. p. 662. Univ. of Toronto Press, Toronto, 1970. @

”

4.2.

465

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

4.2.1. Emitted and Absorbed Radiation List of Symbols Absorptivity Integrated absorptivity Natural, Lorentz, Doppler spectral line half-width Focal ratio Emittance Wave vector Intensity of light Boltzmann constant Length Electron mass Electron density Pressure Reflectance Equilibrium temperature Brightness temperature

Electron temperature Ion temperature Rotational temperature Translational temperature Vibrational temperature Pi

Ion charge Degree of dissociation Absorption coefficient Scattering angle

hulk Light wavelength Debye wavelength Density Radiant flux Plasma frequency

Here we consider temperature measurements based primarily on emission and absorption of light. Only brief mention is made of light scattering techniques. Gas temperature measurements by light scattering are discussed in Sections 4.2.1.8 and 4.2.2, and methods employing electronbeam excited radiation in Section 4.2.3. There is some overlap of methods discussed in Chapter 4.2 with those presented in Parts 2 and 3 in Volume 9 of this treatise. The discussion of laboratory techniques in Chapter 3.4 of Volume 9 may be especially useful to the reader who is setting out to measure temperature in the laboratory. 4.2.1.l.Brightness Temperature. Many optical methods of temperature measurement are based on Kirchhoff’s law

(4.2.1) Here the ratio of the spectral emittance E to the absorptivity a is the Planck functionf(h, T) = ~ ~ h - ~ / [1 Ie ~ ~ ’ ~ ~ It follows from Eq. (4.2.1)that one may determine a temperature if the two values E and a are known. For gases there is also the possibility of obtaining information on the density of the radiating particles as well. This is discussed in Part 3. Sometimes only emittance is measured, with a set equal to 1 , and in this case the quantity “brightness temperature” Tb is measured. This is the blackbody temperature for which emittance is equal to the emittance of

466

4.

MEASUREMENT OF TEMPERATURE

the body under investigation. The brightness temperature is always less than or equal to the true temperature. If the Planck formula in (4.2.1) is replaced by Wien's formula (2c,/h5) e-m'hT,then

h _I _ _l = -In a(h, T ) , (4.2.2) T T b c2 where the radiation constants c1 = 27rc2h = 3.742 x erg cm2 sP1, and c2 = 1.438 cm K. Brightness temperature is determined in cases where the absorptivity measurement is difficult, for example in Ref. 14 where the dynamics of the temperature rise with laser damage to solids are investigated, or in Ref. 15 where radiation from the boundary layer of samples being destroyed by the action of convective and radiation heat fluxes is studied. 4.2.1.2. Line Reversal Methods. Perhaps the line reversal method is the one most widely used. It was suggested by Feryls in 1903 and widely used for flame temperature measurements. More recently with development of shock wave techniques it has been used for precise temperature measurement of gas heated by the shock. The method is illustrated in Fig. 1. The light from the continuous spectrum source S passes through the gas G whose temperature is measured and reaches a spectrometer with a photomultiplier PM. Usually investigations are made in the visible spectrum, on the lines of the metals Na, Ba, Li, Ca, etc., which happen to be present in the gas under investigation. Typically emission and absorption of the Na D-lines is investigated. The question of spectral line choice in the line reversal method is considered in detail in the book by Gaydon and H ~ r l e .It~ is shown that one can use any lines irrespective of whether they correspond to electron transitions, electron-vibrational or rotational transitions, and whether they are reabsorbed or not. These lines can be situated at different regions of the spectrum. In the infrared region of the spectrum, this method was first used by BauerI7 and Schmidt.18 The temperature, whether rotational, vibrational, o r electron excitational is determined in accordance with the transition mechanism of the line. If all these temperatures are equal to each other, the thermodynamic temperature of the gas is measured. N . F. Pilipetski, A . K . Fannibo, and V . A . Epstein, Zh. Prikl. Spektrosk. 15, 33 (1971). E. B. Georg, Yu. K. Rulyev, G . F. Sipachev, and M. 1. Yakushin, Izv. Akod. Nauk S S S R , Mekh. Zhidk. Gaza No. 2, p. 25 (1972). Is C. Fery, C . R . Hebd. Seances Acad. Sci. 137, 909 (1903). I' E. Bauer, C. R . Hebd. Seunces Acud. Sci. 147, 1397 (1908). H . Schmidt, Ann. Phys. (Leipzig) [4] 29, 971 (1909). I'

l5

4.2.

MEASUREMENT OF TEMPERATURE BY RADlATION ANALYSIS

D

0

PM

467

CT

S LI G L2 FIG.1. Experimental setup for measurement of temperature by the line reversal method.

If the temperature of the light source is higher than the temperature of the investigated gas, then the metal lines are seen as dark absorption lines in the continuous spectrum of the source. On the other hand if the source temperature is lower than the gas temperature, the lines appear lighter than the background. In the classical version of the line reversal method one adjusts the source temperature until the gas emission and light absorption are equal as judged visually in a spectroscope. In this case the brightness temperature of the comparison source Tb will be equal to true gas temperature T,. One can show this using Kirchhoff s law. Let a(h, T,) be the absorptivity of the gas in the region of the line under consideration, E(h, T,) be the emittance of the blackbody in this region, and E(h, Tb)be the emittance of the background source. Then at the reversal point one can write Q(h,Tz)E(hj T x )

f

E(X,

Tb)

- Q(X,

Tz)E(h, T b )

=

E(X, T b ) , (4-2-3)

Le., E(h, T z ) = E(A, Tb) and hence T , = T b . Of course in high speed processes (for example, behind a shock front) it is practically impossible to arrange for the condition (4.2.3). With rapid developments in shock tube techniques, new adaptations of the line reversal method have appeared. One of them, developed by Sobolev and his c o - w o r k e r ~ , is ~ ~a -generalized ~~ line reversal method. It is not necessary to observe the reversal point for a gas temperature determination but it is sufficient to measure three values21: ( I ) Gas radiation flux qz in the wavelength region of the spectral line. (2) Radiation flux v,+~from gas and background source. (3) Flux cps from background source

A. G. Sviridov and N. N . Sobolev, Zh. Eksp. 7 i w . Fiz. 24, 93 (1953). N . N. Sobolev, Tr. Fiz. Inst.. Akud. Nuuk SSSR I, 195 (1956). Proc. P . N . Lebedev Phys. 1nJt. [Acad. Sci. U S S R ] . 2' N . N . Sobolev, A. V. Potopov, V . F. Kitayeva, F. S. Faizullov, V . N . Alyamovski, E. T. Antropov, and I . L. Isaev, l z v . Akad. Nuuk S S S R , S r r . Fiz. 22,730 (1958); Bull. Acud. Sci. U S S R , Phys. Ser. (English T r a n s / . ) .22, 725 (1958). 22 F. S. Faizullov, N . N. Sobolev, and E. M. Kudryavtsev, Dokl. Akud. Nuuk S S S R 127, 541 (1959); Sov. Phy.r.-Dokl. (English Trunsl.). 4, 833 (1959). 23 F. S. Faizullov, N . N . Sobolev, and E. M. Kudryavtsev, Opr. Spektrosk. 8,585 (1960); Opt. Spectrosc. ( U S S R ) (English ?runs/.).8, 311 (1960). 24 F. S. Faizullov, N . N . Sobolev, and E. M. Kudryavtsev, Opt. Spekfrosk. 8,761 (1960); O p t . Spectrosc.. ( U S S R ) (English [ r u n s / . ) .8, 400 (1960). ID

2o

468

4.

MEASUREMENT OF TEMPERATURE

(4.2.4)

(4.2.6) where D is the linear dispersion in angstroms per millimeter, 6h the effective width of the spectral line, d2/fithe relative aperture of the objective, s is the width of the entrance slit of the spectrograph, s’ and h’ are width and height of the exit slit, cl, c2 are the radiation constants, and Tb is the brightness temperature of the comparison source. One can express gas temperature T, in terms of the comparison source temperature Tb using equations (4.2.4)-(4.2.6). (4.2.7) A xenon arc lamp with brightness temperature T,, = 4750 K was used as a comparison source. Temperature behind the shock was measured by Na D-lines and by the resonance line of ionized Ba I1 (4554 A). In the experiments one can use shock tube Na contamination in the gas or expressly add a small amount of NaCl. Ba was added to the gas by covering the surface of the shock tube with a small amount of BaCI,. Experiments showed that it is possible to determine gas temperature by observing the Na D-lines over the temperature interval 1500-3000 K, and by observing the Ba II-line at higher temperatures, because in this case the influence of the cold boundary layer is eliminated. For temperatures T > 5000 K one should use high temperature pulsed light sources. The most important question regarding applicability of the line reversal method in investigations of high-speed processes is whether equilibrium exists between impurity and system. This question has been widely disc u ~ s e d . ’ * ~It ~has * ~been ~ shown that in molecular gases electronic exitation of the atom impurity keeps up with vibrational degrees of freedom. This fact allows determination of vibrational gas temperature in the case of no thermodynamic equilibrium. But in monatomic gases such as argon where the cross sections for collisions of the second kind with excited atoms are very small the measured temperature behind a shock is much lower than the temperature calculated according to the fluid dynamical l5

I. R. Hurle, in Proc. S y m p . Low-Tempcruture Plusmu. Moscow, 1965.

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS

469

Shock speed, V (km/s) FIG.2. Gas temperature as a function of shock velocity in argon and air according to Ref. 24, A argon at initial pressure p I = 700 Pa; 0 and 0 , air at p 2 = 270 and 1300 Pa; calculated equilibrium values shown by curves.

laws, especially at densities p 2 < 1 atm, where p2 is density behind the shock front (Fig. 2). To determine temperatures behind shock and detonation waves, Gaydon and his colleagues developed a somewhat different version of the double-beam version of the reversal neth hod.'*^^-^^ Line reversal is achieved by using two sources of continuous radiation with slightly different temperatures. Actually one light source is used, and two light beams are obtained from it. In the path of one beam a neutral filter is placed to lower the brightness temperature a known amount. The beams pass through the gas under investigation at the same place. Gain is adjusted so that on the oscillograph screen the same deflection of the trace is observed on both detectors when there is no neutral filter. If the temperature of the gas under investigation is in the interval between the brightness temperature T l b of the unattenuated beam and the reduced brightness temperature T2bof the beam with filter then the detector of the first beam records a signal - d and the other detects a signal + e upon shock passage. Using linear interpolation the true gas temperature is then

T

= T2b

+ e(T1,

- T2b)/(d -I-e).

(4.2.8)

28 J . G . Clouston, A. G . Gaydon, and I. R. Hurle, Proc. R . S o c . London, Ser. A 252, 143 (1959). " A. G. Gaydon and I. R. Hurle, Proc. R . SOC.London, Ser. A 262, 38 (1961). '* A. G. Gaydon, I . R . Hurle, and G. H. Kimbell, Proc. R . SOC.London, Ser. A 273,291 (1961).

470

4.

MEASUREMENT OF TEMPERATURE

Gaydon and ~ o - w o r k e r s ~used * ~ ~a* method ~~ similar to S ~ b o l e v ' s ' ~ ~ ~ ~ when large temperature fluctuations were observed behind the shock. Losev and GeneralovZ9used a pulsed light source as the background in the generalized line reversal method to measure argon temperature behind a shock. Soloukhin8 used the method in measurements of temperature behind detonation fronts in CzHz-O2mixtures; the C II-line at 4267 A was selected to observe reversal. The sensitivity and accuracy of line reversal methods have been analyzed by Gaydon' and F a i ~ u l l o v . ~ ~ 4.2.1.3. Method of Simultaneous Recording of Absorption and Radiation. This method was developed to investigate in a shock tube electron concentration and temperature behind the shock f r ~ n t . ~Emitted " radiation from and absorption by a xenon plasma are recorded simultaneously behind the shock front in the infrared (A = 10.6 pm). Plasma radiation is recorded on one optical path, and absorption at A = 10.6 pm on another, both paths overlapping in the region whose temperature is to be measured. As a background source of radiation, a COz laser pulsed with an essentially flat top for 600 ps is used. It is known that in the spectral region A = 10.6 p m the main contributions to the radiation and absorption processes are made by free-free transition of electrons in the field of the ions. Indeed the ratio of contributions of bound-free and free-free transitions for all atoms, at frequencies below threshold is approximately ehvlkT- 1 and therefore at a temperature of the order 1 eV the contribution of the bound-free transitions at wavelength 10.6 pm is small and can be neglected. Photoresistors made of gold-doped germanium served as radiation detectors. The resolution time of the sensitive elements was about 1 ps. The transit of the shock front at the measuring point was recorded by a piezoelectric pickup mounted in the same section as the windows for observation of the radiation and absorption. Typical oscillograms are shown in Fig. 3. On each of the oscillograms thedistributions of the absorptivity (lower curve) and of the radiation intensity (upper curve) are given. Since the emission and absorption of the radiation by the plasma behind the shock at A = 10.6 pm are connected principally with free-free transitions of the electrons, it follows, as will be shown below, that the profile of the square of the electron density essentially duplicates the shape of the oscillograms of the emission and absorption. ae S. A . Losev and N . A . Generalov, Prih. Tekh. Eksp. No. 3 , p. 108 (1959);Instrum.E x p . T&h. (English Transl.). No. 3, p. 454 (1959). N . A. Generalov, V. P. Zimakov, and G . I . Kozlov, Zh. E k s p . T w r . Fiz. 58, 1928

(1970); Sov. P h y s . --JETP (English Trunsl.). 31, 1038 (1970).

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION A N A L Y S I S

47 1

FIG.3. Oscillograms of emission (top) and absorption (bottom) of xenon at A = 10.6 pm; p, = 400 Pa; timing trace marks at 66.7 ps. (a)

M

=

11.2; (b) M = 12.7.

The sensitivity of the method does not make it possible to detect the appearance of the electrons during the period they are produced by atom-atom collisions, but the period of cascade ionization, accompanied by a sharp increase in the emission and absorption of infrared radiation, is recorded with sufficient reliability. Since the establishment of a Maxwellian distribution for electrons under the conditions of the experiments takes approximately lo-” s, which is much less than the time required to reach equilibrium ionization, Kirchhoff s law may be used for quantitative analysis of the results. For a homogeneous plasma layer of thickness 1, the relation between radiation intensity I and the absorption coefficient E is given by I ( u , T,) = f ( u , T,)(1

-

e-E’t).

(4.2.9)

Heref(v, T,) is the intensity of blackbody radiation (erg cm-2s-1 sr-l), T, is the electron temperature; E ’ is the absorption coefficient corrected for the

472

4.

MEASUREMENT OF TEMPERATURE

stimulated emission, i.e., E’ = ~ ( -1 e-h”’kTe),where E is the true absorption coefficient. It is known that the quantity 1 - e-E’’is equal, under the conditions of a continuous spectrum, to the absorptivity of the substance a ( v , T,). Therefore expression (4.2.9)can be rewritten in the form (4.2.1). Formula (4.2.1)was used to determine the profile of the electron temperature T, from the data on the emission and absorption of the plasma. The radiation intensity of the plasma was calibrated by means of a source with known temperature. 4.2.1.4. Temperature Measurement by Gas Absorption. The temperature of gases with a strong absorption spectrum (e.g., I,, Br, , Cl,) may be found by measuring the absorption in two spectral regions. To do this it is necessary to know the dependence of absorptivity (1 on temperature and density. For a separate line this value is called total line absorption or equivalent line width (1,

=

loffi (1 -

,p(L’l,T)P/

)

dv,

(4.2.10)

where & ( v i ,T) is the mass absorption coefficient, p is gas density in grams per cubic centimeter and 1 is length of path in the gas in centimeters. Usually when one is concerned with the spectra of physical-chemical processes in a shock tube the quantity mean dispersion is considered since in the spectral interval employed by the instrument a great number of lines is included; some of them overlap forming a quasicontinuous spectrum. This is often the case at high temperatures and densities. A precise calculation of the expression

is very difficult and loses meaning in the case of investigations of relaxation phenomena. Here cp(v)is the instrument function. Evaluation of (4.2.11) is especially difficult for halogens since in their spectra the line density and the level of the continuous background are both very high. In the face of these problems, Sulzer and Wieland31 devised an approximate and very simple determination of E ( V , 7) 1. In the absorption region the upper potential curve is replaced by a straight line. 2. A molecule is approximated by a harmonic oscillator. 3. It is supposed that for all temperatures P. Sulzer and K . Wieland, He/\*. Phys. Actu 25, 653 (1952).

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS

lom +

(4.2.12)

du = const.

As a result the expression for E(U,

7‘) =

egm

(th

E(V,

473

T ) becomes

g)”’exp { - th 52T (-)’},

(4.2.13)

L\u$

where ~f is an experimentally determined quantity, O0 is the characteristic vibrational temperature. vo is the frequency at which absorption is maximum, and Au$ is the natural half width at the temperature T = 0. In Ref. 31 appears a comparison of experimental and calculated E ( U , T ) for iodine bromine and chlorine at room and moderate temperature T d 1300 K . The behavior of the absorption coefficient in a higher range 400 K < T < 3000 K has been studied behind s h o ~ k s as ~~ well - ~as~ under equilibrium conditions. In the referenced work good agreement between theoretical and experimental results has been obtained and this justifies use of Eq. (4.2.13) for temperature measurement (Fig. 4).31,33-35 One of the main assumptions underlying formula (4.2.13) is the existence of a Boltzmann distribution in the vibrational levels. The translational gas temperature of the gas is not necessarily equal to its vibrational temperature. Practically there is no dependence of absorption coefficient on rotational quantum number, because for any 9 the energy level curves between which rotational transitions take place are rising essentially at the same rate as the energy level 9 = 0, and because 9 = $”’ 1 . In view of this, relation (4.2.11) can be written in the form a = 1 - e-d~.T)P1 (4.2.14)

*

By use of (4.2.14)together with (4.2.13)for two wavelengths in the continuous or quasi-continuous spectrum, the expression for the ratio E ~ / beE ~ come~.~~*~’

32 N. A. Generalov, S. A . Losev, V. D. Kosynkin, and V. Ya. Ovechkin, Vesfn. Mosk. Univ., Fiz., Astronomiyu 6 , 29 (1965). 33 H. B. Palmer and D. F. Hornig, J . C h e m . Phys. 26, 98 (1957). 34 D. Britton and N. Davidson, J . C h e m . Phys. 25, 810 (1956). J. K . K. Ip and G. Burns, J . C h e m . Phys. 51, 3425 (1969). 3 8 N . A . Generalov and V . A. Maksirnenko, Z h . Eksp. Teor. Fiz. 58, 420 (1970); Sov. Phy.c.-JETP (English Trunsl.) 31, 223 (1970). 37 N . A . Generalov and V . Ya. Ovechkin, Teor. Eksp. K h i m . 4, 829 (1968); Theor. Exp. Chern. (English Tronsl.). 4, 530 (1968).

474

4.

MEASUREMENT OF TEMPERATURE

00

Temperature, T ( K )

FIG.4. A comparison of experimental and calculated data for the effective absorption coefficient cfT) of bromine as a function of temperature. E ~ ( Tis) the absorption coefficient at T = 293 K . Curve 1: data from Palmer and Hornigsl; Curve 2: theory by Sulzer and Wieland31; Curve 3: data of Britton and Davidson3'; Curve 4: from Ip and Burns.35

If we know u1 and u2 from experiment, then we can find the temperature Tfrom (4.2.15) without gas density measurements. Using this absorption method in two spectral intervals one can determine the gas vibrational temperature during dissociation in case the Boltzmann distribution is undistorted. For a dissociation process the relation (4.2.14) takes the form = 1 -

e-~(u,T,Hl-a~)Pl

(4.2.16)

where a0 is the degree of dissociation, so Eq. (4.2.15) remains valid. The accuracy of temperature measurement by the absorption method in separated wavelength intervals depends on the choice of the intervals and on the gas density p . This method may be used successfully for plasma investigation if there is local thermodynamic e q ~ i l i b r i u m .One ~ ~ cannot apply formula (4.2.13) for a plasma, as it is necessary to perform numerical calculations of E ( U , Te, ne). Billman and S a l l ~ a performed p~~ such calculations for 0. 1H2 + 0.9He in the ranges of T, = (1-22) . lo3 K and of n, = (0.45-7) * lOI7 cm-3 for the visible spectrum taking into account photodetachment of H-, inverse bremsstrahlung in the field of ions and neutrals, photoionization, and resonant excitation. In these calculations, Stark broadening of the hydrogen lines and depression of the ionization potential were also taken into account. The authors propose two variants of the method of Tedetermination by absorption. In the first variant the parameter n, is independently deters~ K . W. Billman and J . R . Sallcap, Recent D e v . Shock Tube R e s . , Proc. Int. Shock Tube Symp., 9th, 1973 p. 218. Stanford Univ. Press, Stanford, California. 1973,

4.2.

475

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS 24

0

t-

FIG. 5. Comparison of plasma temperature T, measured by absorption and Ts calculated from shock velocity. [From Ref. 38.1 A:h = 0.633 pm; 0: A - 1.15 p m ; 0; A = 3.39 p m .

20

I0

12

14

16

18

20

22

24

Plasma temperature, T~ ( lo3K )

mined, for example by laser interferometry, and then the temperature T, is determined using a Beer's type law such as Eq. (4.2.14). Experiments show that measured values of Te agree with good accuracy (+-5 percent) with values calculated from the shock-wave velocity and the conservation laws (Fig. 5 ) . The accuracy of the T, measurement depends on many factors, above all on the accuracy of the E calculations. The main error source is the n, determination, because the depression of the ionization potential is unknown. In the first variant of Billman and Sallcap's paper a He-Ne laser with A's of 0.633, 1.15, and 3.39 pm was used. In the second variant, the authors measured simultaneously the absorption of argon laser radiation at two wave lengths Al = 0.488 Frn and A2 = 0.515 Fm and obtained both the electron temperature and electron concentration in the hydrogen plasma. It appeared to be successful because the absorption coefficients for the wavelengths used are different because Al falls at the fine broadened in the field of the charges. The measured values 'of T, agree within & 5 percent with data obtained by other met hods. In conclusion it should be noted that use of a laser as background source allows almost total elimination of the influence of the plasma radiation on the measurement of the absorption. 4.2.1.5. Two-Path Method. The two-path method suggested by Hattel and Broughton30 in contradistinction to the above considered methods does not require a background source. The true temperature is determined by comparing brightness temperatures indicated by the radiation emitted on two paths of different lengths in the gas. One can measure intensities of impurity Na, Ba, Ca, etc., lines and also of the charac39

H. C. Hattel and F. P. Broughton, Ind. E n g . Chern.. Anal. Ed. 4, 166 (1932)

476

4.

MEASUREMENT OF TEMPERATURE

teristic spectrum of the main gas. The spectrum can be either continuous or discrete. In the case of the continuous spectrum the slit width of the spectrometer and type of instrument do not play any decisive role. With high resolution, when one can record in only the wavelength interval of a single line, this method may be used for line spectra as well. Let us consider the arrangement where a mirror, with reflection coefficient R , is inserted to change the optical path in one of the beams. The emittance ,?(A, TI,,)of a plane isothermal layer of thickness I with absorption coefficient ~ ( hT, ) is E(A, TII,)= JVX, T)(l - e-e(A,T)P’)).

(4.2.17)

If the mirror is inserted in this beam, then E(A, T2J = f ( A , T)( 1

- e-E(hsT)P’)( 1 + Re-E(A*T)Pl, (4.2.18)

The ratio E(X, T,,)/E((h, TI,,) equals E(X, Tz,)/E(X,Tlh) = 1

+ Rt?-E(h*7’)P’.

(4.2.19)

Using Wien’s law for the emittance of the gas, one obtains from (4.2.17) and (4.2.19)

(4.2.20) The true temperature T is determined from this relation. In the case of optically thick gas the relations (4.2.17) and (4.2.19) take the simple form ,?(A, Tb) = f ( A , T ) , i.e., the brightness temperature Tb is the temperature. In the opposite case, when E + 0 and E(X, T2h)/f%h Tlb) = 1 + R , the two path method does not allow measurement of gas temperature. Thus the absence of reabsorption is a condition for the applicability of the relative intensity method, whereas in the two-path method the reabsorption of radiation is a necessary condition for its applicability. The advantage of the method of two paths over the method of relative intensities is that one need not have information about transition probabilities. Nerem ef ~ 1 1 adapted . ~ ~ this method to measure shock-wave temperatures in the velocity range 4-10 km/s in air and xenon at an initial pressure of 1 Torr. It was shown that the sensitivity of the method increases MI R. M. Nerem, J . B. Bader, J. B. Dann, and M. A. Culp, Rrcenf Dev. Shock Tube Res., P m . fnf. Shock Tube Symp.. Y t h , 1973, p. 773. Stanford Univ. Press, Stanford, California,

1973.

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

FIG.6 . Dependence of relative sensitivity of the first path in the two-path method on optical thickness for different ratios of path lengths 1 2 / l , . [From Ref. 40.1

477

Optical thickness, E ~ L

with increasing optical gas density, with decreasing path ratio and falling temperature (Fig. 6 ) . Since the two-path method involves absolute intensity measurement in the two beams at the same wavelength, the data permit in addition to temperature, determination of the gas optical density. Uncertainty in the temperature measurement does not exceed 5 percent. It is possible by this method to study gas cooling phenomena and establishment of equilibrium. If the gas studied and the resolution of the apparatus are such that the intensity of radiation is recorded over a broad spectral interval, then in formula (4.2.19), instead of the absorption coefficient, the equivalent line width ui appears (formula (4.2.10)) depending on the line shape. This It is based on the relation technique was developed by Penner.5~41.42 E(A, Tzt,)/E(A, T1b)

= (P(EmaxX, a’),

(4.2.21)

where uo = (b, + b,)(ln 2)1’2/bD,where b N ,b,, and b, are the natural, Lorentz, and Doppler line halfwidths respectively, x is the partial pressure of the radiating gas p multiplied by the optical length 1. Emax =

const

g,(qk)2

exp

(- 2)

(4.2.22)

where g, is the statistical weight of the upper state, El is the lower energy level of the transition, and (q# is the square of the matrix element of the transition. As an example of the function q(emaxx,aa) Penner5*41.42 gives results of calculations for the OH radical versus E,,, x. Experimental determination ‘I

S. S. Penner, J . Chem. Phys. 20, 1341 (1952). S . S. Penner and E. K . Bjornerud, J . Chem. P h y s . 23, 143 (1955).

478

4. M E A S U R E M E N T

OF TEMPERATURE

of E(A, TZb)/E(A,T l b ) for the spectral line permits evaluation of E,,, x and consequently, a temperature. This method differs from the method of relative intensities discussed in the next section, in that here the influence of reabsorption on the accuracy of the temperature measurement is eliminated. The two-path method was developed for use in flame investigations. It has been applied also to temperature measurement in a propulsive jet.43 In addition Penner and c o - ~ o r k e r have s ~ ~ called attention to the possibility of using this method for investigation of chemical reactions behind the shock front, as E(A, Tzb)/E(A, TI),) is a single valued function of E,,, pl. Using a mirror interrupter in the optical beam allows one to record both wavelengths using a single detector. 4.2.1.6. The Method of Relative Intensities. The intensity of line radiation per unit solid angle can be expressed by the following quantities (4.2.23)

where Aki is the transition probability, 1 is the radiating layer thickness, C is a constant, and Nk is the degeneracy of the upper level. It is rarely of value to measure the absolute intensity of a spectral line, firstly because Nk usually is not known and secondly it is very difficult to determine the geometry of the optical instruments and windows. Nevertheless in a number of paper^^^,^ gas temperature has been measured by the absolute intensity method. In Ref. 45 plasma temperature in an electrodeless plasmotron was obtained by measuring the intensity of 01 5330 A and in Ref. 46 plasma temperature of a mercury discharge at high pressure was obtained through measurements on the doublet A = 57705590 A.

More often, the method of relative intensities is used in which the intensities of two lines from two different levels of the same source are compared and these uncertainties are eliminated from the ratio Iki/Inm =

(Aki/A,m)e-(Ek-Em)’kT.

(4.2.24)

It is sufficient now to have information about I M , I,, , A,, , and Aki . The measurement accuracy depends to a great extent on the level difference C. C. Ferriso, S y m p . (1tzl.) Combmt. [Pruc.],8rh, I960 p. 275. Williams & Wilkins, Baltimore, 1962. W. Hooker, M. Lapp, D. Weber, and S. S. Penner, J . C h e m . Phys. 25, 1087 (1956). F. A . Buyevich, V. M. Nikolaev, Yu. A. Plastinin, G. F. Sipachev, and M. I . Fkushin, Zh. Prikl. Mekh., Tekh. Fiz. No. 6, p. 1 1 1 (1968). J . Appl. Mech. Tech. Phyr. (English Trans/.)9, 727 (1968). W. Elenbaas, “The High Pressure Mercury Vapour Discharge,” p. 36. North-Holland Publ., Amsterdam, 1951.

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS

479

- Em. Therefore, if possible one selects lines from different ionization spectra. The relative intensity method was developed by Ornstein and his and has been successfully used under several conditions. , ~ ~Broida and Shuler50 measured Gaydon and Wolfhard ,48 G a y d ~ n and flame temperatures by this method, and Lochte-Holtgreven and MaeckeP measured discharge temperature. In recent times the relative intensity method has been used to measure shock and detonation temperature by CN band^^*^^,^^ and to measure temperature in the glow discharge in the laser mixture C02 + N2 + He54;in the last case the R2 branch with 9 = 20 to 25 in the 0, 0 band of N2 (transition C3rU- B37r,) was used. It is known that relation (4.2.23)is valid for optically thin layers. Indeed since the intensity I’ recorded by the photodetector is related to the intensity I, determined by formula (4.2.23),by the relation Ek

(4.2.25) then I’ -+ I when ~ p l - 0. For optically thick layers we have =

IIE(Az,T)/IzE(AI,T ) ,

(4.2.26)

so the ratio of absorption coefficients appears in Eq. (4.2.24), i.e., line reabsorption takes place. When investigating high speed processes such as those behind shock fronts one has to deal with the strength of spectrum lines and this increases the possibility of reabsorption. Sobolev et U I . using ~ ~ this method in shock tube investigations employing the resonant doublets of Na and Li concluded that the results were unreliable because of reabsorption. To increase the accuracy of experimental measurement it is advisable to use ~ ~ the Balmer series to more than two lines. For example, J i i ~ g e n sused measure the arc temperature in hydrogen (Fig. 7). 47

48

L. S. Ornstein and W. R . van Wijk, Z . f h y s . 49, 315 (1928). A . G. Gaydon and H. G. Wolfhard, Proc. R . Soc. London, Ser. A 194, 169 (1948).

A . G. Gaydon, “The Spectroscopy of Flames.” Chapman & Hall, London, 1957. H. P. Broida and K. E. Shuler, J . Chem. f h y s . 20, 168 (1952). W. Lochte-Holtgreven and H. Maecker, Z . Phys. 105, 1 (1937). ‘* N . N. Sobolev, A. B . Potapov, V. F. Kitayeva, F. S. Faizullov, V. N. Alyamovski, E. T. Antropov, and 1. L. Isaev, Opt. Spektrosk. 30, 612 (1971). Opt. Spectrosc. (USSR) (English Trunsl.) 30, 332 (1971). 53 W. H . Parkinson and R . W. Nicholls, Cun. J. Phys. 38, 175 (1960). 54 L. F. Erybasheva and V . N . Ivanov, Opr. Spektrosk. 30,612 (1971). Opt. Spectrosc. (USSR) (English Trans/.)30, 332 (1971). 55 G. Jiirgens, Z . Phys. 134, 21 (1952). 48

480

4.

MEASUREMENT OF TEMPERATURE

15000

20000

25000

30000

35000

Wave number, I /.A(crn-')

FIG.7. Measured electron excitation temperature of a stabilized arc in hydrogen by the relative intensity method using Balmer lines and continuum. [From Ref. 55.1

The most frequent use of this method is to measure rotational temperature. In this case the probabilities Akl in (4.2.24) are calculated theoretically whereas in the case of electron transition they are taken from experiment. We can look at the method of temperature determination by observation of transitions between the rotational lines of a separate branch. From (4.2.23) we find (4.2.27)

and plot log Zki/Akiv4 against Ek . A straight line as a result justifies the assumption that there is a Boltzmann distribution in the gas. Then according to (4.2.27) the slope of the line gives us l/Tr (Fig. 8). Since energy exchange between rotational and translational degrees of freedom is very effective, requiring only a few collisions, the rotational temperature coincides, as a rule, with the translational temperature. But there are some exceptions where a nonequilibrium distribution of rotational energy is found. For example the rotational temperature of the OH radical obtained by radiation measurement in flames usually differs from the equilibrium temperature. To determine the rotational temperature experimentally by resolving rotational structure one needs to use a spectrograph capable of high resolution. Since this is not always feasible a technique of rotational temperature determination not requiring resolution of the rotational structure is ~seful.~~-~~ J . A. Smith, Dissertation, Utrecht (1950). P. I . Sommers. Dissertation, Nijemegen (1954). 50 A . M . Gubanov, Z h . Prikl. Sp&rosk. 12, 794 (1970). J8 57

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

FIG. 8. Equation (4.2.27) for two-flame mixtures for the OH band at A = 3064 A . Case I: the mixture CzHz + 2.5OZ, T = 5700 K . Case 2: the oxygen flame of formic acid.

48 1

K'(K'+I)

One can also determine gas temperature from the intensity distribution in the vibrational bands. Strictly speaking, to accomplish this one needs to sum up all the rotational transitions of every vibrational state in order to use Eq. (4.2.27). This is a cumbersome operation and has not been carried out. If there is a common Boltzmann distribution for all the rotational states, then in the absence of line reabsorption one can apply the equation (4.2.28) In this case the probabilities A involve simultaneously the probabilities of rotational and vibrational transitions and to be useful for measurement the rotational temperature must be known. The measurement of the intensity distributions in rotational-vibrational bands of infrared regions thus leads directly to a vibrational temperature. This method is not valid for molecules such as 02,N2,I2 etc. The measurement of vibrational temperature by vibrational transitions has been carried out successfully in arcsJg and in An attempt has been made to apply this method for temperature measurement in the shock 4.2.1.7. The Measurement of Temperature by Doppler Broadening. Random thermal motion of radiating particles leads to line broadening (Doppler effect). If other factors do not influence the spectral intensity contour of a line one can determine translational gas temperature by line L. S. Ornstein and H. Brinkman, Proc. K. Ned. Akad. Wet. 34, 33 (1931). N . Thomas, A . G. Gaydon, and L. Brewer, J . Chem. Phys. 20, 369 (1952). F. Rossler, Proc. I n t . Conf. loniz. Phenom. Gases, 5th, 1961 1 , p. 842 (1962). 62 H . P. Broida, J . Chem. Phys. 21, 34 (1953).

5s Bo

482

4.

MEASUREMENT OF TEMPERATURE

halfwidth. The line halfwidth due to Doppler broadening alone is

"'= 7.16 . 10-7 A

cm, (4.2.29)

where A is the wavelength, R is the gas constant, c is the speed of light, and p is the molecular weight. This method is not sensitive because the temperature appears under the square root sign. For example, to measure temperature with an accuracy of l percent the line halfwidth must be measured to 0.5 percent. As Doppler widths are only some fraction of an angstrom (for example AX,, for H B is 0.061 A at T = 300 K) determination of line shapes with precision requires spectral equipment with extremely high resolution. The method of temperature measurement by Doppler broadening was developed by Gaydon and Wolfhards3 for low pressure flames. They investigated the Doppler width of the radiation line of CH (A = 3900 A) by interferometric means. Doppler broadening is one of the few methods which can be applied to study high temperature low density plasmas as in research on fusion and At these conditions Doppler broadening predominates and all other kinds of broadening play a minor

4.2.1.8. Measurement of Electron Temperature by Light Scattering. Temperature measurement from the character of emitted light gives an average over a geometrical path and one can determine local temperatures only on the basis of some symmetry assumptions. Probes inserted in a reacting high temperature gas alter its properties significantly. Under some conditions only the method of light scattering from electrons allows us to measure the local translational temperature of electrons and ions. Methods have been devised to sense the temperature of regions as small as fractions of a millimeter. The theoretical foundations of the method of light scattering from electrons are given in Ref. 67-71. According to Salpeters7 the scattered radiation intensity is

I,

=

Z,

(21' S(n, -

o)n,

(4.2.30)

A . G. Gaydon and H. G. Wolfhard, Proc. R . SOC. London, Ser. A 199, 89 (1949). A. Pierce and L. Goldberg, AJtron. J . 53, 202 (1950). R. E. Redman, Mon. Nor. R . Astron. SOC. 102, 104 (1942); 104, 99 (1944). 88 H . R. Griem, in "Temperature, its Measurement and Control in Science and Industry" (C. M. Herzfeld. ed.), Vol. 3 ( l ) , p. 615. Van Nostrand-Reinhold, New York, 1962; also Vol. 9 of this series. B7 E. E. Salpeter, Phys. Rev. 120, 1528 (1960); 122, 1663 (1961). J . A . Fejer, Con. J . Phys. 38, 1114 (1960); 39, 716 (1961). ((o J. P. Dougherty and D. T. Farley, P m c . R . SOC. London, Ser. A 259, 79 (1960). K. L. Bowles, Adv. Elecfron. Electron Phys. 19, 55 (1964). '*D. E. Evans and J . Katzenstein, Rep. Prog. Phy.5. 32, 207, 1969. +M

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

483

where 1, is the incident radiation intensity, re = e2/m,c2 is the classical electron radius, n, is electron number density, n = n, - no, no, n, are the wave vectors of incident and scattered radiation, w = o, - oo,and S(n, w ) is a function related with the electron and ion components. For both electrons and ions scattering takes place by electrons, free and bound respectively, so we have S(n, 0) = Se(Xe, a h ,H e , T,) + Si(XiZ, (TJTi), a(n, f i e , Te))? (4.2.31) = = 2kT,/m,, k is the Boltzmann constant, Z where X, = o / n V , , is ion charge, a = (nA,)-l = Ao/4.rrADsin(8/2), 8 is the scattering angle, A. is the wavelength of the incident radiation, and A, = ( k T / 4 ~ m , e ~ is ) ”the ~ Debye radius. When S(n, w ) is integrated over frequency, one obtains72 an expression for the form factor

S,

1

=-----a

1

a4 s.’ = (1 + a2)(1 + za2 + Z(T,/Ti)a2)’

+ a2’

(4.2.32)

The parameter a is proportional to the ratio of wavelength of the scattered radiation to the Debye radius. It has an important role in the scattering theory. Let us consider the most significant cases (Figs. 9 and 101.73 1. a << 1. In this case one can neglect the collective interaction of electrons in the theory of light scattering; the Thomson scattering with cross section uT = 8m,/3 is observed. The line shape corresponds to a simple Gaussian and is determined only by the electrons (vide Eq. (4.2.32)), its ion part being essentially zero. The line halfwidth is given by the relation

AAlI2 = 4Ao sin(8/2)(2kTe In

(4.2.33)

from which one can find electron temperature if Ahllz is known. 2. a >> 1 . In this case S, = l / a 2and Si= Z/(1 + ZT,/Ti); for Z = 1 and thermal equilibrium Si-+ 1/2. Thus for a >> 1 the scattered light has a spectral distribution wholly determined by the collective particle interaction. It consists of a central line whose width depends on the ion temperature and two satellite lines symmetrically situated relative to the center at Ah

A

=2

2r e

(4.2.34)

E. E. Salpeter, J . Geophys. Rrs. 68, 1291 (1963). M o d . O p t . Method3 Gus Dyn. Rrs., Proc. l n t . S y m p . . I970 p. 155 ( I97 I ). 72

73

R. H . Lovberg,

484

4. MEASUREMENT

OF TEMPERATURE

AA = k - k , ( n m )

FIG.9. Measured spectrum of light scattered from a nitrogen theta-pinch at pressure 7 Pa, a = 0.43. [From Ref. 73.1 T, = 7.1 i 2.7 x 10' K; n, = 0.9 t 0.9 x 10l8 ~ m - ~ .

where upis the plasma frequency. From a measurement of AA one can find the electron concentration. 3. For temperature measurement when 1y 1 it is necessary to cornpare the measured shape of the scattered line with the calculated shape. The parameter cy depends not only on the properties of the plasma but on the observation angle 8 as well, and so the shape of the spectrum also depends on 8.

-

The radiation scattering cross section is very small so the intensities of incident which one needs to measure in the laboratory are beam intensities. Only with the advent of lasers have the needed radiation powers of more than lo8 W become available. The question arises

2

4

6

8

10

AX=X-X,(nm)

a

FIG. 10. Measured spectrum of light scattered from a nitrogen &pinch at pressure 7 Pa, = 1.22. [From Ref. 73.1 I; = 5.6 % 0.3 x 1 Q K; n, = 59 t 0.2 x 10'" cm-:j.

4 . 2 . MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

485

whether such light fluxes influence the plasma. This question has been considered both t h e ~ r e t i c a l I y ' ~and * ~ ~e~perimentally.'~.'~It has been shown that when ruby laser radiation passes through a xenon plasma (n, = 9.7 x 10'' ~ m - n, ~ ,= 5.6 x 10I8~ m - at ~ I) > lo8 W/cmZ nonlinear effects exist; first the plasma transparency and then the absorptivity increases. This means that for Z > lonW/cmZ the plasma is perturbed by the measuring radiation. Laser radiation scattering has been studied experimentally under various conditions: on free electron^,^^ in the B - p i n ~ h ,in~ ~ the , ~arc ~ discharge,s1 behind ~ h o c k s ~ and * - ~in ~ pulsed high frequency discharges.86 These examples show the variety; reference should be made to Vol. 9 of this treatise for a complete discussion. 4.2.1.9. The Methods of Two Absorbers. This procedure for high temperature measurement is a double-path method. In each path of far ultraviolet or X-radiation thin foils of different thickness are placed normal to the beam and in these foils a part of the radiation is absorbed. One can calculate the intensity ratio in two passes 11/1287 and plot the ratio as a function of temperature

( 4 . 2 . 35) 74 G . M. Malyshev,Zh. Ehsp. Tuor. Fiz. 35,2129(1965);Sov. Phys. Tech. Phys. (English trans/.) 10, 1633 (1966). 75 L. A . Dushin and 0. S. Pavlichenko, "Issledovaniye Plasmy s Pomosch'yu Lazerov." Atomizdat, Moscow, 1968. 76 N . A. Generalov, G . I. Kozlov, and Yu. P. Raizer, Pis'rna Zh. Eksp. Teor. Fiz. 8, 138 (1968);JETP Lett. (English trunsl.) 8, 82 (1968). 77 N . A . Generalov, G . I . Kozlov, and Yu. P. Raizer, Zh. Prikl. Mekh. Tekh. Fiz. 1, 142 (1970);J. Appl. Mech. Tech. Phyc. (English trunsl.) 11, 144 (1970). 7n G . Ficco and E. Thompson, Phys. Rev. L e f t . 10,89 (1963). E. Funfer, W. H. Kegel, B. Kronast, and H . J . Kunze, Proc. l n t . Conf. loniz. Phenom. Gases, 61h, Paris. 1963 Vol. 4, p. 119. 1964. W. E. R. Davis and S . A . Ramsden, Phys. Lett. 8, 179 (1964). A . W. De Silva, D. E. Evans, and M. J. Forrest, Nature (London) 203, 1321 (1964). R. M . Patrick, Phys. Fluids 8, 1988 (1965). 83 E. T . Gerry and R . M. Patrick, Phys. Fluids 8, 208 (1965). " Y . Jzawa, M. Yokojama, and C. Yamanaka, J . Phys. Soc. J p n . 21, 1610 (1966);23,1185 (1%7). Y . Jzawa, M. Yojama, and C . Yamanaka, Jpn. J . Appl. Phys. 7 , 954 (1968). BB A. A. Besshaposhnikov, Zh. Prikl. Spektrosk. 6, 172 (1967). 87 A . J. Alcock, P. P. Phashinin. and S. A . Ramsden, Phys. Rev. Lett. 17, 528 (1966).

486

4. MEASUREMENT

OF TEMPERATURE

100

2

80

m c .La

e

60

x

5 0 ._ +

40

LT

20

0

Electron temperature, T, ( e V )

FIG.1 1 . Calculated ratio of x-ray intensities in two paths versus electron temperature. [From Ref. 87.1 Foil thicknesses dl = 0.051 mm and dr = 0.127 mm. Solid angles of beams dn, = 0.27, dill = 0.75 sr.

where G is the ratio of the photomultiplier sensitivities, N(A, T,) is the number of photons emitted as a result of recombinational radiation per unit length of gas at temperature T, per unit solid angle per unit time, E and p are the coefficients of gas and foil, I, d, and d2 are the lengths in gas and foils, and 28 is the angle with vertex at the source which the foils subtend. The experimentally measured intensity ratioa7 allows determination of the temperature of a laser spark from the graph of Fig. 11. Beryllium foils were used and T, was found to be -60 to 180 eV. For bremsstrahlung Johode et ul.B8have carried out extensive calculations of thin film transparencies of different materials. The most frequently used absorbers are beryllium, carbon in the form of polyethylene, aluminum, nickel and titanium. 4.2.1 .lo. Concluding Remarks. Throughout Section 4.2.1 mention has frequently been made of the need for the assumption of local thermodynamic equilibrium in order to interpret the measurements. It is often difficult to ascertain whether this assumption is justified, and interpretation must depend.on an understanding of the chemical and electronic processes, and analysis of parallel observations which also give information on whether local thermodynamic equilibrium exists. An excellent general discussion of this problem has been given by L a p ~ o r t h . ' ~ In addition, it is apparent that reliable use of several of the methods is dependent on assumptions concerning the geometry of the radiating F. C. Johode, E. M . Little, W. E. Quinn, G . A . Sawyer, and T. F. Stratton, Phys. Re\,. 119, 843 (1%0). BB K . C. Lapworth, J . Phys. E 7,413 (1974).

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS

487

medium. Whether the observed medium is optically dense, whether it is homogeneous, or whether, if it is not, the structure of boundary layers on windows is sufficiently understood, can be very important. 4.2.2. Temperature Measurement by Analysis of Scattered Light* The measurement of temperature using Raman and Rayleigh scattering has been discussed in Chapter 3.2. The techniques are related to those for the electron temperature measurements described in Section 4.2.1.8, notably providing precise space and time resolution. Here, we recount some of the concepts, and refer the reader to the references here and in Chapter 3.2 for details of the methods. Intensities of Rayleigh and Raman-scattered radiation from a laser beam traversing a fluid are independent of the velocity of the scattering molecules. The scattered intensity S is, however, proportional to the density in accordance with S = const x

Nluj,

(4.2.36)

j

where Nl is the concentration (e.g., moles/cm3) of an observed species and uj is the corresponding scattering cross section for the observed process. For Rayleigh scattering, either under conditions where the scattering cross section does not vary appreciably from species to species, or where no appreciable change in the concentration-weighted mean value of the cross section occurs with chemical reaction, values of gas temperature can be found from density measurements and the equation of state, if one can assume constant pressure or known pressure variations.8ea Additionally, temperature can be found from the width (via the Doppler effect) of the Rayleigh line,”’ although this somewhat difficult method can depend upon density-sensitive corrections at densities near or greater than ambient and/or at small scattering angles. The spectral structure of Raman scattering is fundamentally different from Rayleigh scattering, since each molecule possesses a distinctive signature at wavelengths characteristic of that molecule (with occasional spectral coincidences). Furthermore, the bands of Raman scattering that are observed depend upon molecular excitation (rotational, vibrational, and electronic). Thus, in the case of Raman scattering, the concentration Nl distinguishes the fractional population of a specific type of molecule in R . W. Dibble and R. E. Hollenbach, Symp. ( i n t . ) Combust. [Proc.], 18th. Combustion Institute, Pittsburgh (to appear). BBb R . Cattolica, F. Robben, and L. Talbot, Progr. Astronaut. Aeronaut. 53,575 (1977); R. W. Pitz, R. Cattolica, F. Robben, and L. Talbot, Cambusr. Flame 27, 313 (1976).

* Section 4.2.2 is by

Marshall Lapp and C. Murray Penney.

488

4.

MEASUREMENT OF TEMPERATURE

a particular excited level. Since spectral discrimination can often be used to isolate Raman lines or narrow bands, contributions to the observed Raman scattering strength in Eq. (4.2.36) can often be constrained to those from a particular initial molecular level. This sensitive dependence upon initial state molecular populations for Raman scattering permits it to be utilized for accurate temperature diagnostics of various species. In fact, the Raman effect provides a method for thermometry at the molecular level, since the method is based upon relative energy level populations for the molecule under measurement. The resultant molecular Raman signature can then yield the temperature by means of a variety of methods, including spectral contour analysis (i.e., fitting the spectral shape to theoretically predicted values uniquely dependent upon temperature), ratios of intensities of selected portions of the overall Raman spectrum, the width of portions of the Raman contour, e t ~ . ’ ~ ~ Vibrational Raman scattering has been used successfully to determine temperature in a wide variety of laboratory-scale flows, including combustion systems.Rsd-8smRotational Raman scattering has also been used s ~ ~ ~ e s s f ~ l l yand , ~is~ stronger ” - * ~ ~than the vibrational effect, but suffers from complications introduced by seriously overlapping spectral strucM. Lapp, in “Laser Raman Gas Diagnostics” (M. Lapp and C. M. Penney, eds.), p. 107. Plenum, New York, 1974. M . Lapp and C. M. Penney, eds., “Laser Raman Gas Diagnostics.” Plenum, New York, 1974. R. Goulard, ed., “Combustion Measurements.” Academic Press, New York, 1976. 8gf 9. T. Zinn, ed., “Experimental Diagnostics in Gas Phase Combustion Systems,” Progr. Astronuut. Aeronaut. 53 (1977). S . Lederman, Prog. Energy Combust. Sci. 3, I (1977); Phgs. Ffuids 22, 1065 (1979). M. Lapp and C. M. Penney, in “Advances in Infrared and Raman Spectroscopy” (R. J . H . Clark and R. E . Hester, eds.) Vol. 3, Chapt. 6. Heyden and Son Ltd., London, 1977. L. A. Kennedy, ed., “Turbulent Combustion,” Progr. Astroriciut. Aeronuut. 58 (1977). A. C. Eckbreth, P. A. Bonczyk, and J. F. Verdieck, Appl. Spcctrosc. Rev. 13 (I), I5 (1978); published with revisions in Prog. Energy Combust. Sci. 5 , 253 (1979). ‘ ~ 3 Pi. ~ Lapp, in “Laser Probes for Combustion Chemistry” (D. R. Crosley, ed.). Amer. Chem. SOC.Symp. Series. Vol. 134. Chapt. 17, Washington, D.C., 1980. A. C. Eckbreth, Symp. ( I n t . ) on Combust. [Proc.]. 18th. Combustion Institute. Pittsburgh (to appear). M. C. Drake, M. Lapp, C. M. Penney, S. Warshaw, and B. W. Gerhold, Symp. ( I n r . ) Combust. [Proc.], 18th. Combustion Institute, Pittsburgh (to appear). B8n M. C. Drake and G. M. Rosenblatt, in “Characterization of High Temperature Vapors and Gases” ( J . W. Hastie, ed.), Vol. 1 , p. 609. National Bureau of Standards Special Publication 561/1, 1979. 88” W. D. Williams, H. M. Powell, R. L. McGuire, L. L. Price, J. H. Jones, D. P. Weaver, and J. W. L. Lewis, Progr. Astronuut. Aeronuut. 58, 273 (1977). 89p J . Smith and W. H. Giedt, I t i f . J . Heut M i s s TrmsfPr 20, 899 (1977). J . J . Barrett, in “Laser Raman Gas Diagnostics” (M. Lapp and C. M.Penney, eds.) p. 63. Plenum, New York, 1974.

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

489

tures for many molecular species of interest, and from potential scattered light interferences from the spectrally nearby exciting laser line. The main disadvantages of Raman diagnostics for temperature are the weakness of the Raman effect, which limits applications to moderately “clean” systems, and the care necessary to set up precision optical systems with adequate optical access. Where the value of the measurements warrants even greater experimental complexity, highly luminous and/or particle-laden systems can be probed for temperature by coherent anti-Stokes Raman spectroscopy (CARS), a technique which requires two (or sometimes three) incident laser beams, but which produces intense output signals from which temperature can be extracted.8gJ CARS is described in Section 3.2.3.7 of this volume. 4.2.3. Measurement of Temperature by Analysis of Electron Beam Excited Radiation*

The intensity distributions in the fine structure of electron beam excited emission spectra (refer to Chapter 3.3) can be used to measure vibrational, rotational, and translational temperatures. The methods employed are those described in Section 4.2.1, but it is necessary to relate the distribution of emission intensity observed in, say, the rotational lines of a vibrational band, to the population distribution in the rotational energy levels of the molecules before they were excited by the electrons. In the case of rotation, the question of how the angular momentum of the excited molecule is affected by the electrons must be answered. For a measurement of translational energy distributions the Doppler profile of a line can be used if the linear momentum added to the excited particles by the excitation process is known and not excessive. Consider the case of an energetic electron exciting an atom. It has been noted (cf. Chapter 3.3) that by far the most probable excitation transitions will be those that are optically allowed; i.e., dipole transitions. But how much momentum is desposited in the translational motion of the excited particle as a result of such an excitation collision? If the differential scattering cross sections are known for the particular excitation studied, the momentum transferred to the excited particle can be calculated by simply considering the initial and final electron velocity vectors. It turns out that for high energy electrons the momentum that is transferred is very small, whereas for low energy electrons it is quite significant. Secondary electrons (Chapter 3.3) will, therefore, have the potential of in* Section 4.2.3 is by E . P. Muntr.

490

4. MEASUREMENT OF TEMPERATURE

troducing disturbances. As an example consider the case of helium. For excitation of the 3lP state the maximum transfer, even for low energy electrons, is only a little greater than 4 percent of the thermal energy of the helium translational motion at 300 K.gO The usefulness of electron beam fluorescence as an indication of population distributions is described in the remainder of this section. In each case, the discussion centers around those gases which have been investigated experimentally. Any generalizations about the applicability to other gases of the fluorescent population distribution or temperature measurements is in detail unjustified. 4.2.3.1. Translational Temperature. A method developed by Muntzgo can be used to measure molecular velocity distribution functions in rarefied helium flows. The technique involves the excitation of helium atoms by a beam of energetic electrons. Fluorescent radiation is selected by optical stops from a small segment of the beam length and observed with a high-resolution spectrograph. Under suitable conditions of gas density, electron energy, and beam current, the 501.567-nm (2lS - 3IP) helium line (refer to Fig. 2, Section 3.3) has a Doppler profile that represents the velocity distribution function of the gas atoms in the region selected for observation. The distribution is for the direction of observation, as modified by the solid angle in which the optics accepts light, and averaged over the two coordinates orthogonal to this direction. The technique is described in detail in Munt~.~O Use of the technique was extended to argon by Harnett and MuntzS1 and H01tz.~~A superior computer-controlled instrument has been developed by Cattolica et ~ 1 and. used ~ ~ by Robbene4 and also used by Be~ker.A ~ ~nonequilibrium molecular velocity distribution was observed by H o l t ~ . Distribution ~~ functions in flows have also been measured by Muntz and Harnettes and by Rixen and A d ~ m e i t . ~ ’ Under optimum conditions the technique can measure translational temperatures in He with an accuracy of a few kelvins at room tempera-

P. Muntz, Phys. Fluids 11(1), 64 (1968). N . Harnett and E. P. Muntz, Phys. Fluids IS, 565 (1972). #* Holtz, Ph.D. Thesis, University of Southern California, Los Angeles (1974). g3 Cattolica, F. Robben, and L. Talbot, in “Rarefied Gas Dynamics” (M. Becker and M. Fiebig, eds.), p. B.16. DFVLR Press, Porz-Wahn, 1974. u4 F. Robben, in “Rarefied Gas Dynamics” (M. Becker and M. Fiebig, eds.), p. C . l . 95 M . Becker, F. Robben, and R. Cattolica, A I A A J . 12 (9), 1247 (1974). 88 E. P. Muntz and L. N . Harnett, Phys. Fluids 12(10), 2027 (1969). O7 W. Rixen and G . Adomeit, in “Rarefied Gas Dynamics” (Proc. Inr. Symp., 9th) (M. Becker and M. Fiebig, eds.), p. B. 18. DFVLR Press, Porz-Wahn, 1974. 91

E. L. T. R.

4.2.

MEASUREMENT OF TEMPERATURE B Y RADIATION ANALYSIS

49 I

ture. The accuracy in argon is not as good because of reduced fluorescence intensity and a small Doppler broadening. 4.2.3.2. Vibrational Temperature. There have been a number of gas dynamic studies that have relied on the measurement of fluorescent vibrational band intensity ratios in nitrogen’s first negative system (refer to Fig. 4b in Chapter 3.3) to provide a measure of vibrational temperatures. While the results have generally been consistent (see a correlation of vibrational relaxation measurements by SebachersB)with other vibrational temperature measurements the whole procedure for measuring vibrational temperatures (or level population distributions) was only critically examined in 1977 by Campbell.9s There is still no extensive high temperature calibration available, although Hunterloohas presented a few data to slightly above 1000 K . The theory of electron excitation has been presented by Bates’O’ and Langstroth.Io2 Nitrogen has been studied in great detail since both in aeronomic and gas dynamic applications it provides strong, easily observed emissions. The relationship between electron beam excited emission and vibrational temperature for Nz has been given by Muntz’03 and Lewis and Williams.’04 Consider a ground state molecule with vibrational levels u;’ subject to excitation by energetic electrons as shown in Fig. 4b of Chapter 3.3. Excitation is to an upper electronic state in the vibrational level u’ with subsequent emission to a lower electronic level and vibrational level 4’. When only spontaneous depopulating transitions affect the populations of the v’ levels the intensity of the vibrational transition u’ + u;’ is14*15

(4.2.37)

where x is a constant, vUtu,, the wave number of the emission and the q’s are Franck-Condon factors. If known, the q’s can be replaced by vibrational transition probabilities. Useful lists of Franck-Condon factors and vibrational transition probaD. I . Sebacher, AIAA 1. 3 4 ) . 819 (1967). D. Campbell, in “Rarefied Gas Dynamics” (R. Campargue, ed.), p. 763. C . E . A . , Paris, 1979. loo W. W. Hunter, AIAA J. 8, 959 (1970). D. R . Bates, f r o c . R . Soc. London. Ser. A 196, 217 (1949). loz G. 0. Langstroth, Proc. R . Sot,. Lundon, Ser. A 146, 166 (1934). Io3 E. P. Muntz, AGARDOgruph 132 (1969). IM J . W. L . Lewis and W . D. Williams, AIAA J . 7, 1202 (1969). 4y

4. MEASUREMENT

492

OF TEMPERATURE

i I

.

lz

'/I

I

0

I 1000

(3,4)

I I I I 2000 3000 4000 5000 6000 T

Y

FIG.12. Temperature dependence of the vibrational band intensity ratio for two pairs of the transitions in the first negative system of N;.

bilities have been compiled by Nicholls.lo5 His tables for the excitations N2X'Z + N2C377, N$B2X, and the first negative and second positive systems of N2 can be found in Muntz's review.Io3 Many other band systems are tabulated by Nicholls including a set of Franck-Condon factors for the excitation of the first negative system of oxygen that has been published by Petrie r t ~ 1 . as' ~the~ result of a private communication. If a vibrational temperature T, is assumed for the ground state vibrational levels of nitrogen the predicted ratio of intensities for any two bands of the first negative system can easily be found. In Fig. 12 the ratio of intensities Zo,/Zlo versus T, in the N i first negative system is shown as calculated by Lewis and Williams.1o4 Other bands can be chosen to give more or less sensitivity of the intensity ratio to T, for different ranges of T,. The prediction is relatively straightforward, the real difficulty is to know the accuracy of the Franck-Condon factors or the vibrational tran-

'06

R . W . Nicholls, J . Q u i n t . Spectrosc. & Radicit. Trunqfer 2, 433 (1962). S. L. Petrie. A . A. Boiarski, and S. S . Lazdinis, AFFDL TR-71-30(1971).

4 . 2 . MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

493

sition probabilities. This is discussed in more detail by Muntz,lo3Butefisch and Vennemann,lo7Campbelles and Petrie.lo8 A further difficulty is raised by the work of Lewis and Williams104who found significant variation in the nitrogen first negative ratios Ioo/Il, , Io2/I13, and IOl/Il2 as a function of pressure at room temperature. The variation appears to be important for pressure above about 10 Pa at room temperature. Williams' work was done in a slow flow system. The pressure effect seems to be somewhat less in a high speed flow. Butefisch and Vennemannlo7 show results based on measurements in a hypersonic flow Of

zO1/zl2= 7.65 + (3.7 x

10-17)n,

(4.2.38)

where ng is the total nitrogen number density. The corresponding theoretical ratio is 7.75. Care should be exercised in using this relationship as it is valid for only one temperature. The temperature sensitivity is unknown but probably significant. The role played by secondary electrons in vibrational temperature measurements has not been identified. 4.2.3.3. Rotational Temperature. The direct measurement of temperature in gas flows furnishes an extremely important datum in many circumstances. The measurement of temperature in a moving gas using the fluorescence excited by a beam of energetic electrons in nitrogen was first studied in detail by Muntz.'Og The rotational temperature is valuable because in many cases it can be safely assumed to be equal to the translational temperature. This technique has found wide application in aerodynamic and gas dynamic investigation^.^^^ There are still a number of questions about its use, principally at high density and/or low temperatures. These difficulties are currently under continuing investigation. The only emission that has been studied in detail is the first negative system of nitrogen, primarily the (0, 0) and (0, 1) vibrational bands. These are by far the most intense spectral features in most flows of nitrogen or air. Petrie and LazdinisllO have also investigated the technique for oxygen but the emission spectrum is very complicated. Long before the use of fluorescent emission as a flow diagnostic, auroral emissions in the N: first negative system were being used to indicate atmospheric temperatures in the vicinity of the auroral displays. 10'

'lo

K . A. Biitefisch and D. Vennemann, Prog. Aerosp. Sci. 15,217 (1974). S. L. Petrie, Aeronaut. Res. Lab., R e p . ARL-65-122(1965). E.P. Muntz, Phys. Nuids 5(1), 80 (1962). S. L. Petrie and S. S . Lazdinis, AFFDL TR-68-153(1968).

494

4.

MEASUREMENT OF TEMPERATURE

The major problems associated with the interpretation of the auroral rotational temperature measurements as gas temperatures are uncertainties about the nature of the exciting particles. A good short review of earlier work on this is given by Roesler et nl.ll' Oldenberg112reviewed the early results and suggested certain generalized criteria which, if satisfied, should provide a rotational temperature representative of the translational temperature. He also noted that if conservation of angular momentum applies during the excitation, the excited state should have a Boltzmann distribution in the rotational levels. However, if the internuclear separation is different in the ground state from that in an excited state, the energy associated with the rotational levels will be altered. If K is a rotational quantum number, the rotational energy associated with a rotational level is113B,K(K + 1)hc. For conservation of rotational quantum number, the energy of the distribution will thus be changed proportionately with the change in the rotational constant B,. The rotational constant is representative of the moment of inertia of the state in question. As a consequence, the indicated temperature would be expected to increase if the internuclear distance decreased. Some years later Branscomb114following Oldenberg's suggestion showed that this idea was substantially correct for the second negative system of oxygen. The intensity in a rotational line following the Oldenberg reasoning would be given by (Bran~comb"~ or H e r ~ b e r g , "pp. ~ 126 and 207), ln(lxr/ZoK')= {-- B,$'(K'

+

l)hc/kTd

+ const

(4.2.39)

for the R branches of the first negative system of nitrogen, where K' is the upper state rotational quantum number designating the emission line, TR is the rotational temperature of the gas before excitation, and B,, is the lower state rotational constant. It was at about this time that Muntdoepublished his results for 17.5 keV electrons along with a somewhat different analysis of the excitation as well as an experimental illustration of its use in gas flows. He found additional terms in the relationship for intensity as a function of rotational quantum number by taking into account the rotational transition probabilities. It is generally referred to as the dipole excitation model. For the particular case of the first negative system at 300 K these terms are about F. L. Roesler, C. Y . Fan, and J . W. Chamberlain,J. Armos. T e r r . Phys. 12,200 (1958). IIZ

0. Oldenberg, Phys. R e v . 46, 210 (1934).

113 G. Herzberg, "Electronic Spectra of Diatomic Molecules." Van Rostrand-Reinhold, New York, 1950. II' L. M. Branscomb, Phys. R e v . 79(4), 619 (1950).

4.2.

MEASUREMENT OF TEMPERATURE BY RADIATION ANALYSIS

495

as important (introduce 2 percent change in temperature) as the simple conservation of rotational quantum number use ofB,,, (introduces 3.5 percent change in temperature) such as in Eq. (4.2.39). This is because of the small change in equilibrium internuclear distance from 0.1094 nm for N2XIX to 0.1075 nm for N2+ B2Z. At low temperatures the effect of the terms found by Muntz becomes much larger, representing at 75 K about an 8 percent correction. Muntz's expression for the relative intensities of the rotational lines ( K ' , K;') in one of the bands ( u ' , u;') of the first negative system is

ln[{(lx~xr),~uh'/lo)/(K' + G' + 1)[Glv41 = - K ' ( K ' + l)&;Phc/kTR

+ const.

(4.2.40)

The [GI term has the value

[GI =

(K'

+ 1) exp{-2(K' +

I)Bufk/kTR} (2K' + 1)

+ K'

e~p{2K'B,~~hc/kT,} (4.2.41)

If the theoretical description of the excitation process is correct, the rotational temperature may be obtained by measuring the relative intensities l/10 of the rotational lines in a vibrational band of the first negative system and plotting

ln{[(Z~~~h')Y~u'2~/Z0]/(K' + K;' + l)[G]} versus K'(K' - 1.2

+ 1).

(4.2.42)

-

2.02.2 -

2.42.62.8-

3.0-

I

I

I

I

FIG. 13. Relative rotational line intensities for nitrogen at 300 K, versus K'(K' + 1). K' is rotational quantum number of the excited state. n, = 3.2 x loP1 m-3. TRA= 300 K and TRM= 305 K are the actual and measured rotational temperatures respectively. TRM comes from the data shown in the figure, TRAfrom independent experimental information.

496

4. MEASUREMENT

OF TEMPERATURE

This requires an iteration process, starting with a guess for T,. Details are given in Biitefisch and Vennemann.lo7 A typical plot appears in Fig. 13 where it can be seen that for this particular case the straight line fit is excellent. The interpretation of intensity measurements in other situations is not so clear. Following Muntz’s publicationlog there have been a large number of investigations of the method over a wide range of condition^.^^^-'^^ Discrepancies between the Muntz dipole model and observations have appeared and are greatest at low temperatures and high densities. The origin of these discrepancies for low temperatures is still a matter of some controversy but is almost certainly associated with secondary or possibly with heat release attendant on incipient condensation in the very low temperature flow used to investigate the di~crepancies.’~’The discrepancies at high densities are more conjectural.13’ For more details on this technique the reader should refer to the reviews of Muntz,lo3 Butefisch and Vennemann107and to more recent results presented by Karelov et and Coe et There is little doubt that if used properly, the technique will measure rotational temperature in nitrogen flows with an absolute systematic error not exceeding k 5 K for temperatures below 1000 K. The most recent ~ ~ rpromises k ~to reduce ~ ~this *error~still ~further. ~ W. D. Williams, AEDC TR-68, 265 (1969). F. Robben and L. Talbot, Phys. Fluids 9, 644 (1966). 11’ F. Shelby and R. A. Hill, Phys. Fluids 14, 2543 (1971). S. Lewy, J . Phys. (Puris) 33, 955 (1972). 118 W. C. Ho and G. Schweiger, Phys. Fluids 15, 1447 (1972). I2O D. J. Marsden, in “Rarefied Gas Dynamics” (J. H. de Leeuw, ed.), Vol. 2, p. 566. Academic Press, New York, 1966. 121 B. L. Maguire, in “Rarefied Gas Dynamics” (L. Trilling and H. Wachman, eds.), Vol. 2, p. 1761. Academic Press, New York 1969. Iz2 P. V. Marrone, Phys. Fluids 10, 521 (1967). R. S. Hickman, Univ. South. Calif., Aerosp. E n g . Rep. 104 (1966). H. Ashkenas, Phys. Fluids 10, 2509 (1967). S . L. Petrie and A. A. Boiarski, in “Rarefied Gas Dynamics” (Proc. 6 1 1 . S y m p . , 6rh) 2, (L. Trilling and H. Wachman, ed.) p. 1685. Academic Press, New York, 1969. R. B. Smith, in “Rarefied Gas Dynamics” (L. Trilling and H. Wachman, eds.), p. 1749. Academic Press, New York, 1969. D. C. Lillicrap, AlAA Pop. No. 71-605 (1971). D. C. Lillicrap and L. P. Lee, N A S A Tech. Note D-6576(1971). 129 A. E. Kassem and R. S. Hickman, A l A A J . 13(6), 770 (1975). I3O A. K. Rebrov, in “Rarefied Gas Dynamics” (J. L. Potter, ed.), p. 811. AIAA Press, New York, 1977. l3I D. Coe, F. Robben, and L. Talbot, in “Rarefied Gas Dynamics” (R. Campargue, ed.), Abstr. 160. C.E.A., Paris, 1978. 132 N. V. Karelov, A. K. Rebrov, and R. ci. Sharafutdinov, in “Rarefied Gas Dynamics” (R. Campargue, ed.), Abstr. 135. C.E.A., Paris, 1978. IIfl

4.2.

MEASUREMENT OF TEMPERATURE BY RADl ATION ANALYSIS

497

Additional interpretation of molecular and flow processes is needed to connect the rotational temperature T R ,so determined, to the translational temperature or gas temperature T g , particularly for T, less than 300 K . ACKNOWLEDGMENT

The writing of this article was made possible by partial support from the United States Air Force Office of Scientific Research, No. AFOSR 77-3242.

This Page Intentionally Left Blank

5. MEASUREMENT OF PRESSURE* 5.1. Introduction

List of Symbols Cross sectional area; parameter Parameter Electrical capacitance; parameter Dilatational elastic modulus Component of electric field relative to axes of piezoelectric element emf Fringe count; wave front Gage factor for resistance strain gage Frequency response function Electrical current; light intensity; time dependent input I ( ( ) Exponential transform of input I ( t ) Zero order Bessel function of first kind of imaginary argument Zero order Bessel function of first kind with real argument Relative electrical permittivity; K = (w/c,h)”* in diaphragm theory Electrical inductance Elastic modulus unspecified Time dependent output Resonant period = 2 m / o q ; 9, longest resonant period Electrical charge Electrical resistance Unit step at time I = 0 Pressure sensitivity; surface tension Hold time of gage Time variable in addition to t

Time dependent response to unit step input S ( t ) Electrical voltage Shock wave velocity Volume Single crystal axes Young’s modulus of elasticity radius, or half-width, of elastic element Speed of electromagnetic radiation = 0.3 Gm . s-’ nominal value Velocity of strain propagation unspecified C; = Y / [ 1 2 p ( l - v’)] for linear bending diaphragm cg = Y / p strain wave speed in bar c%= E / p dilation wave speed for one-dimensional wave : C = p / p shear wave speed c: = u o / p for pretensioned diaphragm Diameter, or lateral dimension, of elastic element Stress related piezoelectric constant Frequency in Hertz (complete cycles per second) Weight per unit mass = 9.8 N . kg-’ nominal value Thickness of diaphragm; dimension of sensor in direction of strain propagation Dimension

* Part 5 is by R. I. Soloukhin, C. W. Curtis, and R. J. Emrich. 499 METHODS OF EXPERIMENTAL PHYSICS, VOL. 18B

Copyright 0 1981 by Academic Press. lnc All rights of reproduction in any form reserved

ISBN 0 12-475956-4

5 00 %n

I

P Po r

t U

UO

v

M’,W o

5.

MEASUREMENT OF PRESSURE

Unit vector, component of unit vector in ith direction i = 1,2,3 Pressure Amplitude of pressure step or of periodic pressure variation Spatial coordinate, usually radial, rectangular in case of slit diaphragm Time Longitudinal displacement for propagating strain; displacement in plane of diaphragm Maximum displacement in plane of diaphragm due to pretension Material velocity, usually in direction of propagation of strain wave Displacement perpendicular to plane of diaphragm, maximum displacement

Cartesian coordinates, usually z in direction of strain wave propagation Optical path length Sensing element Strain element Permittivity of vacuum Strain, strain components Wavelength = c/f Viscosity coefficient; shear modulus of elasticity Kinematic viscosity coefficient = p / p ; Poisson’s ratio Mass density; electrical resistivity Stress Stress component in one-, twoindex notation Response time of gage Radian frequency = 2nf

Measurement of pressure is often as simple as reading a pointer on a dial calibrated in pascals, bars, or millimeters of mercury or some other set of units in which pressure is expressed. Before proceeding to describe how such simple pressure gages operate and are used, however, it is worthwhile reviewing the inherent assumptions made in supposing such a variable as pressure has a meaning. There is more than the usual confusion among users in different professions regarding the meaning of the variable and the units in which it is measured. Certain kinds of “pressure” will not be discussed in the current part at all. 5.1.l.Mechanical Concept of Pressure

The existence of contact forces everywhere between contiguous parts of matter is one of the basic concepts of continuum mechanics. Pressure is an especially simple case of this concept. In its most general form, the concept is expressed by postulating the existence of astress tensor, which provides that nine numbers specified at a point in the matter and a specified set of rectangular coordinate axes can describe the contact force per unit area acting across uny small area at the point. Specifically the nine numbers, called components of the stress tensor (uI1,ulz, (TI37 azl, . . . , c ~ ~allow ) , calculation of the three components of the vector force per unit area P with the components (P1, Pz , P3) for an area whose orientation is described by a unit vector ii normal to it with components

5.1. INTRODUCTION

50 I

(nl , n 2 , n3). The calculation is carried out using the three formulas 3

Pi

=

C uonj,

i = 1, 2, 3 .

j= 1

For any of the infinite number of orientations fl may have, the force per unit area P pulling by the material on one side on the material on the opposite side is thus given by the nine components of the stress tensor. For all processes observed in nature, only six of the nine components are needed, because u21= u12,( ~ 3 2= ( ~ 2 3and (TI3 = ~ 3 1 . That is, the stress tensor is symmetric. The six components of the stress tensor provide the means of calculating the forces per unit area at a point. At neighboring points, the stress is in general different, and the full description of the internal contact forces in matter requires a stressjeld. This means that each component is a function of x, y , and z. In this chapter, we will not discuss general methods of measuring the stress field, Indeed, measurement is quite difficult and is accomplished only in very special cases. Two special cases will concern us. The term pressure applies to both, but it is a good idea to recognize that there are two and to be aware which one is under consideration when we speak of pressure. The term “pressure” is sometimes used in older literature to be synonymous with “negative stress,” particularly in cases of uniaxial stress (all other 8 components zero). “Isotropic pressure” or “hydrostatic pressure” was then used for the modern term. The first special case applies to a fluid at rest, for which case ull = ( T = ~ u33 ~ = - p and all other components of the stress tensor are zero. In fact, a fluid, as distinguished from a solid, is usually defined as a substance having this property. The existence of such substances as oils and greases, pitch and structural polymers such as rubber, nylon, and plexiglas, for which “at rest” may demand waiting for very long times, illustrates that the concept of a fluid at rest is only a limiting case. The single number p in this case is called “pressure.” Since the contact forces are expressible in terms of a single number times the unit tensor 1=(;

8 ;)

and the stress field is expressible as a sculurjeld times the unit tensor, pressure is often referred to as a scalar. This can be very misleading in understanding the physical meaning of the concept.

502

5.

MEASUREMENT OF PRESSURE

The second special case applies to a fluid in motion where the fluid has properties of isotropy and forgetfulness of its previous motion sufficient to allow the assumption of linear viscosity. This is the assumption that each stress component is a linear function of all rate of strain components and that a single material constant, called the viscosity coefficient, is sufficient to provide the interdependence of stress and rate-of-strain components. In this special case, although the six components of the stress are in general all different, it is useful to employ the differences of the diagonal stress components from their average and call that average the negative of the pressure: (5.1.1)

This procedure has the advantage that the resulting equations of motion for the fluid-called Navier-Stokes equations-reduce to the hydrostatic equations as the motion ceases. The kinetic theory of gases provides a different concept of pressure from the concept we have presented of a force per unit area between contiguous parts of matter. In the kinetic theory, pressure is thought of as net transport through an element of area of momentum component normal to the area, per unit of area and per unit of time. Since the kinetic theory does not deal with contiguous matter but only with separate molecules moving through empty space and colliding with other molecules, the momentum transport is wholly by the material molecules themselves. Fundamental theoretical problems still exist when momentum transport by long range action-at-a-distance forces is contemplated; the reader is referred to treatises on nonequilibrium statistical mechanics, kinetic theory, and particularly plasma theory. The pressure of gases at rest and equations of state such as the laws of Boyle, Gay-Lussac, and Charles and the ideal gas law provide an elementary range of experience from which much of our thinking about pressure emerges. The kinetic theory elucidates phenomena in low density gases very well, and extensions of the concept of pressure as momentum transport by molecules can provide modified equations of state such as the van der Waals equation, but there is no meaningful relation between molecular transport and contact forces when liquid densities are reached. The lower limit of the range of pressure measurements is in the region where the two concepts overlap, at about 0.01 Pa. Confusion between the mechanical concept of pressure and the kinetic theory concept of pressure is common, and it is helpful to keep in mind that the range of phenomena treated jointly is limited. In particular, the

5.1.

INTRODUCTION

503

equality of the concepts must be limited to surfaces across which there is no net transfer of matter. Thermodynamic equations of state of condensed matter, as well as of gases, employ pressure as one of the thermodynamic variables. Experiments to date have shown that, for a fluid in motion, the quantity defined by Eq. (5.1.1) serves for this purpose so long as the Navier-Stokes equations describe the mechanical properties of fluids. Extension of the concept of pressure to higher values than can be attained with rigid materials using the piston and cylinder gage (Section 5.2.3), by thermodynamics and fluid dynamics, employs shock wave relations in explosively driven metals. Data on the pressure dependence of ruby fluorescence wavelength shift based on shock wave experiments have been used to measure steady pressure achieved by piston and cylinder methods employing diamonds. Steady pressure of 170 GPa is reported to have been measured, representing the current top of the range of pressure measurements. Finally, we call attention to the tendency of scientists to use the word “pressure” to describe physical quantities which are not within the scope of the meaning of the word in this article, namely contact force per unit area between contiguous parts of matter. We list these as a warning to the reader that he needs to look elsewhere for methods of measuring these “pressures.” Category of action-at-a-distance forces on matter: (i) (ii) (iii) (iv)

gravitational pressure. radiation pressure. magnetic pressure. electrostriction pressure.

Category of analogous equations of state: (i) partial pressure. (ii) vapor pressure. (iii) osmotic pressure. Two other uses of the word pressure in the parlance of fluid dynamicists cause conceptual confusion and are discussed in Chapter 5.3. They are listed here along with other “pressures” which are not pressure with the aim of clarifying what the meaning of the word is in this chapter:

(i) impact pressure, also called total pressure, Bernoulli pressure, or stagnation pressure; (ii) dynamic pressure, which is merely two words designating Bpvz at the point within the fluid:

5.

5 04

MEASUREMENT OF PRESSURE

5.1.2. Contact with Gage Element Necessary

A contactless pressure measuring device cannot exist. However, if the thermodynamic equation of state is known and pressure is calculated from this equation and other measured variables, one often says that the pressure has been measured; for example, by measuring molecule number density n and temperature T of a gas, one can calculate the pressure from the equation p = nkT, where k is Boltzmann’s constant. ( k = 1.38 X 10-235

a

K-1).

The insertion of a gage is very likely to change the pressure at a point in a flow from the value that would exist there without the gage element. This classic problem will be dealt with in detail in Chapter 5.3, and one important feature, which is employed in velocity measurement, is dealt with in Section 1.2.2, Pitot Probe. At this point, we emphasize that measurement of pressure with a probe means finding the pressure that would be there in the absence of the probe. Most of this part deals with the gages inserted into fluids for pressure measurement. A wide range of gages is manufactured and sold by commercial companies for industrial and research use. 5.1.3. Calibration and Standards

The most accurate method of pressure measurement employs a piston fitting tightly in a cylinder but not touching. (See Section 5.2.3.) Combining this with the principle that pressure in a homogeneous fluid at rest is uniform throughout if there are no action-at-a-distance forces permits calibration and comparison of pressure gages. Accuracies to within uncertainties of 0.01 percent can be achieved with static fluid calibration. Fidelity of dynamic response of gages having undergone steady pressure calibration is inferred from an understanding of their behavior and from shock tube tests; reliability in the range of 1 percent is rarely achieved, however, as discussed in Chapter 5.8. Extraneous effects on pressure gage readings are numerous and difficult to avoid. Means of intercomparing measurements made by gages operating on different principles are much to be desired. Provision for frequent calibrations of gages under conditions where their behavior is well-understood can be helpful, both in routine monitoring and in research investigations. It is our intention to list and illustrate the types of gage designs that have been recommended or manufactured, especially to clarify the principles of action employed, and to provide recommendations for specific situations.

5.2.

MEASURING CONSTANT AND SLOWLY VARYING PRESSURES

505

5.2. Gages for Measuring Constant and Slowly Varying Pressures The most familiar and widely used pressure measuring devices are U-tube manometers and dial and digital gages. Manometers are easily constructed of equipment found in every laboratory and, for rough measurements, fairly insensitive to errors. Dial gages are cheap, sturdy, and easily connected. A manometer can provide absolute readings, with suitable precautions, but a dial or digital gage must always refer to another gage for calibration. One ordinarily takes it for granted that the reading of a manometer or a dial or digital gage can be carried out at one’s leisure. Both require a few seconds typically to respond to a changing pressure, and the assumption is made that they have had an indefinitely long time to come to mechanical equilibrium with the fluid whose pressure is measured. 5.2.1. Liquid Manometers The liquid manometer, typified by two columns of liquid partly filling a piece of glass tubing bent into a “U” shape with hoses connecting to two reservoirs, employs the hydrostatic law p - pgy = const

(5.2.1)

applicable to a homogeneous fluid at rest with no forces except pressure and weight. The liquid in the manometer typically has a density lo3 times the density of the gas which is connecting the manometer to the reservoirs; correction for the pressure difference associated with the weight of the gas in the connecting tubes can be made, but usually the correction is negligible in comparison with other corrections which can only be estimated. To the extent that Eq. (5.2.1) is valid, the pressure difference between the two reservoirs is measured by p z - p 1 = pg(yz - yl) = p g h , where y z and y1 are the vertical coordinates of the respective surfaces between liquid and gas on the two sides of the U-tube, p is the mass per unit volume of liquid, and g is weight per unit mass, nominally 9.8 N kg-’. One disadvantage of the U-tube manometer is the ease with which the liquid is blown out of the manometer when the pressure difference exceeds the range. A trap to catch the liquid in case of this accident is advisable. Chemical contamination of the reservoirs where the pressure is being measured on either side is avoided by using a liquid with low “vapor pressure” such as mercury or silicone oil. Mercury has the additional advantage, due to its high density, of measuring high pressure; a manometer to

5 06

5 . MEASUREMENT OF PRESSURE

FIG.I . Modification of U-tube manometer to provide increased sensitivity in reading difference in heights of two surfaces.

measure a pressure difference larger than that corresponding to about 1 meter of height difference is seldom used, however. In the direction of small pressures, a liquid of density lower than that of silicone oil (approximately 0.8 the density of water) is impractical. One arm of the manometer may be bent to an almost horizontal position as shown in Fig. 1 to “magnify” the position of the liquid-air interface to aid in the measurement. Elaborate techniques have been developed to aid in precision measurement of the height of the surfaces’; only a few will be mentioned. One commonly used technique is to mount a pointer internally which does not wet the liquid (ivory-mercury) and which is attached to an accurately readable micrometer scale. The pointer is adjusted until the observer does not see a depression in the mirrorlike liquid surface. Another method for mercury uses a steel float carrying a glass mast on which is engraved an accurate scale; the height of the mercury column is obtained in terms of the position of the glass scale read with a microscope (Betz manometer). An ultrasonic pinger and receiver at the base of a mercury column is used in a commercial instrument to detect the time of travel of an ultrasonic pulse from the base to the surface of the mercury. A fringe counting laser interferometer allows measurements of the light reflecting surface to a sensitivity of less than one micr~rneter.~.~ Other methods of pressure measurement are probably more practical than these, however, since so much trouble is required to operate the measuring equipment. Pressure differences smaller than measured by approximately 10 mm of oil (
C. R . Tilford, Rev. Sci. Instrum. 44, 180 (1973). S . J. Bennett, P. B. Clapham, J. E. Daborn, and D. I . Simpson, J . Phys. E 8 , s (1974).

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

507

Reference may be made to a vacuum technology handbook for descriptions of a thermocouple gage, a Pirani gage, a Phillips corona gage and an ionization gage. The first two of these “pressure” measuring methods employ the heat transfer properties of the particular gas and the others are measurements of density rather than of pressure. The most reliable method of measuring pressure in the range below 100 Pa is by means of diaphragm type gages, calibrated by a piston gage, as described below. 5.2.1.l. Sources of Inaccuracy in Liquid Manometers. Liquid manometers fail most frequently as absolute pressure measuring instruments on two accounts: (1) lack of homogeneity of the liquid; and (2) presence of other forces than weight. No recipe can be given for avoidance of error in either category. Many metals dissolve in mercury and change its density; it is similarly difficult to be certain that water or oils obtained for use in a manometer are homogeneous, pure and of known density. Temperature variations in the parts of the manometer will cause expansion of the scale for reading h , and one must also be aware that the density of the liquid depends on its temperature. For very precise measurement, departure of the local value of g from 9.80665 N kg-I must be accounted for. In avoiding the effects of extraneous forces, one must, of course, avoid magnetic forces and electrostatic forces and not use the manometer in an accelerating frame of reference. The most bothersome type of extraneous force is that of surface tension which may prevent the pressures on the two sides of the liquid gas interface being equal as is assumed when the heights of the two manometer columns are measured and formula (5.2.1) used to determine the pressure difference. If the tube is sufficiently small, so that the shape of the interface is approximately a sphere, an additional height of the manometer is present of about

2 Y cos a (5.2.2) (PL - Pc)& where Y is surface tension, a is the contact angle between the liquid-gas interface and the wall, pLand p c are the liquid and gas mass densities, and r is the radius of the tube. a = 0 for a water-air-glass interface, and a = 140” for a mercury-air-glass interface. If the glass is extremely clean and outgassed, the mercury-glass angle may become less than 900.4 If the tube is sufficiently large in diameter, the surface will not be a hemisphere and may become sufficiently plane at the center of the tube so that a correction does not have to be applied. The measure of the distance from the wall at which the curvature of the surface is noticeable is ( 2 Y / ~ g ) ” ~ , e.g., 3.9 mm for water and 2.7 mm for mercury. The tube radius must be y’

=

‘ N . K . Adam, “The Physics and Chemistry of Surfaces,” p. 185.

Dover, N e w York.

5.

SOX

MEASUREMENT OF PRESSURE

TABLE1. Pressure Units ~~

Name of unit

Valueb

1 kgf/cmz

98.07 0.1 100 101.3

kPa Pa kPa kPa 10 Pa (transducers) or 20 @Pa (hearing)

1 dyne/cm2 1 bar 1 atm 0 acoustic decibel (reference level)”

-~

[

~~

Name of unit 1 “in Hg” 1 “in

HzO”

1 “mm Hg” (Torr) 1 “mm HzO” 1 Ib/ft2 1 Ib/inz

Value 3.38 kPa 249 Pa 133.3 Pa 9.81 Pa 47.9 Pa 6.895 kPa

~~

" Sound pressure level, in decibels, is 20 times the logarithm to the base

10 of the ratio of the sound pressure to the reference pressure. The reference pressure should be explicitly stated. Unless otherwise explicitly stated, it is understood that the sound pressure is the rms pressure change. The SI unit of pressure, pascal (symbol Pa), is 1 N . m-*.

of the order of 10 times this value to avoid measurable curvature in the most exacting devices. The effect of the capillary elevations in two arms-both vertical-of a manometer will cancel in determining the difference in heights if the two arms are of equal diameter tubing. Most stock laboratory glass tubing does not have uniform diameter, but uniform bore tubing can be obtained. Small amounts of foreign substances, e.g. grease, on the glass can destroy the compensation hoped for. The manometer is so widely used that a variety of units for pressure has grown up. Some are quite confusing. Table I includes some of the units which have been found in the literature and gives their value in terms of the SI unit pascal ( 1 Pa = 1 newton per square meter). 5.2.1.2. Response Time of Liquid Manometer. A manometer subjected to a time-varying pressure behaves as a damped oscillator. The frequency of oscillation is (5.2.3)

where L is the length of the liquid column, u is its radius and v is the kinematic viscosity of the liquid. Typically, manometers are heavily damped, but not overdamped; in this case, the damping time is of the order of magnitude u2/4v. This tells us, for example, that the time elapsing after a pressure change in a water (v = m2 s-l) manometer of 2-mm radius before we can read it is greater than 1 s. The detailed behavior is more c~mplicated.~

’ W. L. Holley and J . R. Banister, J . Ind. Arrodyn.

1, 139-165 (1975).

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

509

Equation (5.2.3) assumes that there is no lag in the equalization of pressure in the connecting tubing. Such a lag can be appreciable in a 1 mm bore tube many meters in length.6

5.2.1.3. McLeod Gage. Precise measurement of pressure in the range below 100 Pa is difficult to carry out with a U-tube manometer, even with the refinements described in preceding sections. Down to about 0.1 Pa, manometric methods can be extended by isothermally reducing a volume of a gas such as H z , He, or Nz at a low pressure and measuring the pressure by a manometer a t the higher pressure. Since the virial coefficients* of these gases are known with high accuracy, the pressure ratio can be accurately calculated from the volume ratio. The McLeod gage illustrates one method of making a known reduction in volume and of measuring the higher pressure. It has been used for many years but is cumbersome to use in routine pressure measurement. It does not measure pressure of a condensible vapor or of a mixture of gases and vapor. Figure 2 illustrates the gage. A glass bulb of known volume "Irl communicates with the vessel where the pressure is to be measured; at its top, a glass capillary with precision bore and a flat closed end is sealed. Mercury is caused to rise from the lower reservoir, trapping gas at the pressure to be determined p1 in the bulb and compresses it to a small "Irz. As the mercury rises in the bulb, it also rises in the nearby tubing, which includes a side arm capillary of the same size as the capillary sealed to the top of the bulb. If the trapped gas was in the correct range of pressure initially, it is contained entirely in the capillary when the mercury in the side arm reaches the level of the closed end. Then the pressure p z of the compressed gas is measured by the difference in vertical heights h of the two mercury columns in the capillaries. "Irz is given by this same height difference multiplied by the capillary cross section A . The two identical precision bore capillaries are used in the hope that the capillary depression will be the same and cancel in the calculation of the pressure p z . The volume "Ir, is determined before the lower parts of glass tubing are attached to the bulb during manufacture; Sr, is determined by weighing the bulb and capillary when completely filled with mercury, weighing the empty bulb, and knowing the density of mercury. The cross section of the capillary is likewise found during manufacture by moving a mercury drop of known weight (and volume) along inside and measuring its length.' A . L. Ducoffe, J . Appl. Phy.\. 24, 1343 (1953). Strong, "Procedures in Experimental Physics," p. 138. Prentice-Hall, Englewood Cliffs, New Jersey, 1938.

'J.

* See Section 6. I . 1 and Eq. (6.1.1)for further information on the virial equation of state.

5 10

5.

MEASUREMENT OF PRESSURE

be measured

1-l;

capillary

1 roughing Pump

FIG.2. McLeod gage. Pressure of gas trapped in volume V , is “magnified” when it is compressed by mercury rising in the bulb. Both pressure and volume are indicated by the dimension h in the compressed state.

Assuming p l y l = p z 7 f 2 ,i.e., neglecting higher-order terms in the virial equation of state,

where p is the density of mercury. The McLeod gage’s scale is nonlinear; the range is p g A 1 2 / V l , where 1 is the length of the capillary’s precision bore. The bore loses its precision near the joint with the bulb, s o l may not extend that far. 5.2.2. Deformation Gages with Mechanical Readout

A mechanical structure with a cavity connected to a vessel whose pressure is to be measured will deform. Three important structures are in use. The most convenient and most highly developed of these are the Bourdon gage, the capsule (also called sylphon or bellows) gage and the diaphragm gage. The diaphragm has unique properties and is discussed

5.2.

MEASURING CONSTANT AND SLOWLY V A R Y I N G PRESSURES

5I I

separately in Chapter 5.9. Here we discuss the Bourdon and capsule type gages briefly because of their wide use. For details and design information, see Andreeva.8 5.2.2.1. Bourdon Gage. A metal tube of elliptical cross section which is bent to form a nearly complete circle has one end fixed to a base. The elastic displacement of the other end as the circle tries to straighten out is nearly proportional to the pressure inside the tube. By means of a rack and pinion connection, the displaced end of the tube moves a needle whose position is read on a circular dial. This is the common gage one sees so frequentiy on machinery and control panels. Metallurgical skill in fabricating the elliptical cross section tube and watchmaker-type skill in the linkage magnifying the displacement of the tube end has produced gages capable of maintaining remarkable accuracy over many years. Ambient temperature changes, corrosive fluids and damage caused by overload affect the reading; therefore, most such gages are demountable so they may be taken to a calibrating station or replaced easily by a calibrated unit. Variations in the size, shape and material of the tube have been made in a large range of applications. Particularly noteworthy is use of fused quartz tubing, fabricated into a spiral, fixed at one end and suspending a mirror at the other. Rotation when the spiral unwinds is detected and converted into a pressure reading. The rotation may be linked to a mechanism with a digital readout as an operator maintains a null position of a light beam reflected from the mirror. Another readout employs an electromagnetic counter-torque automatically maintained by a photocell detector of the light beam deflection and a digital readout of the current required to balance the spiral torque change. The advantage of fused quartz is its long term mechanical stability, relatively small sensitivity to temperature, and ability to withstand somewhat higher overloads without loss of calibration. 5.2.2.2. Capsule Gage. The capsule, or bellows-type gage, is similar in construction to a diaphragm, but the elastic element is convoluted to magnify the motion of the diaphragm. By combining metallurgical art with mechanical art to translate the motion into the rotation of a needle on a dial without backlash, manufacturers have been able to supply gages with a wide variety of operating ranges, compensated for temperature L. E. Andreeva, "Uprugie elementy priborov," Mashgiz, Gosudarstvennoe NauchnoTekhnicheskoe Izdatel'stvo Mashinostroitel'noi Literatury, Moskva, 1962; English translation, A . Baruch and D. Alster, in "Elastic Elements of Instruments" (H. Schneider, ed.), pp. 194-360. Israel Program for Scientific Translations, Jerusalem, 1966.

512

5.

MEASUREMENT OF PRESSURE

POINTER

CAPSULE CALIBRATION ADJUSTMENT

PINION

GEARED SECTOR BACKLASH ELIMINATOR REVOLUTION INDICATOR FLEXURE

FIG.3. Capsule-type dial gage. The dial on which the pointer is read is not shown. [Courtesy Wallace and Tiernan Division of Pennwalt Corp., Newark, New Jersey.]

change, and retaining precision of the order of 0.1 percent of the range. To achieve this order of reliability, each gage is individually calibrated at several places on its dial since nonlinearities arise in several parts of the mechanism. Such gages are easily damaged by surges and excessive pressures beyond about 125 percent of the intended range, and by mechanical shock such as is sustained by being dropped on the floor. Frequent intercomparison of such gages, particularly with gages which have been tenderly protected from hazards, is necessary to assure reliability. Figure 3 illustrates the construction of a capsule gage.

5.2.3.Piston and Cylinder Gage This gage is the basic standard for pressure measurements above about 100 kPa. Even within the range of usefulness of the U-tube manometer, the piston and cylinder instrument is used for absolute pressure measure-

5.2.

MEASURING CONSTANT A N D SLOWLY VARYING PRESSURES

5 I3

I

2

--

\ I

(a) (b) FIG.4. Piston and cylinder gage. (a) Arrangement for use. (b) Detail of clearance area. 1 : piston; 2: cylinder: 3: bleed hole; 4:jacket pressure inlet. The jacket pressure inlet allows the clearance between piston and cylinder to be maintained at an optimum value and the “effective area” to be accurately evaluated.

ment and calibration of other types of instruments. Also called a “deadweight gage” and a “gage tester,” it is seldom seen outside a standards laboratory because its operation requires skill, care, patience, and conditions of cleanliness. Figure 4 shows the principal parts of the gage. Known masses are stacked on the piston of known mass and cross sectional area. The piston fits with extremely small clearance in a cylinder resting on a firm base. A liquid (usually oil) or a gas (nitrogen or air) fills the cavity in the cylinder below the piston and communicates via tubing and fittings with a dial gage to be calibrated and a plunger pump allowing fine adjustment of the volume. By means of the pump, the pressure is increased until the piston rises and is supported. Fluid escapes through the small clearance region between the piston and cylinder, and the piston gradually settles. While the piston is settling, the pressure in the fluid at the level of the closely fitting portion of the cylinder is known to be the weight of the piston and supported masses divided by the effective area of the piston. The eflectiw area has been determined in principle by measurement with a metrologic microscope of the piston and cylinder diameters at the region of small clearance and is the average of the two area^.^ The temperature at which the areas were measured must be known; in use, the area depends on the temperature of the test together with the coefficient J . L. Cross, Reduction of data for piston gage pressure measurements. N u t / . Bur. Stund. ( U . S . ) ,Monogr. 65 (1970).

514

5.

MEASUREMENT OF PRESSURE

of thermal expansion of the piston and cylinder material. Changes in the areas are also caused by the pressure to which they are subjected. If the pressure on the cylinder is not increased by introducing pressure in the jacket (see Fig. 4) when the pressure being measured is large, the clearance becomes large, and the piston falls too rapidly. The changes in areas of piston and cylinder must be calculated from a knowledge of the elastic modulus of the material and the pressures beneath the piston and in the jacket. For the highest pressures, the piston may be made of sintered tungsten carbide. Piston and cylinder gages are obtainable from commercial manufacturers. The effecfive areas of gages are supplied by the manufacturer, having been obtained by comparing gages with gages built and maintained by national standards bureaus. Because the clearance between cylinder and piston is so small, especially when gases are used, cleanliness is mandatory. The gage must be frequently disassembled, cleaned and reassembled to dispose of fine particles which lodge in the clearance region. The presence of foreign particles is detected by spinning the piston and masses very slowly while they are settling. If the clearance region is unblocked, the supported masses decrease their slow rotation very evenly and gradually. With care, pressures can be measured with errors as small as 0.01 percent in ranges from 100 Pa to 100 MPa (1000 bars). In addition to corrections noted for temperature and pressure to determine the effective areu, corrections must be made for oil buoyancy of the immersed portion of the piston, air buoyancy of masses (0.15 percent), local value of g (up to 0.3 percent), fluid head when oil is used and pressure at a level other than the clearance region between piston and cylinder is sought, and surface tension of oil. The last two corrections mentioned are needed only when pressures in the kilopascal range are measured; in this range, mercury and oil manometers can be used for absolute measurement with about the same care as is required for piston and cylinder gages. 5.2.4. Calibration Procedures for Constant Pressure

Use of absolute instruments-manometers and piston gages-is possible in routine pressure measurement, but usually sufficient care is not taken to avoid errors that may exceed 1 percent. Dirty mercury, dissolved gases in oil, and scratches causing sticking of pistons easily occur to invalidate the readings. In practice, it is convenient to obtain dial gages, or deformation gages with digital readout, from a manufacturer who maintains a “standards laboratory” for calibrating his products. Intercomparison of older gages with newly acquired gages is relatively simple.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

515

Damage to calibrated gages in transport can always be suspected. A new differential pressure gage that does not read “zero” on being unpacked can be assumed to have suffered damage. Most large industries maintain “standards laboratories,” keeping secondary standards traceable to national standards laboratories. Inquiry in a large city will usually locate one or more where a pressure gage can be calibrated . The difficulties of establishing exact pressures in the range below 10 Pa have led some to propose that a prestressed diaphragm gage with capacitance bridge detection of the diaphragm deflection may be the best method of measuring such pressures.lO The readability of the tensioned diaphragm gage is the order of 1 part in lo6, and gages have been built which are linear (at higher pressures) to 1 part in lo4. So one is inclined to think that such a gage calibrated by a piston gage at 100 Pa will be able to measure 0.01 Pa to an accuracy of 0.1 percent. Attempts to calibrate diaphragm gages with McLeod gages have been troubled by mercury streaming in a cold trap, inaccuracies in mercury column readings due to variable capillary depression and unknown “thermal transpiration” (associated with maintaining the diaphragm gage at a different temperature) so that demonstrated accuracy in the 0.01-Pa range is not better than 0.5 percent .I1

5.3. Pressure Measurement in Moving Fluid A principle of mechanics is that forces are independent of the frame of reference for all inertial frames. Thus pressure is the same at a place and time in a fluid whether we imagine that we are at rest or in (constant velocity) motion with respect to the fluid. However, if a stationary probe is inserted in a steadily moving fluid the pressure over the surface of the probe is not constant and in general is different from the pressure in the fluid before the probe was inserted. The highest pressure on the probe is at the stagnation point which is usually the point facing the flow; this highest pressure is often called stagnation pressure, or impact pressure (or “total” pressure) and measurement of its value is one of the measurements required when the Pirot tube is used for fluid velocity measurement (see Section 1.2.2). At other points on the surface of the probe the pressure has values ranging from the stagnation pressure to below that of the fluid without the probe. The purpose being to measure the pressure in the undisturbed fluid, lo

I’

N . G . Utterback and T. Griffith, R e v . Sci. Instrum. 37, 866-870 (1966). J. P. Bromberg,J. V a c . Sci. Techno/. 6, 801-808 (1969).

5.

516

MEASUREMENT OF PRESSURE

probes have been developed to sense the pressure at the intermediate places on their surfaces where one has reason to believe that the pressure is the same as in the undisturbed fluid. Such probes are called stutic probes because they respond in the way any gage would if it were at rest relative to the fluid, i.e., moving with the fluid. The terminology is as confusing as the explanation. Confusion can be lessened if one remembers that “static pressure” is merely pressure. The term is used to emphasize that one is not referring to some of the many other kinds of ”pressure,” such as “stagnation pressure.”

5.3.1.Wall Taps A small hole in the wall of a pipe or duct, or in the surface of an im-

mersed object, connected by tubing to a pressure gage is often used to measure the pressure of a Newtonian fluid at the surface containing the hole. The wall tap method cannot be used for non-Newtonian fluids such as polymers, where stresses are not linearly related to rates of strain. Since even in supersonic flow the gas velocity approaches zero at all solid surfaces (except for very rarefied flows), the measurement of pressure at any surjucr is relatively easy. However, even when the flow is steady, the pressure in the connecting tubing is a bit larger than the pressure at the surface without a hole. When the flow is changing, or turbulent, the pressure in the connecting tubing may be considerably higher than the time average of the pressure at the surface. If there is a protuberance comparable in dimension with the hole diameter, the pressure in the connecting tubing may be either higher or lower than the pressure that would exist at the smooth surface; higher if the protuberance is on the downstream edge of the hole, and lower in the opposite case. The hole-error, assuming no

-

-

3E, c,

FIG.5 . Streamlines in the neighborhood of a hole in a wall or surface of an immersed body.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

5 I7

7 1-

!

/burr

3%d

bob o 6 -

\

Q

-4

5-

L

z L

4 -

I

0

100

zoo

300

Hole size Reynolds number, d(p t~~)'''/p FIG.6. Dimensionless hole-error versus Reynolds number for hole diameter d = 1.6 mm and varying sizes of burr. Wall stress IT,,= p ( a ~ / d y ) ~ - , , .p, are density and viscosity of fluid. u is mean speed of fluid at distance y from surface. Solid curves are dimensionless hole-errors for the indicated size of burr. Dashed curve is hole-error for a sharp edged "well-finished" hole. [From Shaw.'*]

protuberance, is smaller the smaller the hole, and it is assumed that the hole-error would extrapolate to zero for hole size zero; however, the difficulty in avoiding burrs and other mechanical irregularities at small hole sizes requires that the hole-error be calculated for a larger hole in practice. The computed inviscid steady flow pattern in the neighborhood of a wall tap is sketched in Fig. 5 . The streamlines curve at the hole; such curvature is associated with a pressure gradient. In a real flow, there is also an eddy or system of eddies set up in the fluid within the hole, caused by flow separation at the hole leading edge, and an increase in pressure at the stagnation point near the downstream edge. These three factors combine to give the net hole-error.*2 The hole-error in straight pipe flow of air has been measured by Shawl' and Franklin and W a l l a ~ e . ' ~Figure 6 displays results for various flow speeds (Reynolds numbers) and for the effects of burrs of various sizes. Shaw, on the basis of dimensional argu-

l3

R . Shaw, J . Fluid Mech. 7, 550-564 (1960). R. E . Franklin and J . M . Wallace, J . Fluid Mech. 42, 33-48 (1970).

518

5.

MEASUREMENT OF PRESSURE

ments, deduces that the hole-error can be determined most easily in terms of the "wall stress" wo = , U ( L ~ V / L ~ ~ ) , , = ~ as the mean flow speed in the pipe, or the free stream velocity outside the boundary layer, varies. In a long uniform steady pipe flow, wois readily found from the axial pressure drop dp/dx since d u o= (7#/4)(dp/dx) where D is pipe diameter. In a boundary layer, wo is found from an estimate or measurements of (av/ay),=o. It is evident from Fig. 6 that the effect of relatively small burrs is significant and the hole-error cannot be reduced safely merely by using small holes. As an example of the magnitude of hole-error, it may be noted that a hole size Reynolds number of 300 in air at 100 kPa (1 bar) with hole diameter d = 1.6 mm corresponds to a mean pipe flow speed of 66 m s-' in a pipe of 51 mm diameter, and for this case u0 = 9.6 Pa = 0.01 percent of the pressure. The pipe flow Reynolds number pv,,,,D/,~ for this case is 225,000. The dimensionless hole-error reaches its maximum value for hole size Reynolds number 800 and hole depth 1S d and does not increase beyond; it is then 3.8. Again considering a 51-mm diameter pipe, air at 100 kPa and hole diameter d = 1.6 mm, the hole-error of 3 . 8 is~ 260 ~ Pa, or 0.3 percent of the pressure; this case corresponds to a mean pipe flow speed of 200 m s-' and a pipe flow Reynolds number of 700,000.

-

5.3.2.Static Probe in Steady Flow In view of the situation described in the preceding section, one sees that if a very small object with a hole in its wall connecting to a pressure gage is inserted into a steady flow, it can measure the pressure there as disturbed by the presence of the probe. Figure 7 indicates how flow with straight streamlines is disturbed by introduction of a small bent tube.

s

_ I _

3

3

3

-

4

FIG.7. Streamlines about a bent tube immersed in a uniform flow. 1: side view. 2: nose alone. 3: bottom view. 4: stem alone.

5.3. PRESSURE

MEASUREMENT IN MOVING FLUID

519

stem4 effect

FIG.8. Pressure variation along wall of a bent tube due to streamline curvature. Separate effects due to the nose and due to the stem can combine to yield the undisturbed stream pressure at position of holes.

Imagine that this tube is small in comparison with the extent of the flow being probed; if the hole in the side wall of the tube is judiciously located, it may read a pressure which would be the same as the pressure would have been without the probe. The pressure on the wall of a closed round-headed tube aligned with a steady flow has been measured by a series of holes and increases with distance from the shoulder of the tube, but never reaches the pressure in the undisturbed flow. However, the stem of a bent tube causes the pressure in front of the stem to be larger; the effect is less at greater distances in front. For a given tube, the two effects cancel at some intermediate distance as indicated in Fig. 8, and this is the place chosen for the hole(s) in the tube wall. An 8-mm-diameter tube with nose 3d long and with holes 6d back of the shoulder, where d is tube diameter, has been accepted by workers in wind tunnel laboratories for use in subsonic and supersonic flows, up to Mach 2. Mach number corrections must be made above M = 0.7 which are ddpendent on the geometry of the tube.l4-Is A tube with a single hole in the wall making an angle a with the direction of the steady flow can have a higher or a lower pressure than the I‘ D. W. Bryer and R. C. Pankhurst, “Pressure Probe Methods for Determining Wind Speed and Flow Direction.” HM Stationery Office, London, 1971. Is A. N. Petunin, “Methody i tekhnika izmerenii parametrov gazovogo potoka.” Makhostroeniye, Moscow, 1972 (in Russian). S. H. Chue, Pressure probes for fluid measurement. f r o g . Aerosp. Sci. 16, 147-223 (1975).

5 20

5.

MEASUREMENT OF PRESSURE

Undisturbed stream velocity

-20"

FIG.

-10"

,

0"

loo

20°

9. Pressure inside "static probe" inclined at various angles to steady uniform flow.

undisturbed pressure, depending on whether the single hole is in front or in back. Usually multiple holes symmetrically placed around the tube are used. Then the measured pressure pm is always lower as indicated in Fig. 9. On the simple hypothesis that the deviation is proportional to pv:, where u,, = urn sin (I! and p is fluid density, pm - pm = B p ( v , sin The coefficient B is found experimentally to be about - 0.25. An airplane, or other vehicle, in motion through the air requires a knowledge of the pressure outside the vehicle. The pressure outside is needed when a barometer is used as an altimeter, and also when a Pitot tube (see Section 1.2.2) is used for measurement of the vehicle's air speed. General regions suitable for the location of pressure taps, called "static vents," on the sides of an airplane are found from model tests in wind tunnels. In most cases the taps are arranged in pairs, one on each side of the fuselage, either between the nose of the fuselage and the leading edge of the wing or between the trailing edge of the wing and the leading edge of the stabilizer. The pressures measured at these holes may be appreciably affected by the attitude of the airplane and the amount of thrust in the engines. Calibration of the readings of "static vents" is carried out in flight tests near a site where the pressure on the ground is known and radar measurements of height permit corrections for difference in pressure with height to be made." The need for empirical search for the places on the surface of an airplane where the undisturbed pressure may be found characterizes the problem of designing any probe to measure "static pressure," i.e., the pressure in a fluid were it not moving over the probe.

'' F. J . Bailey, Jr.. J . A . Zalovcik, W. H. Phillips, and W. B. Huston, Piloted aircraft testing. "High Speed Aerodynamics and Jet Propulsion," Vol. VIII, Article N,1, page 839. Princeton Univ. Press, Princeton, New Jersey, 1961.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

52 1

5.3.3. Static Probe in Unsteady Flow In an unsteady flow, such as a sinusoidally oscillating sound field, or a fluctuating random field characteristic of turbulent flow, the pressure is a function of time, and probes display a fluctuating output. Whether the indicated pressure is related to the true pressure that would have been there in the undisturbed flow is a question open to debate. An acoustic cavity, driven by oscillating one wall and with calibrated (at steady pressures) diaphragm type gages mounted in other walls is used for "dynamic calibration" of a probe either also mounted in the walls or immersed in the interior of the cavity. Nonlinear responses and generation of harmonics are familiar problems discussed in acoustics handbooks, textbooks and journals,'* and the problems associated with fluctuating flows which have a mean flow component are to some extent an extension of these. However, an additional feature of the problem with a mean flow is the accompanying fluctuation in both the direction and magnitude of the velocity of the fluid impinging on the probe. Elliott19 approached this problem by designing a probe with a shape found to give pressure readings relatively independent of pitch and yaw when mounted in a steady wind tunnel flow. The probe was then immersed in an acoustic "dynamic calibration" cavity and its fidelity tested at various frequencies. In the range from 0.01 Hz to 10 Hz the response was found to be within 20 percent in amplitude and 20" in phase of the pressure at the wall of the cavity. The probe consisted of a 2.5-mm thick circular disk, 45 mm in diameter, mounted on a sting of tubing 0.56 m long connecting to a differential tensioned diaphragm gage with capacitor sensor (see Fig. 22 and Section 5.9.3.) The disk was an ellipsoid of revolution but with the flat sides smoothly indented on both sides to give a thickness at the center of 1.8 mm; 0.5-mm ports in the center of each indentation connected, via extended 0.5-mm channels, to the supporting tube. When the disk is tilted at more than 10" to the main flow in the wind tunnel, the flow over the disk separates and the pressure reading changes drastically. This probe was designed to measure fluctuations in pressure in the turbulent boundary layer in the atmosphere, with minimum interference due to velocity fluctuations. Elliott concludes that it is usable within statistical accuracy of 10 percent in amplitude for wind range of 3 to 10 m s-' in stable atmospheric conditions. Under unstable conditions the angle between the velocity vector and the plane of the probe can attain L. L. Beranek, "Acoustics." McGraw-Hill, New York, 1954. '13

J . A. Elliott, Instrumentation for measuring static pressure fluctuations within the atmo-

spheric boundary layer. Bnundary-Layer Metearol. 2, 476-495 (1972); see also J . Fluid Mech. 53, 351-383 (1972).

522

5.

MEASUREMENT OF PRESSURE

20-30" and the pressure readings would be seriously in error. Obviously this probe could not detect rapid fluctuations, and the extent to which its readings are affected by small scale rapid fluctuations is unknown. The diameter of the connecting tubing from the ports to the diaphragm gage is purposely small to delay, and time-average, rapid pressure fluctuations which, in turbulent flow, arise from small scale eddies carried across the probe. Another attack on the measurement of pressure within a fluctuating fluid flow has been made by Siddon.20 Following earlier workers, Siddon suggests treating steady (time averaged) and unsteady components of the flow variables and of the measured variables: P ( t ) = P + p ( t ) , V d t ) = P, + ul(t), V 2 ( f )= V2 + u2(t) and V3(t)= U3 + u3(t), where V , ( t ) is the component of velocity along the probe axis and V2(t)and V3(t)are components normal to the axis. The measured (indicated) pressure P,(t) = F,,,+ p,(t) is a function of the (true) pressure Pt(t)and of V , ( t ) ,V2(t),and V3(t). Assuming isotropy, and assuming that averages do not change with time, Siddon proposes that separate equations can be written: Steady error: Unsteady error:

p,,, - p , = A p ( P : + 2)+ B p ( p : + p m ( t ) - p t ( t ) = Ap(2v1v1 + u: - $1

z),(5.3.1) -

+ Bp(2V2p2 + 2U3V3 + V i - v:).

(5.3.2) For convenience the notation 8: = + Vi and = u: + ui is used for net cross-stream components. The coefficients A and B are to be determined by a calibration procedure. The form of these equations is analogous to the corrections found to apply in steady flow and illustrated in Figs. 8 and 9. Coefficient A is largely a form factor relating to the imperfect cancellation of the streamline curvature effects of nose and stem; coefficient B represents the cross-stream effects. Eqs. (5.3.1) and (5.3.2) might be expected to apply to a turbulent flow with eddies large in comparison to the size of the probe, for it is assumed that the pressure gage recording p&) reaches a quasi equilibrium with the flow in the time such an eddy passes over the probe. Stated in the conventional language of turbulence researchers, this means that the unsteady flow must have a spatial scale larger than the probe if Eqs. (5.3.1) and (5.3.2) apply. The flow is then approximately uniform over the probe at all times. Siddon built a probe 3 mm in diameter whose construction is shown in Fig. 10 to test these hypotheses. The probe not only had a diaphragm u, T. E. Siddon, On the response of pressure measuring instrumentation in unsteady flow. Univ. Toronto Inst. Aerosp. Stud., Rep. UTIAS 136, Jan. 1969.

5.3.

PRESSURE MEASUREMENT IN MOVING FLUID

523

CROSS SECTION

6 2

4

FIG.10. "Static probe" to measure p ( t ) and cross-stream velocity components v,(r) and u,(t). 1: epoxy mounting of nosepiece on I beam. 2: balsa nosepiece with epoxy surface. 3: four element piezoelectric I beam. 4: pressure sensing slit. 5: leads with u1 and q signals. 6: cotton plug. 7: pressure gage. [From Sidd~n.~']

pressure gage (see Fig. 23 and Chapter 5.9) but also had a piezoelectric crystal beam-supported nosepiece which recorded the fluctuating vz(t) and u3(t) components of the stream flow velocity. The pressure gage was calibrated by placing it in a cavity, one end of which was a piston, driven sinusoidally to produce a 31 Pa rms amplitude. The ug and us sensors were calibrated by placing the assembled probe in a rotating inclined channel flow; the frequency of rotation (5- 150 Hz), gas speed (35 to 65 m s-') and angle of inclination (2"-7.5") could be varied. The pressure gage calibration and the u2 and u3 sensors calibrations were each accurate to 5 percent. The exterior contour of the probe, Fig. 10, was chosen to make A < 0.001 in a steady wind tunnel flow. With A negligible, and the probe inserted into the rotating inclined channel flow with pt(t) = 0, Eq. (5.3.2) becomes p,(t)

=

2Bpvl, sin a sin Bo sin 21rfi,

(5.3.3)

where a is the probe inclination to the axis of rotation, 0, is the angle the channel makes with the axis of rotation and f is the frequency of the channel's rotation. Callingpk the rms value of the sinusoidally varying p m signal, the observedp; divided by q sin S o , whereq = Bpufm,is plotted against sin a in Fig. 11. Three values of q and three values of O0 give overlapping data points indicating the linear dependence, predicted by

5 24

5 . MEASUREMENT OF PRESSURE

0.4

0.3 rn .-C

in U

- E

0 2

a

01

7 -----7 7 ~__1 _ I _ ~ .

0

01

0 2

0 3

04

sin a

FIG. I I . Pressure signal amplitude versus sine of angle of inclination of probe in rotating inclined channel flow. q = &pv:-; 6,-angle of inclination of rotating flow to axis of rotation, p6 is the rms value of measured pressure oscillation. (0)ul0, = 62 ms?: (0) ulo, = 50 ms-'; (0) ulm = 35 ms-I. [From S i d d ~ n . * ~ ]

Eq. (5.3.3), valid up to 0.24 (sin 14") on the abscissa. The value of B = -0.46 5 10 percent obtained by measurement of the slope of the graph of Fig. 11 was shown also to be independent of channel rotation frequency in its range of operation, namely 5 Hz to 150 Hz. The probe was used to explore the pressure variations in three different turbulent flows (channel, jet wake, and grid). Corrections employing Eq. (5.3.2) were applied, and it was found that the correction to rms fluctuation levels was small, generally amounting to less than 20 percent. This last result is surprising, since the corrections themselves were as much as 100 percent of the measured fluctuating pressures. The validity of Eq. (5.3.1) for correcting static probe readings for mean pressure in a turbulent flow can not be evaluated. Showing that Eqs. (5.3.1) and (5.3.2) apply in any given flow is a bootstrap operation. A probe of size smaller than Siddon's probe, i.e., smaller than 5 mm, would have to be used to determine whether the spatial scale of the flow is smaller than 5 mm. It is noteworthy, if Eq. (5.3.1) is valid, that there is a substantial correction due to 3 even if A is made small (by shaping the probe) and the probe is aligned with the mean direction of the flow field (so = 0.)

v:

5.4.

TIME-DEPENDENT PRESSURE MEASUREMENTS: PREVIEW

525

5.3.4. Fluctuating Wall Pressure near Turbulent Flow

Measurement at the wall underneath a fluctuating turbulent flow is possible with miniature diaphragm and stub gages (see Sections 5.9.3 and 5.10.3 where some gages built for this purpose are described). Willmarth and Yangzl describe measurements using 13 gages of 1.5-mm diameter by 0.5-mm-thick barium zirconate and barium titanate mounted flush in the surface of a 100-mm diameter cylinder concentric with a wind tunnel axis. A frequency response flat from 5 Hz to 50 kHz and a sensitivity of 13 p V Pa-', with simultaneous recording of 3 channels on magnetic tape gave both spatial and time data. Other studies with batteries of diaphragm gages in a wind tunnel wall, with optical recording, have also attempted to observe fluctuating pressure^.^^*^^ Despite the small size and close spacing of the pressure gages in these studies, the measurements still suffer from poor spatial resolution; still smaller gages are needed. In an attempt to improve the spatial resolution small pinhole microphones have been used. The pinholes and connecting channels to the microphones disturb the flow, however, and cause serious errors in fluctuating pressure meas~rements.~~ Although one thinks of a wind tunnel as a steady flow with superposed fluctuations, the pressure at the wall is essentially time dependent and the measurement of time dependent pressure is the subject of the remainder of this part.

5.4. Time-Dependent Pressure Measurements: Preview Nearly all of today's gages designed to measure pressure as a function of time consist basically of an element whose deformation varies with the magnitude of the applied pressure and a sensor which continuously converts some aspect of the deformation, such as displacement or strain, into an electrical output. An exception to gages with an electrical output, is one using laser interferometry and photographic recording (Section 5.6.3). Also purely mechanical gages are still being used but most can follow only extremely slow pressure changes and are therefore limited to essentially steady measurements (Chapter 5.2). Various gages differ from one another in the type of sensor employed W. W. Willmarth and C. S. Yang, J . HuiJ Mech. 41,47-80 (1970). Dinkelacker, M. Hessel, G. E. A . Meier, and G. Schewe, P h y s . Fluids 20, S216 (1977). 23 R . I . Soloukhin, Yu. A . Yakobi, and D . I Margulis, Z h . Prikla. M e k h . Tekh. Fiz. 1, 88-92 (1975). *' M. K . Bull and A. S. W. Thomas, P h y s . Nuids 19, 597 (1976). 21

** A .

526

5.

MEASUREMENT OF PRESSURE

and also in the mounting, the size and particularly the shape of the elastic element. A suitable choice depends on the amount and rate by which the pressure is expected to change. The main characteristics for judging the suitability of a gage are considered in the following chapter (Chapter 5.5). A variety of sensors, operating on quite different principles, can be used with different types of elastic element. Since there is little correlation between type of sensor and geometry of the elastic element, sensors are considered first in Chapter 5.6 with little reference to gage application. Following initial development of a signal by a sensor the signal is “conditioned” or “processed” (amplify, filter, digitize, etc.) for purposes of display and recording (Chapter 5.7). Dynamic calibration is considered in Chapter 5.8. With respect to the elastic element, gages for measuring time varying pressures belong to one of six basic types which differ from each other in the shape of the element, the support supplied by the mounting, and the way in which pressure is applied. The schematic diagrams of Fig. 12 illustrate these types which are designated diaphragm gage, stub gage, slab gage, probe gage, bar gage, and dilatational gage. As far as the shape of the elastic element is concerned, the probe and dilatational gage might have the same name, say block gage; they differ in that the strain in a probe gage is a volumetric change whereas during a limited time in which measurements are carried out with a dilatational gage the strain is strictly one-dimensional. The strain is also primarily one-dimensional in the slab gage; in this case it is because of the constraint provided by the mounting

P

t t t t t;t t 1 1 t

TP 1

t t t t;t t t t

f e 1

(a) Element Rigid Mount

=] l;f

FIG.12. Gages for measuring time varying pressures: (a) diaphragm; (b) stub; (c) bar; (d) slab; (e) dilatational; (f) probe. Diaphragm deforms by bending and stretching; other types deform by compression.

5.5.

GAGE CHARACTERIZATION

527

material. There is a major difference between the diaphragm gage and the other types; a diaphragm deforms by bending and stretching whereas the elastic elements of the other types deform primarily by compression. Theory and examples of diaphragm gages are considered in Chapter 5.9. The other types are grouped under the general heading Fast response gages in Chapter 5.10. This is not meant to imply that a moderately fast diaphragm gage cannot be made. Roughly, the time response of which a particular type of gage is capable decreases in the order diaphragm, stub or probe, bar,,slab and dilatational, with very little in favor of the stub or probe over the diaphragm.

5.5. Gage Characterization This chapter is concerned with methods for evaluating gage capabilities and with criteria for determining the suitability of a gage for a particular application. Definitions of terms are given.

5.5.1. Time-Dependent Response One way of indicating a gage’s ability to respond to changing pressure is to give its frequency response function, X(o);another way is to give U(t), its time varying response to a step function load. Other characteristics, such as response time, hold time and resonant period are less informative but useful. These are considered following a brief discussion of X ( w ) and

W t ). The prevalent use of the term frequency r e ~ p o n s e ~ ~to- characterize ~’ the ability of a system to reproduce time dependent changes of the input reflects the mathematical power of Fourier analysis, or more generally, exponential transformation t h e ~ r y . ~By ~ . definition, ~~ a system’s frequency response function, X ( w ) , is the amplitude (and phase) of the output when the input is a steady-state sinusoidally varying function of time having unit amplitude, radian frequency w and zero phase. For a perfect W. Bleakney and A. B. Arons, Pressure measuring manometers and gauges. In “Physical Measurements in Gas Dynamics and Combustion” (R. Ladenburg, ed.), Artic. B2. Princeton Univ. Press, Princeton, New Jersey, 1954. O6 W. W. Willmarth, Unsteady force and pressure measurements. Annu. Rev. Fluid Mech. 3, 147 (1971). L. Bernstein, in “Measurement of Unsteady Fluid Dynamic Phenomena” (B.E. Richards, ed.), Chapter 3. McGraw-Hill, New York, 1977. I* I. N . Sneddon, “Fourier Transforms.” McGraw-Hill, New York, 1951. ** L. W. T. Thornson, “Laplace Transformation.” Prentice-Hall, Englewood Cliffs, New Jersey, 1950.

*’

528

5.

MEASUREMENT OF PRESSURE

system %(a) does not depend on o. In principle, when X ( w ) is known one can determine the output due to any physically realistic input by use of a Fourier series or inversion integral of the form

where 9 ( w ) is the Fourier component (i.e., transform) of the input and 00)is the output. Characterization by specification of X(o)is particularly useful for pressure sensitive systems, such as a microphone, which are designed to respond to a periodically varying pressure of small amplitude and low frequency. For gages whose function is to measure large amplitude, high frequency pressure oscillations such as occur in rocket motors, %(w) is useful but hard to determine.30 Gages planned for these purposes are usually complicated, making mathematical analysis difficult and often unrealistic. On the other hand, %(w) cannot be determined directly by experiment because no satisfactory sinusoidally varying pressure source of large amplitude and high frequency is available.31 Lack of a suitable periodic source for determining X ( w ) also presents a problem for gages whose purpose is to measure large, rapid, nonperiodic pressure changes, such as produced by explosions and projectile impacts. Furthermore, even if a theoretical expression for %(a) is accepted, evaluation of the inversion integral, Eq. (5.5.1) is difficult for any but the simplest types of nonperiodic input. An alternative to using X ( o ) is to specify the system’s response to a nonperiodic input which is simple enough to be mathematically tractable and realistic enough to permit experimental duplication. The input should also contain an abrupt change and be of long duration in order to test both the fast and slow response of the system. A step function meets these requirements. Analytical expressions for the response to unit step function load U ( t )are given for several simple types of gage in following sections. Experimental testing and calibration methods are considered in Chapter 5.8. If U(r) is known, a system’s response to any nonperiodic input can be obtained with the use of a relatively simple superposition (convolution) integral, one form of which is (5.5.2) R. Bowersox, ISA J . 5, 98 (1958). D. S . Bynum, R . L. Ledford, and W. E. Smotherman, “Wind Tunnel Pressure Measuring Techniques,” Advisory Group for Aerospace Research and Development, NATO, AGARD-AG-145-70. Available through NASA, Washington, D.C., 1970. 31

5.5.

GAGE CHARACTERIZATION

529

where O(r)is the response at time t to an input Z ( 8 applied at time F and U(t - .T)is the response at t to a unit step applied at F.32*33 Equation (5.5.2) is usually much easier to evaluate than the inversion integral, Eq. (5.5.1). Other types of input, such as a delta function or a ramp function (linear increase over a finite time to a constant value), are sometimes considered in mathematical analyses, but they are less suitable than a step for use as a standard, since corresponding pressure changes cannot be as satisfactorily produced by experiment. On the other hand, response to such inputs can be readily calculated with the use of Eq. (5.5.2), or an equivalent form, when U ( t ) is known. In finding an expression for U ( t )by mathematical analysis, using transformation theory, the first and easy step is the determination of X ( w ) . On substituting i/o for 9(w), the inversion integral of Eq. (5.5.1) then provides a formal representation of U ( t ) . Usually the most difficult part of the problem is the reduction of the inversion integral to a simple, readily useable form. Many reported analyses are not carried beyond a determination of X ( w ) . When the gage system is too complicated for a reliable mathematical analysis, or, more likely, to provide a check, a numerical representation of U ( t ) can be obtained directly from experimental records. As indicated previously, however, direct measurement of X(w) is often difficult, if not impossible. For cases in which U(t) alone is known, but for which X ( w )is more useful, it has been proposed that a numerical procedure on a computer be used to determine X ( w ) from U(t).30.31*34 One method of calculation is based on the following integral relationship:

(5.5.3)

where S ( t ) is a step function. The denominator reduces immediately to i / w . If the transients of U ( t)decay so that U(t )can be considered constant after a finite time t , , the contribution from the integral in the numerator between the limits t, and 03 is given by ie'""/o. This leaves the finite integral from 0 to t, to be determined by numerical integration for specific values of o. In brief summary either X ( w ) or U(t)furnishes, at least in principle, 3p L. W. T . Thomson, "Laplace Transformation," p. 36. Prentice-Hall, Englewood Cliffs, New Jersey, 1950. 33 F. B. Hildebrand, "Advanced Calculus for Applications," p. 451. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 34 R. B. Bowersox and J. Carlson, Digital computer calculation of transducer frequency response from its response to a step function. Jet Propul. Lab., Prog. Rep. 20-331 (1957).

5.

530

MEASUREMENT OF PRESSURE

complete information for evaluating the time response cabability of a gage. Depending on the purpose of the gage, one or the other of these forms may be more appropriate, with the preference going to W t ) for gages measuring nonperiodic or large pressure changes. Lacking specification of %(a) or U ( t ) ,other characteristics, which can be described by a single number, can provide a partial guide. Among these are response time, hold (or dwell) time, and resonant period. The response time T may be described as the time needed to determine the steady state output of a gage following step function pressure loading and is a measure of the gage’s ability to follow rapid changes. Having reached a steady state value following step function loading the output of some gages begins to decay. The hold time T of a gage is the period during which the decay is acceptably small. To measure slow variations in pressure the hold time must be long and, strictly speaking, it should be infinite for steady measurements. A resonant period P, is the period of a gage’s free or characteristic vibration. The applicability of these characteristics and the exact criteria for assigning numbers depend on the type of response of a particular gage. The drawings of Fig. 13 illustrate typical types of response to step function loading. Drawing (a) resembles records from a dilatational gage (Section

output

U(t’

i^”-.I

:m-

Time - t

(b)

~=.+-‘-ii--output U(t)

I

Time

-

(C)

t

FIG. 13. Typical types of gage response to step function loading. T = response time; T = hold time; 9, = resonant period.

5.5.

GAGE CHARACTERIZATION

53 1

5.10.6); drawing (b), a bar gage (Section 5.10.5); drawing (c), an undamped diaphragm gage (Chapter 5.9). The feature of decaying oscillations, drawing (b), is also typical of records from damped diaphragm, stub (Section 5.10.3) and probe (Section 5.10.4) gages, although details of the oscillations may differ considerably from those shown: The terms “overshoot” and “ringing” are sometimes used to signify this type of behavior. For a response of type (a) the meaning of response time T is clear, but resonant period is not an applicable characteristic since, within the time of measurement, resonance has not yet developed. On the other hand, the record of type (c) consists entirely of resonant oscillations which begin immediately and, without damping, continue indefinitely. For this case response time is a vague concept with no generally accepted, exact definition. For the purpose of comparison with other types of gage, the response time may be considered to be simply some stated fraction or small multiple of the longest resonant period PI.When the spurious oscillations are small and decay with time as in drawing (b), the response time is a more useful characteristic and may be considered to be some specified large fraction of the rise time to the first maximum. The “acceptable decay” needed to determine the hold time T may be taken to be some stated small fraction of what is judged to be the steady state value; a precisely stated value of the fraction is relatively unimportant if an obviously spurious change in the response occurs suddenly as shown in drawings (a) and (b). Response time and resonant period provide good figures of merit for judging the effect of changes in the parameters of a particular type of gage. They also serve as a semiquantitative guide for comparing the capabilities of gages of different types, but for this purpose are much less informative than a knowledge of U ( t ) . Unfortunately, a graph or analytical expression for U(t) is seldom given in specifications for commercial gages.

5.5.2.Sensitivity and Range Disregarding distortions in time, the change in output of a gage due to a change in pressure from p o to p is given by ro

(5.5.4)

where Y ( p ) ,the rate of change of output with respect to pressure ( a O / d p ) , is the sensitivity, which in general is a function of p . The range is the difference between the maximum allowable pressure, pmaxand the lowest (zero), or usually simply pmax. Primarily to simplify data reduction by

532

5.

MEASUREMENT OF PRESSURE

eliminating the need for evaluation of a cumbersome integral, gages are usually designed so that 3 p ) will be independent of pressure over their useable range, thus providing a linear response. PO(p) = Y ( p - P o)

for 9’const.

(5.5.5)

Often changing a parameter to improve one of a gage’s characteristics automatically degrades another characteristic. Sensitivity and range are likely to be competing characteristics. For example, in the case of a diaphragm gage decreasing the thickness of the diaphragm keeping its other dimensions unchanged increases sensitivity but reduces the pressure at which the response becomes nonlinear (or, beyond this, the pressure at which the diaphragm deforms permanently or ruptures) thus reducing pmax. Other characteristics competing with sensitivity are response time or hold time. It is a rule, almost without exception, that cutting down the response time to improve the capability of measuring rapid changes will result in a decrease in sensitivity. Thus scaling down all dimensions of an elastic element and sensor will reduce the response time (or resonant period), but it will also decrease the sensitivity, although it will not change the range. Also gages having very short response times often have short hold times, so they cannot be used for measuring slowly varying or steady pressures and must be calibrated dynamically. Fortunately, in practice, if the requirement for one of a pair of conflicting characteristics is stringent the requirement for the other is often lenient. For example, except during severe storms, changes in atmospheric pressure are small and slow, so that an ordinary barometer must be relatively sensitive and have an essentially infinite hold time but its response time can be long-of the order of seconds and greater-and its range small. On the other hand pressure changes due to sonic booms, explosions, etc., are extremely rapid-occurring within microseconds or less-but they are usually large and do not continue over a long period, so that although a gage must have a very fast response time and large range its sensitivity can be low and its hold time relatively short.

5.5.3. Pressure as a Function of Position A single gage measures average pressure over a finite area. In order that this correspond as nearly as possible to the idealization of pressure at a point, the size of the sensitive element should obviously be as small as possible. A decrease in size of a particular type of gage thus increases its space resolution, as well as its time resolution, but at the expense of a decrease in sensitivity, which sets a practical lower limit on size.

5.5.

GAGE CHARACTERIZATION

533

Arrays, composed of a large number of gages, are used to measure pressure distribution over an extended region in space. A system which records continuously from each gage, and thus simultaneously for all gages, usually requires an inordinate amount of expensive equipment. Two schemes, each involving a compromise between time and space measurement, are used to reduce the equipment requirement. In one, called multiplexing, the electrical or mechanical responses from the individual gages are recorded in succession, each for a short period of time.31 In the other scheme, optical sensors are used to produce a photograph from which the response of all gages at a particular instant can be determined (Section 5.6.3). Either scheme is quite easily adapted to the study of steady state phenomena.

5.5.4.Environmental Effects A part of the problem of pressure gage design is eliminating or compensating for response to changes other than those of pressure. Among these are changes of temperature, of external electric and magnetic fields and vibrations of the base on which the gage is mounted. An unwanted change may occur either during a measurement and cause a spurious response directly, or between measurements and lead to a lack of repeatability (long term stability). For example, a piezoelectric sensor may also be pyroelectric so that a change in temperature accompanying a change in pressure will produce a spurious response during a measurement. But even if the sensor is not pyroelectric a change in ambient temperature between measurements may change the values of the piezoelectric constants and consequently the pressure sensitivity of the gage. Gage properties may also change over a long time due to ageing. Even repeated changes in pressure itself occurring during application can cause changes in proper tie^,^^ particularly mechanical deterioration leading to a short life. 5.5.5. Accuracy The problem of attaining and being assured of a given accuracy is in general the same for a pressure gage as for any measuring instrument. If a pressure gage is unique, it is because its operation is usually based on the deformation of a supposedly purely elastic element. Since no material is strictly elastic, to assume that it is can lead to systematic error. A material can be used only well below its yield o r fracture point and should not behave anelastically, ie strain should depend only on the magnitude of A . H . Meitzler, R e v . Sci. Insirurn. 27, 56 (1956).

534

5.

MEASUREMENT OF PRESSURE

the applied stress and not on the rate of application, otherwise hysteresis or relaxation will O C C U ~ . ~ ~ ,Polymer ~’ solids in particular are suspect and should be carefully chosen and tested. A s for all measuring instruments, the sensitivity must be large enough so that the smallest readable division corresponds to a pressure increment less than the required accuracy; also the sum of the random errors, including noise, should not be appreciably greater than this least count. Probably the best method of assuring that there is no large unsuspected systematic error is to compare readings with those of a gage of an entirely different type. Calibration should ultimately refer to values obtained with a simple, absolute gage, such as a liquid manometer or piston gage (Chapter 5.2). Most gages have an accuracy between 2 and 10 percent. An accuracy of 10 percent is usually relatively easy to obtain but extreme care, including frequent calibration, is needed for an absolute accuracy less than 2 percent.

5.5.6. Ease of Construction, Calibration, and Operation Although there is no quantitative measure of simplicity, its advantages are obvious. Complexity leads to high construction costs, both in time and in money, and is likely to increase the need for lengthy and frequent calibration. Ease of operation decreases the chance of mistakes.

5.6. Sensors This chapter emphasizes physical phenomena basic to sensor operation and does not purport to be an exhaustive compendium of all proposed devices. It includes a selection of commonly used or unique means of measuring the strain within an elastic element or the displacement or velocity of one of its surfaces. Most pressure gages use a sensor which responds directly to strain or displacement. There is at least one important case (Section 5.10.6.2), however, in which the velocity of a surface, rather than its displacement is proportional to the applied pressure and in this case a velocity sensor is the most suitable. There are also accelerometers which are sometimes used as part of a “force transducer” to measure the components of force due to the integrated pressure over the surface of a body such as an artilC . M. Zener, “Elasticity and Anelasticity of Metals.” Chicago Univ. Press, Chicago, Illinois, 1956. Wiley, New York, 1956. 37 J. C. Jaeger, “Elasticity, Fracture and Flow.”

5.6. SENSORS

535

lery shell or an air vehicle. But the sensors employed in an accelerometer measure the instantaneous displacement of an inertial mass working against an “elastic spring” of some type and so involve no new principles. Special problems associated with accelerometers and force transducers are considered in previous review^^^.^^ and are not treated here. Acceleration is important in another way, however. Most pressure gages are attached to a base which supposedly is perfectly rigid and stationary, but which in fact often vibrates. If the mounting does not isolate the gage from vibrations of the base and if the sensor responds to acceleration of the gage, spurious oscillations will appear in the recorded signal (Section 5.10.3). For descriptions of sensors not considered here and for more detailed discussions of those which are, refer to earlier reviews26*27.31 and their bibliographies. Some and a compendium of commercial gages,*O although older, may prove useful. Sensors can be classified as mechanical, electrical, or optical. 5.6.1. Mechanical Sensors

The liquid column (Section 5.2.1) and dead weight tester (Section 5.2.3) fall into this category. A more convenient type consists of a system of levers and gears which translates the displacement of a point on an elastic element (Bourdon tube, capsule) into movement of a pointer along a scale (Section 5.2.2). Construction of sensors of the lever-gear type might best be described as a watch maker’s art and, like the spring-driven watch, they are reliable and simple to operate. Also like the spring-driven watch, and probably for much the same reasons, many gages of this type are still in use and only gradually being replaced by those with electronic mechanisms. But, as previously noted, they are only useful for measuring steady or slowly varying pressures. The fact that large pressures produce plastic flow of soft metals with resulting permanent deformation has been used to measure peak values of nonperiodic pressure changes.25 For blast pressures produced by explosions, a number of holes of different diameters can be drilled through the side of a heavy steel box and covered with a sheet of soft aluminum to form a cluster of diaphragms. After an explosion near the box, the largest unbroken and the smallest broken diaphragm serve, with suitable calibration, to bracket the peak pressure. Or the peak pressure is revealed in 38 H. K . P. Neubert, “Instrument Transducers.” Oxford Univ. Press, London and New York, 1965. 39 K . S. Lion, “Instrumentation in Scientific Research.” McGraw-Hill, New York, 1959. G . F. Harvey, ed., “Transducer Compendium.” ISA/PIenum, New York, 1969.

536

5.

MEASUREMENT OF PRESSURE

a continuous fashion by the amount of permanent bulge of an unbroken diaphragm. For gun pressures, an undersized cylinder or sphere of soft copper can be placed in a cylindrical cavity in the wall of the gun barrel or shell chamber. A nonleaking piston of hard steel touching the copper piece closes the cavity and causes permanent deformation of the copper when pressure is applied. The amount of the permanent set is related to the peak value of the pressure. A purely mechanical method of measuring the amplitude and time variation of an unsteady pressure in the form of a pulse, such as produced by an explosion, was developed shortly after the turn of the century by Hopkinson. It is described in Section 5.10.5.1 because it is ingenious and unique. 5.6.2. Electrical Sensors

Electrical sensors may be characterized in several ways. They may be passive or active depending respectively on whether an external source is or is not required.*’ They may also be either intrinsic or extrinsic.26 An intrinsic sensor (sometimes called “molecular”) depends on an electric or magnetic change (resistive, piezoelectric, magnetostrictive, etc.) which takes place at the atomic-molecular level within a material; the associated property (resistivity, piezoelectric charge per unit volume, etc.) is independent of a body’s size. An extrinsic sensor (sometimes called parametric) operates because of the macroscopic movement or change in shape of a body; the related property (resistance, capacitance, inductance) is a “lumped” electrical parameter whose value is a function of a sensor’s overall dimensions. All sensors are extrinsic, but some operate primarily because of an intrinsic change. A third division is based on the quantity (displacement, strain etc.) to which a sensor directly responds and a fourth depends on principle of operation (Ohm’s law, Faraday’s law of induction, etc.). The following divisions are based roughly on principle of operation. 5.6.2.1. Resistance Sensors: Current-Voltage Dependence. Passive sensors of a large class depend on a change in their current-voltage characteristic. Most are resistance sensors for which current I and voltage V are related by

V = RI

(5.6.1)

and sensing is accomplished by finding the change in the total resistance

R. Some old types of sensor consisted of a rheostat or potentiometer with a mechanically sliding ~ o n t a c t , but ~ ’ nearly all of today’s sensors respond

5.6. SENSORS

537

FIG. 14. Element of a resistance or capacitance sensor. Shaded areas are electrodes, i.e., surfaces of constant voltage.

to a change in geometry of the resistance element, which may or may not be accompanied by a change in the resistivity p of the material. A slab of homogeneous, isotropic material with plane parallel electrodes as shown in Fig. 14 has total resistance given by

R = pl/A,

(5.6.2)

where A is the effective cross-sectional area of the element and 1 is the distance between the electrodes. Sensors for medical applications have been made with a liquid as the resistive material.41 In this case I is made small compared to the dimensions of the electrodes so that the electric field is essentially confined to the region between the electrodes. Provided changes in 1 are not too rapid, the liquid merely moves in or out of the electric field, p and A can be considered constant and R is a linear function of 1 alone. Ordinarily resistance sensors use a solid for the resistive material and are called strain sensors. Particularly in gages designed to measure pressures in the gigapascal range, the sensing element and the mechanical element to which pressure is applied may be physically the same. The value of 1 is comparable to the dimensions of A . For this type it is convenient to define sensitivity 9, in terms of applied pressure instead of resulting strain

ARIR YP = AP ’

(5.6.3)

where Ap is the change in pressure. If the pressure is transmitted equally 41

J . R . Pappenheimer, Rev. Sci. fnsfrum. 25, 912 (1954).

538

5.

MEASUR.EMENT OF PRESSURE

to all faces of the element (hydrostatic stress) and if the element remains elastic,

Y*= c + (3B)-’,

(5.6.4)

where C is a piezoresistive constant (fractional change in resistivity per unit change in stress) and B is the bulk modulus of the material. See the last part of Section 5.10.4 for a method of calibrating resistance sensors at high pressures and for representative values of 9,for several materials. The most common type of resistance strain sensor has the form of a thin filament with electrical connections at the end^.^*-^^ The filament may be a fine wire, a thin foil, a deposited film or a region of impurity concentration within a semiconductor. The filament is attached to or made an integral part of an elastic body and ideally, without introducing constraint, undergoes the same strain as the body, or at least undergoes the same strain as the body along one or two directions. This ideal is almost exactly satisfied in case the body is a pure semiconductor in which the filament is formed by impurity doping and can be closely approximated by a film deposited directly on the body, or on a previously deposited insulating film. For some older type sensors where the filament is embedded in an insulating sheet (e.g., paper) glued to the body, the bond may not hold. Either the assumption of no slip may not be justified or the bond may break during use. Another drawback of stick-on sensors is that they are difficult to apply to extremely small areas. Conventionally, the sensitivity of resistance strain sensors is given in terms of the so-called gage factor

G=-ARIR E

’

(5.6.5)

where E is the strain sensed. Sometimes the expression for the change in resistance AR,is quite complicated. In the case of a single crystal semiconductor, AR depends on the orientation of the filament relative to the crystal axes, on changes in the various components of strain and on accompanying piezoresistive changes in the resistivity tensor. Although bothersome, this complexity can be an advantage to a designer, because of the large number of parameters at his disposal. In other cases, a sensor may be highly sensitive only to one component of strain in the

‘*

C. C. Perry and H. R . Lissner, “The Strain Gage Primer.” McGraw-Hill, New York, 1955. M. Dean and R . D. Douglas, “Semi-Conductor and Conventional Strain Gages.” Academic Press, New York, 1962. H. K. P. Neubert, “Strain Gauges.” Macmillan, New York, 1967.

5.6.

SENSORS

539

body, usually in a direction along the length of the filament, because of the relative strength of the filament and the way it is attached. Often A l / l , where 1 is the effective length of the filament, is used for E in Eq. ( 5 . 6 . 3 , with the understanding that AR depends on other factors (such as transverse constraint) as well. This can, however, be misleading if one or more of the other factors does not remain constant during a measurement or is not directly related to Al/l; A R / R may change even when All1 is zero. Wire strain sensors have a gage factor of about 2 which is mainly attributable to a change in geometry rather than to a change in resistivity. The wire is usually a metal alloy having a low temperature coefficient of resistance to minimize temperature dependence. On the other hand semiconductor resistance sensors (e.g., silicon doped with N- or P-type impurities) have gage factors in the range 50 to 150. This large sensitivity is due to the fact that the material is highly piezoresistive and so undergoes large changes in resistivity; the sensitivity to strain is a function of impurity concentration. The principal disadvantage of semiconductors compared to metals is their much greater temperature sensitivity. The temperature sensitivity of a semiconductor is also a function of impurity concentration; unfortunately it increases as the strain sensitivity increases. To reduce temperature dependence, some strain sensitivity is usually sacrificed and the circuit is designed to provide temperature compensation. The change in resistance is sensed either by providing a constant current (perhaps with a constant voltage source in series with a large ballast resistance) and by measuring the change in terminal voltage or by keeping the terminal voltage constant and measuring the change in current. A dc source is usually preferred for very fast response gages since dc provides continuous recording, but an ac source can be used if its period is considerably less than the response time of the gage. Excitation by ac has the advantage that unwanted pickup is less troublesome. A bridge network can be useful, particularly when two or four sensors are used to increase sensitivity or to provide temperature compensation. For example, consider four sensors attached to a diaphragm as shown in Fig. 15. As indicated by the arrow tips, the sensors are directionally sensitive. As the diaphragm distorts, the resistance of sensors 1 and 2 increases since they sense tension whereas that of sensors 3 and 4 decreases since they sense compression. When connected as shown in Fig. 15c all sensors contribute to a change in output due to pressure change but there is very little output due to equal temperature changes of the sensors. An unusual type of sensor which is not a resistance sensor, but which depends on a change in current-voltage characteristic is a so called piezo-

5 40

5.

;;;,

M E A S U R E M E N T OF PRESSURE

Source

FIG. 15. Disposition of resistance sensors on diaphragm to enhance pressure sensitivity and minimize temperature sensitivity. R , and RPincrease (tension) and RS and R4decrease (compression) as the diaphragm deflection increases. (a) section through diaphragm; (b) plan of diaphragm; (c) connections in bridge circuit.

junction.45 This is an N-P semiconductor junction whose characteristic is sensitive to applied stress. It is used as the emitter-base junction of an N-P-N transistor so that the applied stress controls the current from emitter to collector, thus exploiting the gain of the transistor. The dimensions of the active semiconductor region are only a few micrometers, so that the sensor can be small and the gage resonance frequency high. 5.6.2.2. Capacitance Sensors: Charge-Voltage Dependence. Equations for capacitance are formally the same as those for the conductance of resistance sensors. Thus Eq. (5.6.2) becomes the equation for the capacitance C of a parallel plate capacitor upon substitution of C for the conductance R-', and permittivity KeOfor the conductivity p-'. C = K&oA/I.

(5.6.6)

See Fig. 14. The circuit relation for the operation of the resistance sensor is V = RZ, Eq. (5.6.1): the analogous relation for the capacitance sensor is Q

=

CV,

(5.6.7)

where V is the voltage difference between the electrodes and + Q and - Q are the charges on the electrodes. Both the resistance and the capacitance sensor respond to relative displacement of the electrodes. An important difference is that a resistance sensor must have a material, with mass and usually some strength, between the electrodes, but a capacitance sensor need not. Another difference is that a resistance sensor is a heat generator, whereas a capacitance sensor is not. An air or vacuum spaced capacitor is commonly used to sense the deformation of a metal diaphragm, which serves as one of the electrodes. 45

W. Rindner and R. Nelson, Proc. IRE 50, 2106 (1962).

5.6.

54 I

SENSORS

Although the displacement, w = A I , varies across the face of a diaphragm, its maximum value wo can be used as a parameter related both to C and to the applied pressure. For small values of w o and change in capacitance AC AC = (dC/dl)wo.

(5.6.8)

In other applications, the capacitor has an elastic dielectric solid between the electrodes and is basically a strain sensor. Pressure may be applied either directly to the electrodes or to a larger elastic element in which the capacitor is embedded. Formally, AC is again given by Eq. (5.6.8) with the understanding that K and A may, but usually do not, depend on the strain A l / l . In analogy to the gage factor for a resistance strain sensor, Eq. (5.6.5), the sensitivity Y for a capacitance sensor is

y=- AC/C &

= -

A(l/.C)/(l/C) E

,

(5.6.9)

where E = w / l = Al/I. For a sensor in which the electrodes remain plane and K and A do not change, Y = 1. A capacitance sensor requires an external electrical source. For measuring rapid pressure changes a dc source in series with a large ballast resistor can be used to charge the capacitor. The charge remains essentially constant during rapid changes in capacitance which, according to Eq. (5.6.7),produce changes in voltage across the electrodes. These are amplified and measured. The main difficulty is that the capacitance is usually small and may be the same order of magnitude as the distributed capacitance of connecting cables and associated circuitry; electrostatic pickup and reduction in sensitivity often loom as major problems. To minimize these problems the capacitor plates are connected to a nearby miniature amplifier with a high impedance input which develops a voltage signal across a lower impedance for remote monitoring and recording. Because of charge leakage and low frequency disturbances, the constant charge method is unsuitable for sensing steady and very slowly varying pressures. At this extreme an ac source, usually feeding into a bridge containing one or two capacitance sensors, can be used. To determine capacitance changes, one can either monitor changes in bridge output or use null balancing. Fig. 22b displays the bridge circuit used widely for capacitance sensing of diaphragm deflection. Instead of deriving a voltage change in the constant charge mode, or a change in C when the capacitor is in an ac bridge, sometimes the capacitor is used with a fixed inductor L to determine the resonant frequency of an oscillator. The frequency o = (LC)1’2changes as C changes. The signal

542

5.

MEASUREMENT OF PRESSURE

is transmitted, or radiated by an antenna, to a remote receiver where the value of C is derived by frequency demodulation. The capacitance sensor is simple, can be made with low inertia and little inherent strength, and is not a heat source. 5.6.2.3. Piezoelectric Sensors: Voltage-Charge Generation. The basic element of a piezoelectric sensor is a dielectric lacking a center of symmetry and having the property that an applied stress produces an internal electric field terminating on positive and negative surface charges. Sheets of conducting material placed adjacent to appropriate areas of the element serve as capacitor electrodes whose open-circuit voltage is controlled by the piezoelectric field. No external electric source is needed.27.4s-48 There are three types of piezoelectric sensor. In one type the basic element is a properly cut single crystal of a piezoelectric material, such as quartz, tourmaline, etc. Single crystal plates or bars are naturally polarized in a direction determined by the crystal structure. In a second type, the basic element is a ceramic composed of tiny crystals of a ferroelectric material such as barium titanate, lead zirconate, etc. embedded in clay. The ferroelectric clay mixture is molded into shape and fired. It is then subjected to a strong electric field, its temperature is raised above the Curie point of the ferroelectric and it is allowed to cool slowly to a temperature below the Curie point while the electric field is maintained. When the electric field is then removed, the element is left in a polarized state, with the direction of polarization the same as that of the previously applied field, and it remains in this state unless its temperature is again raised above the Curie point. The third type comes in the form of a thin plastic film made from a polymer such as PVFz (polyvinylidene f l ~ o r i d e ) .Piezoelectric ~~ films of PVFzhave been suggested for practical use only recently and their preparation at present is as much an art as a science. In general there are three steps. First, following polymerization of the vinylidene fluoride monomer, which usually contains impurities whose role is still a subject for debate, the resin is pressed into sheet form. Next it is stretched, either uni- or biaxially, to form an oriented, semicrystalline matrix whose W. G . Cady, “Piezoelectricity.” McGraw-Hill, New York, 1946. J. F. Nye, “Physical Properties of Crystals.” Oxford Univ. Press, London and New York, 1957. W. P. Mason, “Piezoelectric Crystals and Their Applications to Ultrasonics.” Van Nostrand-Reinhold, New York, 1950. * A. L. Robinson, Science 200, 1371 (1978). 47

5.6.

SENSORS

543

crystallites are thought to be mainly in a particular phase, called the pphase. Finally, the film is subjected to a large electric field perpendicular to the plane of the film. If this has been done at an elevated temperature, which is not always the case, the film is allowed to cool and the field is removed. With removal of the field, the film is left with a large permanent electrical polarization across its thickness and will behave in the same way as a poled ceramic. For all three types, the electrical detection system may be a voltage or charge amplifier connected, usually through a network including a cable, to the electrodes of the piezoelectric sensor. In part because of charge leakage between the electrodes of the piezoelectric element in the case of a voltage amplifier or across the feedback capacitor of a charge amplifier, piezoelectric sensors are not suitable for measuring steady pressures. They can only be used to measure changes taking place in an interval much less than the effective time constant (RC) of the circuit, but in some cases this can be quite large (greater than 100 s). They are ideally suited, however, for use in fast response gages (milliseconds and less). The open-circuit voltage V and the short-circuited charge Q developed by a single crystal or electrically poled element, can be calculated by the following equation (the open-circuit voltage and the short-circuited charge are, respectively, the quantities measured by a voltage and a charge amplifier):

where & is the electric field produced by unit voltage applied to the electrodes of a mechanically free element, d& are the stress related piezoelectric constants, c { k are the stress components in an electrically free material and Co is the geometric capacitance of the element. The integral in Eq. (5.6.10) is to be taken over the volume 'V of the element and the summation convention is implied by the repeated subscripts. Because of symmetry not all of the piezoelectric constants are independent and many are zero. As Eq. (5.6.10) indicates, V and Q depend on the detailed distribution of stress, including shear as well as normal components, within the element. They also depend, through &, on the orientation of the conducting electrodes relative to the crystal or polarization axes. In most pressure applications, the piezoelectric element is a slab of thickness 1, between plane parallel conducting plates whose normal will be taken in the direction of the unit vector ii. Since the relative permittivity K of the element is usually large, the direction of E will to a fair approximation be along ii and its magnitude will be I-'. Using for ajkthe

544

5 . MEASUREMENT OF PRESSURE 0x1s of polarizai ion

axis of polarization

OXIS o f polarization

3- x

3- x

z I-Y

I-Y

I-Y

eio

(a) axis o f polarizo tion

3-x

directions of

P

I-Y

(e) FIG.16. Operating modes for piezoelectric elements. ( I , 2, 3) denote axes for a polarized ceramic or polymer. (A', Y , 2)denote axes for a single crystal.

six component notation of Mason,4B,*Eq. (5.6.10) becomes Q=- &ntA

1

v=

- -1 1

I,.

n<&Uj d7f

(j= 1, 2 , 3, 4, 5 , 6 ) (5.6.11)

(i = 1, 2 , 3),

where A is the area of one of the electrodes and the Ki are permittivity constants. Using linear stress-strain relations, Eq. (5.6.11) can be written in terms of strain by replacing ujby E ~ the , strain components, and dg by d t the piezoelectric constants related to strain. For relations between systems using different sets of dependent variables see Mason.5o Several idealized modes of operation are shown in Fig. 16 where the axes are those of a polarized ceramic or polymer (1, 2, 3) or of a single so

W. P. Mason, Bell S y s t . Tech. J . 26, 80 (1947).

* In this notation, stress is represented by a first rank tensor whose six components are related to the components of the more common second rank tensor in the following way: 0 1

=

UII,

US

us = um, =

031

=

0 3

UlJt

= u s s , u, = USBp = ups, = UBpI = ule

5.6.

545

SENSORS

TABLE11. Piezoelectric Components for Electrically Poled Materials” j

i

2

1

3

4

5

6

Entries in the table are values of the components du in the notation detailed in the text.

crystal (X, Y , 2). The axes of the ceramic and crystal are labeled differently because the components of the piezoelectric tensor are given in terms of these axes and by convention the polarization direction of a ceramic is taken along the 3-axis, but for a crystal it is along the X axis, or by number designation the 1-axis. Thus the d33component for a ceramic is the dll component for a crystal and there are similar confusions with other components. Unfortunately, these conventions are too entrenched to be disregarded. The charge sensitivity to stress, defined by 9’ = Q / A u , can be calculated for different modes with the use of Eq. (5.6.1 l), on the assumption that stress does not vary throughout the element. As previously mentioned, the problem is simplified by the fact that, due to symmetries of the material, many of the components of the piezoelectric tensor are zero and relations exist among the nonzero component^.^^,^^ All electrically poled ceramics and polymers have transverse isotropic symmetry and the piezoelectric components have the values shown in Table 11. Quartz, which is the most used of the single crystals, has trigonal rhombohedra1 (D3) symmetry and the piezoelectric components have the values shown in Table 111. The scalar integrand of Eq. (5.6.11) becomes, for a poled material,

YU

=

&d@j

=

nid15V5

+ nzdl@4 - n,(d31ul + daiUz - d33CT3),

(5.6.12)

+ dl4V4) - nz(d14V5 + dIlu6).

(5.6.13)

and for quartz, 9’U = QdUUj

=

nl(d11Vi

-

d11Uz

In pressure gage applications, the shear mode Fig. 16c, which for poled material has 9’= dI5(in this case ul = u2= u3 = u4 = u 6 = 0, u5# 0; n2 = n3 = 0 , nl = l), is seldom used. st W. P. Mason, “Electromechanical Transducers and Wave Filters.” Van NostrandReinhold, New York, 1948.

5.

546

MEASUREMENT OF PRESSURE

TABLE111. Piezoelectric Components for Quartz”

.i 1

2

3

4

5

6

1

4I

-dii

2

0 0

0 0

0 0 0

d14 0

0 -dl4 0

0 -dii 0

1

3 ‘I

0

Entries in the table are values of the components du in the notation detailed in the text.

The hydrostatic mode is used in probe gages which are mounted in the interior of a fluid rather than on a boundary. In this case, Fig. 16d, (ar = (+I = u 6 = 0; u1 = o2= u3= - p ) the pressure sensitivity for a 2 4 , - d,, but for quartz it is, poled material is, from Eq. (5.6.12), Y’= from Eq. (5.6.13), 9’= d,, - dil = 0. Thus, since in general d, # 2ds1, a poled material can be used in the hydrostatic mode but quartz cannot. Single crystal tourmaline, unlike quartz, is hydrostatically sensitive and has been used in probe gage applications. A bimorph, Fig. 16e, which responds to bending, consists of two bonded piezoelectric elements, each of which operates in a transverse mode, Fig. 16b. If the polarization vector of each element had the same sense, bending of the bimorph would produce no voltage between electrodes on its outer faces, since the effect of contraction of one element would exactly balance that due to extension of the other. A signal can be obtained either by bonding the elements so that their polarization vectors are oppositely directed, Fig. 16e, or, with the polarization vectors having the same sense, by using a third electrode in the bonding surface as one terminal and the outer electrodes connected together as the other terminal (parallel connection). For sensors having comparable dimensions, thin bimorphs have an advantage relative to those operating in a thickness or longitudinal mode of higher overall sensitivity, but they have the disadvantage of a lower natural frequency. Fast response gages designed to measure pressure at a wall have customarily used a sensor made from a poled ceramic or X-cut quartz operating in the mode in which the applied pressure produces a compressive strain.” X-cut quartz has the normal to one pair of faces parallel to the X-axis. The element is shown in Fig. 16a with the normals to the other pairs of faces along the Y- and Z-axis. With this orientation, the quartz * The name “thickness mode” usually implies that its thickness is much less than at least one of its lateral dimensions and that its major resonant frequency is determined by its thickness. The name “longitudinal mode” implies that the major resonant frequency is determined by the greater lateral dimension.

5.6.

s47

SENSORS

element will respond in the transverse mode (Fig. 16b) (0,= - p , u3 = 0 or u2 = 0, u3 = - p and u1 = u4= u5= uB= 0; n, = 1, n2 = n3 = 0), only to pressure applied to the faces with their normal along the Y-axis but not to pressure applied along the Z-axis. This is seen from Eq. (5.6.13), with Y = dll when u2 = - p . u3= 0 but Y = 0 when uz= 0, u3= - p . Some other crystals, such as Rochelle salt, when cut with faces having normals along theX-, Y-, Z-axes, will not respond to pressure on any pair of faces, and it is sometimes said that Rochelle salt is sensitive only to shear stresses. However, if a Rochelle salt crystal is cut so that the normal to the electrode surfaces is along the X-axis and the normals to the other two pairs of faces are at 45” with respect to the Y- and Z-axis, it will respond to pressure on either pair of these surfaces. For a discussion of crystal cuts see For the purpose of comparison of ceramic and quartz sensors, nominal values for the compressive sensitivity of two poled ceramics (us= - p , u1 = u2= u4 = u5= cre = 0; nl = n, = 0 , n3 = 1) and of X-cut quartz (u,= - p , u, = u3 = w4 = u5 = u6 = 0; n , = 1, n2 = n3 = 0) are52 barium titanate lead zirconate titanate (PZT-5) quartz

Y = d33

=

Y

= 593 pC

=

d33

149 PC N-’

Y = dll = 2.3

. N-’

pC * N-’

Although the sensitivities of the poled ceramics are much greater than those for quartz, quartz has the advantages that it will respond linearly over a greater pressure range, will withstand higher pressures and temperatures without damage, has a less temperature dependent sensitivity, and in general has a higher resistivity and better long term stability. The piezoelectric property of the polymer PVF, has recently become of interest, although the polymer has for some time been used as a protective coating for metals because, like Teflon, it is chemically inert and electrically i n ~ u l a t i n g .Interest ~~ in its piezoelectric property stems in large part from the fact that it can be formed into very thin (6-50 pn), nonbrittle, highly flexible sheets. For example, ceramic and crystal sensors have long been used in hydrophones designed to detect bodies emitting or reflecting underwater acoustic waves, but they are highly susceptible to breakage under conditions in which they are likely to be used; the nonbrittle characteristic of PVF, provides an advantage. Since thin sheets of PVF, are light, flexible and of small tensile strength, they easily conform to a body of any shape and when attached virtually lose their mechanical sz W. P. Mason, “Physical Acoustics and Properties of Solids.” Van NostrandReinhold, New York, 1958. Values for more recently developed ceramics are found in manufacturer’s literature.

548

5.

MEASUREMENT OF PRESSURE

identity. Unlike a crystal or ceramic they do not appreciably constrain the body or impose their own resonant characteristics. Electrically poled PVFz, like poled ceramics, is not only piezoelectric, but is pyroelectric (a change in temperature produces an electric field) as well and this is a disadvantage for many pressure gage applications. 5.6.2.4. Electromagnetic, Magnetostrictive and Inductance Sensors: Change in Magnetic Flux. Faraday’s law of induction provides the operating basis for a large variety of possible sensors. According to the law, an emf is induced in a coil of conducting material whenever there is a change in the total magnetic flux linking the coil, the magnitude of the emf being proportional to the time rate of change of flux. In some sensors a pick-up coil is placed in a magnetic field produced by a permanent magnet; in others, the field is produced by a dc electric source maintaining a current through an excitation coil. A change in flux through the pick-up coil can be due to a movement of the permanent magnet or of a separate excitation coil, of a piece of magnetic material in a magnetic circuit, or of the pick-up coil as a whole or in part, thus allowing many possible sensor arrangements. Such devices are referred to as electromagnetic In general, the response of this type of device to the movement of a mechanical element to which pressure is applied is proportional to the velocity of the element rather than its displacement, because the measured emf is proportional not to the change in flux but to its rate of change. Since the pressure usually produces a proportional displacement (or relative displacement), as in the case of a diaphragm, electromagnetic sensors, in their primary form, are unsuitable for most pressure gage applications. They cannot be used to measure a steady pressure difference and to measure a pressure change the record must be integrated over time. The magnetic analogue of the piezoelectric sensor is the magnetostrictive sensor. Ferromagnetic materials, particularly nickel and nickel-iron alloys, have the property, when magnetized, that struin produces a change in magnetization. This results in a change in magnetic flux which can be detected with a coil magnetically linked to the material. Although magnetostrictive have been developed to measure pressure changes, they are rarely used today. Compared to piezoelectric sensors, they suffer from low sensitivity, nonlinearity and long term instability. Also, if one senses the induced emf by measuring the open circuit voltage across the terminals of the pick-up coil, the response is proportional to rate of strain rather than to strain and, like an electromagnetic sensor, the device has the disadvantage of being basically a velocity sensor. 53

A. W. Smith and D. K . Weimer, Rev. Sci. fnsrrum. 18, 188 (1947).

5.6. SENSORS

549

Unlike electromagnetic and magnetostrictive sensors, an inductance sensor responds to displacement. Physically an inductance sensor employs Faraday's law as does an electromagnetic sensor; the distinction is that an ac source is used for an inductance sensor, but a dc source for an electromagnetic sensor. The essential aspects are displayed in a simple series circuit consisting of a coil linking a magnetic circuit and having an inductance L , a resistor R , and an electric source 8. Suppose the magnetic circuit linked by the coil contains a part whose position changes with pressure, as in the case of the diaphragm gage shown in Fig. 22a, and that this change in position w causes a change in the self-inductance L of the coil. The equation for the series circuit is dl L-+RZ=-v dt

(5.6.14)

where I is the current and v is the velocity of the moving part. For a dc source, 8 is constant and we have approximately for Z(t), the time varying component of the current, (5.6.15)

where Zo(= 8 / R ) is the steady state component of the current. The driving emf, given by the right-hand side of Eq. (5.6.13, depends on the velocity of the moving part. On the other hand, if 8 varies periodically and L is a slowly varying function of time (v is small), the term in Eq. (5.6.14) containing v can be neglected and we have

L

=+ dt

RZ(t) = 80).

(5.6.16)

The effect of a change in L is to alter the amplitude and phase of the current. In other words, the ac source provides a carrier signal which is modulated by a change in L ; L in turn depends on the displacement rather than the velocity of the moveable part. An inductance sensor can measure a static pressure difference and follow a smooth variation in pressure as a function of time. Its response time can be no smaller than the period of the ac source.

5.6.3.Optical Sensors Specular reflection, the doppler effect, the photoelastic effect and various types of interference phenomena, including holography, have been suggested as a basis for using light to sense displacement, strain, or velocity. Recording is accomplished by visual observation, by pho-

550

5.

MEASUREMENT OF PRESSURE

tography or by means of a photocell. Compared to purely electrical sensors, light sensors usually require more space, are subject to more severe problems of alignment and need more complicated equipment for time-dependent measurements. Because of these drawbacks, light sensors have shown promise of competing successfully with electrical sensors only for special situations, such as in a noisy electrical environment where pickup precludes reliable transmission of electrical signals or for special problems of measurement. One special problem is that of measuring pressure at a particular time as a function of position. For example, one may wish to know how pressure varies over an extended region of a flow about an airfoil. As mentioned in Section 5.5.3, this requires a large array of small sensors; to record electrical signals from all sensing elements simultaneously may involve the use of an inordinate amount of equipment. To reduce the amount of equipment needed, signals from the individual elements are often recorded in sequence (multiplexing), but this in itself complicates the circuitry and at best is merely a trade-off of time resolution for space resolution. If displacement of the individual elements is the quantity to be sensed, interferometric holography23 offers an alternative. The holographic method is described in Section 8.2.4 of Part 8 of this volume. Its basis is illustrated here by Fig. 17. Consider the three pairs of light sources indicated at the left of the figure to be coherent and equally intense. Because of interference, the intensity of the light I producing a particular image at F depends on the spacing A between the corre-

FIG.17. Schematic arrangement for reconstruction of image at three points ( 1 , 2 , 3 ) from double exposed hologram H. Hologram is illuminated by coherent light beam S which is the same as the reference beam used in making the hologram. Images are focused on photographic film F by lens L. Virtual objects are shown at left at equivalent large distances. See Fig. 28 and Section 5.9.3 for method of making hologram.

5.6.

55 I

SENSORS

sponding pair of sources. If the angle 0 between the rays producing the image and the axis of the optical system is small, and if the wavelength A is the same for the two interfering waves,

I = 210 cos2

(F+

$9).

(5.6.17)

where Zo is constant and $9 is a constant phase factor. The photographic densities of the images on a film placed at F thus provide a measure of A for each pair of sources. The sources shown in the figure are not real but are virtual sources formed by illuminating a composite hologram at H by light from S. The virtual sources indicated by full lines in the figure are images reconstructed from one hologram, and those indicated by dashed lines are due to a second hologram made at a different time. The two holograms are superposed on the same film by double exposure so that reconstruction from the composite contains both sets of virtual sources; one set provides reference positions from which to measure displacements of the other. Description of an arrangement for making the hologram, together with an example of a reconstruction, is contained in Section 5.9.3. By using a classical interferometer, described in Chapter 2.4, Section 2.4.1 of Part 2 of this volume, the intermediate step of making the hologram can be eliminated.22 But the classical interferometer has the disadvantage that it requires optical components, including the surfaces whose displacements are to be measured of extremely high precision (defects no greater than a small fraction of A). On the other hand, in the holographic method phase shifts due to distortions resulting from poor optics are practically the same for the two holograms and therefore cancel out on reconstruction from the composite. Another special requirement of some experiments is an extremely short response time, in some cases much shorter than a microsecond. Interferometric sensors both of displacement and of velocity have been developed with response times in this range. These employ photodetectors. For descriptions see Section 5.10.6.2 and a review article by Barker.54 These are called chrono-interferometers because, in contrast to the more common type on which the holographic interferometer is based, they measure movement at a particular place or small region in space mm2) as a function of time. A moving film rather than a photodetector can be used with some interferometric arrangements. The interferometer is adjusted to form straight fringes which are focused on and at right angles to a slit so that a change in optical path length of one of the interfering beams causes a shift in all the 54

L. M. Barker, Exp. Mech. 12, 209 (1972).

552

5 . MEASUREMENT OF PRESSURE

fringes along the slit. The shift is recorded on film traveling perpendicularly to the slit. Cinematographic methods are also available for measuring displacements of objects moving at speeds requiring nanosecond framing rates (See Section 8.2.2.3 in Chapter 8.2 of this volume). While quite elaborate, these can provide both spatial and temporal resolution over limited space-time regions.

5.7. Pressure -Ti me Recording In experimental work, when the manometer or elastic element and the sensor have been chosen, assembled and tested for determining pressure as a function of space and time, an additional important requirement is the reliable recording of results. Many ingenious methods have been devised; we will survey several but omit many details. Many other clever methods and modifications will be omitted. Ease of operation, reliability and cost are changing so rapidly with the advent of miniature solid state electronic devices that our survey cannot be up-to-date even at the time of writing. 5.7.1. Nonelectrical Recording

Mechanical coupling of an inked penpoint to a deformation gage allows it to record on moving paper driven by clockwork. The mechanical requirements are not much different from those required to move a pointer on a dial for visual display. Such gages, once used widely for recording atmospheric pressure and biological pressures, are superceded by more reliable electrical recording methods. Banks of liquid manometers connected by tubing to wall taps in the surface of a model in a wind tunnel have been photographed as a method of recording pressure as the model’s angle of attack, or the tunnel’s speed, changes. Sequences of photographs formed a convenient record of a great deal of data in wind tunnel installations in earlier years, but electrical recording has replaced this method also. Optical sensors described in Section 5.6.3 may be used with photographic recording. When diaphragm deflection is detected by forming interference fringes, the interferometer may be adjusted so that fringes are focused on and at right angles to a slit in front of the moving film, so that fringe-shift versus time is displayed graphically on the developed film. As described in Section 5.6.3, holographic interferometry records information on the deflections of diaphragms distributed over space,

5.7.

PRESSURE-TIME RECORDING

553

5.7.2. Electrical Signal Recording At the present stage of technical development, once the quantity of data recorded exceeds pencil and notebook feasibility, practically all recording is electrical, or electrically facilitated to the stage of obtaining magnetic tape, “hard copy” of an oscilloscope display by photography, or graphical display by xerography or computer plotter actuating pen motion on paper. Once the pressure transducer has delivered the electrical signal, there is nothing unique to pressure measurement in the techniques of signal processing, display and recording. Acquaintance with the terminology and familiarity with the processing and recording units is necessary for one to be able to buy or use the electronic equipment, but it would be out of place to describe the principles here. Manufacturers’ literature and instruction manuals constitute the most common source of education and information, but there are useful textbookP where the terminology and principles of instrumentation are presented. Some examples of useful applications are given in the remainder of this section.

5.7.2.1. Analog Display on Oscilloscope with Linear Sweep. Suitable amplifiers are often available in a bench oscilloscope to display the voltage generated by a piezoelectric sensor or a bridge output from a resistance, capacitance, or inductance sensor. Sweep triggering may be obtained from the signal itself, but an external trigger may be needed to establish a time reference. Oscilloscope cameras are usually offered for sale by the oscilloscope manufacturers. Attention needs to be given to spurious electrical pickup (it may be reduced by use of shielded cable), cable and input capacitance, noise generation in high gain amplifiers, time response of oscilloscope (frequency response), linearity, trace brightness for sweep rate used, and ease of time and deflection calibration. Probably the chief drawback is the limited time of display when sufficient time detail is present. A raster displuy can increase the duration by a factor of 10 or so with accompanying decrease in accuracy of measurement of signal amplitude and danger of overlapping displays. Usually an auxiliary unit must be built or purchased to provide the raster display for a bench oscilloscope. An oscilloscope screen storage mode is advantageous for saving film for the camera and set-up time when the single sweep mode is employed and preliminary adjustments need to be made. The digital memory oscilloscopes provide this along with other advantages, however (see below). Achievement of better than 5 percent accuracy is difficult with analog display on an oscilloscope screen. 55 A . J. Diefenderfer, “Principles of Electronic Instrumentation,” 2nd ed. Saunders, Philadelphia, Pennsylvania, 1979.

554

5.

MEASUREMENT OF PRESSURE

5.7.2.2. Photography of Oscilloscope Spot on Moving Film. The difficulty of limited length of display has been overcome in some laboratories by providing mechanically for sweep. The image is projected by a rotating mirror on a stationary circular loop of film, or directly on to moving film. The film may be on the inside or outside of a rotating drum, or it may be driven by sprockets from one rotating spool to another. By using both x- and y-deflections of the oscilloscope trace, a raster may be recorded on the moving film and time detail at 0.1 ps is displayed for intervals of tens or hundreds of milliseconds. A disadvantage is that the record is not available until the film has been developed, which is inconvenient when many trial adjustments are needed. 5.7.2.3. Digital Memory Bank. Integrated circuit and microprocessor chips promise to make oscilloscope and photographic recording obsolete. Improvements in miniaturization, accuracy, reliability, time resolution, and cost are achieved; long time storage is provided by magnetic tape and magnetic disks, but permanent storage on integrated circuit chips may become even more advantageous. As an example of the current state of the art, a digital memory oscilloscope-not much more costly than analog display instruments -will be described.56 The analog signal, full scale 0.1 V, is sampled every 0.5 ps, and the size measured and converted to a 12 bit binary word (4096 fineness) and the words for 4000 successive time intervals are stored in a digital memory. Until the oscilloscope is triggered, these data are displayed once and continuously replaced by the next 4000 intervals sampled. The trigger “freezes” the memory and the last data sample is preserved. If desired, one may choose to save some data preceding the trigger in time. The captured data can be displayed in up to 64 times the detail seen on the original unexpanded sweep. Since the sweep time is recorded in 4000 individual intervals, the signal can be displayed with a fineness of 1/4000 of the sweep interval; the accuracy of the displayed time, and of the voltage, is 1/400 of the full range. All the data, or the portion it is desired to keep, may be transferred to an analog x - y recorder or digitally to magnetic tape, or the oscilloscope screen may be photographed with the best choice of magnification as the data are repetitively displayed. Panel controls permit more liesurely sampling times (up to many seconds per point) and larger full scale voltage ranges may be chosen before recording; these provide longer total recording time and capacity for recording the largest voltage signal expected from the transducer. Manufacturers provide separate units-analog-to-digital converter (ADC), oscilloscope display, enlarged memory bank, disk and tape 56 “Nicolet Explorer Digital Oscilloscopes,” catalog of Nicolet Instrument Corporation, 5225 Verona Rd., Madison, Wisconsin 53711.

5.8.

DYNAMIC CALlBRATlON

555

recorders-which are compatible with general purpose minicomputers so that data acquisition can be programmed and the resulting data analyzed and correlated. ADC’s with 0.01 ps sampling interval are currently available; few if any transducers can make use of this time resolution. Also none can be calibrated so accurately as to justify the 1/4096 sensitivity of ADC’s, but this sensitivity is quite useful because often the maximum voltage output is not known before recording and a margin of safety can be indulged in without fear of recording too small a signal.

5.7.3. Multiplexing When many pressure (and other) transducers are sensing at the same time, for example many pressure taps in the surface of a wind tunnel model, use of many separate recording channels becomes expensive in space, money, and repair time. A recording system with more rapid response than needed to record single channels may be used to sample and display the multiple outputs in turn. The fast ADC’s mentioned in the preceding paragraph have been used to record pressures from multiple wall-taps by leading the many tubes from the taps via a mechanically rotating valve to a single pressure transduceF so that only one recording channel is needed.

5.8. Dynamic Calibration Calibration of gages for measuring steady and slowly varying pressures was considered in Chapter 5.2. In Section 5.5.1 it was pointed out that the ability of a gage to measure time varying pressures can be judged either from its frequency response function or from its response to a specified nonperiodic input. Correspondingly, there are two general types of dynamic calibration. In one type, carried out in the frequency domain, the amplitude and phase of a periodic output due to a “steady state,” ideally sinusoidal, input is measured as a function of frequency. In the other type, which provides results in the real time domain, the nonperiodic input used is nearly always taken to be a step function, Frequency domain calibrators commonly have a cavity or chamber, in whose walls are placed the gage to be calibrated and usually a monitoring gage, and a source for producing and maintaining pressure oscillations within the contained fluid. A large number of quite different arrangements is possible.31 In one type of arrangement the cavity is excited so that it operates in one of its resonant modes, usually the one of lowest frequency. The exciting driver-a siren, a piezoelectric stack, an electro-

556

5 . MEASUREMENT

OF PRESSURE

statically driven diaphragm, rotating jet-has its frequency tuned to the resonant frequency of the cavity. The frequency is changed by altering the dimensions of the cavity or by using a fluid with a different sound speed. Other arrangements operate in a cavity but at nonresonant frequencies. In some nonresonant calibrators a fixed mass of fluid is periodically compressed, in others, the flow of fluid through a cavity is modulated causing a periodic pressure change. The compression or flow modulation is carried out by the same kinds of periodically driven devices as used with resonant cavity calibrators. An advantage of the nonresonant over the resonant calibrator is that it is easier to change the frequency. The upper frequency limit for the nonresonant calibrator is determined by the onset of strong standing waves within the cavity. For both resonant and nonresonant types, smaller cavities will produce higher frequency limits as will an anechoic configuration. With few exceptions, such as provided by a reciprocity ~ a l i b r a t o r ’ and ~ - ~certain ~~~~ electrostatic actuators and pistonphones, frequency domain calibrators require a reference gage, whose sensitivity has in turn been related to static measurements made by such basic gages as the manometer and piston and cylinder described in Chapter 5 . 2 , to determine the input pressure to the gage being calibrated. A major problem of frequency domain calibrators is the requirement that they produce essentially sinusoidal pressure variations. Because of the inherent nonlinearity of fluid dynamic phenomena, sinusoidal variations have been possible only when the peak-to-peak variation is no more than a few percent of the ambient pressure and the frequency is less than a few tens of kilohertz.31 Consequently, the use of frequency domain calibrators has been limited mainly to gages designed for relatively low pressure level applications in the audio and near audio frequency range. Real time domain calibrators are essential for fast response gages designed to measure transient pressure changes and, with few exceptions, the type of pressure change employed is a step function. Subjecting a gage to a step function pressure change provides a test condition which is either similar to or more severe than encountered in most transient pressure applications. Furthermore, as pointed out in Section 5.5.1, even in applications where the shape of the pressure-time event to be recorded is quite different from a step function-a short rectangular pulse at one extreme, or a gradually rising pressure at the other extreme-the response of a gage can be predicted from a knowledge of its response to a step function by using a superposition integral, such as given by Eq. (5.5.2) of Section 5.5.1. 57

L. L. Beranek, “Acoustic Measurements.” Wiley , New York, 1949.

5.8.

DYNAMIC CALIBRATION

557

A linear shock tube, suitably instrumented for measuring shock strengths, provides an excellent test and calibration facility. Equations pertinent to the design of shock tubes and to the prediction of values for the parameters (pressure, temperature, density, velocity, etc.) of flow induced by a one-dimensional shock wave are contained in numerous arThe gage to be tested and calibrated can be ticles and mounted either in the end plate from which the shock reflects or in the side wall of the channel, with the wall mount having the obvious disadvantage that it takes the shock front a finite time to cross the sensitive face of the gage. Further details are in Chapter 9.2 of this volume. Important gage characteristics were considered in Chapter 5.5. For a gage designed to measure transient pressures they are: (a) its useful pressure range and sensitivity; (b) its hold time T , during which measurements can be made; and (c) its response time T and its freedom from spurious oscillations. The time dependent characteristics (b) and (c) can be determined from records made with shocks of unknown strength, but obviously the pressure increase across the applied shock must be known in order to measure the gage’s sensitivity. The pressure increase across the applied shock is best measured directly with a gage which has been calibrated at steady pressures and is mounted in the wall of the channel close to the gage under test. Although the gage for measuring shock strength should be accurate, have long term stability and have a long enough hold time so that it can be calibrated with essentially steady pressures which in turn can be measured with a manometer or piston and cylinder gage (Chapter 5.2), its response time need not be extremely short, nor are rapidly damped spurious oscillations particularly deleterious, since the pressure behind the shock can usually be arranged to be essentially constant for several hundred microseconds. A diaphragm gage, Section 5.9.3, which has been specially designed for accuracy and stability and which has a relatively short response time, or a stub gage, Section 5.10.3, such as pictured in Fig. 33, are good gages for shock strength determination following calibration at essentially steady pressures. Lacking a suitable gage for measuring shock strength directly, a reasonable value can be calculated under the assumption of ideal gas behavior from a knowledge of the temperature and composition and thus 58 R. I. Soloukhin, “Shock Waves and Detonations in Gases.” Mono Book Corp., Baltimore, Maryland, 1966. 5@ A . R . Hartunian, in “Methods of Experimental Physics” (L. Marton, ed.), Vol. 7, Part B, Chapter 7. Academic Press, New York, 1968. 6o D. E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed., p. 2-275. McGraw-Hill, New York, 1972. W. Bleakney and R. J . Emrich, The shock tube. “High Speed Aerodynamics and Jet Propulsion,” Vol. 8, Artic. J. Princeton Univ. Press, Princeton, New Jersey, 1961.

558

5.

MEASUREMENT OF PRESSURE

2

-3

-~4 -!(a)

FIG. 18. Mechanical devices for calibrating transient pressure gages. (a) Rapidly operating valve. I: test gage; 2: gas seal acting as valve; 3: pressurized chamber; 4: falling weight which breaks seal. (b) Spring loading. 1 : test gage; 2: compressed spring; 3: support wire; 4: wire cutter or electrically initiated explosion; 5: known weight reduces force applied to gage by spring until wire is broken.

of the sound speed and the heat capacities of the gas and either measurements of the initial pressures in the driver and channel sections (Eq. (5.2.4), Ref. 59) or measurements of the shock velocity and initial pressure in the channel (Eq. (7.3.7), Ref. 59). Even if the shock strength is determined indirectly, the accumulated error in calibration of a test gage with a shock tube is usually less than 5 percent. If a higher pressure step than that produced in a shock tube is required, a fast acting mechanical device similar to one of those shown in Fig. 18 can be used for c a l i b r a t i ~ n . ~ However, ~ - ~ ~ . ~ ~such devices do not apply pressure to the gage as rapidly as a shock, so they are unsuitable for measuring microsecond response times and usually provide a poor test of high frequency vibrational characteristics of the gage. The apparatus shown in Fig. 18a employs a falling weight to break a seal allowing pressure on the gage to build up to a known value. The pressure applied to the gage increases to the known value in approximately 70 ps. In the arrangement of Fig. 18b, a force is applied to the gage by a compressed spring. This force is reduced by a suspended body of known weight. When the suspending wire is broken, the force on the gage increases by an amount equal to the weight of the body. This increase occurs over the finite time required to break the wire completely.

'* G . D. Salamandra,T. V . Bazhenova, S. G . Zaitsev, R. I. Soloukhin, I. M. Naboko, and I. K . Sevast'yanova, "Nekotorye Metody Issledovaniya Bystroprotekayushchikh Processov." USSR Acad. Sci. Press, Moscow, 1960 (in Russian). K . Haider, H . Holtbecker, and E. Jorzik, J . Phys. E 3,945 (1970).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDlNG A N D STRETCHING

559

5.9 Diaphragm Gages: Strain by Bending and Stretching Some gages, such as a Bourdon type or those employing a bellows, have elastic elements of quite complicated shape which respond by bending and stretching. These were described in Section 5.2.2 and will not be considered further. Their use is restricted to the measurement of essentially steady pressures. The theory of the steady deflection of elements having complicated shapes is well covered by Andreeva.8 The following considerations deal only with simple diaphragms which are initially flat. 5.9.1. Theory of Diaphragm Deflection

Figure 19 shows the deflection of a clamped diaphragm produced by the application of a uniform pressure. Even when the time dependence of the behavior is not considered, the general problem of predicting the deflection due to a known pressure is nonlinear and cannot be solved exactly. Andreeva8 provides an excellent review of the static problem including approximate equations which are adequate for the design of slow response gages. A comparable review of the dynamic problem is not known, but aspects of the problem are covered by Den Hartoge4 and Timoshenko. 85 5.9.1.l.Static Behavior. As Andreeva8 points out, a useful practical solution to the general problem of predicting the deflection of a simple diaphragm due to a known applied pressure can be obtained by adding the solutions for three idealized situations. With the diaphragm clamped in a perfectly rigid mount, as in Fig. 19, the idealized situations are: (1) the diaphragm is initially unstressed and the maximum deflection is much less than the thickness of the diaphragm (pure bending), (2) the diaphragm is initially unstressed and the maximum deflection is much greater than the thickness of the diaphragm (pure stretching), (3) the diaphragm is prestressed and held by clamping so that it is initially in tension and the initial tension is of sufficient amount so that it is not effectively increased by subsequent stretching (constant tension). (Situation (3) is often referred to as the “membrane” problem.) Equation (5.9.1) relates the maximum deflection w oat the center of the diaphragm to the applied pressure p ; the first, second, and third terms on the left-hand side represent, respectively, J. P. Den Hartog, “Mechanical Vibrations.” McGraw-Hill, New York, 1940. S. Timoshenko, “Vibration Problems in Engineering,” 3rd ed. Van NostrandReinhold, New York, 1955. B(

5 . MEASUREMENT OF

5 60

PRESSURE

Section Circular Diaphragm

m 1 r

+u-i

-a--

.i

X E h

Section Slit Diaphragm

FIG.19. Diaphragm clamped at the boundary where r

= u, w = u =

0, and dw/dr = 0.

the contributions due to pure bending, pure stretching, and initial tension.

.(y)+.(y)”+ c@)(?)

=(a>’$.

(5.9.1)

In this equation (J is the minimum half width of the diaphragm (half width of shorter side of rectangle, radius of circle), h is its thickness, Y is Young’s modulus and uo is the amount by which the diaphragm is stretched before clamping. The dimensionless coefficients A , B , and C depend only on Poisson’s ratio and on the shape of the diaphragm; values range between 1 and 8. Expressions for A , B , C, and uo/a are given in Table IV for the extreme shapes of a circle and an infinitely long rectangular slit. As Eq. (5.9.1) indicates, wo does not in general increase linearly with p . As p increases, wo/h increases, the second term on the left hand side of the equation becomes relatively more important and for sufficiently large values of p the second term dominates. Gages can be designed, however, to operate in an essentially linear range. For a prescribed maximum value of p and with no initial tension, a / h can be made small enough so that the first term on the left of Eq. (5.9.1) dominates and the problem reduces to the idealization of pure bending. TABLEIV. Diaphragm Deflection Coefficients” Diaphragm shape Circle Slit

B

C

(7 - v)

4 ( 1 - v)

A 16 3(1 - 9)

3(1

- V)

4 __3(1 - v’)

uda

2

(1

-

(1

- v ’UOy U O

4)

(1 - 3’-j7

v , Poisson’s ratio; Y , Young’s modulus; N o , prestretch; u,,, initial tensile stress (hue = tension per unit length applied at boundary).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING AND STRETCHING

561

We shall refer to a diaphragm operating in this idealized way as a “linear bending diaphragm.” The displacement w in the case of linear bending is given in terms of the distance from the center of the diaphragm r by

wh (L) A =

(574

h

($) (1 - ! J

(5.9.2)

In terms of the maximum displacement, the sensitivity, .Yo= dwo/dp, is 9 0

=

(3(;)3

(5.9.3)

For some gages operating in the linear bending range, strain is sensed rather than displacement. For a circular diaphragm, the strain components at a surface are E,

= 2

-g)

($)(I

2);();(

(5.9.4)

and Ee

=

a 2 p * (2;i(x) ) (T) (1

-

f).

where E , is the radial strain and E~ is orthogonal to E,. The plus and minus signs apply to opposite faces and indicate, for a particular position, that if one face is in tension, the other is in compression. For a slit diaphragm, there is no strain along the length of the slit, but the strain across the slit has the same dependence as E, of Eq. (5.9.4). The strain sensitivity for E , , .Ye = de,/dp, is, for either a circular or slit diaphragm, YE=

*

(&)(;y

(1

-

g).

(5.9.5)

For a bridge circuit (see Fig. 15), it is useful to have two sensors whose response is the same in magnitude but opposite in sign. As indicated by Eq. ( 5 . 9 . 9 , this can be accomplished for sensors responding to E, by placing them at the same position ( r = 0) but on opposite faces. This has the disadvantage that the sensor on the surface where the fluid pressure is measured is more subject to damage and to sudden temperature changes of the fluid. They can both be on the protected surface if they are small and one is positioned at Y = 0 and the other at Y = (2/3)%. Instead of operating in the linear bending range, many gages use a highly pretensioned diaphragm. The advantage of using prestress, in addition to making linear behavior easier to achieve, is that it reduces erratic

5.

562

MEASUREMENT OF PRESSURE

response due to intitial unevenness of the diaphragm and to residual stresses, which are difficult to eliminate in the manufacture of thin diaphragms. With sufficiently large prestress, the third term on the left-hand side of Eq. (5.9.1)dominates and the problem reduces to the idealization of constant tension, For this case, the displacement is

w =

h 1 a z p c1 (6)(;)3(;) ( 1 - 5) c,(7;) ( a , ) ( l - f). (5.9.6) =

where Co has a value of 4 for a circle and 2 for a slit. The sensitivity in terms of maximum displacement is

(5.9.7) 5.9.1.2. Dynamic Behavior. Following are expressions for the frequency response of a diaphragm (%To),Section 5.5.1). Specifically, the expressions give the amplitude, w(w, r), of the oscillating displacement produced by a sinusoidally varying pressure, p = e’”‘, of unit amplitude and frequency w . For a linear bending circular diaphragm

where K = (w/coh)”’, c%= Y/[12p(l - v’)], Jo(x) and Jl(x) are first-kind Bessel functions of zeroth and first order and Zo(x) and Z,(x) are the corresponding Bessel functions with imaginary arguments. Other symbols have the same meaning as for static loading. For the case of a prestressed circular diaphragm having constant tension

(5.9.9) where ct = a o / p . These transform to equations for a slit diaphragm by replacement of the Bessel functions with analogous trigonometric functions. For a linear bending slit diaphragm sinh(Ka) cos(Kr) + sin(Ka) cosh(Kr) - I] sinh(Ka) cos(Ka) + sin(Ka) cosh(Ka)

(5.9.10)

and for a prestressed slit diaphragm - 11.

(5.9.11)

While a slit is not practical, the slit and the circle are extreme shapes. These expressions are standing wave representations of the dia-

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

563

phragm's contour. For small frequencies they are the same as the contours produced by static loading, i.e., for small o, Eqs. (5.9.8) and (5.9.lo), when multiplied by p , reduce to Eq. (5.9.2) and Eqs. (5.9.9)and (5.9.11), when multiplied by p , reduce to Eq. (5.9.6). As the frequency increases the contour distorts from the shape produced by a steady pressure and the amplitude increases until resonance occurs. At still higher frequencies other resonances appear and the diaphragm vibrates in progressively more complicated modes. For measuring periodic variations, the frequency of the applied pressure should be well below the lowest resonant frequency of the diaphragm, in which case Eq. (5.9.2) or Eq. (5.9.6) with p = poebt, will provide a good approximation for the displacement. For nonperiodic pressure changes, response to a step function provides a better guide to behavior ( U ( t ) ,Section 5.5.1). A formal expression for the response to a step can be obtained by use of the inversion integral, Eq. (5.5.1) of Section 5.5.1. Use i/o (transform of unit step) for 9(o)and w ( o , r) from Eqs. (5.9.8), (5.9.5), (5.9.lo), or (5.9.11) for %lo). The inversion integral can be reduced to a more useable form by use of the Cauchy residue theorem. The form of the resulting expression for the displacement, W ( Y , t), depends upon whether the integrand of the inversion integral is viewed as being composed of traveling waves or standing waves. The traveling wave representation is easier to visualize in cases, such as that of the prestressed diaphragm, where the phase velocity of the waves is independent of the frequency (no dispersion) but is more complicated when it varies with frequency (dispersion), as in the case of the linear bending diaphragm. For consistency and ease of comparison the standing wave representation is used below for the prestressed as well as the linear bending diaphragm. The displacement due to a unit step applied at time t = 0 is given by Eq. (5.9.12)for linear bending circular diaphragm, by Eq. (5.9.13)for a prestressed circular diaphragm, by Eq. (5.9.14) for a linear bending slit diaphragm and by Eq. (5.9.15)for a prestressed slit diaphragm.* * The expressions of Eqs. (5.9.12) through (5.9.15) as well as those of Eqs. (5.9.8) through (5.9.11) are solutions to the following basic differential equations. Lineur bending diuphragm:

where the meanings of the symbols are the same as in Eqs. (5.9.8), (5.9.10). (5.9.12) and (5.9.14).

5.

5 64

MEASUREMENT OF PRESSURE

(5.9.12a) where K, = ( w , / ~ , h )is~ given /~ by J,(K,a)/Jo(K,u) = - z l ( ~ , a ) / ~ o ( ~ & ) ,

(5.9.12b)

where wq is given by Jo(w,a/ct) = 0, or approximately, wq

(5.9.13b)

= (Y -

c

2 tan(K,a) cos(K,r) pc;h3 q=l (aK5,) [cos(K,a)

w ( r , 1) = -

-

cosh(Kqr)] [l cosh(K,a)

-

cos(w,t)], (5.9.14a)

where K,

given by

= (w,/coh)1/2is

tan(K,a) = - tanh(K,a) or approximately, K,a

I :

(5.9.14b)

(4q - 1 ) ~ / 4 ;

where w, is given by cos(w,a/cl) 0,

=

0 or

= (2q -

1)7rct/2a.

(5.9.15b)

In all cases q is a running positive integer. Other symbols are the same as in Eqs. (5.9.8), (5.9.9), (5.9.10) and (5.9.11). The effect of applying a step function load is to produce a periodic moPrestressed diaphragm:

where the meanings of the symbols are the same as in Eqs. (5.9.9), (.5.9.11),(5.9.13) and (5.9.15).

For all equations p ( t ) represents the applied time-varying pressure. For nonvarying pressures, p(t) is constant and azw/dtz is zero.

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

56.5

Time

Position, r Position

Time, t

FIG.20. Prestressed membrane shape dependence with time and time dependence of displacement. (a) Contour at 118 period intervals following step function loading. (b) Time dependence of displacement at particular positions following step function loading. Full lines are exact; dashed lines are contributions of standing wave of lowest freqeuncy. c: = u o / p ;9, = 4a/c,.

tion which is expressed in Eqs. (5.9.12), ( 5 . 9 . 1 3 ) , (5.9.14), or (5.9.15) as a Fourier series whose components are resonant oscillations. The time average of the motion, about which the oscillations are centered, is the displacement that would be produced by a constant pressure having the magnitude of the step.* A good approximation to the exact motion is ob~~

~~

* To obtain displacements due to load p (5.9.13), (59.14). and (5.9.15) hypo.

= poS(t)multiply expressions of

Eqs. (5.9.12),

5 . MEASUREMENT

566

OF PRESSURE

tained by neglecting all terms of the series except the one having the lowest frequency. To illustrate these points, consider the effect of applying a step load to a prestressed slit diaphragm, Eq. (5.9. 15a), noting first that the characteristic frequencies, wq of Eq. (5.9.15b), are the same as the resonant frequencies at which the frequency function, Eq. (5.9.11) becomes infinite. Figure 20a shows the contour of the diaphragm at intervals of one-eighth the longest resonant period and Fig. 20b pictures the time variation of the displacement at several positions. Considering the full-line curves, which represent exact values for the displacement, the central contour of Fig. 20a at time a/ct and the time averages of the curves of Fig. 20b are the same as for static loading, Eq. (5.9.6). The period of motion, g1= 27r/wl = 4a/c,, corresponds to the lowest resonant frequency of Eq. (5.9.15b) and, for this simple case where there is no dispersion, is the time for a wave to travel back and forth across the diaphragm. The dashed curves of Fig. 20a and Fig. 20b were calculated using only the first term of Eq. (5.9.15a), i.e. the term with the lowest characteristic frequency. The small differences between the dashed and full-line curves indicate that the first term is dominant and suggest the following as a good approximation for Eq. (5.9.15a): w(r, t ) = Y p o [ l - c o s ( 2 ~ t / 9 ~ ) ] ,

(5.9.16)

where Yis sensitivity, i.e., displacement at position r due to unit constant pressure, po is the magnitude of the applied pressure step and 9,is the longest resonant period. With appropriate values for 9'and PI, Eq. (5.9.16) provides a good approximation for Eqs. (5.9.12), (5.9.13), and (5.9.14) as well as Eq. TABLEV . Parameters of Approximate Expression for Displacement of Diaphragm Produced by Step Function Loading" ~~

Conditions

Y

9 1 ~~

Linear bending, circular diaphragm Prestressed, circular diaphragm Linear bending, slit diaphragm Prestressed slit diaphragm a

[3( 1; 1 ] [$1 [ Y')

[7][$J[1 (1 - 3)

~~

- f] '

-5j'

~~~

~~

Derived from exact Eq. No.

~~~

~~

az

0.615 -

coh

2.612

ct

1.123- a2 coh 4'

~~~

(5.9.12)

(5.9.13) (5.9.14) (5.9.15)

Cl

Y (sensitivity) and ?PI (longest resonant period) are parameters to use in Eq. (5.9.16).

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

567

(5.9.15). Expressions for Y and Pl for the four cases treated here are given in Table V. For a linear bending diaphragm the expression for Y comes from Eq. (5.9.2) and for a prestressed diaphragm it comes from Eq. (5.9.6). Expressions for the longest resonant period Pl = 27r/ol come from Eqs. (5.9.12b), (5.9.13b), (5.9.14b), and (5.9.15b). 5.9.2. Response Characteristics of a Diaphragm Gage Knowing the response to a unit step function load, the time dependent behavior of a diaphragm for any type of variable loading can readily be determined by the superposition integral Eq. (5.5.2) of Section 5.5.1. A good approximation for U(t - Y), the response at time t to unit step applied at time 9, is given by Eq. (5.9.16) of Section 5.9.1.2. (5.9.17) To illustrate the response of a diaphragm to a less precipitous loading than given by a step function, suppose the applied pressure, of Eq. (5.5.2) of Section 5.5.1, increases as a sine function for a quarter of some period Po,and subsequently remains constant:

Z(n

zcn =

po sin(2.rrT/Po)

for 0 < 9 c P0/4 for P0/4 < T < m

.

(5.9.18)

Equations (5.9.17) and (5.9.18) have been used in Eq. (5.5.2) of Section 5.5.1 to develop the graphs of Fig. 21. Figure 21 shows the response of a diaphragm gage for three values of Po (four times the rise time of the load) relative to PI (the longest resonant period of the gage). The three values of Poare (1) Po= 0, i.e., the load is a step function, (2) Po= 4P1, i.e., the rise time of the load is equal to the longest resonant period of the gage and (3) Po= lop1, i.e., the rise time of the load is 2.5 times the longest resonant period of the gage. As the graphs indicate, the size of the spurious oscillations in the response becomes smaller as goincreases relative to Pl. For Pogreater than 10P1 the spurious oscillations are for practical purposes inconsequential. This behavior is consistent with a diaphragm's frequency response [Eqs. (5.9.8)-(5.9.1 l)]. For a sustained sinusoidally varying load with period Po,the response varies sinusoidally with period Pobut the amplitude is a function of YO/Pl. The amplitude is virtually infinite at resonance (90/P1= 1) and decreases as Po/Pl increases. For Pogreater than a few multiples of B1 the amplitude is essentially independent of Po/Plwith a value determined by the steady state sensitivity. These two cases demonstrate a general rule of thumb. Provided no im-

5.

568

MEASUREMENT OF PRESSURE

2

FIG.21. Decrease in amplitude of resonant oscillations with increase in risetime Poof apis largest resplied pressure. 1: step function load, 9 0 = 0; 2: Po= 4P1 ; 3: go= 109,. 9, onant period of gage.

portant pressure change occurs in a time interval less than a few resonant periods of the gage, the response at each instant is practically equal to the applied pressure multiplied by the gage's steady state sensitivity. Sensitivity and longest resonant period are clearly important characteristics of a diaphragm, particularly since they are parameters of Eqs. (5.9.16) and (5.9.17). However, since step function loading produces large oscillations, response time is a vague concept and its exact definition is arbitrary. For purposes of comparison with other types of gage, the response time of a diaphragm gage will be considered to be its longest resonant period. Except for limitations set by the sensor or recording circuitry, the hold time of all diaphragm gages is practically infinite. To simplify the interpretation of records, resonant oscillations are sometimes reduced by adding mechanical damping or by using an electrical filter. These modifications do not, however, improve a gage's ability to measure rapid pressure changes. Real improvement is obtained only by shortening the resonant period. Of all shapes, a circular diaphragm has, for the same value of the smallest dimension a , the shortest resonant period and in practice most diaphragms have been made with this shape. For all gages the resonant period is inversely proportional to ct or co and thus decreases as these increase. See Table V. In the case of the prestressed diaphragm, where ct = ( ~ ~ / p ) the l ' ~resonant , period decreases slowly as the prestress uois increased, with an ultimate limit set by the condition that cro must be less than the material's yield stress, which is nominally at least two orders of

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

569

magnitude less than Young’s modulus. In the case of a linear bending diaphragm, co = ( Y / [12( 1 - ~ ~ ) p ] ) lwhich ’ ~ , depends only on material properties. Both ct and co are several times smaller than the propagation velocity for compressional strain which is the type of distortion basic to the operation of most fast response gages. The remaining parameters are the dimensions, a and h. The resonant period decreases with a decrease in a and for an unprestressed diaphragm with an increase in h. Changing a parameter to decrease the resonant period always leads to a decrease in displacement sensitivity. See Table V. It is usually advantageous to sensitivity to make a / h as large as feasible since for a prestressed diaphragm an increase in this ratio increases the sensitivity but does not change the resonant period and for a linear bending diaphragm the sensitivity increases as the cube of the ratio whereas the resonant period increases only as the first power. For a given maximum pressure an upper limit on a / h is set by the fact that as the ratio is increased the gage response becomes nonlinear and eventually the diaphragm will stretch beyond yield. For similar diaphragms, having the same value of a / h , both the resonant period and the sensitivity decrease in direct proportion to a so that decreasing the resonant period by miniaturization is eventually limited by inadequate sensitivity. The following values for the resonant period are of the order of the smallest period obtainable with a diaphragm gage using present day technology. Miniature gages using a semiconductor sensing technique have been made with very small, unprestressed silicon diaphragms. Assuming Y = 110 GPa, p = 2400 kg m-“, Y = 0.3 and a circular shape with a = 1 mm, h = 25 pm, the longest resonant period is 12 ps, as calculated from 8, = 0.615 a2/c& [Table V]. A very small gage has also been made with a prestressed, stainless steel diaphragm. Assuming uo = (2/3) yield stress = 500 MPa, p = 7800 kg m-3 and a circular diaphragm with a = 1.25 mm, the expression 8,= 2.612 a/ct [Table V] predicts 13 ps for the longest resonant period. The theory on which these predictions depend presupposes no damping, a perfectly rigid stationary mount and direct contact between fluid and diaphragm at the position where the pressure is to be measured. Damping causes decay of the spurious oscillations, but also increases their period slightly. In practice, the assumption of a stationary mount is often not justified; there are often unwanted vibrations in the base to which the gage is attached. In this case either the diaphragm must in some way be isolated from the unwanted vibrations or a means of compensation must be employed. Also in practice, the diaphragm is sometimes mounted in a cavity which has a tube extending to the position of p. 31). This certainly increases the measurement (Bynum el

-

570

5.

MEASUREMENT OF PRESSURE

response time and may introduce spurious oscillations due to resonance of the fluid in the tube or cavity. Other characteristics, such as accuracy, temperature independence, etc. may be important in selecting a gage for a particular application but no recipe for an optimum choice can be given. Gages offered foresale by several manufacturers offer a variety of compromises. 5.9.3. Types of Diaphragm Gage

Treatises on high fidelity microphones and loudspeakers contain much that is useful in the design of gages for measuring pressure changes in the audio range.18*6s-67 (Figure 22a is a schematic of a diaphragm gage whose electromechanical system is similar to an induction microphone. Fig. 22b pictures a gage which is similar to a condenser microphone. Both types are shown with two sensors symmetrically placed on opposite sides of the diaphragm. The inductance type sensor is considered in Section 5.6.2.4. The particular form shown here consists of a wire coil of many turns linking a core of high permeability which, together with the diaphragm, forms a magnetic loop. Displacement of the diaphragm changes the reluctance of the magnetic circuit and therefore the self inductance of the c ~ i l . * In ~ *the ~~ other type of gage, the diaphragm serves as one plate of a capacitance sensor, such as considered in Section 5.6.2.2. When the diaphragm is displaced the capacitance of the sensor changes, due to change in plate ~ e p a r a t i o n . ’ ~Both * ~ ~ types of sensor are shown as parts of an alternating current bridge whose output is modulated by a change either in inductance or capacitance. With this arrangement measurement of constant as well as changing pressures is possible, the push-pull aspect of the bridge makes the response more linear than that of a single sensor and some temperature compensation is provided. Gages of best longterm stability must be built in a thermostated enclosure to assure freedom from ambient temperature effects. For gages of moderate size, the inductance type sensor can be made with relatively high electrical sensitivity and signal-to-noise ratio, but the capacitance type is simpler and can be made with a mechanically superior diaphragm since it is not restricted to materials of high permeability. In part because a duct or a tube leading to the cavity must be provided, gages of these types are ordinarily useful only for measuring pressure changes in the audio frequency range and lower. McGraw-Hill, BB G . P. Harnwell, “Principles of Electricity and Magnetism,” p. 464. New York, 1938. *‘D.E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed., Sect. 3. McGraw-Hill, New York, 1972.

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

571

A

I

(a I

I

I

(b)

FIG.22. Typical diaphragm gages with cavity and tubing connecting to region where pressure is to be measured. (a) Variable reluctance (inductance)type. (b) Variable capacitance type. Electrical connections to ac bridge provide output signal proportional to pressure change.

Gages with faster response are smaller and usually employ dc sensors and amplifying circuits. An inductance type sensor is not a viable candidate for use in a fast response gage because it is difficult to miniaturize and has unattractive features when operated with a direct current (Section 5.6.2.4). On the other hand, a moderately fast diaphragm gage has been made with a capacitance type sensor.2o A schematic diagram is shown in Fig. 23. The problem of scaling down this type of gage is not only that the sensitivity is decreased directly because the capacitance of the sensor becomes smaller but also, unless the parallel capacitance of the connecting cable is made smaller in proportion, because the fraction of the total capacitance which is independent of pressure becomes larger. Furthermore, as mentioned in Section 5.6.2.2, when the capacitance of the sensor becomes comparable to that of the cable, pick-up of electrical disturbances by the cable becomes important and adversely affects the signal-to-noise ratio. These difficulties associated with the cable were alleviated in the gage of Fig. 23 by placing a small electronic impedance matching unit next to the capacitor of the sensor. A highly prestressed membrane is built on 1

4

6

FIG.23. Miniature capacitance gage, overall diameter 2.5 mm. 1 : Pretensioned diaphragm of stainless steel 2.6 pm thick. 2: Metal backplate. 3: Metal collar for diaphragm support. 4: Conducting cement. 5: miniature coaxial cable connector. 6: to adjacent miniature preamplifier. [From S i d d ~ n . ~ ' ]

572

5.

MEASUREMENT OF PRESSURE

DIAPHRAGM THICKNESS

‘‘

pyjL

(not to %ale)

-

STRESS DISTRIBUTION

\-‘---

\

-&,

FIG.24. Distribution of radial component of strain in a circular linear bending diaphragm clamped at its perimeter. 1 : uniform thickness diaphragm. 2: optimally chosen profile to provide uniformly distributed strain. 3: positions of strain resistive elements.

the end of a coaxial connector directly coupled to a cathode follower. With a prestressed stainless steel diaphragm of 2.5-mm diameter and 0.0025-mm thickness, the gage provided an overall sensitivity of 10 V Pa-’. The calculated resonant period is 13 ps. As noted in Section 5.6.2.2, a capacitance sensor has a distinct advantage in that it does not generate heat. This is a necessity, for example, in most cryogenic experiments. In experiments with helium performed at temperatures near 0.013 K, pressures of 10 MPa have been measured with a detection capability of 20 Pa using a capacitance sensor in conjunction with Be-Cu and polymer diaphragms.‘* Another type of gage uses a linear bending diaphragm and one of a variety of resistance sensors which respond to strain rather than displacement [Eq. (5.9.5)l. In the simplest construction the resistance sensor consists of a single fine wire or thin film stick-on element. This is likely to present a bonding problem, particularly since the strain varies across a diaphragm of constant thickness. Semiconductor elements of single crystal silicon are useful strain elements because of their high sensitivity. They are however limited to strain levels of 0.001 and have correspondingly high temperature sensitivity. Figure 24 shows the distribution of strain over the face of a uniform-thickness diaphragm, and it is seen that the maximum strain is reached at the center while the strain elsewhere is smaller. In a special variable-thickness diaphragmse it is possible to obtain a uniform distribution of radial strain, and the strain produced in such a diaphragm is shown in Fig. 24 together with the positions of cemented-on strain elements. The strain in the central region is positive and near the border it is nega-

*

G . C. Straty and E. D. Adams, Rev. Sci. Instrum. 43, 394 (1972). E. A. Samoletov, “Aerofizicheskiye Issledovaniya,” Annu. Rep., Inst. Pure Appl. Mech., p. 38. Novosibirsk, 1972 (in Russian). dQ

5.9.

DIAPHRAGM GAGES: STRAIN B Y BENDING A N D STRETCHING

573

tive. Several pairs of sensors are cemented to the central and border regions to provide symmetry and averaging of residual nonuniformities in the gage construction. At strain levels of 0.001, electric signal magnitudes of AVIV, --- 0.1 ( V , is bridge supply voltage) are obtained with silicon crystal sensors each several millimeters in extent, and some temperature compensation for ambient changes is provided. In another method of construction better bonding of resistive elements is obtained by evaporating thin resistance films directly onto the diaphragm. The technique of placing pairs of resistive elements on various regions of uniform thickness diaphragms to increase sensitivity and to provide temperature compensation in the bridge connection is generally used; see Section 5.6.2.1 and Fig. 15. A third type of diaphragm gage using resistance sensors employs techniques developed for forming microcircuits on semiconductors. In one version a small area of a silicon chip 250 pm in thickness is etched from one side leaving on the other side a rectangular diaphragm 2.5 mm by 1.25 mm and only 25 pm thick. Working in a vacuum a second chip is bonded to the first so as to form behind the diaphragm a cavity in which the pressure is zero. A tiny resistance bridge is formed in the outer face of the diaphragm by diffusion of an N- or P-type material. The arrangement is shown schematically in Fig. 25. The bridge is centrally located and its resistance arms are identical. The rectangular diaphragm provides a net strain sensitivity, whereas the equal arm bridge compensates for equal temperature changes of the resistors. An independently connected minitransistor placed at the center of the diaphragm serves as a temperature sensor and can be used either as a thermometer or as a thermostat for further temperature compensation. The calculated value for the resonant period of a silicon diaphragm without prestress and having dimensions 2500 X 1200 X 25 pm3 is 17 ps. Gages of this type have been built with different diaphragm thicknesses for pressure ranges from 0- 100 kPa to 0-30 MPa. Using techniques developed for manufacturing integrated circuits, this gage can be mass produced relatively inexpensively .'O A gage which uses a thin polymer membrane both to supply a piezoelectric emf (Section 5.6.2.3) and to provide an array of small diaphragms, each acting as one plate of a variable capacitor, is shown in Fig. 26 (Bernstein," p. 46). The poled piezoelectric polymer membrane is metallized on one side and supported on the other side along a network of lines by a corrugated metal electrode. The electrical system thus consists of a parallel-connected array of small capacitors with an electrical source resulting from the piezoelectric property of the membrane. The electrodes National Semiconductor Corp., Santa Clara, California.

574

5.

MEASUREMENT OF PRESSURE

E!zk3 I - - - - - -

-7

FIG.26 FIG.25 FIG.25. Miniature diaphragm gage with resistance sensor made by semiconductor microcircuit technology. 1: silicon chips bonded together. 2: chip etched to provide diaphragm and evacuated cavity. 3: resistance bridge formed by diffusion of N or P type impurity. 4: diffused P-N junction transistor senses temperature. FIG.26. Diaphragm gage using piezoelectric source. 1: Metallized surface. 2: Poled piezoelectric polymer film. 3: Corrugated metal electrode and diaphragm support.

are connected to a field-effect transistor deposited on the back of the corrugated electrode. The effect of a change in pressure on the membrane is to change the input voltage to the transistor. (The voltage may change due to a change either in capacitance or in the piezoelectric effect. That is, in Eq. (5.6.10) of Section 5.6.2.3 there may be a change either in C, or in the integral.) The sensitivity of the transducer is about 50 pV Pa-' and the quoted band width 3-250 kHz (response time approximately 4 ps). Its attractive features are its fast response time and relative insensitivity to vibrations of the mount, due to low inertia of the membrane, as well as its small size, 5-mm diameter by 1-mm thick. It was designed for use on thin aerofoils and compressor blades. Figure 27 shows a gage which uses a silicon NPN planar transistor as a sensor. See last sensor described in Section 5.6.2.1. A diaphragm is mechanically coupled to a transistor by a small cylinder whose conical tip bears against the pressure sensitive emitter-base junction. Although this gage uses a diaphragm, it might be classified as a stub gage, which is considered in Section 5.10.3, since a change in pressure causes very little movement of the diaphragm. The dynamic behavior is probably determined as much by the size, shape and elastic properties of the coupling cylinder as by the diaphragm. A response time in the neighborhood of 10 ps has been reported for a gage of this type. In order to measure the pressure distribution in a flow about a model in a wind tunnel, use has been made of a large array of small diaphragm sensors whose displacements were simultaneously determined by the

5.9.

DIAPHRAGM GAGES: STRAIN BY BENDING A N D STRETCHING

575

I

FIG.27. Diaphragm gage using transistor as sensor. 1: Diaphragm. 2: Pressure pin. 3: Transistor with pressure sensitive emitter-base junction. [After "Pitran Pressure Transistors," Stow Laboratories Catdog, Hudson, Massachusetts.]

3

double exposure holographic method considered in Section 5.6.3 .23 The hologram used. for reconstruction of the diaphragm deflections, by the method illustrated in Fig. 17 on page 550, was made with the arrangement shown in Fig. 28. The flow and model Ware shown at the left of a rigid plate containing an array of chambers (1, 2, 3, 4,etc.) each 3 mm in diameter and connected to the flow through a hole having a diameter of 0.2 mm. A diaphragm was formed at the end (right) of each chamber by soldering a brass foil of thickness 0.05 mm to the rigid plate. To obtain a hologram, coherent light from a laser is divided into beams 1 and S by a beam splitter. A lens L1 causes beam I to diverge so that after reflection from a mirror it illuminates all diaphragms of the array. The diaphragms have been etched so that they diffusely reflect some of the light from I

1

I

1

Laser

-1

___

H

---

1 .

V...

.L.W

.WL.,LW..W"

V"..'..

Y

I V

y . Y U Y " W

u *."'"6.U

I...

,"W"

a x e .

I ,

1"s

..IWI.I"U

" 1

L I W V I I

structing picture of diaphragms to measure deflections.) W is a model of a wing in a wind tunnel flow.

576

5.

MEASUREMENT OF PRESSURE

FIG.29. Interferometric reconstruction from a double exposure hologram. The intensity variations over an array of diaphragms correspond to differences in deflections, mapping out pressure variations around an airfoil.

onto the film at H where it combines with the diverging light from the reference beam S to produce a hologram. For the reconstruction shown in Fig. 17 on page 550, the light beam S used for illumination must be the same relative to the hologram H as in the diverging reference beam S of Fig. 28. Actually, the virtual sources of Fig. 17, which correspond to diaphragm positions, occur in pairs and thus produce interference at F, because H is a double exposure hologram. The first exposure is made with no flow and the second during the flow under study. Fig. 29 is an interferogram obtained by reconstruction from a double exposure hologram made in a study of flow about an airfoil. Using light of wavelength 600 nm, the detection sensitivity is estimated as Ap,,,,,, = 500 Pa. This is ample for measuring pressure variations in this example, where the total pressure difference is 7000 Pa. Most gages described in this chapter are commercially available.40 They provide a variety of attractive features including the one that both constant and moderately rapid changes can be measured. Gages capable of measuring more rapid changes are considered in the following chapter.

5.10. Fast Response Gages: Compressional Strain The gages to be considered in this chapter differ from those of Chapter 5.9 primarily in that the elastic element deforms by compressional strain rather than by bending and stretching. Most of the gages were developed for pressure measurements in studies of detonation waves ,71-7* underW . Turetric, “Probleme der Detonation.” Berlin Akademie der Luftfahrtforschung, 1941. D. H. Edwards, J . Sci. Instrum. 35, 346 (1958). 73 D . H. Edwards, L. Davies, and T. R . Lawrence, J . Sri. Instrum. 41, 609 (1964). ” M. K . Mclntosh, J . Phys. E 4, 145 (1971).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

577

water blast waves," and gaseous shock waves,58~62~76-* as well as acoustic waves and boundary layer fluctuations.26*81*82 These phenomena often involve large pressures which last only a few milliseconds but the characteristic of main concern in this chapter is that pressure changes may occur within a range from a fraction of a microsecond to several microseconds. Fast response is required. 5.10.1. Basic Elements of Fast Response Gage

Although proposals for gages using compressional strain involve a wide variety of geometries, the gages all contain the same basic elements pictured in Fig. 30. The time dependent pressure" p ( t ) is applied to the surface of the strain element C. Usually X contains a strain sensor q ,although sometimes v' is omitted and instead the motion of the back surface (bc) of Z is recorded. In regions B and C there may or may not be materials which support the strain element. As the pressure p ( f ) changes, it produces strain which propagates (straight dot-dash line, F) with finite velocity into the strain element 8 . Either the strain or the motion of the back surface of 2 is translated into an electrical signal which is recorded as a function of time. Gage types having elastic elements of different shapes were designated by name in Chapter 5.4 and pictured in Fig. 12. In a dilatational gage the dimensions d and 1 in Fig. 30 are usually of the same order but more importantly the recorded strain or surface motion is considered only up to the time that the one-dimensional behavior due to the passage of the front F is complicated either by the arrival of relief strain (dashed curves) from the lateral boundary (ab-dc) between Z and C or by the arrival of a disturbing reflection from the surface (cb) between C and B or, when movement of the back surface (cd) is sensed, by the arrival of the first reflection from the front surface (ad). '5

R. H. Cole, "Underwater Explosions." Princeton Univ. Press, Princeton, New Jersey,

1948.

S. G. Zaitsev, Prib. Tekh. Exsp. 6, 97 (1958). W. W. Willmarth, Rev. Sci. Instrum. 29, 218 (1958). 'I8 R. I. Soloukhin, Prib. Tekh. Exsp. 3, 170 (1961). 7e K. W. Ragland and R. E. Cullen, Rev. Sci. Insfrum. 38, 740 (i967). M. I. Vorotnikova and R. I. Soloukhin, Zh. Prikl. Mekh. Tekh. Fiz. 5 , 138 (1964). *I A. L. Kistler, Phys. Fluids 2, 290 (1959). * A. L. Kistler and W. S. Chen, J . Nuid Mech. 16, 41 (1963). 7e

'I7

* In general, the applied pressure p(r, t ) is space as well as time dependent, but a single pressure gage measures only the average of the pressure on its sensitive surface. For simplicity in analysis the applied pressure is usually assumed to be uniform.

578

5 . MEASUREMENT OF PRESSURE

FIG. 30. Principal parts of fast response pressure gage. p ( r , 1 ) is applied pressure, which changes with time t and may or may not depend on position r across the surface of the gage. For simplicity of analysis it is assumed to be uniform. S is the strain element (region abcd). P is the sensor or strain-sensitive element. B and C are regions which usually contain materials different from that of P. F is front of propagating strain pulse.

For a bar gage I is either ten or more times larger than d or the properties of the material in region B are such that no reflection occurs at the surface (bc) between Z and B. Often the lateral boundary of the bar can be taken to be stress free because region C contains only gas which provides negligible constraint for the bar. A stub gage is essentially a short bar gage backed by a dense, rigid material in region B whose lateral dimension d is much greater than for the strain element Z. Region C contains a gas and is sealed to isolate it from the applied pressure by a very thin diaphragm. Sizeable reflections occur at the back and front surfaces (bc-ad) of the strain element and its lateral surface (ab-dc) is practically stress free. For a slab gage the shape of the elastic element is similar to that of a diaphragm, with the thickness dimension l being much smaller than d. If the material in region B has very large elastic constants so that for practical purposes it can be considered to be perfectly rigid, it will constrain the elastic element laterally. The strain will be basically one dimensional with the displacement in the thickness direction. In a probe gage dimensions 1 and d usually have approximately the same value. The gage is either mounted in the interior of a fluid whose pressure is to be measured or stress due to applied pressure rapidly propagates into regions C and B. In either case the distinguishing feature is that all surfaces of the strain element 8 are rapidly subjected to a normal stress equal to the applied pressure. The steady state strain is volumetric. For the diaphragm gage, which was discussed in the preceding chapter, the dimension 1 is many times smaller than d, there is only gas in region B and a dense, rigid material in region C provides a mount to which the

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

579

elastic diaphragm is clamped. It was tacitly assumed that reflections between the faces of the diaphragm are so rapid that within time intervals of practical interest they can be neglected. Bending and stretching remain as the types of strain to be taken into account. 5.10.2. Theory of Behavior of Elastic Element

Predictions of the strain produced in an elastic element due to a time varying pressure are usually carried out at one of three general levels of exactness. First is a prediction of practical use, but one that provides little more than a rough criterion for satisfactory performance. The prediction requires only a knowledge of the gage’s response time. An order of magnitude value for the response time can be taken as l / c , where 1 is an appropriate dimension of the elastic element or sensor and c is the effective velocity at which strain propagates. Although the best values to use for 1 and c differ with the shape of the elastic element and the boundary conditions, only a rudimentary knowledge of how strain propagates and reflects at discontinuities is needed to choose a reasonable value for 1 and a value for c can be taken as the square root of the ratio Young’s modulus to density with an error less than a factor of two. Finally, if there are no important pressure changes in a time interval less than several, say 10, multiples of the response time, one can assume for practical purposes that no spurious oscillations are excited and that the applied pressure at each instant immediately produces the steady state (static) strain. This might be called the instantaneous steady state approximation. At the next level of approximation the elastic system of the gage is sometimes replaced by a model consisting of one or a number of rigid massive bodies connected to the mounting and perhaps to each other by massless elastic springs. Dashpots can be used to introduce dissinative effects. General mathematical methods are available for handling such coupled systems of massive bodiess4 and predictions of their behavior provide insight into the nature of the dynamic behavior of gages.27 But, like the analogous lumped parameter models for electric circuits, the approximation they provide usually becomes invalid at extremely high rates of change. These models will not be considered here since this chapter is primarily concerned with very fast response. At the level nearest to actual behavior, strain variations within the elastic element are treated as being due to traveling or standing waves. For gages with response times of the order of microseconds analysis of strain waves is usually necessary. For all but a few simple cases, however, exact solutions, even where possible, are too complicated to be useful.

580

5.

MEASUREMENT OF PRESSURE

The simplest type of wave is a one-dimensional, plane wave whose representation is a solution of the following well known equation

(5.10.1) where E is a strain component, c is a constant propagation velocity, z is a Cartesian coordinate and t is time. Alternatively, the dependent variable might be displacement u , material velocity = & / a t , or stress component cr = (elastic const)e. Referring to Fig. 30 and assuming that the pressure p ( t ) is uniform and suddenly applied, the strain at T is initially due only to a one-dimensional plane wave indicated by the wave front F. Later, with dimension 1 less than d/4, the strain due to direct propagation from the stressed surface is in general augmented by reflection from the interface (bc) between X and B. Although this adds to the complexity, the effect of such a reflection can readily be taken into account since the reflected strain is still onedimensional and plane. Later still, if again 1 is less than d/4, the strain at T is affected by the presence of the lateral boundary (ab-cd) between Z and C. This complication ordinarily results from a relief wave which develops at the boundary following passage of the initial compression front F and propagates inward as a three-dimensional wave (dashed curves). These three-dimensional waves cannot be described by a simple analytical expression. However, for the case of a bar where 1 is much larger than d, for a stress free lateral surface and for certain types of end conditions, an exact solution for the strain at q can be formulated.83 In this solution the effect of the lateral surface on the strain at q is not pictured as being due to three-dimensional relief waves which undergo multiple reflections. Instead the solution is given in terms of a Fourier integral representing the superposition of plane sinusoidally varying phase waves propagating along the bar. In this representation, the effect of the free lateral surface is to produce a dependence of phase velocity on frequency. The result is that a strain pulse propagating away from the end of the bar disperses, i.e., changes shape during the travel. Unfortunately, this exact integral formulation is in general too complicated to be useful. But relatively simple, approximate expressions have been developed which are valid when q is more than a few diameters away from the stressed end of the bar. No truly satisfactory description of the behavior near the stressed end of the bar has been proposed for times after the first arrival of relief strain. 83 R. Folk, G . Fox, C. A. Shook, and C. W.Curtis,J. Acousr. SOC.A m . 30,552 (1958); G . Fox and C. W.Curtis, ibid. p. 559.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

58 1

For the dilatational gage, Section 5.10.6, the one-dimensional plane wave representation is exact. Also, this representation provides a first approximation for a bar gage, Section 5.10.5, with a correction as described in Section 5.10.5.1 possible for a remote sensor. For a bar gage with a contact sensor, Section 5.10.5.2, and for a stub gage, Section 5.10.3, the one-dimensional plane wave approximation is usually poor, but it furnishes insight into the nature of the actual behavior and provides semiquantitative predictions. In the case of a probe gage, Section 5.10.4, an assumption of a one-dimensional wave is probably useful for no more than an order of magnitude calculation. In the remainder of the current chapter only the one-dimensional plane wave approximation will be considered. Well known relationships are given for easy r e f e r e n ~ e . ~ ~ * ~ ~ - ~ ~

5.10.2.1. Propagation of a One-Dimensional Plane Wave. Using the stress-strain relation ~ ( z t,) = ME(z, t )

(5.10.2)

between longitudinal strain E(Z, t ) and longitudinal stress u(z, r ) together with the boundary conditions that the applied pressure p( t ) is uniform and equal to longitudinal stress when z = 0, we have for a solution to Eq. (5.10.1) u(z, t )

=

p

(I -

f ) = p(t’)

(5.10.3)

and

(5.10.4)

where t’ = t - z/c. These equations represent a wave traveling in the positive z-direction with constant velocity c. c = (M/p)1’2,

(5.10.5)

where p is density and the value of the elastic modulus M depends on the type of strain being propagated. A . E. H. Love, “On the Mathematical Theory of Elasticity,” 4th ed. Dover, New York, 1944. Bs H. Kolsky, ”Stress Waves in Solids.” Oxford Univ. Press, London and New York, 1953. E. M . Ewing, W. S . Jardetsky, and F . Press, “Elastic Waves in Layered Media.” McGraw-Hill, New York, 1957.

5. MEASUREMENT OF PRESSURE

5 82

For dilatational strain, where there is no displacement perpendicular to M is the dilatational modulus E which is related to Young’s modulus Y and Poisson’s ratio v.

z,

M = E =

Y(l - v) ( 1 + v)(l - 2v)

and

Cd = (5)l”.

(5.10.6)

The velocity c d is called the dilatational velocity. For a bar with a f r e e lateral surface and with the assumptions that only the stress component directed axially (z-direction) is nonzero and that inertia due to lateral expansion can be neglected,*

M

=

Y

and

cg=

(--JY

‘2

.

(5.10.7)

The bar velocity cb is a few percent smaller than c d . In addition to a ( z , t) and ~ ( zt),, other dependent variables are the longitudinal displacement u and the material velocity u , both in the z-direction. u(z, t ) =

JZ 0

a” dz = az

loZ

1 e dz = PC

J

f’

p(t’) dt’

(5.10.8)

0

and (5.10.9)

as before t’ = t -z/c. 5.10.2.2. Time to Travel Through a Sensor. A relation such as given by Eq. (5.10.4) is basic to the operation of gages using a strain sensor; Eq. (5.10.9)plays a similar role for gages using a velocity sensor. The attractive feature of these equations is the prediction that the time variation of strain, or velocity, at any point on a surface for which z is constant is the same as the time variation of the applied pressure. This behavior is exactly realized in the case of a dilatational gage with a velocity sensor, such as described in Section 5.10.6.2, since, with careful alignment, the wave is strictly one-dimensional and velocity is measured at a surface with z constant. However, for a gage with a strain sensor the behavior is less simple, perhaps because the wave is only approximately onedimensional, but, in any case, because the sensor is necessarily finite in size. Consider the effect of measuring strain over a finite distance in the direction of propagation of a wave.

* In this approximation, the lateral displacement is equal to rve, where r is the polar distance from the z-axis.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

583

The general equation for the electrical signal V ( t )produced by a strain sensor having a length h in the propagation direction of a strain wave is V ( t ) = k / =*I ' c ( t - f ) d z = $ [ ' p ( f - ; ) d z .

(5.10.10)

The parameter k depends on the type of sensor, its dimensions, its sensitivity to various components of strain, the elastic properties of the medium through which the wave is traveling and the type of strain being measured. Expressions for k are given for a resistance sensor and for a piezoelectric sensor in Section 5.10.5.1 and for a capacitance sensor in Section 5.10.6.1. For a continuous wave the limits of the integral may be taken to be zi = - h / 2 and zf = h / 2 . But for a pulse with a beginning and an end the limits depend on whether the head, the tail, or neither are in the region covered by the sensor. Assuming that both the head and tail of the pulse are not between the ends of the sensor at the same time and that the pressure is applied initially at time t = 0 and drops to zero at time t = t o , the limits are

zi = 0 zi = 0 zi = ct

and and and

zf = ct zf = h zf = h V ( t )= 0

when 0 < t < h / c , when h / c < t < t o , (5.10.11) when to < t < to + h / c , when to + h / c < r .

For a continuous, sinusoidally varying pressure having a radian frequency w,

~ ( t=) M

po sin [o(t -hl2

-

f ) ] dz =

Pokh

sin

wh (x) sin(wt)

(5.10.12) The amplitude of the response is a function of frequency with large, approximately constant values occurring only for radian frequencies w considerably less than 2 c / h or, stated differently, only if the wavelength A = 27rc/w is much greater than h . For w = 27rc/h << 2 c / h ,

(5.10.13) If values of w are less than one-tenth 2 c / h , the amplitude of V ( t )will differ from pokh/M by less than 1 percent and pokh/M is the magnitude of the signal that would be obtained by applying a constant pressure p o to the

5 84

5.

MEASUREMENT OF PRESSURE

Time, t

-

FIG. 31. Comparison of gage signals V&) and V&) with applied pressure p(r) for two sensor lengths hl and h z . h, is ten times hl leading to a response time T* = hl/c which is also ten times T~ = hl/c. T is time at which relief strain arrives at sensor from lateral surfaces.

elastic element. Decreasing h improves the frequency response of the gage but at the expense of a decrease in sensitivity, which is k h / M . The gage response, calculated using Eqs. (5.10.10) and (5.10.11), is shown in Fig. 3 1 for three types of nonperiodic pressure-time profile and for two lengths of sensor. For step-function loading (middle graph) the signal increases linearly over the interval T = h / c , which is the response time of the gage. As shown by the top and bottom graphs, if the applied pressure varies appreciably over an interval equal to the response time 7, the resulting output of the gage (V, curves) differs considerably from the input and the integral of Eq. (5.10.10) would have to be inverted to obtain a reasonable representation of p ( t ) from V ( t ) . The tedious process of inverting the integral of Eq. (5.10.10) can be avoided without appreciable loss of accuracy provided the response time T is much smaller, say 10 times smaller, than the time during which a significant pressure change

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

585

occurs. In this case, Eq. (5.10.10) with limits zi = 0 and zf = h can be approximated by the simpler relation V(r) =

kh p (I -

%)

(5.10.14)

since p(r - z/c) is assumed to be a slowly varying function which can be removed from the integral. Equation (5.10.14) also follows directly from the assumption that the sensor measures strain at a point. The approximation is equivalent to replacing the V ( t ) curves in Fig. 31 by the p ( t ) curves. For the Vl(r) curves, which apply to the gage with the shorter response time, the approximation introduces an error of only about 5 percent except during the time interval 0 to T~ when the change in pressure is significant. Eqs. (5.10.13) and (5.10.14) are examples of the “instantaneous steady state” approximation mentioned at the beginning of Section 5.10.2. 5.10.2.3. Reflection and Transmission of an Elastic Wave at an Interface between Two Media. In general, a plane dilatational (longitudinal) wave incident on an interface between two elastic media at an oblique angle will spontaneously generate reflected and transmitted shear (transverse) waves in addition to reflected and transmitted dilatational Only for normal incidence are the shear waves missing, but for present purposes this is the case of most interest and is the only one considered here. For normal incidence, expressions for the reflected strain E;’ and the transmitted strain E;” at the interface (z = 0) are

(5.10.15) where single, double and triple primed quantities are respectively associated with the incident, reflected and transmitted waves. (The boundary conditions are that stress and material velocity be continuous across the interface. Use of these conditions, together with the relations of Eq. (5.10.9) leads to Eq. (5.10.15) and also Eq. (5.10.16).) The incident and reflected waves travel in the medium identified by the subscript 1 and the transmitted wave travels in the medium identified by the subscript 2. It is assumed that the incident wave is produced by the applied pressure p ( t ) , which is related to the material velocity u ; , the strain E; and the stress (+I of the incident wave by the relations of Eq. (5.10.9). There are similar expressions for the material velocity associated with the reflected and transmitted waves:

5.

5 86

v;t

=

[1

MEASUREMENT OF PRESSURE

-

[1 +

””1

PlCl

2

p(f)

3

;

PlCl

VLtf =

’(‘)

[1 + E]

.

(5.10.16)

According to Eqs. (5.10.15) and (5.10.16)a reflected wave does not exist if the product of the density p and the wave velocity c , called the acoustic impedance, is the same for the two materials. This prediction has been used as a basis for extending the length of a bar gage and for reducing disturbances produced by a sensor embedded in an elastic element. See Sections 5.10.5.1and 5.10.5.2. In some experimental arrangements, such as considered in Section 5.10.6.2, pressure applied to the front face of an elastic element produces a strain wave which propagates through the element and subsequently reflects from a stress free back face. During reflection the velocity of the back face is measured. Since the reason the back face is stress free is that there is no supporting material in contact with it, the measured velocity is given by v;” of Eq. (5.10.16)when pz is set equal to zero. Then we have the following relation between measured velocity and applied pressure:

(5.10.17) Measurements are discontinued after the arrival at the point of observation of a doubly reflected wave or of a disturbance which has propagated from a lateral boundary. With gages such as the stub and probe types, which are considered in Sections 5.10.3 and 5.10.4, measurements are continued long after the time needed for multiple reflections to occur at the faces of the elastic element. The behavior is then similar to that of a diaphragm gage. Step function loading produces oscillations of the strain which are centered on steady state values. Both the period of the oscillations and the magnitude of the steady state strain depend on the boundary conditions. For a one-dimensional wave, the well-known result of the normal reflection of strain from a free surface is obtained from Eq. (5.10.15)by setting pz = 0: EI’

=

- p(t)/M.

(5.10.18)

For a fixed surface, setting p p equal to infinity, we have El’

= p(f)/hf.

(5.10.19)

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

587

0 y/////fl/fl&//////m

V

V

0

-

1

Time (a 1

0

-

1

Time

(b)

FIG. 32. Normal reflections of one-dimensional strain waves at two parallel surfaces. Waves are produced at left surface by step loading, so stress at this surface is constant after time zero. The right surface is fixed in case (a) and free in case (b). Full line contour and hatching represents actual strain; dashed contour and hatching represents time average.

Equations (5.10.18) and (5.10.19) have been used as a basis for constructing Fig. 32. Figure 32a illustrates the situation in which one end of a bar* is subjected to a step function load and the other end is fixed. For Fig. 32b, one end is again subjected to a step function load, but the other end is free. As illustrated, the period of oscillation for the bar with a free end is 21/c; the time average of the strain varies linearly with distance along the bar and is equal to po[1 - ( z / l ) ] / M ,where p o is the magnitude of * Although this picture is oversimplified for the case of a bar, it illustrates certain general features of the behavior correctly. See Section 5.10.5.1 for a better representation.

588

5 . MEASUREMENT OF PRESSURE

the step, 1 is the length of the bar and z is the distance from the stressed end. For the bar with the fixed end, the period is 41/c; the time average of the strain is uniform along the bar and equal to p o / M . Actually, because of damping and of dispersion due to the free lateral surface, the head of the pulse does not remain a step as shown, but gradually spreads out during travel with the result that oscillatory variations in the strain die out leaving the bar at each position with a steady state strain equal to the time average. 5.10.3. Stub and Slab Gages

There are two fundamentally different approaches to problems resulting from the reflection of strain at discontinuities within a gage. One approach is to locate the discontinuity, such as the back end of a pressure bar, Section 5.10.5, or the lateral surface of a dilatational gage, Section 5.10.6, far enough from the sensor to prevent the arrival of a disturbance from the discontinuity before the pressure measurement is complete. The other approach is to make the strain element very small so that reflections from opposite boundaries follow in rapid succession and tend to cancel each other. This leads to resonance oscillations of high frequency so they are not excited with appreciable amplitude except by an abrupt pressure change, i.e., a change which occurs within less than a few resonant periods. See Fig. 21. As noted in Section 5.5.1 these oscillations are often referred to as “ringing” of the gage. Depending on the damping properties of the strain element and supporting material, as well as the overall geometry, these oscillations may or may not attenuate rapidly. The gages described in this and the next section use the second of the above approaches, so shock wave loading usually produces large ringing oscillations. The appearance of ringing is often removed from the record by an electrical filter, but this is done at the expense of increasing the response time. Essential features of a reliable, fast response stub gage are shown in Fig. 33. The strain element is the central cylinder or stub, which resembles a very short pressure bar, such as described in Section 5.10.5, with nearly rigid back end support. Due to reflections at the front and back faces, either an abrupt application of pressure to the front of the stub, or a sudden change in velocity of the back end, produces periodic changes in the stress and strain. According to the one-dimensional theory considered in the preceding section, the periods of the oscillations are proportional to the distance between the reflecting surfaces and therefore decrease as the size of the stub decreases. The theory also indicates

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

589

FIG.33. Fast response stub gage. (a) Construction: 1: thin diaphragm and seal, 2: front of stub, 3: back of stub, 4: stub support, 5: piezoelectric quartz crystal stack, 6: piezoelectric quartz crystal compensator. (b) Response to spark induced shock wave with no electrical filter. (c) Response with filter.

that the time-average of the stress or strain at every position along the stub is the same as the steady state value at that position. Due to the back end support, the steady state stress produced by pressure applied to the front face is essentially uniform along the length of the stub. On the other hand, changing motion of the base causes the stub to accelerate* and, since the stress induced by acceleration is due solely to inertia, the steady state stress due to uniform acceleration decreases from a maximum at the base to zero at the front because the mass being accelerated by the stress decreases in this direction. This difference in stress patterns provides the operating basis for the unusual arrangement of op-

* Step function pressure loading of the back end of the stub, which was considered in the preceding section, is equivalent to an abrupt change in velocity of the end followed by subsequent changes of twice the initial amount at intervals equal to the time for a strain wave to make a round trip of the stub. The resultant time-average acceleration is uniform along the length of the stub. For the stress or strain produced see Fig. 32b.

590

5 . MEASUREMENT

OF PRESSURE,

posing sensors shown in Fig. 33a. The purpose of the arrangement is to balance out the effects of accelerations due to vibrations of the support and at the same time provide a measurable response to the strain produced by the applied pressure. This is accomplished, at least for steady state conditions, by making the quartz sensor near the front of the stub thicker, and therefore more sensitivie to stress, than the opposing sensor near the back. The combination is sensitive to the essentially equal stresses produced by the applied pressure but is insensitive to the unequal stresses resulting from acceleration. An alternative but less attractive method of acceleration compensation is to use two identical elastic elements with their sensors connected to oppose each other; both are mounted on the same base, but the pressure to be measured is applied only to one, the other being shieided (Bynum et p. 21). Figs. 33b and 33c are schematic drawings showing the response of a tiny stub gage to a pressure pulse consisting of a leading shock front fob lowed by a rarefaction. The direct response of the sensors, shown in Fig. 33b, exhibits prominent ringing oscillations having a period of approximately 2 ps. In the record of Fig. 33c the ringing oscillations have been virtually eliminated by an electrical filter, which has also increased the response time by a factor of two or three. An advantage of this gage over the bar and dilatational gages described later is that its hold time can, with the use of an electrostatic charge amplifier, be made very long-long enough, for example, to permit calibration with essentially constant pressures. Usable pressure ranges extend from 10 kPa (fraction of a bar) to nearly 1 GPa (few kilobars). Gages of this type are made commercially in the United States and Switzerland. For a slab, Fig. 12d, which deforms under pressure in roughly the same way as a stub, a capacitor provides a useful means of sensing strain. Shock pressures from 0.5 to 20 MPa have been measured using capacitors with liquid electrolytess7or dielectric filmse8between the plates. See Fig. 34. It has also been suggesteds0that a capacitor using a thin polymer film between its electrodes be cemented to an airfoil to measure pressure on its surface. A third possibilityoois to use a piezoelectric film between the V. N . Kochnev, Electrokhirnicheskiye datchiki dinamicheskikh davlenii. Absrr., All-Union Conf. Dyn. Pressure M e a s . , Ist, 1973, pp. 9-10. VNIIFTRI, Moscow, 1973 (in Russian). G . V. Stepanov, Izmerenie davlenii v udarnykh volnakh dielektricheskirn datchikom. Absrr., All-Union Conf. Dyn. Pressure Measu., Ist, 1973, pp. 13-14. VNIIFTRI, Moscow, 1973 (in Russian). 88 M. Chatanier, “Capteurs de pression pelliculaires.” Office National d’Etudes et de Recherches ACrospatiales, ONERA, France, 1975 (in French). 00 A. L . Robinson, Science 200, 1371 (1978).

5.10. FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

* AP

AP

&$&--AV=?

\,el:+,cnl;

,I

“

f

’’ Film

59 1

G IT R

“0

AC

!rn -

FIG.34. Pressure sensors which utilize pressure-dielectric effects in a liquid ele~trolyte,~’ and in a solid dielectric filmsa capacitor.

electrodes: with the “plate spacing” only a small fraction of a millimeter, the response time of the unit will be appreciably less than 1 ps. Also, because of small mass, the unit is relatively insensitive to acceleration. 5.10.4. Probe Gage; Volumetric Strain

A variety of gages has been developed to measure pressure within the body of a fluid, rather than at a wall. These are mainly the result of field75,@1*9Z and l a b o r a t ~ r y ’ ~ *studies ~ ~ * ~of, ~open ~ air and underwater explosions. An example of such a gages1-e3is shown in Fig. 35. The piezoelectric element is an x-cut tourmaline crystal, quartz being unsuitable because, as pointed out in Section 5.6.2.3, it is insensitive to a volumetric change. The sensor is embedded in a hard epoxy resin casing which is attached to the end of a hypodermic needle. In a steady state condition, the stress within the elastic element and sensor is a uniform pressure and the fractional change in volume u is given by

AT/T = p/B

=

(5.10.20)

3p(l - 2v)/Y,

where AVis change in volume, p is applied pressure, B is the bulk modulus of the material, Y is Young’s modulus, and v is Poisson’s ratio. Fast response with a minimum of spurious oscillations results from the small size and the damping properties of the casing. As with most gages of this type an exact response time or resonance period is difficult to predict. To the extent that the elastic element can be considered to be a sphere to which pressure is suddenly applied, the resonant periods Pqare given by solutions to the following equation. -

[I

- 2(1

’)

- 2v)

sin(rna) - (ma) cos(ma) = 0, (5.10.21)

where v is Poisson’s ratio, a is radius of sphere, m

OP

=

2?r/cdP = w/cd,

I. B. Sinani, Prib. Tekh. Exsp. 4, 85 (1957). M. 1. Vorotnikova, Zh. Prikl. Mekh. Tekh. Fiz. 2, 110 (1962). M. I. Vorotnikova, V. K. Kedrinskii, and R. 1. Soloukhin, Fiz. Goreniyu Vzryvu 1, 5

(1965).

5.

592 1

MEASUREMENT OF PRESSURE 2

3

1 cm

FIG.35. Tourmaline probe gage for underwater blast measurement of pressure. 1: tourmaline platelet; 2: epoxy resin; 3: hypodermic needle. Size of tourmaline sensing element is 1 x 1 x 0.2 mm3.

and c d is dilatational wave velocity. For a nominal value of 0.3 for v, the longest resonant period is

(5.10.22) The gage pictured in Fig. 35 has a response time of the order of 1 ps and has been used to measure pressures of approximately 100 MPa. Another type of fast response gage for measuring pressure in the interior of a fluid" uses a small tube in the end of which is a tiny pressure bar such as described in Section 5.10.5. The high pressure gage shown in Fig. 36 measures pressure at a wall,

"2 t

v t (a)

(b)

FIG.36. Resistive probe gage for pressure measurement. (a) Water column impact tube for testing and calibratinggage. 1 : sensing element, 2: impacting shock front generated by a water-water collision, 3: thin diaphragm separating colliding water columns, 4: piston, 5 : gas driver chamber. (b) Calibration curves for Ge(Si) and Ge(P-N) samples.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

593

but very shortly after the pressure is applied the sensing element experiences compressive loading on all faces. Its operation depends on a resistance change of a metal such as manganing4or a sernicondu~tor.~~ According to Eq. (5.6.8) of Section 5.6.2.1, a measure of a material’s sensitivity for resistance change is its value for Y,,, defined by

y

1 dR R dp’

=--

(5.10.23)

where R is the resistance and p is pressure. Values of 9,for the metal manganin and for the germanium semiconductors Ge(Si) and Ge(P-N) are, respectively, 0.024, 0.03, and 0.01 MPa-’. Figure 36 shows a resistance sensor mounted in an apparatus for producing shock waves in water. The sensor was a semiconductor, Ge(Si) or Ge(P-N), having dimensions 1 X 1 x 0.5 mm. It was mounted in an epoxy resin case and measured pressures of nearly 1 GPa (10 kbar). Fig. 36 also shows calibration curves for Ge(Si) and Ge(P-N). 5.10.5. Pressure Bar Gage

Use of a long bar for the strain element of fast response gages has been popular since the idea was introduced by Hopkinson in 1905.9s-9sIn these gages the time varying pressure to be measured is applied to one end of a bar and either the resulting strain at some position along the bar or the movement of the far end is sensed. Measurements are made during the first passage of the strain wave and are completed before distortions are introduced by the arrival of a wave reflected from one of the ends. 5.10.5.1. Remote Sensor: Stress-Free Surface. Figure 37 shows a long bar with several types of sensor placed at some distance from the stressed end. It is supposed that effects due to supports or any surrounding material, such as shown in regions B and C of Fig. 30, can be neglected. The oldest and simplest theory of strain propagation along a bar with a stress-free lateral surface is the one-dimensional theory considered in P. W. Bridgman, Proc. A m . Acud. Arts Sci. 43, 347 (1911). V. K. Kedrinskii, R . I. Soloukhin, and S.V. Stebnovskii, Zh. Prikl. Mekh. Tekh. Fiz. 4, 93 (1969). ge H. Kolsky, “Stress Waves in Solids,” p. 87. Oxford Univ. Press, London and New York, 1973. O7 B. Hopkinson, Proc. R. Soc. London, Ser. A 74, 498 (1905). B . Hopkinson, Philos. Trans. R. Soc. London, Ser. A 213, 437 (1914). R . M. Davies, Philos. Trans. R. Soc. London. Ser. A 240, 375 (1948).

5 94

5.

MEASUREMENT OF PRESSURE

-

h

+ . ic

4

A

B

h=o

1

C

FIG. 37. Pressure bar and types of sensors. A: Surface type, B: cross section type, C: end type.

Section 5.10.2.1. Although this theory is only approximate, it predicts correctly for step pressure loading of one end of a bar that the head of the resulting strain pulse travels with the bar velocity c b defined by Eq. (5.10.7) and that the steady state longitudinal strain following passage of the head of the pulse is equal to the applied pressure divided by Young’s modulus. The one-dimensional theory is assumed in the following discussion of sensor types. Modifications due to exact theory are considered later. Three general types of sensor are indicated in Fig. 37: surface sensor, cross section sensor, and end sensor. In all cases the hold time of the gage T is determined by the first arrival of a reflection from one of the ends. Thus T = 21/cb, where 1 is the distance from the sensor to the far end of the bar for the surface and cross section sensors and is the length of the bar for an end sensor. Capacitance, piezoelectric, and resistance sensors have all been used for surface measurements. The capacitor is cylindrical, with the metal bar serving as the grounded e l e ~ t r o d e . It ~ ~responds to radial displacement which is related to longitudinal strain through Poisson’s ratio. The piezoelectric sensor is cemented to the side of the bar. One type is a thin slab of polarized ceramic (e.g., barium titanate) with a metal foil on the outer face for one electrode and the grounded bar for the other. The electrodes act as capacitor plates on which charge or voltage builds up as the element is strained. With the polarization direction perpendicular to the surface of the bar, the sensor response is proportional to change in surface area which is equal to the sum of the longitudinal and radial strains. Stick-on resistance sensors have also been used. These have consisted of a fine metal grid whose resistance changes when the grid expands or contracts in a particular direction. Thus the grid responds to longitudinal or radial strain depending on whether its sensitive axis is parallel or perpendicular to the axis of the bar. Or a fine resistance wire with an insulating coating has been wrapped around the circumference of the bar, in which case it responds to radial strain. On the basis of one-dimensional theory, all of these sensors operate according to Eq. (5.10. lo), or a similar equation for a change in charge or current. The velocity c is the bar velocity and k and M depend on the particular type of sensor and arrangement. The contribution to the response time due to the size of the sensor is given by T = h / c b . As an example, for a stick-on resistance sensor di-

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

595

rected so as to respond only to longitudinal strain, M is Young’s modulus and k = GIoR/h, where G is the gage factor, I, is the constant current through the sensor and R is the sensor’s total resistance. Equation (5.10.10) becomes (5.10.24)

and, for small h, we have for Eq. (5.10.14), (5.10.25)

Heating limits the value of I,,. Since R is proportional to h , the signal V(r),as well as the response time r , increase linearly with h. Of the three sensors the polarized ceramic has the greatest sensitivity, but the sensitivity is more dependent on ambient temperature and less stable over a long period. Lack of long term stability of the polarized ceramic necessitates frequent calibration, which must be carried out dynamically. The need for precise electrode alignment is an annoyance with an air-spaced capacitor whereas bonding problems can be troublesome with cemented sensors. For cases in which pressure is a monotonically increasing function, bonding is much less of a ,problem with the wraparound wire gage than with commercial stick-on resistance gages. A problem peculiar to the cross-section sensor, type B in Fig. 37, is the distortion of strain in the bar due to reflections at the interfaces. According to Eq. (5.10.19, reflections of strain will not occur if the product of density and bar velocity, p c b , is the same for the bar and the sensor. On this basis the following pairs of metal and piezoelectric material provide reasonable matches: aluminum-quartz, zinc - barium titanate, tinlead metaniobate. Use of such corn bin at ion^^^ has been found to minimize reflections greatly, often to the point of no practical importance. A cross section sensor using a piezoelectric material operates as a capacitor with adjacent sections of the bar acting as electrodes. Assuming one-dimensional theory, the voltage developed between the electrodes is given by Eq. (5.10.10), with k of this equation determined from Eq. (5.6.1 1) and Eq. (5.6.12) or Eq. (5.6.13) of Section 5.6.2.3 for either a poled ceramic or x-cut quartz with the direction of polarization along the axis of the bar.

v(t)= Q(r)/co= l z f p( t hCo 5,

-

k)

dz,

(5.10.26)

where the sensor has a thickness h , a cross-sectional area A, which is the same as that of the bar, and a geometrical capacitance Co. The piezoelec-

5 96

5.

MEASUREMENT OF PRESSURE

tric constant d,, is d,, for x-cut quartz and dS3for a poled ceramic. If h is very small,

(5.10.27) Since C, is inversely proportional to h, V(r) increases linearly as h increases but Q(t) is independent of h. End sensors measure displacement or velocity rather than strain. Again relying on one-dimensional theory, the operating equation for an end sensor is Eq. (5.10.17) of Section 5.10.2.3, with c1 equal to the bar velocity c b . This relates the material velocity u ( t ) of the far end of a bar of length I to the pressure applied at the other end:

(5.10.28) Hopkinson’s original experiment^*^*^* were carried out with an end sensor in the form of a short bar placed in contact with the pressure bar. The compression pulse propagating from the stressed end of the pressure bar passes through the contact interface and reflects a; a tension pulse from the free end of the sensor. When the reflected pulse reduces the compression at the contact interface to zero, the sensing bar separates, trapping momentum in the process. See Fig. 38. The trapped momentum, which depends on the pressure-time profile and the length of the sensor, was measured with a ballistic pendulum. The pressure-time profile was determined roughly from several measurements using sensing bars of different lengths. Although this technique is cumbersome and not very accurate, it is unique in that it uses no electrical measurement. Also modifications of this technique have been used in recent experiments not involving a pressure bar. One type of electrical sensor uses the end of the pressure bar as one plate of a parallel plate capacitor. With the other plate stationary, the voltage across the capacitor varies with the displacement of the end of the bar. An advantage of this sensor is that its contribution to the response time 7 is negligible (in effect, h = 0). Its serious disadvantage is that it measures displacement and therefore the voltage-time record must be differentiated to obtain the velocity u of Eq. (5.10.28). In principle, both types of laser interferometer described in Section 5.10.6.2 could be used as end sensors, but €or most pressure bar applications only the displacement interferometer would be suitable since the velocity interferometer would be too insensitive. The principle ambiguity in these analyses of pressure bar gages results from the approximate forms of Eqs. (5.10.4) and (5.10.9) of Section

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

IP

597

v

Tension

I

IP

4 I

q

I

I 1

I------

FIG.38. Reflections at end of Hopkinson sensor bar. Graphs, at successive times, of pressure p and material velocity u as function of spatial coordinate. All parts of pulses are assumed to propagate with unchanged form at speed c. Bottom graphs show conditions after separation. Next to bottom graphs show conditions at instant of separation (p = 0 at joint .)

5.10.2.1, which relate the applied pressure to the measured strain or material velocity. According to Eq. (5.10.4) a step function change in pressure should produce a step function change in strain at any position along the bar. For comparison with this prediction, Fig. 39 shows an actual record of strain some distance from the end of a bar which has been subjected to reflection of an air shock. The slow rise and complex oscillations are due to the lateral free surface, which makes the problem of prediction three-dimensional. Exact solutions to problems of this type have been f o r m ~ l a t e d ,but ~ ~in, ~general ~ they are too complicated to be useful. G. P. DeVault and C. W. Curtis, J. Acousf. SOC.A m . 34, 421 (1962).

598

5.

MEASUREMENT OF PRESSURE

F

S

TIME

-

FIG.39. Strain observed at surface of cylindrical bar subjected to step function end loading. Bar is magnesium with diameter 38 mm. Distance of sensor from loaded end is 1.51 m. A superposition of very high frequency, low amplitude oscillations begins at the time denoted by S. At the sensitivity of this oscillogram there are no observable oscillations following time marked F. [From the reports of Ref. 83.1

They can, however, be replaced by simple asymptotic expressions which are valid for large distances of travel (>10-20 diameter). The expression* most useful for pressure gage design describes the slow rise and the large, low frequency oscillations near the beginning of the pulse; the strain is

EJB) =

9 [k + 1A@) d B ]

=

poU(B),

(5.10.29)

where p o is the magnitude of the pressure step, Ai(B) is the Airy function, and (5.10.30)

where d is the bar diameter, u is Poisson’s ratio, and z is distance of travel. The time t’ = t - z/cb is measured relative to the arrival time for strain traveling with the bar velocity c b . A graph of U(B)versus B is shown in Fig. 40. The beginning strain arrives in the form of a precursor which travels with the dilatational velocity c d but this decays rapidly during travel and is negligible after a few diameters (> 10-20 diameter) beyond which Eq. (5.10.29) becomes a reasonable representation. Since U(B)of Eq. (5.10.29) is the strain produced by unit step function load, p ( B ) = S(B), it can be used in the superposition integral of Eq. (5.5.2) of Section 5.5.1, to predict the strain when the applied pressure p ( B ) changes gradually rather than abruptly with time. Similar to the case of a diaphragm, for which results are shown in Fig. 21, the integra* Other expressions describe the complex behavior in the region between the arrows at S and F in Fig. 39, but this behavior is less important because the oscillations are of much smaller amplitude and higher frequency.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

599

tion of Eq. (5.5.2)has the effect of averaging out the oscillations of U(B). When no significant pressure variations occur within the interval 7B [shown in Fig. 401 the expression for E,(B) given by Eq. (5.5.2)reduces to p(B)/ Y, which is the value predicted by the one-dimensional theory. Thus the rough one-dimensional theory is reasonable provided there are no appreciable pressure changes within an interval 7B. Call T~ the dimensionless response time, which is in addition to the response time due to the length of the sensor. With the assignment of a value of 3.00 to 78 = B in Eq. (5.10.30), the actual response time T = t’ is given by 7 =

3.00 (

(5.10.31)

3 ) 2 / 3 ($;)1’3.

Values for the magnesium bar used to obtain the record of Fig. 39 were d = 3.81 cm, c b = 5.02 km s-’, Y = 0.300, z = 1.51 m, which result in a response time of T = 20 ps. This can be reduced somewhat by measuring the strain closer to the stressed end of the bar. Taking z / d = 20 gives 7 = 16 ps. As z / d becomes smaller than about 20 the oscillations become more complicated than indicated by Eq. (5.10.29) and, contrary to that equation, the amount of overshoot to the first maximum increases. However, the rise time to the first maximum continues to decrease as z ” ~ in accord with Eq. (5.10.31). Edwards et ~ 1 . suggest ‘ ~ that a value of z / d between 2 and 4 provides the best compromise between short rise time and large overshoot. Assuming z / d = 2, 7 is 7.4 ps. A more significant reduction in 7 results from a decrease in the diameter of the bar. If the diameter is decreased to 5 mm, keeping z / d , cb and v unchanged, 7 = 1 ps. Aluminum, steel, and most other practical materials

-

Dilatational Precursor

\ -2

. 2

4

6

8

10

Time, B

FIG.40. Strain versus time predicted by asymptotic theory for a bar subjected to step function end loading. The dimensionless quantity B is proportional to t’ which is time measured from the instant at which strain traveling at the bar velocity cb arrives at the sensor.

600

5.

MEASUREMENT OF PRESSURE

have approximately the same values for c b and v , so 1 ks is about the lower limit for the resolving time of pressure bars with a free surface and a remote sensor. However, a possible exception has been pointed out by Baganoff.101*102 Beryliium has a bar velocity approximately 2.5 times larger and a Poisson's ratio about 10 times smaller than the above values, which could result in a response time of approximately 0.1 ps. Joneslo3 has examined this possibility experimentally. He used a beryllium bar with a thin cross-section sensor made from a poled ceramic (PZT-4) and measured a response time several times less than predicted for a bar having the same dimensions but made from magnesium, aluminum, or steel. The measured time was slightly greater than anticipated, possibly because of a mismatch between the elastic properties of beryllium and the ceramic PZT-4. This mismatch was thought to cause small spurious radial oscillations of the sensing disc. Problems resulting from the mismatch might be reduced or eliminated with further development. Although records from a pressure bar with remote sensor, like those from a stub or probe gage, contain spurious oscillations, the amplitude of the oscillations can in principle be smaller and its time rate of decrease larger in the case of the bar. On the other hand, the hold time can be greater for the stub or probe gages. Unlike the dilatational gage, which is described in Section 5.10.6, the bar can be used to measure average pressure when the distribution over the sensitive surface is nonuniform. This is because the effect of an asymmetrical distribution of pressure over the end of a bar is to produce flexural oscillations which can be cancelled out by the sensor.1oo Thus, a bar gage can be used to measure pressure at a wall as well as at the end of a shock tube. A pressure bar with a remote sensor can be particularly useful in cases, such as occur in plasma experiments, where the pressure to be measured is in an environment of large electrical disturbances, since a remote sensor can be placed outside such a region. For constructional details and typical records of bar gages with remote sensors, see reports of Refs. 72-74. There are alternatives to the pressure bar which also have the attractive feature that the record is not complicated by reflections from the back of the strain element. The alternatives suggested so far, however, have other characteristics which make them impractical or unattractive for general purpose measurements. One alternative is the torsion bar, described by Davies and Owen.lo4 Unlike the longitudinal strain pulse in a pressure bar, a torsional pulse lol lo*

Io3 lo'

D. Baganoff, R e v . Sci. Instrum. 35, 288 (1964). R . K . Hanson and D. Baganoff, R e v . Sci. Instrum. 43, 396 (1972). I. R. Jones, R e v . Sci. Instrum. 37, 1059 (1966). R . M. Davies and J . D. Owen, Proc. R . Soc. London, Ser. A 240, 17 (1950).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

60 I

I------% ___L( FIG. 41. Expansion of thin tube during passage of confined shock wave. a = initial radius, h = wall thickness, w = lateral expansion of wall. Y is a sensor of fine enamelled resistance wire wrapped around the tube. T = response time, V , = shock wave velocity.

propagating in the fundamental mode does not spread out or develop spurious oscillations as it travels along a bar. The propagation velocity is c, = ( ~ / p ) ” where ~, p is the shear modulus. Davies and Owen confirmed this experimentally by observing a torsional pulse produced by the impact of a bullet fired into a notch on the side of a bar. Large pressure changes taking place in a fraction of a microsecond were recorded. The obvious difficulty with this technique for general pressure measurement is translating pressure into torque. Another alternative is to use the wall of a tube as a strain element. For example, the pressure jump across a one-dimensional shock propagating through a fluid contained in a shock tube could be determined by measuring the expansion of the wall. Usually the velocity with which a strain pulse propagates along a solid is determined by the elastic properties of the material, but in this case, the expansion pulse travels with the velocity of the shock as shown in Fig. 41. Both experiment10sand theorylWindicate that, like the head of a longitudinal pulse in a bar, the head of the lateral expansion pulse in a tube is spread out in space and time. An approximate expression for the consequent response time is (5.10.32)

where h is tube thickness, a is tube radius, and V , is shock velocity. This expression is reasonable if h << a and V, << cp, where cp is the plate velocity ( Y / p ( 1 - Y ~ ) ) ~ / * . Even assuming h / a = 0.1 and a = 1 cm, which are unrealistically small for a conventional shock tube, T would be about ‘‘I W . R. Smith, Shock Tube R e s . , Proc. I n t . Shock Tube S y m p . . 8th, London, July 1971, Paper 59. Chapman and Hall, London, 1971. ‘OB S. Tang, Proc. A m . SOC. Civ. Eng., J . Eng. M e c h . Div. 91, 97 (1965).

602

5.

MEASUREMENT OF PRESSURE

20 ps when V , = 750 m s?. This response time is an order of magnitude larger than the response time of a small pressure bar, so the tube is much less suitable for measuring rapid pressure changes. 5.10.5.2. Contact Sensor: Lateral Constraint. A versatile bar gage, with a sensor at the pressure end, has been developed by Turetic, Zaitsev, Soloukhin and, later, by ~ t h e r ~ . ~Details ~ , ~ are ~ J shown ~ J ~ in ~ Fig. 42. Pressure is applied directly to one surface of a small piezoelectric sensor which is backed by a relatively long, matching metal rod. Both sensor and rod are embedded in a pliable material which is contained by a mounting tube. The purpose of the backup rod is to prevent oscillations of the sensor due to reflections of strain at its front and back surfaces. For a onedimensional wave it follows from Eqs. (5.10.15) and (5.10.16) of Section 5.10.2.3 that reflections at the interface between the sensor and the rod can be effectively eliminated by using materials having the same acoustic impedance, pc. With a polarized ceramic such as barium titanate (lead zirconate-titanate, lead metaniobate, etc.) for the sensor, the backup rod is matched by making it of zinc (zinc, tin, etc.). To avoid depolarization of the sensor, which occurs at high temperatures, a low melting point solder, such as Wood’s metal, is used to attach the sensor to the rod; some of the recently developed polymer cements might also be satisfactory. The embedding material has a number of functions in addition to that of supporting the sensor and rod. First, it insulates the sensor from disturbances due to extraneous vibrations of the mounting tube. Second, together with the pressure pulse it sustains, it tends to suppress the lateral expansion of the sensor, thus reducing the inward propagating relief pulse. See schematic pulse profiles in regions ‘c and C of Fig. 30. Third, it causes attenuation of the strain pulse in the backup rod so that reflection of the pulse from the far end of the rod has little effect on the sensor. Beeswax and silicone rubber have been found to be effective embedding materials. Use of a poled ceramic for the piezoelectric sensor has the advantage of high intrinsic sensitivity, thus permitting construction of a very small gage. Its response is linear at low pressures5E~82*76~78~108 but becomes nonlinear above a few MPa (tens of bars).log As mentioned in Section lo’ Yu. E. Nesterikhin and R . I . Soloukhin, “Metody Skorostnykh Izrnerenii b Gazdinarniki i Fiziki Plazmy.” Nauka, Moscow, 1967; English translation available from Federal Scientific and Technical Information, Springfield, Virginia (Doc. AD 682067). Io8 J. P. Huni, R. Ardila, and B. Ahlborn, Rev. Sci. Instrum. 41, 1074 (1970). ‘00 E. K . Dobrer and K . N . Karmen, Zh. Tekh. Fiz. 2, 455 (1957).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

603

FIG.42. Bar gage with piezoelectric end sensor. Two gages are shown, one mounted in the shock tube side wall and one in the end plate. The output signals are shown near each gage and the x - t diagram indicates the shock arrivals at each gage. 1: piezoelectric sensor, 2: acoustic absorbing rod, 3: beeswax potting, 4: supporting and mounting tube. Electrode at pressure loaded surface is connected to grounded mounting tube with a fine wire.

5.6.2.3, the mechanical strength of poled ceramics is less than that of single piezoelectric crystals, such as quartz and tourmaline. As with all gages, miniaturization is the key to fast response. Good performance has been obtained with contact sensors having a bar diameter d in the range 1 to 5 mm and a sensor thickness h of comparable value, 0.5 < h / d < 1. Experiment has indicated that the best compromise for the thickness of the embedding material 6 is S/d 0.5. Tests have also shown that the best position for the sensor is at the very end of the bar. Typical response of this gage to step loading produced by a shock wave is shown by the records reproduced in Fig. 43.* The response is close to the ideal predicted on the assumption of one-dimensional strain propagation. See middle graph of Fig. 3 1. The rise to a constant value is approximately linear and there are no sizeable oscillations. Distortions due to relief at the lateral surface are either suppressed by the embedding material or smoothed out by averaging over the volume of the sensor. If the lateral constraint were perfect, the response time would be T = h / c d . Measured response times are somewhat greater than this, 7 1.5 h / c d , but a value of 1 ps is readily attainable. The hold time depends on the length of the backup rod and is nominally T = 21/cb but is actually greater than this because of attenuation due to the embedding material. Hold times of approximately 100 ps are reasonable.

-

-

A. H. Meitzler, IRE Nutl. Conv. Rec. Part 9, p. 55 (1956); J. Miklowitz and C. R. Nisewanger, J . Appl. Mech. (Trans. ASME, Ser. E ) 24, 240 (1957).

* Exact theory is too complicated to be helpful in predicting the spacetime dependence of strain within the sensor. Near the end of the bar Eq. (5.10.29)does not provide a reasonable description. On the other hand, a tractable solution based on the idea of an inward propagating relief pulse has not been developed quantitatively for a bar whose side is free or partially constrained. Prediction is simple only if it is assumed that the lateral constraint is perfect. For experimental studies of strain near the end of a bar see Ref. 110.

604

5 . MEASUREMENT OF PRESSURE

FIG.43. Response of pressure bar gage with contact sensor and lateral constraint. Oscillographic records of pressure at the side wall of shock tube as incident and reflected shocks pass gage. (a): incident shock M s = 3 and reflected shock in argon, timing trace period 10 ps. (b): incident shock on expanded time scale, timing trace period 1 ps.

The versatility of this piezoelectric gage is illustrated by some of its applications. It has been successfully used (i) to measure pressures behind detonation and shock waves in shock tube experiment^,^^ (ii) to study shock front configurations and to determine pressure distributions behind high Mach number shock waves generated in an electromagnetic shock tube,81,82(iii) to measure pressure profiles of one dimensional shock waves produced in water-to-water impact, as well as (iv) for auxiliary purposes such as triggering and shock velocity measurement. 5.10.6. Dilatational Gage 5.10.6.1. Contact Strain Sensor. Figure 44 shows a gage due to Baganoff"' which has an extremely short response time. It is essentially the same as the basic gage depicted in Fig. 30 except that the length of the strain element is greater than its diameter and the sensor, which in this case is a capacitor, is in contact with the stressed surface. The gage operates on the principle that the strain propagating into the strain ele-

FIG.44. Dilatational gage using an electrical capacitor as sensor. 1: sensor electrode, conducting epoxy. 2: outer electrode, silver paint. 3: polycarbonate.

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

605

ment is purely dilatational, i.e., the material undergoes no lateral extension. This should be true provided the applied pressure is uniform and measurements are completed before strain propagates to the sensor from the lateral boundary or from the back surface after reflection. Under these conditions, the equation for strain propagation is strictly one-dimensional, and the relationships of Section 5.10.2 should be exact. If the capacitor is initially charged with a voltage V , and if there is no leakage of charge during measurement, the voltage signal V ( t )of Eq. (5.10.10) becomes

(5.10.33) where E and c d are, respectively, the dilatational modulus and propagation velocity, Eq. (5.10.6). When all important pressure changes take place in an interval which is large compared to the time for a strain wave to propagate through the capacitor Eq. (5.10.33) then reduces to V(t)= - p E

'

(t - c'-, )

(5.10.34)

Tests of this gage, using air shocks of negligible thickness, have shown that it responds to step function loading as predicted by Eq. (5.10.33), and shown by the middle graphs of Fig. 31. The records show no overshoot or spurious oscillations. With the strain element made of polycarbonate polymer (General Electric's Lexan) and with a separation h = 0.13 mm for the capacitor plates, the response time T = h/cd is about 0.1 ps. The hold time, during which the measurement must be completed, is set by the arrival of relief strain from the lateral boundary. For a gage having the dimensions given in Fig. 44, the hold time is approximately 5 ps. The value of the modulus E calculated by Eq. (5.10.6) from the measured value of c d is reported to be several times larger than the listed static value. This is not unusual for a polymer experiencing such a high rate of strain. Baganoff does not discuss the effect of material relaxation; presumably it is too slow to be important. A unique aspect of this gage is that the underlying propagation theory is both simple and exact. This and its extremely short response time are very attractive features. One of its limitations is that the applied pressure must be uniform across the sensitive surface of the strain element. Thus it is suitable for measuring pressures at the end plate of a linear shock tube, but, unlike a bar gage, not along a side wall. Another limitation is that the hold time is small, so measurements cannot be carried out over a long period. But it is ideal for certain applications, such as the one for

606

5.

MEASUREMENT OF PRESSURE

which it was designed, namely to measure pressure profiles across shock fronts about 1 cm thick traveling in low density argon.

5.10.6.2. Free Surface Motion. Another gage, which has the fastest response of any proposed, also depends on the propagation of dilatational strain. Pressure is applied to one surface of a plate and the subsequent motion of the opposite surface, which is free, is measured with a chr~no-interferometer,~~J~~~~~~ such as referred to in Section 5.6.3. The motion is observed up to the time that regularity of the first reflection is disturbed by the arrival of a signal either from the periphery of the region over which the applied pressure is uniform or from the lateral boundary of the plate or from a second internal reflection. The hold time is given by the smallest of the values given approximately by r / c d , W / k d or 2 1 / c d , where r is the radius of the region of uniform pressure, w is the width of the plate and 1 is its thickness. The measured velocity u ( t ) of the back surface of the plate is related to the applied pressure p (t) by Eq. (5.10.17), (5.10.35)

where the zero of time is 1 / c d after pressure is applied to the front surface. In many high pressure applications, the plate is stressed beyond its elastic limit, where c d is a slowly varying function of the pressure, so it is sometimes necessary to determine c d by measuring the time for stress to propagate through the plate. But a determination of v ( t ) is the basic measurement, which is made with the chrono-interferometer. In a chrono-interferometer the output of a photodetector V ( t )is proportional to the light intensity of two interfering beams Z(t) = 21,

cos2[aF(t)

+ p],

(5.10.36)

where F(t) is referred to as the “fringe count” because the intensity passes through a maximum and a minimum and returns to its original value when F(r) changes by 1. See Chapter 2.4 of Part 2 for more description of interferometers. In general, F(t) is not an integer but it and the intensity are constant provided the optical path difference A for the interfering waves is constant and their wave length A is the same. Two basically different types of interferometer have been proposed. In one type,”’ illustrated by the Michelson arrangement shown in Fig. 45, one of the interfering beams is reflected from a fixed mirror and the other L. M. Barker and R. E. Hollenbach, Rev. Sci. Instrum. 36, 1617 (1%5). L. M.Barker and R. E. Hollenbach, J . Appl. Phys. 41,4208 (1970).

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

suring FIG. 45. surface Chronointerferometer displacement Al(t). for meal:

moving surface. 2: stationary mirror. 3: beam splitter. 4: light source. 5: photodetector.

607

"." all

2

is reflected from a mirror whose movement is to be sensed. If v ( t ) is the velocity of the moving mirror, the frequency f of the reflected wave is Doppler shifted an amount Af given to a good approximation by

(5.10.37) where the velocity of light F is much greater than v ( t ) . The rate of change of the fringe count is the frequency difference, Af, of the interfering waves, i.e., the beat frequency. Thus if u(t) is zero when t is zero,

(5.10.38) where Al(r) is the displacement of the mirror and A(t) is the increase in optical path length. This leads to Eq. (5.6.17) of Section 5.6.3 when substituted into Eq. (5.10.36). A drawback of this type of interferometer is that it is basically a displacement sensor since fringe count is the measured quantity. To obtain the velocity v ( t ) the fringe count must be differentiated with an accompanying loss in accuracy. A second type of chrono-interferometer, which is of more recent origin,54*112 measures velocity directly. Such an interferometer is shown in Fig. 46. In this case light from the source is reflected from the surface whose movement is to be sensed before it is split into two beams, one of which travels a greater optical length A before it arrives at the photodetector where it is combined with the beam which has traveled the shorter path. The distance A is constant and can be made fairly large by using laser light with a long coherence length. The frequencies of the interfering waves are Doppler shifted and by different amounts if the velocity of the moving surface is changing since they would have been reflected at different times. If the wave traveling the shorter path was reflected at time t when the velocity was v(t) it would interfere with the other wave

608

5.

MEASUREMENT OF PRESSURE

FIG.46. Chronointerferometer for measuring surface velocity u ( t ) . 1: moving surface, 2: laser, 3: lens, 4: delay path, 5: beam splitters, 6: photomultiplier, 7: alternate position for photomultiplier.

which had been reflected at time t - T when the velocity was v(t - T ) , where T is A/?, the “delay time” for the wave traveling the longer path. Using Eq. (5.10.37) and the fact that the rate of change of the fringe count is the beat frequency of the interfering waves, we have F(t) =

A

[1;

u ( t ) dt

-

1:

u(t

- T ) dt

1.

(5.10.39)

If t, the time after the reflecting surface started to move, is less than T , the second integral of Eq. (5.10.39) is zero and the expression for F(t) is the same as given by Eq. (5.10.38). For t greater than T , Eq. (5.10.39) reduces to (5.10.40)

or in terms of uav(r), the velocity averaged over the interval from t

-

7

to

(7

(5.10.41) If v ( t ) varies monotonically, ua,(t) can usually be replaced to a good approximation by u(t - 7/2), the instantaneous velocity at the midtime of the interval. For a step change in velocity of amount u , the fringe count increases lin-

5.10.

FAST RESPONSE GAGES: COMPRESSIONAL STRAIN

609

early with time over an interval 7 and then remains constant at a value determined by Eq. (5.10.41) with u substituted for vav(f). An upper limit on the size of a velocity step is set by the fact that if it is too large the time between successive fringes will be so small that individual fringes cannot be resolved; this holds for the velocity interferometer during the interval T and for the displacement interferometer at all times. For an early velocity interferometer having a single delay line as shown in Fig. 46, nonresolution of fringes over the interval T results in a loss of the integer but not the fractional count of fringes. To be able to determine the integer count unambiguously, later velocity interferometers have been built with two, simultaneously operating, delay lines having T ’ S which differ by a noninteger multiple.113 The response time of the velocity interferometer is T = A/i. and its sensitivity is Flu = 2T/h = 2 A / t h . With values of T in the range 1-10 ns feasible, a dilatational gage using a velocity interferometer can provide one of the shortest response times available. With the simple chrono-interferometers pictured in Figs. 45 and 46, the mirrors must be precisely ground to provide essentially pure specular reflection and the source must supply light having a long coherence length. These requirements set a practical upper limit on the length of the delay time that can be employed. Consequently, very low velocities cannot be measured with velocity interferometers of this type because impossibly long delay times are needed to provide several fringes which are necessary for accurate measurement. On the other hand, as the velocity to be measured with a displacement interferometer is increased it becomes progressively more difficult to distinguish and count individual fringes because of an increase in fringe frequency. In Barker’s 1972 review,s4 he suggested the advantage lies with the velocity interferometer for measuring velocities greater than 100 m . s-l, but shifts to the displacement interferometer for velocities below this value. Since this review, however, velocity interferometers have been developed which can operate with a diffuse reflecting surface and for which the requirement of coherent light is much less severe, thus permitting larger values of T to be used. These interferometer^'^^^^^^ operate on the principle that good fringes can ‘I3 R . A. Lederer, S. A. Sheffield, A. C. Schwarz, and D. B. Hayes, The use of a dualdelay-leg velocity interferometer with automatic data reduction in a high explosive facility. In “6th Symposium (International) on Detonation” (D.J . Edwards, ed.), ACR-221, p. 668. Office of Naval Research, Arlington, Virginia, 1976. *I4 L. M . Barker and R . E. Hollenbach, J . Appl. Phys. 43, 4669 (1972). 115 B. T. Amery, Wide range velocity interferometer. In “6th Symposium (International) on Detonation” (D. J. Edwards, ed.), ARC-221, p. 673. Office of Naval Research, Arlington, Virginia, 1976.

610

5.

MEASUREMENT OF PRESSURE

be obtained with practically incoherent light provided the images of the source produced by the two interfering beams appear to be coincident from the point of view of the photodetector. This has been accomplished, and at the same time a nonzero value of T obtained, by the use of lenses in one design115and by the use of a thick plate with a high index of refraction in a n ~ t h e r . ” ~A velocity interferometer of this type has provided 1-2 percent accuracy over a range of u from 8 m s-’ to 400 m s-l. The measured velocity can be related to the applied pressure by Eq. (5.10.35). For example, ifp = 8000 kg mP3and c d = 5 km s-l, the applied pressure is 0.16 GPa (or 1.6 kbar) for u = 8 m SO and 8 GPa (or 80 kbar) for u = 400 m s-’. Considerably higher velocities and pressures have been measured with similar accuracy in other experiments. Chrono-interferometers have been used particularly in shock wave experiments with colliding plates and contact explosives to study the dynamical behavior of solids stressed well beyond their elastic limit. An important result of the measurements made with shocks in metal plates is the establishment of properties of materials which can be used to calibrate “gages” for high pressure measurements. Each of the four metals Cu, Mo, Pd, and Ag has been mixed with ruby crystals and the mixture subjected to steady high pressures in a piston and cylinder press employing diamonds. Pressure-volume relations for these metals at pressures beyond 100 GPa, established by shock wave experiments, were used to “measure” the pressure by determining volumes by x-ray diffraction inside the high pressure cell. Simultaneous optical measurements then provided a pressure calibration of the wavelength shift of the ruby R, fluorescent radiation .”” Subsequently the ‘‘ruby gage” calibration from 6 to 100 GPa was extrapolated and used to measure a sustained pressure of 170 GPa in the piston and cylinder press employing diamonds. This is the highest known pressure measurement. The reading of the ruby R1pressure gage is performed by determining the wavelength shift AA between 100 kPa (1 bar) and the high pressure, and using p = 380.8[(AA/694.2

+ 1)5 - 11,

(5.10.42)

where Ah is in nanometers and p is in gigapascals. The calibration carried out by Mao and B e l P is estimated to have systematic uncertainty of k 10 percent at 100 GPa and below, and to be within + 20 and - 10 percent at the highest pressure of 170 GPa.

li6

H.K.Mao and P. M. Bell, Science 200, 1147 (1978).

6. MEASUREMENT OF COMPOSITION* 6.1. Introduction A complete measurement of the composition of a fluid requires the identification of the chemical species present and the determination of their concentrations. For these purposes, the experimenter has at his disposal the whole of modern analytical chemistry. The range of available methods is so great that a fully comprehensive survey cannot possibly be attempted within the confines of a single presentation. It is therefore the purpose of Chapter 6.1 to give an indication of the range of methods available, to provide the reader with general guidelines towards the selection of an appropriate experimental method, and to show him where further information may be obtained. Subsequent sections will then concentrate on certain methods which are particularly applicable to measurements on fluids. 6.1.l.Description of Composition

The types of species to be measured in a chemical system include atoms, molecules, free radicals, ions, and electrons. The composition of a fluid is most often specified by listing the concentrations of individual components as molarities in moles per liter or as number densities in particles per cubic centimeter. Compositions may also be expressed in mole fraction units, i.e., as moles of component per mole of fluid. For low density gases, compositions are often stated as partial pressures in atmospheres; this is convenient if thermochemical calculations are to be made since most thermochemical tables use atmospheres as pressure units. The total concentration in a fluid can be related to the other state variables through the equation of state. A standard work on this subject is Hirschfelder rt (11. For gases at moderate densities, where the ideal gas equation may not any longer be accurate enough, a number of other equations of state are available (Ref. l , Ch. 3). The virial equation of state (Ref. 1, Ch. 3) ‘J. 0. Hirschfelder, C. F. Curtiss, and R . B. Bird, “Molecular Theory of Gases and Liquids,” corrected ed. Wiley, New York, 1964.

* Part 6 is by

John E. Dove. 61 I

METHODS OF EXPERIMENTAL PHYSICS, VOL. 18B

Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-475956-4

612

6.

MEASUREMENT OF COMPOSITION

PV/RT

=

1

+ B ( T ) / V + C ( T ) / V 2+

*

. .,

(6.1.1)

where V is the volume occupied by 1 mole and B(7‘), C ( T ) , . . . are the second, third, . . . virial coefficients, has the advantage that the coefficients can be theoretically related to intermolecular potential functions. Tabulations of values of second virial coefficients are For mixtures, when direct experimental data are not available, second virial coefficients can be estimated by the use of empirical combining laws (Ref. 1, Ch. 3). For liquids, the Tait equation of state (Ref. 1, Ch. 4) is widely used. Reference may also be made to tabulations’**of densities of liquids and solutions at various temperatures. A convenient approximate method for calculating the equation of state behavior of pure dense gases and liquids is provided by the principle of corresponding states (Ref. 1, Ch. 4). Application of this method is facilitated by the use of Hougen-Watson chart^.^-^^ The method can also be extended to mixtures of dense gases (Ref. 1, Ch. 4). Special formulations12are required for the state parameters of fluids in the critical region. 6.1.2. Identification of Species Present

Clearly, an essential step in analysis is to identify the chemical species present. Even when only the concentration of a single known component is to be measured, one may still need a complete qualitative analysis to be sure that there is n o other substance present which might interfere with

V. B. Leonas and E. V. Samuilov, High Temp. Engl. Trans/. 4, 664 (1966). M. E. Boyd, J. Rrs. Null. Bur. Siund., Seci. A 75, 57 (1971). ‘J . S. Gallagher and M. Klein, J . Res. Nail. Bur. Siand. Srci. A 75, 337 (1971). D. E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed. McGraw-Hill, New York, 1972. ‘ G . W. C. Kaye and T. H. Laby, “Tables of Physical and Chemical Constants.” Longmans, Green, New York, 1973. R. C. Weast, ed., “Handbook of Chemistry and Physics.” CRC Press, Cleveland, Ohio. (Published annually.) * R. Perry and C. H. Chilton, “Chemical Engineers’ Handbook,” 5th ed. McGraw-Hill, New York, 1973. 0. A. Hougen, K. M. Watson, and R. A. Ragatz, “Chemical Process Principles,” Part 11, “Thermodynamics,” 2nd ed. Wiley, New York, 1959. lo 0. A. Hougen, “Chemical Process Principles Charts,” 3rd ed. Wiley, New York, 1964. * I J. B. Maxwell, “Data Book on Hydrocarbons.” Van Nostrand-Reinhold, New York, I95 I . (Contains state data charts generally similar to Hougen-Watson charts.) (Reprinted by Krieger, Huntington, New York, 1970.) J. M.H. Levelt Sengers, W. L. Greer, and J. V. Sengers, J. f h y s . Chem. Ref. Data 5, 1 (1976).

’

6.1.

INTRODUCTION

613

measurement. In fact, some methods of concentration measurement, e.g., chromatography, spectroscopy, and mass spectrometry, provide information about the identity of the components, so that qualitative and quantitative analysis can be combined into a single procedure. Where they cannot be so combined, many specific qualitative tests for individual substances are available and are described in standard references .I3-l5 6.1.3. Classification of Methods of Composition Measurement

In this section are listed briefly the various types of methodL6which can be used to determine composition. The main objective of this listing is to help the experimenter to make sure that he has not overlooked a simple method of solving a problem. For example, when analyzing mixtures of two known substances, a straightforward measurement of density or refractive index may determine composition much more readily than, say, chromatography or mass spectrometry. 6.1.3.1. Chemical Methods. Determination by chemical reaction.

(i) Gravimetric analysis. The component of interest is reacted to form a known substance, usually a solid precipitate, which is weighed. (ii) Volumetric analysis. This measures the volume of a standard reagent solution consumed by the substance being determined. 6.1.3.2. Physical Methods.* Almost every physical property is potentially usable to help to identify a substance, o r to determine the amount present. Examples include:

(i) Mechanical properties: density; viscosity; sound speed; sound absorption. (ii) Thermal properties: vapour pressure; melting point; boiling point; osmotic pressure; thermal conductivity; heat of reaction; transition temperatures. (iii) Electrical properties: electrical conductivity; current-voltage curves; cell electrode potentials; dielectric constant.

’

I. M. Kolthoff and P. J . Elving, eds., “Treatise on Analytical Chemistry.” Wiley (Interscience), New York. (Volumes published from 1959 onwards.) l4 F. Feigl and V . Anger, “Spot Tests in Inorganic Analysis,” 6th ed. Elsevier, Amsterdam, 1972. Z. Rappoport, ed., “Handbook of Tables for Organic Compound Identification,” 3rd ed. CRC Press, Cleveland, Ohio, 1967. l 6 G. W. Ewing, “Instrumental Methods of Chemical Analysis,” 4th ed. McGraw-Hill, New York, 1975.

* See also Volumes 3A and B (Molecular Physics) in this treatise, as well as Volumes 13A and B (Spectroscopy).

614

6.

MEASUREMENT OF COMPOSITION

(iv) Magnetic properties: magnetic susceptibility. (v) Interaction with electromagnetic radiation: Rayleigh scattering: Raman scattering: absorption of radiation: emission of radiation: index of refraction: refractive dispersion; optical rotatory dispersion: circular dichroism; fluorescence; phosphorescence; electron spin resonance; nuclear spin resonance. (vii) Nuclear properties: nuclear mass: radioactivity.

6.1.4. Selection of Method of Composition Measurement

In selecting a method to measure composition, the first essential is to define carefully what information is actually needed. Is only one species to be determined, or is a complete analysis required? Will it suffice to find ratios of concentrations, or are absolute values needed? If absolute concentrations are to be determined, can one measure relative values and then calculate absolute concentrations using a knowledge of the state variables? Or must direct absolute measurements be made? What is the expected range of magnitudes of concentrations? What accuracy is required'? What are the time constraints? If the concentrations vary with time, what time resolution or response rate must the equipment have? How many measurements are likely to be needed, in total? An important related question is whether the composition measurement should be made in situ, or whether the fluid should be sampled for external analysis. Advantages of in situ measurement include: (i) Direct absolute measurement of concentrations is possible. (ii) Disturbance of the flow system by the sampling process is avoided.

However some disadvantages are:

(i) Equipment for composition measurement may have to be specially built or adapted. (ii) Many good analysis methods are not readily applicable in situ. Advantages of taking a sample for external analysis include:

(i) A wide variety of methods is available. (ii) Standard apparatus can be used. (iii) More time may be available for analysis, giving better sensitivity and accuracy. Disadvantages are :

(i) There may be disturbance of the flow system. (ii) The sampling process itself may distort the sample composition.

6.1.

INTRODUCTION

615

The question of calibration also needs careful consideration. Often the best method is to use a known sample of the substance of interest. For chemically reactive species, this can be difficult; in a high temperature system it may, however, be possible to use an equilibrium mixture to obtain a calibration for such species. Sometimes, a reference sample of known composition is not available at all. Then the absolute size of the effect, e.g., the amount of heat evolved when atoms recombine, must be measured carefully and related by calculation and theoretical reasoning to, e.g., the atom concentration in the system under study. When the above points have been clarified, the methods available can be assessed with respect to criteria such as: accuracy; sensitivity; response speed to concentration changes; time required to complete a measurement; certainty (e.g., freedom from ambiguity of species identification); cost; requirement for special skills. A method can then be selected. A number of the points raised in this section on method selection may seem self-evident. Nevertheless experience shows that some of them are easily overlooked. Careful and explicit consideration of these points can be well repaid by results. 6.1.5. Separation Methods

In the case of a sampled fluid, it may be necessary to separate some of the components to avoid interference with the subsequent composition measurement. method^'^*'^,^' available include: precipitation as an insoluble solid; addition of a reagent which forms an inactive complex; solvent extraction; dialysis, i.e., use of semipermeable membranes; fractional distillation; chromatographic methods; electrolysis; ion exchange. 6.1.6. Sources of Informationl8

Chemical Abstracts published by the American Chemical Society, and Analytical Abstracts published by the Analytical Division of the Chemical Society of London, England, provide information in all areas of analytical chemistry. Many organizations can conduct computer searches, based I’ C. L. Wilson and D. W. Wilson, eds., “Comprehensive Analytical Chemistry,” Vol. IIB. Elsevier, Amsterdam, 1968. R. E. Maizell, “How to Find Chemical Information.” Wiley, New York, 1979.

616

6.

MEASUREMENT OF COMPOSITION

on keywords supplied by the requester, of Chemical Abstracts tapes for the past several years. A computer based retrieval service is also provided by the United Kingdom Chemical Information Service of the Chemical Society. Kolthoff and Elving13 provide detailed theoretical and practical information about analytical methods in a massive treatise which is added to periodically and which, at the time of writing, comprises some 30 vol~ umes. Advances in Analytical Chemistry and I n ~ t r u m e n t u t i o n ’provides critical reviews of various techniques. Many journals provide coverage of analytical chemistry and instrumentation including Analytical Chemistry, The Analyst, Zeitschr$t fur Analytische Chemie, Analytica Chimica Acta, Tulanta, Review of Scientijic Instruments, and Journal of Scientijic. Instruments. Each April, Analytical Chemistry publishes an Annual Reviews issue containing authoritative re-

views of most areas of analysis.

6.2. Analysis of Sampled Fluids In this chapter, sampling methods and several analytical techniques for externally sampled fluids will be described. 6.2.1. Sampling Methods and Problems

Here we suppose that the fluid is to be sampled into an external volume and then transferred to analysis equipment. By necessity, therefore, we confine our consideration to relatively nonreactive fluids; if chemically reacting systems are sampled in this way, the reaction is likely to continue after sampling and to change the composition to an unknown extent. This latter statement may well remain true even if a hot fluid is very rapidly cooled during sampling. Most radical-radical reactions, and some radical-molecule reactions, have low activation energies and may continue in the cooled mixture. Sampling of rapidly reacting mixtures requires special techniques that will be described in Chapter 6.5. However for slowly reacting fluids, it may be feasible to withdraw the mixture directly into a reagent which will immediately quench the reaction in a known manner. le “Advances in Analytical Chemistry and Instrumentation.” Wiley, New York. (Published, under various editors, at intervals since 1%0.) P. G . Jeffery and P. J. Kipping, “Gas Analysis by Gas Chromatography,” 2nd ed. Pergamon, Oxford, 1972. 21 “Methods for the Analysis of Fuel Gases,” B S 3156. British Standards Institution, London. 1968. ’’ “Methods for the Sampling and Analysis of Flue Gases,” Part 1, BS 1756. British Standards Institution, London. 1971.

6.2.

ANALYSIS OF SAMPLED FLUIDS

617

It is, of course, essential to obtain a representative sample20-23of the fluid. One must consider whether the flow is intrinsically homogeneous, or whether the composition is time-variable or perhaps stratified. For a homogeneous flow, a spot sample will suffice. Otherwise the gas must be sampled continuously over a long period, or a series of spot samples must be taken. Statistical criteria for samplingz4may need to be considered. If the composition of the flow is stratified, then it must be mixed or spot samples taken from appropriate points. Sometimes the variations in composition will themselves be of interest, in which case a series of spot samples will be taken. For gases, sampling probes will normally be of metal, glass, or a refractory ceramic. Care must be taken that the act of sampling does not distort the composition. The probe diameter is usually chosen to be large enough to ensure viscous flow, otherwise the sample is likely to be enriched in the lighter components. Sampling probes must be kept hot enough to ensure that no component condenses. Small amounts of water are especially hard to sample. Glass, ceramic, and most metal surfaces normally have significant amounts of water adsorbed on to them. If the sampling system is insufficiently degassed, it may introduce water vapor into the sample. But a thoroughly degassed system can remove a substantial fraction of the sampled water vapor. Often it is useful to keep the sampled fluid flowing until it has equilibrated with the sampling volume. Even so, when the sample is later transferred to the analytical system, part of the water from the surface of the sampling volume may also be transferred. Other polar species such as ammonia and sulfur dioxide can also be lost by adsorption. Stopcock grease is to be avoided with condensibles; it will absorb them in varying amounts and release them again to subsequent samples. Rubber will absorb hydrocarbons. Moreover, hydrogen, helium and, to a lesser extent, oxygen will diffuse through natural rubber and some plastics. For sampling liquids, most of the general statements above also apply. Special problems arise when the liquid contains dissolved gas which it is desired to analyze separately. Techniques for separating the gas are described by Jeffery and Kipping.20 6.2.2. Chromatographic Methods 6.2.2.1. Principles of C h r o r n a t ~ g r a p h y . ~Chromatography ~.~~,~~ is a countercurrent separation process which normally takes place by move~

~~

ps “Standards for Petroleum and Its Products,” 2nd ed., Part IV. Institute of Petroleum, London, 1962. *‘ E. B. Wilson, “An Introduction to Scientific Research,” Chapter 7. McGraw-Hill, New York, 1952. 25 E. Heftmann, “Chromatography.” Van Nostrand-Reinhold, New York, 1967.

6.

618

MEASUREMENT OF COMPOSITION

ment of a gas or liquid past a stationary liquid or solid. The actual separation occurs as a result of selective partition between the moving and stationary phases, in a manner roughly analogous to fractional distillation. However the partition process may depend on differing extents of adsorption, on solubility differences, on differing degrees of penetration into molecular-sized holes in the support material, or on differing chemical equilibria between sample components and the stationary phase. When combined with identification and quantitative detection of the separated components, chromatography is a versatile tool of chemical analysis. 6.2.2.2. Gas C h r ~ r n a t o g r a p h y .Gas ~ ~ ~chromatography ~ ~ ~ ~ ~ * ~ ~ has now almost completely replaced the older types of gas analysis equipment. The essential features of a typical gas chromatograph are shown in Fig. 1. The technique can be used for mixtures of gases or volatile liquids. A sample of the mixture to be analyzed is injected into a continuous stream of carrier gas and swept through the chromatograph column. With a suitable choice of column packing, the components of the mixture will be separated and will emerge at different times. Under standardized conditions for a given column, the retention time in the column is characteristic of a substance. With an appropriate detection system, the output signal is proportional to the concentration at the detector. An example of a gas chromatogram is shown in Fig. 2. The stationary phase in the column may be either a granular solid (gas-solid chromatography) or a nonvolatile liquid coated onto a solid support (gas -liquid chromatography); the latter has some advantageszo*28~28 and is much more widely used. A long capillary coated on the Column r - - 7

meter

Oven Carrier

Q Amplifier

lhLhl Recorder

FIG.1 . Basic components of a gas chromatograph.

A. B. Littlewood, “Gas Chromatography.’’ Academic Press, New York, 1970. J . P. Okamura and D. T. Sawyer, Gas chromatography. I n “Physical Methods in Mode m Chemical Analysis” (T. Kuwana, ed.), Vol. 1, p. 2. Academic Press, New York, 1978. 28 S. Dal Nogare and R. S. Juvet, “Gas-Liquid Chromatography.” Wiley (Interscience), New York. 1962. zE 27

6.2.

ANALYSIS OF SAMPLED FLUIDS

619

i\

CYCLOHEXANE

FIG.2. Gas chromatogram of a mixture of vapors of three liquids, obtained on a capillary column of 2-mm diameter and 1.83 m in length, coated with Carbowax@20M. This illustrates the good separation of three substances with very similar boiling points, cyclohexane 81.4”C, benzene 8O.l0C, ethanol 78.5”C. [Courtesy of D. W. Riddle.]

inside with a liquid may also be used as a column; such columns can give very large separation factors and need only small samples. An important factor in gas chromatography is the choice of the stationary phase20.29*30 to give good separation of the components of a particular mixture. The commonest detector is the thermal conductivity detector.31 A heated resistive element immersed in the output gas stream forms one arm of a Wheatstone bridge. If the composition of the gas changes, the resultant alteration in the amount of heat conducted from the hot element alters its resistance and creates an off-balance signal from the bridge. Hydrogen and helium are the best carrier gases to use with this detector because their thermal conductivities are very different from those of any other gas. This detector is simple, robust, and universal, but relatively insensitive. Ionization detectors are very much more sensitive. The flame ionization d e t e c t 0 1 3 ~depends * ~ ~ ~ ~ on ~ the fact that when organic compounds are burnt, they produce significant concentrations of gaseous ions. The effluent from a chromatograph is introduced into a hydrogen-air flame *9 G. Zweig and J . Sherma, eds., “Handbook of Chromatography,” Vols. 1 and 2. CRC Press, Cleveland, Ohio, 1972. Analabs, “Guide to Stationary Phases for Gas Chromatography.” Analabs Inc., North Haven, Connecticut (revised annually). 31 A. E. Lawson, Jr. and J. M. Miller, J . Gas. Chromarogr. 4, 273 (1966).

620

6.

MEASUREMENT OF COMPOSITION

whose electrical conductivity is monitored by measuring the current flow between a pair of electrodes. Formation of ions in the flame leads to an increased current flow. This detector has a linear response over a very g of hydrocarbon. large dynamic range and can detect as little as However the detector is totally insensitive to many common gases such as C02 and Nz. Another type of ionization detector creates ions by collision with metastable Ar or He and can be even more sensitive than the flame detector. Measurement of retention times20*2s.32 is an important aid to identification of components. Tabulated relative retention timeszQare useful here. Note that the retention time is related directly to the partition coefficient between the component and the stationary phase which, of coime, is constant at a particular temperature and pressure. The specific retention volume,32 which may be calculated from the retention time, should be independent of experimental variables such as the column length, size of the packing, and the flow rate of the gas. An excellent, but expensive, identification tool is the mass spectrometer.20*33Part of the chromatograph effluent is led into the mass spectrometer ion source. As each chromatograph peak appears, the characteristic mass spectrum of that component is also recorded. Calibration for quantitative analysis is by use of samples of known composition. For approximate work, the detector peak height can be used as a measure of concentration, but it is better to use the peak area. Nowadays, peak areas are usually obtained by mechanical or electronic integration. With care, concentration measurements can be made to within 1 percent. The time required for a single analysis is typically a few minutes to several hours. Often the time can be shortened and the analysis facilitated by using “temperature programming,” i.e., by systematically increasing the temperature of the column during the separation ~ ~ o c ~ s s . ~ ~ ~ ~ ~ - ~ ~ Gas chromatography can, naturally, be used only for stable species. Nevertheless it is an extremely versatile, effective and sensitive tool for the analysis of mixtures of gases and volatile liquids. It finds wide application in studying the composition in all types of gas flow systems. As an example of the range of its applicability, it can even be used to analyse mixtures of the isotopes and spin isomers of gaseous hydrogen. H . Purnell, “Gas Chromatography,” Wiley, New York, 1962. C . J . W. Brooks and B . S. Middleditch. Gas chromatography-mass spectrometry. Spec.. Period. Rep.: Muss SpcJctrom.2, 302 (1973). 32

6.2.

ANALYSIS OF SAMPLED FLUIDS

62 1

6.2.2.3. Liquid C h r o m a t ~ g r a p h y .The ~ ~ . technique ~~ of liquid chromatography shows many similarities in principle to gas chromatography. Typically a small sample of the solution or liquid mixture is placed at one end of a column packed with a granular solid (which may be coated with a suitable liquid) and is then eluted through the column by a continuous flow of solvent. The mechanisms which can be used for the separation process in the column include ion exchange, adsorption, and variations in solubility. Analysis by liquid chromatography tends to be slow. In order to achieve separations in times comparable to gas chromatography, pressures up to several hundred atmospheres may be used to force the liquid through the column. The most usual detectors are the differential refractometer, in which the refractive index of the liquid from the column is continuously compared with that of the pure solvent, and the ultraviolet absorptiometer in which the absorption of uv (often 250-280 nm from a mercury lamp) is continuously monitored. Other detectors use variable wavelength light absorption, fluorescence, electrical conductivity changes, and radioactivity measurements. 6.2.3. Gravirnetric, Volumetric, and Electrochemical Analysis Methods

Particularly for inorganic substances dissolved in liquids, an extremely wide range of analytical techniques is available. Kolthoff and Elving13 give a comprehensive survey, and there are many monographs on particular methods, e.g., Heyrovsky and Z ~ m a n .There ~ ~ is currently a strong trend to automation of analytical methods, often involvi.ng the use of microprocessors or minicomputers. 6.2.4. Absorption Spectrophotometry of Samples”

Measurements of the absorption of ultraviolet, visible and infrared radiation are widely used in chemical analysis. Below, the principles of such measurements, and methods of application to analysis of sampled fluids, are outlined. 34 J . J. Kirkland, ed., “Modern Practice of Liquid Chromatography.” Wiley (Interscience), New York, 1971. 35 W. J . Price, “Analytical Atomic Absorption Spectrometry.” Heyden, London, 1972. 38 J . Heyrovsky and P. Zuman, “Practical Polarography.” Academic Press, N e w York,

1%8.

* See also Volumes

13A and 13B (Spectroscopy) in this treatise.

622

6.

MEASUREMENT OF COMPOSITION

6.2.4.1. Absorption S p e ~ t r a . l ~ A , ~given ~ - ~ molecular ~ species will absorb radiation at wavelengths characteristic of that molecule, giving rise to an absorption spectrum. The molecule can be identified, and its concentration determined, by measuring that absorption. Consider a thin absorbing layer of thickness dx. The absorbed intensity at wavelength u,, corresponding to a molecular transition from states m to n is - ders =

PnmNmBmnhv,, dx;

(6.2.I )

N , is the number of molecules per unit volume in the lower state, pnmis the density of radiation and B,, is the Einstein transition probability for the absorption. The intensity falling on unit area in unit time is @" = cpnm,so that

- detr, = ~ " N , B , , h ~ , , dxlc.

(6.2.2)

B,, can be calculated from quantum mechanics: (6.2.3) where R"" is the matrix element connecting states m and n. From Eq. (6.2.2),we see that the absorbed intensity is proportional to N , and @". Combining Eqs. (6.2.2) and (6.2.3), we have -deKq

-

(6.2.4)

The preceding applies only to transitions between single states, i.e., between nondegenerate levels. For transitions between levels of degeneracy g, and g,, the Einstein transition probability becomes (6.2.5)

where i a n d j number the degenerate states of the lower and upper levels, 37 E. U. Condon and G. H. Shortley, "The Theory of Atomic Spectra." Cambridge Univ. Press, London and New York, IW. 38 G. Herzberg, "Atomic Spectra and Atomic Structure," 2nd ed. Dover, New York,

1944. 38 G. Herzberg, "Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules." Van Nostrand-Reinhold, New York, 1950. G . Herzberg, "Molecular Spectra and Molecular Structure. 11. Infrared and Raman Spectra of Polyatomic Molecules." Van Nostrand-Reinhold, New York, 1945. G . Herzberg, "Molecular Spectra and Molecular Structure. 111. Electronic Spectra and Electronic Structure of Polyatomic Molecules." Van Nostrand-Reinhold, New York,

1966.

6.2.

ANALYSIS OF SAMPLED FLUIDS

623

and the summation is over all possible combinations of lower with upper states. Note, that, for a fixed total number N of noninteracting or weakly interacting molecules of a given species, the number of molecules in any level m at equilibrium varies with temperature according to the Boltzmann distribution law

(6.2.6) where E , is the.energy of level m and Q is the molecular partition function. 6.2.4.2. Principles of Absorption Spectrophotometry.1s.42-44We have seen that the amount of radiation absorbed in a thin layer is proportional to the incident intensity and to the concentration of the absorbing species. Hence the intensity will fall off exponentially through a thick layer. Therefore when an intensity Zo of a single wavelength passes through a cell which contains absorbing molecules, the transmitted intensity Z is given by A = lOg(Zo/Z)

=

E~C.

(6.2.7)

This is one form of Beer’s law. C is the concentration of absorbing molecules in moles per liter, b is the length of the absorption cell in centimeters, and E is the molar absorptivity in liters per mole per centimeter. A is called the absorbance; it will be seen that it is directly proportional to C. Therefore if A is measured, C can be found if ~b is known. This is the principle of absorption spectrophotometry . For the best accuracy in measuring C, the conditions should be such that A = 0.4343. For a mixture of two substances obeying Beer’s law, it can readily be shown that the absorbances are additive so that the total absorbance A is A

=

b ( ~ l C 1+ EZC~).

(6.2.8)

If A is measured at two different wavelengths, then C1 and C, can be found. Naturally the best accuracy will be obtained if two regions can be found where first one and then the other species does not absorb. The extension to multicomponent systems is obvious in principle, though it may be difficult in practice if the spectra are complex and overlap sub-

42 C. K . Mann, T. J. Vickers, and W. M.Gulick, “Instrumental Analysis.” Harper, New York, 1974. J . W. O’Laughlin and C. V. Banks, Differential spectrophotometry. In “The Encyclopedia of Spectroscopy” ( G . L. Clark, ed.),p. 19. Van Nostrand-Reinhold, New York, 1960. 44 R. P. Bauman, “Absorption Spectroscopy.” Wiley, New York, 1962.

624

6.

MEASUREMENT OF COMPOSITION

stantially. An excellent discussion of precision spectrophotometry has been given by O’Laughlin and Banks.43 6.2.4.3. Deviations from Beer’s Law

6.2.4.3.1. EFFECTOF LARGECONCENTRATIONS. The absorptivity E is normally independent of C . However at large concentrations of absorber, interactions between absorber molecules can cause E to depend on C . Then the value of ~b obtained at lower concentrations cannot be used. However A can still be used to measure C if a calibration curve is constructed using known concentrations of absorber. 6.2.4.3.2. EFFECTOF FINITE B A N D W I D T H . Beer’s law is derived for light of a single wavelength. However it can be extended to the absorbance of light over a finite bandwidth, provided that E is effectively constant over that bandwidth. If E varies significantly over the bandwidth transmitted by the spectrometer, then Beer’s law does not apply. Nevertheless, as above, A can still be used to measure C from a calibration curve. This effect is likely to be most serious for gases, for which line width^^^ are often determined mainly by Doppler b r ~ a d e n i n gand ~ ~ are typically one part in loe of the wavelength. Beer’s law will therefore not be obeyed in such cases unless the spectrometer has an extremely narrow bandwidth. 6.2.4.3.3. OTHERD E V I A T I O N S . Apparent deviations from Beer’s law can also be caused by stray light in the spectrometer, or by a chemical interaction between components in the mixture being studied.

6.2.5. UIt ravio Iet and Vis i bIe Spect rophoto met ry16,39,41,42,44,46 The electronic spectra of molecules generally lie in the ultraviolet and visible regions. Electronic transitions are accompanied by simultaneous changes in rotational and vibrational energy. The electronic spectrum of a gaseous molecule will, therefore, often appear to consist of broad bands, but examination under high resolution will generally reveal a fine structure of individual vibrational-rotational lines. In the liquid phase, however, intermolecular interactions broaden the energy levels so that the fine structure merges. Commercially available uv-visible spectrophotometers cover the spectral region from about 165 t o 1000 nm, i.e., from the early part of the

*’ D. R . Lide, Microwave spectroscopy. In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3 , Part A, 2nd ed., p. 1 1 . Academic Press, New York, 1974. C. W. Mathews, Electronic spectroscopy. In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3, Part A , 2nd e d . , p. 203. Academic Press, New York, 1974.

6.2.

ANALYSIS OF SAMPLED FLUIDS

625

vacuum uv to the near infrared. However the region below about 200 nm, where atmospheric gases begin to absorb strongly, presents many experimental difficulties. In uv spectrophotometry, compounds can be identified by recording their spectra and comparing with standard reference s p e ~ t r a . ~ ’ -Con~~ centrations can be determined from absorbances at specific wavelengths. In principle, this method can be applied to almost all organic, metal organic and inorganic compounds, with the exception of a few gases whose electronic spectra lie wholly below 165 nm. In practice, analysis of multicomponent mixtures may be quite difficult. The components of a typical spectrophotometer, and an example of a spectrum of a liquid, are shown in Figs. 3 and 4. Note that double-beam operation is common, to avoid the necessity of determining and subtracting a “blank” spectrum. A typical spectral bandwidth is 1 nm. Ultraviolet spectrophotometers are mostly used for liquids so that deviations from Beer’s law due to this spectral bandwidth are generally very small, particularly if absorbance measurements are made at spectral maxima. PHOTOTUBE /

CONTROL

,

, ,

@ , k

SAMPLE

GRATING or PRISM

MONOCHROMATOR

‘PHOTOTUBE

FIG.3. Basic components of a single beam ultraviolet spectrophotometer 47 “DMS UV Atlas of Organic Compounds.” Vols. I and 2, 1966; Vol. 3, 1967; Vol. 4, 1968; Vol. 5, 1971. Butterworth, London and Verlag Chemie. Weinheim. L. LAng, “Absorption Spectra in the Ultraviolet and Visible Region.” (Volumes published at intervals: Vol. 23, 1979; Vols. 1-15, published by Akademiai Kiad6, Budapest; Vols. 16-20, Adadtmiai Kiad6 and Academic Press, New York; Vols. 21-23, Akademiai Kiad6 and Krieger, Huntington, New York.) J. P. Phillips, H. Feuer, and B. S. Thyagarajan, eds., “Organic Electronic Spectral Data.” Wiley (Interscience), New York. (Volumes published at intervals; Vol. X, 1974.) 50 J. W. Robinson, “Handbook of Spectroscopy,” 2 vols. CRC Press, Cleveland, Ohio, 1974. J . G. Grasselli, “Atlas of Spectral Data and Physical Constants for Organic Compounds,” 2nd ed., 6 vols. CRC Press, Cleveland, Ohio, 1975.

’*

626

6.

MEASUREMENT OF COMPOSITION

WAVELENGTH (nm)

FIG.4. Ultraviolet absorption spectrum of liquid toluene. Note the very strong and rather featureless absorption at short wavelengths; small amounts of another component absorbing at these wavelengths would be very hard to detect. The more structured band at about 270 nm is much weaker and in an instrument of small dynamic range would be unobserved except in measurements on thick layers. Note that absorption is plotted as increasing upwards. [Adapted, with permission, from a spectrum by H. H. Perkampus and G. Kassebeer in Ref. 47.1

The uv absorption spectrophotometer is often a good and rapid means of checking the identity and purity of a compound and of analysing a simple solution to an accuracy of 1-2 percent. The method can be quite sensitive since strong absorption bands will often represent molar absorptivities of 104-105 liter mole-' cm-'. With a 10-cm spectrometer cell, good accuracy can then be obtained for concentrations of the order of lod6 mole liter-', and the detection limit can be below lop8mole liter1. For complex mixtures, however, overlapping bands often make the spectra hard to unravel, and chromatography is likely to be easier to apply.

6.2.

627

ANALYSIS OF SAMPLED FLUIDS

6.2.6. Infrared Spectrophotometry

a molecule of N atoms 6.2.6.1. Vibrational S p e c t ~ a . In ~ ~general, .~~ has 3N - 6 normal modes of vibration of characteristic frequencies. Usually some, but not all, of these normal modes are infrared active. The resulting spectral transitions between vibrational energy levels generally occur at wavelengths greater than 2 pm. While each normal mode is, in principle, a vibration of the whole molecule, in practice certain specific atomic groupings (“functional groups”) are found to give rise to absorptions at characteristic frequencies. These characteristic frequencies are relatively constant from molecule to molecule, and this can be a valuable aid to identification. For example, the - OH group shows stretching and bending absorptions which almost always fall in the ranges 2.7-2.8 and 7.0-7.5 pm respectively. 6.2.6.2. Methods and A ppI icat io ns.16,42350,52*53 Many commercial infrared spectrophotometers are available. The basic features of a typical instrument are shown in Fig. 5. Note that there is no solid which is transparent to the whole infrared region. Various materials must therefore be used for optical components in different spectral ranges. Some of these materials, e.g., NaCl, are not compatible with water or water vapor.

s RECORDER

REFERENCE

I

WEDGE

I

+ MONOCHROMATOR

I

I

I

\UL

CHOPPER

FIG.5 . Basic components of a double-beam infrared absorption spectrophotometer. The beam from the monochromator is chopped and passes alternately through reference and sample. If the two beams are out of balance, an ac signal is produced and amplified to drive a servo motor which moves a wedge in order to equalize the two beams. As the monochromator scans the wavelength range, the position of the servo motor is therefore a measure of the infrared absorption difference between reference and sample paths. 12 W. E. Blass and A. H. Nielsen, Infrared. In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3. Part A, 2nd ed., p. 126. Academic Press, New York, 1974. J. E. Stewart, “Infrared Spectroscopy.” Dekker, New York, 1970.

62 8

6.

MEASUREMENT OF COMPOSITION

Also, all solvents have absorption bands in the infrared region, and these must be taken into account in the analysis of solutions. Generally speaking, the spectral region from 2 to 8 p m is especially useful for identifying particular functional groups present and thus gaining a general idea of the identity of a molecule. Several very useful tabulations of functional group frequencies have been p ~ b l i s h e d . ~ ~ Absorp-~’ tions in the 8-15 p m region are often complex and reflect interactions between several normal modes; this “fingerprint” region can often be used for almost unambiguous identification of pure substances by comparison with atlases of known ~ p e c t r a . ~ ” . ~ An ~ * ~example * - ~ ~ of an absorption spectrum of a liquid is shown in Fig. 6. Note that quite often infrared spectra are plotted as the transmittance T = Z/Zo, so that absorption regions will appear as dips, rather than as maxima as is usual in uv spectra. Infrared spectrophotometry can be used for quantitative analysis by measuring the absorbance and relating it to the concentration. The con2.5

WAVELENGTH (pm) 4 5

3

6

8

10

15 20 30

a-“ 100 0

z 80 60

5 v7

5

[L

c

40

20

o 4000

3500

3000

2500

2000

WAVE N U M B E R

1500

loo0

500

(ern-')

FIG..6. Infrared absorption spectrum of liquid toluene, showing the functional group region, 2.5-8 pm, and the fingerprint region, 8-15 prn. [Courtesy of D. W. Riddle.]

=* N. B. Colthup, J . Opt. Soc. Am. 40, 397 (1950). F. F. Bentley and E. E. Wolfarth, Spectrorhim. Acta 15, 165 (1959). R. F. Goddu and D. A. Delker, Anal. Chrm. 36, 783 (1960). ST L. Meites, ed., “Handbook of Analytical Chemistry.” McGraw-Hill, New York, 1963. 58 W. W. Simons, ed., “Sadtler Handbook of Infrared Spectra.” Sadtler Research Laboratory Inc., Philadelphia, Pennsylvania, 1978. 58 “Selected Infrared Spectral Data”, Am. Pet. Inst. Res. Proj. 44. Thermodynamics Research Center, Texas A & M University, College Station, Texas. (Tabulations of infrared spectra, updated periodically.) 6o C. J. Pouchert, “The Aldrich Library of Infrared Spectra.” Aldrich Chemical Co., Milwaukee, Wisconsin, 1970. H. A. Szymanski and R. E. Erickson, “Infrared Band Handbook,” 2nd ed.. 2 vols. IFI/Plenum, New York, 1970. JJ

56

6.2.

ANALYSIS OF SAMPLED FLUIDS

629

siderations about Beer’s law, which were stated in Section 6.2.4 apply also to infrared spectra, naturally. In general, infrared spectrophotometry is excellent for identification of pure substances and for rapid analysis of relatively simple mixtures to an accuracy of several percent. Sensitivity is good because infrared absorption is often quite intense, though not as intense as the strongest uv bands. Nevertheless, for complex mixtures the spectra are likely to be hard to interpret, especially where there are unknown substances present, and chromatographic methods will then often be preferable. 6.2.6.3. Fourier Transform S p e ~ t r a . A ~ ~relatively .~~ recent development is the measurement of infrared spectra by the Fourier transform method. The radiation to be analyzed is passed through a Michelson interferometer, and the output intensity is recorded as a function of position of a movable interferometer mirror. The resulting interferogram must then be Fourier transformed by a computer to give the actual spectrum. The major advantage is that all of the available radiation is used throughout the recording of the interferogram. By contrast, only one wavelength is recorded at a time in a standard spectrometer. Since infrared spectrometers are normally limited by detector noise, the Fourier transform method can give a large increase in signal to noise ratio. Commercial spectrometers are available. The technique makes severe demands on the mechanical precision of the equipment and is currently expensive. Nevertheless, continuing rapid development is likely and this method should be considered when high sensitivity is at a premium. 6.2.7. Mi crowave Spectra42,45,62

The rotational spectra of gaseous molecules typically consist of a very large number of lines in the microwave region. The frequencies of these lines can be measured with great precision and are highly characteristic of particular molecular species. Microwave spectroscopy, using sensitive microwave spectrometers which are now commercially available, appears in principle to be a very good method for qualitative and quantitative analysis. Only about a micromole of sample is needed, and a typical detectability limit is of the order of 0.1 percent. The method also seems to be very suitable for monitoring a continuously sampled fluid. However so far it has not become widely used, probably because of expense.

82 W. Gordy and R. L. Cook, Microwave molecular spectra. In “Technique of Organic Chemistry” (A. Weissberger, ed.), Vol. IX, Part 2, Wiley (Interscience), New York, 1970.

630

6.

MEASUREMENT OF COMPOSITION

6.2.8. Laser S p e c t r ~ r n e t r y ~ ~ * ~ ~

The use of lasers is creating major changes in existing spectroscopic techniques, and is also opening up wholly new methods based, for example, on stimulated and multiphoton effects. Continuously tunable lasers, which can already generate lines in the visible and near uv which are much narrower than the typical Doppler width of gas spectra, are likely to become extremely useful for absorption spectrophotometry, especially of gases (cf. Section 6.2.4.3.2). 6.2.9. Other Methods

Particularly for two component fluids, the measurement of a simple physical property such as density, refractive index, or viscosity can be a quick and convenient method of monitoring composition. 6.2.9.1. Radioactive tracer^.^^*^^-^^ Radioactive tracers can be detected at extremely low concentrations and measured with great precision by counting techniques, and have wide analytical applications. Isotope dilution analysis can be very useful for measuring the amount of a substance present when that substance is hard to isolate quantitatively or to measure by other methods. A small amount of the substance in radioactive form, of known activity, is mixed into the system. Then a sample is taken, the substance extracted, purified, and the activity of a known amount measured. From the extent of the dilution of the radioactive material, the amount of the inactive substance in the system can be directly calculated. The method relies on the fact that the active and inactive forms will usually behave identically in the extraction and purification. Note that the efficiency of the extraction and purification does not need to be known. Neutron activation or radioactivation analysis is a highly sensitive method of analyzing for certain elements.

6.3. Analysis of Radiation Absorbed by in Situ Fluid Analytical operations in situ tend to be intrinsically more difficult and less accurate than those on an external sample. Therefore in situ meaR . J . Pressley, ed., “Handbook of Lasers.” CRC Press, Cleveland, Ohio, 1971. Academic Press, New York, 1979. BJ H . A. Strobel, “Chemical Instrumentation,” 2nd ed. Addison-Wesley, Reading, Massachusetts, 1973. G . Friedlander, J . W. Kennedy, and J . M. Miller, “Nuclear and Radiochemistry,” 2nd ed. Wiley, New York, 1964. 83

’‘C. B. Moore, ed., “Chemical and Biochemical Applications of Lasers,” Vol. 4.

6.3.

ANALYSIS OF R A D I A T I O N ABSORBED B Y I N SITU FLUID

631

surements are most often made when the act of sampling would disturb the system, or when studying very fast processes such as those behind shock waves.

6.3.1.General Principles The principles of absorption spectrophotometry were discussed in Section 6.2.4. However for in situ measurements, some additional points should be noted. It is important to realise that while absorptivities are quoted on a per mole basis, in fact only the molecules in one particular level or group of levels absorb at a given wavelength [cf. Eq. (6.2. l)]. If the distribution over molecular energy levels changes, the amount of light absorbed will change. Consequently, absorptivities are generally temperature dependent. In addition, spectral line widths of gases will change with temperature, and this will alter the amount of light absorbed from a narrow line source. In order to allow for these effects, it is wise to calibrate under conditions as similar as possible to the actual experiments. If the absorbing levels are not in thermal equilibrium with the rest of the gas, for example during chemical reaction or relaxation of internal energy, then clearly the measured spectral absorption cannot be converted to a molar concentration by using the absorptivity. However it may be possible to use the absorption to get useful data about the populations of the absorbing levels themselves .68 Where the concentrations to be measured are varying rapidly with time, e.g., in a shock wave experiment, additional problems arise. The need for fast response will restrict the choice of detection systems. Systems using choppers will usually not be fast enough, and the advantages of a double beam system will be reduced; in practice, most systems for in situ spectrophotometry are single beam systems. Another problem is that light emission from flames, plasmas, or shock waves can interfere with absorption measurements. Also, signal to noise problems6Qwill be enhanced by increased detector bandwidth and by shot noise. These problems are particularly severe in the infrared where absorptivities are relatively low. For spectrophotometry in situ, it may not be feasible to use or adapt complete commercial instruments, in which case the equipment must be constructed from separately built or purchased light sources, detectors, etc. For such cases. there are several useful sources of advice about

@’ H. A. C. McKay, “Principles of Radiochemistry,”

Butterworth, London, 1971. J . P. Appleton, J . Chem. Phys. 47, 3231 (1967). N . Davidson, “Statistical Mechanics,” Chapter 14. McGraw-Hill, New York, 1%2.

632

6.

MEASUREMENT OF COMPOSITION

techniques, and about suitability and compatibility of materials and components .42,43.48,50,70 6.3.2. Ultraviolet Absorption

Important applications of in situ uv absorption are to follow very rapid changes in reactant or product concentrations in systems such as shock and detonation waves and to measure atom and free radical concentrations in gas The light source may be a small spectral region isolated from a continuum by a monochromator, o r an emission line or band from a specially chosen lamp. Even when using a special lamp, a filter or monochromator may still be needed to eliminate unwanted emission. For detection of atoms, atomic resonance radiation from a lamp filled with the same element can be used. The detector is almost invariably a photomultiplier. It is often difficult to ensure that Beer's law will be obeyed, so that calibration curves must be obtained. Four applications of these techniques will be described briefly. Jost, Wagner, Troe and co-workers have made a classic series of studies of the kinetics of pyrolysis of gaseous triatomic molecules in shock waves, using absorption and emission spectrophotometry. References and a summary of results are given by Troe and Wagner.'* For example, the pyrolysis of N 2 0 diluted in Ar was ~ t u d i e d 'by ~ uv absorption photometry: NIO

+ M-NS

+0+M

0 + N20 +NZ

0 + NpO

----j

[I1

+02

2N0.

N 2 0 has a quasi-continuous absorption at wavelengths below 307 nm, while NO has a number of sharp uv absorption bands. By appropriate choice of wavelength, the absorption of N,O only or of N 2 0 and NO together, could be measured. It was, incidentally, found that the absorptivity of N 2 0 is strongly temperature dependent (cf. Section 6.3.1). The light source was a xenon arc; a monochromator limited the detected radiation to a wavelength region of about 0.8 nm. In this work, N 2 0 removal could be measured at total gas concentrations of 1.5-15 x mole 70 A. G . Gaydon and 1. R. Hurle, "The Shock Tube in High Temperature Chemical Physics." Chapman & Hall, London, 1963. D. W. Setser, ed., "Reactive Intermediates in the Gas Phase: Generation and Monitoring.'' Academic Press, New York, 1979. 72 J . Troe and H. Gg. Wagner, B e r . Bunsenper. Pliys. Chem. 71, 937 (1967). '3 W . Jost, K . W . Michel, J . Troe, and H . Gg. Wagner, Z . Nuturforsch., Ted A 19, 59 (1964).

6.3.

ANALYSIS OF RADIATION ABSORBED B Y I N SITU F L U I D

633

liter-' and temperatures of 1500-2500 K , with as little as 0.5 percent N,O in the initial mixture. These conditions correspond to time constants, for N 2 0 removal, of the order of 1 ps to more than 1 ms. Figure 7 shows a typical experimental arrangement for absorption spectrophotometry behind a reflected shock wave. Appel and A p p l e t ~ nhave ~ ~ used the atomic resonance absorption spectrophotometry (ARAS) technique75 to measure the rate of dissociation and oxidation of deuterium behind shock waves, e.g., DP+ A r - D

+ D + Ar.

The light source was a deuterium discharge lamp, with a molecular oxygen gas filter to isolate the 121.6-nm line. The system was calibrated by shock heating mixtures containing 0.2 percent NzO and very small amounts of D2; the D2was converted quantitatively to D atoms. The rate of D2 dissociation was measured over the temperature range 1800-4000 K; over this range, the measured dissociation rate constants extend over nearly six decades. The ARAS technique is extremely sensitive for measuring atom concentrations, and has also been applied to H and 0 The concentration of OH radicals can be followed by monitoring absorption due to the Z - r ( O , 0) resonance band system near 308 nm. The light source can be a xenon arc continuum from which a 1-2-nm region is isolated by a m o n ~ c h r o m a t o ror , ~an ~ OH resonance 1amp.78*79Smith and SHOCK SPEED

SHOCK

HIGH PRESSURE SECTION

LOW PRESSURE SECTION

TUBE

L,GHT SOURCE

OSCl LLOSCOPE

FIG.7. Basic components for uv/visible absorption photometric measurement of a chemical reaction behind a reflected shock wave.

'' D. Appel and J . P. Appleton, S y m p . (Inr.) Combust. [Proc.],15th, Tokyo. 1974, p. 701. Combustion Institute, Pittsburgh, Pennsylvania, 1975. 7s A. L. Myerson and W. S. Watt, . I. C h e m . PhyJ. 49, 425 (1968). 76 W. S. Watt and A. L. Myerson. J . C h e m . Phys. 51, 1638 (1969). 77 C. T. Bowman, S y m p . ( I n r . ) Combust. [ P r o c . ] , 15th. Tokyo, 1974, p. 869. Combustion Institute, Pittsburgh, Pennsylvania, 1975. '* T. Carrington and H. P. Broida, J . Mol. Spectrosc. 2, 273 (1958). " C. Morley and I . W. M . Smith, J . C h r m . Soc., Furuduy Trans. 2 68, 1016 (1972).

634

6.

MEASUREMENT OF COMPOSITION

Zellnerso have applied this technique to study reactions of OH produced by flash photolysis, e.g., CO + OH-COZ + H. By working with a known excess of CO, the reaction can be made to be pseudo first-order. Then the rate constant can be obtained from a plot of log[OH] against time, and an absolute calibration for OH is not necessary. This method of detecting OH is very sensitive; Smith and Zellner estimate that their OH concentrations were always less than 1.6 X 10’’ cm-3. In a recent shock tube study of the reaction between CH4 and 02, Bowman?? measured the time-dependent OH concentrations by monitoring absorption of a 1.6-nm region isolated from a Xe-Hg arc continuum. The calibration for the absorption measurements was obtained by shock heating H2-02-Ar mixtures, so that the OH concentration rapidly reached a calculable “partial equilibrium” concentration. 6.3.3. Electron Spin Resonance81-83*

Electrons, and many nuclei, possess an intrinsic angular momentum associated with “spin.” In a strong applied magnetic field, different spin states will have different energies. Transitions among these states can be induced by a second weaker field varying at an appropriate radio frequency. These transitions may be detected in an electron spin resonance (ESR) or nuclear spin resonance (NMR) experiment. Both techniques can be used for measurements on fluids. However ESR is of particular interest because it can be used to measure radical concentrations which would be difficult to study by any other method. In a typical ESR experiment, a microwave field of fixed frequency induces the actual transitions and the spectrum is scanned by varying the strength of the main magnetic field. In general, only species with net spin, i.e., free radicals, have ESR spectra. For solutions, ESR can be applied to a wide range of radicalss4 of I . W. M. Smith and R. Zellner, J . Chem. Soc., Faraday Trans. 2 69, 1617 (1973). A. Carrington and A. D. McLachlan, “Introduction to Magnetic Resonance with Applications to Chemistry and Chemical Physics.” Halsted Press, New York, 1979. P. B. Ayscough, “Electron Spin Resonance in Chemistry.” Methuen, London, 1967. 83 J . D. Memory and G . W. Parker, Resonance studies. In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3, Part B, 2nd ed., p . 465. Academic Press, New York, 1974. BI D. J . E. Ingram, “Free Radicals as Studied by Electron Spin Resonance.” Butterworth, London, 1958. 8o

* See also Section 9.7.3. Volume 2B (Electronic Methods) in this treatise.

6.3.

ANALYSIS OF RADIATION ABSORBED B Y I N SITU F L U I D

635

varying degrees of complexity, of which the Mn2+ion o r the C2H, . radical are simple examples. With a suitably designed cell, the formation and reaction of radicals formed, e.g., by gamma radiation can be studied in sit^.'^ ESR studies of radicals in solution are reviewed regularly, see, e.g., D a v i e P and earlier reports in that series. In gases, atom concentrations of about log cm-3 upwards can be measured. ESR has been widely used t o study the chemical kinetics of atoms in flow systems. Westenberge7 gives references of gas phase kinetic studies up to 1972, and he lists 22 different atoms (including isotopic species) which have been detected in the gas phase. In the case of gaseous diatomic or polyatomic radicals, the completely free rotation of the radicals interacts with the spin to produce a large number of states each with a small population. The ESR spectrum is therefore spread over a large number of lines and becomes hard to detect, so that it is much more difficult to study molecular radicals in gases than in liquids. Nevertheless over 20 diatomic or triatomic free radicals have been detected in the gas phase by ESR,87and successful kinetic studies of the reactions of OH have been made.88~Rg In one equipment design,8gdetection and measurement takes place in a flow system which actually passes through the ESR cavity (Fig. 8).

MAGNET POLE

P

Fic. 8. Experimental arrangement for ESR study of reaction of 0 atoms with Hzand CHI. 0 atoms are produced by reacting N atoms with NO, or by a discharge in 0,. Note that the reactant injector is movable along the flow tube axis, t o allow variation of reaction time. [Adapted, with permission, from Westenberg and de H a a ~ . ~ @ ] K . Eiben and R. W. Fessenden, J . Phys. Chem. 75, 1186 (1971). A . G. Davies, Annu. Rep. Prog. Chem., Sect. B 75,70 (1978), and earlier reports in this series. 87 A . A . Westenberg, Prog. Reacr. Kine!. 7 , 23 (1973). Bs W. E. Wilson and A. Westenberg, Symp. (Int.) Combust. [Prof.],Ilth, Berkeleyy,1966, p. 1143. Combustion Institute, Pittsburgh, Pennsylvania, 1967. ea A . A . Westenberg and N . de Haas, J . Chem. Phys. 46,490 (1967). BB

6.

636

MEASUREMENT OF COMPOSITION

Wagner and co-workerse0 sample directly from the flow system into the cavity (Fig. 9). Incidentally an important point is that atomic ESR transitions are always magnetic dipole transitions, whereas for free radicals they may be either magnetic or electric dipole transitions; the type of cavity to be selected depends on the type of transition to be measured. Current ESR spectrometers tend to have a fairly slow response. Though they can readily be used to monitor fluctuations of radical concentration occurring on a time scale of a fraction of a second, the ESR method has not yet been applied to the much faster changes occurring in, say, a shock tube experiment. Absolute calibration of the concentration sensitivity of the ESR system, in the case of liquids, can be carried out using known amounts of the chemically stable free radical 1,l-diphenyl-2-picrylhydrazyl. In the gas phase, the stable molecular free radicals O2and NO can be used similarly.87~gi~e2 Alternatively, atom concentrations can be measured by “titration” reactions with stable m 0 1 e c u l e s . ~ ~For ~ ~example, ~ titrati~n~~,~~ with NO can be used to measure N concentration or to produce a known amount of 0 atoms: N+NO-N,+0.

Similarly H atoms from a discharge can be determined by measuring the amount of NOz which is just enough to remove them by the fast reaction:

‘N’Td>ug, A PRESSURE

PUMP

INLET +2

MIONq SOURCE E C

INLET 3

+

(movable) P FURNACE T U B E

E.S.R CAVITY

+

PUMP

FIG.9. Schematic diagram of experimental arrangement in use at the University of Gottingen for study of atom and radical reactions by ESR and mass spectrometry. K. Hoyermann, H . Gg. Wagner, and J . Wolfrum, Ber. Eunsmges. P h y . ~Chem. . 71,599 ( 1967). u1

82 83 O4

O5

S. Krongelb and M. W. P. Strandberg, J . Chern. Phys. 31, 1196 (1959). A . A. Westenberg, J . C h m . P h y ~ 43, . 1544 (1965). A . A . Westenberg and N. de Haas, J . Chem. Phys. 40, 3087 (1964). F. Kaufman and J . R. Kelso,J. Chem. Phys. 27, 1209 (1957); 28, 992 (1958). B. Brocklehurst and K . R. Jennings, Prog. Reuct. Kiner. 4, 1 (1967).

6.4.

ANALYSIS OF RADIATION EMITTED B Y IN SITU FLUID

H

+ NO,-OH

637

+ NO.

By using these reactions, the absolute sensitivity of an ESR spectrometer to N , 0, H and OH can be determined.

6.4. Analysis of Radiation Emitted by in Situ The radiation emitted from a fluid can be analyzed to give information about the identity and concentration of excited species present.7*16,37-42.46,52,96-98 This technique is usually applied to gases rather than to liquids and will be discussed in that context. Emission of radiation is also used to measure temperature, as discussed in Chapter 4.2 of this volume. The emphasis in the following discussion is on methods of determining concentrations of species. Examples of systems on which such measurements have been made include flames, shock and detonation waves, gas discharges, arcs, sparks, and systems exposed t o ionizing radiation. 6.4.1. Types of Emission

Three general types of spectral emission may be distinguished: continuous emission, where at least one of the states involved is unbound; banded emission, where the transitions are between bound molecular states; line emission from transitions between bound states of atoms. Measurements of banded and line spectra can therefore be used to obtain information about molecular and atomic species in a fluid. Continuous emission often arises from recombination reactions between neutrals: Br

+ Br

or between charged species: Ar+ + e-

-

BrZ + hv

-

Ar

+ hv

and can be used to determine the concentrations of the recombining species. D6 A. G. Gaydon, "The Spectroscopy of Flames,'' 2nd ed. Chapman & Hall, London, 1974. 97 R. W. B . Pearse and A. G. Gaydon, "The Identification of Molecular Spectra," 4th ed. Chapman & Hall, London, 1976. 98 G. R . Harrison, "M.I.T. Wavelength Tables," 2nd ed. MIT Press, Cambridge, Massachusetts. 1970.

63 8

6.

MEASUREMENT OF COMPOSITION

6.4.2. Principles of Spectral Emission

The theory of emission 6.4.2.1. Emission lntensities.37-41~46~52.99-” spectra has been considered extensively in many texts on quantum mechanics and spectroscopy. Here we will outline only a few topics concerned with the quantitative treatment of emission intensities. Consider a transition from a state n to a lower state m, by emission of a photon of energy hv,, . The intensity of spontaneous emission is proportional to the number N , of atoms or molecules in the upper state:

ZiE

= A,,N,hv,,.

(6.4.1)

The proportionality constant A,, is the Einstein transition probability for spontaneous emission. A,, is related to Rnm,the matrix element for the transition, through A,,

=

(64r44v3,,/3h)lR”m12.

(6.4.2)

For the case of transitions between degenerate levels, this is modified to An, = (64r4~3,,/3h)(CIRnlm~12/gn)

(6.4.3)

with notation as in Section 6.2.4.1. It follows that: B,,

=

A,,/8rhv~,.

(6.4.4)

Particularly for electronic transitions, the intrinsic intensity of a transition may also be expressed as the oscillator strength f”” which is related to B m n by f”” = p h c u , , B , , / r e ~ , (6.4.5) where p and eo are the electronic mass and charge. 6.4.2.2. Frequency Dependence of Intensity. From Eqs. (6.4. I) and (6.4.2), we see that

&?;

fx

U4,,qR”qz.

(6.4.6)

Both emission and absorption intensities [Eq. (6.2.4)]are proportional to the square of the matrix element. Therefore, a transition that is strong in emission will also tend to give rise to a strong absorption. Moreover, other things being equal, intensities of both emission and absorption S . S. Penner, “Quantitative Molecular Spectroscopy and Gas Emissivities.” Addison-Wesley, Reading, Massachusetts, 1959. loo B . H. Armstrong and R. W. Nicholls, “Emission, Absorption and Transfer of Radiation in Heated Atmospheres.” Pergamon, Oxford, 1972. lo’ E. E. Whiting and R. W. Nicholls, Asrrophys. J . 27, Suppl. 235, 1 (1974).

6.4.

ANALYSIS OF RADIATION EMITTED BY I N SITU F L U I D

639

spectra tend to increase with frequency, i.e., with the size of the energy jump. However emission increases much more strongly with frequency than does absorption. For low frequency transitions, therefore, emission will usually tend to be weak. In consequence, while absorption spectroscopic measurements of various types can be made at frequencies extending even down to the megahertz range, emission measurements for transitions among discrete states are not usually attempted at frequencies much below the near infrared (-1014 Hz). 6.4.3. Measurement of Composition from Emission Spectra

In this section, the principles of measurement of composition from emission spectra will be outlined, and some possible problems in such measurements will be briefly discussed. 6.4.3.1. Emission Measurements. From the overall form of the emission spectrum, information can be obtained about the general type of emitting species, e.g., whether atomic, diatomic, or polyatomic. Specific identifications can be made by measuring wavelengths accurately and comparing with published s p e ~ t r a . ~ ~ Observation ~ ~ - ~ ~ ~of~ relative ~ * ~ ~ - ~ ~ intensities will often help to confirm such identification. The concentrations of emitting species can be obtained from measured intensities, using Eq. (6.4.1). For this, a knowledge of A,, is needed; literature data are available for a number of specie^.'^^^^^^ Values of A,, to use in Eq. (6.4.1) can also be obtained from measured emission intensities in carefully designed calibration experiments. However, it is often difficult to know excited state concentrations exactly. Therefore, for transitions involving the ground state, it is usually preferable to determine B,, instead from absorption experiments. Note that A,,,,, B,,, andf", are interrelated through Eqs. (6.4.4)and (6.4.5), so that if one is known the others can be calculated. Values off"" can also be obtained from quantum mechanical calculations which, however, are generally lengthy and expensive. The direct determination of N , from emission spectra requires difficult and painstaking measurement of absolute intensities, even when A,, is known. Some of these difficulties may be avoided by calibrating the equipment using emission from a well-understood system. Nevertheless it is advisable, if possible, to design the experiment in such a way that only relative intensities need to be measured. lo2

B. M. Miles and W. L. Wiese, Bibliography on atomic transition probabilities. N u f l .

Bur. Srand. Spec. Pub/. 320(1970);J . R. Fuhrand W. L. Wiese,ibid. Suppl. l(1971); Suppl. 2

(1973). lo5

H. Klemsdal, J . Quanr. Specfrosc. & Rudiai. Transfer 13, 517 (1973).

640

6.

MEASUREMENT OF COMPOSITION

6.4.3.2. S e l f - A b s o r p t i ~ n . ~ In ~ ~the ~ ~above ~ ~ ~ ~analysis, - ~ ~ * we have assumed that in order to account for measured intensities we need to consider only emission processes. This assumption is generally satisfactory if N , is sufficiently small. However if N , is increased continuously, e.g., by increasing the thickness of the emitting fluid, the observed emission intensity does not increase without limit. As the thickness increases, radiation emitted in one layer will be partly absorbed by the next. This self-absorption effect will become noticeable first for the strongest emission lines. There will be two effects: (a) the ratio of intensities between strong and weak lines will be reduced, and (b) since the strongest part of a line, i.e., the center, becomes self-absorbed first, the line shape is distorted and in fact broadened. At sufficiently large N , , the emissivity at the line center tends to a constant value, namely the value for a blackbody at that temperature. However this does not mean that 4: eventually becomes constant as N , increases, since the emission line will continue to broaden gradually. Nevertheless, whereas the total emission intensity for a given transition is proportional to N , for optically thin systems, the emission tends to increase only as (N,)1’2when self-absorption is important. Accurate measurements of relative or absolute concentrations are most easily made under optically thin conditions. A simple test for optical thinness is to reflect the emitted radiation through the emitting region again, to see whether this changes the intensity ratios. Alternatively, if absolute intensities have been measured, and line widths are known or can be estimated, the optical depth can be calculated to see whether the system is optically thin. If it is not, then the observed intensities can be corrected for self-absorption, but this requires a detailed knowledge of the line profiles which may be difficult to obtain. If emitted radiation is passed through a cooler layer containing the same species, part of the radiation may be absorbed in the cooler layer, and the resultant intensities will not any longer be an accurate measure of concentrations in the emitting region. 6.4.3.3. Total Species Concentration; Nonequilibrium Problems. When N , has been determined, the total number N of molecules or atoms of that species can be calculated by inserting N , into the Boltzmann distribution law, Eq. (6.2.6), which is however obeyed only if thermal equilibrium exists among the various levels. Deviations from equilibrium are lo*

‘OB

lo’

R. D. Cowan and G. H. Dieke, Rev. Mod. Phys. 20, 418 (1948). A. C. Kolb and E. R. Streed, J . Chem. Phys. 20, 1872 (1952). L. Huldt and E. Knaal, Z . Naturjorvch., lril A 9, 663 (1954). C. G. James and T. M. Sugden, Proc. R . SOC. London, S r r . A 221, 312 (1955). H . R. Griem, “Plasma Spectroscopy.” McGraw-Hill, New York, 1964.

6.4.

ANALYSIS OF RADIATION EMITTED B Y IN SITU FLUID

641

not uncommon in the types of system commonly studied by emission spectrometry. Indeed the observation of a line or banded emission is already an indication that the system is not fully at equilibrium! At total equilibrium, the amount of radiation emitted would be the same as that absorbed, at every frequency,96and the overall radiation would be a continuous black body emission. In principle, therefore, if there is a strong banded o r line emission, the emitting levels will tend to be depleted thereby. In practice, the depletion will often not be appreciable. To some extent, the existence of an equilibrium distribution can be checked experimentally, by measuring the populations of several levels. Or if the rates of the radiative and collisional energy transfer processes in the system are known, one can, in principle, make calculations t o determine whether the collisions will be able to maintain level n at its thermal equilibrium population. 6.4.3.4. Chemiluminescence. In chemically reacting systems, excited species may be generated in highly nonequilibrium distributions. The resulting emission is called c h e m i l u m i n e s ~ e n c e . One ~ ~ can, of course, still use the measured emission intensities to obtain information about concentrations in emitting levels. However, the measured emission from one level cannot be used t o infer concentrations in other levels by use of the Boltzmann distribution law. Indeed, the unsuspected occurrence of chemiluminescence is one source of error in measurements of N by emission spectrometry. 6.4.4. Techniques a n d Applications of Emission Spectrophotometry 6.4.4.1. Techniques. An excellent, though old, article by Diekelog on combustion spectroscopy contains much helpful practical information on emission measurements. There are many useful publications about materials and methods for uv/visible ~pectrophotornetry,~~,~~~~ and for infrared emission r n e a ~ u r e m e n t s . ~ ~ ~ ~ ~ ~ ~ ~ * ~ ~ For uv/visible measurements, the first stage is normally to survey the spectrum and select features for further study. A low dispersion spectrograph with photographic plate recording, or a low dispersion scanning spectrometer, will normally be selected. For detailed measurements, a monochromator or spectrograph of higher resolution will often be required. Dieke gives helpful advice about choice of resolving power. Instruments are commercially available. As might be expected, very high resolution usually involves small apertures and therefore gives low intensities. A phototube is normally used for detection, especially if the emisIn “Physical Measurements in Gas lO8 G . H . Dieke, Spectroscopy of combustion. Dynamics and Combustion” (R. W. Ladenburg, B. Lewis, R. N . Pease, and H. S . Taylor, eds.), p. 467. Oxford Univ. Press, London and New York, 1955.

642

6.

MEASUREMENT OF COMPOSITION

sion is time dependent. For intensity calibration, an equilibrium system will often be used. In studies of weak emitters, efficient light gatheringl10 can be important to the success of the experiment. The White cel1111-l14offers a substantial sensitivity increase over conventional collimating systems. Photographic sensitivity extends only a short distance into the infrared, to about 1.1 p m , so that scanning monochromators or spectrophotometers will normally be used for survey of a spectrum. The Golay ~ e 1 1is~a ~good * ~ universal ~ detector which has, however, an intrinsically slow response and cannot be used for studies of rapidly changing systems, e.g., behind shock waves. A variety of solid state infrared detector^^^.^^ is available for specific spectral regions, and some of these have microsecond response times. For emission studies on a system with a simple band structure, a solid state detector, with an interference filter rather than a monochromator, will often suffice. 6.4.4.2. Applications. Gaydon and c o - w ~ r k e r s , in ~ ~a ~series ~ * ~of~ ~ elegant experimental studies, have applied emission spectroscopy and spectrophotometry to the study of structure and reaction kinetics of flames, yielding a great deal of information of fundamental value. Chemiluminescent emission of metal atoms has been employed by Dixon-Lewis"' in an ingenious manner t o obtain information about the concentration of hydrogen atoms in flames. If small amounts of a metal atom X are added to a flame, under appropriate conditions almost all of the emission from that atom comes from excited species formed in the ~eactions"~ HI0 + X*. H +H +X HZ + X*, H + OH + X

-

-

Where [H]/[OH] is constant, the chemiluminescent intensity is a relative measure of [HI2. Continuum emission from radiative recombination reactions such as Br

f

has been used by Palmer"* and

Br -+

Br,

+ hv

to measure atom concentrations

R. A . Young, J . Opt. Soc. Am. 50, 627 (1960). J . U . White, J. Opt. Sor. Am. 32, 285 (1942). 112 H. L. Welsh, C. Cumming, and E. J. Stansbury, J . Opt. Soc. Am. 41, 712 (1951). H. L. Welsh, E. J. Stansbury, J. Romanko, and T. Feldman, J . Opt. Soc. Am. 45, 338-343 (1955). 'I4 P. E. Charters and J . C. Polanyi, Can. J . Chem. 38, 1742 (1960). 115 A. G. Gaydon and H. G . Wolfhard, "Flames, Their Structure, Radiation and Temperature," 3rd ed. Chapman & Hall, London, 1970. 'I6 G . Dixon-Lewis, M. M . Sutton, and A. Williams, Proc. R . Soc. London Ser. A 317,227 (1970). 11' P. J . Padley and T. M. Sugden, Symp. ( I n / . )Combust. [Pror.],7th. Oxford, 1958, p. 235. Butterworth, London, 1959. 110

6.4.

ANALYSIS OF RADIATION EMITTED BY IN SITU FLUID

643

in shock-heated gases; the observed emission intensity is proportional to the square of the atom concentration, and a calibration can be obtained from measurements of the equilibrium emission. By this method, the atom concentrations were measured in experiment^'^^*'^^ on the dissociation of Br,. Infrared emission has been used to measure concentrations in many shock tube experiments. For example, Olschewski, Troe and Wagner121 have studied the unimolecular decomposition of COz between 2800 and 3700 K , using a monochromator with indium antimonide detector. The method is sensitive, and it is possible to work with substantially less than 0.1 percent of reactant in an inert diluent. Chemiluminescence can also be observed in the infrared. For example, the reaction F

+ Hz ----*

HF

+H

is exothermic by 32 kcal mole-’ and produces vibrationally and rotationally excited HF. By studying the resulting emission in a specially designed flow system, which removes reaction products before they can be collisionally quenched, Perry and Polanyi122measured initial distributions of the product HF in vibrational and rotational levels up to u’ = 3 and J‘ = 13. The study of infrared chemil~minescence’~~ of gases has been developed by Polanyi and co-workers into a major tool for investigating the dynamics of elementary chemical reactions. The spontaneous Raman effect can be regarded as the inelastic scattering of a photon, in which the scattering system or molecule changes R. A. Carabetta and H . B. Palmer, J . Chem. Phys. 46, 1333 (1%7). R . K . Boyd, G. Burns, T. R. Lawrence, and J. H. Lippiatt, J . Chem. Phys. 49, 3804 ( 1968). R. K. Boyd, J . D. Brown, G . Burns, and J. H. Lippiatt,J. Chem. Phys. 49,3822 (1968). H. A. Olschewski, J. Troe, and H. Gg. Wagner, Ber. Bunsenges. Phys. Chem. 70,1060 ( 1966). lZ2 D. S. Perry and J . C. Polanyi, Chem. Phys. 12, 419 (1976). lZs T. Carrington and J. C. Polanyi, Chemiluminescent reactions. In “Reaction Kinetics” (J. C. Polanyi, ed.), MTP International Review of Science, Phys. Chem. Ser. One, Vol. 9, p. 135. Medical and Technical Publishing Co./Butterworth, Oxford, 1972. lZ4H. A. Szymanski, “Raman Spectroscopy,” Vol. 1 . Plenum, New York, 1967; Vol. 2, 1970. M. C. Tobin, “Laser Raman Spectroscopy.” Wiley (Interscience), New York, 1971. Iz6 S. K. Freeman, “Applications of Laser Raman Spectroscopy.” Wiley (Interscience), New York, 1974. 12’ D. H . Rank and T. A. Wiggins, Light scattering. I n “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3, Part A, 2nd ed., p. 395. Academic Press, New York, 1974. 11*

llS

* Raman scattering for determination of temperature, density, and composition is also described in Chapter 3.2.

644

6. MEASUREMENT

O F COMPOSITION

from one quantized state to another. The Raman shift, i.e., the change in frequency of the scattered photon, is therefore a direct measure of the molecular energy change. Observed Raman shifts generally result from interaction with molecular vibrations. Raman active vibrations are those in which the polarizability changes as the molecule vibrates. The Raman spectrum may be contrasted with the ordinary infrared spectrum, where the active vibrations are those in which the dipole moment changes. In consequence, Raman shifts will often be different from infrared frequencies, and a Raman spectrum can occur for molecules that do not absorb in the infrared. The Raman spectrum can be used for identification of species and for concentration measurements, in much the same way as the infrared spectrum. Raman lines can be divided into Stokes and anti-Stokes lines, whose frequencies are respectively less than and greater than that of the exciting line. Clearly the anti-Stokes lines arise from molecules in vibrationally excited states, so that the intensity can be used as a measure of the concentration of excited molecules. The Raman effect is generally a weak effect. Theory shows that the scattered intensity is proportional to the fourth power of the frequency of the incident photon. Experimentally, a highly monochromatic light source, preferably also of high frequency, is used to irradiate the sample; Raman sources are usually in the visible or uv. The Raman scattering is generally observed at right angles to the incident light, in order to avoid swamping the detection system. Since the Raman lines lie on either side of the exciting line, the spectrophotometric techniques used are those appropriate to the exciting frequency. Because of generally low intensities, and other experimental difficulties, the Raman effect has beemelatively little used for concentration measurements. This situation is, hbwever, changing rapidly, with the advent of high power lasers, which are nearly ideal Raman sources. Raman spectrometry has potentially some special advantages for measurements in flowing systems. Since observations are made at right angles to the incident beam, one can arrange to receive only the light scattered from a small chosen element in the interior of the fluid. Because the Raman scattered frequency will not usualiy be appreciably absorbed or emitted by the intervening parts of the fluid, observations can be made essentially free of boundary effects, and various regions of the fluid can be probed. This is in contrast to ordinary absorption or emission studies, where the observed intensities represent an effect integrated along the whole light path, and may be strongly influenced by boundary layers, other inhomogeneities, and self-absorption.

6.5.

MASS SPECTROMETRY

645

As an example of the potentialities of this method, an application by Hirschfeld and others (described by FreemanlZ6)of Raman spectroscopy to remote measurement of atmospheric pollutants may be cited. A high power uv laser pulse is sent to a remote sample. The backscattered light is Raman shifted, giving spectral bands characteristic of the components of the atmosphere. This light is collected and spectrally analyzed to measure the scattered intensity corresponding to various components. Because this is a pulsed system, the signal returned from a small chosen slice of the atmosphere can be selectively detected. The N2 signal is used as a reference for calibration. The system described can detect two parts per million of low molecular weight hydrocarbons at a distance of 400 m. By making use of the resonance Raman effect, much lower levels of certain specific pollutants can be detected. Other examples of measurement in fluid dynamics are mentioned in Chapter 3.2.

6.5.Mass Spectrometry" 6.5.1. introduction

The essential feature of mass spectrometry is the separation of ions, according to their mass to charge ratio, by means of electric or magnetic fields. Neutral species must be subjected to ionization before mass analysis. The mass spectrometer is a uniquely versatile and flexible analytical instrument. Any species which can be introduced into the instrument will produce a mass spectrum; the masses of the ions formed will be a definite (though not necessarily unambiguous) indication of their identities; the concentrations of various molecular species present can be related to the heights of individual peaks in the mass spectrum. The mass spectrometer can therefore carry out qualitative and quantitative analysis in a single operation, with excellent sensitivity. Moreover, by the use of suitable sampling systems, free radicals and other unstable species, as well as stable molecules, can be introduced into the mass spectrometer and mass analyzed. Of course, many of the above remarks could be made about other techniques such as optical spectrometry. Nevertheless, different molecules may be best analyzed in quite different regions of the electromagnetic

* See also Part 5 in Volume 3B (Molecular Physics) of this treatise.

646

6.

MEASUREMENT OF COMPOSITION

spectrum, requiring the use of several different instruments. However, in mass spectrometry, any molecule can be analyzed, in principle, in the one instrument, and it is perhaps this factor more than any other which makes mass spectrometry so widely useful. In this chapter, we concentrate on aspects of mass spectrometry which are particularly relevant to studies of flowing gases. For a more general account of mass spectrometry and its applications to a variety of problems of chemical physics, reference may be made to an article by McDowel1128in an earlier volume of this series, and to other books and articles cited by him. A mass spectrometer normally has four essential features: a sampling system, an ion source, a mass analyzer, and a detector. These four will be treated separately, starting with the ion source. 6.5.2. Ion Sources Atoms and molecules can be ionized by the removal of one or more electrons, or by the attachment of a charged species such as an electron or proton. The minimum energy needed to remove an electron is called the ionization potential of that species; the ionization potential^^^^^^^^ of most molecules lie between 8 and 15 eV. 6.5.2.1. Electron Impact Sources.128The commonest method of ionization is by impact with an energetic electron: XY

+e

-

XY+ + 2e.

If the energy of the ionizing electrons is continuously increased from the ionization potential upwards, the efficiency of formation of XY+increases rapidly at first but then reaches a plateau at typically 70 e V , often tending to decline again at higher energies. Fragment ions can also be formed:

+ e-X+ XY + e - X XY

+ Y + 2e, + Y+ + 2e

Usually, the formation of fragment ions begins a few electron volts above the ionization potential, and the proportion of fragments increases with the energy of the ionizing electrons. 1p8 C. A. McDowell, Mass spectrometry. I n “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3, Part B, 2nd ed., p. 575. Academic Press, New York, 1974. lZ9 J . L. Franklin, J. G. Dillard, H. M. Rosenstock, J . T. Herron, K. Draxl, and F . H. Field, “Ionization Potentials, Appearance Potentials, and Heats of Formation of Gaseous Positive Ions,” NSRDS-NBS 26. US. Govt. Printing Office, Washington, D.C., 1969. V. I . Vedeneyev, L. V. Gurvich, V. N . Kondratyev, V. A. Medvedev, and Ye. L. Frankevich, “Bond Energies, Ionization Potentials and Electron Affinities.” Arnold, London, 1966. (Translated from the Russian; originally published, 1962.)

6.5.

MASS SPECTROMETRY

647

The characteristic pattern of fragment ions formed in a mass spectrometer at a given ionizing energy is termed the mass spectrum of that species. Fragmentation is both an advantage and a disadvantage of electron impact ionization. It is an advantage in that the mass spectrum is a characteristic fingerprint for a given molecular species. Thus there are several possible molecular ions that have m / e 28 including C2Ha, CO’ and N:. However these can be distinguished by the fact that only C2H4 will have a spectrum which includes m/e = 27 (C,H,+), and only CO will yield significant amounts of m / e = 16 (O+). A disadvantage is the fact that fragment mass peaks may tend to mask useful information. For example, it will be difficult to detect traces of the CH, . free radical in the presence of large amounts of CHI because the latter will itself tend to produce a large fragment peak of CHZ. The standard electron impact ion source includes an electron gun for producing a well-defined beam of electrons at any chosen energy up to 100 eV, and an electrode system for extracting ions into the mass analyzer, Ion sources are usually operated at 50- or 70-eV ionizing energy, and standard mass spectra are tabulated at these energies. Incidentally, the design and construction of an efficient electron impact ion source is very time-consuming, and anyone contemplating it in preference to purchasing a commercial model should first read the remarks on page 720 of the article by English and Z01-n.‘~~

-

6.5.2.2. Photoionization source^.^^*,^^^ An atom or other species can also be ionized by a photon of sufficient energy: XY

+ hv

-

XY+

+ e.

The required energy generally corresponds to a photon in the vacuum uv region. Highly monochromatic beams of photons can be produced more readily than monoenergetic electron beams (a typical energy spread of an electron beam is of the order of 1 eV), so that photoionization can be made to be highly specific. In particular, mass spectra can be produced with virtually no fragmentation, and this will allow free radicals to be detected in the presence of species which in an electron impact source would produce large amounts of interfering fragment ions. Photoionization mass spectrometers are currently mainly research tools rather than commercially developed instruments, and their use is probably best reserved for special circumstances. Note also that they tend to generate rather low ion currents, in comparison with electron imT. C. English and J. C. Zom, Molecular beam spectroscopy. In “Methods of Experimental Physics” (L. Marton, ed.), Vol. 3, Part B, 2nd ed., p. 669. Academic Press, New York, 1974. W. A. Chupka and J. Berkowitz, J . Chem. Phys. 51,4244 (1969).

648

6.

MEASUREMENT OF COMPOSITION

pact sources, so that they are not very suitable for fast reaction studies or for other applications where high sensitivity is important. 6.5.2.3. Chemical Ion ization Sou r ~ e s . ’ ~ ~Chemical -*~’ ionization is a relatively recent technique for production of mass spectra. Essentially, the gas to be analyzed is mixed with a 1000-fold excess of a gas such as CH, and is then bombarded with electrons in a special ion source which operates at a higher pressure than in earlier, conventional mass spectrometers. The principal primary ions are CH: and CH:, but these undergo rapid ion-molecule reactions to yield mainly CH: and C2H:. The substance of interest, say M, can then react with these ions to form MH’: M

+ CH:

M

+ CZH:

-

MH+ + CH,, MH’

+ CzH,.

Typically, therefore, the mass spectrum will consist of several ions formed from ion-molecule reactions in CH,, together with the ion MH+. The latter ion is formed with very high efficiency, provided that M has a higher proton efficiency than CzH4 or CH4. In practice, the spectra are often a little more complicated than indicated above. Nevertheless, chemical ionization offers very high sensitivity and a very simple mass spectrum with little or no fragmentation of ions from the species to be analyzed. This technique is rapidly becoming a standard analysis method, and chemical ionization mass spectrometers are now commercially available. In considering application to sampling from flowing systems, two points should however be borne in mind. In chemical ionization, the ion source “flow-through” times are typically rather long, so that the method is not suitable for sampling from systems whose composition changes on the microsecond time scale. Second, any free radicals introduced into the ion source will tend to react with other molecules under the relatively high pressure conditions there and therefore will not be directly detectable by chemical ionization. 6.5.2.4. ton Sources for Free Radicals. In studies of flowing systems, it is often desired to measure concentrations of reactive atoms and free radicals. Several points about such measurements have already been mentioned; the following should also be noted. With electron impact sources, the selectivity towards free radicals can

134

F. H.Field, A c c . Chern. R e s . 1, 42 (1968). B. Munson, A n d . Chern. 43 (13) 28A (1971). R. E. Mather and J. F. J. Todd, I n t . J . M m s Specfrom. [on Phys. 30, I (1979).

6.5.

MASS SPECTROMETRY

649

be improved by operating at low electron energies. For example, the formation of CH; from CHI requires a minimum of 14.25 eV: CHI

+e

-

CH:

+ H + 2e,

whereas the ionization potential of CH3 itself is 9.84 eV. Therefore, by operating with electron energies slightly below 14.25 eV, CH3 can be detected and measured in the presence of large amounts of CH,. The disadvantage is, however, that at such low energies, the ionization efficiency is small and there is a substantial loss of overall sensitivity. Normally, ion sources are designed in such a way that molecules reside there for an appreciable length of time and make many collisions with the wall before being pumped away. One reason for operating in this way is to increase the probability of ionization, but at the same time to'avoid too high a pressure in the rest of the mass spectrometer. However if a mixture of stable molecules and reactive free radicals is admitted to such an ion source, the radicals will rapidly react at the wall while the stable molecules remain in the gas phase. Since the ionization takes place in the gas phase, the resultant detected composition will be seriously distorted and will show much less than the sampled proportion of free radicals. To avoid this difficulty a collision-free molecular beam sampling ~ y s t e m ' ~should ~ * ' ~ ~be used. In such a system, discrimination against background gas can be greatly improved by chopping o r modulating the incoming beam and detecting only the in-phase component of the spectrometer output. 6.5.3. Mass

Of the many types of mass analyzer, we will briefly discuss three which have already been extensively applied in the study of fluid systems. 6.5.3.1. Magnetic Deflection Ana1y~ers.l~~ The magnetic deflection mass spectrometer is well known and quite effective; it is, however, probably somewhat less tolerant of high pressure in the analyzer tube than other types, and it is wise not to exceed 1 X Torr there. 6.5.3.2. Time-of-Flight A n a l y ~ e r s . The ~ ~ ~time-of-flight .~~~ mass spectrometer (Fig. 10) is a pulsed instrument. A group of ions is generated by an electron pulse of approximately 1-ps duration, and is immediately accelerated along a flight tube. Because all singly charged ions acquire

Iae

S. N. Foner and R. L. Hudson, J. Chem. Phys. 45, 40 (1966). C. A . McDowell, e d . , "Mass Spectrometry." McGraw-Hill, New York, 1963. W. C. Wiley and 1. H. McLaren, Rev. Sci. Insrrum. 26, 1150 (1955).

6.

650 PULSED

MEASUREMENT OF COMPOSITION

GRID i(PULSED1

OUTPUT GRIDS

2,3 ION

FLIGHT

SOURCE

TUBE

RESISTIVE STRIP ELECTRON MULTl PLlER

FIG.10. Schematic diagram of time-of-flight mass spectrometer

the same energy, ions of different masses will have different speeds. The lightest ions arrive at the detector first, and the mass spectrum is obtained in the form of a time-of-arrival spectrum. As soon as the first group of ions is well on its way to the detector, a second mass spectrum can be initiated. The repetition frequency is limited mainly by the need to avoid overlapping successive spectra, but successive mass spectra can fairly readily be generated at rates as fast as 100,000 per second. Vacuum requirements are somewhat less stringent than for magnetic instruments, and about Torr can be tolerated without excessive mass peak broadening or loss of ions. 6.5.3.3. Quadrupole Mass S p e c t r o r n e t e r ~ . ~The ~ ~ .quadrupole ~~~~~~~ mass spectrometer consists of an array of four rods (Fig. 11) to which are applied rf and dc potentials. Ions to be mass analyzed are injected along the axis of the rod assembly. The highest attainable mass resolution of the instrument is determined by the mechanical accuracy of assembly and by the number of rf periods which the ions spend in the analyzer. A

FIG. 1 1 . Schematic diagram of quadrupole mass spectrometer (quadrupole mass filter). W . Paul, H. P. Reinhard, and U. von Zahn, Z . Phys. 140, 262 (1958).

6.5.

65 1

MASS SPECTROMETRY

useful feature for some applications is that the mass resolution can be adjusted, up to the possible maximum, by simply varying the ratio of rf to dc potential; if necessary, the analyzer can provide a “window” several mass units wide. The instrument is tolerant of relatively high pressures, and up to about Torr is permi~sible.’~~ Other possible advantages are the absence of a magnet and the fact that a very light and compact analyzer can be built. 6.5.3.4. General Remarks. All of the described types are commercially available. Standard instruments are usually “low resolution” types which will give good separation of mass peaks one atomic mass unit apart, up to a mass of several hundred or more. (Spectral peaks generally become more closely bunched at high masses.) This type is used for general analysis and for most applications to fluids and other systems (cf. Fig. 12). For some purposes, a double-focusing instrument of much higher resolution is required. It was mentioned above that a mass peak of nominal mass to charge ratio of 28 can actually be several different ionic species

t

b-

Z W K

a 3

0 Z

i2

I , a I 28 25

I , / , I 8 I 20 18 16 14

I

I

12

-MASS

FIG.12. Part of low resolution mass spectrum of residual gas in a vacuum system at a total pressure of 1 x lo-’ Torr. The complete spectrum extends to mass 250 and has about 100 identifiable mass peaks, many of them due to small traces of hydrocarbon. This figure illustrates the considerable sensitivity of mass spectrometric detection. The principal peaks are 18 (HpO+),28 (N,+) and 32 (0;). Incidentally, from relative sensitivity data, it can be shown that Oz is enhanced in this spectrum, relative to N, ; this is attributed to diffusion of atmospheric O2 into the system through rubber O-ring seals.

652

6.

MEASUREMENT OF COMPOSITION

such as C2H:, CO+ or N:. The same comment applies to any other mass number. However such peaks actually differ slightly in mass, and can be resolved with a suitable mass spectrometer (Fig. 13). Such an instrument can be very useful for analyzing complex mixtures or for identifying a totally unknown compound, but is expensive. It should be realised that mass spectra, i.e., the height ratios of fragment peaks, tend to be more instrument-dependent than, say, optical spectra. Therefore spectra tabulated for one type of instrument cannot be directly used for quantitative evaluation of experimental data obtained on another type of mass spectrometer. Indeed, mass spectra taken at different times on the same instrument can also differ appreciably. Such differences in a single instrument are usually attributable to surface charges; such charges can build up on insulating deposits inside the mass spectrometer, especially if the vacuum system is not clean. The surface charges can alter the energy distribution of the ionizing electrons, and can also appreciably influence the ion trajectories in the source in a massdependent manner. Therefore, for accurate quantitative work, careful attention must be paid to calibrations, which should be checked periodically. 6.5.4. detector^'^'

Ion detectors are usually either electrometer amplifiers or electron multipliers. 6.5.4.1. Electrometer D e t e ~ t 0 r s . lThese ~ ~ comprise a collecting cup for the ions, and a high current gain amplifier. These detectors are

27.994914

28.006148

28.026829

28.031299

MASS-

FIG. 13. Part of high resolution mass spectrum taken with AEI MS902 mass spectrometer. This shows resolution of four different ions, all of nominal mass 28. The numbers are literature values of the masses in atomic mass units. [Courtesy of A. G . Harrison and D. W. Priddle.]

6.5.

MASS SPECTROMETRY

653

relatively robust and immune from surface poisoning effects, and they measure the actual ion current directly. However their sensitivity limit corresponds to a current of several thousand ions per second. Moreover their response time is relatively long, typically of the order of a second at maximum sensitivity. 6.5.4.2. Electron M ~ l t i p l i e r s . ~ ~In~ . an ~~~ electron ** multiplier detector, ions at an energy of several kiloelectron volts are made to strike a cathode and produce one or more secondary electrons. These electrons are then multiplied by a dynode array similar to that of a standard photomultiplier. A gain of lo6 or more can be attained. One practical disadvantage is that the dynodes are exposed to the interior of the mass spectrometer and may be poisoned by excessive amounts of hydrocarbons or certain other gases, causing serious loss of stability and sensitivity. An alternative type of multiplier uses a continuous resistive strip as a multistep dynode, and is very resistive to poisoning effects. Using counting techniques, with careful attention to elimination of noise, currents of a few ions per minute can be measured with electron multipliers. Moreover, the response time of multipliers is typically in the nanosecond range, so that they are very suitable for fast reaction studies.

6.5.5.Sampling Systems Three types of sampling system will be briefly discussed, the molecular leak inlet, the direct viscous flow inlet, and the Kantrowitz-Grey molecular beam inlet. 6.5.5.1. Molecular Leak The pressure inside a mass spectrometer ion source is usually sufficiently low for the outflow to the pumping system to be in the molecular flow regime. The rate of outflow of molecules is then proportional to the respective molecular speeds, so that the lighter component is preferentially lost. In ‘order for the composition in the ion source to be the same as that in the original gas sample, it is necessary to arrange that the inflow also be in the molecular flow regime. This implies a low sample pressure or a small sampling hole; often multiple holes are used in order to ensure a sufficiently large inflow. It is, incidentally, wise to use a relatively large volume of sample on the upstream side of the molecular leak inlet, otherwise this sample itself will soon become significantly depleted in the lighter components. This type of inlet is used in many analysis applications, e.g., for static analysis of gas mixtures or for monitoring a gas chromatograph effluent. G. W. Goodrich and W. C . Wiley, R e v . Sci. Insfrum. 32, 846 (1961).

* See also Section 1 I . I .3 in Volume 2B (Electronic Methods).

654

6.

MEASUREMENT OF COMPOSITION

It is a standard accessory for many commercial mass spectrometers and will not be further discussed here. 6.5.5.2. Direct Viscous Flow lnlet.141-143 Sometimes it is necessary to have a larger inflow than can be provided by a molecular leak inlet. Then direct viscous flow can be used. An uncollimated jet is directed into the electron beam of the mass spectrometer. Because there is bulk flow, the gas composition as it enters the ion source will be that of the original sample. It is, however, necessary to ensure that the background gas in the spectrometer, whose composition will be different from that of the inflowing sample, does not make a substantial contribution to the ion current. This can be arranged by careful alignment of inflowingjet and electron beam, by having an open ion source structure so that the gas flows directly through to the pumping system, and by providing large capacity pumps. There is still the possibility of diffusive separation of gases in the jet; unless the whole jet is uniformly sampled by the electron beam, an apparent distortion of composition may be caused. It is therefore wise to calibrate the system by using known gas mixtures whose composition is as similar as possible to the mixtures to be analyzed. This type of inlet has been used in some shock tube studies and will be discussed again below. 6.5.5.3. Kantrowitz-Grey Molecular Beam lnlet.144-147In this system, a free jet expansion from a miniature nozzle creates a high intensity supersonic gas stream. The central part of this beam is separated and formed into a molecular beam by means of a skimmer (cf. Fig. 14). This type of system has been subjected to intensive investigation as a molecular beam source for studies in rarefied gas dynamics and chemical kinetics. Nevertheless there are still significant uncertainties about some aspects of the behavior of these sources. In this section, there will be summarized the results which are most directly relevant to mass spectrometric sampling and which appear to be reasonably well established. J. E. Dove and D. McL. Moulton, Proc. R . Soc. London, Ser. A 283, 216 (1965). G. P. Glass, G. B. Kistiakowsky, J. V. Michael, and H. Niki, J . Chem. Phys. 42, 608 (1965). 'lS D. Gutman, R. L. Belford, A. J . Hay, and R. Pancir0v.J. Phys. Chern. 70,1793 (1966). 144 A. Kantrowitz and J. Grey, Rev. Sci. Instrum. 22, 328 (1951). I r a H. Ashkenas and F. S. Sherman, Rarefied Gas Dyn. Proc. Int. Symp., 4th, Toronto, 1964 (J. H. de Leeuw, ed.), Vol. 2, p. 84. Academic Press, New York, 1966. K. Bier and 0. Hagena, Rarefied Gas Dyn., Proc. Int. Syrnp., 4th, Toronto, 1964, (J. H. de Leeuw, ed.), Val. 2, p. 260. Academic Press, New York, 1966. I" W. S. Young, Y. G . Wang, P. K. Sharma, W. E. Rodgers, and E. L. Knuth, Rarefied Gas Dyn., Proc. Znt. Symp. 8th, Stanford University, 1972, (K. Karamchati, ed.), p. 253. Academic Press, New York, 1974.

6.5.

655

MASS SPECTROMETRY

11 MASS

SPECTROMETER

1 PUMP

FIG. 14. Schematic diagram of Kantrowitz-Grey sampling system.

The gas which will eventually be separated in order for it to be ionized in the mass spectrometer is that on the centerline of the jet expansion. It is therefore important to ensure that this part of the gas is not contaminated and that any effects influencing its composition are well characterized. If the gas sample is taken from a flowing gas stream, there will be a viscous boundary layer on the plate containing the sampling nozzle. Since this boundary layer may be different in composition from the bulk flow, it is necessary to ensure that the central portion of the nozzle outflow is not significantly affected by the boundary layer. To meet this criterion, the diameter of the sampling orifice must be not less than the boundary layer thickness. This boundary layer thickness can be calculated by standard fluid mechanics formulae, when the flow conditions in the sampled region are known. (The case where a thermal boundary layer is present will be discussed later.) In the expansion beyond the sampling orifice, the translational temperature and the density are both decreasing. Eventually a condition of free molecule flow will be attained, where the rate of collisions is very small. In order to minimize chemical reactions during expansion, which might distort the initial composition, the expansion should be carried out as quickly as possible. A free expansion meets this criterion, and it also helps to ensure that the streamlines during the continuum (viscous) part of the expansion are always divergent. This latter condition will tend to keep the outer layers of the expanding gas, which may originate from the boundary layer in the sampled region, away from the centerline. The center of the flow must now be skimmed in order to eliminate the unwanted outer regions of the flow, and in order to keep the magnitude of

656

6.

MEASUREMENT O F COMPOSITION

the gas flow into the spectrometer within reasonable bounds. The angle and shape of the conical skiinmer must however be carefully chosen. If the internal angle is too small, molecules scattered from the inner walls of the cone will re-enter the beam in large numbers and in serious cases perturb the flow. If the external angle is too large, a detached shock wave in the expanding gas will be formed in front of the skimmer opening; such a shock wave should be avoided. In addition, the edges of the skimmer cone should be as sharp as possible, to avoid shock waves in the core of the skimmed gas. In practice, a cone angle of about 80" seems to be generally suitable. A number of very useful diagrams, and an extensive discussion of these factors, are given by Ashkenas and Sherman145 and Bier and Hagena.146 The distance between nozzle and skimmer is another important parameter. The choice of this distance influences the occurrence of disturbances upstream or downstream of the skimmer.146 Moreover the intensity of the final beam is strongly affected by the nozzle-skimmer distance. Clearly there is a close relationship between this distance and the required pumping capacity in the final compartment. Two other problems should be mentioned. The lighter species in a gas mixture tends to be lost preferentially from the centerline, resulting in a changed composition of the sampled mixture. In principle, this enrichment is calculable under certain circumstance^,'^^ but in practice it is generally wise to calibrate with known mixtures. Second, the very large temperature drop in the free jet expansion can sometimes cause actual condensation to occur.'46 Where this happens, it can be avoided by somewhat heating the sampled gas, if this is feasible. 6.5.6. Applications

Two applications of mass spectrometric measurements by direct sampling will be briefly outlined, namely t o the study of chemical reactions in shock waves, and to investigation of flame structure and kinetics. 6.5.6.1. Shock Wave Studies. Fdlowing the pioneering work of Kistiakowsky and co-workers,14e~14e direct sampling from a shock tube into a mass spectrometer has been used by a number of workers to study the mechanisms and rates of chemical reactions. A time-of-flight mass spectrometer has been used for almost all of these studies. This instrument can provide complete mass spectra at a very high repetition rate, and provides an excellent overview of the kinetic behavior of a reacting system.

14B

G . B. Kistiakowsky and P. H. Kydd, J . A m . C h ~ r nSoc. . 79, 4825 (1957). J. N . Bradley and G . B. Kistiakowsky, J . Chem. Phys. 35, 256 (1961).

6.5.

MASS SPECTROMETRY

657

A single experiment will yield a set of concentration versus time profiles for all major reactant and product species, and will often give at least an indication of the concentrations of reactive intermediates (Fig. 15). A wealth of kinetic information can be obtained in this way. A quadrupole mass s p e c t r ~ m e t e r 'and ~ ~ a magnetic mass spectrometer150 have also been used to monitor concentrations behind shock waves. These types of spectrometer of course cannot be scanned over a mass range during a single experiment. Instead, they are set to display the time variation of a single mass peak. For information about the

I

16

I I I

28 30 32

44

MASS --f

FIG.15. A sequence of 12 time-of-flight mass spectra originally recorded on oscilloscopes at 25-ps intervals. Data are from a shock tube experiment on NIO pyrolysis, and show arrival of shock wave and subsequent reaction, with decrease of reactant NpO ( m / e = 44) and increase of N2. NO, and 0, ( m / r = 28,30,32). [Reproduced by permission of the National Research Council of Canada from the Cunudiun Journal of Chemistry (Barton and Dove151).] Iso

B. Sturtevant, J . Fluid Mech. 25, 641 (1966).

65 8

6. MEASUREMENT

OF COMPOSITION

behavior of other mass peaks, the mass spectrometer can be reset and the experiment rerun. The shock tube/mass spectrometer method has been applied to the study of a wide variety of reactions, e.g., chain oxidation^,^^' pyrolysis reaction^,^^^*^^^ diatomic d i s s ~ c i a t i o n ,exchange '~~ and ionization p r o c e ~ ~ In e ~ these . ~experiments, ~ ~ ~ ~ ~ the ~ sample was generally taken from the reflected shock region, through an orifice in the end plate of the shock tube, directly into the electron beam of the mass spectrometer (Fig. 16). An important question in these experiments concerns the sampling. The sampling method is direct viscous flow. However the sample is taken through a continuously growing thermal boundary layer. Clearly, if a substantial portion of the sampled gas was cooled by the thermal boundary layer some time after shock arrival, the results will be seriously biased. The question of sampling has been investigated by several ~ o r k e r s , ' ~mainly ~ * ' ~ ~by careful study of the mass spectrometer experiments themselves or by comparison of the results with other kinetic data. In each case, the finding has been that, for a limited working time, an adequate sample can be obtained. Nevertheless it is clear that it is also easy to stray into working conditions where a poor sample is being ob-

Tube

FIG.16. Equipment at the University of Toronto for mass spectrometric study of chemical reactions in shock waves, using a quadrupole mass filter. This equipment is actually primarily designed for measurement of positive and negative ions sampled from reacting gas in the shock tube itself. The internal ion source was therefore primarily used for setting-up purposes. However, the same configuration can be used for mass spectrometric studies of neutral species sampled from the shock tube and then ionized inside the mass spectrometer.

15*

S. C. Barton and J. E. Dove, Cun. J . Chem. 47, 521 (1969). R. W.Diesen and W.J. Felmlee, J . Chem. Phys. 39, 2115 (1%3).

R. D. Kern and G. G. Nika, J. Phys. Chem. 75, 2541 (1971). R. Creswell, M. A. Di Valentin, and J. E. Dove, Phys. Fluids 9, 2285 (1966) lSs G. B. Kistiakowsky and J. V. Michael, J. Chem. Phys. 40, 1447 (1964). IU

6.5.

MASS SPECTROMETRY

65 9

tained. As shock tube experiments generally improve in accuracy and precision, the question of sampling needs to be kept under critical review. However in this author’s opinion, the best use of the mass spectrometric method is to provide a good survey of the overall kinetic behavior of a system. This survey can then be used to guide experiments carefully designed to elucidate specific questions, possibly using optical or other observation techniques. The theory of sampling through a transient thermal boundary layer is unfortunately very complex. Dove and M o ~ l t o n ’made ~ ~ a somewhat rough-and-ready treatment by essentially decoupling the sampling from the boundary layer growth. Calculations from this model seem to fit with experimental observations, but it was nevertheless difficult to assess how far such calculations could be relied upon. Recently V01dne1-l~~ has made extensive numerical calculations of the transient sampling flow. The flow region was divided into a mesh of rectangular cells, and the equations for transport of mass, energy, and momentum were approximated by conservative finite difference expressions relative to these cells. The calculation progressed through a sequence of finite time intervals. While a complete solution to the transient flow/transient boundary layer problem proved to be too expensive, nevertheless, the temperature history of sampled gas up to 1 ms could be assessed. The results indicated that the simplified Dove -Moulton approach worked surprisingly well and that rather than being overoptimistic, as had been feared, it was in fact in certain respects somewhat too pessimistic. Voldner’s calculations do show, however, that the outer layers of sampled gas contain boundary layer material. To get a sample unaffected by the thermal boundary layer, it is advisable to skim the central core, as in the Kantrowitz-Grey approach. Unfortunately the flow through a Kantrowitz-Grey system can take several hundred microseconds to become steady after shock arrival, and the most interesting kinetic data are usually those taken near the beginning of the reaction when the transient nature of the flow through the skimmer is at its worst.

6.5.6.2. Flame Studies. Mass spectrometry is widely applied to the study of concentration profiles in flames. However an excellent account of the principles of such studies has been given by Fristrom and Westenberg,I5’ so that only a brief summary will be presented here. It should be pointed out that the structure of a flame represents a complex balance of transport and chemical kinetic processes, so that the

Is’

E. C. Voldner, Ph.D. Thesis, University of Toronto (1975). R. M. Fristrom and A. A . Westenberg, “Flame Structure.” McGraw-Hill, 1965.

660

6.

MEASUREMENT OF COMPOSITION

structure of a flame may be more easily disturbed than that of many other types of flowing system. Consequently, particular care has to be taken when introducing a sampling probe into a flame.15* Both stable and unstable species can be sampled from flames. 6.5.6.2.1. SAMPLING OF STABLESPECIES.Quartz microprobes are generally used; Fristrom and Westenberg give considerable details. For a good recent example of this type of work, see Biordi, Lazzara, and pap^.'^* Discussions of sampling problems have been given by B i ~ r d i , ' ~and ~ * 'by ~ ~Eberius, Hoyermann, and Wagner.lso 6.5.6.2.2. SAMPLING OF ATOMS A N D RADICALS. For such a procedure, molecular beam must be used, cf. Section 6.5.5. A short discussion of application to flames has been given by Fristrom and Westenberg, while sampling problems have been discussed by Hastie.'" For a recent application, see Biordi et al. ls2 6.5.6.2.3. SAMPLING OF IONS. Certain flames, especially hydrocarbon flames, produce relatively large concentrations of ions, evidently by a chemical rather than a thermal process. A reaction which is believed to play a part is CH

+0

-

CHO+ + e-.

Following the pioneering work in this field,163,164 many studies have been made by mass spectrometry and other technique^'^^-^^'; Fontijn16*has reviewed work up to 1971. Sampling problems and procedures are not unlike those for radicals. However ions are in some respects easier to work with since they can be controlled and focused by electric or magnetic fields. J . C. Biordi, C . P. Lazzara, and J . F. Papp, Combust. Flume 21, 371 (1974). J . 6. Biordi, C. P. Lazzara, and J . F. Papp, Symp. ( I n t . ) Comhust. [Proc.], /4th, 1972 p. 367. Combustion Institute, Pittsburgh, Pennsylvania, 1973. I8O K . H. Eberius, K . Hoyerrnann, and H . Gg. Wagner, Symp. ( l n t . ) Combust. [ P r o c . ] , 13th, Salt Luke C i t y . 1970. p. 713. Combustion Institute, Pittsburgh, Pennsylvania, 1971. J . W. Hastie, I n t . J . M u s s Speclrom. Ion Phys. 16, 89 (1975). J. C. Biordi. C . P. Lazzara, and J. F. Papp, Symp. ( I n r . ) Comhusr. [ P r ~ c . ]16rh. , Leeds. 1976, p. 1097. Combustion Institute, Pittsburgh, Pennsylvania, (1977). 163 J. Deckers and A. Van Tiggelen, Symp. ( I n r . ) Combust. [ P r o c . ] , 7th, Oxford. 1958 p. 254. Butteworth, London, 1959. P. F. Knewstubb and T. M. Sugden, Symp. ( h i . ) Combust. [ P r o c . ] . 7th, Oxfwd. 1958 p. 247. Butteworth, London, 1959. H. F. Calcote and J. L. Reuter, J . Chem. Phys. 38, 310 (1963). IWI J . Peeters, C. Vinckier, and A. Van Tiggelen, Oxid. Combust. R e v . 4, 93 (1969). '61 J . M. Goodings, D. K . Bohme, and T. M. Sugden, Symp. ( I n t . ) [ P r o c . ] , 16th, Leuds. 1976 p. 891. Combustion Institute, Pittsburgh, Pennsylvania, 1977. '" A. Fontijn, Prug. Rericr. Kinet. 6, 75 (1971). L58

6.5.

MASS SPECTROMETRY

66 I

6.5.7. Sources of Information on Mass Spectrometry The International Journal of’Mc1s.s Spectrometry and ton Physics, and the various instrument journals, publish articles on advances in techniques and instrumentation. The Mass Spectrometry Bulletin provides monthly abstracts of articles and compilations of mass spectrometric data; it is available on computer tape. Two European conference series publish their proceedings under the titles Advances in Ma.ss Spectrometry and Dynamic Mass Spectrometry; these volumes contain many useful articles on techniques and applications. Specialist Periodical Reports169of the Chemical Society provide authoritative reviews on various areas of application. Extensive compilations of mass spectra have been publ i ~ h e d . ’ ~In~ addition, - ~ ~ ~ the National Institutes of Health and the Environmental Protection Administration maintain a mass spectral data base172which currently contains over 39,000 mass spectra and provides various computer search173options for aiding in the identification of unknown spectra. In the United Kingdom, the Mass Spectrometry Data Centre of the Chemical Society Information Service compiles and evaluates mass spectral data.

Spec. Period. R e p . : Muss Speclrometry. 5 (1979). A. Cornu and R . Massot, “Compilation of Mass Spectral Data,” 2 vols. Heyden, London, 1975. “Selected Mass Spectral Data,” Am. Pet. Inst. Res. E’roj. 44. Thermodynamics Research Center, Texas A & M University, College Station, Texas. (Tabulations of mass spectra, updated periodically.) S. R. Heller and G. W.A . Milne, “EPA/NIH Mass Spectral Data Base,” NSRDSNBS 63,5 vols. 1978. (Available as Order No. PB290661 from the National Technical Information Service, U.S. Dept. of Commerce, Springfield, Virginia 22161 .) 173 V. A . Vinton, G. W. A. Milne, and S. R. Heller, Anal. Chim. Acra 95, 41 (1977). 170

This Page Intentionally Left Blank

7. HEAT TRANSFER GAGES* List of Symbols Area of sensor Specific heat at constant pressure Nullpoint calorimeter hole depth; voltage output Radiative heat flux Surface heat transfer coefficient Current Thermal conductivity Biot number, L = Ih Nusselt number, Nu = H x / K Prandtl number, Pr = C&/K = v / a Total heat (Joule) Resistance; radius Reynolds number, Re = Vx/v Temperature Adiabatic wall temperature Total or stagnation temperature Probe temperature A reference temperature for heat transfer Static temperature Wall temperature Freestream (static) temperature Flow velocity Linear dimension

f h h I m

P

4 r (1

e

Y h P V

P U

e

Frequency Heat transfer function (Chapter 7.2) Surface conductance, h = H/K; enthalpy Thickness of substate material or sensor Thermal parameter for thermometer problem (Chapter 4. I) Pressure Heat flux (W m-*) Recovery factor Thermal diffusivity, (Y = K / p C , Temperature coefficient of resistance Emissivity or absorptivity; error in a quantity Ratio of specific heats; thermocouple voltage-temperature Root of transcendental equation (Chapter 7.2) Viscosity Kinematic viscosity Density Stefan - Boltzmann constant Fourier number, 9 = at/P

7.1.Introduction Heat transfer measurement to surfaces bounding a flow is of interest to the fluid physicist because it can provide information on the thermodynamic, chemical, or mechanical state of the fluid itself (e.g., stagnation enthalpy, degree of dissociation, laminar or turbulent state, respectively). The engineer may be concerned with the heat transfer itself as it affects design and performance parameters of an aircraft wing, reactor heat exchanger, or jet engine blades. * Part 7 is by W. Paul Thompson. 663 METHODS OF EXPERIMENTAL PHYSICS, VOL. 18B

Copyright @ 1981 by Academic Press, lnc. All rights of reproduction in any form reserved. ISRN O.l?-A7505L.A

664

7.

HEAT TRANSFER GAGES

The basic heat transfer measurement techniques to be described are frequently applicable to full scale flight tests as well as to laboratory simulation For example, thermocouple instrumentation (Sections 7.3.1 and 7.3.2) is frequently used in aircraft flight testing, and the thin film gage (Section 7.6.1) has been used on ballistic range models. The instrumentation principles are the same in laboratory and flight, the differences lying in rugged construction or in data transmission and recording systems, The discussion of this chapter will focus on laboratory applications, though the methods have greater generality. Very little interest was shown in aerodynamic heat transfer measurement prior to about 1950, since it had negligible practical effect in subsonic and low supersonic flow. However, the advent of hypersonic atmospheric entry technology, for which heat protection of the vehicle was perhaps the overriding technical challenge, caused a great surge of experiment3s4during the period 1953-1968. Since that time, the field has matured, and significant advances have been less frequent, being characterized usually by the application of exotic sensor technology to established thermal configurations. There are three conceptual methods of measuring heat flux: (1) the flux may set up a temperature gradient in thin material layers, which can be related to the heat flux and material properties. These “sandwich” gages5n6 tend to be slow and insensitive and are little used today in fluid mechanics. (2) the heat may be caught in a thermal mass which acts as a calorimeter whose transient temperature change can be related to the heat flux. (3) the heat input (in steady state) may be balanced against a calibrated heat removal, e.g., a water-cooled heat exchanger or hotwire heating current. The instruments can be designed to operate either in a transient mode, with large thermal mass and long equilibration times compared to the experiment time scale (e.g., the slug and thick film calorimeters): or they may be made small and/or thermally thin, so that they achieve a quasi steady state in times short compared to the variations in heat flux (e.g., the membrane calorimeters and thin film gage). A. F. Donovan, H. R. Lawrence, F. E. Goddard, and R. R . Gilruth, eds., “High Speed Aerodynamics and Jet Propulsion,” Vol. 8. Princeton Univ. Press, Princeton, New Jersey, 1%1.

* M. Letarte and L. E. Moir, in “Advances in Hypervelocity Techniques” (A. M. Krill, ed.), p. 773. Plenum, New York, 1962. J. G. Hall and A. Hertzberg, Jet Propul. 28, 719 (1958). W. C. Nelson, ed., “The High Temperature Aspects of Hypersonic Flow.” Macmillan, New York, 1964. N . E. Hager, Rev. Sci. Instrum. 36, 1564 (1965). E. A. Brown, R. J. Charlson, and D. L. Johnson, Rev. Sci. Instrum. 32, 984 (1961).

7.2.

665

ONE-DIMENSIONAL HEAT CONDUCTION RELATIONS

For development or validation of analytical models, the experimental situation is made as simple, ideal, and analyzable as possible. Hence most experimental methods are particularly suited to sharp flat plates, flat faced or hemispherical stagnation flows, cylindrical pipes, etc. The engineering heat transfer problem is nearly always characterized by a heat transfer coefficient H which appears in Newton’s law of cooling7 4 = H(Tref - Tw),

(7.1.1)

where q is heat transfer per unit area per unit time, T, is wall temperature, and Trefis an appropriate fluid reference temperature, e.g., the stagnation temperature To or the adiabatic wall temperature Tad(cf. Chapter 4.1). It is usually assumed that the coefficient H remains constant in time, and that the heat flux will then vary as wall or stream conditions change. If heating occurs over an extended time such that T , approaches Trer,the structure of the surface layer will change, and a large correction to H results (cf. Chapter 7.4). The coefficient H combines the effects of geometry, fluid state, and fluid thermal properties for a particular situation. Typical values of H range from order 55 W . m-2 . K-’ for air flow, to 5500 for boiling or condensing water.7 The heat transfer coefficient establishes what is often called the “radiation boundary condition” at the surface;* the heat flux to the body is then determined by solution of a heat conduction problem dependent on wall thermal properties.

7.2. One-Dimensional Heat Conduction Relations One-dimensional heat transfer in a planar solid is described by the Fourier equation (7.2.’1)

where the thermal parameters of the material are described by the diffusivity a: ff

=

KIPC,,

(7.2.2)

where K = thermal conductivity, p = density, and C, = specific heat at constant pressure. The heat flux is related to the thermal conductivity

’ E. R . G. Eckert and R. M. Drake, “Heat and Mass Transfer.” McCraw-Hill, New York, 1959. H. S . Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd ed. Oxford Univ. Press (Clarendon), London and New York, 1959.

666

7.

HEAT TRANSFER GAGES

and the temperature gradient according to the definition of K: aT ax

q = -K--.

(7.2.3)

Several geometric cases are of interest; the solutions may be found in Refs. 8 and 9 and are quoted here without proof. It is assumed for simplicity that initial temperature is zero. Case I : Surface temperature T(0, t ) of a semi-infinite solid with constant surface heat flux qo. (Ref. 8, p. 75; Ref. 9, p. 106) (7.2.4)

where erfc p = 1 - e r f p and erf p = ( 2 / ~ l ‘ ~ ) J $ e ( -dX. ~*) T(0, t ) =

2q0P ( T K ~ C , ) *” ~

(7.2.6)

This case is illustrated by the ideal surface temperature rise of a thinfilm gage (Chapter 7.5) on a shock tube wall under turbulent heat transfer, or at the stagnation point of an axisymmetric model in impulsive flow. The solution within the body (7.2.5) describes the multilayered gagelo o r the nullpoint calorimeter with in-depth thermocouple sensor (Section 7.4.3). The error functions are tabulated in Ref. 8, Appendix 11. Case 11: Heat flux at the surface of a semi-infinite solid when surface temperature T(0, t ) is known. (Ref. 8, p. 62; Ref. 9, p. 109)

Let T(0, t ) = F(r). Then (7.2.7) This is a generally useful form for impulsive flows, which can be integrated either by digital computer or with sufficient accuracy by an electronic analog circuit.” It allows for finite starting time or slow flow variations, as may be seen by a thin film gage o r surface thermocouple on a thick-wall model in shock tunnel flow. Review of several possible methods of numerically integrating Eq. (7.2.7) has led to a procedure by R . V . Churchill, “Modern Operational Mathematics in Engineering.” McGraw-Hill, New York, 1944. lo K . Willeke and D. Bershader, Rev. Sci. Insrrurn. 44, 22 (1973). G . T. Skinner, ARS J . 30, 570 (1960). @

7.2.

ONE-DIMENSIONAL HEAT COKDUCTION RELATIONS

667

which discontinuities at endpoints of the integration interval are avoided, and rapid convergence is achieved with great saving in computer time.12 In the special case of constant surface temperature F(t) = F,, and

(7.2.8) This applies to the ideal case of a shock tube wall under laminar boundary layer heat transfer. Case ZZZ: Surface temperature of a semi-infinite solid heated by convection at its surface (radiation boundary condition) (Ref. 8, p. 72; Ref. 9, p. 216 # 1 ) (Fig. 1).

Convective heat flux at a surface (x = 0) is customarily described in terms of the "radiation boundary condition"

d o , t ) = H ( T m - T(0, I)),

(7.2.9)

where H is the heat transfer coefficient, T, is the (constant) source temperature for the heat transfer, and T(0, t ) is the varying surface temperature of the body of conductivity K. The quantity

(7.2.10)

h = H/K

is then an equivalent surface conductance, and surface temperature is given by T(0, t) = [ l

-

(7.2.11)

ech""erfc(h(at)l'z)].

Case ZV: Surface temperature of a finite slab, backface (x = 0) insulated, under constant heat flux qo at the surface (x = 1) (Ref. 8, p. 112, i).

In the more usual case where the heated body is of finite thickness 1, the long-time response of the heat transfer measurement must take penetration of heat to the backface into account. In most practical cases the backface of a calorimeter or heat transfer gage will be insulated so that Cases I and I11 are modified: ,T

FIG. I . Heat transfer to a semi-infinite solid.

W.J. Cook and E. J. Felderman, AIAA J . 4,561 (1966).

T 10.11

x=o

7.

668

HEAT TRANSFER GAGES

where ierfc, the integral error function is defined as ierfc x

=

jzmerfc

A dX

(7.2.13)

and is tabulated in Appendix I1 of Ref. 8. Case V: Surface (x = 1) temperature of a finite slab heated by convection, with backface (x = 0) insulated. (Ref. 8, p. 124, iii, and Section 3.11) (Fig. 2)

where and

h = H/K, L

=

Ih

(the Biot number),

O

=

at/P

(the Fourier number),

(7.2.16)

and A, is a root of A tan A

=

L,

(7.2.17)

tabulated in Ref. 8 (Appendix V). Plots of (1 - Jl)are found in Ref. 8 (Section 3.11). These solutions for a variety of plane, cylindrical, and spherical geometries are plotted in Ref. 13, in useful form for experiment design. The nondimensional Biot and Fourier numbers characterize respectively the TiL,tI

x-I1

1

FIG.2. Heat transfer to a finite slab with insulated hackface. l3

P. J . Schneider, "Temperature Response Charts." Wiley, New York, 1963

7.2.

669

ONE-DIMENSIONAL HEAT CONDUCTION RELATIONS

ratio of conductance of the surface film H which controls surface input heat flux, to the conductance, K / l of the body which removes it; and the time t nondimensionalized by the characteristic time 12/afor diffusion of heat to depth I (i.e., the time at which the temperature at x = I has reached l / e of the surface temperature under constant heat input). At very short times, the boundary conditions of constant flux or convective heating are equivalent; where T(0, 0) = 0

qo = H T , ,

(7.2.18)

for simplicity, but as the body heats up, the flux decreases. Similarly, there is a characteristic time at which the presence of a finite thickness I causes deviation from the simpler semi-infinite solution. Comparing the four cases and expanding the solutions for short time leads to the following expressions for the effects of finite thickness and nonconstant heat flux. Curse I :

Semi-infinite body under constant flux: (7.2.19)

Caw IV: Finite slab of thickness I under constant flux:

The influence of the backface is then felt with error of 1 percent at t 3 t 3 P/2.20a. The effect of the constant flux versus convective heating assumptions is seen from comparison of Cases I and 111: P/3.39a and 5 percent at

Caw If/:

Semi-infinite body with radiation boundary condition: (7.2.21)

or from, e.g., (7.2.9),

where 40

HTW,

then we incur 1 percent q error at t a at t 3 2.5 x w/4h2a.

q / 4 h 2 a , and 5 percent q error

670

7.

HEAT TRANSFER GAGES

Simple data reduction is often performed by assuming that Solution I holds; then the time limits for proper gage operation are given by (7.2.20) and (7.2.22) respectively for finite thickness and nonconstant flux effects. In practice, sufficient supporting facilities are available for integration of the surface temperature by analog or digital computer, and Case I1 or V is used to determine heat flux from arbitrary surface temperature variation.

7.3. Instrumented Models A very useful review of several current wind tunnel heat transfer measurement techniques is given in Ref. 14.

7.3.1.Thin Wall Models The most straightforward way to determine heat transfer to a complex aerodynamic shape is to instrument a thin wall model of thickness / with backface thermocouples and map the transient temperature change of the rn0de1.l~ This method is appropriate for test times of several seconds in typical blowdown wind tunnels or for transient model insertion in a continuous flow. The time response is limited on the short end by the establishment of steady state heat flux normal to the surface, given by a t / / 2> I , for the case of constant heat flux. This defines the model as thermally thin, i.e., the heat penetrates to the back surface, and the model is essentially isothermal normal to its surface. If the heat flux varies during the test, then the model skin conductance K / / must also be much greater than the heat transfer coefficient H in order to track the changes, i.e., H f / K < 0.1. The long time response limit must, of course, be greater than the tunnel starting or insertion time, and is ultimately determined by error due to tangential conduction along the model surface. Assuming that each surface element stores all the incident heat flux as a calorimeter,

(7.3.1) where Tadis the adiabatic wall temperature of the flow, known from facility calibration (cf. Chapter 4. l), and subscript w denotes conditions at the wall. An empirical determination of the presence of tangential conduction error can be made by rewriting (7.3.1) as I4 L. L. Trimmer, R. K . Matthews, and T . D. Buchanan, "IEEE Congress on Instrumentation in Aerospace Simulation Facilities," 1973 (ICIASF Record '73), p. 35. Inst. Elec. Electron. Eng., New York, 1973.

7.3.

INSTRUMENTED MODELS

67 1 (7.3.2)

where Ti is model initial temperature. When a plot of the logarithmic expression versus time deviates from linear, then nonideal conduction errors exist. The optimum thickness I , especially for long tunnel starting or insertion times is quite thick, of order 1 mm for stainless steel. The error due to tangential conduction is given by15 (7.3.3) The use of very thin models in order to reduce response time may introduce severe structural problems; and the use of a filler to support thin walls can cause intolerable backface heat loss errors.'" 7.3.2. Thick Wall Models A variant of the thin wall model used as a calorimeter, is the thick wall model treated as a semi-infinite body. Measurement of the frontface temperature of a thick (-5 mm) stainless steel model can be accomplished using a coaxial thermoco~ple'~ in which the thermocouple junction is formed at the surface, in the process of filing to fit the model contours. In order for the semi-infinite solution to obtain for data reduction, the Fourier number at/P must be less than 0.5; i.e., heat must not diffuse to the backface in time t . (Contrast with the thin wall model requirement at/P > 1 where we desire the backface to track the front surface.) If the model is made of an insulator, e.g., glass, then thin film gages (Chapter 7.5) can be applied directly to the surface, giving very high sensitivity. If the model is treated as a semi-infinite solid, then its surface temperature at each point is given by

(7.3.4) where

P

Ht1I2

= @C,K)1/2'

7.3.3. Surface Temperature Mapping.

The surface temperature can be sensed by several methods. A commercial temperature sensitive paint is available which changes from an opaque l5 l8

A . R. George and W. G . Reinecke, AIAA J. 1, 1956 (1963). M.Cooper and E. E. Mayo, J. Aerosp. Sci. 24, 461 (1957).

672

7.

H E A T TRANSFER GAGES

solid to a transparent liquid at a selectable temperature." The progress of the melt line across the model is followed with a cink camera, and a temperaturz map derived as a function of the time. Alternate methods14 include use of painted phosphors whose intensity of fluorescence under uv light is a function of temperature. Again a photographic record can be analyzed to give a temperature map, and thence a heat transfer history. Some success has also been achieved using direct infrared pyrometric determination of model temperature.18 This technique is limited by poor optical resolution at long wavelengths and tends to underestimate heat transfer. It might profit from application of recent ir image converter camera techniques. All of these surface mapping techniques are limited to an accuracy of about 15 percent, representing poor knowledge of model thermal constants KpC,, and uncertainty in judging the precise time and location of isotherms. Very sensitive model instrumentation techniques have been developedIg for low Re flows at high Mach number. By use of various thin film or thermocouple techniques on models cooled with liquid nitrogen for large (Tad - Ti), sensitivity of order lo2 W m-2 can be obtained.

7.4. Thin Membrane Calorimeters Self-contained calorimeter gages with high sensitivity and rapid response have been developed for insertion into test models. Two variants of similar structure but different operating principles are shown in Figs. 3 and 4 . The cupacitunce culorimerer (Fig. 3 ) is a thermally isolated element, whose temperature rises linearly with time as it stores all incident heat flux. The Gardon, or asymptotic culorimeter (Fig. 4 ) establishes a steady state temperature gradient proportional to heat flux, from the center of the sensing disk t o a peripheral heat sink. Both gage types can be made with diameter of order 5 mm, time response of order 1- 10 ms, and sensitivity of order 0.0 1-0.1 mV m2/kW, using disk elements 0.1 mm thick, which provide about the minimum usable strength for typical supersonic aerodynamic facilities. The thermocouple elements used as temperature sensors are attached by resistance welding, brazing, or swaging t o the backface of the disk, depending on calorimeter element thickness 1. The error due to heat conduction along the thermocouple wires20depends on the ratio l / u (a = wire R . A. Jones and J. L. Hunt, N A S A Tech. R e p . R-230 (1966). H. Thornann and B. Frisk, I n t . J . Heui Muss Transfer 11, 819 (1968). '@ R . S. Hickrnan, H. Tong, and W. H. Giedt, in "Advances in Hypervelocity Techniques" (A. M . Krill, ed.), p. 713. Plenum, New York, 1962. 2o D. R . Burnett, J . Hrur Transfer 83, 505 (1961).

7.4.

T H I N MEMBRANE CALORIMETERS

673

radius), but can usually be ignored for the 0.01 to 0.05 mm thermocouple wires, and 0.1-0.5 mm thick calorimeter disks used in practice. The temperature range of operation is limited by the melt temperature of the cement used to hold the capacitance element to its insulator (450 K), or the solder which attaches the Gardon gage disk t o its peripheral copper heatsink (600 K). Within this range, the temperature variation of the Constantan disk conductivity and the Cu-constantan thermocouple coefficients compensate each other, resulting in a linear EMF-temperature relationship for the Cu-constantan Gardon gage.’l A very nearly linear response curve is also obtained for chrome1 -constantan thermocouples in a copper disk capacitance calorimeter,z2where the thermocouple temperature coefficients are compensated by the decrease in heat flux as the surface temperature rises. When a thin membrane calorimeter is introduced into a model wall, the temperature difference which can develop between the unperturbed model and the gage surface will change the temperature gradient and heat flux in the flow near the wall, and can cause significant errors in heat flux measurement. For the characteristic situation of a flat plate wall boundary layer, analyses have been made for the uniform-temperature capacitance calorimeterz3 and for the Gardon gage with its additional radial temperature gradient .24 Experimentz5has confirmed the calculated errors of 20-50 percent in measured H, compared to the desired isothermal configuration, and 50-140 percent error in q , for temperature differences between gage and model of 350- 1000°F. Although few applications will encounter this large a temperature difference, a careful analysis should be made if accuracy of 5 percent or so is desired. The heat flux limitations of the thin membrane calorimeter can be extended by water-cooling the Gardon gage heatsink,26 or by sweeping the gages rapidly through the high enthalpy core of the tunnel flow. 7.4.1. Capacitance Calorimeter

The capacitance calorimeter (Fig. 3) is described in detail in Refs. 14 and 27. Data reduction is performed by assuming that all incident flux is retained in the gage; R. Gardon, R e v . Sci. Insrrum. 24, 366 (1953). R. L. Ledford, in “Advances in Hypervelocity Techniques” (A. M. Krill, e d . ) , p. 673. Plenum, N e w York, 1962. 23 J . C. Westkaemper, J . Arrosp. Sci. 28, 907 (1961). 24 L . W. Woodruff, L. F. Hearne, and T. J . Keliher, AIAA J . 5, 795 (1967). 25 R . C. Bachmann, J . T. Chambers, and W. H . Giedt, I S A Trans. 4, 143 (1965). l6 F. C. Stempel and D. L. Rall, ISA J . 11, 68 (1964). 27 R . L. Ledford, W .E. Smotherman, and C. T. Kidd, Proc. Inr. Crmgr. Instnrm. Aerosp. Sitnul. Fucil., Znd, 1966, pp. 2-1-2-9. 21 22

674

7. HEAT TRANSFER GAGES COPPER DISK

STEEL SHELL NYLON tNSULATOH

GROUND

9-

FIG.3. Capacitance calorimeter with thermocouple sensor.

plc, dE '=

dt'

y

(7.4.1)

where y is the thermocouple constant (mV/K) and E the thermocouple output voltage. The short-time response limit is identical with that of the thin wall model, a time at/P > 1 being needed to equilibrate the backface and frontface temperatures. More accurately, for any given degree of indicated heat flux error, the response time i d 4 2' 1

(7.4.2)

qlndlcated

The running time is limited by conduction losses to 1 s or less. Variants of the thin wall calorimeter with continuous frontfaces (no insulation discontinuities) have been usedz8 as stagnation heat flux probes in highenthalpy arc jets at fluxes up to 50 MW mF2. Detailed charts for material selection and time response computation are given in Ref. 28. Also introduced is a method of sweeping the probe through the arc, thus avoiding burnout and permitting reuse and recalibration. Models with controlled surface roughness have also been used to investigate roughness-augmented heating.29 The low-level output of the thermocouple can be troublesome if data must be recorded in the presence of the large electrical noise characteristic of many discharge or arc-heated impulse facilities. Careful attention to shielding and grounding of gages, and the use of narrowband ac carrier amplifiers with high common-mode rejectionz2can appreciably reduce the noise. The sensitivity of the capacitance calorimeter gage has been increased by a factor 25 by substituting for t h e thermocouple a sputtered platinum resistance thermometer on an anodized aluminum calorimeter disk as the backface s e n s ~ r . ' ~ , ~ ' K . E. Starner, ISA Trans. 7, 181 (1968). E. Starner and R. L. Vanvig, TR-0066 (5240-10)-1. Aerospace Corporation, El Segundo, California, 1969.

*@K .

675

7 . 4 . T H I N MEMBRANE CALORIMETERS

7.4.2. Gardon or Asymptotic Calorimeter

The Gardon, or asymptotic calorimeter operates by establishing a steady state temperature difference between the center and edge of a constantan disk soldered to a peripheral copper heatsink (Fig. 4). A copper wire welded to the center of the disk completes a thermocouple junction pair at center and edge, and the radial temperature gradient is read directly. The gage is best suited for use in metallic models, where its heatsink temperature is nearly the same as that of the unperturbed model. The radial temperature difference is directly proportional to heat flux:

(7.4.3) The time response of the gage was approximated by GardonZ1 using a simple exponential response assumption for a very thin gage. More accurate a n a l y ~ e shave ~ ~ . allowed ~~ greater leeway in the thickness parameter. A thickness R / l a 10 is sufficient to achieve 99 percent of the ideal sensitivity; and values of the Fourier number a t / R Z 3 0.7 achieve 99 percent of the response parameter kTl/qR2. Empirically, it is found14 that the gage achieves 95 percent of its final response to a steady heat flux in t = 0.75 ( R 2 / a )which agrees with theoretical analysis.31 For large heat flux and low gage radial conductance (i.e., thin membrane) the response may saturate at very short times30 so that a detailed design of the experiment is often desirable. Convenient nomographs for initial gage design are a ~ a i l a b l e .The ~ ~ application ~~~ of an Sb-Bi evaporated thermopile as the backface temperature gradient sensor can increase gage sensitivity by a factor 6 if d e ~ i r e d . ’ ~

COPPER HEAT SINK

FIG.4. Gardon or asymptotic calorimeter with thermocouple sensor.

31

1 b AT

R . L. Ash, AIAA J . 7, 2332 (1969). R. H. Kirchoff, J . Heat Transfer 94, 244 (1972). R. Gardon, J . Heat Transfer CS2, 396 (1960).

676

7.

H E A T TRANSFER GAGES

7.5. Thick Calorimeters 7.5.1. Slug or Nullpoint Calorimeter

For higher heat fluxes, particularly as found in arc heated facilities, and for long run times (necessitated by long tunnel starting transients), the calorimeter must be made thicker (and unfortunately slower) to increase its heat capacity and avoid melt or excessive temperature difference between body and gage. Perhaps the simplest configuration is that of Fig. 5-a thick copper slug, insulated at side and backfaces so that it acts as a one-dimensional slab of thickness 1. The minimum operating time limit tl is set by the need to reach linear response of the backface, rl z= 0.35 P/(Y and the maximum time tz by melting of the front face. tz 6 ~ / 4 ( f = n l C l t / k )

- 91.

The optimum thickness found33 for maximum running time ( t z given 4 level is

I,

=

(7.5.1) -

tl) at a

KTmc1,/1.366q.

(7.5.2)

The sensitivity and time response can be greatly improved by drilling the end of the thermocouple well very close to the slug surface. The optimum location occurs at E = R (Fig. 6 ) . At this “null point” depth within a solid perturbed by the hole, the measured temperature history (after initial transients) is the same as that at the surface of a semi-infinite solid of the same The presence of the thermocouple and its brazing to the slug and protective sheath within the cavity; the requisite insulating airgaps; and nonuniformity of heat flux over the model surface due either to model shape or flow facility gradients; all serve to complicate the actual thermal configuration, and require a detailed numerical model for data reduction if accuracy of better than 15 percent or so is to r

AIR GAP

THERMOCOUPI F

L S T E E L SHELL 33 34

FIG.5 . Slug calorimeter.

R. H. Kirchoff, AIAA J . 2, 966 (1964). J . V. Beck and H . Hurwicz, J . Heut Transfer 82, 27 (1960).

7.5.

677

THICK CALORIMETERS s1I E l MODEL

1

,-

AIH GAP

T C f l P P E R SLUG

-1HERMOCOUPLF Wtlt

-1 -

t

R

FIG 6. Nullpoint calorimeter.

-’

kt

“THERMOCDUPLE

AT

JUNCTION NULL POINT

be obtained. Approximate analysis shows that the slug should be sufficiently long to appear semi-infinite (&/I2 < 0.3); but it can be reduced by half if finite-thickness analysis is used for modeling (Case V, Chapter 7.2). The presumed “null point” will settle at location E = R for time ar/,!? > 30 (commonly taken as the response time for a null point calorimeter). However, numerical modeling of practical shows that at/,!? > 3 is sufficient for 5 percent accuracy, and that positioning of the thermocouple slightly closer to the surface ( R I E = 1.1) can improve the time response. An excellent review of calorimeter error sources and high q operation is given in Ref. 36. The copper null point calorimeter can be operated continuously at heat flux up to 5 GW m-2 with burnout time approaching 0.5 sec. For higher fluxes, up to 20 G W m-2, the gage can be swept through the flow, thus avoiding burnout and providing a map of flow nonuniformities. The minimum practical gage dimensions are controlled mainly by thermocouple cavity size and manufacturing tolerances: R = E = 0.25 mm, 1 = 5 mm, slug radius 3 mm, with time response of order 5 ms. Both the thin and thick slug calorimeters can be calibrated by exposure to an oxygen-acetylene flame22and by exposure to high intensity ir radiation from an oven.28 For copper elements the thermal properties and thermocouple constants are fairly well known, and calculation of the calorimeter sensitivity from first principles is reliable to 5 percent or so.

7.5.2.Thick Film Resistance Thermometer The fast response and high enthalpy requirements of reentry heat transfer simulation in shock tubes prompted the development of a thick film resistance thermometer c a l ~ r i m e t e r . The ~ ~ gage element, a strip of 35 D. C. Howey and V . D. Cristina, Arrodvn. Test. Cotif., 3rd, fY68 AIAA Paper 68-404 (1968). 36 C. A. Powars, W. S . Kennedy, and R . A. Rindal, J . Spacecr. Rockets 9, 668 (1972). 37 P. H . Rose, R w . ScI. Imrrum. 29, 557 (1958).

67 8

7.

I-33

HEAT TRANSFER GAGES

pin PLATINUM STRIP

thermometer.

platinum 30 p m thick, serves to integrate the heat flux over the short test times (50-100 ps) characteristic of shock tube flows. The gage (Fig. 7) is operated as a resistance thermometer, with a bias current of several amperes. Sensitivity is of order 10 V mz/s MW, and the gage performance equation for constant heat flux input, and negligible conduction losses is

(7.5.3) where I is the bias current and E the voltage output. In order to keep the heat loss error through the backface to 5 percent or less in time f, the gage thickness is selected so that r / P < 4 s cm-2 for Pt. The gage is mechanically formed and held against the model backing, but not bonded. Alternatively, the model can be routed out to permit approximate flush mounting of the gage element. Complete absence of edge roughness is almost impossible to achieve, so that in cases where tripping of turbulent flow is of concern, it must be assumed that roughness exists. The gage response time is of order 1 ps. The large gage temperature changes caused by the large required bias currents and by the heat transfer (up to 400 MW m-*) require careful calibration of the gage. Electrical pulse heating3' can be used for calibration, assuming all of the P R loss is stored in the gage element. Measurements at shock tube stagnation enthalpies appropriate to flight at 10-20 km s-' in dissociated3' and ionized39air flows have shown the thick film gage to be a fast, reliable and low-noise instrument. It agrees well with thin-film techniques (Chapter 7.6) where their sensitivities overlapg8and can be used in ionized flow30 if insulated from the conducting flow by a thin layer of evaporated SiO or S O z . At very high gas temperature, the contribution of atom recombination,40 ionization41 and radiation to the measured heat transfer cannot be ignored. P. H. Rose and W. 1. Stark, J . Aerusp. Sci. 25, 86 (1958). P. H . Rose and J . 0. Stankevics, AIAA J . 1, 2752 (1963). 40 J. A . Fay and F. R . Riddell, J . Aerusp. Sri. 25, 73 (1958). J . A. Fay and N . Kemp, AIAA J . 1, 2741 (1963).

38

39

7.6.

679

THIN FILM GAGES

SHUCK IUHE WALL

CARBON FILM

SAPPHlRt WINDOW MIRROH

. I

F I G .8. Carbon film ir bolometer.

A totally different type of calorimeter, the infrared bolometer (Fig. 8) was developed42for use in a high-noise, high-enthalpy electric shock tube environment. A carbon film, evaporated or chemically deposited on an ir-transmitting sapphire or CaF2 window, receives both convective and radiative heat flux. The increased carbon film temperature is sensed by the variation in its longwave ir emission in the 5 - 10 pm band. The calibration and interpretation of the gage must take into account the ir window and sensor bandpass characteristics, as well as the two-layer conduction and grey-body emission problem of the carbon calorimeter layer in thermal contact with a conducting, radiating, almost transparent window. The time response is excellent, of order 1 ps for an 800-nm carbon film. If the gage is reversed so that the carbon layer is isolated from the flow by the window, then it can be used to measure radiative heat flux only. Calibration can be performed dynamically in the shock tube endwall where a known heat flux is available in convenient geometry.

7.6.Thin Film Gages 7.6.1. Construction and Calibration

Probably the most sensitive and fastest response gage is the thin film resistance thermometer o n the surface of an insulating b ~ d y . ~ In ~con. ~ trast to the calorimeter gages discussed previously, we desire the thin film gage to retain as little heat as possible-it should be thermally negligible, simply recording the surface temperature of its substrate without adding additional heat capacity. That is, the sensor is much thinner than the thermal diffusion depth For test times of order 1 ms, a 1-mm thick glass substrate is effectively semi-infinite. Using a typical Pt film therM. Camac and R. M. Feinberg, Rev. Sci. Instrum. 33, 964 (1962). R . J . Vidal, in "Proceedings of the ASME Symposium on Measurements in Unsteady Flow," pp. 90-99. ASME, New York, 1962. R. J. Vidal, Report AD-917-A-1. Cornell Aerosp. Lab., Buffalo, New York, 1956. 42

7.

680

u

HEAT TRANSFER GAGES

METAL PLUG

FIG.9. Thin film gage in shock tube wall.

mometer, gage sensitivity of 1 kW m+ and risetime well under 1 ps are achievable. Figure 9 shows a typical application. The thin metallic resistor films can be applied to glass or quartz by e v a p o r a t i ~ n , ' sputtering47 ~ * ~ ~ ~ ~ or baking of metallic paints onto the surface of the glass s u b ~ t r a t e .Plugs ~ ~ can be made for insertion into metal models, or all glass models are useful when surface discontinuities are to be avoided, as in boundary layer tripping studies. The standard gage has become a 0.1-pm film of Pt pyrolyzed from an organic ester* and deposited on pyrex, then baked to bond it to the glass. The bond is better on pyrex and soft glass than on quartz, apparently because the glasses soften, and there is penetration of the metal into the glass. This is confirmed by interference microscope measurements of gage thickness,48 which show a typical film coat of 0.01 pm, considerably thinner on pyrex than on quartz. In order to use the heat transfer solutions of Chapter 7.2, the thermal properties of the Pt-glass interface must be calibrated individually, since the bulk properties of the materials, including the temperature coefficient of resistance, are not applicable to the diffused layer. The temperature coefficient aR is calibrated in a controlled oven. The effective gage thermal constant ( T K ~ C , )(cf. ~ ' ~Chapter 7.2) can be determined to 5 percent or better using a pulse heating t e c h n i q ~ e .The ~ ~ gage is placed in one leg of a bridge circuit, and pulsed with a known current. The temperature response as the known heat pulse penetrates the gage yields the desired calibration. By repeating the calibration in a thermally conducting chemically inert fluid such as glycerine or silicone oil, the need to measure gage nonuniformity and area is eliminated. When extreme pains T. B. Simpson and C. C. Winding, AlChE J . 2, 113 (1956). R. B. Belser, J . Appl. Phys. 28, 109 (1957). " J . Rabinowicz, M. E. Jessey, and C. A . Bartsch, J . Appl. Phys. 27, 97 (1956). D. J. McCaa, AIAA J . 6, 747 (1%8). 4e G. T. Skinner, ARS J . 31, 671 (1961). 46

@

*

Hanovia 05-x Liquid Platinum Bright, available from Engelhard Chemical Company.

7.6.

THIN FILM GAGES

68 1

are taken, including accounting for two-dimensional heat penetration during ~ a l i b r a t i o nan , ~ ~accuracy of 3 percent is possible. If operation in ionized flows is desired, the gage must be i n s ~ l a t e d ' ~ * ~ ~ , ~ ~ with a thin coating of evaporated SiO (baked to S O z )or MgF,. The gage response time can be kept under 1 ps as long as the overcoat is of order 0.1 p m in thickness. The extreme thickness, 10 pm or more, required to avoid pinholes and gage shorting in highly ionized flows requires data interpretation to be done for a sensor at a finite depth 1 in a glass (SiOz)slab. The effects of finite gage thickness and heat capacity in general cause a lag in the response which needs to be accounted for in careful calibrations where submicrosecond response is important.52 In operation, the gage is placed in series with a large ballast resistor and operated at constant current. For a constant heat flux input, the voltage output is (7.6.1) The gain may be increased up to a point by making thinner gages of higher R o , and by increasing bias current. Eventually, the dissipation causes a perceptible temperature rise in the substrate, and the gage thermal constants, which are functions of T , change. This effect can be calibrated and used to increase s e n s i t i ~ i t y . ~ ~ . ~ ~ The effect of temperature sensitivity of gage thermal properties is particularly acute when high heat transfer is measured for long times. In typical shock tunnel re-entry simulations at 16 MW rn-' for times of 1-5 ms, the glass surface temperature can rise several hundred kelvins. Analysis s h o ~ sthat ~ ~the. ~ variations ~ in K , a , and aRare all significant. The correction to measured q which results from large temperature rise may be found from tables in the references, or the analog computer circuits" devised to perform the inversion from T to q [Eq. (7.2.7)] may be modified to correct for the nonconstant proper tie^.^^^^' Errors in meaV. Krnonicek, I t i t . J . Hctrr Muss Trunsjer 9, 199 (1966). P. V . Marrone and R . A. Hartunian, Phvs. N u i d s 2, 719 (1959). 52 R. K . Hanson, A I A A J . 9, 975 (1971). 53 S. B. Mason and R. L. Varwig, TR-0059 (6240-10)-4. Aerospace Corporation, EI Segundo, California, 1971. '* W. J. Cook, AIAA J . 8, 1366 (1970). s5 R . A . Hartunian and R . L. Varwig, Phys. Hidids 5, 169 (1962); also erratum Phys. Ftuids 5, 637. 56 J . W. Reece Proc. Int. Congr. Instrum. A m q . Sinzul. Fucil., Znd, August 29-31, 1966, Stunfi,rd Utziversiry IEEE/G-AES, 1966. 57 2. A . Walenta, UTIAS Tech. Note 84. University of Toronto, 1964. 51

682

7. HEAT TRANSFER

GAGES

sured q can be as large as 20-50 percent if gage property variations are ignored. It has become accepted practice to calibrate stagnation point heat transfer gages by exposing them to shock tube flows at low Mach number (no dissociation or ionization) and high density (minimal flow nonuniformity due to boundary layer effects). The theory of stagnation point heat transfer is sufficiently established so that accuracy of 5 percent can be as~ u m e d . ~When ~ * ~the ~ gage * ~ ~is to be used for shock tube studies, this is a great time-saver, since all the gage properties are combined into a simple proportion between E and qo and P.

7.6.2. Special Applications Because its construction allows the addition of evaporated thin film overcoatings of insulating or chemically reactive substances, the thin film gage is uniquely suited to the study of transient nonequilibrium phenomena on very short time scales. By placing the gage in the endwall of a shock tube at low density, the shock ~ t r u c t u r eand ~ ~ *accommodation ~~ coefficient may be determined. By comparing stagnation heat transfer to noncatalytic (SiOz) and catalytic (Ag, Cu, Ni) coated gages, the effects of atomic diffusion and surface recombination on heat transfer may be evaluated.51,B0-szThe effects of surface catalysis have also been studied using thin films in glow discharge flow where atomic diffusivity can be meas ~ r e dand ~ ~by appropriately coating calorimeter gages in arc tunnel

Rows.~~-~~ In addition, thin film gages have been used for supersonic wind tunnel ’ in shock tube sidewalls for evaluameasurements of skin f r i c t i ~ n , ~and tion of viscosity.s8 In addition to the accommodation coefficient measurements noted above, the shock reflection from an endwall can also be m J. R. Busing and J . F. Clarke, in “Advances in Aerothermochemistry” (1. Glassman, ed.), AGARD Proc., No. 12, Vol. 1, pp. 165-190. NATO AGARD, 1967. 58 R. K . Hanson, in “Shock Tube Research” (J. L. Stollery, A. G. Gaydon, and P. R. Owen, eds.). Chapman & Hall, London, 1971. 8o R. A . Hartunian and W. P. Thompson, AIAA Prepr. 63-464 (1963). R. A. Hartunian, Phys. Fluids 6, 343 (1963). Ez G. R. Inger, AlAA J . 1, 1776 (1963). R. A . Hartunian, W. P. Thompson, and S. Safron, J . Chern. Phys. 43, 4003 (1965). 64 K. E. Starner and W . P. Thompson, Proc. Heat Transfrr Fluid M r c h . Inst.. 1966, pp. 428-444. Stanford Univ. Press, Stanford, California, 1966. a W. H. Carden, AIAA J . 4, 1704 (1966). 88 N . M. Reddy, AIAA J . 3, 1336 (1965). ‘’ B. J. Bellhouse and D. L. Schultz, G. B . , Aeron. R e s . Counc.. R M No. 3490 (1965). gs R. A. Hartunian and P. V. Marrone, Phys. Nuids 4, 535 (1961).

7.7.

RADIATION HEAT TRANSFER GAGES

683

interpreted to yield thermal ~ o n d u c t i v i t y ~of~noble - ~ ~ gases and dissociated 02.

7.7. Radiation Heat Transfer Gages Radiative heat transfer from high temperature gas can be a significant portion of the flow energy balance at temperatures above 6000 K. Any of the heat transfer gages discussed previously can in principle absorb and register the radiation falling upon them. But the spectral absorption qualities of the surface must be tailored to the important wavelengths, and time response and protection from spurious signals caused by ionization and photoelectric effects must be considered. The measurement of radiation as an important aerodynamic heat transfer component has been of most interest for planetary atmospheric reentry at speeds up to 14 km - s-', which are best simulated in very high energy, often electrically driven, impulse facilities. As the stagnation temperature reaches 8- 12 kK or higher, a significant portion of the radiation is in the vacuum uv, and special windowless techniques are required. The thin film gage (Chapter 7.6) can serve as a radiation gage when covered with absorbing coatings. Rear surface coatings are somewhat more a b ~ o r p t i v eso , ~that ~ direct gage application to a window backface may be useful. A commercial flat black paint, with 0.5 mm thick overlaid sapphire windows has been used7' to cover a spectral range from 0.17-6 pm. The absorptivity E is of order 0.9 but spectral characteristics are not well defined. Sputtered pt gages behind thin quartz windows75 showed E = 0.6 at 1 pm, decreasing to 0.4 at 0.25 pm. The addition of an evaporated carbon film of order 0.1 bm thickness provides E = 0.87 +-9percent from 0.18 to 1 pm. The film has significant heat capacity and its time lag (of order several microseconds) must be calibrated. The correction to heat transfer is given by M. R . Lauver, Phys. Fluids 7, 61 1 (1964). D. J . Collins and W. A. Menard, J . Heat Transfer 88, 52 (1966). C. F. Hansen, R. A. Early, F. E. Alzofon, and F. C. Wittebom. NASA Tech. Rep. R-27 (1959). '' S . S. Penner, in High Speed Aerodynamics and Jet Propulsion, (C. C. Lin, ed.), Vol. 5, pp. 489-501. Princeton Univ. Press, Princeton, New Jersey, 1959. 73 L. Bogdan, NASA Contract. Rep. NASA CR-27 (1964). " R . M. Nerem and G. H. Stickford AIAA J . 2, 1647 (1964). 75 H. Hoshizaki in Hypervrlocity Techniques Symp., 3rd, Univ. of Denver, Denver, Colorado, 1964 p. 245. 'O

684

7.

HEAT TRANSFER GAGES

(7.7.1) The gage is conveniently calibrated using a flash lamp or a chopped high intensity source, whose spectral content is known. For these simple configurations, care must be taken to insure that the window is sufficiently thick to prevent penetration of convective heat flux during the test time (which may limit the spectral bandpass). Conversely, gages with and without absorptive coatings may be compared, the differential heat transfer being attributed to radiation. This requires a highly reflective surface on one gage, which is difficult to achieve at both very short and very long wavelengths-almost all painted or sputtered Pt and Au surfaces have appreciable absorptivity at ir wavelengths beyond 1 pm. In general, any electrically operated gage exposed directly to the flow must be insulated electrically from the ionization which usually accompanies appreciable radiative heat flux. S O , , MgF, coatings have been successfully ~ s e d ,but ~ ~they , ~increase ~ the gage risetime to several microseconds, an appreciable fraction of the 10-50 ps test times in high temperature facilities. The differential heat transfer technique has been successfully applied to an argon arc,76 using water cooled wall-mounted calorimeters (as sensors). The surfaces are coated with flat black paint ( E = 0.98) and evaporated aluminum ( E = 0.07), so that the radiation heat flux G can be found from 41

- q2 =

G(&i -

E2)

(7.7.2)

to an accuracy of about 10 percent. Five major problems plague the measurement of radiation; (1) lack of knowledge of the spectral absorptivity of the gage surface; ( 2 ) the wavelength limitations of windows, particularly in the vacuum uv; (3) the time response, adversely affected by coatings added to mitigate (1 and 2); (4) the photoelectric emission from the gage under uv radiation initiates a discharge which short-circuits the thin film (or any gage with a voltage gradient); ( 5 ) the noise introduced by currents in the facility due to electric discharge energy sources or to highly ionized flow caused by shock reflections in the vicinity of the gage. These effects are eased by the use of low-impedance gages, which unfortunately give low sensitivity. A windowless thin film shock tube radiation7' gage which avoids all of these problems has been developed for stagnation temperatures to I 1 kK L. A . Lukens and F. P. Incropera, A I A A J . 10, 359 (1972). "J. Gruszczynski and W . R . Warren, A I A A J . 5, SI7 (1967). 76

7.7.

RADIATION HEAT TRANSFER GAGES

685

or greater and radiative flux of order lo-’ to 10 MW m-2 sr-l. Slotted glass rings, painted inside with Pt thin film gages, are stacked to form a cylindrical “black body” cavity which is placed at the stagnation line of a cylindrical model. Since the entering radiation is multiply reflected and absorbed internally, the absorptivity need not be precisely known. The gage is filled with He and Kr at 1 atm, (transparent to uv radiation down to 0.088 pm) which serves to scatter any photoelectrons and prevent shorting. The fill pressure is set slightly greater than the test stagnation pressure. A thin latex membrane contains the gas, and is ruptured by a hotwire about 20 ps before shock arrival. The outflow gas is swept away in a few microseconds leaving a windowless, totally absorbing, low impedance, photoelectrically quenched, fast response gage. Alternative methods have been developed7*using sputtered Pt and solid state pyroelectric gages covered with 0.1 p m of A1 black evaporated in a nitrogen atmosphere ( E = 0.98). Using an arrangement of splitter plates and aperture stops in the reflected shock region of a magnetically driven shock tube to isolate the test sample remote from the gage, successful measurements were made both with and without windows from ir through vacuum uv. Pyroelectric gages such as barium titanate, which produce a charge when heated, offer the potential of 50-fold higher sensitivity than thin films, and freedom from photoelectric shorting, since they can be designed to have no voltage gradient. They are sensitive to the total energy deposited in the gage, radiative or convective. However, they require high impedance circuitry for maximum sensitivity, which makes them susceptible to noise and necessitates a careful compromise between achieving sensitivity (high Z , low C ) and time response (low 2). The carbon film ir bolometer42can also be used as a radiation gage, but the relative insensitivity of the ir detection instruments makes the thin film technique better by about three orders of magnitude.

‘I*

A . D. Wood and J . C. Andrews, IEEE Trcrns. Aerosp. Electron. S y s t . aes-3, 356 (1967).

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8. LIGHT SOURCES AND RECORDING METHODS* 8.1. Light Sources 8.1.l.Introduction High speed photographic recording is one of the most important techniques for the study of transient phenomena in the field of technical engineering and scientific research. The development of light sources adapted to the special uses to be described in this part requires attention to spectral distribution, luminous flux, and pulse duration. About one hundred years ago, sparks were used by E. Mach to take pictures of bullets in flight. Since that time, a rapid evolution of the science has provided intense short duration sources with controlled light output. The production of short pulses needs electric energy stored in a high voltage capacitor which can be released suddenly across air gaps, or across gas filled tubes, or into explosive heating and vaporization of thin metal wires. In the case of explosive flashes, luminous energy is obtained from shock waves initiated by chemical reactions. To complete the list of the so-called thermal sources, we should add the plasma focus source and the laser produced plasmas. Neither was developed or investigated for the purpose of high speed photography, but their application is suitable because a relatively large amount of energy is converted to radiation in the visible spectral range. Since 1960, a second group of light sources has become increasingly important: lasers. Their basic properties are: they are monochromatic, of high intensity, are capable of generating short pulses, and have a high degree of spatial and temporal coherence. Solid state lasers (ruby and neodymium lasers) are well suited for diagnostic applications such as photography, interferometry, and holography. Pulse durations in the nanoand picosecond range can be realized by the techniques of Q-switching and mode locking, respectively. Thereby, peak powers of some tens to hundreds of megawatts are easily available. Flash lamp pumped or laser pumped dye lasers provide tunable sources with narrow spectral band-

* Part 8 is by M. Hugenschmidt and K. Vollrath. 687 METHODS O F EXPERIMENTAL PHYSICS, VOL. I8B

Copyright @ 1981 by Academic Pre%. lnc. All rights of reproduction in any form reserved. ISBN 0-12-475956-1

688

8.

LIGHT SOURCES A N D RECORDING METHODS

spectral A energy distribution (relative units)

(a)

I

II

0.1 I

1 dve -Lasers

tpml

FIG.1 . Spectral distribution of the radiation of (a) thermal- (xenon flash lamp) and (b) laser-light sources.

widths which cover the whole range from the uv to the near ir. In some cases, molecular infrared lasers such as CO,(A = 10.6 pm) or HCN(A = 330 p m ) lasers may give a marked increase in sensitivity. Tunability in the infrared is achieved by spin-flip Raman techniques or by rotational line broadening and overlap of electron beam or uv sustained high pressure devices. For the visualization of the ir information, however, rather complicated image converters have to be employed. In Fig. 1 some spectral characteristics of the two groups of light sources are shown. Above, we see the spectral irradiance curve of a xenon flash lamp, a source chosen as an example from among numerous thermal sources. This is characterized by a relatively broad continuum with superimposed strong lines. Most of the laser sources, on the other hand, show narrow lines or relatively narrow bands corresponding to the transitions of the excited and inverted population of energy levels. Only some of the most important types of lasers have been indicated. 8.1.2. Thermal Sources 8.1.2.1. Physical and Photometric Aspects of Light Sources. The part of the spectrum between about 200 to 300 nm, and the far infrared up to several hundred microns which is used for photographic recording occupies only a small portion of the whole electromagnetic spectrum. In the visible range the human eye is the most important detector, so the radiation properties are often described in terms of photometric quantities which are closely related to the spectral response of the eye. These are

8.1.

689

LIGHT SOURCES

TABLEI. Physical and Photometric Quantities

/

of the detector d A 2

surface element of the source d A 1 of the praJected area dA, cos

I,

physical quantities

9

radiation power

Q'

radiant intensity

I* =$

radiance

daQe Le=dQdA,cos

spectral radiance irradiance

photometric quantities

[WI

dt

Le,,=dLe/d

[-$-I A

€,-d~~,/dA,

~, [SJ

I&-[

dt

I

Luminous intensity Luminance spectral luminance

[%]

ev.*

luminous f l u x

illummonce

L

[lml

-&

"-dIl

- d2e~ v - dQ dA, cas c, dL dh

-*I * [

L v , = L E

v - dAI

also used for other sensors such as electrooptical detectors or photographic material. It would be more accurate, however, to use physical quantities such as radiant energy, power, emittance, and irradiance throughout the entire spectrum. Both groups of units are briefly reviewed in Table I . In the case of the physical quantities, all units are related to power, measured in watts. The radiant intensity is the power per unit solid angle; the radiance is the power per unit solid angle and per unit projected area. The unit of luminous flux on the other side is the lumen (lm). This corresponds to & of the luminous flux of a blackbody radiator from an area of 1 cm2 which is at a temperature of 2042 K (the solidification temperature of Pt). The luminous intensity, corresponding to the radiant intensity is the flux per unit solid angle measured in candela (1 cd = 1 Im sr-I). Relating this to the unit projected area yields the luminance (cd m+ = Im m-2 sr-l) which is also often called the brightness. The factor VA describing the physiological influence, connects the luminous flux to the radiant power. (Qv = 6821; V,Q,(A) dh.) The maximum of luminosity of 682 Im W-' occurs at a wavelength of 556 nm (V, = I), @,,(A = 556 nm)[lm] = 682, @,(A = 556 nm)[W]. It should be mentioned that differentiation should be made between two eye response curves, depending upon the sensitivities of the two different receptors, the rods and the cones, for which different V , curves are valid.' F. A . Barnes, "RCA Electro-Optics Handbook." RCA Commercial Engineering, Harrison, New Jersey, 1968.

690

8. LIGHT

SOURCES AND RECORDING METHODS

8.1.2.2. Blackbody Radiation. A blackbody absorbs all the incident radiation of the electromagnetic spectrum. The emission is described by Planck’s law giving the temperature and wavelength dependence of the spectral radiance.

(8.1.1) A is the wavelength, c the velocity of light in vacuum, k the Boltzmann constant, h the Planck constant, and T the temperature (h = 6.6252 X 10-34 J S, k = 1.3846 x 1 0 4 3 J K-1). Following Wien’s displacement law, the maximum power per unit wavelength occurs at A,, which is determined by the relation

A,,,T

= 2898

pm K.

(8.1.2)

Blackbody radiation distributions for three different temperatures are indicated in Fig. 2a. More detailed values are given in the literature.2 The overall radiance is obtained by integrating the spectral radiance over all wavelengths, thus yielding the law of Stefan-Boltzmann [Le(T)lhlackhody

(8.1.3)

=

rn = 56.8 nW mP2K-4 is the Stefan-Boltzmann constant.

8.1.2.3. Real Sources of Radiation. The methods applied for the generation of light include heating of solids, electric discharges in gases at low pressures, and high pressure electric discharges. The distributions of the spectral energy densities of such sources are often approximated by a least square fit to a curve of blackbody radiation distribution. By this method, the spectral luminance may be characterized quantitatively by some temperature. A similar situation occurs for the light of natural sources, e.g., the sun, the moon or the sky. Defining the absorptance a as the ratio of absorbed to incident energy, power or radiance, the real spectral radiance of a source as compared to that of an idealized blackbody radiator is given by Kirchoff s law.3 Lek(A9

T ) = a(A, T ) [Le,(A,

T)lblackhody,

where a depends on A and T. The spectral energy distributions of some typical lamps are shown in Fig. 2b-2d. The tungsten filament lamp for example has a continuum with a maximum near 900 nm at 2854 K. Xenon short-arc lamps are also characterized by a strong continuum with a great number of lines espe-

* M . PivonSky and M. R. Nagel, “Blackbody Radiation Functions.” Macmillan, New York, 1961. W. Elenbaas, “Light Sources,” Philips Technical Library, Eindhoven 1972.

’

8.1.

69 1

LIGHT SOURCES

1000 y

0.5

w 02

1

0 5 - i

2

5

Ipml

2

FIG.2. Spectral output of a blackbody radiator and of different types of lamp sources. (a) Blackbody, (b) tungsten filament lamp, (c) xenon short arc lamp, and (d) mercury lamp.

cially in the near ir between 800 and 1000 nm. Mercury lamps show strong line emission from the uv throughout the visible range. For further types of lamps and details, such as electrical parameters, influence of gas-halogenide or alkali-metal additives, fluorescent screens or others on lifetime, spectral distribution of the radiation, luminous flux and efficiency, reference is made to the l i t e r a t ~ r e . ~ , ~ The luminous efficiency 77 relates the effectiveness of a source to produce a luminous flux to the effectiveness of a monochromatic source of the same power in watt, radiating at the wavelength of the maximum of the eye-response curve ( h = 556 nm, assuming a photopic curve). Following the notations of Table I, (8.1.4)

P. Schulz and H . Strub, Lichrruchnih 10, 364 (1958). A . Bauer, Lichffechnih 16, 118 (1964).

692

8.

LIGHT SOURCES A N D RECORDING METHODS

V his again the spectral sensitivity curve of the human eye. Typical values for a tungsten filament lamp with an input power of 1000 W are q = 16.3 lm W-’, whereas with mercury short-arc lamps (for example 200-W lamps) values of about 47.5 Im W-’ can be achieved.

8.1.2.4. Flash Lamps. Flash lamps are capable of emitting intense, short-duration light pulses. Today portable, small size electronic flash sources are available from a great number of manufacturers. Some of them use rechargeable batteries as a power supply, or they are directly operated from the ac line. The flash lamps of these systems are usually operated with xenon. Typical flash durations are about 200 ps to 1 ms, thus giving exposure times which even in the case of simple nature photography of an amateur photographer, may lead to blurred images. Scientific applications of flash lamps include high speed photography, photochemical studies, solid state, and dye-laser excitation. Lamps can be used for single flash and for stroboscopic applications up to several thousands of hertz. Their spectral emission may resemble daylight or may be more intense in the blue so that they can be adapted to the special type of application. Tubes are in use with krypton, xenon, and argon, the luminous flux of which is dependant upon the gases as well as upon electrical circuitry and energy storage capacitor. Light efficiencies of flash tubes can be increased by choosing gases of high atomic mass, e.g., xenon. Thereby, total efficiencies, as defined by the ratio of light energy integrated over all wavelengths to electrical energy of the storage capacitor, of about 40 percent and more have been achieved. The spectral output of a typical xenon flash, the color distribution of which resembles that of daylight has been indicated in Fig. 1. The maximum radiance occurs at wavelengths of about 4500 A. The distribution of the radiation can be approximated by blackbody distributions corresponding to temperatures of 7000-9000 K. A marked increase in intensity as compared to the blackbody radiance is observed in the near infrared due to the intense lines around 8000 A. The spectral output and the flash durations are strongly depending on electric parameters such as the capacitor C , the inductance L , and the discharge resistance R . To obtain short exposure times with flash duration of less than 1 ps, the discharge circuit must have a small overall inductance, comprising the inductance of the discharge, the inductance of the capacitor, and that of the external circuit. The inductance also determines the peak current which induces strong mechanical stresses so that the limitation is given by the explosion point of the tube. In order to get a damped discharge, the resistance R has to be greater than about 4(L/C)1’2. This condition may be difficult to be met, because R varys in time achiev-

8.1. LIGHT SOURCES

693

ing minimal values of even some tenths of an ohm. When the required condition is valid, the pulse half-width is roughly proportional to C . In other cases the current can be inverted, so that several light maxima following the current oscillations are obtained. The lower part of Fig. 3 shows electric circuit diagrams which are commonly used. The voltage is applied continuously to the electrodes. Triggering can be obtained by ionization due to pulses of some tens of kilovolts which can be applied directly to the electrodes or which are applied to a wire which is attached externally to the tube. The trigger pulses yield corona-like filamentary discharges which initiate the main discharge. This initiation may be influenced by the pressure of the gas (which is normally some 100 mmHg up to 1 atm), by the type of electrodes, the thickness of the lamp walls, the high voltage pulse rise-time, gaseous impurities, or sputtered metal. The maximum energy that can be dissipated by a flash tube is as already mentioned determined by the mechanical forces which are built up when the current is switched on, depending thus on the current rise-time. The energy density that flash tubes may tolerate is of the order of 0.2 J/mm3. A great variety of different types of flashlamps are commercially available. Tubes in linear, helical, or other geometric form are made of hard glass or quartz. Diameters range from about 1.2 to 20 mm with lengths of the interelectrode gap from 3 to 300 mm. Electric storage energies can be supplied up to several thousands of joules with voltages from some hundred volts to some kilovolts. Most of the lamps are convectively

arc length

635mm

50mm

orc diameter

l2mm

Lmm

typicol energy

(001-1) J

(ZOO-9OO)J ( 0 5-21 kV

voltage

( 0 A-2) kV

capacitance

-05pF

typical pulsedurat ion

- 1 ps

(02-2)ms

( 0 7 - 2 ) kV

11

-,lo0 pF

5 ~ 2 2 0 0pF

~ 1 5 ps 0

(1- 2) ms

150mm

; ;o;

165 mm

300mm

13mm

17mm

10,000 J

(8000-19,OOO) J

(1-3) kV

(1.9-4)kV

ry+y+,,n rrLyl -I ~

mode of operation

pwer supply, pulse Po forming network

flash tube

trigger-pulse generator

FIG.3. Technical data of some commercially available types of flash lamps.

694

8.

LIGHT SOURCES A N D RECORDING METHODS

cooled by air, but other cooling techniques can be applied as well. Experiments indicated that in the case of high loading or intermittent operation such as stroboscopic application an overheating of the lamp walls, the electrodes or the lead-in seals can be avoided if the specific load is smaller than 5 W/cm2 for free convection air circulation, 40 W/cm2 for forced air cooling, and about 300 W/cm2 for liquid cooling. Typical data for a few types of xenon flash lamps are summarized in Fig. 3. For a more complete description, the reader is referred to the reference^.^" Because of the small discharge volume with a length of 6.35 mm and a diameter of 1.2 mm, the small size xenon flash tubes approximate a point light source and will be especially useful for applications in optical systems (as they are used for flow visualization by means of shadowgraph and interferometric techniques). According to the different capacitor values (0.01 p F to 1 pF) the flash duration can be varied from 150 ns to 3 ps. High frequency stroboscopic systems can be equipped with such lamps. They can be operated up to 1000 Hz. The duration of such a train of pulses is limited to about a tenth of a second also depending upon the value of the energy storing capacitor. The other types of lamps listed in Fig. 3 are essentially used for laser excitation. Their flash duration is adapted to the radiation life time of the ions or molecules around 1 ms for ruby lasers and some tens to hundreds of microseconds for Nd lasers. These standard linear tubes allow for higher energies to be dissipated. Recent developments led to the construction of coaxial capillary flashtubes designed for the excitation of dye lasers. Using low inductive circuits, light pulses with a rise-time of 50- 100 ns have been obtained with energies of the storage capacitor up to 100 J. Using tubes filled with krypton, it is possible to optimize the spectral radiance for specific laser pumping which can be effective to raise for example the efficiency of Nd-YAG lasers. Pulsed mode of operation has been obtained with high pressure mercury capillary lamps which originally have been designed for cw operation. By this, a considerable increase of brightness can be achieved. As it was shown by Dal Pozo er ~ l .the . ~spectral emission under pulse excitation shows high efficiency in the blue and in the near uv part of the spectrum, thus providing an effective pumping source for dye lasers emitting in the blue. F. Friingel, “High-speed Pulse Technology,” Vols. I , 2, and 3. Academic Press, New York, 1965, 1965, and 1976, resp. ’ H . E. Edgerton, “Electronic Flash, Strobe.” McGraw-Hill, New York, 1970. P. Dal Pozo, R . Polloni, and 0. Svelto, Appl. Phys. 6, 341, 381 (1975).

8.1.

LIGHT SOURCES

695

Another type of very intense short-duration pulsed lamp has been described by F e r r a ~who , ~ uses tubes of quartz that are continuously evacuated, whereby a small air leak provides a residual gas pressure of a few torr. The discharges are primarily occurring in the vapors ablated from the quartz tube walls. No significant difference in light output can be observed if other gases such as N2,Ar, He, or C02 are used. Intense light pulses of several microseconds duration have thus been obtained with electric energies in the range of 10 J to about 1000 J. 8.1.2.5. Spark Light Sources. Sparks are convenient flexible light sources for high speed photographic recording.’O They are obtained by the rapid discharge of electrical energy stored in a capacitor. They provide the possibility to obtain extremely short exposure times which are necessary for the investigation of fast events with velocities up to several 10 km/s or even higher. Sparks in air have been the earliest sources to study such phenomena. The volume of these discharges can be made very small, so that they are especially useful to take shadowgraph pictures. As the sharpness of the images depends on the diameter of the source, the light of the spark is sometimes collected by an objective lens and imaged to a small pinhole which then serves as the actual point source. Another application of sparks is the use in schlieren systems. This was first described by Foucault who visualized rapidly changing disturbances, causing refractive index changes in the atmosphere. Toepler continued the development and application of these methods to investigate changes in refractive index in liquids, gases and even in flames. Further, schlieren and interferometric techniques using spark sources have been proposed and described by Schardin” for the purpose of photographing rapidly varying processes. Such systems are now standard equipment in wind tunnel investigations and for aerodynamic studies of sub- or supersonic flows. 8.1.2.5.1. SPARK-FORMATION, ELECTRICAL A N D HYDRODYNAMIC PARAMETERS OF SPARKS. The mechanisms leading to the electric breakdown of the gas between the electrodes can be described by the Townsend and the canal (streamer) models.12 The Townsend mechanism holds for small overvoltages as compared to the static breakdown voltage and C. M. Ferrar, Re\,. Sci. Instrum. 40, 1436 (1969). K . Vollrath and G. Thomer, eds., “Kurzzeitphysik.” Springer-Verlag, Berlin and New York, 1967. I I H. Schardin, ErRph. E x u k f . Nuturwiss. 20, 303 (1942). I* P. Schulz, “Elektronische Vorgange in Gasen und Festkorpern,” 2nd ed. Verlag G. Braun, Karlsruhe, 1974.

8.

696

LIGHT SOURCES A N D RECORDING METHODS

for low values of the electron density where space charge effects can be neglected. The generation of new electrons, the so-called successors, then occurs essentially by the photo effect at the cathode and is described by Townsend's first coefficient a . The canal mechanism requires higher overvoltages. If no is the number of starting electrons, the avalanche achieves a critical value, if the following conditions holds

In no

+ ad

>

= 20.

(8. I .5)

In distance dCritfrom the starting point, the electron avalanche then causes the formation of anode- and cathode-directed streamers which are strongly influenced by their own space charge. The radius of the streamer discharges are of the order of the diffusion radius of the avalanche. After the streamers have arrived at the electrodes, fast luminous fronts, so-called ionizing waves are generated moving in both directions through the gap, achieving velocities in the order of 108-109cm/s. In a homogeneous electric field these conditions have been studied extensively by Koppitz,I3 who used fast framing and streak image converter cameras to detect the phenomena of streamers and ionizing waves. The velocities of the anode- and cathode-directed streamers have also been investigated by Timm. In his experiments a starting electron cloud of lo6 to loBelectrons has been produced in the interspace of the electrodes by multiphoton processes by a focused giant pulse of a Q-switched ruby laser.I4 The electrical properties of sparks are determined by the charging voltage, the capacitor C , the inductance L , and the resistances R , both of the channel and the discharge circuit. L and R are depending on time. The energy W , dissipated in the spark can be calculated from the following equation^:'^

(8.1.6) with

Assuming the inductivities L to be constant, yields us

J. Koppitz, Z.

N u r i c t j h c h . , Teil A 26, 700 (1971).

*'U . Timm, Dissertation, Universitat Hamburg (1972). R . Grunberg, 2. Nufurforsch.. TeilA 20, 202 (1965).

8.1.

697

LIGHT SOURCES

C = 5500 pF

iIkAl

L =lOnH+LK(t)

0 80

40

120'hsl

t

L la)

L =15nH theory of Braginskii

FIG.4. (a) Electrical and (b) gas dynamic parameters of open spark discharges in air. [Electrical parameters from Andreev and Vanyukov;16hydrodynamic parameters from Andreev el d .I T ]

C UR(t)

UOC

-

$ i dt - LC

1 di = Uo - - $ i dt - L - - ,

C

dt

dt

(8.1.7)

Figure 4 shows a typical example as given by Andreev et u / . ' ~ * ' 'concerning the electrical and hydrodynamic parameters of spark discharges. S. I. Andreev and M. P . Vanyukov, Sov. Phys. -Tech. Phys. (EngI. Trunsl.) 6, 700 (1962). l 7 S. I. Andreev, M. P. Vanyukov, and A . B. Kotolov, Sov. Phys.-Tech. PhyA. (EngI. Truns/.) 7, 37 (1962).

698

8.

LIGHT SOURCES A N D RECORDING METHODS

By measuring the electric current, the voltages uR, uL, u c , the resistance R and the dissipated power or energy, respectively, can be calculated following the above mentioned equations. The value of R is thereby strongly influenced by the temporal variation of the radius p ( t ) of the discharge channel. Apart from the initial stages during the streamer state, the growth of the high-current channel can be described with adequate accuracy by hydrodynamic models as they have been worked out by Drabkinala and Braginskii.lQ Thereby, it was assumed that pressures, temperatures and densities are constant over the cross section of the channel. Starting from shockwave theory, these authors derived expressions for the radius of the spark channel, the velocity of the shockwave and the pressures built up as a function of the electrical energy and as a function of time. The radius of the shockwave reveals to be proportional to Ell4 . P2,the and the pressure proportional to E / p 2 . E is the envelocity to E1/4t-1’2, ergy liberated per centimeter length of the discharge, and p is the shockwave radius depending on time. The lower part of Fig. 4 shows some further results of Andreev ef al., giving the time development of the current, the radius of the shockwave produced by the discharge and the radius of the channel. As compared to the experimental data, it can be seen that the model of Braginskii shows a good agreement, whereas the model of Drabkina gives reliable values only during the first 30-40 ns of the discharge. 8.1.2.5.2. TEMPERATURE A N D DENSITY DISTRIBUTIONS. Temperature distributions in electric sparks have been determined by various authors.20*21Depending upon the gases, this can be done spectroscopically by measuring for example the absolute intensity of a line, the oscillator strength of which must be known, the relative intensities of two or more spectral lines or the intensity of continuum radiation. Measurements performed by Egerova (Fig. 5a) show the distribution of spark temperature in air. First quantitative informations on gas-density profiles have been obtained by Vollrath,22measuring the absorption of soft x rays. Figure 5b shows some results of the density in a spark in argon. Further parameters, for example electron densities, can be measured interferometrically. Some values are also indicated in Fig. 5c as obtained by ir laserS. 1. Drabkina, Z h . Eksp. Teor. Fiz. 21, 473 (1951). S . I. Braginskii, Sov. Phyb. -JETP (Engl. Trunsl.) 7 , 1068 (1958). 2o L. Krauss and H, Krempl, Z . A n g r w . Phys. 16a, 243 (1963). 21 V. F. Egerova, V. J . Isaenko, and A . A. Mak, Sov. Phys. -Tech. Phys. (Engl. Trans/.) 7, 242 (1962). 22 K . Vollrath, Proc. l n t . Congr. High-Sprrd Photogr., 5 t h , Wushington, D . C . . 1960, p. 179. SMITE, New York, 1962. In

l8

8.1.

699

LIGHT SOURCES

-83 ps

0.5

5 (0

1

10 (b)

15imrn1 (Cl

FIG.5. Some fundamental physical properties of electric spark discharges. (a) Radial distribution of temperature of an air spark. [After Egerova ef a/.z1] (b) Radial dependence of the density of an Ar spark (no density at ambient pressure and temperature). [After Vollrathzz] (c) Radial profile of the electron density of an air spark. [After Hugenschmidt and V ~ l l r a t h . ~ ~ ]

interferometric methodsz3using a TEA-C02 laser as infrared source and a liquid crystal-display as an image converter to convert the infrared radiation towards the visible (see Section 2.8.5.2). 8.1.2.5.3. SPECTRAL EMISSION. The optical properties are determined by temperatures and densities in the channel during the discharge. The time dependence of emission can be measured by mutipliers, fast cameras, and spectrometer^.^^ The spectral luminance, thereby, is strongly dependent upon the current density. In most cases, the radiation distribution can be approximated by blackbody radiation. Ar discharges, for example (p = 4 atm) 100 ns after breakdown, resemble a blackbody at a temperature of 31,000 K in the wavelength interval from 4000-9000 A. During the following 300 ns the temperature decays to about 18,000 K.l0 The maximum obtainable values of spectral luminance LvA depend upon the velocity of the energy accumulation which is proportional to dildt. As shown experimentally, LvA is limited because of saturation effects first in the red and subsequently in the violet range of the spectrum. Increasing the electric current leads to higher expansion velocities of the spark channel but not to higher luminance. Table I1 shows some photometric and corresponding electrical data of spark discharges in different gases as given by Vanyukov and Mak.25 8.1.2.5.4. DIFFERENT TYPESOF SPARK DISCHARGES. According to different applications, a great variety of spark sources has been developed. z3 M. Hugenschmidt and K. Vollrath, Proc. Int. Congr. High-Speed P h o f . , IOrh, Nice, 1972 p. 515. Assoc. Nat. Rech. Tech., Paris, 1972. 24 G . Glaser, 2. Naturjbrsch., Teil A 6 , 706 (1951). 25 M. P. Vanyukov and A . A . Mak, Proc. I n f . Congr. High-speed Photog. 5rh, Wushingion, D. C . , 1960, p. 41. SMPTE, New York, 1962.

8.

700

LIGHT SOURCES A N D RECORDING METHODS

TABLE11. Photometric and Electrical Parameters of Spark Discharges in Different Gases" p (kPa)

di/di ( A d ' )

C (pF)

T (K)

L V ( G c d m-*)

lo'=

0.2

70000

370

~~

He

3100

N,

300

1.1

.lo'*

0.1

62000

320

Ar

300

0.3

.lo"

0.2

46000

220

air

100

0.1

40000

I70

0.05~10"

" Data from Vanyukov and Mak.25

Most common is the free spark in air under normal atmospheric conditions. As already mentioned, the duration of the light pulses depends on electrical and geometrical parameters. Low inductivities can be realized, if the capacitor is formed by a circular plate condensor. Plates of Ba-titanate ceramics, for example, have been used by Stenzel for the construction of short duration point-light sources. The ceramic plate has a central hole containing an electrically isolated trigger electrode allowing the main discharge to be initiated at any moment desired. The light is coupled out through a small hole of 1-mm diameter in one electrode. With a capacitor value of 40 n F and a voltage of 10 kV, intense short duration pulses of 0.1 ps duration have been achieved. A more recent development using ceramic 3.3-nF capacitors is shown in Fig. 6.2e Exposure times in the range of a few nanoseconds have been obtained by FischerZ7by using coaxial capacitor discharges. The capacitor is formed by a copper line, isolated externally by a thin Teflon sheet on top with a copperplating which connects into a cap towards the spark gap which is formed between the center electrode and the outer narrow center hole with a short pin. This setup represents thus a very short low inductance transmission line with a distributed capacitance C. The inductance is given by the contributions of the line of about 0.2 nH, of the connecting cap (0.4 nH), and of the gap (0.3 nH). With these nanolites, which are commercially available, light pulses with a rise time of 2 ns and a halfwidth of 8 ns are obtainable with electric energies of 0.01 J. Current densities exceed lo7 A/cmZ and the brightness obtained was measured to be of the order of lo7 cd/cm2. 26 A . Stenzel, LRSL-Note Tech. 17a/57. Dtsch.-Franz. Forschungsinst. St.-Louis, 1957; Pro(,. Int. Congr. High-speed Photog., 8ih, Stockholm, 1968, p. 153, Wiley, New York, 1968. z7 H . Fischer, J . Opt. Soc. Am. 51, 543 (1961).

8.1. LIGHT SOURCES

70 1

With guided sparks (sliding sparks) it is possible to increase the spark length about to the 20-fold of the value of the length of a free spark at the same voltage. With an increase in the length of the interelectrode gap, the spark resistance will also increase, so that a much better adaption of the load to the discharge circuit can be achieved, thus providing a better electro-optic conversion ratio. The electrodes may be connected with an isolating material such as glass, quartz, or glimmer plates o r tubes, or with electrolytic conductive surfaces, such as porous ceramic material drenched with an electrolyte. The pulse durations normally obtained with sliding sparks are in the range of 1 ps. Nearly the double value of intensity with constant pulse half-width can be achieved as compared to free sparks. With a capacitor of C = 0.05 pF and a voltage of 18 kV, Tredwell** has realized light pulses of 0.4 ps duration and a luminous intensity of 5 x lo6 cd. The light output can be increased by increasing the pressure of the gas or by application of rare gases instead of air. Measurements have been performed by FUnferz9 using krypton, argon, and xenon achieving a five-fold light increase. Constricting the discharge by walls as in the case of capillary sparks, higher energy densities are obtained. At the same energies, these discharges are reaching higher temperatures. The optimum diameters of the capillary are depending on the energy and the nature of gas. Spark resistance is determined by the cross section of the bore and by the temperature-dependent electrical conductivity of the plasma. As the rapid cooling of the plasma channel by the gas-dynamic expansion is prevented by the walls, it is not possible to obtain shorter exposure times than with freely expanding sparks. The capillary sparks are spatially well defined so that they are especially suited for schlieren photographic applications. In a certain sense, capillary sparks can be considered as special case of flash-lamp discharges which have already been treated in the preceding section. High light output and short pulse durations are obtained by large area spark discharges, as developed by Schwertl and S t e n ~ e l . ~ A ' schematic diagram of the setup and the temporal shape of the light pulses are both included in Fig. 6 . The radially distributed discharge takes place between a central electrode and a circular concentric electrode in the small volume determined by two glass or quartz plates. Because of its high efficiency, xenon is used as filling-gas at pressures of about 150 Torr. Normally, the discharge is split up in a great number of spokelike filaments. In spite of J. Tredwell, MS Thesis, Elec. Eng. Dept., MIT, Cambridge Massachusetts, 1960. zs E. Fiinfer, 2. Angew. Phys. 1, 295 (1949).

M. Schwertl and A . Stenzel, Tech. Ber. 12/73. Dtsch.-Franz. Forschungsinst. StLouis, 1973; Proc. I n t . Congr. High-speed Photogr.. lOth, Nice, 1972, p. 335, Assoc. Nat. Rech. Tech., Paris, 1972.

702

8.

LIGHT SOURCES A N D RECORDING METHODS

electrodes pin

3300 pF capacitor

outer coating

Fc electrodes

metal support

I

(b)

0 200 COO 600

lnsl

F

discharge volume A n

\

g electrode

cent& electrode (C)

F I G .6. Different types of spark sources. (a) Spark point source. [After StenzeLZ6] (b) Nanolite. [After Fi~cher.~'](c) Large area spark source. [After Schwertl and S t e n ~ e l . ~ " ]

the fact that the location of the individual spokes is not well defined, the light output shows an excellent reproducibility. Using capacitors of C = 0.15 pF and energy values of E = 12 3, pulses of 0.2-ps duration, and a maximum luminous intensity of 2 x lo7 cd have been realized. 8.1.2.5.5. ELECTRICAL A N D OPTICAL TRIGGERING OF SPARKS (JITTER). Triggered spark gaps are often used for example for fast switching of electric circuits or for the generation of high voltage pulses with short rise time. As we are concerned with light sources, these aspects and applications shall not be considered in the present article. To obtain light pulses at any desired moment, the spark source itself can be used as a triggered spark-gap. One reliable system, which is often used, has a central bore in.one electrode which contains, electrically isolated, a trigger electrode. The main discharge can be initiated by applying a steep high voltage pulse to this trigger electrode, thus providing a change of the electrical potential distribution and an uv preionization due to hard photons produced by the

8.1.

LIGHT SOURCES

703

starting trigger discharge. The voltage applied to the electrodes should not be less than about 80 percent of the static breakdown voltage. The time delay between the initiating high voltage pulse and the instant of breakdown depends on the gas, the pressure, geometric form of the electrodes, gap length, and voltage. With typical triggered spark gaps of this s can be realtype, time delays of a few lo-* s with a jitter of some ized. Jitter can be minimized by preionizing the gap with radioactive material. Another possibility to obtain short delay times and small jitter is to use three- or four-electrode sparks, where usually the first gap is also preionized. Since the first demonstration of laser triggered spark gaps in 1963 by Guenther and Griffin,31numerous investigations on the problem of optical triggering have been carried out. Giant pulses of ruby and neodymium lasers have been focused to one of the electrodes, thus providing target-induced plasmas which are completely ionized. The uv emission from these plasmas was shown to ionize the neutral surrounding gas. Nitrogen lasers, emitting in the near uv at a wavelength of 3371 8, are inexpensive and especially suited for the initiation of spark discharges because of the short pulse durations of only a few nanoseconds. Recently, experiments have been carried out by Dewhurst at nl.32with single picosecond pulses gated out from the mode-locked train of a Nd laser. The characteristics are very similar to those using conventional Q-switched pulses. For gaps up to a few centimeters and pressures of some bars, subnanosecond jitter has been obtained.

8.1.2.6. Miscellaneous. Before the advent of lasers the sparks and flashlamps described in the foregoing sections represented 'the most important group of light sources designed for the widespread range of photographic recording of rapid events. The intensities, spectral distribution of the radiance, and pulse duration can largely be adapted to the problems to be investigated. Because of their simplicity and relative low cost of operation, they are still in use today and will be used furtheron. For completeness we shall describe in this section some processes in which emission of intense radiation in the visible or near visible spectral range can also be obtained. Large conversion ratios of stored energy to light output have been achieved with exploding wires, explosive flashes, plasma foci and laser-produced plasmas which have been studied extensively and used mainly with respect to other scientific or technical applications. 8.1.2.6.1. EXPLODING WIRES.Exploding wires, for example, can be used to switch off overloaded electrical circuits, to generate strong cylin31

32

A . H . Guenther and R. J. Bettis, J . Phys. D : Appl. Phys. 11, 1577 (1978). R. J . Dewhurst, G . J. Pert, and S. A. Ramsden, J . Phys. D 5,97 (1972).

704

8. LIGHT

SOURCES AND RECORDING METHODS

drical shock waves or to initiate explosive^.^^ These processes are followed by an intense burst of light. The experimental set up simply consists of a thin metal wire with diameters of several 100 pm through which the energy stored in a high voltage capacitor is discharged in an inductively loaded electrical circuit. High current densities of more than lo5- lo6 A/mm2 are thereby generated causing a rapid increase in temperature and an explosive fusion and evaporation of the heated material within a few ps. The transition from the solid to the liquid phase of the material can be observed experimentally by the sudden variation of the electric current due to the change in resistivity. In the following development, the current will even be switched off completely because the metallic vapours are not conducting in their initial phase. High overvoltages L di/dt are thereby built up which can cause reinitiation and breakdown of the gap by the formation of bright, highly conducting arcs. If the overvoltage is not sufficiently high, reignition can occur after a longer timedelay by thermal ionization processes. Measurements reported by Kruge?4 have been carried out by using 20-40 mm long, 0.1-0.2-mm diameter, Cu-Mg wires. The electrical parameters are: C = 40 pF, U = 10 kV, the oscillation period, 66 ps. The plasmas obtained proved to be optically thick during the first some tens of microseconds. Radial temperature and electron-density profile measurements have only been possible, therefore, in the later stages of the development. Both profiles have a cylindrical shape with a minimum central value, the maximum occurring at a distance r of several millimeters, depending on time. Maximum temperature values 90 ps after the initiation are still as high as 12,000 K, electron densities are in the order of some lo1?cmP3. Further, measurements have been performed using electrolytic liquid jets (for example BaNz solutions). In this case, temperatures of 62,000 K have been determined 15 ps after initiation, which also dropped to about 12,000 K in 100 ps. The duration of the light pulses can greatly be influenced by the amount of electrical energy, the material and the dimensions of the wires and the material surrounding the wire (gases at various pressures, liquids or solids). The light output is typically characterized by a first short peak of low intensity at the end of the vaporization period. The beginning explosion is marked by a dark pause, whereas the main light emission occurs during and after the period of reignition. As an example for the application of exploding wires in optical systems, one could point out the investigations reported by Hornung and S a n d e m a r ~who , ~ ~ studied hyper% W. G. Chaces and H. K . Moore, "Exploding Wires," Vols. 1 , 2, 3, and 4. Plenum, New York, 1959, 1962, 1964, and 1968, resp. 34 R . Kriiger, Z . A n g e w . P h y s . 25, 283 (1968). 35 H . G. Hornung and R . J . Sandeman, J . Phys. D 7, 920 (1974).

8.1.

LIGHT SOURCES

705

sonic flows of argon over blunt bodies by means of multiple-wavelength interferometry. 8.1.2.6.2. EXPLOSIVE FLASH.As already mentioned, short duration light pulses can also be obtained by using chemically stored energy of explosives creating rapidly expanding shock waves propagating in rare gases such as argon. In the most simple case, a milky transparent balloon filled with argon attached to a detonator.of some tens of grams of explosive can be used. The shock waves thus produced are characterized by extremely high luminosity which can be further increased by intersecting two waves initiated from two sides which are producing considerably higher pressures in this plane. The pulse widths of typical explosive flash sources are depending upon the length of the rare-gas-filled volume. They are of the order of some 10-100 ps. These bright sources are especially useful for recording in reflected light using high speed rotating mirror cameras where individual framing exposure times are varying from about 10 ns to several rnicro~econds.~~ 8.1.2.6.3. PLASMAFOCUS. Plasmafocus experiments have been performed for the generation of hydrogen or deuterium plasmas with extremely high electron densities and temperatures. The experimental setup uses coaxial cylindrical electrode configurations in which at one side the energy of a storage capacitor is discharged concentrically. The uniform concentric plasma layer between the outer and the inner electrode is then axially accelerated in z-direction by the Lorentz force K = j x B ( j = current density, B = magnetic inductance). At the other end of the inner electrode which is shorter than the outer one, radial magnetic forces cause a rapid compression of the plasma. Measurements of the plasma parameters yielded values of the electron density of several lo2*cm-3 and electron temperatures from 1 to 8 keV (corresponding to several lo7 K). These values are observed in a focus volume of about 5 x cm in diameter and 1 cm in length. The main application of the plasmafocus would, therefore, be that of a pulsed neutron source. Great efforts are undertaken for high speed neutrographic re~ording.~?Simultaneously x-ray emission and strong light emission from the uv up to the medium ir is also obtained. The exact values depend upon the storage-capacitor energy, ranging in typical experiments from some tens of joules to some tens to hundreds of kilojoules, and upon geometric configurations. The lifetime of the plasma in its compression phase is about 100-200 ns. When

Z. Pressmann, Proc,. I n t . Congr. High-Speed Phohof., 5 t h . Wushingmn. D . C . , 1960, p. 56. SMPTE, New York, 1962. 37 D. Ruffner, Ber. IPF 74-3. Inst. Plasmaforsch., Stuttgart, 1974.

706

8.

LIGHT SOURCES A N D RECORDING METHODS

D2is used as a filling gas, strong emission of neutrons can be observed 10” neutrons per pulse with energies in the range of 2.5 MeV).38*3e Measurements ranging from A = 0.2 pm to 8.6 pm have been carried . a~deuterium-filled ~ 30-kJ focus. The intensity out by Schmidt et ~ 1using drops down first proportional to I/AZ, then proportional to l/A4. Investigations of a small 50-J focus, operated with hydrogen, show spectral luminance corresponding to thermal radiation of a blackbody temperature of 80,000 K. The applicability of the plasma focus as an intense light source in a large spectral range including vacuum uv and x-ray range has been pointed out by some investigators. Direct application to photographic recording of flow or plasma phenomena, however, has not yet been published. 8.1.2.6.4. LASERPRODUCED PLASMAS. The high degree of spatial coherence of lasers to be discussed in the following sections allows for high energy densities to be obtained in the focal plane of an objective lens. Laser induced breakdown in gases, for example, giving optically dense plasmas, can be obtained by starting from multiphoton absorption and inverse bremsstrahlungs absorption processes. Threshold values of the power densities depend on the wavelength of the laser radiation as well as on the type of gas and its initial pressure. Lower thresholds are observed by irradiation of solid state targets. The characteristics of the plasma parameters, the temperatures, pressures, and specific internabenergies have been studied extensively. For details, the reader is referred to the l i t e r a t ~ r e . ~ ’The * ~ ~great effort in this field of research has been stimulated by two facts. The first one was due to the possibility of applying high intensity focused laser beams for material processing, thereby causing fusion or evaporation of the material so that laser technology can be used for cutting or drilling purposes. The main fact, however, was to study laser produced plasmas in thermonuclear fusion experiments. The basic concept involves compression and heating of a mixture of deuterium and tritium. The densities which are necessary to approach the breakeven point, where thermonuclear energy equals the laser input energy, are about 1000 g/cm3, ignition temperatures are of the order of 30-40 x lo6 K. ( 1Oln to

L. Michel, K. H. Schonbach, and H. Fischer, Appl. Phys. Lett. 24, 13 (1974). H . Rapp, P h y s . Lett. A 43, 420 (1973). 4u H. Schmidt and H. Conrdds, Verhandlungen der Friihjahrstagung der DPG, Stuttgart, 1974, Phys. Verlag, Weinheim, 1974. J. Schwarz and H. Hora, “Laser Interaction and Related Plasma Phenomena,” Vols. 1, 2, 3A, and 3B. Plenum, New York, 1971, 1972, 1974, and 1974, resp. $ ’ J . F. Ready, “Effects of High-Power-Laser Radiation.” Academic Press, New York, 1971. 38 30

8.1.

LIGHT SOURCES

707

Estimates are showing that those values of symmetric compression of a material should be possible with laser pulse energies of 104-105 J . Experiments have been carried out with ruby and neodymium lasers, providing pulses in the nano- or picosecond range with energies up to some kilojoules. High pulse energies can also be achieved with COz lasers, one of the most powerful systems of 10 kJ being under construction at the Los Alamos Scientific Laboratories. 8.1.3. Laser Light Sources 8.1.3.1. Fundamental Properties. Lasers are generating and amplifying electromagnetic waves at optical frequencies scanning a broad spectral range from the vacuum uv to the ir. As compared to thermal sources, lasers are normally characterized by extremely narrow bandwidths. Laser oscillation has been observed and reported for a large number of ~ ~ efforts are stimulated neutral and ionized atoms or m 0 1 e c u l e s . ~ ~ -Great to extend the band of wavelengths towards the soft x-ray range. The basic principles of lasers, the interaction of radiation fields and photons, respectively, with atomic systems, are theoretically described by quantum-mechanical formalisms. As we are particularly concerned with optical methods, the spatial and temporal coherence, the speckle properties, the contrast or the visibility of fringes and interference patterns are of importance. These topics can largely be treated by the semiclassical or even the classical theory. Starting from a simplified model with a two-energy level material, the radiative energy balance can be set up by a rate equation in which induced emission is considered to be a negative absorption. Using the definition of a net-absorption coefficient, the derivation exactly leads to Kirchhoff s law, which then proves to be consistent with the laser principle.46 Since induced transition probabilities are equal both for absorption and emission, an amplification can only occur as the number density N z of particles excited in the upper energy level E2 is greater than the density N , in the lower state El, whereby the statistical weights have to be taken into account. The activation of the laser material necessary for obtaining inversion ( N z > N , ) can be achieved by different pumping mechanisms depending on the type of the laser used. Optical excitation by the intense light of flash lamps is used in a D. Ross, “Laser, Lichtverstarker und Oszillatoren.” Akad. Verlagsges., Frankfurt, 1966. 44 W. Kleen and R. Miiller, “Laser.” Springer-Verlag, Berlin and New York, 1969. 45 G . Herziger and H. Weber, “Laser, Grundlagen und Anwendungen.” Physik-Verlag, Weinheim, 1972. A. Bauer, Optik 29, 179 (1969). @

708

8.

LIGHT SOURCES A N D RECORDING METHODS

the case of dielectric solid state lasers, dye lasers, and photo-dissociation lasers. Gas lasers and semiconductor lasers can be excited directly by electric currents. Other excitation mechanisms have been successfully applied including electron-beam techniques and gas-dynamic methods for high power gas lasers. The primary process in an “inverted” laser material is the amplification of an incident flux of light or of spontaneously emitted photons. Providing a suitable feedback, this amplification can exceed the losses (absorption, diffraction, mirror losses, etc.) thus producing self oscillation. The feedback can 8.1.3.1.1. MONOCHROMASY A N D MODE SPECTRUM. be realized by a Fabry-Perot-type or ring-type resonator. The common characteristic of nearly all laser resonators is that they are open resonators that do not require side walls. The actual wavelengths of the laser lines are determined both by the fluorescence profile of the considered transition of the laser medium and by the eigenfrequencies of the resonator modes. The spectral line shape is thereby influenced by different broadening mechanisms. In gas lasers, the Doppler effect or pressure broadening is mainly acting on the line width. In solid state lasers there are the statistical Stark fields of thermal vibrating crystal lattice, inhomogeneities, and impurities. The largest spectral widths are observed with dye lasers due to the strong interaction of the dye molecules with their solvents. The amplitudes and phases can undergo irregular fluctuations. These changes are relatively slow, however, they are depending upon the effective spectral width Av. As Av is much smaller than the central laser oscillation frequency vo (Av << vo), the light output can be described by quasi-monochromatic waves. Figure 7 schematically shows the spectral distribution of the laser emission. The frequency dependence of the transition, which is usually approximated by a Gaussian or Lorentzian profile is characterized by the half-width Sv. This frequency bandwidth is actually reduced by stimulated emission, depending on the laser threshold condition G(R1R2)1’2. L a 1, where G represents the gain, R 1 , R2 the two cavity mirror reflectivities, and L the additional losses. According to the Huygens principle, the modes of the open resonators can be considered to be eigenfunctions of a Fredholm integral equation. Assuming linearly polarized waves this equation can be written in a scalar form. Numerical calculations, as for example performed by Fox and Li4’ and K ~ g e l n i k yield , ~ ~ the spatial structure of the distribution of the electromagnetic field inside the laser cavity. As the field strengths are mainly ” A.

G. Fox and T. Li, Bell S y s t . Tech. J . 40, 453 (1961).

*’ H . Kogelnik and T. Li, Proc. IEEE

54, 1312 (1966).

8.1.

709

LIGHT SOURCES

cavity with plane-parollel Fabry - Perot square aperture

B5G TEM01

L Fresnel number

F

= aZ/(L.X)

resonances

(1, = (El

transverse mode distribution

v ( x , ~ )=

mode

A(+)

2

2

Separation

TEMll

2 +

(7) 2

+

(n+l)*y0 sin - sin 20

=Ti: 1 [(s,-s,)

+=(mi LX

- m:+

n:

- n: 11

FIG.7. Longitudinal and transverse mode spectra.

transverse to the direction of propagation, the modes are termed transverse electromagnetic modes (TEM,,,-modes), where m and n are giving the number of intensity nodes in the mirror planes in (x, y)- or (r, 4)directions, r e s p e ~ t i v e l y . Some ~ ~ typical equations concerning the resonances, the longitudinal and transverse mode distribution and the mode separation are also included in Fig. 7. For simplicity, a plane parallel Fabry -Perot square aperture resonator has been chosen. For large Fresnel numbers F , the modes thus described can thereby be considered to be in good approximation with the real modes. Similar but more complicated expressions including Bessel functions can be derived for circular apertures or for curved mirror configurations. As q are large integer numbers ( L >> A,,,), this index characterizing the longitudinal modes is usually omitted. In most cases transverse fundamental mode of operation (m = n = 0) is preferable. This can be achieved by adequate cavity design. Under normal conditions, the emission will be longitudinally multimode with randomly distributed initial phases, however single longitudinal G. Grau, Optische Resonatoren und Ausbreitungsgesetze fur Laserstrahlen. I n “Laser” (W. Kleen and R. Muller, eds.), p. 49. Springer-Verlag, Berlin and New York, 1969.

710

8.

LIGHT SOURCES A N D RECORDING METHODS

mode operation can be realized. This proves to be important for holographic applications. Simultaneous oscillation of a great number of longitudinal modes with strongly coupled phases leads to the generation of trains of ultrashort pulses with ps duration of the individual pulses. Both cases, i.e., the single-mode and mode-locked operation will be discussed more in detail in the following sections. It should be mentioned that besides the use of stable resonators for laser sources as applied to photography or spectroscopy, lasers can also be operated with unstable resonators which in the case of high power lasers allow the energy in the fundamental mode to be extracted from large volumes.5o 8.1.3.1.2. COHERENCE PROPERTIES. Coherence properties of lasers can be described by second or higher order correlation effects. As already mentioned, it is possible to select the transverse fundamental mode TEMoo by suitable resonator configurations. That means that the phases of the emitted waves over the whole diameter of the beam are well correlated. Such a radiation field is termed as spatially coherent. Experimental evidence can be shown by the visualization of the interference pattern obtained for example in Young’s experiment as indicated in Fig. 8, using two pinholes, separated by a variable distance. Fundamental mode of operation is favorable for a great number of applications because it allows for propagation over long distances with minimum angular divergence and for production of high power densities in the focal plane of an objective lens. Due to this fact, it is possible in optical systems to generate nearly exactly diffraction limited point-light sources which are important for shadowgraph techniques. Furthermore, Fig. 8 shows schematically the relationship between the temporal pulse shapes and their spectral distributions for two different wavetrains, the envelopes of which have been chosen arbitrarily to have a rectangular or a Gaussian form, respectively. The spectra are calculated by Fourier transforms from which the bandwidths 6 v are determined. In both cases S v shows to be proportional to the inverse pulse length A t . If V(r,t ) is the complex analytical signal of the light-field amplitude in a more general way,51 it can be written in the quasi-monochromatic approach in the following form

v(,., t ) = p(,., f ) e - j ( 2 n u o t - ~ ( r , t ) )

51

(8.1.8)

A . E. Siegman, Laser Focus May, p. 42 (1971). M . Born and E. Wolf, “Principles of Optics,” 4th ed. Pergamon, Oxford, 1970.

8.1. LIGHT

SOURCES

71 I

1

coherence time

AT 2 coherence length AL = c . A ? typical measuring devices far temporal coherence spatial coherence

J----fql loser detector

gl-$

Michelson interferometer Young's experiment

FIG.8. Illustrations of the coherence properties of lasers

V ( r , 1) and +(r, t ) are both functions of the space coordinates r and the time C. vo is the central frequency. By means of a Fourier transform it will be stated again that $' and may be considered to be nearly constant during a time AT which is smaller than the inverse spectral width 6v. This time is called the coherence time which is defined by

+

AT

3

1/4dv.

(8.1.9)

Experimentally, the temporal coherence can be measured in a two-beam interference experiment using, for example, a Michelson interferometer (Fig. 8). By increasing the mirror spacing s2 in one arm of the interferometer, the interference fringe visibility which is directly correlated to the coherence length A L = CAT is decreased, so that A L can be determined in this way. During AT, the amplitudes and phases of the wavetrain at different times are linearly correlated. As compared to the monochromatic light of

712

8.

LIGHT SOURCES A N D RECORDING METHODS

s, lasers allow to obtain values up thermal sources where AT is at best to lo-* s. Mathematically, these coherence properties can be described which in the case of two waves by the mutual coherence function r12(7) (V,(rl , t ) and V2(r2,t ) ) can be written as

=

( vi(ril t -t 7)

*

v,*(rz

9

t)),

(8.1.10)

where the bracket notation is used to replace the more complex integral relation. The asterix denotes the complex conjugate. It is more convenient, however, to use the normalized coherence function y&) which is related to r12(T) by the following equation: (8.1.11) y12(7)is also a complex function. As already mentioned, its absolute

value can be measured easily by interferometric techniques. The classical theory of coherence can be extended to higher order correlation effects or even to quantum-mechanical formalism as pointed out by G l a ~ b e r . ~ ~ 8 . 1 . 3 . 1 . 3 . SPECKLES. It is a well known fact that laser photographic recordings such as visual observations are characterized by a high contrast granulation pattern which is superimposed to the image information and which forms a background noise. In fact, this is an interference phenomenon due to the coherence properties of laser sources. It is always obtained when laser light is diffusely reflected or transmitted. Figure 9 shows the main features for the case of a simplified experimental set up, where the scattered light is transmitted through a diffusor screen. In the observation or image-plane, respectively, the light distribution reveals the randomly distributed granulation. The grain size depends strongly upon the free aperture of the beam that is upon the number of scattering centers. This can be evaluated by calculating the intensity correlation function for a given intensity distribution in the diffusor plane where the individual scattering centers are located.53 The multiple interferences including their statistical properties can be described by a two-dimensional autocorrelation-function C(a, 6) of the intensity in the (x, y)-image plane. Using again the already mentioned bracket notation, this can be expressed in terms of 52 53

R. J. Glauber, Phys. Rev. 130, 2529 (1963). J. C. Dainty, Opt. Actu 17, 761 (1970).

8.1.

713

LIGHT SOURCES

initial intensity I [ $ , ? ]

image plane IIx,yl

-jj+ Laser

speckle distribution

centers system,

/Path

&

rr

-1

L

&

I

< d ) % . hLb

(d)nX

-f = X .F DL

Fici. 9. Speckle formation in laser photographic systems.

C(a, 6) = ( I ( & y ) . I*(x

+ u , y + b))

(8.1.12)

(the asterix again denotes the complex conjugate). Averaging over all the phases of the light waves emanating from all the randomly distributed scattering centers in the 5-7 plane yields that an alternative part e t a , b) can be split off the mean value This alternative part is finally responsible for the spatial intensity fluctuations. As the calculation shows, is proportional to the square absolute value of the Fourier transform of the intensity distribution in the scattering plane. In determining the first zero values of this function, one obtains the speckle size which is then only depending on the geometrical form and dimension of the beam aperture in the scattering plane. For rough estimates the values of the mean diameters of the speckles ( d ) are given in Fig. 9. For an optical imaging system using a lens of focal length fand a lens aperture DL,the parameter ( d ) is only limited by the F value F = f/DL. The shape and diameter of the speckle pattern can thus largely be influenced by a suitable choice of these parameters. Rectangular diaphragms are producing long shaped speckle patterns. In normal photographic applications these speckles are disturbing and limiting the high resolution obtainable with lasers. The speckles can be used, however, in photography for the measurements of small-scale distortions of objects

c.

714

8.

LIGHT SOURCES A N D RECORDING METHODS

undergoing mechanical loads, or generally for the possibility of separating a great number of different photographs which are superimposed on a single photographic plate. The photographs have to be taken with different diaphragms so that each picture is characterized by its own speckle pattern. The evaluation and reconstruction of the individual photographs are then simply to be performed by optical spatial filtering t e c h n i q ~ e s . ~ ~ 8.1.3.2. Spectral Ranges Covered by the Most Important Types of Lasers Applied to Fluid Dynamic Research. Table I11 shows a schematic classification for the different groups of lasers. In each group only one or two of the most important characteristic lasers are indicated. The neutral gas lasers, for example, include laser oscillation in 29 elements on about 450 identified t r a n s i t i o n ~including ~~ metal vapors. The most important laser of this group is the He-Ne laser, the strongest lines of which are in the red at 0.6328 p m and in the ir at 3.39 pm. Molecular lasers which are most effective for high power generation are classified to constitute another group of lasers. These lasers can be excited by electrical, optical, chemical or gas-dynamic pumping. Among these lasers we find the COz laser which can be operated in continuous or pulsed mode and which is emitting a large number of rotational vibrational lines ranging from 9.2. to about 1 1 p m . Single-line operation and tuning over the different lines can be obtained using a dispersive element inside the resonator. These lasers have been used successfully for ir recording which requires special techniques such as evaporation of thin films or liquid

In the case of ionized gas lasers, the transitions are originating from energy levels in the ionized state of atoms (or molecules) in gas discharges. The noble gas ion lasers such as the Ar I1 laser belong to this group. As already pointed out, the largest number of experiments conducted in the past in the field of laser photography have been performed particularly with the group of dielectric solid state lasers, with the chromium-doped ruby laser emitting at X = 0.6943 p m or with neodymium-doped glass or YAG lasers emitting in the near ir at 1.06 pm. Together with frequency doubling, this group of lasers provides pico- or nanosecond duration pulses in the ir, visible, and uv part of the spectrum. Rare-earth ions such as neodymium can also be imbedded in several organic components such as chelates. Most of these chelates are soluble in organic solvents. Historically, it is perhaps interesting to remember that Eu chelate dis54 55

U . Kdpf, Siemens Forsch. Entwicklunyshrr. 2, 277 (1973). R . J . Pressley, “Handbook of Lasers.” Chem. Rubber Publ. Co., Cleveland, Ohio,

1971. as

F. Keilmann, Ber. IPP IV/4. Inst. Plasmaphys. Garching, 1970.

8.1.

LIGHT SOURCES

715

solved in alcohol has been the first liquid material exhibiting laser actione5' The emission of semiconductor lasers occurs mainly in the near infrared. In the case of GaAs, the emitted wavelengths are in the range of 0.83-0.9 pm. These values can be shiftet by various dopants, even towards the visible. Semiconductor lasers can easily be modulated so that they are especially useful for optical communication systems, optical radar or for range finding; they are less important for the purpose discussed in this presentation. While the previously mentioned lasers operate at discrete frequencies, the last group of lasers to be discussed in this section, the dye lasers, is characterized by a large band of fluorescence so that the emitted wavelength can be tuned in this range. These lasers have been studied extenThe tunability makes them an attracsively during the last few tive tool for spectroscopists. For photographic recording these lasers are also very suitable. The lasers are optically excited by means of flash lamps or other laser light sources. Pulse durations of a few nanoseconds to some 10-100 p s can be obtained. Such long pulses are used for the monochromatic background illumination for rotating-mirror framing or streak cameras. In a large part of the visible and near ir spectrum mode-locked pulses have been achieved. Using laser pumping, cw operation can be obtained as well. Among the increasing number of dyes, only those belonging to the group of the Rhodamines and Coumarines have been mentioned in Table 111. Rhodamine 6G, for example, shows high quantum efficiency and a large tuning range. By means of other dyes, including frequency-doubling and Raman-type oscillation, the whole wavelength range from about 0.2 pm to the far ir can be covered. It is important to note that the spectral width 6v of these lasers can be made less than 0.01 A by using etalons and gratings or prisms so that great coherence lengths can be realized causing dye lasers to become also excellent sources for holographic recording.

8.1.3.3. Time-Dependent Emission Characteristics. The fundamental characteristics outlined in the previous sections make the laser an attractive light source for photography. Light output can, thereby, be obtained continuously or in different pulsed modes. 8.1.3.3.1. CONTINUOUSEMISSION. The first laser with which continuous-wave (cw) operation has been obtained is the He-Ne laser emitting at 6328 A. Nowadays, such lasers are commercially available with powers up to some tens of milliwatts. Higher powers can be ST

A . Lempicki and H. Samelson, Phys. Lerr. 4, 133 (1963).

58

F. P. Schafer, "Dye Lasers." Springer-Verlag, Berlin and New York, 1973.

716

8 . LIGHT SOURCES A N D RECORDING METHODS

TABLE111. Spectral Characteristics of Several Types of Lasers" ~~

Group of losers

Type

Typical representolive

Strongest tronsitians (pm)

Neutral gas lasers

Losers

e p .He-Ns laser

mcillaiing o i frequencies

Ah

06326

(nm)

0 001

3 39

discrete Molecular gas losers

Llnwldth.

'

''

loser

wntered around

low pressure

0 037

10 6

high prrrwra

I5 ~

Ionized gas losers

e g..Ar

U

loser

0 5145 0.4880

m0.01

0.4579 Rore earih liquid losera

0 . 9 , Nd-doped chdab

Dielectric Bolid siaie

e.g., Ruby laser.

Naadymium glass

lasers

0.001

1.06

laair 0 6943

I

.sm

losar

-

-0.1 10

----__---------------------------------------------Semiconductor lasers Dye losers

Go-As loser

Tunable lasers

(I

Rhodarnlm ( Carmarine I

0.83 0.9

WI

0.54- o m

*I

0.45-054

W l

Linewidths at room temperature

achieved with noble gas ion lasers, the Ar-ion laser of which is best known. The emission of this laser occurs on several lines in the blue and in the green spectral range. Among the solid state lasers, cw emission has been obtained with ruby- and with neodymium-doped YAG crystals. These YAG lasers are favorable because lasing threshold and pumping requirements are relatively low. Its radiation (A = 1.06 pm) can easily be converted to the visible range by frequency doubling using nonlinear optical crystals (KDP, ADP) outside or even inside the resonator. Both, argon and YAG lasers, are frequently used to generate continuous wave tunable emission. The argon lines are mainly used for the pumping of dye lasers,59whereas YAG lasers are often used to obtain parametric oscillation in crystals, the first cw operation having been achieved by Boyd and Ashkin.60 Large tunable ranges can be obtained by controlling the temperature or by angle tuning of the nonlinear crystals (for example KH2P04,NH4H2P04,LiI03, LiNbO,, or BazNaNb5015). 8 . 1 . 3 . 3 . 2 . RELAXATION PULSES.Among pulsed lasers emitting in the

5s

0. G . Peterson, S. A. Tuccio, and B. B. Snavely, Appl. Phys. Leti. 17, 245 (1970). D. Boyd and A. Ashkin, Phys. Rev. 146, 187 (1966).

WJ G.

8.1.

LIGHT SOURCES

717

visible range, solid state lasers, excited by the thermal light of intense flash lamps are of great importance. The pumping pulse duration has to be adapted to the lifetime of the metastable laser levels (1-2 ms in the case of ruby and some hundreds of ,us in the case of neodymium lasers). The laser oscillations are obtained after the inversion threshold has been achieved. Under normal conditions, a large number of modes with different initial phases are appearing; these are further subject to thermally induced changes of the resonance conditions so that the time shapes of such pulses are randomly distributed sets of spikes which last until the end of the pumping pulse. By the use of longitudinal and transverse mode selection, regularly shaped, periodically oscillating, decaying pulses can be obtained. Because of the strong intensity fluctuations and the rather long overall duration of the order of some hundreds of microseconds to 1 ms, however, both types of pulses, the randomly distributed spikes and the periodically modulated pulses, are not well suited for photographic applications. 8.1.3.3.3. GIANTPULSES.The best and most reproducible laser pulses can be obtained by Q-switching the laser cavity. This technique employs some type of (mechanical, electro-optical or intensity-dependent transmitting dye solution) shutters to suppress cavity feedback during the pumping process. Nearly all the stored energy can then be delivered to a single pulse by rapidly establishing a high Q-value of the resonator (Q = quality) so that short duration pulses of some tens of nanoseconds with peak-powers in the range of some 10 to some 100 of megawatts can easily be obtained. Spectrally, the emission of such giant-pulse lasers, under normal conditions, proves to be multimode. Figure 10 shows schematically the experimental setup which allows for the generation of single mode Q-switched laser pulses.61 The actual shutter is formed by a Kerr or a Pockels cell which allows for time synchronization. Longitudinal mode selection is provided both by a saturable absorber (e.g., cryptocyanine in methanol) and by the Fabry -Perot etalon out-coupling mirror. Transverse fundamental-mode operation is obtained by a near confocal resonator configuration supported by a small aperture diaphragm which produces additional losses for the higher order transverse modes. The time dependence of the pulse as measured by a fast photodiode and oscilloscope (TK 5 19) and the spectral output measured by a Fabry-Perot interferometer are also included in Fig. 10. The frequency spacing from one order of interference to the next higher one has been chosen to be 1.5 GHz. This proves to be sufficient to resolve longitudinal modes which are separated by Av = c/2L = 0 . 3 GHz for L = 50 cm. The apA . Hirth, 1SL-Ber. 10/68. Dtsch.-Franz. Forschungsinst. St.-Louis, 1968.

718

8.

LIGHT SOURCES A N D RECORDING METHODS Pockels

dye c e l l

m i r r o r (100%)

cell

/ \

poldrizer

-

etolon reflector

20ns/cm

FIG.10. Monomode giant pulse ruby laser, pulse intensity versus time, spectral distribution.

pearence of only one circle line per order of interference thus indicates real single frequency operation. In measuring the width of the lines by means of a microdensitometer, one obtains a first rough information on the spectral width of the emitted light, thus yielding the coherence length AL = c AT of the wave trains. The output energies of such a system incorporating a 3-in, ruby rod is typically of the order of 10-20 mJ. This is sufficient for most types of optical investigation even when relatively low sensitive holographic plates have to be exposed. PULSES.As already pointed out, the emis8.1.3:3.4. MODE-LOCKED sion spectrum of all lasers consists of a large number of longitudinal and transverse modes. For simplicity, we assume in the following discussion fundamental transverse mode (TEMoo)operation. The number of axial modes is depending on the bandwidth of the laser transition and on the mirror spacing including the refractive indices which are determining the mode separation. Strong coupling of the phases of all simultaneously excited and oscillating modes can be achieved by active and by passive modulation techniques. Active modulation is mainly employed with continuously running gas lasers. This calls for an amplitude, phase or frequency modulator inside the cavity which is driven at a modulation frequency v M . Let us assume first the oscillation on a single mode n that has just reached the threshold of oscillation, characterized by its amplitude a , and initial phase &:

8.1. LIGHT

V,(/) =

719

SOURCES

sin(2.rrunt+ 4,,).

(8.1.13)

(I,

Performing an amplitude modulation a,(/) = uno cos(2.rrvMt + have V,(t) = =

U,

cos(2wMt

we

+ (bM) sin(2?runt+ 4,)

{sin[2.rr(un-

+ sinP.rr(u, +

+M)r

u,)t

VM)f

+ +n

-

+M1

+ 4 n + 4J1,

(8.1.14)

indicating the formation of an upper and a lower side-band frequency. If the modulation frequency corresponds to the frequency spacing of two successive modes vM = Au, then u,, - uM = u, - AU = u,,-~and the two sideband frequencies are identical with resonator eigenfrequencies which are then amplified in phase with the originally oscillating mode. This procedure is repeated until all the modes of the spectrum are oscillating in phase. The time behavior of the laser output can be described mathematically by the inverse Fourier transform corresponding to the characteristic frequency spectrum. This yields the experimentally observed train of mode locked pulses, the duration of which can be shown to be proportional to the inverse number of coupled modes N , whereas the peak intensities are proportional to W . In flash-lamp pumped dye or solid state lasers, this active modulation technique can no longer be used, because of the thermal drift of the optical cavity length causing frequency shifts and variations in frequency spacing of neighboring axial modes. This effect would necessitate automatic frequency pulling of the active modulator during the 1-2 ms pumping pulse. As found by De Maria,62this matching condition can nevertheless be fulfilled automatically by using nonlinear saturable absorbers similar to those applied to Q-switching. These absorbers are characterized by high absorption coefficients in narrow spectral bands for low light intensities, whereas, above a given threshold value, the dye levels are saturated so that the dye is highly transmitting. The relaxation time for the decaying transparency has to be smaller than the cavity round-trip time. Formally, the saturable absorbers can also be considered to cause modulation, the frequency of which is automatically identical with the inverse cavity round-trip time. LetokhoP3 and otherss4 have developed a theory which starts from initial small random intensity fluctuations due to A . J . DeMaria, W. H. Glenn, M. J . Brienza, and M. E. Mack, Proc. IEEE 57,2 (1969). S. Letokhov, Sov. Phys. -.IETP (Engl. Trunsl.) 28, 562 (1969). P. G. Kryukov, Yu. A. Matveev, S. A . Churilova, and 0. B. Shatberashvili. Sov. Phys. -JETP (Engl. Transl.) 35, 1062 (1972). "

gg

8.

720

n

Q

LIGHT SOURCES A N D RECORDING METHODS

madulator

(vbl =Au’

‘U

actively modulated c w loser passively modulated pulsed laser

saturable absorber

pulse train of a mode - Locked ruby Laser 2 0 ns/cm

5 ns/cm

FIG.1 1 . Active and passive mode-locking of cw and pulsed lasers.

spontaneous emission processes which are subsequently amplified traversing the active medium. The selection of a single pulse traveling back and forth in the cavity, originating from the highest fluctuation peak, is provided by the nonlinear transmission characteristics of the saturable absorbers which show higher absorption losses for the smaller intensity peaks. Figure 11 shows schematically the two methods of active and passive modulation as well as an example of the light output of a mode-locked ruby laser. This has been measured by means of a fast photodiode and an oscilloscope which, because of the limited bandwidth of about 1 GHz, cannot follow the actual rise and decay times of the ultra-short pulses. The pulsewidth has to be measured by rather sophisticated methods such as two-photon fluorescence techniques, nonlinear correlation techniques, or by rapid scanning image-converter streak cameras. Reference is made to the l i t e r a t ~ r efor ~ ~details which are out of the scope of the present contribution. 8.1.3.4. Superradiant Light Sources. Laser oscillation usually occurs if a medium is suitably pumped, inverted and if feedback is provided by an appropriate resonator. If stimulated emission dominates, the amplificaBJ

R. Dhdliker,

( 1970).

A. A. Griitter, and H. P. Weber, IEEE J . Quantum Electron. 6, 687

8.1.

LIGHT SOURCES

72 I

tion of a wave propagating in z-direction is given by the following equation: Z(z>

=

I(0)e-a'Z.

(8.1.15)

a is the gain factor, a = cr A N , where cr is the cross section for stimulated emission (correlated with the dipole matrix element of the transition) and AN is the population inversion between the two lasing levels involved. If az is sufficiently high, spontaneously emitted photons propagating in &z direction are amplified by stimulated emission processes thus forming superradiant light pulses without any resonator. In the strict sense, the superradiation therefore proves to be spatially and temporally largely incoherent. The spectral linewidth is approximately identical with that of a fluorescence line. This means that the light output is quasimonochromatic but not restricted to discrete resonator eigenfrequencies. In long lasers or in laser-amplifier chains this effect is disadvantageous because the inversion can be reduced considerably by superradiance. In a large number of cases, however, the superradiant mode of operation is preferentially used. Due to their high gain, most NJasers, for example, emitting in the near uv at A = 0.3371 pm are operated in the superradiant mode. For high-speed photographic applications, special types of superradiant sources have been investigated using solid state materials such as ZnS, ZnO, CdSe, ZnTe, GaAs, CdTe. These materials are showing strong fluorescence when they are irradiated with a high-energy electron beam. Short-duration electron pulses of some tens of nanoseconds with highcurrent densities can be generated by vacuum field-emission discharges by using high voltage Marx surge generators with voltages of some hundreds of kilovolts. The superradiant material is deposited on foils which are positioned near the exit window of the electron beam gun. The gain achieved by this method in these materials is so high that the excited wave provides a very intense light output after a single pass in the amplifying thin layer. The halfwidths of the emitted pulses are of the order of some nanoseconds. By using different materials, a large range of the visible spectrum can be covered.66 It should be pointed out that by removing the superradiant plate the same installation can be used as a pulsed source for electron beam or for x ray recording techniques.

8.1.3.5. Generation of Coherent Radiation Using Nonlinear Optical Methods. The nonlinear behavior of material at optical frequencies, as it can be described mathematically by field dependent dielectric constants BB J. L. Brewster, J. P. Barbour, F. M. Carbonnier, and F. J. Grundhauser, Proc. fnt. Congr. High-speed Phorogr., 9rh, Denver, 1970. p. 304. SMPTE, New York, 1970.

8.

722

LIGHT SOURCES A N D RECORDING METHODS

and magnetic permeabilities, is well known. The polarization, for example, can be expressed as a function of the electric field by a power series in the following form: y =

E+x‘”E.E++X‘:”E.E.E+ . . . (8.1.16) L- higher-order nonlinear susceptibilities Llowest-order nonlinear susceptibility linear susceptibility

x(l) *

L

Using the high field intensities of existing lasers, the nonlinear effects can be used for the generation of a large number of new lines of coherent radiation, see Fig. 12. OF HARMONICS. The nonlinear properties 8.1.3.5.1. GENERATION have first been strikingly demonstrated in 1961 by Franken and coworkersG7by the observation of the harmonic of a ruby laser beam. According to the notations of Bloembergen,B8the lowest order nonlinearity at an angular frequency w3 is given by P L ( w 3 )= x(w3 = w1 + wz) EIEz e‘(k1+k)z-i(W1+W2)I)

where

x is a third-rank tensor.

(8.1.17)

Second harmonics are obtained when

w1 = w2 = w and w3 = 2w. Since k3 = 2k, = 2kz, the propagation velocities at w and 20 are different due to dispersion ((ki/= 2.rr/hi). Thus it is

necessary to match these two phase velocities. This can be done if the nonlinear crystals are birefringent as in the case of ADP or KDP (NH4H2POI,KHzPO,). The fundamental and harmonics have then to be attributed to the ordinary and extraordinary ray so that for specific angular conditions with respect to the crystal axis the color dispersion is compensated for the anisotropy of the two phase velocities. As the nonlinear polarization proved to be proportional to the square of the laser electric field amplitude, the efficiency can be increased by increasing laser intensity. Taking higher order susceptibilities into account, higher order harmonics can be generated as well. Terhune was the first researcher to have observed third harmonic generation.sB 8.1.3.5.2. RAMANLASERS.The investigations of stimulated Raman processes gave rise to numerous applications including the generation of molecular or lattice vibrations, the measurement of the lifetimes of excited vibrational states and the production of intense coherent light at new frequencies. Early laser studies have shown that ruby lasers gain 67

P. A . Franken, A. E. Hill, C. W . Peters, and G. Weinreich. P h y s . Rev. L e f t . 7, 118

( 1961). “ N . Bloembergen, Nonlinear optics. In “Quantum Optics and Electronics,” p. 411. Gordon & Breach, New York, 1965. 6e R. W. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Lett. 8,404 (1962).

8.1.

LIGHT SOURCES

1 p q

723

external e.9. KDP

Zw, operation

Roman octive material le.g.Ht)

pump source

Stokes or Anti-Stokes frequencies frequencies

frequencies cavity resonant far US, or for both q , o n d w,

(C) I

*

I s.y.

pump source

wL- qi+ q

i;,

.

r,

wL = loser * signal

WJ

-

w, idler

7

angular frequency

FIG.12. Generation of coherent radiation using nonlinear optical methods. (a) Harmonic generator. (b) Raman laser. (c) Parametric oscillation. (Vector quantities are indicated by arrows over letters in figure and by boldface letters in text.)

switched with nitrobenzene Kerr cells emitted additional light at 0.767 pm. This means that quanta of energy hwL are absorbed, one part of which is converted to quanta of energy nos. A large number of liquids, solids, and gases are showing typical frequency shifts from the exciting laser line which correspond to the vibrational frequencies of the molecules involved. This can be interpreted as a scattering process. Low laser intensity variations are linearly influencing the intensity of the new line. High laser intensities are capable of generating large numbers of scattered photons which are then amplified exponentially due to a transition from a spontaneous to a stimulated scattering process. By characterizing the molecular vibrational wave by the angular frequency wv , the following equation holds for the angular frequency w, of the new Stokes-shifted Raman line: wL = w s + w v . If both frequencies, i.e., the laser and the Stokes frequency are present, higher order Stokes lines, oL- 2 w v , wL - 3wv, and so on, can be obtained. As there will be a polarization term at the anti-Stokes frequency wAs = 2wL - w s , it is also possible to obtain blue shifted anti-Stokes

724

8.

LIGHT SOURCES A N D RECORDING METHODS

lines. The gain of the Stokes line is proportional to the Stokes susceptibility which can be expressed by the differential Raman scattering cross section. The above gain is proportional to the square of the electric field strength of the laser. The amplification of the scattered wave thus grows exponentially with the incident laser intensity. Providing a feedback by resonant mirrors, such a medium constitutes a Raman laser in which oscillation can start from noise. Suitable frequency selective mirror reflectivities can force oscillation on the first- or higher-order Stokes lines. The excitation is mostly obtained by giant pulses from solid state lasers. The application of tunable dye lasers in the visible thus provides tunability of the Raman laser emission to an extended range in the infrared.'O 8.1.3.5.3. OPTICALPARAMETRIC OSCILLATORS.As compared to normal lasers where amplification is obtained by population inversion, the gain in parametric amplifiers is produced by the interaction of three electromagnetic waves with a nonlinear medium which is characterized by its second-order nonlinear coefficient x'~'. In Raman processes, two electromagnetic fields and a molecular vibrational mode were interacting, whereas in parametric oscillator studies we are concerned with three purely electromagnetic waves, one high frequency pump wave (up)and one pair of lower frequency waves called the signal (us,)and the idler (wi). The three frequencies are related by the formula w, = wSi + wi which corresponds to the energy balance. Above a threshold value of the pump, the signal and idler waves experience a net gain. They can grow in such a way that their fields are comparable to that of the pump. The parametric gain is critically dependent on the amount of momentum mismatch Ak = k, - ksi - k i . In a medium without dispersion Ak would be zero. In practice, Ak may be large making the parametric gain relatively small. As in the case of harmonic generation this can best be compensated by using birefringent nonlinear crystals. Experiments have been performed using double resonance oscillators, where mirrors are used which reflect both for the signal and the idler wave whereas the mirrors are transparent to the pump radiation. Under optimum conditions, one half of the pump power goes into the signal and the idler, one quarter is transmitted and one quarter is reflected. Efficiency can be increased by using ring cavities with which the upper theoretical limit of 100 percent can be attained. Oscillation has also been obtained using single resonance oscillators with mirrors which only reflect for wsl or w i . A third possibility is given by internal parametric oscillators where the nonlinear crystal is incorporated in the cavity of the pump source. The materials used for parametric oscillators are the same as for second 'O

J . Kuhl and W.Schmidt, Appl. Phys. 3, 251 (1974).

8.2.

RECORDING METHODS

725

harmonic generation. They must have a lack of symmetry centers, a large value of the nonlinear second-order electro-optical coefficient, and a large transparency range. They should further be homogeneous, phase-matchable, and resistant to optical damage. Besides, ADP and KDP, LiNbOs, and B%NaNb,015 are frequently used. The first operation has been achieved by Gi~rdrnaine.~’Since then, pulsed and cw operation has been studied successfully under different conditions.‘* Ruby laser pulses and second or third harmonics of neodymium laser emission have been used as pump sources, cw operation was possible by using the Ar 11-5145-i% line or neodymium-doped cw YAG lasers. Tuning of the parametric oscillators can be achieved by varying the index of refraction of the crystal which can be done by temperature changes or by varying the angle between the three waves in the case of noncollinear interaction. The largest tuning range from 0.684-2.36 pm has. been obtained in a double resonance oscillator, pumped by a frequency-doubled neodymium laser, by using three crystals and a set of three mirrors. The tuning ranges obtained are thus considerably larger than those achieved with dye lasers.

8.2.Recording Methods 8.2.1. Introduction A large number of cameras has been developed in the past for the investigation of rapidly varying phenomena. These include single exposure as well as cinematographic techniques. Because of the use of lasers, the range of possibilities of conventional photography has been considerably extended. Holographic methods yield informations both on amplitudes and phases of the wavefronts. A survey of the most important possibilities is schematically shown in Fig. 13. For the sake of clearness, overlapping ranges are not indicated. The single exposure techniques are devided into two groups, one of which applies to short illuminating pulses whereas the other one uses high speed shutters. The cinematographic methods are classified following the mainly applied methods of image separation. It is obvious however, that the different techniques can be combined such as for example in the case of the operation of high speed shutters or periodical pulse trains with mechanical cameras.

’*J . A. Giordmaine and R. C. Miller, Phys. Rev. L e f t . 14, 973 (1965). 72 R. G. Smith, Optical parametric oscillators. In “Laser Handbook (F.T. Arecchi and E. 0. Schulz-Dubois, eds.), Vol. 1 , p. 837. North-Holland Publ., Amsterdam, 1972.

726

8.

LIGHT SOURCES A N D RECORDING METHODS

FIG 13 Schematic classification of recording methods and systems.

The aspects of mechanical cameras, with the exception of cineholography, have mainly been considered by D ~ b o v i k , ’the ~ importante of electron-optical high speed photographic systems by Courtney-Pratt ,74 whereas the applicability and vertisability of spark cinematography and electrooptical methods are treated comprehensively by Vollrath and Th~mer.’~ 8.2.2. High Speed Photographic and Cinematographic Methods.

8.2.2.1. Spectral Sensitivity and Resolution of Photographic M a t e rial. The main parameters characterizing photographic emulsions and their applicability for high speed recording are the sensitivity and the spatial resolution power. The sensitivity (ASA or DIN) gives a measure for the blackening as a function of the amount of illumination (exposure). Each photographic plate has its characteristic curve showing the optical density as a function of the exposure. The light levels involved should be adapted to work in the linear range of this curve. The spectral sensitivi73 A . S. Dubovik, “Photographic Recording of High-speed Processes.” Pergamon, Oxford, 1968. “ J . S . Courtney-Pratt, Phofogr. J . , Sect. B 92, 137 (1952). 75 K . Vollrath and G . Thomer, eds., “Kurzzeitphysik,” p. 76. Springer-Verlag. Berlin and New York, 1967.

8.2.

RECORDING METHODS

727

ties are mainly determined by sensitizations, that means by the addition of small amounts of dyes, so that various distributions can be obtained in different ranges in the uv, visible, and ir part of the spectrum. The material can thus be adapted to the special type of thermal or laser light source used for the investigations. The spatial resolution is limited by the graininess. Improved image quality necessitates fine grain materials in which, however, sensitivity is decreased. In most conventional procedures including laser photography, the spatial resolution is not limited by the photographic material (which is normally of the order of some hundreds of lines per millimeter), but by the optical system itself. In contrast, holographic recording requires higher resolution up to several thousands of lines per millimeter. For this purpose, special materials have been developed such as the Kodak 649 F or the Agfa 10E70 or 10E75. The last type is mostly used for pulsed ruby laser holographic techniques, whereby about 2800 lines/mm can be resolved, and the energy necessary for the exposure is of the order of 50 erg/cm2. 8.2.2.2. Single Exposure Techniques

8.2.2.2.1. APPLICATION OF SHORT DURATION LIGHTPULSES.Singleexposure techniques can be performed by using rather simple cameras and short duration light pulses that can be provided by either type of thermal or laser light source discussed in the previous sections. Pulses are ranging from several picoseconds up to several milliseconds. The pulse duration required is, thereby, only limited by the maximum admissible blur of the recorded image. The relatively inexpensive thermal light sources can be used for the investigation of many fluid dynamic problems. Lasers are more suitable in the field of strongly self-luminous effects such as flames, deflagrations, detonations, or plasmas. The single-exposure technique can be applied to the investigation of objects in the reflected light or, provided the objects are partially transparent, in the transmitted light. In the latter case, the known optical methods such as shadowgraphy , interferometry, or schlieren techniques yield valuable informations. Three-dimensional information can even be obtained by using coherent pulsed radiation sources for holographic techniques such as single-exposure holography or double-exposure holographic interferometry.76 8.2.2.2.2. H I G HSPEEDSHUTTERS. Most optical shutters are based on the linear or quadratic electro-optic or magneto-optic effect. The propagation of a light wave is then influenced by electrically o r magnetically in‘s J .

1971.

C . Vienot, P. Srnigielski, and H. Royer, “Holographie Optique.” Dunod, Paris,

728

8.

LIGHT SOURCES A N D RECORDING METHODS

duced birefringence which is acting on the phase velocities of different linearly or circularly polarized wave components. In image converter type shutters, photoelectrons are set free on photosensitive cathode materials due to the Hallwachs effect. Gating can be achieved by applying suitable high voltage pulses between the cathode and the fluorescence screen. 8.2.2.2.2.1. PolNrizution-Dependent High Speed Shutters. Propagation characteristics of light in nonisotropic media are properly described by the index-ellipsoid representation:” (8.2.1)

Suis the impermeability (inverse permeability fractive indey p by

E)

which is related to the re(8.2.2)

By choosing the Cartesian coordinates xI ,x2,and x3 such that they correspond to the main axis of the ellipsoid, the above equation is simplified to read (8.2.3) for biaxial materials and (8.2.4) for uni-axial materials, where po and pe are indices of the ordinary and extraordinary ray. For isotropic materials p holds for all directions. Distortions of the index ellipsoid can be induced electrically, optically, magnetically, and mechanically. The last case of electro-acoustic effects shall not be considered, however. The distortions can be taken into account by replacing Suby (Sij + ASij). Figure 14 shows the different types of shutters, based on the polarization-dependent velocities of propagation to be discussed in the following sections. Pockels Cell Shutters. In the linear electro-optic effect which is also called “Pockels effect,” the distortion A( l/p2)uis proportional to the electric field E . This dependence can be expressed by a third rank r tensor ” S . H. Wemple, Electro-optic materials. I n “Laser Handbook” (F. T. Arecchi and E. 0. Schulz-Dubois, eds.), Vol. 1, p. 977. North-Holland Publ., Amsterdam, 1972.

8.2.

729

RECORDING METHODS

&

mode-locked laser

(C)

~

carner

beam ident

ps -gated optical E - fieldstrength

(d)

FIG. 14. Schematic representation of polarization-dependent high speed shutters. (a) Pockets cell (longitudinal effect). (b) Kerr cell. (c) Optically gated Kerr cell. (d) Faraday shutter.

A

e;v+

(8.2.5)

In the same way, afu,k tensor can be used to relate A ( ~ / P ~to) the ~ polarization P k . In the reduced index notation, the following abbreviations are used for ij: 1 1 = 1 , 22 = 2, 33 = 3, 23 = 32 = 4, 13 = 31 = 5, 12 = 21 = 6. Depending on the symmetry conditions of the materials, only some nonvanishing matrix elements have to be retained. The field induced birefringence can then be calculated from the ellipsoid distortions AP =

&I

- PI.

The indices “parallel and perpendicular” refer to the polarization of the wave with respect to the electric field direction. The exact notations are depending on the special type of material used and the direction of propagation and polarization of the beams. For shutter applications, the material has to be placed between two polarizers orientated perpendicular to

730

8.

LIGHT SOURCES A N D RECORDING METHODS

one another. Both longitudinal and transverse electric field configurations can be applied. The phase retardation 6 of the two polarization components of the incident beam after the traversal of the medium of length 1 is written as (8.2.6) Let us consider briefly KDP crystals which are well known from laser Q-switching techniques. They are optically uniaxial, and are belonging to the tetragonal symmetry class. If E , for example, is orientated along the crystal axis x3, E = E 3 , the induced index distortion yields7* A

-

-

r12,3

E3 = rwE3,

so (8.2.7) and

po is the refractive index of the ordinary ray, r, the relevant electro-

optical coefficient. The above relations can be used to derive the halfwave voltage U,,,: (8.2.8) which is necessary for a 90-degree rotation of the plane of polarization. Besides the longitudinal effect, transverse effects can also be applied. Kerr Cell Shutters. The induced optical birefringence, in the case of the quadratic electro-optic effect (Kerr effect) proves to be proportional to Ez. The relations are thus more complex (8.2.9) A similar equation can be written relating A( l/pz)uto the polarizations P k and P l , thus defining the polarization optic coefficients Gi,,kl. For simplicity, the reduced notations are preferred again.

’’ R . T. Denton, Modulation techniques. I n “Laser Handbook” (F.T. Arecchi and E. 0. Schulz-Dubois, eds.), Vol. 1, p. 703. North-Holland Pub]., Amsterdam, 1972.

8.2.

73 I

RECORDING METHODS

For the construction of high speed shutters, liquids such as CS, or nitrobenzene are often used. The phase retardation for the two polarization components parallel and perpendicular to the applied electric field which is transverse to the direction of beam propagation, is then given by 6 = 2rBlE2. B is the Kerr constant which is dependent on wavelength and temperature. A halfwave voltage can be defined as in the case of the linear electro-optic effect by Uh,, = ~ / ( 2 B l where ) ~ / ~ a is the width of the Kerr cell ( E = U / a ) and U the applied ~ o l t a g e . ’ ~ As the electrically induced birefringence has extremely short relaxation times, those shutters can even be operated with hf electric fields up to the optical frequency ranges. The application of mode-locked lasers, therefore, yields the possibility of generating ps shutters. This can be achieved by focusing the laser radiation into a cell of CS, or nitrobenzene, respectively, which is placed between two crossed polarizers. By this method, Duguay et al.*O first visualized the spatial shape of ps light bundles. The output pulses of a mode-locked Nd-glass laser, thereby, passed through a nonlinear optical crystal thus generating the second harmonic. This part, the green pulse, was split using a wavelength-selective mirror which, after being optically delayed, traversed a cell of milky water placed in front of the camera Kerr cell shutter configuration. The Kerr cell was gated by the remaining part of the infrared pulse so that the extension of the green pulse can be photographed due to its own straylight. Faraday Shutters. In the magneto-optic effect, the plane of polarization of light is rotated by a certain amount when it is passed through magneto-optic materials such as different types of glasses with a magnetic field orientation parallel to the direction of propagation. The rnagnetization causes a change of the refractive indices (+)* = E k 6 of the two circularly polarized components of the light. We describe the two components by the following expressions:

(8.2.10) a plane polarized incident light beam will be rotated by an angle 4=(p+-p-)-

7T

A

.i=e.i,

4 (8.2.11)

78 W . Miiller, Elektrooptische Verschlusse. In “Kurzzeitphysik” (K.Vollrath and G. Thomer, eds.), p. 207. Springer-Verlag. Berlin and New York, 1967. aa M. A . Duguay, J. W. Hansen, and S. L. Shapiro, IEEE J . Quantum Electron. 6, 725 (1970).

732

8.

LIGHT SOURCES A N D RECORDING METHODS

where 8 is the rotation angle per centimeter length. In some para- or diamagnetic materials, large values of 8 can be observed. 0 is thereby proportional to the applied magnetic field H , where 8 = V . H . V is the Verdet constant which is mostly given in the literature in (deg/(cm Oe)). Because of the high currents which are to be switched to obtain the required magnetic fields, Faraday shutters are operating at lower speeds than do electro-optic shutters. A flint glass, for example, of 2 cm in length with a diameter of 1 cm necessitates magnetic fields of 45,000 Oe ( V = 0.001 degs/(Oe cm)) to obtain a halfwave retardation corresponding to a rotation of the plane of polarization of 900.79 8.2.2.2.2.2. Image-Converter-Type oj' Shutters. Image converter techniques advanced rapidly during the last few years providing electronically controlled shutters which allow to stop motion at precise times with exposures in the nano- or microsecond range. The operation principle involves a photosensitive cathode on which the image is formed by means of conventional optics.*' The emitted photoelectrons are accelerated in the evacuated chamber by using suitable electric fields. The anode utilizes fast response cathode-ray tube-phosphors. The images on this screen are formed by electric, magnetic, electromagnetic, or proximity focusing techniques and can be photographed directly by means of a second lens system. Shutter times are determined by the high voltage pulses applied to the anode or to an extraction mesh grid. The proximity focus biplanar tubes are especially useful because they are virtually distortion free (with the exception of the peripheral region around the edges), and because they allow for obtaining high spatial and temporal resolution. The strong electric fields are generated by a high voltage, applied to the closely spaced parallel electrodes. In commercially available systems, images can be switched on or off with exposure times of 5 ns. Due to the achievable high radiant gain across the image tubes (50- lo4),considerably lower light levels can be tolerated than with Kerr cell shutters. The main features of the long focus tubes including deflection electrodes with their ps capabilities will be discussed in the following sections concerned with cinematographic techniques. 8.2.2.3. Cinematographic Methods. In Fig. 13, a classification of cinematographic techniques has been chosen, following the different methods of high speed image separation. The optical information can, thereby, be obtained in the way, that all the individual elements contribute to the photographic recording or that only a smaller number of raster J . S . Courtney-Pratt, Research (London) 2, 287 (1949).

8.2.

RECORDING METHODS

733

points or lines are to be considered. In the extreme case only one single line will be extracted. The first method is utilized in most types of cameras, in the mechanically driven cameras with intermittant o r optically compensated continuous film transport, in electronic image-converter cameras and in multiple spark cameras using optical image separation. The second way, to split up the information into a large but finite number of raster-elements is utilized in image dissection cameras. The extraction of a single line finally leads to streak records, where the only part of the image of an object which is transmitted through a small aperture slit is temporally smeared by the relative motion of the slit image with respect to the film. This can be obtained mechanically by moving the film (drum cameras), or by mirror scanning o r slit scanning (e.g., rotating mirror cameras), or electronically in image-converter cameras by applying suitable deflection voltages inside the tubes. The streak techniques proved to be especially useful, if fast unidimensional motions of waves in fluids, luminous fronts in plasmas, flames, or detonations have t o be investigated. 8.2.2.3.1. MECHANICAL C A M E R A S . As already mentioned, an extensive description of the large variety of experimental techniques and apparatus has been given by D ~ b o v i k , 'so ~ that only the most important features of these cameras shall be pointed out. Techniques using intermittant film transport are restricted to relatively low repetition frequencies of less than 600 pps. Considerably higher image repetition rates are obtained with continuously moving film. Drum cameras for example, with externally mounted film can be operated up to rotation frequencies of 50-70 rev/s, whereby velocities up to 100 m/s are obtained. This value corresponds to the limit, given by the maximum admissible centrifugal forces that film materials are able to withstand. Higher velocities of about 200 m/s can be achieved, however, if the film is mounted internally. Drum cameras are often operated in the streak mode in the case of self-luminous phenomena or they are combined with periodically emitting light sources such as stroboscopes. The temporal resolution can be increased, if the relative motion of the image with respect to the continuously transported film is compensated, for example optically by means of an additional rotating cube of glass through which the image is transmitted. With rotating mirror cameras, even higher resolutions can be achieved.6Z Figure 15 shows schematically the operation principles of rotating mirror cameras both in the streak and in the framing mode. In the streak mode, an image of the object to be studied is formed in the plane of 82

E. B. Turner, SPIE J . 8, 157 (1970).

734

8.

LIGHT SOURCES A N D RECORDING METHODS

I

lens

lens -

'ob ects

e

FIG.15. Schematic of rotating mirror cameras. (a) Rotating mirror streak camera. (b) Rotating mirror framing camera with optical compensation.

the slit. By means of a relay lens, the final image is subsequently generated in the film plane, after being reflected on the rotating mirror or prism surface. Object movements parallel to the direction of the slit are thus transformed into a component of motion perpendicular to the axis of the slit image. From known scanning speed, the real speed of the object motion can be determined with high accuracy. Streak velocities of 10 mm/ps are currently obtainable with commercially available cameras providing a temporal resolution power of only a few nanoseconds. Transformation of such a camera to the framing mode necessitates the incorporation of some further components. This is indicated in the upper part of Fig. 15. Optical compensation of the mirror scanning is thereby achieved by using a series of aperture stops with an additional array of relay lenses. An intermediate image is formed near the surface of the rotating mirror by means of the field lens. Following the mirror rotation, the light is subsequently transmitted through successive apertures and relay lenses, thereby causing the framing action. The number of discrete images corresponds to the number of relay lenses. Framing rates of several lo6 frames per second are possible, whereby the exposure times of the individual frames are about the half interframe time. Commercially available cameras even allow for simultaneous recording in the streak and in the framing mode. 8.2.2.3.2. IMAGE DISSECTION C A M E R A S . An important technique applied for the construction of high speed cinematographic cameras which is quite different from those discussed in the previous sections uses the principle of image d i s ~ e c t i o n . The ~ ~ main feature is that the images are di-

8.2.

RECORDING METHODS

735

vided into a large number of small elements (typical values are about 620 dots/cm*). As compared to their own diameter, the interspaces between the individual dots on the recording film are relatively large. To obtain a complete separation of two subsequent images, each element has, therefore, to be displaced only by a small amount so that extremely high recording rates can be achieved.84 Different systems have been developed, applying, e.g., simple dissecting plates with a moving film o r by combining dissection plates with rotating mirror cameras. Higher performance of operation has been obtained using lenticular plates to dissect the image and aperture or mirror scanning for sequential recording. Thereby, an image of the object to be studied is formed in front of the lenticular plate which provides the dissection of the original image, as each one of the small lenslets sees only a portion of the whole image. The addition of a movable aperture, for example a rotating disk (Nipkow disk using a spiral row of aperture holes) near the objective lens, allows scanning of different raster elements proportional to the displacement of the aperture. The different pictures thus obtained can finally be restored after processing the film by uniformly illuminating the photographic plate in a holder with the same or a similar movable aperture and lenticular-plate array. Cameras of this type have been designed allowing 3000 pictures to be exposed with rates of lo6 p p ~ More . ~ ~ elaborate systems even include fiber optics or combine image dissection with deflecting image converter tubes. In USSR, cameras with maximum repetition rates up to lo9 pps have been realized.86 8.2.2.3.3. IMAGE CONVERTER A N D I N T E N S I F I E R S . Deflection image converters combined with intensifiers have proved to be one of the most useful tools in high speed cinematography with subnanosecond and even picosecond r e s ~ l u t i o n . ~ In ’ ~ ~present-day ~ cameras of this type, long focus tubes are applying electrostatic or magnetic focusing techniques, whereby the electrostatically focused tubes are mainly used. Diodes, triodes, and even tubes with larger numbers of electrodes have been developed. These tubes provide flexibility, high gain, and deflection capability. Mesh extraction grids which are located only a few millimeters be83 H . Bender, Die Rasterverfahren der Hochfrequenzphotographie. In “Kurzzeitphysik” ( K . Vollrath and G. Thomer, eds.), p. 301. Springer-Verlag, Berlin and New York, 1967 J . S. Courtney-Pratt, J . SMPTE 82, 167 (1973). M. P. Battaglia, SPIE 8, 175 (1970). A. S. Dubovik and N . M. Sitsinskaya, J . S M P T E 80,691 (1971). n7 D. J. Bradley, P r o r . Int. C o n g r . , High-Sprrd Photog., 11th. London, 1974, p. 2 3 . Chapman and Hall, London, 1975. 88 R. Hadland, Lecture presented at the Technical Seminar of the British Electro-Optics and Laser Equipment Exhibition, Tokyo, December, 1975.

8.

736

LIGHT SOURCES A N D RECORDING METHODS

hind the cathode are often incorporated to accelerate the emitted photoelectrons. As the high voltages applied are of the order of 5-20 kV, the initial energies become insignificant. Variations due to the different velocities of the electrons thus do not cause a serious spread in the arrival times on the anode, typical values of which are smaller than 2-4 ps. A schematic diagram of an electrostatically focused tube is shown in Fig. 16 where the electrons traverse the acceleration or gating grid, respectively, the focusing cone and further acceleration and deflection plates. These tubes can be operated both in the framing and in the streak mode. Special sweep circuits have been developed that can achieve final sweep rates on the photoanode of 75 mm/ns. A typical streak recordsg (Fig. 16) reveals the temporal evolution of a laser-supported detonation wave produced by the impact of a 10-15-5 COz laser pulse on an aluminium target surface. The sweep velocity, thereby, achieves a value of 0.25 mm/ns. To obtain higher streak rates or shorter exposure times with a bright enough image on the screen, the brightness has to be in-

objective lens X

I

100

focusing cone

photocathode

I

gdting

grid

-t

200

300

AL-target

acceleration electrodes

def l i c t i o n plates

photoanode

/

film plane

evolution o f lasersupported detonation waves

Ins]

measurements o f picosecond mode Locked laser pulses (scanning speed 63 ps/mm)

--I

1.2 ns

L-

FIG. 16. Long focus tube image converter camera. Application of the camera in the streak mode for the recording of fast luminous events. 8D M . Hugenschmidt and K . Voh'dlh, Proc. Int. Congr. High-Speed Photog. 12th, Toronto, 1976. p. 427. SPIE, Washington, 1977.

8.2.

RECORDING METHODS

737

creased by using further intensifier stages. This will be necessary because the beam current must be kept low to avoid image distortions due to charge repulsion. As an example for the application of an image converter camera including an intensifier, Fig. 16 contains a streak record of the temporal shape of the output-pulses of a mode-locked Rhodamine 6G dye laser. As the streak rate is known to be 63 mm/ns, the evaluation of the photodensitometer traces of such photographs allows for the exact determination of the real pulse durations The image intensification can be obtained by magnetically focused intensifiers with 3 or 4 stages (for example in the ,Imacon 600 system) as it was used to take the photographs shown in Fig. 16 or by microchannel-plate intensifiers. Conventional intensifiers are optically coupled by a lens system, the transfer efficiencies of which are only a few percent, so that the maximum gain of typically lo6 will be reduced to about 1 to 2 x lo4. The development of multichannel plate image intensifiers potentially yields the best solution to this problem.g1 These components allow for obtaining high gain of the order of lo6 by applying electric voltages of about 1000 V. The plates are, thereby, a few millimeters thick. In Livermore, a camera has been built that incorporates a wafer-type channel plate, which is proximity focused and does not require focus cones. Because of technical problems in manufacturing, focused-type channel plate intensifiers are often used. Such types are incorporated in the Imacon 675 image converter streak cameras where, due to the absence of optical coupling, the full gain of the channel plate intensifier is obtained at the film. As compared to the model 600, this camera is more compact, provides improved time resolution and has an increased total recording time from 0.9 to 1.5 ns. In addition, the signal-to-noise ratio is considerably higher. 8.2.2.3.4. MULTIPLE SPARKCAMERAS.A very simple, most effective and relatively inexpensive method for high-speed recording of phase objects in the field of fluid dynamics excluding mechanically driven components uses the optical separation of subsequent images. This was introduced by Cranz and Scharding2in 1929. The operation principle of a multiple spark camera is schematically represented in Fig. 17. The main components are first a series of n light sources, usually open sparks in air, a field lens which can also be replaced by a large aperture spherical mirror, and an equal number n of small objective lenses which are genA. Hirth, Dissertation, ISL-Ber. 26/74. Dtsch.-Franz. Forschungsinst. St.-Louis, 1974. J . Graf and R. Polaert, Acra Electron. 16, 11 (1973). O2 C. Cranz and H . Schardin, Z . Phys. 56, 147 (1929).

738

8.

LiCHT SOURCES A N D RECORDING METHODS

field lens

Laser-produced shock waves

FIG.17. Schematic of multiple-spark cameras. (a) Electrically triggered spark camera. (b) Optically delayed laser system.

erating n images of the object under investigation on the photographic plate. Due to the characteristics of the field lens, the light output of each source is imaged exactly on the aperture of the corresponding objective lens. Subsequent triggering of the individual sources thus allows for obtaining the different temporal phases of the object, geometrically separated on the photographic plate. Information-theoretical considerations and practical experimental aspects led to the development of different types of cameras with 8, 24, or 36 frames. Mostly used is the 24 multiple-spark camera as realized at the ISL.93 The repetition frequencies are, thereby, only limited by the duration of the light pulses. Low inductance spark circuits yielding exposure times of only a few tens of nanoseconds allowed for the realization of framing rates up to 10 MHz. Reference is made to the 1iteratu1-e'~for the large number of modifications including the application of prisms, fiber optics or even combinations with mechanically driven systems. Figure 83

A. Stenzel, ISL-Tech. Mitt. T 26/70. Dtsch.-Franz. Forschungsinst. St.-Louis, 1970.

8.2.

RECORDING METHODS

739

17 further shows a few shadowgraph-records (chosen from a series of 24 pictures) revealing the growth of a COz laser produced shock wave propagated from a plexiglass target surface into the surrounding air. The application of coherent radiation sources to cameras or systems of this type proved to be most vertisile for the investigation of transient fast self-luminous phenomena. The schematic arrangement of such a system which was used for the investigation of laser produced plasmas is also shown in Fig. 17. Starting from a single short duration laser pulse (a few nano- or picoseconds), a series of temporally delayed illuminating pulses is obtained by means of an optical delay line. This includes a row of mirrors, the reflectivities of which are chosen to yield equal intensities of the different reflected parts of the original beam. An interference-filter has to be inserted to suppress self-luminosity of the object. Figure 17 shows four frames of a laser induced gas breakdown in Thereby, a framing rate of 60 MHz has been applied. Higher rates can be achieved, however, up to the GHz-range by using mode-locked lasers. Valuable information can even be obtained in a much simpler way by directly photographing the object with a periodically pulsed source such as by a mode-locked train of ps pulses, with an open camera. This will be possible, if the object movements are so fast, that subsequent exposures do not overlapg5as in conventional low speed stroboscopic systems. 8.2.3. lnterferometric Methods

In the field of fluid dynamics, interferometric methods provide a large number of quantitative information on refractive-index changes or optical path differences introduced by phase objects, respectively. Thermal light sources can be applied as well as lasers. The detection of the interference patterns can be performed either by using photographic or photoelectric recording techniques. Photographic recordings yield the spatial distribution at a fixed time, photoelectrical registrations provide high temporal resolution capabilities along a given optical path. Both techniques are able to give the whole information, however, if optical scanning or combinations with cinematographic methods are used. Then, temporal variations and spatial distributions of refractive-index fields can be determined simultaneously. This is most important for the investigation of transient phenomena. Interference effects are always observed when two or even more light beams, the phases of which are strongly correlated are superimposed. In M. Hugenschmidt, K . Vollrath, and A . Hirth, Appl. Opt. 11, 339 (1972). K . Vollrath and M. Hugenschrnidt, Pro(,. Int. Congr. High-speed P h o t o g . , IZth, Toronto. 1976, p. 407. SPIE, Washington, 1977. 94

740

8.

LIGHT SOURCES AND RECORDING METHODS

the case of two interfering beams, for example, the intensity as a function of the phase difference 6 is described by a cos26 distribution. 6 is related to the refractive index p(r, t ) (which is both depending upon the spacecoordinate r and time t ) by the following equation 6 = (2.rr/h) J &, t ) dl. The integration has to be performed along the direction of the optical path. In the case of a large number of interfering rays, as in a FabryPerot interferometer, the fringes of maximum intensity are becoming considerably narrower than the cos2 6 profiles. This narrowing is depending on the finess F of the Fabry-Perot interferometer which is related with the mirror reflectivities R by the equation F = 4R/(1 - R)2.51 8.2.3.1. Classical lnterferometric Systems. Classical interferometric systems are designed for the use of nonmonochromatic or even white light sources. Thereby, two main groups of interferometers have to be considered: (1) The two-beam interferometer group, represented by the MachZehnder type (Mach-Zehnder, Michelson, etc.) and by the shear type (differential interferometer). (2) The Fabry-Perot types, on which belong to the group of the multiple beam interferometers.

The Mach-Zehnder types have in common that the rays intersecting the phase objects are spatially completely separated from their reference rays. In the shear type, as well as in the Fabry-Perot type, the interfering beams are largely overlapping. In particular, the differential interferometers using two Wollaston prisms have proved their versitility in gas dynamic research.e6 8.2.3.2. Laser Interferometry. Since lasers provide monochromatic and mainly coherent radiation, the adjustment of an interferometric system is greatly facilitated. By the use of small-band interference filters, investigations of self-luminous phenomena (such as flames or plasmas) can be carried out in a straightforward manner without loss of information. Figure 18 shows in the upper part two interferograms of laser-produced, rapidly expanding plasmas one of which was taken with the giant pulse of a ruby laser (7 = 20 ns, P = 5-10 MW), whereas the other picture reveals a spark illuminated exposure. The differences can be seen clearly. In the laser interferogram, the fringe contrast, even for high orders of interference is much greater than in the case of the spark interferogram. Furthermore, the fringe shift can be determined with high accuracy throughout the whole area of the object including the central G. Smeets, ISL-Tech. Mitt. T 21/70. Dtsch.-Franz. Forschungsinst. St.-Louis, 1970.

8.2.

74 1

RECORDING METHODS

spark interferogram

laser interferogram

evaluation of the fringe shift Aa/a along the axis A-B (At = 390 ns)

i U

a

ne

t tCni-9

fP

+2

1.002

0 -2

-4

2.35ps

-iI

-6

6

i i

[mml

0.9 9 6

‘2.35 ps I

1

v

2 rmmi

1 [mml

FIG.18. Laser interferometry. Quantitative evaluation (electron density of a ruby laser produced Xe plasma).

part which is overexposed by the plasma luminosity in the spark photography. The lower part of the figure shows an example indicating the evaluation procedure. This interferogram was taken with a Wollaston prism interferometer. The laser produced plasmas are assumed to expand in a rotational symmetric way around the axis of the incident laser beam. The measured fringe shift Au, normalized to the distance a

742

8.

LIGHT SOURCES A N D RECORDING METHODS

between the undisturbed fringes V ( x ) = A a / a , is then related to the radial profile of the refractive index by an Abelian integral equation. If i different particle groups are concerned, the refractive index p is related to the physical parameters such as the local densities ni by the equation p - I = C 274X)nt.

(8.2.12)

The polarizabilities of the different groups at(X)are dispersive. Their relative influence has to be estimated for each experimental condition concerned. In the case of atoms or molecules in excited states, &$(A) shows pronounced resonances due to anomalous dispersion. By using dye lasers as interferometric light sources, laser frequencies can be tuned to resonance rendering the other terms of (8.2.12) negligible. Laser interferometric techniques are thus able to provide quantitatively partial densities of specially excited particles. In highly ionized plasmas as shown in Fig. 18, the free electron term considerably exceeds the other terms so that, starting from the measured fringe shift, the procedure allows for the calculation of the electron densities.g7 8.2.3.3. Two-Wavelength Interferometry. By applying light sources emitting simultaneously at two or more wavelengths, the above mentioned dispersive behavior of the refractive index can be used to yield more detailed information. However, it must be proved that the wavelength ranges chosen are not affected by anomalous dispersion. In this case, the refractive index is mainly influenced by the group of electrons n, and by the group of heavy particles (essentially neutral particles a,,). eZhZ n, ; m-%Oc2m,

(8.2.13)

e0 is the dielectric constant of vacuum. Frequency doubling of a dye or ruby laser output provides a simple method to generate at the same time two monochromatic short intense light pulses starting with a single laser. Figure 19 shows a schematic a r r a t ~ g e m e n t . ~ *Harmonic *~~ generation is achieved by means of an optically nonlinear crystal such as KDP or ADP. The separation of the two interferograms can be performed by using a wavelength selective mirror. Suitably chosen interference filters again suppress background illumination. The evaluation of fringe shifts proceeds in the same manner as already described yielding the two refractive index profiles from which the neutral particle and electron densities can be determined. A similar setup can be applied using continuous-wave

@' M. Hugenschrnidt, 2. Angew. Phys. 30, 350 (1971). 9*

M.Hugenscbrnidt and K . Vollrath, Opt. Loser Techno/. 3,93 (1971).

ss

A. J . Alcock and S . A . Ramsden, Appl. Phyhys. L e f t . 8, 197 (1966).

8.2. ruby

laser

743

RECORDING METHODS

KDP

crystal interferometer

’

imaging system

L8

FIG. 19. Two-wavelength interferometry. Evaluation of fringe shifts V ( x ) .

lasers, for example two He-Ne lasers, one of which emits in the red (0.6328 pm), and the second in the infrared (e.g., A = 1.15 or 3.39 pm). Accurate time resolution is obtained by the use of high speed photoelectric detectors.loO 8.2.4. Holographic Methods

Wave front reconstruction of images has been discussed by a large number of physicists since about 1920. The first experimental results were obtained in 1949 by Gabor who suggested the name “holography.” This new optical procedure allowed the registration of amplitudes and phases of optical waves,lol which are schematically designed in Fig. 20 by the phase fronts C. The indices 0, Rf, and Rc refer to object, reference, and reconstruction. The method of Gabor can be applied to partially transparent objects. One part of the incident light wave is directly transmitted; the other part is scattered by the object. On the photographic plate these two parts are superimposed thus forming an interference pattern containing the whole information. The reconstruction of such an in-line hologram can be obloo lol

G . Smeets, ISL-Notiz N 608/75. Dtsch.-Franz. Forschungsinst. St.-Louis, 1975. D. Gabor, Electron. & Power 12, 230 (1966).

744

8.

LIGHT SOURCES A N D RECORDING METHODS

Irecordings 7

[-reconstructions-\

,...,',

ti

FIG. 20. Schematic of holographic recording and wavefront reconstruction. P = arbitrary object point, P' = normal image point, P" = conjugate image point. (a) Inline holography. [After G a b ~ r . ~ O ~(b) ] Off-axis holography. [After Leith and U p a t n i e k ~ . ' ~ ~(c) ] Holograms in reflected light. (d) Holograms in transmitted light.

tained by illuminating the developed plate with a parallel beam of monochromatic light. Diffraction effects are responsible for the appearance of the image which is, however, always disturbed in this simple setup by a conjugate image. This difficulty was overcome by Leith and Upatniekslo2 in 1962 by the technique of the off-axis holography which was able to be realized because of the availability of laser light sources. Since then, holography has grown into an expanding field of scientific research and technical application. 8.2.4.1. Basic Principles of Holography. Applying the off-axis technique in the lower part of Fig. 20, the experimental setup is shown both for the registration of objects in the reflected and in the transmitted light, a being the angle between the object beams and reference beams. As indicated, this can be done experimentally by appropriate optical elements such as prisms and mirrors. The wave front reconstruction is simply obtained by illuminating the developed plate as indicated by means of a reconstruction beam. Due to the off-axis condition, the real and virtual images are then geometrically completely separated. Mathematically, the exposure and reconstruction of a hologram can be described by the following simple set of equations. lo*

E. N . Leith and J . Upatnieks, J . Opt. SOC. A m . 52, 1123 (1962).

8.2.

745

RECORDING METHODS

OF THE HOLOGRAM. The complex light ampli8.2.4.1.1. EXPOSURE tudes Vo scattered from an object are superimposed upon a reference wave described by the complex amplitude VRf. For simplicity, Vo is represented by a spherical wave emanating from an arbitrary object point P , whereas V,, shall be represented by a plane wave. The amplitudes incident on a point ( 6 , q) of the holographic plate are then V(t) = Vo(t) + VRXr), and the resulting intensity is

I(?) = W O + VRf)(V,* + V&).

(8.2.14)

After an exposure time 7E, the optical density D of the plate in the consid. ered point will be proportional to the energy E = I ( t ) T ~ Substituting the above relations yields the expression E = { ( V O+ ( ~ lv~rl'+

vov&4-

VzVRf}TE.

(8.2.15)

The transmission T, the ratio of transmitted to incident intensity, is related to the optical density by the equation D = log 1/T.

(8.2.16)

In the linear range of the characteristic curve D versus log E , the transmission will be proportional to the energy, thus yielding T=

T - P ( E - 0.

(8.2.17)

Tand E are mean values of the transmission and energy, respectively. p yields the slope of the curve. Assuming that E is mainly determined by the intensity of the reference beam, E can be approximated by E = IV,f(27, so that the following equation holds: T=

T - /~TE{IVO~~ + VoV& + ViVRf}.

(8.2.18)

Thus T is dependent on a term proportional to V,, and on a second term proportional to the complex conjugate Vg , which both contain information on the amplitude and the phase relations of the object wave. 8.2.4.1.2. RECONSTRUCTION OF THE WAVE FRONT.The process of wavefront reconstruction can be described in a similar way by multiplying the transmission T with the complex amplitude of a reconstruction wave V, which (with respect to the wavelength or the angle of incidence) must not be identical with the original reference wave. For any point (6, q) one obtains Vm . T = VRc

*

T-

P ~ E V ~ ( l v 0 1+' VoV&

+ V,hV,f}.

(8.2.19)

The first term describes a mean attenuation. The second term indicates a further attenuation of the reconstruction wave by diffraction due to 1 VOl2. Both terms are thus concerned with the directly transmitted part of the

746

8.

LIGHT SOURCES A N D RECORDING METHODS

wave VRc. The amplitude and phase information concerning the object wave is contained in the third term directly and in the forth term with inverted polarity of the phases. The spherical wave approximation can easily be extended to describe more complicated objects by summing or integrating the contributions of all object points. The same mathematical formalism can be applied to calculate the geometrical location of the normal and conjugate image points and the magnification. The treatment of these questions, the discussion of orthoscopic or pseudoscopic images, the distinction between Fresnel, Fraunhofer, and Fourier holograms, amplitude and phase holograms, numerically computed holograms, Bragg-Lippmann holography, to mention only a few topics, is beyond the scope of this presentation. Reference is made to the l i t e r a t ~ r e . ~ ~ ~ - * ~ ~ 8.2.4.1.3. APPLICATION OF HOLOGRAPHIC TECHNIQUES. Holographic techniques are well suited for investigations in fluid dynamics.lo6 Simple experimental arrangements can be used if lasers are available, the coherence lengths of which are larger than the maximum optical path differences between the object and reference beam. Ruby lasers have mostly been used for studying transient phenomena (more recently dye lasers have been applied as well). As holograms store the amplitudes and phases, the objects can be reconstructed without loss of their threedimensional character. The images can be observed and evaluated in different planes. Furthermore, holograms can be evaluated following different optical procedures; see Fig. 21. The reconstruction of the wave field of an aerodynamic flow or of a plasma, for example, can be visualized by means of shadowgraph or schlieren techniques, depending on the absence or presence of an edge.lo7 By using a Wollaston prism, the same holographic plate can yield an interferogram which can easily be subject to quantitative evaluation. 8.2.4.2. Holographic Interferometry. As already mentioned, the holographic information can be measured by using classical interferometers. It is most important, however, that due to the storing capabilities of photographic plates, different wave fronts be registrated even in the event

Io3 H. Kiemle and D. Ross, “Einfiihrung in die Technik der Holographie.” Akad. Verlagsges., Frankfurt, 1969. lo‘ J. C. Vienot, Holography. In “Laser Handbook” (F. T. Arecchi and E. 0. Schulz-Dubois, eds.), Vol. 2, p. 1487. North-Holland h b l . , Amsterdam, 1972. M. FranCon, “Holographie.” Masson, Paris, 1969. lO8 E. R. Robertson, “The Engineering Uses of Coherent Optics.” Cambridge Univ. Press, London and New York, 1976. lo’ A. Hirth, C . R . Hebd; Sronces Acod. S c i . , S r r . B 268, 961 (1969).

8.2.

RECORDING METHODS

747

(a)

reference beam

shadowgraph

schlicre n picture

interferogram FIG. 21. (a) Recording of phase objects. (b) Reconstructions of holograms following different optical procedures.

that they expose the hologram at two separate times.108 The two superimposed holograms then can be considered stored independently from one another in the same emulsion. Upon reconstruction, both waves are restituted giving rise to interference effects. In this way, small changes of the optical path length due to variations of the refractive index or due to macroscopic displacements of an object can be measured with high accuracy. Optical path changes of A are just cause the fringes to be shifted by the amount of the undisturbed fringe spacing.'08 A major advantage of the holographic interferometric technique is that accuracy is not affected by using inexpensive schlieren grade windows, mirrors, lenses, or prisms in the optical setup, because these distortions do not change in the time interval between the two exposures so that they are canceled. By varying the time delay between the two exposures, direct temporal variation of the optical path lengths can be obtained. Figure 22 shows some double exposure holographic interferograms of the

lo@

F. Albe, 1SL-Ber. R 114/76. Dtsch.-Franz. Forschungsinst. St.-Louis, 1976. P. Smigielski and A . Hirth, ISL-Ber. 11/71. Dtsch.-Franz. Forschungsinst. St.-Louis,

1971.

748

8.

L I G H T SOURCES A N D RECORDING M E T H O D S

TEA-CO1 laser

ruby laser

cell

cell

etalon

\ diffusor /

% / ; y ; s t

hologram

r uct ion

FIG.22. Investigations of the temporal variation and spatial distribution of refractive index fields by holographic interferometry.

electrical discharge of a pin-type TEA CO, laser, where the infrared laser mirrors have been replaced by simple glass plates. The object beam is transmitted end on through the plasma tube, whereas the reference beam is split off by means of a prism."' The light pulse of the monomode ruby laser is synchronized with respect to the TEA discharge by a Pockels cell inside the laser cavity which is driven by an adjustable delay time. The holographic plates are first exposed without discharge plasma. A few minutes later, the discharge is initiated to obtain the second exposure at the desired moment. The left row of Fig. 22 shows some recordings of the temporal development of the plasma. The lower row demonstrates the capability of spatial scanning the complex locally varying fields of refractive indices. These reconstructions reveal the fringe distributions at a fixed time. To obtain this spatial information, a diffusor sheet has to be placed near the entrance window of the CO, laser discharge tube. The holographic technique proves to be the only method with which such irregular spatial density profiles lacking symmetry can be analyzed. 'Io M. Hugenschmidt and K . Vollrath, ISL-Ber. 21/71. Dtsch-Franz. Forschungsinst. St.-Louis, 1971.

8.2.

RECORDING METHODS

749

Further double-exposure techniques can be applied in a modified form by using lasers emitting two successive pulses. These can be separated in time by some tens of microseconds to a few nanoseconds, so that changes of rapidly varying events which are introduced in short time intervals can be tested dynamically."' Periodically vibrating processes are investigated by real time holography. Thereby, the reconstruction of the hologram is arranged in such a way that the image of the object is superimposed upon the real object which is also illuminated by the reconstruction beam. Distortions due to mechanical stresses or vibrations thus give rise to interference effects which then can directly be seen or photographed. Material testing of periodically vibrating objects can also be carried out by a time averaging analysis, where the object is exposed for time intervals longer than the vibration period. In this case, high fringe contrast is only obtained for the nodes of the vibration, whereas in other parts of the vibrating surface the contrast is eliminated 8.2.4.3. Two-Wavelength Holography. The interpretation and evaluation of holographic interferograms proceed in the same way as in the case of classical interferometry. The refractive index again has to be related to the particle densities involved for objects so that the application of multiple wavelength coherent sources also will be able to provide more information. Experimentally, the pulse of the previously described monomode ruby laser transmitted through a KDP crystal which is properly aligned to hold for the phase-matching conditions is split up to provide the object beam and the reference beam, each of which contains both wavelengths. The sensitivity of photographic plates (Kodak 649 F) proves to be adapted to the whole wavelength range so that the double exposure technique can directly be applied. Thereby, the photographic plate stores four different holograms. Upon reconstructing the two interferograms, the red and the ultraviolet can be separated if certain geometrical conditions concerning the overall dimensions of the object and the angles between object beam and reference beam are met. This is shown schematically in Fig. 23. The image separation is due to the fact that different Bragg conditions are valid for the diffraction of the reconstruction which depend upon the wavelength. As expected, the magnification ratio of the two images corresponds to the wavelength ratio.113 A. Felske and A. Happe, in "The Engineering Uses of Coherent Optics" (E. R . Robertson, ed.), p. 595. Cambridge Univ. Press, London and New York, 1976. R . L. Powell and K . A . Stetson, .I. O p t . Soc. A m . 55, 1593 (1965). A . Hirth, C. R . A r a d . Sci., Srr. B 271, 28 (1970).

750

8. rubv __ a .

LlGHT SOURCES A N D RECORDING METHODS

laser -

,reference beam

KDP

I

*I""'

'

'ogram

object beam

/--

h=691.3 nm

h=347.1 nm

imaging sys tem

FIG.23. Two-wavelength holographic interferometry arrangement.

As an example, Fig. 23 contains two interferograms of an electric spark discharge. 8.2.5. Infrared Imaging

Infrared recording techniques are becoming increasingly important especially in the field of plasmaphysics. The sensitivity in schlieren experiments, the amount of rotation of the direction of polarization in Faraday rotation measurements, and the polarizabilities of the free electrons in interferometric measurements, are revealed to be proportional to h2. The last mentioned fact shall be discussed more in detail. An infrared light source combined with an interferometric technique will thus provide a marked increase in ~ e n s i t i v i t y .As ~ ~ compared to the fundamental waves of a ruby laser, COz lasers emitting at 10.6 p m yield a gain of 250, as compared to their harmonics; this will even imply a gain factor of lo3. Pulsed TEA-COZ lasers represent simple and convenient infrared sources which are emitting short pulses with half-widths in the range of some tens to some hundreds of nanoseconds, and peak powers of the order of several megawatts. Shorter pulses in the nano- and subnanosecond range can be obtained as well by mode locking techniques which require, however, more elaborate laser systems. 8.2.5.1. Infrared Recording Systems. The registration of infrared radiation which can be performed electrically by means of a great number of thermal detectors or quantum detectors covering a large range of spectral sensitivities and detection bandwidths (from slow thermal systems to high speed pulse detection) will not be considered in the present section.

8.2.

75 1

RECORDING METHODS

If photographic recording techniques are required, the main problem will be the conversion of the infrared radiation towards the visible. No infrared photographic materials are available beyond 1.1 to 1.2 pm. Some of the methods used for the visualization are: the Czerny evaporograph, the detection by means of liquid crystals, the evaporation of thin sheets of material (metals or paraffin wax), the Lippmann plates, or the quenching of fluorescence. Some of these methods are indicated in Table IV. Their applicability is dependent upon parameters such as the threshold power or energy density and the spatial resolution limit. Some of these values are also indicated in Table IV.lI4 The highest resolution of 70 lines/mm has been obtained with thin sheets of various materials which are evaporated by the infrared radiat i ~ n , ”whereas ~ the lowest threshold power densities have been achieved with Czernys evaporograph. Liquid crystals are characterized by medium resolution power but relatively low threshold energy densities. They show good performance characteristics: the process of the temperature dependent wavelength selective reflectivity in the case of cholesterinic liquid crystals is reversible, they are easy to handle, and can be adapted to different temperature range^.^^^^^^' TABLE1V. Limiting Values for Different Infrared Recording Techniques” _

_

_

Threshold power

~

~

Energy density

Resolution

( J /ern*)

(lines/mrn)

density (W/crn‘) ~~

~

evaporography

10-e

I

10

liquid erystols

10-I

lo-z

20

metallic sheets

3.10-l

70

paraffin wax

3.10-l

70

10

4

0.06 - 1.8

10

Lipprnann plates quenchlng of fluorescence

“A

power

=

10-I up to 60

10.6 p m .

W. Waidelich, “Kurzzeitphysik,” Vortrag auf Friihjahrstagung, Kiel, 1972. DPGVerhandlungen, Phys. Verlag, Weinheim, 1972. A . Darr, G . Decker, and H . Rohr, Z . Phys. 248, 121 (1971). 118 J . Fergason, A p p l . Opt. 7, 1729 (1968). M . Hugenschmidt and K . Vollrath, C. R . Acud. S c i . , Ser. B 274, 1221 1972).

8.

752

LIGHT SOURCES A N D RECORDING METHODS

Further image converters have been built using different semiconducting materials such as SeCr which provide a temperature and wavelength dependent absorptivity. The optinicon developed by Ulmer*18 uses temperature induced changes of the refractive index of thin liquid film layers. This causes changes in the reflectivities for the additionally incident visible radiation (for example of a He-Ne laser) proportional to the infrared radiation distribution. 8.2.5.2. Applications to Flow and Plasma Diagnostics. An example of quantitative analysis of an infrared interferometric method is shown in Fig. 24. The Michelson interferometer uses two gold coated quartz mirrors and a Ge beam splitter with a 50 percent reflection coating on the front and an antireflection coating on the rear side. The light source con-

.># I

I

camera

.,--

,--0

**-

Michelson interferometer

v(x)=

(b)

%

,i ri-4-

+0.2

-0.2-0.4-

1

/

-0.6-

1.000

o'~-

1 0.998

- 0.80 1 2[

m

m

I ~ 2 ~[mmj~

~

0~

1~ 2 h m 3

FIG.24. (a) Infrared interferometric recording of spark plasmas. (b) Quantitative evolution of the electron density distribution.

W. Ulrner, Infrared Phys. 11, 221 (1971).

8.2.

RECORDING METHODS

753

sists of a TEA-COz laser emitting pulses of about 3 MW (7 = 200 ns) on the P(20) line at 10.56 pm. The output mirror of the laser is given by the plane surface of a plane-convex Ge lens which together with a mirror in a confocal arrangement acts at the same time as a beam expander. The phase object to be studied (an electric spark discharge), is located in one arm of the interferometer. A further mirror serves to form an image on a suitable infrared converter. The sensitivity of the liquid crystal detector (a mixture of cholesterol-oleyl-carbonate and -nonanoate, 75 : 15) can be chosen in such a way that small temperature variations of a few tenths of degrees are spectrally changing the reflected part of a white light illuminating source (flash lamp) from the red to the blue. The color distribution thus directly indicates the ir radiation pattern, which can then be photographed by means of an ordinary camera and electronic flash equipment. The numerical evaluation and calculation of the refractive index and electron density is ~traightforward.'~The results are also given in Fig. 24. It should be pointed out that an increasing number of laser lines in the ir are becoming available by molecular gas lasers, optically pumped lasers, and dye-laser-pumped Raman lasers. In most cases their power is still relatively low. For the investigations of transient phenomena, more powerful systems would be preferable. It seems possible, however, that such systems can be designed in the near future, e.g., by the transversely excited pulsed HCN lasers which emit in the far ir at 337 pm.119

B. Adam, H. J. Schotzau, and F. K . Kneubiihl, f h y s . Left. A 45, 365 (1973)

lln

This Page Intentionally Left Blank

9. APPARATUS

9.0.Introduction* Fluid dynamic apparatus has developed rapidly, especially since World War 11. The changes, which have combined a broad spectrum of other branches of physics into the design and operation of fluid dynamic research and testing equipment, stem both from rapid advances in aerospace technology and, more recently, from increased stress on environmental sciences, weather prediction, and energy conservation research. The existence of compressible flow at high subsonic and supersonic speeds means that the physical equations relating energy and momentum are interdependent, with the result that flow thermodynamics becomes important. In addition, variation of density in the flow field produces a corresponding variation in optical refractivity; that, in turn, has led to the use of powerful optical methods for flow analysis (see Part 2 of this volume) and to changes in facility design to accomodate optical instrumentation. Another feature of supersonic flows, namely shock waves, has led to the use of an important facility, the shock tube, which has taken its place as a standard research tool in fluid dynamics. Again, in hypervelocity flows, kinetic energies become comparable with internal energies of atoms and molecules, with the result that excitation, dissociation, ionization, and radiative properties have to be taken into account in apparatus design, o r themselves become the subject of research studies, especially with the shock tube. Further, as fluid transit times have become comparable to o r shorter than atomic excitation times, facilities to study reaction kinetics have been developed by fluid dynamics researchers. Wind tunnels and shock tubes are useful for study of flows in which viscosity plays no part, i.e., at very high Reynolds numbers, and in flows where viscosity and inertial forces are comparable, notably boundary layers. In low Reynolds number flows, say Re < 0.1, different apparatus is employed, some of which is used for determining the characteristic viscosity coefficients of fluid substances-the so-called viscometers.

* Chapter 9.0 is by R.J. Emrich. 75s M E T H O D S OF EXPERIMENl A L PHYSICS, VOL. 18B

Copyright @ 1981 by Academic Press. Inc All rights of reproduction in any form reserved.

._”., .._. . I

.-rnc,

1

756

9.

APPARATUS

In geophysical flows, as studied by meteorologists and oceanographers, great progress has been made in detailed measurements of the fluid velocities, pressures, temperatures, and compositions over the expanse of the thin layer of air and water covering the earth. Significant help to the interpretation of these measurements has been furnished by studies in rotating tanks of water to simulate the dynamic conditions characteristic of flows in rotating coordinate systems. Finally, progress has been made, but more is needed, in the development of apparatus to increase our understanding of the most challenging of all fluid dynamic problems, namely turbulence. The discussions which follow in this presentation make reference to additional specialized facilities needed for other problems of interest, such as the study of dusty flows, the aerodynamics of wind flow around clusters of high-rise buildings, and the flow of polymers.

9.1. Wind Tunnels and Free-Flight Facilities* Both wind tunnels and free-flight facilities have been utilized to study problems associated with the motion of flight vehicles in the atmosphere for purposes of aircraft design and aerodynamic research. The wind tunnel is a partially or totally enclosed configuration in which, typically, a moving air stream of desired pressure and temperature passes over a suitably scaled and mounted stationary model (Fig. 1). Since it is the relative motion which matters, such variation on the free-flight case does not in itself introduce restriction on interpretation of experimental results. However, the careful researcher will have to account for error-producing effects relating to the geometry of the tunnel and mounting, power source, wind-tunnel flow field and thermodynamic conditions, physical properties of the air or other fluid and, of course, the limitations of particular measuring instrumentation. The last-mentioned consideration is especially important for the ballistic range and other types of free-flight facilities. The wide scope of current problems has led to an increase in the number and types of applications of wind tunnels. As before, measurement of lift, drag, structural stresses, flow interference over adjacent aircraft components, flutter and other stability problems are only some of the aircraft design problems whose study has now extended into the supersonic range with a suitable variation of Reynolds number. An example of a newer application is the simulation of the wind flow around an urban high-rise cluster complex. For this purpose, a special wind tunnel is required in which * Chapters 9.1, 9.2, and 9.3 are by Daniel Bershader.

9.1.

W I N D TUNNELS A N D FREE-FLIGHT FACILITIES

757

Working section

FIG. 1. Typical arrangement for a continuous, closed-return supersonic wind tunnel. Shown schematically is a model mounted in the test section. [From Ref. 1 , Fig. G,lc.]

a suitable flow profile corresponding to the low-altitude atmospheric boundary layer passes over a scaled model of the configuration of buildings and other landscape features. Other applications of wind tunnels include anemometer and other instrument calibration, and several types of more basic studies such as growth and interaction of boundary layers, transition of laminar to turbulent flows, properties of turbulence, stratified flows, and two-phase flows. In what follows, we examine briefly some of the physical guidelines relating to design and use of wind tunnels and free-flight facilities. 9.1.1. Overview of Wind Tunnel Systems

Measurements in a wind tunnel are conducted in what is typically called the “test section” (see Fig. 1). The latter is characterized by a welldefined flow field, together with suitable model mounts, windows, linkages for special instrumentation, such as a force balance, and other adapters for use with monitoring and recording instrumentation systems. The air flow is established by a pressure difference between the sections upstream and downstream, respectively, of the test section. By now, all F. E. Goddard, Supersonic tunnels. In “High Speed Aerodynamics and Jet Propulsion” (F. Goddard, ed.), Vol. 8, Sect. G, p. 491. Princeton Univ. Press, Princeton, New Jersey, 1961.

758

9.

APPARATUS

most any method which the reader can envisage to produce the difference in pressure has probably been utilized. For low speed flows, suitably designed motor-driven fans maintain a continuous flow. Alternatively, compressed air from storage tanks may be fed into an upstream chamber to generate excess pressure for the tunnel flow. This is referred to as an intermittent or “blow-down” wind tunnel. In yet another arrangement, a partial vacuum is generated on the downstream side, again establishing a flow through the test section. The latter system is known as an induction or in-draft wind tunnel.2 The air-intake and working section (Fig. 1) are usually separated by an air storage and settling plenum or “supply” chamber, together with a constricting channel called a nozzle, which is designed with care to accelerate the flow smoothly to test conditions. Downstream of the working section the flow enters a diffuser which provides the transition from test section speed and pressure to exit low speed and pressure. The exit pressure may be atmospheric or not, depending on the type of tunnel. The type is determined in part by the degree of “pressure recovery” required by power considerations (see Section 9.1.2.2). Large scale wind tunnels and also those working at very high pressure and/or temperatures are faced with special technological problems of structural integrity, as well as socioeconomic problems dealing with environmental pollution such as noise, and overall facility cost. Such problems are largely outside the scope of the present discussion. However the requirements for size, in relating to modeling, are discussed in Section 9.1.2. 9.1.2. Classification of Wind Tunnels

Classification of wind tunnels is a multidimensional affair. Important parameters are scale or size, flow duration (intermittent versus continuous) level of background turbulence and general flow quality, degree of control of density (Le., Reynolds number), Mach number range, stagnation temperature or stagnation enthalpy range, and others. The application the tunnel is intended for may be the primary factor. Thus, a wind tunnel to study some features of a propulsion system in flight will be quite different in design from one which is to measure components of forces and moments on an aircraft model. In turn, a tunnel designed to study instability of laminar boundary layers will have different features again. Nevertheless, there are some important general guidelines to classification which deserve special mention in connection with fluid physics studies.

* A . Pope and K. Goin, “High-speed

Wind Tunnel Testing.” Wiley, New York, 1965.

9.1.

W I N D TUNNELS A N D FREE-FLIGHT FACILITIES

759

9.1.2.1. Reynolds Number Basis. As always, the experimentalist will inquire initially into the underlying physical nature of the problem with which he is concerned. In the past and also today a principal class of problems in fluid physics and aerodynamic research has dealt with phenomena which involve the combined action of normal pressure forces and viscous stresses. Dimensional analysis (see Part 10 of this volume) says that the dynamical behavior of the fluid and its interaction with solid surfaces is functionally related to the dimensionless ratio of the momentum flux resulting from normal pressure action to the viscous stress. The ratio is known as the Reynolds number Re:

Re = p d / p = v l / v

(9.1.1)

where p is the fiuid density, u the velocity, 1 a characteristic length, p the coefficient of viscosity, and v the kinematic viscosity. The Reynolds number actually controls the functional form of the relation among the variables in the type of problem just mentioned. For example, suppose one undertakes to verify Stokes’s law for the drag force FDon a sphere of radius R moving with velocity u through a fluid of viscosity p . The law reads FD = 6 ~ p R v .

(9.1.2)

It turns out that Stokes’s formulation is indeed correct for the case of the air force on the droplet in the Millikan oil-drop experiment, but that is because both the Stokes derivation and the oil-drop motion correspond to low values of Re, of order unity or less. On the other hand, experiments performed at higher Reynolds numbers, around Re = 1000, reveal that FD is independent of p and, further, is more nearly proportional to u z than to u ! That is, F , is also a function of Re, and (9.1.2)is a limiting case for low values of Re. The basic need for the experimentalist to have access to a uniform test section flow implies that the latter will be characterized by a sufficiently high Reynolds number; for only then is the nonuniform behavior associated with viscosity relegated to so-called boundary layers in the vicinity of the tunnel walls. Similarly, in the case of a free jet test flow, the spread of the boundary mixing region will be less at high Reynolds numbers, leaving a larger uniform-core volume for test purposes. Including special applications, wind tunnel experiments have by now covered the Reynolds number range from to lo*. 9.1.2.2. Mach Number Basis. Mach number M is the ratio of the fluid speed u to the local sound speed u in the fluid:

M = u/a.

(9.1.3)

760

9.

APPARATUS

When M is nonnegligible compared to unity, changes in flow velocities and pressures are accompanied by density changes; i.e., the flow is compressible. In this case, the experimentalist who wishes to perform dynamic modeling studies will have to simulate both Re and M (again, see Part 10). However, this requirement only begins to indicate the importance of Mach number in wind tunnel experimentation. Of special significance is the change in nature of the flow field when M becomes larger than one. The flow equation then changes from elliptic to hyperbolic, i.e., it becomes a wave equation. Disturbances, instead of spreading everywhere in the flow field, travel along so-called characteristics which, in some circumstances, build up to shock waves. When such waves are generated by the presence of a model, reflections from the walls interfere with the flow field, as do shocks which form on insertion-type measuring probes. Supersonic flow through channels and nozzles takes place when the driving pressure ratio across the throat (minimum area) section becomes large enough. Then, sonic flow is established at the throat (consider the problem in one-dimensional terms for the present) and remains so independent of any further increases in upstream pressure. The gas expands to supersonic velocities downstream of the throat into the test section, and later must be decelerated with an accompanying pressure rise through shock waves to a subsonic condition determined in accordance with the pressure recovery design of the tunnel system. Design of suitable diffusers for supersonic tunnels is a challenging task’ because flow through shock waves is dissipative, and one seeks the particular downstream shock configuration which minimizes the dissipation and thus the power required to keep the tunnel going. The essence of the problem turn out to be the “pressure recovery,” i.e., the degree to which the gas pressure returns to the value it had in the supply section when the velocity returns to near zero again. Also associated with compressible flows are temperature changes which can be appreciable in magnitude. For a gas (treated as ideal) of specific heat ratio y , the temperatures and Mach numbers at two points in the flow field are related by

(9.1.4) For example, a gas such as air, with y = 1.4, expanding from rest ( M , = 0) at a temperature T2 = 290 K to a flow Mach number M 1 = 5l’’ will be cooled t o a temperature T1 = T2/2 = 145 K. At these moderate supersonic Mach numbers, the experimentalist concerned with nontherma1 problems can still avoid heat transfer problems in spite of the con-

9.1.

W I N D T U N N E L S A N D FREE-FLIGHT FACILITIES

76 1

siderable temperature change. The reason is that viscous production of heat together with heat conduction in the compressible boundary layers on tunnel walls and model surfaces produce so-called recovery temperatures which are rather close to the temperature of the supply or stagnation section. However, at still higher Mach numbers, say around 3 o r above, very low free stream temperatures T , may cause actual freezing of the molecular components of air unless heat has been supplied t o raise the temperature of the air in the supply section. When that is done, temperature recovery near walls and model surfaces will produce substantial heat transfer, and the investigator will have to decide about thermal insulation o r wall-cooling techniques, depending on the temperature, geometry and wall materials, and the flow duration. Other Mach number-sensitive features of wind tunnel design and utilization will come up in subsequent sections of this chapter, especially in connection with so-called transonic tunnels. However, the singular importance of M is already evident.

9.1.2.3. Tunnel Flows with Nonideal Gas Behavior. As long as the working fluid can be treated as a continuum with definable transport properties, the flow categorizations given in Sections 9.1.2.1. and 9.1.2.2. above apply with sufficient generality. However, there are circumstances relating to modern flow research problems where behavior of the gas on a molecular scale poses additional problems of experiment design and interpretation of results. The first of these has to do with high temperature behavior associated with energy exchange in very high speed flows. Apart from overall engineering-type problems such as cooling the tunnel walls, attainment of sufficiently high temperatures in the gas means that additional internal energy states, especially molecular vibration, will participate in the energy distribution. The change in specific heat will, in turn, produce a change in the specific heat ratio y which in fact may vary over parts of the flow field. The usual ideal-gas compressible flow formulas such as Eq. (9.1.4) relating Mach number, pressure, temperature, etc., are not then valid with the result that working section conditions may change and unwanted gradients may appear. At temperatures of interest in modern aerothermodynamics internal excitation may be accompanied by molecular dissociation, ionization, and significant amounts of radiation heat transfer. Thus, for example, an experiment simulating conditions near the nose of a vehicle traveling around 6 km s-' at an altitude of approximately 50 km will produce a temperature of 6000 K at about 0.001 sea level density. Under such conditions, the O2 component of air is essentially all dissociated, and the N, is

762

9. APPARATUS

substantially dissociated as well. Reactions then take place which produce the gas NO in the amount of a few mol per cent. Because NO has a lower ionization potential than other components in this high temperature mixture, it is NO which furnishes the principal fraction of electrons present in the boundary layer and shock layer surrounding the vehicle or the model, about 102O m-3 near the model nose in the example discussed above. At still higher speeds, radiation over a wide spectrum may occur and play an important role in the energy balance. Apollo reentry capsules enter the atmosphere at about 11 km s-l, which is approximately the speed regime where radiative heating becomes comparable with convective heating. In the case of entry into Jupiter’s atmosphere, the calculated heating rate is of the order of 50 kW cm-*, over 90 percent of it radiative.*” Planning of experiments must then deal with special problems of model design, opacity of the flow field, absorptive properties of the wind tunnel walls, radiative shielding of probes and radiation-measuring instrumentation. It follows too that many hypervelocity fluid flow experiments are indeed designed to measure radiative characteristics and also radiative heat transfer. Spectroscopic methods are discussed in Parts 3, 4, and 6, and heat flow gages are treated in Part 7. Key quantities determining high temperature fluid-flow behavior are the so-called total or stagnation enthalpy h, (typically nearly equal to the free-stream kinetic energy of a fast-moving fluid), and the mass density p . Laboratory flow studies at moderately high values of h, over a range of densities have been performed with supersonic or hypersonic tunnels in which air or another test gas has been heated before expansion into the test section. Experiments where thermal radiation and possibly also ionization are important have utilized a so-called plasma jet or arc tunnel (see Section 9.1.6.1) when appreciable flow duration has been required, or a shock tube/tunnel device where fast-response measurements are available to provide the necessary data (see Chapter 9.2). Simulation of very high-altitude aerodynamic and fluid-dynamic behavior requires flow studies at correspondingly low ambient densities. At an altitude of 48 km, the atmospheric density is 1.0 x that of the sealevel value of 1.22 kg.mP3 (at 288 K); and the density decreases to 6.5 X of the sea-level value at 85 km a l t i t ~ d e .In ~ the latter case, the mean free path is 1.0 cm, comparable to typical dimensions which specify ** C. Park, Problems of radiative base heating. AIAA/NASA Conf. on Advance Tech. for Future Space Systems, Langley Res. Center, Hampton, Virginia, May 1979. Paper 79-09 19. “U.S. Standard Atmosphere.” US Govt. Printing Office, Washington, D.C., 1962.

9.1.

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763

vehicles, such as nose radius of curvature of vehicles used in satellite, reentry or hypersonic vehicle designs; nose radius may be of primary interest because the nose is subjected to the highest temperatures. Two principal nonideal gas-dynamic effects take place at low densities. The first of these is the so-called slip flow4 which, according to hypervelocity dimensional analysis, depends on both Mach number M and Reynolds number Re. Slip flow occurs when M2/Re > 0.01; for such conditions the velocity along a surface parallel to the flow is different from zero (the flow “slips”). In slip flow, viscosity effects are not relegated to boundary layers, but pervade the flow field. Results for drag, base pressure, heat transfer, and other practical questions applied to various basic geometrical configurations will be different, and experimental programs are needed to measure these quantities. However, design of low density wind tunnels is made difficult by the extended diffusion of viscous effects just mentioned. These effects originate at the flow boundaries, and they introduce unwanted gradients which destroy the desired uniformity of the test section flow. Thus, low density facilities tend to be sizeable with suitably large diameter test sections so as to insure a minimum size of uniform “core” flow. (See also Section 9.1.3.1 on boundary layers.) At even lower densities, slip flow gives way to so-called free molecule flow. That happens as the mean free path for molecular collisions approaches the same order and then surpasses the characteristic length associated with the body, e.g., nose radius of curvature. The ratio of mean free path to reference length is called the Knudsen number K . The latter, in turn, is again related to M and Re by5 K = 1.26y1/2M/Re

(9.1 .S)

where y is the specific heat ratio. Flow problems at Knudsen numbers > 1 overlap with questions of molecular surface physics in that problems of drag and heat transfer depend on accommodation coefficients of surfaces for molecular impacts, parameters which are, in general, not very well known. Prominent features of low density tunnels are the systems of vacuum pumps and vacuum storage chambers; and the combination of windtunnel and molecular beam techniques to produce suitable test section flows. The measuring instrumentation for such work is correspondingly E. D. Kane, G . J . Maslach, and S. A . Schaaf, Low density wind tunnels. In “High Speed Aerodynamics and Jet Propulsion” (F. Goddard, ed.), Vol. 8, Sect. I , p. 576. Princeton Univ. Press, Princeton, New Jersey, 1961. A. Pope and K . Goin, “High-speed Wind Tunnel Testing,” p. 164. Wiley, New York, 1%5.

764

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unique. Many of the developments in this research area are reported in the published proceedings of the biennial international conferences on rarefied gas

9.1.3. Low Speed Tunnels Some general features of wind tunnels and their usage were discussed in Sections 9.1.1 and 9.1.2, Referring again to Fig. 1, we note that the working section is located just downstream of a nozzle section which is designed to establish uniform flow for the tests. The figure indicates a closed-wall test section, but many tunnels are used with a so-called open-jet test section, i.e., one in which the flow-forming nozzle section terminates at an orifice; the latter exhausts the flow into the room or into a large volume test chamber containing test instruments with access to the jet flow. In well-designed tunnels, the test-section flow is uniform with virtually zero pressure gradient. The background turbulence is very low, the rms velocity fluctuation being less than 0.1 percent of the stream speed. In the case of continuous closed-loop tunnels powered by motor-driven fans, there are suitable arrangements for dissipation of the excess heat transferred to the air flow, in order to maintain a constant temperature. Further, there should be good access to the tunnel and good overall facility design which will enable performance not only of mechanical measurements, but optical and acoustical studies as required. For example, measurements of aerodynamic noise must be made in a laboratory of very low acoustic reverberation. Most low-speed tunnels have been used for investigation of aerodynamic applications of fluid dynamics. The principal similarity parameter is the Reynolds number. Measurements made on geometrically similar models with Reynolds number simulation can then be applied to the full-scale vehicle by use of a constant inverse scale factor. Where full simulation of Reynolds number is not feasible, special corrections have been developed.’ Other inherent limitations of any wind-tunnel flow relate to its finite area in ratio to the model size and to residual imperfections in the testsection flow such as unwanted pressure gradients or background turbulence; a few comments on these problems follows. The presence of tunnel walls or a free jet boundary changes the stream-

‘R . Carnpurgue, ed., Proc. h i . Symp. Rarefied Gus Dynumics, Ilrh, Comm. I L’energie Atomique, Paris, 1979. R . C . Pankhurst and D. Holder, “Wind Tunnel Technique,” Chapter 9. Pitman, London, 1952.

’

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lines in the flow around a model from their configuration in the open atmosphere. This implies a change in surface pressure. At subsonic speeds and at a small ratio of model to tunnel dimensions, the pressure change is nearly the same for all points on the body so details of the body shape are unimportant in determining any correction.* The theory for such corrections is well established. For example, the blockage interference produces a free stream velocity correction Au given by Glauert for the case of a square tunnel, as9

Au

=

0.717U~/P~A~’~,

(9.1.6)

where I/ is the free-stream velocity, T the volume of the model, = (1 M2)1’2,and A is the cross-sectional area of the tunnel. The walls also produce so-called lift interference, evidenced by a required correction A a in the angle of attack. Again, for the tunnel of square cross section, this quantity is given bye

p

-

ha = 0.137CLS/A,

(9.1.7)

where CLSis the product of lift coefficient and lifting surface (wing) area S ( C ,being defined as the ratio of lifting force to the product of the stream dynamic pressure p U 2 / 2 and the effective lifting area S.) Variations on these formulas when dealing with open rather than closed wall tunnels, or tunnels of circular cross section, are also discussed by Allen and Spiegel.8 That work further treats the more general case of slotted or porous-wall tunnels, which have special importance for the study of transonic flow. There are other corrections of interest to the wind-tunnel specialist but somewhat beyond the scope of the present discussion, except for a brief mention. They include so-called wake blockage, wall boundary-layer interference (see Section 9.1.3.1), and a series of corrections to flows around aerodynamic models associated with the effect of the tunnel flow boundaries on the model-generated vortices in their interaction with the velocity field. Of special concern also are the effects of model supports. The latter are designed in a variety of configurations, depending on the experiment. A widely used technique is that of a rear-mounted “sting” (Fig. 2). Such mounting causes minimal interference with most model measurements, but has been shown to interfere with measurements of base pressure, even for relatively small ratios of sting to model diameter. H. Allen and J . Spiegel, Wind tunnel measurements. I n “High Speed Aerodynamics and Jet Propulsion” (F. Goddard, ed.), Vol. 8, Sect. K,2, p. 648. Princeton Univ. Press, Princeton, New Jersey, 1961. H. Glauert, Wind tunnel interference on wings, bodies and airscrews. G. B. Aeronaut. Rrs. Corrnc.. Rep. Mem. No. 1566 (1933).

9.

766

t-

APPARATUS

14m

FIG.2. Model support in wind tunnel. Two of many possible positions are shown, one solid and one dashed. The sting is as narrow as rigidity permits. It is fastened to the pod which attaches to the vertical strut. The strut has a thin profile to constitute minimum blockage of the wind tunnel working section, and extends from floor to ceiling of the tunnel. Instrumentation from sensors in the model and pod is conveyed by wires and pressure tubing in the strut. [From Ref. 2, Fig. 2:24.]

The problem has been alleviated by the introduction of the magnetic suspension method which replaces solid struts or sting supports, and is now in use in several research tunnels.10

9.1.3.1. Boundary Layers. In most wind-tunnel experiments, even at relatively low speeds, the Reynolds numbers are high enough so that the vorticity or boundary layers associated with viscous or turbulent drag at walls or free jet boundaries are relatively thin and do not penetrate appreciably into the uniform test region. Typically, such layers may range from a few millimeters to several centimeters in thickness depending on Reynolds number. Even though the effect of the boundary layer on the area of the uniform core flow in the test section is not large, there are several features associated with such layers which are of importance to the experimentalist. Let us first give a few important properties of boundary layer flows. In low speed flow, the boundary layer is simply that region in which the velocity parallel to a surface shows a profile ranging from zero at the surface to essentially the free stream value. It is a result of viscous frictional drag between the moving fluid and the surface. If x = 0 at the leading edge of a thin flat plate immersed parallel to a low speed flow, the thickness 6 of the laminar layer at x = L is given by (Fig. 3) 6 = L(ReL)-1’2,

(9.1.8)

lo R. N. Zapata, Development of a superconducting electromagnetic suspension and balance system for dynamic stability studies. NASA Contract. Rep. NASA CR-132255(1973).

9.1.

767

WIND T U N N E L S A N D FREE-FLIGHT FACILITIES

Free stream flow

Boundary layer

~ _t _

-~ -

Plate surface

!

I

i

0

_

-

--

~ _ _ x_

I

L

FIG.3. Schematic diagram of boundary layer development over a flat surface immersed parallel to the flow in a uniform stream. Thickness of layer is somewhat exaggerated for illustrative purposes. There is also a boundary layer (not shown) on the lower surface of the plate.

where Re, is the Reynolds number based on the length L and free-stream values of v and I/. Since 6 / L is typically << 1 , 6 / L is an indication of the In angular spread of the layer; and we see that 6 / L varies with high speed (compressible) flow, there is also a temperature profile associated with the velocity layer whose width aT is about the same as 6. The concept of a boundary layer consisting of a succession of laminae or surfaces in relative motion is limited to moderate Reynolds numbers, say lo3 < Re,, < lo5 in the case of a flat plate. For higher Reynolds numbers, instabilities develop and transition to turbulent flow takes place in the layer. The flow looks more irregular with “granularities” of the order of 6 or smaller, but the total boundary-layer configuration is still well defined (Fig. 4). It is important for the experimentalist to predict or know what type of boundary layer will occur because several features of the two types of layers are quite different. These include layer thickness, momentum distribution, stability against pressure gradients, especially impinging shocks in the case of transonic and supersonic flows, and also transport behavior such as heat transfer coefficients. Such subjects form an important area of wind tunnel experimentation because boundary layer behavior on aircraft is a key factor in aerodynamic behavior, power requirements, and aircraft stability and control. Boundary layers will form on test models whatever their shape, as they will on any instrument probes inserted into the flow. It is also occasionally important to realize that optical studies of flow fields (see Part 2) may be affected by boundary layers on the test-section windows. The study of boundary layers by other than optical methods o r flush-wall techniques is difficult when the boundary layers are thin, because of the disturbing effects of the probes themselves; this is especially true when shock waves form around probes in a supersonic layer. We note also that in high tem-

768

9.

APPARATUS

FIG.4. Shadowgraph showing boundary layer development and transition to turbulence on a slender body of revolution at 1.2 km SS’. [Courtesy NASA-Ames Research Center.]

perature flows, chemical reactions give rise to concentration gradients and diffusion of species; and radiative behavior of boundary layer flows may become complex as evidenced by spectral variations and spatial gradients of the emissivity.

9.1.3.2. Instrumentation. Here, the reader will want to make reference to other chapters in this volume which discuss specific measurement methods in detail. For the majority of low speed tunnel studies where temperature variation and heat transfer effects are not significant, basic measurements include pressure, velocity (including velocity fluctuations in the case of turbulence research) and forces and moments on models. The information thus gathered can be used to determine drag and lift forces, stall and flutter characteristics, and other stability properties and aerodynamic coefficients. Most instruments produce electrical readouts which are recorded on oscillograms and dial gages and, more frequently today, magnetic tape recorders. Pressure and velocity measurement methods are described in Parts 5 and 1 , respectively, while forces and moments are measured with wind-tunnel balances, which generally use strain gages on beams supporting the model. Strain gages are described in Part 5 . In aerodynamic studies, both the direction and point of application of the resultant force on models are unknown. In rigid body mechanics, the

9.1.

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769

resultant force may be represented by three mutually perpendicular component forces; and the point of application determined by three moments about mutually orthogonal axes. The forces chosen both for analysis and measurement are lift, drag and side (cross wind) force; and the moments are those associated with pitch, roll and yaw. Balances for such measurements range from relatively simple single component balances to very complex six component balances with built-in compensations for model deflection, etc., in order to achieve suitable precision. A simple illustration will show why balance design is a challenging problem. Suppose, in the measurement of lift force a small deflection of the model results in a comparatively small component of the lift force projecting on the side force axis. Since the lift is usually a much larger force, even a very small deviation from real orthogonality between these axes may produce a very large percentage error in the side force measurement. A good overview of different types of balance designs along with their advantages and disadvantages can be obtained in the books by Pope and Harper” and by Gorlin and Slezinger.12 The former authors divide wind tunnel balances into two fundamental types: external balances, which carry the loads outside the tunnel before they are measured, and internal balances, which fit into the models and are arranged to send data out through electrical circuits. The latter are more recent, but the former are widely used, including six-component wire balances, as well as so-called strut-type, platform, yoke, and pyramidal balances. An example of a strut-type balance is illustrated in Figure 5 . The electronic developments in systems have helped with the automation of force and moment measurements by the use of modern data processing and readout techniques. Ever-present problems in balance design have to do with effect of model mounts on the flow field and drag on the mounts themselves. In short, balance design and calibration is one of the more sophisticated subjects in experimental aerodynamics.

9.1.3.3.Flow Visualization. Being able to see a flow pattern can be extremely informative for flow analysis even when the method used is not strictly quantitative. For compressible flows, the variation of density in the flow field makes possible several dramatic and useful refractive techniques (see Part 2). Low speed flows without special heat-transfer effects, on the other hand, are incompressible so the refractive methods do not apply. However, there are methods which are quite well suited in I ’ A. Pope and . I. Harper, “Low-Speed Wind Tunnel Testing,” Chapter 4. Wiley, New York, 1966. S. M. Gorlin and I. I. Slezinger, “Wind Tunnels and Their Instrumentation,” Chapter VI. Israel Program for Scientific Translations, S. Manson, Jerusalem, 1966.

9. APPARATUS

770

Trunnion

Front load member 1 , Windshield _C

Balance turntable

Model compensating weight

FIG.5. Simplified sketch of a strut-type balance which supports the model, provides for vatiation of angles, and transmits loads to a linkage system that separates the latter into components. [From Ref. 1 1 , Fig. 4:3.]

this velocity range. The principal techniques for flow field observation involve use of tracers, i.e., particulates introduced into the stream which can be observed by reflected or scattered light. The tracers must be small enough so that there are no inertial effects to interfere with their following the local direction of the stream. The most familiar example is the use of smoke tracers of 0.2-pm diameter. A popular method for smoke production is the vaporization of such liquids as titanium tetrachloride. The methods are discussed in Section I . 1.2.6. Introduction of the smoke into the air stream has been accomplished by a number of methods depending on whether one wishes, for example, to observe the flow downstream of a body, or in a jet or boundary layer (Fig. 6). A recent interesting application to shear layer separation and transition is described by Arena and Mueller .12a Another widely used technique is that of tufts. These are attached to model surfaces when information on such problems as flow detachment is desired. There is a distinct difference in the shape of the tufts when a smooth flow separates from a model. In another application due to Bird,I3 a tufted wire grid is used behind a model in a wind tunnel. It produces patterns which give information on such problems as roll up of vortices downstream of various airfoil configurations. lza A. Arena and T. Mueller, Laminar separation, transition, and turbulent reattachment near the leading edge of airfoils. A I A A J . 18, 747-753 (1980). l3 A. Pope and J. Harper, “Low-Speed Wind Tunnel Testing,” p. 105. Wiley, New York,

1966.

9.1.

W I N D TUNNELS A N D FR E E - FLI G H T FACILITIES

77 1

FIG.6. Smoke picture of flow around airfoil at angle of attack. [Figure kindly supplied by A. Pope.]

Another method, especially useful for studying transition from laminar to turbulent boundary layer flow along surfaces has been t o coat them with a fluorescent lacquer. The lacquer is painted on the surface in question and then illuminated with ultraviolet light. Those portions adjacent to a laminar boundary layer will show smooth luminosity, whereas in the turbulent case there will be considerably more irregularity in luminosity. 9.1.4. Transonic and Supersonic Tunnels

The term “transonic” does not have a precise definition, but signifies the Mach number range in which compressibility effects first become significant, extending beyond Mach 1.0 to where all parts of the flow field are supersonic. The transonic range is typically chosen to lie between Mach numbers 0.6 and 1.4, and is characterized by the co-existence of subsonic and supersonic regions in the flow field around models. This combination of a flow governed by an elliptic-type differential equation with that of a hyperbolic or wave-equation-type flow in the supersonic case results in features which are extremely sensitive to the presence of the tunnel walls (see remarks in Section 9.1.2.2). Without suitable modification, as discussed, the walls will interfere seriously to the extent that the flow around the model is substantially altered from that under free-flight conditions; further, control of free-stream conditions may become almost impossible. In the next section we discuss some ways in which transonic tunnel designers have coped with these problems. Tunnels operating in the Mach range 1.4 to about 5 are termed “supersonic.” A uniform flow field is obtained by allowing air o r other test gas

772

9.

APPARATUS

to exhaust from the storage chamber through a convergent -divergent nozzle (Laval-type) into a test section and thence to a diffuser. Sonic flow occurs at the throat at some distance ahead of the test section so that the special interference effects of transonic flow do not occur in supersonic flow experimentation. Careful design of the divergent nozzle contours, with corrections for boundary layers, is required to produce uniform flow in the test section. Additional design problems arise from the generation of shock waves by any bodies in the flow, such as the model and instrumentation probes, and their struts or other supports. These waves will reflect from the wall back into the test section. Downstream of the latter, supersonic diffusor design faces the problems of decreasing the flow speed and increasing the pressure in such a way that the shock configuration causes minimal dissipation. Usually, that means replacing normal shocks with oblique shocks at suitable places. The overall problem is not simple but good guidelines are available in the existing literature.14 Finally, we mention the importance of minimizing the moisture content of the air in supersonic tunnel flows because of the large temperature drop associated with the nozzle expansion. Depending on the extent of the latter, the water vapor condenses to droplets or ice pariicles and this produces an unwanted two-phase flow in the test section. The design of suitable manifold systems with compressors, air driers, including heaters and after-coolers, diffuser systems, and control valves is a technological challenge for the wind tunnel experimentalist which involves considerable fluid physics. The underlying transonic tunnel design difficulty mentioned above stems from the phenomenon known as “choking.” What happens is that the projected area of the model, even if relatively small, decreases the total flow area by producing an effective throat. When the free-stream Mach number approaches unity, a normal shock develops at the model and between the model and the walls. Further upstream pressure increase has no effect on the velocity, merely causing a small displacement of the normal shock toward the trailing edge, with some distortion of its shape partly due to the interaction of the shock with the boundary layer on the model. Reflection of the shock from the walls at near-normal incidence causes further interference with the flow field around the model. Meaningful experimentation under these conditions is not possible. Existence of choking can be determined experimentally in several ways. For example, one may note that the ratio of pressure at the test station to total (storage chamber) pressure has become equal to 0.528 (for \

l4 A . Pope and K . Coin, “High-speed Wind Tunnel Testing,” Chapter 9. Wiley, New York. 1965.

9.1

W I N D TUNNELS AND FREE-FLIGHT FACILITIES

773

air or other diatomic gas). Analytically, the tendency toward a sonic choked throat flow for high subsonic Mach numbers is indicated by the one-dimensional steady nozzle flow relation between velocity u and area A at successive positions along the nozzle:

du _ --u

1 dA M2--1A'

(9.1.9)

As M approaches 1, small fractional area decreases produce large velocity increases, leading to sonic choking; or if M is a little larger than 1, the area decrease will also decrease the velocity, again leading to sonic blockage. Of the various special designs for transonic tunnels, perhaps the most viable so far has been the tunnel with ventilated, i.e., slotted or perforated walls. This allows the removal of fluid from the stream which alleviates choking; and it also can reduce wave reflections, since the open part and closed part of the wall reflect with opposite phase, thus partially canceling each other. The ventilated tunnel can be operated continuously through the transonic range. The modification in Eq. (9.1.9)when there is a mass flow rate d m through the walls is

(9.1.10) where m is the tunnel mass flow rate. Extraction of air from the tunnel corresponds to a negative value for d m ; and it is seen that this is equivalent to an area increase. Calculations by Allen and SpiegeP provide guidelines with respect to the use of slotted or porous walls, especially as applied to blockage and lift corrections. Other guidelines for transonic tunnel design depend on the specific study program. Several problem areas are Reynolds number sensitive. Flight Reynolds numbers for many current aircraft based, say, on a characteristic wing dimension, range from 20 to 200 million. Reynolds number effects are important in drag measurements, especially skin friction and base drag. Intimately involved is the laminar-turbulent posture of the boundary layer, which is Reynolds number sensitive. The type of boundary layer also has an effect on aircraft stability, especially at high angles of attack, because the transonic shock intersecting the boundary layer on the wing has more severe consequences in the laminar case, including high probability of boundary layer separation, as well as problems with lift and pitching moment. The last-mentioned example illustrates the need for simulation of Reynolds number, but high Re tunnels in the transonic regime (say 100 million) require enormous power and are just too expensive. What the experimenter asks is whether some modification or interpretation can

774

9.

APPARATUS

permit the test to be made at lower Reynolds number but still be applicable at the higher values. In the boundary layer problem, a so-called trip is used to assure that the model boundary layer is fully turbulent as would be the case for the full Reynolds number or free-flight case. It is found that Reynolds number effects taper off above Re = 10 million for several types of problems, as well as for tests on models of aircraft components. However Reynolds number effects are pronounced in the range 3- 10 million. A concurrent problem at high Re and therefore also high tunnel power is the increase of tunnel turbulence level which is thought (in a not well-understood way) to produce effects which, mistakenly, may be attributed to Reynolds number. The continuing problem of transonic testing at high Reynolds numbers has rather recently led to the increasing popularity of a radically different type of tunnel for intermittent testing, commonly known as the Ludwieg tube.15 This is more of a shock tube than a wind tunnel; it features a relatively long high pressure supply tube which exhausts, after opening of a diaphragm or valve, into a convergent-divergent nozzle and ultimately to a dump tank. In accordance with shock-wave behavior, the flow out of the high pressure section is actuated by an expansion wave which travels upstream through the tube and then reflects from its end wall. The flow period ends when the leading edge of this wave returns to the valve-throat area. The tube can be charged to very high pressures, thus aiding in the achievement of high Re. Further, there are at the time of this writing rather large Ludwieg tube facilities in preparation in order to attain fullscale simulation of flight-vehicle Re. The quality of the test flow, e.g., free-stream turbulence, is evidently better than that of conventional wind tunnels. However, the development of nonsteady boundary layers both in the supply pipe and the expansion nozzle and test section is a problem which has required a t t e n t i ~ n . ' ~ A variation of the Ludwieg tube is the Evans clean tunnel in which the end of the high pressure tube contains a moving piston whose motion cancels the flow-starting expansion wave, and thus allows a longer run time for undisturbed flow. Another approach to high Reynolds number transonic flow is the national facility currently being constructed at the NASA Langley Laboratory in the United States. This will be a pressurized, 13 MPa (130 bars), cryogenic tunnel. Liquid nitrogen at the rate of 500 kg s-l will be poured into the air stream after the flow starts in order to achieve Reynolds numbers up to 110 million. The working section is 2.5 m by 2.5 m and the Mach number range is 0.2-1.2. This high Reynolds number correD.Russell and K.Tong, AIAA J .

11, 643 (1973).

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W I N D TUNNELS AND FREE-FLIGHT FACILITIES

775

sponds to that associated with the flight of a new large transport airplane whose design and construction has been under consideration in the United States. (The corresponding Re for a Boeing 747 is 60 million.) Ferri and SearsI6 have proposed an iterative procedure to correct for the serious wall effects in transonic flow. One sets a flow in the tunnel and measures the conditions at a “control surface” surrounding the model. Then, he computes the conditions at “infinity” based on the measured values at the control surface. One wants to adjust the tunnel wall ventilation (or shape) such that perturbations vanish at large distances. If the calculation does not yield this result, then the wall conditions must be adjusted, and the control surface conditions measured again, etc. In other words the process is iterative. Basically, it makes use of the fact that measurement of two flow-perturbation distributions, namely pressure and flow inclination, provide redundant boundary data for the exterior flow. Application shows that the method is very effective and a logical way to couple experimentation, theoretical work, and computer technology. Persons concerned with this problem may also be interested in the recent application of the fast Fourier transform technique. l7 Many of the parameters determining wind tunnel design in general apply, in particular to the supersonic tunnel. Thus, depending on the type of problem, one will choose between intermittent and continuous operation, scale of the apparatus and pressure level, both of which help to determine Reynolds number; temperature level, moisture-control features, diffuser design, and finally specific test section arrangement to accommodate model and measuring instrument configurations. Supersonic problem areas include problems of applied aerodynamics, but also another related group of more basic problems related to shock wave behavior, including shock interaction with other flow field elements such as viscous or turbulent boundary layers or jet mixing configurations. Some relations based on simplifications valid in low speed steady flow are no longer applicable. For example, the pressure p s at the surface of a thin flat plate inclined at an angle a radians with respect to a uniform flow at Mach number Mand free-stream dynamic pressure q = p u 2 / 2 is, for air,’* Mach-number dependent: W. R . Sears, Modem developments in wind tunnel testing. Isr. J . Techno/. 14, 1 (1976). l7 M. Mokry and L. Ohman, Application of the fast Fourier transform to two-dimensional wind tunnel wall interference. J . Aircrujf 17, 402-408 (1980). la A. Pope and K . Goin, “High-speed Wind Tunnel Testing,” p . 351. Wiley, New York, 1965.

9.

776

APPARATUS

( p , - p ) / q = 2 d M 2 - l)l’z,

(9.1.11)

where p is the free stream pressure. Mach number effects, especially the presence of shocks, play a large role in supersonic experimentation. The reflection of shock waves from the model by the tunnel wall back into the test region adds a new element to the wall, interference problem. Such reflections may be canceled by suitable bends or slots or other porosity in the wall. Shocks generated by strut or sting supports are a parallel problem. In a totally supersonic flow, a shock around a rear-mounted sting flare would not disturb the flow around the model since the disturbance would be limited to the so-called Mach cone downstream of the flare. However, when viscous regions are present, e.g., boundary layer and wake, they contain embedded subsonic regions, so that disturbance signals can, in fact travel up stream. Measurement of Mach number in a supersonic tunnel can be done by pressure measurements. In flow measurements faur “pressures” should be distinguished, namely (1) the stagnation or supply chamber pressure pol (2) the test-section pressure p often called static pressure, measured, for example, on the wall of the test section, (3) the dynamic pressure q defined in terms of the density p and velocity u as pu2/2, or for ideal gases in terms of p , Mach number M and specific heat ratio y as yM2p/2; and finally (4) the total head or recovery pressure p t , which is the pressure registered on a pressure gage connected to a narrow tube, called a pitot tube, which is aligned with the flow and whose opening faces the oncoming flow. See Section 1.2.2 and Chapter 5.3. For incompressible flow with Mach number negligibly small, a form of the Bernoulli equation is often valid: po = pt = p

+ q = p + ipl.42.

(9.1.12)

For compressible flow, relations among these pressures will also involve the Mach number. When no shock waves or other dissipative effects are present, p and p o are related by p/po

=

(1

+ 1 ( y - l)MZ)-Y’(Y-1);

(9.1.13)

and this is a useful relation for determination of M in the high subsonic and low supersonic range (say to M = 1.8) by simultaneous measurement of p and p o . At higher Mach numbers, it is generally more accurate to determine M by measuring p o and the pitot or total-head pressure p t , which are related by

9.1.

W I N D TUNNELS A N D FREE-FLIGHT FACILITIES

777

This relation reflects the loss in recovery pressureaaused by the bow shock generated by the pitot tube. For further details of pressure measurement and instrumentation, see Part 5. A classical method of Mach number measurement makes use of the angles of waves produced by probes or wall disturbances and made visible by shadow or schlieren methods. If the wave is weak, such as one generated by a piece of cellophane tape adhering to the wind tunnel wall, then its angle a with respect to the flow is simply relatedAo Mach number M by

M

=

csc a.

(9.1.15)

At free stream Mach numbers below 2, the detachment distance S of the normal shock in front of a blunt body in ratio to the latter’s nose radius of curvature R is a sensitive function of Mach number.19 There are, .however, practical difficulties in obtaining good precision with these optical-wave angle measurement techniques, and they are not widely used as standard procedures in testing. However, they can be most useful in special research configurations. Asymmetry in the shock wave pattern around test wedges or cones can also be used to survey flow angularity. Pressure taps onanposite sides but fairly close to the cone tip can be used to detect inclination%ofthe cane by observing the differential pressure across the probe. At low supersonic speeds cones of about 15”half angle are quite good; at higher speeds with M = 2.5 or above cone half angles approaching 45” givemore sensitivity. A good supersonic tunnel without a model in the working section exhibits flow angularity of +0.1”, or less. For studies where temperature or thermal effects are of interest, the remarks regarding Mach number in Section 9.1.2.2 are relevant, including Eq. (9.1.4) for the relation between Mach number and temperature. That relation is one for the stream temperature, which is difficult to measure directly and, in any case, is not the temperature achieved at the surface of a model or the wind-tunnel wall. Under the typical flow conditions in supersonic tunnels the viscous or turbulent boundary layers along solid surfaces produce strong temperature gradients near the surfaces. A characteristic parameter here is the Prandtl number u which is a ratio of viscous to thermal diffusivities. Thus, the wall temperature T , achieved on a thermally insulated surface immersed in the flow, expressed in terms of a recovery factor r involving the free stream static temperature T and supply-chamber temperature T o , I @ H. W. Liepmann and A . Roshko, “Essentials of Gasdynamics,” p. 101. Wiley, New York, 1950.

778

9. r = (Tw

APPARATUS

- T)/(To - T ) ,

(9.1.16)

may be determined, in the case of laminar flow, by =

ul/z.

,

(9.1.17)

and, in the case of a turbulent boundary layer, by (9.1.18) The Prandtl number depends only on material properties of the gas, namely the specific heat at constant pressure c,, the viscosity coefficient F , and the heat conduction coefficient K , as U = Cp/L/K.

(9.1.19)

For air, (+ = 0.74 so that the recovery factors are 0.86 and 0.91 for laminar and for turbulent flow, respectively. Typically, then, the surface temperatures will more closely reflect the stagnation or supply-chamber temperature than the free-stream temperature. In contrast to the side-wall recovery temperature, the “total” temperature Tt achieved at a thermally insulated surface facing the flow is more nearly equal to To (for nonviscous flow Tt = To),the difference being due to a viscous layer near the head or stagnation point of the model or probe. Typical recovery factors for determination of Tt run around 0.96 to 0.98. For noninsulated surfaces at temperatures different from the recovery values, heat transfer will take place, the driving agency being the temperature difference To - Tw. Apart from heat-transfer studies in supersonic flow, heat transfer problems arise in any case because of requirements for heating or drying the air to remove moisture or, at higher Mach numbers, say M 3 4, to avoid liquefaction or freezing of the air itself. These problems will be discussed somewhat further in connection with hypersonic tunnels. Measurement of temperature is discussed more fully in Part 4 and heat flow measurements and instrumentation in Part 7 of this volume. Meaningful measurements in supersonic tunnel flows require careful calibration to account for the effects of “parasitic” shock waves at joints in the wall and other places, turbulence level of the free stream (especially if one is looking at laminar-turbulent transition phenomena), and noise radiated from the flow system as enhanced by reverberation of the tunnel and room configuration. We note that aerodynamic noise is currently an important research problem and that some laboratories are providing special arrangements in which tunnel or jet flows are studied (1) in an anechoic chamber to obtain localized and directional analysis of the noise; and (2) in fully reverberant rooms to measure noise energy level. For ex-

9.1.

W I N D TUNNELS A N D FREE-FLIGHT FACILITIES

779

ample, a combined anechoic-reverberant aerodynamic noise facility was constructed in 1976 at the NASA Langley Laboratory, Hampton, Virginia. What corrections can or need to be made for wave interference of probes, radiated noise, background turbulence, interactions with wall boundary layers or other wall effects, sting-mounting effects, and lack of Reynolds number simulation requires, quite clearly, the judgement of the experienced investigator in connection with the problem in question. Each wind tunnel is typically optimized for the study of one class of problems. Although there are combined facilities (e .g., several tunnels using one compressed air supply) at some of the larger laboratories, the researcher in fluid physics may have to design his own small laboratory facility optimized for the problem to be investigated. 9.1.5. Free-Flight Apparatus In free-flight type studies, as the name implies, it is the model itself which is in motion relative to the atmosphere and the laboratory. One can construct a list of relative advantages and disadvantages of this type of apparatus with respect to the wind tunnel. It turns out that free-flight techniques are especially useful in making measurements on drag, lift, and instability of certain specialized shapes; and that a better simulation of the flow field is obtained without the interfering effects of boundary layers on wind-tunnel walls, reflection of waves from the walls, interference effects of supports and other unwanted gradients, and background turbulence which may occur in a wind tunnel flow. “Free flight” can mean different things. Important free-flight tests are made with sizable and suitably instrumented models dropped from aircraft at desired altitudes. Alternatively, such models may be launched from the ground with rocket-type propulsive devices. Of more interest here is the so-called ballistic range facility, a laboratory free-flight range in which a model is fired from some type of gun device into a long flight-test chamber filled with air or some other gas at a specified temperature and pressure. Records of the model behavior during flight are taken by fast-response instrumentation whose nature depends, of course, on the purpose of the test. Before the middle 1950s, major emphasis was given to aerodynamic studies of artillery projectiles and aircraft models, such as drag and stability. Since then, ballistic ranges have been adapted to hypervelocity aerophysics type studies in connection with the interest in atmospheric entry and related energy transfer processes at very high speeds. We have discussed earlier the shortcomings of wind tunnels in this connection.

9.

780

APPARATUS

Major considerations in ballistic-range technology include design of the launcher, matching of the launcher and model, ability of the model, and its instrumentation to withstand extreme acceleration (of the order of los g in some cases), model “catchers,” range pressure, and temperature control, and design of the instrumentation systems. For aerophysics applications, the desire to maximize projectile velocities (small models have been accelerated to 10 km s-l or more in some cases) has led to the development of so-called light gas guns (Fig. 7). Ignition of the powder accelerates the piston through a reservoir chamber containing a gas such as hydrogen with a high speed of sound (high energy per unit mass at a given temperature). The piston and the shock it creates heat and compress the gas in such a way that the gas pressure reaches, the value specified for the base pressure of the model as the second diaphragm or break valve opens. With proper design, the base pressure wiN remain nearly constant for the run of the model through the launch tube. Typically, breech pressures rise to 20,000 bars in order to maintain a base presswe, say, at somewhat over 1000 bars. The projectile velocity squared v 2 is proportional to the base area A, base pressure Pa, and length of launching run L, and inversely proportional to its mass m in accordance with v2 =

2ApbL/m.

(9.1.20)

Note, by the way, that the term “projectile” may include an adapter or “sabot” needed to match some models, such as nonaxisymmetric ones with wings or fins, to the launch tube. Powder chamber

High pressure

Break valve

FIG.7. Schematic drawing of two-stage light gas gun for hypervelocity ballistic range. [From Ref. 20, Fig. 3.1 *O A. C. Charters, The free flight range: A tool for research in the physics of high speed flight. Prog. Astronaut. Rocketry 7 (1962).

9.1.

W I N D TUNNELS A N D FREE-FLIGHT FACILITIES

78 1

Instrumentation for the study of aerodynamic problems will necessarily be of the fast-response type, for example, spark shadowgrams over a series of stations to measure drag, or the use of yaw cards to look at stability problems. In the latter technique, the changes in shapes of the holes made in successive yaw cards gives direct information on such problems as flutter and other oscillations. At hypervelocity conditions, radiation from the shock layer or wake is monitored by special radiation detectors with filters, designed to achieve fast spectral analysis. In the more sophisticated free-flight facilities,21 models are highly instrumented and telemetry is used to obtain information while the model is in flight. The instrumentation includes accelerometers, pressure pickups, and heat transfer gages. Telemetry techniques have been used, but for those facilities where economics and experimental convenience dictate an upper limit to the size of models for the very highest speeds (typically around 1 cm more or less), the challenge to the instrumentation engineer is clearly a severe one. In an extension of the free-flight facility developed at the NASA-Ames Research Center,22the test chamber was replaced by the working section of a wind tunnel, with the model fired upstream against the air flow. Records are then taken of the model’s flight through the working section. Typically, the tunnel can generate a flow speed of 5000 m s-l, and the tunnel flow can be so designed that the density variation is exponential; thus the model passes through an environment simulating that of the exponential atmosphere of a planet. One should add that programming the stagnation enthalpy (see Section 9.1.2.3) remains a concurrent problem. 9.1.6. Hypersonic Experimentation and Facilities

Expansion of a gas to hypersonic Mach numbers, say Mach 5 or higher, is accompanied by very large pressure and temperature drops. Apart from the additional technology required to pressurize and heat the test air or other gas beyond that accomplished in supersonic tunnels, there is a serious difficulty in achieving flow simulation of hypersonic motion in the earth’s atmosphere. The latter can be considered as a “wind-tunnel” flow with pressure and temperature at the ambient values; and therefore the equivalent values of stagnation temperature and pressure are very high indeed. That fact is reflected by the high values of recovery tempera*l P. Clemens and M. Kingery, “Development of Instrumentation for a Hypervelocity Range,” AEDC-TN-60-230. Tullahoma, Tennessee, 1960. 22 A. Seiff, A free flight wind tunnel for aerodynamic testing at hypersonic speeds. Nut/. Advis. Comm. Aeronaut., Rep. 1222 (1959).

782

9.

APPARATUS

tures and pressures (see Section 9.1.4.)achieved in free flight. Thus, for example, if we were to use the relation [see Eq. (9.1.4)] between Mach number and ideal total or stagnation temperature Tt , there would result for a reentering space vehicle at, say, Mach 20, with ambient temperature at 250 K, and y = 1.40, Tt = 20,000 K, a temperature which cannot be simulated by “ordinary” heating of the supply gas in a wind tunnel. One would, in fact, not expect to achieve this temperature because excitation and dissociation of the nitrogen and oxygen would give a much smaller value of T t , say about i that just mentioned (but still very high), due to the correspondingly smaller value of the specific heat ratio y . Such “realgas” effects are a serious consideration in hypersonic studies along with the high temperatures, and we will see that shock tubes and tunnels are able to give better thermal scaling. The same is true of arc-discharge tunnels (plasma jets) to be discussed further below. For hypersonic problems which can be separated from accurate thermal scaling of free-flight phenomena, so-called hypersonic tunnels are used which are basically similar in concept to lower speed tunnels. However, the hypersonic tunnel differs especially in requiring higher stagnation pressures and temperatures, say up to 100 bars and 2500 K, respectively. The latter requirement will give some thermal simulation at moderate hypersonic Mach numbers, but some heating is required in any case to avoid freezing of the air components. The latter problem can be avoided by the use of helium as the test gas, and several helium tunnels have been built and utilized for the study of nonthermal hypersonic fluid dynamic problems. The use of helium, whose boiling point is much lower than that of air, can be understood by noting, for example, that its stagnation temperatures equivalent to those required to prevent condensation of air at Mach 8 and 12 are only about 800 and 1400 K respectively. A good deal of heating technology now exists for hypersonic tunnels. Apart from resistance heaters using graphite or other materials, special mention should be made of the “pebble-bed” storage heaters in which l-cm-diameter pellets of refractory materials such as zirconium oxide are preheated by electric means or by gas-air combustion. Air is then passed over the pellets and attains temperatures, in the case of ZrOz, approaching 2500 K. The containing vessels must be designed both for high pressure and high temperature, so that the overall cost is high. For purposes of temperature control and protection of the pebble beds, temperatures at which a refractory changes its crystal structure should be avoided. Flow calibration in hypersonic tunnels is similar in principle to that in other tunnels, except that more attention must be paid to the very thick nozzle wall boundary layers, and to the fact that real-gas effects men-

9.1.

W I N D T U N N E L S A N D FREE-FLIGHT FACILITIES

783

tioned earlier make the test section Mach number a sensitive function of total temperature. It is, in fact, hard to eliminate temperature and other flow gradients; one has to calibrate the tunnel and live with it. A good approach to tunnel design at a particular Mach number and class of problems is to plot stagnation pressure versus stagnation temperature with lines of constant Reynolds number as parameter. Maximum temperature and pressure limits can be drawn in as well, along with the air liquefaction limit.23 Model testing at hypersonic speeds is affected by the thick boundary layers just mentioned, which in turn are especially Reynolds number dependent. Hypersonic boundary layers will, typically, be laminar at available Reynolds numbers. Transition behavior is also less predictable. Boundary layers on hypersonic rockets have evidently remained laminar at Reynolds numbers well over lo7. Thick boundary layers also lead to flow separation problems, a hard phenomenon to scale. A special complication in hypersonic flow studies, both theoretical and experimental, has to do with the shock-boundary layer interaction. At hypersonic speeds, the shock makes only a small angle with the flow, and therefore practically lies on the surface of a model afterbody, or at least touches the boundary layer if it is not immersed in it. The ability to separate the inviscid outer flow behavior from the viscous boundary layer behavior is lost and the analysis and measurements are more difficult. The problems for blunt-nosed versus slender bodies are quite different in this respect. Again, Reynolds number effects are an important consideration.

9.1.6.1. Arc-Plasma Tunnel. This device uses a high-current arc-type discharge to heat air or other test gas to very high temperatures. Many types have been including magnetically driven arcs, and vortex flows so designed that the swirling, turbulent flow of gas helps to stabilize the arc discharge. The test gas passes through the arc at as high a pressure as the discharge can be maintained (this is a problem). The discharge is maintained by thermal ionization in the arc region. Magnetohydrodynamic forces produce a hot central arc core with correspondingly high gas pressure, and the test gas is ejected into the test section by electromagnetic and thermal pressure forces. The price for the high stagnation temperature or enthalpy is a loss of flow uniformity, contamination from eroded electrodes, difficult cooling problems for nozzle walls and 2s A . Pope and K . Coin, “High-speed Wind Tunnel Testing,” p. 407. Wiley, New York, 1965. 24 W. Warren and N . Diaconis, Air arc simulation of hypersonic environments. Prog.

Astronuut. Rocketry

I (1962).

784

9.

APPARATUS

windows, and rather poorly defined flow fields in view of such phenomena as radiation losses. The latter means that there will be a negative gradient of stagnation enthalpy in the downstream direction, corresponding to temperature losses of, say, 3000 K cm-’ or more in typical facilities. Actually, plasma tunnels are not designed for general hypersonic testing, but are especially useful in obtaining heating rates of an immersed body in sensitive regions such asanearthe stagnation point. They are used especially for the testingof thermal properties of materials, most especially of ablation of surface material, in connection with the design of reentry nose cones whose cooling depends o n ablation. From a research point of view, the ionized gas flow together with its electromagnetic and radiative behavior is of special interest, and plasma jets’for this purpose have been utilized in research laboratories. The fact that arc-plasma flows are generally not in thermal equilibrium introduces an additional complication for hypervelocity testing, but an interesting opportunity for those who study rate processes in high temperature flows. 9.1.7. Low Density Wind Tunnels

This area of study overlaps somewhat with hypersonic wind-tunnel experimentation because of the large pressure drop associated with hypersonic expansion. Also, it turns out that some types.of low density tunnels use arc heaters of the type just discussed in the previous paragraph. In Section 9.1.2.3, we introduced the basic rationale and scaling parameters associated with low density flow, as well as some major features and problems of the associated design and experimentation. Design of both facilities and instrumentation is affected by the characteristically low Reynolds numbers and very thick boundary layers. The achievement of a nonviscous core is a,major task, and that is why large size test sections are desirable. Viscous losses are large, and diffuser design to achieve better pressure recovery is not a major feature. Measurements in low density flows must likewise be corrected for viscous effects. Pitot probes at .Reynolds numbers below 30 (based on tube inner diameter) require substantial correction^.^^ Attempts to calibrate the flows using sphere drag,.for example, are mace difficult than with higher Re tunnels, but some progress has been achieved. At sufficiently low densities, air leakage into the system may cause a substantial change in mass flux, and connectia of measurement systems such as pressure probes must be done with special care. The ultimate “wind tunnel” for very low density or free molecule 25

S . H.Chue, Prog. Aerosp. Sci. 16, 147 (1975).

9.2.

SHOCK TUBES AND TUNNELS

785

studies is the molecular beam. In this regime, perhaps the principal problem still remains the momentum and thermal accommodation resulting from individual molecules interacting with a solid surface. Such problems are, indeed, best studied with molecular beams, and whether the latter can be really suited for other fluid dynamic or aerodynamic studies, e.g., direct measurement of high altitude drag, awaits the development of improved instrumentation; The use of conventiand refractive optical methods for low density studies is precluded because of low sensitivity. However, the afterglow method is a good one to use over a substantial density range (see Part 2).

9.2. Shock Tubes and Tunnels 9.2.1. Basic Flow Regimes

The shock tube and its several variations constitute basically simple yet versatile devices which have had perhaps a greater influence on the development of experimental fluid dynamics during the past few decades than any other facility. Shock tubes have enjoyed wide application in studies of aerodynamics, viscous flows, nunsteady shock interactions, high enthalpy aerothermodynamic testing, chemical kinetics studies, plasma and magnetohydrodynamics investigatians, studies of two-phase flow and radiative properties of high temperatures gases and gas flows. Some of these apphcations will be discussed in the following sections. A shoek tube is basically a pipe or duct which is divided into two sections separated by a diaphragm which permits differential pressurization. When the diaphragm is suddenly removed, a shock wave, that is a near-discontinuity in pressure propagates into the low pressure section while an expansion wave travels into the high pressure chamber. These wave motions provide the mechanism for expansion of the high pressure gas into the low pressure region with subsequent compression of the latter; i.e., the high pressure gas acts as a driving piston. Figure 8 indicates these events schematically. Note the existence of a contact discontinuity which is the plane originally separating the high pressure and low pressure gases. This is a surface across which both the velocity and pressure are constant but which shows a sudden change in density, temperature and entropy. The contact surface exists because the gas originally in the driven section has been processed by a dissipative shock wave while the gas originally in the driver chamber has been processed by a nondissipative expansion wave to bring them to the same speed and pressure according to the simplified theory which neglects wall friction and heat transfer to the wall.

9.

786

APPARATUS

t

incident shock, At

Observation

station FIG.8. An x-f diagram following rupture of diaphragm separating gases in driver and driven sections, respectively. Note that a contact surface is not a pressure front but rather an entropy discontinuity which divides the gases originally located in the separate chambers. [From Ref. 26, Fig. 1.1

The wave phenomena in a constant area shock tube represent a classic example of one-dimensional nonsteady flow. Figure 8 is a so called x - r diagram which illustrates the various regimes or regions between waves or surfaces. The shock wave and expansion waves ultimately reflect from the ends of the tube, then travel toward each other, and may ultimately interact with each other and with the contact surface. By this time, the phenomena become quite complex and wall effects are beginning to predominate, so that two- and three-dimensional flows degrade the usefulness of the shock tube. On the other hand, the incident shock motion is normally extremely constant and uniform over a wide range of conditions, and the flow induced between it and the contact surface will, in turn, be highly uniform. It is even more uniform than any wind-tunnel flow. It is, in fact, that region which is used for aerodynamic and other flow studies in the shock tube. Thus the flow is generated by the shock wave itself, and, at some observation station located at xo, lasts for a time R. A. Hartunian, Shock tubes. I n "Methods of Experimental Physics" (B. Bederson and W. Fite, eds.), Vol. 7B, p. 141. Academic Press, New York, 1968.

9.2. SHOCK

TUBES A N D TUNNELS

787

shown as A t on the x - t diagram of Fig. 8. That time is clearly related to the geometry of the tube and to the strength or speed of the shock. For typical laboratory devices, it may be of the order of several hundreds of microseconds for weak to medium strength shocks, but considerably less than 100 ps for very strong shocks. Elementary shock tube theory is available in the literature.2sj26a A few relations are worth noting here for reference. First, the flow velocity u p behind the shock is related to the shock's strength or pressure ratio p 2 / p 1 by

(9.2.1) where p1 = (yl - l)/(yl + 1) and a t , y1 are the velocity of sound and the specific heat ratio, respectively, in the driven section. A further equation that is basic to the shock tube is the one given below which relates shock strength p 2 / p l and diaphragm pressure ratio p l / p 4 . Note that the ratio of the sound velocities of the different gases on each side of the diaphragm is a parameter in this equation.

The above equations follow directly from the ideal gas flow and shock relations and the requirement that there be no discontinuity of flow velocity and pressure across the contact surface. Additional equations exist to provide temperature ratios and density ratios, but it is evident that by suitable choice of the initial density or pressure p 1 and the diaphragm pressure ratio p1/p4 desired conditions such as suitable values of Reynolds numbers can be obtained in the uniform flow behind the shock. Another useful regime for certain types of studies is that behind the reflected shock. Here the gas has been brought to rest but has experienced an additional increase in pressure, density and temperature. The pressure rise across the reflected shock p s / p z is given in terms of the incident shock strength p 2 / p l by (9.2.3)

Studies involving the reflected shock region are usually concerned with *Ea W. Bleakney and R. J. Emrich, The shock tube. In "High Speed Aerodynamics and Jet Propulsion" (F. Goddard, ed.), Vol. 8, Sect. J, p. 596. Princeton Univ. Press, Princeton, New Jersey, 1961.

788

9. APPARATUS

high pressure or high temperature gases. Such states can be achieved since the gas has been processed twice by shock waves, the reflected shock wave being particularly strong if the incident shock wave has appreciable strength. 9.2.2. Production of Strong Shock Waves Equation (9.22) can be expressed in terms of a relationship between p J p 4 and shock Mach number M, = VsH/al where VSH is the speed at which the shock moves into the driven section. In the limit of very high pressure ratios p4/p1, one obtains a relationship for the maximum Mach number as follows:

(9.2.4) Evidently the velocity of sound ratio across the diaphragm is what determines the shock strength. Thus, stronger shocks are obtained if one has a low molecular weight, high temperature gas driving a higher molecular weight, cooler gas. In this connection, a good deal of driver technology has been developed. Typically, if we go from air driving air to helium driving air the Mach number is approximately doubled; for hydrogen driving air it is more than tripled; and by use of a combustion driver using a hydrogen and oxygen mixture with helium dilution it can be increased five times or more. For example, a combustion mixture at a total pressure p4 of about 7 MPa (70 bars) driving into argon gas at an ambient pressure of approximately 700 Pa produces a shock wave traveling around 5 mm/ps, corresponding to a shock Mach number M , of about 15. Strong shock technology has led to other designs, for example, that of using a shock tube with contraction at the diaphragm section, Lev,the use of a larger diameter driver section compared to that of the driven tube.2’ Another variation has to do with a multiple diaphragm driver using a buffer gas between the pressure drive chamber and the low pressure test section. With suitable design parameters, it can be shown that it requires a smaller diaphragm pressure ratio to obtain a given shock Mach number with such a device.28 In high temperature gasdynamics, it is more often the shock velocity itself or the total or stagnation enthalpy of the gas (see Section 9.1.2.3) which is of special importance. Figure 9 shows values of total temperature versus shock speed, i.e., the values which would be attained at the

’’R. A. Alpher and D. R. White, J . Nuid Mech. 3, 457 (1958). *‘J. N . Bradley, “Shock Waves in Chemistry and Physics,” Chapter 4. Wiley, New York. 1962.

9.2.

789

SHOCK TUBES A N D TU N N ELS

44

40

kPa

/ 5 3 Initial pressure in air ahead of shock, 101 kP

36

32

28

Tt

TI

24

20

16

12

8

4 I

,

,

I

l

l

,

,

1

1

1

1

1

1

1

I

I

0

1

,

MS

FIG.9. Real air stagnation temperature versus shock Mach number M, for a shockgenerated flow. Note that temperature is given in ratio to ambient value [From Ref. 29, p. 90.1

stagnation point of a model or probe immersed in the shock generated flow. These magnitudes are comparable although somewhat smaller than those achieved behind a reflected shock wave. It is evident that very large values of Tt are obtainable compared to typical heated wind-tunnel flows. Interest in obtaining high temperature conditions by shock heating AVCO Re*8 S. Feldman, "Hypersonic Gas Dynamic Charts for Equilibrium Air." search Laboratory, Everett, Massachusetts, 1957.

790

9.

APPARATUS

developed also in connection with the growing interest in plasma physics. It was noticed by Fowler and later others that the behavior of pulsed electric discharge tubes under some conditions resembled that of a type of shock tube and that such shocks may be very strong indeed. In the last two decades, much investigation has been made of the use of electromagnetic drive techniques for ionized gases to produce extremely strong shock waves and extremely high temperatures. The T-tube used by Kolb and others30 is a well-known example of a device where magnetohydrodynamic forces help to accelerate a shock (see Fig. 10). Gross and his group at Columbia University have used electromagnetic drive and acceleration of shocks to produce extremely high temperatures that are of interest in connection with the attainment of thermonuclear f ~ s i o n . ~ ’

Electrode Bockstr’OP

W

\

G

FIG.10. Schematic of a T-shaped electromagnetic shock tube. The switch is opened after capacitor C is fully charged and then spark gap G is triggered (for example by focusing light from a pulsed ruby laser) to deliver most of the capacitor energy to the low pressure gas in the glass T-tube. The return current in the backstrap exerts a Lorentz force to accelerate the plasma along the tube toward wall W. 3o A. Cloupeau, Phys. Fluids 6, 679 (1963); R. H. Lovberg, Plasma problems in electrical propulsion. I n “Methods of Experimental Physics” (R.H. Lovberg and H. Griem, eds.), Vol. 9B, p. 251. Academic Press, New York, 1971. 31 R. A. Gross and B. Miller, Plasma heating by strong shock waves. I n “Methods of Experimental Physics” (R. H. Lovberg and H. R. Griem, eds.), Vol. 9A, p. 169. Academic Press, New York, 1970.

9.2.

SHOCK TUBES A N D TUNNELS

79 1

9.2.3. Aero- and Thermodynamic Testing Apparatus

In the 1950s, shock tube use expanded from the research laboratory into the applications area in connection with the growing interest in the effect of excessive heat transfer on the thermal-structural integrity of the surfaces of vehicles entering planetary atmospheres at high speeds-the “reentry problem.” The difficulty is most severe in the stagnation region near the nose of the vehicle where shock layer temperatures (i.e., temperature between the shock wave and the surface) may approach 10,000 K or higher depending on the reentry speed and vehicle shape. For example, the Apollo reentry capsules entered the earth’s atmosphere at somewhat over 10 km s-l, sufficient to produce temperatures in the general range just mentioned. It turns out that scaling of the stagnation point heat transfer problem requires, apart from well established geometrical factors, simulation of stagnation enthalpy and density. This cannot be done with conventional wind tunnels, as the thermal problems are too severe. However, the shock tube is able to simulate such conditions (Fig. 9), and there is now a good deal of published literature representing the results of such measurements at many l a b o r a t o r i e ~ . It ~ ~has been shown that stagnation point heat transfer is not sensitive to the Mach number of the incoming flow, for this is what the shock tube itself does not simulate. Flows approaching a model set in a shock tube have been generated by the initial shock wave and are thus at a high temperature; thus the flow Mach numbers tend to be rather low. For simulation of flow fields and aerothermodynamic behavior of afterbody flows and wake flows, better simulation of free-stream Mach number is more essential, and was a contributing motivation to the design of so-called shock tunnels. The shock tunnel makes use of the shock tube’s capability of compressing and heating a gas but then includes some sort of nozzle expansion arrangement to permit a more realistic wind tunnel flow. By now, several designs of shock tunnels have been used successfully. For conceptual purposes, the so-called reflected method of shock tunnel drive is simplest. Here, a suitably small hole is made in the end plate of the shock tube against which the shock reflects and a nozzle-diffuser combination is attached. What happens essentially is that the high temperature and high pressure gas in region 5 of the shock tube becomes the stagnation gas which is then expanded to suitably high Mach numbers in the shock tunnel itself. Models inserted into the tunnel test section experience the high stagnation enthalpies but they are also im32

P. Rose and W.Stark, J .

Aerosp. Sci. 25,

86 (1958).

792

9.

APPARATUS

mersed in the flow which more nearly simulates the hypersonic conditions of reentry vehicles (see Fig. 11). A few words are in order about experimental methods in connection with which the reader will want to look at Part 7 and Section 4.2.1 of this volume. It is clear that all measurement methods in shock tubes must use fast-response equipment, often in the microsecond or submicrosecond range. Measurements central to the heat transfer problems discussed in this section make use of heat flow gages suitably placed on models or in some cases on the wall of the shock tube. At speeds comparable to the Apollo reentry speed mentioned above, radiation from the excited gases in the shock layer plays an important role in the total heat transfer rate. Thus, the heat flow gage measurements have been supplemented by spectroscopic studies of the radiation. Fast response spectroscopic techniques for such measurements in shock tubes have been developed in several laboratories. The radiative measurements have given interesting results on radiative species and other physical phenomena in the high speed reactive flows around models of reentry bodies."j 9.2.4. Studies of Chemical Kinetics

The importance of gas kinetics studies as a principal application of the shock tube stems from two sources: (1) the unique capabilities of the device itself for controlled studies of chemical reactions, and (2) the role played by rate processes in high temperature fluid dynamics research and technology. The capacity for depositing a known amount of energy into a specified gas or gas mixture at a well defined instant with suitable geometrical homogeneity is indeed unique. Of recent interest is the parallel capacity, with suitable modification of the device, to generate definable energy transfer processes during rapid expansion of shock heated gases with deviations from thermal equilibrium. The last remark refers to the gas dynamic laser and its variations. Some of the gas kinetics applications of shock tubes are described by Hartunian in Part 7 of Volume 7B of this treatise. Continued strong interest in chemical rate processes is also indicated by the development of powerful auxiliary measurement techniques, especially those associated with laser devices. As illustrations of the many types of rate processes which have been studied in shock tubes, we will make brief reference to the problems of molecular vibrational relaxation, and ionization rate processes in shock heated gases. The inefficiency of vibrational equilibration when energy is suddenly deposited into a gas (at room temperature and pressure, 80,000 33

C . H . Tremor, Radiation at hypersonic speeds.

Prog. A ~ r o ~ 7 t r uRoc!trtry l. 7 11962).

FIG.11. Sketch of Cornell Aeronautical Laboratory (now Calspan Corporation)hypersonic shock tunnel. Use of double wedge type nozzles gives hypersonic Mach number flow while eliminating most of boundary layer. [Courtesy Calspan Corporation. From Ref. 1, Fig. H,156.]

794

9. APPARATUS

collisions between molecules are required per excitation in C O z ) gives rise to a so-called relaxation zone behind the initial shock wave.34 Not only the total length of the relaxation zone but the distribution of physical variables in this zone have been measured by such techniques as time resolved interferometry, and spectroscopic emission and absorption. A technique used recently utilizes three synchronized time histories of both absorption of light from a tunable CO, laser and infrared emissions of two of the characteristic transitions at 15 and 4.3 j~~~ Experiments of this sort yield more specific information on rate constants for intramolecular energy transfer (V-V collisions) as distinguished from the rate process for initial excitation (T-V collisions). Such experiments represent a refinement of earlier work which originally measured only the total relaxation time for all processes. Optical techniques have also been utilized for study of ionization rates, especially in rare gases such as argon. For this purpose it has been possible to utilize the method of two-wavelength i n t e r f e r ~ m e t r ywhich ~~ makes possible the simultaneous determination of heavy particle density and electron density. By use of a beam splitter and two narrow band filters together with a rotating camera streak interferometry technique, one can get fringe shifts as a function of time which result in the determination of the ionization profile behind a shock wave with resolution of appreciably less than 1 ps. Figure 12 shows a typical interferogram of this type. In this way, investigators3’ were able to distinguish the location of maximum ionization as the equilibrium point, and to show that the onset of radiation, which gives a different character to the fringes, takes place earlier and is most likely the onset of recombination. Further, the sloping of fringes downstream of equilibrium indicated that radiation cooling can be measured by these techniques. The studies mentioned above represent only a few examples of sophisticated techniques for measurement of kinetic processes. Others have included such methods as time-resolved microwave interferometry at lower electron densities, pickup coils sensitive to the electrical conductivity changes in the ionizing or recombining gas, and electron beam excitation techniques which generate certain characteristic radiations which are 3 ( W. Bleakney and R. J. Emrich, The shock tube. i n “High Speed Aerodynamics and Jet Propulsion” (F. Goddard, ed.), Vol. 8, Sect. J. Princeton Univ. Press, Princeton, New Jersey, 1961. D. Bershader, Progress in shock tube studies of gas kinetics. I n “Modern Developments in Shock Tube Research” (G. Kamimoto, ed.). Shock Tube Research Society of Japan, Kyoto, 1975. R. A. Alpher and D. R. White, Phys. Fluids 2, 162 (1959). 37 H. Wong and D. Bershader, J . Fluid M e c h . 26, 459 (1966).

9.2.

SHOCK TUBES AND TUNNELS

795

FIG.12. Rotating mirror two-wavelength interferogram showing ionization relaxation and the onset of radiation behind a shock wave at Mach 15.4 advancing into argon at initial pressure 670 Pa. [From Ref. 37, Plate 1.1

studied as a function of time. See Chapter 3.3 and Section 4.2.3 of this volume for a description of electron beam excited radiation methods.

9.2.5.Further Uses of Shock Tubes Shock tube apparatus has been developed for other types of studies such as two-phase flows, an example of which is the interaction of a shock wave with water droplets, another example being the acceleration of dusty gases. In the latter case, studies have been made to look at the “sand blasting” effect on surfaces when a shock heated gas containing small particles impinges on it under a specified set of conditions. Another continuing application of shock tubes for which special apparatus will be further developed has to do with energy transfer problems. Brief mention was made earlier to the use of electromagnetic drive in shock tubes. By now considerable work has also been done on the interaction of shock heated ionized gases with magnetic fields, a phenomenon which has direct relation to magnetohydrodynamic energy conversion. In some experiments energy has actually been fed to an outside circuit by MHD interactions. In the work of L o u b ~ k y , an ~ * ionized shock front was made to impinge on the magnetic field produced by a suddenly 38 W. Loubsky et d., Penetration of a magnetic field barrier by an ionizing shock wave. In “Recent Developments in Shock Tube Research” (D. Bershader and W. C. Griffith, eds.), p. 568. Stanford Univ. Press, Stanford, California, 1973.

796

9.

APPARATUS

energized Helmholtz coil surrounding a fiberglass shock tube section. The reflected and transmitted waves were studied as were the properties of the flow itself. The understanding and development of the gasdynamic laser has been aided by modifications and innovations in shock tube apparatus dealing with the development of more sophisticated optical cavities, gasprocessing nozzles which optimize turbulent mixing, and initial energizing techniques using combined electric discharge and s h o ~ k - h e a t i n g . ~ ~ Future developments in shock tube apparatus may include, in the author’s opinion, the use of shock tubes for the study of intense sounds. Intense sounds are really weak waves by shock tube standards but problems of dispersion, absorption coefficients and other properties of such sounds are not sufficiently well known; and it would appear that a shock tube of suitable size with suitable materials for the walls and end plate could well be designed to work in conjunction with other acoustical apparatus to study current problems relating to noise pollution and better acoustic design of buildings, vehicles and acoustic barriers at highways and airports. One of the early applications of shock tubes, namely, that dealing with personnel safety in coal mines and other underground passages and configurations continues to be of interest. Problems remain requiring further study of shock reflection, refraction and interactions with all sorts of geometrical configurations, as well as the initiation and propagation of coal dust explosions by shock waves in these passages. Finally, an application that will likely see further development in connection with energy and chemical studies is the wave machine. The latter refers to a configuration of several tubes in which energy or pressure is transferred by nonsteady wave processes. Depending on the application, such devices may take the form of superchargers, wave reactors, or superheaters. 40

9.3.Low Reynolds Number Flows In this section we look more closely at experimental methods of measuring flows in which viscous forces predominate over inertial forces, i.e., flows at low Reynolds numbers [see Eq. (9.1.1)]. Examples would be

’* P. H. Rose, Advanced laser technology development in shock tubes. In “Shock Tube and Shock Wave Research” ( 8 . Ahlborn, A. Hertzberg, and D. A. Russell, eds.), p. 508. University of Washington, Seattle, 1978; R. I . Soloukhin, Shock tubes in flow laser research; modeling and applications (ihid., p. 629). P. H. Rose, Potential applications of wave machinery to energy and chemical processes, in “Shock Tubes and Waves” (A. Lifshitz and J . Rom, eds.) pp. 3-30. Magnes Press, Jerusalem, 1980.

9.3.

LOW REYNOLDS NUMBER FLOWS

797

blood flow in capillaries and the flow of glycerine through a medium-small pipe. Such flows are to be distinguished from the higher-speed flows of viscous fluids discussed earlier in which viscosity is negligible in large regions of the field (case of free-stream or jet-core flows); or comparable with inertial effects in other regions (boundary layer or free shear layer flows). Basically, viscosity represents a rubbing-type transfer of momentum in a fluid when velocity gradients are present. Viscous stresses usually appear in the dynamical equations as proportional to the rates of fluid strain. For gases, water and many other liquids, this relation is valid and such fluids are called linear or Newtonian. Thus, in the simple case of a thin flat plate oriented perpendicular to the y axis and immersed in a steady flow in the x direction, the viscous stress T (i.e., tangential force acting on a unit area of the fluid in the x-z plane) at a distance y from the plate surface is given by

(9.3.1) where u is the x component of velocity, and p is the coefficient of viscosity: T and u are functions of y. For y = 0, the above relation gives the surface stress acting between the fluid and solid surface. It serves, as we shall see, as the basis for a method of viscosity measurement. Viscosity is a diffusive type of transport phenomenon, which explains why the fluid at a distance y from the wall “feels” the effect of viscous friction acting at y = 0. Analysis shows that the effective diffusion coefficient v , called the kinematic viscosity, is given by p / p , where p is the fluid density. Values of p and Y for water and air at 20°C and atmospheric pressure are given in the accompanying table. Water

Air

Units“

‘The unit Pa (pascal) is the name for newton per square meter. As a final introductory remark, we point out that the coefficient of viscosity p , while independent of the rate of strain, is quite temperaturedependent. Viscosities of gases increase with temperature while those of liquids generally decrease with temperature. The more exact kinetic theory formulations of temperature-viscosity relations are complicated, but satisfactory semiempirical relations have been developed. For further details the reader is referred to a suitable handbook.*I D. E. Gray, ed., “American Institute of Physics Handbook,” 3rd ed. pp. 2-187 to2-201, pp. 2-232 to 2-248. McGraw-Hill, New York, 1972.

798

9.

APPARATUS

9.3.1. Features of Highly Viscous Flows

In the case mentioned of glycerine flowing through a pipe, the steady motion involves the opposition of the driving pressure differential along the pipe by the viscous frictional resistance of the pipe wall. The result is that the volume flow per unit area per second or average fluid velocity ij. is proportional to the axial pressure gradient; this is the well-known Hagen-Poiseuille law which for a tube of radius a reads (9.3.2) where the minus sign indicates that the flow is in the direction of the negative pressure gradient. Such behavior is typical of “internal” flows such as flows in ducts, tubes, and channels. Similar considerations apply to important applications such as lubrication mechanics, and the percolation of fluids through beds of small particles such as sand or gravel. Another class of flows, termed “external,” deals with bodies immersed in a flow field far from the latter’s own boundaries. Highly viscous flows around such bodies as spheres and cylinders are well-understood problems. For example, the Stokes formula for the viscous drag force on a sphere of radius R, Eq. (9.1.2)shows that such drag force is directly proportional to the velocity u: F = 67rpRv. This formula is used in the oil-drop experiment to determine the charge on the electron (see next section). 9.3.2. Measurements with Viscous Flows

Viscous flow behavior has been utilized to measure both fluid physical properties and flow-field details. We mention first the use of viscous flow between closely spaced parallel glass plates to study, curiously enough, streamline or potential flow of a nonviscous (ideal) fluid in two dimensions. The device is known as the H d e - S h a w apparatus.42 One inserts some model of interest, e.g., a cylinder, with axis perpendicular to the plates and whose length spans the distance between them. The obstacle’s dimension, say the radius of the cylinder, should be appreciably larger than the plate separation. In that case, one can set the z-averaged values of u and u , the x- and y-velocity components, proportional to the corresponding components of the pressure gradient. This result corresponds to a similar proportionality in the Hagen-Poiseuille Law. Thus, we write G. K. Batchelor, “Introduction to Fluid Dynamics,” p. 222. Cambridge Univ. Press, London and New York, 1967.

9.3. u

LOW REYNOLDS NUMBER FLOWS = -C

ap/ax;

v

= -C

splay.

799

(9.3.3)

Substitution of these relations into the equation of continuity for twodimensional steady incompressible flow, a u / d x + au/dy = 0 , yields Laplace’s equation for the pressure: a2p/ax2 + d2p/ay2 = 0. The same relation holds for the velocity potential, which means that the flow is irrotational. The flow satisfies the condition of zero normal velocity at any boundary in the x-y plane, but it differs from a true irrotational flow in that here there is also zero parallel velocity at such boundaries. In any case, the Hele-Shaw apparatus is most useful in visualizing irrotational flows around two-dimensional bodies of arbitrary shape. Injection of dye at appropriate spacing into the flow gives a very fine representation of the stream lines. In such experiments the velocity should be kept quite low, and the curvature of the model should be small enough to avoid appreciable accelerations with possible separation of the flow from the body. For measurements of dynamic viscosity of linear (Newtonian) fluids, the stress is measured by determining the torque needed to maintain a constant rotation rate of an outer cylinder concentric with an inner cylinder at rest. This is called a Couette viscometer, and end corrections are of great i m p ~ r t a n c e . Measurements ~~ can also be made by use of Eq. (9.3.2) and finding the pressure difference required to maintain a steady flow in a circular cross section capillary tube; again a great distance from each end of the capillary must be provided for and end corrections made before the Hagen-Poisseuille formula may be used to calculate the viscosity ~ o e f f i c i e n t .Use ~ ~ of Stokes’s law for slow falling of a sphere in a viscous fluid (Eq. (9.1.2))cannot be used for absolute m e a s ~ r e r n e n t bes~~ cause the effects of walls at surprisingly large distances from the falling sphere destroy the validity of Stokes’s law. For example, the falling rate of a sphere whose radius is 10 percent of the containing cylinder radius is 25 percent in error. Timing the fall of a solid sphere in a glass cylinder filled with fluid is so easily accomplished, however, that empirical relations established by using fluids of known viscosity are used a great deal for specifying properties of oils and solutions for industrial and consumer use. Another method of precise measurement applied to gases is to observe the damping rate of an oscillating horizontal circular disk suspended in the fluid.46 The apparatus constant is determined by using gases whose viscosity has been accurately determined by absolute means. J. A . Bearden, f h y s . Rev. 56, 1023 (1939). J . F. Swindells, J . R. Coe, and T. B . Godfrey, J . Re.?. Nut/. Bur. Stand. 48, 1 (1952). lS J. Happel and H . Brenner, “Low Reynolds Number Hydrodynamics,” p. 320. 44

Prentice-Hall, Englewood Cliffs, New Jersey, 1965. Is J . Kestin and W. Leidenfrost, Physiru (Utrechr) 25, 1033 (1959).

800

9.

APPARATUS

Finally, a further remark about external viscous flow measurements with spheres. We mentioned earlier the Millikan oil drop experiment. In it the velocity of a charged oil drop moving with uniform speed as a result of the balance of viscous, electric, gravitational and buoyant forces can be observed at various values of the applied electric field and the electronic charge deduced therefrom.47 From a more general fluid mechanics point of view, drag measurement on test spheres immersed in laboratory flows has served to reveal the Reynolds number range of the limiting cases of purely viscous flow at very low values of Re, and nonviscous free-stream flow with a thin turbulent boundary layer adjacent to the body at very large values of Re. By elementary application of Newton’s second law for a moving sphere transferring momentum to its surroundings by “pushing” the fluid in front of it, we obtain for the drag force FD, leaving out viscous considerations, (9.3.4)

where A is the cross-sectional area, and CDa factor of proportionality termed the drag coefficient. Now, dimensional analysis (see Part 10) indicates that CD = f(l/Re). If this function is expressed as a power series in 1/Re which (1) approaches unity ]/Re -+ 0 (thus giving the observed v 2 law for drag force at large Re) and (2) approaches 1/Re for very small values of Re, substitution into Eq. (9.3.4) yields Stokes law, Eq. (9.1.2), except for adjustment of the constant whose value is not given by Eq. (9.3.4). At intermediate values of Re, the CDversus Re relation is more complicated because both inertial and viscous effects are playing a role. A plot of CDversus Re reveals48not only the limiting behavior at high and at low Re, but many features in between. The latter include the onset of so-called Oseen flow which is the first step away from the purely viscous case: separation of the boundary layer with an unsteady vortex formation; boundary layer transition to turbulence with partial reattachment of the layer; and changes in the boundary layer profiles and base flows-all these are revealed by a study in the wind tunnel and in free flight of the drag of spheres where viscosity, pressure and inertia play changing roles as Re changes. 9.3.3. Measurements with Non-Newtonian Fluids

Many important fluids do not behave as described by the linear or Newtonian relationship between shear stress and rate of strain, Eq. (9.3.1).

‘’ R. A . Millikan, ”Electrons ( + and -), Protons, Photons, Neutrons and Cosmic Rays,” Univ. of Chicago Press, Chicago, Illinois, 1936. *‘D. E. Gray. ed., “American Institute of Physics Handbook,” 3rd ed., p. 2-268. McGraw-Hill, New York. 1972.

9.4.

ROTATING GEOPHYSICAL F L U I D DYNAMIC STUDIES

801

Examples are blood, motor oil additive and other polymers, glues, bread dough, paper pulp and, presumably, the earth. Such fluids are usually referred to as rheological fluids as distinguished from viscous fluids, which unfortunately violates the etymology of both words. For such fluids, normal stresses (pressures) are generated by shearing and more than one coefficient is required to describe their mechanical behavior. Apparatus to study flow of such fluids includes rotation of a vertical cylinder dipping into a pool of liquid which climbs up the rod in a steady flow pattern,49capillary tubing attached to a plunger which forces fluid out the open end so that the die-swell-effect on emerging fluid can be measured, a metal cone steadily rotating about its symmetry axis perpendicular to and touching (nearly) a flat plate with a sticky fluid between with measurement of total force and torque on the cone or plate and pressure measurements with small gages flush with the flat plate, rotating flat plates with the axes of rotation normal to the plates and parallel but displaced with sticky fluid between and pressure gages flush with one of the plates, and finally several oscillatory methods using equipment as described for steady motion but with one element driven in sinusoidal rotation and recording pressures or total force and torque on the other element. The forces, torques, and pressure distributions on rotating concentric cylinders with the fluid under study filling the annulus are sometimes measured, both for steady and oscillatory drive. Further information and theoretical treatment of the flow in these instruments is given in a vast literature, from which we mention only a book by Walter~.~OIt is worth noting that substantial errors in interpretation are present if holes and manometers are used to try to obtain the pressure distributions in these studies instead of diaphragm gages imbedded in the wall. The stresses in a rheologically complex fluid depend on the history of the deformation. Characterization of such a fluid’s constitutive constants is in general limited to a certain type of motion which may be produced in one apparatus, and little or no information on the same fluid’s behavior in another type of apparatus may be inferred.

9.4. Apparatus for Rotating Geophysical Fluid Dynamic Studies* Laboratory studies of the flow in rotating systems had an early but sporadic start. Many of these early studies have been reviewed by Fultz’ @

G . S. Beavers and C . D. Joseph, J . Fluid Much. 69, 475 (1975). K . Walters, “Rheometry.” Chapman & Hall, London, 1975.

* Chapter 9.4 is by Alan J.

Faller.

802

9.

APPARATUS

and will not be described in detail here. The advent of recent serious and continued studies of the flow in rotating systems began about 1950 with attempts to simulate several complex geophysical circulations. At the University of Chicago, Fultz* began a series of experiments emphasizing at first the rather direct modeling of various aspects of the atmospheric circulation; at the Woods Hole Oceanographic Institution von Arx3 developed a laboratory experiment to simulate the wind-driven ocean circulation; and at Kings College, Newcastle-upon-Tyne, Hide4 began the first “annulus” experiments in which fluid between concentric rotating cylihdrical walls was heated by the outer wall and cooled by the inner wall. This fundamental experiment was initially directed toward an understanding of possible thermally driven circulations in the core of the earth. 9.4.1. Flow in Rotating Systems

The flow relative to a rotating coordinate system often requires a different “intuition” than that arising from familiarity with fluid phenomena in nonrotating (inertial) systems. New types of viscous boundary layers occur, new wave motions and free oscillations are permitted, and the flow patterns even of slow, steady, inviscid flows may at times seem bizarre and counter-intuitive. To facilitate the later discussion we present here a short summary of some of the simpler but unique characteristics of flows in rotating systems. A more rigorous and complete development of this subject has been given by G r e e n ~ p a n . ~ The equation of motion relative to a steadily rotating coordinate system with angular velocity a, assumed to be collinear with the force per unit mass of gravity g, may be written

_ Dv_ -a’ + ( v - v ) v = Dt

at

-2fi x v -

axax

r - -1v p P

+ g + -P1 V

*

p

VV, (9.4.1)

D. Fultz, A survey of certain thermally and mechanically driven systems of meteorological interest. In “Fluid Models in Geophysics” (R. R. Long, ed.), Proc. 1st Symp. Use Models Geophys. Fluid Dyn. John Hopkins University, Baltimore, Maryland, 1953. D. Fultz, A preliminary report on experiments with thermally produced lateral mixing in a rotating hemispherical shell of liquid. J . Mereorul. 6, 17-33 (1949). W. S. von Arx, A laboratory study of the wind-driven ocean circulation. Tellus 4, 311-318 (1952). ‘R. Hide, Some experiments on thermal convection in a rotating liquid. Q. J . R . Mereorol. Sor. 79, 161 (1953). ’ H. P. Greenspan, “The Theory of Rotating Fluids.” Cambridge Univ. Press, London and New York, 1968.

9.4.

ROTATING GEOPHYSICAL FLUID DYNAMIC STUDIES

803

where r is a position vector from the axis of rotation, and in the viscous terms it has been assumed that V v = 0. The apparent accelerations introduced by the use of a rotating reference frame have been placed on the right-hand side of Eq. (9.4.1) and hence may be referred to as forces per unit mass. These are the Coriolis force, -2R x v, and the centrifugal force, - R x R X r. Because the gravity and centrifugal effects can each be expressed as the gradient of a potential, as g = -V4g and - R x R x r = V$Qzr2 = -V&, it is often convenient to combine X r = -V4, where 4 = & + &. these terms in the form g - R X Surfaces of constant 4 are then paraboloids of revolution. In the case of solid rotation v = 0, and with spatially constant atmospheric pressure the free surface of a rotating fluid is a surface of constant 4. It is pertinent to note that introduction of the Coriolis force frequently simplifies the dynamics because the new term is linear in v. When 2 R x v is large compared to the nonlinear term (v V)v, Eq. (9.4.1) becomes essentially linear in v, as is generally the case for large-scale geophysical flows. When the new linear term dominates, a whole new class of linear solutions to the equations can be found. Moreover, as long as (v V)v remains relatively small, the nonlinear effects can be treated by perturbation theory and many new stability problems arise that can be treated by linearization.

-

-

9.4.1 .I. The Taylor-Proudman Theorem and Geostrophic Flow. For relatively slow, steady, inviscid flow without strong gradients, Eq. (9.4.1) simplifies to 1 (9.4.2) 0--2Rxv--Vp-V+. P For a fluid of constant density p the curl of Eq. (9.4.2) leads to the simple vorticity equation

-

(2R V)v = 0.

(9.4.3)

This result indicates that under the prescribed conditions, v is independent of the direction parallel to 42. Taking the z-axis parallel to Q so that 0 = kQ, Eq. (9.4.3) reduces to av/az = 0. This result is known as the Taylor-Proudman theorem. The horizontal components of Eq. (9.4.2),in the plane perpendicular to 0 and g, may be written vH =-k 1 2%

X

VHpl

v,.

(9.4.4)

This is the geostrophic relation. It describes a flow, familiar in meteorology, with the wind direction parallel to the isobars on a weather map

804

9.

APPARATUS

and the speed proportional to VHp'. The left-hand side of Eq. (9.4.4) points out that a real flow is only approximately geostrophic. The right-hand side defines v,, the geostrophic velocity, as a substitute for the pressure gradient. The vertical component of Eq. (9.4.2) is the hydrostatic relation (9.4.5)

a p / a z = -pg.

Thus the Taylor-Proudman result specifies hydrostatic, geostrophic flow. It also demands the more severe constraint a v / d z = 0. We now consider various simple departures from the Taylor-Proudman result that arise when other terms from Eq. (9.4.2)are included in the problem. We will consider, in order, some of the effects of density variations, the nonlinear terms, time dependence, and viscous effects. 9.4.1.2. The Thermal Wind Equation. With density variations the flow can be geostrophic and hydrostatic but still have a v / a z # 0. Taking d/dz of Eq. (9.4.4)and eliminating p by applying VHto Eq. (9.4.5)one can obtain what is known in geophysical fluid dynamics as the thermal wind relation, namely,

(negligible terms in a p / a z ) .

(9.4.6)

Invoking a linear relation between p and T with a thermal expansion coefficient a,we may write Eq. (9.4.6) (9.4.7) which shows a similar relation between av,/dz v, and VHp' in Eq. (9.4.5).

and VHT as between

9.4.1.3. Flow with Variable Depth. Departures from av/az = 0 may occur with a uniform fluid when the nonlinear terms become important. In particular, we consider the case of a flow in a container of spatially variable depth h ( x , y ) = Z z ( x ,y ) - Z l ( x , y ) where Z z and 2, are the upper and lower boundaries. In the limit of the Taylor-Proudman theorem the condition av/dz = 0 requires that columns of fluid move only along contours of constant h . When the nonlinear (inertial) terms begin to become significant the condition avH/dz = 0 is still valid but aw/az = 0 (w = k * v) no longer applies. The fluid is then not confined to move along contours of h and, as it moves, a w / a z = h-'(vH V h ) . This motion can occur for steady flow, a v / a t = 0, and is one of the first important inertial effects expected from the nonlinear terms other than the possibility of turbulence.

9.4.

ROTATING GEOPHYSICAL F L U I D D Y N A M I C S T U D I E S

805

9.4.1.4. Inertial Oscillations. For time-dependent flow we first consider the horizontal balance Dv - + 2n x v = 0, (9.4.8) Dt with parcels of fluid confined to equipotential surfaces. Equation (9.4.8) describes what is called “inertial” motion, but this is not true inertial motion because of the stated vertical constraint. The motion of parcels consists of circular trajectories with radii r = (v,(/R and with periods T = v/a. But in a fluid not all particles on the same equipotential surface can exercise the same oscillation because of geometrical constraints. If oscillations exist, they must build up pressure gradients and Eq. (9.4.8) is no longer applicable. Adding the pressure gradient force to Eq. (9.4.8)and taking the curl of the resultant equation (with constant p ) , we obtain at

+ (2n’ V)v

f

0,

(9.4.9)

where 5 = V x v. This modification of the Taylor-Proudman theorem together with the continuity equation and appropriate boundary conditions admits a very wide class of free oscillations. In systems with variable h or with other asymmetrical effects such as the curvature of the earth, the oscillations may take on the character of progressive Rossby waves. The variety of possible waves and oscillations introduced by the effects of rotation far exceeds the scope of this necessarily brief account. 9.4.1.5. The Ekman Boundary Layer. With geostrophic flow above (or below) a boundary at which a condition on v or dv/az is applied, an Ekman boundary layer will be found. To Eq. (9.4.2)we add the viscous term and substitute the definition of v, from Eq. (9.4.4). The horizontal component of the resultant equation may be written

0

-2n x

(VH

- vg)

+ vv2vH.

(9.4.10)

Under the assumptions that Vkv, << a2vH/az2 and that dv,/az = 0 in the boundary layer, Eq. (9.4.10) reduces to a second-order ordinary differential equation for v H. Equations for the individual components of velocity may be obtained by taking vH = iu + j u . For the boundary conditions vH = 0 at z = 0 and vH = iu, at z = m, the boundary layer flow is

where D

=

u/ug =

I

v/u8

-e-Z‘Dsin z / D ,

=

- e-””

cos z/D,

(v/Q)”* is the characteristic thickness of the Ekman layer.

806

9.

APPARATUS

The Ekman layer has several characteristics in common with the boundary layer over a rotating disk. For example, depth D is independent of flow speed and does not grow in time or space, and there is an integrated boundary layer transport T = lvgl D / 2 perpendicular to the freestream flow. Spatial variations of v, result in “Ekman suction” or “Ekman pumping,” the exchange of boundary layer fluid and momentum with the interior flow. Boundary layers at vertical walls, known as Stewartson layers, also depend upon the rotational constraint, but space does not permit their consideration here. 9.4.2. Rotating Apparatus

For most studies it is possible to divide the total experimental arrangement into three somewhat distinct sections: the rotating turntable and its supporting structure, the drive mechanism, and the fluid experiment itself. A wide variety of fluid experiments are considered in Section 9.4.3 and only the basic rotating system and drive mechanism upon which any experiment might be mounted is considered here. 9.4.2.1. The Rotating Turntable. The configuration of a more or less typical general-purpose turntable is shown in Fig. 1. The heavy steel tripod is supported by three screws for vertical alignment of the rotation axis. At the top of the hollow shaft two steel plates are connected by adjustable push-pull screws for gross leveling of the rotating platform. Fine leveling of the large tank is accomplished by leveling screws which, in this case, can be easily adj,usted to make the water depth constant to within 1 mm. The frame supporting the tank should be rigid enough to avoid deflections of 1 mm under heavy loads and under the changing loads produced by rotation. With water 25 cm deep the load for the apparatus of Fig. 1 is 10 MN (weight of 1 ton). Slip rings for the conduction of power to the rotating table and the conduction of electrical signals from the rotating system are often necessary. The apparatus of Fig. 1 has three sets of Plexiglas mercury troughs, 12 in each set, fastened to the tripod frame. Circular copper strips mounted on a Plexiglas disk and fastened to the rotating shaft dip into the mercury. Wires from the copper strips lead through the hallow shaft to the table above. Copper slip rings with carbon brushes for the transmission of power can be mounted elsewhere on the shaft. In the present case flat circular rings are mounted on the upper bearing plate with the brush block extending outward from the shaft. The rotating shaft is supported by a thrust bearing at the bottom and by

SUP

P

LUGS,

NUTS, 8 LEZLING SCREWS

PLISH - PULL LEVELING SCREWS

>

FLAT COPPER SLIP RINGS 8 BRUSH BLOCK (8)

RCURY TROUGH SLIP1 RINGS (36)

TIMING BELT PULLEY LEVELING SCREWS 40 THREADWN. I

1

,, ,

2%TS

LOWER BEARING

FIG.1 . A general purpose rotating apparatus in use at the University of Maryland. Rotating apparatus other than the open cylindical tank can be mounted inside or on top of the indicated tank. The scale is indicated by the inner radius of the tank.

808

9.

APPARATUS

a self-aligning radial bearing in the upper bearing plate. A timing belt and timing belt gears are used to avoid slippage between the drive system and the rotating shaft. Alternatively, direct drive with smoother transmission of power at low speeds may be arranged by extending the rotating shaft beneath the lower bearing plate and connecting the drive system directly to the shaft. In this arrangement a clutch and flexible coupling are desirable. Depending upon the requirements and sizes of particular experiments, a wide variety of alternatives to the arrangements of Fig. 1 are available. Ring bearings have been used on smaller experiments to provide access to the bottom of the apparatus.' Air or oil bearings6 can provide smoother rotation, especially at very slow speeds where bearing and pulley effects may be noticeable. Floating circular tanks have had a certain aesthetic appeal to fluid dynamicists since one was used by Taylor,' but as a general purpose tool they suffer from several disadvantages. In particular, balance and symmetry must be maintained. The size of rotating turntables varies according to special needs. For a general purpose facility, however, it is desirable to have a large turntable so as to minimize boundary layer effects when desired and to introduce rotational inertia for smooth operation. The disadvantages of large size are frequently compensated by greater flexibility in observational techniques and operation. With the apparatus of Fig. 1, for example, pumps, recorders, transformers, amplifiers, etc., can easily be suspended from the webbed steel frame, and smaller interchangeable experiments can be set inside the general purpose cylindrical tank. 9.4.2.2. Drive Systems. The power train accompanying the apparatus of Fig. 1 consists of the sequence of components: a 1.1 kW Synchrospeed motor, a Graham variable-speed transmission, a gear box (from an old lathe), and a right-angle reduction gear with output to the timing belt pulley. The power train is assembled on a sliding plate that is controlled by a worm screw so that the timing belt tension can be adjusted or the belt disconnected. The entire assembly is rigidly anchored to the floor (no shock mounts!) to minimize vibrations. Since continuously variable transmissions may drift noticeably at very W . W . Fowlis, R . L. Pfeffer, G . Buzyna, J . C. Buckley, and J . W. Rupert, "Measurements of Temperature and Flow Speed Fields in a Rotating and Differentially-Heated Cylindrical Annulus of Liquid," Tech. Rep. No. 33. Geophys. Fluid Dyn. Inst., Florida State University, Tallahassee, 1970. ' G. I. Taylor. Experiments with rotating fluids. Proc. R . Snc. London. S e r . A 100, 114-121 (1921).

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low speeds, the gear box is used to provide different ranges of speed so that the variable transmission can always be operated at a relatively high speed where drift is not a problem. Although a wide variety of feedback mechanisms for servocontrol of the speed can be imagined, in the author’s experience this is not generally necessary if good components are used in the drive train. For the apparatus of Fig. 1 the speed control shaft of the transmission is turned by a stepping motor, the acceleration thus being determined by the rate at which pulses are delivered to the stepping motor. Adjustment of the acceleration is often desirable for control of imbalanced forces during acceleration and for experiments that may make use of continuous acceleration, deceleration, or oscillations of the rotational speed. The positive drive of a timing belt (or chain) avoids slippage between the drive train and the turntable, but at low speeds it may introduce small pulsations in the rotation rate. This disadvantage can be overcome by direct coupling of the drive train to the rotating shaft. Direct coupling, however, requires an appropriate vibration-absorbing coupling and a clutch that will disconnect with excessive torque. 9.4.2.3. Some Considerations of Precision and Control. The apparatus of Fig. 1 will be used to illustrate the required precision of alignment of the rotation axis and the control of external influences in a particular case of interest. Consider a spin-down experiment in which we first attain solid rotation at R = 1 s-l for a layer of water 10 cm deep. We then reduce the tank speed to R = 0.95 s-l, a change that produces an initial relative speed 2rg = 5 cm s-’ at a radius of r = 100 cm. We would like to guarantee that errors due to uncontrolled influences are no more than 1 percent of this maximum speed so that as the flow slows down we can still have reasonable accuracy. It can be shown that a tilt of the axis by an angle @ will result in elliptical oscillations of parcels of fluid with rotational speeds characterized by

AV

=

(R2/h)@

where R is the tank radius and h is the depth. For R = 114 cm and h = 10 cm the minimum allowable error A V = 0.05 em s-’ will occur with @ = 4 x This angle corresponds to a turn of one of the tripod leveling screws (16 threads/cm) through about 30 deg. Therefore, if the tripod, base, and floor are sufficiently rigid, the necessary reduction of @ can be attained. The tilt angle becomes more critical, however, when the basin has internal boundaries with small gaps as in the ocean model experiments of von Arx. He found large periodic oscillations that could not be elimi-

9. APPARATUS

810

nated. Assume that the circular basin is divided in half by a wall with a gap of width W. It can be readily shown that the average speed of flow through the gap, for complete adjustment of the water depth in one-half a revolution, is then

and other parameters as given before, we For W = 10 cm, CP = 4 X find = 0.25 cm s-l. Wind stress on the water surface leads to a surface Ekman layer and compensating interior flows. The wind-driven interior circular flow speeds are given by

v

where ve is the azimuthal component of velocity and where the subscripts a and w refer to air and water.* At r = 100 cm, the interior flow would be ug = - 0.3 cm s-' and the surface flow would be faster. Under these circumstances a cover would have to be provided to eliminate wind stress. Thermal control at all boundaries is often required. Maintaining the fluid at room temperature will eliminate conduction through the sides and the bottom, but evaporative heat loss and heating by illumination must also be controlled. For complete elimination of evaporation, the cover should be tightly sealed, for the wind caused by the rotation may provide significant ventilation even through small holes. Illumination should be minimized and supplied only as needed. Thermal control cannot be overemphasized for observed phenomena that were, in fact, due to evaporative cooling have occasionally been attributed to other mechanisms, and uncontrolled illumination has often been found to lead to conflicting results, especially with fluids like glycerine for which viscosity is a strong function of temperature. Horizontal temperature gradients lead to disturbances associated with the so-called thermal wind effect. The permissible temperature difference may be determined from Eq. (9.4.7) as AT = 2 0 A V L l g a h . If heating near the rim caused a temperature difference AT = 0.05"C in an L = 1 cm wide boundary layer, A V would equal 0.05 cm s-'. Since the thermal expansion coefficient of water at room temperature is much A. J. Faller, An experimental study of the instability of the laminar Ekrnan boundary layer. J . Fluid Mech. 15, 560-576 (1963).

9.4.

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811

smaller than for most other fluids, thermal control with other fluids is even more critical. The permissible variation in R is easily determined in the present example to be AR = 5 x lop4s-l. Variations of R may be high frequency, as might be produced by teeth in a timing belt or rapid voltage fluctuations, or low frequency, such as long term drift due to heating of the motor or transmission. High frequency fluctuations of the tank may not significantly affect the motion of the water, but if the measuring system is fixed to the fluctuating tank, errors will be recorded for the flow speeds. Disturbances to the flow also may arise from probe effects and from the densities of dyes that may be used for visual tracers. In many rotating systems, an exception being that of Fig. 2f, the fluid recirculates without specific provision for the damping of disturbances, and probe effects may be cumulative. A specific example of oscillations caused by a probe has recently been carefully analyzed by C e r a ~ o l i . Another ~ example of the erroneous interpretation of experimental results occurred when water that was injected into a uniform water mass through a wall had a slightly

-

1 -

I

,.--._

FIG.2. Several geometrical configurations of interest. Dashed lines give examples of internal boundaries sometimes used. (a) A right circular cylinder. (b) An annulus. (c) A sphere. (d) Three configurations with the same radial variations of the depth h measured parallel to n. (e) A cylindrical tank (from above) with various internal boundaries. (f) A rotating pipe, duct, or channel. C . P. Cerasoli, Free shear layer instability due to probes in rotating source-sink flows. J . Fluid Merh. 72, 559-586 (1975).

812

9.

APPARATUS

different temperature.lOB1l These examples illustrate the unusual care that must be taken in many rotating fluid experiments. 9.4.2.4. Experimental Configurations. Fig. 2 illustrates some typical experimental geometries of interest in the general area of geophysical fluid dynamics. A wide variety of studies of flows in rotating systems have been conducted for technical applications involving rotating machinery, fans, turbines, missiles, etc. Most of these have been concerned primarily with boundary layers or other special effects due to rotation and have required the construction of specialized equipment. N o attempt is made here to review the varied apparatus and experimental methods that have been developed for these purposes. The significance of variations of the fluid depth h , measured parallel to fl was stressed in Section 9.4.1.3. Many of the examples with added internal boundaries (dashed lines) indicate controlled depth variations of various types. Figure 2c, concentric spheres, has an important geometrical effect in the discontinuity of h. At low Rossby number Ro (a convenient measure of the magnitude of the horizontal fluid acceleration compared to the Coriolis force per unit mass) this has the general effect of causing nearly discontinuous interior flows with strong internal shear layers extending through the entire depth. Similar affects occur when discontinuities of h are produced by an immersed object or by a sharp depression in an otherwise smooth bottom, as indicated in Fig. 2a. Internal boundaries parallel to fl are indicated in Fig. 2e. In a way these examples merely represent tanks that are not circularly symmetrical. A square tank was once used to provide convenient boundaries for a comparative numerical integration of the flow.I2 The 60-deg sector and partial radial walls indicated in Fig. 2e were for fundamental experiments related to the ocean c i r c ~ l a t i o n . ' ~ . ~ ~ Figure 2f represents a straight pipe, duct, or channel in a rotating system. Changes of h can occur by changes in the cross section or, as indicated, by an internal ridge.14 Such flows have unusual entrance and exit requirements if rotation is a dominant effect. If the flow in the duct is to be quasi geostrophic the direction of the pressure gradient must be esl o A. J . Faller, Further examples of stationary planetary flow patterns in bounded basins. Tellus 12, 159-171 (1960). 'I A . J. Faller and D. L. Porter. A note on eastern boundary currents in a laboratory analogue of the ocean circulation. T d u s 28, 88-89 (1976). H. Sundquist, Numerical forecast of fluid motion in a rotating tank and a study of how finite-difference approximations affect non-linear interactions. Tellus 15, 44-58 (1963). l3 H. Stommel, A . B. Arons, and A. J. Faller, Some examples of stationary planetary flow patterns in bounded basins. Tellus 10, 179-187 (1958). D. L. Boyer, Flow over long shallow ridges. Geophys. Nuid D y n . 2, 165-184 (1971).

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ROTATING GEOPHYSICAL F L U I D D Y N A M I C STUDIES

813

sentially U C ~ O S Sthe pipe rather than along it, although a small component along the pipe must exist to overcome viscous stresses in the Ekman and Stewartson layers. The adjustment to fully developed uniform flow from the entrance region can be assisted by the trial and error adjustment of vanes in the entrance region." Irregularities of the entrance flow will tend to set up vortices or waves that damp slowly as the flow moves down the channel. The characteristic distance downstream for damping of disturbances will be approximately given by X

=

Vh/RD,

D = (v/R)l",

where 7 is the average downstream flow speed. For 7 = 1 cm/s, h = 10 cm, R = 1 s-' and D = 0.1 cm, this distance is X = 1 m. A total distance of 3X or more may be required if the initial disturbances are large. In studies where depth variations may be important a precise knowledge of h and its variations is necessary. The author is aware of an instance where it was asserted that certain results, supposedly due only to depth variations, were being observed with a flat bottomed tank at low R where the water surface was quite level. The paradox was explained when it was found that the bottom of the Plexiglas tank had bowed slightly upward in the center, thus producing a small but significant radial variation of h. 9.4.3. Basic Methods of Generating Fluid Circulation 9.4.3.1. Variations of the Fluid Density. Generally speaking, rotating experiments with fluids of variable density have been confined to simple geometries like (a) and (b) of Fig. 2. Examples of variable density experiments with slightly more complex geometries are the early experiments of Fultz' and recent experiments on flow over sills.15 The most well-controlled and most thoroughly studied thermally driven experiments have been the annulus experiments initiated by Hide4 in which the inner cylindrical wall was cooled by circulating cold water and the outer wall was heated by a controlled warm-water bath. These experiments have now been extended over a wide range of rotation rates, temperature differences, aspect ratios, and v i s c o s i t i e ~ . ' ~ *In~ ~some cases Is J . A . Whitehead, A. Leetmaa, and R . A. Knox, Rotating hydraulics of straits and sill flows. Gtophys. Nuid Dyn. 6, 101-125 (1974). l6 W. W. Fowlis and R. Hide, Thermal convection in a rotating annulus of liquid: Effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J .

Afmos. Sci. 22, 541 -558 (1965). I' D. Fultz, R. R. Long, G . V. Owens, W. Bohan, R. Kaylor, and J . R. Weil. Studies of thermal convection in a rotating cylinder with some implications for large-scale atmospheric motions. Mettorol. Monogr. 4 (21) (1959).

8 I4

9.

APPARATUS

elaborate thermistor networks have been used to study the threedimensional thermal structure," but this is done at the expense of significant probe interference effects. Irregular circulations more analogous to atmospheric flows have been produced in open shallow cylinders by heating from below near the rim or by radiative heating from above. In these cases cooling was accomplished by cold water sprayed against the central bottoms of the cylinders, Salt stratification has occasionally been used in place of thermal diff e r e n c e ~ .But ~ ~ it is particularly difficult and time consuming to set up a given salt stratification in a rotating system. Two-fluid systems were extensively studied by Fultz in early analogs of atmospheric flows.' At the low density differences that were required for interesting phenomena, however, difficulties were caused by interfacial tension between the two fluids and between the fluids and the glass wall of the container. From time to time, special methods of generating density differences have been introduced. Turnerz0provided a buoyancy source to maintain a tornado-like vortex by using bubbling from carbonated water. As another unusual example, in an attempt to simulate the release of latent heat due to condensation in a hurricane, Hadlock and Hess21 introduced a sodium hydroxide solution through holes in the base of their cylindrical tank to produce an exothermic reaction with a solution of hydrochloric acid. To experimentally study some aspects of thermal convection analogous to that on the sun as influenced by both rotation and magnetic fields, Nakagawazz placed a rotating cylindrical tank of mercury, heated from below and cooled from above, between the poles of a cyclotron magnet. 9.4.3.2. Moving External Boundaries. The simplest example of flow generated by tangentially moving boundaries is the spin-down experiment mentioned in Section 9.4.2.3. The resultant flow, as simple as it may apR. Pfeffer, G . Buzyna, and W. W. Fowlis, Synoptic features and energetics of waveamplitude vascillation in a rotating, differentially-heated fluid. ./. A m o s . Sci. 31, 622-645 (1974). I* K. Bryan, The instability of a two-layered system enclosed between horizontal, coaxially rotating plates. ./. Mrruorol. 17, 446-455 (1960). 2o J. S. Turner and D. K . Lilly, Carbonated-water tornado vortex. J . air no^. Sri. 5, 468-471 (1963). R. K . Hadlock and S . L. Hess, Laboratory hurricane model incorporating an analog to release of latent heat. J . Armas. S c i . 25, 161-177 (1968). 22 Y. Nakagawa, Experiments of the instability of a layer of mercury heated from below atId subject to the simultaneous action of a magnetic field and rotation. Proc. R . Sac. London, Ser. A 242, 81-88 (1957).

9.4.

ROTATING GEOPHYSICAL F L U I D D Y N A M I C STUDIES

815

pear, can exhibit a number of interesting phenomena including inertial oscillations and boundary layer instabilities. The spin-down of a stratified fluidz3 has been of special interest because of its possible relation to spin-down within the sun as well as because of its relevance to transient circulations in the atmosphere and oceans. Spin-down also has been used to create smooth controlled circular flow over obstacles and rough boundaries .24 Other experiments with moving boundaries have been the Taylor rotating cylinder experiments, studied by many investigator^,^^ and concentric differentially rotating spheres.z6 In open cylinder experiments Bryanz0used a rotating cover in contact with the water to drive the circulation of his salt-stratified system, and Beardsleyz7used a rotating cover to generate flow over a sloping bottom as illustrated in Fig. 2a. In a much more elaborate arrangement with fluid confined between concentric, circular spherical sectors, the upper spherical sector was rotated about an axis not coincident with the basic rotational axis of the system.z8 The effect of a wind on the water surface, as discussed in Section 9.4.2.3, is very similar to that of a rotating cover but lacks the control afforded by a cover with variable speeds. On the other hand, wind from controlled jets, as developed by von A ~ xcan , ~be used to generate a more complex pattern of surface stresses. In this section on moving external boundaries we may also mention studies involving the precession of the rotational axis-systems with two rotational axes. The possible effects of precession of the earth upon motions within the earth’s core and the generation of the magnetic field have been studied by MalkusZ9with a sphere of fluid rotated about a tilted axis. This apparatus itself was precessed by an additional rotation. Hye?O studied oscillations in a 120-cm diameter open cylindrical tank of water 23 K . D. Saunders and R. C. Beardsley, An experimental study of the spin-up of a thermally stratified rotating fluid. Geophys. Fluid Dyn. 7 , 1-27 (1975). 24 A. J. Faller and K . Mooney, The Ekman boundary layer stress due to flow over a regular array of hills. Boundary-Layer Mereurol. 11, 67-91 (1971). 25 D. Coles, Transition in circular couette flow. J . Fluid Mech. 21, 385-425 (1965). K . Stewartson, A weak spherical source in a rotating fluid. Q. J . Mech. A p p l . Math. 6 , 45-49 (1953). 27 R. C. Beardsley, A laboratory model of the wind-driven ocean circulation. J . F h i d Mech. 38, 255-271 (1969). ** D. J . Baker, Jr., and A. R. Robinson, A laboratory model for the general ocean circulation. Philos. Trans. R . Soc. London. Ser. A 265, 533-566 (1969). W . V . R . Malkus, Precession of the earth as a cause of geomagnetism. Science 160, 259-264 (1968). P. V. Hyer, “Second-order Effects in Rotary Tidal Oscillations,’’ Tech. Note BN-594. Inst. Fluid Dyn. and Appl. Math., University of Maryland, College Park, 1969.

816

9. APPARATUS

with an axis of rotation tilted with respect to g and then set his entire apparatus inside the rotating tank of Fig. I . In this case the difference of the two rotational frequencies gave the frequency of what were effectively tidal oscillations. 9.4.3.3. Internally Moving Boundaries and Objects. Many fundamental rotating experiments from those of Taylor onward have been concerned with the motions caused by simple objects moving within rotating fluids. These experiments may be arbitrarily divided into steadily moving objects and oscillators. They may be further divided into motion along or perpendicular to the axis of rotation. A study typical of the steady motion of an object along the rotation axis is that of Long,31and one typical of oscillations is that of F ~ l t z .These ~~ experiments exhibit elastoid-inertia oscillations, Taylor-Proudman effects, or other special effects according to ranges of the significant nondimensional numbers. To study the reflections of Rossby waves in rotating systems, Ibbetson and Phillips33used a horizontally oscillating paddle to generate the waves. Sundquist12used an irregularly shaped, cylindrical-wall "cookie cutter" to initiate a complex circulation. The device was introduced into the rotating tank, twisted, and then withdrawn to avoid further influence on the flow. Absorbent material at the walls damped the gravity wave disturbances to leave only the irregular horizontal motions of interest. 9.4.3.4. Pumping. A variety of rotating experiments have used external pumps to generate circulations by providing sources and sinks of fluid either through the boundaries or in the interior of the fluid. Pumping has the advantage, along with heating and cooling, that the circulation can be maintained indefinitely, and it can be precisely controlled. The most common pumping experiment is the generation of a vortex by withdrawal of fluid from the axis of rotation and introduction of fluid near the rim. As an example, a special large vortex generator for the simulation of tornadoes was described by Wan and C h a r ~ g . In ~ ~shallow cylinders vortex flows have been used to study the stability of the Ekman 31 R. R. Long, Steady motion around a symmetrical obstacle moving along the axis of a rotating fluid. ./. Mateorol. 10, 197-203 (1953). 32 D. Fultz, A note in overstability and the elastoid-inertia oscillations of Kelvin, Solberg and Bjerknes. J . Mereorol. 16, 199-208 (1961). 33 A . lbbetson and N. A . Phillips, Some laboratory experiments on Rossby waves with application to the ocean. Tellids 19. 81 -88 (1967). '3 C . A . Wan and C . C. Chang, Measurements of the velocity field in a simulated tomado-like vortex using a three-dimensional velocity probe. J . Atmos. Sci. 29, 116-127 (1972).

9.4.

ROTATING GEOPHYSICAL F L U I D D Y N A M I C STUDIES

817

boundary layer in wateP and in air.35 At higher pumping rates the nearly fully developed turbulent boundary layer has also been i n ~ e s t i g a t e d . ~ ~ Occasionally fluid has been injected or extracted through slots in vertical walls or through tubes into the interior of a rotating fluid. At low Rossby numbers and with depth variations the flow patterns in such cases can be quite unusual and unexpected.1° Distributed flow through porous boundaries has also been used for analogues of the ocean c i r c ~ l a t i o n . ~ ~ H ~ l t o used n ~ ~ pumping to generate barotropic Rossby waves. 9.4.4. Special Observational Methods

To obtain observations from rotating systems often requires special techniques of data transmission and photography. 9.4.4.1. Data Transmission from Rotating Apparatus. The conventional method of transmitting electrical signals from rotating apparatus is by means of slip rings. Since all slip rings will generate some electrical noise, a careful consideration of the signal-to-noise ratio is advisable before selecting rings and electrical contacts. Copper rings serve well for general power transmission; silver rings may be required for data transmission. Mercury troughs, as indicated in Fig. 1, are relatively inexpensive and satisfactory if there is no danger of mercury spillage at the higher speeds of rotation. The mercury is normally covered with a layer of oil to minimize oxidation and contamination and to eliminate the mercury vapor hazard. In a large rotating tank with no slip rings, von Arx once used a conventional radiosonde transmitter, battery operated, to transmit temperatures from a thermistor probe that was oscillated vertically through a thermally stratified fluid. 9.4.4.2. Photography. The apparatus of Fig. 1 has a synchronized rotating camera mounted separately to the building about 3 m above the bottom of the cylindrical tank. The rotating shaft is mounted in an adjustable frame for careful positioning and alignment, for in many cases it is necessary to have the axes of rotation precisely aligned. 35 P. Tatro and E. L. Mollo-Christensen, Experiments on Ekrnan layer instability. J . Fluid h4ec.h. 28, 53 1-543 (1967). 38 D. R. Caldwell, C . W. van Atta, and K . N . Helland, A laboratory study of the turbulent Ekman layer. Geophys. Nuid Dyri. 3, 125-160 (1972). 37 D. J . Baker, J r . , A source-sink laboratory model of the ocean circulation. Geaphys. Flltid Dyn.2, 17-20 (1971). 38 J . R. Holton, An experimental study of forced barotropic Rossby waves. Geaphys. Nuid Dyn. 2, 323-341 11971).

818

9.

APPARATUS

Rotational syncronism is provided by Selsyn motors transmitting positional signals from the primary drive train to the camera shaft, the Selsyns being geared up to operate at high rotational speeds. Slip rings are provided on the camera shaft for electrical operation of a variety of 35 mm and cine cameras. In many cases it is desirable to mount a camera rigidly on the rotating system for convenience of positioning and to avoid problems of alignment. Electrical operation through slip rings or by radio control may then be necessary, although at low rotational speeds manual operation of a rotating camera may be practical. Streak photographs-time exposures in which moving white particles leave a streak against a dark background-are particularly useful for determining general surface flow patterns and for measuring velocities. Ambiguities in flow direction and speed can be eliminated by providing one or more flashes at precisely timed intervals during the time exposure \ to give bright spots on the streak. The apparatus of Fig. 1 is provided with a set of microswitches attached to the tripod frame that can be triggered by vertical rods in a circular plate attached to the rotating shaft. The rods can be set at precisely machined angular intervals of 5 deg. A combination of rods and microswitches can then open the shutter of the rotating camera, provide precisely timed flashes from electronic flash units, and close the camera shutter. They may also be used to pulse the shutter of a cine camera for time-lapse films. The Rota~cope'~.''is particularly useful for qualitative viewing of rapidly rotating systems. For photographic purposes, however, the difficulties of accurate alignment and the optical distortions in the prism make this technique generally inferior to more direct photographs.

9.4.4.3. Flow Visualization. Because of the complexities of the three-dimensional flows in rotating systems, the difficulties of measuring slow flows, and the large disturbances often produced by probe supports, flow visualization has played a particularly important role in rotating fluid experiments . Free surface tracers have generally been flakes of aluminum, as used by Fultz' in relatively small experiments, or paper dots (from a paper punch) for large experiments like that of Fig. 1.* In general, it is advisable to not use detergents for cleaning the tank unless absolutely necessary as detergent residues can cause undesirable film effects. Detergents will also cause paper dots to absorb water and sink soon after their introduction. Where larger tracers are permitted the author has used small light bulbs, or polyethylene rods weighted at the bottom and tapered at the

9.4.

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819

top so as to float vertically with only a section of small diameter protruding through the surface film. Various dyes have been used effectively for flow visualization of different types. The author has found potassium permangate crystals to be the most useful in experiments with water, particularly for boundary layer processes. Crystals of KM,O, fall rapidly to the bottom and clearly show flow directions by plumes from the dissolving crystals. Convergence patterns often are clearly indicated by accumulations of dye as in the cases of the simulation of atmospheric fronts39and the detection of boundary layer instabilities.8 For the quantitative measurement of interior flows it has always been difficult to use dye tracers effectively. An important exception is the Thymol Blue technique developed by Baker4' for the measurement of slow velocities. An electrical impulse through a wire immersed in a special solution causes a line of coloration that is easily followed and photographed. Two basically qualitative methods of following interior flows have been extensively used. Aluminum flakes immersed in silicone fluid take on characteristic patterns in response to shear flows. Suspended floculent particles or neutrally buoyant plastic beads distributed throughout the fluid can be illuminated by slit lighting to show flow patterns in a particular cross section of the fluid. A laser doppler velocimeter has recently been used successfully with this instrumentation mounted on the rotating Local velocity measurements with a spatial resolution of 0.1 cm and a time resolution of 0.5 Hz were found to be possible. For more details on tracer methods of flow visualization and particle tracking for velocity measurement, see Chapter 1 . 1 of this volume.

38 A . J. Faller, A demonstration of fronts and frontal waves in an atmospheric model. J . Mefeorol. 13, 1-4 (1956). 40 D. J . Baker, A technique for the precise measurement of small fluid velocities. J . Fluid Mech. 26, 573-575 (1966). 41 W. W. Fowlis and P. J . Martin. A rotating laser doppler velocimeter and some new results on the spin-up experiment. Geuphys. Huid D.yn. 7, 67-78 (1975).

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10. DIMENSIONAL ANALYSIS AND MODEL TESTING PRINCIPLES" When investigating a phenomenon in physics it is important to determine which physical properties play an important role and which can be ignored. Furthermore, the significant unknowns in the phenomenon must be identified, together with the number and type of independent variables. Such a preliminary survey of a given problem simplifies its investigation, avoids unnecessary measurements in experimental work, and reduces the expense in theoretical or computational work. Dimensional analysis is an indispensible tool in this process and is used in three ways, classification, measurement, and simplification of physical laws. For the purpose of classification, a few physical quantities are selected as basic and a more general quantity is then analyzed in terms of these. In mechanics the basic quantities usually selected are mass, length, and time. To measure a general quantity units of measurement must be assigned to the basic quantities. Dimensional analysis then gives the units of measurement of the general quantity in terms of the basic units. Simplification of the results of an experimental or theoretical investigation is achieved by expressing these in dimensionless form. Thus, a numerical solution or a set of experimental measurements may apply to a whole class of problems instead of to an isolated configuration. Reduction of data to dimensionless form is particularly important in model testing. It is necessary to ensure that data acquired on a small scale model can be applied to the design of a full scale structure. For this two conditions must be satisfied, identified by geometrical and dynamical similarity, respectively. The first simply means that the test and full scale models must be of the same shape. The second is more exacting and requires that all significant dimensionless physical constants have the same value under both test and full scale operation. The first part of this article is devoted to the basic theory of dimensional analysis, illustrated with a few examples in viscous and compressible fluid flow. The second part gives a brief survey of the method and principles of model testing with emphasis on testing in high speed wind tunnels. * Part 10 is by Maurice Holt. 82 I M E I ' H O D S OF E X P E R I M E N T A L PHYSICS, VOL. 18B

Copyright @ 1981 by Academic Preas, Inc. All rights of reproduction in any form reserved. ISRN n-t?-d7wh-d

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DIMENSIONAL ANALYSIS A N D MODEL TESTING

10.1. Mathematical Foundations of Dimensional Analysis 10.1.1. Units and Dimensions

In studying any process a number of quantities must be measured. These are classified in two general categories, first, those which depend on the scale of measurement, called dimensionul quantities, and second, those which are independent of the scale, called dimensionless quantities. The speed of a fluid particle, the resultant force acting on an aircraft wing, the total energy released by a nuclear explosion, are examples of dimensional quantities. The angle of incidence of an aircraft or space vehicle in flight, the ratio of specific heats of a perfect gas, are typical dimensionless quantities. When measuring the quantities of importance in some phenomenon, certain quantities, usually the simplest and most familiar, are selected as basic or primary quantities, and these are measured in terms of operationally defined units. All other quantities arising in the phenomenon are regarded as derived or secondary quantities and are measured in units formed from combinations of the basic units involving some law or other statement about nature. In fluid dynamics it is usual to select length, mass, and time as the fundamental units. The units preferred today are the meter, kilogram, and second. The SI system of units is superseding the fps (foot, pound, second) system formerly employed in English speaking countries and the technical metric system used elsewhere. The SI (Systbme International) standards are established by international agreement and based largely on atomic properties. The units of measurement of derived quantities are formed from those of the basic units, and the relationship between the units of the derived quantities and the basic units is called the dimensions of the quantity. We shall show that dimensions are always monomial powers of the basic units. Dimensions of derived quantities are implicit in their operational definitions. If a particle moves uniformly its speed is defined as the distance moved in a given time interval divided by the time elapsed. The dimensions of speed are then simply length/time, and this is still true when the motion is nonuniform. To establish the dimensions of the quantities occurring frequently in fluid dynamics we first introduce the symbols L,M,T for the basic quantities length, mass and time, respectively. Further, we use the notation [fl to denote the dimensions of some derived quantity f. The algebra of dimensions follows rules similar to arithmetic.

10.1.

MATHEMATICAL F O U N D A T I O N S

823

If V denotes velocity, [ V ] = LT-'.

The acceleration a is defined as time rate of change of velocity so that [a]

=

LT-'.

In one-dimensional flow momentum equals mass flow per unit time, so that [momentum] = MLT-I. From Newton's second law, the force F equals time rate of change of momentum, and therefore

[F] = MLTP2,

kg m s - ~= N , a newton.

Similarly, if E denotes energy, a force times a length,

[El

=

ML2T-2,

N m

=

J, a joule.

The pressure p is the normal force per unit area, and hence [ p ] = ML-lT-',

N mP2 = Pa, a pascal.

For the density p we have [ p ] = ML-3.

There is no name or special symbol for a kg m-3. The coefficient of viscosity p is usually introduced in discussions of viscous flow in a channel, using the statement tangential stress

=

p(ve1ocity gradient at channel wall).

Then, equating dimensions, we have

ML-'T-'

=

[PIT-',

so that [p] = ML-lT-'.

The unit is Pa s. The kinematic viscosity v

= p/p,

so that

[ v ] = L2T-'.

The unit is m2 s-l. 10.1.2. Scaling and Conversion It is frequently desirable to change the scale used for measuring a physical quantity arising in some process under investigation. For example,

824

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

the basic SI unit of speed, m s-l, is too small when discussing supersonic flight and it is preferable to use km s-l as the unit of measurement. Also, it is more informative, when discussing blast wave pressures, to use megapascals rather than pascals. At the present time conversion to SI requires constant change of units from the English system still prevalent in laboratories and workshops of Britain and the United States. This scaling and conversion is achieved without blunder by using the appropriate dimensional formula. In the case of pressure, for example, [ p ] = ML-lT-'.

In the English system L

=

1 ft, F

=

1 Ib, T

1 Ib

=

4.45 N ,

1 ft

=

0.3048 m.

=

1 s. Now,

Hence

Thus to find the scaling or conversion factor when a given quantity is expressed in terms of new units, simply express the old units in terms of the new in the dimensional formula for that quantity. The simplicity of conversion formulas is due to the property that dimensional formulas always occur as products of powers of the basic dimensions. This is a consequence of the fact that if two values of the same quantity are in a certain ratio in one system of units they must be in the same ratio in a scaled or converted system. Scientists have chosen to describe their observations of nature only in terms of physical quantities that have this property. To illustrate that the dimensions are products of powers. suppose that some quantity is denoted by u. Without loss of generality we suppose that it is a functionf(x) of a single coordinate x . Evaluate u at two points x = x l , x = x2 and then reevaluate u at the same two points when the scale of x is magnified by a factor a. Then, since the ratio u t / u l has the same value in both systems, (10.1.1)

and, since u

=f(x),

(10.1.2)

10.1.

MATHEMATICAL FOUNDATIONS

825

From Eq. (lO.l.l),

(10.1.3)

To determine g consider two different scales e l , at and put a

= a1/a2,

x2 = a z x l . Then from (10.1.3),

(10.1.4) But from (10.1.2)

The last ratio equals g ( a l ) / g ( a z )from (10.1.3). Hence g ( w / a z ) = R(%)/R(az).

(10.1.5)

Differentiate (10.1.5) with respect t o a1(keeping uZconstant) and subsequently put a = a1 = az. Then

'

1 g'(1) = - = ___ g'(a)

a

a

Integrating with respect to

g(4'

(Y

where 1 = g'(1).

(10.1.6)

we find that g ( a ) = Ca'.

C

= 1 since g = 1 when a = 1 , from Eq. (10.1.5). It follows that if x has dimension L, then the contribution to the dimensions of u from the basic parameter x is of the form L'. If u depends on three basic parameters with dimensions L , M , T , respectively, an extension of this argument shows that

[u] = L'Mm71,

(10.1.7)

where l,rn,r are constants. 10.1.3. Statement of Physical Laws

When investigating physical phenomena we seek to know the relationships between important physical variables, such as pressure or temperature, and the essential parameters in the phenomena. Some of these essential parameters are variables, such as coordinates defining position and time, while others are constants. Sometimes the relationships are independent of position and time. For example, in thermodynamics we frequently stipulate that two thermodynamic variables, say enthalpy and entropy, characterize the state of the material and seek algebraic relations defining other variables in terms of these.

826

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

In fluid dynamics we deal with two types of relationships: first, algebraic formulas arising mostly in the expression of thermodynamic properties and second, partial differential equations expressing equations of continuity and force. The ultimate objective of a given investigation in fluid dynamics is to represent the principal fluid variables as functions of position and time in all parts of the flow field. This information can be obtained from experiments or from solutions of partial differential equations. It is clearly desirable that these functional relationships, whether found theoretically or experimentally, should be applicable to a whole class of flow fields rather than to an isolated situation. To achieve this we need to express the relationships in dimensionless form, i.e., in a form independent of the units and scales of measurements used. The transformation of a fundamental relation from dimensional to dimensionless form is achieved by selecting certain parameters in the phenomenon as fundamental or basic parameters and forming dimensionless combinations of these with the remaining physical variables entering the phenomenon. The fundamental parameters need not all be constants; the only requirement is that they have different dimensions. 10.1.4. Dimensional Homogeneity

When discussing reduction to dimensionless form it is important to ensure that each functional relationship dealt with is dimensionally homogeneous. This means that every term in the relationship must have the same dimensions. This requirement is met in all thermodynamic laws expressed individually as physical principles and in equations of motion of a fluid, as they appear in their original form. Dimensional homogeneity can be lost when equations are added and the dimensions of the original equations are not the same; the continuity equation has dimensions ML-3T-1, while the force equation has dimensions ML+ T-2. Governing equations should therefore be reduced to dimensionless form before being combined. 10.15.The Pi Theorem

The reduction of a given functional relationship from dimensional to dimensionless form is the subject of the Pi theorem. This is one of the most important results in dimensional analysis and was first stated explicitly by Buckingham. We shall assume that every functional relationship to be considered is

* E. Buckingham, Phys. Rev. 4,

345 (1914).

10.1. MATHEMATICAL FOUNDATIONS

827

not only dimensionally homogeneous but also complete, i.e., it takes the same form in all respects, whatever system of units is used. A full statement and proof of the Pi theorem for implicit functional relations is given by Bridgman.2 We shall give a simpler statement and proof for explicit relations due to S e d ~ v .The ~ two statements are equivalent. We shall also restrict the number of basic parameters to three, the total arising in fluid dynamics. Suppose that, in some physical problem, one physical quantity, regarded as an unknown and denoted by u , is expressed in terms of three basic parameters, a,b,c,and N other parameters x i , . . . , xN by the relation u

= U(U,

h,

C , XI,

...

, xN).

(10.1.8)

Here, the dimensions of a,b,c are independent and those of the other parameters (which may be constants and variables) are combinations of those of a,b, and c . Then if we form dimensionless combinations of u , x i , . . . , x N with a,b, and c and denote these by 7r, 7 r I r . . . , r N ,respectively, the Pi theorem states that the relation between 7r and 7 r l , . . . , rN has the form 7r = f(1,

1, 1,

7r1,

,

..

3

VN).

(10.1.9)

The proof of this result simply rests on the property of completeness. If a,b,c are constants we simply consider each of these parameters as a unit of measurement and Eq. (10.1.8) immediately takes the form Eq. (10.1.9). If some of a,h,c, are variable, the proof of the theorem requires comparisons of the forms of Eq. (10.1.8) when two different scales of units are used. The details are given in S e d ~ v . ~ A simple result immediately obtained from the Pi theorem is that if u depends only on the three basic parameters a,b,c, then from Eq. (10.1.9)

i.e., 7r =

const

=

k,

so that, in dimensional form * P. W . Bridgman, “Dimensional Analysis.” Yale Univ. Press, New Haven, Connecticut, 1922. L. I. Sedov, “Similarity and Dimensional Methods in Mechanics” (English translation edited by M. Holt). Academic Press, New York, 1959.

828

10.

D I M E N S I O N A L ANALYSIS A N D MODEL TESTING

u = ka'bmcn,

( 10.1.1 0)

by use of the monomial property of dimensional formulas. The Pi theorem has many important applications which are illustrated in later sections.

10.2. Geometrical and Dynamical Similarity Similarity (or similitude) is a property connected with the validity of expressions defining physical variables for a whole class of problems rather than a single special example of physical phenomena. To illustrate this in fluid dynamics, consider the motion of a uniform stream past a wing and suppose that we determine the pressure distribution on the wing either by experiment or by solving a boundary value problem mathematically. Then the pressure is a function of the coordinates defining the position of the point on the wing and of certain constants related to conditions in the undisturbed free stream, including its velocity, pressure, density and other physical properties. If this functional relation is written in terms of dimensionless variables and if the dimensionless physical constants are held fixed, then the local dimensionless pressure depends only on the position, i.e., on the geometrical shape of the wing. Two flow fields which are bounded by surfaces of the same shape but of possibly different scales are said to be geometrically similar. Flow fields which, in addition, have the same distribution of physical variables, expressed in dimensionless form, at corresponding positions, are said to be dyncimiccilly similar. Similarity properties can be used to simplify formulae for distributions of pressure and other physical variables and for force coefficients on given configurations. The fewer the number of physical constants entering the flow field the easier this simplification becomes. A further simplification arises if the governing equations of motion can be linearized since the formulae can then be written in analytical form, while for nonlinear equations the representation is usually numerical. An important class of problems is governed by the property of .sd$ similarity. The best known steady flow fields with this property are supersonic conical flows in which the physical variables have the same value on all radial lines through a given origin. In unsteady flow selfsimilar flows arise for plane, cylindrical, or spherical symmetry when the number of physical constants playing a role in the problem does not exceed two. In such cases the number of independent variables can be reduced by one so that a three-dimensional steady problem reduces to one of plane flow and unsteady symmetrical problems are governed by ordinary differential equations.

10.3.

829

APPLICATIONS I N F L U I D DYNAMICS

There are many problems in which more than two physical constants have to be considered but in which the influence of the additional constants is small. Such problems can be treated by using the self-similar solutions as a foundation and calculating corrections for the small effects of the additional constants. This technique comes under the heading nearly similar solutions or perturbations of similar solutions and provides valuable information about many important physical phenomena.

10.3. Applications in Fluid Dynamics We now discuss the significance of the concept of dimensional analysis and similitude in fluid dynamics. We begin by listing the important dimensionless numbers and then examine the role they play in simplifying the governing equations of fluid flow under widely different conditions, ranging from low speed, highly viscous flow to high speed, rarefied flow. We then derive a number of significant basic solutions in fluid dynamics, using dimensional and similarity properties. 10.3.1. The Principal Dimensionless Numbers in Fluid Dynamics

If we fix attention on the steady flow of a uniform fluid stream past a body of given shape in an unlimited space the constants important in the problem can be listed as follows:

L

a typical length or dimension of the body specific heat of fluid at undisturbed volume (assume ideal gas) CP UO speed of fluid stream far ahead of the body P o pressure of the fluid far ahead Po density of fluid far ahead TO temperature of fluid far ahead PO the coefficient of viscosity of fluid the thermal conductivity of fluid k a0 the speed of sound in the fluid Suppose, for the present, that the density is not low so that the flow can be treated as a continuum. Then, either by experiment or by solving equations of motion, we can obtain an expression for the magnitude of the force F , say, acting on the body. This represents the integrated excess pressure (difference between local and free stream pressure) and can be written (10.3.1)

As a special case, suppose that the effects of viscosity, heat conduction and compressibility (represented by a o )are unimportant. Then

830

10. DIMENSIONAL

F

ANALYSIS A N D M O D E L TESTING

=

4(L3uo, P O ) .

( 10.3.2)

Choose the parameters L , U o , p o as basic (note that U o , p o , po are not independent in dimension). Then from the Pi theorem, we can define a dimensionless force coefficient Cf = F / f P o G L =

1,

( 10.3.3)

so that the force coefficient is the same for all flows with geometrical similarity. Proceeding to a less special case, suppose viscosity is important. Then

F

=

4(L, uo, P O , P O ) .

( 10.3.4)

If we reduce to dimensionless form with the same basic parameters we now have

Cf = f(1.

1, 1, p o l p o ~ o ~ ) .

(10.3.5)

In geometrically similar flows Cf is then the same for all flows in which the dimensionless number ( p oU o L ) / F ohas the same value. We call this the Reynolds number Re and write Re

=

p o U o L / p o= U o L / v o .

_= po/pois called the kinematic viscosity. Further dimensionless numbers are introduced as additional physical properties are considered to be important. Compressibility becomes important as the free stream speed increases in comparison with the speed of sound so its effect is represented by the Mach number, Mo = U o / a o . Heat conduction influences the flow through the dimensionless form of k formed with p and c p . This is called the Prcrndtl number, Pr = p c p / k . This can also be written as v / i where u is called the thermal diffusivity klp, c, . Other physical phenomena play a role in other problems and bring additional parameters, not listed for the first problem, into consideration. For ship motion or motion of submerged bodies gravity force per unit mass g must be considered, leading to the definition of the Froude number Llo/(gL)1’2. When the problem has gravitational forces and thermal effects the Grashof number G = g/3(ATo)L3/v2plays an important role. Here p is the coefficient of expansion of the fluid while ATo is the temperature difference between the body wall and far upstream. In unsteady motion with periodic character the significant dimensionless parameter is the Srrouhnl number S = n L / U o , where n is a characteristic frequency (e.g., that of vortex shedding behind a circular cylinder). In low density flow, the magnitude of the mean free path A is impor-

vo

10.3.

APPLICATIONS IN FLUID DYNAMICS

83 1

tant and the importance of rarefaction effects is represented by the Knudsen number Kn = h / L . In an unbounded low density stream, according to the kinetic theory of gases, the coefficient of viscosity is approximately p = poaoXoso that the Knudsen number can be written Kn = M/Re. Only a few of the dimensionless numbers arising in fluid dynamics have been mentioned, although we have included those occurring most frequently. We next consider the role played by certain dimensionless numbers in simplifying governing equations of motion, with special attention to viscous incompressible flow and inviscid compressible flow. 10.3.2. Simplification of the Navier-Stokes Equations

The motion of a viscous incompressible fluid (without heat conduction) is governed by the Navier-Stokes equations. In the general case these can only be solved numerically, but simple solutions are possible if the Reynolds number based on free stream conditions and a typical body length is either small o r large. In the first case the inertia terms in the equations of motion are negligible compared to the pressure and viscous terms and the equations can be written in an approximate form due to Stokes. An example of the Stokes approximation is given by a sphere of radius a rotating with angular velocity o about a fixed diameter in a dense liquid of large viscosity p. Examination of the governing equations shows that the velocity is in the circumferential direction only, while the pressure is constant. The resultant torque on the sphere is M

= 87rpa3w.

If we define a dimensionless moment coefficient

CM = M / ( p a 5 0 2 ) , then

C,

=

8r/Re

where Re is based on the maximum velocity of the sphere w a ,the radius a and Y. The Pi theorem shows that CM is a function of Re only, but the Stokes analysis is needed to show that CMis inversely proportional to Re. To simplify the Navier-Stokes equations when the Reynolds number is large, we combine the properties of one exact solution with examination of the general equations in dimensionless form. Consider the exact solution defining parallel flow past an infinite flat

832

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

plate impulsively given a uniform velocity U at time t = 0. If x and y are coordinates parallel and normal to the plate with corresponding velocity components u and u , then u is independent of x and the equation of continuity and boundary conditions show that u = 0 everywhere. The Navier-Stokes equation then reduces to au/at =

v

(10.3.6)

azU/ayz

to be solved with boundary conditions u = U for

y =0

and

for y =

u =0

m.

Since v and U are the only parameters in the problem, the velocity u must be of the form =f(Y,

( 10.3.7)

t , v, U ) .

If we take U , t, v as basic parameters, the Pi theorem shows that this has the dimensionless form ic

=

u / u = f ( y / d z , 1 , 1 , 1)

( 1 0.3.8)

so that Eq. (10.3.6)has a similarity solution defined by dzic dz2

1 dii 2 dz

0 for z

= cc., where

-+ - z - - 0 , with ii = 1 for z = 0; ii The solution of this is

=

(10.3.9)

z

=

y/(vt)”*.

u = 1 - e1f{y/2(vt)”~}

(10.3.10)

where (10.3.11) The greatest part of the drop in velocity from the wall value of ii = 1 to i = 0 in the outside stream occurs in a layer of small thickness 6 called the boundary layer. Equation (10.3.10) shows that 6 is proportional to (vt)l/Z*

Rayleigh4 pointed out an analogy between the unsteady impulsive motion of a flat plate in a stream at rest and the steady motion past a semi-infinite flat plate. Essentially the disturbance introduced at the leading edge of the plate is carried downstream with the main stream velocity U o and the role o f t in the impulsive start problem is replaced by the time-like variable x / U o , where x is the distance from the plate leading J. W.Strutt, Lord Rayleigh, Philos. M q . [6] 21, 697 (1911).

10.3.

APPLICATIONS I N FLUID DYNAMICS

833

edge. It then follows that the boundary layer thickness in steady flow is of the form 6/d = (1/Re1/2)lx/d)1'2

(10.3.12)

where d is a typical length and Re = U,d/v. Equation (10.3.12) provides the general basis for the boundary layer approximations to the Navier-Stokes equations, valid for large values of Reynolds number Re. The equation shows that the extent of the region disturbed by viscous effects in the transverse direction y is of the order Re-1/2 times that in the main stream direction x. To investigate viscous flow past a body at large values of Reynolds number we transform the Navier-Stokes equations referred to dimensional variables x', y ' , u ' , v', p ' , p ' to dimensionless form in terms of the variables

x = x'/d,

y = Re1l2y'/d,

u = u'/uQ,

v = Re"2V'/Ug,

p = p'/p'UG,

t =

Ut'/d,

where d is a typical length (the body length for a finite body). That the transverse velocity component u' must be scaled in the same way as y ' is suggested by the equation of continuity. The dimensionless form of the Navier-Stokes equations and the continuity equation then becomes au

-+ at

au + uax

au = ay

0-

=

1 a2u a2u ap -+-+ax

-*+ ay

Re ax2

ay2'

1 a2v + -I a% -Re2 ax2

Re d y 2 '

(10.3.13)

av -au + = o. ax ay

As Re -+ 00, the second equation states that the transverse pressure gradient is negligible, while in the first equation the diffusive term in the xdirection can be dropped, leading to the boundary layer equations. Returning now to the specific problem of uniform steady flow past a flat plate, for boundary layer flow, we solve the equations

(10.3.14) au ax

au ay

-+--0,

834

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

with boundaryconditions u = u = Ofory = Oandu = 1 fory = 03. Rayleigh’s analogy with the impulsive start problem suggests that this problem has a similarity solution of the form u =f(A),

where A is the similarity variable A = y/xl’* (the role of Reynolds number is imbedded in the dimensionless variable y). This is the basis for the Blasius solution: if we writef’(A) = +’(A), then the equation for r$ is the ordinary differential equation

2 4 “ ’ + $4”

=0

with +(O)

=

$ ‘ ( O ) = 0,

4 ‘ ( ~=)

1.

(10.3.15)

This illustration of the use of dimensional analysis to find the form of the equations of fluid dynamics at large Reynolds number and their application to the formulation of the flat plate boundary layer problem suggests how dimensional analysis may be used. An elegant proof of the validity of the Blasius similarity solution, using dimensional arguments, is given in Sed~v.~ 10.3.3. Similitude in Compressible Flow

Similitude plays an especially important role in inviscid compressible flow. The simplest example of this occurs in linearized, steady, subsonic or supersonic flow past a thin airfoil. If the pressure distribution on a thin airfoil of given shape is known at one value of Mach number, the conesponding distribution at a different Mach number can be obtained immediately by a simple scaling known as the Prandtl-Glauert rule. If the flow is transonic a somewhat more complicated similarity law must be used because the governing equations, even though simplified by small disturbance approximations, are nonlinear. The similarity parameters in the transonic case are associated with von KArmBn5 and Spreiter.e Hypersonic small disturbance flow is also nonlinear; this was investigated by Van Dyke.7 In steady supersonic flow past conical bodies, no body length can be defined and the flow variables can be shown to be functions only of the angular coordinates. Thus the number of independent variables is reduced from three to two. Such flows are called superonic conical fields. In unsteady compressible flow with symmetry about a plane, line, or point there are many problems where no characteristic length or time can T. von KBrmBn, 1. Marh. Phys. (Cumbridge, M U S S . 26, ) 186 (1947). J . R . Spreiter, J . Aerosp. Sri. 26, 485 (1959). M. D. Van Dyke, A study of hypersonic small disturbance theory. N A S A Tech. R e p . R-1194(1954).

’

10.3.

835

APPLICATIONS I N F L U I D DYNAMICS

be defined and the unknowns are functions of a single similarity variable formed by combining the space coordinate with some suitable power of the time. Such problems come under the heading of Blast Wave Theory. Conical flows, blast wave solutions and the Blasius solution are all examples of self-similar solutions in fluid dynamics.

10.3.4.Conical Flow In uniform supersonic flow past conical bodies, similarity properties enable us to reduce the number of independent variables by one. In the general three-dimensional case, each unknown (e.g., the x component of velocity) would be given by a relation Lf = f ( x , Y ,

z,

uo,P o , P o , Y ) ,

(10.3.16)

where suffix 0 refers to free stream conditions and y is the dimensionless gas constant. Only two of U o ,p o , po have independent dimensions. No fundamental length enters the problem so, if we select x, U o , and p o as basic parameters the Pi theorem shows that we may write

(10.3.17) Then u and the other unknowns are functions only of the angular variables defined by y / x and z/x. In axisymmetric conical flow the governing equations reduce to ordinary differential equations with the well-known solution of Taylor and Maccoll.8 In flow at angles of attack the flow equations are solved numerically in a cross flow plane.9~10

10.3.5.Self-similar Solutions in Unsteady Flow We consider the unsteady motion of a perfect gas with plane, cylindrical, or spherical symmetry. If f is the time and r the distance measured from the plane, axis, or center of symmetry, the fluid dynamic equations may be written

-aa+vt + - + aa- vr- = op1, ap ar e!!

at

* G . I. Taylor and J .

+ a0 + ( v

ar

- 1) eli r

(10.3.18) = 0,

(10.3.19)

W. Maccoll, Proc. R . SOL..London, S r r . A 139, 278 (1933). M. Holt and D. E. Ndefo, J . Comp. Phy.7. 5, 403 (1970). C. A . J. Fletcher, AIAA .I. 13, 1073 (1975).

836

10.

DIMENSIONAL ANALYSIS A N D M O D E L T E S T I N G

( 10.3.20)

where v denotes velocity, p pressure, p density, y is the ratio of specific heats and v

=

i:) 2

1

in

cylindrical symmetry. (:xicaI

If, in a given problem, not more than two constants with independent dimensions are needed in defining initial and boundary conditions, it can be shown that the solutions to the problem are self-similar. This means that the flow variables are functions of only a single independent variable, formed by combining Y and t , and that the flow pattern has the same qualitative form at all times, only the scale changing with time. This property was first noticed explicitly by Sedov3 and was used by him to work out a series of examples of practical importance. To prove the self-similarity property suppose, without loss of generality, that one of the dimensional constants a , say, contains mass. The other constant, h , can then be chosen to be independent of mass. We then write [h] = L"T".

[u] = MLkTS,

The solution of the problem in question gives a formula for the pressure of the form P = f [ r , t , u, hl.

If we take r , t , and a as basic parameters then, according to the Pi theorem, this relation can be written in dimensionless form (fi+1ts+2/a)p = A1, 1 , 1, b/rmtn),

so that the dimensionless pressure on the left, which we denote by P is a function of the single independent variable r m t n / b . We adopt a modified form of this as the similarity vuriublc A writing

A

=

r/(b1lrnt'),

(10.3.21)

where 6 = - n / m (in all practical applications m # 0). If we express all the flow variables p , u , p in dimensionless form, using a, Y, t as basic parameters, then Eqs. (10.3.18)-(10.3.20)will reduce to ordinary differential equations in P , R , V , where

p=-

rk+lts+!2

U

p,

rk+3ts

R=-

U

p,

t

v = -r u .

10.3.

APPLICATIONS I N F L U I D DYNAMICS

837

If a further dimensionless variable

z

=

yP/R

is used in place of P , the three ordinary differential equations in z, R , and V can be reduced to the following Z{[2(V - 1) - (y

dlnA -dV

+ v ( y - I)V](V - 6 ) 2 -

l ) V ( V - 1)(V

-

6)

z - ( V - 6)2 V ( V - 1)(V - 6 ) + ( K - vV)z ’

(10.3.23)

V ( V - 1)(V - 6)

( v - 8) dd InIn RA = ~

cs + ( k - + 3 1 ~ 1 -

where K

=

{S

z

(K

- ”)’,

( V - 6)2

+

-

(10.3.24)

+ 2 + $(k + l ) } / V .

Equation (10.3.22) only contains z and V . If a solution of this can be found in a given case, the remaining equations (10.3.23) and (10.3.24) can be solved by simple quadratures. In most of the problems governed by self-similar solutions, the flow field does not extend continuously from the origin to infinity but is terminated, either externally or internally, by a shock wave, across which the dimensionless variables are discontinuous. This shock wave will always be a surface A = const (the value of the constant is usually chosen to be unity). To prove this the equation of the shock takes the form r , = g ( t , N , b). If we take a , b, t as basic parameters the Pi theorem gives the dimensionless form of this relation as y,

or A

v,

=

t-sjh’lm

=

!?(I, 1, 1)

A,, a constant. It follows that the dimensionless shock velocity is

= 6.

When the shock equations are written in dimensionless form they become 6)

=

RZ(V2 -

Vi - 6 + z i / y ( V , - 6)

=

Vz

Ri(V1

tcv, -

+ z,/(y

-

- 1) =

-

b(Vz

6 -

a),

+ Z z / y ( V , - a), + z z / ( y - 1).

(10.3.25)

838

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

Here suffices 1 and 2 refer to conditions ahead of and behind the shock, respectively. The last two equations can be solved for V2 and zz in terms of V , and zl. We note the following properties about these relations (i) Points on the parabola

-

z = (V

( 10.3.26)

S)2

transform into themselves. In fact, this curve represents a characteristic through the origin t = 0, r = 0. (ii) Points on the V axis, corresponding to z1 = 0, transform into points on the parabola 2 = 2y/(y -

1)(V - 6 ) Z .

(10.3.27)

(iii) For motion into a medium at rest V1 = 0 and points behind the shock must lie on the parabola z z = -S(Vz -

S)(l

+ [ ( y - 1)/26]V,).

( 10.3.28)

(iv) For a detonation wave moving into an undisturbed medium, a term Q / c 2 (representing the energy addition) must be added to the left of the third term of Eqs. (10.3.25). Also the values of y on the two sides of the wave differ and must be distinguished by suffices 1 and 2. In general, the Chapman-Jouguet condition is satisfied behind a detonation wave. This takes the form

(10.3.29)

z z = (VZ - 6)2.

We consider three important self-similar motions. 1. Spherical detonation wave. 2. Point source explosion. 3. Strong implosion.

10.3.5.1.Spherical Detonation Wave. In this problem, one of the dimensional constants is the constant detonation velocity with dimensions [b] = LT-I and the other is density in the undisturbed medium with dimensions [u] = ML-". Thus k = - 3 , s = 0, 6 = 1 , K = 0. In this case, Eq. (10.3.22) has the following singular points (see Fig. I )

c ( 1 , 0) B {3(y - 1)2/(3y D (a,0 ) A

8

(0, 1 ) ( 1 , 03)

2/(3y - 1))

node saddle point saddle point node saddle point.

The integral curves through these points and the shock point boundaries are shown in Fig. 1 .

10.3

APPLICATlONS IN FLUID DYNAMICS

839

If the Chapman-Jouguet condition is satisfied, the point behind the detonation wave must lie on the branch AC of the parabola z = (1 - V2. The heat released Q is then related uniquely to the state ahead of the wave, represented by z, , while ( V , , z,) have the values

corresponding to a particular point on the branch AC of z = ( 1 - V ) 2 . On the integral curve behind this point A must decrease as we move towards the center of symmetry. Such a curve must originate at the node A . Thus the required integral curve defining the motion behind the detonation wave is the curve through A running below z = ( 1 - V ) 2 and terminating at ( V 2 ,z,) given by Eq. (10.3.30). Now at A , V = 0 but A # 0 so that the expansion of the gas behind the detonation wave only extends up to a certain sphere of finite radius. Inside this sphere we have a core of gas at rest. The distribution of velocity with radius is shown in Fig. 2.

10.3.5.2.Point Source Explosion. In this problem a strong explosion is idealized as resulting from the instantaneous release of a large constant source of energy from a point, line, or plane. One of the constants a is related to this energy with [a] = MLv-'T-2

840

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

WEAK

4”

OETONATION FRONT (SHOCK1

FIG.2. Velocity distribution behind a spherical detonation wave. [u denotes the total energy in the spherical case, the energy per unit length

and per unit area in the cylindrical and plane cases.] The second constant h is found by dividing the constant energy E by the density in the unIt is more convedisturbed medium p l . Then [b] = [ E / p l ] = Lw+2T-2. nient to work with b, = b1lrn,so that [b,] = L T 8 = J ~ T - ~ / ( ”In + ~this ). case, then, 6=2/(+ ~ 2),

k =v

-

1,

s =

-2.

(10.3.3 1)

The similarity variable is A =

y/blt2/(”+2).

( 10.3.32)

The fact that the integrated internal and kinetic energy in the disturbed field must equal the constant energy initially released yields an energy integral. This can be reduced to the form

( [ P / ( y - l)]

+ fRV2)(V - 6 ) + PV = const.

(10.3.33)

In the point source explosion problem the pressure ahead of the shock must be neglected to preserve similarity (this is the strong shock assumption). Since, in addition, the undisturbed medium is at rest, Eqs. (10.3.25) give the values behind the shock

When these are substituted in Eq. (10.3.33), the expression on the lefthand side vanishes identically. Therefore, the energy integral yields the following relation between z and V behind the strong shock wave:

10.3.

84 1

APPLICATIONS IN FLUID DYNAMICS

7

=

(y

1)VYV - 6 ) 2(Sly - V ) .

-

(10.3.35)

This is, in fact, the required integral curve of Eq. (10.3.22) in the point explosion problem. The singular points and integral curves of Eq. (10.3.22) for this problem are shown in Fig. 3. The required integral curve is a segment of CD joining the saddle point C (Sly, m) to the node 3 ( 6 , 0). The segments extend from C to the shock point defined by Eq. (10.3.34) (lying above and to the left of B ) . It can be shown that A + 0 zs V + S / y along the integral curve so that C corresponds to the origin in the ( r , f ) plane. The complete solution representing velocity, pressure and density distributions behind the strong shock wave can now be calculated. Full results are given in S e d ~ v . ~

10.3.5.3.The Strong Implosion. When a spherical shock wave is converging on its center, in a medium initially uniform and at rest, its strength increases rapidly as the center is approached. In the final stages of the motion, shortly before the shock collapses on its center, its motion is essentially independent of disturbances propagated from behind the shock. Under these conditions the motion of the imploding wave is self-similar. The main object in solving the equations governing this self-similar motion is to calculate the value of the index 6 in the similarity variable A = r / P . In the first two problems discussed above, the value of 6 is fixed by the dimensions of the two physical constants characterizing each D

0

2 5Y

FIG.3. The family of integral curves in the strong spherical explosion problem.

842

10.

DIMENSIONAL ANALYSIS A N D MODEL TESTING

case. In the implosion problem the value of 6 is fixed by the condition that the solution for the physical variables must be regular throughout the disturbed region. The motion is again governed by Eqs. (10.3.22)-(10.3.24) with K = 2(1 - S ) / u [only the cylindrical and spherical cases v = 2, 3 are of interest in this problem]. Equations (10.3.22)-(10.3.24) have singular points where z = ( V - 8)' corresponding to the positive or negative characteristics which arrive at the center at the same time as the imploding shock. If we integrate Eqs. (10.3.22)-(10.3.24) outwards from the shock point the singular negative characteristic A = A, (on which z = (V will be encountered. The value of 6 must be chosen so that the solution of Eqs. (10.3.22)-(10.3.24) passes through this point smoothly. The value is determined iteratively and is unique. It was first calculated by Guderley" and later by Butler.'2 Butler's value is 6 = 0.688377.

Figure 4 shows the characteristics and integral curves in the r-t plane. Equations (10.3.22)-(10.3.24) are integrated up to the positive characteristic OB reflected from the center of the implosion. A shock is fitted in the region t > 0 (post reflection) which cuts off the singular behavior on OB.

F I G . 4. The paths of characteristics and shock paths in the strong implosion problem.

K . G. Guderley, Luftfaahrt-Forsch. i9, 302 11942). D. S. Butler, "Converging Spherical and Cylindrical Shocks," ARDE Report 54/54. Fort Halstead, Kent, England, 1954.

10.4.

MODEL TESTING PRINCIPLES

843

10.4. Model Testing Principles 10.4.1. General Considerations

In the design of an aircraft, space vehicle, ship, or submerged vessel information is needed on the characteristics of the disturbance created by the motion of such a body through the atmosphere or the ocean. In particular, we need to know the resultant force and moment acting on the body under all conditions of operation as well as the distributed normal pressures and tangential stresses applied to the body by the surrounding fluid. This information is crucial in assessing the power requirements of the vehicle, its structural strength requirements and its stability characteristics. The force and distributed load data are needed in the preliminary design stage and must therefore be obtained from tests on models. The models are usually small scale versions of the vehicle to be designed and the flow field characteristics associated with them must be determined by simulating the conditions to be experienced by the final vehicle. This can be done in several ways. In the case of aircraft or space vehicles the possible ways of acquiring data are flight tests, sled tests, whirling arms, and wind tunnels. In the first three techniques the model is moved in some way through air at rest and recording instrumentation must either be carried on the model or its motion characteristics must be followed continuously photographically. In the fourth method the model is at rest relative to the observer and the laboratory where he works, and flow or force measurements are easier to control. The disadvantage of wind tunnel testing as compared with flight or range testing is that the flow disturbance must be generated by setting a stream of air in motion past a stationary wall rather than propelling a model through air at rest. This can be expensive and also requires that the extent of disturbance is confined within the walls of the wind tunnel producing it. For ship design, since the main interest is in effects at or just below the water surface most force and load information can be obtained by towing a model through a water tank and attaching all measuring instrumentation to a carriage on which the ship model is mounted. Water tunnels are used more for data on submerged vehicles or hydrofoils (submerged wings) and are restricted in size because of power requirements. In all types of model testing two requirements must be satisfied if the data acquired on the model are to be applicable to the design of the full scale vehicle. First, the model must be of identical shape to the design vehicle, that is, geometrical similarity must be satisfied. Second, the relations between the flow variables (such as force or pressure coefficients)

844

10.

DIMENSIONAL ANALYSIS A N D M O D E L T E S T I N G

and the independent variables, when expressed in dimensionless form, must be the same in both model test and full scale conditions, that is, dynamical similarity must be achieved. The obstacle to achieving geometric similarity is that in full scale flight the flow past the vehicle is effectively unconfined, while in model testing the flow field is generally bounded, by the walls in wind tunnel testing or by the ground in sled or range testing. Corrections must be made for such wall effects. Dynamical similarity is even harder to achieve since this requires that all the dimensionless physical numbers playing a role in the flow phenomenon have the same values in both test and full scale conditions. If several physical properties must be taken into account in a given flow field the only practical way to determine their effect is to assume that each dimensionless number influences the flow behavior independently of all others. 10.4.2. Applications in Fluid Dynamics

To illustrate the main principles of model testing in fluid dynamics we shall briefly discuss two areas ( 1) Determination of aerodynamic characteristics in high speed wind tunnels, (2) Determination of ship loading in towing tanks. 10.4.2.1. Wind Tunnel Testing. When testing models in high speed tunnels such as that shown in Fig. 5 , three principal dimensionless numbers must be considered together with the specific heat ratio y. These are Reynolds number, determining viscous diffusion effects; Mach number, determining compressibility effects; and Prandtl number, representing the effect of heat conduction (in comparison with that of viscosity). Other numbers have to be taken into account when more details of the thermal behavior of the vehicle are investigated. There is no difficulty in matching gas constant values between model and full scale conditions. Moreover, variation in Prandtl number does not have a large effect on overall aerodynamic characteristics and does not vary significantly from an average value of 0.74 in air. Thus Mach number and Reynolds number are the two most important parameters. When investigating high speed phenomena it is important to ensure that model tests are carried out at the full scale value of Mach number. Provided the body is thin corrections for small differences between tunnel and full scale values of Mach number can be made with the PrandtlGlauert or Transonic Similarity rules. For thick bodies at moderate or large angles of attack the aerodynamic characteristics vary with Mach number in a strongly nonlinear manner, especially in the transonic range, and the tunnel Mach numbers should be as close as possible to the full

10.4.

MODEL TESTING PRINCIPLES

845

FIG.5. Test section of 6-ft supersonic wind tunnel at NASA Ames Research Center, Moffett Field, California. [Courtesy NASA Ames Research Center.]

scale value. Changing tunnel Mach number is not a simple matter, especially in the supersonic range, and is usually achieved by changing the nozzle upstream of the working section. The reproduction of full scale Reynolds number presents a problem in wind tunnel tests both at low and high speeds. Since the test model is considerably smaller than the full scale body, the reduction in Reynolds number must be compensated by decreasing the kinematic viscosity. In low speed tests this can be done by raising the density of the tunnel stream; this idea is used in Compressed Air Tunnels. At higher speeds compressibility takes effect and increases in density are accompanied by undesirable increases in pressure-which could produce unreasonable loads on the test model. In this range, however, use can be made of the property that the coefficient of viscosity increases with temperature so higher Reynolds numbers can be attained by using a low temperature tunnel stream-this is the principle of the cryogenic wind tunnel.

10.

846

DIMENSIONAL ANALYSIS A N D MODEL TESTING

10.4.2.2. Scale Effect. In spite of these devices for increasing wind tunnel Reynolds numbers it is not generally possible to attain full scale Mach numbers and Reynolds numbers simultaneously. The influence of the difference between model and full scale Reynolds numbers on experimental data is called the scale effect and corrections for this must be made. When evaluating force coefficients on aircraft it is usual to distinguish between those depending mainly on normal pressure distributions, namely, lift and moment coefficients, from the resistance or drag coefficients depending primarily on integrated tangential stresses. For an aircraft of good aerodynamic design operating near cruising conditions there is very little scale effect on normal pressure coefficient distributions and the main concern is to evaluate the complete viscous drag coefficient. At supersonic speeds there is a contribution to drag from wave propagation but this depends mainly on Mach number. Provided wave drag contributions are first subtracted the scale effect on the remaining viscous drag (profile drag) is essentially independent of Mach number and can be estimated by the same techniques used at low speeds. The main difficulty encountered in determining scale effect is due to change in basic character of viscous flow from mainly laminar at low values to mainly turbulent at high values of Reynolds number. This is clearly illustrated in Fig. 6 showing experimental measurements of the drag coefficient of a flat plate plotted as a function of Reynolds number (using a logarithmic scale). The laminar part of the curve has a steeper slope than the turbulent part, while the drag coefficient rises in the range of transition from the laminar to turbulent regimes. The point of transition on a given plate from laminar to turbulent flow depends on many factors, including free stream Reynolds number, level of free stream turbulence, and plate roughness. 10 9

e

IOOOC,~ 6

EXPERIMENTAL

5 4

3 2.5

2 1.5

1 106

2

3 4 5 6 8

6

10

2

3 4 5 6 8

I07

2

3 4568

8

10

* = =810s

= '

FIG.6. Variation of flat plate skin friction coefficient with Reynolds number.

10.4. MODEL

TESTING PRINCIPLES

847

FIG.7. Sphere-cylinder reentry vehicle with flared afterbody.

The scale effect is more difficult to estimate in the design of re-entry and other space vehicles. Since rapid deceleration is required on re-entry into the earth’s atmosphere, shapes of space vehicles are deliberately chosen to produce high drag both from viscous and compressible sources. A typical shape is shown in Fig. 7. The detached shock wave ahead of the vehicle causes a large increase in pressure. The afterbody is usually flared to induce separation and high pressure on the rear of the body. Finally, the base pressure, as a result of expansion at the shoulder of the vehicle, is considerably below the pressure on the nose of the body. The flow pattern depends on a strong interaction between inviscid supersonic stream and flow in the boundary layer and wake, making it difficult to separate Mach number and Reynolds number effects. In fact, the best data for design of space vehicles are provided by solutions of theoretical models using large scale computers. 10.4.2.3. Tunnel Wall Corrections. In all wind tunnel testing corrections must be made for the confinement of the working section by the tunnel walls. At subsonic speeds the effect of the walls on the aerodynamics of the test model can be found by representing the latter as a local singularity (source, vortex, or line of sources and vortices), determining the image system of the singularity and calculating its contribution to the flow field on the test body surface. This is then subtracted from the observed low field. In supersonic wind tunnels wall corrections (from inviscid sources) can be eliminated by making sure that disturbances originating at the body, on reaching the tunnel walls, are reflected to points downstream of the model. Another way of handling the tunnel interference problem is to use flexible tunnel walls and adjust these, for a given model, so that the stream surfaces generated during the test coincide with those in unconfined flow past the same model. More recently it has been proposed to use fixed walls along which the pressure distribution can be adjusted to correspond to that in unconfined flow. 10.4.2.4. Ship Model Testing and Cavitation. When investigating the forces acting on vessels moving on the ocean or other water surface three

a48

10. DIMENSIONAL ANALYSIS A N D MODEL TESTING

dimensionless parameters are of importance. First, since viscous drag forms a large part of ship resistance, the Reynolds number based on the length of the ship, the ship speed and the kinematic viscosity of water must be considered. Second, the other component of ship drag is due to surface wave resistance, and this depends on a dimensionless combination of ship dimensions, ship speed and gravity called the Froude number U/(dg)1’2. Third, the performance of ship propellers or hydrofoils depends on cavitation effects, represented by the number (pr - p , ) / l p U 2 , where pf is the free stream pressure and pv is the vapor pressure. To assemble data on ship drag the contributions due to viscosity and wave motion are assumed to be independent, the former being proportional to the wetted surface area and the latter proportional to the volume of water displaced by the ship. To estimate the frictional coefficient, data on flat plate resistance are used with the effects of curvature and thickness neglected. The wave drag coefficient is then determined by measuring the total drag in a towing tank and subtracting the estimated friction contribution. To model cavitation effects use is made of water tunnels, in which the vapor pressure can be varied and of hydroballistic tanks (used mainly for water entry investigations) in which the atmospheric pressure can be varied. Reproduction of full scale cavitation numbers in model tests presents no serious problem.

AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. A

Abel, S. J., 436 Adam, B., 753 Adam, N. K., 507 A d a m , E. D., 572 Adomeit, G., 490 Ahlborn, B., 602 Albe, F., 747 Alcock, A. J., 485, 486(87), 742 Allen, H., 765, 773 Alpher, R. A , , 788, 794 Alyamovski, V. N . , 467, 470(21), 479, 481(52) Alzofon, F. E., 683 Amery, B. T., 609, 610(115) Andreev, S. I., 697 Andreeva, L. E., 511, 559 Andreeve, S. I., 697 Anger, V., 613 Antropov, E. T., 467, 470(21), 479, 481(52) Appel, D., 633 Appleton, J . P., 631, 633 Ardila, R., 602 Arena, A , , 770 Armstrong, B. H., 638 Arons, A. B., 527, 535(25), 812 Ash, R. L., 675 Asher, J. A., 408(1), 409, 420(1), 421(1), 42%I ) Ashkenas, H., 436, 496, 654, 656 Ashkin, A,, 716 Attal, B., 431, 433(58) Ayscough, P. B., 634 B

Bachmann, R. C., 673 Bader, J. B., 476, 477(40) 849

Baganoff, D., 600, 604 Bailey, F. J., Jr., 520 Baker, D. J., Jr., 815, 817, 819 Banister, J. R., 508 Banks, C. V., 623, 624, 632(43) Barbour, J. P., 721 Barker, L. M., 551, 606, 607(54,112),609, 6 1O( 1 14) Barnes, F. A., 689 Barrett, J . J., 430, 488 Barton, S. C., 657, 658 Bartsch, C. A , , 680 Batchelor, G. K., 798 Bates, D. R., 491 Battaglia, M . P., 735 Bauer, A,, 691, 707 Bauer, E., 466 Bauman, R. P., 623, 624(44), 643(44) Bazhenova, T.V., 558, 577(62), 602(62) Bearden, J. A,, 799 Beardsley, R. C., 815 Beavers, G. S., 801 Beck, J. V., 676 Becker, M., 490 Belford, R. L., 654, 657(143), 658(143) Bell, P. M., 610 Bellhouse, B. J., 682 Belser, R. B., 680 Bender, H., 735 Benedict, R. P., 460 Benner, R. E., 410 Bennett, S. J., 506 Bentley, F. F., 628 Beranek, L. L., 521, 556, 570(18) Berkowitz, J., 647 Bernstein, L., 527, 535(27), 536(27), 542(27), 548(27), 573, 579(27) Bershader, D., 666, 680(IO), 681(10), 794, 79337)

850

AUTHOR INDEX

Besshaposhnikov, A. A., 485 Bettis, R. J., 703 Bienkowski, G. K., 453 Bier, K., 654, 656 Billman, K. W., 474, 47338) Biordi, J. C., 660 Bird, R. B., 61 1 Bjornerud, E. K. 477 Black, P. C.. 429 Blass, W. E., 627, 629(52), 637(52), 638(52), 641(52), 642(52) Bleakney, W., 527, 535(25), 557, 787, 794 Bloembergen, N . , 722 Bochkajov, A. A., 453 Bogdan, L., 452, 683, 684(73) Bohme, D. K., 660 Boiarski, A. A., 436, 437, 451(45), 452(45), 492, 496 Bonczyk, P. A., 408(8), 409, 421(8), 422(8), 428(8), 431(8), 488, 489(89j) Born, M., 416(16), 417, 710, 740(51) Bowersox, R., 528, 529(30) Bowles, K. L., 482 Bowman, C. T., 633, 634 Boyd, G. D., 716 Boyd. M. E., 612 Boyd, R. K., 643 Boyer, A. G., 431 Boyer, D. L., 812, 813(14) Bradley, D., 461 Bradley, D. J., 735 Bradley, J. N . , 656, 658(149), 788 Bradley, R. B., 431, 433(57) Braginskii, S. 1.. 698 Branscomb, L. M., 494 Brenner, H., 799 Brewer, L., 481 Brewster, J. L., 721 Bribes, J. L., 428 Bridgman, P. W., 593, 827 Bridoux, M., 429 Brienza, M. J., 719 Brinkman, H., 481 Britton, D., 473, 474 Brocklehurst, B., 636 Broida, H. P., 479, 481, 633 Brombacher, W. G., 506

Bromberg, J. P., 515 Brooks, C. J. W., 620 Broughton, F. P., 415 Brown, E. A., 664 Brown, J. D., 643 Bryan, K., 814, 815 Bryer, D. W., 519 Buchanan, T. D., 670, 671(14)67.51 14) Buckingham, E., 826 Buckley, J. C., 808 Biitefisch, K. A., 437, 453, 493, 496 Bull, M. K., 525 Burnett, D. R., 672 Bums, G. 473, 474, 643 Busing, J. R., 682 Butler, D. S., 842 Buyevich, F. A., 478 Buzyna, G., 808, 814 Bynum, D. S., 528, 529(31), 533(31), 535(31), 536(31), 548(31), 555(31)558(31), 569, 581(31), 590

C Cady, W. G., 542 Calcote, H. F., 660 Caldwell, D. R.,817 Camac, M., 444, 449, 450, 679, 685 Campbell, D., 491, 493 Carabetta, R. A., 643 Carbonnier, F. M., 721 Carden, W. H., 682 Carlson, J., 529 Camngton, A., 634 Camngton, T., 633, 643 Carslaw, H. S., 665, 666(8)-668(8) Cattolica, R., 415, 487, 490 Center, R. E., 452 Cerasoli, C. P., 81 I Chaces, W. G., 704 Chamberlain, J. W., 494 Chambers, J . T., 673 Chandra, S . , 431, 432(54) Chang, C. C., 816 Chang, R. K., 410, 429 Charlson, R. J., 664

AUTHOR INDEX

Charters, A. C., 780 Charters, P. E., 642 Chatanier, M . , 590 Chekrnayov, S. P., 453 Chen, W. S., 577, 604(82) Chew, H. W., 410 Chilton, C. H., 612 Chue, S. H., 519, 784 Chupka, W. A,, 647 Churchill, R. V., 666, 667(Y) Churilova, S. A., 719 Clapham, P. B., 506 Clarke, J. F., 682 Clemens, P., 781 Cloupeau, A,, 790 Clouston, J. G., 469, 470(26) Coe, D., 496 Coe, J. R., 799 Cole, R. H., 577, 591(75) Coles, D., 815 Collins, D. J., 683 Colthup, N. B., 628 Compaan, A., 43 I , 432(54) Condon, E.U., 622, 637(37), 638(37) Coney, T. A., 430 Conrads, H., 706 Cook, R. L., 629 Cook, W. J., 667, 681 Cooney, J., 430 Cooper, M., 671 Cornu, A., 661 Courtney-Pratt, J. S., 726, 732, 735 Cowan, R. D., 640 Cranz, C., 737 Creswell, R., 658 Cristina, V. D., 677 Cross, J. L., 513 Crunelle-Cras, M., 429 Cullen, R. E., 577, 602(79) Culp, M . A . , 476, 477(40) Curnrning, C., 642 Cunningham, J. W., 452 Curtis, C. W., 580, 597, 598(83), 600(100) Curtiss, C. F., 611

D Daborn, J. E., 506 Dandliker, R., 720

85 1

Dam, A., 751 Dainty, J. C., 712 Dal Nogare, S., 618, 620(28) Dal Pozo, P., 694 Danberg, J. E., 461 Dankert, C., 437 Dann, J. B., 476, 477(40) Davidson, N., 473, 474, 631 Davies, A. G., 635 Davies, L., 576, 599(73), 600(73) Davies, R. M., 593, 594(99), 600 Davies, W. E. R., 437 Davis, W. E. R.,485 Dean, M . , 538 Decker, G., 751 Deckers, J., 660 de Haas, N., 635, 636 de Leeuw, J. H., 437 Delhaye, M . , 429 Delker, D. A,, 628 DeMaria, A. J . , 719 Den Hartog, J. P., 559, 579(64) Denton, R. T., 730 De Silva, A. W., 485 DeVault, G . P., 597, 600(100) Dewhurst, R. J., 703 Diaconis, N., 783 Dibble, R. W., 417, 487 Diefenderfer, A. J., 553 Dieke, G. H., 640, 641 Diesen, R. W., 658 Dillard, J. G., 646 Dinkelacker, A., 525, 551(22) Di Valentin, M. A . , 658 Dixon-Lewis, G., 642 Dobrer, E. K., 602 Dougherty, J. P., 482 Douglas, R. D., 538 Dove, J. E., 654, 657-659 Drabkina, S. I., 698 Drake, M. C., 42 I , 422(23), 429(23), 430, 488 Drake, R. M . , 459, 460(3), 665 Draxl, K., 646 Dnscoll, J . F., 437, 451, 452 Druet, S. A. J., 431 Dubovik, A. S., 726, 733, 735 Duckett, R. J., 436 Ducoffe, A . L., 509

852

AUTHOR INDEX

Duguay, M. A , , 731 Dushin, L. A., 485 E Early, R. A,, 683 Eberius, K. H., 660 Eckbreth, A. C.. 408(8,10), 409, 421(8), 4 2 W . 428(8), 431, 432(53), 433(56), 488, 489(89j) Eckert, E. R. G., 459, 460(3), 665 Edgerton, H. E., 694 Edwards, D. H., 576, 595(72), 599, 600(72,73) Egerova, V. F., 698, 699 Eiben, K., 635 Elenbaas, W., 478, 690 Elliott, J . A , , 521 Ernrich, R. J., 557, 787, 794 English, T. C., 647 Epstein, V. A., 466, 491(14) Erickson, R. E., 628 Erybasheva, L. F., 479 Evans, D. E.. 482, 485 Evans, R. D., 406 Evans, R. G., 408 Ewing, E. M.,581, 585(86) Ewing, G. W., 613, 61316). 617(16), 618(16), 622(16)-624(16), 627(16), 630(16), 637(16), 643(16)

F Fdizullov, F. S., 463( lo), 464, 467, 468(24), 470, 479, 481(52) Faller, A. J., 810, 812, 814, 815, 817(8,10), 818(8,19), 819 Fan, C. Y., 494 Fannibo, A. K., 466, 491(14) Farley, D. T., 482 Farrow, R. L., 433 Fay, J . A., 463, 678 Feigl, F., 613 Feinberg, R. M., 679, 685 Fejer, J . A., 482 Felderrnan, E. J., 667 Feldman, S., 789 Feldrnan, T., 642

Felrnlee, W. J., 658 Felske, A., 749 Fergason, J., 751 Ferrar, C. M., 695 Ferriso, C. C., 478 Fery, C., 466 Fessenden, R. W., 635 Ficco, G., 485 Field, F. H., 646, 648 Fischer, H., 700, 702, 706 Fisher, C. H., 452 Fkushin, M. I . , 478 Fleming, J. W., 431 Fletcher, C. A. J., 835 Folk, R., 580. 597(83), 598(83) Foner, S. N . , 649, 660(136) Fontijn. A., 660 Forrest, M . J., 485 Fowlis, W. W., 808, 813, 814 Fox, A . G., 708 Fox, G., 580, 597(83), 598(83) F r a q o n , M., 746 Franken, P. A,, 722 Frankevich, Ye. L., 646 Franklin, J. L., 646 Franklin, R. E., 517 Freeman, S. K., 643, 645 Friedlander, G., 630 Frisk, B., 672 Fristrom, R. M., 659 Friingel, F., 694 Funfer, E., 485, 701 Fultz, D., 801(1), 802, 808(1), 813, 814. 816, 818

G

Gabor, D., 743, 744 Gadamer, E. O., 435 Gallagher, J. S., 612 Gardon, R., 673, 675 Gaufres, R.,428 Gaydon, A. G., 463, 464, 466, 468(7), 469, 470, 479, 481, 482, 632. 637, 639(96,97), 640(96), 641(96), 642 Generalov, N . A., 470, 473, 474, 485 Georg, E. B., 466, 491(15) George, A. R., 671 Gerry, E. T., 485

AUTHOR INDEX

Giedt, W. H., 430, 488, 672, 673 Giordmaine, J . A , , 725 Glaser, G., 699 Glass, G. P.. 654 Glauber, R. J., 712 Glauert, H., 765 Glenn, W. H., 719 Goddard, F. E., 757, 760(1), 793(1) Goddu, R. F., 628 Godfrey, T. B., 799 Goin, K., 758, 763, 766(2), 772, 775, 783 Goldberg, L., 482 Goldman, L. M., 420, 421(20), 425, 427(26) Goodings, J. M.,660 Goodrich, G. W., 653 Gordy, W., 629 Gorlin, S. M., 769 Goss, L. P., 431, 433(57) Grabner, L. H., 430 Graf, J., 737 Grase, F., 429 Grasselli, J . G., 625, 628(51) Grau, G., 709 Greenspan, H. P., 802 Greer, W. L., 612 Grey, J., 462, 654 Griem, H. R., 482, 640 Griffith, T., 515 Gross, R. A., 790 Griin, A. E., 436, 437, 441(3), 445, 446(34), 449 Griinberg, R.,696 Griitter, A. A , , 720 Grundhauser, F. J . , 721 Gruszczynski, J., 684 Gubanov, A. M.,480 Guderley, K. G., 842 Guenther, A. H., 703 Gulick, W. M., 623, 624(42), 627(42), 629(42), 632(42), 637(42), 641(42) Gurvich, L. V., 646 Gutrnan, D., 654, 657(143), 658(143)

H Haasz, A. A., 437 Hadland, R., 735 Hadlock, R. K., 812, 814 Hagena, 0.. 654, 656

853

Hager, N. E., 664 Haider, K., 558 Hall, J. G., 664 Hall, R. J., 431 Hall, T. A , , 408 Hansen, C. F., 683 Hansen, J. W., 731 Hanson, R. K., 600, 681, 682 Happe, A., 749 Happel, J., 799 Harbour, P. J., 453 Harnett, L. N., 490 Harnwell, G. P., 570 Harper, J., 769, 770 Harrison, G. R.,637, 639(98) Hartley, D. L., 419 Hartunian, A. R.,557 Hartunian, R. A., 681, 682, 684(51), 786, 787(26) Harvey, A. B., 420, 431 Harvey, J. K., 436 Harvey, W. N., 437, 451 Hastie, J. W., 430, 660 Hattel, H. C., 475 Hay, A. J., 654, 657(143), 658(143) Hayes, D. B., 609 Hearne, L. F., 673 Heftmann, E., 617 Helland, K. N., 817 Heller, S. R., 661 Herron, J. T., 646 Hertzberg, A., 664 Herzberg, G., 425, 494, 622, 624(39,41), 627(39,40), 637(38-41), 638(38-41), 639(39-41), 643(41) Herziger, G., 707 Hess, S. L., 812, 814 Hessel, M., 525, 551(22) Heyrovsky, J., 621 Hickman, R. S . , 436, 437,442, 496, 672 Hide, R., 802, 813 Hildebrand, F. B., 529 Hill, A. E., 722 Hill, R. A., 419, 436, 496 Hilliard, M. E., 451 Hirschfelder, J. O., 611 Hirth, A., 717, 737, 739, 746, 747, 749 Ho, W. C., 436, 496 Holder, D., 764

854

AUTHOR INDEX

Hollenbach, R. E., 417, 487, 606, 607(112), 609, 610(114) Holley, W. L., 508 Holt, M., 835 Holtbecker, H., 558 Holton, J. R., 817 Holtz, T., 490 Hooker, W., 478 Hopkinson, B., 593, 5%(98) Hora, H., 706 Hornig, D. F., 473, 474 Hornung, H. G., 704 Hoshizaki, H., 683 Hougen, 0. A., 612 Howey, D. C., 677 Hoyermann, K., 636, 660 Hudson, R. L., 649, 660(136) Hugenschmidt, M., 699, 736, 739, 742, 748, 751, 753(23) Huldt, L., 640 Huni, J. P., 602 Hunt, J. L., 672 Hunter, W. W. Jr., 436, 437, 451, 491 Hurle, I. R., 463(7), 464, 466, 468, 469, 470(7,26,28), 632 Hurwicz, H., 676 Huston, W. B., 520 Hyer, P. V., 815 I Ibbetson, A., 816 Inaba, H., 429 Incropera, F. P., 684 Inger, G. R., 682 Ingram, D. 1. E., 634 Ip, J. K. K., 473, 474 Isaenko, V. J., 698, 699(21) Isaev, I. L., 467, 470(21), 479, 481(52) Ivanov, V. N., 479

J Jacobs, P. F., 462 Jaeger, J. C., 534, 665, 666(8)-668(8) James, C. G., 640 Jamet, F., 408 Jardetsky, W. S . , 581, 585(86) Jeffery, P. G., 616, 617. 618(20), 619(20), 620(20)

Jennings, K. R.,636 Jessey, M. E., 680 Johnson, D. L., 664 Johode, F. C., 486 Jones, I. R.,600 Jones, J. H., 430, 488 Jones, R. A., 672 Jorzik, E., 558 Joseph, C. D., 801 Jost, W., 632 JUrgens, G . , 479, 480(55) Juvet, R. S., 618, 620(28) Jzawa, Y., 485 K Kane, E. D., 763 Kantrowitz, A., 654 Karelov, N. V., 496 Karmen, K. N., 602 Kassem, A. E., 436, 442, 4% Katzenstein, J., 482 Kaufman, F., 636 Kaye, G. W. C., 612 Kedrinskii, V. K., 591, 593 Kegel, W. H., 485 Keilmann, F., 714, 750(56) Keliher, T. J., 673 Kelso, J. R.,636 Kemp, N., 463, 678 Kennedy, J. W., 630 Kennedy, W. S., 677 Kerker, M., 410 Kern, R. D., 658 Kestin. J., 799 Key, M. H., 408 Kidd, C. T., 673, 674(27) Kiemle, H., 746 Kimbell, G. H., 469, 470(28) Kingery, M., 781 Kipping, P. J., 616, 617, 618(20)-620(20) Kirchoff, R. H., 675, 676 Kistiakowsky, G. B., 654, 656, 658 Kistler, A. L., 577, 604(81,82) Kitayeva, V. F., 467, 470(21), 479, 481(52) Kleen, W., 707 Klein, M., 612 Klemsdal, H., 639 KIBckner, H. W., 429 Kmonicek, V., 681 Knaal, E., 640

855

AUTHOR INDEX

KneubOhl, F. K., 753 Knewstubb, P. F., 660 Knox, R. A., 813 Knuth, E. L., 654, 656(147) Kochnev, V. N., 590, 591(87) Kopf, U . , 714 Kogelnik, H., 708 Kolb, A. C., 640 Kolsky, H., 581, 585(85), 593, 596(85) Komar, J. J., 437, 451(44) Kondratyev, V. N., 646 Koppitz, J., 696 Kosynkin, V. D., 473 Kotolov, A. B., 697 Kozlov, G. I., 470, 485 Krauss, L., 698 Krempl, H., 698 Kronast, B . , 485 Krongelb, S ., 636 Kriiger, R., 704 Kryukov, P. G., 719 Kudryavtsev, E. M., 467, 468(24), 470(22- 24) Kuethe, A. M., 458 Kuhl, J., 724 Kunze, H. J., 485 Kydd, P. H., 656 L

Laby, T. H., 612 L h g , L., 625 Langstroth, G. O., 491 Lapp, M., 408(1,6,9), 409, 412(6), 414(6), 419(6), 420, 421, 422(6,22,23), 423(24), 424, 425, 426(22), 427(24, 26). 428, 429, 478, 488 Lapworth, K. C., 486 Lauver, M. R., 683 Lawrence, T. R., 576, 599(73), 600(73), 643 Lawson, A. E., Jr., 619 Lazdinis, S. S . , 437, 451(45), 452(45), 492, 493 Lazzara, C. P., 660 Lederer, R. A., 609 Lederrnan, S., 408(5), 409, 422(5), 488 Ledford, R. L., 528, 529(31), 533(31), 535(31), 536(31), 548(31), 555(31)558(31), 569(31), 581(31), 590(31), 673, 674(22,27), 677(22) Lee, H. F., 437

Lee, L. P., 436, 496 Leetmaa, A ,, 813 Leidenfrost, W., 799 Leinhardt, T. E., 451 Leith, E. N., 744 Lempicki, A., 715 Leonard, D. A., 428 Leonas, V. B., 612 Letarte, M.,664 Letokhov, S., 719 Lewis, B., 463 Lewis, C. L. S., 408 Lewis, J. W. L., 430, 488, 491-493 Lewy, S., 496 Li, T., 708 Lide, D. R., 624, 629(45) Liepmann, H. W., 458, 777 Lillicrap, D. C., 436, 451, 496 Lilly, D. K., 814 Lion, K. S., 535, 548(39) Lippiatt, J. H., 643 Lissner, H. R., 538 Little, E. M., 486 Littlewood, A. B., 618, 619(26), 620(26) Lochte-Holtgreven, W., 463(12), 464, 479 Long, R. R., 816 Losev, S. A., 463(6), 464, 470, 473 Loubsky, W., 795 Lovberg, R. H., 483, 484(73) Love, A. E. H., 581, 585(84) Lucquin, M., 429 Lukens, L. A , , 684 Lunney, J. G., 408 M McCaa, D. J., 452, 680 Maccoll, J. W., 835 McDonald, J. R., 430 McDowell, C. A., 646, 647(128), 649, 650( 128) McGuire, R. L., 430, 488 Mclntosh, M. K., 576, 591(74), 592(74), 600(74) Mack, M. E., 719 McKay, H. A. C., 630(67), 631 McLachlan, A. D., 634 McLaren, I. H., 649 McRonald, A. D., 437, 451, 452 Maecker, H., 479 Maguire, B. L., 436, 437, 439, 496

856

AUTHOR INDEX

Maizell, R. E., 615 Mak, A. A., 698-700 Maker, P. D., 722 Maksimenko, V. A., 473 Malkus, W. V. R., 815 Mallin, J. R., 437 Malyshev, G. M., 485 Mann, C. K., 623, 624(42), 627(42), 629(42), 632(42), 637(42), 641(42) Mao, H. K., 610 Margulis, D. I., 525, 550(23), 575(23) Marrone, P. V., 436, 496, 681, 682, 684(51) Marsden, D. J., 436, 496 Maslach, G. J., 763 Mason, S. B., 681 Mason, W.P., 542, 544, 545, 547 Massot, R., 661 Mather, R. E., 648 Mathews, C. W., 624, 632(46), 637(46), 638(46), 641(46) Mattern, P. L., 433 Matthews, K. J., 461 Matthews, R. K., 670, 671(14)-675(14) Matveev, Yu. A , , 719 Maxwell, J. B., 612 May, A . D., 419 Mayo, E. E., 671 Medvedev, V. A., 646 Meier, G. E. A . , 525, 551(22) Meitzler, A. H., 533, 603 Memory, J. D., 634 Menard, W. A,, 683 Michael, J. V., 654, 658 Michel, K. W., 632 Michel, L., 706 Middleditch, B. S., 620 Miles, B. M., 639 Miller, B., 790 Miller, J. M., 619, 630 Miller, R. C., 725 Millikan, R. A,, 800 Milne, G. W. A., 661 Moffat, R. J., 460 Moir, L. E., 664 Mokry, M., 775 Mollo-Christensen, E. L., 817 Monan, R., 428 Mooney, K., 815 Moore, A., 408 Moore, H. K., 704 Morley, C.. 633

Moulton, D. McL., 654, 658(141), 659 Muller, R., 707 Mueller, T., 770 Miiller, W., 731, 732(79) Munson, B., 648 Muntz, E. P., 436, 437, 442, 446(48), 451 453, 454(63), 490-494, 496 Myerson, A. L,, 633 N

Naboko, 1. M., 558, 577(62), 602(62) Nagel, M. R., 690 Nakagawa, Y., 814 Ndefo, D. E., 835 Neal, T., 408 Neely, G. 0.. 420 Nelson, L. Y., 420 Nelson, R., 540 Nerem, R. M., 476, 477(40), 683 Nesterikhin, Yu. E., 463(9), 464, 479(9), 602 Neubert, H. K. P., 535, 538, 548(38) Nibler, J. W., 431 Nicholls, R. W., 479, 492, 638 Nielsen, A . H., 627, 629(52), 637(52), 638(52), 641(52), 642(52) Nika, G. G., 658 Niki, H., 654 Nikolaev, V. M., 478 Nolen, R. L., Jr., 430 Nye, J. F., 542 0

Ocheltree, S. L., 437, 446(36), 451 Ohman, L., 775 Okamura, .I.P., 618, 620(27) O'Laughlin, J. W., 623, 624, 632(43) Oldenberg, O., 494 Olschewski, H. A., 643 Ornstein, L. S., 479, 481 Osipov, A. I., 463(6), 464 Ovechkin, V. Ya., 473, 474 Owen, J. D., 600 Owen, J. F., 410 P Padley. P. J., 642 Palmer, H. B., 473, 474, 643

857

AUTHOR INDEX

Pancirov, R., 654, 657(143), 658(143) Pankhurst, R. C., 519, 764 Papp, J. F., 660 Pappenheimer, J. R., 537 Park, C., 762 Parker, G . W., 634 Parkinson, W. H., 479 Patrick, R. M., 485 Paul, W., 650, 651(139) Pavlichenko, 0. S., 485 Pealat, M., 431, 433(58) Pearse, R. W. B., 637, 639(97), 642(97) Peeters, J., 660 Penner, S. S . , 427, 463, 464, 477, 478, 638, 640(99), 683 Penney, C. M., 408(1,6), 409, 412(6), 414(6), 419(6), 420, 421, 422(6,22,23), 424(22), 425, 426(22,26), 428, 429, 488 Perry, C. C., 538 Perry, D. S., 643 Perry, R., 612 Pert, G. J., 703 Peters, C. W., 722 Peterson, 0. G., 716 Petne, S. L., 436, 437, 451, 452, 492, 493, 496 Petunin, A. N., 519, 570(15) Pfeffer, R. L., 808, 814 Phashinin, P. P., 485, 486(87) Phillips, N . A,, 816 Phillips, W. H., 520 Pierce, A,, 482 Pilipetski, N . F., 466, 491(14) Pina, M., 430 Pitz, R. W . , 487 Pivonsky, M., 690 Placzek, G., 425, 427(27) Plastinin, Yu. A., 478 Polaert, R., 737 Polanyi, J. C., 642, 643 Polloni, R., 694 Pope, A., 758, 763, 766(2), 769, 770, 172, 775, 783 Potapov, A. B., 479, 481(52) Potopov, A. V., 467, 470(21) Pouchert, C. J., 628 Powars, C. A,, 677 Powell, H. M., 430, 488 Powell, R. L., 749 Press, F., 581, 585(86) Pressley, R. J., 714

Pressmann, Z., 705 Price, L. L., 430, 488 Price, W. J., 621 Purnell, H., 620

Q Quinn, W. E.. 486

R Rabinowicz, J., 680 Ragland, K. W., 577, 602(79) Rahn, L. A., 433 Raizer, Yu. P., 485 Rall, D. L., 673 Ramsden, S. A., 485, 486(87), 703, 742 Rank, D. H., 410, 643 Rapp, H., 706 Ready, J. F., 706 Rebrov, A. K., 453, 496 Reddy, N . M., 682 Redman, R. E., 482 Reece, J. W., 681 Regnier, P. R.,43 1 Reinecke, W. G., 671 Reinhard, H. P., 650, 651(139) Reuter, J. L., 660 Riddell, F. R., 463, 678 Rindal, R. A . , 677 Rindner, W., 540 Rixen, W., 490 Robben, F., 415, 436, 487, 490, 496 Robertson, E. R.,746 Robinson, A. L., 542, 547(49), 590 Robinson, A. R., 815 Robinson, J. W., 625, 627(50), 628(50), 632(50), 639(50), 641(50), 643(50) Rodgers, W. E., 654, 656(147) Rohr, H., 751 Roesler, F. L., 494 Ross, D., 707, 746 Rossler, F., 481 Roh, W. B., 431, 432(55), 433(57) Romanko, J., 642 Roquemore, W. M., 431, 433(57) Rosasco, G. J., 430 Rose, P., 791

858

AUTHOR INDEX

Rose, P. H., 463, 677, 678, 681(39), 682(38,39), 796 Rosenblatt, G. M., 430, 488 Rosenstock, H. M., 646 Roshko, A., 458, 777 Rothe, D. E.. 437, 445 Royer, H., 727 Rubins, P. M., 428 Ruffner, D., 705 Rulyev, Yu. K., 466, 491(15) Rupert, J. W., 808 Russell, D., 774 S

Safron, S., 682 St. Peters, R. L., 429 Salamandra, G. D., 558, 577(62), 602(62) Sallcap, J. R., 474, 475(38) Salpeter, E. E., 482, 483 Salzman, J. A., 430 Samelson, H., 715 Samiulov, E. V., 612 Samoletov, E. A., 572 Sandeman, R. J., 704 Saunders, A . W., 420 Saunders, K. D., 815 Savage, C . M., 722 Sawyer, D. T., 618, 620(27) Sawyer, G. A , , 486 Schaaf, S. A , , 763 Schafer, F. P., 715 Schardin, H., 695, 737 Schetzer, J . D., 458 Schewe, G., 525, 551(22) Schmidt, H., 466, 706 Schmidt, W., 724 Schneggenberger, R., 430 Schneider, P. J . , 668 Schonbach, K. H., 706 Schdtzau, H. J., 753 Schopper, E., 437, 445, 446(34) Schrieber, P. W., 431, 432(55), 433(57) Schrotter, H. W., 429 Schultz, D. L., 682 Schulz, P., 691, 695 Schumacher, B. W., 435, 437, 446(34) Schwarz, A. C., 609 Schwarz, J., 706 Schweiger, G., 436, 496

Schwertl, M., 701, 702 Sears, W. R., 775 Sebacher, D. I., 436, 437, 446(35), 49 1 Sedov, L. I., 827, 834, 836, 841 Seiff, A, 781 Sengers, J. M. H. Levelt, 612 Sengers, J. V., 612 Setchell, R. E., 428 Sevast' yanova, I. K., 558, 602(62) Shapiro, S. L., 731 Sharafutdinov, R. G., 453, 496 Sharma, P. K., 654, 656(147) Shatberashvili, 0. B., 719 Shaub, W. M., 431 Shaw, R., 517 Sheffield, S. A., 609 Shelby, F., 436, 496 Sherman, F. S., 654, 656 Sherman, M. P., 462 Shirley, J. A., 431 Shook, C. A., 580, 597(83), 598(83) Shortley, G. H., 622, 637(37), 638(37) Shuler, K. E., 479, 481(50) Siddon, T. E., 522-524, 571(20) Siegman, A. E., 710 Simpson, D. I., 506 Simpson, T. B., 680 Sinani, I. B., 591 Sipachev, G. F., 466, 478, 491(15) Sitsinskaya, N. M.,735 Skinner, G. T., 666, 680, 681(11) Slezinger, I. I., 769 Smeets, G., 740, 743 Smigielski, P., 727, 747 Smith, A. W., 548 Smith, I. W. M., 633, 634 Smith, J., 430, 488 Smith, J. A., 437, 451, 452, 480 Smith, R. B., 436, 453, 4% Smith, R. G., 725 Smith, W. R., 601 Smotherman, W. E., 528, 529(31), 533(31), 535(31), 536(31), 548(31), 555(31)558(31), 569, 581(31), 590(31), 673, 674(27) Snavely, B. B., 716 Sneddon, 1. N., 527 Snyden, T. M., 660

859

AUTHOR INDEX

So, R. M. C., 424 Sobolev, N . N., 467, 468(24), 470, 479, 481(52) Sochet, L. R., 429 Softley, E., 453, 454(63) Soloukhin, R. I., 463(8,9,13), 464, 470, 479(9), 525, 550(23), 557, 558, 575(23), 577, 591, 593, 602, 604(58), 796 Solukhin, R. I., 577, 602(78) Sommers, P. I., 480 Spiegel, J., 765, 773 Spreiter, J. R., 834 463, 678, 681(39), Stankevics, J. 0.. 682(39) Stansbury, E. J., 642 Stark, W., 791 Stark, W. I . , 463, 678, 682(38) Starner, K. E., 674, 677(28), 682 Stebnovskii, S. V., 593 Stempel, F. C., 673 Stenzel, A., 700-702, 738 Stepanov, G. V., 590, 591(88) Stephenson, D. A., 413 Stetson, K. A., 749 Stewart, J. E., 627, 641(53), 642(53) Stewartson, K., 815 Stickford, G. H., 683 Stoicheff. B. P., 410 Stommel, H., 812 Storey, R. W., 451 Strandberg, M. W. P., 636 Stratton, T. F., 486 Straty, G. C., 572 Streed, E. R., 640 Strobel, H. A., 630 Strong, J., 509 Strub, H., 691 Strutt, J. W., Lord Rayleigh, 832 Stryland, J. C., 419 Stupochenko, E. V., 463(6), 464 Sturtevant, B., 657 Sugden, T. M., 640, 642, 660 Sulzer, P., 472, 473(31), 474 Sundquist, H., 812, 816, 818(12) Sutton, M. M., 642 Svelto, O., 694 Sviridov, A. G., 467, 470(19) Swindells, J. F., 799 Switzer, G. L., 431, 433(57) Szymanski, H. A., 628, 643

T Talbot, L., 415, 436, 487, 490, 496 Tang, S . , 601 Taran, J . P. E., 431, 432(55), 433(58) Tatro, P., 817 Taylor, G. I . , 808, 835 Terhune, R. W., 722 Thomann, H., 672 Thomas, A. S. W., 525 Thomas, K . M., 437 Thomas, N., 481 Thomer, G., 408, 463(11), 464 Thompson, E., 485 Thompson, W. P., 682 Thomson, L. W. T., 527, 529 Tilford, C. R., 506 Timm, U., 696 Timoshenko, S . , 559 Tobin, M. C., 643 Todd, J. F. J., 648 Tong, H., 672 Tong, K., 774 Treanor, C. H., 792 Tredwell, J., 701 Trimmer, L. L., 670, 671(14)-675(14) Troe, J., 632, 643 Tuccio, S. A., 716 Turetric, W., 576, 602 Turner, E. B., 733 Turner, J. S., 814

U Ulmer, W., 752 Upatnieks, J., 744 Utterback, N . G., 515

V van Atta, C. W., 817 Van Dyke, M. D., 834 Van Tiggelen, A., 660 van Wijk, W. R., 479 Vanyukov, M. P., 697,699,700 Varghese, G., 419 Vanvig, R. L., 674, 681 Vedeneyev, V. I., 646 Vennemann, D., 437, 453, 493, 496

860

AUTHOR INDEX

Verdieck, J. F., 408(8),409,421(8), 422(8),

428(8),431(8), 488,489(89j) Vickers, T. J., 623,624(42), 627(42), 629(42), 632(42),637(42),641(42) Vidal, R.J., 679,680(44)-682(44) Vienot, J. C., 727,746 Vinckier, C., 660 Vinton, V. A , , 661 Voldner, E. C., 659 Vollrath, K., 408,463(1 I), 464,698,699 736,739,742,748,751,753(23) von Arx, W. S., 802,815 von Elbe, G., 463 von KBrrnBn, T., 834 von Zahn, U., 650,651(139) Vorotnikova, M. I., 577,591 W

Wagner, H. G g . , 632,636,643,660 Wagner, R. D., Jr., 437,446(36) Waidelich, W., 751 Walenta, Z. A., 681 Wallace, J. E., 437 Wallace, J. M., 517 Walters, K., 801 Wan, C. A., XI6 Wang, D. S., 410 Wang, Y. G . , 654,656(147) Warren, W., 783 Warren, W. R., 684 Warshaw, S., 421,422(23), 429(23), 488 Watt, W. S., 633 Weaver, D.P.430,488 Weber, A., 425,427(28) Weber, D., 478 Weber, H., 707 Weber, H.P., 720 Weirner, D.K., 548 Weinreich, G., 722 Weinstein, L. M., 437,446(36) Welsh, H. L., 642 Wemple, S. H., 728 Westenberg, A. A., 635,636,659 Westkaemper, J. C., 673 White, D.R.,788,794 White, J . U . , 642

Whitehead, J. A.. 813 Whiting, E. E., 638 Wieland, K., 472,473(31), 474 Wiese, W. L.,639 Wiggins, T. A., 410,643 Wiley, W.C., 649,653 Willeke, K., 666,680(10), 681(10) Williams, A., 642 Williams, W. D., 430,437,488.491-493,

496 Willrnarth, W. W., 525, 527,535(26),

536(26), 548(26),577 Wilson, E. B., 617 Wilson, W. E., 635 Winding, C. C . , 680 Witteborn, F. C., 683 Wolf, E., 416(16),417,710, 740(51) Wolfarth, E. E., 628 Wolfhard, H.G., 463,479,482,642 Wolfrurn, J., 636 Wong, H., 794,795(37) Wood, R.D., 462 Woodruff, L. W., 673 Woodward, L. A., 425,427(30),428(30)

Y Yakobi, Yu. A., 525,550(23), 57323) Yakushin, M. I., 466,491(15) Yarnanaka, C., 485 Yang, C. S., 525 Yojama, M., 485 Yokojama, M., 485 Young, R. A., 642 Young, W. S . , 654,656(147) Z

Zaitsev, S. G., 558,577,602 Zalovcik, J. A., 520 Zapata, R. N ., 766 Zellner, R.,634 Zener, C. M., 534 Zirnakov, V. P.,470 Zorn, J. C., 647 Zuman, P., 621

SUBJECT INDEX This is a combined index for Parts A and B of Volume 18. A

Abel inversion, 394, 742 Absorption of radiation atomic attenuation coefficient, 406 for chemical composition, 621 -634 for density measurement, 405-408 linear attenuation coefficient, 406 mass attenuation coefficient, 406 Absorptivity, spectral, 465, 472-475 Acoustic anemometer, 315-318 Acoustic Doppler velocimeter, 317 Acoustic flowmeter, 337-340 Adiabatic wall temperature, 458-459, 665 Aerodynamic force principle, 254-256 on vane anemometer, 254-258 on whirling arm anemometer, 256-259 Aerodynamic noise, study in wind tunnel, 779 Ambiguity noise. 104, 126, 136, 160. 162. 166 effect on LDV signal processing. 162 Anechoic chamber, 778 Antenna theorem, 126 Anti-Stokes Raman line, 422, 723 Aperture broadening, SCY Antenna theorem Apparatus, for fluid dynamic research, 755-819 Arc-plasma tunnel, 462, 784

B Ballistic range. 779-781 Bar gage for pressure measurement, 593, 602 Barium titanate pressure gage sensor, 542 Basset -Boussinesq-Oseen (BBO) equation, 8 Beam, SCC Light source

Beam absorption densitometry, 405-408 Beam splitter, 379 Beer's law, 407, 475, 623-624, 629, 632 Bellows gage for pressure measurement, 51 1 Bernoulli formula, 243-245, 325, 333, 776 Bernoulli pressure, 503 Bias, in particle tracking, 5 , 174, 195 Biot number, 668-669 Blackbody radiation, 465-466, 690, 699 Blast wave solutions, by self-similarity, 835-842 Blow-down wind tunnel, 758 Boltzmann distribution, 464, 473-474, 480, 640-641 Bond (chemical) density, 413, 419 Boundary layer recovery factor for temperature probe, 459 study in wind tunnel, 766-768 Bourdon gage for pressure measurement, 51 I Bow shock wave, 358 BOXCARS (variant of CARS), 432 Bragg cell, for frequency shifting in LDV, 193-194 Brehmsstrahlung, 407, 486, 706 Brightness of light source, 689 Brightness temperature, 465 -466 Brillouin scattering, 415-417 Broad crested weir, 334-336 Brownian motion, effect on tracer method, 39

C Calibration camera, in chronophotography , 84-86 electron beam fluorescence system, 45045 I , 453 flowmeter, 322 861

862

SUBJECT INDEX

Calibration (continued) heat transfer gage, 677, 680, 682, 684 hot-wire anemometer, 285, 297 pressure gage, 504, 507, 509, 512, 514, 555, 592, 606, 610 Raman scattering diagnostic system, 429 Calorimetry applied to heat transfer measurement, 664, 670-679 capacitance calorimeter, 672-674 tangential conduction error, 670-671 Canal mechanism of spark formation, 696 Candela, 689 Capacitance sensor diaphragm gage, 570-572 method, 540-542 Capillary correction to manometer, 507 Capsule gage for pressure measurement, 51 1 CARS (coherent anti-Stokes Raman scattering), 43 1-433, 489 Cavitation, dimensional analysis, 843, 848 Centrifugal force, 803 Ceramic capacitor spark light source. 700 Channel flow metering, 332-336 Chapman-Jouguet condition, 838 Chemical composition measurement, scc Composition measurement Chemical kinetics, study in shock tube, 656-659, 792-795 Chemiluminescence, use in composition measurement, 641-643 Choked flow, 330, 772 Chromatography, for composition of sampled fluid, 617-621 Chrono-intetferometer, 551, 607 Chronophotography calibration of camera, 84-86 camera requirements, 79-83 compared with other velocimeters. 64-66 dark and bright field illumination, 76-79 data analysis, 86-87 definition, 64,66 directional information, method, 67 error analysis, 87-89 illustration of system design, 89-93 interrupted illumination, 67-76 measuring volume, 83-84 rotating flow apparatus, 818-819 system elements, 67 Cinematography, high speed, 726, 732-739

Clausius-Mosotti relation, 348 Coal mine dust explosions, 796 Coherence lateral. 125 spatial, 406, 707, 710 temporal, 125, 129, 707, 710 Coherence function, 118-131, see ulso Heterodyne efficiency Coherence length definition, 71 1 light source, 130-132, 715 measurement, 71 1 Coherence time definition, 130, 711 measurement, 7 11 Color interferometry, 742-743 Color schlieren, 367, 370 Combustion driver, shock tube, 788 Composition, method of description, 61 1 Composition measurement absorbed radiation by in . T i m fluid, 630637 absorption spectrophotometry of sampled fluid, 62 1-630 analysis of emitted radiation by in ~ i t u fluid, 637-643 analysis of sampled fluids, 616-630 classification of methods, 613-616 electron beam fluorescence, 434, 445 mass spectrometer, 645-661 methods, 611-661 sampling methods, 616-617 species concentration by molecular scattering, 408-433, 643-645 Compressible flow, in wind tunnel, 759-761 Compressible flow field, density by light refraction, 346 Compton effect, 407 Conrad probe, 254 Constant current anemometer, hot-wire or hot-film basic circuitry, 277 calibration, 285 compensation, 283 square wave test, 285 Constant temperature anemometer, hot-wire or hot-film basic circuitry, 290-292 calibration, 297 characteristic frequency, 295

863

SUBJECT INDEX

cutoff frequency, 292-293, 301 damping coefficient, 295 higher-order system response, 300 linearization of signal, 302 offset voltage, 291 square wave test, 297 unbalance parameter, 292, 294, 296 Convection of heat at surface. 667 role in hot-wire and hot-film anemometer. 269 Conversion of units, 823-825 Coriolis force, 803 Couette viscometer, 797 Cranz-Schardin camera. 737 Critical flow liquid in channel, 333 nozzle throat, 330, 772

D Data analysis chronophotography, 86-86 interferometry, 205 -398 Dead weight pressure gage, 512 Decibel, 508 Density gradient, by Raman scattering, 424 Density measurement beam absorption technique, 405-408. 705 electron beam excited radiation. 434-455 interferometer technique, 345-403 Raman scattering technique, 418-433 Rayleigh scattering technique, 414-418 schlieren method, 363 Depth of modulation, 121 Detonation, 838-&39 Detonation wave, temperature measurement. 470 Diaphragm pressure gage, .we Pressure gage, diaphragm Differential pressure flowmeter, 324-331 Diffraction, effect on schlieren method. 364 Diffraction grating interferometer, 380 Diffraction-limited point light source. 710 Diffuser orifice flowmeter. 326 wind tunnel, 758. 760: 772, 784 Dilatational pressure gage, .see Pressure gage, dilatational Dimensional analysis

examples, 832-842 mathematical foundations, 821 -828 nature, 821 Dimensional and dimensionless quantities, 822-825 Dimensional homogeneity, 826 Dimensionless numbers in fluid dynamics, 829-831 Dimensions, 822-823 Directional ambiguity in LDV. removal of, 186-190 Direct spectrum analysis, in LDV illustrations of use, 220-227 image converter use, 208-209 method, 194-227 streak camera use, 217-218 synchronous detection use, 205-208 Discharge coefficient, flowmeter, 328-330, 335 Distortion of solid, measurement by laser speckle. 713 Division, of amplitude or wave front, LDV configuration, 124 Doppler ambiguity, scc Ambiguity noise Doppler bursts, 154, 175, 180. 190 Doppler shift formulas, 99- 104, 342 Doppler velocimeter, acoustic, 317-318 Drag coefficient, sphere, 8. 10-15, 759. 800 Drag force, measurement, 768-769 Dropout, S Y P Signal dropout Drum camera, 733 Dust. acceleration by shock wave, 31 -33, 795 Dye, marker for flow visualization, 819 Dye laser applications, 715-716. 746 pump lamp, 693-694 Dynamical similarity. 828-829 Dynamic pressure, 247, 503 Dynamic response, .wc Frequency response

E EBF, ..we Electron beam fluorescence Ekman boundary layer, 805-806. 810, 817 Electromagnetic anemometer, 3 18-321 Electromagnetic flowmeter, 337, 340 Electron beam fluorescence beam generation, 452 beam spreading, 452

864

SUBJECT INDEX

Electron beam fluorescence (conrinued ) calibration of system, 450-45 I chemical composition measurement, 434, 445

compared with laser light scattering technipue, 435 density measurements, 450-451 Row visualization, 399, 437, 453 general description, 434-438 intensity relation to gas density, 441 -450 role of gas motion, 438, 446-447 role of secondary electrons, 443-445 selection mles. 438-441 temperature measurement, 489-497 Electron density, measurement, 698, 704, 705, 742, 750-753. 794 Electron gun. EBF system, 451-452 Electron spin yesonance, for species concentrations, 634-637 Electron temperature, 464, 472, 475, 705 Electro-optical shutter, 727-732 Elliptic Row equation. 760 Emittance of light sources. 465, 689 Emitted characteristic radiation, for velocity measurement, 341 -345 Equations of state, 612 Equivalent surface conductance, 667 Error analysis chronophotography, 87-89 particle tracking methods, SO Error functions, definitions, 666, 668 Etalon, Fabry-Perot, Fizeau-Tolansky. 198, 202, 214, 708, 717. 740 Excitation cross section. electron beam. 438-447 Explosion diagnostics, X-ray Rash, 408 Exposure times. photographic, 692-702

F Fabry-Perot etalon, use in laser, 708, 717. 740 Fabry-Perot filter, S C P Direct spectrum analysis, in LDV Fabry-Perot interferometer. 105, 195-227. 708. 717, 740 Faraday shutter, use in high speed photography, 73 1-732 Fast luminous fronts, 696 Fast response pressure gages, 576-610

Fermat’s principle, 352 Field absorption as visualization method. 389 - 392 Finesse, 197 Fizeau-Tolansky interferometer. use in LDV, 198-199, 214 Flame composition by emission spectroscopy, 641 -643 by mass spectrometry, 659-660 Flame front velocity, measured by schlieren method. 369 Flame temperature, 421 -425, 466-470 Flash lamp characteristics, 692-695 dye laser pump, 719 Flash radiography, 408 Flight testing apparatus, 779-781 heat transfer, 664 Flow disturbance by electron beam fluorescence diagnostics, 45 1 , 453 by hot-wire probe. 308 tracer particles, 38, 41, 49-51 by Pitot probe, 243, 250 by Raman scattering diagnostics. 419 Flow meter acoustic, 337-340 bundle of capillaries, 336-337 calibration, 322 definition. 241 electromagnetic, 337, 340 float meter, 331 Rume, 332, 334 orifice, 324-330 positive displacement, 323-324 power loss, 328 sonic nozzle. 330 turbine, 324 variable area, 33 1 Venturi. 324. 331 weir, 332-336 wet-gas, 323 Flow straightener, 326 Flow tracing particles advantages and disadvantages, 3-4, 97 definition, 2. 6 dynamic characteristics, 32-34, 221 -222, 225 effect on Row field (loading error), 38-41

SUBJECT INDEX

effect of sedimentation, 41 -43 equation of motion. 8 generation and dispersal, 43-50 hydrodynamic resistance. 8- 15. 795 interaction effects, 223-224 light scattering, 52-60. 64 limit of sensitivity in velocity measurement. 38-41 location in measuring volume. effect on LDV, 126 motion of, effect of size and density. 15- I6 optical characteristics. 51 -60 refractive index data. 53-54 response time determination, 26-32, 795 response time effect on turbulence measurements. 34-38 selection of, illustration, 60-64 size and density, measurement of, 16-26 size effect on LDV performance, 124 system for velocity measurement, 2-4, 6-7. 201-206, 23.5-240 use in rotating flow apparatus. 818-819 Flow visualization electric glow discharge, 402 electron beam fluorescence. 399. 437 Hele-Shaw apparatus, 798-799 high speed photography, 725-753 infrared. 750-753 interferometer. 377 jet, 359 light source. 694 phase contrast. 389 radiation emission. 398 rotating flow apparatus. 818-819 schlieren. 365 shadowgraph. 358 shock waves, 36.5. 386 smoke, 6-8. 770 tufts. 241, 770 wind tunnel. 769-771 Fluid, definition. SO1 Fluid dynamic equations in rotating coordinate system, 802-806 Flume. 332, 334 Fluorescence dye laser, 715-716 infrared sensor, 672. 751 meaning, 411-412 quenching, 412

865

relation to resonance scattering, 413 use for density, temperature, composition diagnosis, 410-414 Fluorescent lacquer, visualize transition to turbulence, 771 Fluorescent radiation Doppler shift to measure velocity. 343 Force balance, aerodynamic model in wind tunnel, 768-770 Force balance, aerodynamic model in wind tunnel, 768-770 Forcing function, role in hot-wire and hot-film circuit response, 278 Fourier heat conduction equation, 665 Fourier number, 668-669. 675 Fourier transform spectra, for species concentrations. 629 Framing camera application. 696. 733-734 light source, 705. 715 Franck-Condon factors, 491 -492 Free flight apparatus, 779-781 Free-molecule flow. 763 Frequency counting as LDV signal processing technique, 174- 175 Frequency domain signal processing in LDV, 161 Frequency response. ,wt’ crlso Response time calibrator for pressure gage, 555 density measurement by Rayleigh scattering, 418 diaphragm pressure gage, 562 flow tracing particles, 32-34, 221. 225 function. 527 hot-wire and hot-film probe, 273. 278. 282. 293. 297 pressure bar gage. 599 Raman scattering diagnostics. 420, 42 I Rayleigh scattering diagnostics. 418 stub pressure gage, 587 vane anemometer. 257-259 Frequency shifting in LDV. 163- 164. 185. 187-189, 193-194 Fringe anemometer. 109. S P P t r f s o Optical heterodyne detection Fringe distortion methods. 369 Fringe interpretation, of LDV operation. 117

866

SUBJECT INDEX

Froude number definition, 830 role in ship fluid dynamics, 848

G Gage factor, resistance sensor, 538 Gardon heat transfer gage, 672-675 Gaussian line profile, distortion by Brillouin scattering, 414-416 Geometric similarity, 828, 843, 844 Geophysical flow apparatus, 801 -819 Geostrophic flow, 803-806 Gladstone -Dale constant electron gas, 350 ionized gas, 350 Gladstone-Dale relation, 348. 351 Grashof number definition, 830 hot-wire and hot-film convection. 269

H Hagen-Poisseuille formula, 336, 798 Head (of fluid), meaning, 333 Heat conduction relations, one-dimensional, 665 -670 Heat transfer coefficient, definition, 459 Heat transfer gage asymptotic type, 672-675 balanced heat removal type, 664 calorimeter type, 664 capacitance calorimeter, 672-674 construction and principles, 663-685 Gardon type, 672-675 high heat flux, 676-677, 681 high temperature gas flows, 683-685 infrared bolometer, 679 membrane calorimeter, 664 multilayer gage, 666 radiation type, 683-686 sandwich type, 664 shock tube and shock tunnel, 666. 677-679. 682-685, 792 thick film type, 677-678 thin film type, 664, 679-683 thin membrane calorimeter, 672-675 use in arc-plasma tunnel. 784 use in free-flight model, 781 Heat transfer measurement conceptual methods, 664 gages, 663-685

shock tube technique, 791 wind tunnel technique, 674, 778 Heat transfer hypersonic tunnel, 762 radiation loss in arc-plasma tunnel, 674-677, 784 Hele-Shaw apparatus, 788-789 Heterodyne detection, bee Optical heterodyne detection Heterodyne efficiency, I19 High speed recording methods, 725-753 High temperature gases. produced in shock tube. 787 - 790 Hold time definition, 530 diaphragm pressure gage, 568 free surface motion pressure gage, 606 pressure bar gage with end sensor, 603 stub pressure gage, 590 Holographic interferometry, 381, 550. 746-750 Holography combined with interferometer method, 38 I combined with schlieren method, 368 methods, 715, 743-750 principle, 744-746 reconstruction. 550, 743-746, 747. 748, 750 Homodyne detection, see Optical homodyne detection Hook method, 351 Hopkinson pressure bar, 596 Hot-film anemometer, see c t l s o Constant current anemometer: Constant temperature anemometer; hot-wire anemometer calibration, 258-289. 297-303 compensating circuit, 283 construction. 266-268 external coating, 282 frequency response, 282 heat conduction, 273, 382 linearized energy balance equation, 275 physical characteristics, 267 -268 resistance. 268 Reynolds number, 269 substrate, 267, 273 temperature sensitivity, 313-314. 461 theory, 268-276 thickness, 267 time constant, 281-297

867

SUBJECT INDEX

Hot-wire anemometer. .we o h Constant current anemometer; Constant temperature anemometer aspect ratio, 271, 304, 307, 309 calibration. 285-289, 297-303 compensating circuit, 283 constant current, 276-289 constant temperature, 289-303 construction, 261 diameter, 261, 284 directional dependence, 306-308 effective cooling velocity. 306 effect of temperature variation along length. 303 flow interference effects, 308 frequency response, 278, 283, 297 heat conduction, 272 linearized energy balance equation, 275 multiple probe arrays, 310-313 physical characteristics, 260-267 resistance of wire. 268 Reynolds number, 269 sheath, 261, 265 supports, 265 temperature sensitivity, 313-314. 461 theory, 268-276 time constant. 278-297 X-probe, 310 Hydrostatic correction. 244, 332 Hydrostatic law. 505 Hydrostatic pressure, 501 Hyperbolic flow equation, 760 Hypersonic apparatus, 781 -784. 791 793 Hypersonic atmosphere entry, 664

I Image converter camera description. 732, 735-737 use in LDV , 208 - 209 Image converter streak camera, 720 Image dissection camera, 734-735 Image intensifier camera, 735-737 Impact pressure, 247, 503 Implosion. strong, 841 -842 Index of refraction, see Refractive index Inductance sensor. use with diaphragm gage. 570 Induction wind tunnel, 758 Infrared interferometer diagnostic method, 750-753 Infrared pyrometer, 672

Instrumented heat gage models thin wall, 670-671 thick wall, 671 surface temperature mapping, 671 -672 Intensifying screen, for x-ray detector, 407 Interference fringe visibility, 71 1 Interferometry diffraction grating, 380 evaluation procedures, 392-398 high speed recording, 739-743 holographic, 38 I , 550, 746- 750 infrared, 750-753 measurement of electron density, 698 multiple-beam resolving power, 196 principles. 196. 374-392, 739-743 reference beam, 375 shearing, 375 two color, 742-743, 794 Ionization chamber, 407 Ionization rate, study in shock tube, 792-794 Ionized gas heat transfer measurement. 673-679 lrradiance by light sources, 689 Isotope dilution analysis, for species concentrations. 630

J

Jet open, use as wind tunnel. 759. 764 shadowgraph visualization. 359 Jitter. spark triggering. 702-703

K Kerr cell, 717. 723, 729-731 Kiel probe. 249 Kinematic viscosity, 797 King's law, 271. 297 Kirchhoff radiation law. 465. 467. 471, 690, 707 Knudsen number, 763. 831 Kolmogorov length scale, 265 L Lagrangian and Eulerian mean square velocities. 38 Lambert-Beer relation. Beer's law

868

SUBJECT INDEX

Laminar boundary layer heat transfer, 667 in hypersonic tunnel, 783-784 similarity solution, 832-834 study in wind tunnel. 766-767 Laser active modulation, 718-719 argon ion, 7 I6 carbon dioxide, 714-716, 750-753, 794 coherence properties, 710-712, 717 continuous emission, 715-716 dye, 715-725 energy output, 718, 750-753 fundamental properties, 707-708 gasdynamic. 796 generation of harmonics, 722-725 giant pulses, 717-718 helium-neon , 7 14-7 16 inversion of levels, 707-708 line shape, 708-709, 718 mode spectrum, 708-710 mode-locked pulses, 718-720 neodymium-doped glass. 714-716 nitrogen, 721 nonlinear optical methods, 721 -725 parametric amplifier, 724-725 pumping by flash lamp. 692-693. 707 Q-switch, 717-718 Raman, 722-724 recording interferometry, 739-743 relaxation pulses, 716-717 ruby, 714-716, 717, 725, 746 saturable absorber. 717-719 speckle, 707. 712-714 spectral ranges. 714-716 superradiant, 720-72 1 YAG, 714-716, 725 Laser anemometer, ~ e Laser r Doppler velocimeter Laser Doppler anemometer, .w Laser Doppler velocimeter Laser Doppler velocimeter characteristics of, 96 choice of technique, 232-235 combined with Raman scattering diagnostics, 431 combined with schlieren method, 369 compared with probe methods, 97 design calculation, illustration of, 235-240 illustrations of signal, 155

optical configurations, 108- 110 optimization of performance, 235 photodetector output current, 110-1 15 principle, 97 rotating flow apparatus, 819 signal analysis. 229-232 signal processing methods. 227-228 Laser triggered spark gap, 703 Lava1 nozzle, in wind tunnel, 772 LDA, see Laser Doppler velocimeter LDV, scc Laser Doppler velocimeter Lift force, aerodynamic, 768-769 Light beating, see Optical mixing Light distribution function, in LDV measuring volume, 119- 120 Light gas gun, 780 Light path lengths, effect on LDV performance, 131 Light recording methods, 725-753 Light scattering, see ulso Raman scattering; Rayleigh scattering by flow tracing particles, 52-60, 64,410 Light sensor Image intensifier camera. 735-737 photographic material, 689, 715, 726-727 phototube. 689 spectral response, 689, 751 Light source absorptivity, 690 beam, 406 broad source, 406 chemical explosive, 705 coherence length, 130-132 diffraction-limited point source, 710 duration, 692 -695, 700-702, effect of size on LDV performance, 124-126 energy. flash lamp, 693-695 exploding wire, 703-705 flash lamps, 692-695 flow visualization, 694. 695-703 general, 687-725 infrared, 750 laser, 707-725 laser pump lamps, 693-694 laser spectral ranges, 714-716 luminous efficiency. 691, 692. 698-703. 705 nonlinear optical methods, 721 -725 physical and photometric aspects. 688-689

869

SUBJECT INDEX

plasma focus, 705-706 point source, 356. 405, 710 Raman scattering diagnostics. 419. 425 shadowgraph. 694, 695-703 short duration pulse, 710, 717-721, 727. 750 measurement. 720 spark. 695-703 triggering, 702-703 spatial coherence, 125. 406 spectral characteristics. 688, 691 spectral luminance, 690 spectral output of lamps, 691 -695 of spark. 699 temporal coherence. 125, 129 thermal, 688-707 units of output, 688-689 xenon flash lamp, 693-694 Light spectroscopy, application in LDV, 105-106 Line reversal method of temperature measurement, 466-470 Liquid crystal, infrared sensor. 751 -753 Liquid manometer. 505 Low density gas flows. S P C Rarefied gas flows Low density wind tunnel, 784-785 Ludwieg tube, 774 Luminance, 689 Luminous efficiency, 691,692,698-703,705

M Mach number definition, 246, 830 measurement, 776-777 wind tunnel, 758. 759-761, 771 -779, 844-847 Mach-Zehnder interferometer. 377. 379, 740 McLeod gage. 509 Magnetohydrodynamic Row studies, in shock tube, 795 Marx surge generator. for nitrogen laser, 72 I Mass spectrometry advantages in composition diagnostics. 645-646 composition measurement. 645-661

detectors, 652-653 flame composition measurement, 659-660 fragmentation, 645-647 free radicals, 648-649 ion sources, 646-649 Kantrowitz-Grey molecular beam inlet, 654-656 quadrupole, 650 sampling systems. 653-656 time-of-flight , 649-650 use in shock tube, 656-659 Measuring volume, w e u/so Spatial resolution dimensions in LDV, 64, 83-84, 96, 126, 13I - 134 distribution of light in, in LDV. 119- 120 Membrane calorimeter, 664 Membrane pressure gage, see Disphragm pressure gage Metering nozzle, 324-331 Michelson interferometer diagnostic uses, 606-610. 740 infrared, 752-753 measure coherence time, 71 1 Micromanometer, 506 Mie theory of light particle scattering. 51 Mode-locked laser, 718-720 Model testing principles, 821 -848 Molecular light scattering. advantages of diagnostics, 409, 418-421 Multiple-beam interferometer (FahryPerot). 195-227, 708, 717, 740 Multiple spark camera, 737-739 Multiplexing, 555 Mutual coherence function, 712-714

N Nanolight, 700, 702 Negative absorption. 406. 707, 720-725 Neutron absorption, for density measurement, 705 Newtonian fluid, 502, 797-800 Newton's law of cooling, 665 Noise, s i v idw Signal-to-noise ratio Johnson. 143 optical. 141 photodetector in LDV, 142-143 shot. 143. 185

870

SUBJECT INDEX

Nonequilibrium system composition measurements, 631 -634, 637-643, 645-649 level population by Raman scattering, 420, 424, 430 temperature measurements, 463-465, 472-497 Nozzle, wind tunnel, 758. 760, 764, 772 Nullpoint calorimeter, 676-677 Nusselt number, 460, 461 0

Open channel liquid metering, 332-336 Optical characteristics of flow tracing particles, 51-60 Optical delay line, 738-739 Optical filter, multiple-beam interferometer, 105, 201, 195-227, 708, 717, 740 Optical heterodyne detection in LDV, 109, see ulso Heterodyne efficiency Optical homodyne detection, in LDV, 116-118 Optical interferometer. see Interferometry Optical mixing, 106- I IS Optical multichannel detector, 429 Optical radiation absorbed, 405-408, 621-634 emitted, 341-345, 641-645, 687-725 scattered, 408-433, 643-645 Optical sensor for surface displacement or velocity, 549, 606 Orifice flowmeter. 324-331 Overheat ratio, hot-wire and hot-film probes, 269

P Paint, temperature indicating, 671 -672, 771 Parametric oscillations, laser, 724-725 Partial pressure, use in description of composition, 61 1 Particle tracking. see Flow tracing particles Particulates. acceleration by flow, 8- 15. 795 Partition function, 427 Pascal (pressure unit). 508 Pebble-bed storage heater, 782 Pedestal, in LDV signal, 114, 121, 156, 159, 184

Period counting. LDV signal processing, 174-180 Phase contrast as visualization method, 389 Phase object, 345 Photodetector, .see ulso Light sensor output, LDV statistical character, 154 Photoelectric effect, 407 Photography high speed, 725-753 recording material, 726-727 short duration light sources, 727 Photometric aspects of light sources, 688-689 Photomultiplier, 104-107, 407 Photon counting correlation. LDV signal processing, 180- 186 Piezoelectric scanning interferometer, 201 Piezoelectric sensor, 542 Piezometric head, 333 Pitch, aerodynamic, 768-769 Pi theorem, 826-828 Pitot probe angular sensitivity, 248 calibration, 243, 248, 253. 515-524 corrections for turbulence, 252 corrections for viscous effects, 251 general, 240. 242-254, 515-524 principle, 243 tube construction, 248 use in low density wind tunnel, 784 use in velocity gradient, 250 use near wall, 250 wind tunnel, 776-777 Pitot-static probe, 249 Pitot tube. see Pitot probe Planck radiation law, 465-466, 690 Pockels cell, 717. 728-729 Point light source, 356, 405, 710 Point source explosion, 839-841 Poiseuille formula, 336, 798 Polarizability electronic, 347 infrared measurement, 750-753 molecular, 41 I Polarization vector, 347 Poled ceramic pressure gage sensor, 542 Polyvinylidene fluoride (PVF,) piezoelectric sensor, 542 Positive displacement flowmeter, 323-324 Prandtl number definition, 459, 830

87 1

SUBJECT INDEX

wind tunnel, 777. 844 Pressure relation to stress tensor, 501 units, 508 Pressure bar gage, 593 Pressure concept extension by thermodynamics, 503 kinetic theory. 502 mechanical, 500, 801 Pressure gage bar gage, 593, 602 Bourdon tube type, 51 I calibration, dynamic, 555 calibration at gigapascal range, 592, 606 calibration at kilopascal and megapascal range, 507. 512, 557 calibration at 100 gigapascals, 610 calibration below 10 pascals, 509. 515 calibration standards, 504, 514 capsule type, 51 1 characterization, 527 deformation type. 510 diaphragm, 559-576 diaphragm below 10 pascals, 515 diaphragm types, 570 dilatational gage, 604 dynamic calibration, 555 fast response, 576 free surface sensor, 606 frequency response function, 527 hold time, 527, 531, 557 holographic method of recording many diaphragm gages, 575 Hopkinson bar, 596 inductance sensor. 548 McLeod, 509 meaning. 504 miniature bar gage. 602 miniature capacitance, 571. 573, 575 miniature probe, 592 miniature stub. 588 optical sensor, 549 peak pressure, 535 piezoelectric sensor on bar gage, 593. 603 on stub gage. 588 piston and cylinder. 512 probe, 59 I range, 531, 557 recording methods, 552-555 reluctance sensor. 548. 571

resonant period, 527, 531 response, to step function, 527 response characteristics of diaphragm gage, 567-570 response time, 527. 531, 557 sensitivity, 531. 557 diaphragm type, 561, 562. 566 sensors, 534-552 slab type, 588 standards, 504 static calibration, 504, 514 steady or slowly varying pressure, 505-5 15 stub type, 588 theory of diaphragm gage, 559-570 of fast response gage, 579-588 thin polymer piezoelectric, 573 types, 505, 526, 577 use in free-flight model, 781 use in non-Newtonian flow, 801 U-tube manometer, SO5 wall taps, 516-518. 801 Pressure measurement above 100 kilopascals, 5 12 below 10 pascals, 515 general, 499-610 in moving fluid, 515 static probe, 516, 518, 521 Pressure probe in moving fluid, 515 Pressure recovery, wind tunnel, 758, 760 Pressure-time recording, 552 Pressure transducers. .set’ Pressure gage, sensors Probe gage for dynamic pressure measurement, 591 Probe methods for pressure measurement, 515-516, 518, 521. 591 for temperature measurement. 457-463 for velocity measurement, 240-341 Propeller anemometer, 254 Pulsed Doppler ultrasonic velocity meter, 317-318 Pyroelectnc temperature sensor, 685

Q Q-switched laser, 717-718 Quartz piezoelectric pressure gage sensor. 542

872

SUBJECT INDEX

Quenching role in electron beam fluorescence, 438, 447-448 insensitivity of Raman scattering, 413

R Radiant energy of light sources, 689, 706-707, 718-719 Radiation boundary condition, 665,667, 669 Radiation constants, blackbody, 466, 468 Radiation detectors, 407 Radiation source. .see Light source Radiative heating, study in hypersonic tunnel, shock tube, 679, 683-685, 762, 79 1 -792 Radiative loss, temperature sensor, 461 Radiography, 407 Raman laser, 722-724 Raman scattering advantages over other density measurement techniques, 414, 418-421 basic features, 412-414 calibration, 429 density, temperature, composition diagnosis, 408-455, 643-645 light sources, 419 line intensity use for density, concentration measurement, 428, 431 meaning. 41 1-414 molecular rotation, 41 1 molecular vibration, 41 1 nitrogen vibrational line contour, 421 -428 pulsed laser illumination, 419-421, 425 rotational line contribution, 425-430, 488 scattering amplitudes, 413 Stokes and anti-Stokes line, meaning, 422 temperature effects on density measurement, 421 -425 Raman shift, 413, w e ulso Raman scattering Rarefied gas flow density measurement by EBF. 434-455 visualization, 398 wind tunnel, 762-763, 784-785 Rayleigh scattering advantages over emission and absorption spectroscopy, 414 basic features, 412-414 line intensities used for diagnostics, 417-418

line shape used for diagnostics, 414-417 meaning, 41 1-414 Real gas effects, in hypersonic apparatus, 782 Receiving aperture size, effect on LDV performance, 138, see also Antenna theorem Recording methods infrared, 750-753 light, 725-753 pressure gage, 552-555 wind tunnel, 768-769 Recovery factor, temperature Couette flow, 459 definition, 459 flat plate boundary layer, 459 wind tunnel, 777-778 Recovery pressure, wind tunnel, 776 Reference beam interferometer, 375-383 Refractive behavior of fluids, 346, 347. 41 1 Refractive index density dependence, 348 flow tracing particle materials. 58-59 gas mixture, 349 Reradiation after interaction with medium, 407, 411-412 Resistance, temperature coefficient, 262. 264 Resistance thermometer. 313-314, 461 Resisting vane anemometer, 254. 258 Resistivity. hot-wire material. 262, 264 Resolving power of multiple-beam interferometer, 196 Resonance scattering, 413 Response time, w e c r l s o Frequency response dilatational pressure gage. 604 flow tracing particles, 26-32. 38 free surface motion pressure gage. 609 heat transfer gage, 668-669, 674-678 hot-wire anemometer, 293-295 liquid manometer, 508 Pitot probe, 252-253 pressure bar gage, 599 with end sensor, 603 radiation scattering diagnostics. 409 tracer particle, 26-32, 38 Reynolds number definition, 830 hot-wire and hot-film convection, 269 low. apparatus, 796-801

873

SUBJECT INDEX

temperature sensor, 461 wind tunnel, 758-759. 773-774. 844-847 Rheological fluid, 801 Rise time, .WP Response time Rochelle salt pressure gage sensor, 547 Roll, aerodynamic, 768-769 Ronchi schlieren, 370 Rotameter. 322, 331 Rotating flow apparatus construction. 806 -809 data transmission and photography. 817-819 examples of studies. 801-819 experimental configurations, 81 1-813 moving boundaries, 8 14-8 I6 precision and control requirements. 809-8 12 pumping, 816-8 I7 Rotating mirror camera. stw Framing camera; Streak camera Rotational temperature EBF technique, 436, 493-497 measurement .457,464.466.480,493-497 spectral emission technique, 480 Ruby high pressure gage, 610 Ruby laser giant pulse, 717-720 properties, 714-716. 725. 746 pump lamp, 693-694

S

Sabol. 780 Sampling error, 4-6 Sandwich heat transfer gage, 664 Scale effect, 846 Scanning multiple-beam interferometer, 201 -220 Schlieren interferometer. 384 Schlieren method combined with holography, 368 combined with laser Doppler. 369 dephasing schlieren system. 373 effects of diffraction. 364 sharp focusing. 368 Schlieren systems color. 367 construction and principles. 361 -374 double knife edge. 367 double pass, 367

infrared, 750 schlieren head, 361 spark light source, 695-703 Toepler, 361 Scintillating crystal radiation detector, 407 Sedimentation, effect on flow tracing particles, 41 Selection rules. in electron beam excitation, 438-440 Self-absorption, effect on intensity of emitted radiation, 640 Self-similarity 828 Sensor light, 689, 715, 726, 735 temperature probe, 460-463 pressure gage. 534-552 Servo frequency tracking, applied to LDV, 169-174 Servo multiple-beam interferometer. applied to LDV, 21 1-220 Settling plenum, wind tunnel, 758 Shadowgraph high speed frames, 738 light source, 694, 695-703 method, 355. 356-360, 407, 738 Shearing interferometer, 375, 383-389, 740 Ship flow dynamics, dimensional analysis, 847-848 Shock strength. 787 Shock tube chemical kinetic studies, 792-795 combustion driver. 788 composition measurement in reactions, 632-634. 656-659 description as research apparatus. 785-796 electric driver, 790 gas and sound speed measurement by ultrasound, 339 gas temperature measurement methods. 467-487 measurement of surface heat transfer, 667. 677-685 modified as shock tunnel, 791-792 pressure gage test and calibration, 557 production of high temperature gases. 787-790 reflected shock region, 787-788 .r-r diagram, 786 Shock tunnel, research apparatus, 462,666. 79 1 -792

.

874

SUBJECT INDEX

Shock wave recorded by interferometer, 741 by schlieren method, 365 by shadowgraph, 358, 738, 768 by shearing interferometer, 386 Short duration light sources, 717-718 Shrouded thermocouple stagnation temperature probe, 463 Shutter, for single exposure photography, 727-732 SI system of units (Systeme International), 822 Signal analysis classification, 161 definition. 154 Signal conditioning, 162- 165 Signal dropout in LDV, 157-164, 172-173, I75 Signal filtering, 163 Signal processing classification, 161 definition, 154 effect of signal-to-noise ratio, 155 heat transfer gage, 666, 670 hot-wire anemometer linearizer, 302 Signal spectra, in LDV, 159 Signal-to-noise ratio effect of refractive index variations, in LDV, 141 heterodyne configuration, 144 homodyne configuration, 145 multiple particle effects in LDV, 147- 154 photodetector, 140, 146 requirements in LDV, 140-154, 234, 235 Similarity, 828-829 Similarity solution. examples, 832, 834, 835-842 Sing-around type flowmeter, 339 Skimmer, for molecular beam, 654-656 Slab pressure gage, 588 Slip flow, 763 slug calorimeter, 676 Sonic anemometer, 315-318 Sonic flow, 760 Sonic nozzle flowmeter, 330 Sound speed, 246, 315-317 Spark discharge, electrical and fluid dynamical parameters. 695-698, 752-753 Spark formation, mechanism, 695-698 Spatial coherence, see Light source Spatial resolution, see ulso Measuring

volume dimensions density measurement by Rayleigh scattering, 418 in LDV, 134, 137-138, 235 pressure gages, 525, 573-576 in radiation scattering diagnostics, 409, 420 Species concentration, see U/SO Composition measurement electron beam fluorescence technique, 434, 451 mass spectrometer, 645-661 Raman scattering diagnostics, 428, 6 4 3 -645 Speckle, laser, 712-714 Spectral broadening, effect in LDV resolution, 133-140, see ulso Ambiguity noise Spectral line shape distortion by Brillouin scattering, 41 5 -4 17 Raman and Rayleigh scattering, 412 use for temperature measurement, 414-417, 490 Spectral radiance, 690 Spectral response of the eye, 688-689, 692 Spectral width of light, effect on LDV performance, 137 Spectrometer slit function effect on spectral line shape, 412 Raman scattering analysis, 427-428 Spectrum-scanned LDV, requirements, 201 -203 Spectrum scanning, applied to LDV, 154, 159-162, 165-174, 201-211 Spin-down. 809. 814-815 Spontaneous emission requirements in EBF, 438. 441, 442-447 temperature measurement, 465-482 Square wave test hot-wire and hot-film probe, 279, 285. 297-299 pressure probe, 563-570 system frequency response, 528-531 whirling vane anemometer, 259 Stagnation enthalpy, 245, 458, 788 Stagnation pressure meaning, 244, 246, 503, 515 produced for shock tunnel, 791 ratio across shock, 247 wind tunnel, 776 Stagnation temperature

875

SUBJECT INDEX

hypersonic apparatus, 782 meaning, 245, 457, 665 measurement, 460-463 produced for shock tunnel, 791 relation to gas speed and temperature, 458 in shock tube flow, 788 Static pressure, meaning, 247, 516 Static pressure probe in steady flow, 243. 518 in unsteady flow, 521 Static temperature, meaning, 457 Static vents on airplane, 520 Stefan-Boltzmann constant, 690 Step function hot-wire and hot-film response testing, 279, 285 loading bar pressure gage, 598, 603 diaphragm pressure gage, 563 dilatational pressure gage, 605 response function, 527 Stewartson layer, 806, 813 Sting, wind tunnel mount, 765 Stokes drag formula, 10-15, 759, 798, 799 Stokes number, 28 Stokes Q-branch, 421 Stokes Raman diagnostics for temperature measurement, 422-425 Stokes Raman line, 422, 723 Strain pulse dispersion, 580 reflection at end or interface, 585 theory of one-dimensional wave, 581 Strain sensitivity of diaphragm, 561, 562. 566 Strain sensor, 536-540, 542-549, 572-574 Stratified fluid, optical visualization, 355 Streak camera image converter recording, 696 light source, 705, 715 use with interferometer, 795 using rotating mirror, 733-734 Stress tensor, 500 Strouhal number, definition, 830 Stub pressure gage, 588 Superradiant light sources, 720-721 Supersonic flow, 330, 760 Supersonic wind tunnels, 771 -779 Surface temperature sensor fluorescent paint, 672 infrared pyrometer. 672

light transmitting paint, 671 thermocouple- thermopile, 666, 672-673, 675 thin fllm resistance, 664, 666, 672-685 Swept oscillator wave analyzer, LDV signal processing, 165

T Taylor-F’roudman theorem, 803, 805 Temperature fluctuations, measurement by Rayleigh scattering, 417 Temperature gradient measurement, Raman scattering, 424 Temperature in moving fluid, 457-460 Temperature measurement by analysis of emitted and absorbed radiation, 465-487, 698-699, 704 behind detonation front, 470 by Doppler broadened line shape, 481 -482, 490 electron beam fluorescence, 489-497 emittance on two paths, 475-478 hot-wire probe method, 313, 461 infrared pyrometer, 672 line reversal methods, 466-470 method of absorption in two spectral regions, 472-475 molecular scattering of radiation, 409, 414-418, 421-428 in moving fluid by probe, 457-463 by paint transparency and phosphorescence, 671 -672 probe methods, 457-463 radiation analysis methods, 463-497 Raman scattering, 487-489 by Rayleigh scattered spectral lineshape, 412, 414-417, 482-485, 487-489 relative intensities, 478-481 by simultaneous detection of radiation emission and absorption, 470-472 simultaneous with velocity measurement, 313 in sparks, 698-699, 704 two path absorption in thin foils of x-rays, 485-486 vibrational temperature by analysis of emitted radiation, 473-474, 481 Temperature sensors probe type, 460-463 resistance film, 460, 677-685 thermocouple, 460-461, 671-676

876

SUBJECT INDEX

Temporal coherence, see Coherence, temporal Test section, wind tunnel, 758, 764 Thermal conductivity. 665-666 Thermal diffusivity, 665 Thermal wind relation, 804, 810 Thermocouple, 460-461, 671, 672, 675-676 Thin film heat transfer gage, 664, 666-685 Time constant hot-wire and hot-film anemometers, 293 -295 Raman scattering diagnostics, 420, 42 I -425 Time dependent response of pressure gage, 527 Time domain signal processing, LDV, 161 Time resolution, S P ~ J Frequency response Time response, see Hold time; Response time Toepler schlieren system, 361 Torr, 508 Total enthalpy, see Stagnation enthalpy Total head (fluid), 333 Total pressure, see Stagnation pressure Total temperature, .we d s o Stagnation temperature probe. 462-463 Towing tank. dimensional analysis, 843, 847 - 848 Townsend mechanism of spark formation, 695 -696 Tracer particle tracking, .\re Flow tracing particles Tracking bandpass filter, LDV signal processing, 169- 174 Tracking multiple-beam interferometer for LDV signal, 211-220 Transducer, see Sensor Transition probability, in electron beam fluorescence, 438-441 Translational temperature, 457, 464, 490 Transonic wind tunnels, 771 -779 T-tube, electrically driven shock tube, 790 Tufts, for visualization, 241, 770 Tunnel wall corrections, 773, 847 Turbulence level, wind tunnel, 764, 774 Turbulence application of LDV for measurement, 103-104 effect on LDV spectrum, 161, 166. 221

effect on photon counting correlation in LDV, 184 effect on pressure probe, 252-253 temperature fluctuations in flame, 417 visualize transition, 771 wind tunnel flow, 764 Turbulent boundary layer, 459, 767 heat transfer. 666

U Ultrasonic flowmeter. 337-338 Unbalance parameter. hot-wire and hot-film probe compensation, 294, 296 Units conversion, 823-824 photometric , 688-689 pressure, 508 SI (Systeme International). 822 V

Vane anemometer. principle, 255-257 Variable area flowmeter, 331 Velocity components measurement by chronophotography 67 measurement by hot-wire probe, 306-3 1 I measurement by LDV, 190-195 Velocity head. 333 Velocity gradient, measurement by hot-wire probe. 312 Velocity measurement by chronophotography, 64-93. 818-819 direction by chronophotography, 67 direction by hot-wire anemometer, 306-312 direction by LDV. 190- 195 direction by Pitot probe, 254 direction, 241 by Doppler shift of emitted characteristic radiation, 34 1 - 345 of scattered light, 93-240. 342 of scattered sound from tracers, 317-318 electromagnetic method, 3 18-321 fluorescent radiation Doppler shift, 343 by Hall voltage, 318 by heat loss probe method, 259-314 hot-wire and hot-film probes, 259-314

.

877

SUBJECT INDEX

laser Doppler from tracing particles, 96-240 by laser Doppler velocimeter. 93-240 LDV with direct spectrum analysis, 194-227 Pitot probe, 242-254 by pressure probe, 242-254 probe methods, 240-341 propeller anemometer. 254. 256 resonant absorption of Doppler shifted radiation. 344 rotating flow apparatus. 817-819 sensitivity of measurement using tracer methods, 36 simultaneous with temperature measurement by hot-wire probe, 313 by timed sound pulses, 315-318 tracer,methods. 1-240 tracer particle loading error. 38 vane anemometer. 254-259 Ventilated wall. wind tunnel, 773. 775 Venturi flowmeter. 324, 33 I Vibrational spectral line contour analysis, 425-428 Vibrational temperature. 436. 457. 464, 466. 473-474. 481, 491-493 Virial equation of state, 61 1-612 Virtual fringes, in LDV. I18 Viscometry. 798-801 Viscosity of a particulate suspension. 38 Viscous fluid. 502, 797-800 Visualization. .A('(' Flow visualization Vortex generator. 816 Vorticity meter. 241. 312 W

Wall temperature discontinuity, effect on heat transfer measurement. 673 Wave machine. 796 Weir. 332. 334 Weir block. 334

Wet-gas meter, 323-324 Whirling cup anemometer, 256-257 Wien displacement law, 690 Wien radiation formula, 466 Wind tunnel blockage interference. 765 classification, 758-764 dimensional analysis applied to, 844-847 heat transfer techniques. 670-672 flow visualization, 769-771 lift interference. 765 low speed, 764-771 model testing principles, 843-849 open-jet test section, 764 research apparatus. 756-785 supersonic, 77 1-779 transonic, 771-779 turbulence, 764 wall corrections. 773, 847 Wollaston prism shearing interferometer, 387 Working section, wind tunnel, 758

X X-probe, hot-wire anemometer, 310 X-ray radiation, 407-408. 705 .r-i diagram, shock tube. 786 Xenon flash lamp, 693-694

Y Yaw aerodynamic moment, 768-769 card, 780 hot-wire probe correction, 306-308 meaning, 248 Pitot probe correction. 248-249 total temperature probe correction. 462 Young's experiment. measure spatial coherence. 710-71 I

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