The Performance of Chemical Propellants

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AGARDograph Number One Hundred and Twentynine

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Chemical Propellants

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The Advisory Group for Aerospace Research and Development, NATO.

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Authors

I. GLASSMAN Professor of Aerospace Sciences Guggenheh Aerospace Propulsion Laboratories Department of Aerospace and Mechanical Sciences Princeton University

R.F. SAWYER Associate Professor, Thermal Systems Division Department of Mechanical Engineering University of California, Berkeley

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Printed and published by

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Technivision Services Slough, England. A Division of Engelhard Hnnovia International Ltd.

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Copyright January 1969 The Advisory Group for Aerospace Research and Development. NATO.

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Standard Book Number 85102. 018. 6 Library of Congress Catalog Card No. 74-82418

To Our G i r l s :

9

Preface

Too often, extensive calculations and development tests are made in order to evaluate the performance of chemical propellants. However, reasonable analysis of the fundamental factors which govern the performance can reduce the effort nece s s a r y for this evaluation and give greater insight as t o what one should expect from a proposed propellant system. This thesis has been the basis of material that the senior author has presented in parts of two graduate courses: Chemical Rocket Engines and Combustion Processes in Jet Propulsion, taught at Princeton University. In 1965 the Propulsion and Energetics Panel of AGARD agreed to sponsor the preparation of this material in the form of a monograph. In order to make coverage of the subject matter more complete, additional material particularly on non-equilibrium effects in the rocket combustion chamber was added, contributed for the most part by the junior author. The first part of Chapter I1 is tutorial in nature and lays the background thought to b e necessary for the discussion'and analysis in the latter chapters. The material on non-equilibrium effects included in this chapter and Chapter III is perhaps one of the more complete coverages of this subject to be presented in a book. The performance of chemical propellants is then analyzed in the last two chapters. In particular, the authors hope that the readers will find the Postface a useful and stimulating summary of most of the major points made throughout the text. To be acknowledged are the patient and careful typing of the manuscript by Mrs. Daisy Guest, the drawing of the figures by Mrs. Margot Van Horn, and the detailed review of the original draft by Dr. W. R. Maxwell (Rocket Propulsion Establishment, England). Irvin Glassman Robert F. Sawyer Princeton, N.J. Berkeley, Calif.

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Table of Contents

Page

LIST O F FIGURES

I.

INTRODUCTION

15 19 21 25

11.

THEORETICAL CALCULATIONS

27

A.

PERFORMANCE PARAMETERS

27

(1) Thrust (2) Effective exhaust velocity (3) Specific impulse (4) Thrust coefficient (5) Characteristic velocity

27 32 34 35 36

THERMODYNAMICS, COMBUSTION CHAMBER EQUILIBRIA AND FLAME TEMPERATURE

38

(1) Basic thermodynamic concepts (2) Chemical equilibria (3) Condensed phases in equilibrium combustion mixtures (4) Determination of the product composition (5) Calculation of the adiabatic flame temperature

38 44 51 52 54

NOZZLE EXPANSION

60

(11 The question of equilibrium (2) Isentropicity of nozzle flow processes (3) Performance under isentropic (equilibrium and frozen) expansion conditions (4) Non-equilibrium performance (5) Approximations to non-equilibrium perforniance (6) Non-equilibrium effects due to condensible products

60 61

LIST O F TABLES SYMBOLS

B.

C.

111.

81

NON-EQUILIBRIUM CHAMBER EFFECTS A.

IV .

64 67 69 75

EXPERIMENTAL EVIDENCE OF NON-EQUILIBRIUM COMBUSTION

81

B.

PREDICTION O F COMBUSTION CHAMBER KINETICS AND NONEQUILIBRIUM PRODUCTS 82

C.

CONSEQUENCES OF NON-EQUILIBRIUM COMBUSTION

D.

SUMMARY

'

85 85

PROPELLANT SELECTION

91

A.

92

THERMODYNAMIC AND SYSTEM CRITERIA (1) Liquid Propellants (2) Solid Propellants

93 104

MORPHOLOGY OF CHEMICAL SYSTEMS

107

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12 TABLE O F CONTENTS (CONT. ) C.

D.

E.

V.

MONOPROPELLANTS

108

(1) Monopropellant characteristics (2) Monopropellant decomposition

108 109

MULTICOMPONENT PROPELLANTS

109

(1) High combustion temperature reactants (2) Optimum combinations of several propellants (3) Introduction of the metal

109 110 110

PROPELLANTS FOR NON-COMBUSTION ROCKETS

113

(1) Thermal rockets (2) Electrostatic rockets (3) Electromagnetic rockets (4) Other propulsion systems

113 117 117 117

PARAMETER EFFECTS

119

A.

MMTURERATIO

119

(1) Combustion temperature, molecular weight, specific impulse (2) Effect of pressure on optimum mixture ratio (3) Effect of expansion ratio on optimum mixture ratio (4) . , The effect of mixture ratio on characteristic velocity and thrust coefficient (5) Propellant density considerations

119 120 120 125

PRESSURE

125

(1) Effects at.fixed nozzle pressure ratio (2) Effects at fixed area ratio, significant ambient pressure (3) Other effects of chamber pressure (4) Temperature limited systems

126 126 126 127

TEMPERATURE

127

(1) Relation of enthalpy to temperature (2) Diabatic nozzle flow (3) The performance of hydrogen

127 127 127

EXPANSION RATIO

128

(1) The relation between pressure and area ratios (2) Limitation on expansion ratio

128 128

ASSUMED PRODUCTS

128

(1) Identification of probable species (2) Effect of dominant species

131 131

THERMOCHEMICAL DATA

131

(1) Results of statistical thermodynamic calculations (2) Enthalpies of formation

131 131

B.

C.

D.

E.

F.

125

13

.

TABLE OF CONTENTS (CONT.) G.

INDUCTION ENTHALPY

132

(1) Effect of propellant temperature and phase (2) Positive enthalpy of formation propellants

132 132

POSTFACE

135

REFERENCES

141

15

L i s t of Figures Figure II.A.l. II.A. 2. II.A.3. II. A. 4.

page Propulsive duct of arbitrary shape with appropriate symbols

28

Ideal thermodynamic processes in the combustion chamber and nozzle of a rocket motor.

28

Variation of nozzle area ratio with pressure ratio.

33

Variation of effective exhaust velocity with exit

area. II.A. 5. 11. B. 1.

II. B.2.

33

Variation of rocket thrust coefficient with nozzle area ratio and pressure ratio Pc/P for Y = 1 . 2 . a Reaction paths showing heats of reaction at different temperatures.

37 40

The heat available and heat absorbed in a reaction mixture as a function of temperature

40

The enthalpy - entropy variation in the recombination of dissociated species - a three-dimensional plot.

62

Variation of composition in a nozzle to show transition to frozen flow.

62

Effect of oxidizer-fuel ratio on calculated H,-0, rocket specific impulse for various nozzle flow conditions.

72

Variation of calculated H,-0 rocket specific impulse with pressure for ratios of 5 and 8 for various nozzle flow conditions

72

Enthalpy-Entropy variation in the recombination of dissociated species - Two dimensional plots permitting specific impulse comparisons.

74

Comparison of theoretical and experimental combustion performance of the hydrazine/nitrogen tetroxide propellant combination.

83

III.A.2.

Comparison of hydrazine reaction rates.

83

lTI.A.3.

Comparison of PEC and equilibrium combustion propellant performance of hydrazine/nitrogen tetroxide. Partial equilibrium nozzle expansion from a chamber pressure of 1000 psia to one atmosphere pressure.

a4

11. c.1.

II. c.2. II. c. 3.

II. c.4.

11. c. 5.

III.A.l.

0,h

16 m.A. 4.

N.A.l.

Comparison of PEC and equilibrium combustion propellant performance of hydrazine/nitrogen tetroxide. Frozen composition nozzle expansion from a chamber pressure of 1000 psia to one atmosphere pressure.

84

Heat of combustion of the elements as a function of atomic number. Oxidizer-Oxygen.

94

N.A. 2. Dissociation of rocket combustion products as a function of temperature.

94

Heat of combustion of the elements as a function of atomic number. Oxidizer-Fluorine.

105

Heat of combustions of the elements as a function of atomic number. Oxidizer-Chlorine.

105

IV. B. 1.

Morphology of storable propellant systems.

106

IV. D. 1.

Theoretical performance of the beryllium hydride/ ozone/hydrogen propellant system. Composition: 3BeH2 + Os/Ha. Chamber pressure, Pc = 1000 psia.

106

lV.A.3.

N.A. 4.

N.E.l. Enthalpy of equilibrium hydrogen, (ho-hoz 98), for

-

N.E.2.

V. A. 1.

V. A. 2.

V.A. 3.

V. A. 4.

various pressures. The large enthalpy rise from dissociation, H2 2H, depends strongly upon pressure.

112

Frozen flow efficiencies of some potential heat transfer rocket propellants.(49).

112

Hydrogen/oxygen combustion products characteristics and propellant performance. The maximum specific impulse lies at a mixture ratio between the mixture ratios of minimum molecular weight and maximum combustion temperature. Pc = 1000 psia, optimum equilibrium expansion to one atmosphere ambient pressure.

121

Theoretical equilibrium expansion performance of hydrogen/ozygen showing the effect of chamber pressure, Pc, on optimum mixture ratio. (50).

121

Theoretical frozen composition expansion performance of hydrogen/oxygen showing the effect of chamber pressure, Pc, on optimum mixture ratio. (50).

122

Theoretical equilibrium expansion performance of hydrogen/oxygen showing the effect of expansion ratio on optimum mixture ratio. Chamber pressure, Pc = 1000 psia. (50).

122

17

V. A. 5.

V. A. 6.

V. B. 1.

V. B. 2.

V. B.3.

v. c. 1.

v. c. 2. V.D.l.

V. D. 2.

V.E. 1.

Theoretical frozen composition expansion performance of hydrogen/oxygen showing the effect of expansion ratio on optimum mixture ratio. Chamber pressure, Pc = 1000 psia. (50).

123

Comparison of the performance parameters, specific impulse (I,,) characteristic , velocity (c*), and thrust coefficient (CF),and their dependence on mixture ratio. Hydrogen/oxygen, equilibrium expansion to one atmosphere pressure, Pc = 1000 psia. (50).

123

The effect of chamber pressure upon specific impulse at fixed expansion pressure ratio. Hydrogen/ oxygen; I,, OPT = optimum expansion for a pressure ratio of Pc.$e = 340; I,, V A C = expansion for an area ratio of Ae/At = 100 and &cuum ambient conditions. Equilibrium expansion flow. (50).

124

Effect of chamber pressure upon specific impulse at fixed expansion pressure ratio. Hydrogen/oxygen; Is, OPT = optimum expansion for a pressure ratio of PJPe = 340; I,,, V A C = expansion for an area ratio of Ae/At = 100 and vacuum ambient conditions. Frozen expansion flow. (50).

124

Effect of chamber pressure upon specific impulse with significant ambient pressure. Hydrogen/ optimum expansion to one oxygen; Isp, atmosphere pressure; I,, sL = expansion through , an a r e a ratio oT Ae/At = Id0 and one atmosphere ambient pressure. Equilibrium expansion flow. (50).

124

Enthalpy of equilibrium methane showing the nonlinear relation between enthalpy and temperature. P = 1.0 atm. The highly non-linear relationship is associated with hydrogen decomposition and carbon phase change with increasing temperature.

129

Performance of hydrogen as a heat transfer rocket propellant. PC= 1000 psia. Vacuum specific impulse to an area ratio of 100.

129

The relation between nozzle area ratio and pressure ratios for various combustion chamber pressures, Pc. Equilibrium hydrogen, Tc = 3000°K, equilibrium expansion, vacuum ambient conditions.

.

130

Vacuum specific impulse as a function of nozzle area ratio. Hydrogen/fluorine, Pc = 1000 psia, equilibrium expansion.

130

Performance of hydrogen/fluorine showing the effect of improper identification of product species. Sea-level specific impulse, optimum expansion Pc = 1000, psia, equilibrium expansion.

133

18

V. G. 1.

Relative performance for hydrogen/oxygen propellants as cryogenic liquids and as gases. P, 7 1000 psia, vacuum specific impulse, Ae/At = eqmlibrium expansion.

133

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L i s t of Tables Table

Page

II. c. 1.

Specific impacts with particle lag.

80

III.A.l.

Comparison of the equilibrium and kinetic decomposition of hydrazine monopropellant.

87

Comparison of the energetics of nonequilibrium reactions with their equilibrium alternatives.

87

Partial equilibrium combustion (PEC) model for the reaction o€ the hydrazine/nitrogen tetroxide propellant combination.

88

Hydrazine/nitrogen tetroxide combustion product composition and properties according to equilibrium combustion (EC)and partial equilibrium combustion (PEC). Pc = 1000 psia.

89

III. A. 2.

III. A. 3.

III.A. 4.

IV.A.l. Heats of dissociation reactions. N . A . 2. I

I N . A . 3.

Performance and reaction products of gasoline and liquid oxygen at 300 psia with shifting equilibrium to 14.7 psia.

98

Performance and reaction products of hydrazine and liquid oxygen at 300 psia with shifting equilibrium to 14.7 p i a .

99

Performance and reaction products of liquid hydrogen and liquid oxygen at 300 psia with shifting equilibrium to 14.7 psia.

100

Performance and reaction products of liquid hydrogen and liquid fluorine at 300 psia with shifting equilibrium to 14.7 psia.

101

Summary of maximum shifting specific impulse data.

101

Specific impulse and combustion temperature of several monopropellants. Equilibrium decomposition and expansion, Pc = 1000 psia, sea level specific impulse.

111

Iv.c .2.

Characteristics of monopropellants.

111

N.c . 3.

Decomposition of hydrazine monopropellants, equilibrium and experimentally observed compositions, mole fractions.

111

N.A. 4.

N . A . 5.

N . A . 6.

N.C.1.

~

98

20

W.E. 1.

IV. E. 2.

V. G. 1.

V. G.2.

Enthalpy content of some candidate heat transfer rocket propellants. Pure gaseous substances, frozen composition (no dissociation, recombination, or phase change).

114

Enthalpy content of some candidate heat transfer rocket propellants, equilibrium composition, one atmosphere pressure (includes enthalpies of dissociation and phase changes).

114

Effect of reactant enthalpy of formation on propellant performance (selected comparisons).

134

Effect of hydrocarbon enthalpy of formation and hydrogen to carbon ratio on performance with liquid oxygen as rocket propellants (sea level specific impulse, Pc = 1000 psia, equilibrium expansion, mixture ratio of maximum specific impulse).

134

21

Likt of Mathematical Symbols A A A

Area Arbitrary reactant Molar multiplier

a

Stoichiometric coefficient of arbitrary reactant A or arbitrary element Z Arbitrary reactant

B

Stoichiometric coefficient of arbitrary reactant B or arbitrary element Y Specific heat of a particle Stoichiometric coefficient of arbitrary element X Effective exhaust velocity Characteristic velocity Specific heat at constant pressure, cal/gm OK Specific heat at instant volume, cal/gm OK Thrust coefficient

b CS

C C

C* CP

cv CF

CS CP E EC F F f g

go ge, H h h IS,

K k k k L L* M

h m m

I

Total specific heat of particle, cal/OK Total specific heat at constant pressure, cal/"K Molar internal energy Equilibrium combustion Thrust Molar free energy Fugacity Acceleration due to gravity Conversion constant 32.2 lbm/lb, sec2 number of atoms of an element e in a molecule of species i Molar enthalpy Specific enthalpy Heat transfer coefficient Specific impulse Equilibrium constant Specific reaction rate constant Thermal conductivity Error in equilibrium reaction equation in arbitrarily assuming partial' pressures Latent heat of transition vc /A,

Mach number Molecular weight Mass M a s s flow rate

22

List of Mathematical Symbols (Contd.) n

Number of moles

Nu P P

Nusselt number Total pressure Pressure

P

Partial pressure

Q Q

Heat Arbitrary function

q

Arbitrary variable

q R

Electrical charge

R R Re r r

Gas constant

r r

S S S S

T t U

V V V

W W W W

X X X X

Y Y

Universal gas constant Arbitrary product Reynolds number Rate of change of concentration Radius Arbitrary variable Stoichiometric coefficient of arbitrary product R Molar entropy Arbitrary product Specific entropy Stoichiometric coefficient of arbitrary product S Temperature Time Velocity Volume Voltage Velocity of vehicle Mass flow r k e per unit reference area Weight Weight flow rate Rate of formation Arbitrary element Moles per cent mass of mixture Fraction r eassociated A coordinate M a s s fraction Arbitrary element

23

List of Mathematical Symbols (Contd. ) A coordinate Ratio of the square of atom concentration to molecule concentration Arbitrary element A coordinate Compressibility factor Ratio of specific heats All terms in reaction equation which can vary with gas flow in nozzle Frozen flow efficiency Total number of moles Chemical potential Fugacity coefficient General stoichiometric coefficient Density Mass of solid per unit volume of gas Equivalence ratio, oxidizer to fuel ratio divided by the stoichiometric oxidizer to fuel ratio

Subscripts a B b

f

Ambient conditions Backward reaction Backward reaction Combustion chamber Critical Exit of nozzle Equilibrium flow Forward reaction Force Formation

f

r. froz

Flow with finite ratio reactions controlling Frozen flow

g i

Gas Inlet React ants Products of reaction Gaseous products at exit of exhaust nozzle Molar Mass Maximum

C

crit

e eq F f

i

1 k m m max

24

Subscripts (Contd. ) 0

Zero degrees Kelvin

0

Empty Optimum

opt

-

Constant pressure

P P

Propellant

R

Reaction

S

Constant entropy

S

Particle

t

Throat

T

Constant temperature

ult

Ultimate

V

Volume

v.p.

Vapor pressure

i 1

I

I

I

Superscripts 0

The standard state

N

Products side of reaction

/

Reactant side of reaction Average

-

1

1

I

~

1

25

I

Introduction

This monograph is not an attempt to review all chemical propellants and their performance. On the contrary, the whole point of this endeavor is that such a review is not necessary and that by adapting fundamental thinking with respect to the basic thermochemistry, kinetics and fluid mechanics it is possible to characterize w & r-yl &particular propellant for a particular pr_opulsion scheme (1). Unfortunately many aspects of the field of rocket propulsion have developed almost as an empirical science. There are few exceptions to this rather critical statement. Much excellent fundamental work on droplet burning (2)(3), combustion instability (4), nozzle defect problems (5) (6) (?) (8), etc. , seems to have escaped the propulsion development engineer. In many instances he appears to have taken solace in the computer and extensive development tests. Having thought of, o r having had suggested to him, a propellant combination which may have promise, he simply inserts the appropriate data into his computer program and awaits an answer. If the resulting specific impulse is sufficiently attractive, development tests are begun. It is realized that the processes taking place in a rocket are much too complex to be amenable to complete analytical design, and that development tests will always be necessary in the propulsion field. However, many fundamental concepts have evolved from a knowledge of the basic sciences and requirements of propulsion systems to reduce appreciably the length of time requiredto evaluate a propellant system from its original concept and to introduce it to practice.

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Those entrusted with the development of propellant systems frequently have attemptedto gaininsight from new analyses, only to find the fundamental conceptual results of the analyses obscured in non-understandable detail. Correspondingly, many books which have appeared report and analyze the performance of chemical propellants (9 through 13). Yet rarely is an attempt made to explain why the performance results of the analyses are oriented as they are. This monograph is an attempt to state clearly the important concepts of propellant evaluation and to answer the important question of 'why?' Why is one propellant better than another, why does the point of maximum specific impulse shift towards stoichiometric mixture ratio as the chamber pressure is increased, why doesn't one obtain equilibrium chamber product concentrations with certain monopropellants, etc ? Necessarily, the authors have had to provide the background for what they believe to be a clear statement of the important concepts and the answers to 'why?' This procedure can, of course, lead to the same critical comment that the authors have used with respect to non-understandable detail. It is hoped that the detail is presented with clarity but in order to circumvent possible criticism the authors will adopt the unusual practice of underlining all key concepts and statements and sunimarizing them out of context in the Postface. In essence, the authors may be seeking the same objectives that Spalding sought in the combustion field when he put forth his most astute 10 half-truths of combustion (14). It must be emphasized that in or out of context some of the concepts may sound as gross generalizations. True, generalities will be given, but they will never be 'gross' in the sometimes popular use of this word. There will be exceptions to the many 'rule of thumb' concepts given, but the details in the text should make evident when exceptions should be expected. Partly in this vein, the authors quote Poincar??, It is better t o foresee with a little uncertainty than not to foresee at all. ' With respect to the structure of this monograph one finds that the purpose of the second chapter is to develop by ideal rocket theory, and to discuss, the rocket per formance parameters and, in particular, the significance of these parameters. The third chapter is a review of the pertinent chemical thermodynamics. Of importance a r e the thermochemical nomenclature, the discussion of the equilibrium constant and the handling of condensed phases, the methods of determining the combustor

26

product composition and the procedures for calculating the adiabatic combustion or flame temperature. The last section of Chapter III is most important and discusses the nozzles process of chemically reacting mixtures and how these processes determine the explicit values of the performance parameters. The emphasis is on the difference of the frozen, equilibrium and kinetic rate controlled flow situations. Detailed discussion of methods of making approximate kinetic rate calculations is given. The last chapters are addressed to the main theme as explained at the beginning of this introduction. In contrast the beginning chapters give the basis for the explanations in these later chapters. There is no attempt to give detailed performance information, although some is given. Only that which is necessary to prove the validity of a specific point is presented. There can be many approaches to the systematic evaluation of the performance of chemical propellants. The one presented here has been used effectively by the authors and it is their hope that the readers will benefit accordingly.

27

I1 Theoretical Calculation The parameters most commonly used to evaluate the performance of rocket engines are introduced first. From these, the significant parameters which determine the performance of propellants are derived. Similar to the approach in (15) simplified expressions will be derived theoretically in terms of the thermodynamic and other properties of the system in order to give insight into the fundamental significance of the individual performance parameters. These simplified expressions are derived from the so-called ideal rocket motor analysis. More accurate derivations are deferred until the next sections where the appropriate aspects of chemical thermodynamics a r e developed. A. PERFORMANCE PARAMETERS

The performance analysis of a rocket motor comprises the calculation of:

- the thrust c - the effective exhaust velocity c, - the thrust coefficient c* - the characteristic velocity F

and another parameter derived from c, I,,

- the specific impulse.

Many choose to list the adiabatic combustion chamber temperature Q , the average molecular weight of the gases leaving the combustion chamber fi, and Y , the ratio of the specific heats of these gases, in describing a given propellant combination. By considering the so-called ideal rocket motor, advantage can be taken of the simple one-dimensional isentropic flow relations to describe the performance paraIn meters in t e r m s of Tc, #IY and the appropriate pressures and nozzle areas. accurate theoretical evaluations, the /n and y described above lose their significance except when the flow is frozen; that is, when the composition of the combustion chamber gases remains the same throughout the nozzle. However, Tc r e mains significant in that it establishes the thermodynamic properties of the gases to a large degree establishes the extent of the entering the nozzle. Further, ‘I& heat transfer to the nozzle walls. The heat transfer is of fundamental importance in nozzle desifn. Tc is listed in many cases as a performance parameter together with c, c, , c , and I,,

.

(1) Thrust.

In any such engine performance analysis the thrust equation is the fundamental starting point. In general, the thrust exerted on a duct of arbitrary shape can be calculated from the momentum equation written in integrated form appropriate for one-dimensional problems. Following Figure II. A. l., one has: piAi (1 + yiM: 1

(stream thrust), =

&Ui

+ piA,)

(stream thrust), =

@,U,

+ peAe) = peAe (1 + y,Mi)

=

the total force on the external surface of the duct:

= F + p,(A,

- A,)

28

m e t A,

Fig. II. A. 1

Propulsive duct of arbitrary shape with appropriate symbols

A

A h

P

I

e

c

Fig. II.A.2 Ideal thermodynamic processes in the combustion chamber and nozzle of a rocket motor

-

29

Equating the forces to the stream thrust change:

The external forces on the duct are expressed as if the pressure on the external surface were identical with the ambient pressure of the atmosphere, although of course in actual flight this is not so. Therefore, for a duct in flight, this equation implies a certain arbitrary separation between the thrust F and the aerodynamic drag D. Separation in this manner is justified by its convenience, because the thrust measured in a ground test of the propulsion system is closely equal to the thrust F thus calculated.

Since his a constant, F = h u e +

be - pa) A,

(Eq.II. A. 1)

The ideal rocket motor analysis r e s t s on the following simplifications: 1) The combustion gases obey the perfect gas laws. This assumption is a good one at present day pressures of 500-800 psia for solids; however, proposed operating pressures of 2000-3000 psia even at the

temperatures of concern, may require some correction to the perfect gas law. Under these conditions, one should use the Beattie-Bridgeman or van der Waal's equation for the state equation and fugacity coefficients in the equilibrium calculations. 2) The average specific heat is constant. This is the overriding assumption in ideal gas theory and perhaps the worst. Com-, position and temperature changes drastically in the nozzle for most high temperature systems; however, the low temperature decomposition of monopropellants can be t r e d e d as ideal.

3) There is one-dimensional flow. This assumption is required even in accurate theoretical calculations. drastic of an assumption. 4) No friction or other dissipative losses. This assumption is required in order to assume isentropic flow. erally small for most practical rocket motors.

It is not too

Losses a r e gen-

5) No heat transfer losses. Siace most rockets a r e regeneratively cooled, this loss is not felt. There is more loss in solid propellant rockets but even here with modern case bonded propellants the losses are almost entirely confined to the nozzle except at the very end of burning and are of no great significance. 6) Flow velocity at nozzle entrance zero. In both ideal accurate theoretical calculations, one first calculates the temperature and then deals with the expansion process. For comparison's sake in evaluating propellants this simplification is required. One can correct readily for the velocity at the entrance of the nozzle.

30 7) Combustion is completed in the chamber and takes place at constant pressure. This simplification must be made because these factors depend upon motor design and not on the propellant system. The pressure drop can be calculated readily when necessary. 8) The process is steady in time.

For performance calculations one does not consider any type of time variation in pressure or mass flow. The thermodynamic process can b e indicated both on a P-V (pressure-volume) diagram and on a h-S(enthalpy-entropy) diagram as shown in figure II. A. 2. The propellants enter the chamber at point i and are gasified. They react as a constant pressure, pc , and then they are expanded isentropically through the nozzle to The throat conditions are noted with the subscript t. the exhaust pressure pe The h-S diagram becomes most convenient in following rocket motor processes and this is the reason for i t s introduction. The conveniences obtained are generally hidden by machine computation programs which essentially deal with the enthalpy-entropy process for the expansion process. h is the sensible enthalpy only. Theoretical performance calculations are performed in terms of the total enthalpy which is here defined as the sum of the sensible and chemical enthalpies only.

.

Under the ideal assumptions discussed above the combustion temperature is determined by the heat of reaction at constant pressure per unit mass as follows: AhR = Cp(Tc - T i )

(Eq. II. A. 2 . )

Ti is the temperature of the propellants entering the chamber.

At any station in the nozzle, the entropy, pressure, temperature, velocity, and Mach number are given by the following relations:

(Eq. II. A. 3. )

is always sufficiently large in rockets to Since the overall pressure ratio p,/p, establish sonic flow at the throat, then: @q. II.A.4.)

31

(Eq. II. A. 4. ) The specific heat ratios of typical rocket exhaust gases range between 1.1 and 1.3. The first figure corresponds to mixtures at very high temperatures with large concentrations of water vapor and large effective specific heats due to strong dissociation; the latter figure applies to moderate temperatures with moderate concentrations of H,O and CO,. With Y = 1.2, it can be seen that the drop in pressure from the chamber t o the throat is approximately half the chamber pressure, a n d the drop in temperature is only about one-tenth the chamber temperature. The mass flow through the nozzle can be expressed in terms of flow conditions at any station: rj,=

puA=pcA

;m= P U

= pc

I* Y - l

1%

h RTc

.]I

(E)?[1-(E) y - l i

6 M (E)

2

71- (6) -.-]I y-l

$

(Eq. II.A. 5 . ) (Eq.II.A.6.)

2

It is readily seen by substituting equation 11. A. 4. into equation II. A. 6. that: (Eq. II.A.7.)

A plot of mass flow per unit area (m/A) against static pressure ratio @/p, ) exhibits a maximum at the throat of the deLaval nozzle. By dividing equation 11. A. 7. by II. A. 6., one obtains:

(Eq. 11.A.8.)

By inserting A, .and p, for A and P, the nozzle area ratio, E = A, / A t , can be expressed as a function of pe /p, as shown below and in figure II. A. 3.

The maximum exit velocity is obtained by setting p, /p, equal to zero in equation II. A. 3.

The original thrust formula can now be rearranged by substituting for A, and for m:

32

Y-1

Y+l

(Eq. II.A.9.)

From this expression it is seen that the thrust does not depend at all on the combustion temperature T, , but depends mainly on the dimensions of A, and A+ and on the chamber pressure Pc In other words, the thrust that a rocket motor develops does not depend upon the particular choice of pcopellants, but upon the chamber pressure. The designer controls the pressure level of operation byamount of propellants chosen to be inje-. In fact, if it were not for the slight dependence of y on the combustion products, then the thrust level of a rocket engine could be considered entirely independent of the propellants used.

.

.

For given values of Pc, Pa,and A,, it can be shown that F reaches a maximum As will be seen later the specific impulse is also a maxivalue when P, = Pa. mum when p, = pa and in fact theoretical calculations which evaluate propellants always are performed for this condition. The proof of maximum F for pe = pa is as follows: One differentiates equation II. A. 1. to obtain d F=

U,

dm

+

&due

+ (p, - pa) dA, + A, dp,

Since in a rocket d m = 0 and the momentum equation gives: m due =

- A,

dp,

one has:

b e - pa) d A,

dF = Since

4 and pe are interrelated, it follows immediately that:

dF d 'e

= (p, - p a ) = 0

or for F to be a maximum pe = p a . (2) Effective exhaust velocity

Since the thrust reaches a maximum value when pe = p a , a well designed rocket exhaust nozzle should exhaust gases at an exit pressure nearly or exactly equal t o the ambient pressure. Near the point of maximum thrust the second term in the thrust equation (Eq. II. A. 9.) is like a small correction term. Thus it is appropriate to define another parameter, the effective velocity c, whose significance will be examined in more detail: C E

+L

=U,

+

Pe-pa c

(Eq. II.A. 10. )

e'

It follows from this definition and the maximum conditions for F, that c also reaches i t s maximum value at pe = p a , and, of course, at this point c = ue

.

Both

U,

and the term [$,

- p,)/G]A,

vary strongly with A,,

but in opposite

i

C

Ve

//#-

"e-

/ I

-0 0-

/

/

\

LOPTIMU

\ '\

EXIT AREA \

34

directions, so that the sum near the point of maximum thrust is practically insensitive to A, Jl?ig. II. A. 4. ) As a result, the effective exhaust velocitv as determined by the ratio of F to can be taken to be the optimum value of U, even if . the actual experimental nozzle is somewhat off desigq. Herein lies the practical significance of the concept of the effective e;xhaust velocity.

.

+

c may be expressed in t e r m s of the thermodynamic properties by substituting e for p into equation II. A. 3. and obtaining:

4s.

II.A.ll.)

The above expression for Uc is substituted in Equation II. A. 10 and one obtains:

From equation II. A. 12. one s e e s that c is a function of the chamber temperature Tc and the average molecular weight of the combustion products, as well as pc and the dimensions of the nozzle. Since and fi are determined from the given propellant combustion, c is a parameter used for comparing various propellant combinations. However. such comparisons should be under optimum conditions; that is, for pe =pa,. For this reason: r

(Eq. II.A. 1 3 . ) (3) Specific impulse

Perhaps one of the most important performance parameters in rocket technology from point of usage is the specific impulse. It is defined as the propulsive impulse.

Beingsimply the quotient of the thrust and the total w e i g h m w , thespecific imparameter readily measured experimentally with Food accuracy. This fact accounts for its pcpular acceptance. With regard to convenience there is no greater merit in the use of I,, instead of c. As for the effective exhaust velocity, the specific impulse is evaluated for optimum conditions when theoretical comparisons a r e made between various propellant combinations. For p, = pa then :

L

J

(Eq. 11. A. 14.) This equation shows that thespecific impulse varies directly with the square root of the chamber temperature and inversely with the square root of the average molecular weight of the combustion products. Thus from ideal engine theory one seeks a propellant combination which gives the highest temperature and the lowest molecular weight of the combustion products. Of course, it should be remembered that although @/fi ) is a good figure of merit for propellant comparisons, it is not explicitly correct under real conditions and for a few cases could be misleading. The units of specific impulse are fieconds. There is some confusion with respect to these units when the English system is used. As stated above the specific impulse is defined as the thrust divided by the weight flow rate:

35 I,,

=

F T

W

One pound mass (1 lb, ) under a one g (32.2 ft/sec2) acceleration at sea level exerts a weight of one pound force (1 lb, ). Thus: 1 lb, =

-

1 lb,,, g go

=

h(ft/secz)

lib, ft/lbf sec?

=

lbf

where go is the conversion constant 32.2 lb, ft/lb, sec2. It follows then:

-A the weight flow rate i n the definition of specific impulse should be specified as at sea level and thus go = p and I,, = (F/m) numerically. (4) Thrust coefficient

It is convenient to non-dimensionalize the thrust equation and obtain a dimensionless ratio F/P, A t . This ratio is defined as the thrust coefficient: CF

=

F

(Eq. II.A.15.)

P c t

From equation II. A. 9. one obtains: Y +1

y-1

@q.11. A. 16. ) Sometimes as a convenience two functions

r and r' are defined:

Equation II. A. 16. is then rewritten as:

From equation II. A. 8. it is seen that pe/po is a function of E. cp is then dependent only on the three independent variables y , po/pa and E. Plots of cF are given in figure II. A. 5. It follows from the development for F that cF should reach a maximum at p, = p a and that a plot of cF v s E at various values of p,/ pa should give curves which exhibit maxima. These maxima occur at values of pe = pa and the nozzle area ratio at this condition is called E, t. A nozzle having an area ratio less than E o p t is said to be underexpanded, ancfone having an area ratio greater than E o p t is overexpanded. It follows from figure 1I.A. 5. that nozzles that are either under-or-over-expanded produce less thrust than a proDerly expanded nozzle. The maximum values of cF are found by dropping the second term in equation II.A. 17:

36 (Eq. II. A. 18. ) L

One can replace pa /pc by cOpt through equation II. A. 8. A plot of c, v s Copt gives a curve which goes through the maxima in figure II. A. 5. This curve approaches an asymptotic value at infinite expansion ratio or (pa /pc ) = 0. This asymptotic value is the ultimate + obtainable and is given by: (Eq. II. A. 19. )

.

Conseauently It is significant that C, is completely @pendent of T, and fi a i g u r e of merit, it is insensitive to the efficiency of combustion, but sensitive to the nozzle design, In practice the test engineer compares the measured C, , which is determined from actual measurements of pc , A t , and F, with the theoretical C, computed from equation II. A. 17. or figure II. A. 5. to determine whether the nozzle is functioning efficiently. In this way he can localize to some extent the cause of an unexpected defect in the specific impulse. If no defect is found in CF then the loss must be in the combustion process. In the next paragraphs a parameter solely dependent upon the combustion efficiency is defined. (5) Characteristic velocity

The characteristic velocity c* is defined as: c*

*gP At

(Eq. II.A. 20. )

Thus one sees that c* is,readily determined from experimental measurements. It is immediately evident that: (Eq. II.A. 21.) Many possible expressions may then be developed for c*

.

Some forms are:

From equation II. A. 23., it is seen that c* depends mainly on conditions in the combustion chamber; that is, the flame temperature and combustion product composition through fi and y . The chamber pressure only indirectly influences c* through i t s effect on T,

.

With c* sensitive to the combustion process and C, sensitive to the nozzle process, a defective specific impulse may be more readily located by not only measuring F and \6, but pc and At as well. Since the chamber pressure is always measured and the throat area known, but generally checked before and after operation, information to calculate I,, , c, and c* is always on hand.

37

THRUST COEFFICIENT CF

Fig. II. A. 5 Variation of rocket thrust coefficient with nozzle area ratio and pressure ratio PJP, for Y = 1 . 2

38

B.

THERMODYNAMICS, COMBUSTION CHAMBER EQUILIBRIA AND FLAME TEMPERATURE

Before discussing the procedures for calculating the flame temperature reached in the combustion chamber, a brief review of chemical thermodynamics is in order. No attempt is made, of course, to review all of chemical thermodynamics. Only those aspects necessary to the proper understanding of the calculation of flame temperature and product composition a r e reviewed. (1) Basic thermodynamic concepts (15) (16)

The internal energy of a given substance is dependent upon i t s temperature, pressure, and state of aggregation and is independent of the means by which this state was brought about. Likewise the change in internal energy, AE , of a system which results from any physical change or chemical reaction depends only on the initial and final state of the system. The total change in internal energy will be the same whether or not the energy is evolved or absorbed in any form of heat, energy, or work. For a flow reaction proceeding with negligible changes in kinetic energy, potential energy and with no form of work beyond that required for flow, the heat added is equal to the increase of enthalpy of the system: &=AH

(Eq. II. B. 1.)

For a non-flow reaction proceeding at constant pressure the heat added is also equal to the gain in enthalpy: Qp = A H and if heat is evolved: Q, = - A H For a non-flow reaction proceeding at constant volume the heat added is equal to the gain in the internal energy of the system: Q, = A E In dealing with chemical reactions the stoichiometry is best represented in terms of molar quantities. In the previous chapter as a convenience in dealing with flow problems, all properties were evaluated on a mass basis.

One of the important thermodynamic facts to know about a given chemical reaction is the change in energy or heat content associated with the reaction at some specified temperature with each of the reactants and products in an appropriate standard state. This change is known either as the energy or heat of reaction at the specified temperature. The standard state means that for each state a reference state of the aggregate exists. For gases, the thermodynamic standard reference state is taken to the ideal gaseous state at one atmosphere pressure at each temperature. The ideal gaseous state is the case of isolated molecules which gives no interactions and which obey the equation of state of a perfect gas. The standard reference state for pure liquids and solids at a given temperature is the real state of the liquid o r solid substance at a pressure of 1 atmosphere. The thermodynamic symbol which represents the property of the substance in the

39

standard state at a given temperature is written, for example, as, H; , E; , etc., where the superscript specifies the standard state and the subscript T the temperature. A number such as 298 or 0, written in place of the subscript T represents an actual temperature. Statistical calculations permit the determination of E; - E: , which is the energy content at a given temperature referred to the energy content at 0°K. For one mole of molecules in the ideal gaseous state the following relationships hold: pV = RT H;

=

E; + (pV)"

= E;

+ RT

At 0°K these equations reduce to

and thus the heat content at any temperoatwe referrzd to the heat or energy content - H i = (E; - Eo ) + RT H, - H i also can be determined at 0°K is known, J$ for known % d%a from 0°K to temperoature T. Consequently one finds convenient tabulations of HT - eo and E; - E o at various temperatures for many substances.

.

For any specified reaction, the differences between the values of AE and AH may be evaluated from the definition of H: E +pV

H

A H = AE+A(pV)

In the above equation, A (pV) is simply the sum of product of pressure and volume for each of the products l e s s that of the reactants for the specified reaction. Since the pV product for liquid and solids is small compared to the p V product of gaseous substances, one would expect that there would be little difference in A E and A H for reactions in which all constituents were liquids or solids. In reactions in which ideal gases are involved, the difference in A H and AE is readily evaluated.

Since: A (pv) = ( A n ) R T

Then:

AH

= AE

+ RT A n

where: An =

I

'"products

- 'nreactants,

and n is the number of moles of gas. Generally the A (pV) term is small and very rarely more than 5 percent, usually about 1 or 2 percent of the total.

From the definition of the heat of reaction,

Qp

will depend on the temperature T at

40

i I I

t

( 1 ) I--,-I

I

I REACTANTS

+ pATH I

*---+

t---I

HTo

TO

I

PRODUCTS

Fig. II. B. 1 Reaction paths showing heats of reaction at different temperatures

T

I

Fig. L B . 2 The heat available and heat absorbed in a reaction mixture as a function of temperature

41

.

which the reactants and product enthalpies are evaluated. The heat of reaction at one temperature T o can be related to that at another temperature T, by Kirchoff's Law, as follows:

sT

T1

TX

( A C p ) d T + (ALIT + JT

0

X

(ACp)dT

X

(Eq.II. B. 2. )

where: A C = ~ Eni

c,, -

C nj -

cpj

j

i

and

The subscripts i and j refer to products and reactants respectively, and L represents the latent heats of transition. Equation II. B. 2. is easily developed from the diagramatic representation in figure II. B. l., which for convenience omits phase transitions. According to the first law of thermodynamics the heat changes in going from reactants at temperature To to products at temperature T, by either Path A o r Path B shown must be the same. Path A raises the reactants from Path B reacts at To and raises the temperature To to T,, and reacts at T,. products from To to T,. Thus:

(Eq. II.B.3.)

from which equation 11, B. 2. follows. It is tacitly assumed in the above that all phase transitions take place at the same temperature. This assumption is merely one of convenience since the explicit evaluations where transitions are present a r e obvious. If the heats of reaction at a given temperature a r e known for two separate reactions, the heat of reaction of a third reaction at the same temperature may be determined by simple algebraic addition. This statement is the Law of Heat Summation. For example, given below are two reactions which can be carried out conveniently in a calorimeter at constant pressure:

C

(graphite) + 0 2

k) +CO,

CO(g) + $ 02(g)-COz(g)

Qp

(g) Qp = + 97.7 kcal

(Eq. II.B.4.)

=+67.7kcal

(Eq. II. B. 5. )

Subtraction of equation SI. B. 5. from eqmtion 11. B. 4. gives: Cgraphik + $ o,(g)--Cco(g)Qp=+26.7 kcal

(Eq. II.B.6.)

. .

One notes that this last reaction would be difficult to carry out in a calorimeter since the carbon would burn to CO,, not CO, and thus the heat release of the CO reaction is more conveniently determined a s above.

42 It is of course not necessary to have an extensive list of heats of reaction to determine the heat absorbed or evolved in every possible chemical reaction. A more convenient and logiczdprocedure is t o list what are known as the standard heats of formation of chemical substances. The standard heat of formation is the enthalpy of a substance in its standard state referred to i t s elements in their standard states at the same temperature. From the above definition it is obvious that the heats of formation of the elements in their standard states are zero, The value of the heat of formation of a given substance from its elements may be the result of t h e determination of the heat of one reaction. Thus from the reaction represented by equation II. B. 4., at 25°C or 298°K it is apparent that heat of formation of carbon dioxide is: =

- 97.7 kcal/mole

As before, the superscript to the heat of formation symbol, A H f , represents the standard state, and the subscript number the base o r reference temperature. Similarly from equation U. B. 6., the heat of formation of carbon monoxide is: 0

(AH,)

29E

=

- 26.7 kcal/mole

It is evident that by judicious choice, the number of reactions whose heats must be measured will be about the same as t h e number of substances whose heats of formation are to be listed. The logical consequence of the above is, of course, that given the heats of formation of the substances which make up the reaction, one can determine directly the heat of reaction or heat evolved at the reference temperature, as follows: (Eq. II.B. 7. ) There exist extensive tables of standard heats of formation but all are not at the same base or reference temperature. The most frequently used base temperature is 25°C or 298'K. All the extensive published data by the National Bureau of Standards (17)are referenced to this temperature as are the more recent JANAF thermochemical tables (18). Another convenient base temperature is that evaluated at 0°K. The convenience arises from the fact that all values of heat of formation at this base are positive. The heat of formation of a compound at any temperature T is related t o the heat of formation at 0°K by the following relationship:

-

c

"j

j elements

0%"

-

(Eq. II.B.8.)

Thus one can pee a difficulty in use of the 0°K base. Any change of the low temperature specific heat data would cause a change in H T " - H," and would require recalculation of all (AHf )o data. One must remember that the (AHf qT values are obtained from the heat of reaction measured at a convenient calorimeter temperature.

For cases in which the products are measured at a temperature T, different from the reactants temperature TI, the heat of reaction becomes:

43

(Eq. II.B.9.)

-

nj

1O

j, r e a c t

- H;) - (

H o~ -

H~OII

+ (AH,O),

0

lj=

.

-Q~

0

where To is the base temperature at which the heats of formation are available. The enthalpies are written in t e r m s of Ho - Hoo because most tables of enthalpy are tabulated in this form. When all the heat evolved is used to heat up the product ga-s, the product temperature T, is called the flame o r adiabatic combustion temperature a n d

- HOo)

"i [{(HT,

-

@Too

.. Hgo)} + (AHfo)Tb]i =

i, p r o d

(Eq. II.B.lO.)

Ideally equation 11. B. 10. is the expression to be used for the calculation of the flame temperature in rocket motors.

If the products of the reaction ni are known equation 11. B. 10. can b e solved for the flame temperature. For reactants whose product temperature is less than 1250°K, the products are the normal stable species CO,, H,O, N,, etc. However in high energy combustion, such as in rocket motors, the temperatures are appreciably greater than 1250°K and dissociation of the stable species occurs. Since the dissociation reactions are quite endothermic, a few percent dissociation can lower the flame temperatures substantially. Consider for example reactants which are made up only of the atoms of C, H, and 0. The stable products for a C - H - 0 reaction can dissociate by any one of the following reactions:

H,O H,O

CO,

eH e iH,

+ 80, + OH +

OH

CO + 80,

Each of these reactions helps specify a definite equilibrium concentration of each product at a given temperature. Whereas in heat of reaction experiments or low temperature combustion experiments, the products could be specified from the chemical stoichiometry, one sees now that with dissociation the specification of the product concentrations becomes much more complex and the ni in equation II. B. 10. are unknown, as well as the temperature T,. Before various possible solutions for the ni and T, are discussed, it will be beneficial to examine the equilibrium conditions which govern the relationship that one product concentration has to another.

44 (2) Chemical equilibria

A system is in equilibrium when its state is such that it can be changed by an infinitesimal variation i n any direction, reversibly. For such a reversible process then one has for a system in equilibrium:

TdS = dE

+ pdv

(Eq. II.B.ll)

For a constant pressure and temperature, equation II. B. 11. may be rearranged to:

-

TS),,,

TS),,

,= 0

d(E + PV

= 0

or : d (H

-

By definition:

F

t

E + PV

-

TS

H

-

TS

where F is called the free energy. Thus the condition for equilibrium at constant temperature and pressure is that the'change in free energy be zero, i.e. : WIT,

,= 0

(Eq. II.B.12.)

Since the rocket motor combustion process takes place at constant pressure with gases in equilibrium at the flame temperature, this above criterion is of the greatest interest here. It is readily shown that the criterion for equilibrium in a constant volume, adiabatic system is: (Eq. II.B.13.) and for a constant pressure, adiabatic system:

(a)s,, = 0

(Eq. II.B. 14.)

For the equilibrium condition at constant temperature and pressure, it is now possible to determine the relationship between the free energy and the equilibrium partial pressures of the combustion mixture. One deals with perfect gases so that there are no forces of interaction between the molecules except at the instant of reaction, and thus each gas acts as if it were in a container alone. Let F, the total free energy of the mixture be represented by: I

niFi

F = i = A,B..

.R,

(Eq. II.B. 15.)

S

for an arbitrary equilibrium reaction:

a A + b B +.

. .+ r R

+ sS +

. .

(Eq. II. B. 16.)

In the strict sense one cannot refer to reactants and products since the reaction is proceding in both directions, but by convention the substances on the left are called r, 8 , ' are the stoichioreactants and those on the right products. 'a, b,

...

45

metric coefficients which govern the proportions by which the different substances appear and disappear. The ' ni ' a r e the instantaneous number of moles of each compound. Under the ideal gas assumption the f r e e energies are additive as shown above. This assumption permits one to neglect the f r e e energy of mixing. Thus: F @, T ) = H

0") -

TS (P, T )

(Eq. II.B.17.)

Since the standard pressure state for a gas is po = 1 atm, one may write: F" (Po, T ) = Ho 0")

-

TS" (Po, T )

'

(Eq. II.B. 18.)

Subtracting equation II. B. 18. from equation II. B . 17., one obtains: F

- F"

= @ - H oI)

-

T (S - So)

Since H is not a function of pressure, then H-Ho must be zero and: F

-

-

F" = -T (S So)

(Eq. II.B.19.)

Equation II. B. 19. gives the difference in free energy from the standard state of a gas at any temperature and pressure. The relationship of the entropy to the pressu r e is given by:

S

- So

= -R

In (P/po)

Hence, one finds: (Eq. II.B.20.)

F 0",P) = Fo + R T l n $/po)

An expression for the total free energy of a gas mixture now can be written.

in this case is the'partial pressure pi of the particular gas, thus: n p , = - L p

'p'

Cni i

where P is the total pressure.

(Eq. II.B.21.) The criterion for equilibrium being (dF) T,

c

A, B,

...R, S Fi"%

+

RT

chi In

= 0, one forms (dF) T,

(Pi/po) + R T x n i

dpi

and obtains:

= 0

pi

Evaluating the last term, one has:

since the total pressure is constant and thus c d p i = 0. Now consider the first term in equation 11. B. 21: C F P d n i =dn, FAo+dnB -dnR

FRO-

FBo

dn, Fso

+

S

.

.

- ..... I .

(Eq. II. B. 22.)

46 By definition of the stoichiometric coefficients: dn,

- a,

,

dn, = ka,

Hence: Z F P d n , = k{aFA0+bFBo+. ,

. -rFRo-SFso-. . . }

where k is a proportionality constant. Since the right side of equation II. B. 21. is zero and since k cannot equal zero, then:

... -r.FRo- sF,

aFAo+ b F B o +

0

-

.. = R T

The fact that the standard pressure p o is one atm will be used now. defines: - A F o = a F A o + b F B o + . . . - r F R o- SFSo -

Also one

....

A F o is called the standard free energy change, which is a reasonable name since A F o i s the change in the free energy if the reaction took place at standard condit-

ions and went to completion to the right. Hence the condition at equilibrium becomes: r

s

PS PA PBb where the pressures are measured in atmospheres. AFO

= -RT I n

(Eq. II.B.23.) One can define: (Eq. II.B.24.)

as the'proper quotient of partial pressures at equilibrium. K, is called the equilibrium constant. IC, is not a function of total pressure, but is a function of temp%ature alone. This statement is clear since A T i s a function of T only. It is a little surprising that the free energy-ge at the standard pressure po determines the equilibrium conditions at all other pressures. Sometimes it is easier to work with mole fractions than with partial pressures: pi = x i P

,

"i

xi =

(Eq. II. B.25.)

Xfii * X I is called the mole fraction. 24., one obtains:

AFO

=

- RT

In K,

(gn, )

r

where:

Substituting equation 11. B. 25. into equation II.B.

+

...

SI+

r

+s+

- a

..,

-a

- b - ...

- b - ...

(Eq.II. B.26)

47 r

s

K n = "R * "s nA

b

nB

From equation II. B. 26. above one sees that the proper quotient of molar concentration or mole fractions does depend on pressure. For flame temperature calculations, it is most convenient to write K, in t e r m s of the ni . The special case of:

r + s + . . . - a - b

.... =

O

is called a pressure insensitive reaction.

CO, + H,

+ CO=

H,O

H,

Consider the following two reactions:

- 2H

the first is pressure insensitive and the second is pressure sensitive. Increasing the pressure suppresses dissociation, as given by Le Chatelier's principle. The actual values of K, found in the tables depends on the reaction equation since r, s, a, b, . appear as exponents. If one uses:

...

1

-

xH~--instead of: H,

H

- 2H -

a much different value of K, is found. Care must be exercised to note the reaction The K, of the second hydrogen equatequation specified when using tables of K,. ion is the square of the first.

How the equilibrium constant varies with temperature is of great importance. sider: (Eq. XI. B. 27. )

ai constant pressure. dF

m

::

- -

+

From the definition of F one obtains: P

dV -S -T dS dT

At equilibrium:

T -dS dT

= -dE

dT

+ p dV -

dT

Thus:

Hence equation II. B. 27. becomes: d(F/T) dT

=

-TS-F T2

- H - -p

Con-

48

This expression is valid for any substance at constant pressure. action with each substance in i t s standard state, one obtains:

Applied to a re-

(Eq. II.B.28.)

AHo is the heat of reaction for the reaction: aA

+

+ s S

b B-rR

at temperature T and pressure of 1 atm. AFo = - R T

It is known that:

In K,

Substituting in equation II. B. 28., one obtains: d (In - - KP)

dT

AHo

-3

If it is assumed that

l1

InK,

2

is a slowly varying function of T:

AH

--

=

AHO

R

(1

TZ

-

1) Tl

Hence: (Eq. II. B. 29. ) For small changes in T:

@(,Iz

>

where T 2

> Tl

It is now possible to find K, at any temperature if i t s value is known at one temperat u r e and AHo (”) can be computed. Increasingly higher pressures are being used in rockets and calculations of equilibria and flame temperatures must necessarily follow this trend. As mentioned previously, at these high pressures gases deviate from the ideal condition. Consequently the functions defining equilibrium that have been, presented no longer hold, but must be modified. G.N. Lewis (19)the renowned thermodynamicist defined a function called the fugacity to simplify the mathematical relationships describing the equilibrium state no matter what the ideality of the situation. He defined the fugacity f as follows:

F =RTln f

The relationship is of the exact same form as equation 11. B. 20. in which pressure was used for f at low pressures. What then is the relationship between f and p ?

It can be shown that at constant temperature, it follows from: F = E that:

+ PV

-

TS

49

Therefore:

For non ideal gases V = z RT/p where z is the compressibility factor.

Thus:

Z

alnf = - ap = z a I n p P This relationship can be extended further to give:

a l n f -a I n p

-

= za In p

alnp

a l n (f/p) = ( 2 - 1 ) a l n p 1n@/p) =

-I,"(.-

i)a

lnp

(f/p) is called the fugacity coefficient v and f o r gases is also the activity coefficient. r h e condition for equilibrium is now: r

-

AFO

. = RT I n

fR

s f S

for the characteristic equilibrium reaction. Thus:

where K is the equilibrium constant defined previously. The fugacity coefficients are tabufated as functions of the reduced pressure (P/Pcrit ) and reduced temperature ( T/Tcr i t ) in various thermodynamic texts (see Hougen and Watson (20) ). One may not necessarily be able to find the K, for every reaction of concern; therefore the determination from sources of thermodynamic data can become important.

K, can be calculated once the standard free energy change is known: AF' = AH' A

-

T AS'

Ho can be determined from the tabulation of the heats of formation:

AH' =

C i, prod

ri(~Hfo)i-

aj j , react

AH^')^

50

From tabulation of entropies one arrives at: ASO

r,~:

=

-

i

ai S ; j

where S o is the absolute entropy. AFO

={E r i

AH,")^ -

i

Hence:

CI aj (AH~")JI - T {E r i si. i

ajSj "} 1

Sometimes the f r e e energy of formation is tabulated and: AFo =

r i (AFfo)i

- C a j (AF:)j j

i

Tables of ( F o - H )/T are listed since this quantity can be determined directly from the partitionhnction (21). This parameter is useful since:

-A F O-

-

T

Erir-

FO-H,O

T

i

+

- jCa j [ Fo - Hoo

(AH:)~ +i

+-

(AH:), T

lj

That the above relation holds follows from: F = E

+ RT

-

TS,

Hence at 0°K: ( A F ~ " ) ~= ( A E ~ " ) =~

AH^")^

One also knows that: r

AFo =

(AF:)i

- E

i

Thus one can write: ( A F f ) i (r) = (AFf,

L\FO=

ri

T

["'

aj (AFfO)j

j o ) i+

-HOO

Fi

-

+

Fq

=

AHf,oo

A2P~]-

+ Fi

- Ho0 Fy

aj

[

- Hoo

+

T

3

If the equilibrium constant for a given reaction is not available, it is also possible to calculate i t s value at anyxmperature by simple algebraic manipulation, if the gquilibrium constants of formation of the substances present relative to their elements in their standard states are known at that temperature. All the latest, convenient thermodynamic tabulations in fact list equilibrium constants of formation and this procedure has become the standard for the calculating K, of a general equilibrium reaction system, For example, consider the determination of the K, of the reaction: H,O

K,

&

H + OH

PH ,o

= PH POH

The K ,Is of formation (K , f ) must constitute the above K, as seen from the following:

51

$0, +

i H,

OH,

=

Kp, f (OH)

Po H 1 ' PO ZZPH2'

$ ) KP,f(OH) = PH POH = Kp KP ,fGI 2 0 ) PH 20 As an added convenience most thermodynamic tabulations list the log K ,f , so that K , may be calculated most readily. Note the K , 'S. as shown in the example always are written so that one mole of the 'product' is formed. KP,f

(3) Condensed phase in equilibrium combustion gas mixtures.

In many modern rocket propellant systems, condensable oxides are always present. Further the equilibrium constant of formation of carbon containing substances must contain graphite, since this is the form of carbon in its standard state. Thus, the condition presented by condensed phases must be considered. The equilibrium constant of formation of CO, follows from the simple reaction: 'graphite

-t * Z

'OZ

In a given complex reaction system where a substance is in the condensed form, the substance actually exerts a partial pressure in the gas phase equal to its vapor pressure at the temperature of concern. Thus the CO, reaction above is represented by:

where p,. g r a hite is the vapor pressure of carbon. However the vapor pressu r e is a thermo&namic property and is not affected by the reaction system. Thus the last reaction may be written as: #

KP,f

= K p , f pv.p.,

graphite

=

Pco 2

It is above K i , which are list$ for the equilibrium constants of formation when a condensed phase is present. Kp, f is determined from the standard free energy change of the gaseous system and the vapor pressure of the condensed phase. In order to determine whether a substance will condense o r not, one first determines the partial pressure without assuming condensation. If this partial pressure & greater than the vapor pressure, then in an equilibrium situation condensation must have taken place. Because most equilibrium reactions have their K '8 referenced to the elements in the standard states as, for example, carbon discussed above, it is difficult to determine the partial pressure of the carbon since the K Is, when carbon is not condensed are not readily available. Say, then, one has'carbon as a product and he wishes to determine the physical state. First he calculates the number of moles of carbon as condensed. Then, taking the same number of moles as gaseous, he determines the hypothetical partial pressure these number of moles of gas would exert. This partial pressure must be greater than the vapor pressure for the initial assumption that condensed phase is present to be

52 valid. When carbon is present as a product and is found to be condensed, then the exact quantity present is determined from the atom conservation equations. (4) Determination of the product composition

In the preceding sections it was seen that the product concentrations are only functions of temperature and pressure and that the original source of the atoms was not important. Thus for a C, H, 0 system one can specify that the products are CO,, H,O and their dissociated products. Some dissociated species have been listed but a more complete product list would include the following:

CO,, H,O, CO, H,,

02,

OH, H, 0, 0 3 ,

c,

CH,

For a C, H, 0, N system it is necessary to a d d N,,

N, NO, NH,,

and CN (at high pressures)

For a normal composite solid propellant which contains C, H, 0, N, C1, and Al, many more product compounds would have to be considered. In fact, in this case the possible number of products would b e so extensive that solution of our flame temperature problem would be almost prohibitive even for some digital computers. However knowledge of thermodynamic equilibrium constants and kinetics allows one to eliminate many possible product species. Consider a C, H, 0, N system, for example. For an overoxidized case, there is an excess of oxygen and the principal products would be CO,, H,O, 0, and N,. AS the temperature of the flame increases, dissociation begins, and if T, > 2200'K at P = 1 atm or T, > 2500°K at 20 atm, one must take into account the dissociation of CO, and H,O by the following reactions: CO, H,O H,O

-e

E CO + i o , 2

Qp = -27.8 kcal

H, +%O,

Qp = -57.8 kcal

i H 2 + OH

Qp =

- 67.1 kcal

The equilibrium constants show that there is at least one percent dissociation under these conditions and this criterion is the one used to specify presenck? of a given specie. Since the reactions are quite endothermic, even this small percentage must be considered. If one initially assumes that certain products of dissociation are absent and then calculates a temperature which would indicate such products, the flame temperature must be re-evaluated by including in the product mixture these products of dissociation; i.e. the presence of CO, H, and OH as products is now indicated: by the equilibrium reactions shown above.

> 2400°K at P = 1 atm or T ? > 2800°K at P = 20 atm then the dissociation of 0, and H, becomes important, viz: If T,

2

2H

Qp

=-103.8 kcal

0 2 720

Qp

=-117.2 kcal

H,

7

2

These reactions are highly endothermic and even very small percentages of dissociation can affect the final temperature. The new products are H and 0 atoms. Actually the pressure of 0 atoms could come about from the djssociation d water at this higher temperature according to the equilibrium:

53 H2O

+ 0

H2

Qp

= -116.9 kcal

From Le Chatelier's principle there is basically no preference in the reactions leading to 0 since the heat absorption is about the same in each. Thus in an overoxidized flame water dissociation introduces the species H,, 0,, OH, H, 0. At even higher temperatures, the nitrogen begins to take part in the reactions.

T

>

At

3000"K, NO forms mostly from:

i N 2 + $0, -NO

Qp

= -21.5 kcal

rather than: $N2

+ H2O

4 NO

+ H,

Qp =

-79.3 kcal

The first of the NO reactions is, of course, pressure insensitive. If T > 3500'K at P = 1 atm or T > 3600'K at 20 atm, nitrogen starts to dissociate according to:

N,

@

Qp

2N

=&

-225.1 k C a l

another highly endothermic reaction. At very high temperatures in highly oxygen rich (lean) flames ozone possibly could form from:

203

302

Qp

= -69.2 kcal

However very lean flames never give high temperatures and the existence of ozone in a C, H, 0, N system is most improbable. In an underoxidized (rich) C, H, 0, N system, there is not enough oxygen to burn The principal products are now: all the carbon t o CO, and hydrogen to H,O. H,O,

H,,

CO,, CO, N,

These products are linked by the pressure insensitive water gas reaction:

CO, + H,

-).

H2O + CO

Qp =

-9.8 kcal

A s the temperature rises, according to Le Chatelier's principle, the formation of H,O is favored since the reaction is endothermic in that direction. At temperat u r e s high enough to cause dissociation products, rich flames will dissociate according t o the reactions given earlier and: CO,

--- CO + io,

Qp = -27.8 kcal

Only at low temperatures (about 1200 - 1300%) are traces of C (graphite), CH, and NH3found. These species axe formed according to the exothermic reactions represented by:

2C0

+ 2H2-

-

N, + 2 CO

* I

CH,

3_

2

H,-

CO, + H,O

Qp = 59kcal

NH,

Qp = 11 kcal

Ciraphite +CO,

Q,,= -11 kcal

Cgraphite + 2H2

%

= -18 kcal

54

Equilibrium concentrations of carbon or ammonia are not found in short combustion chambers used in rocket motors. The reason for this non-equilibrium situation is that the rate of formation of soot is very slow and carbon does not have time t o form. Similarly the dissociation of NH, is very slow. Thus in ethylene oxide monopropellant rocket motors one finds very little carbon, whereas equilibrium considerations predict carbon as a predominant product; and in hydrazine decomposition chambers one finds an excess of NH, over that predicted by equilibrium considerations. In ethylene oxide motors carbon forms from the decomposition of methane, not the reaction represented above, thus both non-equilibrium situations give higher performance than expected, since the endothermic reactions do not have time to take place. Of course, carbon also could form in cool reactions which take place in boundary layers along the walls where velocities are slow.

For a rich mixture (CHON), a quick approximate method for determining the flame temperature and molecular weight of the products is as follows. All the carbon is burned to CO and all the hydrogen is left as H,. If any oxygen remains, the hydrogen is burned t o water and the CO is left unchanged. If less than stoichiometric proportions of oxygen are available then as much hydrogen is burned to water as possible. If more oxygen remains, then as much CO is converted to CO, as possible. All CO could not be converted since one is dealing with a rich mixture. Of course the nitrogen becomes N,. It should be emphasized that although the procedure leads to good estimates of the product temperature and average molecular weight, the actual product composition found is not correct. The discussions in the preceding paragraphs guide one in the choice of possible products for a C H 0 N system. Thus for reactants which would give a flame temperature of about 2500°K at 20 atm only CO,, H,O, CO, H,, OH, 0,, and N,, would have to be considered instead of the fifteen compounds enumerated for this system. Similar reasoning can be used to develop the logical set of products for systems containing A l , B, F, C1, etc. (5) Calculation of the adiab.atic flame temperature. The calculation of the flame temperature proceeds from equation II. B. l o . , which now is rearranged slightly as follows:

Q absArbed

I

I

I

Q available (Eq. II.B.30)

T, is the temperature at which the reactants enter the system. To is the base temperature at which heat of formation data are available, as noted before, principally 25°C. Most non-cryogenic systems have their theoretical temperature evaluoated with propellzpts added at the base temperature, consequently the term @TI -HOo)' @To - Hoo ) becomes zero. Cryogenic systems should be evaluated with' the cryogenic propellant added at its normal boiling point, thus this

55

enthalpy term is no longer zero but negative. If the ni products total a number p then one has 1.1 + 1 equations to solve for the Further one has a mass balance p ni and T2. The energy equation is available. equation for each atom in the system. If there are (Y atoms, then p Q! more

-

equations are required. These p - a equations come from the equilibrium relations and generally are referred to as the equilibrium equations. The equilibrium equations are basically non-linear. For the C H 0 N system it is necessary to solve 5 linear equations and ( p -4) non-linear equations simultaneously, i n which one of the unknowns T, is not present explicitly. These are a very difficult set of equations. Expressions already have been given for the energy and the equilibrium equations. To complete the system the mass balance equations will be written for a typical system. Consider the reaction between octane and nitric acid. nC 8H Is'' 8 '18 +

+ nH N 0

co

%O

HN03

---)

nC 0 2

+ %,02+nNzN2+%H

+ nNo N O + +

nIi 20 H2°

+

n H 2 H2

OH

C +nH H

Since the mixture ratio is not specified for the general expression, no effort was made to eliminate products. The mass balance equations then are:

+ 2nH2o +

NH = 2n, NO

q , ,

+ n,

= 2%2+nH20 +2%Oz+nC0

N, = 18 nc

+nNO

+nc

Ne = * o , + n c o NN = 2nN where:

+%€I '"0

,+ nNO ,

8 18

+n,No3

No = 3 n 1 . 1 ~ 0 ~ NC

=

NN =

8nC8H18 ~

H

N

O

~

nc 8H and nH N 0 are specified by the problem. the reacting fuel is taken as equal to 1.

Note that generally the n for

It is obvious that the system of equations that has been described can be programmed for a digital computer. There have been many treatises on how to accomplish computer solutions. One approach will b e described later. However, for purposes of insight, a reiterative method that will allow solutions to be calculated without digital computers will be discussed. Equation 11. B. 30. is so written that the t e r m s on the left hand side can be considered the Q absorbed and the terms with right the Q available. For a given mixt u r e and pressure one can proceed as follows: 1.

Estimate a temperature.

I

56

.

It is assumed that 2. Select the principal products and solve for their ni the ni of the minor products are zero. This calculation will involve at most one non-linear equation. From the remaining equations and the principal ni calculated in 2, solve 3. for the ni of the minor products. Return and satisfy the complete mass balance equations using the n 4. minor products determined in 3. Calculate the new principal ni.

of the

5.

Recalculate the ni for the minor products using the principal ni found in 4.

6.

Recheck the mass balance equations.

If the valuesof ni a r e repeating themselves within reasonable accuracies, 7. check the energy equatiob with these ni and the trial temperature estimated. If the energy equation is not properly satisfied, select another trial temper8. ature and repeat steps 2-7. 9.

Interpolate to obtain correct T,.

The energy equation was written i n a convenient way for interpolation. Let the trial temperatures be called T' , T " , T"' , etc. Using the compositions found for each temperature, calculate Q available and Q absorbed. These data are available from the last form of the energy equation. Plot Q vs T as shown in figure 11. B . 2. The general shapes of the curves in figure II. B . 2. can be deduced. At low temperature Q available is independent of T since there is no dissociation. As T increases, dissociation occurs and Q available decreases. Q absorbed will increase steadily with temperature since it requires more heat to raise the products to T, a s T, increases. Of course, the choice of trial temperature must be close enough to T, so that linear interpolation is possible. If not, mother trial temperature should be selected. If the composition of the product gas is required then steps 2-6 must be repeated for the T, obtained by the interpolation.

As the mixture ratio approaches stoichiometric proportions, the dissociation products become more important and the distinction between maj or and minor products becomes indistinguishable. In this range of mixture ratio, one needs more refined analytical techniques than that just described. A method due to Huff et n l (22) will be described and is particularly appropriate for digital machine computation. In order to show the procedure, the method will be adapted to the case where an initial temperature estimate is made and the problem is to find the product composition. Using the nomenclature of Huff, one can write any chemical equation as follows:

(Eq. I I . B . 3 1 . )

where: Z

,'

Y

, and

X a r e the elements in their standard states,

ex. :.

The approach to be followed is slightly different from that of Huffin that the elements a r e used as the reference states throughout rather than the corresponding

57 w

atoms. Assuming ideal gases, a volume is chosen so that V = RT.

i

Thus, since pV = nRT:

pi = ni

(Eq. II.B.32.)

The approach begins by arbitrarily assuming values of A and the ni for the given chamber pressure and original mixture composition. Corrections to the original estimates are found until the exact solution is obtained.

.

For the equilibrium relations, one chooses a s explained previously, the equilibrium reactions of formation. In the general form to correspond to the representation above, these a r e written as:

a, Z + b i Y + c , X +ZaiY,,,XCi

(J3q. .XII.B.33.)

Thus: Pi

K, =

P i i P Y b i Px c i

Or, in a form that will be more convenient later: log K, = log pi

- a, log P, - b

log PY

- C i log Px

(Eq. II.B. 34. )

The K, here a r e of course the equilibrium constants of formation. Because the assumed composition (initial estimate) may not correspond to that at chemical equilibrium, one writes: log k, = log pi

- a, log p

- b, log p,

- c, log P,

- log Ki

(Eq. II-B. 35.

As one comes closer to the correct pi, k,

__L

I

1

Since a composition is initially assumed, the mass balance equations may be written

as:

~

I

l

-

Again, as the solution to the problem is approached

I

a, b, c,

a,,

bo

9

C O

The total pressure equation is quite simple and is written as: p = c Pi

(J3q.II.B. 37.)

i 1

P also approaches the given pressure, Po, as the proper solution is approached.

The form of the above equations permits ready solution for k, , a, b, c, and p from estimates of ni and A. From these values corrections can be made t o ni and A. I

_-

58 e

These adjustments a r e made from a scheme of correction equations derived from the ones above. These correction equations adjust estimates by the Newton-Raphson method for simultaneous equations. The method can be illustrated by the following example. If Q1 and Q2 a r e functions of q and r, one writes:

By taking estimated values, for example qo and i o , each function may be expanWhen derivatives of higher than ded i n a Taylor series about the point q, , r, first order a r e neglected

.

AQ,

=

aQz as

A ~ +aQL

ar

~r

Therefore the equations for log k,, a, b, c, Taylor series. From equations II. B. 35: A log k, = A log pi -ai A log pz

- b,

... can be expanded similarly in a

A log p,

- c, A log p,

(Eq. D.B. 38.)

!

because:

a log k, a logp,

= 0-ai-0-0-0

, etc.

A log K, equals zero, since K i is aknown quantity. 1

:Aa=-

A

a,An,-C i

i

Similarly:

%!L A A A2

The corresponding equations for b and c a r e obvious.

From pi = ni

, one has:

A log pi = A log n, A Taylor's expansion of the logarithm of a variable log q yields: A q = q A log q

when terms of higher order than first a r e dropped. Thus equations II. B. 38. then become: A Iog n,

- a, A log nz - b, A log n,

- c,

A log n, =

- log k,

(Eq. II.B.39.)

because: A log

k, = log 1 - log k, =

- log k,

k, = 1 is, of course, the correct value being sought.

-log k, 'can be calculated

59

from equation 11. B. 35. and the original estimate, K, and the fact n, = p p Equation II. B.36. and the equations for b and c become of the form: 1

A

But

2 a, ni A log n, -

2 a, n, = a A,

a. n f i

A log A = a log a

A2

so the equations 11. B. 36. become:

a, n, A log ni - Aa A log A = Aa log (a,/a) b, n, A log n,

- Ab A log A = Ab log (bob)

ci ni A log n, - AC A log A

(Eq. II. B. 40. )

= AC log (c,/c)

Again, recall that: A log a = log a,

- log a = log (a,/a)

a, b, and c can be calculated from the assumed value of A and ni in equations I I , B . 36.

Also, it is required t o write:

2 n, A log ni = P A log P = P log

P P

(9)

(Eq. II.B. 41.

where the value of P is calculated from the original estimates of n, and the expressionni = p i

.

Equations II. B . 39-41 can be readily solved by determinants for A log n, and A log A. From these results one obtains An, and hA: A n, = n, A log ni,

A A = A A log A

The second estimates are obviously:

A, = A

+

AA

One reiterates until ki-+l,

a -.. a,, etc, within sufficient accuracy.

With the final n, and the assumedvalue of T, one checks the energy equation, or as stated in a previous section, Q avail.and Q absorbed. If the incorrect T is assumed the complete process is repeated for another T. The energy equation was not included in the above discussion for sakeof clarity in explaining the approach. However, since the above procedures are normally performed on digital computers, the energy equations could have been included as well and the temperature calculated directly. The complexity increases, but not outside the realm of capabilities of modern computers. When this procedure is followed the thermochemical data must be supplied to the computer as well as the equilibrium constants. There are various means of representing such data for computer language. In fact there are other procedures to solve the temperature problem than the reiterative process described above. What was attempted here was to give some understanding of how the problem is'solved. Not only is it possible to extend this approach to the calculation of the temperature, but also to the complete

60

rocket specific impulse problem. Such complete computer programs a r e available either on tape o r punched card inputs (23). Since the energy equation does not contain the final temperature implicitly, but through the enthalpy content of the product composition, the enthalpy values of the products a r e represented as power series in terms of the temperature for the computer systems. Explicit means of handling these aspects of the problem a r e discussed in detail in (22) and (23). C.

NOZZLE EXPANSION

Procedures for calculating the theoretical flame, o r product temperature and product composition of a propellant mixture were discussed in the previous section. What remains to be analyzed is the nozzle expansion process. Since most thermochemical performance calculations a r e made in order to compare various propellants or propellant combinations, certain ideal assumptions as discussed in Section II. A. are made. These ideal assumptions, however, only relate to the physical processes which actually occur in the rocket motor. A s explained, normal dissipative losses such as friction and heat transfer a r e ignored. The gasses are assumed to enter the nozzle at zero velocity at the temperature and product composition calculated theoretically for the given mixture ratio of the propellant combination. (1) The question of equilibrium

The thermal energy, or stagnation enthalpy, of the combustion gases is high and the stagnation internal energy of the gases is stored in the various degrees of freedom, translation, rotation, vibration, dissociation, and perhaps electronic. Temperatures high enough for appreciable electronic excitation and ionization are rarely attained with chemical reacting systems and thus are ignored here. However, temperatures are reached where a significant portion of the internal energy is stored in the higher degrees of freedom, such as vibration and dissociation. Unlike translation and molecular rotation, vibration and dissociation generally require a larger number of collisions to attain energy equilibration. Thus as the product gas expands in the nozzle and sensible enthalpy is converted into kinetic energy, the question is whether these internal modes of energy can lag. If there should be a lag in the equilibration process, lower performance results would be obtained. The time it takes for an internal degree of freedom to adjust to the change in state, caused in the rocket nozzle by the expansion, is called a relaxation time. For rocket motors of less than 50 lbs. thrust which have nozzles l e s s than 2 inches long, expansion times of 10-4 sec. or less occur and there can be appreciable dissociation or chemical lag. For motors of 100 lbs. thrust or more, the effects a r e not as important until one deals with high energy or low chamber pressures as may occur in motors to be used for space applications. Thus chemical lags are of concern and will be treated here. Fortunately, vibrational relaxation times do not further complicate the picture. They a r e generally much faster than the chemical times and thus the chemical times control the equilibration process. The vibration relaxation time of H, and 0,is of the order of 10-8 s e c s at combustion temperatures. The vibrational relaxation times decrease with temperature in proportion to exp ("-l/3). Further it is well established that the more complex the molecule the shorter the relaxation time. Thus for most propellant product mixtumtheassumption that vibrational lags a r e not of concern, particularly when dissociation lags a r e present, is apparently a good one. If the reaction times taking place in the reacting mixture a r e extremely fast compared to the expansion time, then chemical equilibrium will be maintained at all instances during the ewansion process; this flow process is referred to as e q u E brium flow. However, expansion in the nozzle may occur so rapidly that the reactions may not be fast enough to maintain equilibrium. In fact t b x p a n s i o n can be

61

so fast that the chemical composition does not change effectively from the c o m p a tion which entered the nozzle. This tvpe of flow is called frozen flow; i.e. the composition is frozen at the chamber conditions throughout the nozzle expansion process. The frozen and equilibrium cases represent two limits in the performance to be obtained from the system. The latter of course gives the higher performance because the dissociated species recombine in the nozzle and release chemical energy which can be converted into kinetic energy. Approximate procedures have been evolved which permit one to determine the state of the expansion process for a given system. In fact these procedures permit the performance to be calculated when the chemical r a t e s are finite and thus do not correspond to frozen, essentially zero chemical rate o r equilibrium, essentially infinite chemical rate,flow. As one would expect intuitively, the results of these finite rate determinations show that the flow remains nearly in chemical equilibrium at the beginning of the expansion process, and at a given temperature or point in the nozzle the composition becomes frozen and remains so throughout the expansion process. Finite rate performance calculations are very complex and a r e presently limited to only a few systems due to lack of kinetic data at the temperatures of concern. Thus most performance' calculations are made for either or both equilibrium and frozen flow and it is kept in mind that the actual results must lie somewhere between the two. For most systems equilibrium calculations are very satisfactory. (2) Isentropicity of nozzle flow processes

For an adiabatic process the equilibrium and frozen composition expansion processe s a r e both isentropic, whereas the finite rate process is not. The following thermodynamic development following (24) explicitly verifies this statement. For the energy equation, one has: dh

+ duZ/2

= 0

from which it follows for no dissipative losses such as heat transfer and friction:

A generalized energy equation, including chemical effects can be written: dh = TdS

+ */p + csp

dY,

where p ,is the chemical potential of species s and Y is the mass fraction. Combining the last two equations, the resulting equaticn is: (Eq. II. c. 1.) When the composition is frozen, i. e. , when dY, therefore the flow is isentropic.

= 0, dS must equal zero and

The rate of formation of substance S in a reversible, one step chemical reaction can be expressed in the form:

62

I/

PLANE

B

hfinite

-I

1

S

/

1?

Fig. 11. C . 1

The enthalpy - entropy variation in the recombination of dissociated species - a three-dimensional plot

a i chamber a.

'

SPECIFIC

FROZEN FLOW CONDITION

frozen

DISTANCE

Fig. 11. C . 2 Variation of composition in a nozzle to show transition to frozen flow

63

If a given single reaction R in the system is assumed to proceed to a small extent during a time dt, then the change i n Y, can then be written as:

where S, contains all the t e r m s in the reaction equation which can vary as the gas flows through the nozzle. The overall change in Y can be written as:

If this expression f o r dY, is substituted in equation ion for TdS is:

II. C. l., the resulting express-

The order of the summation can be changed so that the last equation takes the form:

Observe that the indicated summation over S is actually the change in f r e e energy occurring during the infinitesimal reaction step which takes place in the small time interval dt. This summation may be written as A F, so that the preceding equation may be written: (Eq. II.C.2.) It can be seen that the entropy will vanish, that is the process will be isentropic if d% = 0 or if A F R = 0 . Since dS, contains the specific reaction rate as a factor, this term will be zero if the composition is frozen through the nozzle, since it implies that the reaction r a t e itself is zero. On the other hand the term A F, will vanish if the flow through the nozzle is in equilibrium at all points since the general criterion for equilibrium is that the free energy change for infinitesimal variation of the system shall vanish. It should be noted that the entropy considered here is the total entropy and includes the entropy of formation.

As discussed earlier, in any nozzle flow it is conceivable that three ranges of values for reaction r a t e s should exist. In the first the temperature would be high enough and the rates fast enough to maintain local equilibrium. A s the flow expands and the temperature drops, the r a t e s become too slow to maintain equilibrium but reaction still proceeds in the nozzle. Further expansion lowers the temperature to a point where the reaction r a t e s become negligible and the composition can be considered to be frozen through the remainder of the nozzle. The two extreme cases are represented by AF, = 0 and dS, = 0, respectively, and are therefore isentropic. No information can be deduced about the entropy variation in the intermediate range of reaction rates: however the process is not isentropic because equation 11. C. 2. does not go to zero. Considerations from irreversible thermodynamics show that the entropy must always rise in a closed thermodynamic system when irreversible reaction processes take place (25). Figure II. C. 1. is a three-dimensional graph of enthalpy, entropy and composition. For simplicity only one composition coordinate is considered. In this figure two constant entropy planes are shown. In plane A the flow proceeds from point (C) to point (I) while chemical equilibrium is maintained. In plane B, which is set at a higher entropy, the flow goes from point (G) to point (A) with no change in composition. Between points (I) and (G) the reaction r a t e is not fast enough for equilibrium, but does proceed at some finite r a t e and is not frozen. Thus while the curves are defined adequately between points (C) and (Iand ) between points (G) and (A), the region between (Iand ) (G)is undefined

64

essentially. Such definition could be obtained from the complete finite rate calculation. However, this discussion, and an explanation of the other lines in figure II. C. 1 . , are deferred until sections II. C. 4 and II. C. 5. because of the somewhat limited application of such procedures to all practical systems and overall complexity. With the isentropicity of frozen and equilibrium flow established, explicit performance calculation procedures will be enumerated for the case of isentropic expansion. (3) Performance under isentropic (equilibrium and frozen) expansion conditions

It is reiterated that no effort is made to correct theoretical performance calculations for the case of heat loss and the fact there is a finite velocity of the gases entering the nozzle. Most theoretical performance calculations are essentially for comparison purposes, that is, to compare the potential of a given chemical system with what is available. This type of calculation is most meaningful because rocket design is an important factor in establishing the order of the losses and such considerations should not reflect upon the potentialities of a propellant. Similarly, for theoretical purposes, when condensed phases a r e present$ is assumed that the phases are in thermal and kinetic equilibrium with the expanding gases. It would be most difficult if not impossible at the present time, to predict lag and heat loss effects of particles theoretically when the size of particles attained cannot be predicted from purely theoretical considerations. With the assumptions thus stated, and following the approach of (15), the theoretical calculation of the specific impulse, c, c * , and cF then proceeds from the isentropic statement of the nozzle expansion process:

S e (exhaust products)

= S,

(combustor products) where S is the total

entropy. It is most convenient to carry out the determination of the performance parameters i n terms of the total enthalpy of the reacting mixture and the total entropy S; both quantities are computed for a definite amount of mixture. The total enthalpy is the sum of the sensible enthalpy and the chemical enthalpy. Since energy must be conserved and there is no kinetic energy change in the combustion chamber part of the motor, the total enthalpy of the incoming propellants must be equal t o the total enthalpy of the product gas at the product temperature.

and also:

= the absolute molar entropy of component j at temperature T, and where Ss a standar8’pressure: one atmosphere.

partial pressure (in atmospheres) of gaseous component j, 3eG gas.

n

= the number of moles of gaseous specie j.

considered an

65

n ' = the number of moles of condensed phase, either solid or liquid. It is seen that the total entropy is the sum of the standard state entropy and a logarithmic-pressure term. This log term contains the entropy of mixing. The condensed phases in their ideal states have entropies that are independent of pressure. In considering an expansion process in which it is possible that the mixture initially contains all gases and then as the temperature drops condensation takes place, one should remember that there is an entropy of mixing change. However, the definition of total entropy given above and which is used throughout this chapter automatically accounts for this change. The expansion with complete equilibrium is governed by the following entropy equation:

Where the k ' s designate the gaseous components at the nozzle exhaust and k' Is the condensed phases. From the product temperature and composition one can calculate the total entropy in the chamber. The unknowns in equation 11. C. 4. are T, and n k . Recall: pk =

"k -

e'

nk

where Pe is the total pressure at the exhaust. One solves for the unknowns in the same manner as for the temperature and product combination in the combustor, except that the energy equation is replaced with the above entropy equation. When one has obtained the temperature and product composition at the exhaust, he can proceed to calculate the total enthalpy at the exhaust. Recall that the total enthalpy is the sum of the sensible and chemical enthalpy and follows equation II. C. 3. evaluated at the exit conditions. The total enthalpy of the exit differs, of course, from the total enthalpy of the chamber by the amount of energy which was converted to kinetic energy i.e. : $ u e 2 E n k h k = H, - H ,

The mean molecular weight, density andarea per unit mass flow at the exit are given by:

nk

P, -Ae = lil

fi =-

P,

R

"e

1 P,

U,

If thqre are condensed phases present, one sees thatbl,

one is not dealing with an ideal gas.

and Pe are fictitious and The condensed phases give the gas an artific-

66

ially high density and molecular weight; of course, if no condensed phases are present then the above equations a r e familiar ones, which could be written in t h e form: =

CnkM k nk

e'

- =

1

m 7iaxT An important problem ~ generally not discussed - in texts is the calculation of the nozzle area ratio E for an eauilibrium flow system in which the chemical composition. Y and h a r e c h a n g i i . To determine the proper nozzle arearatio, the method outlined above can be utilized. A s e r i e s of pressures P is constructed to range from P, down to P, , and Ae/ni solved for each P as above. A curve is plotted of A/m versus P. The minimum corresponds to the throat of the nozzle and is called A /m. The desired nozzle ratio E o p t is then:

,

The above procedure gives more information than the optimum area ratio. In the E .,,determination, A /ni is obtained. Since c * = P, A /& the value of c * is calculated readily from the given value of Pc. The theoretical thrust coefficient is obtained as U, /c *. .

Whenever there is a condensed vapor present in the system, a two phase problem can occur and certain difficulties in comparing theoretical and experimental values of c* and c can be introduced. In particular the dilemma arises when a slight deviation from equilibrium conditions may cause substantial differences between measured and theoretical values of c * and thus c+ For example, it is quite possible that an experimental specific impulse efficiency greater than a c* efficiency can be obtained. If a condensable species is present and if theoretical calculations based on equilibrium show that the condensable species should condense before the throat, but in actuality due to a slow reaction or condensation process the specie does not condense until after the throat but still in the nozzle, then the specific impulse efficiency will remain high. However, since the heats of condensation are large generally, the measured c* will be low, because the effects downstream of the throat cannot be felt upstream. Thus the mean pressure generated in the rocket for a given mass flow will be lower, yet there is essentially no loss in the energy obtained from the system as a small c * might indicate. In fact, since thermal and kinetic equilibria are assumed between the gas and the particles, a large difference in theoretical, and experimental for that matter, c * will be found if a species condenses at slight distance downstream of the throat instead of upstream. If. i n a practical case, w r y predicts condensation in the contraction portion of the nozzle and it actually takes place in the expansion Dart, then a c * substantially lower than theoretical will be measured and a c p greater than theoretical can be obtained. The specific impulse efficiency will be normal since the condensation energy is recovered and c, is obtained throughJsp g/&.

.

The scheme for calculating the various performance parameters given can be programmed, of course, on a digital computer. If the machine is so programmed as to determine the area ratio as well, then it essentially calculates a complete Mollier diagram in the process. Because of the rapidity of machine calculations, there is no necessity for plotting a Mollier diagram. However, once a Mollier

67

diagram is available, it is convenient in rocket test programs when it may be necessary to correct for heat or other losses in order to determine how close a given propellant combination approaches theoretical performance. For a given number of gram atoms of each kind, the product composition is determined for various temperatures at the chamber pressure. From this composition and the appropriate temperature the total enthalpy and entropy may be calculated. Of course, the total enthalpy of the reactants must be equal to the total enthalpy of the product composition in the chamber before expansion; this value of total enthalpy determines the true chamber temperature and entropy before expansion providing there a r e no losses. The process is repeated at various pressures down to at least the exhaust pressures desired. The data which result permit the plotting of a complete H-S diagram. Thus for any system having the same number of gram atoms of each kind the specific impulse and other parameters may be calculated. The total enthalpy of the reactants establishes where one enters the enthalpy coordinate. Proceeding t o the appropriate value of the chamber pressure, the total entropy is established and since it must remain constant, one proceeds along the constant entropy coordinate until the exhaust pressure is reached. This point establishes the total enthalpy at the exhaust, and the difference of the total enthalpy entering and this value determines the exhaust velocity and consequently the specific impulse. It is obvious from previous descriptions how the other parameters would be determined. If there are losses known to be associated with a rocket motor performance value test, that value may be handled explicitly to determine the appropriate performance value. For example, if a rocket motor chamber loses heat, then one would subtract the enthalpy loss from the total .nthalpy, enter the diagram at the lower enthalpy and proceed as before. It should be realized that each point on the Mollier diagram is for the proper equilibrium mixture at the temperature and pressure of that point. Thus a specific impulse determined from an H-S diagram of this type is for equilibrium flow. Unless one is going to make many performance determinations for a fixed mixture ratio, a Mollier diagram is not fruitful. Performance values determination for frozen flow a r e much simpler. Since the flow is frozen the composition at the exhaust is the same a s that in the chamber and therefore known. For the given exhaust pressure and composition, the entropy equation (Eq., II. C. 4. ) i s solved to determine the exhaust temperature. Since one does not have to make a product composition determination, the entropy equation is solved quite easily for the temperature and thus the total enthalpy at the exhaust determined just as readily. The exhaust temperature for the frozen composition case is always less than the exhaust temperature for equilibrium flow. The difference will depend on the amount of dissociation existing in the chamber, the pressure and the temperature level. However the enthalpy of the eauilibrium exhaust state will always be less than that of the corresponding frozen situation. This fact is true because the total enthalpy is the warameter of concern and as -species recombine they release energy which can be converted to kinetic energyform products with heats oformation lower than chamber species. (4) Non-equilibrium performance.

A s stated earlier, most practical cases do not follow either frozen or equilibrium conditions, but a condition governed by the rate of the reactions taking place in the nozzle, i.e. a non-equilibrium condition. =typical non-equilibrium flow in nozzles, one begins more or less in equilibrium at high temperatures, because the_ chemical rates a r e very temperature sensitive and are the fastest at high temperatures. The flow e ands rapidly with the chemistry at first keepkg Dace with the -expansion, then falling behind in a transition zone, and finally virtually s t o p p a (frozen flow). In order to solve the rocket problem for this case, one must know the specific reaction rate constants for all the reactions taking place in the nozzle. Since the problem is non-isentropic, no direct method of solution is available and a

l

68

finite-difference numerical analysis solution for a many-equation problem must be used. Perhaps one of the clearest expositions on this problem is that given by Westenberg and Favin (a), and their approach i s the one followed here. The characteristic nozzle flow assumptions a r e made, i.e. the flow is laminar, steady, one-dimensional and there are no dissipative o r external forces of any kind. The reacting gas is considered to be composed of p chemical species, each of which is present at a concentration xi (moles per unit mass of mixture). The usual flow variable temperature T, density p , pressure P and velocity U then make a total of L./ + 4 variables. The cross-sectional a r e a ratio E, is generally specified as a function of the axial distance z ; and the axial distance z along the flow direction becomes the independent variable. A mass flow rate W per unit reference area is chosen. The reference area is usually taken as the minimum area and is the reference area for E as well. With this description of the variables the flow equations can be written as: '

(Eq. I I . C . 5 . ) (Eq. 11. C. 6.) (Eq. I I . C . 7 . )

w

= pu€

(Eq. 1 1 . C . 8 . )

where the H i are absolute molar enthalpies. If there are CY chemical elements in the system and the number of atoms of an element e in a molecule of species i is called g,, then similar to the discussion in section 11. B. 5. there will be (Y element conservation equations of the form:

,

C g,,

xi = constant

(e = 1 , 2 ,

. ..

1

(Eq. 11. C. 9. )

There are p + 4 unknowns; however, there are 4 flow equations as listed above ahd N element conservation equations. Just as in the solution of the equilibrium flame temperature problem discussed in section 11. B. 5 . , p - (Y additional equations are required. Except instead of using the equilibrium equations, one must adopt the chemical kinetic rate equations. The form used with the present problem is: ai= --

dz

r i [i = 1 , 2 ,

....

~JL-CY)]

(Eq. I I . C . 1 0 . )

where ri is the net rate of change in concentration of species i due to all the chemical reactions in which i takes part in the complex reacting system. r , has units of moles per unit volume per unit time and, of course, includes the various concentration variables X i, P , T and the specific reaction rate constants of the reactions which determine the concentration of Xi. The first four equations are reduced by 2 by eliminating P and U as variables. Westenberg and Favin make use of the fact that for ideal gases the enthalpy Hi is a function only of T so that:

'

69

where

is the molar heat capacity.

Then the two new expressions are:

and:

(Eq. II. c. 11. )

Thus for numerical solution, the equations are the ( U )equations II. C. 9., the &-a) equations II. C. lo., II. C. 11. and 11. C. 12. f o r the p + 2 variables T, P, and pXi . With all quantities known at some starting point z = 0, a computing machine can be programmed to calculate the derivatives in equations II. C. 10-12. Various machine integration routines are then available t o solve simultaneous, first order differential equations. Such routines should have a variable step-wise feature for automatically doubling o r halving the internal to satisfy a chosen precision index. The starting point for the problem is the equilibrium chamber conditions of the rocket. Thus the static temperature, static pressure and initial composition are known. The only remaining quantity to be established is W, the mass flow parameter. There is only one value of W allowed for the complete nozzle problem; i.e. one which will give a monotonic decrease in pressure and density throughout the nozzle. This eingenvalue W for the specified conditions must be formed by reiteration until the solution satisfies the condition that t h e frozen Mach number is unity at a point just downstream of the physical throat. The best procedure when trouble arises from starting from a state of equilibrium is to empirically adjust the initial Xi slightly from their equilibrium values to make the densities approximately correct. This procedure works because the subsequent solution is negligibly affected by small initial perturbations. (5) Approximations to non-equilibrium performance

The overall method described above can be quite complex when many reactions are considered and consequently can be quite time consuming with respect to machines and thus at times quite expensive to calculate. However, within the error introduced by the uncertainty in the rate constants, one can adopt simplified procedures. The most useful of the s i m p w p r o c e d u r e s is an adaption of Bray's technique for handling a recombination reaction in nozzles of heated supersonic tunnels (7). &ay points out simply that the transition r e m from equilibrium to frozen flow is very narrow. One can now make the same assumption for the multi-reaction nozzle problem. However, the question arises since there can be many transition regions, is it possible t o handle this problem in the same manner? But this difficulty can be circumvented by fundamental knowledge of chemical kinetics and thermodynamics. Generally there is one reaction in the complex scheme one writes for combustion gases that is the main energyleasing step. T h i s reaction then becomes the reaction of concern in the Bray

-

-

_approach.

A complex reacting mixture with which one must deal in propulsive devices may have ten to fifteen reactions to be considered. However, it is known that a reacting

I 4

1

70

I

mixture can maintain itself nearly in chemical eauilibrium, as the pressure and i f the three body recombination reactions follow fast enough. They are the controlling&eps in the same way a non-equilibrium mixture -approaches the equilibrium state, Two-body reactions merely exchange radicals back and forth, and may be individually nearly in equilibrium, even though, because of the three body reactions, the mixture as a whole is not.

temperature change, &y

I

I

What is particularly h p o r t a n t from a propulsion point of view, in this regard. is that the three-body recombination reactions are t h e major energy releasine: reactions as well. Thus if the Bray freezing point criterion is applied to only one step, it must be applied to a controlling three body recombination steR@ t h e reaction scheme. As one would perhaps expect such a result would agree well with the performance given by the complete finite rate solution, but would not predict the . other properties with great accuracy (8). I

Thus, to repeat, when the selected three-body reaction freezes, one can, for impulse purposes, consider the system frozen. Now it is possible t o find the point at which the system freezes in the same manner Bray used for the less complicated supersonic tunnel problems.

I

Bray obtained his approximate condition for dissociation-recombination reactions merely by estimating the sizes of the various terms in the rate equation. Essentially he writes the r a t e equation:

I where r F is the forward reaction rate and rB is the backward reaction rate. X, is the mass fraction of the species i of interest as before. There are now three regions to be considered 1) When the flow is near equilibrium, as at the beginning of the nozzle expansion:

r.F

FJ

rB

Thus: /U

<
2) In the transition region r F , rB and magnitude.

I u(dX,/dz) I

are all of the same order of

I

3) When the flow is near frozen, either rF<< rB or rB << rF and I u(dxi!dz) is the same order of the magnitude as the larger of.the two terms, i.e. either:

or :

Graphically, if Xi is a specie which is disappearing (and releasing energy as it reacts) the above situation can be represented as shown in figure 11. C. 2. Since the transition zone is very small, the intermediate freezing point condition is

I

71

therefore determined by either:

depending on whether rF or r B becomes small in frozen flow. When freezing occurs suddenly, at the freezing point, again for a narrow transition zone, the quantities may be evaluated in near equilibrium flow, and the freezing condition then becomes approximately:

where (eq) designates equilibrium flow conditions (26). Perhaps the concept can be clarified further by repeating the analysis according to the approach of Burwell, rt a/ (27), for the important three body recombination reaction which is most likely controlling. An arbitrary reaction is considered as follows: kF J + J + M J , + M kB where NI is the third body whose concentration is the sums of the concentration of all gaseous species present. The rate of formation of J, follows from kinetic considerations:

Parentheses denote concentratwhere a = k, (Jz) @ I) and y = Q),/(J,). ions. For near equilibrium flow, it is clear that y - ye, << yep, s o that:

In frozen flow the concentration of ( J ) is much greater than its .equilibrium value (Jeq) since none have recombined. Thus near frozen flow: yep <
The freezing point criterion then becomes:

as previously given. In terms of flow parameters:

The problem is then solved by first determining the equilibrium result for composition, pressure, temperature, velocity, etc., throughout the whole nozzle. Then one ] down proceeds to calculate r [= k, 0 ) eQ @ I)the nozzle and compares it with

,

480 MODIFIED BRAY

460 -

/

EQUILIBRIUM /--

440 I

I s p vac - sec

H t OH t M e =

420 -

\

\ \ \

400 -

\ \

0

I

KINETIC POINT I

I

I

I

Effect of oxidizer-fuel ratio on calculated HZ10,rocket specific impulse for various nozzle flow conditions

Fig. 11. C. 3

----

460

\

a

} O/F= 5.0 c--

440 O/F= 8.0

420

I I

SPvac - sec

/*#-

,/ -- EQUILIBRIUM / I

400

a 4)’

/ y

380

5

/I I

100

2

I

---- MODIFIED BRAY I

FROZEN

1000 2 Pc -PSlA

5

5 10,000

Fig. II. C. 4 Variation of calculated H,-0, rocket specific impulse with pressure for O/F ratios of 5 and 8 for various nozzle flow conditions

73 Since in practically the product of the concentration gradient (aX,,/a) andau. all cases freezing occurs just after the throat, comparisons can be begun at the throat in order to save time. When the equilibrium equality given above is satisfied the freezing point is found, the composition, entropy, enthalpy, temperature, etc., is noted and a frozen flow calculation is performed through the remaining nozzle section to the specified exhaust pressure. This calculation for frozen flow is isentropic, of course, as the equilibrium flow calculated up to the freezing point is isentropic. The final enthalpy difference is found by calculating both the equilibrium and frozen flow enthalpy differences. From the sum of these differences the exit velocity is calculated. Thus the specific impulse is determined. The applicability of this modified Bray approach under various conditions is discussed in detail by Burwell and his coworkers. Their extensive work warrants further discussion. Figure II. C. 3. taken from (27) shows the excellent agreement between the modified Bray approach and the complete finite kinetic scheme for H,-0, at P, = 300 psia. The critical reaction which gives this result is H +OH + M =HzO + M, which is the obvious appropriate choice from termolecular and energy considerations. Its only competitor H + H + M e H, + M is less energetic. Figure 11. C. 4. also takenfrom (27) shows the effect of the chamber pressure on the results. Again excellent agreement is noted between the complete kinetic and Bray methods. However, notice further that as the chamber pressure increases the kinetic solutions approach the equilibrium solution. This result would be expected because at higher chamber pressures there is less dissociation and higher possible kinetic recombination ratesdue to the high mixture densities. Similarly m y low chamber pressures, sav about 1 psia, the results would approach that pj frozen flow. The point in the nozzle at which the flow is frozen is also of great interest. In the range of moderate pressures, the freezing point is always found just downstream of the physical throat. u t , for the fuel-rich mixture ratios used to obtain maximum specific impuue, the freezingpoint is almost precisely at the physical throat ( E ) . *higher chamber pressures the freezing point will move further downstream. -Except at the hipressures, about 800-1000 pssa, where one would expect the kinetic and equilibrium results to be the same, the following verv simplified approach should give a very good approximation to the kinetic specific impulse result. A n q u i l i b r i u m calculation is performed from the chamber to the throat and then a frozen flow calculation from the throat to nozzle exit. Such an approach requires only a very minor modification to an equilibrium-performance machine calculation program, since such programs read out the throat conditions. The procedures for calculating the performance are also represented on fig. II. C. 1. For the modified Bray process the flow proceeds from the chamber condition (C) and follows the equilibrium result to a point just past the throat condition to the freezing point (J3). Since equilibrium prevails from (C) to @), the process is isentropic. From (B), the flow expands to (R) and remains frozen during the process. Again this step takes place in Plane A since the flow is frozen. In the quick estimate procedure or the approximate Bray technique for determining kinetic performance, the flow remains in equilibrium until the throat and is frozen from the throat to the exit. The process is completely isentropic, takes place entirely in Plane A, and is represented by (C) to (l”)to @). The relative entropy changes for all the processes discussed, pure equilibrium (C-E), pure frozen (C-F), kinetic rate controlling (C-I-C-A), modified Bray (C-B-R) and approximate Bray (C-T-S), a r e all shown on Fig. 11. C. 1. In order to see somewhat more clearly the relative order of the performance for each procedure, the three-dimensional graph given in figure II. C. 2. is represented by

74 h SPEC. TOTAL ENTHALPY

,

=CHAMBER CONDITION =EXIT COND.,EO. FLOW =EXIT COND..FROZ. FLOW l=EXlT COND..BRAY CRIT. 4 X I T COND.,ACT.FlNlTE RATE CALC.

1

/

/

h=THROAT COND.

X FRACTION

S Fig. 11. C. 5

I

I

RE-ASSOCIATED

SPECIFIC TOTAL ENTROPY

Enthalpy-Entropy variation in the recombination of dissociated species - Two dimensional plots permitting specific impulse comparisons

75

two 2-dimensional graphs in figure II. C. 5. In one of these graphs the enthalpy versus fraction reassociated for the critical reaction is plotted and in the other the enthalpy is plotted as a function of the entropy. The second plot is artificial in that each exit condition, althoughat the same exit pressure, must be plotted differently. Nevertheless, the performance order is shown more clearly. The order of frozen to approximate Bray to modified Bray to kinetic is quite readily understood the more recombination, the more energy release. The question a r i s e s why the kinetic result is less than the pure equilibrium. Here, obviously, the nonisentropicity of the kinetic approach plays a part as well. This effect is clearly shown in the h - s part of figure II. C. 5. (6) Non-equilibrium effects due to condensible products

Another non-equilibrium effect arises when the product composition contains a condensible substance. Solid propellant formulations based upon potassium perchlor ate form solidpotassium chloride and the acetylenic monopropellants upon decomposition form large quantities of carbon particles, as do very fuel-rich mixture ratios of hydrocarbon propellant systems. More recently metal and metal compounds have been used as fuels and form product oxides which are very high boiling point compounds that condense to varying degrees in the rocket chamber and nozzle. For example, estimates indicate that the normal boiling points of Li,O, BeO, A1,0, and MgO are approximately 2800, 4000, 3300 and 3800'K respectively. These boiling point temperatures of the metal oxides increase with pressure and thus can be seen to be appreciably greater than the corresponding equilibrium combustion chamber temperature of propellant systems. If the oxide product does not condense in the chamber prior t o expansion, then it is necessary to know the rate and point of nucleation and the rate of growth of the condensed nuclei. These rates must be fast in order for equilibrium to hold in the expansion process. If they are not then the problem must be treated much like the chemical kinetic rate problem discussed in the previous sub-sections. Unfortunately very little is known about nucleation and growth rates of even the most simple compounds and this problem has never been treated. Fortunately saturation temperatures of the metal oxide products of interest in rocket propulsion are so high that one generally assumes that practically all condensation and growth are complete in the chamber. However, given condensed phases entering the nozzle, the expansion process can be affected further by the rate of heat transfer from the particles to the gas stream and any velocity lag between the suspended particles and the expanding gas. Specific impulse defects occur when the heat transfer is slow and the velocity lag is great.

This problem of performance defects due to he& transfer and velocity lag when condensation is completed in the chamber is now treated. Very interesting limiting cases which have been detailed by Altman and Carter (5) are reviewed first. Then, the more general case of the effect of simultaneous heat transfer and velocity lag on nozzle performance is developed. Altman and Carter show that only very small particles (< 0.1 p ) have essentially no lag and that particles greater than l o p lag considerably behind the gas velocity, ' even upstream of the nozzle. They treat the heat transfer problem in three parts: 1.

Convective heat transfer from the particle to the gas.

2.

Conductive heat transfer within the particle.

3.

Radiative heat transfer between the particle and surroundings.

76

It is estimated that to maintain a temperature differential between particle and gas of no greater than 100°C, the particle radius should be less than 1 p . This result is based on rate of change of gas temperature in the nozzle of 107'C/sec. Assuming the heat transfer process at the surface of the particle to be fast, the limitation on thermal transfer between particle and gas can then result from conductivity within the particle itself. It would appear for materials having a thermal diffusivity of 10-3 cmZ/sec, that thermal transfer within the particle is fast for particle s i z e s less than 0 . 1 p and slow for sizes greater than l p . For conductive particles with thermal diffusivities of 10-1 cmZ/sec, the borderline for thermal transfer would be between 10 and lp. Particle radiation losses to the surroundings in an essentially transparent gas would only be of importance when the rocket motor was not regeneratively cooled. For uncooled rockets then, it has been estimated that for particle s i z e s greater than loop, very little energy is lost by radiation; however, for particles less than 0. l p , the rate of radiant energy loss can exceed the convective exchange to the gas during adiabatic expansion.

The limiting values for particle radii given above are, of course, dependent upon assumptions made as to the properties of the particles and gas streams. The values should be interpreted only as giving ranges of concern. The reader should refer to (5) for details. The actual distribution of kinetic and thermal energy between a gas and a particle during expansion can be calculated by procedures to be shown at the end of this section. However, it is interesting to determine first the effect of certain other limiting cases. Consider, for example, the case of no velocity lag and complete thermal equilibrium between gas and particles; i.e. :

For the nozzle expansion process the energy equation becomes:

dH = (C,,

+ C , ) dT

= Vdp

(Eq. II.C.13.)

and C, are the total heat capacities in cal/',K of the gas and partiwhere the C, cle respectivbfy. Assuming the'gas t o be perfect, one has for the equation of state: pV = % R T

(Eq. II.C.14.)

For the case of gases not chemically reacting and with constant specific heats, one can combine equations II. C. 13 and II. C. 14 and obtain:

(Eq. II.C.15.) The enthalpy change between T, and T, is then: AH

=

(Cp,g + C,) p,

- Te) = 4 mu,z

(Eq. II.C.16.)

I

77

This impulse expression may be written in the same form as that given in section 1I.A. 3. The terms in equation 11. C. 17 are now written as (30):

m = ng h ,

+“s fis

9

where Y is the mass fraction of solids. By defining:

~

17r

(1 (1

- Y) c p , g + Yc, -- Y) C”, + Yc,

and : ngfig+nsMs f-i = ---

i

I

(Eq. 11. C. 18. )

“g

Equation II. C. 17 becomes: 2

(Eq. 11. C. 19.) Thus an expression exactly similar to equtition 11.A. 14 has been obtained with the average molecular weight and average specific heat ratio redefined.

For the limiting case in which the particles travel with the gas but remain thermally insuiated:

~

U, I

,

I

-I

(Eq. 11. C. 20. )

Tc

It follows for the case U, = 0 and T, = T, that: I,,

I

1

T, = T,

- -2g --yr - 1 -ifR

I

l

,

The C, dT, term in equation 11. C. 13 now becomes zero. Using the redefined terms for the average ratio of specific heats and the average molecular weight, one finds that the impulse expression becomes (30): I,,

I

= ug

and f o r U,

=

I2-

=

17-1

-9

(77-1)

0 and T, = T, that:

Tc [l

-():

-1

‘ ]I’

(Eq. 11.C.21.)

78

In any real system, there always exists the possibility that the solid particles will lag to some degree and will not be in thermal equilibrium with the gas as well. If us/ug and T,/T, are known there the specific impulse can be calculated from:

1 E 2 g

[ (1-Y) us

2

+ Y

1

U,,

= Cp,g (Tc-T,)g

+

c, (Tc-Te)s (Eq. 11. C. 23. )

and: (Eq. 11. C. 24. ) For the following set of conditions the relative importance of the effects of nonequilibrium may be shown: T,= 3000°K (Pc/P,) = 20.7 Y = 0.20

ln

=

1g (total mass)

CP,+= Or6 cal/gm

/N

= 20g/mole

q g = 0.07 mole

C, = 0.1 cal/deg

In table II. C. l., the effect of various particle velocities, expressed as fractions of the gas velocity, on the specific impulse is illustrated for both complete thermal equilibrium and lack of thermal equilibrium. What is of practical interest is that thermal equilibrium betWeen particle and w is of far lesser imwrtance than particlevelocityhg. One should not be misled by the results for the no solids and the (us/ug) = 1 cases. These results are for the hypothetical considerations of fixed total mass in which one case has a fractional solid content of 0.20. The results show simply that when a fraction of the mass cannot be expanded there is a performance loss. Combustion gases with particles can be considered to have an artifically high molecular weight given by equation II. C. 18. For practical propellant systems containing metals, it should not be interpreted that is desirable to prevent condensation in order to eliminate the performance defects due to velocity lag, because for metal systems most of the energy release is related to the condensation step. The energy gain by condensation in metal systems far overrides the performance defect due to the particles formed. Although the developments for the limiting cases above give insight into the performance defect caused by condensed phases, it is also possible to deal with the more general case of simultaneous heat transfer and velocity lag in the nozzle expansion process and its subsequent effect on performance. It has been found that by choosing a restricted thermodynamic model for the gas phase, a one-dimensional, two phase analysis, which includes the proper energy exchange based upon the lags as they develop, will give a relatively reliable estimate of the performance loss (28) (29). This procedure has been developed by Kliegel (28) and Nilson, .dnl and is reproduced her e. The assumptions in the one-dimensional analysis are: 1.

The particles are spheres of constant and uniform size and do not collide.

2.

The particles occupy negligible relative volume.

79

3.

No phase change (condensation or fusion) occurs in the nozzle.

4.

The total energy is fixed.

5.

There a r e no viscous effects in the gas except that which exerts a drag force upon the particles.

6.

Heat transfer between the particles and the surroundings is by convection to the gas alone.

7.

The particle has a uniform temperature and a constant specific heat.

The continuity equations for the gas particle flow is then written as: G m : (1-Y)& = G U g A

Particle : Y & = P

I I

sg

(Eq. II.C.25.)

us A

where the cross-sectional area of the nozzle, A, can be known as a function of the axial distance z. The nomenclature is as given previously, but it is important to note that p,, is defined as mass of solid per unit volume of gas and the subscript s is to designate the particle.

I

The momentum, energy and state equations take the form: (Eq. II.C.26.)

(Eq. II.C.27.)

(Eq. II.C.28.)

'

' '

where c, is the specific heat of the solid. Since the drag coefficient may be represented as a function of the particle Reynolds number, i.e. (C, Re/24)=f(Re) the equation of particle motion may be written as:

where p is the actual density of the particle. Since Re=[Zr ,(u6 -U s) P / p g ] and for Stokes flow f @e) = 1, it follows the equation for particle motion cans, written

as: dus

us

-

9 clg. - 2p,r,Z

(Ug

-

us)

(Eq. II.C.29.)

The Stokes flow assumption is a good one for most metallized propellant situations. n / (29) whose explicit analysis is being followed, carries through the Nilson, function f (Re). C

B

~

The equation describing the loss of heat from the particle to the gas is simply:

80 4

nr,3 p, C,

U

dT ,

- Tg)

- h2 n r,2 (T,

s d z -

Since the particles are so small in most cases, the Nusselt number based on the diameter reaches its lower limiting value of 2, i.e., Nu = (h2r ,)/k = 2 when k is the thermal conductivity of the gas and h is the heat transfer coefficient. Then he preceding equation becomes: (Eq. II.C.30.) Equations II. C. 24, 11. (3.25, 11. C. 26 and 11. C. 28 are combined to eliminate pss, p s s and p: U ( g -!-) ug R dTg dz

du, dz

Y + -

= RT L A

dus

U

1-Y

6

dz

t

(Eq. II.C.31.)

dA dz

After rewriting equation II. C. 27 as: dT 1 L -- c [ u s =dug dz = P, 6

+

Y dT, 1-y ( C S T

du, + U , - - - ) dz ]

(Eq. II.C.32.)

then equation 11. C. 31 can take the form: du,

Y

dz

- 7 Y

g d z -

ug

dT, _-Rc c, ,, ~ dz

- -E-. CP, g

)

-

U

]I

Ru, c , , , ~ du, dz

(Eq. II.C.33.) 6

Equations 11. C. 29, II. C. 30, 11. C. 32 and II. C. 33 constitute-the system of differential equations to be solved. The equation must be integrated to find values of U . , U ,, Tg and Ts at the nozzle exit. Complications arise due to the fact that the h a c h number of the gas @ at the I)nozzle throat is l e s s than 1 (28). The denominator of equation II.C.33 is [(M2-l)/u6]. When (dA/dz) = 0, the physical throat, the numerator of equation 11. C. 33 is negative since the term in brackets remains positive. Thus the location of points where numerator and denominator vanish is not known in advance. The initial conditions upstream o r the cross-sectional area of nozzle must be modified to bring the zeros into coincidence. In order to hold the mass flow fixed as a reference point upon which to compare performance, Nilson, c*/ ctl (29) corrected the cross-sectional area. They have shown the change in crosssectional area generally required is quite small. Table II. C . 1. - Specific Impulse Variation with Particle Lag U,

/us 0

0.25 0.50 1.00 No Solids

I,, Cr,=Ts) sec. 206 217 224 230 254

I,,

Cr,=T,)

sec. 203 214 221 227 254

81

I11 Non Equilibrium Chamber Effects Most, if not all, solutions of the nozzle expansion problem have used equilibrium composition chamber conditions as the initial condition for nozzle solution. The feature is common to all of the nozzle flow solutions; that is, the equilibrium composition expansion, frozen composition expansion, Bray freezing model, and kineti c rate solutions have all invoked the assumption of equilibrium composition at the begihing of the expansion process. While the failure to obtain equilibrium composition predicted performance, in terms of experimental characteristic velocities, has suggested a departure from equilibrium in the combustion chamber, only recently have non-equilibrium compositions been measured directly (31). The existence of non-equilibrium combustion products is important t o at least two considerations. Firstly, the observed propellant performance may depart substantially from the predicted level. This departure may result in performance either less than or greater than the equilibrium predicted level. A striking example of greater than equilibrium performance is that of hydrazine monopropellant decomposition, table ID-A-1. Another is that of ethylene oxide monopropellant, as mentioned in section II. B. 4. , in which the equilibrium quantities of condensed carbon never are formed. Secondly, the non-equilibrium composition may have significant effects on the expansion process. In particular, nozzle kinetic calculations based on an assumed equilibrium composition initial condition may diverge significantly from expansions occurring from non-equilibrium initial conditions.

A.

EXPERIMENTAL EVIDENCE OF NON-EQUILIBRIUM COMBUSTION

Greater than equilibrium concentrations of intermediate species have been observed in the combustion products of several reactant systems. Examples are the concentrations of ammonia in the products of the decomposition of hydrazine (32), the concentration of CH, in ethylene oxide decomposition (33), nitric oxide and ammonia in the products of the reaction of hydrazine and nitrogen tetroxide (34), and chlorine monofluoride in the products of the reaction of hydrazine and chlorine pentafluoride (35). Direct evidence may be found both in laboratory and rocket engine experiments that the kinetics of the hydrazine/nitroq?n tetroxide reaction controls the composition of the react ion p r o d u c k x e n early observations of the reaction of hydrazine and nitrogen tetroxide in rocket combustion chambers contain evidence of the role of chemical kinetics in the production of non-equilibrium combustion products (36). The results of several rocket engine investigations are summarized as the variation of characteristic velocity with mixture ratio and are compared with the predicted values based on equilibrium combustion in figure III-A-1. Greater than theoretical performance is obtained at fuel rich mixture ratios while considerably less than theoretical performance is reported at oxidizer rich mixture ratios. The results cannot be dismissed as the consequences of poor injection technique, poor mixing, or insufficient reaction time (L *), especially with the observation of greater than theoretical performance. At near stoichiometric mixture ratios and at chamber pressures of about 300 psia, performance in terms of characteristic velocity is near the theoretically predicted value. The explanation of the p a p e r f o r m a n c e at high oxidizer to fuel ratios lies in the failure of the excess nitrogen tetroxide to decompose to molecular n i t r o g e m gen as would be predicted from equilibrium considerations. This effect has been identified in other experiments to be the freezing of nitric oxide. The effect is a substantial one a s is shown in table III-A-2. The decomposition of liquid nitrogen tetroxide to nitric oxide and oxygen is 43.2 kcal/mole'of nitrogen tetroxide more

a2

endothermic than the decomposition to nitrogen and oxygen as predicted from equilibrium considerations. Performance in excess of the predicted value is due to the well-established decomposition behavior of hydrazine in which ammonia, nitrogen, and hydrogen a r e the decompositions products rather than only nitrogen and hydrogen as predicted from equilibrium considerations. The effect in this case amounts to the liberation of about 11 kcal/mole of hydrazine more energy in the actual decomposition than in the equilibrium predicted decomposition. Other observations of the reaction of hydrazine and nitrogen tetroxide substantiate the production of non-equilibrium combustion products. Non-equilibrium product concentrations were found in combustion gases extracted from a small rocket combustion chamber through a molecular beam sampling device with direct mass spectrometric analysis (31) (39). Under oxidizer rich conditions excessive amounts of nitric oxide were found; under fuel rich conditions excessive amounts of ammonia were found. A correlation between the experimentally observed characteristic velocity and nitric oxide concentration exists (40). Related kinetic effects a r e postulated to account for the two stage flame observed in the burning of hydrazine droplets in nitrogen dioxide atmospheres (41) (42). The premixed vapor phase reaction of hydrazine and nitrogen dioxide has been observed to occur in two distinct steps, the reduction of nitrogen dioxide to nitric oxide followed by the slower reaction of nitric oxide (34). The results of this investigation are summarized in figure IU-A-2. The first step of this reaction takes place more rapidly than hydrazine decomposition. The second Step, the reaction of hydrazine and nitric oxide, however, is slower than the decomposition of hydrazine. In general, either the prior decomposition of hydrazine to ammonia, nitrogen, and hydrogen or the prior decomposition or nitrogen dioxide to nitric oxide and oxygen w a s observed to slow its subsequent reaction. The results of these kinetic investigations are consistent with the rocket combustor experiments in that they both show ammonia and nitric oxide, once formed, to remain in concentrations cont r a r y to equilibrium predictions.

B. PREDICTION O F COMBUSL'ION CHAMBER KINETICS AND NON-EQUILIBRIUM PRODUCTS The technique for coupling the chemical kinetic rate equations to the combustion process taking place in a rocket combustion chamber has not been devised. A detailed solution of the combustion chamber kinetics problem requires combination of the relations governing mixing, droplet burning, chemical reaction r a t e s and combustion chamber flow characteristics. It is neither obvious that the complete solution to the complex combustion kinetics problem is possible nor that the efforts in this direction are wisely undertaken on the basis of present understanding of the more fundamental processes. Certain crude approaches are available to predict overall results, that is, nonequilibrium compositions. More refined techniques are available for the analysis of simplified models. Solution of the reaction kinetics of homogeneous gas phase combustion is possible through numerical solution of the rate equations. With the exception of the role of an overall highly exothermic reaction, the procedures are similar to those described in the preceding section on nozzle processes. The solution of the droplet burning problem including the role of chemical reaction rates, while not particularly tractable, is feasible.

~

With the current marked increase in both the quantity and quality of reaction rate data, the prospects for valid prediction of kinetic processes and their effect on combustion product compositions will improve. Direct measurements of compos-

I

83

/ 4000 f

3000

'\

-

2ooo

A CHILENSKI AND LEE 33 B ROLLBUHLER AND TOMAZIC 34 C WANHAINEN De WITT, AND ROSS35

1000

I

l

l

1

2.5

-

RATE CONSTANT, LOG,o k I

(k, [ SEC-']

'.'\.

)

1

\

\

1

1

1

,

d$

\

\

~

~

k[N*H41

N,H4 DECOMPOSITION ( EBERSTEIN)

\ \ \

\

\ I .o

-

\

\ '(WET*

\

N2H4 DECOMPOSITION (EBERSTEIN)

,

1

1

,

84

280-

EOUlLlBRlUM COMBUSTION

260 PARTIAL EOUlLlBRlUM 240 -

220 -

1

2001

I

Fig. III. A. 3

I

-$L +

I

1

1

2 I I+

3

5 EOUIVALENCE RATIO

I5

IO

Comparison of PEC and equilibrium combustion propellant performance of hydrazine/nitrogen tetroxide. Partial equilibrium nozzle expansion from a chamber pressure of 1000 psia to one atmosphere pressure

SPECIFIC IMPULSE , sec 280 FOUlLlBRlUM COMBUSTION 260

240

220 I '

200'

;

1

I

I

I

3

2

I

, ,++ ~ I

I

1

5

IO

EOUIVALENCE RATIO

Fig. In. A. 4

Comparison of PEC and equilibrium combustion propellant performance of hydrazine/nitrogen tetrodixe. Frozen composition nozzle expansion from a chamber pressure of 1000 p i a to one atmosphere pressure

I

15

85

itions provide another means of establishing combustion product compositions. Unfortunately, both lack of facility in the carrying out of such experiments and the low precision of the results limit the information available through this source. C.

CONSEQUENCESOF NON-EQUILIBRIUM COMBUSTION

Two important effects of non-equilibrium combustion are departures from propellant performance predicted from equilibrium considerations and the introduction of e r r o r s in kinetic nozzle flow solutions which similarly are based upon the assumption of equilibrium combustion products. The magnitude andcharacter of the first effect can be demonstrated by a gross model for the non-equilibrium character of the hydrazine/nitrogen tetroxide reaction. The following 'partial equilibrium combustion, ' PEC, model for the composition of hydrazine and nitrogen tetroxide combustion products and their nozzle expansion was employed to make theoretical predictions of the PEC performance of these propellants, @able III-A-3). Performance calculations were accomplished through the expediency of renaming the "frozen" species and thereby arbitrarily preventing their participation in equilibrium reactions while retaining their contribution t o the properties of the mixture, satisfying the energy equation, and participating in the still isentropic frozen and part equilibrium nozzle expansion processes. Performance calculations are based upon the technique and program developed at the NASA Lewis Laboratories (23) which follows the formalism presented in chapter II. The performance of the hydrazine/nitrogen tetroxide propellant combination as based on the PEC model is summarized in figures III-A-3 and III-A-4. Four cases are presented equilibrium combustion with equilibrium and frozen composition expansion and partial equilibrium combustion with partial equilibrium and frozen expansion. At large equivalence ratios, fuel rich, PEC performance is greater than equilibrium combustion performance. At small equivalence ratios, fuel lean, PEC performance is less than equilibrium combustion performance. At equivalence ratios in the region of the stoichiometric value, the PEC performance is less than the equilibrium combustion performance. Combustion product composition and properties a r e compared in table 111-A-4.

D. SUMMARY Even t h o u u x p l i c i t reaction data may not be available, knowledge of certain classes of reactions is useful in jLdpig when to expect non-equilibrium effects in the rocket chamber. The mean stay time of a small elemental volume of a species in a rocket combustion chamber is normally relatively long with respect to chemical reaction time. Characteristically, then, the effect of chemical kinetic processes in rocket reactors has been ignored. The recent extensive use of hydrazine propellants brought to the fore the chamber equilibrium problem and the emphasis in this Section has been related to the hydrazine propellants. It has shown that the non-equilibrium effects in hydrazine combinations has been related to the presence of NH, and NO. It has been long realized in the field of chemical kinetics that the decomposition times of NH, and NO are long. Thus, it should not have been too surprising, considering the stay time in rocket motors, that when these compounds are formed as intermediates the possibility of non-equilibrium effects exist. Similarly reactions leading to condensed p,haseproducts are'also very slow. Thus, as mentioned in section II. B. 4., equilibrium concentrations of carbon a r e not found in ethylene oxide decomposition, and it is not likely equilibrium concentrations of carbon would be found in other rocket systems, particularly low temperature ones. The slow condensed phase reaction may be the reason for some of the efficiency PE-

86

blems which arise in the use of metallizeampalants. The metals must form the condensed metal oxide to obtain the large energy release characteristic of the metallized systems. One must now conclude that the reaction kinetics of a rocket propellant system should be examined in order to predict or explain performance anomalies.

87

Table m. A. 1

I

-

Comparison of the Equilibrium and Kinetic Decomposition of Hydrazine Monopropellant Equilibrium .Decomposition

I

P, psia T, "K

,

A

gm/gm mole c * ft/sec

I

Kinetic Decomposition

1000

1000

906 10.934 4054

16.029 4388

1456

.647 .329

.250 .250 .500

.024 ISp*

i

~

ISp* sec frozen expansion

*

i I

sec equilibrium expansion

198.5

215.3

,189.2

215.2

optimum expansion to one atm pressure

Table III. A. 2

-

Comparison of the Energetics of Non-Equilibrium Reactions with their Equilibrium Alternatives

FUEL RICH MIXTURES equilibrium non-equilibrium

+ 12.1 [kcal] N2H4 Q) N, + H2 N2H4 Q) N H 3 + %N2+ $Hz + 23.1 [kcall

OXIDIZER RICH MIXTURES equilibrium non-equilibrium

NzO, Q) Nzo4 0)

N, + 202 2NO +O,

- 6.9 [kcall - 50.1 [kcall

88

Table IIX. A. 3

(1)

-

Partial Equilibrium Combustion P E C ) Model for the Reaction of the Hydrazine/Nitrogen Tetroxide Propellant Combination

Hydrazine (a) All Hydrazine in excess of the stoichiometric amount decomposes according to the kinetic stoichiometry N2H4 --+

NH, + i H 2 + $N2

(b) All ammonia s o formed undergoes no further reaction, i. e., does not

participate in equilibrium reactions. (2)

Nitrogen Tetroxide (a) All nitrogen tetroxide in excess of the stoichiometric amount decomposes according to the kinetic stoichiometry N,04

+ 2NO + 0,

(b) All nitric oxide so formed undergoes no further reactions, i.e., does

not participate in equilibria reactions.

(3)

Equilibrium Considerations The composition of the mixture containing all species not so formed and 'frozen' as in (1) and (2) above is determined by equilibrium considerations. The equilibrium part of the product may contain additional concentrations of ammonia and nitric oxide which are unrelated to the frozen concentrations of these same species.

(4)

Nozzle m a n s i o n (a) Frozen composition nozzle expansion performance is based on the PEC composition. (b) Equilibrium composition nozzle expansion performance 1s based on a partial equilibrium condition in the nozzle. Those species frozen in the PEC composition remain frozen in the nozzle expansion. Those species in equilibrium in the PEC composition remain in equilibrium in the nozzle expansion.

89

Table Ill. A. 4

-

Hydrazine/Nitrogen Tetroxide Combustion Product Composition and Properties according to Equilibrium Combustion (EC) and Partial Equilibrium Combustion (PEC) (P, = 1000 psia)

Combustion

Equivalence ratio, Chamber temp. T, , OK Molecular weight, /M gm/gm mole Characteristic velocity c*, ft/sec

EC

PEC

0.47 2680

0.47 2322

24.89 4798

ECAND PEC 1.0 3260

25.00 4416

21.34 5791

EC

4.65 1785 13.501 5154

PEC

4.65 2157 18.06 5052

Composition in chamber, mole fraction H2

0.001

0.

0.052

0.505

0.169

H2O

0.373

0.382

0.473

0.136

0.185

N2 NH, frozen

0.384

0.285

0.404

0.356

0.308

--

--

--

--

0.217

--

0.338

NO frozen

---

NH, equil. NO equil.

0.

0.

0.

0.

0.

0.022

0.007

0.012

0.

0.

02

0.201

0.104

0.012

0.

0.

H OH

0.

0.

0.009

0.

0.

0.016

0.005

0.033

0.

0.

0

0.002

0.

0.003

0.

0.

I s p , asec, equil. expansion a sec, frozen expansion a

--

235.3

215.0

290.9

246.0

249.5

230.9

213.5

277.8

245.9

249.4

optimum expansion to one atmosphere pressure

91

IV Propellant Selection Current common practice is to divide chemical propellants into four classes: liquids, solids, hybrids and thixotropes o r gels. Traditionally the liquid systems are considered to be either bipropellants or monopropellants although as discussed in later sections multicomponent systems are feasible. The most common liquid systems are the bipropellant ones in which the fuel and oxidizer are introduced separately into the rocket combustion chamber. The monopropellants are liquids which can undergo controlled exothermix decomposition, o r combustion, reactions. As the prefix implies, only one liquid is injected into the combustion chamber. Various types of monopropellants are:

1) Single compounds which decompose exothermically such as hydrazine, hydrogen peroxide, ethylene oxide, acetylene, etc. 2) Single compounds which contain both fuel and oxidizing elements in a stable form at most ambient conditions but which decompose at elevated temperature into oxidizing and reducing fragments which subsequently react exothermically. Propyl nitrate falls into this class.

3) Liquid fuels and oxidizers which are mutually soluble at ambient conditions. The liquid reactant undergoes reaction, either gas or liquid phase, at elevated temperatures. A methyl nitrate-methyl alcohol solution fits this category.

The various fuels and oxidizers that comprise the bipropellants may be classified somewhat differently. If the fuel or oxidizer has a critical point such that it must b a e p t at low temperatures in order that it remain liquid throughout the system, then it is termed a cryopenic propellant e.g., oxygen, fluorine, hydrogen, etc. If it can be stored as a liquid under a pressure equal to i t s own vapor pressure o r less, then essentially it is noncryogenic; such as hydrazine, perchloryl fluoride, hydrogen peroxide, hydrocarbon fuels, nitric acid, etc. Those non-cryogenics which remain liquid over a wide temperature.ranga p e l y -60" to 160°F, which have vapor pressures less than 500 psia at the upper temperature limit, and which a r e .structurally stable over long periods are termed the storable prowllants. Thus the first storable criterion eliminates hydrazine, the second.perchlory1 flouride, and the third hydrogen peroxide; so the best example as possible storable propellants a r e nitric acid, unsymmetrical dimethyl hydrazine (UDMH),hydrocarbon fuels, etc. There are basically two classes of solid propellants -- homogeneous (double base) and heterogeneous (composite). The double base propellants are like the smokeless powders. The principal ingredients are nitrocellulose and an explosive plasticizer, usually nitroglycerin (43). The mechanical and ballistic properties of double base propellants may be varied by varying the nitrogen content of the nitrocellulose and the percentage of nitroglycerin. Actual propellants contain many additives to acheive specific purposes. For example, a non-explosive plasticizer, such as dibutyl pathalate, is added to permit machining of the final grain. A stabilizer, such as ethyl centralite, is added to absorb the oxides of nitrogen decomposition products of the nitrocellulose. These oxides are autocatalytic to continued decomposition of the nitrocellulose. Added as well is an opacifier - either carbon black o r a nigrosein dye - to prevent radiant penetration to the internal part of the otherwise clear grain, wax for extrusion lubricating qualities, and a flash suppressor such as potassium sulfate. Composite propellants a r e made by uniformly impregnating a finely divided solid oxidizer into a plastic, resinous o r elastomeric matrix. The matrix usually

1

92

I

serves as the fuel. Metal particles and solid metal compounds are added to increase the performance of composites. Minor additions such as burning rate modifiers and opacifiers a r e added a s well. Of importance is the fact that a double base material may be used as a binder in a composite structure, i.e., with an additional solid oxidizer and metal particles. Such a double base formulation is called a modified double base. More discussion on composites follows at the end of this chapter. The hybrids consist of solid and liquid substances in the same system. The most widely used hybrid concept has a hollow solid fuel grain, much like a solid propellant grain, but with no solid oxidizer present. The fuel grain may contain other . energetic solid substances. The oxidizer is generally a storable liquid propellant such as nitrogen tetroxide o r chlorine trifluoride. The terms thixotrope and gel a r e not synonymous. A gel is a system in a semisolid state without any supernatant liquid. Thioxotropicitv is the uropertv whereby a gel is liauified by internal mechanical stress, and usually returns to the pel form when the stress ceases. It is the thixotropic gels which are useful as propellants in rocket application. Such thixotropic gels are commonly called thixotropes. Normal fuels and oxidizers a r e converted into thixotropes by the addition of small quantities of particular gelling agents in order to be ableto suspend solid additives (usually metal particles) during storage. Each of the various chemical systems discussed has i t s own advantages and disadvantages. Each undoubtedly will have its place in the power plant spectrum. The morphology from a specific impulse point of view is given in a later section. Extensive analyses a r e required before a proper selection can be made for a given mission. In the following section the fundamental and technological factors which prevail in the choice of specific propellants will be reviewed so that an effective screening of all possible compounds can be made. Although much of the fundamental discussions will pertain to solid propellants, the emphasis will be upon liquid systems. A.

~

I

I I

I I

1

THERMODYNAMIC AND SYSTEM CRITERIA

Generally propellants are compared on the basis of their specific impulse. However, it can be shown readily that other properties of the propellant combination may play an important role in the system as well. Ideally, for zero drag and gravity forces, the velocity increment imparted to a missile by the power plant is (10):

As given in previous chapters, m i is the initial mass of the rocket power plant and m, is the empty mass. If now V is the volume of the propellant tanks, then: mi = Vpp

+ m,

I

where pp is the average bulk density of the propellants.

Thus:

A V = g ISP In (1+vp,) m0 The argument of the logarithm may be treated in two limiting cases:

I

93

,

which is the case of small boosters and volume limited systems in general. the above equation reduces to:

Thus

A v s k I s p p,

In this limit then the bulk density approaches the specific impulse in importance. 2)

(vPp

/ m,)

>>

1,

I

which is the case of long range stages comprised mostly of fuel.

1 i

In this limit the specific impulse is of prime importance. (1) Liquid propellants.

Eopellant selection then should consider density, as well as specific impulse. For pure terrestrial weapon svstems other considerations such as reliability, r e a d i b i z ty, cost. are also of importance. Other desirable properties of propellants are detailed below.

l

I 1

I 1

Thus:

a)

Liquid range.

It is desirable for the propellants to be liquids over as wide

a temperature range as possible, ideally between -60 to 160°F. b) Heat transfer properties. Most liquid propellant rockets are regeneratively cooled; that is, one or both propellants pass through a coolant jacket before they are injected into the combustion chamber. In this way, the rocket can operate for short o r long periods of time without burning out, there are no overall heat losses, and the wall of the combustion chamber can be kept at sufficiently low temperatures so that relatively thin structural members can be used. Viscosity. Since the propellants are used as coolants and must pass through coolant and other flow passages it is preferred that the viscosity of the fluids be low in order t o reduce the pressure drop through the system. c)

d)

Toxicity.

Both propellants and products preferably should be non-toxic.

e) Corrosivity. Both propellants and products should be non-corrosive. Further it is desirable that the products be non-abrasive as well.

I 1

'

I

Hypergolicity. When a fuel and oxidizer ignite spontaneously under ambient f) conditions, they are termed hypergolic under those conditions. Since hypergolic propellants do not require secondary ignition sources, they have a decided advantage over other propellants. Under very low pressure conditions, ignition by secondary sources becomes difficult, and spontaneous ignition by liquid or gas phase chemical reaction becomes most attractive. In the previous chapters, it w a s observed that in a real sense the highest specific impulse was obtained for those propellants which gave the highest heat release per unit mass rate of consumption of propellants. In a more ideal sense, maximum specific impulse is obtained when T,fl is maximized. For conditions of correct expansion specific impulse increases with increasing chamber pressure for two reasons: first, the pressure ratio across the nozzle increases (ambient kept constant) and second according to LeChatelier's principle there is less dissociation at

94

HEAT OF COMBUSTION K CAL /GRAM (FUEL+OXIDIZER

IO

0

1

30

20 ATOMIC NUMBER

OF

40

ELEMENT

Fig. IV. A. 1 Heat of combustion of the elements as a function of atomic number. Oxidizer-Oxygen I

I

I O h

DISSOCIATED

2000

4000

6000

8000

10000

TOF DISSOCIATION AT 5 0 0 PSlA

Fig. IV. A. 2 Dissociation of rocket combustion products as a function of temperature

50

95

higher pressures and consequently higher chamber temperatures. With these basic and simple facts as a background, it is now appropriate to examine which chemical compounds are suitable for rocket propellants. There are over a hundred elements in the periodic table and thus the possible fuel combinations become extremely large. However, the basic purpose of specific compounds in propellant systems is simply to introduce certain elements into the combustion process. It is rare that the heat of formation of a propellant influences the performance of an oxidation-reduction reaction system. Monopropellants, which undergo decomposition reactions, are not included, of course, in this generalization. The amount of energy released by a given combustion reaction is equal to the differences in the heat of formation of the products and reactants stated in equation form, one has: 03:s. l V . A . 1 . )

where, as befose, Qp is the energy release in the chamber, n the number of moles and AH, the standard state heat of formation. The heats of formation of the products are large negative values and even the introduction of the most energetic reactants only affects the overall energy release slightly; that is, more correctly, the first term of equation N.A. 1. is always much larger than the second. For the heat of formation of a reactant to affect the performance, it must have a value greater than 4 . 5 kcal/gm to increase the performance noticeably or a value less than -0.5 kcal/gm to decrease it. There a r e very few compounds indeed that meet these requirements, ozone (+) ammonia (-), and acetylene (+). A rapid method to determine the most energetic fuels would be to evaluate their heats of combustion, which is essentially equation IV. A. 1. evaluated for no product dissociation. For rocket considerations one must realize that the heat of combustion of concern is that per unit weight of fuel and oxidizer since the oxidizer must be carried in a rocket system. The normal heat of combustion considered is that per unit weight of fuel. An interesting plot facilitates the heats of combustion comparison. If one plots the heat of combustion of the elements in kcal/gm of fuel and oxidizer as a function of atomic number he will note a certain periodicity (44). Figure N. A. 1. is such a plot based upon liquid oxygen as the oxidizer. The choice of the atomic number aa the abscissa is strictly a convenience. It is t o be noted as well that for the elements the ordinate is simply the negative of the heat of formation of the corresponding product oxide. Further, compounds can be placed on this figure as well. Of particular interest here will be the hydrides and the hydrocarbons. In general all such compounds must fall between the elements from which they are constituted. Thus all hydrocarbon heats of combustion f a l l between C and H, the various hydrides between the particular parent element and hydrogen, etc.

Figure IV. A. 1. immediately predicts that of elements with atomic numbers g r m than 10 only Mg, U , and Si and their hydrides would be worth considering from a specific i m p u p o i n t of view. Mg, Al, and Si being metals, can be adopted i n a liquid system, only as slurries. Some of their alkylates may be liquids however, no matter in what form they are introduced i n liquid propellant systems their performance will not be higher than hydrogen. The reason is that these metals do not have a heat release appreciably greater than hydrogen and also form solid oxide products. Such products cannot be expanded through the nozzle and thus a specific impulse penalty is paid. Whenever there are condensed phases present in the rocket system certain unique

96

problems arise. To analyze these problems in an exact way it would be necessary t o know where nucleation occurs, the rate of growth of the condensed nuclei, the rate of heat transfer from the hot particles to the cooler gas stream and the velocity lag between the suspended particles and the gas. Specific impulse defects can occur if the heat transfer is slow and the velocity lag is great as discussed in section II. C. 6. Generally the heat of vaporization of most condensible species of concern is sufficiently&ge that it is desirable t o have condensation take p m p r o v i d e d , of course, the presence of particles would not induce large losses for the reasons mentioned above. The best performance occurs when all condensation takes place in the combustion chambers, i.e., at the highest pressure level to do expansion work. If one compares two similar systems at the same temperatures, one with particles, the other without, then the one with particles will have the lower specific impnlse since the particles cannot do expansion work. This concept was discussed in section II. C. 6. Returning t o the selection of appropriate propellants then, it appears for liquid systems that the choice is narrowed somewhat to Cy By Li, Be and H compounds. The toxicity of Be almost precludes it from consideration, b& its uniquLposition on the grspph_givenin f i m e lV. A. 1. will always make it of interest. Li as a metal is so reactive(it is pyrophoric)that the convenient introduction of this element in liquid or solid systems is difficult. The hydride, of course, could possibly be used in solid propellants. The boron compounds form glassy (vitreous) solids and the accumulation of such solids in nozzles is a serious detriment t o the use of boron compounds. In fact the handling of boron compounds, the vitreous product, and the slow burning characteristics of the elemental metal itself makes it extremely doubfful whether boron compounds will ever be used with cxy-gen systems. Consequently the screening procedure suggests the hydrocarbons, hydrogen and hydrazine as the best fuels in oxygen systems. From product molecular weight considerations hydrogen is obviously the best liquid fuel; hydrazine since it forms no CO,, which dissociates, or CO, the next best and the hydrocarbons the least. Thus, one sees that relative position of compounds on figure IV. A. 1. does not reflect that dissociation of products can alter this position; for example, hydrazine is always better than the hydrocarbons. Not every C, H, 0, N compound has been considered as a fuel in figure IV. A. 1. and the question naturally arises for non-cryogenic applications 'is it possible to find a fuel much superior to the normal hydrocarbons or hydrazine 7' Essentially the answer to this question is no, for there is an upper temperature limit for CHON systems. This limit is a result of the onset of several reactions which result in the dissociation or conversion of the primary combustion products into unstable or less energetic species. Some of these reactions along with the energy absorption for the process are given in table IV. A. 1. The values listed in this table take on greater significance when compared to the typical release values found for the CHON systems of about 1 t o 1.5 kcal/gm of propellants. Recall then that even the slightest dissociation of the stable combustion products can have a large effect on the overall energy release. The relative amount of dissociation in various systems is graphically represented in figure lV. A. 2. In fact this figure shows quite readily that carbonaceous fuel systems with oxygen suffer great dissociation and as the temperature is raised more and more dissociation takes place. Thus if a very energetic hydrocarbon fuel is found, the temperature will be limited by the great heat absorbing dissociation reactions. An interesting exception to this CHON case is cyanogen-oxygen in stoichiometric proportions to CO and N, which only dissociates at very high temperatures. Although it is a high temperature system, it is also one of high molecular weight ( 28) and thus low performance. Most CHON systems have a product molecular weight of about 22. Tables IV. A. 2 to 5 show typical performance calculation results and chamber compositions of hydrogen-ozone, hydrazine-chlorine pentafluoride, hydrazine-

97 I

Table IV. A. 1

-

Heats of Dissociation Reactions

-AH Kcal/mole

! '

H 2 1 2H H , O Y 1/2H + OH 1/2N2 + H20 + H, + NO CO, + H 2 d CO + H 2 0

Kcal/gm

103.8 67.9 79.4

51.9 3.8 1.7

9.8

0.2

I

Typical Heat Releases 1.

Table W.A. 2

j

Mixture

I

MR

1

-

-

1.5 kcal/gm Propellant

Performance and Reaction Products of Hydrogen and Ozone at 1000 psia. (Shifting Equilibrium and Frozen Expansion to One Atmosphere (optimum area ratio) and to Vacuum (1OO:l area ratio) )

B

2.0 3.97

3.0 2.65

4.0 1.98

5.0 1.59

6.0 1.32

2145 6.05 .0004

2857 8.01 .0094 .6147 .3741 .0017

3600 11.46 .0527 .3525

2750 12.88 .0613 .2577 .5866 .0752 .OlOl .0090

Chamber Properties T, "K

1

H,

1

H2O OH 0

.OOOO .OOOO

.OOOO

3323 9.84 .0322 .4760 .4772 .0138 .0006

0 2

.ooaa

.oooo

.0002

.5481 .0410 .0035 .0021

408 478

422 502

424 511

418 512

409 507

408

418

413

496

493

401 482

387

478

I

MW, gm/gm-mole H, mole fraction

.7476 .2519

Performance Isp, eq, 1 atm I

I I

Isp, e% vac Isp, fr, 1 atm

Isp, e% vac

466

98 Table IV. A. 3

-

Performance and Reaction Products of Hydrazine and Chlorine Pentafluoride at 1000 psia. (Shifting Equilibrium and Frozen Expansion to One Atmosphere (optimum area ratio and to Vacuum (1OO:l area ratio) )

Mixture MR

B

4.52 0.60

2.26 0.80

2.71 1.00

3.39 1.20

1.81 1.50

Chamber Properties T,

OK

M W , gm/gm-mole

c1 C1F C12

F F2 H HC1 HF N2 H2

N Performance I s p , eq, 1 atm 1 s p , eq, vac

I s p , fr, 1 atm I s p , fr, vac

3151 4018 4101 4167 23.15 21.2.8 23.58 22.23 .1167 .lo14 .0378 .0631 0000 .0005 .OOOl .0153 .006C .0005 .0002 .OOOl .Olll .1954 .0777 .0284 .0003 0000 0000 0000 .01151 0512 .0407 0000 .0751 .0040 .0342 .0610 .5543 .6071 .5941 .5289 .2035 .1332 .1644 .1867 .0667 .0025 .0256 0000 .0000 .OOOl .0002 .OOOl

3762 20.08 .0180 0000 0000 .0033 0000 .0458 .OB11 .4923 .2230 .1364 0000

258 313 241 278

306 362 294 344

.

.

.

.

.

..

.

296 357 283 329

313 374 293 342

311 369 295 344

. . .

.

99 Table IV. A. 4

-

Performance and Reaction Products of Hydrogen and Chlorine Trifluoride at 1000 psia (Shifting Equilibrium and Frozen Expansion to One Atmosphere (optimum area ratio) and to Vacuum (1OO:l area ratio) )

Mixture MR

4

8.0 2.87

10.0 2.29

12.0 1.91

14.0 1.64

16.0 1.43

Chamber Properties T, gc mw, gm/gm-mole

C1, mole fraction F

H H2 HCl HF c12

2887 13;38 .0017 0000 .0092 .4764 .1269 .3858 0000

.

.

3224 3667 3474 18.09 15.22 16.77 .0057 .0229 .0128 .0002 .0021 .0009 .0222 ,0481 .0364 .2215 .3790 .2938 .1440 .1596 .1546 .5458 .4488 .5016 0000 0000 .OOOl

.

.

3818 19.22 .0360 .0042 .0556 .1619 .1595 .5827 0001

319 3 74 312 364

320 376 309 360

319 378 302 352

.

Performance I s p , eq, 1 d m I s p , eq, vac

I s p , fr, 1 atm

I s p , fr, vac

317 370 314 366

319 377 306 356

100 Table W. A. 5

-

Performance and Reaction Products of Hydrazine and Nitrogen Tetroxide at 1000 psia Bhifting Equilibrium and Frozen Expansion to One Atmosphere (optimum area ratio) and to Vacuum (100:l area ratio) )

Mixture MR

d

0.50 2. a 7

1.00 1.44

1.50 0.96

2.00 0.72

G. 48

.300

2232 15.14

3048 18.96

3248 21.60

3076 23.06

2697 24. a i

Chamber Properties T, OK mw, gm/gm-mole H, mole fraction

HZ HZO NZ NH3

OH NO 0 0 2

NO2

.0005 .4101 .2194 .3698 0001

. ..0000 0000 .0000 .0000 .0000

.0093 .la03 .4030 .39a3 0000 .0074 .0012

.

.0002, .0002 0000

.

.0079 .0425 .4747 .4037

.0025 .0113 .44a9 .3953

.0361 .0135 .0040 .0176

.03637' .0232 .0049 .0774 .OOOl

.0000 .oooo

.0003

.0015

.ma

.3846 0000 .0173 .0222 .0019 .1952 0001

.

.0000

.

Performance Isp, erl, 1 atm

ell, vac Isp, fr, 1 atm I, , fr, vac lsp,

za 7

264 311 264 3 10

343 283 338

2aa 351 2 76 331

267 323 259 3 10

237 283 232 277 r

Table lV. A. 6

- Summary of Maximum Shiftihg Specific Impulse Data P, P,

Fuel Oxidizer:

NZH 4

03

363 345 322 313 312 295 293 291 286 283 265 244

FZ FZO NF3 0 2

ClF,

C103F ClF, N2°4 I

= 1000 psia = 14.7psia

%A NH ,Cl0 BrF,

HZ 422 410 411 351 391 343 344 3 ia 342 322 325 287

UDMH 344 351 309 3 10 298 290 280 286 283 2 76 2 59 23 1

I 101

chlorine trifluoride, and hydrazine-nitrogen tetroxide (44). Examination of these tables shows the degree of dissociation and the molecular weight variation with mixture ratio. It is evident then why all propellant combinations are adjusted fuel rich.

I

, I

'

1

The hydrocarbon fuel most frequently used today is R. P. fuel, rocket propulsion fuel. It differs from JP-4, which was developed for air-breathing jet engines, but was used as one of the initial hydrocarbon rocket fuels. Both differ from gasoline in that they are blends of low grade cuts of petroleum, whereas gasoline is a highly refined product. JP-4 is a wide range blend of kerosene, naphtha, and gasoline in order t o achieve a low boiling point and vapor pressure. Certain components in the kerosene lead to coking in cooling jackets causing heat transfer difficulties. In order to eliminate these difficulties, a pearl oil is used as a base in order to eliminate the aromatics. Further the use of pearl oil decreases the density range variations found from batch t o batch on JP fuels. This blend based on pearl oil is R. P. fuel. The necessity of small density variation in various batches for missile tank design is obvious.

It is evident that liquid hydrogen is the best fuel for use with liquid oxygen when one wishes to obtain high specific impulse. The hydrogen-oxygen system has the disadvantages of low bulk density and of being completely cryogenic. Of course no system which uses liquid oxygen as the oxidizer can be placed in the storable class. Now it is well to examine other oxidizers with regard to performance and their storable characteristics. Oxidizers can in fact be listed in order of decreasing performance, with a given reducing substance containing hydrogen predominantly, in the following manner: 03,

1

~

1

1 ,

I

F,,

F,O,

NF,,

02,

C103F, C1F3, H202

Table IV. A. 6 helps to confirm this order (45). Comprehensive performance data are given in (10)Ch. 9, Table 9. Properties of propellants to be discussed are detailed as well in (9) (Ch.9, Table 8). and only those pertinent t o the discussion at hand will be mentioned in the text. Liquid ozone which has a large positive heat of formation still must be proven because of its present sensitivity to explosive decomposition in high concentrations. I t s relatively high density is to advantage but its performance and bulk density with hydrogen may not make it much superior to fluorine. Fluorine chemistry is worth examining, for there is a possibility that fluorine may be one of the oxidizers for space travel. It is not likely that it would ever be used for a terrestrial weapon or a first stage, because of the corrosive HF exhaust. Its hypergolicity properties with practically all fuels is an added advantage, particularly under vacuum conditions which exist at high altitudes. Whereas 0, - H, has a 390 sec impulse, H, F, has a 410 sec impulse under the same pressure conditions.

-

Figure N. A. 3. is a heat of combustion plot of the elements with fluorine as the oxidizer. One can c a r r y through the same arguments as those made for the oxygen plot except it is well to remember that many of the metal fluorides are gaseous under the combustion chamber conditions that prevail. Since BF, is a gas, t& significance of boron compounds as fuels with fluorine oxidizers particularly changes for the better. Comparison of figures IV. A. 1. and IV. A. 3. will show that H, 0, and H, - F2 have the same standard state heat release. Yet above it was stated that the specific impulse with hydrogen-fluorine was the greater. This comparison again points out the limitations of the plots, which do not take into account the diss-

102

ociation at combustion chamber temperature conditions. Figure IV.A. 2. reveals why the fluoride system is better than the oxygen system. The product HF is strongly bonded and dissociates only at the very highest temperatures. If the H,-F, were not run H, rich there would be practically no dissociation in this system. As for the 0,system the optimum mixture ratio is quite fuel rich and the product molecular weight is about 9. Like the H, - 0,system the H, - F, system is one of low bulk density. To increase the bulk density and to relieve the cryogenic problems associated with hydrogen, hydrazine was considered as a fuel. Hydrazine has a density and freezing point close to water. It too is appropriate for F, because it forms stable products, unfortunately one of them is N, which has a high molecular weight that decreases the performance. Since carbon containing fuels form the higher molecular weight halocarbon exhausts, their performance with pure fluorine or other halogen systems is not as great as would be expected. If a halogen or interhalogen is t o be used with a carbon-containingfuel, then it is

e t o mix the compound with an oxidizer containingoqgen to allow the formation of CO and CO, rather than the halocarbon. Thus with carbon-containingfuels r20,F , 9, mixtures and C10,F take on added significance. For example, in table M with UDMH as the fuel, F,O becomes superior to F, and C10,F superior t o NF,. Of course, ammonia, NH,, becomes an attractive low-cost fuel for use with the pure halogens. Hydrazine is not only best with fluorine, but also shows better performance than the hydrocarbons with oxygen, peroxide, N,O, or nitric acid. Hydrazine however does not meet the important military specification of low freezing point. In order t o circumvent this difficulty, unsymmetrical dimethyl hydrazine (UDMH) was suggested as a fuel. It meets the physical property requirements of a military rocket fuel but because it contains carbon it gives a lower performance than hydrazine. Mixt u r e s of the two have been used t o obtain suitable physical properties and increased performance over pure UDMH. Both hydrazine and UDMH are hypergolic with N,O, and nitric acid--a decided advantage. As would be expected, fuel components which lower the freezing point of a mixture generally lower the density as well. So, UDMH - hydrazine has a lower density than the pure hydrazine. Increased density can be obtained by the addition of diethylenetriamine @ETA) to the mixture without appreciably lowering the performance. Hydrazine and the modified hydrazine components are monopropellants and thus can be used in gas generator turbine drives for auxiliary power, control, etc. The next oxidizer in the list given previously is F,O, fluorine monoxide. F,O which has a density about that F, has been considered because it is not as corrosive as F,. But the presence of oxygen defeats the purpose of the fluorine oxidizer; water forms, dissociates and causes a decrease in the performance. As discussed earlier, with carbon containing fuels the presence of the 0 atom is beneficial. & should be pointed out that there is no advantage of 60 over an appcopriate molar mixture of F, and 0,, which would be appreciably less expensive. Chlorine trifluoride was introduced and considered because of its high boiling point for the halogens (11.8"C); further it has a high specific gravity (1.7). But it produces high molecular weight exhaust products (HCl) and thus lower performance. It is, however, one of the few non-cryogenics which are hypergolic with practically all solid and liquid fuels. In order t o alleviate the high molecular product gas HCL, NF, was introduced, but it is again cryogenic and of lower density. For volume limited, storable systems, B r F 5 with a high specific gravity (2.46) and boiling point (40.5OC) is probably the best alternative. Due to the formation of the high molecular weight products HBr and Br,, the specific impulse with this oxidizer is low.

103 A recent addition to the interhalogen propellants, first having been reported in 1963,

is chlorine pentafluoride (ClF,). The high fluorine content of this oxidizer, its relatively high boiling point (about -2O"C), and high density (1.7 gm/cc) make it the most attractive of the storable halogens. Chlorine trifluoride has been known for a longer period and, therefore, is more widely used than chlorine pentafluoride.

Perchloryl fluoride (C10,F) initially looks most attractive as an oxidizer. First, its performances with most fuels rank close to that obtained with liquid oxygen. Second, it can be used with carbon-containing fuels for reasons discussed earlier. Lastly it is much less corrosive than the interhalogens. However, the critical point of perchloryl fluoride is such that it has a very high coefficient of expansion and thus tanks containing it require great ullage. Although it can be stored under its vapor pressure as a liquid, the vapor pressure required is very large and it cannot be classified as a storable propellant. In order to evaluate the interhalogens, ammonium perchlorate and oxygen on a comparative basis and particularly in regard to metal additives, figure IV.A. 4, is given. This figure gives a heat of combustion plot based upon chlorine as the oxidizer. The remaining oxidizers in the list a r e the non-cryogenics, non-halogen materials. Some of these are storable propellants, which have high thermal stability and respectable freezing and boiling points. In fact, in order for liquids to compete with solid propellants, a trend towards pre-packaged liquid propellant systems has been started. Packaging would eliminate H,O, as a storable because it undergoes slow decomposition. The attractiveness in hydrogen peroxide has been its density and ability to undergo catalytic initiation with the noble metals. Thus, a motor using peroxide as an oxidizer does not necessarily require an ignition source since the fuel generally ignites when injected into the high temperature decomposition products of the peroxide. Ordinarily peroxide is used in various strengths ranging from 86 to essentially 100 percent. The other constituent is always water. It has been reported that the 100 percent peroxide is more stable to surface initiation than the other grades. It is however more difficult to make and more expensive. The remaining oxidizers on the list given previously are then Nz04, nitric acid and tetranitromethane. Tetranitromethane, though comparatively very dense, h a s never been given serious consideration due to its very poor stability characteristics. Consequently the two oxidizers which have predominated as storables have been nitric acid and N,O,. Of the two N,04 is preferred, because its performance is higher and its handhng properties, cost and availability are favorable for its use in storable and prepackaged propulsion systems. Its major disadvantages are its vapor pressure and its high freezing point of 12'F. In order to depress the freezing point mixtures of N,O, and nitric oxide (NO) are made. These mixtures are called mixed oxides of nitrogen (MON). Somewhat higher performance is obtained with the mixed oxides, but they have the disadvantage of lower density. The nitric acid most commonly used in propulsion systems is inhibited red fuming nitric acid (IRFNA), which is 100 percent nitric acid with about 20 percent dissolved NO,. The inhibitor is used to The color comes from the reddish-brown NO,. prevent corrosion in containers and is a small amount of hydrogen fluoride dissolved in the acid. The H F passivates nickel containing vessels by forming nickel fluoride which is resistant to the acid. IRFNA is preferred because of its low freezing point and high density. A maximum density acid (MDNA) can be made by dissolving 52 percent NO,, however the vapor pressure rises due to the large amount of dissolved gases. This oxidizer h a s not seen much usage.

104

(2)

Solid propellants

The best projected solid propellants approach the performance of current operable storable liquid systems. Current solids have a performance somewhat lower than that of the storable liquids, but density and other advantages give them a unique position in the power plant field. Reliability, readiness, ease of handling, weight savings, etc., are the basic merits of the solid propellant systems that led to the phenomenal rise in their usage. These facts emphasize that which was stated earlier, specific impulse cannot be the sole basis for comparison of various systems. Much of what has been said in regard to liquid propellant philosophy can be carried over to solids; however, certain unique points do arise. Before they are discussed, a historical trend in composite solid propellant development will be given. The first composite solid propellants were made of asphalt @hefuel binder), oil (essentially a plasticizer), and potassium perchlorate (the oxidizer); however, the ballistites, the nitrocellulose-nitroglycerine homogeneous combinations, preceded and gave higher performance. For the composite (heterogeneous) propellants, the solid ammonium salts followed to eliminate the smoke due to KC1 in the exhaust of the asphalt propellants. Ammonium perchlorate (NH,ClO,), although more expensive than the nitrate, contains more available oxygen, gives faster burning rates, and consequently has been the solid oxidizer most predominantly uskd. Under humid conditions smoke mists are observed with the perchlorate, but not with the nitrate. Most other solid oxidizers, NaClO,, Mg(C10,),, Li,ClO,, are hygroscopic and are thus not suitable for propellant usage. Early in the development of solid propellant, the asphalt composites were found to have poor physical properties, such as cracking under normal temperature cycling, poor tensile characteristics, etc. They were replaced with the elastomeric polymers which have become the present-day binders. The first of these was Thiokol rubber, a polysulfide rubber, whichgives the propellant with good physical properties. The presence of the sulfur atom in the Thiokol rubber decreases the performance compared to a CHO polymer; thus the most frequently used binders are polyurekhane, polybutadiene acrylic acid (PBAA), epoxy resin, etc. The choice of the latter binders is made with regard t o physical properties rather than performance. Most solid propellants are limited in their performance by the amount of solid oxidizer that can be added. The addition of too much solids makes the cured propellant brittle and causes complicated manufacturing difficulties because of the very viscous mix in the uncured state. The blending of the oxidizer in the viscous polymer before curing can be an expensive plant operation. In order to achieve higher performance, aluminium metal is added to the solid mix. Examination of figure IV. A. 1. reveals that this metal is the best choice. Beryllium would be better, but again the toxicity question arises. Li and B are difficult t o add as in the metallic form because of their activity and the reasons outlined in the discussion of liquids. Undoubtedly, the light metal elements offer the greatest promise for increased performance in both solids and liquids. For the solid propellants, the best additive, when NH,C10, the oxidizer listed on the chart given earlier in this section is used, would be an alloy of Be and Li. This conclusion is drawn from examination of figure N. A. 4. and figure IV. A. 1. From figure N. A. 4. it is seen that Li is the best fuel with chlorine, whereas Be is the best with oxygen, The free energies of formation of the metal products are approximately proportional to their heats of formation. The entropy change in two competing systems will be about the same and small compared t o the heat of formation even at combustion temperatures. Thus in the competition between various reducing agents for various oxidizing agents, the compounds with the highest heat

105

0

Fig. IV. A. 3

IO

20 30 ATOMIC NUMBER OF ELEMENT

40

50

.

Heat of combustion of the elements as a function of atomic number. Oxidizer-Fluorine

HEAT OF COMBUSTION K CALIGRAM (FUEL+OXIDIZER) OXIDIZER -CHLORINE

0

10

20 30 ATOMIC NUMBER OF ELEMENT

40

50

Fig. 1V.A. 4 Heat of combustion of the elements a s a function of atomic number. Oxidizer-Chlorine

PRESENT GENERATION

/

I

NEXT GENERATION

FUTURE GENERATIONS

/

*

I INCREASING SPECIFIC IMPULSE

I

Fig. IV. B.1 Morphology of storable propellant systems

SPECIFIC IMPULSE , sec 550

-

500

-

450

-

4000

I

2

3

4

5

6

MIXTURE RATIO, (3BeH2*0~)/(H2)

Fig. IV.D. 1 Theoretical performance of the beryllium hydride/ozone/ hydrogen propellant system. Composition: 313eH2+ O,/H,. Chamber pressure, P, = 1000 psia

107 of formation will form. Thus an alloy in proportion to the C1 and 0 ratio in NH,C10, would then be the best a s has been verified in (46). Solids which require high density impulse, the product of density and the specific impulse, usually add the heavy metals such as Ti o r Zr.

Of course, the addition of metal to the propellant imposes a problem with respect to total solids added. In most cases, the amount of oxidizer is reduced. For aluminum addition, one nevertheless obtains an increase in performance because aluminum reduces the water and any CO, to aluminum oxide and lower molecular weight products. One then obtains a large energy release and the presence of the solid oxide is compensated by the larger amounts of H, present. In order to alleviate the total solids problem, there a r e great efforts to introduce the energv-bearing fuel atom in the polymer o r by substituting appropriate oxidizer groups such as nitrate in the pure p m m e r . Both approaches tend to decrease the amount of solid oxidizer required and thus the total solids content.

B. MORPHOLOGY OF CHEMICAL SYSTEMS High specific impulse always has been the primary reason for developing new propellant systems. The discussion in the previous sections has shown that one of the best means of augmenting the performance of a propellant system is by addinggreater quantities of the light metallic e h " t s . It has been this desire to add m ets-a which has brought the recent innovations in rocket developments to the forefront.

As one reflects on the suggestions in recent years for various rocket configuration changes, he appreciates that there has been and continues to be a morphological development, particularly in the storable propellant area, which appears to be centered about the concept of metal addition. This morphology is best shown by figure IV. B. 1. which is a histogram that plots the specific impulse as the abscissa and arbitrarily the type of rocket configuration a s the ordinate. In this figure it is seen that the present generation of systems includes the storable liquids (A) and the solid propellants (B), which contain aluminum. The performance of the storable liquid systems is better than the solid propellant systems because the liquids contain an oxidizer (generally N204) which is superior toNH,C104 and they operate at optimum mixture ratio. Further, the fuel, the standard hydrazine mixture, is superior to the polymeric fuel of the solid propellant. In order t o increase the performance, one needs to introduce greater quantities of metal and oxidizer into the system. Recall however that this is not possible in a normal system because of structural considerations. A method that was proposed to accomplish the objective was simply to make the solid propellant a very viscous medium by the addition of large quantities of plasticizer (C). It is then possible to incorporate the optimum quantity of oxidizer and of metal, but the final product does not have a physical structure. With the optimum quantities of metal and oxidizer one obtains a performance greater than the storable liquids, that is in figure N. B. l., I (C) > I sp (A). Operationally, it was proposed to place the plasticized propeflants in trays and to introduce the trays into a normal solid propellant rocket combustion chamber. However, heat transfer problems and obvious difficulties of design render this approach impossible. Another method to obtain high performance with solid propellants i s to remove the solid oxidizer in favor of a larger quantity of metal. But, of course, an oxidizer is necessary, s o one proposes to use a liquid a s an oxidizer and thus one has a hybrid @), which uses as the fuel, a singly perforated hydrocarbon polymeric grain containing larger quantities of metals. The dxidizer usually N,O, or C1 F, is injected according to the normal techniques of injecting liquids, ignites and burns with the grain. Thus one has a plausible systemwith excellent structural characteristics

108

which operate at optimum mixture ratio of metal and oxidizer. Consequently from a performance view this system is in theory better than the normal storable liquid system using the same oxidizers. In order to compete, the storable liquid systems must introduce high energy fuels; that is, metals. However, means of suspending the metal particles must be found. One method is to make a part of the fuel components a polymer and to suspend the metal particles in a polymerized solid body. But in the morphological plan this brings one again to the hybrid @). There is, however, another approach which one can use with liquid propellants. The metal particles could be suspended in the liquids by gelling them. Then it is possible to use the same oxidizers, but fuels such as hydrazine which are superior to the hydrocarbon fuels. This system of gelled propellants (E), called thixotropes, must have a specific impulse greater than the hybrids which use polymeric fuels. Say, for example, it is not possible to suspend the necessary amount of particles in a thixotropic system, or that similarly i n the hybrid it w a s not possible to burn the metal efficiently because the large quantities of metal particles must be projected into the flame from a decomposing surface creating only a small amount of gaseous components. Then it is possible to place small quantities of metal in both the liquid and solid to obtain higher combustion efficiency and thus performance. Figure N . B . 1. represents this approach a s the gelled hybrid (F). Figure N. B. 1. concentrates on the storable propellant systems. The morphology obviously can be extended to include the cryogenics. Again, the method by which the metal particles a r e introduced determines the approach. Both the hybrid and the gelling procedure would be considered. The only perturbation would be in the case of the hybrid i n which both fuel and oxidizer a r e injected as liquids and the binder is used simply as a vehicle to carry the metal particles. The morphology as described i n the preceding paragraphs encompasses all the known or suggested chemical rocket systems. The important aspect is that it is controlled by the necessity of adding the high energy, liquid metallic elements. C.

MONOPROPELLANTS

The advantages of a monopropellant over a bipropellant combination result primarily from a substantial reduction in the number of components in the tankage and flow hardware. The attractive simplications in the propulsion system resulting from the use of monopropellants a r e obtained only at the expense of a reduced specific impulse. The resulting implied trade-off between simplicity and propellant performance limits the attractiveness of monopropellants to propulsion systems where a simplicity and the usually associated reliability which comes with simplicity a r e premium desired characteristics. Typical applications have included attitude control rockets, vernier rockets for mid-course trajectory corrections, 'and other low thrust propulsors, especially those having a requirement for pulsed operation or repeated restarts. Monopropellants also find application as a source of relatively low temperature working fluids, as for driving gas turbines. (1) Monopropellant characteristics.

Any compound or mixture of compounds capable of undergoing an exothermic reaction is a potential monopropellant. In this sense all solid propellants a r e monopropellants. The term monopropellant however generally is reserved for liquid propellants. The monopropellant may be a single compound such as hydrogen peroxide or propyl nitrate. For hydrogen peroxide, the exothermic reaction is in the form of a decomposition. The exothermic reaction associated with propyl nitrate is better characterized as a fuel-oxidizer reaction, the fuel and oxidizer in

109

Table IV. C. 1

-

Specific Impulse and Combustion Temperature of Several Monopropellants. (Equilibrium decomposition and expansion, P, = 1000 psi, sea level specific impulse).

Monopropellant H202 N2H4 CH3N,H3

(CH3),N,H,

Specific Impulse (sec)

(MMH) (UDMH)

Table IV. C. 2

-

"K

165 198 209 200

1278 906 1191 1154

Characteristics of Monopropellants (P, = 300 psia, P, = 14.7 psia)

Ethylene Oxide Decomposition Products

Combustion Temperature

.481 .404 .077 .038

CO CH, H, C,H,

Hydrazine .656 H, .331 N, .013 NH,

Nitromethane .057 .277 .277 .223 .166

CO, CO

H20 H,

N,

ROPY1 Nitrate .385 .023 .431 .044 .079 .038

CO

CO, H, H,O

8 7%

Hydrogen Peroxide 0.666 H,O 0.334 0,

N,

CH4 Trace C T, O I SP

1860 160

F

Table m. C. 3

-

1125 170

3950 218

1840 167

1310 126

Decomposition of Hydrazine Monopropellants (J3yilibrium and Ekperimentally Observed Compositions, Mole Fractions). N2H4

MMH

UDMH

.647 .329 .024

.421 .2a8 .002 .222 .067

.359 .212 .002 .246 .180

.30 .45 .25

---

.42

.41 ---

.24 .10 .51

.32 .21 .06

Equilibrium Composition

----Observed Composition H,

N,

a,

HCN Note:

--

--

Experimental compositions (32) are approximate only and do not balance.

110

this case being joined in the same molecule. A second type of monopropellant is the composite monopropellant, composed of a mixture of a fuel and an oxidizer which are unreactive at the storage conditions, for example, a mixture of nitric acid and amyl acetate. The limited performance of monopropellant? results from a basic incompatibility of the requirements of smooth propellant combustion and high exothermicity. High performance in monopropellants always has been accompanied by increased tendency toward detonation and its resulting severe difficulties in application in propulsion systems. While one cannot categorically state that there will never be a high performance monopropellant, the outlook for the development of a monopropellant with performance comparable to bipropellant combinations is not optimistic. The most attractive monopropellants are those whose exothermic reaction or decomposition can b e catalytically initiated. The employment of monopropellants requiring thermal ignition results in the undesirable addition of an ignition device to the propulsion system. The recent development of a low temperature catalyst for the decomposition of hydrazine has made hydrazine and hydrazine based compounds among the most attractive monopropellants for propulsion purposes. (2) Monopropellant decomposition. Because of the relatively low temperatures of decomposition of monopropellants, the attainment of non-equilibrium products is observed in some cases. The specific impulse and equilibrium combustion temperature of several monopropellants are summarized in table IV. C. 1 and IV. C. 2. The non-equilibrium character of hydrazine decomposition was discussed previously in section III. A. The methyl substituted hydrazines are observed to behave in the same manner (32). These characteristics are summarized in table IV. C. 3. The non-equilibrium decomposition of the hydrazine is particularly interesting because it results in combustion temperatures and performances which are higher than those which are predicted from equilibrium considerations.

D. MULTICOMPONENT PROPELLANTS The attainment of high temperature combustion products is not necessarily consistent with the attainment of low molecular weight products. The reaction of fluorine and lithium, for example, results in combustion products at considerably higher temperature than the reaction of fluorine and hydrogen (5620 versus 3990'K at the mixture ratio of maximum specific impulse). The lower product molecular weight resulting with the second combination, however, yields a higher specific impulse for the fluorine/hydrogen combination. The extension of the propellant combination from two comp-s, a bipropellant system, to more than two compounds, i.e. a multicomponent propellant system, offers the potential of increasing-performance above level obtainable with propellant systems limited t o two propellants. However, this increase in performance results simpjy from the possibility of conveniently introducing more elements into the combustion process.

u.,

I

(1) High combustion temperature reactants. '

I

The highest combustion temperatures have been noted to be associated with the reaction of metals with oxygen or fluorine containing oxidizers. The combustion products uniformly have higher molecular weights than the products of the combustion of hydrogen with these same oxidizers. The introduction of a propellant component which would result in the production of low molecular weight species in the product gas would be anticipated t o result in an increased level of performance. The lowest molecular weight species are associated with hydrogen or hydrogen com-

111 pounds. While the addition of other low molecular weight species such as helium can be shown to improve the specific impulse of high temperature propellant systems, hydrogen is identified as being the best source of molecular weight reducing species. The maximum decrease in molecular weight is obtained i f the hydrogen is added as pure hydrogen, rather than as a hydrogen compound. For qualitative examination of the concept of multicomponent propellant systems, the hydrogen and enthalpy producing combination may be separated and their contributions to performance considered individually. The reacting propellants, then, perform much the same function as does the thermal energy source in a heat transfer rocket, for example the reactor in a nuclear rocket. The hydrogen in the multicomponent propellant system has much the same function as the hydrogen propellant of a nuclear rocket, that of a working fluid. The situation is not quite.this simple in the multicomponent propellant rocket, of course, in that the hydrogen in general can react chemically with other species in the propellant combination. Metals, however, are the main element in multicomponent systems and those added are usually much stronger reducing agents than hydrogen. Thus, the hydrogen does not participate chemically to any great extent. The highest levels of performance are to be obtained through the addition of hydrogen to the reaction products of lithium and fluorine and through the addition of hydrogen to the reaction products of beryllium and oxygen. If ozone is considered-p&ntial oxidizer with berxu m , the addition of hydrogen can result in theoretical performance at h i g m pansion ratios near 600 seconds specific impulse. The effect of hydrogen addition to beryllium/ozone reaction products is summarized in figure W . D . 1. In this case the hydrogen is added both through the metal hydride and through direct addition. The important consideration, however, is the elemental composition and not the form in which the elements are introduced. (2) Optimum combinations of several propellants

Although no rigorous proof has been proposed, the use of more than three propellants is not expected to produce a further increase in theoretical performance. Three propellants should be sufficient to introduce in a convenient manner any desired combination of elements. The optimum combination of a tripropellant system is predictable through the systematic variation of the concentration of each of the propellants. The weight ratio of the oxidizer and metal components are always in stoichiometric proportion to produce the metal oxide o r metal halide. The low molecular weighLspecies is then added in an amount determined by the trade-off between decreasing molecular weight and decreasingLmperature of the combustion products. The results of such calculations are reported in the literature (47). (3) Introduction of the metal.

The tripropellant combination is not necessarily obtained through introduction of hydrogen to the combustion products of a bipropellant reaction of a metal and an oxidizer. If a preferential and undesirable reaction of the hydrogen and the oxidizer is anticipated, then addition of the hydrogen following the formation of the metal/oxidizer products is indicated. More often, the problem becomes one of introducing the metal into an otherwise bipropellant system comprised of hydrogen and an oxidizer. Colloidal suspensions, gelled fuels o r oxidizers, hybrid grains, and metal containing compounds, such as the metallic hydrides, have been investigated as methods of introducing the metal. Use of a metallic hydride does not, cs -~ might first a p p E , reduce the tripropellant system to a bipropellant system. general. the optimum hydrogen to metal ratio will not b e that correspondinp to the metallic hydride hydrogen to metal ratio.

112 'TEMPERATURE, T (OK)

ENTHALPY, h-hZ9* ( kcol /mole H p )

Fig. IV. E. 1 Enthalpy of equilibrium hydrogen, (hD-h0298),for various pressures. The large enthalpy rise from dissociation, H, -2H, depends strongly upon pressure

FROZEN -FLOW EFFICIENCY

1 . 0

.8

.6

.4

2 CS-20677

0

I

600

I

I

800

I

I

1000

I

I

1200

l

l

1400

1

1

1600

1

1

1800

1

1

2000

FROZEN SPECIFIC IMPULSE. IspF sec

Fig. IV. E. 2

Frozen flow efficiencies of some potential heat transfer rocket propellants. (49)

113

E.

PROPELLANTS FOR NON-COMBUSTION ROCKETS

The criteria for selection of propellants for non-combustion rockets are independent of any consideration of a chemical heat of reaction. In fact, for many non-combustion rockets, even the requirement for a low molecular weight exhaust gas may no longer exist. The only generalization which may be drawn in the selection of propellants for this c a t e g g y of rocket engines is that the selection process is related to the particular manner in which energy is imparted to the propellant. It is convenient therefore to classify non-combustion rockets according to the means by which energy is transfered from an energy source to the propellant. Three broad categories of energy transfer processes a r e sufficient to include most advanced rocket concepts which a r e under development or have been proposed. The terms thermal, electrostatic and electromagnetic are chosen to describe the primary means by which energy is imparted to the propellant. In all thermal rockets a subsequent conversion of internal thermal energy to kinetic energy is r e quired. In electrostatic or electromagnetic rockets, body forces provide for the direct conversion of electric or magnetic energy to propellant kinetic energy. In many examples, of course, more than one energy transfer process may play a role. Nuclear particle emitters and photon rockets are additional advanced concepts which do not seem to fit into the three categories proposed. Other exceptions, no doubt, may be added. (1) Thermal rockets.

Thermal propulsion devices may be divided further according to the means by which thermal energy is imparted to the propellant. Chemical rockets, which have been discussed previously, comprise one means by which propellant heating can occur. Since thermal rockets generally require, for an efficient conversion of thermal energy to kinetic energy, expansion through a nozzle, many considerations applicable to propellants for chemical rockets also apply to thermal rockets in general. The one important difference which separateschemical rockets from other thermal rockets is the high heat of reaction required of the chemical propellants. The thermal rockets, other than chemical rockets, currently at the furthest state of development are surface heat transfer rockets. The term surface heat transfer is used to imply that thermal energy is transferred to the propellant through a material wall. Many sources of the thermal energy are possible and include solid core nuclear reactors, radioisotopes, electrical resistance heaters, and solar heaters. As for chemical thermal rockets, the performance of the rocket vehicle depends strongly upon propellant specific impulse which i n turn is dependent upon the enthalpy change during the expansion process. The specific impulse is given by:

where Ah is the enthalpy change per unit mass of propellant in expanding from the "heat transfer chamber" to the nozzle exit. The simplified relation assumes optimum expansion, that is, that the nozzle exit plane and ambient pressures are equal. The maximization of specific impulse is then concerned with maximization of the enthalpy change experienced by the propellant. The initial propellant enthalpy is determined by the propellant specific heat and temperature. The temperature, in turn, in heat transfer rockets, is limited by the w a l l material properties. Note that in heat transfer rockets, i n direct opposition to,chemical rockets, the wall must be hotter rather than cooler than the maximum propellant temperature. While such propellant characteristics as compatability with the wall, nuclear properties,

114 Table IV. E. 1

-

Enthalpy Content of Some Candidate Heat Transfer Rocket Propellants (Pure Gaseous Substances, Frozen Composition (no Dissociation, Recombination, or Phase Change). ) Enthalpy

(h'n",,) kcal/gm

Species

Molecular Weight

2000°K

3000°K

4000°K

H H2 He N N2

1.008 2.016 4.003 14.008 28.016 17.032 18.016 20.183 6.940 10.82 9.013 12.011 16.043

8.46 6.33 2.11 .60 .48 1.34 .96 .42 1.22 .78 .94 .71 1.85

13.43 10.61 3.35 .96 .79 2.39 1.68 .67 1.95 1.24 1.50 1.13 3.31

18.39 15.16 4.59 1.32 1.11 3.49 2.43 .91 2.72 1.70 2.07 1.57 4.85

NH3

H2O Ne Li B Be C CH4

Table IV. E. 2

-

Enthalpy Content of Some Candidate Heat Transfer Rocket Propellants, Equilibrium Composition, One Atmosphere Pressure (includes Enthalpies of Dissociation and Phase Changes).

Specie

2000°K

H2

6.33 2.11 6.76 1.53 .98 2.14 .99 .42 .48 1.72 .44

He Li Be B H2O Ne N2 B5H9 0 2

Enthalpy (h" -hig8) kcal/gm 3000°K 4000°K

14.79 3.35 7.49 10.17 2.10 4.01 2.86 .67 .79 1.72 .95

49.14 4.59 8.29 10.72 13.95 10.62 11.29 .91 1.11 19.86 3.32

115 density, storability, and heat trans e r properties are necessarily of concern, attention will be focused upon the capaci y of the propellant to sustain a large enthalpy change in the expansion process. A large enthalpy change in the propellant requires first the capacity to absorb a large quantity of energy and second, the ability to release this energy during the expansion process. Storage of energy within the propellant is not limited to internal thermal modes, that is translational, rotational, vibrational, and electronic excitations, but also includes energy absorbed by the propellant 'in molecular dissocation, ionization, and phase changes. The recovery of energy so invested during the heating process depends largely upon the kinetics of the recombination and condensation occuring in the nozzle. The effect of such possibly lagging kinetic processes as discussed in chapter ID is equally applicable to the present considerations.

The enthalpy contents of some candidate heat transfer rocket propellants are presented in table JY. E. 1. These enthalpies a r e referenced to the enthalpy of the same specie at the standard conditions of one atmosphere pressure and 298°K. This case represents frozen composition expansion in the nozzle. The enthalpy contents of some candidate heat transfer rocket propellants in their respective equilibrium compositions at a pressure of one atmosphere and selected temperatures are recorded in table lV.E.2. The enthalpies of this table a r e referenced to the enthalpy the propellant would have at its equilibrium composition at the standard conditions of one atmosphere pressure and 298°K. This case is that of equilibrium composition expansion in the nozzle. Examination of these tables reveals several guidelines to be followed in selecting heat transfer, or temperature limited, ' rocket propellants on the basis of maximum specific impulse. For both frozen and equilibrium expansion hydrogen is expected, because of its low molecular weighfto deliver the greatest specific impulse for a given initial temperature. The particularly high performance potential of hydrogen in equilibrium flow expansion is attributed to recovery of a large investment in dissociation energy. Since the degree of dissociation is strongly dependent upon the pressure, the energy absorption capacity of hydrogen similarly depends strongly upon the pressure at which it is heated, (Fig.IV.E. 1.). Hydrogen containing compounds such as ammonia and water similarly reflect high equilibrium expansion performances which may be traced to the combination of low molecular weight and large energy absorption in dissociation processes. The light metals also have high energy absorption capacities which, however, may be traced to their high heats of evaporation rather than dissociation processes. The metallic hydrides, such as pentaborane (B5JI9), lithium hvdride (LLH), and beryllium hydride (BeH2) are attractive candidates if the tem evaporation of the metal will occur (43). Such propellants would have a definite advantage over hydrogen in their storability and relatively high densities. As discussed previously, in many propulsion systems the recovery of a large fraction of the dissociation energy in the nozzle expansion through recombination is difficult to achieve. While the assumption of frozen flow with respect to recombination reactions appears necessary for many heat transfer rocket nozzle expansions, it is possible that condensation phenomena are sufficiently rapid to provide near equilibrium flow with respect to phase changes. For this special possibility, phase equilibrium in the presence of frozen dissociation, i t s been shown theoretically(58)that the performance in terms of specific impulse of propellants containing light metallic elements can exceed the performance of hydrogen. If a large fraction of the energy absorbed by the propellant is trapped by frozen flow expansion in dissociated or vaporized species, then it is convenient to consider the efficiency of energy utilization. A frozen flow efficiency is defined as:

116

v

=

( A H) nozzle (A H) input

As the specific impulse is a measure of the efficiency with which the propellant is utilized, the frozen flow efficiency-is a measure of the efficiency with which the power source is used. Frozen flow efficiencies are increasingly important as the power source becomes a large fraction of the total vehicle weight. Specific impulse is the important performance parameter for vehicles in which the propellant is a large fraction of the total vehicle weight. The frozen flow efficiency becomes of importance, then, in electrothermal rockets (resistojets o r arcjets) which have characteristically large power supply weights. The apparent advantage of helium and the light metal compounds over hydrogen in t e r m s of frozen flow efficiency, figure IV.E. 2 . , is realized only at temperatures greatly in excess of the capability of heat transfer rockets.

i I

Direct heating thermal rockets are distinguished from heat transfer thermal rockets in that the energy transfer does not occur through a material wall. Such devices include the arcjet, which already has been mentioned, liquid and gaseous core nuclear rockets, and nuclear bomb propulsion. Most of the previous discussion applies directly to the selection of a propellant for an arcjet propulsion device. In reality even the arcjet performance is temperature limited--except in this case the propellant acts to heat the wall rather than the reverse. Propellant selection for arcjet propulsion devices is likely to be made on the basis of availability rather than maximum performance. Arcjet propulsors have the uncommon characteristic of operational capability using practically any material as the propellant. Because of the high temperatures of operation, all propellants which might be used would be reduced largely to atomic species in the a r c chamber. A seeding impurity, often an alkali metal, is sometimes used to improve a r c characteristics. Oxidizing propellants are to be avoided to improve electrode Life. An important additional consideration in the selection of propellants for liquid and

gaseous core nuclear rockets relates to the loss of fissionable material from the reactor. Since the propellant will either p a s s through o r over the surface of the fissionable material, the loss of the fissionable material is related to the drag produced by the propellant. The advantages of a high density in reducing fissionable material loss may favor a propellant other than hydrogen, As for the solid core nuclear rocket, the nuclear properties of the propellant must be considered for applications in 1iquid.or gaseous core nuclear rockets. In general the requirement that the propellant be a good neutron moderator is consistent with the advantages of low molecular weight. In a nuclear bomb propulsion device, the primary purpose of the 'propellant would be to absorb heat. To serve this purpose, the propellant' wouId probably be chosen for its ablative properties. As a consequence of such thermal energy absorption, the ablative material would be expelled and in thus contributing to the thrust, may be considered to be a 'propellant. '

I

ii i ~

i 1

iI I

i

A thermal rocket of recent development which produces low thrust and specific impulse for satellite control purposes is the subliming propellant rocket. In this rocket the propellant is ordinarily a high vapor pressure solid. Propellant flow rate is controlled by the addition of heat to the subliming propellant. Desirable properties of propellants for such rockets is stability in the solid phase, high vapor pressure, and, as for all thermal rocket propellants, low molecular weight of the vapor produced.

1 I

117 (2)

Electrostatic rockets.

If the propellant is to be accelerated by electric body forces a primary requirement is that the propellant be a charged particle. While interest h a s centered on positively charged atomic ions, the use of both negatively and positively charged colloids has been considered.

i I

A simple model of an electrostatic accelerator allows equating the kinetic energy of the ejected propellant with the electrical energy expended, to produce the ejection velocity:

while maximization of the exhaust velocity and, hence, specific impulse, might appear a goal, concommitant increases in electric power supply weights indicate that a limitation on specific impulse is desirable for many missions. The accelerating potential generally is fixed at a large value by a need to maximize the thrust produced per unit area and hence to minimize the size of the propulsion device. The most attractive approach to reducing the specific impulse of electrostatic accelerators and thereby to maintain reasonable electrical power system requirements is to choose a propellant with a small charge to mass ratio. In direct contrast to electrothermal rockets, electrostatic rockets require propellants with large molecular weights for efficient operation. An attendant requirement is, of course, that the ionized specie be readily produced. Species with low ionization potentials are therefore required. Of the atomic species, cesium with an atomic mass of 133 A M U and an ionization potential of only 3.89 ev has been the favored propellant. Colloidal species can be either solids or liquids. The criterion of selection of the colloidal material i s based on the facility with which colloids of uniform mass and charge can be produced.

'

(3) Electromagnetic rockets.

1

Electromagnetic rockets a r e based on propellant acceleration through magnetic body forces. Such forces a r e produced through the application of magnetic fields on .moving charged particles, that is, electrical currents. A requirement of the propellant will be then, that it have a high electrical conductivity. Such electrical conductivity is generally obtained through thermoionization. A propellant with a low ionization potential is therefore favored. In practice the high conductivity is obtained through thermoionization. A propellant with a low ionization potential is obtained through use of a seeding material, generally an alkali metal, which is present in concentrations less than one-tenth of one percent. In contrast to the propellant in electrostatic rockets, the bulk of the propellant is not accelerated directly by the applied body forces, but through collisions with the current carrying species. The requirement of strong magnetic fields is likely to indicate the use of a cryogenic propellant which can be used to cool the associated electromagnet to aid in reducing electromagnet weight and magnetic field power. In practice, electroniagiietic acceleration often is obtained in conjunction with arcjet propulsion so that the coniments related to arcjet propellants should be consideredin selectingelectromagnetic propulsion propellants. (4)

Other propulsion systems.

Several other types of propulsion systems have been proposed which do not f a l l logically into the preceding categories. Photon rockets would produce thrust at

118

low levels but at very high specific impulse levels through the emission of radiant energy. The quanta of electromagnetic radiation from such devices would comprise the 'propellant. Schemes have been proposed for the direct use of the emission of high velocity particles in nuclear fission or fusion reactions. The propellant for these devices would be, then, the nuclear particles.

119

V Parameter Effects The performance of a particular propellant combination has been established to depend primarily upon the nature of the combustion products, that is, upon their enthalpy and molecular weight. More precisely, the performance of a given propellant combination is determined directly by the extent of the conversion of internal energy t o kinetic energy as the combustion products expand through a nozzle. A number of variables characteristic of the particular propellant combination and rocket design under consideration affect the delivered performance. Some of these variables, for example, the selection of the propellant combination itself, the mixture ratio, the chamber pressure, and the expansion ratio are available as design parameters. In designing the system for maximum performance it is reasonable and necessary to ask how variations in these parameters affect performance. Other possible variables, such as the combustion or initial temperature, assumed products, thermodynamic data, and induction enthalpy, while not generally available as design parameters, do affect predicted and/or delivered performance. The previous chapter was devoted to the development of guidelines for the selection of a suitable propellant combination. It was assumed implicitly, for the purpose of predicting and comparing the performance of different propellant combinations, that other parameters affecting performance were fixed. The effects of system parameters, other than the propellant combination itself, a r e now considered. A.

MIXTURE RATIO

The propellant combination is not completely specified by selection of the propellants, The reactant composition o r mixture ratio remains as a primary and independent design parameter. The selection of mixture ratio may be influenced byconcerns other than propellant performance as measured by the specific imp-. In volume limited applications, high density propellant combinations are favored and some appropriate trade-off between performance and density is established. In a truly volume limited system as shown in section IV. A. I., the appropriate performance parameter is the product of the specific impulse and the propellant bulk density, a quantity usually labeled the density impulse. Conceivably, mixture If a new ratio may be determined by yet other vehicle system considerations. propellant combination is to be utilized in an existing vehicle, the optimum mixture ratio may be influenced by such considerations as existing pump flow rate capacities, tank volumes, and structure load carrying capacities. Even other system considerations, such as the desirability of operating at equal fuel and oxidizer volume flow rates to allow interchange of fuel and oxidizer flow hardware, may determine the propellant mixture ratio. (1) Combustion temperature, molecular weight, and specific impulse.

The present consideration is limited to the effect of mixture ratio upon performance as measured simply by specific impulse. Since maximum enthalpy of reaction and minimum product molecular weight will not occur at the same mixture ratio one would predict, n p r i w i that the optimum mixture ratio should fall between the mixture ratio of maximum enthalpy of reaction and the mixture ratio of lowest product density. The maximum enthalpy of reaction occurs at the stoichiometricmixture ratio, that ratio at which there is theoretically just sufficient oxidizer to completely oxidize the fuel elements. Any excess fuel or oxidizer essentially acts as a diluent. The maximum temperature thus should fall at the stoichiometric p m . In most cases it does, but, according to the character of the elements present, can f a l l somewhat to the fuel rich or oxidizer rich side. In most Dropellant systems the maximum temperature falls just on the fuel rich side. The molar specific heat of the products determine this small effect. On the fuel rich side of stoichiometric, most propellant systems form less of the non-linear triatomic water

120

molecule which has a high molar specific heat. In any case, the maximum enthalpy of reaction occurs at the stoichiometric mixture. The minimum product density occurs at some point away from the stoichiometric mixture ratio. The trade-off between the enthalpy of reaction, as represented by the chamber temperature, and the product molecular weight in determining the mixture ratio of maximum specific impulse is demonstrated in figure V. A. 1. It is generally the case that a reduction in product molecular weight is effected by increasing the concentration of hydrogen or low molecular weight hydrogen containing compounds. Since hydrogen generally is associated with the fuel, the mixture ratio of maximum specific impulse generally falls on the fuel rich side of the stoichiometric mixture ratio. An exception to this observation would be the uncommon propellant combination in which the oxidizer is the hydrogen containing compound, for example, the boron/hydrogen peroxide combination. (2)

Effect of pressure on optimum mixture ratio.

The optimum mixture ratio for a given propellant combination is in turn influenced by the other primary design parameters, chamber pressure and expansion ratio. The effect of chamber pressure upon equilibrium expansion specific impulse is presented in figure V.A. 2. For a fixed expansion ratio, expressed in terms of the ratio of the chamber pressure to the exit plane pressure, the effect of chamber pressure on specific impulse is small. The nature of this effect is discussed in the following section of this chapter. It is interesting to note, however, that t& mixture ratio of maximum specific impulse shifts toward the stoichiometric mixture ratio with increasing chamber pressure. The higher pressures of combustion and expansion reduce losses associated with energy 'trapped' in dissociated species. This effect in turn favors mixture ratios closer to the stoichiometric value and hence at the higher combustion temperatures, where there is greater dissociation. Without the higher pressure, the higher combustion temperatures would be penalized by excessive and unrecovered deposition of energy in dissociated species. Even though recombination tends to increase the mean molecular weight, the increased enthalpy overrides. An even more pronounced effect of chamber pressure would be expected and is observed in the frozen flow expansion of these same propellants, as is shown in figure V. A. 3. Unlike the equilibrium expansion case, no recovery of the energy of dissociation is possible so that the positive effect of the inhibition of dissociation in the combustion chamber brought about by higher pressures is evidenced strongly in the predicted specific impulse. The shift of optimum mixture ratio toward the stoichiometric value with increasing chamber pressure again is observed.

(3) Effect of expansion ratio on optimum mixture ratio.

The effect of increasingLxpansion ratio at constant chamber pressure upon the OPtimum mixture ratio is much the same as is the previously discussed effect of increasing chamber pressure at constant expansion ratio. The optimum mixture ratio shifts for much the same reason in the two cases. The gain from greater recovery of energy in dissociated species overrides molecular weight effects. The specific impulse of the hydrogen/oxygen propellant combination at a fixed chamber pressure but for varying expansion pressure ratios is presented in figure V. A. 4. for equilibrium expansion flow and in figure V. A. 5. for frozen expansion flow. For the case of equilibrium expansion flow, expansion t o higher pressure ratios allows greater recombination andtherefore greater recovery of dissociation energy results in a shift of the optimum mixture ratio toward the stoichiometric value with an attendant increase in dissociated species in the chamber. For frozen expansion flow no recovery of dissociation energy in the expansion process is possible. A significant effect of the expansion ratio upon the optimum mixture ratio is neither

121

TEMPERATURE,

MIXTURE RATIO, O / F

Fig. V. A. 1 Hydrogen/oxygen combustion products characteristics and propellant performance. The maximum specific impulse lies a t a mixture ratio between the mixture ratios of minimum molecular weight and maximum combustion temperature. P, = 1000 psia, optimum equilibrium expansion to one atmosphere ambient pressure 440

MAXIMA SPECIFIC IMPULSE, sec 430

420

410

2

4

6

e

IO

MIXTURE RATIO, OIF

Fig. V.A. 2 Theoretical equilibrium expansion performance of hydrogen/ oxygen showing the effect of chamber pressure, P,, on optimum mixture ratio. (50)

OK

122

SPECIFIC IMPULSE sec '

Fig. V. A. 3

,

Theoretical frozen composition expansion performance of hydrogen/oxygen showing the effect of chamber pressure, P,, on optimum mixture ratio. (50)

- 064 440

420

400 SPECIFIC IMPULSE ,sec

380

360

3401

I

I

I

4

6

8

MIXTURE RATIO, O / F

Fig. V. A. 4

Theoretical equilibrium expansion performance of hydrogen/oxygen showing the effect of expansion ratio on optimum mixture ratio. Chamber pressure, P, = 1000 psia. (50)

123 SPECIFIC IMPULSE,

sec

440

\

420

'c' 'e

-1

400

380

360

340

I 2

4

6

8

IO

MIXTURE RATIO, O/F

Fig. V. A. 5 Theoretical frozen composition expansion performance of H,/O, showing the effect of expansion ratio on optimum mixture ratio. Chamber pressure, P, = 1000 psia. (50)

THRUST

SPECIFIC

Fig. V. A. 6 Comparison of the specific impulse (I, ) , characteristic velocity ( c * ) , and thrust coefficient and dependence on mixture ratio. H,/O,, equilibrium expansion to one atmosphere pressure, P, = 1000 psia. (50)

(4),

124

500

Isp,voc

Fig. V. B.1. Effect of chamber pressure on specific impulse at fixed expansion pressure ratio. Hz/Oz;I,, o p t = optimum expansion f o r pressure ratio'of P,/P, = 340; I v a c = expansion for a r e a ratio of A,/A, = f8b and vacuum ambient conditions. Equilibrium expansion flow. (50)

I

450 ISP,OPl

400 SPECIFIC IMPULSE, Sec

500

1

I

35?00

1000

10000

specific impulse at fixed expansion pressure ratio. Hz/Oz; IsPo + = optimum expansion for pressure ratio bfPc/P, = 340; I, V a E expansion for a r e a ratio of AJA, = 16b and vacuum ambient conditions. Frozen expansion flow. (50) 400 SPECIFIC IMPULSE, sec

1

35?h0

1000

I loo00

CHAMBER PRESSURE'(psi01

Fig. V. B. 3. Effect of chamber pressure on specific impulse with significant ambient pressure. Hz/Oz;Isp, optimum expansion to one atm. pressure; I , expansion through a r e a ratio of A,?& = 100 and one atm. ambient pressure. Equilibrium' expansion flow. (50)

CHAMBER PRESSURE (psi01

,

125

The observations that the thrust coefficient has its maximum value near the stoichiometric mixture ratio, see figure V.A. 6.,is consistent with the foregoing expectations. Since specific impulse is proportional to the product of the character-

~

The effect of chamber pressure has been considered in relation to the role of chamber pressure in determining the optimum mixture ratio, as was discussed in the previous section. An important effect of increasing chamber pressure is to elevate t h e heat of reaction and adiabatic flame temperature through inhibition of endothermic decompositions. The undesirable increase in product molecular weight is not of sufficient importance to overcome the advantages associated with decreasing

126

endothermic decompositions or increasing exothermic recombinations. If the rocket is to be operated within an atmosphere, then increasing chamber pressures poses the attendant possibility of increasing the expansion ratio without the losses due to overexpansion which would be encountered at lower chamber pressures. __For a fixed vehicle diameter and thus a fixed maximum nozzle exit a r e a , h i g k area ratios are obtained at higherpressures simply because the throat area decreases for fixed mass flow. This is the major effect of pressure and, indeed, contributes to increase the performance for both atmospheric and space operation. (1) Effects at fixed nozzle pressure ratio.

In reviewing the effect of chamber pressure upon propellant performance, in this case as expressed in terms of specific impulse, one must be careful to identify those parameters which are held constant. If the pressure ratio across the nozzle is a constant, the effect of chamber pressure is a minor one. As is shown in figu r e V.B. 1. increasing the chamber pressure from 100 to 10000 psia produces less than a 2 second change in the theoretical specific impulse for the hydrogen/oxygen propellant combination, for the case of equilibrium expansion and fixed expansion pressure ratio. The effect for frozen expansion, which is more strongly dependent upon dissociation losses, is somewhat greater but still of minor importance (Fig. V.B.2.). (2) Effects at fixed area ratio, significant ambient pressure. A more meaningful presentation of the effect of chamber pressure is obtained by examination of sea level specific impulse at a fixed expansion ratio in t e r m s of the area ratio rather than the pressure ratio of the nozzle. As shown in figure V. B. 3. the advantage of increasing chamber pressure is pronounced. The advantage of high chamber can be demonstrated on a different basis by considering the variation of optimum specific impulse for expansion to sea level conditions. That is, in this case thq exit pressure is fixed but the expansion ratio is allowed to increase as the chamber pressure is increased. To summarize, the advantage of high chamber pressure at a fixed area ratio in t e r m s of specific impulse is obtained only in operating under atmospheric conditions, that is, with a significant ambient pressure. Under vacuum ambient conditions, that is, space operation, any apparent increase in specific impulse associated with increased chamber pressure must be associated with increasing the nozzle expansion ratio and is in reality nearly independent of the chamber pressure. (3) Other effects of chamber pressure.

Advantages of high chamber pressure other than those reflected in specific impulse are associated with reduction of propulsion system size and weight with increases in chamber pressure. Such gains are not obtainable indefinitely with increasing chamber pressure but are expressible in t e r m s of an optimum chamber pressure defined by a minimum propulsion system weight. Increased heat transfer at high pressures presents a serious limitation on high pressure combustors. A chamber pressure effect of probable significant importance but as yet ill-defined is related to the acceleration of reaction kinetics at elevated pressure. Increased pressures in the combustion chamber should speed reaction kinetics and favor production of equilibrium combustion products which in turn, generally yields increased performance. Similarly, the gases in the nozzle will be at higher pressures and, thus, the exothermic three body recombination reactions will be accelerated. The transition from equilibrium to frozen flow i n the nozzle, as discussed in Chapter 111, should thereby be delayed and specific impulse increased. In comparing the effects of operation at very high and very low chamber pressures, the changes in reaction kinetics are likely to play an important role.

127 (4) Temperature limited systems.

The chamber pressure exercises a particular control upon the performance of temperature limited systems. The extent of highly endothermic dissociations o r phase changes can be determined by selection of the chamber pressure. If the expansion kinetics makes recovery of energy so invested in endothermic dissociations o r phase changes impossible, then dissociative thermal inputs can be reduced by heating at elevated chamber pressures. On the other hand, if the energy transfered to the propellant can be recovered in the nozzle expansion process, then the specific impulse is increased by increasing the dissociative thermal inputs. In the latter, or equilibrium expansion case, low chamber pressures a r e favored. The proposed use of light metals in temperature limited rockets, a s discussed in Chapt e r N ,requires low chamber pressures to make possible the utilization of the high heat of vaporization of the metal containing propellant. C.

TEMPERATURE

In combustion rockets, the combustion temperature is not directly available as a design parameter but rather is determined by the propellant selection, mixture ratio and combustion pressure. In heat transfer rockets however, the initial temperature of the propellant, that is, the stagnation temperature of the propellant, is available as a design parameter. It is probably sufficient to say that the propellant stagnation temperature should be maximized for maximum performance. The limitation upon propellant stagnation temperature in heat transfer rockets is associated with material limitations or, possibly power limitations. (1) Relation of enthalpy to temperature.

It is of course the propellant stagnation enthalpy per mass flow rate that determines the performance of the propellant rather than the temperature. Since the specific heat is not a constant and the effect of changing composition with temperature is important, the relation between enthalpy and temperature is not a simple one. The enthalpy is dependent upon both the temperature and the pressure through composition changes. As previously suggested, the selection of the propellant temperature is dependent upon whether the invested enthalpy is likely to be recovered o r not. Large enthalpy increases at only moderate temperature increases a r e associated with decomposition reactions. The value of increasing the propellant temperature to produce dissociation for the purpose of taking advantage of large propellant enthalpies will be great if the enthalpy so invested can be recovered in the expansion process. If the enthalpy is lost due to a frozen expansion process, however, the net effect of the higher propellant temperature is a waste of available power. An example of such a non-linear relation between enthalpy and temperature was presented as figure N.E. 1. for hydrogen. An even more pronounced non-linearity between enthalpy and temperature is shown in figure V.C. 1. for methane, a possible heat transfer rocket propellant. (2) Diabatic nozzle flow.

An additional advantageous possibility in heat transfer rockets is the use of a diabatic nozzle in which propellant heating continues during the expansion process. While difficult to achieve in practice, such heating extends the potential propellant performance beyond the limitation associated with a maximum, pre-expansion temperature.

(3) The performance of hydrogen. The significance of the propellant stagnation temperature is further illustrated by the performance of hydrogen a s a heat transfer rocket propellant, figure V.C.2.

128

For most operating conditions, hydrogen, because of its low molecular weight, offers the highest performance potential and, therefore, is the most likely candidate as a heat transfer rocket propellant. Large losses a r e associated with a failure to recover decomposition energies in frozen expansion nozzle flow. Most three body recombination reactions freeze very quickly once expansion begins. The freezing point is never much past the rocket throat. Thus, the hydrogen atom recombination in heat transfer rockets is ignored and theoretical performance is based on frozen flow expansion. The importance of this effect is evident upon comparison of equilibrium and frozen expansion specific impulse as presented in figure V. C. 2. D.

EXPANSION RATIO

Within the limitations imposed by atmospheric operation, expansion to larger expansion ratios allows conversion of additional propellant enthalpy to kinetic energy and higher performance. The employment of high expansion ratio nozzles is a necessary adjunct to the development of higher pressure combustors.

(1) The relation between pressure and area ratios The dependence of the pressure and ratio upon the area ratio is a function only of the ratio of the specific heats. Since, for a given propellant combination, the specific heat ratio does not vary greatly during the expansion process, the relation between the a r e a and pressure expansion ratios takes a particularly simple form. At high expansion ratios, the relation is:

The expansion ratio may be expressed, therefore, either in terms of the area ratio or the pressure ratio. It should be noted, however, that the pressure ratio rather than the a r e a ratio determines the performance. Larger pressure ratios a r e obtained at a given area ratio ad the propellant specific heat decreases. This effect is demonstrated in the relation between the area ratio and the pressure ratio for hydrogen expansion from different initial pressures as shown in figure V. D.1. Because of the higher dissociation of hydrogen at lower pressures and the resulting higher effective specific heat, larger nozzle area ratios a r e required to provide a given pressure ratio at lower rather than higher initial pressures. (2) Limitation on expansion ratio.

The limitation on expansion ratio with a significant ambient pressure is fixed by the losses associated with nozzle overexpansion. Under vacuum ambient operating conditions, the expansion ratio is limited by considerations of nozzle size and the resulting weight penalty. Since rockets operating under vacuum conditions generally are carried as upper stages, the weight penalty from large nozzle area ratio comes largely from the interstage structure and does not reflect the nozzle weight alone. While ideally, performance continues to increase with increasing area ' ratio, the gain at a r e a ratios in excess of 100 or 200 is small and area ratios greater than these values a r e not of practical importance, (Fig. V. D. 2. ). E.

ASSUMED PRODUCTS

As w a s noted in Chapters Il and IV, the available enthalpy for a given propellant combination is determined p r i m ~ i l yby the product species which are formed in the combustion of the propellants. The failure to include a significant uroduct specie

129 20

I

I

I

I

.I5 -

IO

-

ENTHALPY [kcal /gm

5-

0-

-5.

I

I

I

I

Fig. V.C. 1 Enthalpy of equilibrium methane showing the non-linear relation between enthalpy and temperature. P = 1 . 0 atm. The highly non-linear relationship i s associated with hydrogen decomposition and carbon phase change with

2000

I

I

I

I

SPECIFIC

Fig. V. C. 2 i

Performance of hydrogen as a heat transfer rocket propellant. P, = 1000 psia. Vacuum specific impulse to an area ratio of 100

130 1000

AREA RATIO, A,

I

IO

10000

1000

100

PRESSURE RATIO pC /Pe

Fig.

V. D.1 The relation between nozzle area ratio and pressure ratios for various combustion chamber pressures, P,. Equilibrium hydrogen, T, = 3000°K, equilibrium expansion, vacuum ambient conditions

50(

SPECIFIC IMPULSE, sec

45(

3%

1

I

IO

100

1

lobo

NOZZLE AREA RATIO, A e ’ A ,

Fig. V. D.2 Vacuum specific impulse a8 a function of nozzle area ratio, Hydrogen/fluorine, P, = 1000 p i a , equilibrium expansion

I

131 in calculations of propellant performance has a pronounced effect and results in an erroneous prediction of the theoretical performance.

(1) Identification of probable species. The identification of probable combustion product species for all but the most exotic propellant combinations has been well established. In conjunction with machine calculations which consider large numbers of possible product species, the likelihood of introducing e r r o r into equilibrium performance calculations through omission of significant product species is small. With the completion of the JANAF Thermochemical Tables (18) which currently contain some 800 possible species and continues t o be expanded, the task of takingall possible significant product species into consideration is well defined, although not necessarily simple. The problem of identification of probable species has been reduced to the identification of possible species which for most propellant combination is a routine task. (2) Effect of dominant species.

A s an example of the possibie consequences of omitting a significant product specie in calculation of theoretical propellant performance, the hydrogen/fluorine combination is considered for three cases. The complete list of all possible product species for this combination is short: H, H,, HF, F, and F,. The consequences of omitting the atomic species as product species a r e presented in figure V. E. 1. The severe e r r o r s introduced by failing to include all significant product species is evident. F.

THERMOCHEMICAL DATA

The thermochemical data for the product species are critical in the calculation of propellant performance. The methods of statistical thermodynamics have provided a reliable technique of establishing most of the thermochemical data required for propellant performance calculations. Again, the JANAF Thermochemical Tables (18)represent an extensive and widely used compilation of these data. (1) Results of statistical thermodynamic calculations.

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest t o rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb's function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15"K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data a r e not associated generally with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. (2) Enthalpies of formation.

The critical thermochemical quantities for prediction of propellant performance are the enthalpies of formation of the product and to a degree the reactant species. The enthalpy of formation is essential to the calculation of the enthalpy of reaction and since it also appears in the expression for the equilibrium COMtant, as it is the basis for relating the Gibbs functions of different species, influences the calculated product equilibrium compositions. It is most desirable to measure enthalpies of formation directly from calorimetric experiments, but often the enthalpies must be

132 inferred from considerations of bond energies and assumed molecular configurations. The enthalpies of formation of high temperature species remain difficult t o establish with the same accuracies associated with the other thermochemical data. Since the enthalpy of formation plays a critical role in the prediction of propellant performance, one is not surprised to see a strong variation of specific impulse with product heat of formation of the dominant, stable products. Uncertainties in enthalpies of formation have been greatest for the metal containingxmpimpcrtanceto many hiph energy-propellant combinations. G.

INDUCTION ENTHALPY

Since performance primarily depends upon the properties of the products of combustion, the induction enthalpy (or enthalpy at which the propellants enter the combustion process) generally plays a minor role in determining propellant performance. A few notable exceptions exist which are associated with two factors which may influence the propellant induction enthalpy: (a) the temperature and phase of the propellant and @) the enthalpy of formation of the propellants. (1) Effect of propellant temperature and phase.

The incoming temperature and phase of the propellants are generally determined by considerations of handling, particularly the storability of the propellants. While one generally does not have the option of adjusting the temperature and phase of the propellants to maximize the performance in t e r m s of specific impulse, it is interesting to note the magnitude of these effects. The performance of the hydrogen/ oxygen combination for two induction states, one the cryogenic state and the other the room temperature gaseous state, are compared in figure V. G. 1. (2) Positive enthalpy of formation propellants. A few potential propellants have abnormally high enthalpies of formation which re.sult in significant propellant performance gains over similar propellants resulting in the same products but with 'normal' enthalpies of formation. Ozone, the hydrazines, the oxides of nitrogen, some hydrocarbons, and the ephemeral 'free radical propellants' are among this relatively small group of propellants with abnormally high enthalpies of formation. The potential gain in the extreme case of ozone, as compared t o oxygen, is evident in table V. G. 1. In those c a s e s where the induction enthalpy increase is not as pronounced as for the ozone--oxygen comparison, other factors may make even small gains in specific impulse attractive. The possible substitution of new propellants in existing missile systems without alterations in hardware makes substitutions resulting in only modest performance gains attractive. The substitution of positive heat of formation hydroc'arbons for the commonly used RP-1 (kerosene) propellant falls in this category. A summary of this possibility is presented in table V.G. 2. Optimum sea level specific impulses are presented as they depend both upon the induction enthalpy and upon the hydrogen to carbon ratio. Comparing RP-1, with an induction enthalpy of -6.222 kcal/mole (carbon atom basis) and H/C = 2, to ethylene, with an induction enthalpy of +3.975 kcal/mole (carbon atom basis) and H/C = 2, shows an increase in performance by using ethylene of about 11 seconds of specific impulse. The performance of any hydrocarbon with oxygen may be evaluated merely by considering i t s induction enthalpy and hydrogen to carbon ratio. Tohave a noticeable effect, however as noted earlier the change in propellants must bring about an inc r e a s e in the i n d u w l p y of at least 0.5 kcal/gG.

133 420 I

I

I

I

I

1

ALL PRODUCT SPECIES CONSIDERED SPECIFIC IMPUL.SE, sec

MIXTURE RATIO, O/F

Fig. V. E. 1

Performance of hydrogen/fluorine showing the effect of improper identification of product species. Sea-level specific impulse, optimum expansion P, = 1000 psia, equilibrium expansion I

I

I

I

I

SPECIFIC IMPULSE, sec

MIXTURE RATIO, O / F

i*

Fig. V. G. 1 Relative performance for hydrogen/oxygen propellants as cryogenic liquids and as gases. P, = 1000 p i a , vacuum specific impulse, A,/At = equilibrium e x p n sion

134

Table V. G. 1

-

Effect of Reactant Enthalpy of Formation on Propellant Performance (Selected Comparisons) Oxidizer

Fuel H, (lis.)

-1.89 kcal/mole (-0.95 kcal/gm)

0, (lis.)

-3.08 kcal/mole (-0.10 kcal/gm)

391 sec

H, (lis. )

-1.89 kcal/mole (-0.95 kcal/gm)

0,

(lis.)

+30.4 kcal/mole (4.63 kcal/gm)

424 sec

Table V. G. 2

-

Effect of Hydrocarbon Enthalpy of Formation and Hydrogen to Carbon Ratio on Performance with Liquid Oxygen a s Rocket Propellants (Sea Level Specific Impulse, P, = 1000 psia, Equilibrium J!kpansion, Mixture Ratio of Maximum Specific Impulse.) Specific

Impulse

(sec) for

CH, /02

X

ind

kcal/C ;om

1.0

1.4

1.8

2.2

2.6

3.0

4.0

-20 -15 -10 - 5 0 5 10 15 20 25 30

268 2 73 2 79 284 291 297 304 311 319 327 335

276 283 286 292 298 304 3 10 317 323 330 3 40

283 288 293 298 304 3 10 316 322 329 336 3 43

289 294 299 304 309 315 321 327 333 3 40 347

295 299 304 309 314 320 325 330 337 3 43 3 50

300 304 309 314 3 18 324 329 334 340 346 353

310 314 319 323 327 332 337 3 42 347 353 359

135

I

Postface 'It i s better t o forsee with a little uncertainty, than not to forsee a t all. ' PoincarB 1. . . . adapting fundamental thinking with respect t o the basic thermochemistry, kinetics and fluid mechanics, it is possible to characterize quite readily a particular propellant for a particular propulsion scheme (1).

11. A. 1. With Y = 1.2, it can be seen that the drop in pressure from the throat is

approximately half the chamber pressure, and the drop in temperature is only about one-tenth the chamber temperature.

.

.. .

. . the thrust does not depend on the combustion temperature T,, but depends mainly on the dimensions of A, and A, and on the chamber pressure P,. In other words, the thrust that a rocket motor develops does not depend upon the particular choice of propellants, but upon the chamber pressure. The designer controls the pressure level of operation by the amount of propellants chosen t o be injected. I

I

I I

... the effective exhaust velocity as determined by the ratio of F to i can be taken t o be the optimum value of U, even if the actual experimental nozzle is somewhat off design. Herein lies the practical significance of the concept of the effective exhaust velocity.

II. A. 2.

11.A. 3. Being simply the quotient of the thrust and the total weight flow, the speci-

fic impulse i s a performance parameter readily measured experimentally with good accuracy. This fact accounts for its popular acceptance. In regard to convenience there is no greater merit in the use of I,, instead of C.

I

.

. . the specific impulse varies directly with the square root of the chamber temperature and inversely with the square root of the average molecular weight of the combustion products. The units of specific impulse are seconds. Actually the weight flow rate in the definition of specific impulse should be specified F as at sea level and thus go = g and Isp = - numerically.

i

II.A.4. . . . nozzles that are either under-or-over-expanded produce less thrust than a properly expanded nozzle.

I

It is significant that cF is completely independent of T, andh. Consequently a s a figure of merit, it is insensitive to the efficiency of combustion, but sensitive to

I

I I

m

the nozzle design. 11. A. 5. . . . c * depends mainly on conditions in the combustion chamber: that is, the flame temperature and combustion product composition throughh and 1'.

The chamber pressure only indirectly influences c* through its effect on T,. 11.B. 1. It is of course not necessary t o have an extensive list of heats of reaction t o determine the heat absorbed o r evolved in every possible chemical reaction. A

*

136

more convenient and logical procedure is to list what are known as the standard heats of formation of chemical substances. The standard heat of formation is the enthalpy of a substance in i t s standard state referred to i t s elements in their standa r d states at the same temperature. From the above definition it is obvious that the heats of formation of the elements in their standard states are zero. When all the heat evolved is used to heat up the product gases, the product temperature T, is called the flame o r adiabatic combustion temperature..

.

.. .

II. B.2. the condition for equilibrium at constant pressure is that the change in free energy be zero.

..

K, i s not a function of total pressure, but is a function of temperature alone.

.

. . the free energy change a t the standard pressure po determines the equilibrium conditions at all other pressures. If the equilibrium constant for a given reaction is not available, it is also possible to calculate i t s value a t any temperature by simple algebraic manipulation, if the equilibrium constants of formation of the substances present relative to their elements in their standard states are known at that temperature. 11. B.3. In order to determine whether a substance will condense o r not, one first determines the partial pressure without assuming condensation. If this partial pressure is greater than the vapor pressure, then an equilibrium situation con-

densation must have taken place.

..

.knowledge of thermodynamic equilibrium constants and kinetics allows one to eliminate many possible product species.

11. B.4.

Equilibrium concentrations of carbon o r ammonia are not found in short combustion chambers used in rocket motors. II. C. 1. If the reaction times taking place in the reacting mixture are extremely fast compared t o the expansion time, then chemical equilibrium will be maintained a t all instances during the expansion process; this flow process is referred to as equilibrium flow. , . . the expansion can be so fast that the chemical composition does not change effectively from the composition which entered the nozzle. This type of flow is called frozen flow.. .

11. C. 2. For an adiabatic process the equilibrium and frozen composition expansion process a r e both isentropic, whereas the finite rateprocess is not.

No information can be deduced about the entropy variation in the intermediate range of reaction rates; however, the process is not isentropic because equation II. C. 2.

does not go to zero. Considerarions from irreversible thermodynamics show that . the entropy must always rise in a closed thermodynamic system when irreversible reaction processes take place ( 2 5 ) .

II.C. 3. An important problem generally not discussed in texts is the calculation of the nozzle area ratio E for an equilibrium flow system in which the chemical composition, Y and h are changing. If, in a practical case, theory predicts condensation in the contraction portion of the nozzle and it actually takes place in the expansion part, then a c* substantially lower than theoretical will be measured and a c F greater than theoretical can be

137

obtained. The specific impulse efficiency will be normal since the condensation energy is recovered and cF is obtained through I,, g/c*

.

...

the enthalpy of the equilibrium exhaust state will always be less than that of the corresponding frozen situation. This fact is true because the total enthalpy is the parameter of concern and as species recombine they release energy which can be converted to kinetic energy and form products with heats of formation lower than chamber species. II. C. 4. In typical non-equilibrium flow in nozzles, one begins more or less in equilibrium at high temperatures, because the chemical rates are very temperature sensitive and are the fastest at high temperatures. The flow expands rapidly with the chemistry at first keeping pace with the expansion, then falling behind in a transition zone, and finally virtually stopping (frozen flow). II. C. 5. The most useful of the simplified procedures is an adaption of Bray's technique for handling a recombination reaction in nozzles of heated supersonic tunnels (7). Bray points out simply that the transition region from equilibrium to frozen flow is very narrow.

Generally there is one reaction in the complex scheme one writes for combustion gases that is the main energy releasing step. This reaction then becomes the reaction of concern in the Bray approach.

...

a reacting mixture can maintain itself nearly in chemical equilibrium, as the pressure and temperature change, only if the three body recombination reactions follow fast enough. They are the controlling steps in the same way a non-equilibrium mixture approaches the equilibrium state. What is particularly important from a propulsion point of view, in this regard, is that the three-body recombination reactions are the major energy releasing reactions as well. Thus if the Bray freezing point criterion is applied to only one step, it must be applied to a controlling three body recombination step in the reaction scheme.

...

when the selected three-body reaction freezes, one can, for impulse purposes, consider the system frozen.

...

notice further that as the chamber pressure increases the kinetic solutions approach the equilibrium solution. Similarly at very low chamber pressures, say about 1 psia, the results would approach that of frozen flow.

.

. . for the fuel-rich mixture ratios used to obtain maximum specific impulse, the freezing point is almost precisely at the physical throat (27). At higher chamber pressures the freezing point will move further downstream. Except at the highest pressures, about 800-1000 psia, where one would expect tlic kinetic and equilibrium results to be the same, the following very simplified approach should give a very good approximation to the kinetic specific impulse result. An equilibrium calculation is performed from the chamber to the throat and then a frozen flow calculation from the throat to nozzle exit.

..

. thermal equilibrium between particle and gas is of far lesser iniportance than particle velocity lag.

11. C. 6.

138 Combustion gases with particles can be considered to have an artifically high molecular weight given by equation II. C. 18. III. A. Direct evidence may be found both in laboratory and rocket engine experiments that the kinetics of the hydrazine/nitrogen tetroxide reaction control the composition of the reaction products.

The explanation of the poor performance a t high oxidizer to fuel ratios lies in the failure of the excess nitrogen tetroxide to decompose to molecular nitrogen and oxygen as would be predicted from equilibrium considerations. III. D. Even though explicit reaction data may not be available, knowledge of certain classes of reactions is useful in judging when to expect non-equilibrium effects in the rocket chamber.

It has been long realised in the field of chemical kinetics that the decomposition times of NH, and NO are long.

... reactions leading to condensed phase products are also very slow. The slow condensed phase reaction may be the reason for some of the efficiency problems which rise in the use of metallized propellants.

IV. If the fuel o r oxidizer has a critical point such that it must be kept at low temperatures in order that it remain liquid throughout the system, then it is termed a cryogenic propellant.

..

Those non-cryogenics which remain liquid over a wide temperature range, possibly -60" to 160°F, which do not have extreme vapor pressures at the upper temperature limit (less than 500 psia), and which are structurally stable over long periods are termed the storable propellants. Thixotropicity is called the property of a gel to be liquified by internal mechanical stress, and usually to return to the gel form when the s t r e s s ceases. Propellant selection then should consider density, as well as specific impulse. For pure terrestrial weapon systems other considerations such as reliability, readibility, cost, are also of importance. IV. A. 1.

..

. the basic purpose of specific compounds in propellant systems is simply to introduce certain elements into the combustion process. It is r a r e that the heat of formation of a propellant influences the performance. For the heat of formation of a reactant to affect the performance, it must have a value greater than + 0.5 kcal/gm to increase the performance noticeably o r a value less than -0.5 kcal/gm to decrease it. There are very few compounds indeed that meet these requirements, ozone (+), ammonia (-), and acetylene (+) Figure IV.A. 1. immediately predicts that of elements with atomic numbers greater than 10 only Mg, Al, and Si and their hydrides would be worth considering from a specific impulse point of view.

.

. . the heats of vaporization of most condensable species of concern are sufficiently large so that it is desirable to have condensation take place. ,

The toxicity of Be almost precludes it from consideration, but its unique position on the graph given in figure 1V.A. 1. will always make it of interest.

139

..

. the handling of boron compounds, the vitreous product itself and the slow burning characteristics of the elemental metal makes it extremely doubtful whether boron compounds will ever be used with oxygen systems. Since BF3 is a gas, the significance of boron compounds as fuels with fluoride oxidizers particularly changes for the better. If a halogen o r interhalogen i s to be used with a carbon-containing fuel, then it is best to mix the compound with an oxidizer containing oxygen to allow the formation of CO and CO, rather than the halocarbon. Thus with carbon-containing fuels F,O F,-0, mixtures and C103F rake on added significance.

It should be pointed out that there is no advantage of F,O over an appropriate molar mixture of F, and 0,, which would be appreciably less expensive. IV. A. 2. In order to alleviate the total solids problem, there are great efforts to introduce the energy-bearing fuel atom in the polymer o r by substituting appropriate oxidizer groups such as nitrates in the pure polymer.

.

IV. B. .. one of the best means of augmenting the performance of a propellant system is by adding greater quantities of the light metallic elements. It has been this desire to add additional metal to the system which has brought the recent innovations in rocket developments to the forefront. IV. C. The most attractive monopropellants are those whose exothermic reaction o r decomposition can be catalytically initiated. IV. D. The extension of the propellant combination from two compounds, ie., a bipropellant system, to more than two compounds, ie., a multicomponent propellant system, offers the potential of increasing performance above levels obtainable with propellant systems limited to two propellants. However, this increase in performance results simply from the possibility of conveniently introducing more elements into the combustion process. IV. D. 1. If ozone is considered as a potential oxidizer with beryllium, the addition of hydrogen can result in theoretical performance a t high expansion ratios near 600 seconds specific impulse. IV. D. 2. The weight ratio of the oxidizer and metal components are always in stoichiometric proportion to produce the metal oxide o r metal halide. The low molecular weight specie is then added in an amount determined by the trade-off between decreasing molecular weight and decreasing temperature of the combustion products.

IV.D. 3. Use of a metallic hydride does not, as might first appear, reduce the tripropellant system to a bipropellant system. In general, the optimum hydrogen to metal ratio will not be that corresponding to the metallic hydride hydrogen to metal ratio. N.E. The only generalization which may be drawn in the selection of propellants f o r this category of rocket engines is that the selection process is related to the particular manner in which energy is imparted to the propellant. IV. E.'1. The metallic hydrides, such as pentaborane (BSH9),lithium hydride (LiH), and beryllium hydride (BeH,) are attractive candidates if the temperature and pressure of operation fall into a range where evaporation of the metal will occur (48). Such propellants would have a definite advantage over hydrogen in their storability and relatively high densities.

140

...

it has been shown theoretically (48) that the performance in terms of specific impulse of propellants containing light metallic elements can exceed the performance of hydrogen. V.A. The selection of mixture ratio may be influenced by concerns other than propellant performance as measured by the specific impulse. V.A. 1. The maximum enthalpy of reaction occurs at the stoichoimetric mixture ratio.. . The maximum temperature thus should fall a t the stoichiometric point. In most cases it does, but, according to the character of the elements present, it can fall somewhat to the fuel rich or oxidizer rich side. In most propellant systems the maximum temperature falls just on the fuel rich side. The molar specific heat of the products determines t h i s small effect.

..

V.A. 2. . themixture ratio of maximum specific impulse shifts toward the stoichiometric mixture ratio with increasing chamber pressure. V.A.3. The effect of increasing expansion ratio at constant chamber pressure upon the optimum mixture ratio is much the same as is the previously discussed effect of increasing chamber pressure a t constant expansion ratio. V.A. 4. The failure of these two performance parameters, C' and C, to have maximum values a t the same propellant mixture ratio is traceable to their differing dependencies upon the specific heat ratio. V. B. For a fixed vehicle diameter and thus a fixed maximum nozzle exit area, higher area ratios are obtained at higher pressures simply because the throat area decreases for fixed mass flow. This is the major effect of pressure and, indeed, contributes to increase the performance for both atmospheric and space operation.

V.E. The failure to include a significant product specie in calculations of propellant performance has a pronounced effect and results in a n erroneous prediction of the theoretical performance. V. F.2. Uncertainties in enthalpies of formation have been greatest for the metal containing compounds of importance to many high energy propellant combinations. V. C.2. To have a noticeable effect, however, as noted earlier, the change in propellants must bring about an increase in the induction enthalpy of at least 0.5 kcal/gm.

141

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NASA