Advanced Computational Methods in
Heat Transfer IX
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NINTH INTERNATIONAL CONFERENCE ON ADVANCED COMPUTATIONAL METHODS IN HEAT TRANSFER
HEAT TRANSFER IX CONFERENCE CHAIRMEN B. Sundén Lund Institute of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK
INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE R. Amano A. Buikis L. De Biase G. De Mey S. del Giudice K. Domke J. Gylys
P. Heggs C. Herman D. B. Ingham Y. Jaluria P. S. Larsen X. Luo A. Mendes K. Onishi H. Oosthuizen
B. Pavkovic W. Roetzel B. Sarler S. Sinkunas A. C. M. Sousa J. Szmyd S. Yanniotis
Organised by Wessex Institute of Technology, UK Lund University of Technology, Sweden Sponsored by The Development in Heat Transfer Book Series
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Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email:
[email protected]
WIT Transactions on Engineering Sciences Editorial Board B. Abersek University of Maribor Slovenia
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H Azegami Toyohashi University of Technology Japan
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N Ishikawa National Defence Academy Japan
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A S Kobayashi University of Washington USA
T Krauthammer Penn State University USA
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M Langseth Norwegian University of Science and Technology Norway
Y-W Mai University of Sydney Australia
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G Manara University of Pisa Italy
T Matsui Nagoya University Japan
H A Mang Technische Universitat Wien Austria
R A W Mines The University of Liverpool UK
A C Mendes Univ. de Beira Interior Portugal
T Miyoshi Kobe University Japan
A Miyamoto Yamaguchi University Japan
T B Moodie University of Alberta Canada
G Molinari University of Genoa Italy
D Necsulescu University of Ottawa Canada
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H Nisitani Kyushu Sangyo University Japan
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P O'Donoghue University College Dublin Ireland
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M Ohkusu Kyushu University Japan
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P H Oosthuizen Queens University Canada
W Perrie Bedford Institute of Oceanography Canada
G Pelosi University of Florence Italy
D Poljak University of Split Croatia
H Pina Instituto Superior Tecnico Portugal
H Power University of Nottingham UK
L P Pook University College London UK
I S Putra Institute of Technology Bandung Indonesia
D Prandle Proudman Oceanographic Laboratory UK
M Rahman Dalhousie University Canada
F Rachidi EMC Group Switzerland
T Rang Tallinn Technical University Estonia
K R Rajagopal Texas A & M University USA
B Ribas Spanish National Centre for Environmental Health Spain
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K Richter Graz University of Technology Austria
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S Walker Imperial College UK
T X Yu Hong Kong University of Science & Technology Hong Kong
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M Zamir The University of Western Ontario Canada
K Zakrzewski Politechnika Lodzka Poland
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Advanced Computational Methods in
Heat Transfer IX Editors B. Sundén Lund University of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK
B. Sundén Lund Institute of Technology, Sweden C. A. Brebbia Wessex Institute of Technology, UK
Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail:
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[email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 1-84564-176-0 ISSN: 1746-4471 (print)
ISSN: 1743-3533 (online) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2006 Printed and bound in Great Britain by Athenaeum Press Ltd., Gateshead. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.
Preface Research and developments of computational methods for solving and understanding heat transfer problems continue to be important because heat transfer topics are commonly of a complex nature and different mechanisms like heat conduction, convection, turbulence, thermal radiation and phase change may occur simultaneously. Although heat transfer might be regarded as an established and mature scientific discipline, its role and relevance in sustainable development and reduction of greenhouse gases as well as for micro- and nanoscale structures and bio-engineering have been identified recently. Non-linear phenomena may, besides the momentum transfer, appear due to temperature-dependent thermophysical properties. In engineering design and development works, reliable and accurate computational methods are requested to replace or complement expensive and time comsuming experimental trial and error work. Tremendous advancements have been achieved during recent years due to improved numerical solution approaches of non-linear partial differential equations and computer developments to achieve efficient and rapid calculations by parallelised computations on, e.g., PC-clusters. Nevertheless, to further progress computational methods in heat transfer, developments in theoretical and predictive procedures, both basic and innovative, and applied research are needed. To validate the numerical calculations accurate experimental investigations are needed. This book contains the edited versions of the papers presented at the Ninth International Conference on Advanced Computational Methods and Experimental Measurements in Heat Transfer and Mass Transfer held in the New Forest, Ashurst Lodge, Ashurst, UK in July 2006. The objective of this conference series is to provide a forum for presentation and discussion of advanced topics, new approaches and application of advanced computational methods and experimental measurements to heat and mass transfer problems. All papers have been reproduced directly from material submitted by the authors but an attempt has been made to use a unified outline and methods of presentation for each paper. The contributed papers are grouped in appropriate sections to provide better access for readers. The selected sections show the wide range of applied and fundamental problems in the heat and mass transfer field. The editors would like to thank all the distinguished and wellknown scientists who supported our efforts by serving in the International Scientific Advisory
Committee, reviewing the submitted abstracts and papers. The excellent administrative work of the conference secretariat at WIT is greatly appreciated and the efficient cooperation and encouragement by the staff at WIT Press contributed significantly in producing the conference proceedings. The Editors, May 2006
Contents Section 1: Natural and forced convection A discussion on finite-difference schemes for low Prandtl number Rayleigh-Bénard convection X. Luo & W.-K. Chen ............................................................................................3 A numerical study of the convective heat transfer between a room and a window covered by a partially open plane blind with a gap at the top P. H. Oosthuizen .................................................................................................13 An analytical solution to the Graetz problem with viscous dissipation for non-Newtonian fluids R. Chiba, M. Izumi & Y. Sugano .........................................................................23 Heat transfer by unsteady laminar mixed convection in 2-D ventilated enclosures using the vorticity-stream function formulation S. Boudebous & Z. Nemouchi .............................................................................33 Effect of roughness shape on heat transfer and flow friction characteristics of solar air heater with roughened absorber plate A. Chaube, P. K. Sahoo & S. C. Solanki .............................................................43 Study of conjugate heat transfer accompanying mixed convection in a vertical tube submitted to a step of entry temperature O. Kholai, M. Kadja & T. H. Mai .......................................................................53 Section 2: Advances in computational methods An interaction of a sonic injection jet with a supersonic turbulent flow approaching a re-entry vehicle to atmosphere D. Sun & R. S. Amano .........................................................................................65
Simulation of coupled nonlinear electromagnetic heating with the Green element method A. E. Taigbenu.....................................................................................................77 Lattice Boltzmann simulation of vortices merging in a two-phase mixing layer Y. Y. Yan & Y. Q. Zu............................................................................................87 A numerical solution of the NS equations based on the mean value theorem with applications to aerothermodynamics F. Ferguson & G. Elamin ...................................................................................97 Solution of the radiative transfer problems in two-dimensional participating cylindrical medium with isotropic scattering using the SKN approximation N. Döner & Z. Altaç ..........................................................................................109 Presentation of the hemisphere method P. Vueghs & P. Beckers ....................................................................................121 Temperature identification based on pointwise transient measurements A. Nassiopoulos & F. Bourquin ........................................................................131 Section 3: Heat and mass transfer Heat transfer in 3D water and ice basins S. Ceci, L. De Biase & G. Fossati.....................................................................143 Coupled heat and moisture transport in a building envelope on cast gypsum basis J. Maděra, P. Tesárek & R. Černý....................................................................153 Spray water cooling heat transfer under oxide scale formation conditions R. Viscorova, R. Scholz, K.-H. Spitzer & J. Wendelstorf ..................................163 How to design compact mass transfer packing for maximum efficiency H. Goshayshi .....................................................................................................173 Autowave regimes of heat and mass transfer in the non-isothermal through-reactors A. M. Brener & L. M. Musabekova ...................................................................181
Experimental studies of heat transfer between crystal, crucible elements, and surrounding media when growing large-size alkali halide ingots with melt feeding V. I. Goriletsky, B. V. Grinyov, O. Ts. Sidletskiy, V. V. Vasilyev, M. M. Tymoshenko & V. I. Sumin .....................................................................191 Modelling of heat and mass transfer in water pool type storages for spent nuclear fuel E. Fedorovich, A. Pletnev & V. Talalov............................................................199 Section 4: Modelling and experiments Experimental study of in-line tube bundle heat transfer in vertical foam flow J. Gylys, S. Sinkunas, V. Giedraitis & T. Zdankus............................................213 Use of graphics software in radiative heat transfer simulation K. Domke...........................................................................................................221 On heat transfer variation in film flow related with surface cross curvature S. Sinkunas, J. Gylys & A. Kiela .......................................................................231 Experimental investigation of enhanced heat transfer of self-exciting mode oscillating flow heat pipe with non-uniform profile under laser heating F. Shang, H. Xian, D. Liu, X. Du & Y. Yang.....................................................241 Investigation of heat transfer in the cup-cast method by experiment, and analytical method F. Pahlevani, J. Yaokawa & K. Anzai...............................................................249 Atwood number effects in buoyancy driven flows M. J. Andrews & F. F. Jebrail ..........................................................................259 Testing of the vapour chamber used in electronics cooling A. Haddad, R. Boukhanouf & C. Buffone .........................................................269 Modeling a real backdraft incident fire A. Tinaburri & M. Mazzaro ..............................................................................279 Numerical and experimental studies in the development of new clothing materials E. L. Correia, S. F. C. F. Teixeira & M. M. Neves ...........................................289 Sensitivity analysis of a computer code for modelling confined fires P. Ciambelli, M. G. Meo, P. Russo & S. Vaccaro.............................................299
Conservative averaging as an approximate method for solution of some direct and inverse heat transfer problems A. Buikis ............................................................................................................311 Section 5: Heat exchangers and equipment Estimating number of shells and determining the log mean temperature difference correction factor of shell and tube heat exchangers S. K. Bhatti, Ch. M. Krishna, Ch. Vundru, M. L. Neelapu & I. N. Niranjan Kumar ....................................................................................323 The re-commissioned thermosyphon reboiler research facility in the Morton Laboratory A. Alane & P. J. Heggs .....................................................................................337 Analysis of water condensation and two-phase flow in a channel relevant for plate heat exchangers J. Yuan, C. Wilhelmsson & B. Sundén ..............................................................351 Numerical heat transfer modelling of staggered array impinging jets A. Ramezanpour, I. Mirzaee, R. Rahmani & H. Shirvani .................................361 The re-commissioning of a vent and reflux condensation research facility for vacuum and atmospheric operation J. C. Sacramento Rivero & P. J. Heggs ............................................................371 Heat transfer modelling in double pipes for domestic hot water systems I. Gabrielaitiene, B. Sunden & J. Wollerstrand................................................381 Convective heat transfer investigations at parts of a generator circuit breaker T. Magier, H. Löbl, S. Großmann, M. Lakner & T. Schoenemann ...................391 Simplified 3-D FE model of thermal conditions inside a shoe H. Raval, Z. W. Guan, M. Bailey & D. G. Covill..............................................401 Section 6: Energy systems Radiative heat transfer in a model gas turbine combustor M. C. Paul & W. P. Jones .................................................................................413 Thermo-economics of an irreversible solar driven heat engine K. M. Pandey & R. Deb ....................................................................................423
Analysis of a new solar chimney plant design for mountainous regions M. A. Serag-Eldin..............................................................................................437 Section 7: Micro and nano scale heat and mass transfer Sinusoidal regime analysis of heat transfer in microelectronic systems B. Vermeersch & G. De Mey.............................................................................449 Viscous dissipation and temperature dependent viscosity effects in simultaneously developing flows in flat microchannels with convective boundary conditions S. Del Giudice, S. Savino & C. Nonino.............................................................457 Experimental study of water evaporation from nanoporous cylinder surface in natural convective airflow S. Hara ..............................................................................................................467 Author index ....................................................................................................477
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Section 1 Natural and forced convection
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Advanced Computational Methods in Heat Transfer IX
3
A discussion on finite-difference schemes for low Prandtl number Rayleigh-Bénard convection X. Luo & W.-K. Chen Institute of Thermal Engineering, University of Shanghai for Science and Technology, People’s Republic of China Dedicated to Prof. Dr.-Ing. Wilfried Roetzel on the occasion of his 70th birthday
Abstract The natural convection in a horizontal fluid layer heated from below has complex dynamic behaviour. For the Rayleigh-Bénard convection of low Prandtl number fluids, the calculated flow and temperature fields are very sensitive to the truncation error of numerical algorithms. Different kinds of finite-difference schemes might yield different numerical results. In the present work the error analysis of the upwind scheme and QUICK scheme for the Rayleigh-Bénard convection of low Prandtl number fluid was conducted. It shows that the upwind scheme will introduce numerical dispersion. This effect enlarges the viscosity term of the momentum equations and therefore no oscillation could be predicted. The QUICK scheme has higher calculation accuracy. However, it introduces an additional third-order differential term which might overestimate the oscillation effect. Keywords: Rayleigh-Bénard convection, low Prandtl number fluid, two-dimensional roll, finite-difference scheme, QUICK scheme.
1
Introduction
The natural convection in a horizontal layer confined by two rigid boundaries and heated from below is well known as Rayleigh-Bénard convection. This phenomenon reveals series non-linear characteristics and complex dynamic behaviour and has been well investigated [1−2]. The studies of low Prandtl WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060011
4 Advanced Computational Methods in Heat Transfer IX number Rayleigh-Bénard convection have been motivated not only by its astrophysical applications but also by its special flow and heat transfer characteristics because in this case the non-linear inertial terms become significant. Clever and Busse [3] used the Galerkin technique to obtain the Nusselt number for low Prandtl number convection at various Rayleigh numbers. According to the linear stability analysis, the critical Rayleigh number Rac and critical wave number kc for two rigid boundaries are given as Rac = 1707.762 and kc = 3.117, respectively [1]. Bertin and Ozoe [4] calculated the problem with a finite-element method and found that Rac increases with the decrease of Pr. The computed critical Rayleigh number for Pr = 0.01 and k = 3.14 is given as Rac = 2095.0. Later Ozoe et al. [5] used a more accurate second-order central difference scheme to solve the problem and the calculated Nusselt number agreed closely with those of Clever and Busse [3]. The computed value of Rac for all values of Pr also agreed well with the theoretical value. Their results show that the Rayleigh-Bénard convection of low Prandtl number fluid is very sensitive to the numerical algorithm. As the Rayleigh number is larger than its critical value, the fluid begins to move and forms a steady flow pattern of two-dimensional rolls parallel to each other. For higher Rayleigh number the rolls become unsteady and the bending of the rolls propagates along the roll axis in time. By means of stability analysis for steady convection rolls Clever and Busse [6] discussed the oscillation instability and the critical Rayleigh number Rat for the onset of oscillation. Later they found that the transition from thermal convection in the form of rolls in a fluid layer heated from below to travelling-wave convection occurs at Rat = 1854 in the limit of low Prandtl numbers and in the presence of rigid boundaries [7]. Ozoe and Hara [8] carried out numerical computations with a second-order central difference approximation to predict Rat. The computing region is 4 times as large as the height, i.e., the aspect ratio A = 4. For the grid size of 0.02, the oscillation occurs at Rat ≈ 4500 for 4-roll pattern and Pr = 0.01. By extrapolation to zero grid size they inferred that the critical oscillatory Rayleigh number Rat was less than 2000. Yang et al. [9] and Wang et al. [10] used the SIMPLE algorithm with QUICK (quadratic upwind interpolation of convective kinematics) scheme to solve the same problem as that of Ozoe and Hara [8] with A = 4 and Pr = 0.01 and found that the fluid flow and heat transfer is steady and stable for Ra ≤ 2200. Their numerical calculation showed that the oscillation occurs at Ra = 2500. As has been mentioned by Yang et al. [11], for low Prandtl number fluid the calculated Nusselt numbers with different numerical schemes are quite different. They suggested a possible explanation that the problem might have bifurcations and the results from different numerical schemes might lie at different branches of the solution. In the present work, this problem was solved numerically with the upwind scheme, power law scheme and QUICK scheme. The Taylor series expansion was used to analyse the truncation errors of the schemes. It was found that the upwind scheme would introduce fictitious viscosity and underestimate the value of Nusselt number. The QUICK scheme was more suitable for the low Prandtl
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
5
number natural convection. However, it introduces a third-order differential term which might affect the oscillation characteristics of the physical model.
Figure 1:
2
Schematic diagram of the horizontal fluid layer heated from below.
Mathematical model
Let us consider a horizontal fluid layer with two rigid boundaries at its upper and lower surfaces, as shown in fig. 1. The fluid layer is heated from below and cooled from above. The temperatures at the upper and lower rigid boundaries are Th and Tc, respectively, and Th > Tc. The fluid expands when it is heated, therefore it suffers an upward buoyancy force. The problem is simplified with the following assumptions: (1) The flow pattern is assumed to be two-dimensional; (2) All properties of the fluid are constant except the density in the buoyancy term of the momentum equation which is a linear function of the fluid temperature; (3) The left and right boundaries are two symmetric boundaries, that means that the wave number in this model is fixed. The following dimensionless variables and parameters are used for the modeling: X = x H , Y = y H , τ = UR t H , U = u UR , V = v UR , A = L H , UR =
a H
RaPr , DV =
Pr , DT = Ra
1
RaPr
, Pr =
ν a
, Ra =
gβ (Th − Tc )H 3 . νa
We assume that at first there is no motion in the fluid, and the temperature distribution is in a steady state. The steady-state temperature and reduced pressure distributions can be expressed as Tτ =0 = Tc + (Th − Tc )(1 − Y )
(1)
pτ =0 = ρU R2Y (1 − Y 2) .
(2)
and
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
6 Advanced Computational Methods in Heat Transfer IX It is convenient to introduce the dimensionless temperature and pressure perturbations,
θ=
T − Tτ =0 T − Tc = − (1 − Y ) , Th − Tc Th − Tc
(3)
P=
p − pτ =0 p Y = − Y 1 − . 2 ρU R2 ρU R2
(4)
The dimensionless governing equation system can then be written as: ∂ 2U ∂ 2U ∂U ∂U ∂U ∂P +U +V =− + DV + 2 ∂τ ∂X ∂Y ∂X ∂Y 2 ∂X ∂ 2V ∂ 2V ∂V ∂V ∂V ∂P +U +V =− + DV + 2 ∂τ ∂X ∂Y ∂Y ∂Y 2 ∂X ∂ 2θ ∂ 2θ ∂θ ∂θ ∂θ +U +V = DT + 2 ∂τ ∂X ∂Y ∂Y 2 ∂X
,
(5)
+ θ ,
(6)
+ V ,
∂U ∂V + = 0, ∂X ∂Y
(7)
(8)
with the initial and boundary conditions
τ = 0:
θ =P=0;
X = 0 and X = A :
U=
Y = 0 and Y = 1 :
U =V =θ = 0.
∂V ∂θ = =0; ∂X ∂X
(9) (10) (11)
The average Nusselt number is calculated at the lower plate, A
Nu = 1 −
1 ∂θ A ∫0 ∂Y
dX
(12)
Y =0
The zero solution of the governing equation system (5)−(11) corresponds to the motionless state which is stable for Ra < Rac. Even the fluid is stirred before, the disturbance will gradually vanish and finally the fluid approaches the steady state again. However, If Ra > Rac, this zero solution becomes unstable. Any WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
7
small disturbance will be enlarged gradually and finally the Rayleigh-Bénard convection will be set up. For small value of Ra − Rac, the motion is twodimensional. With the increase of the Rayleigh number, the flow becomes steady three-dimensional flow, unsteady flow, and finally, turbulent flow [12]. In this paper eqns. (5)−(11) for Ra > Rac were solved with finite-difference methods. The research was emphasised on the oscillatory characteristics of the two-dimensional rolls, which reflects some special features of the upwind scheme and QUICK scheme.
3
Comparison of finite-different schemes
Ozoe et al. [5] used the vorticity and stream function to eliminate the pressure gradient terms in eqns. (5)−(11) and solved the problem by central differences for the derivatives in space and alternating-direction implicit differences for the derivatives in time. They calculated the Nusselt number for Ra ≤ 3000 with the aspect ratio A = 1. They found that the grid size had significant effect on the calculated Nusselt number and had to extrapolate their results to the case of (∆X, ∆Y) → 0. As shown in table 1, their results agree well with those of Clever and Busse [3] obtained by Galerkin method. No oscillation is found in their calculation. Table 1:
Ra
2000 2300 2500 3000 4000 5000 6000 10000
The calculated Nusselt numbers from different sources.
Clever and Busse [3] k = 3.11
Nu (Pr = 0.01) Ozoe and Ozoe and Ukeba [5] Hara [8] k = 3.14 k = 3.14 (1 roll) (4 rolls)
1.01955 1.17335 1.33978 1.59614 1.89397 2.22264
1.105 ** 1.18 ** 1.35 ** 1.259 1.289 * 1.502 *
Yang et al. [9] and Wang et al. [10] k = 3.14 k = 3.93 (4 rolls) (5 rolls) 1.018 1.094 1.104 1.167 * 1.176 * 1.269 * 1.355 * * 1.511 1.609 * 1.793 * 1.651 * * 1.764 1.933 * 2.052 * 2.316 *
(*: The solution is oscillating; **: Data are extrapolated to zero grid size.) Ozoe and Hara [8] further calculated the Nusselt number for higher Rayleigh numbers, 4000 ≤ Ra ≤ 2.8×105. The data listed in table 1 are taken from fig. 2 of [8]. These data are calculated under the grid size of 0.02. Their
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
8 Advanced Computational Methods in Heat Transfer IX calculation shows that the oscillation appears when Ra ≥ 6000. However, their predicted values of Nu are much lower than those of Clever and Busse [3]. We calculated the problem by the SIMPLE algorithm [13] and upwind scheme with Ra = 5000, Pr = 0.01, A = 4 and found that the grid size of 0.02 would yield Nu = 1.29 which is a little higher than that of Ozoe and Hara [8]. This value is not correct because the grid size of 0.01 would yield another value of Nu = 1.46. No oscillation can be found in the numerical solutions. The power law scheme yields the same results as the upwind scheme. Applying the Taylor series expansion to the upwind scheme, we can find that the upwind scheme introduces an additional dimensionless viscosity U∆X/2. In the above example, because of the low Prandtl number, we have DV = 1.414×10−3. The maximum dimensionless velocity Umax = 0.7776. Thus, for ∆X = 0.02, this fictitious viscosity could be 5.5 times as large as the real viscosity and therefore leads to a lower value of Nu. Yang et al. [11] compared the QUICK scheme with the power law scheme and found that for natural convection of low Prandtl number fluids the QUICK scheme provides higher accuracy. Therefore, the QUICK scheme was used in the calculations of Yang et al. [9] and Wang et al. [10]. Some results for Pr = 0.01, A = 4 and ∆X = ∆Y = 0.02 are shown in table 1. The calculated values of Nu for 4 rolls are close to those of Clever and Busse [3]. Applying the Taylor series expansion to the QUICK scheme, 3 7 φ ( X ,τ ) − φ ( X ,τ − ∆τ ) U 3 + φ ( X ,τ ) + φ ( X + ∆X ,τ ) − φ ( X − ∆X ,τ ) ∆τ ∆X 8 8 8
1 φ ( X + ∆X ,τ ) + φ ( X − ∆X ,τ ) − 2φ ( X ,τ ) + φ ( X − 2∆X ,τ ) = D , 8 ∆X 2
(13)
we have, ∂φ ∂φ ∂ 2φ + O(∆τ ) + U =D + R ( ∆X 2 ) , ∂τ ∂X ∂X 2
(14)
in which R (∆X 2 ) =
∆X 2 ∂ 4φ ∂ 3φ D + O(∆X 3 ) . − U 24 ∂X 4 ∂X 3
(15)
It shows that the QUICK scheme does not introduce the fictitious viscosity term and has the truncation error of O(∆X 2), therefore the predicted Nu values are more accurate. Wang et al. [10] also used the QUICK scheme to predict the roll oscillation. They found that for Pr = 0.01, A = 4 and ∆X = ∆Y = 0.02 the oscillation occurs at Ra = 2500, which is much higher than the theoretical value of the critical WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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oscillatory Rayleigh number, Rat =1854. However, they neither tested the effect of the grid size on the oscillation characteristics nor analysed the stability characteristics of the QUICK scheme. For low Prandtl number fluid, we have D/U ~ 10−2. Thus, eqn. (15) is simplified as, R (∆X 2 ) = −
U∆X 2 ∂ 3φ + O(∆X 3 ) . 24 ∂X 3
(16)
In such a case, the QUICK scheme will introduce an additional term into the governing equations which affects the oscillation characteristics of the RayleighBénard convection. In fact, the stability of the QUICK scheme is conditional. The stability condition is given as [14], P∆ =
U∆X 8 ≤ . D 3
(17)
For Ra = 2500, Pr = 0.01 and ∆X = 0.02, the maximum dimensionless velocity Umax = 0.867, which yields P∆ = 8.67 and the condition (17) can no longer be valid. Although an oscillation solution can be obtained with the QUICK scheme, but it is not sure whether this oscillation is caused by the instability of the numerical scheme or by the characteristics of the physical problem or by both of them. To verify the numerical schemes, two critical values should be considered. One is the critical oscillatory Rayleigh number Rat, which is difficult to be determined with direct numerical simulation. By extrapolation to zero grid size, Ozoe and Hara [8] estimated that Rat < 2000, which is near to the theoretical value, Rat = 1854. We tested a calculation with QUICK scheme for Ra = 2000, Pr = 0.01 and ∆X = 0.005. And contrary to our expectation, after several thousand hours of computing time of a PC with a 2.8 GHz CPU, we obtained a steady convection (The relative velocity disturbance ∆U/Umax was less than 10−9). When we added a small disturbance in Rayleigh number, an oscillatory disturbance was set up. The disturbance of the maximum vertical velocity component is shown in fig. 2. At τ = 30, there is some change in the frequency. After that point the oscillation becomes violent. Then, at about τ = 55, we increased the iteration accuracy and set the minimum iteration times to 3. And the oscillation gradually vanished. The solution reached the steady convection again. The other criterion is the oscillation frequency. According to fig. 12 of Clever and Busse [6], the theoretical frequency of the oscillatory disturbance for Pr = 0.01 and Ra = 2500 is about 0.01 (after the conversion between different definitions of dimensionless time). But the QUICK scheme of Yang et al. [11] with the grid size of 0.02 resulted a frequency of 0.3125 (see fig. 2 of [11]). Our calculation with the QUICK scheme gave a frequency of 0.3344. Both of them are much larger than the theoretical value given by Clever and Busse [6]. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
10 Advanced Computational Methods in Heat Transfer IX
Figure 2:
Velocity disturbance due to a disturbance in Rayleigh number (Ra = 2000, Pr = 0.01, A = 4 and ∆X = 0.005).
These examples show that the Rayleigh-Bénard convection of low Prandtl number fluid is very sensitive to the oscillation characteristics of the finitedifference schemes. Therefore, more accurate schemes should be developed for the direct simulation of such a problem.
4
Conclusions
The Rayleigh-Bénard convection of low Prandtl number fluid is difficult to be solved because of its nonlinear properties and special dynamic characteristics. The two-dimensional transient numerical calculations with SIMPLE algorithm were carried out to simulate such a problem. Different numerical schemes such as upwind scheme, power law scheme and QUICK scheme were used in the calculation. We find that the upwind scheme and power law scheme are not suitable for the direct simulation of low Prandtl number Rayleigh-Bénard convection because they will introduce a fictitious dispersion term. The QUICK scheme provides an accuracy of the second order and will not introduce the numerical dispersion into the problem to be solved. The predicted values of Nusselt number are reasonably accurate. However, it fails to predict the critical oscillatory Rayleigh number and the oscillation frequency because of the stability and oscillation characteristics of the QUICK scheme itself. New numerical schemes should be developed to simulate the low Prandtl number Rayleigh-Bénard convection. In the present work we have restricted our calculation with a fixed aspect ratio A = 4, i.e., the wave number k = 3.14. In fact, the critical Rayleigh numbers also depend on the wave number k and the number of rolls to be considered. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Therefore, there remains a lot of unsolved problems in the numerical simulation of the Rayleigh-Bénard convection.
Acknowledgements The present research was sponsored by the National Natural Science Foundation of China (No. 50478113), Shanghai Leading Academic Discipline Project (No. T0503) and Shanghai Pujiang Program (No. 05PJ14078).
Nomenclature a A DT
thermal diffusivity, m2/s Aspect ratio, A = L/H dimensionless thermal diffusivity, DT = 1 / RaPr
DV g H k
dimensionless viscosity, DV = Pr / Ra acceleration constant due to gravity, m/s2 height of the fluid layer, m wave number; heat conductivity, W/m2K width of the computing region, m Nusselt number, Nu = qH/[(Th − Tc)k] reduced pressure, N/m2 dimensionless pressure perturbation, eqn. (4) Prandtl number, Pr = ν /a heat flux, W/m2 Rayleigh number, Ra = gβ (Th − Tc)H 3/(νa) the critical oscillatory Rayleigh number time, s temperature of the upper plate, K temperature of the lower plate, K velocity component in the x direction, m/s dimensionless velocity component in the x direction, U = u/UR
L Nu p P Pr q Ra Rat t Tc Th u U
reference velocity, U R = a RaPr / H , m/s UR v velocity component in the y direction, m/s V dimensionless velocity component in the y direction, V = v/UR x horizontal spatial coordinates, m X dimensionless horizontal spatial coordinates, X = x/H y vertical spatial coordinates, m Y dimensionless vertical spatial coordinates, Y = y/H Greek symbols β volumetric coefficient of expansion, 1/K ν kinematic viscosity, m2/s θ dimensionless temperature perturbation, eqn. (3) τ dimensionless time, τ = URt/H Subscript c critical value WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
12 Advanced Computational Methods in Heat Transfer IX
References [1] Getling, A.V., Rayleigh-Benard Convection—Structures and Dynamics, World Scientific: Singapore, 1998. [2] Koschmieder, E.L., Benard Cells and Taylor Vortices, Cambridge University Press, 1993. [3] Clever, R.M. & Busse, F.H., Low-Prandtl-number convection in a layer heated from below, Journal of Fluid Mechanics, 102, 61−74, 1981. [4] Bertin, H. & Ozoe, H., Numerical study of two-dimensional natural convection in a horizontal fluid layer heated from below, by finite-element method: influence of Prandtl number, International Journal of Heat and Mass Transfer, 29(3), 439−449, 1986. [5] Ozoe, H., Ukeba, H. & Churchill, S.W., Numerical analysis of natural convection of low Prandtl number fluids heated from below, Numerical Heat Transfer, Part A, 26, 363−374, 1994. [6] Clever, R.M. & Busse, F.H., Transition to time-dependent convection, Journal of Fluid Mechanics, 65, part 4, 625−645, 1974. [7] Clever, R.M. & Busse, F.H., Convection at very low Prandtl numbers, Physics of Fluids, Series A, 2(3), 334−339, 1990. [8] Ozoe, H. & Hara, T., Numerical analysis for oscillatory natural convection of low Prandtl number fluid heated from below, Numerical Heat Transfer, Part A, 27, 307−317, 1995. [9] Yang, M., Chui, X.-Y., Tao, W.-Q. & Ozoe, H., Bifurcation and oscillation of natural convection in a horizontal layer of low Prandtl number fluid, Journal of Engineering Thermophysics, 21(4), 461−465, 2000 (in Chinese). [10] Wang, J.-G., Yang, M., Zhao, M., Cui, X.-Y. & Zhang, L.-X., Bifurcation of natural convection for low Prandtl number fluid heated from below, Journal of Engineering Thermophysics, 24(1), 76−78, 2003 (in Chinese). [11] Yang, M., Li, X.-H., Tao, W.-Q. & Ozoe, H., Computation and Comparison for heat and flow using a QUICK and other difference schemes, Journal of Engineering Thermophysics, 20(5), 593−597, 1999 (in Chinese). [12] Krishnamurti, R., Some further studies on the transition to turbulent convection, Journal of Fluid Mechanics, 60(3), 285-303, 1973. [13] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere: New York, 1980. [14] Tao, W.-Q., Numerical Heat Transfer, Xi’an Jiaotong University Press: Xi’an, pp. 220−231, 1988 (in Chinese).
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A numerical study of the convective heat transfer between a room and a window covered by a partially open plane blind with a gap at the top P. H. Oosthuizen Department of Mechanical and Materials Engineering, Queen’s University, Canada
Abstract An approximate model of a window covered by a partially open plane blind has been considered. The window is represented by a vertical isothermal wall section which is exposed to a large surrounding room in which the mean temperature is lower than the window temperature. The blind is represented by a thin straight vertical wall which offers no resistance to heat transfer across it. The top of this thin section is aligned with the top of the heated wall section. There is a thin horizontal wall section at the top of the “blind”. This horizontal section does not fully reach to the vertical wall with the result that there is a small gap between the blind system and the vertical wall. The main purpose of this study was to determine the effect of the size of this gap on the heat transfer rate from the “window” to the room. The length of the thin vertical wall section is, in general, less than the height of the window and thus represents a partially open blind. Attention has only been given to the convective heat transfer from the window. The governing equations, written in dimensionless form, have been solved using a commercial finite-element based code. The solution has the following parameters: Rayleigh number, Prandtl number, dimensionless horizontal distance between the window and the blind, dimensionless distance of the bottom of the blind above the bottom of the window and dimensionless size of the gap at the top of the window. Results have only been obtained for a Prandtl number of 0.7. Keywords: heat transfer, convection, windows, shading, blinds, numerical.
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14 Advanced Computational Methods in Heat Transfer IX
1
Introduction
Blinds and other forms of covering can be used to reduce building energy use and improved models for the effects of blinds on window heat transfer are needed to assist in the development of systems that make the maximum use of blinds for this purpose. The situation considered in the present study is an approximate model of a window covered by a partially open plane blind. The present work was undertaken as part of a wider study of the effect of window coverings on the heat transfer rate from windows, particularly for the case where the window is hotter than the room air, i.e., for the case where air-conditioning is being used. The situation considered is shown in fig. 1. In this situation, the window is represented by a vertical isothermal wall section with parallel adiabatic wall sections above and below the heated section. This heated wall section (the “window”) is exposed to a large surrounding room in which the mean temperature is assumed to be known and lower than the window temperature. The plane blind is represented by a thin straight vertical wall which offers no resistance to heat transfer across it and in which conductive heat transfer is negligible. The top of this thin wall section (i.e., of the “blind”) is aligned with the top of the heated wall section (i.e., with the top of the “window. There is a thin horizontal wall section at the top of the “blind” which is thus normal to the “blind”. This horizontal section does not fully reach to the vertical wall with the result that there is a small gap between the blind system and the vertical wall.
Figure 1:
Situation considered. The two limiting cases of a fully open (H=1) and a fully closed (H=0) “blind” are shown on the right.
The main purpose of this study was to determine the effect of the dimensionless size of this gap on the heat transfer rate from the “window” to the room. The length of the thin vertical wall section (i.e., of the “blind”) is, in WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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general, less than the height of the window and thus represents a partially open plane or roller blind. Attention has only been given to the convective heat transfer from the window, i.e., radiative heat transfer and the effects of solar radiation have not been considered. Although the model used here is only an approximation of the real situation, the results obtained with this model will give an indication of the effect of the governing parameters on the convective heat transfer rate from an actual window. The present work was undertaken as part of a wider study of the effect of window coverings on the heat transfer rate from windows particularly for the case where the window is hotter than the room air, i.e. where air-conditioning is being used, see for example Collins et al. [1–3], Machin et al. [4], Shahid et al. [5]. These studies and those described by Duarte et al. [6] and Phillips et al. [7] have concentrated on Venetian blinds. Some studies involving vertical blinds have also been undertaken, e.g., see Oosthuizen et al. [8–10]. Some studies of situations involving plane blinds have been undertaken, e.g., see Oosthuizen [11–13]. However these studies have not considered the effect of a gap between the wall and the top of the window-blind system. The present study, as is the case in many of the previous studies mentioned above, considers only the convective heat transfer. In window heat transfer situations the radiant heat transfer can however be very important and can interact with the convective flow, e.g. see Collins et al. [1] and Phillips et al. [14].
2
Solution procedure
The flow has been assumed to be laminar and two-dimensional. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. The covering over the heated wall section (the “blind”) has been assumed to offer no resistance to heat transfer and to have negligible thickness so that conduction along it is negligible. The effects of radiative heat transfer have been neglected. The governing equations have been written in terms of dimensionless variables using the height, L’, of the heated wall section (the “window”) as the length scale and the overall temperature difference (Tw – Ta) as the temperature scale, Ta being the air temperature in the “room” to which the window is exposed. It has been assumed that the “window” temperature, Tw, is higher than the temperature of the air in the “room”. The resultant dimensionless equations have been solved using a commercial finite-element based code, FIDAP. Only the mean heat transfer rate from the isothermal surface (the “blind”) will be considered here. This has been expressed in terms of a mean Nusselt number, Nu, based on the window height, L’, and on the overall temperature difference (Tw – Ta).
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3
Results
The dimensionless governing equations have the following parameters: 1. 2. 3. 4. 5.
the Rayleigh number based on the window height, L’, and on (Tw – Ta), Ra the Prandtl number, Pr the dimensionless horizontal distance between the window and the blind, W = W’ / L’ the dimensionless distance of the bottom of the blind above the bottom of the window, H = H’ / L’. the dimensionless gap between the horizontal top section of the blind system and the top of the window, G =G’ / L’.
Here, as shown in fig. 1, G’ is the size of gap between the top of “blind system” and “wall”, H’ is the height of bottom of thin wall section (i.e., of the “blind”) above bottom of heated wall section (i.e., of the “window”) and W’ is the distance of thin vertical wall section (i.e., the “blind”) from the heated wall section (i.e., the “window”). Because of the application being considered, results have only been obtained for a Prandtl number of 0.7. Rayleigh numbers of between 10 3 and 10 8, H values of between 0 and 1, W values of between 0.02 and 0.12, and G values of between 0 and W have been considered. An H value of 0 corresponds to a “fullyclosed blind” while an H value of 1 corresponds to a “fully-open blind”, these two cases being shown in fig. 1. Results were also obtained for the no-blind case and the mean Nusselt numbers given for this case were found to be in excellent agreement with values given by empirical equations for the mean natural convective heat transfer rate from an isothermal vertical flat plate. The effect of the dimensionless gap G on the mean Nusselt number for various values of H and for Rayleigh numbers of 105, 106 and 107 is shown in figs. 1, 2 and 3 respectively. These results are all for W = 0.1. It will be seen from these figures that the gap size has the biggest effect on the heat transfer rate when the Rayleigh number is low and the gap size is small. For example, it will be seen from fig. 2 that for a Rayleigh number of 105 the value of G has quite a significant effect on the Nusselt number at all values of G whereas it will be seen from fig. 4 that for a Rayleigh number of 107 the value of G only has a significant effect on the Nusselt number when G is less than greater than about 0.03. This is because at low Rayleigh numbers the thickness of the boundary layer on the “window” is relatively large and significantly greater than the gap size whereas at high Rayleigh numbers the thickness of the boundary layer on the “window” is relatively small and can be significantly less than G with the result that the gap size has only a small effect on the flow and therefore on the heat transfer rate. This is illustrated by the typical streamline patterns shown in fig. 5.
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Figure 2:
Variation of mean Nusselt number with dimensionless gap size G for various values of H for W = 0.1 and Ra =105.
Figure 3:
Variation of mean Nusselt number with dimensionless gap size G for various values of H for W = 0.1 and Ra =106.
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18 Advanced Computational Methods in Heat Transfer IX
Figure 4:
Variation of mean Nusselt number with dimensionless gap size G for various values of H for W = 0.1 and Ra =107.
Figure 5:
Streamline patterns for W = 0.1, H = 0.8 and Ra =107 for G values of from left to right of 0.02, 0.04, 0.06, and 0.08.
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The effect of the dimensionless “blind – window” spacing W on the mean Nusselt number is illustrated by the results given in figs. 6 and 7. It will be seen from these results that as W increases the mean Nusselt decreases but it then passes through a minimum and then increases with increasing W. This form of behaviour arises because when W is small then is very little convective motion in the air between the blind and the window and the heat transfer is mainly by conduction and heat transfer rate decreases as the thickness of the air layer increases. However as W increases further, convective motion begins between the blind and the window and this tends to increase the heat transfer rate and thus leading the minimum in the Nu – W variation. It will also be seen from the results given in figs. 6 and 7 that the value of W at which the minimum heat transfer rate occurs increases with H at the smaller value of H considered. The form of the variation of the Nusselt number with dimensionless blind opening, H, is illustrated by the results given in fig. 8. It will be seen that as the result of the presence of the top gap, the mean Nusselt number does vary linearly with H as is sometimes assumed.
Figure 6:
Variation of mean Nusselt number with dimensionless window-blind spacing, W, for a Rayleigh number of 107 and H = 0.4 for two values of G.
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20 Advanced Computational Methods in Heat Transfer IX
Figure 7:
Variation of mean Nusselt number with dimensionless window-blind spacing, W, for a Rayleigh number of 107 and H = 0.8 for two values of G.
Figure 8:
Variation of mean Nusselt number with dimensionless blind opening H for a Rayleigh number of 107 and W = 0.1 for two values of G.
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21
Conclusions
The dimensionless blind opening, H, has been shown to have a very significant affect on the mean Nusselt number, the results given here allowing an estimate to be made of this effect for various values of Ra and W.
Acknowledgment This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References [1]
[2]
[3] [4] [5]
[6] [7] [8]
[9]
Collins, M., Harrison, S.J., Oosthuizen, P.H. & Naylor, D., Sensitivity analysis of heat transfer from an irradiated window and horizontal louvered blind assembly. American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) Trans., 108, pp. 1-8, 2002. Collins, M.R., Harrison, S.J., Naylor, D., & Oosthuizen, P.H., Heat transfer from an isothermal vertical plate with adjacent heated horizontal louvers: Numerical analysis. J. Heat Transfer, 124(6), pp. 1072-1077, 2002. Collins, M.R., Harrison, S.J., Naylor, D., & Oosthuizen, P.H., Heat transfer from an isothermal vertical plate with adjacent heated horizontal louvers: Validation. J. Heat Transfer, 124(6), pp. 1078-1087, 2002. Machin, A.D., Naylor, D., Oosthuizen, P.H., & Harrison, S.J., Experimental study of free convection at an indoor glazing surface with a Venetian blind. J. HVAC&R Research, 4(2), pp. 153-166, 1998. Shahid, H., Naylor, D., Oosthuizen, P.H., & Harrison, S.J., A numerical study of the effect of horizontal louvered blinds on window thermal performance. Paper SH2. Proc. of the 2nd Int. Conf. on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT), ed. J.P. Meyer, pp. 1-6, 2003 Duarte, N., Naylor, D., Oosthuizen, P.H. & Harrison, S.J., An interferometric study of free convection at a window glazing with a heated Venetian blind. Int. J. HVAC&R Research, 7(2), pp. 169-184, 2001. Phillips, J., Naylor, D., Harrison, S.J., & Oosthuizen, P.H., Free convection from a window glazing with a Venetian blind: Numerical model development. Trans. CSME, 23(1B), pp. 159-172, 1999. Oosthuizen, P.H., Sun, L., & Naylor, D., A numerical study of the effect of normal adiabatic surfaces on natural convective heat transfer from a vertical isothermal plate. Progress in Transport Phenomena, Proc. of the 3rd Int. Symposium on Transport Phenomena, eds. S. Dost, H. Struchtrup, & I Dincer, Elsevier: Paris, pp. 327-331, 2002. Oosthuizen, P.H., Sun, L., & Naylor, D., The effect of inclined vertical slats on natural convective heat transfer from an isothermal heated vertical WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
22 Advanced Computational Methods in Heat Transfer IX
[10]
[11]
[12]
[13]
[14]
plate. Proc. Of the 10th Annual Conf. of the CFD Society of Canada, pp. 515-519, 2002. Oosthuizen, P.H., Sun, L., & Naylor, D., The effect of heat generation in inclined slats in natural convection from an isothermal heated vertical plate. Proc. Of the 11th Annual Conf. of the CFD Society of Canada, pp. 1-8, 2003. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment with heat generation in the attachment. Proc. Of the 1st Int. Conf, on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT), Skukuza, Kruger National Park, South Africa, Vol. 1, Part 2, pp. 702-707, 2002. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment with nonuniform heat generation in the attachment. Progress in Transport Phenomena, Proc. of the 3rd Int. Symposium on Transport Phenomena, eds. S. Dost, H. Struchtrup & I. Dincer, Elsevier: Paris, pp. 29-34, 2002. Oosthuizen, P.H., Natural convection in a square enclosure with a partially heated wall section covered by a blind-like attachment in which there is a linearly varying heat transfer rate, Paper IMECE2002-32968, Proc. On CD of IMECE2002, Vol. 1, 2002. Phillips, J., Naylor, D., Harrison, S.J., & Oosthuizen, P.H., Numerical study of convective and radiative heat transfer from a window glazing with a Venetian blind. Int. J. of HVAC&R Research, 7(4), pp. 383-402, 2001.
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An analytical solution to the Graetz problem with viscous dissipation for non-Newtonian fluids R. Chiba1, M. Izumi2 & Y. Sugano3 1
Miyagi National College of Technology, Japan Department of Mechanical Engineering, Ishinomaki Sensyu University, Japan 3 Department of Mechanical Engineering, Iwate University, Japan 2
Abstract Forced convection heat transfer in a non-Newtonian fluid flow between parallel plates subjected to convective cooling on the external surfaces is investigated analytically. Fully developed laminar velocity distributions obtained by a power law fluid rheology model are used, and viscous dissipation is taken into account. The effect of heat conduction in the direction of fluid flow is considered negligible. The physical properties are assumed to be constant. We approximate the smooth change in the velocity distribution between the plates as a piecewise constant velocity. The theoretical analysis of the heat transfer is performed using an integral transform technique—Vodicka’s method. An important feature of the approach is that an arbitrary distribution of the temperatures of the surrounding media in the direction of fluid flow and an arbitrary velocity distribution of the fluid can be permitted. A comparison with the existing results provides a verification of this technique. The effects of the Brinkman number, Biot number and rheological properties on the distributions of the fluid temperature and the local Nusselt number are illustrated. Moreover, the effects of these parameters on the length of the freeze-free zone are discussed in the case where the temperatures of the surrounding media are below the solidification temperature of the fluid. Keywords: heat transfer, forced convection, non-Newtonian fluid, analytical solution, viscous dissipation, Graetz problem, channel flow.
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24 Advanced Computational Methods in Heat Transfer IX
1
Introduction
An understanding of the convection heat transfer in non-Newtonian fluids inside conduits is of great importance in the design of several types of thermal equipment. From this viewpoint, heat transfer problems of this type have been investigated by a large number of researchers since the 1880s. The problem pertaining to the derivation of the local Nusselt number in the thermal entry region when an incompressible fluid flows through a conduit with a fully developed velocity distribution is of particular interest; this problem is referred to as the Graetz problem. It has attracted the interest of not only engineers but also applied mathematicians because of the difficulties involved in deriving the solution. The Graetz problem is normally solved by using a quasi-analytical method. That is, the separation of variables approach is applied to the governing equation, and only the resultant eigenvalue problem is solved numerically. It is very difficult to obtain an exact solution to the eigenvalue problem except in some special cases. Therefore, a completely explicit analytical solution to the heat transfer problem for a fluid with an arbitrary velocity distribution has not been reported thus far. The objective of this study is to solve mathematically the forced convection heat transfer problem in a conduit between parallel plates subjected to heat loads from the surrounding by Vodicka’s method [1], which is a type of integral transform method, and to derive completely explicit analytical solutions of the fluid temperature and local Nusselt number. Heat conduction in the direction of fluid flow is considered negligible since the present study focuses on heat transfer with a sufficiently large Peclet number. However, viscous dissipation is taken into account. Numerical calculations are used to illustrate the effects of the Brinkman number, Biot number and rheological properties on the distributions of the fluid temperature and local Nusselt number. Moreover, the effects of these parameters on the length of the freeze-free zone are discussed for the case where the temperatures of the surrounding media are below the solidification temperature of the fluid.
2
Analysis
2.1 Analytical model and formulation Figure 1 shows the physical model and coordinate system. A non-Newtonian fluid with a fully developed velocity distribution u(y) flows into a conduit between parallel plates. The fluid temperature at the entrance is T0(y). The conduit of width 2L contacts the surrounding media with temperature T1∞(x) and T2∞(x) at y = L and y = –L, respectively. The heat transfer coefficients on the external surfaces of the plates are h1 and h2. In this study, the flowing fluid has no analytical restriction in its velocity distribution form. With regard to the type of fluid flowing inside the conduit, a power law fluid, which can approximate the non-Newtonian viscosity of many WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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types of fluids with a good accuracy over a wide range of shear rates, is considered here. The shear stress acting on the viscous fluid τyx is given as follows: τ yx = κ
du dy
ν −1
du , dy
(1)
where κ and ν are the power law model parameter and power law model index, respectively. ν < 1 indicates a pseudoplastic fluid, ν > 1 indicates a dilatant fluid and ν = 1 is equivalent to a Newtonian fluid. The fully developed velocity distribution is expressed as follows [2]: 1+ν 2ν + 1 y ν u ( y ) = um 1− , ν + 1 L
(2)
where um is the mean velocity. y
T1∞(x)
h1
u = u(y) T = T0(y) O
2L
x Thin parallel plates
h2
T2∞(x)
Figure 1: Physical model and coordinate system. The following assumptions are introduced: (i) material properties are independent of temperature and are therefore constant, (ii) heat resistance of the parallel plates is negligible, (iii) heat conduction in the direction of fluid flow is negligible, (iv) mode of flow is always laminar. In this case, the steady-state heat balance taking viscous dissipation into account is expressed as follows: ρ cu
∂T ∂ 2T du = λ 2 + τ yx , ∂x dy ∂y
(3)
where ρ, c and λ are the density, specific heat and thermal conductivity, respectively. The boundary conditions are given as follows: T (0, y ) = T0 ( y ) ,
λ
∂T ( x, L) + h1 T ( x, L) − T1∞ ( x) = 0 , ∂y
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(4) (5)
26 Advanced Computational Methods in Heat Transfer IX λ
∂T ( x, − L) − h2 T ( x, − L) − T2∞ ( x) = 0 . ∂y
(6)
Taking the generality of the analysis into account, the expressions of eqns (3)–(6) in dimensionless form yield the following equations: ν +1
U (η )
∂θ ∂ 2θ 1 +ν = + Br ∂ξ ∂η 2 ν θ (0,η ) = θ 0 (η ) ,
η
ν +1 ν
,
(7) (8)
∂θ (ξ ,1) + H1 θ (ξ ,1) − θ1∞ (ξ ) = 0 , ∂η ∂θ (ξ , −1) − H 2 θ (ξ , −1) − θ 2∞ (ξ ) = 0 , ∂η
(9) (10)
+ν 1−ν where U(η) = (2ν + 1)/(ν + 1)[1–|η|(1+ν)/ν], Br = κ u1max L /[λ (T0b − Ts )] , umax = (1 + 2ν)um/(1 + ν), θ = (T–Ts)/(T0b–Ts), η = y/L, ξ = λx/(umL2ρc) and H = hL/λ. Eqn (7) is a partial differential equation with variable coefficients; therefore, it is very difficult to obtain the exact solution. In order to solve eqn (7), we divide the conduit into n regions in the η direction and approximate U(η) as a constant Ui in each region, as shown in fig. 2. In this case, the dimensionless energy equation in the ith region (i = 1, 2, …, n) is obtained as follows:
Ui
∂θi ∂ 2θi = + Q(η ) , ∂ξ ∂η 2
(11)
where Q(η) = Br[(1+ν)/ν]ν+1|η|(1+ν )/ν. The continuous conditions at the imaginary interfaces and boundary conditions are expressed by the following equations: θi (ξ ,ηi ) = θi +1 (ξ ,ηi ) ,
(12)
∂θi (ξ ,ηi ) ∂θi +1 (ξ ,ηi ) = , ∂η ∂η θi (0,η ) = θ0 (η ) ,
(13) (14)
U(η) Region number 1
2
3
i
i+1
n-1 n
U3 U2
U1 η0=-1 η1 η2
ηi
ηn-1ηn=1
η
Figure 2: Virtual division inside the conduit. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX ∂θ n (ξ ,1) + H1 θ n (ξ ,1) − θ1∞ (ξ ) = 0 , ∂η ∂θ1 (ξ , −1) − H 2 θ1 (ξ , −1) − θ 2∞ (ξ ) = 0 . ∂η
27
(15) (16)
2.2 Vodicka’s method The solution to eqns (11)–(16) is obtained as ∞
2
m =1
j =1
θi (ξ ,η ) = ∑ φm (ξ ) X im (η ) + ∑ L j (η ) Pj (ξ ) ,
(17)
P1 (ξ ) = −θ 2∞ (ξ ) , P2 (ξ ) = θ1∞ (ξ ) .
(18)
where
Lj(η) (j = 1, 2) is the solution to the boundary value problem expressed as follows: d2L j dη
2
=0,
dL j (−1) dη
− H 2 L j (−1) = H 2 ⋅ (2 − j ) ,
dL j (1) dη
+ H1L j (1) = H1 ⋅ ( j − 1) . (19)
Xim(η) is the solution to the eigenvalue problem expressed as follows: d 2 X im dX im (ηi ) dX (i +1) m (ηi ) + γ m2U i X im = 0 , X im (ηi ) = X (i +1) m (ηi ) , = , dη dη dη 2 dX 1m (−1) dX nm (1) − H 2 X 1m ( −1) = 0 , + H1 X nm (1) = 0 . dη dη
(20)
Lj(η) and Xim(η) are given as L1 (η ) =
H1H 2η − (1 + H1 ) H 2 H H η + (1 + H 2 ) H1 , L2 (η ) = 1 2 , H1 + H 2 + 2 H1H 2 H1 + H 2 + 2 H1 H 2
(21)
X im (η ) = Aim cos( U i γ mη ) + Bim sin( U i γ mη ) .
(22)
The conditions necessary to determine the unknown coefficients Aim and Bim can be obtained by substituting eqn (22) into the continuous and boundary conditions in eqn (20). Eigenvalues γm (m = 1, 2, ...) are obtained from the condition under which Aim and Bim are both non-zero and are therefore positive roots of the following transcendental equation: G e1 ⋅ Ee1 ⋅ Ee2 " Ee( n−1) ⋅ aen = 0 ,
where G e1 = d1m sin( d1m ) − H 2 cos( d1m ) d1m cos( d1m ) + H 2 sin(d1m ) , dim = U i γ m , cos[d (i +1) mηi ] sin[d (i +1) mηi ] d cos(dimηi ) − sin(dimηi ) Eei = im , −d sin[ d ] d η d sin( d ) cos( d ) η η ( i +1) m i ( i +1) m cos[ d ( i +1) mηi ] (i +1) m im i im i im WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(23)
28 Advanced Computational Methods in Heat Transfer IX d cos(d nm ) + H1 sin(d nm ) aen = nm . d nm sin( d nm ) − H1 cos(d nm )
(24)
By substituting eqn (17) into eqn (14), the following equation is obtained: ∞
2
m =1
j =1
G (η ) = ∑ φm (0) X im (η ) = θ 0 (η ) − ∑ L j (η ) Pj (0) .
(25)
The eigenvalue function Xim(η) obtained from eqns (20) and (22) has an orthogonal relationship with discontinuous weight functions; this is expressed as follows: n
const. (m = k ) . (m ≠ k ) 0
η ∑ U i ∫η X im (η ) X ik (η )dη = i
i −1
i =1
(26)
Lj(η), Q(η) and G(η) can be expanded into an infinite series by Xim(η) as follows: ∞
L j (η ) = ∑ lmj X im (η ) ,
(27)
m =1
∞
Q(η ) = U i ∑ qm X im (η ) ,
(28)
m =1
∞
G (η ) = ∑ g m X im (η ) ,
(29)
m =1
where n
n
η
η
lmj = M m−1 ∑ U i ∫η i L j (η ) X im (η )dη , qm = M m−1 ∑ ∫η i Q(η ) X im (η )dη , i −1
i =1
n
i −1
i =1
n
η
η
g m = M m−1 ∑ U i ∫η i G (η ) X im (η )dη , M m = ∑ U i ∫η i [ X im (η )]2 dη . i −1
i =1
i −1
i =1
(30)
Taking eqns (19) and (20) into account, we substitute eqns (17), (27) and (28) into eqn (11). This yields a first-order linear ordinary differential equation for φm(ξ) as follows: 2 dPj dφm + γ m2 φm = qm − ∑ lmj . dξ dξ j =1
(31)
Solving eqn (31) with the condition φm(0) = gm, which is obtained from the comparison between eqns (25) and (29), we obtain φm(ξ) as φm (ξ ) = g m e−γ mξ + 2
qm
γ m2
ξ
2
dPj (t )
j =1
dt
(1 − e−γ mξ ) − e−γ mξ ∫0 eγ m t ∑ lmj 2
2
2
dt .
(32)
Finally, the temperature solution for the ith region inside the conduit θi(ξ, η) is derived as follows: ∞
θi (ξ ,η ) = ∑ φm (ξ )[Aim cos( U i γ mη ) + Bim sin( U i γ mη )] m =1
[(1 + H1 ) H 2 − H1H 2η ]θ 2∞ (ξ ) + [(1 + H 2 ) H1 + H1H 2η ]θ1∞ (ξ ) + H1 + H 2 + 2 H1 H 2 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
.
(33)
Advanced Computational Methods in Heat Transfer IX
29
The dimensionless bulk temperature of the fluid is given as θ B (ξ ) =
1 1 ∫ U (η )θ (ξ ,η )dη . 2 −1
(34)
Consequently, the local Nusselt numbers at η = 1 and –1 are expressed by the following equations: ∂θ n (ξ ,1) −1 ∂η Nu1 (ξ ) = = = λ θ n (ξ ,1) − θ B (ξ ) θ n (ξ ,1) − θ B (ξ ) h 1L
(35)
H H [θ (ξ ) − θ 2∞ (ξ )] ∞ × ∑ φm (ξ ) U n γ m [Anm sin( U n γ m ) − Bnm cos( U n γ m )] − 1 2 1∞ , H1 + H 2 + 2 H1H 2 m =1 ∂θ1 (ξ , −1) −1 ∂η Nu2 (ξ ) = = = λ θ1 (ξ , −1) − θ B (ξ ) θ1 (ξ , −1) − θ B (ξ ) h 2 L
−
(36)
∞ H H [θ (ξ ) − θ 2∞ (ξ )] × ∑ φm (ξ ) U1 γ m [A1m sin( U1 γ m ) + B1m cos( U1 γ m )] + 1 2 1∞ . H1 + H 2 + 2 H1H 2 m =1
3
Numerical calculation
As a numerical example, we consider the case of θ0(η) = 1, θ1∞(ξ) = θ2∞(ξ) = 0 and H1 = H2 = H. In this case, the local Nusselt numbers of eqns (35) and (36) equate with each other since the temperature field in the conduit is symmetric with respect to the x-axis (ξ-axis). The number of terms in the infinite series in eqn (33) is 500, unless otherwise specified. Note that this value is used under the verification of a sufficient convergence of the numerical results.
4
Results and discussion
4.1 Examination of the number of partitions In order to estimate the accuracy and usefulness of the present analytical solution, which is obtained from the approximation of continuous change as a piecewise constant in the fluid velocity distribution, we first consider the most basic Graetz problem of Br = 0, H = ∞ and ν = 1, or the case in which a Newtonian fluid flows without viscous dissipation between parallel plates that are maintained at a constant temperature. This case has already been analyzed quasi-analytically by Rosales et al. [3]. Figure 3 shows a comparison of the fluid temperature distribution calculated by the present solution with that in reference [3]. A smaller value of ξ has a larger effect of the number of partitions n on the temperature distribution. For n = 5, the temperature distribution at ξ = 10–4 fluctuates widely. The temperature distribution obtained from the present analytical solution with n = 20 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
30 Advanced Computational Methods in Heat Transfer IX is in good agreement with that of reference [3] for all ξ. Table 1 shows a comparison between the local Nusselt numbers of the present solution and reference [3]. For n = 20, the difference between them is less than 0.1% in the range of 10–4 < ξ < 100. Consequently, as long as the local Nusselt number is discussed in this range, n = 20 provides sufficiently accurate results. Note that all of the following results are obtained from the analytical solution with n = 20. Table 1:
Comparison of local Nusselt numbers.
ξ
Number of eigenvalues used
10–4 10–3 10–2 10–1 100
500 200 200 200 200
Nu [Present (n = 20)] 16.661 5 7.749 2 3.692 6 2.047 66 1.885 14
[Rosales et al.] 16.668 7 7.751 3 3.693 4 2.047 82 1.885 17
1 10-4
0.8 n=10,20
0.6 θ
n=5,10,20 n=5 n=10 n=20 Rosales et al. [3]
0.4 0.2 0
n=5,10,20
0
0.2
10-3
10-2 0.1
ξ=1
0.4
0.6
0.8
1
η
Figure 3: Relationship between the number of partitions n and convergence of dimensionless temperature θ. 4.2 Effects of parameters on local Nusselt number distribution Figure 4 illustrates the effects of the viscous dissipation, heat transfer coefficients of the external surfaces and rheological character of the fluid on the local Nusselt number around the entrance of the conduit. The Nusselt numbers at ξ = 10–4 are the largest in figs. 4(a) and (b). For the Brinkman number Br = 0, the Nusselt numbers decrease simply with an increase in ξ and finally converge on the Nusselt numbers in the developed temperature field. On the other hand, for Br ≠ 0, the Nusselt numbers do not necessarily decrease monotonously, and they show a change from a decreasing to an increasing trend, especially for small values of Br. However, they converge to a certain value, regardless of the value of Br. While the Nusselt numbers at ξ = 10–4 increase with a decrease in the Biot number H, the convergent value of the Nusselt number for Br ≠ 0 is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
31
independent of the Biot number, only depending on ν. Smaller values of ν tend to result in higher local Nusselt numbers throughout the conduit. 40
25 H=1
35
15
20
Nu
Nu
25 1
15 10 5 0 -4 10 40
10-3
10-2
10-1
100
0 -4 10 25
101
H=10
0.5
10-3
10-2
10-1
Br=0
100
101
H=10
20
25
15
20
Nu
Nu
1
Br=0
30
15 1
10-3
10-2
10 5
5
10 5
10-1
5
0.5
Br=0
100
0 -4 10 25
101
H=∞
0.5
1
10-3
10-2
Br=0
10-1
100
101
H=∞
20 15
Nu
Nu
10 5
0.5
35
0 -4 10 40 35 30 25 20 15 10 5 0 -4 10
H=1
20
30
5
10
-3
10
-2
ξ
1
10
-1
Br=0
10
10
0
10
1
0 -4 10
(a) ν = 0.3 Figure 4:
5 1
5
0.5
10
-3
10
-2
ξ
10
-1
0.5 Br=0
10
0
10
(b) ν = 3
Nusselt number around the channel entrance.
4.3 Critical fluid velocity diagram The critical fluid velocity for avoiding solidification within the conduit is obtained in the case where the temperatures of the surrounding media are below the solidification temperature of the flowing fluid; this highlights the applicability of the present analytical solution in the field of industry. Figure 5 shows the relationship between the dimensionless solidification temperature and the dimensionless length of the freeze-free zone with different Biot numbers and Brinkman numbers. This figure makes use of the temperature distribution in the direction of the ξ coordinate at η = 1. For Br = 0, it can be seen that θf is asymptotic to 1 as ξf approaches 0 and θf is asymptotic to 0 as ξf approaches infinity. The extent of the approach depends on the value of H. This result is the same as that reported by Sadeghipour et al. [4], who investigated the critical fluid velocity of a pipe flow. For Br ≠ 0, the effect of Br becomes significant as H decreases and ξf increases. The effect of ν on the critical fluid velocity is considerably smaller than that of H or Br.
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1
32 Advanced Computational Methods in Heat Transfer IX 1.5
1.5 Br=0 Br=0.5 Br=1
Br=0 Br=0.5 Br=1
0.1
0.1
1
1
0.5
θf
θf
0.5 1
1
5
0.5
0.5
5
10
10
H=50
0 -4 10
H=50
-3
10
10
-2
ξf
10
-1
0
10
1
10
0 -4 10
10
(a) ν = 0.3 Figure 5:
5
-3
-2
10
-1
ξf
10
0
10
1
10
(b) ν = 3
Variation of freeze-free zone with dimensionless solidification temperature for different Biot numbers and Brinkman numbers.
Conclusions
The forced convection heat transfer problem with viscous dissipation in a conduit between parallel plates subjected to heat loads from the surrounding has been solved mathematically by Vodicka’s method, which is a type of integral transform method, and completely explicit analytical solutions of the fluid temperature and local Nusselt number have been derived. The conclusions obtained through numerical calculations are summarised as follows: (1) With regard to the Graetz problem in the case of a conduit between parallel plates, the number of partitions in the conduit should be over 20 in order to obtain a sufficiently accurate local Nusselt number. (2) The local Nusselt number in the thermal entry region tends to increase with a decrease in the Biot number and power law model index. (3) The effect of the power law model index i.e., velocity distribution of the fluid, on the critical fluid velocity for avoiding solidification is considerably smaller than that of the Biot number or Brinkman number.
References [1] Vodicka, V., Linear heat conduction in laminated bodies. Mathematische Nachrichten, 14(1), pp. 47-55, 1955. (in German) [2] Mikhailov, M.D. & Ozisik, M.N., Unified analysis and solutions of heat and mass diffusion, Dover Publications: New York, pp. 344, 1994. [3] Rosales, M.A. & Frederick, R.L., Semi analytic solution to the Cartesian Graetz problem: results for the entrance region. International Communications in Heat and Mass Transfer, 31(5), pp. 733-740, 2004. [4] Sadeghipour, M.S., Ozisik, M.N. & Mulligan, J.C., Transient freezing of a liquid in a convectively cooled tube. Trans. ASME Journal of Heat Transfer, 104(2), pp. 316-322, 1982.
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Advanced Computational Methods in Heat Transfer IX
33
Heat transfer by unsteady laminar mixed convection in 2-D ventilated enclosures using the vorticity-stream function formulation S. Boudebous & Z. Nemouchi Département de Génie Mécanique, Université Mentouri de Constantine Algéria, Algeria
Abstract In this work, a numerical study is presented of mixed laminar convection in ventilated enclosures. The left vertical wall of enclosure is maintained at a constant temperature greater than that of the fluid at entry, while the other walls are adiabatic. Two cases of ventilation are considered. In the first case, the fluid enters from the bottom left corner and leaves the domain through the right upper corner. In the second case, the fluid enters from the bottom left corner and leaves the domain through the middle of the upper wall. The equations governing the phenomenon are discretised using the finitedifference method. A computer programme is developed to simulate the flow behaviour and the heat transfer in the enclosure. Velocity and temperature fields are obtained. These numerical simulations are performed for a Grashof number of 106, a Prandlt number of 0.7 and three values of the Richardson number (0.5, 5, and 25) Keywords: mixed convection, vorticity-stream function formulation, square enclosure, numerical method.
1
Introduction
An interesting review on natural convection in enclosures can be found in Fusegi and Hyun [1]. Concerning the subject of the present work, numerous investigations have been reported. We can cite as examples: Unsteady 2-D hot water flow for energy extraction from a storage system, Cha and Jaluria [2] with cold water inlet at the bottom of the left vertical wall and hot water exit at the top of the same or opposite wall. Unsteady 2-D flow of air, Raji WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060041
34 Advanced Computational Methods in Heat Transfer IX and Hasnaoui [3] with inlet at the top of the left vertical wall and exit at the top or bottom of the opposite wall. Transient mixed convection, Omri and Ben Nasrallah [4] with air inlet at the top (at the bottom in a second case) of the hot left wall and exit at the bottom (at the top in the second case) of the right cold wall. Air flow with openings at the top of the vertical walls with one or two heat sources of zero thickness imbedded on a vertical board of finite thickness placed on the bottom wall, Hsu and Wang [5]. In the present study, laminar mixed convection flow of air in a ventilated cavity has been investigated numerically. The physical model under consideration and coordinates chosen are depicted in figure 1. Outlet
Outlet Adiabatic wall
Y,V
g
Adiabatc wall
Case A
Hot wall
Adiabatc wall
Hot wall
Case B
X,U
Inlet
Inlet
Adiabatic wall
Xo
Xo
Figure 1:
2
Adiabatic wall
Geometrical configuration.
Governing equations
The flow and heat transfer phenomena to be investigated here are described by the complete Navier-Stokes and energy equations for two-dimensional laminar incompressible flows. The viscous dissipation term in the energy equation is neglected and the Boussinesq approximation is invoked for the buoyancy induced body force term in the Navier-Stokes equations. From the governing equations of mass, momentum conservations, the vorticity-stream function formulation may be obtained by defining the stream function and vorticity, as, respectively, ∂ψ ∂ψ ∂V ∂U − ω= (1) U= V =− ∂Y ∂X ∂X ∂Y Hence, the equations in dimensionless form can be written as follows. Stream function equation ∂ 2ψ ∂X
2
+
∂ 2ψ ∂Y 2
= −ω
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(2)
Advanced Computational Methods in Heat Transfer IX
35
Vorticity-transport equation ∂θ ∂ω 1 ∂ 2 ω ∂ 2 ω ∂ω ∂ω = + + Ri +V +U 2 2 ∂X ∂Y Re ∂X ∂X ∂τ ∂Y
(3)
Energy equation 1 ∂ 2θ ∂ 2θ ∂θ ∂θ ∂θ = + +V +U ∂Y Re Pr ∂X 2 ∂Y 2 ∂X ∂τ
(4)
where Re, Ri and Pr denote, respectively, Reynolds number, Richardson number and Prandlt number. They are defined as Re =
Vo L
Pr =
ν
ν α
Ri =
Gr
Re 2
Gr =
gβ (Tw − To )L3
(5)
ν2
Here Gr is the Grashof number. The other dimensionless parameters are defined as follows:
X =
x L
Y=
y L
U=
u Vo
V=
v Vo
θ=
T − To T w − To
τ=
Vo t L
(6)
where L is the characteristic length of the cavity, Vo is the inlet velocity, ν is the kinematic viscosity, α is the thermal diffusivity, g is the acceleration of gravity, β is the thermal expansion coefficient, Tw is the wall temperature, To is the temperature of the fluid at the entry and t is the time. For the problem geometry, the following boundary conditions are specified:
U = 0. V = 0. θ = 1. Ψ = 0. at X = 0.
U = 0. V = 0.
∂θ = 0. Ψ = Ψ xo ∂X
at
X = 1.
0 ≺ Y ≺ 1.
0 ≺ Y ≺ 1.
U = 0. V = 1. θ = 0. Ψ = − X ω = 0. at Y = 0. 0 ≺ X ≺ XO
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(7)
(8)
(9)
36 Advanced Computational Methods in Heat Transfer IX ∂θ = 0. Ψ = Ψ xo ∂Y
U = 0. V = 0.
U = 0. V = 0.
at Y = 0.
∂θ = 0. Ψ = 0. at ∂Y
∂ 2Φ ∂Y 2
= 0. at
Y = 1.
Y = 1.
X O ≺ X ≺ 1.
(10)
0 ≺ X ≺ 1. − X O
(11)
1. − X O ≺ X ≺ 1.
(12)
where Φ stands for θ , ω , ψ , U and V . The width of the inlet and the outlet Xo is equal to 0.1. In general, the value of vorticity on a solid boundary is deduced from Taylor series expansion of the stream function around the solid point and can be expressed mathematically as
ω wall = −
∂ 2ψ
(13)
∂n 2
where n is the outward drawn normal of the surface. The convective heat transfer from the heated wall can be characterized by an average Nusselt number, Num, defined as Num =
3
1
∂θ
∫ − ∂X 0
dY X =0
Numerical procedure
The governing equations (2-4) along with the boundary conditions (7-13) are solved numerically, employing finite-difference techniques. The Alternating Direction Implicit (A.D.I.) method of Peaceman and Rachford in [6] is used for time marching. The buoyancy and diffusive terms are discretized by using central differencing while the use of a third-order upwind scheme [7] is preferred for convective terms. Convergence of iteration for the stream function solution is obtained at each time step. The resulting set of finite difference equations is then solved by using the Non Linear Over Relaxation (N.L.O.R.) method [8]. All computations are performed using non-uniform grids with denser clustering near the walls where boundary layers develop and high gradients are expected [9]. The solutions were initially tested with mesh sizes of 81 x 81, 101 x 101, 121 x 121 and 151 x 151. It was found that variations in the solution fields were not significant (of the order of 1% in the mean Nusselt number obtained) between
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Advanced Computational Methods in Heat Transfer IX
37
mesh sizes of 121 x 121 and 151 x 151. As a compromise between accuracy and CPU time, the mesh size of 121 x 121 is used for all calculations. The validity of the computer code developed has been checked for the sudden expansion of an oblique velocity field in a cavity [10] and the buoyancy-driven cavity flow [11]. The various sets of results compare very well and are nearly identical, confirming the credibility of the code.
4
Numerical results and discussion
The main characteristics of the flow and energy transport for each Richardson number (Ri) and for each case of ventilation will be shown in the following. The mean Nusselt numbers (Num) are shown as plots versus the time (figure 2). Flow and temperature fields are shown in terms of stream traces, isotherms, velocity and temperature profiles (figures 3-6). 40
40 Ri=0.5
Case A
Ri=0.5
Case B
Ri=5.0
Ri=5.0
Ri=25.0
Ri=25.0
Num
30
Num
30
20
20
10 0E+0
2E+5
4E+5
6E+5
Figure 2:
8E+5
10 1E+6 0E+0
2E+5
4E+5
6E+5
8E+5
1E+6
Mean Nusselt numbers versus time.
In both cases A and B, figure 2 shows that as Ri increases, Num decreases implying that the more the forced convection is dominant, the more important is the heat flux from the hot wall. Figure 3 shows the stream traces and the profiles of the velocity components U and V in the X and Y directions respectively. As Ri increases, a boundary layer forms along the hot vertical wall. The relatively fast circulation in the center of the enclosure for Ri=0.5 becomes slower for Ri=5 and 25. Consistently with this result, the isotherms and the temperature profiles (figure 4) show that an essentially conductive heat transfer for Ri=25 takes over a forced convection for Ri=0.5. It is clear that in case B the effect of Ri on the flow and the heat transfer is very similar to that in case A (figures 5 and 6)
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38 Advanced Computational Methods in Heat Transfer IX 1.0
0.9
0.9
0.8
0.8
0.7
0.7
X=0.25
0.6
0.6
X=0.50
0.5
0.5
X=0.75
Y
1.0
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
V
0.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X
-0.1 -0.2
Y=0.25
-0.3
Y=0.50
-0.4
Y=0.75 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U
Ri=0.5 1.0
1.0
0.9
0.9
0.8
0.8
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0.7
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Y
V
1.1
0.4
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0.0
-0.1 -0.1
Ri=5
0.0
0.1
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0.6
0.7
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0.9
1.0
0.0
0.1
0.2
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0.5 U
0.6
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0.8
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1.0
1.1
1.5 1.4 1.3 1.2 1.0
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0.7
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V
1.1
0.4
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0.6
0.7
-0.2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 U
Ri=25 Figure 3:
Stream traces (left) and Velocity profiles (right) case A.
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Advanced Computational Methods in Heat Transfer IX
39
1.0 X=0.25 0.9
X=0.5 X=0.75
0.8 0.7
Y
0.6 0.5 0.4 0.3 Y=0.25 0.2
Y=0.5 Y=0.75
0.1 0.0
Ri=0.5
0.0
0.1
0.2
0.3
0.4
0.5 X
0.6
0.7
0.8
0.9
1.0
θx
1.0 0.9 0.8 0.7
θy
Y
0.6 0.5 0.4 0.3 0.2 0.1 0.0
Ri=5
0.0
0.1
0.2
0.3
0.4
0.5 X
0.6
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0.8
0.9
1.0
0.0
0.1
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0.5 X
0.6
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0.8
0.9
1.0
1.0 0.9 0.8 0.7
Y
0.6 0.5 0.4 0.3 0.2 0.1 0.0
Ri=25 Figure 4:
Isotherms (left) and Temperature profiles (right) case A.
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V
40 Advanced Computational Methods in Heat Transfer IX 1.0
0.9
0.9
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0.8
0.7
0.7
0.6
0.6
0.5
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Y
1.0
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0.0
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X
-0.1 -0.2 -0.3
-0.4 -0.6-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U
Ri=0.5
1.1 1.0 0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
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Y
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0.6
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1.0
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U
Ri=5 1.6 1.5 1.4 1.3 1.2 1.0
0.9
0.9
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Y
V
1.1 1.0
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-0.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X
-0.2 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 U
Ri=25 Figure 5:
Stream traces (left) and Velocity profiles (right) case B.
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Advanced Computational Methods in Heat Transfer IX
41
1.0 0.9 0.8 0.7
Y
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
Ri=0.5
0.1
0.2
0.3
0.4
0.5 X
0.6
0.7
0.8
0.9
1.0
θx 1.0 0.9 0.8 0.7
Y
θy0.6 0.5 0.4 0.3 0.2 0.1 0.0
Ri=5
0.0
0.1
0.2
0.3
0.4
0.5 X
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5 X
0.6
0.7
0.8
0.9
1.0
1.0 0.9 0.8 0.7
Y
0.6 0.5 0.4 0.3 0.2 0.1 0.0
Ri=25 Figure 6:
Isotherms (left) and Temperature profiles (right) case B.
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42 Advanced Computational Methods in Heat Transfer IX
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Fusegi, T. & Hyun, J.M., Laminar and transitional natural convection in an enclosure with complex and realistic conditions, Int. J. Heat Fluid Flow. 3, pp. 258-268, 1994 Cha, C. K. & Jaluria, Y., Recirculating mixed convection flow for energy extraction. Int. J. Heat Mass Transfer. 27, pp. 1801-1812, 1984 Raji, A. & Hasnaoui, M., Correlations en convection mixte dans des cavités ventilées. Rev. Gen. Therm. 37, pp. 874-884, 1998 Omri, A. & Ben Nasrallah, S., Control volume finite element numerical simulation of mixed convection in an air-cooled cavity. Numerical Heat Transfer, Part A, vol.36, pp. 615-637, 1999. Hsu, T. H. & Wang, S. G., Mixed convection in a rectangular enclosure with discrete heat sources. Numerical Heat Transfer, Part A, vol.38, pp. 627-652, 2000. Peaceman, D.W. & Rachford, H.H., Numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3, pp. 28-41, 1955. Kawamura, T. & Kawamura, K., Computations of high Reynolds number flow around a circular cylinder with surface roughness, AIAA Paper No 84-0340, 1984. Sewell, G., The numerical solution of ordinary and partial differential equations, Academic Press, INC, New-York, 1988. Manole, D.M. & Lage, L.L., Nonuniform grid accuracy test applied to the natural convection flow within a porous medium cavity, Numerical Heat Transfer, Part B,vol.23,pp. 351-368, 1993. Song, B., Liu, G.R. & Lam, K.Y., Four-point interpolation schemes for convective fluxes, Numerical Heat Transfer, Part B,vol.35,pp. 23-39, 1999. Rahman, M.M., Miettinen, A. & Siikonen, T., Modified SIMPLE formulation on a collocated grid with an assessment of the simplified Quick scheme, Numerical Heat Transfer, Part B,vol.30,pp. 291-314, 1996.
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Advanced Computational Methods in Heat Transfer IX
43
Effect of roughness shape on heat transfer and flow friction characteristics of solar air heater with roughened absorber plate A. Chaube1, P. K. Sahoo2 & S. C. Solanki2 1 2
GEC, Jabalpur, India IIT Roorkee, India
Abstract A 2-D computational analysis is carried out to assess the comparative performance of the absorber plate of a solar air heater with different roughness elements using commercial software package Fluent 6.1. The assessment is based on heat transfer enhancement with minimum pressure penalty. Ten different rib shapes (viz. Rectangular, Square, Chamfered, Triangular, Semicircle etc) are investigated at the Reynolds number range from about 3000–20000, in which solar air heaters normally operate. The SST kω turbulence model is selected by comparing the predictions of different turbulence models with experimental results available in the literature. Using the selected turbulence model a computational analysis is carried out to predict the heat transfer performance and flow friction characteristics of absorber plates with 10 different rib roughness elements. The analysis is carried out for both the flow regimes i.e. transitional flow regime (5 ≤ e+ ≤ 70) and fully rough regime (e+ ≥ 70). Keywords: turbulence, roughness Reynolds number, aspect ratio, friction factor, heat transfer coefficient, Nusselt number, chamfer angle, pitch etc.
1
Introduction
Artificial roughness up to laminar sub-layer to enhance heat transfer coefficient is used in various applications like gas turbine blade cooling channels, heat exchangers, nuclear reactors and solar air heaters. A number of experimental studies [1, 2] in this area have been carried out but very few attempts of numerical investigation have been made so far due to complexity of flow pattern and computational limitations. In the present work, an attempt is made to predict WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060051
44 Advanced Computational Methods in Heat Transfer IX numerically, the details of both the velocity and temperature fields responsible for heat transfer enhancement. The presence of rib may enhance heat transfer because of interruption of the viscous sub layer, which yields flow turbulence, separation and reattachment leading to a higher heat transfer coefficient. The enhancement of heat transfer by flow separation and reattachment caused by ribs is significantly higher compared to that by the increased heat transfer area due to ribs (fin-effect) [3]. The heat transfer measurements results for two different steps, p/e = 14 and p/e = 8, indicate the importance of roughness geometry [4]. Liou et al. [6] have performed both the numerical analysis and experimental study to investigate the heat transfer and fluid flow behavior in a rectangular channel flow with stream wise periodic ribs mounted on one of the principal walls. They have concluded that the flow acceleration and the turbulence intensity are two major factors that influence the heat transfer coefficient. The combined effect is found to be optimum for the pitch to rib height ratio equal to 10, which results in the maximum value of average heat transfer coefficient. Rau et al. [5] experimentally found optimum pitch to rib height ratio to be equal to 9. Hence these investigations reveal that not only the rib geometry but also its geometrical arrangement play vital role in enhancing the heat transfer coefficient. Karwa [7] has reported an experimental investigation for the same configuration for the Reynolds number range of 4000-16000. Tanda [9] has reported experimental investigation of heat transfer in a rectangular channel with transverse and V-shaped broken ribs using liquid crystal thermography. He concluded that features of the inter-rib distributions of the heat transfer coefficient are strongly related to rib shape and geometry; a relative maximum is typically attained down stream of each rib for continuous transverse ribs (due to flow reattachment). The main aim of the present analysis is to investigate the flow and heat transfer characteristics of a 2 dimensional rib roughened rectangular duct with only one principal (broad) wall subjected to uniform heat flux by making use of computer simulation. The ribs are provided only on the heated wall. The other three walls are smooth (without ribs) and insulated. Such a case is encountered in solar air heaters with artificially roughened absorber plate.
2
Solution domain
The solution domain shown in Fig. 1 has been selected as per the experimental details given by Karwa [7]. A rectangular duct with the duct height(H) of 40 mm, rib height (e) of 3.4 mm, rib width of 5.8 mm and pitch (p) of 34 mm has been taken for analysis. The uniform heat flux of 4 kW/ m2 is given on ribbed surface. A 2-D CFD analysis of heat transfer and fluid flow through a high aspect ratio (7.5) rectangular duct with transverse ribs provided on a broad, heated wall and other three walls smooth and insulated, is carried out using Fluent 6.1 software. A non-uniform rectangular mesh with grid adoption for y+=1 at adjacent wall region is applied as shown in Fig. 2. Similar analysis is carried out for a smooth duct of same dimensions for similar range of Reynolds number 3000 to 15000 to
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Advanced Computational Methods in Heat Transfer IX
45
find out the ratio of Stanton number for ribbed duct and smooth duct as experimentally determined by Karwa [7].
Figure 1:
Duct geometry and solution domain as per experimental details of Karwa [7].
Figure 2:
Rectangular mesh with grid adoption for y+=1 at top and bottom wall.
2.1 Selection of turbulence model To select the turbulence model, the previous experimental study is simulated using different low Reynolds number models such as Standard kω model, Renormalization-group kε model, Realizable kε model and Shear stress transport kω model. The results of different models are compared with experimental results. The shear stress transport kω model is selected on the basis of its closer results to the experimental results as shown in Fig. 3. 2.2 Thermo hydraulic assessment of different roughness elements After finding out satisfactory simulation capability of the SST kω turbulence model, the performance of different rib shapes as shown in Fig.4 have been WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
46 Advanced Computational Methods in Heat Transfer IX assessed. The similar flow domain used for the predictions is as shown in Fig.1. The other parameters used are as follows (for transitionally rough flow regime): Reynolds Number =2900 to e+ Range = 7 to 84 Range 19500 Duct height (H) =20 mm Pitch (p) = 10 mm Inlet length =225 mm Rib Height (e) = 1 mm Length of test section =121 mm Aspect Ratio of Duct = 5 (AR) Outlet length =115 mm Uniform Heat Flux = 1000W/m2
2.3 For fully rough flow regime e+ Range
= 35 to 400
Rib Height
= 3 mm
Turbulence Model Selection 0.006 0.0055 Stanton No.
0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0
5000
10000 15000 20000 25000 Re
Experimental Results[7] Standard kω Realizable kε Figure 3:
Shear Stress Transpot kω Renormalization-group kε
Comparison between experimental and computational predictions from different low Re turbulence models.
2.4 Boundary conditions The following boundary conditions are given through the boundary conditions panel: (i) Velocity at inlet (ii) Turbulence intensity at inlet (20% taken from literature) WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
(iii) (iv) (v) (vi)
Figure 4:
3
47
Hydraulic diameter of duct Out let pressure (Atmospheric pressure) Constant heat flux on broad bottom surface of test section No heat transfer from other walls of the duct.
Different roughness elements under investigation.
Results and discussion
The comparative performance of different roughness elements are obtained on the basis of heat transfer enhancement at constant pumping power requirement or performance index as described below:
η=
St Sto
( ) f fo
1
3
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48 Advanced Computational Methods in Heat Transfer IX Performance index Vs Re in Transitionally Rough Flow Regime 1.5 1.4
Performance Index
1.3 1.2 1.1 1 0.9 0.8 0
5000
10000
15000
20000
25000
Re Square 1x1
Rectangular1.5x1
Rectangular2x1
Rectangular2.5x1
Chamfered 9
Chamfered 11
Chamfered 13
Chamfered 15
Semicircular r=1
Triangular h=1, b=1
Figure 5:
Variation of heat transfer enhancement for constant pumping power requirement with Reynolds number.
Performance index Vs Re in Fully Rough Regime 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0
5000
10000
15000
20000
25000
Re
Rectangular 2x3
Square 3x3
Rectangular 4x3
Rectangular 5x3
Chamfered 11
Chamfered 13
Chamfered 15
Chamfered 17
Semicircular r=3
Triangular h=3, b=3
Figure 6:
Variation of heat transfer enhancement at constant pumping power with Reynolds number.
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The comparison of the performance by different rib geometries for equal pumping power would be more useful for engineering applications. The performances of different roughness elements on the basis of equal pumping power for transitionally rough flow regime and fully rough flow regime are shown in Figs. 5 and 6 respectively. In transitionally rough flow a substantial enhancement in heat transfer is found along with increase in friction factor, but in fully rough flow a marginal enhancement in heat transfer over that in the transitionally rough flow is found with sharp increase (more than double in comparison to transitionally rough flow regime) in friction factor. Therefore a low value of performance index is found in fully rough flow regime in comparison to transitionally rough flow regime. In the transitional flow regime rectangular rib of 1 mm height and 2 mm width shows best performance among all rib shapes under investigation. Among the rest of ribs, the triangular rib of 1mm height and 1 mm base gave the best performance. In the fully rough flow regime rectangular rib of 3 mm height and 4 mm width is found as best performer under constant pumping power conditions. Although the square rib (3x3) gives the best heat transfer characteristics for constant mass flow rate condition. The effects of various flow and roughness parameters on heat transfer and friction characteristics for flow of air in a rectangular duct of aspect ratio 5 under the present investigation are being discussed below: 3.1 Variation of Nusselt number with Reynolds number In all cases Nusselt number increases with the increase of Reynolds number. The rate of increase of Nusselt number with Reynolds number is substantially higher in roughed duct in comparison to that of in smooth duct of similar dimension. The effect of roughness on variation of Nusselt number with the Reynolds number may be explained as under: At low Reynolds number in transitionally rough flow the roughened surface Nusselt numbers are nearly those of smooth surfaces. It is because the roughness elements lie within the laminar sub layer, which is the major component of the heat transfer resistance. As the Reynolds number increases, the roughness elements begin to project beyond the laminar sub layer because the boundary layer thickness decreases with an increase in Reynolds number. This reduction in boundary layer thickness increases the heat transfer rate. In addition to this, there is local contribution to the heat removal by the vortices originated from the roughness element. Thus the Nusselt number curve deviate from the smooth duct turbulent Nusselt number curve. When the Reynolds number further increases, the roughness elements project deeper in to the turbulent region. Finally, with the increase of Reynolds number, the thickness of laminar sub layer becomes very small and energy loss due to the vortices now attains a constant value and is independent of viscous effect.
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50 Advanced Computational Methods in Heat Transfer IX 3.2 Variation of friction factor with Reynolds number The shedding of vortices also causes additional loss of energy resulting in increased friction factor. Thus the friction factor curves deviate from the smooth duct friction factor curve. Except semicircular rib, in all the cases friction factor increases with increase of Reynolds number in both the flow regimes. In almost all the cases friction factor becomes independent of Reynolds number at high flow rates. The Reynolds number corresponding to the start of the region of independence of friction factor is termed as critical Reynolds number. 3.3 Effect of rib width In the transitionally rough flow regime increasing the width of square rib enhancement of heat transfer and reduction in friction factor is observed. Where as in fully rough flow regime also the best performance index is found for 4x3 rib. Hence for every flow condition there is an optimum w/e ratio. In case of transitionally rough flow 2x1 rib gives the best performance index. The further increase in width gives negligible or adverse effect. 3.4 Effect of chamfer angle It is observed that both the flow regimes, transitionally rough and fully rough, the effect of chamfer angle in enhancement of heat transfer is very small in comparison to the enhancement of friction factor. That is why chamfered rib roughness did not exhibit the comparable performance index with other types of roughness.
4
Conclusions
The following conclusions are drawn from the present analysis: The Shear Stress Transport kω turbulence model predicted very close results to the experimental results, which yields confidence in the predictions done by CFD analysis in the present study. In transitionally rough flow a substantial enhancement in heat transfer is found along with increase in friction factor, but in fully rough flow a marginal enhancement in heat transfer over that in the transitionally rough flow is found with sharp increase (more than double in comparison to transitionally rough flow regime) in friction factor. The reattachment point and point of maximum heat transfer coincide, which shows the great influence of reattachment of flow on convective heat transfer coefficient. The results show that in rectangular ribs, there is an optimum width of rib, at which it gives maximum heat transfer with minimum pressure drop penalty.
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In continuous transverse ribs chamfering is not much effective because although it gives small enhancement in heat transfer but at the cost of steep rise in friction factor. The thermohyraulic analysis for constant pumping power shows that rectangular rib (1mm x 2mm) gives the best performance in comparison to other shapes under consideration.
References [1]
[2] [3] [4] [5] [6]
[7]
[8] [9] [10] [11]
Muluwork, K.B., Solanki, S.C., and Saini, J.S., 2000, “ Study of heat transfer and friction in solar air heaters roughened with staggered discrete ribs,” Proc. 4th ISHMT- ASME Heat and Mass Transfer Conf., Pune, India, pp.391-398. Karwa, R., Solanki, S.C., and Saini, J.S., 2001, “ Thermo-hydraulic performance of solar air-heaters having integral chamfered rib roughness on absorber plates,” Energy 26, 161-176 (2001). Lee, C.K., and Abdel-Moneim, S.A., 2001, “ Computational Analysis of Heat Transfer in Turbulent Flow Past a Horizontal Surface with 2-D Ribs,” Int. Comm. Heat Mass Transfer, Vol. 28 No.2, pp. 161-170. Slanciauskas, A., 2001, “ Two friendly rules for the turbulent heat transfer enhancement” Int. Jr. of Heat and Mass transfer 44, 2155-2161. Rau, G., Cakan, M., Moeller, D., and Arts, T., 1998, “The Effect of Periodic Ribs on the Local Aerodynamic and Heat Transfer Performance of a Straight Cooling Channel,” ASME Vol. 120, pp. 368-375. Tong-Miin Liou, Jenn-Jiang Hwang and Shih-Hui Chen, “Simulation and measurement of enhanced turbulent heat transfer in a channel with periodic ribs on one principal wall” Int. Jr. Heat Mass Transfer, Vol. 36, pp. 507-517 (1993). Karwa, R., 2003, “ Experimental Studies of Augmented Heat Transfer and Friction in Asymmetrically heated Rectangular Ducts with Ribs on the Heated Wall in Transverse, Inclined, V-Continuous and V-Discrete Pattern,” Int. Comm. Heat Mass Transfer, Vol. 30, No. 2, pp. 241-250. Fluent 6.0 User’s Guide Vol. 2. Tanda, G., “ Heat transfer in rectangular channels with transverse and Vshaped broken ribs” Int. Jr. Heat Mass Transfer, Vol. 47 pp. 229-243, (2004). Davidson, Lars, 1997 “An Introduction to Turbulence models,” Chalmers publication. 11.Versteeg, H.K., and Malalasekera, W., 1995, “An Introduction to Computational Fluid Dynamics”, Longman publication.
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Advanced Computational Methods in Heat Transfer IX
53
Study of conjugate heat transfer accompanying mixed convection in a vertical tube submitted to a step of entry temperature O. Kholai1, 2, M. Kadja2 & T. H. Mai1 1
Laboratoire de Thermomécanique, Faculté des sciences, Reims, France Département de Génie Mécanique, Université de Constantine, Constantine, Algeria 2
Abstract The paper presents the results of a numerical study on transient mixed laminar convection for the case of an ascending flow inside a vertical tube in which the wall conduction is significant. The outer surface of the tube is submitted to a convective heat exchange. The transient regime is provoked by a step type perturbation in temperature at entry of the tube. The governing equations are solved using a finite volume method. Results are presented for water (Pr=5) and for two values of Grashof numbers of 104 and 105. This method shows that no matter what the value of the imposed step is, it provokes the birth of a big recirculation cell next to the wall, and gives rise to an instability situated between the cool and the hot regions of the flow. Keywords: transient convection, temperature step, convective heat exchange, finite volume method, instabilities.
1
Introduction
Conjugate heat transfer (mixed convection + conduction) in transient regime in internal flows is present in many industrial installations: such as compact heat exchangers, solar collectors, cooling of electronic components etc. As a matter of fact, the importance and complexity of this phenomenon and the consequences have stimulated the interest of many researchers: so that a very important number of studies have been done in this field. Nguyen et al. [1] performed a numerical study on transient mixed convection in vertical tubes having a wall, which is submitted to variable heat flux. Their WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060061
54 Advanced Computational Methods in Heat Transfer IX results indicate that the increase in heat flux gives rise to a recirculation zone, which increases in intensity and size as a function of time. Experimentally Morton et al. [2] have confirmed the existence of a recirculation cell near the tube axis. For conjugate heat transfer problems (conduction in the wall and convection in the fluid, inside a tube), Bilir and Ates [3] performed a parametric study in a vertical tube submitted to convective exchange at the outer surface. Bernier and Baliga [4, 5] studied the problem of mixed convection by taking into account wall conduction for an ascending flow where the tube is submitted to a uniform heat flux. They have shown that axial diffusion of heat through the tube wall becomes very big for high conductivity and thickness of the wall and for a low Peclet number. Recirculation zones and instability have been shown to occur by Zgal et al. [6] and Su and Chung [7]. More recently Mai et al. [8], Popa and Mai [9] and Mai et al. [10] have focussed attention on a more complex problem of heat transfer instability. They have shown that a perturbation of mixed convection in a vertical tube via a step rise in temperature (of mass flow rate) at entry, gives rise to instabilities of thermo-hydraulic flow structure and to the creation of recirculation cells inside the fluid. The present study consists in analysing the phenomenon of transient instability of a flow in mixed laminar convection in a circular vertical duct when the entry and the outer surface of the duct are submitted to a step change in temperature (positive or negative) and to a convective exchange with ambient air, respectively. The influence of these two conditions and also the ratio of thermal diffusivities (wall/fluid) on the transient behaviour of the flow are studied in detail.
2
Problem formulation
The considered problem consists in studying conjugate heat transfer for a laminar flow inside a vertical tube submitted to convective exchange with air through a section of the outer wall. The wall thickness is taken into consideration and is equal to δ=0.1D. Adiabatic sections have been added upwind and downwind of the cooled section so as to permit the study of longitudinal thermal diffusion in the fluid and the wall. The configuration and the coordinate system of the tube are given in Fig. 1. The governing equations of this problem are: the continuity equation, the Navier-Stokes equations and the equation of energy. The study is based on the following simplifying assumptions: -The fluid is Newtonian and incompressible -The thermo-physical properties of the fluid and the solid are constant -The fluid obeys the Boussinesq approximation: density is constant except in the natural convection generating terms (density varies linearly with temperature) -The flow is laminar and in transient regime -Viscous dissipation and radiation heat transfer are negligible. By taking these assumptions into account and adopting the following dimensionless variables:
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Advanced Computational Methods in Heat Transfer IX
R=
r , D
Z=
z , D
U=
55
T−T u v 0 , V= , θ= U U T −T av av in 0
p + ρ gz t α 0 , τ= , and α * = 2 D U α ρ U av f 0 av where u and v are the velocity components in the z and r directions respectively, T is the temperature, p is the pressure, D is the diameter of the tube, t is the time, Uav is the mean axial velocity at inlet. g is the acceleration due to gravity. T0 is the ambient temperature. ρ0 is the fluid density at T0. The index ‘in’ stands for inlet and ‘f’ for fluid. The governing equations can be written in the following form in cylindrical coordinates: ∂U 1 ∂RV + (1) =0 ∂Z R ∂R P=
(
)
∂U ∂U ∂U ∂P 1 ∂ 2 U 1 ∂ ∂U Gr +U +V =− + + θ R + ∂Z ∂R ∂Z Re ∂Z 2 R ∂R ∂R Re 2 ∂τ
(2)
∂V ∂V ∂V ∂P 1 ∂ 2 V 1 ∂ ∂V V + +U +V =− + R − ∂τ ∂Z ∂R ∂R Re ∂Z 2 R ∂R ∂R R 2
(3)
∂ 2 θ 1 ∂ ∂θ (4) R 2 + R ∂R ∂R ∂Z where U, V, P, θ are the adimensional variables (adimensional velocity components, adimensional pressure and adimensional temperature), R and Z are the adimensional coordinates and Re, Pr, Gr and α* are the dimensionless parameters controlling the problem: U D Reynolds number Re = av ν ν Pr = Prandtl number α gβ ∆T D 3 Gr = Grashof number based on the heat flux ν2 α α* = Ratio of thermal diffusivities solid/fluid αf where β is the thermal expansion coefficient and ν is the kinematic viscosity of the fluid. Initially (i.e. at τ=0), the common boundary conditions are based on a parabolic profile of velocity and a uniform temperature profile at entry, a symmetry of the hydrodynamic and temperature fields with respect to the tube axis and a completely developed profile at exit. ∂θ ∂θ ∂θ α* +U +V = ∂Z ∂R RePr ∂τ
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56 Advanced Computational Methods in Heat Transfer IX On the other hand the thermal condition on the wall (r=D/2+δ) is: ∂θ =0 If Z
If
Z>(L1+L2 ) then
wI wZ Adiabatic . L3
Axis
Figure 1:
3
Geometric configuration of the studied case.
Numerical procedure
The equations presented above have been integrated and discretized on a staggered grid following the finite volume approach explained in [11]. We have chosen the SIMPLE algorithm in order to solve for pressure. The power law-differencing scheme has been employed to calculate the convective fluxes in the transport equations for momentum and energy . Temporal discretization has been achieved using an implicit unconditionally stable scheme. The physical domain (solid + fluid) has been treated as a heterogeneous medium, the kinematic viscosity being supposed equal to infinity in the solid region. At the interface, the thermal diffusivity was evaluated using the harmonic mean between that of the fluid and that of the solid [11]. The resulting algebraic equations have been solved using the iterative line-by-line method associated with the tri-diagonal matrix algorithm (TDMA). Convergence is considered to be reached when the maximum relative change in velocities and temperatures WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
57
during two consecutive time increments is less than 10-4. On the basis of many tests, the mesh adopted is non-uniform and consists of 102 nodes in the axial direction and 38 nodes in the radial direction, from which 31 nodes are in the fluid and 7 nodes in the solid wall.
4 Results and discussions In this study, the results obtained correspond to the mixed convection of water (Pr=5) in a vertical tube submitted on one hand to a convective exchange and on the other hand perturbed at entry via a step change in temperature (positive or negative). Many numerical simulations have been done. They concern the influence of the step change in temperature at entry on the stability of the thermal and hydrodynamic fields for two different Grashof numbers: 104 and 105, and also the influence of the thermal diffusivity ratio (wall/fluid) on the temperature of the interface. 4.1 Validation The results of the model used in the present study have been compared with the corresponding analytic results obtained by Kakac and Yener [12]. Fig. 2 illustrates the axial evolution of the Nusselt number Nuz for the case of forced convection (Gr=0) in a tube submitted to a uniform heat flux. One can notice a very good accord between the model and the analytic solution. 24 Analytical Solution Used Model
20
Nu
16 12 8 4 0
0
20
40
60
80
100
120
140
160
Z
Figure 2:
Validation of the model for an average Nusselt number (forced convection).
4.2 Effect of the thermal diffusivities ratio The effect of the thermal diffusivity ratio α* (wall/fluid) on the axial evolution of the interfacial temperature is presented in Fig. 3. One can notice that the increase of this ratio improves the heat transfer by conduction through the wall, which causes a remarkable decrease of the temperature at the interface. The value of the latter tends to that of the ambient temperature when the ratio becomes equal to infinity. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
58 Advanced Computational Methods in Heat Transfer IX 1,0 0,8 *
α =0,1
0,6
*
α =1
Tint
*
α =10
0,4
*
α =100
0,2 0,0 -0,2
0
1
2
3
4
5
6
7
8
9
10
Z
Figure 3:
Effect of the ratio of thermal diffusivities on the temperature at the interface.
4.3 Structure of the thermal and flow fields Fig. 4(a) shows the variation of the dynamic and thermal fields as a function of the increase in the temperature step for a given time (τ=200) and a Grashof number of 104. One can notice the presence of a recirculation zone stuck on the wall. This zone is due to the increase in ascending forces caused by natural convection after having perturbed the flow with a positive temperature step at entry. The appearance of this zone is due to the fact that the flow is accelerated in the central region of the tube and decelerated next to the wall. In effect, due to reheating, a density differential establishes itself inside the fluid, the density of the fluid situated in the central zone diminishes with respect to that near to the wall (Boussinesq hypothesis). As the temperature step increases, the recirculation cell intensifies and its size becomes bigger and ends by occupying the totality of the tube. In Fig. 4(b), corresponding to the thermal field, one can notice a stratification of the isotherms. They are thinner near to the entry of the tube and become larger and larger as one approaches the fluid exit. On the wall one can notice that the radial temperature gradient is negligible due to the importance of the thermal diffusivity of the material with respect to that of the fluid. On the other hand one can clearly see a significant axial temperature gradient, as shown by the isotherms, which are parallel. The increase of the temperature step causes a more condensed stratification and a high heat transfer between the middle of the tube and the wall. The transient behaviour of the flow is shown in Figs. 5 and 6 for the case of a positive step of 0.6 (reheating) or a negative step of -0.6 (cooling). If a negative step is imposed at entry, one notices that the intensity of the recirculation zone becomes weaker as time increases. This causes a decrease of the important fluid current at the centre of the tube. The dynamic instability equally entrains a WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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59
thermal disequilibrium inside the flow. This is due to the interaction between the fluid cooled next to the wall and the fluid heated next to the tube axis. This instability is clearly remarked for the times τ= 12, 19, 29 and 39.
(a) Figure 4:
Effect of the temperature step on the streamfunction and the isotherms, Gr=104, ∆θ=0.2, 0.4, 0.6, 0.8, 1.
(a) Figure 5:
(b)
Evolution of the dynamic field (a) and thermal field (b) as a function of time τ=0, 12, 19, 29, 39 and 130 for Gr=105 and ∆θ=-0.6.
(a) Figure 6:
(b)
(b)
Evolution of the dynamic field (a) and thermal field (b) as a function of time τ=0, 8, 16, 24, and 39 for Gr=105 and ∆θ=0.6.
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60 Advanced Computational Methods in Heat Transfer IX After a sudden perturbation of the flow via a positive step (reheating) of fig. 6, the fluid undergoes a local acceleration at entry and especially near the axis of symmetry, consisting of the reverse current next to the tube wall so as to obey the law of conservation of mass. This can be explained by the fact that convective exchange applied to the outer surface cools the entire region neighbouring the wall. On the other hand, the central region of the tube is heated due to the temperature step. The increase of time τ=0, 8, 16, 24 and 39 shows that the region touched by the instability is situated between the cooled region and the heated one inside the fluid.
5
Conclusion
In this paper a numerical study has been performed of the transient behaviour of a laminar flow inside a vertical tube submitted to a sudden change of temperature at entry. An implicit method has been used to solve the governing equations of mixed convection. The results allowed the assessment of the influence of a step change in the imposed temperature at entry of the tube and also of the ratio of thermal diffusivities on the velocity and temperature fields. The existence of a transient instability region was put in evidence, situated in the middle of the tube, between the cold and the hot fluid. The results equally show the axial diffusion in the wall.
Acknowledgments The authors are deeply grateful to both the Algerian and the French higher Education Ministries for the BAF grant within the framework of which the present research was done.
References [1] [2] [3] [4] [5]
Nguyen, C. T., Maïga, S. B., Landry, M., Galanis, N., & Roy, G., Numerical investigation of flow reversal and instability in mixed laminar vertical tube flow. Int. J. Therm. Sc., 43(8), pp.797-808, 2004. Morton, B. R., Ingham, D. B., Keen, D. J., & Heggs, P. J., Recirculating combined convection in laminar pipe flow. ASME. J.Heat Transfer, 111, pp.106-113, 1989. Bilir, S. & Ates, A., Transient conjugated heat transfer in thick walled pipes with convective boundary conditions. Int. J. Heat Mass Transfer, 46(14), pp.2701-2709, 2003. Bernier, M. A. & Baliga, B. R., Conjugate conduction and laminar mixed convection in vertical pipes for upward flow and uniform wall heat flow. Num. Heat Transfer part A, 21, pp.313-332, 1992. Bernier, M. A.& Baliga, B. R., Visualization of mixed convections flows in vertical pipes using a thin semi-transparent gold-film heater and dye injection.. Int. J. Heat Fluid Flow, 13(3), pp.241-249, 1992. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
[6] [7] [8] [9]
[10] [11] [12]
61
Zgal, M., Galanis, N., & Nguyen, C. T., Developing mixed convection with aiding buoyancy in vertical tubes: a numerical investigation of different flow regimes. Int. J. Therm. Sc., 40, pp.816-824, 2001. Su, Y. C. & Chung, J. N., Linear stability analysis of mixed convection flow in vertical pipe. J. Fluid Mech., 422, pp.141-166, 2000. Mai, T. H., Zebiri, R. & Lorenzo, T., Convection mixte en régime transitoire de couche limite laminaire sur une plaque vertical. C. R. Acad. Sci. Paris, t.329, Série IIb, pp.627-631, 2001. Mai, T. H. & Popa, C. V., Numerical study of transient mixed convection in vertical pipe flows. Proc. Of the Conf. On Advances in Fluid Mechanics IV, eds. M. Rahman, R. Verhoven & C. A. Brebbia, WIT Press, pp.75-84, 2002. Mai, T. H., Popa, C. V. & Polidori, G., Transient mixed convection flow instabilities in vertical pipe. Heat Mass Transfer, 41, pp.216-225, 2001. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, D. C., 1980 Kakac, S. & Yener, Y., Convectif Heat Transfer. CRC Press, Boca Raton, 1995.
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Section 2 Advances in computational methods
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65
An interaction of a sonic injection jet with a supersonic turbulent flow approaching a re-entry vehicle to atmosphere D. Sun & R. S. Amano Department of Mechanical Engineering, University of Wisconsin-Milwaukee, USA
Abstract A shock-interaction flow field generated by transverse sonic injection into a supersonic flow was simulated by solving the Favre-averaged Navier-Stokes equations using the weighted essentially non-oscillatory (WENO) schemes. Three-dimensional results indicate that there appear four pairs of vortices around the secondary injection. In the upstream of the square injection there exist two main recirculation regions and the primary vortex induces the horseshoe vortex region. After the secondary injection flow ejects from the square hole, it is forced by the supersonic main flow and then it becomes a pair of counter rotation vortices towards downstream. In the downstream there is a low-pressure region that conduces a pair of helical vortices. Keywords: supersonic flow, bow shock wave, horse-shoe vortex, CFD.
1
Introduction
Shock-interaction flowfield generated from a sonic gaseous flow injected transversely into a supersonic freestream is encountered in practical applications such as space shuttle reentry atmosphere, rocket motor thrust vector control systems, supersonic combustion, high-speed flight vehicle reaction control jets, and gas-turbine cooling systems [1-3]. One example is the case when re-entry vehicles or reusable rockets enter atmosphere, its attitude has to be controlled to endure large aerodynamic heating. Because of a high angle of attack on re-entering, the flow separates from the control surface. If the jet is injected into a hypersonic flow, complicated interaction between the jet and the flow occurs. It results in boundary layer separation, shock waves, and vortices, which are WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060071
66 Advanced Computational Methods in Heat Transfer IX schematically shown in Fig. 1. The mixing flow field is very complex, which includes a bow shock wave in front of the injection, boundary layer separation and vortices. Therefore a higher-order scheme is needed when computing such a flow field. In 1980s, a new high-order scheme, essentially non-oscillatory scheme (ENO), was introduced by Harten et al. [4] Later, Wilcoxson and Manousiouthakis [5] and Jiang and Shu [6] developed the weighted ENO schemes based on ENO schemes, which are simpler and more efficient. In 3-D flowfield, a three-dimensional bow shock forms ahead of the injection and interacts with the approaching boundary layer, resulting in a separation bubble. A barrel shock also occurs as the under expanded jet accelerates into the free stream. Acceleration of the jet core flow continues until a normal shock, or Mach disk, forms. Directly downstream of the jet plume another separated zone develops in the region between the jet exit and the boundary layer reattachment point. A pair of counter-rotating vortices generated within the jet fluid and a horseshoe vortex region also forms near the jet exit and wraps around the injector as illustrated in the schematic. In this paper, the fifth-order WENO scheme of Jiang and Shu [6] and the k-ε turbulent model are used to calculate the supersonic flowfield with secondary injection. The freestream Mach number is 3.7 and 3.0 in two and threedimensional flowfield, respectively, and the injections in both flowfield are sonic. The slot width of the injection in two-dimensional flowfield is 1mm and the orific of the injection in three-dimensional flow is a cubic whose width is 1mm. The conditions of the two-dimensional flowfield is the same with reference 4.
Figure 1:
Bow shock in a mixing jet-stream.
The schematic of the transverse jet injected into a supersonic cross flow is shown in Fig. 1. A three-dimensional bow shock is formed ahead of the injected stream and it interacts with the approaching boundary layer, resulting in a separation bubble. A barrel shock also occurs as the under-expanded jet accelerates into the cross flow. Acceleration of the jet core flow continues until a normal shock, or Mach disk, forms. Directly downstream of the jet plume another separated zone develops in the region between the jet exit and the boundary layer reattachment point. A pair of counter-rotating vortices generated within the jet fluid and a horseshoe vortex region also forms near the jet exit and wraps around the injector as illustrated in the schematic. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
2
67
Governing equations
The three-dimensional Favre-averaged Navier-Stokes equations and the low-Reynolds number k-ε turbulent model are given as follows: ∂Q ∂ ∂ ∂ + (E − Ev ) + (F − Fv ) + (G − Gv ) = H ∂z ∂y ∂t ∂x
ρu 2 ρu + p ρuv E = ρuw u (e + p ) ρuk ρuε 0 τ xx τ xy τ xz Ev = uτ xx + vτ xy + wτ xz − qx µt ∂k µ + σ k ∂x µt ∂ε µ + σ ε ∂x
ρ ρu ρv Q = ρw e ρk ρε
(1)
ρv ρuv ρv 2 + p F = ρvw v(e + p ) ρvk ρvε
0 0 τ xz 0 τ yz 0 τ zz uτ + vτ + wτ − q 0 yz zz z Gv = xz H = 0 µt ∂k µ + µt P − ρε σ k ∂z 2 c f µ ε P − c f ρ ε µt ∂ε 1 1 t 2 2 k k µ + σ ε ∂z
ρw ρuw ρvw G = ρw 2 + p w(e + p ) ρwk ρwε 0 τ xy τ yy τ yz Fv = uτ xy + vτ yy + wτ yz − q y µt ∂k µ + σ k ∂y µt ∂ε µ + σ ε ∂y
where P represents the production of kinetic energy and the following form is used: ∂u ∂u 2 ∂u j 2 ∂u k − P = µt i + δ ij − ρkδ ij i 3 ∂x j ∂xi 3 ∂xk ∂x j
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(2)
68 Advanced Computational Methods in Heat Transfer IX µ t = Cµ f µ ρ
[
{
f µ = 1 − exp − 0.0165 R y
k2
ε
, C µ = 0.09 ,
]} (1 + 20.5 R ) 2
(3)
t
c1 = 1.44 , c2 = 1.92 , σ k = 1.0 , σ ε = 1.3
Rt = ρk
2
µε , R y = ρ k y µ , f1 = 1 + (0.05 f µ ) ,
(
f 2 = 1 − exp − Rt2
Figure 2:
3
(4)
3
)
(5)
A flow field around an injected jet.
Numerical method
3.1 Spatial discretization The semi-discrete form of Eq. (1) can be written as:
[
∂Q = − (E − Ev )i +1 2 , j ,k − (E − Ev )i −1 2 , j ,k ∂t − (F − Fv )i , j + 1 2 ,k − (F − Fv )i , j −1 2 ,k
[ − [(G − G )
v i , j ,k + 1 2
]
] ]
(6)
− (G − Gv )i , j ,k −1 2
The spatial differencing of numerical fluxes adopts the fifth-order accurate WENO scheme of Jiang and Shu [6] for the inviscid convective fluxes and the fourth-order central differencing for the viscous fluxes. The key idea of the WENO is to use a combination of all the candidate stencils to approximate the fluxes at the boundaries to a higher-order accuracy and at the same time to avoid spurious numerical oscillations near shocks instead of using only one of the candidate stencils. By adopting the WENO scheme, we split the physical fluxes (say, Fˆ ) locally into positive and negative parts as ˆ = Fˆ + Q ˆ + Fˆ − Q ˆ (7) Fˆ Q
()
()
()
where ∂Fˆ + ∂Qˆ ≥ 0 and ∂Fˆ − ∂Qˆ ≤ 0 . In this paper, the local Lax-Friedrichs flux splitting method is used.
() (()
ˆ = 1 Fˆ Q ˆ ± ΛQ ˆ Fˆ ± Q 2
)
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(8)
Advanced Computational Methods in Heat Transfer IX
69
where Λ = diag ( λ1 , λ2 , λ3 , λ4 , λ5 ) , and λ1 , λ 2 , λ3 , λ4 , λ5 are the local eigenvalues. We first consider the one-dimensional scalar conservation laws. For example, u t + f (u )x = 0 (9) Let us discretize the space into the uniform interval of size ∆x and denote x j = j∆x . The spatial operator of the WENO schemes, which approximates
− f (u )x at x j , will take the conservative form L=− ~
~
(
~ 1 ~ f j +1 2 − f j −1 2 ∆x
)
(10) ~
~
where f j +1 2 and f j −1 2 are numerical fluxes. Designate f j++1 2 and f j−+1 2 , ~
respectively, the positive and negative parts of numerical flux, f j +1 2 , we have
~ ~ ~ f j +1 2 = f j++1 2 + f j−+1 2 (11) ~+ Here we only describe how to compute f j +1 2 on the basis of the WENO.
~+ f j +1 2 can be shown symmetrically as follows: ~+ 7 11 + 2 f j +1 2 = ω0+ f j+− 2 − f j+−1 + fj 6 6 6 5 + 2 + + 1 + + ω1 − f j −1 + f j + f j + 1 6 6 6 5 1 2 + ω 2+ f j+ + f j++1 − f j++ 2 6 6 6
(12)
where ω k+ = α 0+ =
and
(
1 ε + IS0+ 10
)
−2
α 0+
α k+ , k = 0 ,1,2 + α 1+ + α 2+
, α 1+ =
(
6 ε + IS1+ 10
(
)
(
(
)
(
(
)
(
)
−2
α 2+ =
2 1 13 + f j − 2 − 2 f j+−1 + f j+ + f j+− 2 − 4 f j+−1 + 3 f j+ 4 12 2 2 1 13 + IS1+ = f j −1 − 2 f j+ + f j++1 + f j+−1 − f j++ 1 12 4 2 1 13 + f j − 2 f j++1 + f j++ 2 + 3 f j+ − 4 f j++1 + f j++ 2 IS2+ = 4 12
IS0+ =
(
3 ε + IS2+ 10
)
−2
)
2
)
(13)
)
2
Equation (13) represents the smoothness measurement of stencils. Through the smoothness measurement, the interpolation polynomial on each stencil is assigned a weight from which we can construct a polynomial to approximate the numerical fluxes by combining all the polynomials.
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70 Advanced Computational Methods in Heat Transfer IX 3.2 Time discretization The time discretization of the WENO scheme can be implemented by the third-order Runge-Kutta method. To solve the ordinary differential equation du = L(u ) dt
(14)
where L is a discretization of the spatial operator, the third-order Runge-Kutta is simply u (1) = u n + ∆tL u n 3 1 1 u (2 ) = u n + u (1) + ∆tL u (1) (15) 4 4 4 2 2 1 u n +1 = u n + u (2 ) + ∆tL u (2 ) 3 3 3
( )
( )
( )
4 Presentation of results and discussion 4.1 Cross-flow over two-dimensional flat surface The computed flow field resulting from the transverse injection of two-dimensional sonic jets into a supersonic turbulent flow at a Mach number of 3.71 and a unit Reynolds number of 2.01×107 are simulated by employing the WENO scheme. The result is compared with the experiment and published calculations at the same condition [7]. The computational domain is 88mm×50mm with the slot width of 1mm. The free-stream Mach number is 3.71 and total pressure is 1atm. The jet is sonic and the total pressure is 0.31atm. The law of the wall coordinates y+ for the first mesh point has been maintained as y+<1 in all cases. Mach number, density and pressure contours are shown in Figs. 3(a) and (b) for 3-D and 2-D, respectively. The upstream separation shock and the induced bow shock are clearly presented in Fig. 3. The jet normal shock and the recompression shock downstream are also captured. In this figure the presence of two recirculation zones in the upstream and a recirculation zone in the downstream sections are clearly demonstrated. The surface static pressure distributions are compared with the test data of Aso and Okuyama [7] in Fig. 4. The present computations are also compared with the simulations obtained by Toda and Yamamoto [2]. The agreement of the present computation is better that the prediction by Yamamoto along the upstream location due to the improved prediction method for the approaching flow field. However, in the downstream location, both computations seem to be nearly identical. There appear two peaks upstream of the jet that corresponds to the separation and the stagnation line between the two counter rotating vortices upstream. The pressure peak in the downstream flowfield corresponds to the reattachment. It shows excellent agreement.
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Advanced Computational Methods in Heat Transfer IX
(a) Figure 3:
71
(b)
Mach number, density and pressure distributions. (a) 3-D, (b) 2-D.
4
Present comp
3.5
Comp [2]
3
Exp [7]
2.5 2 1.5 1 0.5 0 -40
Figure 4:
-20
0
20
40
Pressure distribution along the plate.
4.2 Three-dimensional results The configuration is a square hole in a flat plate. The computational domain is 37mm long, 22mm wide, and 25mm high. There are 69 points in the streamwise direction and 59×51 in a cross plane. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
72 Advanced Computational Methods in Heat Transfer IX Free-stream Mach number is 3.0 and total pressure is 1atm. Jet is sonic and total pressure is 0.5atm. Boundary conditions are: supersonic inlet, supersonic extrapolated exit, extrapolated upper boundary, and adiabatic no slip on the flat plate. The jet exit is set with uniform conditions and oriented normal to the surface. Velocity vector plots and Mach contours in symmetrical plane are shown in Figs. 5 and 6, respectively. An induced bow shock is clearly presented. From the Mach number contour plot, the barrel shock and Mach disk are presented. As the jet is injected, it expands through a Prandtl-Meyer fan centered at the nozzle lip, compresses through the barrel shock, and then passes through a Mach disk.
Y
5
0
-5
0
Figure 5:
5
X
Velocity vectors in symmetrical section.
Frame 001 31 Dec 1999
Z
15
10
5
0 -5
Figure 6:
0
X
5
10
15
Mach contours in symmetrical section.
Figure 7 presents the wall pressure distribution in symmetrical plane. There are two main vortices in upstream of the jet, which are called the primary vortex and the secondary vortex. The primary vortex is induced by the boundary layer separation whereas the secondary vortex is induced by the injection. Furthermore, between the primary vortex and the wall, another vortex is shown, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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which is called the tertiary vortex. In the symmetrical plane, the wall pressure jumps up with the boundary layer separation and in the stagnation zone between the primary and the secondary vortices, which form the second pressure peak. Frame 001 04 Jan 2000
11 10 9 8 7
Z
6 5 4 3 2 1 0 -5
Figure 7:
0
X
5
Wall pressure distribution.
Frame 001 04 Jan 2000
Figure 8:
Streamlines around the jet.
Figure 8 presents the vortex structure by streamlines. There are four pairs of vortices in the flowfield. Horseshoe vortices form in the near-wall region from the vorticity within the cross-flow boundary layer and the vorticity is generated due to the wall pressure gradient resulting from the jet/freestream interaction. The shear layer vortices are developed from the vorticity contained in the jet boundary layer, and roll up into the free-stream. The counter-rotating structures are also formed from the vorticity presented in the jet boundary layer, and oriented in the streamwise direction. In this paper, other vortices are observed in the wake region downstream of the injector, which is called the helical vortex.
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74 Advanced Computational Methods in Heat Transfer IX Figure 9 presents the vortex structure by streamlines. There are four pairs of vortices in the flowfield. Horseshoe vortices form in the near-wall region from the vorticity within the cross-flow boundary layer and the vortices is generated due to the wall pressure gradient resulting from the jet/freestream interaction. The shear layer vortices are developed from the vorticity contained in the jet boundary layer, and roll up into the free-stream. The counter-rotating structures are also formed from the vorticity presented in the jet boundary layer, and oriented in the streamwise direction. In this paper, other vortices are observed in the wake region downstream of the injector, which is called the helical vortex (see Fig.10).
Figure 9:
Mach number contour at different stream locations. Frame 001 31 Dec 1999
Y
5
0
-5
-5
Figure 10:
0
X
5
10
Surface pressure contours around jet.
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75
Frame 001 04 Jan 2000
6
5
Z
4
3
2
1
0 -2
Figure 11:
0
Y
2
Velocity vectors in transverse section downstream of jet. Frame 001 31 Dec 1999
Figure 12:
Velocity vectors upstream of jet.
Figure 11 presents the velocity vectors in cross-section where the helical vortices exist. The primary vortex is induced by the boundary layer separation whereas the secondary vortex is induced by the injection. Furthermore, between the primary vortex and the wall, another vortex is shown, which is called the tertiary vortex. As shown in Fig. 12, in the symmetrical plane, the wall pressure jumps up with the boundary layer separation and in the stagnation zone between the primary and the secondary vortices, which form the second pressure peak.
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76 Advanced Computational Methods in Heat Transfer IX
5
Conclusion
Applications of the WENO scheme and the low-Reynolds number k-ε turbulence model can accurately simulate the supersonic flow field with a transverse injection. As the flow passed around the injector port, the boundary layer in front of the jet was separated and the bow shock was formed. In front of the jet, a boundary layer separation formed horseshoe vortices. Counter-rotating vortices were formed from the vorticity presented in the jet boundary layer, and oriented in the stream-wise direction. In the wake region downstream of the injector, helical vortices were observed.
Acknowledgment The computations were performed in ORIGIN2000 at National Center for Supercomputing Applications under NSF grant CTS000003N.
References [1] Gruber, M. R. and Goss, L. P., “Surface Pressure Measurements in Supersonic Transverse Injection Flowfields,” Journal of Propulsion and Power, Vol. 15, No. 5, September-October 1999. [2] Toda, K. and Yamamoto, M. “Computation of Supersonic Turbulent Flowfield with Secondary Jet normal to Freestream,” AIAA 98-0944, 1998. [3] Roger, R. P. and Chan, S. C., “Parameters Affecting Penetration of a Single Jet into a Supersonic Crossflow,” AIAA 98-0425, 1998. [4] Harten, A., Engquist, B, Osher, S., and Chakravarthy, “Uniformly high-order accurate nonoscillatory schemes,” J. Comp. Phys. 71, 231, 1987. [5] Wilcoxson, M. and Manousiouthakis, Vasilioss, “On an Implicit ENO Scheme,” J. Comp. Phys. 115, 376-389, 1994. [6] Jiang, Guang-Shan and Shu, Chi-Wang, “Efficient Implementation of Weighted ENO Schemes,” J. Comp. Phys. 126, 202-228, 1996. [7] Aso, S. and Okuyama, S., “Experimental Study on Mixing Phenomena in Supersonic Flows with Slot Injection”, AIAA Paper 91-0016, January 1991.
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Simulation of coupled nonlinear electromagnetic heating with the Green element method A. E. Taigbenu School of Civil and Environmental Engineering, University of the Witwatersrand, Johannesburg, South Africa
Abstract The nonlinear coupled differential equations that govern the problem when materials are electrically heated are solved with a flux-based Green element formulation that has significantly enhanced computational features in comparison to previous Green element formulations. The flux-based Green element formulation takes advantage of the ability of the boundary element theory to correctly calculate the normal derivative of the primary variable by implementing the theory in a finite element sense so that enhanced accuracy is achieved with coarse discretization. The complete solution information (temperature, electric potential and their normal derivatives) are made available by the flux-based formulation in each element so that refined solutions at any point, when needed, are calculated using only the element in which the point is located. The closure problem associated with having more unknowns than discretized equations at internal nodes is addressed in a novel manner by a compatibility relation for the normal derivatives of the primary variable that has universal appeal. The computational accuracy of the flux-based Green element formulation is demonstrated with a numerical example of nonlinear electromagnetic heating problem. Keywords: electromagnetic heating, nonlinear diffusion-advection, nonlinear Poisson equation, Green element method.
1
Introduction
The food and related industries are very interested in addressing the problems associated with the heating of food substances by electrical currents. Of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060081
78 Advanced Computational Methods in Heat Transfer IX particular interest is the distribution of the temperature field (indicative of the effectiveness of the applied heat) that is generated for a given electric potential that is applied to the food material. For the temperature field, the governing equation closely resembles the diffusion-advection-reaction equation that has many applications in many engineering production processes. Its nonlinear nature poses a challenge especially when the advective term is dominant and the equation then has a predominantly hyperbolic nature. This equation is couple with the nonlinear electromagnetic equation which essentially has an elliptic character. These equations were traditionally solved by the finite difference and finite element methods. A boundary element solution has been provided by Crann et al. [1] who used the Laplace transform formulation to simplify the temporal derivative and the dual reciprocity method to ensure that the solution is carried out only on the boundary. That approach, referred to as the Laplace transform dual reciprocity method (LTDRM), is closely similar to that of Satravaha and Zhu [2] for the solution of nonlinear heat conduction problems. Linearization of the differential equations has to be done to be able to apply the Laplace transform dual reciprocity BEM. Here the Green element method (GEM), which retains the normal derivatives at every nodal point, is used to solve the electromagnetic heating problem [3–5]. It is therefore referred to as the flux-based GEM. The integral equations that result from the application Green’s identity are solved in each element. The integrations are evaluated analytically, and the only approximation that is required is done when interpolating the primary variable and its normal derivative in the element. High level of accuracy is thus achieved with coarse discretization of the computational domain and this compensates for the large number of degrees of freedom at each node. The closure problem at the internal nodes that is as a result of a fewer number of integral equations than unknowns is resolved in a novel manner by generating an additional equation from numerically implementing the integration of the normal fluxes around the internal node. An example solved by the LTDRM of Crann et al. [1] which employed 40 nodes is solved by the flux-based GEM with 10 nodes to achieve comparable accuracy.
2
Governing equations
The electromagnetic heating problem that is addressed in this paper is governed by the coupled nonlinear equations that are given by Please et al. [6] ∂ 2 ∇ ⋅ (k∇T ) = (αT ) + v ⋅ ∇(αT ) − σ ∇φ (1) ∂t and ∇ ⋅ (σ∇φ ) = 0 (2) where T and φ represent the temperature and electric potential fields of the medium on which ohmic heating is applied. The material properties of the medium are: k is the thermal conductivity and σ is the electrical conductivity which are both dependent on the temperature field, α is the heat capacity, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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v = iu + jv is the velocity field in two dimensions. The solution to eqns (1) and (2) in a coupled manner can be achieved when the boundary and initial conditions are specified. The first-, second- and third-type boundary conditions are admissible, while the condition for the temperature and potential fields are assumed known at the initial time t 0 . Eqns (1) and (2) are rewritten as Poisson-type equations: 1∂ 2 ∇ 2 T = −∇K ⋅ ∇T + (αT ) + v ⋅ ∇(αT ) − σ ∇φ (3) k ∂t
∇ 2φ = −∇Φ ⋅ ∇T (4) where K = ln k and Φ = ln σ . The solutions to eqns (3) and (4) are achieved in a Green element sense.
3
Green element formulation
The Green element formulation that is employed for the solution of the coupled equations (3) and (4) uses the fundamental solution G = ln(r − ri ) to the Laplacian operator ∇ 2 G = ∂ (r − ri ) in the infinite space to derive integral equations within a spatial element Ω e with closed boundary Γ e that constitutes one of the elements used in discretizing the entire computational region Ω . The integral equations are given by [7]
− λTi + ∫ (T∇G ⋅ n − G∇T ⋅ n )ds, Γe
(5) 1 ∂ (αT ) 2 + ∫∫ G − ∇K ⋅ ∇T + + v ⋅ (αT ) − σ ∇φ dA = 0 k ∂t Ωe
−λφ i + ∫ (φ∇G ⋅ n − G∇φ ⋅ n )ds − ∫∫ G (∇Φ ⋅ ∇φ )dA Γe
Ωe
(6)
where λ is the nodal angle from integrating the dependent variable with the Dirac delta function in a Cauchy sense, and n is the outward pointing normal on the elemental boundary. It should be noted that so far no approximation whatsoever has been introduced in the formulation in arriving at eqns (5) and (6). Approximations are introduced by prescribing an interpolation for the distribution of the dependent variable and their normal derivatives in the element. The Lagrange-type interpolation functions are used. Using either linear triangular or linear rectangular elements and carrying out the integrations results in the matrix equations
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80 Advanced Computational Methods in Heat Transfer IX v u qy q q RijT j + Lij + Χ imj α m T + K m x + Υimj α m T + K m k j k j k j k j k 1 d (αT ) αu q x αv q y j − + Simj − k dt m k m k j k m k
, j
py p p =0 Rijφ j + Lij + Χimj Φ m x + Υimj Φ m k j k j k j
where
Rij = ∫ N j ∇G ⋅ n ds −δ ij λ , Γ
(
(8)
Lij = ∫ N j G ds, Γ
∂N ∂N Χ imj = ∫∫ G m N j dA, Υimj = ∫∫ G m N j dA, Ω ∂x Ω ∂y S imj = ∫∫ G N m N j dA. Ω
(7)
σ 2 − ∇φ = 0. k j j m
(9)
)
q = − k∇T ⋅ n,
q x = −k∇T ⋅ i,
p = −σ∇φ ⋅ n,
p x = −σ∇φ ⋅ i,
q y = −k∇T ⋅ j p y = −σ∇φ ⋅ j
(10a) (10b)
In eqn (9), N j represents the interpolation function with respect to node j . It should be noted that the integrations in eqn (9) are done in each element, and it is for simplicity that the index e has been excluded. All the integrations are done analytically for the two types of elements, namely rectangular and triangular elements. Carrying out the matrix manipulations in eqns (7) and (8) results in two unknowns at every nodal point for each of the equations. These are the temperature T and its normal flux q for the first differential equation, and the electric potential φ and its normal flux p . Eqns (7) and (8) are aggregated for all elements used in discretizing the computational domain and simplified to dT n +1 =0 (11) A n T n +1 + B n q n +1 + S n dt C n φ n +1 + E n p n +1 = 0 (12) We have introduced a new index n , the iteration number, into eqns (11) and (12) to indicate that the elements of the matrices are evaluated with known iterates, while the unknown quantities are to be computed at the current iteration level of n + 1 . Essentially, this iteration process is the Picard algorithm. The time derivative in eqn (11) is simplified by the generalized finite difference scheme with weighting factor θ to become T n +1, 2 − T n +1,1 = 0 (13) ∆t where the indices 2 and 1 , respectively, are indicative of the current time t 2 = t1 + ∆t and previous time t1 , and ω = 1 − θ . Introducing the initial
θA n T n +1, 2 + ω (AT )1 + θB n q n +1, 2 + ω (Bq )1 + S n
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conditions into eqn (13), and the known boundary conditions on external boundaries into eqns (12) and (13) simplify the coupled equations to (14) F n Tn +1, 2 + H nq n +1, 2 = R C n φ n +1 + E n p n +1 = M
(15)
Sn , ∆t
(6)
where F n = θA n +
H n = θB n
The vectors on the right hand side of eqns (14) and (15) are known; their values come from the prescribed boundary and initial conditions. The solution procedure within each time step is herein outlined. (i) For the known constitutive relations for k and σ , and a known distribution
{
} }.
of the temperature field T n, 2 , q n,2 , the matrix equation (14) is solved to
{
n +1, 2
n +1, 2
obtain updated values T ,q (ii) Those updated solutions from (i) are used in eqn (15) to solve for the electric potential field φ n +1 , p n +1 . (iii) The mean deviation of the iterates from the temperature and potential fields is calculated and compared with a prescribed accuracy tolerance ε . Convergence is said to be achieved when the mean deviation is less than ε . If convergence is not attained, steps (i) through (iii) are implemented using the refined solution iterates. (iv) When convergence within a time step has been met, another time increment is made and the above three steps are then repeated.
{
}
4 Compatibility of the internal fluxes Because the current flux-based Green element formulation calculates the dependent variable and as well as its normal derivative (referred to as the flux) at each nodal point, the number of generated integral equations is short by one the number of degrees of freedom at internal nodes. This is not the case with the external nodes where the prescribed boundary conditions make up for the shortfall in the number of generated discrete equations. This is the closure problem that has been recognized in boundary element circles. In the past one approach of resolving the closure problem has been to artificially create additional nodes and relocate them by small distances from the original location of the internal node along the internal segments [8]. Such an approach is a numerical artefact that usually reconfigures the geometry at the internal nodes to suit the numerical formulation. That approach is not followed here but rather an additional equation is generated to make up for the shortfall. Two previous presentations on this additional equation indicated that this additional equation is statement of the continuity of the normal fluxes at the internal node [4, 5]. We have now found this to be in error, and special acknowledgement goes to Dr. Elliot who prompted us to re-examine that statement. We have now carried WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
82 Advanced Computational Methods in Heat Transfer IX out a rigorous proof of this compatibility condition of the internal normal fluxes. It is too lengthy to present here. The proof will be presented in details in manuscripts that are currently being prepared. Suffice to say that the proof is based on integrating the normal derivative of the primary variable around a circle of small radius centred at the internal node. In the limit as the radius tends to zero, the integral equals the value of discontinuity of the primary variable at the internal point. Where no discontinuity exists, the integral equals zero. This can be stated as ∂T dn = T + − T − = 0 (17) ∫k ∂n To implement eqn (17) numerically, 2π M ∂T dn = − Limit 2π ∫ q dβ ≈ ∑ q i ∆β i = 0 (18) ∫k ζ →0 ∂n i =1 0 where M is the number of elements at the internal node, ζ is the radius of the circle at the internal node (See Figure 1). One may ask why excellent results were obtained in the previous publications with a wrong compatibility condition. The reason is that all the simulations were carried out with rectangular elements and, as can be observed from eqn (18), ∆β i = π / 2, i = 1, 2, 3, 4. We have since tested the condition of eqn (18) with triangular elements where different angles of ∆β i are encountered and excellent results were obtained.
3 q
3-2
1
q
2
∆β
3-1
q
4-2
q5-1 ζ
q5-4
5 Figure 1:
4
Normal fluxes at an internal node.
Two unique characteristics of the flux-based formulation are that it provides the complete solution information for each element, and high accuracy is achieved with coarse discretization. The latter compensates for the escalation in the number of degrees of freedom due to the evaluation of the primary variable WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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and the flux at every node. The former allows for calculations of the solution at a point to involve only integrations within the element in which the point is located. No reference is made to other elements in the computational region. That represents enormous computational savings when solutions are required at points of interest other than grid points.
5
Numerical example
The numerical example used by Crann et al. [1] in their Laplace transform dual reciprocity boundary element formulation is also used here to validate the current flux-based Green element formulation. The example is essentially one dimensional in the spatial dimension for which the analytical solutions: T = x − x 2 / 2 2 − e −t and φ = x + x − x 2 e −t , 0 ≤ x ≤ 1 (19) are proposed for the differential equations ∂ 2 ∇ ⋅ (k∇T ) = (αT ) + v ⋅ ∇(αT ) − σ ∇φ + f1 ( x, t ) (20) ∂t and ∇ ⋅ (σ∇φ ) = f 2 ( x, t ) (21) The choice of the functions f1 ( x, t ) and f 2 ( x, t ) has been made so that the proposed solutions satisfy the differential equations. The boundary conditions are: ∂T ( x = 1, t ) T ( x = 0, t ) = 0 , = 0 , φ ( x = 0, t ) = 0 and φ ( x = 1, t ) = 1 (22) ∂x The parameters for the medium and the advection velocity field are: k = 1 + T , σ = 1 + T , α = 1 and v = i . Using only four linear rectangular elements, the Green element simulations are carried out in a two dimensional domain of 1 × 1 so that the size of each element is 0.25 × 1. Effectively, 10 nodes are used to discretize the computation domain, representing a coarse discretization in comparison to the 40 nodes used by Crann et al. [1]. The fully implicit scheme with θ = 1 is used for the differencing in time, while the time step is varied starting with 0.025 for 0 ≤ t ≤ 0.1 , then 0.1 for 0.1 ≤ t ≤ 1 and 0.25 for 1 ≤ t ≤ 5 . The accuracy tolerance value used is ε = 10 −5 and convergence was achieved within 3 iterations in each time step. The spatial distributions of the temperature and electric potential fields for times of 0.1, 0.5, 1, and 5 are presented in Figures 2 and 3, while their temporal distributions at x = 0.25 , x = 0.5 , and x = 0.75 are presented in Figures 4 and 5. There is good agreement between the Green element solutions and the analytical solution. It should be pointed out that the solutions at points other than grid points were generated using only the solutions obtained for the element in which the grid points are located. For instance, the solution at x = 0.1 was carried out on the first element {( x, y ) : x ∈ [0,0.25], y ∈ [0,1]} . The high level of accuracy that is achieved with such a coarse grid is as a result of the fact that the only approximation in this formulation arises from the interpolation of the primary variable and its normal derivative.
(
)(
)
(
)
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84 Advanced Computational Methods in Heat Transfer IX 1 0.8
T(x,t)
0.6 0.4
Exact GEM(t=0.1) GEM(t=0.5) GEM(t=1) GEM(t=5)
0.2 0 0
Figure 2:
0.2
0.4
x
0.6
0.8
1
Analytical and GEM solutions for the spatial temperature distribution at various times. 1
φ(x,t)
0.8 0.6
0.4
Exact GEM(t=0.1) GEM(t=0.5) GEM(t=1) GEM(t=5)
0.2
0 0
Figure 3:
6
0.2
0.4
x
0.6
0.8
1
Analytical and GEM solutions for the spatial electric potential distribution at various times.
Conclusion
The nonlinear coupled differential equations which govern heating of food materials by electrical current have been solved using a flux-based Green element formulation that solves not only for the temperature and electric potential fields but also their normal derivatives. The high level of accuracy achieved by the formulation arises not only because the solution procedure retains the nonlinear nature of the differential equations, but also due to the fact WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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that the normal derivatives of the temperature and electric potential are directly calculated. The increased number of degrees of freedom at each node is compensated by the coarse grid with which high accuracy is achieved. 1
x=0.75
T(x,t)
0.8
x=0.5
0.6
x=0.25 0.4
Exact GEM (x=0.25) GEM (x=0.5) GEM (x=0.75)
0.2 0
Figure 4:
1
2
t
3
4
5
Analytical and GEM solutions for the temporal temperature distribution at various positions. 1
Exact GEM (x=0.25) GEM (x=0.5) GEM (x=0.75)
0.8
φ(x,t)
x=0.75 0.6 x=0.5 0.4 x=0.25 0.2 0
Figure 5:
1
2
t
3
4
5
Analytical and GEM solutions for the temporal electric potential distribution at various positions.
The novel feature of the current Green element formulation is that the elemental solution is complete in the sense that the solution at any point in an element is obtained by carry out the boundary and domain integrations within that element. No reference is made to other elements that have been used to discretize the region. Those solutions at points other than the grid points are equally second-order accurate as those at the nodal points. In this paper, the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
86 Advanced Computational Methods in Heat Transfer IX current flux-based GEM has used 25% of the grid points of LTDRM formulation of Crann et al. [1]. That is typical of the flux-based GEM which generally provides accurate solution with very coarse discretization.
Acknowledgement Special thanks to Dr. L. Elliot of Leeds University who prompted a rethink of compatibility equation of the fluxes at the internal node at the end of the 5th UK conference on Boundary Integral Methods in Liverpool in September 2005.
References [1] Crann, D., Davies, A.J. & Christianson, B., The Laplace transform dual reciprocity boundary element method for electromagnetic heating problems. Proc. In Advances in Boundary Element techniques VI, Eds. A.P. Selvadura, C.L. Tan & M.H. Aliabadi, EC LTd, UK, pp. 229-234, 2005. [2] Satravaha, P. & Zhu S., An application of the LTDRM to transient diffusion problems with nonlinear material properties and nonlinear boundary conditions, App. Maths Comp., 87, pp. 127-160, 1997. [3] Pecher, R., Harris, S.D., Knipe, R.J., Elliot L. & Ingham D.B., New formulation of the Green element method to maintain its second-order accuracy in 2D/3D, Engrg. Anal. with Boundary Elements, 25, pp. 211-219, 2001. [4] Taigbenu, A.E., The flux-correct Green element method for linear and nonlinear potential flows, Proc. In Advances in Boundary Element techniques VI, Eds. A.P. Selvadura, C.L. Tan & M.H. Aliabadi, EC LTd, UK, pp. 245-250, 2005. [5] Taigbenu, A.E., Improvements in heat conduction calculations with fluxbased Green element method, Proc. Of the 5th UK conf. on boundary integral methods, Ed. Ke Chen, The University of Liverpool Press, pp. 190-199, 2005. [6] Please, C.P., Schwendenman, D.W. & Hagan, P.S., Ohmic heating of foods during aseptic processing, IMA J. of Management Maths., 5(1), pp. 283–301, 1993. [7] Taigbenu, A.E. The Green Element Method, Kluwer, Boston, USA, 1999. [8] Liggett, J.A. & Liu, P.L-F, The boundary Integral Equation Method for Porous Media Flow, George Allen & Unwin, 1983.
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Lattice Boltzmann simulation of vortices merging in a two-phase mixing layer Y. Y. Yan & Y. Q. Zu School of the Built Environment, University of Nottingham, UK
Abstract Mixing layers are commonly observed in flow fields of many types of industrial equipment such as combustion chambers, chemical reactors, and fluid ejectors. It is important to effectively mix two co-flowing gaseous/fluids streams in such equipment. As the mixing of two fluids is an interface and surface tension dominated process, numerical simulations of the mixing process are generally very complex. The present study is concerned with using the lattice Boltzmann method (LBM) to study vertices merging in a two-dimensional two-phase spatial growing mixing layer. Different velocity perturbations are forced at the entrance of the flow field of a rectangular mixing layer; the initial interface between twophases is evenly distributed around the midpoint in a vertical direction. By changing the strength of surface tension and combinations of perturbation waves, the effects of surface tension and velocity perturbation on vortices merging are investigated. Some interesting phenomena, which have not occurred in a singlephase mixing layer, are observed and the corresponding mechanism is discussed. Keywords: lattice Boltzmann method, vortices merging, mixing layer, numerical simulation.
1
Introduction
Vortices behaviours including the formation and merging are commonly observed in engineering applications, such as in combustion chambers, premixers for gas turbine combustors, chemical lasers, propulsion systems, flow reactors, micro mixers, etc. Controlling the formation and evolution of the coherent structure in the mixing layer can improve efficiency of combustion or chemical reaction processes; therefore studies on vortices behaviour in mixing layers have been carried out both experimentally and computationally. Ho and WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060091
88 Advanced Computational Methods in Heat Transfer IX Huang [3] experimentally studied vortices merging in single phase mixing layer. Inoue [4, 5] numerically simulated vortices behaviour in double and triple frequency forced two-dimensional spatially growing single-phase mixing layers. By direct numerical simulation (DNS) and large-eddy simulation (LES), Silvestrini [9] have investigated the dynamics of coherent vortices in mixing layer. Lifshitz and Degani [6] proposed mathematical model for turbulent mixing layer with harmonic perturbations; the model reduced numerical complexity, but the surface tension dominated mixing of two-phase fluids was not highlighted. In the present study, the LBM is employed to simulate vortices behaviour in two immiscible fluids mixing layer. The aims of the study are to understand the effects of surface tension and the perturbation waves on vortices merging, and to obtain an insight to the vortices behaviour at the interface.
2
The lattice Boltzmann model
In recent years, the LBM has become an established numerical scheme for simulating multiphase fluid flows. The key idea behind the LBM is to recover the correct macroscopic fluid motion by incorporating the complicated physics into simplified microscopic models or mesoscopic kinetic equations. On simulating multiphase flow problems, four basic LBM models have been reported to date, namely: the chromodynamic model (Andrew et al. [1]), the pseudo-potential model (Shan and Chen [8]), the free energy model (Swift et al. [9, 10]) and the index function model (He et al. [2]). In this article, the basic method of the index function model for tracking the interface between different fluids is employed. In this model, the velocity and pressure field are given by the distribution function equations of index functions and the pressure, which are given as fα ( x + eα δt , t + δt ) − fα ( x, t ) = 2τ − 1 (eα − u ) ⋅ ∇ψ (φ ) Γα (u )δt τ 2τ RT 1 gα ( x + eα δt , t + δt ) − gα ( x , t ) = − [ gα ( x , t ) − gα(eq ) ( x , t )] + −1
[ fα ( x , t ) − fα(eq ) ( x , t )] −
τ
2τ − 1 (eα − u )[Γα (u )( F + G ) − (Γα (u ) − Γα (0))∇ψ ( ρ )]δt 2τ
(1)
(2)
where u is the macroscopic velocity, ρ the density; G the gravity, and F the surface tension; T the background temperature, x the spatial position vector, and t the time; f α is the newly introduced distribution function of index function φ and gα is the newly introduced distribution function of pressure; f α(eq ) and g α(eq ) are the equilibrium distribution functions in the αth direction and can be expressed respectively as f α( eq ) ( x, t ) = Γα (u )φ ,
(3)
g α ( x, t ) = Γα (0) p + [Γα (u ) − Γα (0)]ρ RT ;
(4)
( eq )
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where 3 9 3 (eα ⋅ u ) + 4 (eα ⋅ u ) 2 − 2 u 2 ] ; 2 c 2c 2c 4 / 9, α = 0 ωα = 1 / 9, α = 1，3，5，7 1 / 36, α = 2， 4，6，8
Γα (u ) = ωα [1 +
(5) (6)
δ t denotes the length of time step; p the pressure; τ the dimensionless relaxation time; ωα is the weighting coefficient; eα the discrete velocity vector, which, in two-dimensional 9-velocity (D2Q9) model, as shown in Fig. 1, is expressed as
α =0 0, eα = {cos[(α − 1)π / 4], sin[(α − 1)π / 4]}c, α = 1，3，5，7 2{cos[(α − 1)π / 4], sin[(α − 1)π / 4]}c, α = 2， 4，6，8
(7)
where, c ≡ 3RT is the lattice constant. In equations (1) and (2), ψ ( ρ ) = p − ρ RT to ensure that pressure satisfies the Carnahan-Starling equation: p = ρRT
1 + bρ / 4 + (bρ / 4) 2 − (bρ / 4) 3 − aρ 2 ; (1 − bρ / 4) 3
(8)
where, both a and b are equal to 12RT and RT = 1 / 3 ; thus,
ψ (ρ ) = ρ 2
4 − 2ρ − 4ρ 2 . 3(1 − ρ ) 3
(9)
The index function φ , the macroscopic quantities, u and p are calculated from
φ = ∑ fα
(10)
α
1 u ⋅ ∇ψ ( ρ )δt 2 α RT ρRT u = ∑ eα g α + (F + G)δ t 2 α p = ∑ gα −
(11) (12)
where the surface tension force F is represented as F = kφ∇∇ 2φ
(13)
where k is the surface tension coefficient, which is related to intermolecular pair1 wise potential, u attr , and can be expressed as k = − ∫ r 2u attr (r )dr , in which 6 r >σ σ is the effective diameter of molecular;.
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
90 Advanced Computational Methods in Heat Transfer IX The kinematical viscosity ν = (τ − 1 / 2) RTδt and density ρ are calculated according to the index function:
φ − φl ( ρ − ρl ) φh − φl h φ − φl ν (φ ) = ν l + (ν − ν ) φh − φl h l ρ (φ ) = ρ l +
(14) (15)
where, ρ h and ρ l are the density of heavy and light fluids respectively. ν h and ν l denote the kinematical viscosity of heavy and light fluids respectively. φ h and φ l are the maximum and minimum values of the index function, respectively.
Figure 1: Discrete velocity set of 2-D nine-velocity (D2Q9) model
3
Figure 2: Initial velocity distribution in side the computational domain
Numerical simulation
To simulate vortices merging in a two-phase spatially growing mixing layer, a rectangular domain of the flow-field is considered as: D = [0, Lx] × [-Ly/2, Ly/2], which is surrounded by an inflow boundary at the left, a free outflow boundary at the right, and a slip boundary at two other upper and lower sides. Unless otherwise mentioned, the channel length Lx and width Ly are set at 250 and 50 respectively. The initial interface between two phases is evenly distributed around the midpoint in vertical direction, the corresponding values of index function are given as φ h = 0.259 and φ l = 0.04 , and the initial velocity field in side the computational domain consists of a hyperbolic tangent profile (see Fig. 2) defined as u ( x, y ) = 1 + Ra ⋅ tanh( y / 2) ; v( x, y ) = 0 The velocity profile at the left boundary (x = 0) is given as u ( y ) = 1 + Ra ⋅ tanh( y / 2) + u * ( y ) ; v( y ) = v * ( y )
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(16) (17)
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where, u and v are the velocity components in the x and y directions, u* and v* are the corresponding velocity perturbations; while, Ra is the velocity ratio. Assuming that U1 and U2 are the free-stream velocity in the upper and lower layers, respectively, the average velocity, U = (U1+U2)/2, and the momentum thickness at x = 0, named as θ 0 here, are used as the reference scales for the velocity and length, respectively. Henceforth, all quantities will be normalised with respect to appropriate combinations of U and θ 0 . Reynolds number for the mixing layer can be defined as Re = Uθ 0 / ν , in which ν is kinematical viscosity. It is shown that the influence of Reynolds number upon the kinetic characteristic of the large-scale coherent structures is small enough if Reynolds number is larger than 200. Thus, Reynolds number is set at 200 in the following simulations. The velocity perturbation u* and v* can be expressed as u* = ∑ An Fn' ( y ) cos(α n x − ω n t + ϕ n ) / α n
(18)
n
v* = ∑ An Fn ( y ) sin(α n x − ω n t + ϕ n )
(19)
n
where A, α 1 , ω1 , ϕ1 and Ai (i ≥ 2) , α i (i ≥ 2) , ω i (i ≥ 2) , ϕ i (i ≥ 2) are amplitudes, wave number, frequency and phase of the basic wave and subharmonic waves, respectively. Fn(y) are normalized characteristic mode determined from a linear stability theory. The frequency of basic wave that can lead to the most unstable mode of mixing lay is 0.225 with the velocity ratio Ra = 0.5 (Monkewitz and Patrick, 1982). So, ω1 is given as 0.225, meanwhile Ra = (U1-U2)/(2U)=0.5, Fn(y) = 1/(1+y2), A1 = 0.02 , Ai = 0.01 (i ≥ 2) , ω 2 = ω1 / 2 , ω 3 = ω1 / 3 , ω 4 = ω1 / 4 , α n = ω n , ϕ1 = 0 . In what follows, for simplicity, the subharmonic waves with the frequencies of ω 2 , ω 3 and ω 4 are donated as the second, the third and the fourth subharmonic waves respectively. By superimposing multi-harmonics and changing their phase shifts, one can study the effects of the velocity perturbation to mixing layer.
4
Results and discussion
Because the basic wave and several sub-harmonic waves can be forced synchronously, there are many choices of the parameters in equations (18) and (19). Therefore, some typical combinations of perturbation waves are chosen for the simulation. Under some specific initial and boundary conditions, the instabilities will grow and vortices will appear; two or more vortices start to spiral around one another and then merge into a new vortex. The spiralling behaviour and the merging behaviour may repeat itself with the newly formed vortices. In the following, the results of simulation will be visualized by means of plots for phase distribution, vortices contour and corresponding frequency spectrum.
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92 Advanced Computational Methods in Heat Transfer IX 4.1 Non-vortex-merging In this section, the forced perturbation is only a basic wave with the parameters of ϕ1 and ω1 . Fig. 3 shows the distributions of two immiscible fluids flow at t = 350 for two different values of surface tension. It shows that the vortices appear clearly but no merging takes place under a single-frequency perturbation. This is also be proved by the plots of corresponding frequency spectrum shown in Fig. 4, in which the frequency spectrum for three different cases at x=120 is presented. Similar plots are obtained at the other location of the flow-fields, which indicates that the vortices do not experience the merging anywhere. The effects of surface tension can be judged in Fig. 3. With a zero surface tension, the interface is elongated at upstream. The elongated interface rolls up and forms vortices during the flow. The newly formed vortex spins and migrates continuously towards downstream and sweeps more and more layers of interfaces into it. For k = 0.01, the flow is qualitatively similar to the zero surface tension case except for the shape of interface in the corresponding vortices; the rollup of the interface in the vortices is slower because the extension of the interface is limited by the surface tension. When k increases to 0.1, the rolling up of interface at up- and mid-stream is similar to the previous two cases, but the interface evolution is delayed more by the much stronger surface tension. Differences can be identified at the downstream of the mixing layer, where the interface is pinched and broken.
(a) k = 0
Figure 3:
(b) k = 0.01
Phase distributions with different values of surface tension at t = 350.
(a) k=0
Figure 4:
(c) k = 0.1
(b) k=0.01
(c) k=0.1
Frequency spectrum plots with different values of surface tension at x = 120.
4.2 Two-vortex-merging The interaction of the basic wave and its second sub-harmonic wave of ϕ 2 = 0 in the mixing layer is first considered. The surface tension parameter, k, is chosen as 0 and 0.1 respectively. The phase distribution and corresponding frequency spectrum for these two cases with different surface tension are shown in Figs. 5 and 6 respectively. It can be seen that the vortices formed at the beginning of the mixing layer are nearly of the same frequency as that of the basic wave. Under WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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the action of the second sub-harmonic wave, every two of these vortices will wind with each other and merge into a bigger vortex. As shown in Fig. 6, the frequency of the big vortices becomes half of the small ones. The merged vortices start to shear when they move downstream and no further merging takes place. However, with the effect of surface tension, although the rollup of the interface in vortices is slower, the interfacial ends are blunter and the interface pinching and breaking behaviours take place; nevertheless, the basic mode of vortices merging is the same as that of zero surface tension case.
(a) k = 0
Figure 5:
(b) k = 0.1
Phase distributions with different values of surface tension at t = 350.
Fig. 6 shows the frequency spectrum of different surface tension at different position. At x = 40, the main frequency approximates to ω1 , which indicates that no merging takes place. When vortices move to x = 120, the frequency of ω1 / 2 substitutes ω1 to dominate over the flow-field, while the basic frequency, ω1 , is still obvious. Such a phenomenon reveals that two vortices are merging at this location and the structures of small vortices have not disappeared completely. Some frequencies which have never been added to the perturbation, such as 3ω1 / 2 and 2ω1 , can be identified in Fig. 6. They are the sums or differences of the frequency between the basic wave and sub-harmonic waves. Moreover, the sameness of the frequency spectrum evolution for different cases can further prove that the strength of surface tension does not affect vortices forming, the mode of merging and the migration velocity of the vortices. (a) k=0
(b) k =0.1
Figure 6:
Frequency spectrum plots with different values of surface tension.
4.3 Three-vortex-merging Fig. 7 shows the results in which the perturbation consists of the basic wave and the third sub-harmonic wave of ϕ 3 = π / 3 . With the interactions of the basic wave and sub-harmonic wave, three vortices merging takes place in the mixing layer. Two downstream vortices merge in the first place, and then the newly WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
94 Advanced Computational Methods in Heat Transfer IX formed vortex merges with the third one upstream. In fact, as it is shown, no matter what parameter of surface tension is chosen, the basic mode of vortices merging is almost the same. Furthermore, as shown in Fig. 8, the evolutions of frequency spectrum in three cases of different surface tension are quite synchronous.
(a) k = 0 Figure 7:
(b) k = 0.1
Phase distributions with different values of surface tension at t = 350. (a) k=0
(b) k = 0.1
Figure 8:
Figure 9:
Frequency spectrum plots with different values of surface tension
Interface distributions and vortices contours with different surface tension.
Fig. 9 shows a comparison of corresponding vortices contours with the same velocity perturbation and three different values of surface tension within the region of x ∈ [200, 250] at t = 350; in each case, three main vortices merge into a larger one. It is shown that, with surface tension effect, more small vortices appear in the region of main vortices; and with the increase of surface tension, many small vortices are formed, and the cores of the three main vortices can even disappear. In addition, with zero surface tension at the two fluids interface (indicated by the dished line), the high vortices concentrate in the cores of the vortices; the structures of vortices match the corresponding phase distributions very well. However, with surface tension effect, the vorticity field is disturbed; the vorticity concentrations appear on the interfaces. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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4.4 More vortex-merging Further to the previous discussion, Fig. 10 shows the vortex-merging in a triple-frequency forcing mixing layer. The channel length Lx and width Ly are given as 375 and 125 respectively. The perturbation consists of a basic wave, a second sub-harmonic wave and a fourth sub-harmonic wave; and all phase shifts equal to zero. Based on this type of perturbation, a phenomenon of four vortices merging takes place in the mixing layer. In the first place, smaller basic vortices are formed upstream in the mixing layer. Thus, every two of these vortices will finish the primary merging and form to a bigger vortex. Then in the flow field downstream, the secondary merging takes place, i.e. every two newly merged vortices repeat the merging process. In this way, the whole process of four-vortexmerging is completed and no further merging behaviour takes place. It is noted from the frequency spectrum plots shown in Fig. 11 that, during the process of vortices merging, the main frequency transfers from ω1 to ω1 / 4 via ω1 / 2 .
Figure 10:
Figure 11:
5
Phase distributions with zero surface tension at t = 400.
Frequency spectrum plots with zero surface tension.
Conclusion
The behaviour of vortices merging in two-dimensional two-phase spatially growing mixing layer is numerically studied by the LBM. By changing the strength of surface tension and the combinations of perturbation waves, the effects of the surface tension and the velocity perturbation on the vortices merging are investigated. With a single-frequency forcing, vortices appear clearly but no merging takes place in the mixing layer. When the mixing layer is forced by the two- or threefrequency perturbation, the vortices start to merge. The results show that the lower frequency of sub-harmonic wave applied, the more vortices are merged. The scale of the large vortex is directly proportional to the number of basic WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
96 Advanced Computational Methods in Heat Transfer IX vortices being involved in the process. Through vortices merging, the interfacial surface area is enlarged quickly downstream in the flow-field. In addition, the effect of surface tension strength on vortices formation, the pattern of merging, and the migration velocity of the vortices is limited. Although the rollup of the interface in vortices is slow, the surface tension does have effect on interfacial ends, pinching and breaking. Based on this, it can be assumed that, no matter how strong of the surface tension is, the vortices evolution in two-phase mixing layer should be controlled positively by forcing the suitable perturbation upstream in the flow-field and obtaining expectant flow patterns consequently.
Acknowledgement The project is partly supported by Royal Society international joint project Ref. 15127 and British EPSRC under grant EP/D500125/1.
References [1] [2]
[3] [4] [5] [6] [7] [8] [9] [10] [11]
Andrew, K. Gunstensen, Daniel H. Rothman, Stéphane Zaleski and Gianluigi Zanetti. 1991. Lattice Boltzmann model of immiscible fluids. Physical Review A, 43(8): 4320-4327. He, X., Chen, S. and Zhang, R. 1999. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. Journal of Computational Physics, 152(2): 642-663. Ho C-M., Huang L-S. 1982. Subharmonics and vortex merging in mixing layers. Journal of Fluid Mechanics, 119: 443-473. Inoue, O., 1992. Double-frequency forcing spatially growing mixing layers. Journal of Fluid Mechanics, 234: 553-581. Inoue, O., 1995. Note on multiple-frequency forcing on mixing layers, Fluid Dynamics Research, 16(2/3): 161-172. Lifshitz, Y. and Degani, D., 2004. Mathematical model for turbulent mixing layer with harmonic perturbations. European Congress on Computational Methods in Applied Sciences and Engineering, 1-19. Monkewitz, P.A. and Patrick Huerre. 1982. Influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids, 25(7): 11371143. Shan, X. and Chen, H. 1993. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 47(3): 1815-1819. Silvestrini, J.H. 2000. Dynamics of coherent vortices in mixing layers using direct numerical and large-eddy simulations. Journal of the Brazilian Society of Mechanical Sciences, 22 (1): 53-67. Swift, M.R., Osborn, W.R. and J. M. Yeomans. 1995. Lattice Boltzmann simulation of nonideal fluids. Physical Review Letters, 75(5): 830-833. Swift, M.R., Orlandini, E., W. R. Osborn, and J. M. Yeomans. 1996. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Physical Review E, 54(5): 5041-5052. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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A numerical solution of the NS equations based on the mean value theorem with applications to aerothermodynamics F. Ferguson & G. Elamin Department of Mechanical Engineering, North Carolina A & T State University, USA
Abstract An innovative and robust algorithm capable of solving a variety of complex fluid dynamics problems is developed. This so-called, Integro-Differential Scheme (IDS) is designed to overcome known limitations of well-established schemes. The IDS implements a smart approach in transforming 3-D computational flowfields of fluid dynamic problems into their 2-D counterparts, while preserving their physical attributes. The strength of IDS rests on the implementation of the mean value theorem to the integral form of the conservation laws. This process transforms the integral equations into a finite difference scheme that lends itself to efficient numerical implementation. Preliminary solutions generated by IDS demonstrated its accuracy in terms of its ability to capture complex flowfield behaviours. In this paper, the results obtained from the application of the IDS to two problems; namely, the hypersonic flat plate problem, and the shock/boundary layer interaction problem, are documented and discussed. In both cases, the results showed very good agreement with the physical expectation of these problems. In an effort to this new algorithm, IDS solution to the shock/boundary layer interaction problem was compared to the experimental findings described in NASA Mem., No., 2-1859W, March, 1959. The results obtained by IDS show excellent agreement with the experimental data. Keywords: Integro-Differential Scheme, mean value theorem, hypersonic boundary layer, finite volume, control volume, numerical scheme.
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98 Advanced Computational Methods in Heat Transfer IX
1
Introduction
The Navier-Stokes equations governing fluid flows can either be highly elliptic or highly hyperbolic, or both, depending on the applicable boundary conditions. As a result, the Navier-Stokes equations are very complicated and, in general, do not lend themselves to analytic solutions. In addition, aerospace designers are currently demanding solutions to fluid flow problems under conditions that cannot be duplicated with existing experimental facilities. Hence, the only way to obtain reasonable, complete information on fluid flows and their characteristics lies in computational fluid dynamic (CFD) methods. A literature survey indicated that there are many well-established numerical schemes available to aerospace designers. Anderson [1] and Chung [2] presented a wide variety of these schemes in their books. Akwaboa [3] used MacCormack technique to solve the supersonic flow over a rearward-facing step problem. Chang et al. [4], Zhang et al. [5] and Changh [6] introduced different versions of the space-time conservation element and solution element method for solving fluid flow problems. Even though, these schemes have led to significant improvements in the state of the art in CFD, they have many drawbacks, and therefore still not adequate to handle certain CFD demands. 1.1 Research objective This research focuses on the development of a robust, efficient, and accurate numerical framework that is capable of solving complex fluid flow problems, and one that is capable of overcoming most of the limitations generated by existing schemes. The proposed scheme is based on a clever approach to the merging of the traditional finite volume and the finite difference schemes. In the process of creating a new numerical scheme, the mean value theorem is used to evaluate the rates of change of fluxes at the center of the control volume.
2
The governing equations
When defining any numerical solution to a fluid dynamic problem, the conservation laws must be satisfied for an appropriate set of boundary conditions. As known in fluid dynamics, the conservation laws can be applied in two basic forms; the differential form and the integral form. However, experience has shown that when the integral form of the conservation laws is applied to fluid dynamics problems, high fidelity numerical solutions can be obtained. It is therefore no surprise that the Integro-Differential Scheme (IDS) is based on the integral formulation of the conservation laws described in subsections 2.2, 2.2 and 2.3. 2.1 Conservation of mass equation Consider the conservation of mass equation in the following form,
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Advanced Computational Methods in Heat Transfer IX
∫∫∫ v
∂ρ dv + ∫∫ ρ V d s = 0 . ∂t s
99 (1)
In eqn (1) ρ , v, t , represent density, volume, and time, respectively. The symbols, ds and V , represent the surface of the control volume and the fluid velocity, respectively. These quantities are defined through the use of the following vectors:
ds = dydzi + dxdz j + dxdyk
(2)
and V =ui +
v j+
wk .
(3)
2.2 Conservation of momentum equation Consider the conservation of momentum equation in the following form, ∂ ρ V dv + ∫∫ ρ V .d s V = − ∫∫ Pd s + ∫∫ τˆd s ∂ t ∫∫∫ v s s S
(
)
(4)
where the symbol, P, represents pressure and the symbol, τˆ, is the tensor that defines the various components of the local viscous stresses. This tensor can be described by the following equation:
τ xx τˆ = τ yx τ zx and the symbols,
τ xy
τ xz
τ yy
τ yz
(5)
τ xx ,τ xy ,τ yy ,τ yx ,τ zx ,τ zy and τ zz ,
are the local shear stress
τ zz
τ zy
components. 2.3 Conservation of energy equation Consider the conservation of energy equation in the following form, ∂ ∂t
∫∫∫ ρEdv + ∫∫ ρEV .ds = − ∫∫ PV .ds + ∫∫τˆ.V ds + ∫∫ q ds v
s
s
s
(6)
s
where the symbol, E, represents the total energy per unit mass of fluid. The vector, q , represents the rate of heat conducted per unit area through the surface of the control volume. In general, the vector, q , can be written in Cartesian coordinate format, such that q vis = q x i + q y j + q z k (7) where q x , q y , and q z represent the rate of heat conducted per unit area in x ,y, and z coordinate directions, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
100 Advanced Computational Methods in Heat Transfer IX
3
The Integro-Differential Scheme
The Integro-Differential Scheme (IDS) combines two schemes; namely, the finite volume scheme and finite difference scheme. IDS relies on the coupled behavior of discretized cells and their corresponding nodes. The numerical process is conducted in two alternating fashions, and for the sake of simplicity, only the two-dimension form of the IDS is explained in this paper. A typical control volume, illustrated in Figure 1, describes the numerical details associated with the finite volume formulation. Similarly, numerical details associated with the finite difference formulation are described through the use of Figure 2. 4’
3’
out
i,j+1
i-1,j+1 4
dy
3
1’
in
d 2’
1
2
dx Figure 1:
i-1,j
a
out dz
a
i-1,j-1
i+1,j+1
c i,j
b
i,j-1
i+1,j
i+1,j-1
in Finite volume model.
Figure 2:
Finite difference model.
Moreover, as illustrated in Figure 2, the center of any four neighboring control volumes, namely, control volumes; a, b, c, and d, is defined by the indices i and j. Any control volume will be defined locally by the nodes (1, 2, 3, and 4) as shown in Figure 1, and globally by its relative location to the point i,j as in Figure 2. 3.1 Application of the conservation laws to the control volume To demonstrate the utility of this numerical approach to fluid dynamic problems, consider a typical flow through the surfaces of an infinitesimal control volume, as illustrated in Figure 1. Even though the IDS has the potential to solve any 2D or 3D fluid-flow problem, for the purpose of simplicity, the discussions conducted in this paper are limited to 2D fluid flow problems. However, when describing the 2D approach, a major challenge involves the conversion of the naturally 3D conservation laws into their 2D counterparts that maintain the integrity of the 3D flowfield and its associated effects. To achieve this goal, the control volumes are chosen as infinitesimal rectangular prisms, with unit normal, ñ, in the x, y, and z directions. Also it was assumed that, the dimension, dz, of a typical control volume is always a single unit. These assumptions led to the fact that the fluid properties in the z-direction across any control volume are constants and the net flow of mass, momentum, and energy in the z-direction is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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101
always zero. Armed with these assumptions, the algebraic forms of the rate of change of mass, momentum, and energy at the center of each control volume are formulated as follows: ∂ρ ∂t
average
=
((ρ u )1
average
)3 )
+ (ρ u
+
2∆x
((ρ v )1
∂E ∂t
)4 ) − ((ρ u )2
+ (ρ u
(ρ Eu )1 + (ρ Eu = 2∆x
+ (ρ v )2 ) − ((ρ v )3 + (ρ v )4 ) 2∆y (ρ Eu )2 + (ρ Eu − 2∆x
)4
(ρ Ev )1 + (ρ Ev 2∆y 2 ∞
2 ∞
1 (τ Re L 1 τ xy Re L
( )lower
1 τ Re L
( yy )lower
1 τ Re L
)2
(ρ Tv )3 + (ρ Tv − 2∆y
)4
xx
v3 + v4 2∆y
− τ
( yy )upper
Re
∂ (ρ u ∂t
) average
=
L
(
2 ∞
Pr M
((ρ u ) 2
1
)left
− (q xx
(ρ u ) ) − ((ρ u ) 2
+
2
4
u1 + u 4 + 2∆x lower
yy
∆x
+
2
2∆x
(ρ u ) ) 2
∂ (ρ v ∂t
) average
=
((ρ v ) 2
1
((τ
L
+
xx
)left
3
− (τ
xx
(ρ v ) ) − ((ρ v ) 2
2
2
3
(
+ ρv
2
2∆y
((ρ uv )1
+
(ρ uv )4 ) − ((ρ uv )2
1 2 ∞
1 Re
L
((ρ T )1
((
τ
+
(
− τ ∆y
yy
)3
− (τ
xx
)upper
)
∆y
4
(11)
(ρ uv )3 )
(ρ T )2 ) − ((ρ T )3
)lower
(10)
+ ) −
+
2∆y yy
) ) + +
2∆x
γM
(9)
)
)4 )
lower
xy
)upper
+
) + ((τ )
)right
∆x
(
− q yy ∆y
((ρ vu )1 + (ρ vu )2 ) − ((ρ vu )3 + (ρ vu 2∆y 1 ((ρ T )1 + (ρ T )4 ) − ((ρ T )2 + (ρ T 2∆x γ M ∞2 1 Re
−
−
v1 + v 2 2∆y
) + ((q )
)right
+
u1 + u 4 − 2∆x v1 + v 2 − 2 ∆ y
)right
( xy )upper
− τ
(q xx
+
)3
v3 + v4 2∆y
)4
(ρ Tu )2 + (ρ Tu − 2∆x
( xy )left u 22 ∆+ xu 3 − (τ xy )right
1
+
)4
u2 + u3 − (τ 2∆x
)left
xx
)3
(ρ Ev )3 + (ρ Ev − 2∆y
(ρ Tv )1 + (ρ Tv 2∆y
1
γM
)2
(ρ Tu )1 + (ρ Tu 2∆x
1
γM
(8)
)upper
+
( ρ T )4 )
) + ((τ
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
xy
)left
−
(
− τ ∆x
xy
)right
)
102 Advanced Computational Methods in Heat Transfer IX The non-dimensional total energy, E, is E =
T γ (γ − 1 )M
2 ∞
+
u
2
+ v2 . 2
(12)
3.2 Flowfield Construction A careful examination of the governing eqns (8) – (11), indicates that the system is closed relative to four unknown variables, namely ρ , u , v, and T . These unknowns are included in a solution vector, U m , such that U
U U = U U
1 2 3 4
ρ = ρu ρv E
(13)
Using Taylor’s expansion, the solution can be constructed based of the following time marching scheme: dU m (14) (U m ) it ,+j∆ t = (U m ) ti , j + ∆t . dt i , j 3.3 IDS Marching Steps Eqn (14) represents a typical explicit time marching scheme. Like most established numerical schemes the IDS uses eqn (14). However, the major differences in the IDS as compared to the so-called established explicit schemes, is the way it handles the right side of eqn (14), namely, the old values of the m solution flux vector, (U m ) ti , j , the time derivative vector, (dU dt ) , and the i, j
time step, ∆t . 3.3.1 Evaluation of the time derivative The evaluation of the time derivatives, (dU
dt
)m , , is accomplished through the i, j
use of the mass, momentum, and energy equations. Eqns (8) – (11) are m implemented globally to obtain the time derivative (dU dt ) , at the center of i, j
each cell, a, b, c, and d. In another consistence averaging process, the time derivative at node, (i,j), is obtained as an arithmetic average of the time derivatives at the cell centers. 1 dU m dU m dU m dU m dU m = + + + dt 4 dt dt i, j a b dt c dt d i , j
(15)
3.3.2 Evaluation of the solution vector As indicated in Figure 3, information at the point of interest, (i, j), is updated solely based on the values of the point in question along with all its eight immediate neighbors. All required fluxes and derivatives are evaluated based on arithmetic averages of the primitive variables. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
i-1,j+1
i,j+1 d
i+1,j+1 c
i,j
i-1,j a
i+1,j b i+1,j-1
i,j-1
i-1,j-1 Figure 3:
103
Illustration of the IDS stencil.
3.3.3 Evaluation of the time increment Since the IDS is an explicit scheme, the time increment, ∆t , is subject to a stability criterion. To determine the size of the time step, the Courant-FriedrichsLewy (CFL) criterion, documented in Anderson [1], is used.
4 Results and discussions In this paper, the IDS is employed to solve two problems; namely, the hypersonic flat plate problem, and the shock/boundary layer interaction problem. Grid(101x101)
Grid(151x151)
Grid(201x201)
0.8 0.7 0.6
y/L
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
u/u_inf
Figure 4:
Grid independence studies.
4.1 The supersonic flow over a flat plate problem The hypersonic flow over a flat plate is a classical fluid dynamics problem, and in the past it has received considerable attention [1, 3, 7]. However, it has no exact analytical solution. The IDS solver was used to solve the flat plate problem under a variety of conditions, ranging from incompressible to compressible to hypersonic. The results provided in this study are for a Reynolds Number, Re = 1000, and a Mach number of 4.0. The results of validation studies conducted, using grid densities and residual errors are indicated in Figures 4 and 5. Grid studies were conducted over the following grid sizes; namely, 101x101, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
104 Advanced Computational Methods in Heat Transfer IX 151x151, and 201x201. Convergence studies were conducted on a Dell, Intel based PC until residual errors were in the range of 10-14 – 10-15. The plot in Figures 5 indicates the horizontal velocity profile obtained from the grid density studies. In Figure 6, the maximum residual obtained from the mass, momentum, and the energy fluxes is plotted as a function of the time step. To further strengthen the validity of the algorithm, the reference temperature method was used to evaluate the skin friction coefficient, Cf , and the wall heat transfer coefficient, and Stanton number, Ch (Rasmussen [8]). Data obtained from these studies were also positive.
Figure 5:
Figure 6:
Residual error studies.
Illustration of the shock boundary-layer interaction problem.
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Advanced Computational Methods in Heat Transfer IX
Figure 7:
105
Illustration of IDS obtained y-velocity component distribution. Present
Data in NASA Memor. 2-18-59W
P/P_inf
1.4
1.2
1.0 0.40
0.50
0.60
0.70
0.80
0.90
X/L
Figure 8:
Pressure distribution along the wall.
4.2 The shock/boundary layer interaction problem In 1959, Hakkinen et al. [9] studied the shock wave/boundary layer interaction problem experimentally. This problem is illustrated in Figure 6. More recently, due to the massive increase in computer capabilities, studies, [10, 11, 12], investigated this problem numerically. Using the IDS Solver, the inlet, outlet, and far field boundary conditions were set to be same as those of the flat plate problem. However, the flow on the top boundary is specified to form an oblique shock impinging on the wall. The bottom boundary consists of freestream and solid wall boundaries, whose lengths are 0.2 and 0.8 respectively. The flow Reynolds was set to 296000 and the Mach number set to 2.0. Figure 7 illustrates WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
106 Advanced Computational Methods in Heat Transfer IX the carpet plot of the y-velocity component obtained from this study. Figures 8 and 9 compare the pressure and the friction coefficient along the solid wall with the experimental data obtained by Hakkinen et al. [9]. The present results are in good agreement with the experimental data. Present
Data in NASA Memor. 2-18-59W
5
Cf X 1000
4 3 2 1 0 -1
0.2
Figure 9:
5
0.4
0.6
0.8
1
X/L
Skin friction distribution along the wall.
Conclusion
A new numerical scheme for solving equations that govern fluid dynamics problems was developed. This innovative scheme is called the ‘integrodifferential scheme’ and abbreviated as IDS. The scheme name depicts exactly what it says, by combining the integral form of the conservation laws to formulate the governing equations and transforming them in a suitable differential form for appropriate finite difference representation. The concept of the control volume was considered when calculating the integrations and the finite difference held for the numerical implementation of the scheme. In this paper the new scheme is employed to solve the viscous flow over a flat plate problem and the shock/boundary layer interaction problem. In both cases, the results showed very good agreement with the physical expectation of the flow, the empirical formulas, and the experimental data. This agreement solidified the belief that the scheme is robust, efficient, and capable of solving a variety of complex fluid dynamics problems.
References [1]
Anderson, Jr., J. D., “Computational Fluid Dynamics-The basis with applications”, McGraw-Hill, Inc., 1995.
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[2] [3] [4]
[5]
[6]
[7] [8] [9] [10] [11] [12]
107
Chung, T. J., “Computational Fluid Dynamics”, Cambridge University Press, 2002. Stephen Akwaboa, “Navier-Stokes Solver for a Supersonic Flow over a Rearward-Facing Step”, M. S. Thesis, (Department of Mechanical Engineering, North Carolina A & T State University, Greensboro, 2004). Chang, S. C., Wang, X. Y., & Chow, C. Y., “The space-time Conservation Element and Solution Element method: A new high-resolution and genuinely multidimensional paradigm for solving conservation law”, Journal of Computational Physics, Vol. 156, 1999, PP 89-136. Zeng-Chan Zhang, John Yu, S. T. & Sin-Chung. Chang, “The space-time Conservation Element and Solution Element Method for Solving the Two and Three Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes”, Journal of Computational Physics, Vol. 175, 2002, PP 168-199. Sin-Chung. Chang, “The Method of space-time Conservation Element and Solution Element- A New Approach for Solving the Navier-Stokes and Euler Equations”, Journal of Computational Physics, Vol. 119, 1995, PP 295-324. MacCormack, R. W., “Current Status of Numerical Solutions of the Navier-Stokes Equations”, AIAA paper no. 85-0032, 1985. Maurice Rasmussen, & David Ross Boyd, “Hypersonic flow”, John Wiley & Sons Inc., 1994. Hakkinen, R. J., Greber, I., Trilling, L. & Abarbanel, S. S., “The interaction of an oblique shock wave with a laminar boundary layer.”, NASA Memor. 2-18-59W, March (1959). Reyhner, T. A. , & Flugge-Lotz, I., “The interaction of a shock wave with a laminar boundary layer”, Int. J. Non-linear mechanics, Vol. 3, PP. 173199. Grasso, F., & Marini, M., “Analysis of Hypersonic Shock-Wave Laminar Boundary-Layer Interaction Phenomena”, Computers and Fluids Journal, Vol. 25, No. 6, PP 561-581, 1996. Moujin Zhang, John Yu, S. T. , & Sin-Chang, “Solving the Navier-Stokes Equations by the CESE Method”. AIAA paper no. 2004-0075, 2004.
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Advanced Computational Methods in Heat Transfer IX
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Solution of the radiative transfer problems in two-dimensional participating cylindrical medium with isotropic scattering using the SKN approximation N. Döner1 & Z. Altaç2 1
Mechanical Engineering Department, Dumlupınar University, Kütahya, Turkey 2 Eskişehir Osmangazi University, Eskişehir, Turkey
Abstract The SKN (Synthetic Kernel) approximation is applied to a two-dimensional homogeneous cylindrical participating medium with isotropic scattering. The SKN equations are tested against benchmark problems consisting of cold homogeneous participating medium. The solutions are compared with those obtained by various methods available in the literature. The SK3 approximation results for geometries of very large and very small aspect ratios are in excellent agreement with those of benchmark solutions. The SK3 solutions with moderate aspect ratios are accurate within several percent. Keywords: synthetic kernel method, participating medium, isotropic scattering, two-dimensional cylindrical medium.
1
Introduction
The SKN approximation which was developed to solve neutron transport equation can also be applied to radiative integral transfer equation (RITE) [1]. The significance of solving the integral equations is that one has to deal with the spatial variables rather than spatial and angular variables that exist in radiative transfer equation (RTE)-Boltzmann’s equation. Thus the solutions obtained from the integral equations do not exhibit ray effect which is a phenomena that plagues discrete ordinates and similar methods. Ray effect is the result of discretization of angular variables. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060111
110 Advanced Computational Methods in Heat Transfer IX The SKN method is applied to the RITE by substituting approximations to its radiative transfer kernels through a sum of exponentials, as in the exponential kernel approximation. Then RITE can be reduced into a set of coupled second-order partial differential equations, which will be referred to as the SKN equations. Physical boundary conditions are embedded in the force functions (surface integrals) of the RITE which are exactly preserved in the SKN equations; however, boundary conditions (BCs) of mathematical in nature are required to solve the SKN equations. Recently, the applicability and validity of the SKN approximation to thermal radiative transfer problems of one dimensional plane parallel and spherical geometries, and two-dimensional participating homogeneous and inhomogeneous medium have been investigated [2-4]. Solutions obtained with the SKN method are very accurate and certainly superior to P1 and differential or modified differential approximations. It has been also demonstrated that in twodimensional geometries the SKN method is a high order approximation which contains no ray effect [4]. Solving second order elliptic differential equations with simple boundary conditions lead to less computational efforts and cpu time. In this study, the SKN method with Gauss quadrature set has been applied to radiative transfer in two-dimensional cylindrical participating homogeneous medium. The accuracy and convergence of the method are investigated.
2
Derivations of SKN equations
The SKN equations for two-dimensional cylindrical medium can written as [9] 2 ∂ 2 ∂ 2 1 ∂ + 2 + 1 Gn (τ r ,τ z ) = S (τ r ,τ z ), − µ n 2 + ∂τ r τ r ∂τ r ∂τ z
n = 1, 2,..., N
(1)
where (τ r ,τ z ) are the optical coordinates, Gn (τ r ,τ z ) is a function representing the nth component of the synthetic kernel function, S (τ r ,τ z ) is the isotropic source function for a cold medium and is defined as S (τ r ,τ z ) = ω G (τ r ,τ z )
(2)
and ω is the scattering albedo. The incident radiation and heat flux can be defined as N
G (τ r ,τ z ) = F1 (τ r ,τ z ) + ∑ wn Gn (τ r ,τ z )
(3)
n =1
N
q(τ r ,τ z ) = F2 (τ r ,τ z ) + ∑ wn q n (τ r ,τ z )
(4)
n =1
where ( µ n , wn ) are the Gauss quadratures for µ ∈ (0,1) , q n (τ r ,τ z ) is a vector function defined as q n (τ r ,τ z ) = − µ n2 ∇Gn (τ r ,τ z ) [4], F1 (τ r ,τ z ) and F2 (τ r ,τ z ) WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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are the force functions containing physical boundary conditions (BCs). Then the net heat flux components can be also be expressed as N ∂Gn (τ r ,τ z ) (5) qr (τ r ,τ z ) = F2 r (τ r ,τ z ) − ∑ wn µ n2 ∂τ r n =1 N
qz (τ r ,τ z ) = F2 z (τ r ,τ z ) − ∑ wn µ n2 n =1
∂Gn (τ r ,τ z ) ∂τ z
(6)
The SKN equations are subject to the following mathematical BCs regardless of the physical BCs: For τ r = 0 ,
∂Gn (0,τ z ) =0 ∂τ r
(7)
For τ z = 0 ,
∂Gn (τ r , 0) 1 = G (τ , 0) ∂τ z µn n r
(8)
For τ z = L ,
∂Gn (τ r , L) 1 =− G (τ , L) ∂τ z µn n r
(9)
For τ r = R ,
∂Gn ( R,τ z ) K ( R / µn ) =− 1 G ( R, τ z ) ∂τ r µn K0 ( R / µn ) n
(10)
where K 0 ( x) and K1 ( x) are zeroth and first order modified Bessel functions. These BCs are adapted from the exact boundary conditions of the onedimensional SKN derivations. The source term of the SKN equations is given by Eq. (2). If the medium is pure absorber, Eq. (1) yields zero solution for Gn . Then the RITE is no longer an integral equation and the solutions for the incident energy and the net radiative heat flux; respectively, are simply F1 (τ r ,τ z ) and F2 (τ r ,τ z ) which are the exact solutions. As the scattering albedo increases, source coupling becomes stronger. If an iterative scheme is used in the numerical solution of the SKN equations, computation time for pure scattering cases is the highest. Previous studies revealed that the errors in the SKN method are also the highest for the pure scattering medium [2–4].
3 Results and discussions In order to test the applicability and the accuracy of the SKN approximation, we have adapted the following benchmark problems [9]: Benchmark Problem 1 (BP-1): A Short cylindrical medium with large optical radius, aspect ratios of 10 to 20, can be considered as a plane parallel geometry. The medium is homogeneous, cold and pure scattering ( ω = 1 ). The two-dimensional, solid cylinder solutions of SKN method for a combination of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
112 Advanced Computational Methods in Heat Transfer IX optical radius of 0.2 to 25 mean free path (mfp) and height of 0.002 to 5 mfp are compared with the plane parallel geometry solutions of Crosbie and Viskanta [5] for the incident energy at the top and the incident energy and the net radiative heat flux at bottom. Benchmark Problem 2 (BP-2): Long cylindrical geometry with small aspect rations for a cold medium with scattering albedos of ω = 0.3, 0.5 and 0.9 are considered. The two dimensional cylindrical SKN solutions for the incident energy at the center and the incident energy and the net radiative heat flux at the surface are compared with the one-dimensional cylindrical RITE solutions [6]. Benchmark Problem 3 (BP-3): Radiative transfer in a two-dimensional cylindrical medium (Figure 1) which is subject to collimated unit irradiation on top surface was considered [7]. The other surfaces are cold and transparent. The medium is also cold, homogeneous with pure isotropic scattering ( ω = 1 ). The incident energy and net radiative heat flux solutions using SKN approximation are compared with those of Wu and Wu [7] and Hsu et al. [8]. For BP-3, the surface integrals of the incident energy and the net radiative heat flux in the RITE yield [7]: F1 (τ r ,τ z ) = exp(−τ z )
(11)
F2 r (τ r ,τ z ) = 0
(12)
F2 z (τ r ,τ z ) = exp(−τ z )
τz
(13)
τz = 0
τr
φ τr θ I τz τz = L τr = 0 τr = R
Figure 1:
The geometry and the coordinate system.
This study was carried out on a Pentium III 800 MHz processor with 512 Mb RAM. Computation time with SK1, SK2 and SK3 approximations naturally increase with the order of approximation since the number of differential equations to be solved are also increased. In Table 1, the effects of grid refinement and CPU time on numerical calculations are depicted. It is clear that as the scattering albedo is increased for a WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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fixed grid size and the same order of approximation, cpu time increases due to strong coupling in the SKN equations which requires more iterations for convergence. However, SKN equations can be solved directly using a block tri-diagonal solver in which case the CPU time remains nearly the same. Several cases of grid combinations were considered to ensure grid independence of the presented solutions within four significant digits. The grid structure was increased with the optical dimensions such as using M r × M z =50×100, M r × M z =100×200 and M r × M z =200×200 for cylinders with R=0.125 and L=0.25 mfp, R=0.5 and L=1.0 mfp, and R=2.0 and L=4.0 mfp, respectively.
Table 1:
The effect of cpu time and grid refinement for cylinder of R=0.125 mfp and L=0.25 mfp.
ω = 0.5 Mr × Mz
10×10 10×20 25×50 50×50 50×100
SK1 0.016 0.184 1.414 4.75 33.7
SK2 CPU (sec) 0.034 0.635 2.765 9.69 72.5
Table 2:
R
Z =1.0 L J
0.2 0.002 2 0.2 5 0.5 5 1 10 2 12 2.5 25 5.0
20 20 20 10 10 10 10
ω = 1.0 SK3
SK1
0.062 0.671 4.215 31.85 149.3
0.189 0.283 2.084 9.26 68.6
SK2 CPU (sec) 0.262 0.431 4.814 20.98 173.9
SK3 0.412 0.876 8.0 39.12 432.0
Comparisons of the solutions for BP-1.
G ( L)
1.0452 1.2645 1.4968 1.7574 2.0706 2.1729 2.4552
1.0253 1.0742 1.0538 0.9661 0.7807 0.7028 0.4504
SK3
SK2
Exact (Ref. [5])
G (0)
q ( L) G (0,0) G (0, L) q (0, L) G (0,0) G (0, L) q (0, L) 0.9901 0.9087 0.7975 0.6587 0.4825 0.4240 0.2613
1.0284 1.2499 1.5021 1.7672 2.0776 2.1791 2.4645
1.0085 1.0637 1.0659 0.9757 0.7790 0.7002 0.4505
0.9907 0.9089 0.7974 0.6586 0.4829 0.4249 0.2654
1.0343 1.2669 1.5008 1.7569 2.0721 2.1756 2.4673
1.0145 1.0783 1.0572 0.9634 0.7806 0.7033 0.4516
0.9907 0.9089 0.7976 0.6594 0.4834 0.4252 0.2652
In Table 2, the relative errors of G(0,0), G(0,L) and q(0,L) for BP-1 are compared with the exact plane parallel solutions [5]. The agreement of the exact and the SKN solutions, with increasing orders, is excellent. For example, the SK3 solutions for pure scattering medium (γ=20 and L=0.5 mfp) yield relative errors of –0.26%, –0.32% and –0.01% for G(0,0), G(0,L) and q(0,L), respectively. The errors for the same cylinder are 0.07%, 0.10% and 0.03% in—though not given in Table 2—Wu and Wu [7]’s study. For γ=10 and L=1 mfp, the errors for G(0,0), G(0,L) and q(0,L)are 0.03%, 0.27% and –0.12%, respectively, when using SK3 approximation. In Wu and Wu [7]’s study, the errors are found to be 0.28%, 0.52% and 0.23%, respectively. In other cylinder configurations, the errors when using the SK3 approximation have the same order of magnitude. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
114 Advanced Computational Methods in Heat Transfer IX Comparisons of the exact solutions [6], of BP-2 with the SKN solutions are given in Table 3 for various scattering albedos. For a medium with ω=0.9 and R=0.5 mfp, the relative errors of G(0), G(R) and q(R) with SK3 approximation are found to be –0.99%, 0% and –0.63%, respectively. For the same scattering albedo and R=1.0 mfp, the errors yield 0.50%, 0.02% and –0.67% and for R=2.5 mfp, –0.15%, –0.19% and –1.77% for G(0), G(R) and q(R), respectively. The SK2 solutions are in excellent agreement yielding 2 to 3 significant digit accuracies, however, as the medium becomes strongly scattering, higher order, such as SK3 approximation, becomes necessary. Table 3: R=0.5
ω
Comparisons of the solutions for Benchmark Problem 2. L=5
G (0) / 4π G ( R ) / 4π − q ( R ) G (0) / 4π G ( R ) / 4π − q ( R ) G (0) / 4π G ( R ) / 4π − q ( R )
0.3 0.6011 0.5 0.6828 0.9 0.9183 R=1.0 0.3 0.3644 0.5 0.4571 0.9 0.8247
0.7891 0.8348 0.9588 L=5 0.6944 0.7475 0.9249
1.4890 0.6059 1.1707 0.6917 0.2939 0.9388
0.7891 1.4886 0.6026 0.8352 1.1704 0.6858 0.9609 0.2947 0.9274
0.7889 1.4900 0.8346 1.1723 0.9589 0.2957
2.1647 0.3658 1.8014 0.4608 0.5453 0.8440
0.6937 2.1666 0.3636 0.7465 1.8054 0.4554 0.9246 0.5523 0.8205
0.6943 2.1660 0.7474 1.8036 0.9247 0.5490
0.3 0.0811 0.5 0.1266 0.9 0.5298
0.6030 0.6510 0.8548
2.7308 0.0807 2.4317 0.1254 1.0522 0.5238
0.6028 2.7370 0.0812 0.6507 2.4421 0.1268 0.8547 1.0713 0.5306
0.6035 2.7384 0.6518 2.4439 0.8565 1.0709
In Figure 2, the radial distribution of the incident radiation at the top and bottom surfaces of BP-3 are given comparatively with those of Refs [7] and [8] for cylinder with optical dimensions of R=0.125 and L=0.25 mfp. The maximum relative errors with SK3 approximation on the top surface are -3.06% and –1.87% in comparison to the solutions of [7] and [8], respectively. On the other hand, these errors with SK3 approximation on the bottom surface are –2.4% and – 2.1% in comparison to the solutions of [7] and [8], respectively. In Figure 3, the axial distribution of the incident radiation at the center and outer surface are depicted comparatively with those of [7] and [8] for cylinder with optical dimensions of R=0.125 and L=0.25 mfp. The maximum relative errors with SK3 approximation at the centerline are –2.4% and –2.06% in comparison to the solutions of [7] and [8], respectively; whereas, these errors for the outer surface are –2.32% and –1.92%. In Figure 4, the radial distributions of the incident radiation at the top and bottom surfaces are comparatively presented with those of [7] and [8] for cylinder with optical dimensions of R=0.5 and L=1.0 mfp. The maximum relative errors with SK3 approximation on the top surface are –5.36% and – 3.96% in comparison to [7] and [8] solutions, respectively; while the maximum errors with SK3 approximation on the bottom surface are –8.5% and –3.9% when compared to those of [7] and [8] solutions, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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1.2
TOP
G
1.1
1.0
Wu&Wu Hsu&Tan SK3
0.9 BOTTOM
0.8 0.0
Figure 2:
0.2
0.4
r/R
0.6
0.8
1.0
Radial distribution of the incident radiation at the top and bottom surfaces for cylinder with optical dimensions of R=0.125 mfp and L=0.25 mfp. 1.2
1.1
G
CENTERLINE 1.0 RADIAL SURFACE
Wu&Wu
0.9
Hsu&Tan SK3
0.8 0.0
0.2
0.4
0.6
0.8
1.0
z/L
Figure 3:
Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=0.125 mfp and L=0.25 mfp.
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116 Advanced Computational Methods in Heat Transfer IX 1.4
TOP
1.2
1.0
Wu&Wu
G
Hsu&Tan SK3
0.8
BOTTOM
0.6
0.4 0.0
Figure 4:
0.2
0.4
r/R
0.6
0.8
1.0
Radial distribution of the incident radiation at the top and bottom Surfaces cylinder with optical dimensions of R=0.5 mfp and L=1 mfp. 1.6
1.2
G
CENTERLINE
RADIAL SURFACE
0.8
Wu&Wu
0.4
Hsu&Tan SK3
0.0 0.0
Figure 5:
0.2
0.4
0.6 z/L
0.8
1.0
Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=0.5 mfp and L=1 mfp.
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In Figure 5, the axial distribution of the incident radiation at the center and outer surface using SK3 approximation are shown comparatively with those of [7] and [8] for cylinder with R=0.5 and L=1.0 mfp. The maximum relative errors with SK3 are –5.4% and –3.97% at the center and –4.3% and –3.0% at the outer surface with respect to the solutions of [7] and [8], respectively. In Figure 6, the radial distributions of the incident radiation using SK3 approximation at the top and bottom surfaces are comparatively depicted with those of [7] and [8] for R=2 and L=4 mfp cylinder. The maximum relative errors of SK3 approximation on the top surface are –3.03% and 3.14% in comparison to the solutions of [7] and [8], respectively. On the other hand, the maximum errors with SK3 approximation on the bottom surface are –8.5% and 8.5% with respect to those of [7] and [8], respectively.
2.0 TOP
1.6
1.2
G
Wu&Wu Hsu&Tan SK3
0.8
0.4 BOTTOM
0.0 0.0
Figure 6:
0.2
0.4
r/R
0.6
0.8
1.0
Radial distribution of the incident radiation at the top and bottom surfaces for cylinder with optical dimensions of R=2 mfp and L=4 mfp.
In Figure 7, the axial distribution of the incident radiation using SK3 approximation at the center and outer surface are depicted comparatively with those of [7] and [8] for cylinder with R=2 and L=4 mfp. The maximum relative errors with SK3 approximation at the centerline are –3.03% and 3.14% in comparison to the solutions of [7] and [8] respectively; whereas, these errors for the outer surface are –4.31% and 0.83%, respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
118 Advanced Computational Methods in Heat Transfer IX 2.5 Wu&Wu Hsu&Tan SK3
2.0
1.5 G 1.0
CENTERLINE
0.5
RADIAL SURFACE
0.0 0.0
Figure 7:
0.2
0.4
z/L
0.6
0.8
1.0
Axial distribution of the incident radiation at the center and outer surface for cylinder with optical dimensions of R=2 mfp and L=4 mfp. 0.5
0.4
TOP
0.3 q
Wu&Wu Hsu&Tan SK3
0.2
BOTTOM
0.1
0.0 0.0
0.2
0.4
0.6
0.8
1.0
r/R
Figure 8:
Radial distribution of the radiative heat flux at the top and bottom surfaces for cylinder with optical dimensions of R=2 mfp and L=4 mfp.
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The incoming and/or outgoing radiative heat fluxes using SK3 also compare well within several percent errors, generally yielding underestimated values, with the referred studies. To illustrate the behavior of the radiative heat fluxes, we considered the distribution of cylinder with optical dimensions of R=2 and L=4 mfp. In Figure 8, the radial distribution of the outgoing radiative heat flux using SK3 approximation at the top and bottom surfaces are comparatively depicted with those of [7] and [8]. Solutions at the top and bottom surfaces from Wu and Wu [7] and Hsu et al. [8] almost coincide, and thus two the lines are not distinguishable. The maximum relative errors with SK3 approximation on the bottom surface are –1.92% and –1.87% in comparison to those of [7] and [8]], respectively, while these relative errors are –3.21 and –3.98%, respectively.
4 Conclusions The SKN approximation solutions for various two-dimensional homogeneous cylindrical mediums with isotropic scattering are compared with the solutions available in the literature. This study concludes the following: (i) the method is very accurate yielding 2-3 significant accurate solutions for mostly absorbing medium while the highest errors occur in pure scattering medium, (ii) the SKN equations can be numerically solved very easily with no numerical complexities, (iii) the approximation requires much less then computational effort when compared to the cpu time requirements of the exact RITE of the same grid configuration, (iv) the approximation can be improved especially at low orders by the selection of the synthetic kernel quadratures, (v) the method for moderate aspect ratios yield relative errors of several percent for pure scattering medium; however, medium with some absorption yield less errors, (vi) the method is free of “ray effect”.
References [1] Altaç Z. and Spinrad B.I., The SKN method I: A high order transport approximation to neutron transport problems, Nuclear Science and Engineering, 106, pp. 471-479,1990. [2] Altaç Z., The SKN approximation for solving radiative transfer problems in absorbing, emitting, and isotropically scattering plane-parallel medium: part 1, ASME Journal of Heat Transfer, 124, pp. 674-684, 2002 a. [3] Altaç Z., The SKN approximation for solving radiative transfer problems in absorbing, emitting, and linear anisotropically scattering plane-parallel medium: part 2, ASME Journal of Heat Transfer, 124, pp. 685-695, 2002 b. [4] Altaç Z. and Tekkalmaz M., The SKN approximation for solving radiation transport problems in absorbing, emitting and scattering rectangular geometries, JQSRT, Volume 73, No. 4, pp. 219-230, 2002. [5] Crosbie A.L. and Viskanta R., Nongray Radiative Transfer in a Planar Medium Exposed to a Collimated Flux, JQSRT, Volume 10, pp. 465-485, 1970.
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120 Advanced Computational Methods in Heat Transfer IX [6] Altaç Z., Radiative Transfer in Absorbing, Emitting and Linearly Anisotropic-Scattering Inhomogeneous Cylindrical Medium, JQSRT, Volume 77, pp. 177-192, 2003. [7] Wu S.C. and Wu C.Y., Radiative Heat Transfer in a Two-Dimensional Cylindrical Medium Exposed to Collimated Radiation, International Comm. Heat & Mass Transfer, Volume24, No 4, pp. 475-484, 1997. [8] Hsu P.F., Tan Z.M., Wu S.H. and Wu C.Y., Radiative Heat Transfer in Finite Cylindrical Homogeneous and Nonhomogeneous Scattering Media Exposed to Collimated Radiation, Numerical Heat Transfer, Part A, Volume35, pp. 655-679, 1999. [9] Döner N., Calculation of Radiative Heat Transfer in Cylindrical Participating Medium Using the SKN Method, Ph. D. Thesis, Osmangazi University (in Turkish), 2003.
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Presentation of the hemisphere method P. Vueghs & P. Beckers Aerospace and Mechanical Engineering Department, University of Liege, Belgium
Abstract In the field of image synthesis or thermal radiation, the challenge is to calculate the radiative exchanges between the surfaces of the model. To quantify the interactions between the surfaces, we define a number called view factor, noted Fij. The view factor represents the fraction of the diffusely emitted power from a surface Ai which reaches a surface Aj. The calculation of the view factors is a very difficult problem. A method has been designed in image synthesis and also applied to thermal problems. This method, called hemi cube, uses a cube as a screen onto which all the scene is projected. Here, we present another method that we have named hemisphere, where the surface of projection is the unit sphere surrounded by the hemi cube. This projection, based on the Nusselt’s Analogy, is simpler and more natural than the hemi cube’s projection. We show in this paper that the hemisphere method is faster and more efficient than the hemi cube. In this paper, we present a tessellation of the hemisphere into cells characterized by an equal elementary view factor. This allows us to avoid over sampling and to save computation time. We show that this method is faster and more efficient than the classic hemi cube. We present also some results that illustrate the efficiency of the hemisphere method. Keywords: view factor, hemisphere, projection, spherical coordinates, thermal radiation, Nusselt’s Analogy, ray casting.
1
Definition of the view factors
In the field of image synthesis, it is necessary to calculate the inter reflection of light between all the surfaces that compose the scene [9]. In the same way, in thermic, we have to evaluate all the heat transfers between all the surfaces of the model.
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122 Advanced Computational Methods in Heat Transfer IX The exchanges of energy between two surfaces depend on the geometrical configuration of these surfaces. Mathematically, this geometrical dependency is expressed by a function called view factor [2]. The view factor expresses that the exchange of energy between two surfaces depends on their sizes, their relative orientations and the distance between them. The view factor does not depend on the radiative properties of the surfaces, such as absorbance and emittance. By definition, the view factor between two surfaces Ai and Aj is the fraction of the energy emitted diffusely by the surface Ai which reaches the surface Aj. We can note here the ambiguity of this definition based on the notions of emitted and absorbed energy although only the geometrical parameters appear in the definition, the surface properties do not appear. According to [14], if we assume that two surfaces Ai and Aj are fully visible to each other, the diffuse view factor between these surfaces is given by
FAi − Aj =
1 Ai
∫ ∫ Ai
cos θi cos θ j
Aj
π rij2
dAi dAj
(1)
JG
where θ is the angle between the normal vector at the surface and the vector rij
which joins points on the two surfaces. This expression has no analytical solution, except in the case of particular geometrical configurations [3, 6, 7]. It implies a numerical resolution of the integration involved in the computation of the view factors. Several techniques have been designed. If we consider the view factor between a point on the surface i, noted dAi, and the surface Aj, eqn (1) can be simplified. It is the limit of the usual view factor when the area Ai goes to zero. This new view factor is called pointwise view factor and is the inner integral of eqn (1)
FdAi − Aj = ∫
cos θi cos θ j
Aj
π rij2
dAj
(2)
We can evaluate this expression by different analytical methods. The most usual in image synthesis is called hemi cube [13]. The method presented in this article is mainly based on the hemi cube method. The principle is the same. It is also linked to the idea presented by F. Sillion to decompose the projection surface into cells of equal view factors [4]. F. Sillion proposed to project all the environment onto a single plane which takes the place of the top face of the hemi cube. He introduced a quadrangular mesh of this plane. The cells of this mesh are characterized by approximately equal elementary view factors. This idea has also been developed by Vivo et al. [11] and Lluch et al. [16]. They used the polar coordinates to obtain cells of equal view factors in the single plane. With this method, they conserve the axial symmetry of the formula of the view factor. The main disadvantage of the single plane methods is that the portion of the environment situated near the horizon cannot be taken into account. However, if the size of the plane is sufficient, we can reduce the neglected portion to a given threshold.
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Here, we choose to project the scene onto the hemisphere because it is easier to mesh this surface than a plane of a cube, if we want to obtain cells of equal view factors. We obtain a very regular mesh which respects the axial symmetry of the view factor. We can also guarantee that the entire scene is taken into account in this method, unlike in the single plane methods.
2
The hemi cube
The hemi cube was designed and presented in 1985 by Donald P. Greenberg and Michael F. Cohen [13]. It was first used in image synthesis and after to solve thermal analysis [10]. This method consists of drawing a cube centered around the origin. The
JG
z axis coincides with the normal vector ni . The construction is represented in figure 1 according to references [8, 13].
Figure 1: Principle of the hemi cube. The cube is discretized into a number of cells. Each cell is associated with an element of view factor, called delta view factor. These elements are contributions to the pointwise view factor. Each value depends on the position of the cell on the hemi cube. The cells near the z axis are associated with the highest values and the cells located near the edges are characterized by lower values. This remark explains why we have over sampling. This method presents drawbacks: • an irregular sampling of the space, which implies that some areas are over sampled to guarantee a minimal accuracy. • the contributions to the pointwise view factor are not equal. We need to compute the value and to store it for each cell of the hemi cube. Nevertheless, the main advantage of this technique is that we can easily take occlusions into account by performing a test based on the distance of the surfaces projected onto the hemi cube. If many surfaces are projected onto the same cell of the hemi cube, we only keep the nearest one, the others will not be visible from the origin. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
124 Advanced Computational Methods in Heat Transfer IX
3
Distribution of elements of equal view factors
In this section, we will present the solution we chose to obtain a regular mesh of cells with equal view factors. First, we will call back the definition of the view factor and the Nusselt’s Analogy. After that, we will mesh the hemisphere in spherical coordinates, in colatitude θ and in longitude ϕ. 3.1 Definition of the view factor and Nusselt’s Analogy According to Puech and Sillion [5], the view factor between a point xi on a surface Ai and a surface Aj (fully visible from xi) is noted FdAi − Aj and is given by eqn (2). Note that to establish this equation, we assumed that patch j is fully visible from xi. In other formulations, the visibility function appears explicitly. This function is defined between two points, it is equal to 1 if the points are mutually visible and 0 otherwise. This function is discontinuous and increases the difficulty of the computation. If we use the Nusselt’s Analogy (cfr [15] cited by [5]), the pointwise view factor FdAi − Aj can be considered as the result of two successive projections: •
a first projection onto the unit sphere centered on xi. This step corresponds to the factor
cos θ j rij2
in the relation (2). The solid angle
subtended by the surface dAj is given by d ω j =
cos θ j dAj rij2
.
•
a second orthogonal projection down onto the plane of the surface Ai, which corresponds to the factor cos θi. Now we can transform the relation (2). We obtain the following expression
FdAi − Aj = ∫
cos θi
Ωj
where Ωj is the solid angle subtended by Aj.
π
dω j
(3)
3.2 Mesh of the hemisphere Now, with the help of the Nusselt’s Analogy, we will decompose the hemisphere into solid angles. Each solid angle will be characterized by the same view factor. The decomposition will be performed following the meridians and parallels, i.e. in terms of latitude and longitude. The total number of cells N will be obtained by multiplying the numbers of subdivisions into the longitude and latitude directions, nlon and nlat.
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3.2.1 In terms of colatitude In figure 2, if we consider an elementary surface dAi and a spherical cap (of unit radius) defined between θ=0 and θ=θS, the view factor is calculated by
FdAi − Aj =
Figure 2:
1
2π
θS
0
0
π∫ ∫
cos θ sin θ dθ dϕ = sin 2 θ S
(4)
Spherical cap of angular aperture θS.
By extension, we can determine a sequence of values θ which correspond to spherical caps larger and larger. These caps delimit rings centered on the normal vector at the origin. The rings are characterized by a same value of the view factor with respect to the center of the emitting surface. If we fix the resolution in colatitude nlat, we can subdivide the hemisphere in spherical rings with equal view factors. 3.2.2 In terms of longitude After that, we can chose the discretization of the hemisphere in longitude simply by dividing the rings in a determined number nlon of sectors. All the sectors must have the same angular aperture. 3.2.3 Resulting mesh So we have subdivided the hemisphere along the two spherical coordinates θ (colatitude) and ϕ (longitude). The view factor associated to each cell is equal to
1 . Figure 3 shows the hemisphere subdivided in longitude and latitude. nlat nlon
The cells are quadrangular, except the cells which have a vertex vertically above the center of the hemisphere and which are triangular.
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126 Advanced Computational Methods in Heat Transfer IX
Figure 3:
4
Wireframe view of the hemisphere.
Calculation of the view factor
The origin dAi is placed in a more or less complicated scene. The complexity of the scene is linked to the number of surfaces which compose the scene. The number of view factors to compute is equal to the number of surfaces in the scene. After meshing the hemisphere, we cast a ray along each direction {θ, ϕ}. For each ray, we first compute all the possible intersections with all the surfaces of the scene. If no intersection is detected, this means that the model presents an aperture. For example, this is the case of a satellite which orbits around the Earth and evacuates the power dissipated by its equipments to the deep space at 3K. The contribution to the view factor is added to a node which does not belong to the model. This supplementary node represents the deep space and gathers all the energy fluxes emitted by the spacecraft. When only one intersection is detected, the contribution to the view factor is assigned to the intersected surface. If several intersections are detected along the same direction, we store the distance of each intersection. At the end of the process, we only keep the nearest intersected surface. This technique allows us to solve the visibility problem very easily and quickly.
5
Error analysis
The hemisphere method has been programmed and tested. The results are shown below. Among the several tests we have performed, we chose to present the following one. For 2 perpendicular rectangles sharing a common edge, the aspect ratio of the rectangles is D and E, with respect to the common edge. In this case, it is easy to obtain an analytical solution of the view factor between the two surfaces. In accordance to [6], the resulting formula is given below
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Advanced Computational Methods in Heat Transfer IX
FAi − Aj =
1 1 1 E arctan + D arctan πE E D
1 − D 2 + E 2 arctan 2 2 D +E
(
)
E 2 1 + D2 + E 2 2 2 2 1 + E E + D
(
)(
)
E2
(
)(
2 2 1 1 + E 1 + D + ln 2 2 4 1 + D + E
(
)
D2 1 + D2 + E 2 2 2 2 1 + D D + E
(
)(
)
D2
127
) (5)
Here, the two geometrical parameters are equal to 1. The exact value is 0.200044. We used analytical solutions to verify the hemisphere method (both point-toarea and area-to-area formulations). We utilized the Gauss quadrature formula to integrate the elementary view factors from the hemisphere onto the area of the square. Here we used 5 Gauss points in each direction (25 points in total). We made use of the norm L1 to compute the relative error, i.e. the difference between the exact solution (VFanalytical=0.200044) and the integrated hemisphere’s view factor VFhemisphere.
ε=
VFanalytical − VFhemisphere
(6)
VFanalytical
We performed this calculation for nlat and nlon varying from 20 to 100 (it is not useful to plot the results for lower resolutions, the calculated error will be too high). The results are plotted at figure 4(a).
(a) Figure 4:
(b)
(a) Error for different resolutions. (b) Theoretical error for different resolutions.
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128 Advanced Computational Methods in Heat Transfer IX The abscissa x and y are the values of the parameters nlon and nlat respectively. The z axis represents the logarithm of the relative error. In this figure, the error varies between -2.5 and -4, i.e. 10-2.5 and 10-4. We can observe that the error decreases when the resolution increases. The error can be approximated by ε ≈ 1 . This error is plotted at figure 4(b). For nlon nlat 10 000 casted rays (nlon =100 and nlat =100), the error will be around 10-4 10−4 . Remark: during this test, we found some difficulties with this particular geometry. The vertical edges of the square are projected onto the hemisphere along meridians. When the mesh of the hemisphere and the projected edges are superimposed, nearly all the cells in the meridian are wrong. The error is independent of the colatitude resolution, around
1 . This is called nlon
discretization error. It gives oscillations in the error graphics. To prevent this case, we implemented a variant of the hemisphere, where the rays are randomly casted through each cell. We compute a mean value for each cell and we find an error close to
1 . The mathematical theory which supports this method is nlon nlat
not presented in this article but is developed in the internal report [17]. This random method has an evident drawback. The results are affected by the noise that characterizes all random processes. Nevertheless, this method is more reliable since it is not affected by the discretization error. If we consider the corresponding error plots, the oscillations disappear. Such error plots are shown in reference [17].
6
Results
In order to illustrate the hemisphere method, we implemented the well known scene called the Cornell box [1]. Picture 5 was obtained with a mesh of around 2 000 triangular cells. We used the random version of the hemisphere, with 4 rays per cell. To display this view, we used the tone mapping method called mean value mapping [12] and linear interpolation. The inter reflection of light is correctly modeled. The penumbra caused by the obstacles is also automatically computed. Note that a colour copy of all these pictures is accessible from [18].
7
Conclusions
In this article, we have presented a simple method which allows us to compute the view factors from a point to surfaces of finite dimensions. This method allows us to keep a good physical comprehension of the problem.
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Figure 5: Cornell box using the hemisphere’s method. Compared to the hemi cube method, this method is faster and requires less memory. This method needs only one projection, the algorithm is simpler. All the elementary view factors are equal, which prevents useless over sampling and the storage of all the contributions to the view factors. Compared to the single plane method, the hemisphere method is as fast. Nevertheless, we can study the whole hemisphere, without neglecting the surfaces close to the horizon.
References [1] [2] [3]
The Cornell Box, http://www.graphics.cornell.edu/online/box/. Donald P. Greenberg, Benneth Bataille, Cindy M. Goral, Kenneth E. Torrance, Modeling the interactions of light between diffuse surfaces. Computer Graphics, 6(2), pp. 213-222, 1984. W. R. Morgan, D. C. Hamilton, Radiant-interchange configuration factors, Technical report 2836, National Advisory Committee for Aeronautics, 1952. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
130 Advanced Computational Methods in Heat Transfer IX [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18]
Claude Puech, François Sillion, A general two-pass method integrating specular and diffuse reflection, Computer Graphics, 23(3), pp. 335-344, 1989. Claude Puech, François Sillion. Radiosity and Global Illumination, Morgan Kaufmann Publishers, Inc., 1994. John R. Howell. A Catalog of Radiation Heat Transfer Configuration Factors, McGraw Hill, 1982. http://www.me.utexas.edu/~howell/. H. Y. Wong. Handbook of Essential Formulae and Data on Heat Transfer for Engineers. Longman, 1976. Stéphane Jardon, Première approche sur l’étude des facteurs de vue, 2003, Mechanical Department, University of Liège, Belgium. James T. Kajiya, The rendering equation, SIGGRAPH, 20(4), 1986. T. G. Gonda, K. R. Johnson, A. R. Curran. Development of a signature supercode, Technical Proceedings 1938: Advances in Sensors, Radiometric Calibration, and Processing of Remotely Sensed Data (SPIE International Symposium on Optical Engineering and Photonics in Aerospace and Remote Sensing): Orlando, FL, 1993. Roberto Vivo, Maria José Vicent, Javier Lluch. Un método para el calculo del factor de forma alternativo al hemi-cubo. Universidad Politécnica de Valencia. Kresimir Matkovic. Tone mapping techniques and color image difference in global illumination. Master’s thesis, Technischen Universität Wien, February 1998. Donald P. Greenberg, Michael F. Cohen, The hemi cube – a radiosity solution for complex environments, Computer Graphics, 19(3), pp. 31-40, 1985. John R. Wallace, Michael F. Cohen. Radiosity and Realistic Image Synthesis, Morgan Kaufmann Publishers, Inc, 1993. W. Nusselt, Graphische Bestimmung des Winkelverhaltnisses bei der Wärmestrahlung, Zeitschrift des Vereines Deutscher Ingenieure, 72(20), pp. 673, 1928. J. Lluch, R. Molla, P. Jorquera, R. Vivo, M. J. Vicent, Study of the form factor calculation by single polar plane. Department of Computer Systems and Computation, Universidad Politécnica de Valencia. P. E. Vueghs, Méthode de l’hémisphère – validation de la méthode, technical report, University of Liège, Belgium, 2005. LTAS – Infographie, Université de Liège, http://www.ulg.ac.be/ltas-cao/.
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Temperature identification based on pointwise transient measurements A. Nassiopoulos & F. Bourquin French Public Works Research Laboratory (LCPC), Paris, France
Abstract The inverse problem of temperature identification based on discrete transient measurements is considered. An iterative procedure combining the conjugate gradient algorithm with Tikhonov regularization is used to determine unknown boundary heat fluxes. We show numerically and theoretically that the choice of an H 1 -type space for the minimization gives much better results than a L2 -type one. A dual formulation of the problem coupled with a model reduction method is used to construct a fast and accurate algorithm suited for real time computations in the three-dimensional case. Keywords: temperature reconstruction, inverse heat conduction, optimal control theory, adjoint method.
1 Introduction Various methods for damage detection and structural health monitoring exist. Among them, the output-only techniques consist in identifying the low frequency spectrum of a given structure in order to detect modifications with respect to the non-damaged reference state. However, in civil engineering, thermal loading due to environmental factors induces mechanical stress on structures and can cause eigenfrequency shifts of much larger order of magnitude than those caused by structural damage [1]. Thus, the knowledge of the exact thermal state of a given structure and its effects on vibrational properties appears as a bottleneck in this field [2]. The need for temperature identification arises from the partial information on the thermal state of a structure one can get. The most commonly used temperature sensors can only provide local or pointwise measurements. In addition, some parts of a structure can simply be inaccessible to direct measurements. This is the case WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060131
132 Advanced Computational Methods in Heat Transfer IX in civil engineering structures, although the number of sensors that equip modern ones is rapidly increasing. The thermal state at any point of a given structure can only be obtained through a suitable reconstruction procedure. Among possible methods, one will eliminate Kalman filtering techniques in continuous time which would lead to the solution of Riccati equations of great dimension, and that seems inappropriate without preliminary reduction. The temperature identification problem can be seen as an inverse problem consisting in inverting the heat equation. This class of problems are in general mathematically ill-posed and regularization techniques are needed to solve them. They have been studied extensively in literature during the last few decades together with many techniques for their numerical computation [3]. Inverse problems concerning heat conduction have been the topic of numerous works due to the wide range of applications that are concerned [4–7]. A review of literature can be found in [8–10]. This paper deals with the inverse problem of temperature identification based on discrete measurements. In view of easy implementation in general purpose finite element software, the approach is that of optimal control theory [11]. The problem is written in a least squares setting. The adjoint technique [12, 13] is used to determine the gradient of an error functional together with Tikhonov regularization [3]. When standard gradient-based algorithms are put to work, each computational step entails solving the same forward heat equation with changing data. A dual formulation of the problem and a model reduction technique are put to work to obtain a high speed accurate algorithm suited for real-time applications.
2 Problem statement Consider a solid in a multidimensional domain Ω with boundary ∂Ω. Assuming absence of internal heat sources and zero initial temperature, the temperature field inside the solid is given by the heat equation: ∂θ ρc ∂t − div(K grad θ) = 0 in Ω × [0, T ] (K grad θ) · n + αθ = Φ on ∂Ω × [0, T ] θ(x, 0) = 0 in Ω
(1)
Here, Φ(x, t) = g(x, t) + αθext where g denotes an inward heat flux with FourierRobin conditions and θext is the external temperature, n is the outwards normal vector on the boundary, x ∈ Ω is the space variable, t ∈ [0, T ] the time variable, ρ the mass density, c the heat capacity and K the scalar conductivity coefficient of the material taken to be homogeneous and isotropic for the sake of simplicity. Assume that m sensors are available inside the structure at locations xk , k = 1..m. They deliver the data {θkd (t)}m k=1 , t ∈ [0, T ]. Based on these measurements, the aim is to reconstruct the temperature field over the time interval [0, T ], focusing on the accuracy of the temperature field reconstruction at the final instant T . WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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One way to achieve this is to seek the boundary flux Φ responsible for the measurements. The problem of recovering the boundary condition Φ can be viewed as an operator inversion problem. Consider operator T defined by T :
U →M T Φ = {θ(xk , t)}m k=1
(2)
where θ(x, t) is the temperature field verifying the heat equation (1). In the above, U is the control space (such as Φ ∈ U ) and M = L2 ([0, T ])m is the measurements space (such that {θkd (t)}m k=1 ∈ M ). Note that we have not explicitly defined so far the nature of space U as this will be the object of section 3. Under these notations, the temperature reconstruction problem consists in finding Φ such that T Φ = {θkd }m k=1
(3)
For the sake of simplicity, only the case of the boundary condition Φ reconstruction is considered in this paper. It is possible to extend all that follows to the case where one wants to reconstruct a couple of functions {θ0 , Φ} with θ0 (x) = θ(x, 0) an unknown initial thermal state in (1). Due to the smoothing properties of operator T , the latter is not invertible and the inverse problem (3) is mathematically ill-posed in the sense of Hadamard. The solution of (3) can only be derived in a least squares sense through a functional minimization procedure. The functional can be for instance a quadratic form of the residual term E = θ(xk , t) − θkd (t), (k = 1..m), which measures the distance between the data θkd (t) and the values at sensor locations of the temperature field determined by Φ. J(Φ)
= =
m 2 1 T θ(xk , t) − θkd (t) dt + Φ2U 2 2 k=1 0 1 d m 2 2 T Φ − {θk }k=1 M + ΦU 2 2
(4)
The last term stands for the Tikhonov regularization, being a small coefficient that guarantees numerical stability even with noisy input data. · 2U and · 2M are suitable norms in U and M respectively.
3 Minimization procedures The results of the minimization process strongly depend on the a priori assumptions on Φ that determine the choice of space U . In this section two different possibilities are considered, showing that better results are obtained within the framework of an H 1 -type space. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
134 Advanced Computational Methods in Heat Transfer IX 3.1 Minimization in L2 The most natural choice consists in looking for a solution Φ belonging to the space of square integrable functions, namely L2 (∂Ω×[0, T ]). This space can be equipped with the scalar product: T
u, v U = uv dγ dt, ∀u, v ∈ U (5) 0
∂Ω
The adjoint technique is employed to write J as a quadratic form. To this end, we introduce the adjoint operator T ∗ whenever it exists: T∗ :
M →U T ∗ {ϑk (t)}m k=1 = ϕ(x, t)
(6)
where φ is the restriction over ∂Ω of the so-called adjoint state p: M −ρcpt − div(Kgrad p) = ϑk (t)δxk in Ω × [0, T ] (Kgrad p) · n + αp = 0 p(x, T ) = 0
k=1
on ∂Ω × [0, T ] in Ω
(7)
ϕ(x, t) = p(x|∂Ω , t) Operators T and T ∗ verify the duality relation which writes m M T Φ, {ϑk }k=1 M
=
U Φ, T
∗
{ϑk }m k=1 U
(8)
In the above, U and M are the dual spaces of U and M respectively and angle brackets stand for the duality pairing. To formally prove this relation, one has to rewrite equations (1) and (7) in variational form. After an integration by parts of the time derivative term and combination of the last two expressions, one obtains m k=1
0
T
θ(xk , t)ϑk (t) dt =
0
T
Φϕ dγ dt
(9)
∂Ω
which is exactly relation (8). Using (8), the functional J can be written as a quadratic form of Φ: J
= = =
1 2 2 T Φ − {θkd }m k=1 M + ΦU 2 2
1 d m 2 2 T Φ2M − 2 T Φ, {θkd }m k=1 M + {θk }k=1 M + ΦU 2 2
1 AΦ, Φ U − Φ, b U + c 2
(10)
d m 2 with A = J T ∗ T + I, b = J T ∗ {θkd }m k=1 and c = {θk }k=1 M . J is the Riesz isomorphism between U and U while I stands for the identity operator. It follows
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directly that the gradient of J is given by: ∇J(Φ) = J T ∗ (T Φ − θkd (t)) + Φ
(11)
This expression means that the gradient of the functional is the restriction on the boundary ∂Ω of the adjoint field obtained by (7) with a source term {ϑk (t)}m k=1 equal to the residual E = {θ(xk , t) − θkd (t)}m k=1 . This method of obtaining the gradient allows for a gradient-type method to be used. Each iteration will consist of a computation of the direct equation (1), followed by a computation of the adjoint equation (7) with {ϑk (t)}m k=1 = E. The gradient will then be given by p|∂Ω + Φ. Here, the classical conjugate gradient algorithm was used giving very satisfactory results [7]. Note that equations (1) and (7) have the same structure, so that the same numerical procedure can be used to solve them. The whole algorithm can thus be easily implemented in any classical general purpose scientific software, this being one of the main advantages of the method. The results below show the disadvantages of the approach described so far. They concern a one-dimensional beam of length L to which a given flux is prescribed at each end. All material constants are set to 1. Some direct simulations with an arbitrary flux input give measurements on sensors located at L5 , L2 and 4L 5 respectively. These measurements are then used to simulate the reconstruction algorithm. The model is discretized with P1 finite elements and an implicit Euler scheme is used for time integration. In figure 1, the reconstructed flux (dotted line) is compared to the prescribed one (solid line). The flux is well reconstructed on almost the entire time interval, but the accuracy of the results drops near the final instant: the curve exhibits some oscillations and moves away from the target curve to reach zero at t = T . As a consequence, the reconstruction of temperature field is of acceptable accuracy far from t = T , while the reconstruction of θ(x, T ), which is the most interesting output in view of applications, is very unsatisfactory. This phenomenon is due to the property assigned to the adjoint field p to be null at t = T (equation (7)) and is well known in the literature [8, 13]. This final condition on p has to be prescribed in order to verify relation (8). An alternative definition of the adjoint field can be considered, but this implies that the space U has also to be modified. This is the topic of the next section. 3.2 Minimization in H 1 A change of framework for the minimization procedure overcomes this final instant problem. It consists in choosing an H 1 -type space instead of the L2 -type one. Take U = H 1 ([0, T ], L2 (∂Ω)) with the scalar product
u, v
U
=
0
T
uv dγ dt + ∂Ω
0
T
∂t u∂t v dγ dt, ∀u, v ∈ U ∂Ω
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136 Advanced Computational Methods in Heat Transfer IX
Figure 1: Reconstruction in L2 : flux (left) and final temperature reconstruction (right). In order to verify relation (8), the operator T ∗ has to be defined in a different manner. One can easily verify that if ϕ in (6) is defined by 0
T
φp dt =
0
T
φϕ dt +
0
T
∂t φ∂t ϕ dt, ∀φ ∈ H 1 ([0, T ] × ∂Ω)
(12)
where p verifies (7), then operator T ∗ verifies again the duality relation (8) with the new choice of U . As can be seen, the computation of T ∗ needs one more step involving the solution of (12) with p known. This last equation is of the form L(φ) = a(ϕ, φ), ∀φ, a(·, ·) being a quadratic and L(·) a linear form. It can be easily implemented with the finite element method. The great advantage of this is that the end condition on ϕ with respect to time is not a Dirichlet one anymore. As a consequence, the value of Φ at t = T is not fixed and the accuracy of reconstruction near the final instant is much better as can be seen in figure 2 corresponding to the same case as previously. The curves of reconstructed and target final temperatures are identical.
Figure 2: Reconstruction in H 1 : flux (left) and final temperature reconstruction (right). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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4 Dual formulation Consider the self-adjoint operator T T ∗ . Consider next the problem of finding ∗ m d m {Xk (t)}m k=1 ∈ M such that T T {Xk (t)}k=1 = {θk (t)}k=1 . If we set Φ = ∗ m T {Xk (t)}k=1 , then solving (3) is equivalent to finding {Xk (t)}m k=1 ∈ M such that d m (13) T T ∗ {Xk (t)}m k=1 = {θk (t)}k=1 This new problem is called the dual of (3). It exhibits an ill-posed nature and some form of regularization is needed. Applying Tikhonov regularization, one will consider the problem m d m T T ∗ {Xk (t)}m k=1 + {Xk (t)}k=1 = {θk (t)}k=1
(14)
Due to the self-adjoint nature of T T ∗ , the latter is equivalent to minimizing the dual functional J : 1 m A{Xk (t)}m J = − {Xk (t)}m (15) k=1 , {Xk (t)}k=1 k=1 , b 2 M M with A = T T ∗ + I and b = {θkd (t)}m k=1 . The dual formulation exhibits many features of interest. Firstly, the problem is set over space M which is of much smaller size than the corresponding space U . As a consequence, fewer search directions are involved in the minimization procedure and the conjugate gradient method is expected to converge much faster. Secondly, all scalar products that need to be computed are scalar products in M rather than in U so that the complexity of the algorithm depends more on the number of sensors than on the size of domain Ω. This new formulation results in dramatic reduction of computational costs and in greater adaptability to different geometries. Furthermore, it allows some parallelization of the algorithm, this issue being however beyond the scope of this paper.
5 Model reduction ∞ 1 We introduce two distinct function bases {ξi }∞ i=1 , {χi }i=1 that span H (Ω), and a ∞ 2 third function basis {ηi }i=1 that spans the space L (∂Ω) such that we can approximate the fields θ, p and Φ by a reduced order linear combination:
θ p Φ
∼ θr (x, t) = ∼ pr (x, t) = ∼ Φr (x, t) =
αi (t)ξi (x)
i=1 ˇ
βi (t)χi (x)
i=1 ˜
(16)
γi (t)ηi (x)
i=1
These approximations result in a model reduction that enables a considerable speed-up of the computational procedure, since the size of the systems resulting of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
138 Advanced Computational Methods in Heat Transfer IX the finite element discretization of (1) and (7) only depends on the three parameters ˜ Under these notations, the objective of reconstruction is to recover the , ˇ and . modal coefficients γi (t), i = 1..˜ of the approximate flux Φr . The quality of the results depends on the accurate description of the actual variables θ, p and Φ by the approximate ones. Another crucial point is the choice of the basis functions in order to ensure observability and controllability of the physical system.
6 Numerical test case Hereafter is shown a numerical example on a three-dimensional case based on this model reduction. A solid of dimensions Lx = 1, Ly = 0.5 et Lz = 0.4 and material properties all set to 1 is subject to an unknown heat flux on the face of coordinate x = 0 and zero heat fluxes on all other faces. All boundary conditions are of the Fourier-Robin type. Four sensors are located at points (0.1, 0.4, 0.1), (0.1, 0.1, 0.2), (0.2, 0.1, 0.3) and (0.2, 0.4, 0.4). Like in the previous 1D example, a direct simulation with a prescribed flux on face x = 0 gives measurements that are taken afterwards as input data for a simulation of the reconstruction algorithm. The prescribed flux has the form shown on figure 3 and can be decomposed into two functions η1 (x) and η2 (x) with arbitrary time variation so that Φ = γ1 (t)η1 (x) + γ2 (t)η2 (x). Here, η1 (x) and η2 (x) are the first two eigenvalues of the Laplace operator on a rectangle with Neumann boundary conditions.
Figure 3: Reconstruction of flux coefficients γ1 (t) and γ2 (t) (on the left); prescribed (solid line) and reconstructed (dotted line) flux on face x = 0 (on the right). Figure 3 shows the very accurate reconstruction of the two corresponding coefficients γ1 (t) and γ2 (t). A very accurate reconstruction of the different components αi of the temperature field obtained under this flux input was also observed. Figure 4 shows the isovalues of the final temperature on two sections on the xy-direction and the isovalues of the difference between reconstructed and target fields: there is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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a ratio of 104 between the corresponding orders of magnitude. The reconstruction took 4.4s CPU for 20 iterations.
Figure 4: Isovalues of the temperature field on planes z = 0 and z = 0.2, at t = T : target field (left); difference between target and reconstructed fields (right).
7 Conclusions The adjoint technique for the minimization of a least squares error functional has been applied with success to the problem of temperature reconstruction based on pointwise measurements. The use of the H 1 space has proved to be the best-suited framework for the minimization procedure, and has permitted us to overcome the final instant problem of the classical technique. The dual formulation of the problem and the model reduction using well chosen basis functions enables us to carry very fast computations with great accuracy. The algorithm can thus be adapted for the real time monitoring of structures. In such a case, the reconstruction procedure can be carried out at given intervals with overlapping time domains: the initial temperature condition for each computation will be known from the previous reconstruction computation and added to the reconstructed field. The entire procedure could be compared with an observer of great efficiency and speed.
References [1] Farrar, C., Hemez, F., Shunk, D., Stinemates, D. & Nadler, B., A review of shm literature: 1996-2001. Los Alamos National Laboratory Internal Reports, 2003. [2] Kullaa, J., Elimination of environmental influences from damagesensitive features in a structural health monitoring system. Proceedings of the First WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
140 Advanced Computational Methods in Heat Transfer IX
[3] [4]
[5]
[6]
[7]
[8] [9] [10] [11] [12]
[13]
European Workshop on Structural Health Monitoring, Paris, July 10-12, 2002, Onera. DEStech Publications, pp. 742749, 2002. Engl, H.W., Hanke, M. & Neubauer, A., Regularization of Inverse Problems. Kluwer Academic Publishers, 1994. Beck, J.V., Blackwell, B. & Haji-sheikh, A., Comparison of some inverse heat conduction methods using experimental data. Int J Heat Mass Transfer, 39(17), pp. 36493657, 1996. ¨ Neto, A.J.S. & Ozisik, M.N., Two-dimensional heat conduction problem of estimating the time-varying strength of a line heat source. J Appl Phys, 71(11), pp. 53575362, 1992. Videcoq, E. & Petit, D., Model reduction for the resolution of multidimensional inverse heat conduction problems. Int J Heat Mass Transfer, 44, pp. 18991911, 2001. Prudhomme, M. & Nguyen, T.H., Fourier analysis of conjugate gradient method applied to inverse heat conduction problems. Int J Heat Mass Transfer, 42, pp. 44474460, 1999. Alifanov, O.M., Inverse Heat Transfer Problems. Springer-Verlag, New York, 1994. Beck, J.V., Blackwell, B. & Clair, C.S., Inverse heat conduction, illposed problems. Wiley Interscience, New York, 1985. ¨ Ozisik, M.N. & Orlande, H.R.B., Inverse Heat Transfer. Taylor and Francis, 2000. Lions, J.L., Optimal control of systems gouverned by PDEs. Dunod, 1968. ¨ Jarny, Y., Ozisik, M.N. & Bardon, J.P., A general optimization method using adjoint equation for solving multidimensional inverse heat conduction. Int J Heat Mass Transfer, 34(11), pp. 29112919, 1991. Huang, C.H. & Wang, S.P., A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method. Int J Heat Mass Transfer, 42, pp. 33873403, 1999.
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Section 3 Heat and mass transfer
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Heat transfer in 3D water and ice basins S. Ceci, L. De Biase & G. Fossati University of Milano Bicocca, Department Environmental Sciences, Italy
Abstract Studying heat transfer in 3D water or ice basins involves the solution of Navier Stokes or Stokes systems of PDE, coupled with a scalar equation for thermal energy transport. The method used in this paper is based on a Finite Volume technique where physical quantities and boundary conditions are approximated by means of high order formulae and the time advance is dealt with by a fractional step technique. This paper is mainly concerned with applications, but, with respect to a preceding version of the method, an evaluation of performance of techniques dealing with the turbulent viscosity in water basins is presented and a study of the temperature field in a glacier is included. The method is applied to a thermal discharge in a water basin of lower temperature and to a portion of the Priestley Glacier (Antarctica). The results are very accurate and coherent with the physical theory and with measured data. Keywords: fluid dynamics, glacier modelling, glacier temperature, thermal discharges.
1
Brief description of the method
In this paper some interesting applications of a method presented in Deponti et al. [1] aimed at modelling meso-scale mass and energy flows in glaciers are presented, together with applications of a similar method (Hagos et al. [2]), modelling thermal discharges in water basins. The method for the study of flow and thermal energy transport in water basins is based on Navier-Stokes equation, coupled with the temperature transport equation.
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144 Advanced Computational Methods in Heat Transfer IX In glaciers, convective terms are negligible (LeB et al. [3], Hutter [4], Paterson [5], Colinge and Blatter [6], Blatter et al. [7]). Therefore the problem is modelled by Stokes system, in vector, unsteady conservation form: ∇ ⋅ u = 0 (1) ∂u − ∇ ⋅ ν (∇u) = −∇p + g + (∇ν ) ⋅ (∇u)T ∂t T
where (∇ν ) ⋅ (∇u) is a viscous forcing term. Although an acceleration term appears in momentum equation, the final solution verifies a steady Stokes system since, at steady state, the temporal derivative vanishes.
2
Non-dimensionalization
Equations and boundary conditions are non-dimensionalized by means of factors defined on the basis of the velocity norm, the glacier or basin depth and the amplitude of the temperature range.
3
Temporal evolution
A Fractional Step technique deals with the temporal evolution. At time instant n+1, at first, provisional velocities are computed on the basis of a hydrostatic pressure and a surface elevation η relative to the preceding time instant. Subsequently the full pressure is calculated and velocities are corrected. Finally the surface elevation is computed by Saint-Venant equation and the temperature field is found.
4
Space discretization
The domain is subdivided into a set of cell-centred rectangular control volumes with faces orthogonal to the coordinate axes. A Finite Volume method is used to grant both local and global conservation of each physical quantity; the profiles approximating diffusive and convective terms are defined in such a way that a uniform distribution of volume dimensions is not needed. Boundary conditions are therefore approximated by Generalized Finite Difference formulae (De Biase et al. [8]). The vertical dimension of the surface volumes evolves in time.
5
Viscosity
The turbulent viscosity in water flows is modelled by our method in several possible ways (as an evolution of Hagos et al. [2]): constant viscosity, Prandtl method, standard k − ε and low-Reynolds k − ε methods (Mohammadi and Pironneau [9]) In glaciers, molecular viscosity is considered (Gudmundsson [10]). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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145
Domain definition
The reference system is defined in such a way that the free surface and the bedrock or bottom are, in the average, approximated by planes parallel to the x-y coordinate plane, while inflow and outflow boundaries are parallel to the y-z coordinate plane and lateral boundaries are parallel to the x-z coordinate plane.
7
The thermal energy equation in a glacier
This equation is written as in Anderson [11], on the basis of the temperature field: DT ρc − ∇ ⋅ ( K ∇T ) = S (2) Dt where ρ is the fluid density, K is the thermal conductivity and S, the heat production rate for unit volume, generated by ice deformation, is written on the basis of stress components, as: 2 2 + εyy + εzz2 + 2 ( εxy + εxz + εyz ) + ε02 = 2µ 2εII + ε02 S = 2µ εxx (3) where µ is the dynamic viscosity and εII is the second invariant of the stress tensor ε.
8
Case test 1: thermal discharge
The domain for case test 1 simulates the outfall from an industrial plant: in the channel, water of a given temperature flows toward a water basin of lower temperature. The domain is a 3D rectangle of total length 20 m, width 1 m and total height 2 m, with a backward-facing step of length 10 m and height 1 m; it is discretized into 10750 volumes of dimensions ∆x = ∆z = 0.2 m, ∆y=0.1 m . Initial conditions are of null motion and pressure fields on the whole domain and unperturbed free surface. At the inflow boundary, homogeneous Neumann conditions for hydrodynamic pressure and for components v,w of velocity are assigned, while for the u component a value of 5 m/s is given and
T = 26D C ; k = 0.003uˆ 2 ; ε = cµ3 4
k3 2 L3
, with initial viscosity of 3*10-4 m2/s.
At the outflow boundary homogeneous Neumann conditions for velocity, temperature, k and ε and homogeneous Dirichlet for the hydrodynamic pressure are imposed. On lateral walls friction conditions are assigned, with Chezy coefficient of 50 m1/2/s, homogeneous Neumann conditions for temperature, k and ε. On the free surface null stress is assumed with homogeneous Neumann conditions for temperature, k and ε (Barkley et al. [12], Beaudoin et al. [13]).
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146 Advanced Computational Methods in Heat Transfer IX
Figure 1:
Motion fields with constant viscosity, Prandtl technique and a k − ε method.
In figure 1 the motion fields obtained by imposing constant viscosity of 10-6 m2/s, by Prandtl approximation and by a standard k − ε method are compared. Chezy coefficient is normally in the range between 30 and 60 m1/2/s, depending on the substrate of the channel. Since a constant density hypothesis has been formulated in this work, the coefficient must be divided by 1000, the water density.
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°c
°c
Figure 2:
Temperature distribution after 132 and 600 time steps.
In the application presented above it is evident that the constant viscosity model is not an option for the problem at study and that the best results are obtained by means of the k − ε technique. In figure 2 the temperature distributions obtained by this technique after 132 and 600 time steps are shown.
9
Case test 2: the Priestley Glacier
For the glacier bedrock, the interface ice/rock is implicitly defined as B ≡ −h − z = 0
The kinematic boundary condition imposes the vertical velocity to equal the z component of the melting/freezing rate. On the lateral boundaries, orthogonal to the y axis, the kinematic condition is the same, but the y component must be considered for lateral velocity; dynamic boundary conditions for velocity are:
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
148 Advanced Computational Methods in Heat Transfer IX ∂u ∂v 2 = Cµ 2 + u = Cσ xy ∂y ∂x v = Cσ
2 yy
2
∂v = C −P + 2µ ∂y
2
∂v ∂w 2 = Cµ 2 + w = Cσ yz ∂z ∂y
where ^ is the slip coefficient. (4) 2
On the free surface, the kinematic boundary condition is determined by the accumulation/ablation rate vector, while dynamic B.C. are: ∂u ∂w ∂v ∂w ∂w + = 0, + = 0, = 0. (5) ∂z ∂x ∂z ∂y ∂z On the bedrock, a geothermal flux is assigned: ∂T q geo = (6) ∂n K where q geo , the geothermal flux, varies between 46 mWm-2 and 77 mWm-2 depending on the rock age. On lateral solid walls, at ice/ice interfaces, a null flux condition is imposed, while at rock/ice interfaces, the condition ∂T q geo =± (7) ∂y K is imposed. On the free surface Dirichlet conditions can be assigned (Paterson [5], Luthi and Funk [14, 15]), or a Neumann condition of the form (Hutter, [4]): ∂T q atm = (8) ∂n K where q atm is the atmospheric heat flux. n +1 n +1 n +1 ∂T S n +1 + ∇ ⋅ u i T − κ , i=1,2,3 (9) = ∆t c ∂xi where S is the heat production term, κ is the thermal diffusivity and c is the thermal capacity. The area of study, built by means of the RAMP database and of aerial images, has dimensions L = 14000 m, D = 7000 m and H = 1000m, discretized into 5466 volumes. Dimensions of internal control volumes are: ∆x=∆y=600 m and ∆z=100 m. Components of the gravity acceleration vector are: gx=0.169 m/s2, gy=0.007 m/s2 and gz=9.909 m/s2. Homogeneous Neumann conditions are imposed for velocity and pressure at the inflow boundary, free exit for velocity and homogeneous Neumann for pressure are assigned at the outflow boundary. The free surface can evolve in space and time. On the bedrock and on the lateral
T n +1 − T n
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walls a slip condition is imposed, with slip coefficients C =5x10-18 ms-1Pa-2 and C =10-18 ms-1Pa-2, respectively. The variable viscosity is computed by Glen’s constitutive law, with parameter n=3 (Pettit and Waddington [16]). Initial conditions are null motion and pressure fields on the entire domain.
Figure 3:
Velocity field in a longitudinal section and a horizontal section; surface elevation in the glacier.
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Pas
kPa
Figure 4:
Viscosity field, temperature distribution and hydrodynamic pressure in a longitudinal section of the Priestley Glacier.
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10 Conclusion Results presented in this paper are in full agreement with the theory and with physical observations and measurements (Barkley et al. [12], Beaudoin et al. [13] for water applications, Kamb [17], Pattyn [18] for the Priestley application). In future some more work should be done in order to allow a more flexible assignment of boundary conditions on the volumes belonging to two different boundary surfaces. Indeed the rectangular shape of the volumes creates a steplike border of the computational domain and this makes it difficult to follow irregularities of the physical domain and to assign boundary conditions on some volume faces.
References [1]
[2] [3] [4] [5] [6] [7] [8]
[9] [10] [11]
Deponti A., Pennati V., De Biase L., A fully 3D finite volume method for incompressible Navier-Stokes equations, Int. J. for Numerical Methods in Fluids, (in press; available online in Wiley Interscience, DOI: 10.1002/fld.1190), 2006 Hagos S., Deponti A., Pennati V., De Biase L., Applicazione di un nuovo metodo ai volumi finiti a problemi di inquinamento termico, La Termotecnica, Aprile 2005 LeB. Hooke R., Principles of Glacier Mechanics, Prentice Hall, Upper Saddle River, New Jersey 07458, 1998. Hutter K., Theoretical Glaciology, Material Science of Ice and the Mechanics of Glaciers and Ice Sheets, D. Reidel Publishing Company, Dordrecht, Holland, 1983. Paterson W.S.B., The Physics of Glaciers, Third Edition, Pergamon, 1994. Colinge J., Blatter H., Stress and Velocity fields in glaciers: Part I. Finitedifference schemes for higher-order glacier models, Journal of Glaciology, Vol. 44, N. 148, pages 448-456 (a), 1998. Blatter H., Clarke G. K. C., Colinge J., Stress and Velocity fields in glaciers: Part II. Sliding and basal stress distribution, Journal of Glaciology, Vol. 44, N. 148, pages 457-466 (b), 1998. De Biase L., Feraudi F., Pennati V., A Finite Volume Method for the solution of Convection-Diffusion 2D problems by a Quadratic Profile with Smoothing, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 6, N. 4, pages 3-24, 1996. Mohammadi B., Pironneau O., Analysis of the K-Epsilon Turbulence Model, John Whiley and Sons, 1994. Gudmundsson G. H., Basal-flow characteristics of a non-linear flow sliding frictionless over strongly undulations bedrock, Journal of Glaciology, Vol. 43, N. 143, pages. 80-89 (b) 1997. Anderson J.D., Introduction, Von Karman Institute for Fluids Dynamics Lecture Series: Introduction to Computational Fluid Dynamics, pages 18-22, 1983 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
152 Advanced Computational Methods in Heat Transfer IX [12] [13] [14] [15] [16] [17] [18]
Barkley D., Gomes M., Henderson R.D., Three-dimensional instability in Flow over a Backward Facing Step. International Journal for Fluid Mechanic, Vol. 473, pages 167-190, 2002. Beaudoin J.F., Cadot O., Aider J.L., Centrifugal Instability in the Backward Facing Step, International Journal for Fluid Mechanic, 2003. Lüthi M., Funk M., Dating ice cores from a high Alpine glacier with a flow model for cold firn, Annals of Glaciology N. 31, pages 69-79, 2000. Lüthi M., Funk M., Modelling heat flow in a cold, high-altitude glacier: interpretation of measurements from Colle Gnifetti, Swiss Alps, Journal of Glaciology, N. 47(157), pages 314-324, 2001. Pettit E.C., Waddington E.D., Ice flow at low deviatoric stress, Journal of Glaciology, vol. 49, N. 166, pages. 359-369, 2003 Kamb B., Sliding motion of glaciers: Theory and observation, Reviews of Geophysics and Space Physics, N. 8, 4, pages 673-728, 1970. Pattyn F., Ice-sheet modelling at different spatial resolutions: focus on the grounding zone, Annals of Glaciology, N. 31, pages 211-216, 2000.
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Coupled heat and moisture transport in a building envelope on cast gypsum basis J. Maděra, P. Tesárek & R. Černý Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic
Abstract Flue gas desulphurization (FGD) gypsum can be potentially used as a material for load bearing structures. In this paper, a computational assessment of hygrothermal performance of a building envelope based on several modifications of FGD gypsum is presented. In the computer simulations of temperature and relative humidity fields, three variations of FGD gypsum wall, based on the raw material and on two types of hydrophobized gypsum, with the thickness of 300 mm, are solved. The thermal insulation function of the wall is achieved by exterior thermal insulation boards with the thickness of 100 mm, which are considered in four variants. Insulation I is hydrophilic material with a low value of hygroscopic moisture content on mineral wool basis, Insulation II capillary active material with higher value of hygroscopic moisture content on calciumsilicate basis, Insulation III hydrophobic material with a low value of water vapor resistance factor on mineral wool basis and Insulation IV hydrophobic material with higher value of water vapor resistance factor on polystyrene basis. Lime plaster with the thickness of 10 mm is used on the exterior wall surface. The computational analysis reveals that use of hydrophobization admixtures in the gypsum element does not lead to any improvement of hygrothermal behavior of the envelope provided by an exterior thermal insulation. Therefore, the application of a gypsum element without any hydrophobization seems to be a more favorable solution. The common hydrophobized thermal insulation materials on the basis of polystyrene or mineral wool are found to be satisfactory from the point of view of hygrothermal performance of the analyzed castgypsum based envelope. Keywords: building envelope, flue gas desulphurization (FGD) gypsum, thermal insulation boards, heat and moisture transport. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060151
154 Advanced Computational Methods in Heat Transfer IX
1
Introduction
Calcined gypsum is a historical binder, which was, used already several thousands years ago. Gypsum was called gatch in Persia, gypsos in Greek, and gypsum in Latin. Iranians, Egyptians, Babylonians, Greeks, and Romans were familiar with the art of working with gypsum plasters; decorated interior walls were found for instance in Pompeii. Gypsum was found in the binder of buildings in the territory of today’s Syria dated 7000 B.C.; it was also used in Cheops pyramid 2650 B.C, in the palace of Knossos etc. Nowadays, calcined gypsum is used in many technological modifications, which should improve its properties, in particular as binder of rendering mortars, for the production of stuccowork and also for plasters [1]. In the second half of the 20th century, new technologies for desulfurization of flue gases in power stations and heating plants appeared which were based on the reaction of sulfur (II) oxide formed during combustion of brown coal with high content of sulfur with limestone. Although these technologies were definitely very suitable from the point of view of the protection of environment, one problem appeared from the very beginnings, namely the very high amount of flue gas desulfurization (FGD) gypsum as waste product. The utilization of FGD gypsum as secondary raw material remained insufficient considering the amount of its production until these days. For instance, in Czech Republic calcined gypsum is produced from FGD gypsum only in one power station (Počerady), the remaining production ends with gypsum that is used only partially as additive retarding the setting of cement. Calcined gypsum is mostly used for the production of gypsum plasterboards [2]. That part of produced gypsum, which is not utilized, is deposited as waste. However, FGD gypsum can be potentially used as a material for load bearing structures as well. Modifications of this material can enhance its original properties and increase its service life. In this paper, a computational assessment of hygrothermal performance of a building envelope based on several modifications of FGD gypsum is done.
2
Materials and building envelopes
In the computer simulations of temperature and relative humidity fields we have solved three variations of FGD gypsum wall, based on the raw material and on two types of hydrophobized gypsum, with the thickness of 300 mm (Fig. 1). The thermal insulation function of the wall was achieved by exterior thermal insulation boards with the thickness of 100 mm, which were considered in four variants. Insulation I was hydrophilic material with low value of hygroscopic moisture content on mineral wool basis, Insulation II capillary active material with higher value of hygroscopic moisture content on calcium-silicate basis, Insulation III hydrophobic material with low value of water vapor resistance factor on mineral wool basis and Insulation IV hydrophobic material with higher value of water vapor resistance factor on polystyrene basis. On the external side of the wall, lime plaster with the thickness of 10 mm was used. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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1
155
3
1 – external plaster 2 – thermal insulation 3 – load-bearing structure
10
300
100
Figure 1:
[mm]
Composition of building envelope used to computer simulation.
The basic FGD gypsum material (we will denote it S0 in what follows) was β-form of calcined gypsum with purity higher than 98% of FGD gypsum, produced at the electric power station Počerady, CZ. The water/gypsum ratio was 0.627. After classification according to the Czech standard ČSN 72 2301, the gypsum was categorized as G-13 B III [3]. The first modification of FGD gypsum (S3) contained the admixture IMESTA IBS 47 produced by Imesta Inc., Dubá u České Lípy, CZ. The other (S4) contained the admixture ZONYL 9027 produced by Du Pont, USA. The water/gypsum ratio was the same as for S0. The composition of gypsum materials is shown in Table 1. Table 1: Material S0 S3 S4
Water/gypsum ratio 0.627 0.627 0.627
Composition of gypsum materials. Admixture none IMESTA IBS 47 ZONYL 9027
Concentration of the admixture none 0.5% by mass 5.0% solution
The material properties of non-modified and modified gypsum were measured in the Laboratory of Transport Processes (LTP), Faculty of Civil Engineering, Czech Technical University in Prague [4]. They are given in Table 2, where ρ is the bulk density, c the specific heat capacity, κ the moisture diffusivity, µ the water vapor diffusion resistance factor, θsat the saturated moisture content, θhyg the maximum hygroscopic moisture content, λ the thermal conductivity. The properties of insulation materials and lime plaster were partially obtained from the material database of Delphin computer code [5] and
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156 Advanced Computational Methods in Heat Transfer IX partially measured in LTP. The material properties of insulation boards are given in Table 3. Table 2:
S0 S3 S4
c ρ 3 [kg/m ] [J/kgK] 1019 840 942 840 941 840 Table 3:
Basic materials properties of gypsum. κ [m2/s] 2.63e-7 1.47e-7 7.32e-9
µ [-] 5.4 5.4 5.4
θsat θhyg λ 3 3 3 3 [m /m ] [m /m ] [W/mK] 0.6 0.23 0.47 0.61 0.181 0.41 0.62 0.166 0.38
Material parameters of insulation materials.
c ρ 3 [kg/m ] [J/kgK] I 150 840 II 230 1000 III 280 840 IV 30 1300
κ [m2/s] 1.10-7.e0.0485. θ 2.10-8.e0.0523. θ 5.10-13.e0.1486. θ 2.10-11.e0.0475. θ
µ [-] 2 2.5 3 50
θsat [m3/m3] 0.95 0.88 0.31 0.97
θhyg [m3/m3] 0.006 0.22 0.0073 0.001
λ [W/mK] 1.1 0.4 1.2 0.56
3 Numerical solution by TRANSMAT For the calculations we employed the computer simulation tool TRANSMAT 4.3 [6] which was developed in the Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague. The construction of the code is based on the application of the general finite element computer simulation tool SIFEL (SImple Finite ELements) developed in the Department of Mechanics, FCE CTU. The moisture (1) and heat balance (2) equations were formulated according to the Künzel’s model [7], dρ v ∂ϕ = div Dϕ gradϕ + δ p grad (ϕp s ) dϕ ∂t dH ∂T = div(λgradT ) + Lv div δ p grad (ϕp s ) dT ∂t
[
]
[
]
(1) (2)
where ρv is partial moisture density, ϕ the relative humidity, δp the water vapor permeability, ps the partial pressure of saturated water vapor in the air, H the enthalpy density, Lv the latent heat of evaporation of water, λ the thermal conductivity and T is the temperature. The liquid water transport coefficient is defined as Dϕ = κ
dρ v . dϕ
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The proper initial and boundary conditions of the model are crucial factor affecting the reliability of the calculations. In our computer simulations, the analyzed building envelopes were exposed from inside to constant conditions (temperature equal to 21°C and relative humidity equal to 55%) and from outside to the climatic conditions corresponding to the reference year for Prague. The 1st of July was chosen as the starting point in the calculations. We have chosen two characteristic profiles in the assessment of the hygrothermal performance of the envelope, A-A´, B-B´ (Fig. 2), where the profile A-A´ was between the insulation board and the load-bearing structure (the distance of 110 mm from the exterior), the profile B-B´ was the cross section of the wall from the exterior to the interior. In these profiles we have compared relative humidity and temperature calculated for the analyzed envelopes.
Exterior TRY for Prague
Interior
A
Constant temperature T = 294.15 K
relative humidity temperature wind rain short wave
Constant relative humidity ϕ = 55% A´ Figure 2:
4
Scheme of typical envelope.
Computational results and discussion
4.1 Non modified gypsum S0 Fig. 3 shows an example of the relative humidity profile in the wall based on non-modified gypsum (S0) for December 15, which can be considered as characteristic for the winter period.
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158 Advanced Computational Methods in Heat Transfer IX 1
Relative humidity [-]
0.9 0.8 0.7 0.6 0.5 0.4 0
50
100
150
200
250
300
350
400
Distance [mm] Insulation I
Figure 3:
Insulation II
Insulation III
Insulation IV
Relative humidity, non modified gypsum (S0), B-B´ profile.
Fig. 4 presents the history of relative humidity in the A-A´ profile from January 1 to December 31 for four years simulation.
Relative humidity [-]
0.62 0.6 0.58 0.56 0.54 0.52 0.5
1 500
1 550
1 600
1 650
1 700
1 750
1 800
Time [days] Insulation I
Figure 4:
Insulation II
Insulation III
Insulation IV
Relative humidity, non modified gypsum (S0), A-A´ profile.
Fig. 5 shows an example of the temperature profile in the wall based on the non-modified gypsum (S0) for December 15, which can be considered as characteristic for the winter period.
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Temperature [K]
290 285 280 275 270 0
50
100
150
200
250
300
350
400
Distance [mm] Insulation I
Figure 5:
Insulation II
Insulation III
Insulation IV
Temperature, non modified gypsum (S0), B-B´ profile.
4.2 Modified gypsum S3 Fig. 6 shows an example of the relative humidity profile in the wall based on the modified gypsum (S3) for December 15 analogous to Fig. 3. The results obtained for modified gypsum (S3) and non-modified gypsum (S0) were very similar, so that the effect of gypsum hydrofobization was very small. The same similar results were also achieved in the analogs to Figs. 4 and 5. 1
Relative humidity [-]
0.9 0.8 0.7 0.6 0.5 0.4 0
50
100
150
200
250
300
350
400
Distance [mm] Insulation I
Figure 6:
Insulation II
Insulation III
Insulation IV
Relative humidity, modified gypsum (S3), B-B´ profile.
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160 Advanced Computational Methods in Heat Transfer IX 4.3 Modified gypsum S4 Fig. 7 shows an example of the relative humidity profile in the wall based on the modified gypsum (S4) for December 15. Here, some differences in relative humidity (mainly in the insulation layer and partially also in the gypsum element, the highest for Insulation II) compared to the results for S0 in Fig. 3 were observed but they were not very significant because they did not change the overall character of the hygrothermal performance of the gypsum wall. 1
Relative humidity [-]
0.9 0.8 0.7 0.6 0.5 0.4 0
50
100
150
200
250
300
350
400
Distance [mm] Insulation I
Figure 7:
Insulation II
Insulation III
Insulation IV
Relative humidity, modified gypsum (S4), B-B´ profile.
0.62
Relative humidity [-]
0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 1 500
1 550
1 600
1 650
1 700
1 750
1 800
Time [days] Insulation I
Figure 8:
Insulation II
Insulation III
Insulation IV
Relative humidity, modified gypsum (S4), A-A´ profile.
Fig. 8 presents the relative humidity history in the A-A´ profile from January 1 to December 31 for four years simulation. The differences observed in comparison with Fig. 4 for non-modified gypsum wall were most pronounced for WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Insulation II similarly as in Fig. 7 but in the assessment of the overall hygrothermal performance of the wall, also here they could not be considered as very significant. Fig. 9 shows an example of the temperature profile in the wall based on the modified gypsum (S4) for December 15. A comparison with the corresponding Fig. 5 reveals that the differences from the temperature profiles in the wall based on non-modified gypsum were almost negligible.
Temperature [K]
290 285 280 275 270 0
50
100
150
200
250
300
350
400
Distance [mm] Insulation I
Figure 9:
5
Insulation II
Insulation III
Insulation IV
Temperature, modified gypsum (S4), B-B´ profile.
Conclusions
The computational analysis in this paper revealed that the use of hydrophobization admixtures in the cast-gypsum element of building envelopes provided by any of the four very different exterior thermal insulations did not lead to significant improvements of the hygrothermal behavior of the envelope. The hygrothermal performance of the studied envelopes was satisfactory in all analyzed cases. Therefore, an application of a gypsum element without any hydrophobization seems to be the preferential solution, particularly taking into account the substantially lower price. The common hydrophobized thermal insulation materials on the basis of polystyrene or mineral wool were found to be satisfactory from the point of view of hygrothermal performance of the analyzed cast-gypsum based envelope. Therefore, they are supposed to be the preferred materials in this respect.
Acknowledgements This research has been supported partially by the Czech Science Foundation, under grant No. 103/06/P021 and partially by the Ministry of Education of the Czech Republic, under contract No. 6840770003. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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References [1] Wirsching F., Calcium Sulfate. In: Ullmann’s Encyklopedia of Industrial Chemistry, Weinheim, 1985, pp. 555-583. [2] Hanusch, H., Übersicht über Eigenschaften und Anwendung von Gipskartonplatten. Zement-Kalk-Gips, No. 5, 1974, 245-251. [3] ČSN 72 2301 Gypsum binding materials, Czech standard (in Czech). Vydavatelství Úřadu pro normalizaci s měření, Praha 1979. [4] Tesárek P., Kolísko J., Rovnaníková P., Černý R., Properties of Hydrophobized FGD Gypsum. Cement Wapno Beton 10/72 (2005), 255264. [5] Grunewald J., DELPHIN 4.1 - Documentation, Theoretical Fundamentals, TU Dresden, Dresden, 2000. [6] Maděra J., Černý R. TRANSMAT – a Computer Simulation Tool for Modeling Coupled Heat and Moisture Transport in Building Materials. Proceedings of Workshop 2005 - Part A,B, Prague: CTU, 2005, pp. 470-471. [7] Künzel H.M., Simultaneous Heat and Moisture Transport in Building Components, Ph.D. Thesis. IRB Verlag, Stuttgart 1995.
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Spray water cooling heat transfer under oxide scale formation conditions R. Viscorova1 , R. Scholz2 , K.-H. Spitzer1 and J. Wendelstorf1 1 Institute
of Metallurgy, Clausthal University of Technology, Germany for Energy Process Engineering and Fuel Technology, Clausthal University of Technology, Germany
2 Institute
Abstract Spray water cooling is an important technology used for the cooling of materials from temperatures up to 1800K. The heat transfer coefficient (HTC) in the so-called steady film boiling regime is known to be a function of the water mass flow density. Below a specific surface temperature TL , film boiling becomes unstable and the HTC shows a strong dependence on temperature (Leidenfrost effect). The HTC was measured by an automated cooling test stand (instationary method). Compared to the previous state-of-the-art, an additional temperature dependency in the high temperature regime was found. A new analytic fit formula for the dependence of the heat transfer coefficient on temperature and water impact density is proposed and discussed. Spray water cooling of steel materials at temperature levels above 1000K introduces additional effects due to the formation of oxide layers (scale). These effects and experiments under scale formation conditions will be presented and discussed. Keywords: scale formation, spray water cooling, continuous casting, hot rolling, heat transfer coefficient.
1 Introduction Spray water is used for cooling in steel materials production processes, e.g. as part of the casting and rolling procedure. Due to the high temperatures the steel surface is oxidized. A lack of knowledge about the influence of this oxide layer (scale) on heat transfer conditions may prevent a quantitative prediction of the cooling procedure. For predictable homogenous cooling, the dependence of the heat transfer coefficient (HTC) on its principal parameters, the surface temperature WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060161
164 Advanced Computational Methods in Heat Transfer IX and the water mass flow density, is required even in the case of oxidized steel surfaces. The interrelationship between oxide scale formation and heat transfer also determines the temperature of the metal surface. The layer of the oxide scale forms a resistance for the heat transfer. Additionally, due to the cooling process, mechanical tensions arise between the oxide layer and the bulk material which lead to spallation of the scale and thus again to changes of the heat transfer conditions. For quantitative description, the concept of the Heat Transfer Coefficient (HTC, α) is used, which is defined through its relation to the heat flow density to the surface q [W/m2 ] (see also Figure 1): q = α · (TS − TW )
(1)
For spray water cooling, TW is the water temperature, while TS is the (local) surface temperature. This approach is most suitable for situations were α is constant, i.e. not depending on ∆T ≡ TS − TW . It can also be used for any general heat flow q by using a function for α depending on the same parameters as q. In the so-called steady film boiling regime, α is known to be a function of the water mass flow density [1, 2, 3, 4]. Below a specific surface temperature TS,L , the heat transfer coefficient shows a strong dependence on temperature (Leidenfrost effect [5]). In this paper, the heat transfer coefficient was measured by an automated cooling test stand (instationary method, see Figure 2 and the discussion in [1]) under oxidizing and non-oxidizing sample surface conditions. Additionally, the measurement precision was taken into account (see e.g. [1]). This paper continues the work started in [4].
spray water
Tw surface
S scale heat flow
B q
metal U thermocouple
Figure 1: Heat transfer to an oxidized sample.
2 Experimental procedure The thin sheet specimens of cold rolled metals with a thickness from 1.0-1.5 mm were milled to discs with 70 mm in diameter. Up to 5 thermocouple pairs, one in WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Magnetic valve
Cooling water
Volume flow measuring instrument
Furnace (7,5kW / 1500°C) Nozzle field Steel sample
Electricity supply and furnace temperature control
Sample supporting stand Sample-cross bar D Measuring and control system
Gas supply unit
- temperature measurement - cross bar control
Air Argon
Figure 2: Sketch of the experimental set-up.
the center and the others perpendicular at a radius of 10 mm, were spot welded to the lower side of the sample discs. The temperature measurement was carried out with Ni-CrNi thermocouples with a wire diameter of 0.5 mm. The thermocouple wires were isolated by thin ceramic tubes. The experimental set-up sketched in Figure 2 was used for heating, oxidizing the samples and measurement of the temperature during spray cooling. After installation in the furnace sample holder, the disks were heated up to 1200 ◦ C under protective atmosphere. For the measurements with scale, the samples were oxidized in the furnace by supplying air instead of protective gas for a specific time and temperature. Reaching the test conditions (temperature, oxidation time), the sample was moved automatically from the furnace under the full cone nozzle which takes approximately 4 seconds and subsequently cooled with spray water. The spray water mass flow density VS was determined experimentally and varied in the range of about 4 ± 1 to 30 ± 1 kg m2 s−1 . The water temperature TW was approximately 18 ◦ C. The materials investigated were commercially pure Nickel (99.2% Ni) and Iron (low-carbon steel, AISI 1008, 99.7% Fe).
3 Measurement of the HTC at elevated temperatures For the investigation of the HTC from a non oxidized surface, Nickel material was used, because surface oxidation is minimized, and the physical properties are known within the temperature range being studied. The samples were annealed WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
166 Advanced Computational Methods in Heat Transfer IX at 1200 ◦ C and consequently cooled. The cooling curves were measured at three points for each sample as described above. Figure 3 shows typical cooling curves of the Nickel specimens for different water mass flow densities. The sample reaches the spray water cooling position after about 4.2 s. The cooling starts in the range of stable film boiling. The primary effect of the water mass flow density VS can be seen by comparing the different cooling curves. When increasing VS from 4 up to 30 kg m2 s−1 the sample cooling rate increases. At approximately 300-450 ◦ C, dependent on VS , the cooling curves bend. At this point the Leidenfrost temperature is reached and more rapid cooling begins (unstable film boiling).
1200 Sample: d=1mm, Nickel 99.3
1100 1000 temperature T[°C]
900 800 700 600 500 400 300 200
30
25
18
12
water mass flux density Vs [kg /(m2s)]
8
4
100 4
5
6
7
8
9
10
11
12
13
14
time t[s]
Figure 3: Cooling curve for different water mass flow densities VS (spray water cooling of pure Nickel).
For each cooling curve (VS -value), α(TS ) is obtained from the following equation:
α(TS ) ≈ −
ρ · cp (TU ) · d ∂TU · TU − TW ∂t z=0
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Advanced Computational Methods in Heat Transfer IX α
heat transfer coefficient
[W m−2 K−1 ]
cp
specific heat capacity of the sample (= f (T )!)
[J kg K−1 ]
d
thickness of the sample
[m]
z
vertical coordinate (bottom: z = 0,U; top: z = d,S)
[m]
t
time
[s]
ρ
mass density of the sample
[kg m−3 ]
TU
temperature measured at the lower side (U, Fig.1)
[K]
TW
temperature of the spray water
[K]
167
Equation (2) is valid for thin samples such as in this case (d = 1 mm) because of the small temperature difference between the cooled top and the measurement point at the bottom (≈10 ◦ C in the stable film boiling regime). The specific heat capacity cp of the sample materials depending on temperature was calculated with the Thermocalc Software based on the chemical analysis. A fit function describing the α(∆T, VS ) dependence was calculated from the measurement data VS VS · ∆T α(∆T, VS ) = erf × 245 · VS 1 − + (3) 5 58223 ∆T 4.3 · ∆T 2 1 − tanh 115 and is shown in Figure 4. In the open literature, the HTC in the range of stable film boiling (i.e. above ∆T = 600K) is assumed to be independent of the surface temperature. As shown in Figure 4, for the higher spray water densities, VS > 15 kg m2 s−1 there is a decrease in the measured HTC even in the stable film boiling regime. This effect may be explained with an increasing vapor layer thickness and yields to α-contours no longer parallel to the ∆T -axis.
18000
30 a [W/(m K)] 2
20000 15000 10000 5000 0
20
30
250
10
500 DT [K]
Da=2000
10
750 1000
1000
2
Vs [kg/(m s)]
0
DT [K]
0
200
400
600
800 1000
Figure 4: Heat transfer coefficient after eq. (3) in dependence on temperature difference and water mass flow density. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
168 Advanced Computational Methods in Heat Transfer IX
4 The HTC for an oxidized surface Following the initial considerations and the basic knowledge on scale properties (e.g. [4]), the determination of the influence of scale on the heat transfer conditions introduces some difficulties due to the following reasons: • Thin scale layers (< 10 µm) may show enough adhesion for measurement but imply a very small heat resistance. • Small gas layers between the scale and the substrate (blistering) imply a big effect on heat resistance (random effect). • The adhesion of medium thickness scale layers (10 . . . 200 µm) may be not sufficient to withstand spray water – spallation and descaling occurs (instationary effect). • Thicker scale layers (> 200 µm) are removed rapidly by the spray water – the thermal effect becomes inhomogeneous. Regardless these difficulties, the practical importance remains and thus the influence of an oxide layer on heat transfer was investigated in a second set of experiments in which the low alloy material AISI 1008 (99.7% Fe) was used. In order to get well defined scale formation, the oxidation kinetics was determined in the first step [6]. These investigations allow to calculate the scale layer thickness from the oxidation temperature and time. 4.1 Modelling of the HTC for an oxidized surface For understanding the influence of an oxide layer on the heat transfer during spray water cooling, the mechanism of heat transfer at an oxidized surface is theoretically described first. Since calculations often can not include thin layer effects, an effective heat transfer coefficient containing all scale layer effects is introduced. The heat flow is thus described by an effective heat transfer coefficient which is using the temperature difference between the steel surface and the temperature of the water (see Figure 1): q = αeff,∆T =TB −TW · (TB − TW ) q
heat flow density
[W m−2 ]
αeff
effective heat transfer coefficient
[W m−2 K−1 ]
TB
temperature at the steel-scale interface
[K]
(4)
The heat transfer from the surface to the spray water is given by q = α∆T =TS −TW · (TS − TW )
(5)
and the heat transfer through the oxide layer can be approximated by q=
λsc · (TB − TS ) δsc
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Advanced Computational Methods in Heat Transfer IX λsc
effective heat conductivity of the scale
[W m−1 K−1 ]
δsc
thickness of the scale layer
[m]
TS
temperature at the surface of the oxide scale
[K]
169
Under quasi-stationary conditions and for uniform cooling, the heat flow (4) from the surface to the cooling water (5) is equal to the heat flow through the scale layer (6) and we get the effective HTC defined by (4): αeff,∆T =TB −TW =
1 α∆T =TS −TW
+
δsc λsc
−1 (7)
The effective heat transfer coefficient αeff contains the influence of the oxide layer (thermal insulation, temperature drop). It can be calculated by (7) using the HTC α(∆T = TS − TW , VS ), which does not depend on bulk material properties. So accurate measurements of α(∆T = TS − TW , VS ) without scale can be used for surfaces with well defined oxide layers. Additionally, a measurement using an oxidized surface and (2) for analysis will yield to a measured αeff . The function αeff (∆T = TB − TW , VS ) calculated from (7) using the fit-formula α(∆T = TS − TW , VS ) from (3) is shown in Figure 5 and discussed below.
15000 2
a [W/(m K)]
-3.5
320µm
10000
-4
100µm
5000
-4.5
0
32µm 200
-5
lg(dz[m])
10µm 400 -5.5
3µm
600
DT [K]
800 -6
1µm
Figure 5: Calculated effective heat transfer coefficient depending on temperature difference ∆T = TB −TW and oxide layer thickness (logarithmic scale).
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9000 8000
-2
-1
HTC [W m K ]
7000 6000 5000 4000
Dfrom eq.(3) Deff from eq.(7) & (3) with Gsc=150µm
experimental scatter
170 Advanced Computational Methods in Heat Transfer IX d=1.5mm, AISI-1008 VS=4 kg m-2 s-1
experimental Deff after eq. (2):
oxidation time in air at 1000°C (mean D):
Deff from eq.(7) & (3) with Gsc=300µm or gas gap
300s Ö Gsc~140µm 105s Ö Gsc~83µm 0s Ö Gsc<1µm
3000 2000 1000 0 100
200
300
400
500
600
700
800
900
'T [K]
Figure 6: Predicted and measured αeff in dependence on ∆T = TB − TW for different oxidation times or scale layer thicknesses.
4.2 Measurement of the effective HTC The predicted effective HTC for an oxidized sample is compared with experimental data in Figure 6. Low alloy steel (AISI 1008) samples and a water mass flow density of VS = 4 kg m−2 s−1 were used. The measured cooling curves are analysed by calculating αeff using eq. (2). The first observation is a larger experimental scatter for oxidized samples, especially in the unstable film boiling regime. For this reason all available cooling curves were individually analysed and the mean values α¯eff are plotted (dashed lines). The corresponding scale layer thicknesses were calculated from the given oxidation times using a parabolic growth law [6]. The heat transfer in the stable film boiling regime is not influenced by the scale layers, as expected. At lower temperature differences ∆T = TB − TW , the heat transfer increases, the Leidenfrost phenomenon starts at higher ∆T values, but the maximum αeff values are smaller. Assuming a scale layer thickness of 150 and 300 µm and a heat conductivity of λsc =3 W m−1 K−1 , the calculated αeff is also plotted (solid lines). The agreement is better by assuming a larger scale layer thickness than calculated from the oxidation time. This finding is in accordance with the assumption of a thin gas/vapour layer between scale and metal surface. Due to the much lower heat conductivity of this film, the apparent effect is that of a much thicker scale layer. These theoretical and experimental investigations indicate a change of αeff with increasing surface oxidation mainly in the unstable film boiling regime. The apparent temperature corresponding to the Leidenfrost point TS,L shifts to higher values. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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In the range of stable film boiling, the αeff values are approximately equal to α, as sc is much greater than α. long as λδsc The experimental findings show qualitative agreement with the calculations. The local change in the sample surface state yields a more random local heat transfer. The smaller heat conductivity of the scale layer implies its more rapid cooling and thus a faster onset of unstable film boiling for oxidized surfaces (increasing αeff ). Especially in the stable film boiling regime, the isolation of the scale layer from the metal surface by a small gas or vapour film may imply a decreasing αeff . As a consequence, spatial inhomogeneous heat removal from the surface will cause heat flows by conduction parallel to the surface plane. These αeff measurements are thus not as accurate as the measurements without surface oxidation. The current measurements are estimated to be accurate within ±25% for the determination of the HTC while the literature data is estimated to have about ±40% accuracy in the stable film boiling regime. Below ∆T ≈ 500K, approaching the Leidenfrost point, the measurements are somewhat less accurate. For oxidized surfaces showing descaling during cooling, the local HTC fluctuates randomly. The statistics of this behaviour needs to be investigated in greater detail. Without blistering, stable adhesive scale surface layers need to grow thicker than ≈ 150 µm in order to influence the local heat transfer coefficient (7).
5 Summary and conclusions In this study, spray water cooling of different materials from initial temperatures up to 1200 ◦ C was investigated. The (low alloy) AISI-1008 steel and – for comparison – Nickel was used for the experiments. The heat transfer coefficient was measured by an automated cooling test (instationary method). This allows for a determination of the HTC depending on the surface temperature TS , within a single experiment. The second parameter, the water mass flow density VS , was varied from 4 to 30 kg m−2 s−1 . Furthermore, spray water cooling of steel materials introduces additional effects due to the formation of oxide layers (scale). For the investigation of these effects, steel samples were oxidized in air at different temperatures and for specific times. Finally, the HTC was measured for oxidized samples. For the assessment of the scale layer effect and application purposes, an effective heat transfer coefficient αeff was defined. Summarizing, the following effects where observed • An additional temperature dependency in the high temperature (stable film boiling) regime was found, thus the HTC is also temperature-dependent above the Leidenfrost temperature - simple α(VS )-relations are very inaccurate. • A new fit formula for the recommended α(∆T, VS ) is provided. • For the quantitative description of the oxide scale formation, spallation has to be taken into account. • In the lower ∆T (unstable film boiling) regime, scale layers can dramatically influence the apparent HTC. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
172 Advanced Computational Methods in Heat Transfer IX • The oxide layers can impede heat transfer by random formation of an isolating gas gap between the oxide and the bulk material (blistering). • Spallation of the oxide scale and thus scale plates moving around, additionally influence local and global cooling conditions. As a conclusion, the heat transfer coefficient (HTC, α) introduced by (1) and approximated from experimental data (eq. (3) and Figure 4) is a complex function of the surface temperature, the water mass flow density and the surface condition.
Acknowledgements This work was partially supported by the German Ministry of Commerce (BMWA) under the supervision of the AiF member society VDEh Gesellschaft zur F¨orderung der Eisenforschung under Ref. No. 13933N. We express our gratitude to the supporters of the project, especially to Mr. Hillebrecht with his outstanding experience in the field of HTC measurement.
References [1] N Lambert and M Economopoulos. Measurement of the heat-transfer coefficients in metallurgical processes. Journal of the Iron and Steel Institute, 208:917–928, 10 1970. [2] U Reiners, R Jeschar, R Scholz, D Zebrowski, and W Reichert. A measuring method for quick determination of local heat transfer coefficients in spray water cooling within the range of stable film boiling. steel research, 56(5):239–246, 1985. [3] M Bamberger and B Prinz. Determination of heat transfer coefficients during water cooling of metals. Materials Science and Technologie, 2:410–415, 4 1986. [4] C K¨ohler, R Jeschar, R Scholz, J Slowik, and G Borchardt. Influence of oxide scales on heat transfer in secondary cooling zones in the continuous casting process, part i.&ii. steel research, 61(7):295–301 & 302–311, July 1990. [5] B S Gottfried, C J Lee, and K J Bell. The leidenfrost phenomenon: Film boiling of liquid droplets on a flat plate. International Journal of Heat and Mass Transfer, 9:1167–1187, 1966. [6] R Viscorova, R Scholz, K H Spitzer, and J Wendelstorf. In AISTech 2006, May 1-4, 2006, Cleveland, Ohio, USA. AIST, 2006.
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How to design compact mass transfer packing for maximum efficiency H. Goshayshi Department of Mechanical Engineering, Azad University of Mashhad (IAUM), Iran
Abstract The effect of form with corrugated packing on mass transfer and pressure drop characteristics in atmospheric cooling towers has been studied experimentally. The results showed that the mass transfer coefficient decreased with increase in packing pitch and increase in the ratio of rib pitch to rib height. Friction factors were expressed by a dimensional equation which included pitch and distance between the packings, for both smooth and rough surface. From these results, the relationship between packing mass transfer coefficient and pressure drop was deduced. The correlations were verified with additional experimental data taken with 1.1< P/D <1.70 and 1 ≤ p/e ≤ 5. This provides a useful semi-experimental relation, in the area generally lacking in design and performance data. Keywords: packings, mass transfer and pressure drops.
1
Introduction
In general, the design of an efficient, compact mass transfer pack for gas/liquid applications is based on the optimisation of the passage diameter and passage length. Also from a number of recent studies it is apparent that the choice of material plays a major role in packing design, the ideal material being highly formable in order to provide a high specific surface area, Egberongbe [3]. Heat and mass transfer between a falling liquid film along a vertical wall and upward flowing air contacting directly with the film is an important and interesting phenomenon in industrial apparatus such as cooling towers. While 96% of the cooling towers use PVC packing with smooth and cross ribbing, no data on the flow of liquid over a flat vertical wall with cross ribbing have been published. Only some of the features of their operation in contact heat exchangers have WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060171
174 Advanced Computational Methods in Heat Transfer IX been investigated by [1, 2, 4–6]. Major aspects that remain to be studied include: the geometry and layout of the main corrugation with and without the cross ribbings, the pattern of flow of the liquid film and interaction between phases. In this paper the mass transfer and pressure drop characteristics of many types of corrugated packing, including smooth and rough surface corrugated packings, are investigated, and the relationship between packing mass transfer coefficients and pressure drops are discussed. Mass transfer performance of rough corrugated packing is increased by 1.5 to 2.5 times the smooth packing values, but the pressure drop of packings also increases with the increase in heat transfer performance.
2
Experimental apparatus and procedure
The experimental apparatus for the heat transfer experiments, consisted of a counterflow forced draft cooling tower, as shown in fig. 1. Water stored in a tank at the base was pumped into the spray nozzles. The supply water velocity was regulated by a valve. The cross sectional test area was A= 0.15 x 0.15 m. Inlet and outlet air and water temperatures were measured by mercury in glass thermometers with a range of 0-50°C and an accuracy of 0.2 K. Packing pressure drop was measured by an APM 2000 (0 to 2000 Pa) micromanometer with an accuracy of ±1% FSD (i.e. maximum of 1.2 Pa error in our measurements). Measurements of mass transfer and pressure drop were carried out in the steady state. The mass transfer coefficients and pressure drops were measured for a range of L/A ( L′ ) from 0.45 to 2.22 kg/m2s and G/A ( G ′ ) 0.20 to 1.50 kg/m2s. A series of perimeter deflector plates was installed around the inner perimeter of the column, made in the laboratory of clear Poly Carbonate plastic to allow observation of the water flow. These deflector plates removed the water film from the wall of the tower’s column and redistributed the water in the packing zone. As a result of deflection, most of the water was transferred to the packing surface from the outer wall, forming descending thin films, while air was blown vertically upward, counter current to the water by a fan at the base. The packings tested were of two types, smooth and ribbed, both of PVC. The smooth packing had horizontal corrugations and the ribbed had horizontal corrugations with ribbing set at an angle to the main corrugations. The cross ribs were separated by distance p, ranging from 2 mm to 10 mm, for the six sample packings, and the height e of the ribs ranged from 1 mm to 3 mm. The main corrugation pitch, P, ranged from 30 mm to 70 mm. The thickness of packing was negligible. The forms of corrugated packings used in the experiments are listed in table 1, and typical shapes are shown in figs. 2, 3 and 4. The column packed height, Z, was 160 cm and the water level in the sump was about 1.2 m below the top of the packing. Water inlet and outlet temperatures were 37 °C and 27 °C respectively.
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Advanced Computational Methods in Heat Transfer IX
Table 1: Test Grou p C1 C2 C3 C4 C5 C6 C7
175
Shapes of corrugated packing used.
Type of corrugation sinusoidal sinusoidal triangular triangular hexagonal sinusoidal triangular
P/D
p/e
θ (deg)
1.40 1.65 1.13 1.43 1.32 1.50 1.50
1 3 4 5 4 5
45 0 0 0 45 0
.
Figure 1:
3
Outside view of forced draft cooling tower in laboratory.
Experimental results
3.1 Heat transfer characteristics Cooling tower packings typically have quite complex surface geometries, for which the mass transfer co-efficient, k, cannot be analytically predicted. Because manufacturers treat such data as proprietary, the k relation should be derived from test data, specific to the packing geometry. Fig. 5 shows values of measured mass transfer coefficient k, plotted against the ratio of water flow rate to air flow rate (L/G) for existing packings. The values of k for corrugated packing were 1.5 to 2.5 times higher than comparable smooth packing k values when the water to air ratio was 1.0. The k values for rough and smooth corrugated packings decreased with the increase in pitch, and had a maximum value when P/D = 1.5 and the ratio of distance between repeated ribs to height of rib was 4, and the angle, θ, 45°. (Packing C6). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
176 Advanced Computational Methods in Heat Transfer IX P
R
D
Figure 2:
Typical shape of smooth corrugated packing used in our experiment.
Figure 3:
Figure 4:
Single cross ribbed sheet.
Typical shape of rough corrugated packing used in our experiment.
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0.08
0.07
0.06
0.05
Rough Packings
C6 C7 C5
k
Smooth Packing
0.04
C4 0.03
C2 C1
0.02
0.01
0 0.3
C3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Water/Air flowrate
Figure 5:
Heat transfer characteristic of packing with different spacing and surface roughness.
As it can be seen from ,fig. 5, k increases with (i) decrease of the spacing between the sheets, all other parameters being constant. (ii) increase in the value of L′ , for G ′ = constant. (iii) increase in the ratio of the pitch of the corrugation to the spacing. P/D should be of the order of 1.5 and p/e should be of the order 5 to have maximum heat transfer. (iv) decrease of the ratio of distance between repeated ribs and height of the rib. (v) decrease in θ. It can see that mass transfer increases with the decrease in spacing, but higher mass transfer in packing C6 compared with C7 and C5 is likely to be due mainly to the difference in the effect of the packing wall roughness[factors (iv and v) above]. The resultant correlation k of the Nos 1 to 7 was determined from these experiments with the most susceptible to error of ±4% by; k = c1 ( L ′ )0.45 ( G ′ )0.6
(1)
c1 is an experimental constant. The constant for type No. 3, (smooth surface), is 1.20 while for type Nos 1, 2, 4–7 the constant is 1.75, 1.83, 190, 1.98, 2.20, 2.10 respectively. Using the smooth sample, No 3 as reference, the relative increases due to ribbing were, for No. 1 = 1.45, for No. 2 = 1.52, for No. 4 = 1.58, for No. 5 = 1.65, for No. 6 = 1.83, and for No. 7 = 1.75 respectively.
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178 Advanced Computational Methods in Heat Transfer IX 3.2 Pressure drop characteristics Another aspect of the investigation of the performance of packings concerned pressure drop characteristics. As it can be seen from fig. 6 ∆P increases with (i) decrease in spacing between the sheets when all the other parameters being constant. (ii) increase in L', for G' = constant. (iii) increase of the ratio of distance between repeated ribs and height of rib. (iv) decrease in θ. The resultant of pressure drop for the packing No. 1 to No.7 is expressed with the most susceptible to error of ±3% by; ∆P = c2 ( L ′ )0.35 ( G ′ )0.55
(2)
c2 is an experimental constant. The constant for type No. 3 having a smooth surface is 17.7 while those for types Nos. 1, 2, 4–7, are 20.5, 22.6, 25.6, 27.8, 30.7, 32.5, 35.2, respectively. The result in fig. 6 shows that the pressure drop of packing C7 is about 70% higher than that of C1. This difference appears to be caused by difference in the height of the corrugations and the different surface created by the ribs. The only exception is for C6 of the present investigation (spacing of 20 mm). The pressure drop is lower by about 15% than the pressure drop for the C7 (spacing of 20 mm). This difference can be attributed to the difference of the turbulent flow condition caused by the wall roughness of the packing created by the lower distance between the plates. 140
C7
120
C6 C5 100
C4 C2
∆p
C1
(pa)80 Rough Packings
60
40 C3
Smooth Packing
20
0 0.5
0.65
0.8
0.95 1.1
1.2
1.4
1.55 1.70 1.85
2.0
2.15 2.3
2.4
2
Water mass flux (kg/ms)
Figure 6:
Pressure drop characteristic with different spacing and surface roughness.
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4
179
Discussion
In relatively narrow, corrugated packings flow separation takes place near the ridge of every corrugation, and flow re-attachment takes place upstream of the next ridge in the flow directions. The troughs of the corrugations are partly filled with re-circulating fluid. It was found that a packing of particular interest was the packing C6, which has a vertical main corrugation with the cross ribbing making an angle of 45°. Flow separation enhances the turbulence of the flow (compared with corresponding flow between smooth, straight wall) and thereby increases mass transfer rate and pressure drop, (e.g. Packing C6). A packing with high turbulence in combination with a relatively low fluid velocity is more economic than a fairly smooth and straight packing in combination with a high fluid velocity. The results showed that mass transfer performance of the corrugated packing is increased by up to 1.5 to 2.5 times compared to the smooth packing, C3. In order to have the maximum mass transfer, the ratio of the pitch to spacing of the corrugation, P/D should be of the order of 1.36 to 1.50. In this study packing mass transfer coefficients, k, of corrugated packings were expressed by Eqn. (1). It was found that, for the effect of pitch on the Nusselt number (Nu), the value C was approximated to by (P/D)-0.15.
5
Conclusions
Experiments were conducted to investigate the effect of the spacing and surface roughness on the mass transfer and pressure drop in PVC packing for which no comprehensive investigations had previously been reported. The experiments were carried out for comparative types of packing in a counterflow cooling tower. From the experimental results and discussion on the performance characteristic of seven vertical parallel packings arrangement in forced draft counterflow cooling tower the following conclusions may be drawn; (1) Overall mass transfer coefficients and pressure drops of ribbed corrugated packings increase considerably compared with smooth packing and are affected by spacing of the packing and the distance between the ribs. (2) It was found that the shape and configuration of the roughness projections are as important as the height of those projections in determining their effect on Fanning friction factor and mass transfer coefficient. It was found that a packing of particular interest was the packing C6, which had maximum mass transfer value at P/D = 1.5 and the ratio 4 for the ratio of distance between repeated ribs to height of rib with the cross ribbing making an angle of 45°. (3) Packing mass transfer coefficients vary in proportion to (P/D)-0.15 and C the value decreased with increase in P/D. (4) Friction factors of corrugated packings vary in proportion to (P/D)-0.94 and (p/e)-1.52. (5) Mass transfer coefficients of corrugated packing vary in proportion to the 0.41 power of pressure drop per unit height. This value of 0.41 is smaller than smooth packing value 0.46.
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180 Advanced Computational Methods in Heat Transfer IX
Symbols A D e G G' E L L' k p P Pr Z Rew ρa ua uw θ
Surface area per unit volume Distance between the tower packing Height of roughness Flow rate (air) Mass flux (air) Height of corrugation Flow rate (water) Mass flux (water) Mass transfer coefficient Distance between repeated ribs Pitch of packing Prandtl Number Packed height 2 L ′ D/µw Air density Air velocity inside the packing Water velocity inside the packing Angle of cross ribbing with the horizontal
m-1 mm mm kg/s kg/m2s mm kg/s kg/m2s kg/m2s mm mm dimensionless m dimensionless kg/m3 m/s m/s °
References [1] Bernier, M. A., Thermal performance of cooling towers, ASHRAE Journal, 4, pp. 26 31, 1995. [2] Bukowski. J., Taking the heat off industrial processes, Consulting Specifying Engineer, 10, pp. 32 38, 1995. [3] Egberongbe, S. A., How to design compact mass transfer packing for maximum efficiency, Process Engineering, 9, pp 5 11, 1990. [4] Kranc, S. C., Performance of counterflow cooling towers with structured packings and maldistributed water flow, Numerical Heat Transfer, 23, pp 15 23, 1993. [5] Marselle, A., Progress in Heat and Mass Transfer, ASHRAE Journal, 9, pp 56 62, 1991. [6] Nabhan, B. W & Anabtawi, M., An investigation into a falling film type cooling tower, International Journal of Refrigeration, 18(8), pp 19 25, 1994.
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181
Autowave regimes of heat and mass transfer in the non-isothermal through-reactors A. M. Brener & L. M. Musabekova State University of South Kazakhstan, Kazakhstan
Abstract The paper deals with autowave processes in non-isothermal reacting diffusion systems. The mathematical model of heat and mass transfer in a tubular through-reactor has been submitted. The model consists of the diffusion-kinetic equations for two reagents under the first-order reversible reaction and heat transfer equation with allowance for a heat of reaction. Reaction rate constants for direct and reverse stages are related to temperature according to the Arrhenius law. The cases of both the adiabatic and non-adiabatic reactor have been considered. The stationary regimes and conditions of their stability have been investigated by methods of hydrodynamic stability theory and numerical experiment. As a result, the simultaneous governing transfer parameters including thermal, diffusion and kinetic characteristics of both reagents with allowance for the two reaction stages were detailed. The existence conditions for dissipate structures which can be identified as running circular cells or wave fronts have been obtained. Results of the investigations can be applied to calculating the heat and mass transfer intensity in chemical reactors and heat exchangers.
1
Introduction
Studies of autowave processes in chemical systems with diffusion are of great interest for scientists and engineers of different specialties now. The problem to describe such processes in every details is too complex and far from the complete solution. At the same time, the dynamics of dissipate structures has been investigated for isothermal multiphase systems sufficiently well [1–4]. But regularities of generating and propagating autowaves in non-isothermal systems with chemical reactions accompanied by heat effects have been studied WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060181
182 Advanced Computational Methods in Heat Transfer IX to an insufficient extent. It is particularly correct in reference to multistage or reversible chemical reactions for which we can observe an interaction between different trends in the process evolution [5–7]. In the present work an attempt has been to prove the possibility of existence and to investigate conditions for arising dissipate structures in the simple case of the non-isothermal reversible first-order chemical reaction.
2
Theoretical details
Let us consider the chemical conversion which is going on in a tubular through-reactor according to the scheme of a reversible first-order reaction: k
X
1 →
k2
←
Y.
(1)
Here k1 and k 2 are the reaction rate constants for direct and reverse stages. For an adiabatic reactor the system of mass and heat transfer equations reads ∂C X ∂ 2C X j ∂C X = DX + − k1C X + k 2 CY , 2 ∂t S ∂z ∂z ∂CY ∂ 2CY j ∂CY = DY + + k1C X − k2CY , 2 ∂t S ∂z ∂z
∂T ∂ 2T j ∂T ∆H =χ 2 + + , ∂t S ∂z ρ c p ∂z
(2) (3)
(4)
where C X , CY are the concentrations of components X and Y ; C X 0 is the inlet concentration of X; D X , DY are the diffusion coefficients of reagents; t , z are the time and space coordinates; j is the total consumption of reagents through the reactor; T is the temperature; χ is the average temperature conductivity; ρ is the average density of the reagents mixture; c p is the average specific heat; ∆H is the total heat of the reaction; S is the reactor crosssection surface. We assume that X is the only inlet reagent, and all parameters and properties of reagents are constant except rate coefficients. Such assumption seems to be correct by reason of a prevailing strong dependence of rate constants on temperature according to the Arrhenius law: k1 = k10 exp(− E1 ( RT ) ) ,
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(5)
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183
k 2 = k 20 exp(− E 2 ( RT ) ) , (6) where k10 , k 20 are the standard values of the rate constants for direct and reverse stages; E1 , E 2 are the activating energies for direct and reverse reaction stages; R is the constant of ideal gases. The total heat of the conversion reads ∆H = H 1k1C X + H 2 k 2 CY ,
(7)
where H 1 and H 2 are the heats of direct and reverse stages. The suitable variables are given by t, η = z +
j t. S
(8)
Thus instead of (2), (3), (4) we obtain: ∂C X ∂ 2C X = DX − k1C X + k 2 CY , ∂t ∂η 2
(9)
∂CY ∂ 2 CY = DY + k1C X − k 2 CY , ∂t ∂η 2
(10)
∂T ∂ 2T H 1k1C X + H 2 k 2 CY =χ + . ρc p ∂t ∂η 2
(11)
− k As rank of the matrix 1 k1 a non- trivial fixed points if
k2 equals 1, the system of (9), (10), (11) has − k 2 H1 + H 2 = 0 .
(12)
The condition (12) corresponds with the Hess law. Thus we obtain a balance of fixed concentrations in the system with the ratio: (E − E1 ) C X 0 k 2 k 20 , = = exp − 2 CY 0 k1 k10 RT0
where T0 is the stationary temperature which depends on initial conditions.
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(13)
184 Advanced Computational Methods in Heat Transfer IX Linearizing the system (9)–(11) in a neighbourhood of the fixed point and using the approximation E E E 1+ exp − θ, ≈ exp − 2 RT RT0 RT0
(14)
we can write the system of perturbed equations as follows:
where
EC E C ∂2x ∂x − K1 x + K 2 y − K1 1 X 0 θ + K 2 2 Y 0 θ , = DX 2 2 ∂t RT0 RT02 ∂η
(15)
EC E C ∂2 y ∂y + K1 x − K 2 y + K1 1 X 0 θ − K 2 2 Y 0 θ , = DY 2 2 ∂t RT0 RT02 ∂η
(16)
∂ 2θ H (K1 x − K 2 y ) HK1 (E1 − E 2 )C X 0 ∂θ + + θ, =χ ρc p ∂t ∂η 2 ρ c p RT02
(17)
x = C X − C X 0 , y = CY − CY 0 , θ = T − T0 .
(18)
With allowance for (13) we obtain: ∂x ∂2x = DX − K1x + K 2 y + K 3θ , ∂t ∂η 2
(19)
∂y ∂2 y = DY + K1 x − K 2 y − K 3θ . ∂t ∂η 2
(20)
∂θ ∂ 2θ − h(K1 x − K 2 y ) + K 3 hθ . =χ ∂t ∂η 2
(21)
Here: E E K 1 = exp − 1 , K 2 = exp − 2 RT0 RT0
K (E − E1 )C X 0 , K3 = 1 2 , RT02
H = H1 = − H 2 , h = −
H
ρc p
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.
(22)
(23)
185
Advanced Computational Methods in Heat Transfer IX
Let us look for dissipate structures in the form of running cells [7]: mπ x = α1 exp(λt )sin η , L
(24)
mπ y = α 2 exp(λt )sin η , L
(25)
mπ η , L
(26)
θ = α 3 exp(λt )sin
where L is the characteristic longitudinal scale of reactor; α i is the amplitude of wave structures ; m is the mode number, λ is the increment. As a result we obtain the following linear system for perturbations amplitudes: 2 2 λ + D X m π + K1 α1 − K 2α 2 − K 3α 3 = 0 , L2
(27)
m 2π 2 − K1α1 + λ + DY + K 2 α 2 + K 3α 3 = 0 , 2 L
(28)
m 2π 2 hK1α1 − hK 2α 2 + λ + χ − hK 3 α 3 = 0 . L2
(29)
The above system has non-trivial solutions if and only if 2 2 λ + D X m π + K1 L2
− K2
− K3
− K1
2 2 λ + DY m π + K 2 L2
K3
hK1
− hK 2
=0.
2 2 λ + χ m π − hK 3 L2
(30) Determinant (30) can be exposed in the form:
λ1λ2 λ3 − λ1λ2 hK 3 + λ1λ3 K 2 + λ2 λ3 K1 = 0 , WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(31)
186 Advanced Computational Methods in Heat Transfer IX where m 2π 2
λ1 = λ + D X
λ2 = λ + DY
λ3 = λ + χ
L2 m 2π 2 L2
m 2π 2 L2
,
(32)
,
(33)
.
(34)
As equation (31) is cubic concerning the increment λ it certainly has real roots. Therefore, existence of space dissipate structures is conditioned either by that one real root of the equation (31) equals to zero and real parts of other two roots are non-positive (case 1), or by that one real root is negative and other two roots are imaginary (case 2). Let’s consider both cases. 2.1 Case 1 From the condition for the real root it follows: m 2π 2 2
L
=
K K − 1 + 2 χ D X DY
K 3h
.
(35)
It is obvious that (35) can be realized only if H 1 < 0 (i.e. the direct reaction must be endothermic), and the following inequality is correct: K 3h
χ
K K > 1 + 2 . D D Y X
(36)
From this it follows the expression for the length of solitary wave front: Λ =
π K 3 h K1 K 2 − + χ D X DY
.
(37)
The second condition for complex roots reduces to the system of inequalities:
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Advanced Computational Methods in Heat Transfer IX
χ χ ≥ K 2 − 1 , K1 1 − DX DY K1 χ K2 χ
≥ 1 − − 1 . D X D X DY DY
187 (38) (39)
It is easily seen that the following conditions together with (35) are sufficient for the above system correctness: K1 + K 2 > K 3h ,
(χ − DY )(D X
− DY ) < 0 .
(40) (41)
Let’s start using the dimensionless parameters: A1 =
χ DX
; A2 =
χ DY
; A3 =
K1 K ; A4 = 2 . K 3h K 3h
(42)
Thus, in most cases corresponding to the probability values of media properties we can display the conditions (36), (40), (41) in a graphic form. Shaded areas in the Fig.1 contain points with coordinates satisfying the system (36), (40), (41). 2
A3
1
A1
( A3, A4 )
0.5
•
•
( A1, A2 ) 0.5
Figure 1:
A2
A4 2 1 Domain of governing parameters (42) meeting the conditions (36), (40), (41).
As it follows from (36) the dot product between vectors ( A1 , A2 ) and ( A3 , A4 ) must be less than 1. Therefore, the boundary position of the point
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188 Advanced Computational Methods in Heat Transfer IX ( A3 , A4 ) about the point ( A1 , A2 ) is determined by inversion concerning to the circle of radius 1 in Fig. 1. The minimum length of reactor, in which the space dissipate structures corresponding to case 1 may be generated, is given by
Lmin = Λ =
π K 3 h K1 K 2 − + D X DY χ
.
(43)
2.2 Case 2 In this case analytical investigations entail cumbersome computations. Our attempt to obtain simple sufficient conditions for existence of dissipate structures corresponding to case 2 has failed. At the same time there are reasons to believe that under the condition (40) generation of structures corresponding to case 2 will be impossible. On the other hand it is highly probable that the following conditions will be sufficient for springing up the mentioned dissipate structures: K1 + K 2 < K 3h ,
(44)
hK 3 (D X + DY ) < χ (K1 + K 2 ) + K 2 D X + K1 DY .
(45)
For a non-adiabatic reactor the system of mass and heat transfer equations differs from the system (2)–(4) only in the last equation: k ∂T ∂ 2T j ∂T ∆H + + − T (T − Tm ) , =χ 2 S z ρ c ρ cp ∂ ∂t p ∂z
(46)
where k T is the ambient heat transfer coefficient and Tm is the ambient temperature. For existence of fixed points under the condition (12) it is necessary now to propose that T0 = T m .
(47)
Thus instead of (21) we obtain: k ∂θ ∂ 2θ =χ − h (K1 x − K 2 y ) + K 3h − T 2 ρc p ∂t ∂η
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θ ,
(48)
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It is clear that qualitative analysis of dissipate structures in a non-adiabatic through-reactor hasn’t differences of principle from the case of an adiabatic reactor. However there is yet another governing parameter kT ρ c p .
3
Results and discussion
Figure 2 depicts some results of numerical tests carried out under the realistic values of physical and chemical parameters satisfying the case 1: 1) K 1 = 3.02 ⋅10 −7 , K 2 = 7 ⋅10 −10 , K 3 h = 2.5 ⋅10 −7 , 2) K 1 = 4.0 ⋅10 −7 , K 2 = 8.0 ⋅10 −10 , K 3 h = 3 ⋅10 −7 , 3) K 1 = 2.0 ⋅10 −7 , K 2 = 1.0 ⋅10 −9 , K 3 h = 1.8 ⋅10 −7 , 4) K 1 = 2.0 ⋅10 −7 , K 2 = 1.0 ⋅10 −9 , K 3 h = 9.5 ⋅10 −8 , 5) K 1 = 2.0 ⋅10 −7 , K 2 = 1.0 ⋅10 −9 , K 3 h = 6.5 ⋅10 −8 ,
χ
DX
χ
4
0.90
2
C X C X 0 0.72 0.54
5
= 0.25,
DX
χ χ χ DX
DY
χ DY DY
χ
= 0.3,
DX
χ
χ
= 0.3,
DX
2 3 1.0
= 0.50 ,
DY
χ
= 0.3,
DY
5
= 0.6 . = 0.6 .
0.8
3
0.6 T T0
4
0.18
0.2 0.4
= 0.6 .
1.0 1
0.4
0.2
= 0.5 .
1
0.36
0
= 0.67 .
0.6
0.8
1
η / Lmin Figure 2:
Heterogeneous distributions of concentration (—) and temperature (·–·–·) along a tubular reactor in the case 1.
Heterogeneous distributions of concentration and temperature along a tubular reactor spring up in consequence of perturbing the stationary regime. Dissipate WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
190 Advanced Computational Methods in Heat Transfer IX structures of this type are unstable, and they constitute an initial stage of generating running autowave fronts [6]. Dynamics of autowave structures formation and propagation in every detail can be described only by methods of non-linear stability theory [6, 7].
4
Conclusion
Formation of stable dissipate structures, such as running cells and autowaves, in non-isothermal reactors is controlled by set of diffusion and heat parameters. Our theoretical analysis allowed to detail the governing parameters which can be used for determining regimes of dissipate structures generating. In particular, it has been clearly defined a role of rate constants dependencies on temperature with allowance for direct and reverse stages of the first-order chemical reaction. The main types of possible dissipate structures induced by these factors in a non-isothermal tubular through-reactor as well as conditions of their formation have been determined. The results of investigations are likely to be useful for calculating intensity of heat and mass transfer processes in chemical apparatuses.
References [1]
[2] [3] [4]
[5] [6] [7]
Alamgir M., Epstein I.R., Systematic Design of Chemical Oscillators. 17. Birhythmicity and Compound Oscillation in Coupled Chemical Oscillators: Chlorite-Bromate-Iodide Systems. Jour. Amer. Chem. Soc., 105, 1983, P. 2800. Ortoleva P., Ross J., Theory of Propagation of Discontinuities in Kinetic Systems with Multiple Time Scales: Fronts, Front Multiplicity and Pulses, Jour. Chem. Phys., 63, 1975, P. 3398. Schlöglm F., Chemical Reaction Models for Non-Equilibrium Phase Transitions, Z. Phys., 253, 1972, P.147. Baetens D., Van Keer R., Hosten L.H., Gas-liquid reaction: absorption accompanied by an instantaneous, irreversible reaction, Proc. Computational Modelling of Free and Moving Boundary Problems, 1997, P. 185-195. Tyson J.J., Oscillations, Bistability and Echo Waves in Models of the Belousov-Zhabotiskii Reaction., Ann. New York Acad. Sci., 316, 1979, P. 279. Showalter K., Noyes R.M., Turner H., Detailed Studies of Trigger Wave Initiation and Detection., Jour. Amer. Chem Soc., 101, 1979, P. 7463. Serdukov S.I., About existence of dissipate structures and running fronts for system of reactions with heat effect in a limited volume, Jour. of Phys. Chemistry, 59, No9, 1985, P. 2292-2296. (in Russian)
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Experimental studies of heat transfer between crystal, crucible elements, and surrounding media when growing large-size alkali halide ingots with melt feeding V. I. Goriletsky, B. V. Grinyov, O. Ts. Sidletskiy, V. V. Vasilyev, M. M. Tymoshenko & V. I. Sumin. Institute for Scintillation Materials NASU, Kharkiv, Ukraine
Abstract A series of time stages of heat transfer between the growing crystal and the growth furnace inner elements (different parts of the crucible, water-cooled walls of the growth chamber) have been revealed at growth of АI ВVII crystals. Experimental data on temperature changes of these objects have been obtained for the first time using non-contact pyrometry methods. Some causes of structure defects formation connected with intensification of heat transfer from the growing ingot to the water-cooled walls of the growth chamber and leading to unavoidable melt supercooling have been determined. The causes of crystallization front shape changes initiated by thermal flux from the crucible elements to the melt surface have been established. Keywords: crystal growth, condensate, heat transfer, temperature distribution, pyrometry.
1
Introduction
Until now, in comparison to the Czochralsky growth, a lack of direct experimental studies concerning heat transfer processes inside the vacuum water cooled growth camera has been observed for automated continuous feed method developed by Eidelman et al. [1]. In many instances, this fact is connected with the condensate (layer of evaporated melt components) precipitating on growing crystal and growth furnace inner elements. Utilization of IR-pyrometry is, probably, the one suitable method of measurements at the growth of alkali halide WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060191
192 Advanced Computational Methods in Heat Transfer IX crystals in vacuum, when the crucible and growing ingot continuously rotate. So, a procedure of direct non-contact pyrometric temperature measurements of crystal surface and crucible elements at different growth stages has been developed by Sidletskiy et al [2].
Figure 1:
General view of the growth furnace inner surface and the surface of the 500 mm dia. NaI(Tl) growing crystal. 1 – lower part of the water cooled crystal holder; 2 – condensate collector; 3 – telescopic protective shield of the water-cooled crystal holder rod; 4 – transport tube holder; 5 – inner surface of the furnace case; 6 – transport tube; 7 – coupling nut of the seed; 8 – upper butt of the growing crystal; 9 – vertical wall of the crucible; 10 – melt. The left photo is made after the unloading of the crystal; the right one is made directly during the growth process.
A growth furnace of “ROST” type setup consists of two essential parts. A thermal camera with crucible is placed in the lower part, and growing crystal are situated in the upper part. Thus, this construction predetermines the direction of heat flux from the heaters through the crucible, melt, and crystal to the water-cooled case of the furnace upper part. A free melt surface, not barred by the crystal, always exists during the crystal growth on a seed. The square of this surface, other conditions being equal, determines the quantity of the precipitating condensate. This fact is of great importance for the growth of scintillation crystals such as NaI(Tl), CsI(Tl), where the dopant is a volatile substance. Following these considerations, cooled inner walls of the growth furnace are the preferable place for condensate precipitation. However, condensate is also observed on the more heated surfaces, as one can see in fig. 1, where the general view of the growth furnace inner surface and growing ingot are presented. The role of the condensate seems weighty, because it is an insulator that substantially influences the heat transfer in the system. Non-uniformity of condensate thickness makes heat transfer dynamics even more difficult.
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193
Peculiarities of heat and mass transfer at radial growth and body growth
The three more important stages of the large single crystal growth with melt feeding can be emphasized, fig. 2: a – radial (cone) growth; b – growth on height (body growth) till the moment when crystal upper butt leaves the crucible limits; c – growth on height after the moment when crystal upper butt leaves the crucible. Let us consider step-by-step the peculiarities of heat exchange between the crystal, crucible elements, inner atmosphere, and furnace case.
Figure 2:
Scheme of the large scintillation single crystal growth with melt feeding: 1 – side heater; 2 – bottom heater; 3 – crucible; 4 – periphery circular vessel (PCV) of the crucible; 5 – upper edge of the crucible vertical wall; 6 – water-cooled case of the furnace.
In accordance with the experimental data, at radial growth till the preset diameter, the crystal is covered with a condensate layer, and its thickness is proportional to the growth time. The exception is a small area near the seed where condensate is absent. The upper butt temperature at this stage, fig. 3, smoothly decreases with time; it is lower in the crystal centre and higher near the butt edge. So, at the diameter 270 mm, temperature in this area decreases down to 105°С in comparison to the crystallization point (621°C). One can see in, fig. 3, that in this area the condensate layer is the thickest (under illumination the condensate is lighter, and without illumination it is darker) providing an insulation of this crystal part from radiation passing from the heaters, through the melt and crystal (the latter are high transparent in IR band). The character of heat exchange on the radial growth stage is seen in fig. 4, where common dynamics of the following parameters are presented at the different growth stages:
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194 Advanced Computational Methods in Heat Transfer IX -
bottom heater temperature (tbot) providing automated temperature correction by signals from the melt level probe (see fig. 2); program controlled side heater temperature (tside); temperature of the PCV bottom (tper) measured by the pyrometer; temperature of the upper edge of the crucible wall (twall) measured by the pyrometer.
Figure 3:
Temperature distribution at the upper butt of CsI(Na) crystal for current values of crystal radius 100, 110, 120, and 135 mm. Above and below are photos of growing crystals with additional illumination (left parts of the photos) and without it (right parts of the photos).
Figure 4:
Dynamics of tbot, tside, tper and twall at growth of 300 mm dia. CsI(Na) crystal. The three stages of growth are separated by vertical lines.
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One can see from fig. 4 that at this growth stage initiated by decrease of tbot and tside, the decrease of tper and twall is also observed. This fact indicates the presence of radial temperature gradient in melt taking part in formation of a convex crystallization front (CF). Herein, it is obvious that the crucible elements above melt (see fig. 2, stage a) partly suppress a heat transfer in the direction from the crystal to the water cooled case. Such role of the crucible elements in common heat exchange is proportional to the distance from the melt surface to the crucible upper edge (Hfree). At growth with melt feeding, Hfree remains constant during all the process, and this parameter is not investigated in the present work. The next stage of the growth – body growth when all the crystal is still inside the crucible, fig. 2b – is characterized by the start of twall increase and stabilization of tper with a simultaneous continued decrease of the control parameters tside and tbot (see fig. 4). The cause of the crucible vertical wall temperature increase can be described unambiguously: heat emitting from the crystal side surface. Here we must note that a part of the crystal side surface near the melt surface is not covered with condensate and is capable to transmit a significant part of radiant heat flux. Moreover, at the start of constant diameter crystal growth, the square of the emitting surface substantially increases proportionally to the speed of pulling of the ingot from melt. Thus, if the vertical crucible wall temperature in the cone growth stage is determined, in general, by the tside parameter which must be decreased for radial growth, twall increases proportionally to crystal cylindrical part length Hcyl (see fig. 4) on stage b. On the other hand, with Hcyl increase, the crystal migrates towards the crucible wall upper edge where the influence of the cooled furnace walls increases. The picture of heat transfer presented is confirmed by the measurements of temperature changes on the crystal upper butt, fig. 5. Analyzing them, the following conclusions can be made: - the axial temperature gradient in crystal decreases due to increase of Hcyl and heat removal from the side ingot surface; - the representative minimum at the temperature distribution curve at Hcyl = 5–15 mm, hereinafter, slowly transforms into the plateau at R=100 – 135 mm (fig. 5, Hcyl= 235 mm) due to the increase of the temperature gradient in the crystal. Now considering the third growth stage, let us look back again to figs. 2c and 4. The main difference of this stage from the previous one is that the crystal (i.e. its upper part) is already situated outside the crucible limits stimulating even more powerful heat removal from the ingot. As a result, the melt overcools, and an automated system provides the sign reversal of the dtbot / dH cyl derivative. Then the growth process acquires the more stationary character. As the conclusion of this chapter, it should be noted that the slower tbot increase, fig. 4, and the decrease of the axial temperature gradient at Hcyl above 200 mm, fig. 5 inclusion, can indicate changes in heat removal conditions from the growing crystal to the water cooled furnace walls. The authors plan further studies of heat and mass transfer process in this direction. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
196 Advanced Computational Methods in Heat Transfer IX
Figure 5:
Temperature distribution on the CsI(Na) crystal upper edge at cylindrical part lengths Нcyl = 0, 15, 70, 85, 155, 235 mm. In addition, the corresponding changes of axial temperature gradient on the crystal upper butt (on 150 mm radius) are shown.
Figure 6:
Scheme of CF transformation at crystal growth with additional tside correction and without it: 1 – growing crystal; 2 – melt; 3 – CF with tside correction; 4 – CF without the correction.
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3 Heat and mass transfer and control above the properties of the growing crystal Substantial changes in CF volume at crystal elongation are observed experimentally, fig. 6, at growth of crystals with extremely large diameter (the ratio of crystal diameter to crucible diameter Dcrys/Dcruc > 0.83). Herein, melt level falls, and this reflects in the corresponding increase of the dopant concentration in the crystal with Hcyl increasing in accordance with Goriletsky et al. [3]. Close proximity of the crystal side surface to the crucible vertical wall, as shown above, increases the temperature in the PCV, which is accompanied, in turn, by the increase of volatile dopant evaporation and leads to changes in thermal conditions. So, when Hcyl increases, tper must be continuously controlled by tside temperature in order to stabilize the thermal conditions in an area near CF. For this reason, the additional contour of automated control system (ACS) of PCV bottom temperature (tper) was developed. Referring to fig. 4, we can see the results of such control: at constant tper (its stabilization starts at body growth stage) the tside parameter continuously decreases, thus, compensating heat input from the crystal to the PCV. As mentioned above, heat exchange substantially increases on the stage c, and ACS must react on this by adequate increase of the tbot parameter. Macrodefects represented by the foreign phase inclusions, or dendritic structures that situated along the CF, fig. 7, sometimes can be found just in this part of the grown ingots (at Hcyl = 40 – 60 mm). The represented local defects are situated along the CF shape and in the areas where crystal diameter increase is observed. This effect is accompanied by an increased rate of crystallization above the preset value. Thus, insufficient sensitivity of the ACS to changes in general heat exchange in the growth furnace is one more probable cause of the macrodefects formation.
Figure 7:
Macrodefects (denoted by the ellipses) along the CF (dashed lines) found on the longitudinal cut of CsI(Na) single crystal.
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4
Conclusions
1. Three time phases of heat transfer corresponding to the three general stages of crystal growth are established: 1.1. Thermal field limiting the heat removal from the crystal is formed on the radial growth stage. Primary, it is formed due to condensate layer precipitation, then, due to the close proximity of the crucible PCV creating temperature “barrier”. This phase is accompanied by decrease of the control (tside, tbot) and measured (twall, tper) parameters. 1.2. The moment of initial body growth coincides with the start of heat removal from the side crystal surface where condensate layer has not yet formed. This stage fixes the start of twall increase under stable tside and decreasing tbot. 1.3. More active heat transfer between the crystal, crucible walls, and water cooled case starts when conic part of crystal leaves the crucible limits. This fact is confirmed by the continuing twall increase and increase of tbot by the ACS. During all the body growth stage, the condensate does not precipitate on the conic part and on the area of 5–10 mm width near melt. Its thickness smoothly increases at 10–20 mm Hcyl height, and, further, condensate is distributed uniformly. 2. The main cause of dopant concentration fluctuations in single crystals of extreme diameter and different composition is the CF volume changes caused by the absence of tside correction. The effect is manifested in changes of thermal conditions near the CF as a result of heat exchange between crystal cylindrical part and PCV. 3. The increase of heat transfer from the growing crystal to the surrounding media, and inadequate ACS reaction, lead to the formation of local short-term throws of foreign phase admixtures at the CF due to melt overcooling and a corresponding increase of local mass rate of crystallization. The evidence of this effect is the crystal diameter increase.
References [1] Eidelman, L.G., Goriletsky, V.I., Protsenko, V.G. et al, Automated pulling from the melt – an effective method of growing large alkali-halide crystals for optical and scintillation applications. J. Crystal Growth, 128, p. 1059, 1993. [2] Sidletskiy, O.Ts., Goriletsky, V.I., Grinyov, B.V., Sumin, V.I., Sizov, O.V., Tymoshenko, M.M., Monitoring of thermal fields on surface of alkali halide single crystals grown from the melt. Functional Materials, 12(4), p. 591, 2005. [3] Goriletsky, V.I., Grinyov, B.V., Sumin, V.I., Tymoshenko, M.M., Changes in crystallization conditions when growing large single crystals at melt feeding. Functional Materials, 11, p. 806, 2004.
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Modelling of heat and mass transfer in water pool type storages for spent nuclear fuel E. Fedorovich, A. Pletnev & V. Talalov Department of Physics and Mechanics, Saint-Petersburg State Polytechnic University, Russian Federation
Abstract The organization of heat removal from spent nuclear fuel in “wet” type storages is discussed. The heat- and mass-transfer processes (natural and mixed connection of water coolant, evaporation of water) under normal and accident situations in storages are considered. Special attention is devoted to the mathematical and physical modelling of the above mentioned processes. Keywords: heat mass transfer, nuclear spent fuel, water pool type storage, mixed convection, evaporation, accident situation, mathematical and physical modelling.
1
Introduction
From thermophysical point of view each storage is a heat generating system (due to residual energy release in radioactive decay processes) which has mass and heat exchange with environment. “Wet” type storage represents pools in which SNFA are immersed into. For the additional protection of the pool water from the direct contact with the cladding of highly radioactive fuel rods, the SNFAs can be placed in metal cans, which are also water filled. These cans are opened from above, which allows water free expansion and adding water to them by its evaporation. Such a construction of WS is used at Leningradskaya NPP near S. Petersburg, Russia (fig. 1). Pool water with the temperature of 30–400C is pumped from the bottom of pool compartments to the external system of cooling and purification and returns cooled up to 20–300C flowing into the regions near upper water level. The water of pool evaporates into ventilating air flow, sucked into the space between upper water level and metal floor of a storehouse. Maximum WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060201
200 Advanced Computational Methods in Heat Transfer IX temperatures of pool’s water are fixed in its upper part (fig. 1). Fact of temperature stratification leads to conclusion, that the natural convection is a dominant process in mixed convection heat transfer in water volume. T0C 37 36.6 36.2 35.8 35.4 35 34.6 34.2 33.8 33.4 33
0
1
2
3
4
5
6
7
8
9
10
h, m Figure 1:
Water pool type SNF storage at Leningradskaya NPP and water temperatures distribution along the height of the pool.
The magnitude of heat removal from SNF by water is defined by simultaneous accomplishment of the following processes: - mixed convection in the close vertical bundles, formed by rows of heat generating fuel rods (or cans) and forced water flow from above to down or in opposite direction; - evaporation from upper water levels (in pool and in cans, if last ones are used); - “pure” free convection inside cans, when directed movement of water is absent, because cans are closed from beneath.
2
Physical modelling of the counter flow mixed convection in a pool
2.1 Test section and experimental facility The body of the test section (fig. 2) is manufactured from the organic glass. It presents itself as a vertical box of rectangular cross section 132×118 mm and height 1070 mm. In upper part of body the water inlet collector (13) is placed. The cans with SNFA (or fuel rods inside can) are modeled by the electrically heated 56 stainless steel tubes with diameter 10 mm, arranged at corridor order (8×7). Upper part of the tube bundle with length of 360 mm is made from brass, has negligible electrical resistance and joule heat generation and forms a hydraulic stabilization part of a channel. The temperature of tubes is measured by thermocouples (5), placed inside tubes. Maximal electric power of the heater – 5 kW. The distance between tubes axises – 4 mm, so the narrowest gap between them is 4 mm. The flow rate through test section is controlled by the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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valve (19). It is possible to heat inlet water flow by electric heater in upper water tank (20). The outer surface of the body is thermally isolated , but isolation can be easily removed for visual observation of water flow.
1 – shell of test section (organic glass with outer sizes 132 x 118 x 1070 mm); 2 – heated tubes; 3 – digital millivoltmeter for thermocouples signals registration; 4 – thermocouples switch; 5 – moved thermocouple for measuring temperature of tubes; 6 – 8 moved thermocouples far water temperatures measuring along and across of test section; 7 – Dewar with constant temperatures of “cold” contacts of thermocouples; 8 – voltmeter of electrical power supply system; 9 – amperemeter of that system; 10 – primary controlled autotransformer; 11 – secondary power supply transformer; 12 – transformer of electrical current; 13 – water supply collector with flow uniform distribution nozzle device; 14, 18 – thermometers for water inlet and outlet temperatures measuring; 15 – moved water vessel, providing necessary level of water in test section and supplied by electrical heater of water; 16 – reversed U-type reometer for water flow rate measurements; 17 – filter for water purification; 19 – drainage valve; 20 – tank-damper for water dearation and flow rate stabilization; 21 – drainage valve; 22 – computer equipped with ADT card. Figure 2:
Scheme of test section and experimental facility.
2.2 Experimental methodics The stability of water flow through model has been provided with high accuracy ( ±1 %) due to use of the head tank with constant water level. The entering water was undergone constant cleaning. The local tubes and water temperatures have been registered at four cross sections along model’s height. The physical properties of water during the data treatment have been taken at mean ariphmetic values from water inlet and outlet temperatures. The values of local heat fluxes densities depend on the local joule heat releases in the electrically heated tubes. That’s why the special measurements of the each tube electrical resistance were WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
202 Advanced Computational Methods in Heat Transfer IX conducted and their temperature dependence was found. Additionally threedimensional computer program has been developed for the calculation of electrical potential distribution in the heater elements (upper brass tubes, stainless steel heating tubes, electrical soldered contacts) and in the water between tubes. Usage of this program allows to take into account the role of electrical conductivity of water (and consequently part of heat, generating in water volume) in whole heat flux production and to evaluate the role of reactivity electrical power losses in whole test section heat balance.
T, 0 C +1
Tw τ, s
−1
τ
τ + 40
“cold” mode test mode Figure 3:
Picture of flow in gap between heat generating rods (tubes) and water temperature oscillations inside gap.
The experimental data processing program allows also the possibility to determine the temperature drops across tubes walls thickness. With heater powers level less than 500 W this drop was less than 0.015 K, but with higher powers (close to 4-5 kW) temperature drop in steel wall can form up to 10% from overall temperature difference and consequently should be taken in account in data processing. By each experiment performance the heat balance of the test section was reduced with taking into account heat release in heater elements and water, the amount of heat, absorbed by water flow and heat losses to environment. The accuracy of heat balance reducing was not worse, than 2%. Special attention was given to the uniform flow distributions at water supply on the upper water level. With this aim the upper distributing collectors was used in a form of 14 tubes with 3 openings in each tube. The temperature difference between tubes and water were measured by hyperdifferthermocouples. Their signals were directed to analogous-figure transformer through an amplifier for the following treatment. During experiments the local water temperature oscillations were registered. Their amplitudes have reached about 50% of the “wall–water” temperature differences (fig. 3). It shows the significant turbulence in a space between heated rods. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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2.3 Experimental results The results for mixed convection heat transfer are plotted in coordinates [lg (Nu/Nu0); lg(Raq/Re)] (fig. 4). They related to the next dimensionless parameters ranges: Re = 28-125; Raq = 4,5·105- 1,6·107; Pr = 2,5-8,8. In these ranges the correlation between them can be presented in a convenient for the practical use form: Nu
Ra = 1 ⋅ q Nu 0 3 Re
1
3
,
(1)
which is found by the least square method. lg(Nu/Nu0)
2 1,8 1,6
y = 0,333x - 0,478 1,4 1,2
+20%
1 0,8
-20%
0,6 0,4 0,2 3,6
3,8
4
4,2
4,4
4,6
4,8
5
5,2
5,4 lg(Raq/Re)
, ∆, ◊, ○ – different experiments series; correlation [10]. Figure 4:
-Petukhov and Strigin
Heat transfer by counter flow mixed convection of water in vertical bundle of heat generating rods.
As also can be seen from fig. 4 our data are in good agreement with Petukhov and Strigin’s correlation formula [3, 7, 10]. Their data have been obtained for the counter-flow mixed “turbulent” convection of the water inside vertical tube and their correlation was recommended for diapasons: 3·102 ≤ Re ≤ 2,5·104; 5·103 < Raq < 1,3·107; 2 ≤ Pr ≤ 6. It is interesting to remark, that this agreement is discovered, despite of the considerable difference in Reynolds numbers minimum of Re ≅ 300 in the work [10] and maximum of Re ≅ 120 in our work. That circumstance can be explained by the similarity in physical processes of both cases, when a role of turbulent mass and heat transfer is a dominant one in overall transfer mechanism despite of low values of a mass flux. In our case WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
204 Advanced Computational Methods in Heat Transfer IX we falled up to Reynolds numbers ~ 30. Such flow rate velocities are typical for SNF storage pools. This supposition is supported by our visual observations of the flow inside the inter-tube gaps (see fig. 3). It was seen the visible vortexes formation in the gap space, which is disappeared with model heating stop. It is interesting to compare received data with a theoretical value of Nusselt number for the laminar flow in rod bundle Nulam. According to the work [15] the Nulam number for the square type arrow with the ratio S/d = 1,4 is equal Nulam = 7,7 (boundary condition q = Const). For Raq numbers corresponding to temperature oscillation rise (Raq ~ 2,1·106) the Nusselt numbers in our experiments were equal to ~ 30. It shows, that the heat transfer augmentation effect due to thermogravitational forces at the boundary of vortexes generation was about 30/7,7 ≅ 3,9.
3
Evaporation from pools and cans
Intensity of evaporation is defined by the rate of vapor extraction from the water surface. Vapour flow density g in the boundary layer near the surface is defined by diffusion and Stephen flow [17, 19] and expressed by the formula: G MPD 1 MPD g=− ⋅ ⋅ gradY = ⋅ grad[ln (1 − Y )] (2) RT 1 − Y RT where M – molecular weight of vapour; P – total pressure (atmosphere); D – diffusion coefficient of vapour in the air; R – absolute gas constant; T – absolute temperature; Y = pn/P, where pn – partial vapour pressure. The full heat flux from water per unit of area is G G G q full = g 0 ⋅ r + q 0
G
(3)
G
where g 0 and q 0 are mass flux and heat flux at the surface: r – specific heat of evaporation. Considering the processes in the boundary layer it’s necessary to introduce the values of mass transfer β and heat transfer α coefficients defined by the following expressions: g q and α = (4) β= " T0 − Tflow ρ red ⋅ (Y0 − Yflow ) p" p "0 " ; Yflow = flow , p 0 - partial vapour pressure near the water P P surface equal to the saturation pressure at the temperature surface T0; pflow – MP vapour partial pressure in the flow with the temperature Tflow; ρ"red = – R⋅T vapour density, reduced to the pressure P and defining temperature of the
where Y0 =
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boundary layer T . It often happens when only one of the coefficients is known (α or β). Then for the calculation of the second coefficient it’s possible to use heat and mass exchange process similarity, expressed by the equation: α λ = ⋅ f (Le) β D
(5)
D is Lewis number (λ – heat conductivity of mixture, a a – air-vapour mixture thermal diffusivity coefficient). In the simplest case, when λ λ = = ρ ⋅ c p and f(Le) is also equal to 1. For conditions heat Le = 1, the value D a removal of the water surface in the pools which are close to the conditions of evaporative water cooling in cooling towers [17] Le ≈ 1,2 and f(Le) = Le0,4. By increasing the temperature of the supplied air (hot season) the role of convective heat flux will be lower, and the part of the evaporating heat removal will be risen.
where
4
Le =
Accidental process in “wet” storage after water calculation termination
The scheme of the thermal two – dimensional computation model, corresponding to this problem, is shown at fig. 5. In this scheme the elementary model cell of WS pool is, in fact, reduced to one SNFA “micro storage”, which contents all participating in heat mass transfer elements of actual storage, e.g. fuel rods, can with SNFA, water inside a can, pool’s water and pool’s constraining constructions (walls, bottom, metal list above water level). So, one can expect, that after determination specific parameters of nonstationary process, namely amounts of heat removal and evaporated water related to one cell and after following multiplication these specific parameters on overall number of SNFAs in storage, these parameters for whole storage will be found. In this model all sizes of the can are kept as real ones, but intercan space are modeled by annular channel, which is more simple for the consideration, but similar in heat transfer process development. All sizes of the volumes, external in relation to the single can is chosen in a way, when the cross-sections areas of the intercan space and concrete side restrictions in model and in real storage would be equal. The wideness of two heat generating rings, modeling two rows of fuel rods inside SNFA (such is geometry of SNFA of RBMK – type reactor), is chosen in a way, when evaporation mirrors inside can in model and in natural storage also would be equal. The conditions of thermal modelling consist in the equality of Bio criteria for model and real object and in the recalculation of Fourier criteria, taking into account the difference between thicknesses of restriction concrete walls in reduced to single SNFA model and real pool.
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206 Advanced Computational Methods in Heat Transfer IX
Y, мм
12800 12600 11900
1
8500
2
3 6
7
4
5
1500 8
0
51
81,4 97,7
X, мм
1- central supporting rod; 2, 3 – rows of fuel rods; 4 – water inside canister; 5 – canister wall; 6 – water outside canister; 7 – wall of pool Figure 5:
Scheme of the thermal two-dimensional computation model of a pool (accident situation analysis).
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In above described model the boundary “water – air” is movable one and is beneathing step by step due to evaporation from pool and from cans. System of heat mass-transfer equations including equations of movement, written in cylindrical coordinates with taking into account physical properties of media with temperature, have appearance: (x – radial coordinate, y – axial coordinate) с⋅ρ⋅
∂T 1 ∂ ∂T ∂ ∂T − cρVT + Q v ; = ⋅ xλ − cρUxT + λ ∂τ x ∂x ∂x ∂y ∂y ∂С 1 ∂ ∂C ∂ ∂C − VC ; − UxC + D = ⋅ xD ∂τ x ∂x ∂x ∂y ∂x
(11)
∂U ∂P ∂ ∂U ∂U ∂U 1 ∂U U = − + 2µ − +2 ρ ⋅ +U +V µ + ∂x ∂x ∂x ∂x ∂y x ∂x x ∂τ +
∂ ∂U ∂V + ρFx ; + µ ∂y ∂y ∂x
∂V ∂ ∂V ∂P µ ∂V ∂U ∂V ∂V + + 2 µ = − + − ρ ⋅ +U +V ∂y ∂y ∂y x ∂x ∂y ∂x ∂y ∂τ ∂U ∂V + ρFy ; + µ ∂y ∂x G G 1 ∂ ( xU) ∂V + = 0 ; F = g ⋅ β ⋅ (Ta − T ) . x ∂x ∂y +
∂ ∂x
System (11) is solved by finite differences numerical method at following boundary conditions: - at the solid surfaces – conditions of “sticking “ (U = 0; V = 0); - at the boundary line “water – moistured air” the normal velocity components are equal to zero (V = 0) and tangential stresses are absent ( ∂U ∂y = 0 ); that corresponds to the absence of the above – water level space ventilation (conservative evaluation); - at the boundary line “water-steam-air-mixture” and near the upper metal floor, which is cooled by the natural air convection inside storehouse, the steam is saturated, but at the final stages of accident process the steam pressure near floor may be less than saturation pressure; - at the outer boundary of concrete wall of a pool the boundary condition of third type is specified: ∂T −λ⋅ = α env ⋅ (T − Tenv ) ; (12) ∂x
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208 Advanced Computational Methods in Heat Transfer IX - the temperature at the lower boundary of a computational region is constant and is equal to the average through year temperature of a soil under bottom (fundament) of a pool. The solution the system (11) is reduced by integer processes to the integral heat balance equation for storage and for each moment of time: Q SNFA = Q h + Q ev + Q tr − Qcond , (13)
∑
where
∑Q
SNFA
– sum of all SNFAs heat generations, Qh – power of all
constructions and media heating; Qev – heat removal from SNFS with evaporated water; Qtr – heat transmitted to the environment through boundaries due to heat conductivity, convection, thermal radiation and with ventilating air; Qcond – returned heat due to steam condensation on the colder in comparison with water solid surfaces. In expression (13) the value of Q SNFA is specified and members Qh, Qev,
∑
Qtr and Qcond are calculated. For system (11) solving and finding of equation (13) components it is necessary to have closing empirical correlations for the heat-mass-transfer coefficients. For that in several cases the literature data is possible to use, but sometimes is necessary special experiments to perform with specific processes conditions modelling. Such specific form of pool’s water flow for normal operation conditions and for the initial moment of the accident process is a combination of natural and forced convection (mixed convection which is above discussed).
5
Nomenclature and abbreviations
d – outer diameter of a can with SNFA or heated tube of experimental model, mm; deq = 4F , where F – cross section area of the elementary cell, formed by P four neighboring cans (tubes, rods), P – wetted heated perimeter of tubes outer surfaces part, which is in contact with water in one cell,m; g – gravity acceleration, m/s2; ρ – water density, kg/m3; α – heat-transfer coefficient, w/(m2·K); λ – water heat conductivity, W/( m2·K); v – kinematic viscosity, m/s; q – heat flux density, w/m2; β – water thermal expansion coefficient, 1/K; ∆t – gβ ⋅ ∆t ⋅ d 3eq – Grashof number; temperature difference “wall-liquid”, K; Gr = v2 4 gβ ⋅ d eq ⋅q Grq = – modified Grashof number; a – thermal diffusity coefficient, 2 v ⋅λ m/s; Pr = v – Prandtl number; Ra q = Grq ⋅ Pr – modified Ryleigh number; a w ⋅ d eq w – water velocity, m/s; Re = – Reynolds number; v WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
Nu =
209
α ⋅ d eq
– Nusselt number. Abbreviations: SNF(A) – spent nuclear fuel λ (assembly); SNFS – spent nuclear fuel storage; WS – “wet” type storage; NPP – nuclear power plant.
References [1] [2]
[3] [4]
[5] [6] [7]
[8] [9] [10]
[11] [12] [13]
Lambert R.W. (EPRI) Storage Experience, United States of America, BEFAST-III Conference, Toronto, Canada, Oct. 19-23, report RWL-P9901, 1992. Lambert J.D.B., Lambert R., Spent Fuel at commercial U.S. Reactors – an overview (ANL,EPRI), Proceedings of Advanced Research Workshop “Safety-related issues of spent nuclear fuel storage”, September 26-29, Almaty, Republic of Kazakhstan, p. 24, 2005. (in English). Kutateladze S.S., Leontjev A.I. Heatmasstransfer and friction in the boundary turbulent layer. Energia: Moscow: 342 pp, 1972. (in Russian). Thermal and hydraulic design of heat exchange equipment of Nuclear Power Plants. Methodical directions (RD 24.035.05.89). Ministry of heavy and power equipment. – I.I.Polzunov Central Boiler – Turbine Research and Design Institute: Leningrad, 210 pp, 1990. (in Russian). Design of Fuel Handling and Storage Systems for Nuclear power Plants: Safety Guide, Safety Standards Series No NS-G-1,4, IAEA: Vienna, August 2003. Quality Assurance for Safety in Nuclear Power Plants and other Nuclear Installations: Safety Series No 50-C/SG-Q, IAEA: Vienna, 1996. Petukhov B.S., Strigin B.K. Turbulent flow and heat transfer in tubes under the significant influence of the thermogravitational forces. Proceedings of the International Seminar on turbulent free convection, Dubrovnik, Yugoslavia, p.701, 1976. (in English). Kutateladze S.S. Fundamentals of Heat Transfer. Atomizdat: Moscow, 5th ed, 415pp, 1979. (in Russian); E. Arnold Publishers and New York, Academic Press Inc.,: London, 3d ed, 1963. (in English). Petukhov B.S., Poljakov A.F. Heat transfer by mixed turbulent convection. Nauka: Moscow, 192 pp, 1986. (in Russian). Petukhov B.S., Strigin B.K. Experimental study of heat transfer by viscous – inertia – gravitational flow of liquid in vertical tubes. Teplophysica vysokih temperature (High temperature thermophysics), vol.6, No 5, p. 933-937, 1968. (in Russian and in English). Martynenko O.G., Sokovishin Yu. A. Free – convectional heat transfer (Hand book). Nauka I Tehnika: Minsk, 388 pp, 1983. (in Russian). Petukhov B.S., Kirillov V.V. Heat transfer by the forced convection in tubes. Teploenergetica (Thermal energetics), No. 4, p 63-67, 1958. (in Russian and in English). Jackson J.D. Influences of buoyancy on velocity, turbulence and heat transfer in ascending and descending flows in vertical passages. Proceedings of 4th Baltic Heat Transfer Conference, Kaunas, Lithuania, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
210 Advanced Computational Methods in Heat Transfer IX
[14] [15]
[16]
[17]
[18] [19]
Aug. 25-27, 2003.“Advances in Heat Transfer Engineering”, ed. B. Sunden, J. Velimas, p. 57-78. (in English). Petukhov B.S., Genin L.S., Kovaljov S.A. Heat transfer in nuclear power installations. Energoatomizdat: Moscow, 470 pp, 1986. (in Russian). Kirillov P.L. et al. Methodical directions and recommendations for the thermohydraulic design of the fast reactors active zones (RTM 16.04.008.88) A.I. Leypunsky Institute of Physics and Energetics: 435pp, 1989. (in Russian). Gotovsky M.A., Fedorovich E.D., Fromzel V.N., Shleifer V.A. Heat transmission of the vertical heat generating rods bundle by the absence of coolant circulation. Inzhenerno-physichesky zhurnal (Journal of Engineering Physics), Minsk, vol. XVI No. 4 p. 363-370, 1984. (in Russian and in English). Arefiev K.M., Averkiev A.G., Physical peculiarities of the heatmasstransfer by evaporative water cooling. Izvestia VNIIG (Proceedings of All – Union Hydrotechnics Institute), vol.115, p. 81-86, 1977. (in Russian). Tveiten B., Kersting W., Karpov A. Russian Weapons Plutonium and the Western Option. Nuclear Disarmament Forum AG. Zug, Switzerland: 199pp., 2002. (in English). , Arefiev K.M., Fedorovich E.D. Problems of heatmasstransfer study in water pools for the nuclear power plants spent fuel storage. Proceedings of the XIII School – Seminar leading by Prof. A.I. Leontjev, Moscow Power Institute: Moscow, vol.2, p. 412-419, 2001. (in Russian and in English).
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Section 4 Modelling and experiments
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213
Experimental study of in-line tube bundle heat transfer in vertical foam flow J. Gylys1, S. Sinkunas2, V. Giedraitis2 & T. Zdankus1 1
Energy Technology Institute, Kaunas University of Technology, Lithuania 2 Department of Thermal and Nuclear Energy, Kaunas University of Technology, Lithuania
Abstract Foam has an especially large inter-phase contact surface which allows using it as a coolant. Characteristics of one type of foam – statically stable foam – demonstrated its perfect availability for this purpose. Our previous investigations of heat transfer processes in statically stable foam flow showed that large heat transfer intensity may be reached at a small mass flow rate of the foam. Statically stable foam flow is the two–phase system that has number of peculiarities: drainage of liquid from foam, diffusive gas transfer and destruction of inter-bubble films. Those phenomena are closely linked with each other and make extremely complicated an application of analytic methods for the study of heat transfer in foam. Thus experimental method of investigation was selected in our work. Experimental investigation of the heat transfer process from the in-line tube bundle to the vertical statically stable foam flow was performed. Dependency of heat transfer intensity on flow parameters and on tube position in the bundle was determined. The results of the experimental investigation are presented in this paper. Keywords: vertical foam flow, void fraction, heat transfer, experimental channel, in-line tube bundle.
1
Introduction
Foams are suitable for a lot of different purposes. It can be applied for heat and mass transfer performance as well, but usage of foam as coolant in heat exchangers or in foam apparatus depends on capability to “control” foam flow. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060211
214 Advanced Computational Methods in Heat Transfer IX Wide scientific researches of foam generation [1, 2], formed bubbles structure [1, 3], stability and disintegration [2, 4], foam flow hydrodynamics [3] and so on are necessary for its employment in mentioned cases. However, heat transfer process are investigated insufficiently especially in the regime of statically stable foam. Statically stable foam is such type of foams, which keeps its initial structure and bubbles' dimensions within broad limits of time intervals, from several seconds to days, even after termination of the foam generation. Our previous investigations showed the availability to use a statically stable foam for heat transfer. Heat transfer of different tube bundles to one-phase fluids was investigated enough, but practically there are not data of tube bundles heat transfer to foam flow. Heat transfer of alone cylindrical tube and of tube line to upward statically stable foam flow was investigated in our previous works [3]. The experimental series with staggered tube bundle in upward and downward foam flow followed as well [5–7]. It was determined dependence of heat transfer intensity on the following flow parameters: velocity, direction of foam flow, volumetric void fraction and liquid drainage from foam. Apart of that, influence of tube position in the bundle on heat transfer intensity was investigated also. Presently experimental investigation of heat transfer process from the in-line tube bundle to the vertical upward moving statically stable foam flow was performed. Results of investigations were generalized using relationships between Nusselt number and Reynolds number and volumetric void fraction of foam. The obtained generalized equation can be used for the designing of foam heat exchangers and calculating of heat transfer intensity of the in-line tube bundle.
2
Experimental set–up
The investigations were performed on the experimental set–up consisting of foam generator, vertical channel and bundle of the horizontal tubes, fig. 1. Cross section of the channel had square profile with a dimension of each side 140 mm. Tubes of the bundle were located in five vertical lines with six tubes in each of them, fig. 2. Outside diameter of all tubes was d=20 mm. Spacing between rows of tubes was s1=30 mm and spacing between tubes in vertical row was s2=30 mm too. Volumetric void fraction and velocity of foam flow was controlled by the changing of the air and liquid rates. Experiments were performed within Reynolds number diapason for gas 190÷440 and foam volumetric void fraction – 0.996÷0.998. Tube was heated electrically. An electric current value was measured by ammeter and voltage by voltmeter. The temperature of foam flow was measured by two calibrated thermocouples: one in front of the bundle and one behind. The temperature of heated tube surface was measured by eight calibrated thermocouples. Six of them were placed around central part of heated tube and two of them were placed in both sides of the tube at 50 mm distance from the central part. Water solution with detergents was used in experiments. Concentration of detergents was kept constant and it was equal 0.5%.
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215
1
9 8 2 V 7
A mV
10 11
~ 6
5
4 Liquid
Figure 1:
Foam
3
5
4
Gas
Experimental set–up scheme: 1–liquid reservoir; 2–liquid level control reservoir; 3–liquid receiver; 4–gas and liquid control valves; 5–flow meter; 6–foam generation riddle; 7–experimental channel; 8–tube bundle; 9–thermocouples; 10–transformer; 11–stabiliser.
The foam flow volumetric void fraction can be expressed by the eqn (1) Gg β= . (1) G g + Gl The temperature of the heated tube surface and the foam flow, electric current and voltage were measured and recorded during the experiments. The preliminary investigations showed that hydraulic and thermal regime stabilizes completely within 5 minutes after the change of experiment conditions. Therefore measurements were started not earlier than 5 minutes after adjustment of foam flow parameters. After registration of electric current and voltage the heat flux density on the tube surface qw was calculated. After record of heated WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
216 Advanced Computational Methods in Heat Transfer IX tube surface and foam flow temperatures, the difference of temperature ∆T (between the mean temperatures of the foam flow T f and tube surface T w ) was calculated. The average heat transfer coefficient was calculated as qw α= . (2) ∆T The Nusselt number was computed by formula Nu f =
αd λf
.
(3)
where λf is the thermal conductivity of the statically stable foam flow, W/(m·K), computed from the eqn (4) λ f = βλ g + (1 − β )λl . (4) The gas Reynolds number of foam flow was computed by formula Gg d Re g = . Aν g
B6
C6
A5
B5
C5
A4
B4
C4
A3
B3
C3
A2
B2
C2
A1
B1
C1
d
A6
s2
s1
Foam flow Figure 2:
Tube bundle in vertical foam flow.
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(5)
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217
All experiments and measurements were repeated in order to avoid measurement errors and to increase reliability of investigation results. The experimental uncertainties [8] in the range of test data variation: ±8.0% for ±8.1% for Nu f , ±2.2% for Re g .
3
α,
Results
The experimental results showed that the heat transfer intensity of the in-line tube bundle for the foam flow is much higher than for the one-phase airflow under the same conditions (flow velocity). Data of heat transfer intensity as a function of Re g for the first tube (B1) of the middle line and for comparison in one-phase airflow is shown in figure 3. With increasing of the gas Reynolds number for the foam flow Re g within the limits 190÷440, heat transfer intensity ( Nu f ) of the first tube (B1) increases by 2.5 times for the foam with volumetric void fraction β=0.996 and by 2.3 times for β=0.997, and by 1.9 times for β=0.998. So, the heat transfer intensity of the first tube (B1) to the wettest foam flow (β=0.996) depends on Re g more than foam flow with volumetric void fraction β=0.997 and 0.998. There is (fig. 4) shown the comparison of heat transfer intensity of the first (B1) and the third tube (B3) in the middle line. The heat transfer of the third (B3) tube is worse than of the first-frontal (B1) tube for the whole interval of the gas Reynolds number.
0.996 0.997 0.998 air
800 600
Nu f Re g
400 200 0 150
Figure 3:
200
250
300
350
400
Heat transfer of the first tube (B1): β=0.996, 0.997 and 0.998, and in airflow.
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218 Advanced Computational Methods in Heat Transfer IX
0.996 (B1) 0.997 (B1) 0.998 (B1) 0.996 (B3) 0.997 (B3) 0.998 (B3)
900 700 500
Nu f Re g
300 100 150
Figure 4:
200
250
300
350
400
Comparison of heat transfer intensity of the first (B1) and the third tube (B3) in the middle line.
In one-phase flow case the heat transfer intensity of frontal tubes are equal to about 60% of the third tubes heat transfer intensity, heat transfer of the second tubes are equal to about 90% of the third tubes heat transfer intensity, and the heat transfer intensity of the fourth and furthered tubes in the in-line tube bundles are like of the third tubes [9]. Our experimental investigation shows that the heat transfer intensity of the third tube (B3) is equal to in average 90% of the first tube (B1) heat transfer intensity for the wettest and wetter foam flow (β=0.996 and 0.997) and to in average 92% for the driest foam flow (β=0.998). The comparison of heat transfer intensity for the middle line at the volumetric void fraction β=0.997 is shown in figure 5. The heat transfer of the first tube is better than that of the second tube, heat transfer of the second tube is better than that of the third tube and the heat transfer of the third tube is better than that of the fourth tube. The heat transfer intensity of fifth and sixth tubes is different from previously mentioned order. The heat transfer intensity of the fifth tube is better than that of the fourth tube and less than that of the third and the sixth tubes in whole interval of gas Reynolds number for foam flow. The heat transfer intensity of the sixth – the last tube is higher than that of the third tube when Re g <330 and less when Re g increases from 330 to 440. This phenomenon can be explained by the fact that structure of foam flow changes while it passes the tube bundle. The large bubbles of foam are divided into smaller bubbles, some of foam bubbles collapse. So, the real void fraction and the intensity of the liquid drainage process are not the same along the experimental channel.
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Advanced Computational Methods in Heat Transfer IX
B1 B2 B3 B4 B5 B6
600 500 400
219
Nu f Re g
300 200 150
Figure 5:
200
250
300
350
400
Heat transfer intensity of the tubes in the middle line, β= 0.997.
Experimental results of the heat transfer of the in-line tube bundle to upward statically stable foam flow were summarised by criterion equations using the dependence between the Nusselt number and gas Reynolds Re g number for the foam flow. This dependence within the interval 190 < Re g < 440 for the in-line tube bundle in upward foam flow with the volumetric void fraction β=0.996, 0.997, and 0.998 can be expressed as follows: n
m
Nu f = cβ Re g . (6) On average, for the entire middle line in the bank c=6.64, n=305, m=–95(β–1.006) and on average, for the whole in-line tube bank c=7.6, n=328, m=–95(β–1.006).
4
Conclusions
An experimental investigation of the heat transfer intensity was performed for inline tube bundle under the upward cross flow of the statically stable foam. The experimental results showed that the heat transfer intensity of the in-line bundle tubes for the foam flow is from 20 to 80 times (dependent on tube position in the bundle) higher than for the one-phase airflow under the same conditions. The experimental investigation showed that the heat transfer of the frontal tubes to upward foam flow is the best. It is different in comparison with one-phase fluid flow, case. Exceptional case is the heat transfer of the last and the fifth tubes in the middle line of the tube bundle. The peculiarities of foam as two–phase system take place in this occasion. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
220 Advanced Computational Methods in Heat Transfer IX The experimental results were generalised by criterion equation, which can be used for the calculation and design of the statically stable foam heat exchangers.
Nomenclature A – cross section area of experimental channel, m2; c, m, n – coefficients; d – outside diameter of tube, m; G – volumetric flow rate, m3/s; Nu – Nusselt number; q – heat flux density, W/m2; Re – Reynolds number; T – average temperature, K; α – average coefficient of heat transfer, W/(m2⋅K); β – volumetric void fraction; λ – thermal conductivity, W/(m⋅K); ν – kinematic viscosity, m2/s.
Indexes f – value referred to foam flow; g – value referred to gas.
References [1] Sadoc, J. F., Rivier, N., Foams and Emulsions, Nato ASI Series, 1997. [2] Tichomirov, V., Foams. Theory and Practice of Foam Generation and Destruction, Chimija: Moscow, 1983. [3] Gylys, J., Hydrodynamics and Heat Transfer Under the Cellular Foam Systems, Technologija: Kaunas, 1998. [4] Fournel B., Lemonnier H., Pouvreau J., Foam Drainage Characterization by Using Impedance Methods. Proc. 3rd Int. Symp.on Two–Phase Flow Modelling and Experimentation, p. [1–7], 2004. [5] Gylys J., Jakubcionis M., Sinkunas S., Zdankus T., An Experimental Study of Upward and Downward Foam Flow in Small Test Bundle. Proc. 12th International Heat Transfer Conference, Grenoble, France, pp. 399–404, 2002. [6] Gylys, J., Jakubcionis, M., Sinkunas, S., Zdankus, T., Description of tube bundle heat transfer in foam flow. Proc. of the 4th Baltic Heat Transfer Conference, eds. B. Sunden & J. Vilemas, LEI: Kaunas, pp. 541–548, 2003. [7] Gylys J., Sinkunas S. and Zdankus T., Experimental Study of Staggered Tube Bundle Heat Transfer in Foam Flow, 5th International Symposium on Multiphase Flow, Heat Mass Transfer and Energy Conversion, Xi’an, China, p.[1–6], 2005. [8] Schenck H., Theories of Engineering Experimentation, Mir, Moscow, 1972. [9] Zukauskas A., Convectional Heat Transfer in Heat Exchangers, Nauka: Moscow, p. 472, 1982.
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221
Use of graphics software in radiative heat transfer simulation K. Domke Institute of Industrial Electrical Engineering, Poznan University of Technology, Poland
Abstract A base equation of radiative heat transfer, along with typical calculation methods, has been presented. Goals and base equation of computer graphics programs have been described. A quick analysis of graphics programs has been made, and their functions have been compared with the functions of other programs for radiative heat transfer modeling. Conditions that should be fulfilled by graphics programs in order to be used for modeling radiative transfer have been given. Formulas transforming heat quantities to radiant quantities, as used for Dirichlet’s and Neumann’s boundary conditions, have been presented. Keywords: modeling of radiative heat transfer, graphics software.
1
Introduction
Radiative heat transfer is one of three basic ways of heat transfer. It is a common phenomenon, consisting in heat energy (power) transfer through electromagnetic (infrared) radiation, mainly with a wavelength of λ∈(0,78−1000)µm, between nontransparent (or translucent) surfaces of temperature above 0K. Even in temperature conditions of (0−50)oC range, for non-vacuum systems, the share of this method of heat transfer is ca (15−20)% of the total heat transfer, and increases significantly in higher temperatures. It is also the only method of heat transfer in vacuum systems. In many electrotechnical devices determining temperature fields related to normal work and operation of these devices is a basic condition for verifying the correctness of their construction and it determines their admissible loads. Hence the significance of methods allowing one to calculate temperature distributions and power fluxes related to heat transfer, including radiative heat transfer. Omitting radiative heat transfer in WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060221
222 Advanced Computational Methods in Heat Transfer IX general heat calculations causes at least (15-20)% of errors while determining values of heat quantities.
2
Radiative heat transfer, principles and base equation, calculation methods
Each body of temperature above 0K emits electromagnetic radiation into hemisphere. In case of a black body (indices “bb”), it is described by Planck’s law [6, 8]:
mbb,λ
c = c1 λ λT2 − 1 e
−1
−5
(1)
which shows the dependence of monochromatic radiance mbb,λ (or, in other words, monochromatic power surface density) of a black body from its temperature T. Radiance of a real body is described by the following dependence (2) mλ = ε λ mbb,λ . It is monochromatic radiance density of primary radiation (measured in [W/(m2 sr µm)]), emitted from the surface of temperature T and emissivity ε. Radiation should usually be considered while taking into account the direction into which it propagates. In such a case a notion of monochromatic radiance density is introduced, defined as monochromatic radiance density l measured in a solid angle dω emitted from a perpendicular surface Sp dmλ (3) lλ = dω or for total quantities which do not take into consideration the dependence on wavelength λ dM (4) L= dω which may also be presented by this formula [1, 2, 3, 6] d 2Φ d 2Φ (5) L= = dω ⋅ dS p dω ⋅ dS ⋅ cosη where L is radiance, measured in [W/(m2 sr)]. Total quantities are derived by averaging monochromatic volumes after dλ within range λ∈(0−∞). The eqn. (5) is illustrated in fig. 1. Taking an initial assumption on the non-transparent nature of boundary surface of radiative heat transfer system, the following base equation may be written down for each point of this surface [6, 8]
G G G G G G G G G G L(r0 , sref , λ) = ε (r0 , sref , λ)Lbb(r0 , λ) + ∫ ρkk (r0 , sref , sin, λ)Lp (r0 , sin, λ) cosη dω
(6)
Ω
The equation results directly from the law of conservation of energy. In eqn. (6) ρkk is a bidirectional reflected distribution function (BRDF) describing reflexive properties of a surface, “in” refers to incident and “ref”
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reflected radiation. Angle dω indicates an elementary solid angle, and integration is carried out on the whole hemisphere Ω. dω dSp=dS cosη η dΦ
dS
Figure 1:
Definition of radiance according to eqn. (5).
Eqn. (6) means that for any point on the boundary surface Si, whose position G G is determined by vector r0 , radiance L in the direction determined by vector sref consists of the sum of primary radiation radiance (first component of eqn. (6)) G and the sum of reflections in direction sref of radiation incident from direction G sin (second component – integral of eqn. (6)). Fig. 2 depicts it visually. total radiation: G G L(r0 , sref , λ ) reflected radiation: G G G G G ρ ( r , s ∫ kk 0 ref , sin , λ ) L p (r0 , sin , λ ) cosη dω
Ω
primary radiation: G G G G G L pr (r0 , sref , λ ) = ε (r0 , sref , λ ) Lcc (r0 , λ )
Figure 2:
G G Radiance L in point r0 and direction sref (eqn. (6)).
The complete system of radiative heat transfer consists of numerous surfaces Si, which most often create a closed system. For each point of every boundary surface of such a system, an integral eqn. (6) should be written. The supplement of the system of eqns (6) should be the provision of boundary conditions determining the energy-heat status of the system’s boundary surfaces. These conditions are defined either as Dirichlet’s or Neumann’s conditions. In the former case, a function of temperature distribution on the examined surface Si should be specified [3, 6, 8] G (7) t (r0 ) = t ( x0 , y0 , z0 ) . r0 ∈S i
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224 Advanced Computational Methods in Heat Transfer IX In the latter case, surface power density p proportional to the derivative of the sought function t, penetrating to the system from outside through boundary surface, is determined [3, 6, 8] G ∂t . (8) p (r ) = p ( x , y , z ) = − λ 0 r0∈Si
0
0
0
∂r
Si
In that case, as can be seen from (8), only the derivative of function t is set, which means that the temperature is determined with accuracy to the constant. Thus, in order to unconditionally determine temperature distribution in the system, it is necessary to specify Dirichlet’s condition for at least for one boundary surface. A system of Fredholm’s n integral eqns (6) thus obtained, with boundary conditions, is practically algebraically unsolvable. There are practically three methods of searching for a solution based on determining temperature and heat fluxes penetrating boundary surfaces. Either a series of reductions allowing to transform a system of integral eqns (6) into a system of linear equations is introduced, or a bounded system of eqns (6) is solved by using methods of numerical quadrature of integrals, or radiative heat transfer is modeled on the basis of eqns (6) thus obtaining the solution by means of simulation. The first method, after making assumptions on radiation diffusivity and after dividing boundary surfaces into areas of finite size, leads to a system of linear equations, usually presented as follows [6, 8]: N N 1− ε j δ ij ) p zw , j = ∑ (δ ij − ϕ ij )σ T j4 for i=1,2,....,N, (9) ∑ ( − ϕ ij
εj εj j =1 where N is the quantity of examined areas, ϕij – configuration factor of area ith into jth, and σ is Stefan-Boltzman constant. Eqns (9) along with boundary conditions (7) and (8) form a uniform, determined system of linear equations, which may be solved either by classical (for small N), or iterative methods (for large N). The second method consists in making an assumption that eqn. (6) refers not only to a point but also to a certain area around it. Thanks to the use of numerical quadrature to integral equations (Newton-Cotes, Gauss and others methods) it is possible to transform them into a system of algebraic equations. These can be then solved by means of standard numerical methods. The third method consists in modeling radiative heat transfer. Knowing the distribution of the surfaces of the system of radiative heat transfer, as well as emissive and reflexive characteristics of the surface, it is possible to model the propagation of radiation by tracing each ray starting with its emission, through reflections on boundary surfaces, and ending with the ultimate absorption. This is illustrated in fig. 3. This method can be used to trace heat powers between the system’s surfaces, and later on, determine the sought temperature distributions. The classical form of the above is realized with Monte-Carlo method, which theoretically allows one to represent each heat transfer system. This kind of universality (any possible system can be modeled) constitutes an unquestionable advantage of such solutions. Unfortunately, due to the necessity to trace each j =1
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emitted ray, and, subsequently, each reflected ray, their number increases dramatically (see fig. 3) and even state-of-the-art computers are not powerful enough to fully realize this idea. Pa reflection point: reflected rays and absorbed power
Pa point of emission of primary radiation
Figure 3:
Radiation traced according to classical Monte-Carlo method.
Within this framework, research focuses on developing methods allowing one to limit the required number or rays to be examined, while retaining the required accuracy of simulation. One of such methods is backward ray tracing. The method consists in assuming rays’ reverse courses, in relation to their natural courses, and examining only the ones which contribute significantly to the energy balance of selected, significant points (areas) of boundary surfaces.
3
Computer graphics: goals and base equation
The main goal of computer graphics software is to create a maximally realistic image of virtual reality (a so-called scene) on a monitor’s screen or in the form of graphics files. This image should give the observer an impression of seeing real objects. As distinguished from many image recording technologies (photography, film, TV), computer graphics technologies do not require the recorded objects to exist in reality. It is only virtual reality that is needed, namely mathematical and physical description of objects’ surfaces, and the knowledge of laws determining the generation and distribution of light (visible radiation). In computer graphics programs, the image of virtual reality is obtained by modeling and simulating, in a determined virtual space (created by boundary surfaces and light sources), of propagation of light and visual effects experienced by a virtual viewer. For the purposes of computer graphics, Kajiya [5] introduced an equation describing the propagation of light from point x in the direction of point y with illumination incident on point x from direction z. The equation was later called the visualization governing equation: (10) L λ ( x, y ) = V ( x, y )[ L pr ,λ ( x, y ) + ∫ Lin,λ ( z, x ) ρ kk , λ ( x, y, z ) dz ] Ω
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226 Advanced Computational Methods in Heat Transfer IX where Lλ(x,y) is the luminance of visible radiation (light) emitted from point x in the direction of point y, V(x,y) − geometrical factor allowing to avoid the examination of surfaces obstructing each other in calculations, ρkk(x,y,z) − see eqn. (6). Indices “pr” and “in” constitute primary emission and incident radiation. Fig. 4 illustrates eqn. (10). It represents visually when and how an observer in point z “sees” point x illuminated from direction z.
z
y
primary light: L pr,λ(x,y)
.
illumination Lin,λ(z,x)
.x
b
reflected light: L (z,x)ρ kk − λ(x,y,z)dz ∫ in,λ
Ω
reflexive properties: ρ kk − λ ( x , y , z )
Figure 4:
Illustration for visualization base eqn. (10).
Eqn. (10), although formulated in computer graphics terminology, contains the same physical sense as eqn. (6) expressing the law of conservation of energy for heat radiation. This can also be seen when comparing fig. 2 and 4.
4
Computer graphics programs
Although the goals of computer graphics programs (visualization) and of programs modeling radiative heat transfer (determining heat transfer) are different, one can see a significant concurrence of certain stages of such programs. This applies mainly to the phase of generating and tracing (history) of individual rays. This is shown in fig. 5. Not all computer graphics programs can be adapted for heat calculations. First programs developed in late 1960s did not take into account physical phenomena related to light propagation at all, and their only purpose was to generate simple images. These programs are totally unsuitable for heat calculations. Later programs (1970s) took into account only local illumination. The distribution of luminous flux diffused on examined surfaces and reaching the observer was specified in these programs as a function of a luminous flux incident from light sources and diffusing characteristics of observed surfaces. Surface characteristics took into consideration both diffusive and specular reflections. Eqn. (10) was the mathematical base, although interreflections were not taken into account. Instead of indirect radiation (i.e. interreflection), an artificially determined constant illumination level was introduced, whose sole purpose was to imitate effects connected with indirect illumination. In spite of these solutions, the resulting images based on models which only took into WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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account direct radiation were “artificial” and considered unnatural. Due to the omission of the phenomenon of interreflection, programs based on such assumptions could not be used for modeling radiative heat transfer.
Data on scene geometry and surface emissivity Simulation of radiation emission Data on surface reflexivity Simulation of interreflection The same procedures for graphics and radiative heat transfer programs
Absorption analysis
Calculating absorbed power
Illumination analysis
End of illuminating task: determined luminances Observer’s data
End of heat transfer task: power and temperature transfers calculated
Visualisation
End of visualisation task: scene image known
Figure 5:
Functional diagram of the process of modeling radiative heat transfer and visualization.
The 1980s saw the coming of illumination programs that took into consideration both direct illumination and indirect illumination (interreflections). These are called global illumination models. Indirect lighting is caused by multiple reflections from neighboring surfaces. To take this into account, it is necessary to model the total luminous radiation distribution from the source, through interreflections and to the final absorption on a specified surface or when the observer’s eye has been reached. These programs fully reflect phenomena related to generation and diffusion of light and may be used for radiative heat transfer modeling. Such programs include Radiance, Spectr System, Lightscape Visualization System, Helios and others [1, 3].
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228 Advanced Computational Methods in Heat Transfer IX
5
Usability conditions
Not every computer graphics program can be adapted for radiative heat transfer modeling. Conditions making the above possible can be divided into two groups: necessary conditions and desired conditions. The first group determines the very usability of a program for adaptation for radiative heat transfer simulation, and the second considerably facilitates this task and increases the potential accuracy of future calculations. Necessary conditions will definitely include the following: − basing the modeling of illumination on eqn. (10), − interreflections modeling, − possibility to export intermediate results as files including data (but not bitmaps!) on radiance (or equivalent quantity) on determined surfaces. Desired conditions include: − possibility to define reflective characteristics of surfaces as a BRDF function, − relation of a graphics program to CAD programs, − possibility to freely shape the emissive characteristics of radiation sources, − a graphics program is based on modeling radiant quantities, not light quantities.
6 Transformation of heat quantities into radiant quantities Heat quantities (temperature, power density) are used for the description of radiative heat transfer. Computer graphics programs are based on processing light quantities (luminance) or radiant quantities (radiance). Adapting the latter to model and simulate radiative heat transfer requires heat quantities to be transformed into radiant quantities. This applies particularly to boundary conditions describing the system of radiative heat transfer. The paper of [3, 4] demonstrated that boundary surface S for which Dirichlet’s condition (7) was determined, i.e. temperature T=t+273, can be replaced by energetically equivalent radiation source surface with radiance L and emissivity ε described by the following formula:
L( x0 , y0 , z0 ) =
σ ε T 4 ( x0 , y0 , z0 ) π cosη
(11)
where σ is Stefan-Boltzman constant, and angleη − see fig. 1. Similarly, for surfaces for which Neumann’s condition (8) of equivalent surface of emissivity ε was determined, radiance L can be assigned [3, 4]:
L ( x0 , y 0 , z 0 ) =
p ( x0 , y0 , z 0 ) + E sum ( x0 , y0 , z 0 )ε
π
(12)
where Esum is total irradiation of examined surface originating from the remaining surfaces of the system.
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Therefore, using eqns (11) and (12) the task of radiative heat transfer can be transformed into a task within the scope of light technology. Fig. 6 presents it visually. Radiative transfer task; determined temperatures and powers
Transformation into radiant quantities
Computer graphics program: ray tracing
Power absorption procedures
End of radiative transfer task
Figure 6:
7
Stages of modeling radiative heat transfer matters with the use of computer graphics programs.
Test calculations on the basis of Radiance
As far as computer graphics is concerned, there are many program packages for scene visualization, and Radiance is one of such programs [2, 7, 9]. Utilizing the system’s procedures for ray tracing and adding own, necessary procedures, a tool (program) was developed which allows to model radiative heat transfer systems. A series of calculations was done, with particular attention paid to testing simple systems for which precise analytical solutions are known. Detailed results have been provided in a series of publications [3, 4]. In all cases, a significant concurrence of simulation results (simulations, nevertheless, included a certain degree of randomness) with precise calculation results was achieved.
8
Conclusions
Several computer graphics programs fulfill conditions allowing them to be used not only for the simulation of illumination, but also for solving the most complex WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
230 Advanced Computational Methods in Heat Transfer IX matters of radiative heat transfer. These programs contain procedures for modeling emission and procedures for light interreflections. Fulfillment of conditions discussed in the paper allows such programs to be used for performing the basic function of a modeling program – examining the propagation of rays in a determined system. Adding additional functions consisting in transforming heat quantities into light quantities (and vice versa) as well as functions for the calculation of power transmitted with each ray, produces a fully functional tool for modeling radiative heat transfer. Performed control calculations confirm a significant consistency of results of radiative heat transfer simulation done on the basis of Radiance package with analytical calculations.
References [1] Ashdown I., Radiosity – A Programmer’s Perspective, John Wiley & Sons Inc, New York, 1994. [2] Companion R., RADIANCE: A Simulation Tool for Daylight Systems, University of Cambridge, Department of Architecture, 1997. [3] Domke K., Modelowanie symulacja i badanie radiacyjnej wymiany ciepła w środowisku Radiance, ser. Rozprawy nr 378, Wyd. Pol. Pozn., Poznań, 2004, (in Polish). [4] Domke K. & Hauser J., Application of RADIANCE procedures for radiative heat transfer modeling: in :Computer aid design of electroheat devices ed. Hering M., Sajdak Cz. & Wciślik M., Wyd. Pol Śląskiej, Gliwice, pp 32-49, 2002. [5] Kajiya J., The Rendering Equations, Computer Graphics, 20(4), 1986. [6] Modest M. F., Radiative Heat Transfer, II ed., Academic Press Amsterdam, Boston, London, N. York and Sydney, 2003. [7] Sillon F. X. & Puech C., Radiosity and Global Illumination, Morgan Kaufmann Publishers Inc., San Francisco and California, 1994. [8] Siegel R. & Howell J. R., Thermal radiation heat Transfer , Mc-Graw Hill Book Co., N. York, 1972. [9] Ward G.L. & Shakespeare R., Rendering with RADIANCE - The Art and Science of Lighting Visualization, Morgan Kaufmann Publ., San Francisco, 1998.
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On heat transfer variation in film flow related with surface cross curvature S. Sinkunas1, J. Gylys1 & A. Kiela2 1
Department of Thermal and Nuclear Energy, Kaunas University of Technology, Lithuania 2 Department of Technology, Kaunas College, Lithuania
Abstract This paper is concerned with heat transfer calculation in a liquid film falling down a vertical surface. Heat transfer in laminar film flow is influenced by the cross curvature of the wetted surface, heat flux on the film surface and boundary conditions on the wetted wall. The analytical study evaluating the influence of cross curvature on heat transfer in laminar film has been carried out. Equations using correction factors for the calculation of heat transfer in laminar liquid film with respect to the cross curvature for different boundary conditions were established. An experimental investigation of heat transfer in the entrance region of the turbulent film has also been performed. The description of the experimental set-up is presented in the paper. The research has been carried out on a water film flowing down a surface of vertical tube with the Reynolds number ranging from 9.2·103 to 10.5·103. The results of experiments are discussed with respect to the local heat transfer dependence on the Reynolds number and initial velocity of the film. Heat transfer stabilization length was determined experimentally. Keywords: heat transfer, laminar film, cross curvature, correction factor, turbulent film, entrance region.
1
Introduction
In many technological processes that deal with heat and mass transfer, gravity driven liquid films are widely used. With liquid falling films one can obtain comparatively high heat transfer coefficient. This approach is employed for cooling and heating processes in chemical, food, pharmaceutical and other WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060231
232 Advanced Computational Methods in Heat Transfer IX industries. Prediction of heat transfer in liquid film flow is important from various engineering aspects. Even small improvements in heat transfer during technological processes may lead to significant energy and money savings. The growing awareness of environmental problems recently draw the efforts of many researches, including the field of engineering, to evaluate systems on the basis of heat transfer phenomena. A lot of the heat exchange equipments are designed of vertical tubes with falling films on their external surfaces. In the case when liquid film flows down a vertical tube the curvature of its surface and the film itself effects heat transfer characteristics on the surface and correspondingly thickness of the liquid film [1]. Simultaneously the intensity of the heat exchange between a wetted surface and liquid film is influenced. The heat transfer mechanism of a liquid film flowing down profiled horizontal tubes was studied in several works [2–3]. A method of its enhancement based on the breakdown of the thermal boundary layer by longitudinal fins and grooves was developed. The results of heat transfer research in evaporating falling films on a structured and smooth surface respectively are presented in [4–5]. The obvious change of the wavy film structure by using a profiled heating surface showed the improvement of heat transfer coefficient. Hydrodynamic and thermal liquid film flow study on vertical plate with grids was performed in [6]. The conditions for an optimum heat transfer in evaporating water film driven by gravity down a vertical heated plate were established. It was evaluated the influence of wetted surface geometry and operating parameters on the heat transfer enhancement.
2
Analytical method
2.1 Heat transfer in laminar film flow on a vertical surface Heat transfer across the liquid film takes place due to the temperature difference between the wetted surface and the film. In the case of laminar film flow the conduction may be viewed as a main mode of the heat transfer only. Thus, the equation known as Fourier’s law can be used to compute the amount of energy being transferred per unit time q = −λgradT .
(1)
By substituting the expression of the temperature gradient for eqn. (1) and integrating, we obtain Tw − T =
qw
λ
y
∫ q dy . 0
q
w
The heat flux from a liquid film can be determined as follows WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(2)
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233
δ
∫ cTρwbdy .
Q = cT f ρw bδ =
(3)
0
The mean temperature can be defined as δ
∫ wTdy
Tf =
0
.
wδ
(4)
By defining the temperature difference between a wetted surface and the mean temperature of the film as qw
λ Tw − T f =
w
δ
y
∫ ∫ 0
0
q dy dy qw
wδ
,
(5)
we obtain the following expression for calculation of local heat transfer coefficient
α=
w δλ δ
w
y
∫ ∫ 0
0
q dy dy qw
.
(6)
In order to define the regularity q q w = f ( y ) , we use the energy equation cρw
∂T dq + =0. ∂x dy
(7)
By integrating the eqn. (7) within the limits from 0 to y, we obtain the ratio of heat flux densities in the film q cρ = 1− qw qw
y
∫
w
∂T dy . ∂x
(8)
0
Assume that ∂T ∂x = dT f dx [7]. Then the ratio dT f dx can be determined by the equation of heat balance in the film WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
234 Advanced Computational Methods in Heat Transfer IX dT f
=
dx
qw . cρw δ
(9)
Substituting of eqn. (9) by eqn. (8) leads to the following relationship y
∫
wdy q = 1− 0 . qw wδ
(10)
By substituting eqn. (10) for eqn. (6) and by employing the Nusselt number Nu d , we obtain w δd
Nu = δ
w
y
∫ ∫ 0
0
y wdy 0 1 − w δ dy dy
.
(11)
∫
Velocity in the laminar film flow one can calculate by the following equation w=
gδy y 1 − 0.5 , ν δ
(12)
gδ 2 . 3ν
(13)
and the mean velocity respectively w=
Taking into account eqns. (12) and (13), we obtain the following expression from eqn. (11) Nu d =
8wsδ 2 δ 3 w 0
y
∫ ∫ 0
y 3 wdy 0 1 − 2w dy dy s
∫
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.
(14)
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235
The above-mentioned equation could be solved numerically and the following result is obtained Nu d = 8.2353 .
(15)
In case of heat transfer calculations, it is more reasonable to use modified Nusselt number Nu M . Therefore, the following equation can be used Nu M = 2.27 Re −1 3 .
(16)
As it is shown in [8], the application of eqn. (16) can be expanded using the corresponding multipliers. The curvature of the film flowing down a vertical tube surface and the heat transfer between film surface and surrounding medium of a gas or vapour, can be evaluated by correction factor C Rq . Multiplier ε Pr may estimate the variability of liquid physical properties. Then, the modified Nusselt number can be calculated by the following equation Nu M = 2.27 Re −1 3C Rqε Pr .
(17)
The correction factor C Rq can be determined by equations: when qw = const C Rq =
(
)
(18)
(
)
(19)
136 + 0.52 − 0.03ε q ε R , 136 + 39ε q
and when Tw = const C Rq =
56 + 0.58 − 0.05ε q ε R . 56 + 10ε q
The variations of correction factor C Rq upon the range of relative cross curvature and external heat exchange between the film surface and surrounding medium according to eqns. (18) and (19) are shown in figure 1. The variation of liquid physical properties across the film must be taken into account in a case of high value of temperature difference. The multiplier ε Pr in eqn. (17) can be determined as an exponential function of the following ratios Pr f Pr w or µ f µ w . The results of numerical calculations showed that ratio Pr f Pr w unambiguously does not evaluates the influence of physical properties
variation for different liquids and the character of this influence practically is not dependent upon the values of ε q and ε R . This research showed that for −0.5 ≤ ε R ≤ −1 and 0.4 ≤ Pr f Prw ≤ 2 the multiplier WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(Pr
f
Prw
)
0.25
can be
236 Advanced Computational Methods in Heat Transfer IX used. The multiplier ε Pr must be determined as a function of parameters µ f µ w for more accurate evaluation of liquid physical properties variation. This parameter evaluates practically the influence of physical properties variation for water, transformer oil, fuel oil and compressor oil. The multiplier ε Pr in such case can be calculated by equation
ε Pr = (µ f µ w )n , where n = 0.315(2 + ε R )
−0.49
(20)
, for 0.1 ≤ µ f µ w ≤ 1 and n = 0.325(2 + ε R )
−0.24
,
for 1 ≤ µ f µ w ≤ 10 . The similar data in [8] for ε R = 0 have showed that boundary conditions on the wetted surface have no influence to the character of heat transfer dependence on variability of the liquid physical properties.
CRq
CRq
(a)
5
5
1.4
1.4
4 1.2
4 3
3
1.2
2 1.0
(b)
2
1
1.0
1
0.8 0.0
Figure 1:
3
0.2
0.4
0.6
0.8
εq
0.8 0.0
0.2
0.4
0.6
0.8
εq
Variations of correction factor on film cross curvature and external heat transfer: (a) boundary condition qw = const and (b) boundary condition Tw = const 1–5 – εR = 0; 0.25; 0.50; 0.75 and 1.0 correspondingly.
Experimental method
3.1 Experimental set-up For heat transfer research in the turbulent film falling down a surface of vertical tube the experimental set-up (figure 2) was applied. The arrangement was composed of a closed circulating loop that included liquid reservoir with feed and exhaust electric pumps and liquid tank. Working liquid with the aid of feed pump was supplied to the liquid tank provided with a slot distributive mechanism. The liquid from the distributor fell down a vertical tube (calorimeter). The stainless steel tube 30 mm in outside diameter with the length of 1000 mm was used in the experiment as a calorimeter. The fixing bolts WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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at the end of tested tube allowed the possibility to regulate and to guarantee verticality of the tube. After flowing down the test tube, water was gathered back to the reservoir. The gutter at calorimeter end ensured a smooth falling of the water into the reservoir. The surplus water was discharged to the sewerage by exhaust pump while the fresh water was supplied from water-supply directly. The location of thermocouple in the liquid distributor ensured the measurement of film temperature at the inlet. The thermocouple installed at the end of calorimeter had determined a film temperature at the exit correspondingly. The electric circuit consisted of calorimeter, voltage regulator, and shunt with milivoltmeter, rectifier and voltmeter respectively. The electric current supplied for the calorimeter provided a steady heat flux on the experimental section. 220 V 14
6
3
2
15
1 12
11 8
13 7
4
from water supply
9 10
Figure 2:
5
Schematic diagram of experimental set-up: 1 – calorimeter; 2 – liquid tank; 3 – slot distributive mechanism; 4 – feed-pump; 5 – exhaust-pump; 6 – inlet thermocouple; 7 – outlet thermocouple; 8 – centering bolts; 9 – gutter; 10 – liquid reservoir; 11 – voltmeter; 12 – shunt; 13 – millivoltmeter; 14 – voltage regulator; 15 – rectifier.
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238 Advanced Computational Methods in Heat Transfer IX 3.2 Local heat transfer for a turbulent film flow on a vertical surface The experiments were provided for Reynolds number ranged from 9.2·103 to 10.5·103. The temperature of the tube (calorimeter) surface and the film, electric current, voltage were measured and recorded during the experiment. After registration of electric current I and voltage U, the heat flux density on the calorimeter surface was calculated dividing of heating power I×U by surface area A. When records of heated tube surface and film flow temperatures were performed, the difference of temperature ∆T (between the mean temperatures of film T f and tube surface Tw) was calculated. Local heat transfer coefficient was computed by formula
α = q w ∆T .
(21)
12
α ·10-3,W/(m2 ·K)
10 8 6 1 2 3
4 2 0
0.2
0.6
0.4
0.8
1
x, m Figure 3:
Variation of local heat transfer coefficient in the entrance region of the film flow down a vertical surface: 1 – Re = 9240, ε = 0.89; 2 – Re = 9300, ε = 1.15; 3 – Re = 10540, ε = 1.2.
Experimental data are presented in figure 3. Three different regions may be distinguished along the length of film flow in a case when initial average velocity of the film in the liquid distributor is less or exceeds an average velocity of stabilized flow. The significant decrease of local heat transfer coefficient while reaching minimal value at some distance from liquid distributor is seen at the first region. This phenomenon one can explain by the development of thermal boundary layer and its laminar nature. In the second region, fluid fluctuations begin to develop while heat transfer increases to a maximum value. The beginning of heat transfer stabilization takes place in the third region of the film flow. Augmentation of a thermal boundary layer terminates with the film thickness. The variation of local heat transfer in the entrance region, when initial average velocity of the film exceeds an average velocity of stabilized film is not so high. As we can see from figure 3, in all cases the heat transfer stabilization is WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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not larger than 0.5 m from the liquid distributor when Reynolds number ranged from 9.2·103 to 10.5·103.
4
Conclusions
The cross curvature and external heat exchange of the film has a significant influence on heat transfer rate from the wetted surface in laminar gravitational liquid film flowing down the outside surface of vertical tube. In this case equation for heat transfer intensity calculation in laminar liquid film falling down a vertical plane surface must be supplemented with multiplier CRq evaluating curvature of film and with multiplier εPr evaluating the variability of liquid physical properties respectively. The turbulent flow can be useful in the sense of providing increased heat transfer intensity. However, the turbulent film flow is very complicated and difficult for the theoretical study. In this case, the only possible way is the experiment. The experimental data revealed that Reynolds number and initial velocity has a significant influence on local heat transfer. It is estimated that heat transfer stabilization takes place at 0.5 m distance from the liquid distributor when Re > 9·103.
5
Nomenclature
a – thermal diffusivity, m/s2; b – elementary width of the film; c – specific heat, J/(kg·K); C Rq – heat transfer correction factor; d – hydraulic diameter of the film, m; g – acceleration of gravity, m/s2; Nu – Nusselt number, αδ/λ; Nu d – Nusselt number, αd/λ; Nu M – modified Nusselt number, (α/λ)(ν2/g); Pr – Prandtl number, ν/a; q – heat flux density, W/m2; Re – Reynolds number of liquid film, 4 Γ/(ρν); T - temperature, K; w – film velocity, m/s; x – longitudinal coordinate; y – distance from the wetted surface, m; α – heat transfer coefficient, W/(m2·K); Γ – wetting density, kg/(m·s); δ – liquid film thickness, m; ε – relative film velocity, wd wstab ; ε R – relative cross curvature of the film, δ/R; ε Pr – multiplier for liquid physical properties; ε q – ratio of heat flux densities, (qs/qw); µ – dynamic viscosity, (Pa·s); λ – thermal conductivity, W/(m·K); ν – kinematic viscosity, m2/s; ρ – liquid density, kg/m3; Subscripts: d – distributor; f – film flow; s – film surface; stab – stabilized flow; w – wetted surface.
References [1] Sinkunas, S., Kiela, A., Adomavicius, A. & Gudzinskas, J., Hydromechanical parameters of laminar liquid film falling down a vertical surface. Mechanics, 4(36), pp. 24–29, 2003. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
240 Advanced Computational Methods in Heat Transfer IX [2] Rifert, V., Sidorenko, V., Usenko, V. & Zolotukhin, I., Methodic of measurements and experimental results of local heat transfer at liquid film flow on the horizontal finned tubes. Proc. of the 5th World Conf. On Experimental Heat Transfer, Fluids Mechanics and Thermodynamics, Edizioni ETS: Pisa, Vol. 3, pp.1997–2000, 2001. [3] Rifert, V., Putilin, J. & Podbereznyi, V., Evaporation heat transfer in liquid films flowing down horizontal smooth and longitudinally profiled tubes. J. of Enhanced Heat Transfer, 8, pp. 91–97, 2001. [4] Lozano Aviles, M., Maun, A.H., Iversen, V., Auracher, H. & Wozny, G., High frequency needle probes for time-and-space characteristics measurements of falling films on smooth and enhanced surfaces. Proc. of 6th Word Conf. On Experimental Heat Transfer, Fluids Mechanics and Thermodynamics, Matsushima, Miyagi, Vol. 1, pp. 1–5, 2005. [5] Takase, K., Yoshida, H. & Ose, Y., Predicted thermal-hydraulic of liquid film flow on ribbed surface. Proc. of 1st Int. Forum On Heat Transfer, Kyoto, Vol.1, pp. 207–208, 2004. [6] Mitrovic, J. & Raach, H., Hydrodynamic and thermal film flow study on vertical plates with grids. Report On The Progress of Easy MED Workpage 2 For The Tip, Institute for Energy and Process Engineering University of Paderborn, pp. 1–4, 2004. [7] Kays, W.M., Convective Heat and Mass Transfer, Energy: Moscow, 1972. [8] Gimbutis, G., Gimbutyte, I. & Sinkunas, S., Heat transfer in a falling liquid film with large curvature. Heat Transfer Research, Scripta Technica, 2, pp. 216–219, 1993.
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Experimental investigation of enhanced heat transfer of self-exciting mode oscillating-flow heat pipe with non-uniform profile under laser heating F. Shang1,3, H. Xian1, D. Liu1,2, X. Du1 & Y. Yang1 1
North China Electric Power University, Beijing People’s Republic of China 2 Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, People’s Republic of China 3 Changchun Institute of Technology, Changchun People’s Republic of China
Abstract The aim of this paper is to detail the heat transfer characteristics and transport power of the Self-Exciting Mode Oscillating Flow (SEMOS) Heat Pipe. The enhanced heat transfer performance of a non-uniform cross-section closed loop SEMOS HP is tested and compared with the uniform profile SEMOS HP under same heating conditions. As a result, the heat transfer rate of the SEMOS Heat Pipe with a non-uniform profile could be increased under certain conditions, which forms the basis for future research and development of new SEMOS Heat Pipes. Keywords: laser heating, non-uniform profile SEMOS Heat Pipe, heat transfer enhancement, experimental investigation.
1
Introduction
A new era in heat pipes began with the Pulsating Heat Pipe, invented by H. Akachi in 1994, and later called the Self-Exciting Mode Oscillating-Flow Heat Pipe [1, 2] (SEMOS Heat Pipe). Although it has only been known about for ten years, the SEMOS Heat Pipe is attracting worldwide attention because of its WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060241
242 Advanced Computational Methods in Heat Transfer IX immeasurable application potential. A lot of research about the SEMOS Heat Pipe has been done in Japan, U.S.A, Germany, Russia, Ukraine and China etc. However, research is being done is to understand the working principles of the SEMOS Heat Pipe, as it is still in its initial stage. This new highly effective heat transfer device is attracting attention from all over the world in the field of heat transfer due to phase transition. Therefore the main challenge is to eliminate the contradiction between downsizing the cooling device and increasing the generated heat and to improve the heat-dissipating performance to a higher lever for all of the researchers.
Figure 1:
2
Experimental parts of a non-uniform profile of a SEMOS Heat Pipe.
Heat transfer characteristics of the SEMOS Heat Pipe
According to the working principles of the SEMOS Heat Pipe, there are two basic methods for improving the heat transfer performance: one is enhancing the heat transfer between the interior surface of the pipe and the working gas-liquid medium in the pipe either by increasing the phase-change frequency and intensity between liquid evaporation and gas clot or by improving the heat convection process between the working liquid and working surface because of the fact that the thermal conduction of the SEMOS Heat Pipe is between the pipe interface and two non-steady phases of working liquid in the pipe. The other method is increasing the pulsating frequency and reliable circulating power, which could be done by increasing the difference in temperature between the hot and cold fluid or by increasing the pulsing frequency inside the pipe. By enhancing the heat transfer rate of the SEMOS Heat Pipe, this experiment will
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validate the application of field cooperation theory on the heat transfer field with phase-change.
3
Experiment setup
The objects used in this experiment are SEMOS Heat Pipes with closed loops. One is a SEMOS Heat Pipe with a uniform profile of 3mm in inner diameter, while the other heat pipe based on the uniform one with elliptic non-uniform profile consists of a vertically intervened heating section and insulating section of the pipe as shown in Figure 1. The pipe is made of brass, the working fluid in the pipe is distilled high-purity water with a filling rate φ = 42%. The inclination angle of the heat pipe is θ = 55°, and the pressure in the heat pipe is P = 1.8×10-3Pa.
8 5 4 6 1
3 7 2
1—Unit of refrigeration cycle, 2—Power supply, 3—Laser supply, 4—Experimental table of SEMOS heat pipe, 5—Water tank, 6—Flow meter, 7—Data acquisition system, 8—Personal computer. Figure 2:
Experimental system of SEMOS Heat Pipe heat transfer enhancement.
Figure 2 shows the experimental setup for this thermal performance measurement. It consists of the main test apparatus: a laser supply and cooling system and the power supply system and a data acquisition system combined with a personal computer to show the data collected. As shown in Figure 2, the laser heater consists of 8-channel Quantum Well Laser Diode Arrays, while the maximum output power of every single channel is 50W. The heating electrical current range is 5~40A and the wave-length of the laser is 940nm. The heat input comes continuously from the eight laser heating channels, which could work individually or together as a heat source. A total of 20 K-type thermocouples of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
244 Advanced Computational Methods in Heat Transfer IX 1mm diameter were also added, which were mounted on the surface. Accordingly, the data acquisition frequency is 1Hz and the data acquisition precision is at 0.01°C.
4
Results and discussion
4.1 Comparing and contrasting Heat Pipes with uniform and non-uniform profile Through the experiments mentioned above between the heat pipes with uniform profile and non-uniform profile, Figures 3, 4 and 5 show the change in efficiency in transport power under different heating powers (heat electrical current I). The efficiency transport power is given by the following equation: [W] Po = Gc （ p T2 − T1）
(1)
where G denotes the cooling water (kg/s); c p is the specific heat of water under constant pressure; J/(kg·K) and T2 and T1 denote the input and output temperatures of cooling water. 50
uniform profile non-uniform profile
49 48 47 46
P0/W
45 44 43 42 41 40 39 38 37 0
100
200
300
400
500
Time/s
Figure 3:
Comparison of transport power when heat electrical current is 14A.
As shown in Figures 3, 4 and 5, the transport power of the heat pipe with the non-uniform profile is lower than that of the heat pipe with the uniform profile when the heating electrical current is relatively low, and the rate of the heat pipe with the non-uniform profile would exceed that of the heat pipe with the uniform profile as the heating electrical current of the laser supply increases. It is demonstrated from the trend of the transfer rate Po ~I graph that the greater the heat input, the greater the difference in transfer rate between the heat pipe with the non-uniform profile and the heat pipe with the uniform profile becomes. The transferred power of the heat pipe with the non-uniform profile is 13.6% higher WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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than that of the heat pipe with the uniform profile at the maximum heat input (which, in this experiment is about 25.5W and the heating electrical current of every channel is 23A. 76
uniform profile non-uniform profile
74 72
P0/W
70 68 66 64 62 0
100
200
300
400
500
Time/s
Figure 4:
Comparison of transport power when heat electrical current is 18A. uniform profile non-uniform profile
102 100 98
P0/W
96 94 92 90 88 86 0
100
200
300
400
500
Time/s
Figure 5:
Comparison of transport power when heat electrical current is 23A.
4.2 Comparison of effective thermal conductivity of uniform and nonuniform profiles and a pure conductor Figure 6 shows the relationship between the heat input and the effective thermal conductivity under three different conditions, SEMOS Heat Pipes with non-uniform profiles, with uniform profiles and the heat pipe without any working fluid which makes it a pure conductor. Under different outputs, and taking the difference in temperature between the hot and cold ends of the heat pipe, the effective thermal conductivity is shown below: WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
246 Advanced Computational Methods in Heat Transfer IX λd =
(T
h
Po l
)
− Tc F
[W/(m·K)]
(2)
where Po denotes transport power; W; l is the distance between the hot and cold end; m; Th and Tc are the mean of the temperatures at the hot end and the cold end °C; F is the total heat transfer area (m2), which is the total circulate cross section area for heat transfer and the total cross section area for the pure conductor. 11000
uniform profile non-uniform profile pure conductor
10000 9000 8000
l d/W(m.K)
-1
7000 6000 5000 4000 3000 2000 1000 0 8
10
12
14
16
18
20
22
24
I/A
Figure 6:
Comparison of effective thermal conductivity in varying cases.
As shown in Figure 6, the effective thermal conductivity increases with heat input in all three cases. The greatest increase in effective thermal conductivity is with the non-uniform heat pipe and the smallest increase is in the pure conductor heat pipe. When the heat input is low, the effective thermal conductivity of the uniform profile heat pipe is higher than that of the non-uniform one. As the heat input increases, the effective thermal conductivity of the non-uniform profile increases more than that of the uniform one and the difference between them becomes greater. It is suggested that under certain conditions, the heat transfer rate could be improved by changing the form of the profile of the SEMOS Heat Pipe. Figure 6 also shows that the function of heat transference of heat pipes especially SEMOS Heat Pipes is better than the performance of the heat pipe with the pure conductor. Within this heat input range, the effective thermal conductivity of the heat pipe is 4~15 times that of the pure conductor, and this difference increases with heat input. The effective thermal conductivity of the non-uniform profile heat pipe is 19% higher than that of the one with the uniform profile and 14 times higher than that of the pure conductor at the maximum heat input (23A for every channel) in this experiment.
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4.3 Heat transfer performance of the Heat Pipe with the non-uniform profile Theoretically, there are two reasons why a non-uniform heat pipe consisting of a heating section and an insulating section could improve the heat transfer rate. One is that the portrait eddy is formed on the particular section of the non-uniform profile, so that the vertical component of the velocity on the pipe surface increases, which means the temperature grads field and velocity vector field co-operate, which could improve the heat transfer rate. Meanwhile, the resistance on the working fluid increases because of the non-uniformity of the profile, so a certain amount of heat input is needed for the SEMOS Heat Pipe with the non-uniform profile to improve the heat transfer performance. The advantage of the SEMOS Heat Pipe with the non-uniform profile at improving heat transfer is easily demonstrated when the portrait eddy becomes stronger and circulating power declines, the SEMOS Heat Pipe with the non-uniform profile is more suitable for high density working fluid. On the other hand, as shown in Figure 7, it is possible to conclude by monitoring the temperatures at the hot and cold end surface, that the oscillating amplitude of the non-uniform heat pipe of the surface temperature at the hot end is lower than that of the uniform one, while the oscillating frequency is higher. In the SEMOS Heat Pipe with the non-uniform profile, the circulating power increases because of the alternation of liquid evaporation and gas clot. The accelerated motion inside the pipe due to this unsteady expansion and contraction process along with the vertical velocity on the interface could enhance the heat transfer performance. 110
non-uniform profile uniform profile
105
o
T/ C
100
95
90
85 0
100
200
300
400
500
Time/s
Figure 7:
Comparison wave with the hot end surface temperature.
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248 Advanced Computational Methods in Heat Transfer IX
5
Conclusion
a.
The SEMOS Heat Pipe with the non-uniform profile can improve the heat transfer performance when the input heat is high enough to overcome the resistance brought on by the non-uniform profile. Both the transmitted heat power of SEMOS Heat Pipes with non-uniform profile and uniform profile is much higher than that of the pure conductor under the same experimental conditions. The SEMOS Heat Pipe with the non-uniform profile is more suitable for high density heat current because of its particular advantage at high input levels. The temperature oscillating frequency of the SEMOS heat pipe with the non-uniform profile becomes higher than that of the SEMOS Heat Pipe with the uniform profile as the heat input increases.
b.
c.
References [1] H. Akachi, Looped capillary tube Heat Pipe. Proceedings of 71st General Meeting Conference of JSME, pp. 940-950, 1994. [2] H. Akachi, F. Polasek, P. Stulc, Pulsating Heat Pipe. Proceedings of the 5th International Heat Pipe Symposium: Australia, pp.17-20, 1996. [3] Qu Wei, Ma Tongze, Experimental investigation on flow and heat transfer of Pulsating Heat Pipe. Journal of Engineering Thermo physics, 23(5), pp. 596-598, 2002. [4] Cui Xiaoyu, Weng Jianhua, M. Groll, Experimental investigation of heat performance for the copper/water Pulsating Heat Pipe. Journal of Engineering Thermo physics, 24(5), pp. 864-866, 2003. [5] Meng Jian, Chen Zejing, Li Zhixin, and Guo Zengyuan, Field coordination analysis and convection heat transfer enhancement in Duct. Journal of Engineering Thermo physics, 24(4), pp. 652-654, 2003. [6] Guo Zengyuan, Physical mechanism and control of heat convection: Speed and temperature Field Synergy. Chinese Science Bulletin, 45(19), pp. 2118-2122, 2000. [7] Yasushi KATO, Takao NAGASAKI, and Yutaka ITO, Study on looped Heat Pipe with non-uniform cross section. Proceedings of the 40th Heat Transfer Conference: Japan, pp. 313-314, 2003.
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Investigation of heat transfer in the cup-cast method by experiment, and analytical method F. Pahlevani, J. Yaokawa & K. Anzai Department of Metallurgy, School of Engineering, Tohoku University, Sendai, Japan
Abstract Semi-solid metal alloys have a special microstructure of globular grains suspended in a liquid metal matrix that cause a significant number of benefits. A new semi-solid casting method (cup-cast method) that has been recently developed has eliminated all of the difficulties associated with other semi-solid casting processes and makes semi-solid casting as easy as pouring water from a pitcher into a drinking glass. This method is based on the nucleation and growth of solid-particles. Heat transfer phenomenon is one of the most important factors in this method that governs the shape, size, and fraction of solid particles. In this study, the heat transfer phenomena in the cup cast method has been studied experimentally and theoretically. First, the temperature distribution was measured at a semi-solid casting of Al-A356 with the cup-cast method at different points. Finally, a model for heat transfer phenomena was proposed. The final model was in good agreement with the experimental data. Keywords: semi-solid, cup-cast, heat transfer, analytical equation, modelling.
1
Introduction
The semi-solid processes of Al-A356 alloy produced globular instead of dendritic alpha phase and extraordinarily fine fibrous aluminum-silicon eutectic structure, which apparently gives exceptionally high ductility [1-4]. Semi-solid processing offers several significant process and product cost advantages, for example: 1- reduced cost of porosity-related scrap, rework and impregnation of die cast type products; 2- reduced material content of squeeze, low pressure and permanent mold type products by casting to near-to-net-shape and by casting thinner sections of high integrity material; 3- long tool life, the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060251
250 Advanced Computational Methods in Heat Transfer IX result of substantially less heat of fusion released to the die and a reduced ∆t between tool and casting, both delaying the onset of heat checking and other thermal stress induced tool failures [1-4]. Conventional semi-solid casting processes face many potential problems but the cup-cast method that has been recently developed is a novel process that makes semi-solid casting as easy as pouring water from a pitcher into drinking glass, and avoids all of the problems and difficulties associated with other semi-solid casting processes. In this method, on pouring, nucleation may take place both on the wall and throughout the zone that is super cooled by contact with the cold mould. It is likely that this zone would have a thickness of the order of millimeter for a pouring temperature close to the melting point and thinner for a higher temperature. All nuclei that survive at this time are able to grow; they will “drift” away from the interface by convection motion of the liquid and then moved to the other part of the cup that contains low temperature and composition gradient melt. Modelling and simulation are an important part of process development, providing insight into the mechanisms controlling process performance. On the other hand, one of the specialties of this method (cup-cast method) is its isothermal holding stage that is caused by the heat transfer phenomenon. The aim of this paper is to describe the modelling of the heat transfer phenomenon of the cup cast method and comparing this with the experimental result that shows good agreement between analytical equation of heat transfer and experiment after saturation (isothermal holding).
2
Temperature measurement
The Al-A356 alloy with chemical composition as shown in table 1 was melted at the electrical resistance furnace in the graphite crucible at 720°C and then under certain conditions poured into the cup (shown schematically in Fig. 1). The temperature was measured at the center of the cup, inside and outside of the cup’s wall by 0.1mm thickness positive Chromel wire and negative Alumel wire thermocouple. The temperature was recorded during pouring of the melt to the cup and rest time, 20 per second to exact temperature profile could be achieved. Table 1: Si 7.12
Mg 0.45
Chemical composition of used Al-A356 alloy. Fe 0.11
Cu 0.007
Zn 0.007
Mn 0.006
Sn 0.006
Ni 0.003
At the center of the cup after the pouring start, because it is located at the melt’s stream, the temperature climbed to its optimum temperature rapidly. At the inside of the cup’s wall the temperature increase was delayed, but reached the melt’s flow during turbulence to the point where it increasede up to the same temperature as the center of the cup (there is just a small difference between the temperature of the melt at the center of the cup and inside of the cup’s wall so it shows uniform temperature distribution in the cup). By transferring the heat from WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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the melt to cup, the temperature of the melt will decrease gradually but uniformly at the semi-solid state (between liquidus “620 °C” and Solidus “575 °C” line) and the temperature of the cup will rise up to reach to its saturation point that will be followed by a constant temperature at the cup’s wall.
Furnace
A Figure 1:
Schematic figure of experimental cup-casting method.
From these curves (Fig. 2(a)) it could be said that after pouring, the heat of the melt transfered only to the cup. Then, by heat transfer the temperature of the melt and temperature of the cup approached a saturation temperature and after some time the cup and melt will keep this saturation temperature and after rest time, decrease gradually. From Fig 2(b) that shows temperature deference at the different height of the cup it could be assumed there is one dimensional heat transfer through melt to the cup. 700
700 Ceneter of the cup
600
600 Inside of cup's wall Outside of the cup's wall
Temperature (C)
Temperature (C)
500
400
300
500
400
300
200
200
100
100
0 0
10
20
30
40
50
a
0
60
70
80
0
10
20
30
Time (sec)
Figure 2:
3
40
50
60
b
70
80
90
100
Time (sec)
(a) Temperature profile at different point of the cup during CupCast method. (b) Temperature deference in the melt at different height of cup during Cup-Cast method.
Analytical model for heat transfer
Based on temperature measurement and some assumption (such as constant material properties, one dimensional heat transfer and no-superheating) an analytical model was proposed for heat transfer in cup cast method. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
252 Advanced Computational Methods in Heat Transfer IX Fig 3 shows the way of heat transfer through the melt to the air schematically, as it was mentioned temperature was measured at three point (center of the cup, inside and outside of the cup’s wall) so analytical model also was done for these three points.
θ hi
θ0 θa
θi
ha
D
2
X Figure 3:
Schematic of heat transfer in cup-cast method.
3.1 Center of the cup, and inside of the cup’s wall From heat conservative low and heat balance following equation could be written
ha (θ 0 − θ a ) =
λ (θ i − θ 0 ) X
∂θ hi (θ − θ i )πD = − ∂t If
θ0
λ (θ i − θ 0 )
(1-a)
X
= hi (θ − θ i )
ρH f ρc + θL −θS
(1-b)
D π 2
2
(2)
was calculated from eqn (1-a) and introduce to (1-b) eqn (3) could be
obtained
θi =
hiθ + hi +
θ −θi =
λ
ha (λ X ) θa ha + (λ X ) −
(λ
(3-a)
X)
2
X ha + (λ X ) ha X
λ
(1 + ha X λ )(1 + hi X λ ) − 1
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(θ − θ a )
(3-b)
Advanced Computational Methods in Heat Transfer IX
By substitution the definition of
θ −θi
253
from eqn (3) into eqn (2) and
introducing β (eqn 4) temperature profile at the center of the cup as function of time will be determine as equation 5
hi 4
β=
{(
D
(ha x λ )
)(
) }
ρH f ρc + 1 + ha x 1 + hi x − 1 λ λ − θ θ L s 1 βdt = − dθ ⇒ θ = θ a + (θ L − θ a )e − βt θ −θa
(4)
(5)
Thus, eqn (3-a) together with eqn (5) yields the temperature distribution at inside of cup’s wall
ha X Xhi 1 + λ λ (θ L − θ a )e− βt θi = θ a + Xh Xh 1 + i 1 + a − 1 λ λ
(6)
3.2 Outside of cup’s wall For the temperature profile of the outside of cup’s wall, because at the early stage here is not a uniform temperature distribution in the cup, the temperature profile for the first stage (before uniform temperature distribution) is in the form of eqn (7) (that is for semi-infinite solid was located in a uniform temperature media but suddenly on surface of it will heat up by a high temperature fluid), and after that condition it will given by eqn (8)
θ + = erfc
where θ
+
=
x+ − exp x + + t + erfc + t + + 2 t+ 2 t x+
(
)
(7)
θ0 −θa + x at + , x = , and t = θL −θa λ h (λ h )2
θ0 = θa +
Xhi
λ
1 + Xhi 1 + Xha − 1 λ λ
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(θ L − θ a )e − βt
(8)
254 Advanced Computational Methods in Heat Transfer IX As it was mentioned, this equation is for semi-infinite solid and because the cup is not as thick as could be considered the same as semi-infinite solid, this equation could not be a suitable analytical solution for this problem, so for finding a more realistic solution a slab 0 ≤ x ≤ L would be considered by mathematical formulation of the problem the same as:
∂ 2θ 1 ∂θ in 0 < x < L, t > 0 = ∂x 2 α ∂t θ ( x, t ) = θ a at x = 0, t > 0 ∂θ = 0 at x = L, t > 0 ∂x θ ( x, t ) = θ L for t = 0 in 0 ≤ x ≤ L
(9-a) (9-b) (9-c) (9-d)
And solve this problem by using the integral method. For the case δ (t ) < L (the thermal layer thickness is less than the slab thickness) the eqn (9-a) was integrated over the thermal layer thickness and obtain
−α
∂θ 0 ∂x
= x =0
d (γ − θ aδ ) dt
(10-a)
δ
where
(10-b)
γ ≡ ∫ θ 0 ( x, t )dx x =0
A cubic profile for the temperature as given by eqn (11) was chosen and by applying the conditions given by eqn (12) the coefficients were determined and utilize eqn (10-b) to determine the thermal layer thickness.
{θ (x, t ) = a + bx + cx
2
0
θ
x =0
= θa , θ
x =δ
= θL ,
∂θ ∂x
+ dx 3
=0, x =δ
}
∂T ∂x 2
(11)
2
=0
(12)
x =0
The resulting temperature profile becomes
θ 0 ( x, t ) − θ a x x = 1− 3 + 1 2δ 2δ θL −θa where
δ = 8αt
This solution is valid for 0 ≤ x ≤ L as long as (8) shows the temperature profile.
δ ≤L
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3
(13-a) (13-b) and after that eqn
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4
255
Comparing the experiment and analytical temperature profile
Fig 4 and 5 shows the difference between the temperature profile from the analytical equation and the experiment. From Fig. 4 it could be summarized that there is good agreement between the experiment and analytical equation for temperature distribution at the center of cup and inside of cup’s wall. Center of cup 800
Temperature (C)
700
Equation 5
600
ha=100W/m2k , hi =1000W/m2k
500
Analytical Equ.
400
Experiment
300 200 100 0 0
20
40
60
80
100
120
140
160
Time (sec)
(a) Inside of cup's wall 700
Equation 6
Temperature (C)
600 500
ha=100W/m2k , hi =1000W/m2k
400
Analytical Equ. Experiment
300 200 100 0 0
20
40
60
80
100
120
140
160
Time (sec)
(b) Figure 4:
Comparing of analytical temperature profile and experiment.
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256 Advanced Computational Methods in Heat Transfer IX Outside of cup's wall 600
Temperature (C)
500
Equation 8
400
Equation 7
Analyical Equ.
300
Experiment
ha=100W/m2k , hi =1000W/m2k
200
100
0 0
20
40
60
80
100
120
140
160
180
Time (sec)
(a) Outside of Cup's wall 600
Equation 13 Temperature (C)
500
Equation 8
400
Experiment
ha=100W/m2k , hi =1000W/m2k
300
Analytical Equ.
200
100
0 0
50
100
150
200
250
300
350
Time (sec)
(b) Figure 5:
Comparing of analytical temperature profile and experiment.
Fig. 5(a) shows a combination of equation (8) and (9) could not be a good solution for this problem, but equation (13) for early stage and equation (9) after saturation time (uniform temperature distribution) are in good agreement with experimental result (Fig 5(b)).
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257
Conclusion
Cup-cast method is the most convenient method for semi-solid casting that has just been developed by the author. This method is based on the nucleation and growth of solid particles, and sufficient rest time (holding the isothermally in semi-solid state), heat transfer phenomena will considerably influence on the semi-solid microstructure of this method. Based on the facts of this method those driven out from the temperature measurement a model for heat transfer in cup cast method was proposed. This analytical equation is in good agreement with experiments at the center and inside of the cup’s wall and is in acceptable agreement with experiment at the outside of the cup’s wall.
6 Index A Table 2:
List of symbols and their definition in analytical model.
θ
Temperature of the melt
D
Diameter of cup
X
Thickness of the cup
h
Heat transfer coefficient Thermal conductivity of cup Density of cup Specific heat coefficient of cup Thermal diffusivity of cup Slab thickness
λ ρ c a L
θa θ0 θi δ
Room temperature Temperature of the cup outside Temperature of cup inside Thermal layer thickness
subscript i a
Between melt and cup Between cup and air
References [1] J. L. Jorstad, Q. Y. Pan, D. A. Pelian, “Solidification microstructure affecting ductility in semi-solid-cast products”, Materials Science and Engineering A, 413-414 (2005) pp. 189-191. [2] D. Brabazon, D. J. Browne, A. J. Carr, “Experimental investigation of the transient and steady state rheological behavior of Al-Si alloys in the mushy state”, Materials Science and Engineering A 356 (2003) pp. 69-80.
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
258 Advanced Computational Methods in Heat Transfer IX [3] B. P. Gautham, P. C. Kapur, “Rheological model for short duration response of semi-solid metals”, Materials Science and Engineering A 393 (2005) pp. 223-228. [4] J. L. Wang, Y. H. Su, C-Y. A. Tsao, “Structural evolution of conventional cast dendritic and spray-cast non-dendritic structure during isothermal holding in the semi-solid state”, Scripta Materialia (USA), No. 12, 37 (1997), pp. 2003-2007.
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Atwood number effects in buoyancy driven flows M. J. Andrews & F. F. Jebrail Los Alamos National Laboratory, USA
Abstract Consideration is given to Atwood number (non-dimensional density difference) effects in buoyancy driven flows. Buoyancy driven (natural convection) flows may be treated as Boussinesq for small Atwood number, but as Atwood number increases (>0.1, i.e. large temperature differences) the Boussinesq approximation is no longer valid and the distinct “bubble” and “spike” geometry of Rayleigh– Taylor buoyant plumes is formed. Aside from asymmetry in the flow the Atwood number also affects key turbulent mix parameters such as the molecular mix, and heat transfer coefficients. This paper will present recent experimental work being performed in the buoyancy driven mix laboratory at Texas A&M University with air/helium as mixing components. Corresponding numerical simulations performed at Los Alamos are presented for the experiments, and future directions for the research discussed. Keywords: buoyancy, Boussinesq, Atwood number, natural convection.
1
Introduction
This paper describes experiments and corresponding simulations to investigate non-Boussinesq effects at high Atwood (At) number (At ≡ (ρ1 − ρ 2 ) (ρ1 + ρ 2 ) a non-dimensional density ratio) in buoyancy driven turbulence (Rayleigh-Taylor mixing). The Boussinesq assumption for buoyancy driven flows states that density difference effects need only be accounted for in the gravitational terms, and density may be taken as a constant elsewhere in the equations. At small Atwood number (At<<1) the Boussinesq approximation is valid, however, at large At (~1) it implies a symmetry to the flow that is contrary to the familiar bubbles and spikes of water falling out of a glass. Here we explore the limits of the Boussinesq approximation. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060261
260 Advanced Computational Methods in Heat Transfer IX An incompressible experiment is described that employs air/helium for At=0.035 (small At), and for At=0.26 (high enough to see some asymmetry). Simulations have also been performed on the computer using the MILES (Monotone Integrated LES) method, Youngs [1], and compared with corresponding experiments. Additional simulations have been performed to explore At effects up to At=0.9. At small At the buoyancy driven mix develops symmetrically indicating the Boussinesq approximation is valid. However, at high At number (>0.1) asymmetries in the turbulent mixing become apparent and indicate that the Boussinesq approximation is not appropriate for At>0.1. Consequences for natural convection and high temperature gas heating/cooling problems are discussed.
2
Experiments
2.1 Experimental facility A schematic of the experiment is shown in Figure 1. The experimental facility was a wind tunnel, with a splitter plate that separates an upper air stream ( ρ1 ) from a lower air/helium stream ( ρ 2 ). The stream velocities were kept the same to avoid shear, and great care was taken to obtain a constant and controllable helium flow rate. Thus, by pre-mixing air and helium for the lower stream the density difference between the upper and lower streams could be varied from At=0 (air top and bottom), up to At=0.75 (air on top, and helium on the bottom). More details of the experimental set-up can be found in Banerjee and Andrews [2]. For future reference, the vertical depth of the channel is 1.2m, and out-of-plane width is 0.6m. By introducing smoke into the upper channel and using a calibration wedge the density profile can be measured from digital photographs, see Banerjee and Andrews [2]. Splitter Plate Test Section Vents
Exit Plenum
Meshes
Air In
1.2m 2.0 m
1.0 m He- Air Mix In
Wooden ribs
y Air blowers
x
He from Flow metering unit
Adjustable dampers Side View
Figure 1:
Schematic of the experimental facility.
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2.2 Experimental details Two experiments are reported in this paper corresponding to At=0.035 and At=0.26. Table 1 reports the experimental conditions (note the stream velocity, U, is higher for At=0.26 to keep the spread angle small, and hence a parabolic flow). Use of Taylors’ hypothesis relates distance downstream (x) and time (t) as t=x/U, and this time is also reported in Table 1. Table 1: At 0.035 0.035 0.035 0.26 0.26
3
Experimental parameters.
U (m/s) 0.6 0.6 0.6 1.2 1.2
x (m) 0.75 1.4 1.75 0.5 1.5
t (s) 1.25 2.33 2.92 0.42 1.25
Governing equations and numerical details
3.1 Governing equations The incompressible Euler equations are used in conjunction with the MILES (see below for more details) modeling technique: Volume conservation: Scalar transport: Momentum:
∇•u = 0
Df =0 Dt
D (ρ u ) = −∇p + ρ g Dt
(1) (2) (3)
with the fluid velocity u = (u, v, w ) , density, ρ , pressure, p , and gravity, g = (0,0, g z ) , and scalar f . There are six independent variables and five equations, the seventh equation is a linear equation of state for density such as ρ = L( f ) . In the present work we take f to be the non-dimensional density, or mixture fraction, defined as f = (ρ − ρ 2 ) (ρ1 − ρ 2 ) . 3.2 Numerical solution procedure 3.2.1 Overview For the present experiment we have used MILES, namely, Monotone Integrated LES. MILES modeling involves solving Euler governing equations and using numerical diffusion to model turbulent diffusion. Success with this modeling technique for buoyancy driven flows has been reported by Youngs [1]. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
262 Advanced Computational Methods in Heat Transfer IX The governing equations presented above are a coupled set of partial differential equations for which there exist several solution procedures. The present work solves the governing equations using the RTI3D code described by Andrews [3]. In particular, a fractional time step technique is used in which for each time step an advection calculation is followed by a Lagrangian source term update. The Lagrangian update is presented next, and this is followed by a brief description of the advection step for the scalar f (details may be found in Andrews [3]). 3.2.2 Lagrangian momentum source term updates The Lagrangian w momentum equation is: ∆t w*n = wnn +1 2 + pPn − pTn + g z (4) ρt ∆y The n+1/2 superscript refers to a value from the advection calculation, and * to an intermediate value that does not necessarily satisfy continuity. The subscripts refer to spatial position (north face), typical of the SIMPLE method (Patankar [4]), and a staggered arrangement of momentum and mass cells is used. Following the SIMPLE practice, velocity corrections are defined so that uin,e+1 = ui*,e + ∆ui ,e (and similarly for the other velocities) and a new pressure
(
)
p Pn +1 = p Pn + ∆p p where ∆p is a pressure correction. By substituting these
expressions for n+1 into the volume conservation equation and then subtracting equation (4) evaluated with the * we arrive at the usual Poisson equation for pressure corrections: a P ∆pP + a E ∆pE + aW ∆pW + a N ∆p N + aS ∆pS = − Div
(5)
with Div the divergence of the * velocity values. The Poisson equation (5) is solved using a Full Multi-Grid method, and the pressure corrections are used in a SIMPLE style to provide updated n+1 velocities and pressures that simultaneously satisfy the momentum equations (3) and volume conservation. 3.2.3 Transport procedures The 3D transport procedures are split into x/y/z-steps, this fractional splitting simplifies the calculation to one-dimensional updates that lends itself to high order calculation of cell fluxes with the Van Leer [5] method. There follows a brief description of the scalar x-step advection, the y and z steps being similar, and similar advection steps are performed for the momentum. The x-step advection for the scalar is given by: f P* = f Pn + ∆y∆z∆t (ue f e − u w f w )
(6)
where P refers to the center of a control volume, e the east face, and w the west face. The face values for the u velocities are available, and the face values for the scalar are computed using a second order approximation with Van-Leer limiting to prevent non-physical oscillation as:
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f e = f upwind + sign(ε e )
263
(1 − ε e ) ∆xD
(7) e 2 where ε e = ∆t ue ∆x , and upwind values are taken according to the sign of ε e . The derivative is evaluated following Van Leer as: 2 ∆w 2 ∆e (8) , De = S min D , ∆x ∆x 1 if ∆ e and ∆ w > 0 where ∆ w = f Pn − fWn , ∆ e = f En − f Pn and S = − 1 if ∆ e and ∆ w < 0 . 0 otherwise Van Leer limiters have been used in equation (8) to limit the gradient of the volume fraction profile, thereby preventing spurious oscillations. The representation for the gradient of the cell profile D determines the accuracy of the representation. In the present work D = (∆ e + ∆ w ) (2∆x ) , so the gradient is computed with a central difference so this scheme is referred to as “2nd order”. 3.3 Computational details for simulation of the experiments The experiments are performed in a statistically steady gas channel, but are modelled on the computer using transient simulations that are related to the experiment through the Taylor hypothesis described above. The computational domain is taken to be 1.2m high (z), and 0.6m square (x & y), with a computational grid of 64x64x128 (x-y-z). The computational time step is selected by the computer program to keep the Courant number below 0.25. Initial conditions for the simulations are prescribed to fit the density interface off the splitter plate. Here we use the following initial density interface perturbations: h( x, y ) = hw + ∑ a k cos(k x x )cos k y y + bk cos(k x x )sin k y y + k x ,k y (9) ck sin(k x x )cos k y y + d k sin(k x x )sin k y y
( )
( )
( )
( )
where the spectral amplitudes are chosen randomly but give an rms amplitude of 0.6m/100, and the wave numbers range from modes 8 to 16 (Dimonte et al. [6]). The wake off the splitter plate is modelled as an initial perturbation, hw , where hw = aw sin( k w x ) cos(k w y ) ; the wavelength associated with the wave number k w is 1cm and is taken from inspection of the experimental photograph in Figure 2, and the amplitude is 0.5cm, again taken from Figure 2.
4
Results and discussion
Figure 2 is a photograph taken from the At=0.26 experiment. On the right at the mid-plane is the splitter plate. The air/helium streams move from right to left, and the buoyancy driven mix is seen to develop downstream. On the far left of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
264 Advanced Computational Methods in Heat Transfer IX the photograph (2m from the splitter plate) the mixing has expanded to the top and bottom of the channel (i.e. a total depth of 120cm). The photograph shows a well mixed region with a wide range of mixing length scales. Close inspection of the region close to the end of the splitter plate, reveals initial perturbations of wavelength about 1cm, and amplitude 0.5cm, associated with wake shedding.
Figure 2:
Photograph from At=0.26 experiment.
Figure 3 below shows plots of initial density interface (on the left) used for the At=0.035 simulations (and is the same for all the simulations), and it is evident that there are both short and longer wavelength disturbances corresponding to conditions at the end of the splitter plate. The right side of Figure 3 shows the mix edges interfaces at f=0.01 (lower surface) and f=0.99 (upper surface), and corresponds to a distance downstream of 175cm (near the left side of the photograph in Figure 2). Comparison of the computed and experimental disturbances in Figure 2 and the right of Figure 3 shows more fine scale in the experiment, and this is because of computational grid resolution.
Figure 3:
Initial conditions and late time (t=2.92s) mixing edges for At=0.035.
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Calibration of digital photographs provided time averaged non-dimensional density profiles (the scalar fraction, f) and these are shown in Figures 4 and 5, with the corresponding results from the simulations using MILES. Comparison in Figure 4 for an At=0.035 of the experimental density widths and profiles with the corresponding simulations is good, and in particular the almost linear density profile across the mixing region is well captured by the MILES simulations. Also of note is that the experimental mix is symmetric around the centreline, indicating that the Boussinesq approximation is valid for an At=0.035. Similarly, in Figure 5, comparison of experiment with simulation for the At=0.26 show they also agree quite well. However, the experiments and the simulations show a slight asymmetry, associated with the higher At=0.26, that perhaps best seen by inspecting the penetration at the edges of the mix in Figure 5.
At=0.035 Experiment Figure 4:
At=0.035 Simulation
Comparison of mix profiles for At=0.035. h2
At=0.26 Experiment Figure 5:
h1
At=0.26 Computation
Comparison of mix profiles for At=0.26.
To explore higher At (i.e. large density differences) two additional simulations have been performed for At=0.5 and At=0.9. To facilitate comparison, the product At*gz was held constant at the value for the At=0.035 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
266 Advanced Computational Methods in Heat Transfer IX case, so At*gz=-0.34335 ms-2, and the value of gz then computed for each value of At, so for At=0.5 a value of gz=-0.6867 ms-2 was assigned, and for At=0.9 the value was gz=-0.3815 ms-2. This scaling comes from consideration of the gravitational term in the governing equations. To gauge asymmetry the bubble penetration, h1 , and spike penetration, h2 , were computed as the distance from the centreline to the values of f = 0.99 and f = 0.01 respectively (see Figure 5 for a graphical representation at t=1.25s), and their ratio is plotted in Figure 6. Inspection of Figure 6 reveals that there is practically no asymmetry for At=0.035, however, asymmetry is clearly seen for At=0.5, with the spike/bubble penetration being about 1.2 near the end of the calculation at t=2s. Indeed, there is a strong asymmetry for At=0.9 with h2/h1=1.7 by t=2s. Figure 7 plots the edges of the mix region, in a similar fashion to Figure 5, for each of the cases at t=2s. The At*gz scaling ensures a comparison at similar development of the mix. Comparison of the mix edges for At=0.035 with At=0.5 in Figure 7 shows little difference in bubble and spike penetration. However, for At=0.9 the asymmetry is clear, with bubbles clearly shown at the top, and finger shaped spikes at the bottom.
Figure 6:
Spike to bubble penetration (h2/h1) asymmetry for increasing At
Results from the simulations show that significant departures from the Boussinesq approximation occur by At=0.5, but are not particularly evident at At=0.26. This suggests that the Boussinesq approximation is reasonable for pure buoyancy flows at least for At<0.1, and perhaps as high as At<0.3. In terms of hot/cold gas flows, for an ideal gas at constant pressure the density ratio ρ1 ρ 2 corresponds to a temperature ratio of cold to hot, and for At=0.1 the density ratio is 1.22/1, or temperature ratio of 1000K/1220K; for At=0.3 the density ratio is 1.86/1, or 1000K/1860K. Thus, it would appear for many free convection problems the Boussinesq approximation would be valid. However, for WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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combustion problems and other problems associated with large temperature differences, the full set of equations should be solved when in the presence of a body force (e.g. gravity or centrifugal). We close with a few words of caution – for accurate comparison of experiment and simulation we need better characterization of the initial conditions, i.e., the initial spectrum being shed from the splitter plate. In addition, our use of a 64x64x128 grid should be considered a coarse resolution, and additional calculations should be performed for higher resolution to confirm the results.
Figure 7:
5
Mix edges as a function of At.
Conclusions
A buoyancy driven mix experiment has been described, with corresponding simulations on the computer. Results from experiments at Atwood (At) numbers of 0.035 and 0.026 have been compared with corresponding MILES simulations and found satisfactory. Additional simulations at high At of 0.5 and 0.9 reveal a strong asymmetry in the buoyancy driven mix that indicates the Boussinesq approximation is no longer valid. However, the results suggest that the Boussinesq approximation is valid at least for At<0.1, and perhaps as high as At<0.3. Additional work is required to match the initial conditions of the experiment, and to perform additional simulations with higher grid resolutions to provide further refinement on the At criteria proposed.
Acknowledgements This paper is dedicated to the memory of Sara Boyce Andrews (1926-2006).
References [1] Youngs, D.L., “Application of MILES to Rayleigh-Taylor and RichtmeyerMeshkov mixing”, 16th AIAA Computational Fluid Dynamics Conference, 23-26 June 2003 AIAA 2003-4102.
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268 Advanced Computational Methods in Heat Transfer IX [2] Banerjee, A., and Andrews, M.J., “A Gas Channel Facility to Investigate Statistically Steady Rayleigh-Taylor Mixing at High Atwood Numbers,” accepted to Physics of Fluids, December, 2006. [3] Andrews, M.J., “Accurate Computation of Convective Transport in Transient Two-Phase Flow,” International Journal for Numerical Methods in Fluids, Vol. 21, No. 3, pp. 205-222, 1995. [4] Patankar, S.V., “Numerical Heat Transfer and Fluid Flow”, Hemisphere, 1980. [5] Van Leer, B., “Towards the Ultimate Conservative Difference Scheme, IV. A new Approach to Numerical Convection”, J. Comp. Phys., Vol 23, pp 276-299, 1977. [6] Dimonte, G., Youngs, D.L., Dimits, A., Weber, S., Marinak, M., Calder, A.C., Fryxell, B., Biello, J., Dursi, L., MacNeice, P., Olson, K., Ricker, P., Rosner, R., Timmes, F., Tufo, H., Youns, Y.-N., Zingale, M., Wunsch, S., Garasi, C., Robinson, A., Ramaprabhu, P., and Andrews, M.J., “A Comparative Study of the Turbulent Rayleigh-Taylor (RT) Instability Using High-Resolution 3D Numerical Simulations: The Alpha Group Collaboration,” Physics of Fluids A, Vol. 16, No. 5, pp. 1668-1693, May 2004.
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Testing of vapour chamber used in electronics cooling A. Haddad1 R. Boukhanouf 1 & C. Buffone2 1 2
University of Nottingham, Nottingham, UK Thermacore Europe, Ashington, UK
Abstract This paper presents experimental measurement of performance of a vapour chamber (VC), also known as a flat plate heat pipe, for electronics cooling. The vapour chamber, with square sides of 40 x 40mm and thickness of 3mm, was sandwiched between a heater block and a cooling plate located on the evaporator and the condenser surface respectively. The performance of the vapour chamber was investigated by determining the thermal resistance over a heat input range of 10 to 100W with the condenser held at constant temperature. Test results for two vapour chambers with sintered and mesh type wicks were presented, and then compared to results obtained from tests on identical solid copper samples of 1 and 3mm base thickness. The experimental results show that the vapour chamber with sintered wick material performed markedly better than solid copper base at high heat fluxes, with vapour chamber orientation having minimal effect. On the other hand, the vapour chamber with a mesh wick showed no improvement compared to the 3mm solid copper base and its performance decreased at high heat fluxes, particularly when operated against gravity. The test-rig set up as well as the experimental results will be presented in detail. Keywords: vapour chambers, flat plate heat pipe, planar heat pipe, thermal management, electronics cooling, heat spread, high heat flux.
1
Introduction
Research and development on Heat pipes began in the early 1960s when it was mainly oriented towards space applications operating under micro-gravity conditions. Since then, heat pipe development has undergone steady progress WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060271
270 Advanced Computational Methods in Heat Transfer IX with application in thermal management of many engineering systems such as spacecraft, chemical engineering, and electronics cooling [1, 2]. In its simplest form, a heat pipe is a vacuum sealed vessel with a porous metal wick lining its interior wall and saturated with a working fluid such as water, ethanol, methanol, etc. In operation, heat is applied to one end of the vessel, the evaporator, to evaporate the working fluid from the wick, the vapour then travels to the cooler end of the vessel, the condenser, where it gives up its latent heat and condenses to liquid state. Under the capillary force of the wick, the liquid is returned back to the evaporator and the cycle continues. This cycle of a two-phase working fluid process allows high heat fluxes to be conducted through the heat pipe. Heat pipes exist in simple configurations such as straight pipes or in more complex embodiments such as vapour chambers. In electronics cooling, vapour chamber heat pipes are becoming established thermal elements capable of transferring high heat fluxes from a single to multiple heat generating sources. The inadequacy of existing heat removal techniques (air cooling using fan-heat sink) for use in modern high speed microprocessors means that thermal management has become a critical technology, and it directly influences cost, reliability, and performance [5]. Vapour chambers are commonly integrated into the base of heat sinks as depicted by Figure 1 [3]. This combination of a vapour chamber and a heat sink into a cooling assembly would maintain multiple heat generating components mounting plate isothermal whereas standard heat sinks or cold plates would develop uneven heat distributions, dissipate high heat fluxes, eliminate hot spots and decrease weight of heat sink (by reducing the thickness of heat sinks base) as this arrangement would no longer require the base material to spread the heat.
Figure 1:
Vapour chamber used to spread heat across base of a heat sink (courtesy of Thermacore International).
The aim of this work is to measure performance of a mini-flat plate heat pipe approximately 40x40mm and 3mm base thickness. The study will involve benchmarking a simple copper block, a vapour chamber with a sintered wick material and a screen mesh wick structure. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Experimental setup
The VC heat pipe experimental rig assembly consists of a vapour chamber heat pipe, cold plate with liquid chiller unit (providing water flow at 35oC), an electric heater, as shown in Figure 2. Spring loaded thermocouples
Corner 2
Heater block
Condenser Heater
Corner 1
Test unit Cold plate
Figure 2:
Test unit setup assembly.
Figure 3 shows a simple diagram of the vapour chamber heat pipes used as test samples in this experiment. The units were made from copper material in which the evaporator and condenser have a rectangular surface shape of 40x40mm, with a 3mm thick base. In this experimental arrangement, heat from the heater block is supplied from the top surface (i.e., the evaporator) and hence efficient capillary design of the wick materials is essential to overcome gravity forces acting on returning liquid. The sintered VC inner surface of the evaporator is covered with a sintered powder wick to provide capillary forces for the working fluid to be drawn into the heat supply region. Posts span the evaporator and condenser surface and serve as structural supports against externally applied forces. These maintain the small gap between the two surfaces. The mesh VC uses a coarse mesh at its core with a fine mesh wrapped around it. The fine mesh forms the wick material while the coarse mesh acts as a structural support to maintain the vapour space, as well as exert a pressure force to keep the fine wick pressed against the copper walls of the chamber. Heat Input Evaporator
Liquid return
Vapour flow
Condenser Support post
Figure 3:
Capillary wick
Heat Output
A simplified schematic diagram a) Sintered wick b) Mesh wick.
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272 Advanced Computational Methods in Heat Transfer IX Figure 4 shows the experimental set up and associated data acquisition instruments including thermocouples, a flow meter, and a data logger. For a comparative study, tests were conducted on solid copper samples, as well as both sintered and mesh wick vapour chambers to determine relative performance enhancement.
Figure 4:
Experimental test rig.
The experimental procedure involved clamping the test unit between a cold plate and a heater block. Power was delivered though a controlled power supply to two electric heater cartridges placed inside an Aluminium block of 10x10mm dimension. To reduce contact thermal resistance, a layer of 0.5mm thick thermal interface material of 6W/mK thermal conductivity was inserted between the base of the aluminium heater block and the VC evaporator surface, as well as between the condenser side and the cold plate. The performance of the VC was evaluated for various heat loads by using a variable electric power source to supply heat to the heater block. A wattmeter was also used to accurately measure the power supplied. Initial heat input to the VC was set at 10W and then increased at equal increments up to 100W. At each power level, the experiment was allowed to run for about 5 minutes so that heat transfer steady state in the VC was reached. The maximum heat supply to the test unit was limited to the VC surface temperature approaching the 100°C threshold, above which internal pressure becomes too high and the chamber could be damaged. A filling/venting system was used to accurately fill the required amount of working fluid. The test sample was also mounted on a rotating beam which allows testing to be carried out at tilt angles ranging from 0° to 180°. In this work, the VCs were tested at 3 orientations 0°, 90°, and 180°. Charging of the vapour chambers was achieved by first venting air from a dry unit using a vacuum pump. A fill syringe containing deionised water was attached to the vapour space of the VC through a valve, which when opened allows liquid into the evacuated chamber. The heater was then switched on so that the working fluid is brought to boiling to purge any non-condensable gases from the chamber. After that the fill tube was crimped and the camber sealed without losing its vacuum. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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273
Results and discussion
The experimental results of both sintered and mesh VCs at 0°, 90°, and 180° orientations, and solid copper blocks with 1mm and 3mm base thickness are presented in graphical format. The heat load applied to the VCs was ramped up to a maximum of 100W at all orientations. The graphical plots show the test units surface temperature over a period of time and at various levels of heat input, as well as, the temperature difference between the heater and condenser/corner temperatures. The graphs also depict the transient regime of heat transfer in the test unit at the designated monitoring points on the surface of the test unit sample. The recorded data subsequently used to determine the spreading resistance of all test unit samples (i.e., a sintered VC, mesh VC, and solid copper). 3.1 Solid copper Two solid copper equivalents to the vapour chambers were built and tested in order to provide benchmark results. The copper units tested had 1mm and 3mm base thickness. Figures 5(a) and (b) below show the temperature plots from those units. As can be seen, the thicker 3mm copper base produced lower temperature difference to the 1mm unit. This can be expected as the thicker material will improve heat spread across the base, therefore improving heat input into the adjacent cold plate. Results are linear across the full power range, demonstrating the materials’ constant conductivity. 1mm Copper Base
Temperature (°C)
100 80 60 40 20
10W
20W
30W
50W
40W
60W
70W
80W
0 0
500
1000
1500
Time (s) heater ° Condenser
Corner 1 dT Heater Condenser
Corner 2 dT Heater av-corners
Temperature (°C)
3mm Copper base 80 70 60 50 40 30 20 10 0
10W
0 heater Condenser
Figure 5:
20W
30W
500
40W
Time (s)
50W
60W
70W
1000
Corner 1 dT Heater Condenser
80W
1500 Corner 2 dT Heater av-corners
1mm and 3mm base copper transient state over several heat inputs.
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274 Advanced Computational Methods in Heat Transfer IX 3.2 Sintered VC The sintered wick vapour chamber was tested at various orientations with graphical results illustrated on figures 6(a), (b), and (c). The overall performance demonstrated a reduced temperature difference over copper, particularly at the higher power levels. The performance is not significantly affected by the orientation, as can be expected from the higher capillary forces in sintered wicks [1]. A drop in heater temperature can be seen in all three curves at the highest power level. Overall this translates into improved performance as the temperature difference between the heater and the condenser/corners is reduced as a result. It is likely that the fill volume may not have been optimal. An overfill situation would have resulted in flooding which will impair performance up to a certain higher power level when this excess liquid is evaporated. Further investigation will be needed to determine the exact nature of this occurrence. Temperature (°C)
Sintered VC - 0 Deg 80 70 60 50 40 30 20 10 0 0
200
80W
60W
40W
20W
400
600
100W
800
1000
1200
Time (s) heater Condenser
Corner 2 dT Heater av-corners
Corner 1 dT Heater Condenser
Sintered VC - 90 Deg
Temperature (°C)
100 80 60 40 20
20W
40W
60W
80W
100W
0 0
200
400
600
800
1000
1200
Time (s) heater Condenser
Corner 1 dT Heater Condenser
Corner 2 dT Heater av-corners
Temperature (°C)
Sintered VC - 180 Deg 80 70 60 50 40 30 20 10 0
20W
0
40W
60W
500
80W
1000
100W
1500
Time (s) heater Condenser
Figure 6:
Corner 1 dT Heater Condenser
Corner 2 dT Heater av-corners
Sintered VC transient state over several heat inputs – 0°, 90°, and 180° orientation.
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3.3 Mesh VC Test results for the mesh wick vapour chamber are illustrated in figures 7(a), (b), and (c). The performance curves show a temperature difference between heater and corner/condenser temperatures being markedly higher than the sintered vapour chamber. Furthermore response to the orientation was more sensitive, with a clear dry out condition taking place at 80W in the 90° orientation (fig. 7(b)). Mesh VC - 0 Deg
Temperature (°C)
100 80 60 40 20
10W
40W 50W
20W 30W
70W
60W
80W
90W
0 0
500
1000
1500
2000
2500
Time (s) Corner 1 dT Heater Condenser
heater Condenser
Corner 2 dT Heater av-corners
Mesh VC - 90 Deg
Temperature (°C)
100 80 60
80W
40 20
20W 10W
30W 40W
60W
50W
70W
0 0
1000
2000
3000
Time (s) heater Condenser
Corner 1 dT Heater Condenser
Corner 2 dT Heater av-corners
Mesh VC - 180 Deg
Temperature (°C)
100 80 60 40 20
10W
20W
30W
40W 50W
60W
70W
80W
90W
0 0
500
1000
1500
2000
2500
Time (s) heater Condenser
Figure 7:
Corner 1 dT Heater Condenser
Corner 2 dT Heater av-corners
Mesh VC transient state over several heat inputs – 0°, 90°, and 180° orientation.
3.4 Thermal and spreading resistance In order to gain further insight into the performance measurements of the test units, thermal and spreading resistance were evaluated. Thermal resistance was
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276 Advanced Computational Methods in Heat Transfer IX taken as the temperature difference between the heater and the condenser divided by the power input (1).
RT =
THeater − TCondenser Qin
(1)
Spreading resistance was taken as the difference between the heater and the average of the two corner temperatures, divided by the power input (2).
RS =
THeater −
TCorner1 + TCorner 2 2 Qin
(2)
The Thermal and spreading resistance for all test units are presented graphically in Figures 8(a) and (b). SpreadingResistance(°C/W )
Spreading Resistance 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0
20
40
60
80
100
Power (W) S0
Cu1mm
Cu3mm
S90
S180
M0
M90
M180
Therm al Resistance(°C/W )
Thermal Resistance
0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0
20
40
60
80
100
Power (W) Cu3mm
Figure 8:
Cu1mm
S0
S90
S180
M0
M90
M180
Thermal and spreading resistance for test units across full input power range.
As can be seen, 3mm copper performs significantly better than 1mm copper. The Mesh is slightly better than 1mm copper but is unable to surpass the performance of 3mm copper. All mesh curves are roughly equal, but as can be expected, the 90° unit did not continue to perform as well as the 0° and 180° orientations. Dry out occurred at approximately 70W. This condition is more clearly seen in the corresponding graph in figure 7b. The sintered units were far better than the mesh with curves equalling those of 3mm copper at 40W and surpassing them from that point on. At 80W the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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performance received a sudden surge which ultimately resulted in a resistance of 1.5C/W less than 3mm copper. A further insight here was that these units did not indicate any dry out condition and showed little effect to gravity. The 90° unit was marginally worse in resistance, as the curves indicate. The experimental results presented here did not take into account heat loss from test units through convection and radiation and the electric power supplied to the heater block was assumed to be fully converted by Ohmic effect into heat and transferred through the body of the test unit to the liquid cold plate. More refined experimental measurements which take heat losses into account through the use of a heat balance will be conducted. The heat balance can be obtained from measurement of liquid flow rate and temperature difference between the input and output flow of the cold plate. Additionally, tests with varying degrees of liquid fill, different grades of sintered wick, and methanol as working fluid will also be conducted to expand the scope of this research.
4
Conclusion
The main objective of this study was to investigate performance enhancement of integrating a small vapour chamber, approximately 40x40mm in span, into an electronics cooling application. The heat source was approximately 10x10mm in size delivering a maximum of 100 Watts heat energy. Subsequent testing of sample units was conducted, at power levels of 10W to 100W. These implemented a specially designed heater block and liquid cold plate, taking temperature readings at various surface points. Both mesh and sintered wick vapour chambers were tested with 3mm base thickness against solid copper units with a 1mm and 3mm base thickness. Performance was measured by reading the temperature difference between measurement points and calculating the corresponding thermal resistance. Overall it became evident that an improvement could be gained from the sintered units, exceeding the performance of the copper samples, increasing power input lead to further improvements. After 40W power input the sintered units marginally outperformed the 3mm copper. After 80W power input the sintered units significantly outperformed the 3mm copper with consistent improvement over the full power range and no dry out condition. Orientation did not affect the performance significantly. Mesh units did not show an improvement over the 3mm base copper, furthermore, their performance degraded as power input increased and orientation turned against gravity. Dry out was experienced with mesh at 70W at an orientation of 90°.
References [1] Dunn P.D. and Reay D.A., 1994, Heat Pipes, 4th ed., Pergamon Press Ltd., New York. [2] Groll M., Heat pipe research and development in Western Europe. Heat Recovery Systems and CHP Vol. 9 No. 1 (1989) WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
278 Advanced Computational Methods in Heat Transfer IX [3] Lee, S., Calculating Spreading Resistance in Heat Sinks, Electronics Cooling, Vol. 4 No. 1, 1998, pp. 30-33. [4] Kaplan H., 1993, Practical Applications of Infrared Thermal Sensing and Image Equipment, O’Shea, Series Editor, Georgia Institute of Technology, USA. [5] Moore, G. E., Cramming More Components Onto Integrated Circuits, Electronics, Volume 38, Number 8, April 19, 1965
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Modeling a real backdraft incident fire A. Tinaburri1 & M. Mazzaro2 1
Central Direction for Prevention and Technical Safety, Firefighters, Public Rescue and Civil Defence Department, Ministry of the Interior, Italy 2 Central Direction for Emergency and Technical Rescue, Firefighters, Public Rescue and Civil Defence Department, Ministry of the Interior, Italy
Abstract This paper reports the fire conditions that occurred in a townhouse and the results of software simulations that were performed to provide insight on the peculiar thermal conditions that developed. Fire rescue was called by the occupants of the house when they became aware that smoky conditions were developing on the second floor. Two of the firefighters entered the bedroom where the smoke was originating from and a short time after a flame front invested them while propagating them backward into the stairwell. The post fire investigation determined that the fire started as a consequence of the smoldering combustion of material contained in a wardrobe located adjacent to one of the room walls. A simulation scenario was developed based on the information obtained on the actual building geometry, material thermal properties and the fire behaviour. The calculations that best represented the actual fire conditions indicate that the partial opening of the window and bedroom door provided outside air (oxygen) to a pre-heated, under ventilated fire compartment. Keywords: smouldering combustion, backdraft, fire models, fire investigation.
1
Introduction
Part of the mission of the department of the firefighters, public rescue and civil defence is to conduct basic and applied fire research, including fire investigations, for the purposes of understanding fundamental fire behaviour and to reduce losses from fire. On February 3, 2003, a fire in a townhouse near Lucca WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060281
280 Advanced Computational Methods in Heat Transfer IX caused two firefighters to be injured. Investigation on fire causes were conducted by the local fire brigade and police station. Computer simulations were made to provide insight on the fire development and thermal conditions that may have existed in the townhouse during the fire.
2
Fire scenario
The fire occurred in a masonry dwelling of ordinary construction. The building was inserted between two others that were not involved and contained only one apartment on three stories, connected through an enclosed stairway, with the basement entrance faced on the access road (Figure 1). The occupants of the house, while staying in the first floor rooms, became aware that, starting from a closed bedroom, smoky conditions were developing in the second floor. They exited the residence via the front door at ground level and called for the fire rescue. The first engine arrived on the fire scene in approximately 20 minutes and entered in the house via the front door. Conditions on the second floor were described as smoky, with smoke coming from the doorway and the window of a room located in the second floor. To lessen the smoke level and provide ventilation, the firefighters started to open partially the front window.
Figure 1:
External view of the 2nd floor room originating the fire and of the window that vented out the backdraft flame (pointed by the arrow) All the pictures have been taken about one hour later the fire event.
After some minutes, two of the firefighters, entered the house and through the stairs, reached the second floor room. They reported that the second floor was at that time fully charged with smoke. They opened the door, entered the bedroom and while orienteering to reach the window, one of them saw in the dark a small flame. Soon after a flame front invested them, while propagating backward to the doorway and then into the stairwell. The flame vented out through a window left WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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open located in proximity of the first floor on the front of the house and then extinguished after few seconds. Thanks to the personnel protection equipment, the two firefighters did not suffer appreciable burn injuries. An amateur video has recorded the scene and became an important source of information when later reviewed by the fire department. The tape showed the flame venting out through the only fully open window, without appreciable flame persistence. Damage to the house was mainly limited to the bedroom originating the fire – a closed door prevented fire spreading. There was no fire extension to the other rooms or to the adjacent buildings and no structural damage observed.
Figure 2: Detail of the 2nd floor stairway lamp showing cover destruction.
Figure 3: 2nd floor stairway. The room originating the fire is on the left side.
The stairway from the first to the second floor showed only few appreciable effect of the backdraft flame impingement on the ceiling and walls except for the destruction of the lamp cover located on the second floor ceiling (Figure 2), the partial melting of the light switch located on the left side, near the bedroom doorway (Figure 3). The ladder shown in Figure 3 was used by the firefighters to access the wooden roof structure, after extinguishing the fire, to verify that no fire propagation and smoldering combustion had occurred. The bedroom had significant deposits of soot throughout, with limited thermal damage. The gypsum board walls and ceiling remained intact. A wardrobe and some furniture located adjacently two walls were completely destroyed by the fire (Figures 4, 5 and 6).
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282 Advanced Computational Methods in Heat Transfer IX
Figure 4:
Detail of one wall in the room originating the fire. The wardrobe gone completely destroyed was located near this wall.
Figure 5: Detail of one room wall (next to Figure 4).
Figure 6:
Detail of the bedroom door (next to Figure 5).
The bed showed signs of pyrolysis and limited burning on the upper portions of the back cushions (Figure 7). The triangular shape of the combusted part was attributed to the fire propagation from adjacent furniture, gone completely carbonized.
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Advanced Computational Methods in Heat Transfer IX
Figure 7:
3
283
Detail of the bed.
Cause and origin
The information provided by the firefighters involved and the evidence described was of a classic backdraft, usually persisting only few seconds before exhausting its fuel supply. The post fire investigation determined that the fire started as a consequence of the smoldering combustion of the material contained in a wardrobe, located adjacent to a wall that contained inside a flue gas duct of an adjacent house chimney. A slow soot combustion fire occurred in the flue gas duct. This initial event provided the heat source that, by thermal conduction through the poor insulating wall, originated the smoldering combustion inside the wardrobe. It was witnessed that the wardrobe got usually warm even during normal chimney operation. The occupant and the firefighters confirmed that the bedroom door and the window were initially kept closed, so that there were negligible external source of combustion air. After the firefighters opened slightly the bedroom window, acting from the outside, an amateur video shows an increase in the smoke production. However, when the firefighters opened the door, the resulting fire gases, rich in carbon monoxide, flowed downward into the stairwell with high velocity and exited through the only open window located in proximity of the first floor.
4
Model results
NIST has developed a computational fluid dynamics (CFD) fire model using large eddy simulation (LES) techniques, called Fire Dynamics Simulator (FDS). The CFD model requires that the room or building of interest be divided into small rectangular control volumes or computational cells. The CFD model computes the density, velocity, temperature, pressure and species concentration of the gas in each cell based on the conservation laws of mass, momentum, and WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
284 Advanced Computational Methods in Heat Transfer IX energy to model the movement of fire gases. FDS utilizes material properties of the furnishings, walls, floors, and ceilings to simulate fire spread. A complete description of the FDS model is given in reference [1]. FDS requires as inputs the geometry of the building, the computational cell size, the location of the ignition source, the ignition source, thermal properties of walls, furnishings and the size, location, and timing of vent openings to the outside which critically influence fire growth and spread. The timing of the vent openings, used in the simulation, are based on an approximate timeline of the fire fighting activities and on a real time video taken by an amateur. The floor plan of the second floor of the townhouse and of the stairwell are shown in Figure 8. The placement and size of the interior walls, doorways, and windows were taken from the dimensioned floor plans drawn by personnel of the Fire Brigade. access road window
window
bedroom Fluegas duct
bed
wardrobe
Furniture
Stairway 2nd floor
Figure 8:
2nd floor plan view.
As in the fire incident, the bedroom window and door were kept closed during the simulation, till the firefighters opened them. The front window on the second floor was kept closed while the one located on the first floor, where the backdraft was vented out (see Figure 1), remained opened during the entire simulation. For the FDS simulation, a specified heat flux from the wall adjacent to the wardrobe was used to start the fire growth as the ignition source. Starting the simulation with a flaming ignition enabled fire development to be modeled within a reasonable computational time. The actual fire may have taken several hours to develop to the flaming stage, if eventually reached. As the simulated fire spreads from the ignition source, and then to other items in the room, it depletes its supply of oxygen for combustion. When a fire occurs in a closed room where the only ventilation is due to leakage, it can become limited by the available oxygen and produce large amounts of unburned fuel. If the leakage rate is low enough WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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the fire may enter a smoldering stage. Temperatures within the bedroom has been estimated much lower than flashover temperatures but significantly higher than the ambient. Upon venting, a gravity current carries fresh air into the compartment. This air mixes with the excess pyrolyzates producing a flammable pre-mixed gas, which can be ignited by a flame or a glowing ember, generating a flame front moving through the compartment toward a vent. This entire process: the accumulation of unburned gaseous fuel, the propagation of an oxygen rich gravity current creating a mixed region and carrying it to the ignition source, the ignition and propagation of an eventually turbulent deflagration vented out, altogether constitutes a backdraft [2, 3].
Figure 9:
Perspective view of townhouse.
The FDS calculations supported the hypothesis that unburned fuel and CO accumulated in the bedroom, resulting in a backdraft after few minutes of venting the bedroom. Reported conditions such as smoke accumulation in the second floor and the “exotic” propagation of the deflagration downward the stairway were reproduced by the model (Figure 10). Some assumptions were necessary in performing these calculations, which may have an impact on the model’s predictions. The results are sensitive to the volume of the apartment and to the size and locations of ventilation openings. All of these were known by actual measurements taken by the fire department during their investigation and by witness report and a real time amateur video. It was reported that the bedroom door and window were closed, up to firefighters arrival. So the assumption of no additional leakage was justified. The combustion was predominately ventilation controlled, making the results insensitive to fuel loading and the specific burning characteristics of the fuel. The generation rate of unburned fuel and CO in poor ventilation conditions should be affected by energy feedback from the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
286 Advanced Computational Methods in Heat Transfer IX environment and any flames present during the time of ventilation controlled combustion. The FDS model does not contain such a self-consistent combustion model, so the quantity of unburned fuel and CO production has been estimated and subject to a sensitivity analysis. The field data of the heat release per unit volume during backdraft evolution are reported in Figure 10.
Figure 10:
Field data of the heat release per unit volume (kW/m3) during backdraft evolution.
A temperature profile inside the bedroom during backdraft is shown in Figure 11. The values are in the range of 200-250°C, much lower than flashover temperatures as expected. This figure is compatible with the observed fire scenario and are indeed necessary to support the production of relatively dense unburned fuel, prior to cold air mixing, necessary to allow downward flame propagation from second floor to the first floor window, were the deflagration was vented out.
5 Lessons learned The fire service community has long recognized the hazards associated with backdrafts. Typically, firefighters are involved in initial search and rescue or suppression operations when the backdraft occurs. Current tactics for reducing the backdraft hazards are to vent the structure prior to entry. However, the ventilation process is often a second priority to the rescue operation. Indeed, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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unless it is restricted to roof venting, ventilation may facilitate rather than prevent a backdraft. Even if neither the occupant nor the firefighters get seriously injured, it is fundamental to analyze the incident to increase the overall consciousness of safety culture and avoid the risk that valuable information get lost unreported as because no one had been injured. It has also pointed out the benefits of the use of modern, computer fire modeling in the reconstruction of fire incidents to understand critical factors for mitigating their impacts.
Figure 11:
Temperature profile inside the bedroom (°C) during backdraft.
References [1] Mc Grattan, K.B., Baum, H.R., Rehm, R.G., Hamins, A. & Forney, G.P., Fire Dynamics Simulator – Technical Reference Guide, NIST Special Publication 1018, Nat. Inst. Stand. Tech., Gaithersburg, MD, USA, 2004. [2] Fleischmann, C.M., Backdraft phenomena, NIST-GCR-94-646, Nat. Inst. Stand. Tech., Gaithersburg, MD, USA, 1994. [3] Fleischmann, C.M., Pagni, P.J. & Williamson,R.B, Quantitative backdraft experiments. Proceedings of the International Association for Fire Safety Science 4th International Symposium, July 13-17, 1994, Ottawa, Canada, IAFSS, Boston, MA, USA, T. Kashiwagi, ed., pp. 337-348, 1994.
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Numerical and experimental studies in the development of new clothing materials E. L. Correia1, S. F. C. F. Teixeira1 & M. M. Neves2 1 2
Department of Production and Systems, University of Minho, Portugal Department of Textile Engineering, University of Minho, Portugal
Abstract In order to enhance the quality and added value of their products, Portuguese industry must react to consumer demands. Amongst the main criteria, thermal comfort is becoming a pressing issue. Such new requirements open new fields in research and development, in which computer simulation plays an important role. In the present work a transient model for heat and mass transfer in a fabric, which is a simplified version of the Gibson and Carmachi model, has been implemented. In order to make the model user friendly, a software application has been developed. This makes it easier to introduce the input data and visualize the results. In parallel, new materials made from natural fibres have been used to produce new knitting. Their thermal properties have been measured in the laboratory and were used to validate the numerical model. The model can be integrated into a wider model for the thermal regulation of the human body. Such a tool could be very useful for designing new fabrics for clothing applications. Keywords: thermal human comfort, heat and mass transfer, numerical models, new clothing materials, computer interface, physiological comfort.
1
Introduction
People tastes and concerns have changed over time. Nowadays, customers tend to give less importance to the technical specifications of the fabrics used for clothing and focus more on their appearance and handling characteristics. Wear comfort of clothing is presently one of the major concerns for customers. But it is also a very broad concept, as it involves all the aspects of comfort. It is a balanced mix of psychological, physiological and sensorial comfort [1].
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290 Advanced Computational Methods in Heat Transfer IX The wear comfort is given mostly by the fabrics specifications both functional (related to the physiological and sensorial comfort) as aesthetics (related to the psychological comfort). The aesthetics specifications are the domain of the textile designers and they must satisfy the cultural and social environment of the user, its individuality and fashion trends. The textile design engineer is responsible for the functional specifications. To achieve higher quality products with a greater added value, European producers must respond to the consumer needs, and new demands of wear comfort opened a wide variety of areas to look into as well as, new fields in research and development. The present work is part of a wider effort [2–4] to couple experimental data with numerical tools for the development and testing of functional knitting.
2
Numerical model and interface
Mathematical modelling of the clothing materials has been developed [5–8] and it is a valuable tool to understand the complex mechanisms of the coupled heat and moisture transfer. 2.1 Mathematical equations In order to simulate the transient behaviour of simultaneous heat and moisture transfer in a fabric, a simplified version of Gibson and Chamarchi [5] model has been implemented in the present work. The model includes effects such as heat and mass diffusion through the fabric thickness and also, the sorption phenomena. There is no liquid phase present and the model does not include gas or liquid phase convection in the fabric but it assumes heat convection to the environment. Two partial differential equations are solved for the temperature and vapour density. One of the governing equations is the energy equation, eqn. (1):
ρ cp
∂T ∂ ∂T − (QL + ∆hvap ) m sv k eff = ∂t ∂ x ∂ x
(1)
which includes the transient term in the left side and those on the right hand side are, respectively, the diffusion and sorption. The energy equation is coupled with the gas phase diffusion equation, eqn. (2), which models the phenomena associated with water vapour component:
εγ
∂ (ρ v ) = ∂ Deff ∂ ρ v + m sv − ρ v ∂ (ε γ ) ∂t ∂t ∂ x ∂x
(2)
The transient term appears on the left hand side of eqn (2) and the first term on the right hand side describes the diffusion transfer ( Deff is the effective gas phase diffusivity), the second term accounts for the mass rate of desorption from solid phase to vapour phase and the last term describes the changes of the volume fraction ( ε γ ) occupied by the gas phase (air and water vapour) in the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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control volume. Apart from the gas, the dry solid fibre and the water bounded in the solid phase occupy the remaining part of this volume. In consequence, ρ (density), c p (heat capacity) and k eff (effective thermal conductivity) are average properties depending on the volume fraction of each component (air, water, vapour and fibre) and they had to be updated in each time step. The source term in eqns (1) and (2) due to the vapour sorption is modelled assuming that the material at the fibre surface immediately come into equilibrium with the relative humidity of the gas phase at that point, and so, the mass flux into and out the fibre is calculated by:
m sv =
Dsolid ρ ds (Rtotal − Rskin ) d 2f
(3)
which depends on the difference between the instantaneous fibre regain and the equilibrium regain at relative humidity. The two partial differential equations, eqns (1) and (2), constitute a system of parabolic equations which are solved in time and space. They can be integrated to obtain the temperature and water vapour concentration profiles in the fabric thickness and along the time, after knowing the initial conditions ( T ( x, t = 0 ) and
ρ v ( x, t = 0 ) )
and the boundary conditions at each side of the fabric
thickness for temperature and vapour density. Numerical methods have been used to integrate the system of parabolic equations. The finite volume method (Versteeg and Malalasekera [9]) has been used for space integration and an implicit scheme was used in time integration. Because the temperature is also a function of the vapour density, an iterative scheme has been implemented. At each step in time, the temperature field is calculated assuming a given water vapour concentration in all control volumes across the fabric thickness; with this new temperature profile the vapour density field is calculated; the physical properties are updated and a new temperature field is calculated and this iterative process is repeated until convergence is achieved. More details on the mathematical model, algorithm and numerical solution can be found in Correia [10]. 2.2 Computer interface Considering the large amount of physical constants in the model and in order to make the model user friendly, a software application to store all this information and visualise results has been developed. Using a computer interface, the user can introduce the numerical parameters and to choose the fabric to be tested. The interface TECIDOS was designed in Pascal (Delphi 6) for Windows environment and it allows four main menus: numerical parameters; fabric choice; simulation (Fig. 1) and way out. In the menu “numerical parameters”, the user can choose the time integration step and the total time for simulation as well the results precision. Then, the user can select from a large database a natural fibre and run the numerical code to WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
292 Advanced Computational Methods in Heat Transfer IX obtain the main results at the screen. At the moment, the user will have access to temperature and vapour concentration profiles as well as the amount of water in the fabric at each time step. The application includes also a menu to leave the application and return to the main system.
Figure 1:
3
Main page of the interface.
Experimental results
The raw materials were selected in order to provide the better combination of the following properties: humidity transport, anti bacteria resistance, hypo allergic ability, low thermal conduction. Some ecological considerations were also taken into account: how fibres were produced and recycling possibilities at the end of its life. Thus, the following raw materials were selected: hydrophilic fibres (cotton, corn, soybean and bamboo) and hydrophobic fibres (polypropylene and polyester). Table 1 shows the different raw materials and yarn count used for knitting production. These raw materials were used to produce knitting in order to validate the numerical model. Double face knitting was also produced and is now being characterized. The design of double face knitting tries to answer the questions of humidity transport and maintenance of optimal temperature of the foot, avoiding the formation of micro organisms/fungi. The application of this knitting must contribute, in a substantial way, for an increase in comfort, removing the humidity released by skin and reducing the heat exchange with the environment. To achieve this performance, different structures and raw materials were used. Three double face-knitting structures with eight different combinations of raw materials, in a total of twenty-four samples were made. The different WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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knitting structures, with the hydrophilic fibre on one side and the hydrophobic fibre on the other, are shown in table 2. Table 1:
Raw materials.
Raw Material
Designation
Yarn Count
100% Cotton
CO
30/1 Ne
100% Corn
PLA
30/1 Ne
100% Soybean
SPF
30/1 Ne
100% Bamboo
BAM
30/1 Ne
100% Polypropylene
PP
200 dtex
85% Polyester, 15% Cotton
85% PES, 15% CO
60/2 Ne
Table 2:
Structure 1
Knitting structures.
Structure 2
Structure 3
The 8 different hydrophilic/hydrophobic fibres combinations to produce the 24 samples, at the loom, were PLA/PP, PLA/PES, SPF/PP, SPF/PES, BAM/PP, BAM/PES, CO/PP and CO/PES. After the production, each sample was characterized and some properties are currently being evaluated. For the thermo physiological comfort, the following critical properties are suggested: thermal resistance, air permeability, water vapour permeability and liquid water permeability. To evaluate the air permeability of the different samples, the TEXTEST FX 3300 permeability meter and the EN ISO 9237 standard (100 Pa of pressure) were used. The knitting samples produced with structure 1 and combination PLA/PES have higher permeability than the others. The water vapour permeability of materials is an important property to maintain the thermal equilibrium of the user. It is one of the parameters that have a bigger contribution for the textile thermal physiological comfort. The experiments were made with the PERMETEST permeability meter and according ISO 11092 (see Fig. 2). The samples with higher values were made with structure 1 and combinations PLA/PES, SPF/PES and BAM/PES. A thermal manikin robot is being used for thermal resistance evaluation. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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(%)
Permetest 50,0 45,0 40,0 35,0 30,0 25,0 20,0 15,0 10,0 5,0 0,0
PLA/PP PLA/PES SPF/PP SPF/PES BAM/PP BAM/PES CO/PP CO/PES
Structure 1
Figure 2:
4
Structure 2
Structure 3
Water vapour relative permeability.
Numerical results
The knitting made from natural fibres was tested with the numerical model and some results are now presented. Two different types of fibres can be compared in terms of water retain: a natural hydrophilic fibre (cotton) and a hydrophobic fibre (polyester). The number of control volumes across the knitting thickness and the time integration step, have been changed and optimal values have been obtained. The model sensitivity to the physical properties was tested in special, the convective heat and mass transfer coefficients. These parameters have been investigated [11] because of the results dependence on their values. Two different environments have been considered. Firstly, the dry fabrics with initial temperature of 20ºC were placed into contact with a relative humidity of 100% and at the ambient temperature of 20ºC. With this test, the temperature change due to the water vapour entrance into the fibres can be observed. Fig. 3 shows the temperature profile along the cotton knitting thickness for three different times (10, 150 and 3600 s). The knitting thickness is 0.75 x 10-3 m and the grid points m1 and m11 are the boundaries (ambient conditions at each side of the knitting). It can be observed the expected symmetry. Because cotton fibre is hydrophilic, as soon as, the sorption of water vapour occurs, the temperature rises uniformly across the fabric thickness. Then, evaporation takes place and the temperature decreases again until it reaches the ambient temperature. The amount of water dissolved in the fabric fibres, along the time sv ), can also be calculated. The water vapour entrance will ( ρ v d ε bw d t = − m continue until the equilibrium regain is achieved (Fig. 4). The polyester knitting is also tested with the same boundary conditions. The dry knitting of 1.1 x 10-3 m thickness and initial temperature of 20ºC is placed into contact with a relative humidity of 100%.
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40
Temperature (ºC)
35 30 25 20 15 10 t = 10 s
5
t = 150 s
t = 3600 s
0 m1
m2
m3
m4
m5
m6
m7
m8
m9
m10
m11
Grid points
Figure 3:
Numerical temperature profile across cotton knitting thickness for different times. 0.10 0.08
ε bw
0.06 0.04 0.02 0.00 0
5
10
15
20
25
30
35
40
45
50
55
60
Time (min.)
Figure 4:
Volume fraction of water at the centre point of the cotton knitting.
Because polyester fibre is hydrophobic, the temperature rise due to the sorption mechanism is smaller (Fig. 5) compared with the cotton fabric case (Fig. 3). The fabric made of hydrophilic fibre (cotton) was then tested with a different set of boundary conditions as more suitable to the knitting application: one side of the knitting is usually with contact with the human body skin (at 32ºC and with some moisture – m1) and the other side in contact with the environment (an atmosphere of 20ºC and a relative humidity of 65% – m11). Such imbalance yields non-symmetry in the volume fraction of water profile (Fig. 6), after a 10 s period. The water vapour concentration along the knitting thickness is illustrated in Fig. 7.
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296 Advanced Computational Methods in Heat Transfer IX 32
Temperature (ºC)
30 28 26 24 22 20 m1
m2
m3
m4
m5
m6
m7
m8
m9
m10 m11
Grid points
Figure 5:
Temperature profile for polyester knitting after 10 s.
0.030 0.025
ε bw
0.020 0.015 0.010 0.005 0.000 m1
Figure 6:
m2
m3
m4
m5
m6
m7
Grid points
m8
m9
m10 m11
Volume fraction of water dissolved into the cotton fibres across the knitting thickness after 10 s contact. 0.040 0.035
3 ρ v (kg/m )
0.030 0.025 0.020 0.015 0.010 0.005 0.000 m1
m2
m3
m4
m5
m6
m7
m8
m9
m10 m11
Grid points
Figure 7:
Water vapour concentration profile along the thickness.
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Conclusions
A computer application was developed in order to correctly model the heat and mass transfer phenomena across the knitting thickness. Some experimental work has been taken in order to produce double face knitting, which will contribute in a substantial way, to an increase in the human physiological comfort. Further work will extend the numerical model to simulate these functional knitting and integrate a thermal regulation of the human body [12–14] and it will be applicable to the design of new clothing applications.
Acknowledgment The authors would like to acknowledge the financial support from the Portuguese Foundation for the Science and Technology (FCT) through the research project POCTI/EME/62786/2004.
References [1] [2] [3] [4]
[5] [6]
[7]
[8]
Fourt, L. & Hollies, N. R. S., Clothing: Comfort and Function, Marcel Dekker, Inc., New York, USA, 1970. Teixeira, S.F.C.F., Ferreira, M.L. & Costa, L.G., A Computer interface for human comfort calculations. Proceedings of IMC22, 31 Ago-2 Set, p. 131-137, 2005. Neves, M. & Cunha, J., Total design of function oriented textile product, 5th International Istanbul Textile Conference, Istanbul, Turkey, May 2005. Epifânio, P., Silva, A., Teixeira S.F.C.F. & Teixeira, J.C.F., Modelo de conforto térmico baseado na distribuição da temperatura do corpo humano. Métodos Numéricos en Ingeniería V (CD-ROM), SEMNI, Espanha, 2002. Gibson, P.W. & Chamarchi, M., Modelling convection/diffusion processes in porous textiles with inclusion of humidity-dependent air permeability. Int. Comm. Heat Mass Transfer, 24(5), pp. 709-724, 1997. Kwon, A., Kato, M., Kawamura, H., Yanai, Y. & Tokura, H., Physiological significance of hydrophilic and hydrophobic textile materials during intermittent exercise in humans under the influence of warm ambient temperature with and without wind. Eur. J. Appl. Physiol., 78, pp. 487-493, 1998. Fan, J., Luo, Z. & Li, Y., Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation. International Journal of Heat and Mass Transfer, 43, pp. 2989-3000, 2000. Chang, W. & Weng, C., An analytical solution to coupled heat and moisture diffusion transfer in porous materials. International Journal of Heat and Mass Transfer, 43, pp. 3621-3632, 2000. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
298 Advanced Computational Methods in Heat Transfer IX [9] [10] [11] [12] [13] [14]
Versteeg, H.K. & Malalasekera, W., An Introduction to Computational Fluid Dynamics, the Finite Volume Method, Longman Scientific & Technical, 1995. Correia E. L., Modelo Térmico Aplicado à Caracterização do Conforto Proporcionado pelo Vestuário, MSc Thesis, University of Minho, 1995. Dear, R.J., Arens, E., Hui, Z. & Oguro, M., Convective and radiative heat transfer coefficients for individual human body segments. Int. J. Biometeorol, 40, pp. 141-156, 1997. Berger, X. & Sari, H., A new dynamic clothing model. Part 1: Heat and mass transfers. Int. J. Therm. Sci., 39, pp. 635-645, 2000. Ghaddar, N., Ghali, K. & Jones, B., Integraded human-clothing system model for estimating the effect of walking on clothing insulation. International Journal Thermal Sciences, 42, pp. 605-619, 2003. Salloum, M., Ghaddar, N. & Ghali, K., A new transient bio-heat model of the human body. Proc of HT2005, ASME Summer Heat Transfer Conference, July 17-22, San Francisco, USA, 2005.
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Sensitivity analysis of a computer code for modelling confined fires P. Ciambelli, M. G. Meo, P. Russo & S. Vaccaro Department of Chemical and Food Engineering, University of Salerno, Italy
Abstract Full scale experiments of tunnel fires are expensive and difficult to be carried out while tunnel fires simulation by computer modelling is cheaper and faster. Therefore, such a tool can replace the experiments if simulation results are recognized to be reliable and reflecting the reality. A computational fluid dynamics (CFD) code AIR was employed for the description of the transient behaviour of confined fires. The code solves the balance equations for the conservation of mass, momentum, energy and gas species within the physical domain of interest and yields local predictions of temperature, velocity, smoke, species concentration, etc, as a function of time. Firstly a sensitivity analysis of the computer code with respect to its parameters was performed, then experimental data from literature were employed to test the computer code performances. Simulations were obtained for a small scale steady-state tunnel fire and for an unsteady-state tunnel fire. AIR’s performances in simulating tunnel fires were fair. Results depend on code parameters (grid fineness, number of iterations and step time interval) and on initial and boundary conditions such as temperatures, ventilation, heat release rate and radiant and convective heat transfer at the walls. The main AIR’s limit is that it cannot manage radiative heat exchange with the walls and time variable boundary conditions as those encountered in transient tunnel fires. Keywords: tunnel fires, CFD modelling, sensitivity analysis, temperature profiles, CO and smoke concentration profiles.
1
Introduction
Fires developing in enclosures constitute a terrible threat for lives. The danger of enclosure fires, besides to the temperature increase, derives mainly from the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060301
300 Advanced Computational Methods in Heat Transfer IX presence of high concentrations of smoke, which can significantly reduce the visibility and then the possibility for people to reach escape routes, and of CO, which attenuate reaction faculties up to a total unconsciousness. Indeed, most of the deceases caused by fires are due to inhalation of carbon monoxide and consequent poisoning [l, 2]. Among the various enclosure fires those in road and railway tunnels are particularly dangerous because the amount of firing material can be huge and the length of the way to escape outside the enclosure can be in some cases several kilometres. As a result, road and railway tunnel fires where many people lost their lives represent tragic real examples [3]. To prevent the occurrence or at least to mitigate the consequences of such events, existing tunnels should be upgraded and new tunnels should be equipped with efficient fire protection systems. To this aim, faithful predictions of the fire-induced air velocity, temperature, CO and smoke concentration in enclosure fires and of their evolution with fire protection systems such as ventilation are fundamental. Fire modelling has been made by zone models [2, 4-6] and by field or computational fluid dynamics (CFD) models [7-9]. CFD models are computationally more intensive than zone models, which uses assumed smoke layer functions. However, the former yield more accurate solutions for the actual space and a better mean of comparing system changes. Therefore, CFD allows the evaluation of the effectiveness of fire fighting strategies, the assessment of relative benefits and the comparison of the effects of differing approaches [10]. However, enclosure fire modelling is difficult. Indeed, an enclosure fire is a complex phenomenon involving mass transfer, combustion, turbulence and radiative, convective and, possibly, conductive heat transfer. Such phenomena interact with each other during fire. Therefore, besides to mass and momentum balance equations further submodels for buoyancy, compressibility, turbulence, and thermal radiation are, generally, employed [11]. Obviously, to be really reliable for safety studies CFD models must be able to reproduce closely not only the overall known behaviour of fires in tunnel, but also measured values from controlled tunnel fire experiments. Therefore, CFD models and the associated physical sub-models need to have been validated by means of sensitivity analysis against simple cases and to be suitable for the particular application. Such a validation plays an important role in generating confidence in the application of a CFD fire model to tunnel problems, and in understanding the critical parameters and limitation of this model [12]. The aim of the present study is to examine the performance of a relatively simple field model (AIR) developed by D’Anna and Kent [10] and Novozhilov et al. [13-14] in the description of tunnel fires. A sensitivity analysis of the computer code with respect to its parameters was performed and experimental data from literature were employed to test the code. In particular, simulations results were obtained for a small scale steady-state tunnel fire [11], and for an unsteady tunnel fire [15-18]. Such results are compared and evaluated versus the pertaining experimental data.
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2 CFD The computational fluid dynamics model employed in this work [10, 19] uses given boundaries, heat sources and air inlets to predict three-dimensional distributions of conditions within the space. The code checks mass and energy balances and checks for inconsistencies in input specifications. Solutions can be steady state or time dependent. The governing equations are solved using the finite volume method in a Cartesian computational grid with variable cell size. Embedded fine grids to any level may be placed within coarse grids to improve resolution and efficiency. The “HYBRID first order upwind” scheme is generally used to represent the convection-diffusion terms in the conservation equations, although high-order discretization schemes are also available in the code. The SIMPLER algorithm is used to calculate the pressure field. The algebraic equations are solved using the line-by-line TriDiagonal-Matrix Algorithm (TDMA). AIR employs the buoyancy-augmented k-ε model to represent turbulent transport, and the Eddy Break-Up combustion model. Wall boundary conditions are treated using the wall function approach to eliminate the need to resolve the laminar sub-layer. The standard expressions for momentum and convective heat/scalar fluxes are modified to include the effect of wall roughness. The model does not account for radiation, as well as for walls heating. As suggested in the literature [20-22], a solution to the first lack can be found by empirically setting in the code an Heat Release Rate (HRR) lower than the actual, which would represent the real energy available for gas heating by convection (in this way HRR becomes a sort of convective heat release rate). In other terms, a fraction of the total heat generated by the fire is assumed to be transmitted to the tunnel walls via radiation by flame, smoke and hot gases. This radiative fraction depends, besides to fire size and tunnel geometry, on fuel type and ventilation, and in the literature it is reported to vary between 0.2 and 0.4 [20-22].
3
Experimental data from the literature
A typical feature of CFD models is the ability to simulate both full and reduced scale events, either steady or unsteady. As mentioned in the Introduction, simulations results were obtained for two different tunnel fires of which experimental data were available in the literature, i.e. a small scale steady-state tunnel fire [11], and a full scale unsteady fire in the Runehamar tunnel (UPTUN project) [16-19]. 3.1 Small scale fire The small scale tunnel was made of insulating materials (perspex, silica glass) and of thermo-resistant material (alumina) near the fire. It was 6m long, with rectangular cross section (0.3m high and 0.9m wide). The fire source was located 1.5m from the inlet section (x=0) in the middle of the floor. The burner dimensions were 0.18x0.15m2 with the longer side along the tunnel axis. Liquefied petroleum gas (LPG) was used as fuel and longitudinal ventilation was simulated by an axial fan. In the experimental study, two steady-state heat release rates, 3.15 and WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
302 Advanced Computational Methods in Heat Transfer IX 4.75kW, were employed under four different ventilation flow velocities, i.e. 0.13, 0.31, 0.52 and 0.61m/s. Available data refer only to few conditions, and are expressed as temperature profiles along verticals on the tunnel centreline at three cross-sections, one located upstream the fire (at x=0.9m) and two located downstream the fire (at x=3.3m and x=5.1m). Simulations were carried out with HRR of 3.15kW, and ventilation velocity of 0.13m/s. 3.2 Full scale fire The experimental results of the full scale fire are relevant to tests carried out in September 2003 by SP Swedish National Testing and Research Institute in the context of UPTUN [12]. The tests were performed in Norway in an abandoned two way asphalted road tunnel 1650m long, 6m high and 9m wide, with a slope varying between 1% and 3%. Tests were performed with a fire of a HGV cargo set-up, located at the centre of the left hand lane, loaded with mixtures of different cellulose and plastic materials or with furniture and fixtures. The commodities were placed on boards on a rack storage system to simulate a HGV measuring 10450mm by 2900mm. The total height was 4500mm. The height of the platform floor was 1100mm. The CFD simulations were focused on a test carried out with wood pallets and PE plastic pallets (total weight 10911kg) in the ratio 82/18wt%, yielding a HRR peak of about 200MW after 20min. In order to create a longitudinal flow inside the tunnel, two mobile fans, able to create a longitudinal flow of 3m/s to minimize the risk for backlayering, were used. The centre of the fire was located 563m from the tunnel entrance. Ignition took place at the upstream side of the HGV-trailer. For safety reasons in the fire zone the tunnel was protected with a passive fire protection system made of fire-resistant and insulating PROTOMATECT®-T boards to prevent the tunnel ceiling rocks from falling down. A tunnel length of 75m was protected with boards.
4 Simulations 4.1 Small scale fire The AIR simulations [23] of the small scale steady-state tunnel fire were made by setting the calculation domain just the same of the experimental volume (6m long, 0.9m wide and 0.3m high), and dividing it in 180,000 cells (150x30x40). The grid was made finer along the vertical axis to better simulate smoke buoyancy. To reproduce the experimental ventilation, a uniform 0.13m/s airflow was supplied at the x=0 section. The convective coefficient U was set to zero (adiabatic walls) along the whole tunnel. This is an acceptable approximation thanks to the wall materials, to the fact that the system is under steady-state conditions and to the likely high wall temperatures, especially around the fire. As mentioned, AIR does not account for radiation, and then, simulations were performed in this case employing values of the radiant fraction of the actual HRR of 0.1 and 0.2.
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4.2 Full scale fire The Runehamar tunnel was exemplified as a rectangular tube, 9m wide and 6m high, but in the tunnel zone with the walls coated by protecting boards for which a reduced section 7,2x5m2 was employed. The calculation domain was set 700m long, from 140m upwind the fire centre up to the downstream tunnel portal. Tunnel slope was simulated and a uniform 3m/s airflow was imposed at the x=0 section to reproduce the experimental longitudinal ventilation [23]. At first, sensitivity analysis of the effects of some code parameters was carried out by launching several runs with different values of such parameters. Specifically, the influence of grid fineness, integration time step and maximum number of iterations per time step on the stability of the solution and on the accuracy and the behaviour of the results was studied. For each run the simulation results were compared with experimental data and the required computing time was recorded. These simulations for sensitivity analyses were carried out with a strong approximation, to quickly and simply set boundary conditions values: the heat transfer convective coefficient U was set to zero (adiabatic walls) along all the tunnel length. Finally, the optimization of the code parameters led to these values: The simulated tunnel was divided into 175, 9 and 15 cells along the three directions (base grid), and embedded finer grids were used in the fire area, with a total of about 170,000 cells; The maximum number of iterations per time step was set to 100; The transient simulations were taken up to 50min with time step of 0.5min. With this set of parameters, approximately 9h of computing time was required for the simulation, and the results were stable and followed the experimental behaviour.
5
Results and discussion
5.1 Small scale fire The quality of the simulation can be appreciated comparing (fig. 1) the available experimental data of temperature profiles, measured along verticals at the centreline of the tunnel upstream (x=0.9m) and downstream (x=3.3m) the fire, and simulations results. From fig. 1(a) it is evident that the supplied airflow (0.13m/s) is not able to prevent hot gases backlayering since upstream gases warm up, as shown by both measured and calculated temperatures. In addition, figs 1(a) and 1(b) show that the calculated upper layer hot gases temperatures are always overestimated and that such temperatures oscillate with the number of iterations with amplitude of about 50°C. As a result the solution does not converge even after several thousands iterations. This is probably due to the flow pattern set, intrinsically instable (as proved by backlayering), which does not allow AIR code to get convergence and stability with such boundary conditions. For these tests experimental measurements of pollutants and oxygen concentrations were not WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
304 Advanced Computational Methods in Heat Transfer IX available so, unfortunately, it is not possible to assess the consistence of simulated values of such variables. 250
T [°C]
200
exp 100 iter 500 iter 1000 iter 3000 iter
b)
a)
150 100 50 0 0
0.05
0.1
0.15
0.2
0.25
H [m]
Figure 1:
0. 0
0.05
0.1
0.15
0.2
0.25
0.3
H [m]
Experimental and calculated (90% HRR) temperature profiles along the tunnel height at the centreline. Distances from the tunnel entrance: (a) x=0.9m, (b) x=3.3m.
5.2 Full scale fire The comparison between experimental and calculated CO and O2 concentrations profiles, shown in fig. 2 in the case of O2, is good at relatively short distances from the floor (fig. 2(b)) whole it becomes poor as the distances approach the tunnel height. Moreover calculated CO and O2 concentrations vary little with the actual HRR set, as shown for O2 in fig. 2.
O2 concentration (%)
25
experimental data
experimental data
20
no radiation
no radiation
radiation 20%
radiation 20%
15
radiation 30%
radiation 30%
10
a)
b)
5 0 0
Figure 2:
10
20
t (min)
30
40
50 0
10
20
t (min)
30
40
50
Experimental and calculated O2 concentrations, 458m downstream the fire at the tunnel centreline, at different heights: (a) z=5.1m, (b) z=1.8m.
This result derives, on the one hand, from the fact that the computer code employs both HRR and the net calorific value of the fuel to estimate the evolution with time of smoke, CO and O2 concentrations and, on the other hand, from the fact that during the experimental tests HRRs were measured using oxygen consumption calorimetry. Therefore, in the simulations where the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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assumed HRR was a portion (i.e. 80% or 70%) of the experimental value, the net calorific value of the fuel was decreased proportionally to maintain the same ratio between HRR and the net calorific value, in order to keep the fuel combustion rate constant and consequently to allow the correct computation of pollutant concentrations independently of the assumed HRR value. In contrast with what observed for O2, the time profiles of temperature along the tunnel depend strongly on the HRR value (fig. 3). Results in fig. 3 also show that the mere change of HRR does not allow the correct reproduction of the experimental data, neither in the tunnel zone protected by insulating and refractory panels (temperatures are underestimated: figs. 3(a) and 3(b)), nor downstream in the zone with rock walls, where temperatures are instead greatly overestimated (figs. 3(c) and 3(d)). 1500
experimental data
a)
no radiation
T (°C)
1000
experimental data
b)
no radiation
radiation 20%
radiation 20%
radiation 30%
radiation 30%
500
0 1500 experimental data
c)
no radiation
T (°C)
1000
experimental data
d)
no radiation
radiation 20%
radiation 20%
radiation 30%
radiation 30%
500
0 0
Figure 3:
10
20
30
t (min)
40
50 0
10
20
t (min)
30
40
50
Experimental and calculated temperature profiles at the tunnel centreline, 0.3m under the ceiling: (a) fire centre, (b) 40m downstream the fire, (c) 150m downstream the fire, (d) 458m downstream the fire.
In particular, figs. 3(a) and 3(b) suggest that insulating panels really make adiabatic the region where fire develops, being the reduced HRR insufficient to heat the gases at the measured value. Instead, well downstream such a region the same HRR appear too high leading to a huge overestimation of the gas temperature (figs. 3(c) and 3(d)) and suggesting that relevant convective heat exchange between gases and tunnel walls occurs. Differences take place also when the fire is practically extinguished (≈ 40min). In this case, indeed, whatever the distance from the fire but close to the tunnel exit portal (fig. 3(d)), the gas temperatures are underestimated (fig. 3). This likely means that when the fire is off, tunnel walls, heated up by the gas when the fire was developing, give WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
306 Advanced Computational Methods in Heat Transfer IX back part of the heat making gas temperature higher than that expected. However, with respect to the observed discrepancies between experimental and computed results, it is worth noting that the modelling of the chosen tunnel fire is particularly difficult because of the presence of the insulating and refractory panels mounted on the walls in the region where fire develops and for further 50m downstream. Indeed, their presence changes the geometry and the transport characteristics of the system influencing both fluid dynamics and heat transfer properties. On the basis of the above analysis, some changes have been made to the parameters and boundary conditions. Specifically, such changes were addressed to take into account heat losses to the walls via radiation and convection as long as the fire was fully developed and the reverse heat transfer from hot walls to gases after the fire extinguishment. However, as AIR cannot model wall heating with time during fire, actual convective heat transfer at walls may be reproduced only if wall temperature (Twall) is changed cell by cell in space and step by step in time. Obviously, this complicated and tiresome method cannot be both accurate and useful in simulations. Recognised that radiation and convection in tunnel fires are very important, a more detailed simulation of the heat transfer phenomena was implemented. The actual HRR was used and the tunnel length was divided into three zones: the first comprising the board coated region, and the other two each covering part of the remaining tunnel length. All the heat losses to the walls were modelled using lumped heat transfer coefficients and wall temperatures that accounted for both radiation and convection, and were determined as time and space averages. In addition, the total fire duration was split into shorter time intervals. Finally, using these settings, two longitudinal ventilation velocities (3m/s and 2.5m/s) were tried, in order to best reproduce the experimental airflow generated by fans. The results of such simulations show a good agreement with experimental measurements either for gaseous species concentrations (fig. 4) or for gas temperatures (fig. 5). CO and O2 concentrations downstream the fire depend on the supplied airflow: the greater the ventilation velocity, the lower the oxygen concentrations and the higher the carbon monoxide concentrations (fig. 4). 2500
experimental data
20 15
experimental data
v=3m/s
v=3m/s
v=2.5m/s
v=2.5m/s
2000 1500
b)
a)
10
1000
5
500
CO concentration (ppm)
O2 concentration (% vol)
25
0
0 0
10
20
30
40
50 0
10
20
t (min)
Figure 4:
30
40
50
t (min)
Experimental and calculated time profiles of O2 (a) and CO (b) concentrations. 100% HRR, variable U and Twall, 458m downstream the fire at the tunnel centreline; distance from the floor: z=5.1m.
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Moreover, the run carried out with 3m/s air velocity gives results that seem to come closer to experimental profiles. Instead, 2.5m/s air velocity seems more appropriate for the description of temperature profiles both in the tunnel zone protected by boards and in the zone with rock walls, as shown in fig. 5. 1500
experimental data
a)
experimental data
b)
v=3m/s
v=3m/s
v=2.5m/s
v=2.5m/s
T (°C)
1000
500
0 1500
experimental data
c)
v=3m/s
d)
experimental data v=3m/s
v=2.5m/s
v=2.5m/s
T (°C)
1000
500
0 0
10
20
30
t (min)
Figure 5:
6
40
50 0
10
20
30
40
50
t (min)
Experimental and calculated time profiles of temperature at the tunnel centreline, 0.3m below the tunnel ceiling. 100% HRR, variable U and Twall: (a) fire centre, (b) 40m downstream the fire, (c) 150m downstream the fire, d) 458m downstream the fire.
Conclusions
On the whole, simulations results of pollutants and O2 concentrations and of gas temperatures describe fairly well the experimental values. However, such results were obtained after some modifications of boundary conditions, not possible in the normal AIR settings. Therefore, improvements of the code, especially concerning the boundary conditions management, are needed to accurately predict the behaviour of tunnel fires and to allow the use of the model for reliable fire safety engineering design.
Acknowledgement The authors wish to thank Professor J. Kent for his helpfulness and for making available AIR code.
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308 Advanced Computational Methods in Heat Transfer IX
References [1] [2] [3]
[4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16]
Pitts, W.M., The global equivalence ratio concept and the formation mechanisms of carbon monoxide in enclosure fires, Progress in Energy and Combustion Science, 21(3), pp. 197-237, 1995. Duong, D.Q., The accuracy of computer fire models: some comparisons with experimental data from Australia. Fire Safety Journal, 16, pp.415-431, 1990. Vuilleumier, F., Weatherill, A. & Crausaz, B., Safety aspect of railway and road tunnel: example of the Lötschberg railway tunnel and Mont-Blanc road tunnel, Tunnelling and Underground Space Technology, 17(2), pp. 153-158, 2002. Jones, W.W., A review of compartment fire models, NBSIR 83-2684, pp. 1-38, 1983. Dembsey, N.A., Pagni, P.J. & Williamson, R.B., Compartment fire experiments: comparison with models, Fire Safety Journal, 25(3), pp. 187-227, 1995. Peacock, R.D., Reneke, P.A., Forney, C.L. & Kostreva, M.M., Issues in evaluation of complex fire models, Fire Safety Journal, 30(2), pp. 103-36, 1998. Yang, K.T., Recent development in field modelling of compartment fires, JSME International Journal Series B, 37(4), pp. 702-17, 1994. Beard, A.N., Fire models and design, Fire Safety Journal, 28(2), pp. 117-38, 1997. Cox, G. & Kumar, S., Fielding modelling of fire in forced ventilated enclosures, Combustion Science Technology, 52(1), pp. 7-23, 1987. D’Anna, A. & Kent, J., Modelling of enclosed fires and smoke dispersion by computational fluid dynamics, La rivista dei combustibili, 56(3), pp. 95-103, 2002. Xue, H., Ho, J.C. & Cheng, Y.M., Comparison of different combustion models in enclosure fire simulation, Fire Safety Journal, 36(1), pp. 37-54, 2001. Miles, S.D., Kumar, S. & Andrews, R.D., Validation of a CFD model for fires in the Memorial Tunnel, Proc. of the 1st Int. Conf. on Tunnel Fires and One Day Seminar on Escape from Tunnels, Lyon, France, pp. 159-168, 5-7 May 1999. Novozhilov, V., Moghtaderi, B., Fletcher, D.P. & Kent, J.H., Computational Fluid Dynamics Modelling of Wood Combustion, Fire Safety Journal, 27(1), pp.69-84, 1996. Novozhilov, V., Harvie, D.J.E., Green, A.R. & Kent, J.H., A Computational Fluid Dynamic Model of Fire Burning Rate and Extinction by Water Sprinkler, Combustion Science Technology, 123(1-6), pp. 227-245, 1997. Ingason, H. & Lönnermark, A., Large Scale Fire Tests in the Runehamar tunnel – Heat Release Rate (HRR), Proc. of the Int. Symp. on Catastrophic Tunnel Fires, Borås, Sweden, pp. 81-92, 20-21 November 2003. Lönnermark, A. & Ingason, H., Large Scale Fire Tests in the Runehamar tunnel - Gas temperature and radiation, Proc. of the Int. Symp. on WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
[17] [18] [19] [20] [21] [22] [23]
309
Catastrophic Tunnel Fires, Borås, Sweden, pp. 93-103, 20-21 November 2003. Lemaire, T., Runehamar tunnel fire tests: Radiation, fire spread and backlayering, Proc. of the Int. Symp. on Catastrophic Tunnel Fires, Borås, Sweden, pp. 105-116, 20-21 November 2003. Lönnermark, A. & Ingason, H., Large Scale Fire Tests in the Runehamar tunnel, Proc. of the Int. Symp. on Catastrophic Tunnel Fires, Borås, Sweden, Interactive CD-ROM, 20-21 November 2003. Kent, J.H., AIR 3.4 Manual. Mégret, O., Vauquelin, O., A model to evaluate tunnel fire characteristics, Fire Safety Journal, 34(4), pp. 393-401, 2000. Woodburn, P.J. & Britter, R.E., CFD Simulation of a Tunnel Fire - Part I, Fire Safety Journal, 26(1), pp. 35-62, 1996. Tewarson, A., Generation of heat and chemical compounds in fires (Section 3, Chapter 4), The SFPE Handbook of Fire Protection Engineering, 2nd edition, SFPE and NFPA, 1995. Meo, M.G., Simulazione numerica di incendi in galleria, Thesis in Chemical Engineering, University of Salerno, 2004.
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Advanced Computational Methods in Heat Transfer IX
311
Conservative averaging as an approximate method for solution of some direct and inverse heat transfer problems A. Buikis Institute of Mathematics, Latvian Academy of Sciences and University of Latvia, Latvia
Abstract The conservative averaging method was developed as an approximate analytical and/or numerical method for solving partial differential equation or its system with piece-wise constant (continuous) coefficients. The usage of this approximate method for separate relatively thin sub-domain or/and for subdomain with al large heat conduction coefficient leads to a reduction of domain in which the solution must be found. To apply this method for all sub-domains of layered media, a special type of spline was constructed: the integral averaged values interpolating parabolic spline. The usage of this spline allows diminishing the dimensions of initial problem per one. It is important that in all cases the original problem with discontinuous coefficients from Rn+1 transforms to problem with continuous coefficients in Rn. A method of conservative averaging for ill-posed inverse problems in some cases allows transforming them to wellposed inverse problems. Keywords: heat transfer, piecewise constant (continuous) coefficients, conservative averaging, non-classical conditions, integral spline, mesh (dimension) reduction, direct problem, inverse problem.
1
Introduction
By modeling practically interesting processes, e.g. heat transfer processes in non-homogeneous media, very often we need to consider the situation, when the medium has an organized structure, i.e. it is not fully chaotic. For example it often has a layered structure. In addition some of these layers are relatively thin in comparison with adjacent layers and have strongly different physical properties. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060311
312 Advanced Computational Methods in Heat Transfer IX Mathematically speaking such a situation can be described by a partial differential equation (or its system) with piecewise constant/continuous coefficients, i.e. the domain in which the solution is defined, consists of several sub-domains. In each sub-domain the solution of the partial differential equation can be considered in the classical sense: the solution has continuous all highest partial derivations into the sub-domain. But it is not true on the contact surfaces S of adjacent layers: on these surfaces special additional conditions, which in the literature are often called conjugation or junction conditions, have been formulated:
∂T = 0 . Here [T ] = T ∂n
[T ] = 0, k
S+
−T
S−
is the difference
of one-sided limit values (jump) of function T on surface S. In the case of non-ideal contact one of the conditions is the continuity of heat flux (energy conservation). The second junction or conjugation condition is usually written without detailed deduction. E.g., [1] mentions “temperature has discontinuity when passing through the boundary of non-ideal contact, with the height of the step being proportional to the heat flow, i.e.
k ∂T , ( x, y , z ) ∈ S , (1) α ∂n where the coefficient of contact heat transfer α is associated with the contact [T ] =
conditions”. Similarly, in [1] the so-called “concentrated heat capacity” condition relation on the surface S is written as follows
∂T ∂T (2) k ∂n = cs ρ ∂t . Additionally, it commented that: “ cs is the lumped heat capacity of the contact”. In this paper we will show how these and other conditions and their generalizations can be obtained by our original method of conservative averaging (CAM) [2, 3]. This approach allows us to eliminate some separate sub-domains and reduce partial differential equations for these sub-domains to boundary conditions. This means that we reduce the definitions domain of problem for its analytical or numerical solution [4]. This means we can consider this approach using the mesh reduction method. To apply CAM procedure for several layers, it was necessary to construct a special type of spline: the integral averaged values interpolating parabolic spline [5]. In [6] such an approximation for convectionconduction heat transfer in a layered system was demonstrated. For the approximation of boundary layers we introduced rational spline [7] and in [8] we showed its effectiveness. It is important that in all the cases the original problem with discontinuous coefficients transforms from problem a in Rn+1 to problem a in Rn with continuous coefficients. This method for ill-posed inverse problems in some cases allows transforming them to well-posed inverse problems, e.g. [9, 10].
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Advanced Computational Methods in Heat Transfer IX
2
313
The conservative averaging method for separate boundary sub-domain
Let us assume that in suitable Cartesian coordinates x ∈ R, y ∈ R the domain n
of definition D ⊂ R
n +1
of the solution is represented as consisting of two
sub-domains: D = G ∪ G0 . Here the finite or infinite sub-domain G has form
G = {( x, y ) x ∈ (0, ∆( y )), y ∈ Γ( x)} and
the
second
sub-domain
G0 = {( x, y ) x ∈ (−δ , 0), y ∈ Γ 0 } is a cylinder of finite height δ . The base of
Γ 0 ∈ R n is the bounded/unbounded domain in R n under the condition Γ 0 ⊆ Γ(0) . The differential equation in G0 has the form: the cylinder
∂ ∂U 0 k0 + L0 (U 0 ) = − F0 ( x, y ), ∂x ∂x where L0 is the linear differential operator with respect to the argument
(3)
y and
coefficient k0 = k0 ( y ) . The differential equation in G has a similar form:
∂ ∂U k ∂x ∂x
(4) + L(U ) = − F ( x, y ) , but the operator L with respect to the vector argument y now in the general case can be non-linear and the heat conduction coefficient is k = k ( x, y ) . It should be mentioned that one of the y vector components may be time t , thus the equations (3) and (4) allow describing both the steady-state and transient
Γ 0+ the part of the hyper-plane x = 0 :
processes. Let us denote by
Γ 0+ = G ∩ G0 . On this surface the conjugations conditions must be fulfilled: U0
k0
x =−0
∂U 0 ∂x
=U
x =+0
=k x =−0
,
∂U ∂x
(5) (6) x =+0
On the second base Γ 0 = { x = −δ , y ∈ Γ 0 } of the cylinder, a typical boundary −
conditions for the heat transfer processes is given (we will specify it later). On the other hand we will not specify the conditions of the remaining parts of both sub-domains, because their form is not substantial for the description of the method. We assume that the original problem (3)–(6) with all the necessary boundary or/and initial conditions have a unique and stable solution. In particular it should be emphasized that under the solution of this problem we imply a solution in a WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
314 Advanced Computational Methods in Heat Transfer IX slightly revised classical sense: 1) it is continuous in closure of definition domain D ; 2) the solution has all necessary continuous highest derivations in open sub-domains G0 and G ; 3) the first derivative with respect to argument x of the solution has bounded one-sided limit values which fulfil the second junction condition (6) on surface
Γ 0+ ; 4) it fulfils all additional conditions on
boundary of definition domain. We start the description of the CAM by introducing the integral averaged value of the function U 0 ( x, y ) of the solution of the problem (3)–(6) on G 0 : 0
u0 ( y ) = δ −1 ∫ U 0 ( x, y )dx .
(7)
−δ
The integration of the differential equation (3) over the interval x ∈ (−δ , 0) and the utilization of second junction condition give us the basic relation
k
∂u ∂x
− k0 x =+0
∂U 0 ∂x
+ δ L0 (u0 ) = −δ f 0 ( y ) .
(8)
x =−δ
Firstly, we denote the solution on sub-domain G by
u ( x, y ) instead of the
U ( x, y ) because the solution of the new statement of the problem in general will differ from initial solution. Secondly, we denote by f 0 ( y ) according to (7) the averaged value of source function F0 ( x, y ) . A further
function
transformation of the basic relation (8) depends on the type of boundary −
conditions on Γ 0 . We will consider the boundary condition of second type (other types of boundary condition were considered in [2, 3]):
− k0
∂U 0 ∂x
= ϕ 0 ( y) .
(9)
x =−δ
Assuming the linear approximation for solution U 0 ( x, y ) for each fixed
y (nevertheless it may be different for various y !) we easily find the representation for u0 ( y ) by means of the junction condition (5) and condition (9):
u0 ( y ) = u (0, y ) +
δ ϕ 0 ( y) 2 k0 ( y )
.
It remains to then insert this expression in (8), to use boundary condition (9) and +
we have obtained the non-classical boundary condition on boundary Γ 0 :
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Advanced Computational Methods in Heat Transfer IX
0 δ2 ϕ0 ∂u + δ L0 (u ) = − ϕ ( y ) + δ f 0 ( y ) + k L0 ( ) 2 k0 ∂x for the solution u ( x, y ) of the equation (4) on the reduced domain G : ∂ ∂u k + L(u ) = − F ( x, y ) . ∂x ∂x
(10)
(11)
Of course all additional conditions of the original problem on the remaining part of boundary of domain G must be added to main differential equation (11) and +
new boundary condition (10) on the surface Γ 0 . When this new problem is solved it is easy to obtain a posteriori error estimation ∆U 0 ( x, y ) for the linearly approximate (relatively to x ) solution U 0 ( x, y ) in following form:
∆U 0 ( x, y ) ≤
x × k0
δ ϕ0 ∂u (0, y ) δ 0 × k + ϕ ( y ) + L0 (u0 ) + L0 ( ) + δ 2 f 0 ( y ) ∂x 2 k0 2 We can increase the order of approximation for solution U 0 ( x, y ) up to two by means of both junction conditions and condition (9). We get the representation:
U 0 ( x, y ) = u (0, y ) +
k x ∂u (0, y ) x 2ϕ 0 ( y ) (1 + ) x + . k0 2δ ∂x 2δ k0
(12)
We obtain from (12) following expression for the averaged integral value of the function U 0 ( x, y ) :
u0 ( y ) = u (0, y ) +
δ ϕ 0 ( y)
3k0
2
−k
∂u (0, y ) . ∂x
This means that the new boundary condition on surface
(13)
Γ 0+ consists of two
equations. One of them is a consequence of equation (8):
k
∂u ∂x
x =+0
+ δ L0 (u0 ) = − ϕ 0 ( y ) + δ f 0 ( y ) ,
(14)
the second one is expression (13). These two equations allow finding two unknown functions u0 ( y ) and
u (0, y ) on the boundary Γ 0+ . So the new
problem consists of the main differential equation (11) together with a system of two non-standard boundary conditions (13), (14). The definition domain for the
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316 Advanced Computational Methods in Heat Transfer IX function
u ( x, y ) is sub-domain G ; the function u0 ( y ) is given only on the +
surface Γ 0 . So the CAM in this situation can be interpreted as the mesh reduction method. We finish this part of paper with some remarks. Quite often after the application of CAM with piecewise constant coefficients it is possible to solve the problem analytically. Our experience concerning the numerical methods for other problems with non-classical boundary conditions is as follows. Firstly, we have solved a large number of important practical problems (mostly by the method of finite difference). Secondly, in practice and in theory the stability criterion for classical boundary condition (9) is more stiff than for the non-classical boundary condition (10) (or for the system (13) and (14)). Unfortunately still we haven’t succeeded in proving the solvability of problems with non-classical additional conditions in general (for general operators L0 and L ).
3 The conservative averaging method for separate inner sub-domain (layer) Now we will consider the definition domain
D which consists of three
sub-domains: D = G ∪ G0 ∪ G1 , where finite or infinite sub-domain G1 has the form
G1 = {( x, y ) x ∈ (−∆1 ( y ), −δ ), y ∈ Γ1 ( x)} . We add to equations
(3) and (4) in sub-domain G1 equation
∂ ∂U1 k1 + L1 (U1 ) = − F1 ( x, y ) ∂x ∂x
(15)
and to conditions (5) and (6) add further junction conditions on surface
U1 x =−δ − 0 = U 0
x =− δ + 0
, k1
∂U1 ∂x
= k0 x =−δ − 0
∂U 0 ∂x
Γ0− : .
(16)
x =− δ + 0
The basic relation for this problem looks as follows:
k
∂u ∂x
− k0 x =+0
∂u0 ∂x
+ δ L0 (u0 ) = −δ f 0 ( y ) .
(17)
x =−δ + 0
To exclude thin interlayer – cylinder G0 – and to obtain the new non-standard junction conditions on surface
Γ 0+ we shift the sub-domain G1 to
the right: x 6 x + δ and additionally assume the linear approximation for solution U 0 ( x, y ) for each fixed y . Then from (6) and (16) immediately +
follows the first junction condition on Γ 0 :
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Advanced Computational Methods in Heat Transfer IX
k1
∂u1 ∂x
=k x =−0
∂u ∂x
.
(18)
x =+0
Γ0+ follows from (17) by taking into consideration linearity of function U 0 ( x, y ) : The second junction condition on
k
∂u ∂x
− x =+0
k0
δ
[u ] +
δ 2
L0 (u + u1 ) = −δ f 0 ( y ) .
(19)
If we neglect in (19) both the operator L0 and source term then this condition reduces to the non-ideal contact condition. In addition, we have explicated the expression for the coefficient of contact heat transfer α in (1) through physical and geometrical properties of the interlayer:
α=
capacity”
the
condition
(2)
follows
from
k0
δ
. The “concentrated heat
basic
relation
(17)
with
∂ , f 0 ≡ 0 and equality (18). The explicit expression for ∂t the lumped heat capacity cs derives from (17) and is as cs = c0δ .
operator
4
L0 = c0 ρ 0
The coefficient inverse one-dimensional problem for twolayer system
One of most popular experimental methods for thermal physical properties of homogeneous media (solids, fluids and gases) with low electrical conductivity is the transient hot strip (THS) method developed by Gustafsson [11]. Mathematically this method was formulated as coefficient inverse heat equation with constant coefficients for two-dimensional semi-bounded zone. In our publications [9, 10] we generalize this ill-posed problem for two-layers, solving it by use of Green’s function and reducing it to a system of two transcendent equations. Nevertheless, the numerical solution of system of transcendental equations is an ill-posed problem. It would be important to offer some well-posed method for finding approximate values for coefficients as initial data for the iteration process. In this section we propose such an approach based on conservative averaging. The one-dimensional model for the THS method can be formulated as follows:
∂U 0 ∂ 2U 0 = k0 + f 0δ (0), 0 < x < H 0 , 0 < t , ∂t ∂x 2 ∂U ∂ 2U c1 1 = k1 21 , H 0 < x < H 0 + H1 = H , 0 < t ∂t ∂x c0
with homogeneous second type boundary conditions
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(20) (21)
318 Advanced Computational Methods in Heat Transfer IX
k0
∂U 0 ∂x
= 0, x =0
with conjugation conditions
U0
x = H 0 −0
= U1
k1
x = H0 +0
, k0
∂U 0 ∂x
∂U1 ∂x
x = H 0 −0
=0, x=H
= k1
∂U1 ∂x
x= H 0 +0
and with homogeneous initial conditions
U0
t =0
= 0, U1
t =0
=0.
Additional information is given:
U0
x =0
= T (t )
(22)
in the form T (tk ) = Tk , tk = k ∆t , k = 0, N with N >> 1 . The aim of the solution of the inverse problem is to find the unknown constants c0 , c1 and functions U 0 ( x, t ) and U1 ( x, t ) . Let us introduce the integral averaged values of functions U 0 ( x, t ) and
U1 ( x, t ) : 1 u0 (t ) = H0
H0
∫U
H
0
1 u1 (t ) = U1 ( x, t )dx . H1 H∫0
( x, t )dx ,
0
(23)
Further, we will approximate both unknown functions U 0 ( x, t ), U1 ( x, t ) according to argument x with expressions, which fulfil the boundary and conjugation conditions and equalities (23). Finally we have: x
x
− eG0 (e H 0 + e H 0 + e −1 − e)v(t ) , U 0 ( x, t ) = u0 (t ) − 2(G0 + G1 )
U1 ( x, t ) = u1 (t ) +
eG1 (e 2(G0 + G1 )
where v (t ) = u0 (t ) − u1 (t ),
x−H H1
+e
H −x H1
(24)
+ e −1 − e)v(t ) ,
(25)
Gi = H i ki−1 , i = 0,1 .
Now we integrate the main equations (20) and (21):
c0
duo ∂U 0 = Go−1 dt ∂x
+ f0 , x = H0 −0
c1
du1 ∂U = G1−1 1 dt ∂x
Finally we receive a system of two ordinary differential equations:
du0 du = −α v(t ) + f 0 H 0 , β1 1 = α v(t ), dt dt with homogeneous initial conditions u0 (t ) = u1 (t ) = 0 .
β0
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. x = H0 +0
Advanced Computational Methods in Heat Transfer IX
Here
α=
319
e2 − 1 and β i = ci H i , i = 0,1 . The solution for v(t ) is: 2(G0 + G1 )
v(t ) = f 0 H 0 ( β 0γ ) −1 (1 − e−γ t ) , γ = αδ , δ = β 0−1 + β1−1 . And the solution for u0 (t ) :
u0 (t ) = f 0 H 0 β 0−1 ( β 0γδ ) −1 (1 − e−γ t ) + (1 − ( β 0γ ) −1 ) t .
We have from additional information (22):
Tk = u0 (tk ) + α 0 v(tk ), α 0 = 2−1 (e2 − 2e − 1)G0 (G0 + G1 )−1 . The first finite difference may be written as follows:
∆Tk = f 0 H 0 ( β 0γ ) −1 (α 0 + ( β 0δ ) −1 ) (1 − e −γ∆t ) e−γ tk −1 + (γ − ( β 0 ) −1 ) ∆t
and we have similar expression for the second finite difference:
∆ 2Tk = − f 0 H 0 ( β 0γ ) −1 (α 0 + ( β 0γ ) −1 ) (1 − e−γ∆t ) e−γ tk −1 . 2
Finally, we obtain expression for the sum of the unknown coefficients:
γ=
1 ∆ 2Tk −1 ln 2 . ∆t ∆ Tk
(26)
Evidently we have found the unique solution of the inverse problem, but it is easy to see that solution (26) strongly depends on the errors of the measurements and in this sense we have unstable algorithm. But we can propose the modification of this algorithm by summing up the sub-sequences of the measurement data.
References [1] [2] [3] [4]
[5]
Samarskii, A.A., Vabishchevich, P.N., Computational Heat Transfer. Vol.1, Mathematical Modelling. John Wiley&Sons Ltd., 1995. Buikis, A. Aufgabenstellung und Loesung einer Klasse von Problemen der mathematischen Physik mit nichtklassischen Zusatzbedingungen. Rostock. Math. Kolloq., 1984, 25, pp. 53-62. (In German.) Buikis, A., The change of formulation of problems of mathematical physics with discontinuous coefficients in compound domains. The Electronic Modelling, 1986, Vol. 8, No. 6, pp. 81-83. (In Russian.) Buikis, A., Buike, M., Guseinov, Sh., Analytical two-dimensional solutions for heat transfer in system with rectangular fin. Advanced Computational Methods in Heat Transfer VIII. WIT Press, 2004, pp.35 – 44. Buikis, A., Interpolation of integral averaged values of piecewise smooth function by means of parabolic spline. Latvian Mathematical Yearbook, 1985, No. 29, pp. 194-197. (In Russian.)
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
320 Advanced Computational Methods in Heat Transfer IX [6] [7] [8] [9]
[10]
[11]
Buikis, A., Ulanova, N., Analytically-numerical method for temperature fields in multilayered system. Advanced Computational Methods in Heat Transfer VI. WIT Press, 2000, pp. 445-453. Buikis, A., Rational integral values interpolating spline and its properties. Latvian Mathematical Yearbook, 1988, No.32, pp. 173-182. (In Russian.) Buikis, A., Jegorovs, J., Application of the conservative averaging for the filtration problem with large velocity. Mathematical Modelling and Analysis, 2001, Vol.6, Nr.2, pp. 251-261. Guseinov, Sh., Buikis, A., Inverse heat transport problems for coefficients in two-layer domains and methods for their solution. Mathematical Modelling and Analysis, 2002, Vol. 7, N 2, Vilnius, ”Technika”, pp. 217228. Buikis, A., Guseinov, Sh., Conservative averaging method for solutions of inverse problems of mathematical physics. Progress in Industrial Mathematics at ECMI 2002. A. Buikis, R. Ciegis, A. D. Fitt (Eds.), Springer, 2004, p. 241-246. Gustafsson, S. L., Karawacki, E., Khan, M. N., Transient hot-strip method for simultaneously measuring thermal conductivity and diffusivity of solids and fluids. J. Phys. D: Appl. Phys., 1979, Vol. 12, pp. 1411-1421.
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Section 5 Heat exchangers and equipment
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323
Estimating number of shells and determining the log mean temperature difference correction factor of shell and tube heat exchangers S. K. Bhatti1, Ch. M. Krishna2, Ch. Vundru3, M. L. Neelapu4 & I. N. Niranjan Kumar5 Department of Mechanical and Marine Engineering, Andhra University, Visakhapatnam, India
Abstract This problem aims at developing a practical and computational tool for LMTD Correction Factor ‘F’ and approximate number of shells in the process of designing a shell and tube heat exchanger. In order to analyze the performance of heat exchanger the available approaches are LMTD and effectiveness ε . Since the expressions for Log Mean Temperature Difference (LMTD) Correction Factor ‘F’ are difficult to evaluate, the traditional analysis methods rely on the heat exchanger charts. Apart from being applicable only to a particular heat exchanger, these charts are highly nonlinear and expressed in terms of the non-dimensional parameters P and R. The LMTD correction factor ‘F’ charts are particularly difficult to read in the steep regions. In this study, a simplified analytical approach for determining LMTD correction factor ‘F’ in terms of two non-dimensional parameters W and R is presented. This new approach results in a single general algebraic equation for determining the LMTD correction factor of multi-pass shell and tube heat exchangers with any number of shell passes and even number of tube passes per shell. The correction factor thus obtained is a function of number of shells N and terminal temperatures of the heat exchanger. The value of ‘F’ obtained from this new equation is in agreement with the value obtained from the graph. For a given correction factor ‘F’ and terminal temperatures of the heat exchanger, an equation for the estimation of number of shells has been derived, which is a function of correction factor ‘F’ and terminal temperatures of the heat exchanger. This eliminates all the former cumbersome iterative methods in which values are to be assumed for some factors. Keywords: LMTD correction factor, shell and tube heat exchange, multipass heat exchanger. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060321
324 Advanced Computational Methods in Heat Transfer IX
1
Introduction
Heat exchangers are the devices to recover heat between two process streams. The use of heat exchangers is extensive in power plants, refrigeration and air-conditioning systems, space applications, chemical, nuclear, petrochemical, cryogenic industries etc., Heat exchangers appear in variety of shapes and sizes. It can be as huge as a power plant condenser transferring hundreds of megawatts of heat or as tiny as an electronic chip cooler which transfers only a few watts of thermal energy. 1.1 Multi shell and tube heat exchangers In this type of exchangers, the fluid flows in the tubes is known as tube side fluid and the other fluid which flows outside of the tubes in a shell is known as shell side fluid. The tube bundle is arranged inside the shell. To make shell side fluid flow at a higher turbulence and to induce higher heat transfer, baffles are placed in the shell. In order to increase the heat exchange rate between two streams, industries generally use multi shell heat exchangers. If more than one shell is used in a heat exchanger it is called a multi shell heat exchanger. Shell fluid enters the shell, flows across the tube bundle and leaves the shell. This may not serve the purpose of transferring large quantity of heat. But it can be possible in single shell heat exchanger either by increasing the length of the tubes or by increasing the number of tubes. Both the above arrangements are not practically possible due to higher-pressure drops and cost problems. In the multi shell and tube exchangers the shells are connected either in series or in parallel. Fluid enters the shell flows across it and enters into the next shell. The flow conditions are neither parallel flow nor counter flow type. In these exchangers flow is a combination of both parallel and counter flow types. Hence a correction factor ‘F’ must be introduced in the general heat equation and the equation is modified as Q = UA (F) LMTD. This correction factor ‘F’ depends on the number of shells of the heat exchanger and on the terminal temperatures of the heat exchanger. As far as design aspects of heat exchanger are considered, the number of shells in a multi shell-pass heat exchanger is having prime importance. For a given design correction factor F, one has to estimate the number of shells in order to compute the required amount of heat transfer. So far some iterative methods are available for the estimation of number of shells, which are very laborious methods. Over all heat transfer coefficient ‘U’ can be estimated from correlated heat transfer coefficients and LMTD is calculated from terminal temperatures of the heat exchanger. But LMTD obtained from terminal temperatures in case of multi pass heat exchangers will not give true temperature difference. Hence to determine the actual mean temperature difference the LMTD of a counter flow heat exchanger with the same terminal temperatures as those of the exchanger under consideration is multiplied by a correction factor, F ≤ 1 . It provides the effective temperature difference of the heat exchanger under consideration. It is a measure of heat exchanger’s departure from the ideal WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
325
behaviour of a counter flow heat exchanger having the same terminal temperatures.
Figure 1:
Shell and tube heat exchanger.
2 Theory Wide literature is available on the calculation of Log Mean Temperature Difference correction factor used in the analysis of heat exchangers. Bowman et al. [2] in their pioneering paper compiled the available data, and presented a series of equations and charts to determine F for different heat exchangers, including multi-shell and tube ones. They expressed the correction factor F, in terms of two non-dimensional variables P and R which are defined as P =
t 2 − t1 T1 − t1
and R =
T1 − T2 t 2 − t1
They provided equations for correction factor F, for one shell-two tube pass exchanger and two-shell – four tube pass exchanger. They also provided charts for determining the correction factor for two, four, six shell passes and multiple of tube passes. They have given an equation to calculate P for an N shell-side and 2N tube-side passes (PN,2N) in terms of P for a one-shell-side and two-tube side passes (P1,2) and R. Fakheri [3] in his recent paper introduced two new non-dimensional variables, which are defined as
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326 Advanced Computational Methods in Heat Transfer IX T1 − t 2 (T1 − T2 )2 + (t 2 − t1 )2 φ = 2[(T1 + T2 ) − ( t1 + t 2 )] T2 − t 1 He used these two variables for heat exchanger analysis and presented correlations for the determination of F in these two parameters for single shell and tube heat exchanger. Kern [10] provides correction factor charts for different number of shells and even number of tube passes. He presented the correction factor F, as a function of two variables R and S, which depends on the inlet and exit temperatures of the heat exchanger of both the fluids. Roetzel and Nicole [11] recognized the potential usefulness of explicit representations of LMTD Correction factors in developing computerized packages for heat exchanger design. Taborek [12] interpreted the above parameters P and R as the capacity ratio and a measure of heat exchanger effectiveness respectively. Tucker [13] recently discovered an error in one of the charts for cross flow heat exchangers correction factor. He found for a cross flow heat exchanger with a capacity ratio of unity (R=1) and P=0.6 the LMTD correction factor F read off the chart is 3 percent and extreme combination of low values of R and values of P approaching unity the error is considerably greater. Wales [14] defined correction factor by introducing a new parameter G as T2 − t 2 G = T1 − t 1 and expressed F as a function of G and R. ρ=
3 Analysis for the determination of LMTD correction factor Bowman et al. [2] developed an equation to determine correction Factor ‘F’, for 2 shell and 4 tube pass heat exchanger, Besides that they also presented series of charts for the determination of the correction factors for various arrangements of shell and even number of tube passes. To express F, they have chosen two nondimensional variables, P =
t 2 − t1 T1 − t 1
and
R =
T1 − T2 t 2 − t1
,
The above two variables are interpreted as the capacity ratio and a measure of heat exchanger effectiveness. Reading of charts for the determination of F, particularly in the steep regions of the curves reduces the accuracy. The present analysis is to derive a single expression for the determination of correction factor F, for any shell and multiple of tube passes. The breakthrough paper by Bowman et al. [2] provided the following expression to determine the correction factor for shell and tube heat exchangers with one shell and two tube pass exchangers as
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Advanced Computational Methods in Heat Transfer IX
1 − P1, 2 ln 1 − P1,2 R
2
R +1
F1− 2 =
(R − 1)
ln
2 −1 − R + R2 + 1 P1, 2 2 2 −1− R − R +1 P1, 2
327
(3.1)
F2−4 = 1 − P2,4 ln 1 − P2,4R
2
R +1 2(R − 1)
2 2 (1 − P2,4 )(1 − P2,4R) + R 2 + 1 −1− R + P2,4 P2,4 ln 2 2 2 (1 − P2,4 )(1 − P2,4R) − R + 1 −1− R + P2,4 P 2,4
(3.2)
Bowman et al. [2] also showed that Eq. (3.1) can also be used for N shell 2N tube pass exchanger provided P1,2 is related to the P for a multi-shell multi-tube heat exchanger (PN,2N) by the following expression. N
1 − P1,2 R 1− 1 − P1,2 PN,2 N = N 1 − P1,2 R R− 1 − P1,2
(3.3)
By rearranging the Eq. (3.2) 1 − P1,2 R 1 − P1, 2
N
1 − PN ,2 N R = 1 − PN, 2 N
(3.4)
Solving for P1,2 1
1 − PN ,2 N R N 1− 1 − PN ,2 N P1,2 = 1 1 − PN ,2 N R N R− 1 − PN ,2 N Substituting Eq. (3.4) into Eq. (3.1) and after simplification
(3.5)
FN ,2 N = R 2 + 1 1 − PN,2 N R ln R −1 1 − PN ,2 N
1
N
1 1 N R2 +1 R 2 + 1 1 − PN,2 N R N 1 − PN, 2 N R 1 − + + R −1 R − 1 1 − PN ,2 N 1 − PN,2 N ln 1 1 N R2 +1 R 2 + 1 1 − PN,2 N R N 1 − PN, 2 N R − + 1 + 1 − P R −1 R − 1 1 − PN ,2 N N , 2 N
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(3.6)
328 Advanced Computational Methods in Heat Transfer IX Substituting W of Eq. (3.6) in place of FN, 2 N =
1 − PN , 2 N R 1 − PN , 2 N
Γ ln[W ]
1
N
1 + [W ]1 N − Γ + Γ[W ]1 N ln 1 + [W ]1 N + Γ − Γ[W ]1 N
(3.7)
Eq. (3.7) is used to determine the LMTD correction factor in case N shells 2N tube pass heat exchanger. The correction factor is a function of two variables W, Γ which are in turn functions of the terminal temperatures of the heat exchanger. Special Case For R=1 Eq. (3.1) becomes indeterminate. Lim F1− 2 R →1
1 − P1, 2 ln 1 − P1,2 R R +1 × Lim = Lim R →1 2 R 1 → ( R − 1 ) −1− R + R2 +1 P1, 2 ln 2 2 −1− R − R +1 P1, 2 P1, 2 2 1 − P1, 2 F1, 2 = 2 −2+ 2 P1, 2 ln 2 −2− 2 P1, 2 2
(
)
(3.8)
By rearranging Eq. (3.2) PN , 2 N =
(1 − P1,2 )N − (1 − P1,2R )N R (1 − P1, 2 )N − (1 − P1, 2 R )N (1 − P1,2 )N − (1 − P1,2R )N R →1 R (1 − P )N − (1 − P R )N 1, 2 1, 2
Lim PN, 2 N = Lim R →1
PN,2 N =
P1,2 =
( )
N P1,2
( )
(1 − P1,2 ) + N P1,2
PN.2 N N − NPN,2 N + PN,2 N
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(3.9)
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329
By substituting Eq. (3.9) into Eq. (3.1) FN, 2 N = 2
PN, 2 N N − NPN , 2 N
FN,2 N = 2 W1
4
1 N − NP N,2 N PN , 2 N ln N − NPN , 2 N PN , 2 N
1 + 2 1 − 2
1 1 1 1+ 2 ln W 1 1 1− 2 W
(3.10)
Analysis for the determination of number of shells
Industries require multi shell heat exchangers in order to carry out the required heat transfer processes. The designer has to design a heat exchanger for a specified design value of ‘F’. When multi shell arrangements are necessary it requires estimating the number of shells that are required. Several methods are available so far for determining the number of shells. But all these are iterative methods and values are to be assumed for some factors. To select the values for these factors there is no definite criterion. Traditionally a tedious trial and error procedure is used in which number of shells is changed continuously and a new F value is determined until a required solution is arrived at. Ahmad et al. [1] gave an analytical expression for calculating the number of shells directly. N=
1 − PR ln 1− P R + 1 + R 2 − 1 − 2RX P ln 2 R + 1 + R − 1 − 2X P
(4.1]
In using Eq. (4.1) the designer has to assume a value for Xp to determine number of shells. Then F can be determined and Xp should be modified to get the required value of F, in an iterative process. Gulyani [6] proposed another equation to determine number of shells. He used another parameter Y. N=
1 − GR ln 1− G R 2 + 1 (R + 1 − RY ) − (R + 1)(R − 1 − RY ) ln 2 R + 1 (R + 1 − Y ) − (R + 1)(R − 1 − Y )
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(4.2]
330 Advanced Computational Methods in Heat Transfer IX In Eq. (4.2) also some value is attributed to (Y must be assumed some value) determine the number of shells. In this process also Y should be changed iteratively to determine the desired F. Fakheri [4] developed a general expression for determining the correction factor F, in terms of number of shells (N) and non-dimensional parameters φ, λ N , λ 1 which depends on the inlet and exit temperatures of both the fluids. This single expression, which is a simple form, can be used to determine correction factor of multi-pass shell and tube heat exchangers with any number of shell passes and even number of tube passes per shell. The equation for correction Factor, F developed as φ λN N λ1 F= φ λN 1 + 2 N λ 1 ln 1 − 2 φ λ N N λ1 4
1 λN
(4.3)
By using Eq. (4.3) an approximate expression for the determination of number of shells can be arrived without using iterative method. Equation for number of shells: Eq. (4.3) can be rearranged as φ λN N λ1 λNF = φ λN 1 + 2 N λ 1 ln φ λN 1 − 2 N λ 1 4
φ λN N λ1
By replacing
2
(Assuming
φ λN N λ1
2
(4.4)
of Eq. (4.4) by K
1+ K ≤ 1) and using Taylor’s series of ln 1− K 15 + 5K 2 + 3K 4 =
15 λNF
(4.5)
By converting Eq. (4.5) into polynomial form 1 5 2 4 51 − + K +K =0 λ NF 3 This equation has two imaginary and two real roots. The real roots are: φ λN 5 25 1 =± − + − 51 − N λ1 6 36 λ N F Leaving negative sign, as K is positive. 2
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(4.6)
(4.7)
Advanced Computational Methods in Heat Transfer IX
331
By rearranging the terms of Eq. (4.7) on condition that the values in the square root are positive results that F≤ 1.125
N=
λ φ N λ1
(4.8)
155 5 5 + − − λ 576 16 F 24 N
By using Eq. (4.8), N can be calculated for the given design value of F, φ can be calculated from terminal temperatures of the heat exchanger if • If the resultant N ≥ 2φ • If F ≤ 1.125 which is always true. The value calculated from Eq. (4.8 rounded to the nearest integer will give the number of shells for the specified correction factor. Eq. (4.8) is a direct and simple expression to determine the number of shells in a multi-shell tube heat exchanger. This equation avoids earlier iterative methods for the determination of number of shells to meet the design correction factor for a multi-shell tube heat exchanger.
5
Results and discussion
By considering the equations given in reference [2], a simplified expression for the correction factor F, in terms of non-dimensional parameters W , Γ and number of shells N of the heat exchanger has been given through mathematical analysis. This expression can be used accurately to determine the correction factor F, for heat exchanger having any number of shells. So far, expressions for determining the correction factor for one shell and two shell heat exchangers are available. But, for heat exchangers having more than two shells only charts are available. Charts are also drawn for heat exchangers having different shells. The correction factors obtained are almost equal to the values given in reference [2]. Comparison of results obtained from the new expression against the previous expression: For P=0.32, R=3 F2-4 = 0.56085 (using equation (3.2)) F2-4 = 0.56057 (using equation (3.7)) For P=0.7 and R= 1.2, F4-8= 0.8599 (using equation (3.2)) F4-8= 0.86 (using equation (3.7)) The errors between the two values are negligible. From the above two comparisons, the new expression obtained in the present work is in agreement with graphs which are already existing in the literature. Hence a single expression is sufficient for any number of shells of the heat exchanger. Using this new expression for N the designer can estimate number of shells by adjusting the value obtained to its nearest integer. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
332 Advanced Computational Methods in Heat Transfer IX
6
Conclusions
In this study, a new expression for determining Log Mean Temperature Difference Correction Factor F, in terms of two non-dimensional parameters P, R has been analyzed. This new expression presents a single equation to find out the correction factor F, exactly for any number of shells and even number of tube passes. Owing the development of new equation for F, reading of F from charts can be avoided. An approximate equation for estimating number of shells for a design value of F has also been analyzed. This equation avoids earlier iterative methods which are used to determine the number of shells of a multi-shell tube heat exchanger, to meet a given correction factor.
Nomenclature T1 T2 t1 t2 A F
= = = = = =
G
=
T2 − t 2 (dimensionless) T1 − t1
LMTD
=
Log Mean Temperature Difference, K
=
hot fluid inlet temperature, K hot fluid outlet temperature, K cold fluid inlet temperature, K cold fluid outlet temperature, K Surface, m2 Correction factor, dimensionless
∆T1 − ∆T2 ∆T ln( 1 ) ∆T2
φ λN N λ1
K
=
2
N
=
P
=
Number of shells t 2 − t1 T1 − t 1
R
=
XP
=
Y
=
W1
=
W
=
T1 − T2 t 2 − t1 parameter defined by Ahmed et al. defined by Eq. (4.1), dimensionless Parameter defined by Gulyani defined by Eq. (4.2), dimensionless PN,2 N N − NPN,2 N
1 − PN , 2 N R 1 − PN ,2 N
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Advanced Computational Methods in Heat Transfer IX
Γ
R2 + 1 R −1
=
Greek symbols
ε
=
φ
=
λ1
=
λN
ρ
=
=
Effectiveness of the heat exchanger
(T1 − T2 )2 + (t 2 − t1 )2 2[(T1 + T2 ) − ( t1 + t 2 )]
2
(ρ − 1) 1 (ρ + 1) ln ρ
1N ρ − 1 1 2 1 N ln ρ 1 N ρ + 1 T1 − t 2 T2 − t 1
Appendix Table 1:
Correction factor for a two-shell heat exchanger.
R = 3.0 P 0.321 0.320 0.280 0.200 0.150
R = 1.0 F 0.50 0.56 0.92 0.98 0.99
Table 2:
P 0.730 0.600 0.5 0.4 0.3
F 0.50 0.89 0.95 0.98 0.99
P 0.990 0.980 0.90 0.8 0.5
F 0.5 0.71 0.93 0.97 0.99
Correction factor for a three-shell heat exchanger.
R=3 P 0.332 0.30 0.20 0.16
R = 0.2
R=1 F 0.5 0.94 0.98 0.99
P 0.825 0.70 0.60 0.49
R = 0.4 F 0.5 0.88 0.95 0.99
P 0.986 0.98 0.80 0.60
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F 0.5 0.63 0.96 0.99
333
334 Advanced Computational Methods in Heat Transfer IX Table 3:
Correction factor for a four-shell heat exchanger.
R = 2.5 P 0.398 0.39 0.30 0.20
R = 1.0 F 0.5 0.87 0.98 0.99
Table 4:
P 0.842 0.80 0.66 0.50
Table 5:
From Graphs 0.93 0.92 0.84 0.70
Table 6:
From Graphs 0.93 0.87 0.85 0.76
0.72 0.75 0.8 0.85
0.4 0.43 0.45 0.47
Calculated F 0.9269 0.91 0.83 0.7527
From Graphs 0.92 0.88 0.82 0.75
P 0.5 0.52 0.55 0.57
Calculated F 0.926 0.9 0.832 0.7326
From Graphs 0.92 0.88 0.82 0.72
F values from the expression and from the graph.
R = 1.0 P
P
R = 1.6 Calculated F 0.934 0.878 0.842 0.78
0.42 0.45 0.46 0.47
F 0.5 0.96 0.98 0.99
F values from the expression and from the graph.
R = 2.0 P
P 0.962 0.8 0.7 0.6
R = 1.8 Calculated F 0.948 0.9345 0.851 0.685
0.3 0.32 0.35 0.37
F 0.50 0.8 0.96 0.99
F values from the expression and from the graph.
R = 2.5 P
R = 0.6
R = 0.8 Calculated F 0.97 0.96 0.91 0.82
From Graphs 0.978 0.966 0.93 0.81
P 0.7 0.78 0.82 0.89
Calculated F 0.99 0.97 0.96 0.92
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From Graphs 0.986 0.98 0.97 0.909
Advanced Computational Methods in Heat Transfer IX
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References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Ahmad, S., Linnhoff, B., and Smith, R., “Design of Multi-pass Heat Exchangers: an Alternative Approach”, Journal of Heat Transfer, 1988, Vol 110, PP. 304-309. Bowman, R.A., Mueller, A.C., and Nagel, W.M., “Mean Temperature Difference in Design” , Trans. ASME, 1940, Vol. 62, PP.283-294 Fakheri, A., “Log Mean Temperature Correction Factor: An alternative Representation”, proceedings of the International Mechanical Engineering Congress and Exposition, 2002, Nov. 17-22. Fakheri, A., “ An alternative approach for determining the log mean temperature difference correction factor and the number of shells of shell and tube heat exchangers”, Journal of Enhanced Heat Transfer, 2003, Vol 4, PP.407-420 Gulyani, B.B., and Mohanty, B., “Estimating Log Mean Temperature Difference in Multi-pass Exchangers”, Chemical Engineering, 1996, PP. 127-130. Gulyani, B.B., “Estimating number of shells in shell and tube heat exchangers: A new Approach based on Temperature Cross”, Journal of Heat Transfer, 2000, Vol 122, PP. 566-571. Holman, J.P., “Heat Transfer”, Mc Graw-Hill Book Company, 1992. Incropera, F.P., and David P. DeWitt., “Fundamentals of Heat and Mass Transfer”, John and Wiley publishers ISBN: 0-471-38650-2. Kays, W.M., and London, A.L., “Compact Heat Exchangers”, McGrawHill Book Company, 1958. Kern, D.Q., “Process Heat Transfer”, McGraw – Hill Company, Roetzel, W., and Nicole, F.J.l., “Mean Temperature Difference for Heat Exchanger Design – A General Approximate Explicit Equation”, Journal of Heat Transfer, 1975, Vol. 97, PP. 5-8. Taborek, J., “Hand Book of Heat Exchanger Design”, Hemisphere, 1990. Tucker, A.S., “The LMTD Correction Factor for Single-Pass Cross flow Heat Exchangers with Both Fluids Unmixed”, Journal of Heat Transfer, 1996, Vol 118, PP. 488-490. Wales, R.E., “Mean Temperature Difference in Heat Exchangers”, Chemical Engineering, 1981, Vol 88, PP. 77-81.
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The re-commissioned thermosyphon reboiler research facility in the Morton Laboratory A. Alane & P. J. Heggs School of Chemical Engineering and Analytical Science, The University of Manchester, UK
Abstract Vapour generation through boiling is one of the most ubiquitous industrial processes. Vertical thermosyphon reboilers are frequently used to generate vapour at the base of distillation columns. Present design methods have the major emphasis on the process side (boiling). The heating of the vertical boiler is decoupled from the system. Additionally, most academic research has considered a single tube flow arrangement with controlled and uniform electrical heating. However, many applications in distillation are now using sub-atmospheric pressure operation (higher thermodynamic efficiency, reduced energy consumption, prevents thermal degradation, cheaper materials of construction and safe operation). The literature does not contain many references to this mode of operation and existing design techniques do no adequately cover sub-atmospheric pressure operation. The thermosyphon reboiler research facility at the University of Manchester in the Morton Laboratory comprises 50 tubes of 3 m thermal lengths (19.86 mm ID) and 3 segmental baffles (TEMA E type shell and TEMA A type header) with steam condensing on the shell side, 2 large horizontal condensers with 106 and 196 tubes, both TEMA E type shells and B type headers. Vacuum is pulled on the process side and the shell side by means of two separate liquid ring pumps. The process fluid is water flowing in the tubes counter-current to the condensing steam in the shell side. The primary objective is to study the operation of the thermosyphon reboiler over the pressure range 0.1 bar – atmospheric. New additional instrumentation for temperature, pressure and flow measurements have been calibrated and installed. At the present time, the equipment is fully instrumented, re-insulated and has been successfully commissioned. This paper describes the equipment in detail, its configuration, instrumentation (control, safety and scientific), modifications from the previous arrangement (Emerson DeltaV computer control software and data logger, new instrumentation) and the effect on errors in the mass and energy calculations. A brief reference will be made to the complications encountered during commissioning and the solutions adopted. Keywords: boiling, vacuum, thermosyphon reboiler, research facility, coupled problem, instrumentation, commissioning. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060331
338 Advanced Computational Methods in Heat Transfer IX
1
Introduction
Of all reboiler types, thermosyphon reboilers are the most commonly used in the chemical industry Arneth and Stichlmair [1]. These heat transfer units are characterised by high heat transfer rates and relatively low fouling tendencies. The residence times tend to be reduced in thermosyphon reboilers, which minimises the risk of thermal degradation. Thermosyphon reboilers are regarded as reliable means of heat input and removal into and from industrial processes when used in their proper range of operation. These type of reboilers are highly efficient as they combine high heat transfer rates and low capital and operating costs. They have no moving parts that require maintenance or mechanical seals prone to leaks. When they are designed properly they can be used in a wide range of process conditions, that is temperatures, pressures, flow and heat loads Sloley [2]. However, lack of experimental data and robust design methods deterred many plants, and in some instances entire industries to avoid their usage. It is important therefore to understand both the steady state design of the system and its potential dynamic behaviour, and the interactions between fluid dynamics and heat transfer Sloley [2]. The prediction of the heat transfer and the thermally-induced circulation rates are the primary requirements for the successful designs. Ali and Alam[3] and Kamil et al. [4] used different fluids (acetone and ethylene glycol, distilled water, methanol, benzene, toluene and ethylene glycol respectively) to demonstrate the effects of heat flux and submergence on circulation rates in a single vertical tube thermosyphon reboiler, which was heated electrically and also developed empirical correlations. The circulation of fluid through the thermosyphon is established by the differential head, which exists between the cold and hot legs of the loop. The hydrostatic head in the cold leg (down-flow from disengagement tank or returning fluid) depends upon the liquid submergence, which can have a maximum value that corresponds usually to the top tube sheet or the top end of the test section. This is referred to as 100% submergence and corresponds to conditions at the bottom of a distillation column. Kamil et al. [4] considered other submergence levels (notably, 75%, 50% and 30%). The hydrostatic head in the hot leg (reboiler leg) was provided by the process fluid, which consists of a two-phase mixture, the quality of which changes with boiling and vapour generation as the fluid flows upwards in the heated channel. This generates the buoyancy forces to drive the circulation of fluid in the loop. Flow boiling systems combine complex interfacial heat and mass transfer behaviour in addition to the complexities associated with turbulent two-phase flow. The design of reboilers must take account of the fact that regardless of the temperature of the heating medium, the maximum heat flux is limited by the hydrodynamics of the boiling process. In recent years there has been a growing interest in systems operating at reduced pressures and many applications in distillation are now using sub-atmospheric pressure operation which provides higher thermodynamic efficiency, reduced energy consumption, safe operation,
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prevents thermal degradation and enables the use of cheaper materials of construction. Literature on boiling has concentrated on pressures above atmospheric rising to high pressures encountered in modern steam raising plants. Boiling at sub-atmospheric pressure is one that lacks systematic data. In addition, it is not possible to extrapolate methodologies developed to describe high pressure boiling due to the following factors that make the system unique Webb et al. [5]: At vacuum, dissolved gases in potential boiling nucleation sites are desorbed, which results in delayed nucleation and unsteady or even no operation. In the case of fixed outlet pressure of the boiling system, the pressure at the entrance to the boiling channel could be several times higher. This difference is due to the static head modified by the pressure drops due to friction and acceleration along the boiling channel. This implies that it is no longer possible to assume constant thermo-physical properties and the saturation temperature may vary significantly along the channel, which affects the boiling process (Figure 1). At sub-atmospheric pressures, pressure fluctuations have a greater impact on the boiling temperature of the circulated liquid. An increase of pressure of 0.1 bar is only 10% increase at atmospheric pressure, though at 0.1 bar this same change is a 100% Moore et al. [6]. As a result, the ratio of the static pressure head to the total pressure at the bottom of the liquid phase increases by decreasing the process pressure. This increases the effect of pressure fluctuations and results in a larger area required for sensible heating, which can be up to 90% of the total length of the tubes (Figure 1). This leads to a drop in the overall heat transfer coefficient. Sub-atmospheric boiling systems are especially prone to instabilities as a result of high vapour velocities and more profound impact of pressure fluctuations. This instability can be a whole-loop type, whereby all the tubes undergo the same fluctuating behaviour, or a parallel-channel type, where the behaviour is not uniform across the bundle. In either case the fluctuating pressure and flow may severely hinder the performance of the plant. In addition, excessive fluctuations in the tubes may lead to periodic dryout at the top tube surface, which may result in the deposition of dissolved materials, excessive fouling and operation failure. High wall superheats are required to initiate and sustain nucleate boiling at low pressure. High pressure drops in the high velocity two-phase boiling region causes the velocity at the entrance of heating channel to be lower. This reduces the overall heat transfer in the subcooled heating zone and the fraction of heat transfer area used for boiling. In many industrial examples of thermosyphon reboiler design, recirculation in the thermosyphon loop has not started under the conditions determined in the design. This problem is often solved by an increase in the pressure of the steam, but in modern times, the use process integration, such flexible operational flexibility is hard to obtain. In addition, most of the academic studies on boiling WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
340 Advanced Computational Methods in Heat Transfer IX processes and their stability have considered a single tube flow arrangement of simple geometry with controlled and uniform electrical heating. However, the vast majority of industrial reboilers associated with the chemical industry are shell and tube design heated by condensing steam. As a result, the available data are limited since they do not account for parallel channel instability. Electrical heating also provides a constant heat flux along the tube length as opposed to steam, which provides a constant temperature. A full scale thermosyphon loop has been constructed at the University of Manchester for the purpose of studying sub-atmospheric boiling and stability. This paper describes the facility in detail and some of the initial results so far produced.
Figure 1:
2
Temperature profile along a vertical heating tube of length L in the thermosyphon reboiler.
Equipment
A schematic of this facility is depicted in Figure 2 and comprises a vertical thermosyphon reboiler with the process fluid on the tube side and steam condenses on the shell side, a disengagement tank, two horizontal condensers, a holding tank, a delivery pump and a liquid ring vacuum pump. The flow of fluid through the thermosyphon loop depends primarily on the pressure differential generated by the head difference between the inlet and the outlet ends Sloley [2]. 2.1
Reboiler details, schematics and configuration
The dimensions of the thermosyphon reboiler are listed in Table 1. The reboiler is constructed from stainless steel to avoid corrosion and provide clean heat transfer surfaces. The inlet header is a TEMA A header design. No attempt was made to provide flow straighteners or any device to improve liquid distribution between the tubes of the bundle. The outlet header is of a conical shape to provide minimum disturbance of the two-phase flow as it passes to the separator. The present equipment is instrumented with the objective of providing an indication of the occurrence of instabilities and the key features required for the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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inception of stable operation. Thermocouples are installed at the entrance of a total of 20 tubes at three different depths in the subcooled region. Future work will focus on determining the extent to which flow instabilities might affect performance. Table 1:
Reboiler dimensions.
Shell / Tube
/ mm
Outer diameter, OD Thickness Inside diameter, ID Heat transfer Length Tube pitch (square pitch) Tube outer diameter, OD Thickness Tube inside diameter, ID
323.80 4.57 314.66 3063 31.75 25.40 2.77 19.86
/ mm Tube length Tube heat transfer length Number of tubes Heat Transfer Areas Inside area per tube Total inside area Outside area per tube Total outside area
3135 3063 50 / m2 0.191 9.55 0.244 12.22 m al 2
System 2 Cooling Water Supply
Test Condenser
Q1
Ts 2
m al1
m s 2
System 3
a2 m
Total Condenser
T1
Cooling Water Return
Tc1 m c1 m s1 m w1
Q 2 Cooling Water Return
Tc 2 m c 2
System 1
VENT
V145
PT 103
PCV 103
PIC 103
Disengagemen t Tank
V143
i 0
T m s 0
Seal Fluid Tank
PIC 200
PCV 200
Liquid Ring Pump
PT 600
Thermosyphon Reboiler
PCV 600
Steam Supply
PT 200
P123
E112
PIC 600
Cooling Water Supply
s 3 m a 3 Ts 3 m
Q 0 T0o m s 0
E114
Twr m wr
Cooler
Inlet Gate Valve DPT
V150
T wf m wf
Holding Tank
P126
Figure 2:
System 4
Schematic of the thermosyphon reboiler research facility at the University of Manchester.
2.1.1 Steam supply The process liquid is heated in the tube side of the vertical thermosyphon reboiler by the condensing steam in the shell side. Steam is supplied at 4.5 bar (147.9oC) and passes through a mild steel vessel fitted with knitted wire mess pad to act as a demister to separate the condensate from the steam via a steam trap and stabilise the pressure. The dry steam then passes to the shell of the reboiler via two control valves, the first of which acts to regulate the steam pressure upstream while the second is used to maintain a constant steam flow to WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
342 Advanced Computational Methods in Heat Transfer IX the reboiler. The valve openings were set as a percentage of the full scale and their characteristics are such that flow is linear with the applied pressure drop for incompressible flow. The steam flows correspond to heat loads of approximately 50–840 kW and automatic control is available for both valves but was not utilised. The steam outlet manifold may either be vented to atmosphere in the case of operation above atmospheric or drawn via a small orifice leading to the suction of the liquid ring vacuum pump. However, a positive steam flow through the reboiler is essential when the unit is in operation to ensure that the entire heat transfer area is free from air blanketing. This is verified in each run by monitoring the outlet temperature of the steam, which is required to be at or above its saturation value. Condensate is separated from the vapour at the bottom of the vertical line and flows through a steam trap to be collected in a calibrated vessel for measurement of its flow. 2.1.2 Process side Water used in the process side is stored in a holding tank with a capacity of approximately 500 ℓ (Figure 2). During operation, it is pumped from the holding tank to the disengagement tank above the reboiler at rate of 15–30 ℓ/min, which corresponds to the highest expected boiling rate. Any excess liquid flows back to the holding tank via a U-shape tube mounted on the liquid feed line to the reboiler, which can be set at different levels corresponding to different imposed static pressures. Only the uppermost level of the U bend was used, which corresponds to the tube outlet of the reboiler tube bundle. The liquid level inside the reboiler tubes is confirmed to correspond to the top tube-sheet in all runs by means a gauge used as a level indicator. A process vapour/liquid stream leaves the reboiler outlet to the overhead separator of 0.9 m diameter. At this stage, the liquid circulating around the thermosyphon loop is separated from the vapour, which then passes to the 14″ test condenser. The cooling duty is achieved using cooling water and the condensation rate is measured using a rotameter. An air-cooled heat exchanger is used to remove the low grade heat from the cooling water before it is circulated through the cooling loop. The air-cooled heat exchanger comprises two bays containing two tube bundles (140 finned tubes per bundle) and two fans each. This is designed to accommodate a duty of approximately 2200 kW in each bay. Uncondensed vapour passes to the 18″ total condenser, which is also water-cooled with the highest possible coolant flow to give the lowest possible vapour outlet temperature. Both condensers have a TEMA E shell and B header configurations. The condensate produced from both condensers flows back to the holding tank through an adequately vented dip leg (Figure 2). The holding tank contains two compartments to ensure that the returned condensates and the liquid overflow from the thermosyphon loop do not mix in the tank. These two flows have different temperatures as the mixed condensate is subcooled. The circulation pump draws the liquid from both compartments, though it draws all the liquid overflowing the thermosyphon loop so as to reduce the amount of sub-cooling of the feed, which is essential when the heat load is small.
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2.1.3 Vacuum system The reboiler/condenser rig is brought to and maintained at sub-atmospheric pressure by means of two liquid ring pumps and the associated seal fluid circuits. As the rig runs below atmospheric pressure, some gas inevitably leaks into the system, which is drawn into the liquid ring pump before it is expelled together with the seal fluid into the seal fluid tank. The process pressure is adjusted by means of recycle of some of gas to the liquid ring pump through valve PCV 103. The pressure in the system is found to be constant without automatic control action. The liquid ring pump requires a continuous supply of liquid to sustain the sealing liquid ring revolving concentric to the pump casing. This seal liquid is cooled by means of a cooler to remove the heat generated by compression and the latent heat of condensed vapour. The temperature and hence the vapour pressure of the seal liquid should not increase to prevent cavitation. Some problems are likely to occur, however, as a result of direct venting of the reboiler shell steam to the pump.
3
Instrumentation
3.1
Control instrumentation
Control and monitoring of the plant areas as well as data logging is achieved using DeltaV. This control system provides continuous PID control (if automatic control is activated) and is very flexible with respect to the operating characteristics and the setting of alarms and warnings. Three control loops are suggested for the operation of the reboiler-condenser rig as it is illustrated in the schematic in Figure 2 3.1.1 Steam pressure regulation This is meant to eliminate fluctuations in the steam supply pressure to the reboiler. The control is achieved using a Camflex regulating valve. A pressure transmitter situated downstream of the valve on the steam line sends an input signal, which upon comparison with the set point, acts on the control valve to maintain the supply pressure at the required value. 3.1.2 Rate of boiling The rate of boiling in the reboiler (steam flow and pressure) can be adjusted and maintained using a second Camflex control valve, which is connected a pressure transmitter on the shell side of the reboiler. A set point is selected for the control loop to establish a temperature difference between the shell and tube sides of the reboiler. Based on the input signal from the transmitter relative to the set point, the control system acts on the control valve to allow more or less flow of steam to the shell side of reboiler. 3.1.3 Process pressure A pressure transmitter is located at the disengagement tank (process side) and is used to monitor the pressure of the process side. The signal it generates is collected by the control system, which is compared to the set point before action WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
344 Advanced Computational Methods in Heat Transfer IX is taken on the third Camflex control valve to allow more or less gas recycle around the liquid ring pump. This in turn acts to change the system pressure as the liquid ring pump extracts the gas phase at a constant volumetric flow. 3.2
Safety instrumentation
For safe operation, the steam supply can be interrupted in case of an emergency using the emergency shutdown button on the shutdown panel. This panel is designed in-house and is fail-safe to prevent operation in the event of failure of plant services. This implies that the steam supply is suspended automatically upon failure of electrical supply, instrument compressed air and control computer (integrated on the panel). At the highest level of the control parameters (T, P and steam flow), a programmable logic controller (PLC) secures automatic shutdown of the plant. This system works entirely independently of the process control system. The operation of the reboiler-condenser rig will be shut down in any one of the alarm conditions determined by the HAZOP study that was performed on the rig, notably: High coolant temperature in knockout condenser Low coolant flow to knockout condenser High pressure in process side of the reboiler High temperature of the seal fluid around the liquid ring pump 3.3
Scientific Instrumentation
The pressures are measured using pressure transmitters calibrated over different pressure ranges corresponding to different locations within the facility. Fluid temperatures are measured using constantan-copper T-type thermocouple with a temperature range of −200–400oC and a signal output of –100 to 100 mV. The thermocouples have an outside diameter of 1.50 mm, which gives them a small heat capacity and thus a quick response to temperature changes. Temperatures are measured at 65 locations around the process and cooling system. All the thermocouples have been characterised in DeltaV to account for the deviations obtained during their calibration. The flow of the cooling water is measured by means of orifice plates designed in accordance to the criteria and limits specified in BS1042, the pressure drop across which is measured using strain gauge DP cells, which produce signals in the range of 4–20 mA. Flows around the process side are measured at different locations using rotameters, which will be replaced soon by electromagnetic flow meters. An additional DP cells is fitted across the inlet gate valve to the reboiler bottom header to obtain a reading of the pressure drop and convert it into the corresponding measurement of flow in the thermosyphon loop. Provision has been included in the apparatus for water to be pumped across the valve so that it can be calibrated against a rotameter. The flow is calculated using eq. (1), which has been obtained from the calibration of the inlet gate valve and describes the relationship that exists between the flow, the number of turns of the valve and the signal output: (1) V = k (I − 3.92) n
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where V is the volumetric flow in ℓ/min, kn is a constant related to the number of turns by which the valve was opened and I is the current or signal output (mA), where 3.92 mA is the signal output corresponding to no flow. The Bernoulli’s equation is reflected in this expression by the square root dependence of the flow on the signal output. The constant kn responds to a quadratic expression of the number of the gate valve turns, that is: (2) k n = an 2 + bn + c where, n is the number of turns open of the valve and a, b and c are coefficients determined from the different flows and signal outputs using the matrix method to solve multivariate sets of equations or the Solver option in Excel. Throughout tests that were conducted to check the reliability of eqs. (1) and (2), a maximum deviation of ±6.50% was obtained between the calculated flow and that obtained from the rotameter. The calibration equation is different from the one obtained from the previous configuration Webb et al. [7]. 3.4
DeltaV control and data logging software
The analogue signals from the measuring devices (mV from thermocouples and mA from pressure transmitters) are conditioned in the control panels in the laboratory. These signals are then digitised using Emerson 8 CH, TI blocks for the thermocouples and Emerson AI, 8CH, 4-20 mA for the pressure transmitters and DP cells. These are analogue-to-digital converter cards located in the control panels. The resultant digital signals are then sent through data cables to the computers where they are characterised and converted into numerical readings using the DeltaV control software. The DeltaV system is also used as a data logger, whereby the data is continuously recorded using Historian and logged into an Excel spreadsheet. Alternatively, the data can be continuously recorded using a Macro created to export the data from DeltaV. In both options, readings are taken at regular time steps, which are selected to the requirements of the user.
4
Planned experiments
Theses are water tests at low pressure. The first set of tests on the rig would establish the limits of operability, which would be followed by detailed performance tests over the range of operability. This involves taking measurements of the overall heat transfer coefficients, circulation rates, pressure drops across the circuit and fluid temperatures. It is intended that these investigative studies would cover both steady state and unstable operation and the work will be validated through mass and energy balances. This work will be carried out over a range of system pressures at and below atmospheric (0.1 bar– atmospheric). The range of operation will be defined by the imposed temperature difference at each pressure and the corresponding saturation temperature. This implies that a minimum temperature difference exists to promote stable recirculation in the thermosyphon loop and an upper limit of the imposed temperature difference will be defined by either the available steam pressure corresponding to 125oC, the available heat load upper limit, which is estimated at WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
346 Advanced Computational Methods in Heat Transfer IX about 820 kW or simply by some form of instability. In the case when unstable operation is the limiting factor, throttling the flow into the reboiler should be considered Webb et al. [5] and Moore et al. [6]. The thermocouples are inserted at three different depths in 20 of the reboiler tubes and should provide some indication of the nature of instability through the observed variations of the flow as indicated by the readings from these thermocouples and the pressure transmitters. The depths of insertion of the thermocouples in the tubes were estimated from the expected lengths of the subcooled region over the pressure range of interest.
5
Problems during commissioning
During the early stages of commissioning, the temperature of the seal fluid in the liquid ring vacuum pump on the process side increased considerably, even though it was circulated around a secondary cooler, see Figure 2. The tubes of the cooler were found to be very fouled and almost blocked. The problem was resolved by cleaning using mechanical means. The seal fluid feed line to the pump was fitted with a strainer to ensure that any residual solids were contained. Samples taken from the closed cooling water loop indicated the presence of solid impurities of sizes up to 240 microns, which is likely to foul the two condensers and reduce their thermal efficiencies. The cooling water was treated with sodium nitrite (NaNO2) to neutralise the oxygen present in the cooling water lines and reduce oxidation, which is at the origin of the solid impurities. A filter which is fitted with a 125 micron mesh was also used to filter the cooling water.
6
Initial results
In order to assess the reliability of the measurements, a statistical analysis using linear regression was conducted on the results obtained from the calibration of the inlet gate valve. The flow registered on the variable area flow meter (rotameter) was considered as a function of the flow calculated using eq. (1). Figure 3 depicts the data points and the extrapolation of the regression line to the range of operation with 95% confidence intervals drawn about the regression line. The experimental system is operated under vacuum and so air can leak into the process through the numerous joints and flanges. It is important to quantify the air leakage rate and a method is devised for that purpose. The graph depicted in Figure 4 is a plot of the square root of the manometer height difference and the corresponding air flow. A liquid ring pump connected at the end of the process line pulls the vacuum within the system and so the quantified leakage rate would be for the entire process. The leakage rate was quantified daily using a nozzle box with orifices for different air flows installed in the process line. The entire system is evacuated and all the orifices on the nozzle box were closed. The air is then forced to flow through a vent line. The pressure drop across the vent line was measured using a water manometer and flow of air was varied through a combination of 5, 10, 20 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Advanced Computational Methods in Heat Transfer IX
and 40 lb/h orifices. The negative offset shown in Figure 4 represents the leakage rate of air into the entire process. The research facility is located within a glass-walled laboratory and so it is the atmospheric and process pressures that are driving the leakage rate. A typical air leakage plot obtained for the first day of tests is depicted in Figure 4, where the leakage rate was 0.78 kg/h (offset of 1.73 lb/h) compared to the 2.5 kg/h air leakage rate obtained from the previous work Webb et al. [7]. 675 600 525
y = a.x + b / l/min
450 375 Regression yy+ Actual Data
300 225 150 75 0 0
60
120
180
240
a & b obtained by means of linear regression
Figure 3:
300
360
420
480
540
600
660
x = flowrate / l/min
Regression line for the circulation loop with 95% confidence intervals. 8
7
6 y = 0.2635x + 0.4554 R2 = 0.9919
5
(∆h)
1/2
4
Air Leakage Rate Linear (Air Leakage Rate)
3
2 Offset = leakage rate
1
0 -3
0
3
6
9
12
15
18
21
24
27
Air Flow / lb/h
Figure 4:
Air leakage test on reboiler-condenser rig on 01-02-2006.
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348 Advanced Computational Methods in Heat Transfer IX 6.1
Consistency checks
6.1.1 Vapour generation Individual calculations were carried out on the reboiler and the main condenser to evaluate the rate of vapour generation independently and obtain a basis for comparison as depicted in Figure 5. The graph in Figure 5 indicates that the early tests produced an experimental reliability of around ±20% with respect the flows of vapour from the reboiler and the condenser. In these calculations heat losses have been neglected as the equipment is very well insulated. Previous work by Webb et al. [7] indicated that a permanent condensation rate of around 4 kW was obtained from the main 14″ condenser. However, when compared to heat loads in the experimental work (predominantly above 100 kW) this loss lies within the reliability limits in Figure 5. Later tests have produced lower discrepancies typically within ±5%. 26 24
Produced vapour condenser / kg/min
22 20 18 16 14
Experimental Data bi-section -20% -10% +10% +20%
12 10 8 6 4 2 0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
Produced vapour from reboiler / kg/min
Figure 5:
Comparison of produced vapour rate.
6.1.2 Overall heat load The overall control volume (system 4 in Figure 2) is used to check the overall energy balance (Figure 6). This check is expected to be less reliable since sections of the holding tanks are not insulated. The check is a simple comparison of the enthalpy supplied in kW by the condensed steam ( Q 0 ) with the heat gain of the cooling water from the two condensers, that is Q1 + Q 2 . The obtained data are depicted in Figure 6, which indicates that the heat losses are predominantly within ±20%. Analogous to the mass balance, the later tests produced maximum discrepancies within ±5%.
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900 850 800 750 700 650
Cooling water load / kW
600 550 500 450
Experimental Data bi-section -20% -10% +10% +20%
400 350 300 250 200 150 100 50 0 0
60
120
180
240
300
360
420
480
540
600
660
720
780
840
900
Steam heat load / kW
Figure 6:
7
Overall energy balance.
Conclusions
This paper describes the re-commissioned thermosyphon loop research facility at the University of Manchester, its configuration, instrumentation and the set of planned tests at reduced pressure using water. Some data from the initial stages is also discussed. Future work on the rig will focus on establishing datasets on the operating characteristics of the thermosyphon reboiler research facility.
Acknowledgements This work is supported principally by the School of Chemical Engineering and Analytical Science at the University of Manchester. The commissioning and experimental investigations have been performed in the Morton Laboratory at the University of Manchester and thanks are due to Alan Fowler, Michael Royle and Anthony Diggle for their invaluable help in running the reboiler research facility and building the safety control system respectively. Particular thanks are due to Gary Burns and the workshop personnel for their efficient involvement in modifications and commissioning the equipment. The contribution of John Cuffe and Patrick Kirby to the HAZOP study on the rig is also acknowledged.
References [1] Arneth, S, Stichlmair, J., Characteristics of a thermosiphon reboiler, Editions scientifiques et médicales Elsevier SAS, 2000.
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350 Advanced Computational Methods in Heat Transfer IX [2] Sloley, A. W., Properly Design Thermosyphon Reboilers, pp. 52–64, Chemical Engineering Progress, 1997. [3] Ali, H., Alam, S. S., Circulation Rates in Thermosiphon Reboiler, International Journal of Heat and Fluid Flow, Vol.13, pp. 86-92, 1992. [4] Kamil, M., Alam, S. S., Ali, H., Prediction of Circulation Rates in Vertical Tube Thermosiphon Reboiler, International Journal of Heat Mass Transfer, Vol.38, pp. 745-748, 1993. [5] Webb, D. R, Benson, H, Heggs, P. J., The new thermosyphon reboiler at UMIST, HTFS Research Symposium, paper RS 1108, 2001. [6] Moore, M. J. C, Keys, M. H, Plumb, G.R., Design of vertical thermosyphon reboilers for operation under vacuum conditions application in nuclear fuel reprocessing. In: 2nd UK National Heat Transfer Conference, Vol. 2, pp. 1157 – 1169, Glasgow, UK, 1988. [7] Webb, D. R, Benson, H, Heggs, P. J., Ooi, H., Schnabel, T., Cottrell, C., The UMIST Dataset on Operation of Vertical Thermosiphon Reboilers at Subatmospheric Pressures, HTFS Research Symposium, paper RS 1113, 2002.
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Analysis of water condensation and two-phase flow in a channel relevant for plate heat exchangers J. Yuan, C. Wilhelmsson & B. Sundén Department of Energy Sciences, Faculty of Engineering, Lund University, Sweden
Abstract Water vapor condensation and two-phase flow appear in plate heat exchangers being used as condensers. Analysis of water phase change and flow dynamics is an important but complicated task due to large change in water physical/transport properties across the water liquid-vapor interface boundary. In particular, a singular-link behaviour in Navier-Stokes (N-S) equations is present due to the large step change in the density when computational simulation methods are applied. Conventional methods using ensemble averaged parameters such as void fraction cannot be applied to cases where high-resolution calculations and detailed analysis are required. In this study, a computational fluid dynamics (CFD) approach is presented for analysis of water vapor condensation and twophase flow in a channel relevant for plate heat exchanger parallel plates. The developed model is based on the governing equations which are directly solved for the entire single- and two-phase fields. The water phase change and twophase flow are treated by employing a water liquid-phase fraction factor based on the total enthalpy in each computational cell. The factor is defined as the ratio of the total enthalpy differential to the latent heat of condensation. The density, viscosity and conductivity of the two-phase region are calculated and updated based on the calculated value of the liquid-phase fraction factor until a converged result is reached. It is revealed that, among others, the inlet vapor velocity has significant effects on the water phase change and two-phase flow in the channel, in terms of liquid-water fraction factor distribution. Keywords: water, condensation, two-phase flow, model, analysis.
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352 Advanced Computational Methods in Heat Transfer IX
1
Introduction
It is clear that new cooling strategies are required for many applications such as electronic devices because they are becoming more compact and rejected heat flux is increasing. Condensation and boiling occur in compact heat exchangers, e.g., plate heat exchanges (PHEs), which can remove larger amounts of heat. In these cases, it is necessary to gain a fundamental understanding of phase change and two-phase flow phenomena in PHE channels [1]. The traditional concept of a PHE is the plate-and-frame heat exchanger, which consists of the plates, the gaskets, the frames and some additional components (the carrying and guiding bars, the support column, etc.). The two streams flow into alternate channels between plates, entering and leaving via ports at the corner of the plates. A stream exchanges heat with the streams in adjacent channels. Because of their structures, PHEs offer a number of advantages over conventional shell-and-tube heat exchangers. These advantages are compactness, high effectiveness, flexibility, easy cleaning, cost competitiveness, etc. PHEs have been used for condensation applications for ammonia, ethanol, hydrocarbons, water vapor, etc. In general, the gas enters at the top of the plate and moves downwards over the heat exchanging surface. The condensate leaves at the bottom, while the cooling medium is flowing in neighbouring channels either in counter flow or parallel flow. Water vapor condensation is one of the most common two-phase applications for PHEs [1]. In two-phase applications, the major efforts have been made in the field of experimental investigations with phase change. Due to the small corrugated channels, the flows and heat transfer in PHEs are complicated, and relevant models are very rare.
2
Problem statement
It is known that the condensation performance is dependent on many factors, such as fluid properties, plate geometry, system pressure, mass flow rate, etc. The water vapor condensates because the plate temperature is below the water vapor saturation temperature, and consequently forms a thin film. The latent heat released during condensation must be transferred to the plate through the conduction of the condensate film. Heat transfer during the condensation from the liquid-vapor region is very important for predicting performance of twophase PHEs. In all heat transfer problems involving liquid-vapor phase change, there is a sharp jump in the value of the density and other physical/transport properties across the liquid-vapor interfacial boundary. This sudden change in the properties across the interface challenges the convergence of all traditional numerical schemes that have been successful for single-phase systems [2–6]. There are several CFD algorithms developed during last decades to simulate two-phase flow processes, such as the two-fluid model, the step-function model and the volume of fluid (VOF) method, as summarized in table 1 [7–9].
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Table 1: Model Two-fluid model Stepfunction model Volume of fluid (VOF) model
353
Various CFD models for two-phase flow [7–9].
Features Using two sets of phase N-S equations plus a set of empirical relationships for the interfacial mass transport and wall to fluid heat transport. Using a set of N-S equation for both phases plus a set of step-functions for the liquid, vapor and two-phase regions. Directly simulating and tracking of the two-phase interface boundary
Remarks Very limited to the cases within the range of the experimental correlations. Very often leading to prediction of unrealistic interface. May apply to simple cases involving phase change by interface reconstruction schemes.
This paper presents a CFD approach for analysis of water vapor condensation and two-phase flow between parallel plates. The model is developed by directly solving the governing equations for the entire single- and two-phase flow regions. Tracking of water phase change and two-phase flow is reached by employing a water liquid-phase fraction factor, defined as the ratio of the total enthalpy differential to the latent heat of condensation. The physical and transport properties of the two-phase region are calculated and updated based on the calculated value of the liquid-phase fraction factor until a converged result is reached. It is revealed that, among others, the inlet vapor velocity has significant effects on the water phase change and two-phase flow in the channel.
vin Vapor
Two phase y
Liquid x
Figure 1:
Physical model of water condensation in a channel of plate heat exchangers.
The water liquid-vapor phase interface due to the condensation of water vapor is investigated in this study for a geometric configuration of two parallel plates. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
354 Advanced Computational Methods in Heat Transfer IX Because of its symmetry, only the left half of the channel is solved as shown in Fig. 1. The saturated vapor enters the channel with a plate length of L. Condensation takes place on the wall of the channel since the wall temperature, Tw, is below the saturation temperature of the water vapor, Tsat. The condensate water flows along the main flow direction due to the effects of gravity, shear force and surface tension. In a first attempt to implement the phase change and two-phase flow in a model, the following assumptions are employed during the modeling development: flow both in vapor and liquid phase is laminar; the fluid within the liquid-vapor interface is saturated; surface tension force is neglected (with including surface tension, the condensation length decreases, see [5]).
3
Governing equations and source terms with phase change
The conservation equations for transient, fluid flow in two dimensions are formulated in this section. In order to simplify the solution procedure, one set of governing equations is written for both the water liquid and vapor regions. The mass continuity equation or mass conservation equation is written as ∂ρ (1) + ∇ • (ρ v) = 0 ∂t
The momentum equations read ∂(ρu) ∂p (2) + ∇ • ( µ∇ u ) + ∇ • ( ρ uv ) = − ∂t ∂x ∂(ρ v) ∂p (3) + ∇ • ( ρ vv ) = − + ∇ • ( µ∇ v ) − ρ g ∂t ∂y where V is the velocity vector. The energy equation can be expressed as ∂ ( ρ c pT ) (4) + ∇ • ( ρ c p vT ) = ∇ • ( k ∇T ) − ST ∂t Equation (4) balances the convected energy, the heat conduction, and a source term ST. The heat source term ST in eqn (4) is associated with water phase change (condensation /vaporization), which can be expressed as [7] ∂ ( ρ∆H ) (5) ST = + ∇ • ( ρ∆Hv ) ∂t where H is the total enthalpy. For a pure substance undergoing evaporation or condensation, the total enthalpy is a discontinuous function of the temperature. However, from a computational viewpoint, discontinuities are difficult to track and often the phase change is smeared out over a small temperature range to attain numerical stability [7]. The following relation is thus formulated for the total enthalpy H: 1 1 f ρv ( hv + h fg ) + (1 − f ) ρl hl = [ f ρv hv + (1 − f ) ρl hl ] + f ρ v h fg (6a) ρ ρ ρ where f represents the liquid water fraction factor. The two terms on the right-hand side of eqn (6a) represent contributions of water sensible enthalpy, h, and latent heat, ∆H, respectively, to the total enthalpy H=
1
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Advanced Computational Methods in Heat Transfer IX
h=
1
ρ
[ f ρv hv + (1 − f )ρl hl ]
355 (6b)
1 (6c) fρ h ρ v fg The total enthalpy in eqn (6) can then be rewritten in terms of sensible enthalpy and latent heat (7) H = h + ∆H ∆H =
It should be pointed out that H is the enthalpy identical to that obtained from the steam table. It is equal to sensible enthalpy, hl, for the liquid phase. For the vapor phase, H is the summation of sensible enthalpy hv and latent heat of condensation hfg. Substituting eqn (6c) into eqn (5) yields ∂ ( ρ v fh fg ) (8) + ρ v h fg ∇ • ( fv ) ST = ∂t For incompressible flow, the water vapor density is a constant, and eqn (8) is simplified [5] ∂f (9) ST = ρv h fg + ∇ • ( fv ) ∂t The liquid water and vapor regions are distinguished by the liquid water fraction factor f(x,y,t). The value of f is zero in the vapor phase and unity in the liquid phase. A formulation expressing the liquid phase fraction as a function of the total enthalpy, rather than the temperature, is employed in this study. According to the classical thermodynamic definition, the relationship between the liquid phase fraction f and the enthalpy H is H − Hl (10) f = 1− Hv − Hl
The time and coordinate dependence of the factor f follows the continuity law, and is expressed as ∂f m (11) + ∇ • ( fv ) = ρl ∂t where m is the mass production rate of condensate. It is clear that eqn (11) does not explicitly include the diffusive terms that smear the sharp gradient of the liquid water fraction factor f across the interface. The equation is added to the other governing equations with the purpose to predict the local and instantaneous value of f and other dependent parameters, and in the entire region smooth out the singularity encountered at the interface while preserving the phase discontinuity. By comparing eqns (9) and (11), it is clear that the liquid water mass production rate due to condensation is associated with the source term in the energy equation according to m =
ρl
h fg ρv
ST
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(12)
356 Advanced Computational Methods in Heat Transfer IX
4
Fluid properties
The fluid properties at the interface vary with time as the interface position changes relative to the computational grid. The density, viscosity and thermal conductivity are defined as functions of f according to the following relations [6]: (13) ρ = f ρl + (1 − f ) ρ v (14) µ = f µl + (1 − f ) µv k = fkl + (1 − f )kv
(15)
The specific heat is defined using the weight fraction of water liquid and vapor 1 (16) c p = f ρl c pl + (1 − f ) ρ v c pv ρ
5
Boundary conditions
The inlet conditions can be written as u=0, v=-vin, T=Tin=Tsat, f=0 at the top of the channel (y=L)
(17)
while the boundary conditions at the wall (x=0) are u=v=0, T=Tw, f=1 at x=0
(18)
The boundary conditions at the exit of the channel is written in a generalized form, i.e., ∂φ ∂y = 0 at y=0 for u, v, T or f. Due to the symmetric character, only one half of the channel is considered. Along the line of symmetry (x = W) the boundary conditions can be expressed as ∂φ ∂x = 0 .
6
Numerical solution methodology
The governing differential equations are discretized into algebraic equations by a truncated Taylor series approach using the finite volume method (see, e.g., [10]), and then they are solved by an iterative method. A general two-dimensional CFD code, SIMPLE_HT [11], is employed to solve the governing eqns (1–4, 11), together with the boundary conditions. The finite-volume (or control-volume) technique, which is based on the conservation of a specific physical quantity, is used in the code. The code is designed for convection-diffusion problems, e.g., Navier-Stokes equations, the temperature field and other variable equations. As implemented in the code, each of the differential equations can be cast into the general equation form (see, e.g., [11])
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∂ (ρφ ) + ∂ (ρuφ ) + ∂ (ρvφ ) = ∂ Γ ∂φ + ∂ Γ ∂φ + S ∂τ ∂x ∂y ∂x ∂x ∂y ∂y
357 (19)
where φ denotes any of the dependent variables, Γ is the diffusivity and S is a source term. Once in this form, the equations are integrated over the control volumes defined on a staggered grid. The boundary conditions are introduced as source terms in control volumes neighboring boundaries whenever appropriate. The resulting system of algebraic equations is solved using an iterative method. The pressure-velocity coupling (when φ = u, v ) is treated by the SIMPLECprocedure with the incompressible form of the pressure-correction equation. The convection-diffusion terms are treated by the power-law, hybrid or upwind (used in this study) schemes. Each variable is solved with the TDMA algorithm combined with a block-correction method. As shown above, the equations needed for the calculation are coupled by the liquid phase fraction factor f, temperature, condensation mass generation via source terms and variable physical/transport properties. It should be noted that the mass generation rate is zero in most of the region, and non-zero only in the region of two-phase flow, where phase change occurs. As mentioned earlier, the physical/transport properties are variable. These parameters depend on the position in the channel, and the liquid phase fraction factor as well. All the parameters are calculated and updated during iterations of the calculation procedure. In this investigation, a uniform grid distribution is applied. In order to evaluate the performance of the numerical method and code, test calculations considering grid sensitivity were carried out. It is found that the predictions do not change significantly in terms of the liquid phase fraction factor distribution, when the number of grid points is increased beyond 18×70 (70 for the main flow direction y). It should be mentioned that water liquid and vapor temperatures are continuous at the interface. The temperature at the liquid-vapor interface is therefore expressed as T = Tsat, where Tsat is the saturation temperature. If a fixed value is desired for a variable anywhere in the computational domain, this can be achieved by introducing a fictious source term for the cells with f between 0 and 1. (20) ST = 1010 Tsat − 1010 T By doing this, the source term will dominate and the interface temperature T is set to the saturation temperature Tsat.
7
Results and discussion
Parameters of a PHE configuration, appearing in the common literature, are applied as a base case in this study. Channel geometries are employed as follows: length of the channel L = 20 cm, and width of the channel 2W = 4 mm. The inlet vapor is assumed to be saturated, and the inlet conditions are: temperature Tin = 100oC and velocity vin = 0.6 m/s. It should be noted that all the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
358 Advanced Computational Methods in Heat Transfer IX results presented hereafter are for the base case condition unless otherwise stated, such as the inlet velocity. The velocity vector and liquid phase fraction factor f distributions are shown in Fig. 2. Figure 2(a) shows the velocity profile along the main flow direction, in which the scale of the vector plots (i.e., 0.5 m/s) is a reference value of the maximum velocity. As shown in the figure, a parabolic profile is observed in the flow channel. On the other hand, the velocity in the liquid region close to the plate is very small. In terms of liquid water fraction factor f, water vapor condensation and two-phase flow phenomenon in the channel are shown in Fig. 2(b). It can be observed that water vapor with f = 0 appears in the central part of the channel along the main flow direction, while generated liquid water (f = 1) due to condensation is found in the region closest to the plate. The liquid-vapor interface with 0
(a) Figure 2:
(b)
(a) Velocity vector, and (b) contour of liquid phase fraction factor for the base conditions.
Fig. 3 shows the distribution of water liquid phase fraction factor f at different vapor inlet velocities. For a small vapor inlet velocity (vin = 0.3 m/s) as shown in Fig. 3(a), all the water vapor supplied to the channel condenses, and the generated liquid water fully occupies the flow region at the exit of the channel. On the other hand, for big vapor inlet velocity as shown in Fig. 3(b), both the condensate liquid water layer and the two-phase interface are very thin, and most of the supplied water vapor flows out the channel exit. For this case, a modest phase change (condensation) occurs and weak effects on the two-phase flow are expected. It is obvious that a big downward vapor velocity will tend to thin the liquid water film, and this is the reason why most condensation applications occur in a downward style [1].
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Advanced Computational Methods in Heat Transfer IX
(a) Figure 3:
8
359
(b)
Contours of liquid phase fraction factor for the vapor inlet velocity vin of: (a) 0.3 m/s, and (b) 1.2 m/s.
Conclusion
In this investigation, water condensation and two-phase flow interface are numerically simulated for a channel between two parallel plates relevant for PHEs. The model is based on the total enthalpy of water, and water liquid phase fraction factor is employed to describe the water vapor, liquid and two-phase interface. The results show that the condensation phase change and two-phase interface are very sensitive to the vapor inlet velocity. It should be pointed out that the surface tension is not included in this study, and its effects on the distribution of liquid water fraction factor will be evaluated later.
Acknowledgements The Swedish Energy Agency (STEM) and the Swedish Research Council (VR)/SIDA -Swedish Research Links provided financial support.
References [1]
[2] [3]
Wang, L. and Sundén, B., Thermal and Hydraulic Performance of Plate Heat Exchangers as Condensers, in Proceedings: Compact Heat Exchangers and Enhancement Technology for the Process Industries, Shah, R.K. (ed.), pp 461-469, 2003. Panday, P.K., Two-dimensional Turbulent Film Condensation of Vapours Flowing inside a Vertical Tube and between Parallel Plates: a Numerical Approach, Int. J. Refrigeration, 26, pp 492-503, 2003. Srzic, V., Soliman, H.M. and Ormiston, S.J., Analysis of Laminar Mixedconvection Condensation on Isothermal Plates Using the Full Boundarylayer Equations: Mixtures of a Vapor and a Lighter Gas, Int. J. Heat Mass Transfer, 42, pp. 685-695, 1999. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
360 Advanced Computational Methods in Heat Transfer IX [4] [5] [6] [7]
[8]
[9]
[10] [11]
White, A.J., Numerical Investigation of Condensing Steam Flow in Boundary Layers, Int. J. Heat Fluid Flow, 21, pp. 727-734, 2000. Zhang, Y., Faghri, A. & Shafii, M.B., Capillary Blocking in Forced Convective Condensation in Horizontal Miniature Channels, ASME Transaction J. Heat Transfer, 123, pp. 501-511, 2001. Zhang, Y. & Faghri, A., Numerical Simulation of Condensation on a Capillary Grooved Structure, Num. Heat Transfer (part A), 39, pp. 227243, 2001. Anghaie, A., Chen, G. & Kim, S., An Energy Based Pressure Correction Method for Diabatic Two-Phase Flow with Phase Change, in Proceedings: Trends in Numerical and Physical Modeling for Industrial Multiphase Flows, Institut d’Etudes Scientifigues de Cargese, France, Sept. 27-29, 2000. Wang, L. & Sundén, B., Numerical Simulation of Two-phase Fluid Flow and Heat Transfer With or Without Phase Change Using a Volume-of Fluid Model, in Proceedings: 2004 ASME Int. Mechanical Engineering Congress and Exposition (CD-rom), IMECE2004-59380, UAS, 2004. Wang, L. & Sundén, B., Numerical Simulation of Two-phase Flows Using a Volume-of-Fluid Model with Various Interface Reconstruction Schemes, in Proceedings: 3rd Int. Symposium on Two-Phase Flow Modeling and Experimentation, Pisa, 22-24 Sept. 2004. Versteeg, H.K., and Malalasekera, W.M., An Introduction to Computational Fluid Dynamics, the Finite Volume Method, Longman Scientific & Technical, England, 1995. Sundén, B., Rokni, M., Faghri, M. and Eriksson, D., The Computer Code SIMPLE_HT, ISRN LUTMDN/TMHP-04/3015-SE, Lund Institute of Technology, Lund, Oct. 2004.
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361
Numerical heat transfer modelling of staggered array impinging jets A. Ramezanpour1, I. Mirzaee2, R. Rahmani1 & H. Shirvani1 1 2
Faculty of Science and Technology, Anglia Ruskin University, UK Faculty of Engineering, Urmia University, Iran
Abstract A numerical study of flow field and heat transfer rate in external flow of a novel heat exchanger (Anglia Ruskin University, 2001) was conducted. The design comprises of confined impinging jets from a staggered bundle of tubes in which the fluid flows in an opposite staggered arrangement array after impingement. The RNG k-ε model and enhanced wall treatment near wall turbulence modelling was applied to model a three-dimensional computational domain. The accuracy of the model was validated in two- and three-dimensional cases for single impinging jets with available experimental results. The arrangement of the staggered array was fixed Sn/D=2.1 and Sp/D=1.6 where Sn and Sp are the distances between tubes transverse and parallel to fluid flow respectively and D was the hydraulic diameter of tubes. The dimensionless tubes to impinging surface distance (H/D) were in the range of 0.2, 0.5, 1.0, and 2.0 and the Reynolds number based on the tubes’ hydraulic diameter and average fluid velocity at the exit of tubes in the range of 1000, 5000, and 20000 were studied. The global heat transfer rate on both impinging and confinement plates increased with decreasing of H/D and increasing of Reynolds number; however, the slope of increasing Reynolds number was sharper in low H/Ds. The friction factor increased with a decreasing H/D and an increasing of the Reynolds number. The local Nusselt number was studied on both impinging and confinement plates. The temperature contours and velocity vectors are also presented. Keywords: bundle of impinging jets, jet to jet interaction, RNG k-ε, enhanced wall treatment.
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362 Advanced Computational Methods in Heat Transfer IX
1
Introduction
The bundle of impinging jets has applications where high heat flux over a small area is needed such as electronic cooling systems and turbine blade cooling. This concept was used to develop a novel compact heat exchanger as external flow in combination with staggered tube bundle (Figure 1). External flow, impinging jet
Internal flow, Staggered tube bundle cross flow Figure 1:
Novel compact heat exchanger (Anglia Ruskin University, patent 2001).
Carcasci [1] experimentally studied the single air impinging jet and bundle of jets and their interactions with and without cross flow. For a pair of impinging jets, when the main vortices of two jets interacted with each other, two vortexes were generated in the space between the main vortexes and the bottom flat plate (lower adverse vortexes). The interaction between the main vortices and the topmost flat plate (confinement plate) determined a second series of vortexes (upper adverse vortexes). The experimental heat transfer study of single confined impinging jets, inclined slot nozzle impinging jets, and influence of Prandtl number were carried out by Jung-Yang et al. [2], Beitelmal et al. [3], and ChinYuan and Suresh [4] respectively. In numerical turbulent studies of impinging jets, although more costly Reynolds stress model due to taking into account turbulence anisotropy effects leads to more satisfactory results than eddy viscosity models (Morris et al. [5]) however satisfactory modelling of the heat transfer rate on impinging surface depends mostly on the near wall turbulence modelling (Craft et al. [6]). The research by Durbin [7] showed that the overestimation of heat transfer rate on the impinging surface by eddy viscosity k-ε model is due to excessive growth of the turbulent kinetic energy in the stagnation region. Behnia et al. [8] by using v2f model studied the influence of confinement, nozzle exit velocity profile and turbulence intensity on the average and local heat transfer of the impinging surface for a single round nozzle. Furthermore Park and Sung [9] obtained satisfactory results by modifying a near wall turbulent model. A study of slot nozzle impinging jets by Ramezanpour et al. [10] used enhanced wall treatment near wall turbulent modelling in Fluent and led to a satisfactory prediction of local heat transfer rate on the impinging surface in either vertical or inclined impinging jets. San and Lai [11] and Su and Chang [12] experimentally studied the bundle of impinging jets for staggered and inline arrays respectively. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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This study is concerned with the numerical flow and heat transfer rate modelling with no equilaterally staggered bundle impinging jet in which flow exits through the opposite geometrical staggered tubes after impingement. The RNG k-ε model and enhanced wall treatment near wall turbulence modelling is applied.
2
Numerical modelling
2.1 Governing equations The steady state continuity, momentum, and energy equations for incompressible flow and constant variables may be presented as:
∂ui =0 ∂xi
∂ ( ρui u j + δ ij p − τ ij ) ∂x j
=0
ρc p u i
(1)
where
∂u
∂u
τ ij = µ i + j ∂x j ∂xi
∂T ∂ 2T =λ 2 ∂xi ∂x j
(2) (3)
The turbulent kinematic energy, k, and its dissipation rate, ε, in the RNG k-ε model are modelled as:
µ t ∂k + µ + G − ρε k Prt ∂x j 2 µ t ∂ε G ∂ε ∂ ε * ρε = + + − ρu µ C G C 2ε k i ∂x ∂x Prt ∂x 1ε k k i j j G ∂k ∂ = i ∂x ∂x i j
ρu
where
∂u
j
,
(4)
(5)
Cµ ρη 3 (1 − η /ηD ) 1 + βη 3
(6) G = 2µ t S k ij ∂x i and η=S(k/ε)is function of strain rate. The extra term in ε equation in RNG k-ε model reduces turbulent eddy viscosity, µt=ρCµk2/ε, in high strain rate regions. The constants of the model are:
C2*ε = C2ε +
C µ = 0.0845 , C1ε = 1.42 , C2ε = 1.68 , η0 = 4.38 , β = 0.012 The energy equation is solved by approximating the extra temperature fluctuation term as diffusion term with diffusion coefficient equal to the eddy viscosity. In the near wall region enhanced wall treatment is used in which the inner region (Rey=ρy√k/µ<200), momentum and k equations are solved and eddy viscosity as well as dissipation rate are obtained algebraically. In the inner region WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
364 Advanced Computational Methods in Heat Transfer IX the eddy viscosity and dissipation rate are calculated based on blending function in which its output near the wall leads to the algebraic values of one-equation turbulence model and in boundary of outer region (Rey=200) leads to the values of RNG k-ε equation to be solved in the outer region (Fluent [13]). The temperature and velocity near the wall are obtained by blending the laminar sublayer and turbulent logarithmic laws (Kadar [14]): + + u + = eΓulam + e1 / Γuturb + + du + Γ dulam 1 / Γ duturb = e + e dy + dy + dy +
+ + T + = e ΠTlam + e1/ ΠTturb
a( y + )4 1 + by +
where
Γ=−
where
Π=−
(7)
a(Pr y + ) 4 (8) 1 + b Pr 3 y +
where a=0.01, b=5.0, and κ=0.41 is von Karman constant. The generation term in solving k equation near the wall is obtained using equations (7). 2.2 Computational domain The three dimensional computational domain by symmetrising the external flow in the novel heat exchanger is shown in Fig. 2. The boundaries include tube walls, impinging and confinement surface walls, inlet flow, outlet flow and symmetric boundary all around the domain. The arrangement of tubes is fixed, Sp/D=2.1, Sn/D=1.6. The inlet and outlet tubes have fixed length L=1cm, tube diameter d=1cm, and H/D=0.2,0.5,1.0,2.0 where D is the hydraulic diameter of the tubes. The inlet Reynolds number based on average tube exit velocity and hydraulic diameter of tubes is Re=1000, 5000, 20000. The velocity in the inlet boundary has linear profile and turbulent intensity and length scale are set to 3% and 0.01cm. On the walls no-slip condition is imposed and constant heat flux 100 W/m2 on impinging surface and 50 W/m2 on the confinement and tubes are considered. Also to solve the turbulence kinematic energy equation, ∂k/∂n=0 is considered on the walls. In the symmetric boundaries, the gradient of all variables (zero diffusion) as well as velocity component normal to the boundary (zero convection) are set to zero. Solving the near wall regions, instead of using wall functions, demands fine mesh near to the walls. A 16 layer boundary layer grid with a first grid height of 0.02 mm, growing ration of 1.1 was generated on the impinging and confinement walls. The tetrahedral grid was used to cover the computational domain. The dimensionless wall parameter in the stagnation point adjacent cell was less than 0.5 in whole range of study (Ramezanpour et al. [10]) slightly less than 1.0 which was recommended (Fluent [13]). 2.3 Solution parameters A finite volume based discretization and segregated and explicit method was used to solve the governing equation. SIMPLE predictive-corrector algorithm WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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was used for velocity-pressure coupling and the momentum, energy and turbulence equations were discretized by second order interpolation scheme. The under-relaxation factors were 0.3 for pressure and 0.8 for other parameters. The solution was considered converged when the scaled residual errors were reached 10-6 for energy equation and 10-4 for other equations.
Figure 2:
Computational domain, boundaries, and geometrical parameters. 140
Re=23000, single round im pinging jet
Local Nusselt numbe
130
Experimental Results, Yan [15]
120 110
Numerical Modelling RNG K-Epsilon EWT
100 90 80 70 60 50 40 0
Figure 3:
3
1
2
3
4
r/D
5
Comparison of numerical and experimental results for single round impinging jet.
Results and discussion
The numerical model in two dimensions was validated by Ramezanpour et al. [10]. A three dimensional computational domain of a single round nozzle impinging jet in case of H/D=2 and Re=23000 was solved and the local Nusselt number was compared in the experimental study of Yan [15] (Fig. 3). Although the results of numerical model is not satisfactory in predicting the local Nusselt number where second peak happens in the transitional location to wall jet region, however the average Nusselt number shows acceptable results. It is noted that the local trends of the Nusselt number in stagnation region depend strongly to the inlet turbulence and velocity profile (Behnia et al. [8]). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
366 Advanced Computational Methods in Heat Transfer IX The global heat transfer rate on the impinging and confinement plates, as well as pressure drop versus H/D in different Reynolds numbers were presented in Fig. 4. The global Nusselt number is more sensitive to H/D when H/D<1 and it increases sharper in this region by reduction of H/D in both impinging and confinement plates. By increasing the H/D, the global Nusselt number on confinement plate reduces as the upper adverse vortexes becomes weaker near the surface, however on impinging surface it slightly increases, particularly in high Reynolds number, which may be due to weaker lower adverse vortexes and more effective impinging area. The pressure drop increases in lower H/Ds however this increase is sharper when H/D<0.5. Global Nusselt number, Impinging Surface
350
Re=1000
250
Re=20000
200
Global Nusselt number, Confinement Surface
300
Re=5000
Global Nusselt Numbe
Global Nusselt Numbe
300
150 100 50
Re=1000 Re=5000
250
Re=20000
200 150 100 50 0
0 0
0.5
1
1.5
H/D
2
0
0.5
1
1.5
H/D
2
Re=1000
10000
Re=5000 Re=20000
Pressure Drop
1000
100
10
1 0
Figure 4:
0.5
1
1.5
2
H/D
2.5
Global heat transfer rate and pressure drop.
The local Nusselt number in the jet to jet direction on impinging and confinement plates are shown in Fig. 5. On the impinging surface the local heat transfer rate reaches to a maximum in the stagnation point where two wall jet flows interact and at the same time there are secondary peaks due to transition point before reaching to the maximum point. This is sharper in high Reynolds number however when Re=1000, the local heat transfer rate almost remains constant in jet to jet interacting region. The interaction is also more effective on local heat transfer rate in lower H/Ds. The trends of the local Nusselt number on confinement surface is strongly depend on H/D. For H/D=1 and 2, the main vortexes are weak near to the confinement surface and the heat transfer rate is low and almost constant. When H/D=0.5, the interaction of the wall jets leads to single high momentum flow impinging to the middle of the two jets on confinement surface and therefore there is a single maximum heat transfer rate point. For H/D=0.2, the interaction happens in more confined space and lower adverse vortexes are stronger while upper adverse vortexes are to be shaped. Therefore the main vortex of each jet impinges to the confinement surface slightly near to the jet and two maximum local heat transfer rate occurs. The Nusselt number in the middle of the jet to jet WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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distance is reached to a minimum due to weak upper vortexes forming. The small maximums detected near the jet exits are due to the weak vortexes on the edge of entering flow. 400
500
IMP plate-Jet to Jet-H/D=0.5
300
Re=1000
450
Re=5000
400
Re=20000
250 200 150 100
Local Nusselt numbe
Local Nusselt numbe
350
50
Re=1000 Re=5000 Re=20000
350 300 250 200 150 100 50
0 0.005
200 180 160 140 120 100 80
0.01
0.015
IMP plate-Jet to Jet-Re=5000
r
H/D=0.2 H/D=0.5 H/D=1.0 H/D=2.0
60 40 20 0 0
Figure 5:
0.005
0.01
0.015
r
0 0.005
0.02
0.02
500
Local Nusselt numbe
0
Local Nusselt numbe
CONF plate-Jet to Jet-H/D=0.2
0.007
0.009
0.011
CONF plate-Jet to Jet-Re=20000
0.013 r
0.015
H/D=0.2 H/D=0.5
450 400 350
H/D=1.0 H/D=2.0
300 250 200 150 100 50 0 0.005
0.007
0.009
0.011
0.013 r
0.015
Local Nusselt number in jet to jet direction and on impinging and confinement surface.
Figure 6 shows velocity vectors for H/D=0.2 and Re=20000 in jet to jet section as well as symmetric boundaries in Sn (shorter inlet to outlet tube distance) and Sp (longer inlet to outlet tube distance) directions (see Fig. 2). In the jet to jet section, two main vortexes as well as lower adverse vortexes were detected while two upper adverse vortexes are to be shaped. The velocity vectors in symmetric boundaries shows influence of outlet flows on main vortexes. The main vortex on confinement surface is stronger in Sn direction however the wake flow on the edge of entrance to outlet tube is weaker when distance between inlet-to-outlet tubes is longer (Sp direction). Thus one expects higher local Nusselt number on confinement plate in Sn direction and on outlet tube in Sp direction.
Figure 6:
Velocity vectors (jet to jet, Sn, and Sp directions).
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368 Advanced Computational Methods in Heat Transfer IX
Figure 7:
Temperature contours on impinging and confinement surfaces.
The temperature contours on impinging surface shows the curved maximum local Nusselt number location as well as secondary peaks in the interaction region of wall jets (Fig. 7). In H/D=0.5 the single maximum local Nusselt number line (almost at right angle to the jet to jet line) is seen on the confinement surface. These maximums are higher in the intersection of direct jet to jet line however they become weaker as distance from this line increases.
4
Conclusion
A three dimensional numerical flow and heat transfer study was conducted to investigate the characteristics of external flow of a novel patented compact heat exchanger comprising staggered array of impinging jets flowing in to opposite staggered array of tubes after impingement. The global heat transfer rate and pressure drop shows H/D=0.5 as an optimum design point. The local Nusselt number on impinging surface reaches a maximum with secondary maximum locations on each side however the heat transfer rate on the confinement surface depends strongly on H/D. The velocity vectors show different vortexes due to interaction of jets and temperature contours confirms results of local Nusselt number.
References [1] [2] [3] [4]
Carcasci C., An Experimental Investigation on Air Impinging Jets Using Visualization Methods. Int. J. Thermal Science, 38, p. 808-818, 1999. Jung-Yang S., Chin-Hao H., Ming-Hong S., Impingement Cooling of a Confined Circular Air Jet. Int. J. Heat Mass Transfer, 40(6), p. 1355-1364, 1997. Beitelmal A. H., Saad M. A., Patel C. D., The Effect of Inclination on the Heat Transfer between a Flat Surface and an Impinging Two-dimensional Air Jet. Int. J. Heat Fluid Flow, 21, p. 156-163, 2000. Chin-Yuan L., Suresh V. G., Prandtl-number Effects and Generalized Correlations for Confined and Submerged Jet Impingement. Int. J. Heat Mass Transfer, 44, p. 3471-3480, 2001.
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[5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15]
369
Morris G. K., Garimella S. V., Fitzgerald J. A., Flow-Field Prediction in Submerged and Confined Jet Impingement Using the Reynolds Stress Model. J. Electronic Packaging, 121, p. 255-262, 1999. Craft T. J., Graham L. J. W., Launder B. E., Impinging Jet Studies for Turbulence Model Assessment-II. An Examination of the Performance of Four Turbulence Models. Int. J. Heat Mass Transfer, 36(10), p. 26852697, 1993. Durbin P. A., On the k-ε Stagnation Point Anomaly. Int. J. Heat Fluid Flow, 17, p. 89-90, 1996. Behnia M., Parneix S., Shabany Y., Durbin P. A., Numerical Study of Turbulent Heat Transfer in Confined and Unconfined Impinging Jets. Int. J. Heat Fluid Flow, 20, p. 1-9, 1999. Park T. S., Sung H. J., Development of a Near-Wall Turbulence Model and Application to Jet Impingement Heat Transfer. Int. J. Heat Fluid Flow, 22, p. 10-18, 2001. Ramezanpour A., Shirvani H., Mirzaee I., Heat Transfer of Slot Jet Impinging on an Inclined Plate, Eighth Int. Conf. on Adv. Comp. Meth. in Heat Transfer, Lisbon, Portugal, 2004. San J. Y., Lai M. D. Optimum Jet-to-Jet Spacing of Heat Transfer for Staggered Arrays of Impinging Air Jets. Int. J. Heat Mass Transfer, 44, p. 3997-4007, 2001. Su L. M., Chang S. W. Detailed Heat Transfer Measurements of Impinging Jet Arrays Issued from Grooved Surface. Int. J. Thermal Science, 41, p. 823-841, 2002. Fluent User Guide, Fluent Inc., 2005. Kadar B. A., Temperature and Concentration Profiles in Fully Turbulent Boundary Layer. Int. J. Heat Mass Transfer, 24(9), p. 1541-1544, 1981. Yan X., A preheated-wall transient method using liquid crystals for the measurement of heat transfer on external surfaces and in ducts. PhD Thesis, University of California, Davis, 1993.
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The re-commissioning of a vent and reflux condensation research facility for vacuum and atmospheric operation J. C. Sacramento Rivero & P. J. Heggs School of Chemical Engineering and Analytical Sciences, The University of Manchester, UK
Abstract The vent and reflux condensers are condensers where the primary mode of operation is condensation. Double-pipe vertical condensers are commonly used for their study. On the process side, a vapour mixture enters from the bottom of a vertical tube and as it flows upwards it condenses at the cooled tube wall, forming a falling film flowing counter-current to the vapour. On the other side of the tube, the coolant flows downwards, counter-current to the rising vapour but co-current to the falling condensate film. This paper discusses the re-commissioning of the vent/reflux condensation research facility in the Morton Laboratory at the University of Manchester. The experimental facility was encountered in an inoperable condition after the physical refurbishment of the Morton Laboratory. The sequence of re-commissioning actions taken is detailed and discussed, including instrumentation calibration, the revision and writing of Standard Operation Procedures and modifications to the P&ID. Most of the instrumentation has been connected to the Emerson DeltaV® control system, allowing run-time data acquisition and automated control of the main process variables. A number of operational problems were encountered during the re-commissioning stages and the solution approach is described. The effect of the new and additional instruments/measurements on the mass and energy balances over the facility for previous and preliminary experimental investigations are detailed. Keywords: vent condenser, reflux condenser, re-commissioning, instrumentation, control system.
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372 Advanced Computational Methods in Heat Transfer IX
1
Introduction
The vent and reflux condensers are condensers where the primary mode of operation is condensation. Vent condensers achieve the highest possible condensation of the vapours, whereas reflux condensers only aim for a given separation, letting the most volatile component remain in the vapour phase. Therefore the difference between these two devices is just the operation conditions under which the same physical apparatus is run. Typical applications of reflux condensation are in overhead condensers of distillation columns, U-tube steam generators and two-phase closed loop thermosyphons. Vent condensers are often used to knock-out the vapour content from wet-air streams, such as those found in stirred-tank reactors or in the vent cooling section of aircooled steam condensers. For the research of reflux/vent condenser mechanisms, double pipe heat exchangers are often used (see fig. 1). On the process side, a vapour mixture enters from the bottom of a vertical tube and as it flows upwards it condenses at the cooled tube wall, forming a falling film flowing counter-current to the vapour. On the other side of the tube, the coolant flows downwards, counter-current to the rising vapour but co-current to the falling condensate film. Hence, a reflux/vent condenser is a coupled system of 5 regions (see fig. 1): (1) the vapour flowing upwards, (2) the condensate film flowing downwards, (3) the heat transfer surface, (4) the coolant flowing downwards and (5) the control loop to provide the inlet temperature of the coolant. The design information available for vent and reflux condensers presented in ESDU [1] normally leads to inadequate overdesign factors which can be as large as 70% or more, Jibb and Drogemuller [2]. This uncertainty in the design stage exists because the mechanisms of heat and mass transfer are not well known. Furthermore, the oversized condensers incur high capital and operational costs. Another concern in vent and reflux condensers operation is flooding. Flooding occurs when liquid is pushed upwards and exits along with the vapour at the top of the condenser. The research facility was designed to study the coupled heat and mass transfer effects in the reflux condensation phenomena and allows the analysis of the main process variables using run-time data acquisition through the Emerson DeltaV® Distributed Control System (DCS). It also allows the experimenter to investigate the flooding conditions over a range of pressures up to atmospheric conditions. The main purpose of this research facility is to produce experimental data to support the development of design and performance evaluation methodologies which will reduce the large safety factors used so far in the design of this type of condenser and consequently reduce capital and operational costs, as well as energy consumption.
2
Description of the research facility
The research facility in the Morton Laboratory at the University of Manchester can be run in three different modes: reflux/vent condensation, single phase WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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heating and single-phase cooling. Before starting a research programme it must be ensured that the proper configuration is adopted. The re-commissioning works were centred only on the reflux/vent condensation mode. The pipework connecting to irrelevant sections of the rig were duly isolated and tagged. Vapor Out Coolant In
Tube Wall [4]
[5]
Annulus Coolant Recycled
[1] [3]
[3] [2]
[2]
[4] Coolant Out
Figure 1:
Vapor In Condensate Out A sectional sketch of the vent and reflux condenser (not to scale) showing the 5 regions of interest.
The main section of the facility is the reflux/vent condenser or test section. This consists of a 3 m long vertical copper tube with an internal diameter of 0.028 m. The annulus is a stainless steel jacket designed to allow the cooling water to enter in three different locations, so the overall length of reflux/vent condensation can be adjusted to 1 m, 2 m or 3 m. Before the vapour mixture hits the test tube, it enters to an inlet pot that distributes evenly the vapour over the tube surface. The test tube is tapered at the bottom end in a 30° angle. This should allow an increase of 5% in vapour velocity before the system floods compared to the same tube without a taper (Palen and Yang [3]). Temperature measurements of the coolant are taken at the annulus inlet and outlet. Also, temperature measurements of the coolant and the tube wall are taken at 6 equally spaced points along the test tube length, allowing the validation of the local heat WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
374 Advanced Computational Methods in Heat Transfer IX transfer coefficients. Furthermore, the heat flux profile on the coolant side can be evaluated. The apparatus can operate using multicomponent mixtures and air. Fig. 2 shows a diagram of the rig. Initially, the selected mixture is fed to the boiling tank. Here, it is heated to its saturation temperature, which will vary with the pressure of the system. Normally, the rig is operated under vacuum, so boiling of the mixture is achieved at lower temperatures, although atmospheric operation is possible. The vacuum is generated by a two-stage Liquid Ring Pump (LRP) which uses water as the sealant liquid. The vapours generated in the boiling tank enter directly to the reflux condenser. There, the enthalpy of the rising vapours is removed by the cooling water flowing downwards in the annulus section. The cooling water flow can be manipulated to achieve partial or total condensation of the mixture. If used as a total condenser, all the vapours condense in the test section and are recycled to the boiling tank. If run as a partial condenser, the uncondensed vapours leaving at the top of the reflux condenser are driven to the after-condenser and knocked out. The two streams of condensate leaving both condensers are collected in the condensate collection pot and recycled by gravity to the boiling tank. 2.1 The instrumentation The research facility is fully instrumented, covering all the variables required to close the mass and energy balances. The following paragraphs detail the different types of instrumentation currently in use. 2.1.1 Temperature Most temperature readings are made using copper/constantan T-type thermocouples. These generate higher mV output per °C than other types of thermocouples in the operational range (0–350°C). All the existing thermocouples needed to be re-calibrated since they were non-operational for a period of 3 years. The calibration procedure is described below. Resistance Temperature Detectors (RTD) are used to measure the inlet and outlet temperatures in the oil heaters and the vapour temperature in the boiling tank. The output of the RTD model TC® Pt-100 is a 4–20 mA analog signal proportional to the sensed temperature. The RTD has a longer term stability but a slower response than the thermocouples. Despite the robustness of these meters, a linearity response check was made. 2.1.2 Pressure Where visual pressure readings are desirable, normal manometers are placed, with operational ranges between 0 to 2 bar. These field readings are only to support the online values recorded through the DCS using Pressure Transmitters (PT). The PT consists of a membrane that distorts when pressure/vacuum is applied. This distortion generates a current (4–20 mA) proportional to the applied pressure. The models of the PT used in this facility are DRUCK® PT600 and PT500 with operational ranges varying within 0–1.6 bar, to allow the operation of the rig under vacuum and just above atmosphere. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Figure 2:
375
A diagram of the heat transfer research facility showing the main instrumentation.
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376 Advanced Computational Methods in Heat Transfer IX To measure the pressure drop across the test section, a Differential Pressure (DP) cell model Rosemount® 1151-E is used to double-check the readings retrieved by two PTs located at the bottom and at the top of the test section. It has been observed that the DP cell reading is more accurate than that of the difference of the PTs readings. 2.1.3 Flows All cooling water flows are measured with rotameters. These provide a visual reading of the volumetric flow. Since the measured fluid is water no further calibration of the sight glass scale was needed and the supplier calibration curves can be used. When the measured fluid is air, a correction for pressure is required so a pressure gauge was fitted at the air inlet of the rotameter. For the gravity make up in the test section cooling water loop, a scaled glass vessel (13-T303 in fig. 2) is used to manually read the volumetric flow. For flow measurement of the process condensate, which vary in composition during normal operation, it was necessary to account for density changes. Coriolis flow meters provide not only accurate readings for small mass flow but also density measurements, which can be translated into composition estimates when the condensate is a multicomponent mixture. An orifice plate is used to measure the hot oil flow. The differential pressure across the orifice plate is measured by a DP cell similar to that used to measure the pressure drop across the test section. The equation and algorithm used to estimate the flow from the measured differential pressure was taken from the relevant British Standards [4, 5] and it is of the form: V = C d Ao
2 ρ∆P 1− β 4
(1)
where, V is the volumetric flow, Cd is the coefficient of discharge, Ao is the sectional area of the orifice, ρ is the density of the fluid, β is the orifice to pipe diameters ratio and ∆P is the measured differential pressure. 2.2 The Distributed Control System (DCS) Most of the instrumentation is administrated by the Emerson DCS DeltaV®. It allows continuous recording of compatible instrumentation and manipulating controllers. Every plant instrument sends a discrete or analog signal (current or voltage) and it is received by an I/O card. The signal is converted to digital signal and sent to a Controller, within which all data will be processed and actions will be taken, in the case control valves. Both I/O cards and controllers are inside Local Control Panels (LCP), located in the plant area close to the equipment. All signals received by the controllers are sent to operator stations within the Morton Laboratory control room. From there, they can be monitored and controlled using a graphical interface.
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377
The re-commissioning stages
The experimental facility was in an inoperable condition after the physical refurbishment of the Morton Laboratory. The LRP was physically displaced, almost all the insulation was torn up and several flanges were loose. The schedule of the re-commissioning works comprised the following stages: 1. An integrity check of the experimental apparatus. 2. Updating the documentation of the research facility. 3. The set up of the instrumentation. 4. The set up of the control system. 5. The insulation of the apparatus. 6. The uncertainty analysis. 3.1 Checking the integrity of the experimental apparatus The first activity of the re-commissioning was a full pipe tracing, comparing the physical apparatus with the Process and Instrumentation Diagram (P&ID). Several corrections had to be made to the P&ID since various modifications were made during the refurbishment of the Morton Laboratory, i.e. additional valves and instrumentation. During the pipe tracing all the flanges and valves were checked so any loose bolts and blocked pipes were identified. The sections that were not relevant to the reflux/vent condensation experiments were isolated and the isolation valves were tagged to prevent accidental opening. Once the line tracing was complete a hydraulic test was carried out in the cooling water side. Filling the cooling water system revealed several leaks and pertinent actions were taken. Finally, a “drop test” was made to first locate the air leaks locations and then quantify the residual air leakage. The LRP was used to generate vacuum (0.1 bar) in the rig and a careful inspection of all flanges in the rig revealed the air leaks location. After tightening all the guilty flanges, the drop test was carried out. The test consists in putting vacuum into the system, closing the vacuum pump line and recording the time, t, that the system takes to reach a certain higher pressure. If the system volume, V, is known, the ideal-gas law is used to calculate the air mass flow, M as follows: 29 V∆P (2) M = t RT where, ∆P is the pressure difference along the test, 29 is the molecular mass of air and T is the system temperature. Eqn. (2) assumes that air leaks at sonic velocity, i.e. at a constant rate over the whole test. Therefore, both initial and final pressures should be less than approximately 0.53 times atmospheric pressure (Ryans and Roper [6]). The air leakage was estimated at 0.12 kg/h and it was considered acceptable so no further leak tests were made. 3.2 The set up of the instrumentation In the case of new instrumentation, a selection procedure is needed to choose the most appropriate instrument type for the application. In the case of the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
378 Advanced Computational Methods in Heat Transfer IX thermocouples, the nature of the process, the operation range and the signal quality were the most important parameters, as stated above. Special attention was given to the selection of the coriolis meters. The occasional presence of two-phase flow in the condensate lines made it difficult to size the meter and most vendors warn about high instability of the output signal when two-phase flow is present. The recommended transmitter was the Foxboro® CFT50 which can cope with this situation. The meter was also slightly oversized in such a way that the presence of the gaseous phase will not cause the mass flow to exceed the meter range. Once the selection is made, the installation of the instrument follows. If the instrument is to be connected to the DCS, the signal wire is selected and run from the terminal side of the instrument itself to the LCP and attached to its I/O card. When the DCS shows some reading from the instrument, it is ready for in situ calibration. The calibration of any instrument consists in adjusting the output signal of the instrument to the desired range. For example for a DP cell, a known pressure is applied to the high-pressure terminal (leaving the low pressure terminal open to atmosphere) and the known differential pressure is compared to the reading retrieved from the DCS. If the values do not match, the pertinent corrections to the Zero and Span values in the DP cell are made and the test is repeated several times until the repeatability of the readings is acceptable. This procedure applies to all instrumentation with linear output, such as PTs, RTDs and Coriolis meters. The thermocouples output is not linear, so a different approach is required. The calibration of the thermocouples is made using a Signal Characterisation Block (SCB) in the instrument configuration in the DCS. The SCB stores a comparison table between the recorded value of each thermocouple and a reference value considered to be accurate. In this case the reference temperature was taken from a Platinum Resistance Thermometer (PRT) which was calibrated by the vendor to an accuracy of ±0.03°C for the used range. The readings of the PRT showed to be in good agreement with an alcohol thermometer as well. Each thermocouple was exposed to different steady temperatures within the range of the process conditions and it was compared against the PRT reading. The results were averaged and these calibration parameters were stored in a SCB for each thermocouple. During normal operation the DCS will interpolate the value the PRT would have using the thermocouple sensed value and the calibration parameters. 3.3 The set up of the control system After all the instrumentation is properly connected to their respective LCP, each instrument needs to be configured in the DCS. Care must be taken to (1) ensure each reading in the DCS front end shows the correct instrument signal, (2) check the read scale in the DCS configuration matches the calibrated scale of the instrument and (3) check that the calibration parameters are correctly input in the SCB in the case of instruments with non-linear output. It was also necessary to re-draw the front end graphics to accommodate for the changes in the P&ID, that is new pipework and additional instrumentation. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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3.4 Updating the documentation of the research facility After all modifications were made, the facility P&ID was corrected and updated. The equipment and valves were re-numbered to conform to the general Morton Laboratory numbering system. New Standard Operation Procedures (SOP) were written for the calibration of the instrumentation and the existing ones were updated to conform to the new configuration of the facility and the university’s new safety policy. Before any physical change was made to the rig, a Management of Change (MOC) form had to be submitted, explaining the nature of the job and the key safety issues that may arise from the modification, as well as the impact the new addition may have in the normal operation of the apparatus. Safety documentation also includes Risk Assessments (RA) and COSHH forms, since some research using water/methanol mixtures is programmed. Finally, a full HAZOP study is programmed to evaluate the hazards the operation of the rig may involve. 3.5 The insulation of the apparatus The insulation thickness of the pipework was calculated in such a way the external temperature do not exceed 50°C for safety purposes. The test section was insulated to minimise the heat losses as those temperature measurements are critical for the condenser performance evaluation. The calculation of the insulation thickness contemplated the following resistances: the forced convection inside the pipe, the pipe wall resistance, the insulation material resistance, the external radiation to the ambient and the external free convection. 3.6 The uncertainty analysis All the acquired data have an embedded uncertainty that comes from the nature of the instrument used. When correlations are used to calculate certain parameters an additional uncertainty adds to the final result. To evaluate the overall uncertainty in the presented results the following expression must be used: 1/ 2
2 δX 2 δX 2 δX 2 δX n 3 2 1 + " + N + c + b (3) = a R X 1 X 2 X 3 X n where δXi is the uncertainty of the measured variable Xi expressed as a percentage. Eqn. (3) assumes that the result R is a variable that can be expressed in a pure product form, i.e. R = X 1a X 2b X 3c " X nN . Moffat [7] describes three types of uncertainty analysis: zeroth-, first and nth-order uncertainties. Following Moffat’s nomenclature, the results presented here correspond to the first order uncertainty for single sample experiments. This is intended to describe the scatter that should be expected in a set of observations using the given apparatus and measuring system. This estimate includes all sorts of variable errors, but does not include fixed errors of any kind. It is reported to
δR
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380 Advanced Computational Methods in Heat Transfer IX compare set of data of experiments done in the same apparatus with the same instrumentation, but should not be used to compare experiment results from different rigs, as the fixed errors are not accounted for. The first order uncertainty is obtained using the chi-squared statistic applied to the data retrieved from an auxiliary experiment. From this analysis it can be expected a scatter of the results in the mass and energy balances between 3 and 6%.
4
Future work
With the research facility as it is now, the researcher work splits between the manual data acquisition of some instruments and processing the data of all instrumentation using spreadsheets. A great time saving for the researcher would be to implement a real-time data analysis in DeltaV® in such a way that the mass and energy balances were calculated as the instrument data are read. It will also permit the identification of steady-state operation of the system based on the energy balances rather than in just isolated values of some variables. To achieve this, it would only require writing the same equations in the spreadsheet in Calculator modules within DeltaV®. However and obviously all rotameters and visual instruments have to be substituted by instrumentation with an output signal, which is expensive. If all the instrumentation in the rig is connected to the DCS it would also be possible to include the data uncertainty analysis in run-time, as explained by Moffat [7]. This would allow identifying which measurements are the major source of error during certain experimental run and would prevent the experimenter of doing potential unusable runs.
References [1] ESDU, Engineering Sciences Data Unit (Data Item 89038). Reflux Condensation in Vertical Tubes, 1989. [2] Jibb, R.J. & Drogemuller, P., Design and Application of Reflux Condensers for Separating Vapour Mixtures. 3rd International Conference on Process Intensification, 1999. [3] Palen, J. & Yang, Z.H., Reflux Condensation Flooding Prediction: Review of Current Status, Chemical Engineering Research and Design. Trans IChemE, 79 Part A, pp. 463-469, 2001. [4] British-Standard. Measurement of fluid flow by means of differential pressure devices, BSi. BS1042-1.1:1992 renumbered BS EN ISO 51671:1997, 1997. [5] British-Standard. Measurement of fluid flow in closed conduits, BSi. BS1042-1.4:1998, 1998. [6] Ryans, J.L. & Roper, D.L., Process Vacuum System Design & Operation, ed. McGraw-Hill, 1986. [7] Moffat, R.J., Describing the Uncertainties in Experimental Results. Experimental Thermal and Fluid Science, 1, pp. 3-17, 1988.
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Heat transfer modelling in double pipes for domestic hot water systems I. Gabrielaitiene, B. Sunden & J. Wollerstrand Department of Energy Sciences, Lund University, Sweden
Abstract The purpose of this work is to numerically investigate the heat transfer in a pipeline layout for domestic hot water systems. This layout, called double pipe, includes two adjacent counter-flow pipes, placed one above another in a common insulation (i.e., there is no insulation layer between the two pipes). However, when using this layout, the prediction of fluid temperatures in the pipes becomes complicated as thermal coupling occurs between the two pipes resulting from the presence of a temperature difference and physical contact of the two (copper) pipes. This coupling phenomenon is difficult to account for using analytical approaches and a numerical method, namely a finite volume method, was therefore applied in this study. It was found that a realizable k-epsilon, two-equation, turbulence model with non-equilibrium wall functions showed the best performance in terms of heat transfer prediction. The validation was carried out against the empirical Nu number correlation developed at uniform heat flux conditions. Since this condition is not relevant for the flows in the double pipe, these were simulated as being placed in separately insulated pipes. The results from modelling the double pipe layout showed that the heat flux increases compared to a single pipe arrangement. Keywords: double pipe, domestic hot water systems, coupled thermal regime, low Re number turbulent regime, two-equation turbulent models.
1
Introduction
In domestic water systems, where hot water is distributed around a building from central supply, temperature control becomes an important issue due to restrictions associated with various health risks. The risk is connected with the pathogenic bacterium Legionella, which grows at temperatures below 46°C [1]. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060371
382 Advanced Computational Methods in Heat Transfer IX Another risk is associated with scolding of skin, which typically limits the upper water temperature to 65°C [2]. Based on the above-mentioned restrictions, the requirements for hot water systems [3] state that the outgoing service temperature should be of 55–60°C and a return temperature not lower than 50°C. To meet the lower temperature bound, hot water recirculation is used. Since water recirculation system requires a closed loop between hot water generation and consumption points, return pipelines are added to existing supply lines. They are connected directly and are covered with common insulation forming the double pipe arrangement. In the study described in this paper, no insulation layer was provided between the supply and return pipes which distinguishes our layout from those previously investigated [4]. Our considered layout provides several advantages over layouts in [4], such as compactness and simple installation, which are achieved at a competitive price. It also offers the flexibility in selecting different pipe combinations, since the simple assembling of the pipes can occur locally. Using this layout however adds complication to fluid temperature predictions as thermal coupling occurs between the two pipes resulting from the presence of a temperature difference and the physical contact of the two (copper) pipes. This coupling phenomenon is difficult to account for using analytical approaches and a numerical method, namely the finite volume method, was therefore applied in this study. The double pipe was represented by a 3D numerical model and analysed with the general-purpose code Fluent. Reynolds averaged Navier-Stokes and energy transport equations were employed for the fluid flow and heat transfer solution. For the turbulence modelling, the main emphasis has been placed upon high Reynolds number, two-equation models, augmented by the wall-functions approach. Two-layer-zonal models, which do not utilize the wall functions approach, have also been considered for comparison purposes. It was found that realizable k-epsilon turbulence model with non-equilibrium wall function showed the best performance in terms of heat transfer prediction. In the present study, consideration was given to the heat transfer analysis including the heat transfer coefficient and temperature distribution in the pipe. Attention was focused on the low Reynolds number turbulent flow (Re= 8 ⋅ 103 − 11 ⋅ 103 ), because high water velocity can cause local erosion in copper pipes and is therefore generally avoided.
2
Problem statement
In this work, a new layout was analysed, which included two adjacent counterflow pipes, placed in a common insulation, as seen in Fig. 1. The copper pipes were covered with fibreglass insulation and a layer of aluminium foil on the exterior [5]. The supply pipe (with external diameter 18 mm) was located above the return pipe (external diameter 12 mm) and Reynolds numbers were 8350 and 11453, respectively. Flow velocities were 0.25 m/s and 0.64 m/s for the 18 mm and 12 mm pipes, respectively, which satisfied the condition for velocity being
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below 0.7 m/s to avoid local erosion. A recirculation regime was only considered, implying that the flow volume was the same in both pipes. Front View
View A-A
A
Symmetry plane
Qconv
u1,T1
Supply pipe
'
'
u1 ,T1
Return pipe 12
110mm
1
18
Insulation
y
x A
Figure 1:
Insulation
z Qconv
L
Schematic diagram of the double pipe layout
The two thermal regimes selected for the study accounted for the most typical regimes present in domestic hot water systems: i) The inlet temperatures were T1=60°C and T1' =50°C (according to Fig.1) ii) T1=55°C and T1' =50°C. The pipe was assumed to be placed in indoors with temperature 20°C. The contact area between the supply and return pipes was modelled by assuming a small piece of copper material inserted between the pipes (dimensions x=0.5 mm, y=0.01 mm), because from the modelling viewpoint it was not possible to represent the contact area as a single point, which would be the case when two circles are adjacent. To simplify the analysis, the flow was assumed to be fully developed. In defining the material properties of the water, constant values were prescribed, which corresponded to an inlet temperature according to the selected thermal regime. As the fluid temperature changed along the pipe by less than 2°C (pipe length L=1 m), neglecting temperature-dependent water properties did not distort the results of this study. The thermal insulation properties were assumed to be independent of temperature and were selected according to the average temperature. The variation of the thermal conductivity corresponding to the maximum and minimum insulation temperatures is less than 10% (k(30°C)=0.0335 W/mK, k(60°C)=0.0366 W/mK [5]), and could therefore be neglected without introducing significant errors.
3
Mathematical and numerical modelling
For the numerical investigations, the general-purpose CFD code Fluent [6] has been used, which applies a finite volume method to discretize the governing equations. Ensemble-averaged continuity, Navier-Stokes and energy transport WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
384 Advanced Computational Methods in Heat Transfer IX equations have been solved for incompressible 3D turbulent pipe flow. 3D heat transfer by conduction has been also solved for the copper pipe walls and insulation. Different two-equation turbulence models, namely the standard k-epsilon model [7], the RNG k-epsilon [8], and the realizable k-epsilon model [9] have been used. The modelling of the near-wall region is especially important in convective heat transfer problems. Several formulations including standard [10] or non-equilibrium wall-function [11] (for bypassing this region adoption logarithmic wall-functions) and two-layer zonal methods (adoption low Reynolds number amendments [12] to accurately resolve the near-wall regions) have been considered. The high Reynolds number models offer the advantage over two-layer-zonal model in reduced computational overhead. Two-layer-zonal methods on the contrary require a very fine grid resolution of the near-wall region, which increases the computational overhead considerably, especially in transient problems. As the transient conditions will be considered in the future research, modelling by high Reynolds number models augmented by the wall-functions approach were the focus in this work, whereas two-layer-zonal method was only considered for comparison purposes. The solution domain represents the 3D double pipe shown in Fig. 1, where only the left part from the symmetry plane is considered for minimizing computational efforts. At the inlet, constant velocities (u1, u1' ) and temperatures (T1, T1' ) were applied assuming spatially constant profiles. Inaccuracies due to an uncertainty in the shape of inlet velocity and temperature profiles do not play an important role, since the pipe length was more than 60-pipe diameters. The conditions for the turbulent quantities were derived assuming a turbulence intensity of 5%. Convective boundary conditions were applied on the external surface of the double pipe, where the heat transfer coefficient includes convective and radiative terms. The convective term was estimated from the empirical relation for Nu number developed by Churchill and Chu for free convection from horizontal cylinders [13]. The radiative term was determined from an equation developed for a hot convex object in large enclosures, such as a room [13]. The solution domain was discretized by an unstructured grid. The number of computational cells was adjusted in an optimal way according to the Reynolds number for achieving grid independent results. Figure 2 shows the used computational mesh in the double pipe cross-section. For the standard k-epsilon model, the non-dimensional wall distance describing the location of the near-wall cell is recommended to be y + ≥ 30 . However, the investigation in [14] indicates that the accuracy of the wall-function starts to show a remarkable deterioration only beyond y + < approx.12 . Therefore, the values of y+ were allowed to be 20 for the extreme case of low Re number in the pipe with 12 mm diameter.
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Advanced Computational Methods in Heat Transfer IX
Figure 2:
385
Example of grid in the double pipe cross-section.
In order to investigate the effect of grid size, the simulations were carried out with grid size variation between 1.5 million cells and 3 million cells. The grid was refined near the inlets of supply and return pipes to capture the locally high axial gradients in the entry region of the developing flow. This was followed by a smooth grid expansion towards the middle of the pipe. It was found that there is practically no change in the outlet fluid temperature when grid size was increased beyond 1.5 million cells. To attain a high numerical accuracy, second order schemes were used for the spatial discretization of the governing equations.
4
Numerical results
4.1 Single pipe arrangement Before beginning the double pipe analysis, validation was carried out against the empirical Nu number correlation developed at a uniform heat flux condition around the wall circumference. Since this condition is not relevant for the flows in the double pipe arrangement, these were simulated as being placed in separately insulated pipes. The flow characteristic and inlet temperatures (T1=60°C and T1' =50°C, for the supply and return pipes, respectively) were the same as for the flows in the double pipe. Both pipes were simulated with different turbulence models, as described in the previous section, to find a model with the best agreement with empirical values. For empirical Nu numbers, the standard Dittus-Boelter correlation summarized in [13] was considered: Nu = 0.023 ⋅ Re 0.8 ⋅ Pr 0.4 (1) The empirical heat transfer coefficient was estimated from the following equation: (2) h = Nu ⋅ k / Dhyd
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386 Advanced Computational Methods in Heat Transfer IX The predicted heat transfer coefficients using different turbulence models, for Re=8351 and Re=11453 are presented in Table 1, and the percentage deviations from the empirical values are indicated in brackets. It can be observed that all models over-predict the empirical values of the heat transfer coefficient, however with different magnitudes. For the Re=8351, the RNG model (with standard wall function) showed highest deviations from empirical values, while predictions obtained for Re=11453 indicate that the standard k-epsilon model produces the highest deviation. Furthermore, it can be observed that the use of “standard” or “non-equilibrium” wall functions has a substantial influence on the results. Table 1:
Predicted and empirical heat transfer coefficients.
Empirical heat transfer coefficient:
Re=8351 (D18 mm)
Re=11453 (D12 mm)
2004
4360
Predicted heat transfer coefficient according to the turbulence model: Standard k- ε , standard wall f. Standard k- ε , non-equilibrium wall f. Standard k- ε , two-layer zonal near-wall model RNG k- ε , standard wall funct. RNG k- ε , non-equilibrium wall f. Realizable k- ε , standard wall f. Realizable k- ε , non-equilibrium wall f.
2199 (9.7%) 2188 (8.1%) (9.9%)
4754 (9%) 4599 (5.5%) (8%)
2210 (10.3%) 2163 (7.9%) 2126 (6.1%) 2092 (4.4%)
4751.1 (8.9%) 4569 (4.8%) 4640 (6.4%) 4431 (1.7%)
Based on the comparison presented in Table 1 it may generally be concluded that the realizable k-epsilon turbulence model with non-equilibrium wall function performs better than others, with a deviation of approximately 5% from the empirical value. 4.2 Double pipe arrangement The effect of thermal coupling between two copper pipes is presented in Figs. 3 and 4 in terms of heat flux and temperature distribution. They are shown at the double pipe longitudinal location z=-0.5 m for two thermal regimes i) inlet temperatures were T1=60°C and T1' =50°C (for the 18 mm and 12 mm pipes, respectively), and ii) T1=55°C and T1' =50°C. It was found, that non-uniform heat flux prevailed around the perimeter of the supply and return pipes, and was considerably greater at the contact interface of the two pipes than their average value. It is increased in the above-mentioned area by approximately 60% for the first thermal regime, where the positive heat flux is assumed for the pipe gaining heat. The difference in heat transfer rates between the supply and return pipes is depicted in Fig. 3 as a function of pipe perimeter length. The total heat transfer
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D18
pipe 12mm (T1=60'C, T1'=50'C)
0.25
pipe 18 mm (T1=55'C, T1'=50'C)
14000
0.5
8
0
pipe 18 mm (T1=60'C, T1'=50'C)
D12
Difference in heat transfer rates between pipes, %
0 0.25 0.5
9000
Dimesion. length of pipe perimeter
7 6 5
4000 4 3
-1000
2 -6000 1 -11000
0 0.0
Dimensionless temperature
Figure 3:
0.1
0.2
Differ. in heat transfer rates between
Heat FLux [W/m2]
pipe 12 mm (T1=55'C, T1'=50'C)
pipes , [%]
rate for the 18 mm pipe differs from the 12 mm pipe by 2%, which is due to the heat losses to the surroundings.
0.3 0.4 0.5 Dimensionless length of pipe perimeter
Heat flux distribution in the double pipe cross-section for two thermal regimes: i) T1=60°C, T1' =50°C and ii) T1=55°C, T1' =50°C.
1.5 1 0.5 0 -0.5
D18
Pipe 12 mm (T1=60'C, T1'=50'C)
-1
Pipe 18 mm (T1=60'C, T1'=50'C)
-1.5
-2.5 -1.2
Figure 4:
0 -1 D12
Pipe 18 mm (T1=55'C, T1'=50'C) Pipe 12 mm (T1=55'C, T1'=50'C)
-2
1
1 0 -1
Dimesion. length
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8 1 1.2 Dimensionless length
Temperature distribution in the double pipe cross-section for two thermal regimes: i) T1=60°C, T1' =50°C and ii) T1=55°C, T1' =50°C.
Due to non-uniform heat flux around the pipe circumference, the anisotropy of turbulence in the wall region might be more pronounced, than in case with WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
388 Advanced Computational Methods in Heat Transfer IX uniform heat flux. Since the turbulence models employed in the study neglect that phenomenon, the more advanced turbulence model, which account for the anisotropy of turbulence, can be recommended for future research. The temperature profile of the fluid in the double pipe is presented in Fig. 4 in dimensionless form, which is defined from eq. (3): T − Twall (3) Tdim = Tcentre − Twall where Tcentre – temperature at the pipe centre; Twall – wall temperature, taken at the location, where the dimensionless length equals 1 (see Fig. 4) for the 18 mm pipe, while for the 12 mm pipe the wall temperature is taken at the location, where the dimensionless length is equal to –1. As can be seen from Fig. 4, the dimensionless temperature profile, which is independent of pipe length (except for the entrance effect), has practically the same form irrespective of considered thermal regime. The temperature profile is non-symmetric and the temperature peak is shifted from the pipe centre towards the outer pipe surface (where dimensionless length is equal to 1, Fig. 4). Furthermore, for the 12 mm pipe the wall temperature is greater than the fluid temperature and the cooling regime prevails. The heat transfer through the contact interface causes the fluid temperature in the supply line to drop more than in a case of single pipe arrangement. In the return pipe, on the contrary, the pipe is gaining heat and the fluid temperature increases with length, as can be seen in Fig. 5. 4.3 Simplified estimation of fluid temperatures in double pipe arrangement The temperatures of the warm and cold fluids (the 18 mm and 12 mm pipes, respectively) in the inlet and outlet of the double pipe can be estimated on the basis of equations developed for counter flow in recuperators [15]. A conduction shape factor, which is a prerequisite for solution of these equations, was determined on the basis of the results obtained from the numerical modelling. The defining equation for conduction shape factor is recalled as: S = q / k ⋅ ∆Toverall (4) where k – thermal conductivity [W/mK]; q – heat flow [W]. The outlet temperatures ( T2 and T2' , for warm and cold fluid, respectively) are estimated from equations (5) and (6): T1 − T2' = (T1 − T2' ) / [1 − 1 / exp(kS / C )] (5) kS / C T1 − T2 = (T1 − T1' ) ⋅ 1 − 1 / exp (6) 1 + kS / C where C – product of fluid specific heat and mass flow rate [W/K]; S – conduction shape factor [m], T and T’ - temperature of warm fluid (18 mm pipe) and cold fluid (12 mm pipe), respectively. Subscripts 1 and 2 indicate inlet and outlet temperatures, respectively. After the outlet temperatures were estimated, the longitudinal temperature distribution is obtained from equations (7) and (8): WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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T1 − T = (T1 − T2' ) ⋅ [1 − 1 / exp(ks / C )]
(7)
T2'
(8)
'
−T =
(T1 − T2' ) ⋅
[1 − 1 / exp(ks / C )]
where s – conduction shape factor [m] is presented as a function of longitudinal coordinate and was estimated based on the numerical results. Longitudinal temperature distributions obtained from the numerical solution and from eqs. (7) and (8) are presented in Fig. 5 for the supply and return pipes (diameters 18 mm and 12 mm, respectively). The difference between the numerical and simplified solutions is larger in the flow entrance area. 53
Pipe 18 mm, from eq. 6 Pipe 18 mm, from numerical solution
60
52.5
Pipe 12 mm, from numerical solution Pipe 12 mm, from eq. 7
59.5
52 51.5
59
51
58.5
50.5
58 57.5
Temperature for pipe 12 mm ['C]
Temperature for pipe 18 mm ['C]
60.5
50
57
49.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless length
Figure 5:
Longitudinal temperature distribution in the supply and return pipes (diameters 18 mm and 12 mm, respectively).
From the practical viewpoint the described methodology is useful for estimating fluid temperature in the double pipe arrangement. The presented results are applicable for the considered thermal regimes and Reynolds numbers.
5
Conclusions
In the present work, a numerical study has been performed to investigate conjugated forced convection in two adjacent counter-flow pipes, placed one above another in a common insulation – double pipe layout. For the considered thermal regimes (temperatures 60°C and 55°C in supply pipe, and temperature 50°C in return pipe) the effect of thermal coupling is significant. The enhanced heat transfer at the contact interface between two pipes modified greatly the temperature profile and caused non-uniform heat flux distribution in both pipes. It was found, that a realizable k-epsilon, two-equation, turbulence model with non-equilibrium wall function showed the best performance in terms of heat transfer prediction. This prediction is highly sensitive to the turbulence model WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
390 Advanced Computational Methods in Heat Transfer IX selection for the low Reynolds numbers turbulent regimes (Re=8351 and Re=11453). The simplified equations developed for counter flow in recuperators were applied to estimate the fluid temperatures in the double pipe arrangement. A conduction shape factor, which is a prerequisite for solution of these equations, was determined based on the results obtained from numerical modelling.
References [1] [2] [3]
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
Brundrett, G.W., Legionella and building services, ButterworhtHeineman: Oxford, 1992. Standards and regulations for engineering systems, BFS 1993:57, 2002:19 (in Swedish, Boverkets byggregler BBR) District heating systems: installation and design. FVF F:101, Svenska Fjärrvärmeföreningens Service Ab, 101 52 Stockholm. (in Swedish, Fjärrvärmecentralen – Utförande och installation, Svenska Fjärrvärmeföreningen). Jonson, E., Heat losses from district heating network – influence of pipe geometry (in Swedish), Thesis of Licentiate of Engineering, Lund University of Technology, 2001. Properties of fibreglass insulation, http://sg-isover.dk Fluent 6.1, User’s Guide, Fluent: Lebanon, 2003. Launder, B.E., Spalding, D.B., The numerical computation of turbulent flows, Computational Methods in Applied Mechanical Engineering, 3, pp. 269-289, 1974. Yakhot, V., Orszag S.A., Renormalisation group analysis of turbulence: I. Basic theory, Journal of Scientific Computing, 1, pp. 1-51, 1986. Shih, T.H., Liou, W.W., Shabbir, A., Zhu, J., A new k-epsilon eddyviscosity model for high Reynolds number turbulent flow, Computational Fluids, 24, pp. 227-238, 1995. Launder, B.E., Spalding, D.B., The numerical computation of turbulent flows, Computational Methods in Applied Mechanical Engineering, 3, pp. 269-289, 1974. Kim, S.E., Choudhury, D., A near-wall treatment using wall functions sensitized to pressure gradient, in: Separated and Complex Flows, in ASME FED, vol. 217, ASME, 1995. Wolfstein, M., The velocity and temperature distribution of onedimensional flow with turbulence augmentation and pressure gradient, International Journal of Heat Mass Transfer, 12, pp. 301-318, 1969. Holman, J. P., Heat transfer, 9th ed., pp. 268-269, McGraw-Hill, 2002. Benim, A.C., Arnal, M., A numerical analysis of the labyrinth seal flow, Proc. of the 2nd European Fluid Dynamics Conf., eds. S. Wagner, J. Hirschel, J. Periauz, R. Pive, Wiley: Chichester, pp. 839-846, 1994. Hausen, H., Heat Transfer in counterflow, parallel flow and cross flow, McGraw-Hill: New York, 1983.
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Convective heat transfer investigations at parts of a generator circuit breaker T. Magier1, H. Löbl1, S. Großmann1, M. Lakner2 & T. Schoenemann2 1
Dresden University of Technology, Laboratory of Electrical Power Systems and High Voltage Engineering, Dresden, Germany 2 ABB Switzerland AG, High Voltage Technology High Current Systems, Zürich, Switzerland
Abstract Current carrying equipment for power engineering is getting smaller and more compact in order to meet customer demands. In spite of small dimensions it should be able to carry still growing currents, which lead to high current densities in the current carrying parts and therefore in high heating of these parts. Due to small dimensions the heat transfer from these parts is restricted too. To assure that the maximum temperature rise in equipment parts stays under the allowed temperature rise fixed in the standards, the temperatures in these parts should be calculated first (for instance with thermal networks). The power losses produced in the current carrying parts are transferred through convection, radiation and conduction from these parts to the ambient. For better cooling effectiveness several heat sink types, for example on the parts of generator circuit breakers can be used. To improve the accuracy of convective heat transfer calculation with thermal networks, the n1, c1 factors for affinity function Nu = c1(GrPr)n1 for natural convection, depending on the angle of the heat sink to the airflow, have been determined experimentally. Another point of thermal investigation is heat sink cooling with air, while the air is pre-warmed by other hot parts below the heat sink. This kind of interaction appears particularly in very compact devices and should therefore be investigated for a better accuracy of temperature calculation with thermal networks. Investigations have been carried out for different average temperatures of the heat plate, which was placed below the heat sink and for different heat sink positions. Keywords: thermal networks, heat transfer coefficient, affinity function, natural convection. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060381
392 Advanced Computational Methods in Heat Transfer IX
1
Introduction
As a result of still increasing operating currents and decreasing dimensions of the current currying equipment, which results in high temperatures of the current path and the enclosure, a correct thermal design for this equipment is more important than ever. All temperatures along the current path should stay under temperature limits given in adequate standards (for generator circuit breaker [1]). For a generator circuit breaker operating at maximum ambient temperature ϑ0 = 40°C the maximum allowable temperature is ϑmax = 105°C at silver-coated contacts and ϑmax = 70°C (in some cases 80°C) on the enclosure. Because of small dimensions and compact design of such devices the current densities in the current path are comparatively high and heat transfer from these device’s parts to the ambient is restricted. In order to increase the heat transfer from these parts to the cooling medium, additional heat sinks have to be used. They can be located on the current path (fig. 1a) or on the enclosure (fig. 1b).
Figure 1:
Heat sinks located on: a) current path and on b) enclosure
To improve the current carrying capacity of generator circuit breaker or other current carrying devices and for accelerating the development process of new devices thermal network models are used to predict the behaviour of the device. These models allow temperature calculation of device’s parts and examination of several changes in device’s design as well. In order to use thermal network models for development and improvement of electrical devices fundamental research on convective heat transfer were carried out. The following article deals with experimental convective heat transfer investigations at parts of generator circuit breaker. A method will be shown how to determine the n1, c1 parameters for affinity function Nu = c1(GrPr)n1 (free convection) for two heat sink types, depending on heat sink’s position in the air flow.
2
Temperature calculation with thermal networks fundamentals
The thermal network method, based on the analogy between electrical and thermal field, allows one to calculate the temperature distribution in electrical WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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devices. The thermal heat transfer between a device’s parts and from these parts to the ambient can be calculated as well [2, 5]. For the temperature calculation the device’s model has to be “imaginary” divided into n – “slices”. Each slice (fig. 2) is represented by thermal sources (thermal power losses in current path and in an enclosure PL and PK), convective (RK1, RK2, RK3), radiative (RS1, RS2) and conductive (RL) resistances and for forced convection by the thermal resistance of the coolant stream RV . For dynamic temperature calculation the heat capacities CL and CK have to be included.
Figure 2:
Thermal network for one section (slice) of generator circuit breaker.
Thermal power losses produced in the current path with temperature ϑL are convoyed partially via convection (RKo1) to the interior air with a temperature ϑai and from there to the internal side of the enclosure (RKo2) with ϑKi. Another part of thermal power losses produced in current path is convoyed via radiation (RS1) directly to the internal side of the enclosure and via conduction (RL) across the current path to the neighbouring sections (layers). The power losses arising from the current path to the internal side of enclosure sum up with power losses produced in this enclosure (PK) and are convoyed via conduction (RLK) to the outer side of the enclosure. From there, the power losses are transferred via convection (RKo3) and radiation (RS2) to the ambient air with the temperature ϑ0. In case of cooling via forced convection, the volume stream of cooling air ( RV ) convoys a major part of thermal power losses from current path directly to the ambient air.
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394 Advanced Computational Methods in Heat Transfer IX Each element of the thermal network can be described with adequate equations (Table 1). The accuracy of temperature calculations with thermal networks depends significantly on the accuracy of input parameters in equation (1) to (9) used for calculation. Assuming that the geometrical dimensions (l, d, lw, A, OKo, OS) are known, other parameters and physical constants depending on material, temperature and so on can be in most cases found in the literature. In some cases, especially for seldom-used geometries, or for common geometries under special conditions some parameters must be determined experimentally. In the following, experiments for determination of c1, n1 parameters at free convection (eqn. (4)) and c2, n2 parameters at forced convection (eqn. (5)) on heat sinks under special conditions will be described. Table 1:
Equations describing thermal network elements.
Thermal network elements Power losses in current path PL and in enclosure PK
Equations
PL = kI
2
ρ 20 l A
Thermal conduction resistance RL between parallel surfaces Convective resistance RKo Convective heat transfer coefficient αKo for free convection: Convective heat transfer coefficient αKo for forced convection: Radiation resistance RS Heat transfer coefficient of radiation αS Thermal resistance of the coolant stream by forced convection R V Thermal Capacity CW
3
[1 + α (ϑ − 20°C )]
α Ko =
λ lW
α Ko =
λA 1 = α Ko O Ko
Nu =
λ lW
(2)
d
RL =
R Ko
(1)
T
λ
c1 (Gr Pr ) 1 n
lW
Nu =
(3)
λ
c2 Re n2
lW 1 RS = α S OS
RV =
1
2c P ρ V CW = cW m
(5) (6)
T2 − T1 ϑ 2 − ϑ1 4
α S = ε 12 C S
(4)
4
(7) (8) (9)
Determination of c1, n1 parameters depending on heat sink’s angle to horizontal position (c1 = f(αH), n1 = f(αH))
A heat sink on the current path or on the enclosure of a generator circuit breaker can be placed with different angles to the ascending, cooling air (fig. 1). This can result in different values of the convective heat transfer coefficient αKo and c1, n1 parameters for each heat sink position. To improve the accuracy of temperature calculation with thermal networks, experimental thermal investigations to determine these parameters on two different heat sinks were carried out. The WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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smaller heat sink (fig. 3a), hereinafter called “HS1”, has a characteristic length lw = 107 mm, a convective surface OKo = 0,098 m² and a radiating surface OS = 0,034 m². The bigger one (“HS2” fig. 3b) has a characteristic length lw = 330 mm and respectively OKo = 0,399 m² and OS = 0,09 m². Both heat sinks were mounted on separate heat plates with controllable electric power. The power was measured with a power meter LMG 95. The heat plates were thermally insulated from the ambient with mineral wool, so that the major part of the supplied power was delivered to the ambient via the heat sink. The heat sink-, heat plate-, insulation- and ambient-temperatures were measured with thermocouples and all data were transferred to the PC data acquisition – system. The heat sinks angle to the horizontal position αH was changed from 0° to 120° for HS1 and from 0° to 90° for HS2 (fig. 3c).
Figure 3:
Test arrangement for thermal investigations on heat sinks HS1 (a) and HS2 (b) by different angle αH to horizontal position (c)
The convective dissipation of heat PKo (fig. 4) can be calculated from: PKo = Pel – (PIs - PS)
(10)
where Pel is the supplied electrical power, PIs is the dissipated heat through the thermal insulation and PS is the dissipated heat via radiation given by: PS = ε12 CS OS 10 -8(T14 – T24).
(11)
Resultant emissivity ε12 in equation (11) was determined by dint of a pyrometer to ε12 = 0,96 for HS1 and ε12 = 0,3 for HS2. CS is the radiation capability of the “black body” (Cs=5,67 Wm-2K-4) and T1 and T2 are absolute temperatures of the heat sink and the ambient respectively. The heat leak through the thermal insulation PIs was determined by calibration. For calibration each heat sink was replaced by an insulating plate and the temperature difference ∆ϑ between the heat plate and the insulation was WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
396 Advanced Computational Methods in Heat Transfer IX measured at different supplied power values. This heat dissipated through the thermal insulation, for an ascertained temperature difference ∆ϑ, was then subtracted from the supplied power in experiments conducted with heat sinks (fig. 4). After the calculation of PS and the determination of PIs the convective heat flow PKo could be calculated. For known PKo, the Nußelt – number Nu in eqn. (4) given by: Nu = c1 (Gr Pr) n1
(12)
can also be calculated from: Nu =
PKolw OKo ∆ϑ λ
(13)
For determination of c1 and n1 parameters in eqn.(4 and 6) the Grashof – number Gr must be known too. It is given by: (14) GrPr = k ∆ϑ l 3 s
Figure 4:
w
Supplied electrical power Pel, convective dissipated heat PKo, via radiation dissipated heat PS, and through the thermal insulation dissipated heat PIs versus temperature difference between heat plate and outer side of the thermal insulation (HS1, angle to the horizontal position αH = 45°).
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Because of the two unknown parameters c1 and n1 in eqn. (12) this eqn. can’t be calculated. By logarithm each side of eqn. (12) it can be rewritten as: log Nu = log c1 + n1 log (GrPr)
(15)
and then be solved graphically via double-logarithmic diagram [3, 4]. The substitution of log Nu by A, log (GrPr) by B and log c1 by C results in a linear equation: A = n1B + C (16) in which n1 and C can be determined graphically. Putting the investigation results for free convection in a diagram where log Nu is the ordinate and log (GrPr) the abscissa, the results build a straight line so that the n1, c1 parameters can be determined. In this way the determined parameter n1 was approximately n1 = 0,33 in the whole range of αH angle. In affinity theory this value of αH indicates a turbulent airflow and therefore it has been set up for further calculations for each αH angle. With this value of n1 parameter the c1 parameter for both HS1 and HS2 was calculated (fig. 5) from: Nu (17) c1 = (GrPr ) n In fig. 5 is clearly to see, that c1 parameter and therefore convective heat transfer from the heat sink to the ambient depends for HS1 and HS2 on αH angle. Both heat sinks have the best cooling performances for αH in range between αH = 60° and αH = 90°.
Figure 5:
Parameters c1 by n1 = 0,33 for different heat sink angle αH to the horizontally position.
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398 Advanced Computational Methods in Heat Transfer IX Furthermore it can be seen, that c1 parameter for HS1 achieves much higher values then c1 parameter for HS2. The c1 = 0,15 given in [3] for αH = 90° lies between the values of the c1 parameters for HS1 and HS2. This difference is caused by different characteristic length lw for each heat sink. After air passes a certain critical length between fins, its temperature and therefore pressure drop increases. Thus decreases the heat sink cooling performance. It is therefore advantageous if the characteristic length lw of the heat sink isn’t too long.
4
Determination of c1, n1 parameters depending on heat sink’s angle to horizontal position by heat sink cooling with pre-warmed air
In some cases heat sinks in current carrying devices are placed above other hot parts and therefore they are cooled with air that is warmer then ambient air. In order to estimate that effect on the heat sink’s cooling performance thermal investigations have been performed. The test arrangements described above were modified so that 1000 mm x 250 mm stainless steel heat plates below each heat sink were integrated (fig. 6). Measurements analogous to point 3 for two different heat plate power levels (HP1: 40 W; HP2: 80 W) were carried out. In order to achieve uniform current distribution in the heat plates and therefore uniform temperature distribution in the heat plate DC current was used. The heat plate temperature rise was approximately 26 K for HP1 and 46 K for HP2 power level. Reference temperature was the ambient temperature ϑ0.
Figure 6:
Test arrangement for thermal investigations on heat sink HS1 (a) and HS2 (b) for different angle αH to horizontal position by cooling with pre-warmed air.
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For small αH angles (0° to 30°) heat plates below heat sinks have only a little influence on c1 factors for both HS1 and HS2 (fig. 7). In that case the major part of ascending warm air flows over the heat sink fins. For αH equal or bigger than 60° the heat plates directly below the heat sink can even slightly increase the heat sink’s cooling performance. The ascending air, which was pre-warmed by a hot plate, is warmer than the surrounding air and therefore can pass between cooling fins faster than cold air. That can also cause air turbulences by passing cooling fins, which improve heat transfer between heat sink and the ambient air. The increased temperature of the cooling air is compensated by a higher air velocity. The heat plate temperatures examined in experiment were smaller than the heat sinks temperature for each power level. If the heat plate placed below the heat sink is too warm (approximately as warm as heat sink temperature or warmer) the cooling air can be too hot for efficiently cooling the heat sink even with higher air velocity. In such a case the cooling performance of the heat sink placed above heat plate can decrease.
Figure 7:
5
Parameter c1 for different heat sink angles αH to the horizontal position and different power levels of the heat plates below heat sinks.
Summary
In order to calculate the temperatures of current carrying devices with thermal networks accurately, thermal investigations by free convection on heat sinks, WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
400 Advanced Computational Methods in Heat Transfer IX used in such devices, were carried out. These investigations showed that in addition to the geometry and the dimensions of the heat sink its position to the airflow must be considered too. Further, the influence of the presence of other hot parts below the heat sink was investigated. The results for this experiment shown in fig. 7 indicate only a slight influence of such parts on the cooling performance of the heat sink. In case of heat plates placed below heat sinks, whose temperatures are lower than the heat sink temperature and with heat sink angles to horizontal position αH equal or bigger than 60° the heat sinks cooling performance can even be slightly improved.
References [1] IEEE C37.013-1997: IEEE Standard for AC High-Voltage Generator Circuit Breakers Rated on a Symmetrical Current Basis [2] Löbl, H.: Zur Dauerstrombelastbarkeit und Lebensdauer der Geräte der Elektroenergieübertragung, Dissertation B, TU Dresden, 1985 [3] Löbl,H., Stoye, H.J.: Beiterag zur Optimierung elektrotechnischer Schaltund Verteileranlagen hinsichtlich ihrer thermischen Dauerstrombeanspurchung, Disseration A, TU Dresden, 1972 [4] Michejew,M.A.: Grundlagen der Wärmeübertragung, VEB Verlag Technik Berlin, 1964 [5] Schoeneman,T., Schenk, M., Löbl, H., Pleines, M., Magier, T., Optimal design of generator circuit breakers up to a capacity of 2000 MVA using thermal networks under consideration of electrical and thermal contact resistances, 50 th IEEE Holm Conference on Electrical Contacts, Seattle, 2004
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Simplified 3-D FE model of thermal conditions inside a shoe H. Raval1, Z. W. Guan2, M. Bailey1 & D. G. Covill1 1 2
Micro Climate Research Unit, University of Brighton, Brighton, UK Department of Engineering, University of Liverpool, Liverpool, UK
Abstract To design thermally comfortable shoes, the knowledge of thermal conditions inside the shoes and the variables affecting those conditions is necessary. A simplified 3-D thermal numerical model of a shoe has been developed. A new approach was adopted to construct the mesh. The model was developed to consider the dry heat transfer in the shoe and convective heat loss from the outer surface of the shoe. The foot was the source of the heat in the model. Model’s predictions were compared with the results obtained during the experiments. The predicted in-shoe temperatures correlated reasonably well with the measurements although they were higher than the measurements in some cases. Probable reasons behind some inconsistency between predictions and the measured temperatures have been discussed. The paper concludes that the model’s predictions can be improved by incorporating the effect of other variables. Keywords: shoe climate, thermal comfort, thermal model of a shoe.
1
Purpose
It is important for the body to maintain its core temperature around 37°C (Bazett [1]). Depending on the environmental conditions and level of physical activity, the thermoregulatory system of the body controls the blood flow and hence heat being supplied to the skin and the periphery of the body to maintain the core temperature (Bazett [1]). Normally, the released heat from foot’s surface can vary from 3W to 30W depending on the activity level at the particular time (Oakley [2]). The body also controls the generation of sweat and the resulting wet heat transfer. Both types of responses can have a significant effect on the heat loss from the body because of large surface area of the feet. If sweat is not WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060391
402 Advanced Computational Methods in Heat Transfer IX allowed to evaporate, it can become uncomfortable and affect the heat transfer by increasing thermal conductivities of the skin, sock and shoe even during the situations where minimal heat loss is desired. Therefore, the material properties of the sock and the shoe are important to enable the body to achieve the intended heat loss from the feet and to remove sweat from the feet quickly. It is vital to understand the process of heat transfer taking place from the foot to the environment via the sock, in-shoe air and shoe in order to design thermally comfortable shoes. Experiments to understand shoe climate are useful in obtaining in-shoe thermal conditions and correlating these conditions with the perceptions of the subjects. However, the number of experimental conditions which can be evaluated may be limited due to the time and resources involved. Computer models, once validated, could be used to numerically study the effects of different materials, ambient conditions, heat flux and changes in design shoe climate.
2 Methodology 2.1 Experimental measures Static experiments were conducted at ambient temperatures of 20°C and 32°C with 50% Relative humidity (RH) with nine subjects for each ambient temperature. Prior to the experiments, the subjects bathed their feet in water warmed at 33°C to minimise differences in skin temperatures prior to test. The shoes were conditioned in the environmental chamber for one hour prior to the experiments to attain similar temperatures as that of the environment. Sensors were implanted in the upper shoe that recorded the temperatures of the in-shoe air close to the inner boundary of the shoe. Experiments were carried out for 35 minutes with in-shoe temperatures being measured at every 30 seconds. 2.2 Modelling A plaster cast of a foot was taken from a volunteer, who normally wore a UK size nine shoe. The plaster cast was then positioned inside a UK size nine shoe, so the shoe adopted the shape it would in wear to stabilise the shoe to allow it to be cut into coronal plane sections. The plaster cast and the shoe were cut using a hacksaw at approximately 25 mm intervals apart as shown in the Figure 1. The sections were used to create a series of 2-D planes by digitising coordinates onto graph paper with key points plotted on the circumference of the inner and the outer boundaries of the shoe and on the boundary of the foot to generate 2-D plots. Geometric conditions of all slices were implemented in a finite element programme through ABAQUS (ABAQUS Inc, Providence, Rhode Island) to create the geometric mesh. The coronal planes provided the outline geometry of the outer and inner shoe, foot and in-shoe air at regular intervals. Subsequent layers between the planes were generated at 6.25 mm intervals along the coronal axis using eight-node brick elements. The ends of the foot and the shoe had been slightly simplified to minimise the geometrical difficulties although the inner shape of the shoe was undisturbed. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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A shoe with plaster cast of a foot cut into slices in at 25 mm interval.
To represent the in-shoe air, 8-node linear brick and 6-node linear triangular prism elements were generated between the meshed foot and shoe. The air was meshed by filling the gap between corresponding nodes of the foot and shoe in different sections. Some air-pockets were left un-meshed because of the unavailability of eight or six distinct free nodes to create an element. These were considered to encompass a negligibly small volume. Figure 2 shows 3-D mesh of a shoe as it would be expected to appear in wear, the foot and the in-shoe air. The ends of the shoe and the foot were simplified to overcome the geometrical difficulties faced by ABAQUS. However, the boundary of the inner shoe was undisturbed by this simplification. The foot was the source of heat in the model. In this simplified model, the foot released heat to the system uniformly from its surface at a rate of 13W. Material properties were assigned to the three different materials. The shoe material was to be homogenous and its thermal conductivity to be isotropic. The foot was also assigned uniform material properties. Table 1 shows the material properties used in the model. The model calculates the dry heat transfer taking place in the shoe due to conduction. It also incorporates heat loss to the environment from the outer surface of the shoe due to convection by defining the film coefficient (h = 20.0 W/m2·K). Surface interactions between different surfaces in contact ware defined to allow the model to calculate the heat conduction taking place from the foot to in-shoe air, from the in-shoe air to shoe and from the foot to shoe in the absence of in-shoe air. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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(i) Shoe
(ii) Foot
(iii) In-shoe air Figure 2:
3-D meshes representing a shoe, foot and in-shoe air.
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Thermal Conductivity (W/m·K) 0.16 0.20 0.026
Specific Heat (J/Kg·K) 2010 1260 1005
Density (Kg/m3)
Source
998 810 1.2
Covill [3] Covill [3] Covill [3]
To simulate with the experimental data, the initial temperature of the foot in the model was taken as 33°C. The model also assumed that the initial temperatures of the shoe, in-shoe air and environment were the same, as those of the environmental chamber where the shoes were conditioned prior to the experiments.
3
Results
To validate the model, the predicted temperatures at in-shoe locations were compared with measured in-shoe temperatures. Figure 3 shows the comparison between the predicted and the measured in-shoe temperatures for the toe-region under two different ambient conditions. The temperature for the toe-region was obtained by averaging four different in-shoe temperatures over the front foot. Figures 4 and 5 show the measured and predicted temperature for the arch and midfoot under two different controlled ambient conditions. Temperatures for these two locations were measured at single locations unlike the toe region.
4
Discussion
The predicted results have showed similar trends in changes of temperatures at particular in-shoe locations although the magnitude of these changes, sometimes, was different from the measured results. For the toe region at 20°C, the predictions from the model fall within one standard error of the mean temperatures obtained experimentally after 15 minutes. However, in the hot environment of 32°C, the predicted results are about 6% lower than those measured. For the arch and midfoot, in Figures 4 and 5, the predictions generally fall within the standard error of the mean temperature at the environmental temperature of 32°C. At 20°C, the predictions overestimate the mean measured temperature by around 7% and 16% for the arch and midfoot respectively.
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406 Advanced Computational Methods in Heat Transfer IX Toes; Ambient Temperature - 20 C 40
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Various factors may have affected the predictions. Although the shoe was conditioned in a controlled environment for an hour at 20°C, the initial temperature for all three locations was not exactly the same as that of the environment. In the model, the heat flux was constant irrespective of ambient conditions and was applied uniformly over the surface of the foot; however, the heat flux is a function of ambient conditions and the locations on the foot. To apply the heat to the foot accurately, the knowledge of relationship between the environmental conditions and the heat released from the different regions of the
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foot is essentially required. Since the geometry of the tow-region was simplified and thus the surface area reduced, it could also be one of the reasons for lower heat loss from the toe-region at 20°C. This preliminary model does not consider the effect of wet heat transfer on the in-shoe thermal conditions. It may have a noticeable effect on the shoe climate especially at higher ambient temperatures and/or relative humidity. The material properties of the shoe used in experiments may not have been the same as those of the shoe used in the model.
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408 Advanced Computational Methods in Heat Transfer IX Midfoot; Ambient Temperature - 20 C 40
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Comparison between the predicted and measured in-shoe temperature for the midfoot at ambient temperatures of 20°C and 32°C.
Although great care was taken while cutting the shoe and plaster cast in the slices, some of the material of the plaster cast was lost during cutting. The deformed shoe also partially regained its earlier un-deformed shape once it was cut into the slices as some tension was removed. Also the absence of the sock in the model means its effect on the rate of heat transfer is ignored. The model, however, showed promising results by describing the nature of changes taking place in the shoe, including the initial rapid rise in the temperature, with time in static conditions. It also predicted the lowest temperature for the toe-region of all three locations through out 35 minutes of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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experiment at ambient temperature of 20°C, which was the case during experiments. The model also successfully predicted temperatures for the arch and midfoot at ambient temperature of 32°C.
5
Conclusion
A simplified 3-D thermal model was developed. The mesh was constructed manually using the measurements of the outline of the shoe, foot and in-shoe air. The model incorporated the effect of dry heat transfer in the shoe and convective heat loss from the shoe surface in terms of shoe climate. The model was validated against the corresponding experimental results and reasonable correlation was obtained. The predictions showed a similar pattern to the changes of the in-shoe temperatures with time for some conditions and locations. Investigation of mesh parameters, heterogeneous material properties and realistic variations in heat flux from the foot could yield better predictions of test results by the model.
Acknowledgement The authors would like to thank Clarks International for funding the research and providing the shoes for the experiments.
References [1] Bazett H.C., the regulation of body temperatures. Physiology of Heat Regulation and the Science of Clothing, ed. L.H. Newburgh, Hafner Publishing Co.: London, pp. 109-192, 1968. [2] Oakley E.H.N., the design and function of military footwear: a review following experiences in the South Atlantic. Ergonomics, 27, pp. 631-637, 1984. [3] Covill D.G., Experimental and numerical modelling of the temperature and moisture distributions in footwear, PhD thesis, University of Brighton, 2005.
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Section 6 Energy systems
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Radiative heat transfer in a model gas turbine combustor M. C. Paul1 & W. P. Jones2 1 Department
2 Department
of Mechanical Engineering, University of Glasgow, UK of Mechanical Engineering, Imperial College London, UK
Abstract In this paper we have carried out a three-dimensional numerical study to investigate the radiative heat transfer in a model gas turbine combustor, representative of the Rolls-Royce Tay combustor. The Discrete Ordinate Method (DOM/Sn) in general body-fitted co-ordinate system is applied to solve the filtered Radiative Transfer Equation (RTE) for the radiation modelling and this has been combined with a Large Eddy Simulation (LES) of the flow, temperature and composition fields within the combustion chamber. The radiation considered in this work is due only to the hot combustion gases notably carbon dioxide (CO2 ) and water vapour (H2 O) also known as non-luminous radiation. The instantaneous results of the radiation properties such as the incident radiation and the radiative energy source or sink as the divergence of the radiative heat fluxes are computed inside the combustion chamber and presented graphically. Keywords: discrete ordinates method, large eddy simulation, radiative heat transfer, turbulent combustion.
1 Introduction A large part of the total heat transfer in a combustion chamber, whether it is a gas turbine engine, a car engine or a furnace, occurs by radiation from the flame. This radiation has two components: (i) the ‘non-luminous’, which emanates from the combustion gases notably carbon dioxide (CO2 ) and water vapour (H2 O), and (ii) the ‘luminous’, which is mainly due to the soot formed in the flame. The prediction of wall temperatures is an important aspect in the design of practical engine combustors and this clearly requires that the radiative heat fluxes be predicted accurately. An inability to predict the wall temperatures may lead to an excesWIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060401
414 Advanced Computational Methods in Heat Transfer IX
75mm
50mm
80mm
80mm
Figure 1: The feature of a model Tay gas turbine combustor.
sive amount of the combustor airflow being used for cooling the liner wall and this is likely to lead to a reduced combustion efficiency and an increased emission of the pollutants such as carbon monoxide (CO), N Ox formations and unburned hydrocarbons (U HC). In addition, excessive combustor wall temperatures have a deleterious impact on combustor ‘life’. The Discrete Ordinates Method (DOM) was first proposed by Chandrasekhar [1] in his work on one-dimensional stellar and atmospheric radiation. Subsequently Carlson and Lathrop [2] developed the DOM for multi-dimensional radiation problems employing the finite volume approach. More recently, the DOM has been widely used on various different problems [3–5] where the major emphasis has been on solving the Radiative Transfer Equation (RTE), which is the steady state representation of the radiative transfer. The radiative transfer in high-temperature combustion devices requires a simultaneous solution of the RTE and the governing flow equations such as Navier-Stokes, enthalpy and species concentration conservation equations, etc, [6–8]. With respect to combining the RTE with the LES, to the authors’ knowledge only a small amount of work has been done to date. Desjardin and Frankel [7] studied soot formation in the near field of a strongly radiating turbulent jet flame involving LES and a simplified two-dimensional treatment of radiation involving gray and non-scattering medium was considered. Recently, Jones and Paul [8] have investigated the radiative heat transfer in a three-dimensional model of a gas turbine combustor, where the S4 approximation of the DOM was applied in conjunction with LES. In this paper we have extended the DOM including its various lower and higher order approaches, ie S2 , S6 , S8 , and applied into the model Tay gas turbine combustion chamber for further investigation of the radiative heat transfer and to see how the radiation results are affected with the various order of approximations of the DOM. To the best of our knowledge, there have been no other attempts that combine the three-dimensional DOM with LES for computing the turbulent flame radiation in a gas turbine combustor.
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2 Governing equations The main features of the model gas turbine combustor are shown in fig. 1, which is representative of the Rolls-Royce Tay gas turbine [9]. The combustor walls are made of transply, a laminated porous material. The geometry of the combustor includes a relatively small swirler at the head of the combustor in the centre of which the fuel injector is located and a hemispherical head section attached to a circular barrel of 75mm diameter. This barrel contains a set of six primary ports/holes of 10mm diameter each at the front and another set of six dilution ports/holes of 20mm diameter each at 80mm downstream of the first set. A circular-torectangular nozzle is attached to the end of the barrel. High purity gaseous fuel comprising over 95% propane (C3 H8 ) was injected into the combustion chamber through the centre of the swirler. The equations of motion in LES are obtained after applying a spatial filter [10] and a density weighted Favre filter [11] to the continuity, the Navier-Stokes, and the mixture fraction equations. In DOM the discrete ordinates representation of the radiative transfer equation is filtered using the spatial filter. Finally these equations take the following forms [8]: ˜j ∂ ρ¯ ∂ ρ¯u + =0, ∂t ∂xj ∂ ρ¯u ˜i ∂ ρ¯u ˜i u ˜j ∂ p¯ ∂ 2 ¯ ∂τij ¯ + =− + , 2µSij − µSkk δij − ∂t ∂xj ∂xi ∂xj 3 ∂xj µ ∂ f¯ ∂ ρ¯f˜ ∂ ρ¯u ˜j f˜ ∂ ∂Jj + = , − ∂t ∂xj ∂xj Sc ∂xj ∂xj αm
∂ I¯m ∂ I¯m ∂ I¯m + βm + γm + κIm = κIb . ∂x ∂y ∂z
(1)
(2) (3) (4)
where ρ is the mixture density, t is the time, xj = (x, y, z) is the coordinate vector, uj is the velocity vector, p is the dynamicpressure, µ is the coefficient of viscosity,
∂ui Sij is the strain rate defined as Sij = 12 ∂x + ∂xji , δij is the Kronecker delta, j f is the mixture function and Sc is the Schmidt number. In eqn (4), Im represents the radiative intensity along the angular direction, where m = 1, 2, ..., M (see [8, 10] for a detailed angular representation), thus the equation represents a set of M different directional radiative intensities from a computational grid node. The terms αm , βm and γm in eqn (4) represent the direction cosines of an angular direction along the coordinates and Ib is the blackbody ˜4 intensity at the temperature of the medium which is defined as σπT where σ is the Stefan-Boltzmann constant and T˜ is the flame temperature. κ is the absorption coefficient, and for a non-luminous radiation, which is considered in this work, it is a function of the mole fractions of H2 O and CO2 (see [8]). For the radiation modelling we have also assumed that the enclosure contains an absorbing-emitting, non-scattering and radiatively gray medium. ∂u
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416 Advanced Computational Methods in Heat Transfer IX In eqns (2)–(3) the sub-grid-scale stresses, τij , and the sub-grid-scale scalar fluxes, Jj , are modelled using the standard Smagorinsky model [13] and a gradient model [14] respectively. For the unknown terms, κIm and κIb , in eqn (4), which are the nonlinear correlations between turbulence and radiation, we have ignored the sub-grid-scale turbulence interaction with radiation. A future study is required to incorporate these interactions and to investigate their effects in the radiative heat transfer predictions in a turbulent flame. The detailed boundary conditions for solving the filtered eqns (1)–(3) are presented in [15], and while for the filtered RTE (4) these are given in [8] and will not be repeated here.
3 Numerical procedures The filtered equations (1)–(4) are rewritten in general boundary/body fitted coordinates system using the approach introduced by Thompson [16], where the governing differential equations in the Cartesian coordinates are transformed into the curvilinear coordinates system. The details of the numerical procedures in the LES approach to solve eqns (1)-(3) have already been presented in [15, 17] and will not be repeated. The numerical procedures in DOM to solve the RTE (4) can be found in Jones and Paul [8]. Also a benchmark problem was considered in [8] to assess accuracy of the numerical results of the DOM in a general body fitted co-ordinates system, and a very good agreement was obtained compared with the results available in literature, for further details the readers are referred to [8].
4 Results and discussion In the present computation we employed a total of about 105 control volumes with 40 × 60 × 40 grid nodes in the x, y and z directions respectively. The time step in the simulation was chosen to be dt = 5.53 × 10−7 (sec) based on the consideration of the maximum Courant number which never exceeded 0.1 throughout the computations. Overall the LES code is second order accurate in both space and time domains and for the DOM we used the scheme described in [8]. The results presented in figs. 2-4 are at 6.5 × 105 time steps, which is at the real clock time of t ≈ 0.036sec. Instantaneous results of the flame absorption coefficient and the M temperatures, ¯ total radiative intensity, I¯ = m=1 Im , at various horizontal locations of the combustion chamber are presented in fig. 2. The radiation results shown in this figure are obtained applying the highest order approximation of the DOM, S8 , and the emissivity of walls was kept at 0.5. We note that the results of temperature and absorption coefficient are obtained initially without considering any radiation effects. These results are then feed into the radiation solver, DOM, to solve the RTE (4). These are required; as the sources of radiation, e.g. the black body intensity (I¯b ), depend on the flame temperature; and the absorption coefficient, κ ¯ , which is considered to be a function of the mole fractions of H2 O and CO2 . WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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9 364
1990
13 20
20
229 2
5283
22
3456
2894
6100
¯ Figure 2: Instantaneous contour plots of T˜(K), κ ¯ (cm−1 ), I(kW m−2 ) at (a) y = 20mm, (b) y = 95mm and (c) y = 165mm.
(kW/m−3 )
∇.¯ q (a) . 52
-1 . 4
9
52.9 -0.9
- 3.
3
0. 7
-1.
6.5
-5.1
6
-1.4
7
4
66.
-1.
23.9
(b) -1. 2
66.5
37.1
1.7
-1.2
-1.2
-1. 2
.5 13
-2.1
1. 7 25.3
-2.1
-3.0
3 -6.
-3 . 0
37.1
37
.1
Figure 3: Contours of the divergence of the radiative heat fluxes on (a) the midhorizontal and (b) the mid-vertical planes of the combustor.
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418 Advanced Computational Methods in Heat Transfer IX 700
1500
(a)
650
(b)
1400
S2 S4 S6 S8
600 550
S2 S4 S6 S8
1300 1200 1100
G
G
500
1000
450
900
400
800
350
700
300 250
600
-0.03
-0.02
-0.01
0
0.01
0.02
500
0.03
z(m)
-0.02
-0.01
0
0.02
0.03
0.01
0.02
0.03
(d)
600
S2 S4 S6 S8
S2 S4 S6 S8
1050
0.01
z(m)
(c)
1200
-0.03
G
G
500 900
400
750
600
300 450
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
-0.03
-0.02
-0.01
0
z(m)
z(m)
Figure 4: Radial profiles of the incident radiation, G(kW m−2 ), at (a) y = 20mm, (b) y = 95mm, (c) y = 130mm and (d) y = 165mm on the midhorizontal plane of the combustor.
A good agreement was achieved when di Mare et al. [15] compared these computational results of T˜, Y˜H2 O and Y˜CO2 against experimental measurements of Bicen et al. [9], which will not be repeated here. In this figure we can also see that the total radiative intensity attains a maximum at the region where both the temperature and absorption coefficient are maximum. Thus it provides clear evidence that the medium is highly dominated by the hot H2 O and CO2 gases. The radiative heat flux distributions also become maximum at the position where the radiative intensities are also maximum. Fig. 3 shows the net loss or gain of the energy due to the radiation as a divergence of the radiative heat fluxes, ∇.¯ q, which have been calculated using the following relation κ ¯ I¯ dΩ . (5) ∇.¯ q = 4π¯ κI¯b − 4π
In this figure, the dashed lines represent the negative contours. The results obtained using S8 version of the DOM and the wall emissivity remains the same, ie w = 0.5. In eqn (5), the first term on the right hand side represents the emitted/outgoing radiation from a computational control volume, while the second term represents WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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the total incident radiation into that control volume. Thus, ∇.¯ q gives the rate of the generation of energy by radiation and this must be coupled in the overall energy conservation, which turns to be an improved temperature prediction inside the combustion chamber. Further work is required to investigate this effects on the flame temperature and combustion species. However this would require a substantial increase in computer resources as, at every time-step, the radiation intensities would have to be calculated by the radiation solver for updating the radiative heat fluxes and finally obtaining a new temperature for the next time step. In fig. 4 we show some results of the incident radiation, G, at various radial positions on the horizontal plane of the gas turbine combustor. To judge the accuracy of the various orders of DOM (Sn ) in the present work, the results of incident radiation are plotted for S2 , S4 , S6 and S8 . The wall emissivity is still remain unchanged here. It is also interesting to mention here that the incident radiation is an important radiation property related to the radiative energy density, by which the total radiation energy is stored in each computational node, so each frame in this figure shows a distinct variation of the energy storage inside the combustion chamber. Prediction of the incident radiation is also an essential task which allows us the radiative energy transfer to be coupled with the global energy conservation (for example, see eqn. (5)). The incident radiation is calculated using the relation, G=
4π
I¯ dΩ ≈
M
ωm I¯m .
(6)
m=1
In fig. 4(a)-(d) we can see how the results obtained using the most lower order approximation, S2 , of the DOM diverts from those with the higher orders. But the higher order Sn (S4 , S6 and S8 ) results appear to converge together at a level of G. It is worth to mention here that the DOM with S2 approximation was also tested considering a relatively smaller wall emissivity, w = 0.1, and a poor convergence rate was achieved in the radiation solver due to the very oscillatory nature of the radiative intensity solutions. In fig. 4(a), G is calculated in a position of the combustor head and it is found to be lower compared with those in other fames. This is expected as the radiative intensity, absorption-emission rate and flame temperature in the combustor head region are predicted lower (see fig. 2). However, in the combustor barrel (at y = 95mm, fig. 4(b)), G is predicted to be the highest, because this part of the combustor houses the extremely hot gases. At the dilution ports (at y = 130mm, fig. 4(c)) the effect of the large amount of cooling air on the prediction of the incident radiation is clear, and at the downstream (at y = 165mm, fig. 4(d)) G decreases again because of the effect of the cooling air through the dilution ports and no combustion occurs downstream of these ports.
5 Conclusions The Sn approximation of the discrete ordinate method has been implemented to investigate the radiative heat transfer inside a model gas turbine combustor. The WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
420 Advanced Computational Methods in Heat Transfer IX DOM has been combined with a Large Eddy Simulation of the flow, temperature and composition fields within the combustion chamber. A gray-gas and nonscattering approximation to the RTE has been assumed, and the absorption coefficient for both H2 O and CO2 gases is calculated. The instantaneous results of the radiative heat fluxes, incident radiation, and divergence of radiative heat fluxes have been calculated. A coupling of this radiative heat gain/loss is likely to yield accurately predicted wall temperature and this will aid combustor design by allowing an optimum amount of air to be used for wall cooling. The beneficial effects will be a reduction in the emission of pollutant gases by maximising the combustion efficiency and to allow a longer liner life. The present study excludes the effects of the soot on radiative heat transfer. Soot is likely to enhance the radiation field and ultimately the coupling of soot formation and consumption to heat radiation is an important requirement. The soot concentrations are required and this in turn requires the solution of the appropriate conservation equations for the soot properties. Research is currently underway on this.
References [1] Chandrasekhar, S., Radiative Transfer. Dover Publications, 1960. [2] Carlson, B.G. & Lathrop, K.D., Transport theory-the method of discrete ordinates. Computing Methods in Reactor Physics, Gordon and Breach, New York, pp. 165–266, 1968. [3] Jamaluddin, A.S. & Smith, P.J., Predicting radiative transfer in rectangular enclosures using the discrete ordinates method. Comb Sci and Tech, 59, pp. 321–340, 1988. [4] Liu, J., Shang, H.M., Chen, Y.S. & Wang, T.S., Prediction of radiative transfer in general body-fitted coordinates. Num Heat Tran Part B, 31, pp. 423– 439, 1997. [5] Kayakol, N., Selcuk, N., Campbell, I. & Gulder, O.L., Performance of discrete ordinates method in a gas turbine combustor simulator. Exp Thermal and Fluid Sci, 21, pp. 134–141, 2000. [6] Kaplan, C.R., Baek, S.W., Oran, E.S. & Ellzey, J.L., Dynamics of a strongly radiating unsteady ethylene jet diffusion flame. Combustion and Flame, 96, pp. 1–21, 1994. [7] Desjardin, P.E. & Frankel, S.H., Two-dimensional large eddy simulation of soot formation in the near-field of a strongly radiating nonpremixed acetylene-air turbulent jet flame. Combustion and Flame, 119, pp. 121–132, 1999. [8] Jones, W.P. & Paul, M.C., Combination of dom with les in a gas turbine combustor. Int J Eng Sci, 43, pp. 379–397, 2005. [9] Bicen, A.F., Tse, D.G.N. & Whitelaw, J.H., Combustion characteristics of a model can-type combustor. Tech. rep. fs/87/28, Imperial College London, 1987. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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[10] Leonard, A., Energy cascade in large-eddy simulations of turbulent fluid flows. Advances in Geophysics, 18A, pp. 237–248, 1974. [11] Favre, A., Statistical equations of turbulent cases in problems of hydrodynamics and continuum mechanics. Technical report, Society of Industrial and Applied Mathematics, Philadelphia, 1969. [12] Modest, M.F., Radiative heat transfer, second edition. Academic Press, 2003. [13] Smagorinsky, J., General circulation experiments with the primitive equations. i. the basic experiment. Monthly Weather Review, 91, pp. 99–164, 1963. [14] Schmidt, H. & Schumann, U., Coherent structure of the convective boundary layer derived from large eddy simulation. J Fluid Mech, 200, pp. 511–562, 1989. [15] di Mare, F., Jones, W.P. & Menzies, K., Large eddy simulation of a model gas turbine combustor. Combustion and Flame, 137, pp. 278–294, 2004. [16] Thompson, J.F., Thames, F. & Mastin, C., Automatic numerical generation of body-fitted curvilinear coordinates system for field containing any number of arbitrary two-dimensional bodies. J Comp Phys, 15, pp. 299–319, 1974. [17] Branley, N. & Jones, W.P., Large eddy simulation of a turbulent nonpremixed flame. Combustion and Flame, 127, pp. 1913–1934, 2001.
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Advanced Computational Methods in Heat Transfer IX
423
Thermo-economics of an irreversible solar driven heat engine K. M. Pandey & R. Deb Department of Mechanical Engineering, National Institute of Technology, Silchar, Assam, India
Abstract Thermo-economic optimization has to be carried out for an irreversible solar driven heat engine using finite-time/finite-size thermodynamic theory. In the considered heat engine model, heat transfer from the hot reservoir is assumed to be radiation mode and the heat transfer to the cold reservoir is assumed to be convection mode. The power output per unit total cost is taken as the objective function. The steps of problem formulation are rightly performed and all valid assumptions are taken into consideration. The effects of the irreversibility parameter, economical parameter and the design parameters on the thermo-economic objective function have been investigated. Keywords: finite-time/finite-size thermodynamics, irreversible, solar-driven heat engine, thermoeconomic optimization.
1
Introduction
India is endowed with abundant solar energy for about 70% of the yearly period. Solar driven heat engine systems, which consist of a solar collector and a heat engine, have a large potential for saving fossil fuel and decreasing environmental pollution. The schematic diagram of a solar driven heat engine is shown in fig.1. The energy from solar radiation is collected and utilized to generate steam to run turbines. As temperature required for steam generation is considerably high (200˚C), for obtaining reasonably high efficiencies, concentration type of collectors are used when steam is used as working fluid.
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424 Advanced Computational Methods in Heat Transfer IX
Figure 1:
2
Schematic diagram of an irreversible solar driven heat engine.
Literature review
Power optimization studies of heat engines using finite time thermodynamics were started by Chambadal [1] and Novikov [2] and were continued by Curzon and Ahlborn [3]. The upper bound on the efficiency of a reversible engine is the so-called Carnot efficiency that is given by
Firstly, Curzon and Ahlborn [3] studied the performance of an endoreversible Carnot heat engine at maximum power output. Using convective type linear heat transfer processes through finite temperature difference in both the hot and cold reservoirs, they showed that an upper limit to the endoreversible engine efficiency is
The study of irreversible thermodynamic cycles has been undertaken by many researchers after Curzon and Ahlborn’s work. Sahin et al. [4] studied the efficiency of a Joule-Brayton engine at maximum power density with consideration of engine size. Their results show that the efficiency at maximum power density is always greater than that presented by Curzon and Ahlborn [3]. The principle of operation of a solar thermal power plant is presented by Lund [5] in terms of finite heat transfer rates and an internally reversible heat engine by presenting some parametric equations. Medina et al. [6] extended the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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work of Sahin et al. [4] to a regenerative Joule–Brayton cycle where the optimal operating conditions at the engine were expressed in terms of the compressor and turbine isentropic efficiencies and of the heat exchanger efficiency. Many authors investigated the effects of radiative heat transfer law on the performance of heat engines. Jeter [7], De Vos and Pauwels [8] applied the Stefan–Boltzmann thermal radiation law to the performance analysis of solar powered energy systems. De Vos [9], Chen and Yan [10] and Gordon [11] discussed the effect of a class of heat transfer laws (linear and non-linear radiation) on the performance of endoreversible cycles. They revealed the dependence of performance on the heat-transfer law and the heat-transfer coefficient and derived a universal expression for different common heat transfer laws and concluded that the value of maximum power depends on both the source temperatures and the heattransfer coefficient. Gordon [12] illustrated that the optimal operating temperature for solar-driven heat engines and solar collectors is relatively insensitive to the engine design point. Bejan [13, 14] performed a power optimization study for a solar driven power plant model. He examined the optimal design parameters and optimal distribution of the heat transfer areas at maximum power output conditions. In recent years, the performance of a solar driven heat engine using the technique of finite time thermodynamic analysis has been investigated. In these studies, the objective functions chosen for optimization are usually power output. Chen [15] investigated the optimal performance and design parameters for solar driven heat engine consisting of a solar collector and a heat engine. In the study, he determined the optimum operating temperatures of the working fluid and solar collector. Goktun et al. [16] investigated the design parameters of an endoreversible radiative heat engine at maximum power output conditions. As a major result, they showed that the ratio of the cold to the hot reservoir temperature must be less than 0.2 for an optimal design. The work carried out by Goktun et al. [16] for an endoreversible radiative heat engine model has been extended to an irreversible radiative heat engine model by Ozkaynak [17]. He obtained the design parameters at maximum power output for radiative and convective boundary conditions. He also discussed the effects of internal and external irreversibility parameters on the maximum power output and thermal efficiency at maximum power conditions. Badescu [18] proposed a model of a space power station composed of an endoreversible Carnot heat engine driven by solar energy. He obtained the maximum power output and the optimum ratio between the solar collector and radiator areas. Erbay and Yavuz [19] performed an analysis of an endoreversible Carnot heat engine with the consideration of combined radiation and convection heat transfer between the working fluid and hot and cold heat reservoirs. They showed that the power output is strongly dependent on the temperature and emittance ratios of the heat reservoirs. Badescu et al. [20] performed a power optimization for solar driven endoreversible Carnot heat engine model. They obtained optimum solar collector surface area and temperature under maximum power conditions. Sahin [21] carried out optimization study based on maximum power criterion for an endoreversible solar driven heat engine with radiation mode heat transfer from the hot reservoir and convection mode to the cold WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
426 Advanced Computational Methods in Heat Transfer IX reservoir. He also developed his model in [21] by considering the collective role of radiation and convection heat transfer from hot reservoir [22]. In these studies, he investigated the optimal working fluid temperatures and the thermal efficiency at maximum power conditions. He also discussed the effects of the ratios of the reservoir temperatures and the heat transfer coefficients on the optimal performances. Koyun [23] carried out a comparative performance analysis based on maximum power and maximum power density criteria for a solar-driven heat engine with external irreversibilities. He compared the optimal performances and design parameters at the maximum power and power density conditions. Sahin and Kodal [24] have recently reported a new performance analysis based on an objective function defined as the power output per unit total cost. Using this performance optimization criterion, they performed a finite-time thermoeconomic optimization for endoreversible [24] and irreversible [25] heat engines with linear heat transfer modes. They investigated the optimal performances and design parameters under the maximum thermoeconomic objective function conditions and then they examined the effects of technical and economical parameters on the global and optimal performances. Sahin et al. [27] have recently carried out thermoeconomic optimization for an endoreversible solar driven heat engine using finite time/finite size thermodynamic theory. They investigated the effects of the technical and economical parameters on the thermoeconomic performances. In this paper, the finite-time thermoeconomic optimization technique introduced by Kodal and Sahin [25] for an irreversible heat engine model with linear heat transfer modes has been applied to a solar driven irreversible heat engine model to perform thermo-economic optimization.
3
Aim of the work
The aim of this work is to perform thermo-economic optimization of an irreversible solar driven heat engine using finite-time thermodynamics. 3.1 Theoretical model Solar powered heat engine is considered to operate according to the Rankine cycle given in Fig. 2. The considered Rankine cycle operates between a heat source of temperature TH and a heat sink of temperature TL. In order to simplify the analysis, the Rankine cycle (1-2-3-4-5-1) can be modified by using an entropic average temperature defined by Khaliq [26] to a Carnot cycle (1-a-b-51). Since the area under the process 2-3-4 in the T-S diagram of Fig. 2 represents the amount of heat added to the Rankine cycle, we can make this area equal to the area under the horizontal line with an entropic average temperature of heat addition. The entropic average temperature can be written as,
T X = ∆Q ∆s = (h4 − h2 ) ( s 4 − s 2 )
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Advanced Computational Methods in Heat Transfer IX
Figure 2:
4
427
The components of a solar driven heat engine and its T-S diagram.
Problem formulation
The irreversible Carnot-type solar driven heat engine operating between temperature limits (TH and TL) is shown in fig.3.
Figure 3:
T-S diagram of a Carnot-type solar driven heat engine.
Heat transfer from the hot reservoir is assumed to be radiation dominated and the heat flow rate QH from the hot reservoir to the heat engine can be written as •
(
Q H = U H AH T H4 − T X4
)
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(1)
428 Advanced Computational Methods in Heat Transfer IX where AH is the heat transfer area of the hot side heat exchanger. UH is the hot side heat transfer coefficient. On the other hand, convection heat transfer is assumed to be the main mode of heat transfer to the low temperature reservoir and therefore the heat flow rate QL from the heat engine to the cold reservoir can be written as •
Q L = U L AL (TY − TL )
(2)
where UL is the cold side heat transfer coefficient and AL is the heat transfer area of the cold side heat exchanger. From the first law of the thermodynamics the net power output of the solar driven heat engine is •
•
•
W = QH − QL
(3)
Using eqns (1) and (2) in (3), we get • T4 −T 4 W = U L AL µ H 3 X − (TY − TL ) TH
(4)
where µ is the product of ratio of heat transfer areas (R) and the heat conductance parameter (δ) and is defined as µ = R× δ
(5)
R=AH/AL
(6)
and and
δ=
UH 3 TH UL
(7)
From the second law of thermodynamics for an irreversible cycle, the change in the entropies of the working fluid for heat addition and heat removing processes yields, •
∫
δQ
•
•
Q Q = H − L <0 T TX TY
(8)
One can rewrite the inequality in eqn (8) as •
•
QH Q = I L , 0 < I < 1. TX TY where I is irreversibility parameter. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(9)
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With the above definition I becomes •
I=
Q H TY •
Q L TX
=
T X ( s 3 − s 2 )TY s 3 − s 2 = TY ( s 4 − s1 )T X s 4 − s1
(10)
From eqns (1), (2) and (9), we get
TY =
TL µ (TH4 − T X4 ) 1− IT X TH3
(11)
The thermal efficiency of the irreversible heat engine is •
η =1−
QL •
=1−
QH
TY IT X
(12)
In thermoeconomic analysis, an objective function is defined as power output per unit total cost in order to account for both investment and fuel costs. In this study, the objective function has been defined as the power output per unit investment cost due to no fuel consumption cost in a solar driven heat engine. In order to optimize power output per unit total cost, the objective function is defined as F=W/Ci
(13)
where Ci refers to annual investment cost. The investment cost of the plant is assumed to be proportional to the size of the plant. The size of the plant can be taken proportional to the total heat transfer area. Thus, the annual investment cost of the system can be given as
C i = (hAH + lAL )
(14)
where the investment cost proportionality coefficients for hot and cold sides h and l are equal to the capital recovery factor times investment cost per unit heat transfer area. Substituting eqns (4) and (14) into eqn (13), we obtain
(TH4 − T X4 ) − (TY − TL ) µ 3 TH F= UL k l R + 1 1 − k
(15)
where k is relative investment cost parameter of the hot side heat exchanger and defined as WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
430 Advanced Computational Methods in Heat Transfer IX
k = h (h + l )
(16)
Dimensionless thermo economic objective function is defined as
F = lF (U L TL )
(17)
Dimensionless power output is defined as •
•
W = W (U L AL TL )
(18)
Using eqn (15) in (17), we get
(
)
TH4 − T X4 − (TY − TL ) µ 3 TH F= k 1 − k R + 1TL
(19)
Using eqn (4) in (18), we get
(
)
• T 4 −T 4 (T − TL ) W = µ H 3 X − Y TL TL TH
(20)
Using eqn (11) in (12), we get
TL TH3 η =1− IT X TH3 − µ TH4 − T X4
(
Putting
θ = T X TH
(21)
and τ = TH TL in eqn (19), (20) and (21) we get
(
)
µ 1−θ 4 F = µτ 1 − θ 4 − Iθ − µ 1 − θ 4
(
)
)
(
(
)
W = µτ 1 − θ 4 −
(
)
k 1 − k R + 1
µ (1 − θ 4 ) Iθ − µ (1 − θ 4 )
)
Iτθ − µτ 1 − θ 4 − 1 η= Iτθ − µτ 1 − θ 4
(
)
(22)
(23)
(24)
To maximize the dimensionless thermoeconomic objective function, eqn (22) is differentiated with respect to θ and the resulting derivative is equated to zero. After a lengthy calculation, it is found that the optimum value of θ must satisfy the following equation
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(a)
Figure 4:
(b) Variation of the maximum thermoeconomic objective function with respect to R for various (a) δ values (I=0.8, k=0.8, τ=5) (b) k values (I=0.8, τ=5, δ=2) (c) τ values (I=0.8, δ=2, k=0.8) (d) I values (τ=5, δ=2, k=0.4).
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432 Advanced Computational Methods in Heat Transfer IX
(c)
(d) Figure 4: Continued.
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4 µ 2τθ 11 + 8µτIθ 8 − 8µ 2τθ 7 + 4τI 2θ 5 − (8µτI + 3I )θ 4 + 4 µ 2τθ 3 − I = 0 (25) Solution of eqn (25) can be done numerically.
5
Results and discussion
The variations of maximum thermoeconomic objective function with respect to different technical and economical parameters are shown in figure 4.From the figures, it is clear that the optimal value of ratio of heat transfer areas(R) for which the maximum value of thermo-economic objective function becomes maximum depends on δ, k, I and τ parameters. It is seen from figure 4(a) that the optimal value of ratio of heat transfer areas (R) decreases with increase in the value of heat conductance parameter (δ). Also with increase in the value of δ, the optimal value of objective function also increases. On increasing the value of δ beyond unity, the optimal objective function increases sharply for R lying below 1. From figure 4(b), it can be observed that the optimal value of ratio of heat transfer areas (R) decreases with increase in the value of economical parameter (k). Also the optimal value of objective function decreases with increase in the value of k. From figure 4(c), it is seen that the maximum value of thermoeconomic objective function increases sharply with increase in the value of τ for the value of R lying below 0.5.Also the optimal value of R is not get affected with increase in the value of τ. From figure 4(d), it observed that the optimal value of R decreases slightly with decreasing I. Also there is severe fall in the value of maximum objective function with slight decrease in the value of I. Also the optimal value R lies below unity.
6
Conclusion
A thermo-economic optimization has been carried out for an irreversible solar driven heat engine. The objective function has been defined as the ratio of power output to the total investment cost for setting up the plant. The effects of irreversibility parameter, economical parameter and the design parameters on the thermo-economic objective function have been investigated. By optimizing the objective function the optimum ranges for various parameters have been determined.
References [1]
Chambadal P. Les Centrales Nuclearies. Paris: Armond Colin; 1957. p. 41–58.
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434 Advanced Computational Methods in Heat Transfer IX [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Novikov II. The efficiency of atomic power stations (a review). Atom Energy 1957; 3(11):409. Curzon Fl, Ahlborn B. Efficiency of a Carnot engine at maximum power output. Am J Phys 1975; 43:22–4. Sahin B, Kodal A, Yavuz H. Efficiency of a Joule-Brayton engine at maximum power density. Journal of Physics, D: Applied Physics 1995; 28:1309-13. Lund KO. Applications of finite-time thermodynamics to solar power conversion. In: Sieniutycz S, Salamon P, editors. Finite-time thermodynamics and thermoeconomics. London: Taylor & Francis, 1990. p. 121. Medina A., Roco J.M.M., Hernandez A.C., Regenerative gas turbines at maximum power density conditions, J. Phys. D: Appl. Phys. 29 (1996) 2802–2805. Jeter S. Maximum conversion efficiency for the utilization of direct solar radiation. Sol Energy 1981; 26:231–6. De Vos A, Pauwels H. On the thermodynamic limit of photovoltaic energy conversion. J Appl Phys 1981; 25:119–25. De Vos A. Efficiency of some heat engines at maximum power conditions. Am J Phys 1985; 5:570–3. Chen L, Yan Z. The effect of heat transfer law on performance of a twoheat-source endoreversible cycle. J Chem Phys 1989; 9:3740–3. Gordon JM. Observations on efficiency of heat engines operating at maximum power. Am J Phys 1990; 58:370–5. Gordon JM. On optimized solar-driven heat engines. Sol Energy 1988; 40:457–61. Bejan A. Advanced engineering thermodynamics. New York: Wiley; 1988. Bejan A. Heat transfer. New York: Wiley; 1993. Chen J. Optimization of a solar-driven heat engine. J Appl Phys 1992; 72:3778–80. Goktun S, Ozkaynak S, Yavuz H. Design parameters of a radiative heat engine. Energy 1993; 18:651–5. Ozkaynak S. Maximum power operation of a solar-powered heat engine. Energy 1995; 20:715–21. Badescu V. Optimum design and operation of a dynamic solar power system. Energy Convers Manage 1996; 37:151–60. Erbay LB, Yavuz H. An analysis of an endoreversible heat engine with combined heat transfer. J Phys D: Appl Phys 1997; 30:2841–7. Badescu V, Popescu G, Feidt M. Model of optimized solar heat engine operating on Mars. Energy Convers Manage 1999; 40:1713–21. Sahin AZ. Optimum operating conditions of solar driven heat engines. Energy Convers Manage 2000; 41:1335–43. Sahin AZ. Finite-time thermodynamic analysis of a solar driven heat engine. Energy Int J 2001; 2:122–6.
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[23] [24] [25] [26] [27]
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Koyun A. Performance analysis of a solar driven heat engine with external irreversibilities under maximum power and power density condition. Energy Convers Manage 2004; 45:1941–7. Sahin B, Kodal A. Performance analysis of an endoreversible heat engine based on a new thermoeconomic optimization criterion. Energy Convers Manage 2001; 42:1085–93. Kodal A, Sahin B. Finite size thermoeconomic optimization for irreversible heat engines. Int J Therm Sci 2003; 42:777–82. Khaliq A. Finite-time heat transfer analysis and generalized power optimization of an endoreversible Rankine heat-engine. Appl Energy 2004; 79:27–40. Sahin B., Ust Y., Yilyaz T., Akcay I.H. Thermoeconomic optimization of a solar driven heat engine. Renewable Energy xx (2005) 1-10.
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Analysis of a new solar chimney plant design for mountainous regions M. A. Serag-Eldin Mechanical Engineering Department, American University in Cairo, Egypt
Abstract The paper presents a new design for solar chimney plants to be erected on mountainous terrain. The objective is to exploit the height of nearby mountains to replace the conventional vertical chimney by a duct built over the steep side of the mountain. The concept is evaluated here employing a computational model involving three-dimensional governing equations, describing mass, momentum and energy balances, in addition to transport of two turbulence quantities. A typical application is demonstrated for a 450 sloped mountain. The concept is shown to be plausible and offers many advantages over conventional chimney designs. Keywords: solar chimneys, solar energy, renewable energy, buoyancy driven flows, solar power plants.
1
Introduction
The present paper introduces a modification to the conventional solar chimney plant in order to exploit the available heights of any nearby available mountain. The new design is evaluated by means of a suitable mathematical model. The paper starts in this section by describing the conventional solar chimney plant and then introducing the suggested design modifications. The next section presents the adopted mathematical model. The model is then applied to predict the flow field and performance of a demonstration case, and the results are displayed and discussed. The advantages of the new design are listed, and finally, a summary and conclusions is presented.
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438 Advanced Computational Methods in Heat Transfer IX 1.1 Conventional solar-chimney plant Solar-chimney power plants are renewable energy devices which generate mechanical power from solar energy after first converting it into wind energy. They comprise a tall chimney stack located at the center of a giant solar energy collector displaying an elevated glass roof, covering a large area of ground clad with a suitable absorber material. Figure 1 displays a cross-sectional sketch of a typical plant; the arrows point to the main direction of flow, and the label “W.T.” points to the location of an installed wind-turbine.
Figure 1:
Conventional solar chimney plant.
Solar radiation penetrates the collector roof to warm the collector bottom, by virtue of the so called “green-house effect”. The latter warms the air immediately above it, which then rises inside the chimney stack driven by buoyancy forces, and creating a partial vacuum near the ground surface. The difference in pressure between the outside atmospheric air and the low pressure below the chimney stack causes the outside air to be drawn into the collector and towards the stack. Upon reaching the stack the air turns upwards and flows towards the stack outlet, driving an enclosed wind turbine along its way. The turbine is an axial pressure-staged one, more akin to Kaplan hydraulic turbines than to common wind turbines. The performance of solar chimney plants has been investigated by several researchers [1-5]. Among others, Serag-Eldin [6] has shown that, for a given collector area, the performance increases rapidly with increase of height of stack. This is why commercial designs consider stack heights of 500-1000m. Technological difficulties and high costs restrict stack heights above this level. 1.2 Proposed solar chimney mountain design Given a proposed solar chimney site in the northern hemisphere which features a nearby northern highly sloped mountain; considerable advantages may be gained by locating the collector in the valley immediately adjacent to the mountain side WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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and replacing the chimney with a duct running upwards along the sloped side of the mountain, following the shortest convenient path towards the mountain top. In the southern hemisphere, a southern mountain is the preferred choice. The duct should display a smooth progression from the collector crosssection into a horizontal tube (shaft), necessary to accommodate a horizontal axis wind-turbine, and then evolve into any arbitrary shape, optimized to cut construction costs and reduce flow losses. Obviously in this case the collector cover and absorber plate will no longer be circular, and they have to be shaped for optimum land usage and minimum flow losses, as will be revealed later on for the demonstration case. The duct may be built over the ground surface of the valley and mountain; however, to cut down costs, and or possibly mitigate environmental hazards, the duct could even be built as an underground tunnel or covered trench. Aesthetic considerations may also favour burying it. Although the success of the design is highly site specific, so are many types of renewable energy systems; wind energy and tidal energy being notorious examples. Moreover, many potential sites may be found that meet the criteria.
2
Mathematical model
The model comprises three dimensional differential equations expressing the conservation of mass and energy, balance of momentum, and transport of the kinetic energy of turbulence, k, and its rate of dissipation, ε, according to the (k-ε) model of turbulence, Launder and Spalding [7]. Employing the Cartesian system of coordinates (x,y,z) the governing equations may all be cast in the following concise form: ∇ . ( ρ V φ ) = ∇ . ( Γφ ∇φ ) + Sφ
(1)
where ∇ ≡ ∂/∂x i + ∂/∂y j + ∂/∂z k is the Nabla operator, φ denotes any dependent variable , V = u i + v j + w k is the velocity vector, Γφ and Sφ are the diffusion coefficient and source term, respectively , for the variable φ. The mass conservation equation may also be expressed in the form of eqn (1) with φ≡1 and Γφ=Sφ=0. For the velocity components Γφ = µe, whereas Γh=µt/σh , Γk=µt/σk and Γε=µt/σε . The Sφ expressions are presented in Table.1, where p denotes the static pressure; µe is the effective viscosity, µ is the molecular viscosity, and µt is the eddy diffusivity.. The turbulence model constants C1,C2, Cµ ,σk, σε and σh are those proposed by Launder and Spalding [7]. The local air-density, ρ, is calculated from the ideal gas equation of state, employing the standard atmospheric pressure and the locally computed air temperature. The latter is deduced from the solved for static enthalpy, h, of the air. The external prevailing air density, ρa, is assumed constant and calculated according to the ideal gas equation of state at standard atmospheric pressure and the prevailing atmospheric temperature at ground level.
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440 Advanced Computational Methods in Heat Transfer IX Table 1: φ
Source term expressions.
Sφ
u - ∂p/∂x+∂(µe∂u/∂x)/∂x+∂(µe∂v/∂x)/∂y+∂(µe∂w/∂x)/∂z v -∂p/∂y+∂(µe∂u/∂y)/∂x+∂(µe∂v/∂y)/∂y+∂(µe∂w/∂y)/∂z+(ρa-ρ)g w - ∂p/∂z+∂(µe∂u/∂z)/∂x+∂(µe∂v/∂z)/∂y+∂(µe∂w/∂z)/∂z k Gk* +Gb**- ρ ε ε C1 ε/k.(Gk*+Gb**) - C2 ρ ε2/k h Dp/Dt *
Gk = µt {2[(∂u/∂x)2 + (∂v/∂y)2 +(∂w/∂z)2] + (∂u/∂y+∂v/∂x)2 + (∂u/∂z+∂w/∂x)2 + (∂v/∂z+∂w/∂y)2}, where µt ≡ ρ Cµ k2 /ε , µe = µ + µt
**
Gb= -µt/σh{g / ρ . ∂ρ /∂z }
The governing equations presented here adopt the Cartesian system of coordinates for sake of clarity and simplicity. However, the equations actually solved are the counterparts of the above in boundary-fitted-coordinates, BFC, Hedberg et al. [8]. The coordinate system is selected such that one of the coordinates tracks the local flow direction closely, while the other two coordinates are nearly orthogonal to it. The advantages of using this system are both accurate presentation of the boundary geometry, and minimization of numerical diffusion effects caused by the oblique intersection of local flow direction with grid-line directions. The axi-symmetric counterpart of this model was reported in detail for a conventional solar chimney plant, by Serag-Eldin [6, 9]. Here, by necessity, three boundary fitted coordinates are employed. The wind-turbine is introduced as an actuator disk and its effect on the flow is introduced primarily as a drag force.
3
Demonstration case
The demonstration case is sketched in Fig. 2. It displays an approximately semicircular solar collector of 2000 m radius. The height of the collector cover at inlet is 8 m and it increases gradually towards the centre of the collector. The outlet of the collector develops gradually into a rectangular cross-section, 60 m high by 160 m wide, at a distance of 3200 m downstream the inlet. It develops further into a circular cross-section of 80 m diameter. The later forms the inlet of a 300 m long tube (shaft) which harbours the wind-turbine. Downstream the turbine shaft, the duct forms a 45o vertical bend and rises up the mountain side at an inclination of 45o. Right after the bend the crossWIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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section develops gradually from the circular cross-section into a rectangular one of 42 m height and 120 m width. Thereafter, the rectangular duct continues to expand laterally and linearly to reach a width of 360 m at exit. The height of the duct centre-line at exit is approximately 1000 m above ground level. Three different exit section heights were considered; namely 14 m, 42 m and 84 m. They correspond to constant area, a three folds and six folds increase, respectively.
Figure 2:
4
Sketch of proposed solar chimney design for mountainous regions.
Results
Figure 3 reveals part of the predicted velocity vectors in the collector region, midway up the collector-cover height. The vectors reveal in both magnitude and direction the predicted local velocity, the magnitude being revealed by the displayed arrow scale at the bottom of the diagram. The computational grid covered one half of symmetry plane, employing 20x40x106 cells in the quasi vertical, cross-stream and flow directions, respectively. The vector shows that the external air is sucked in smoothly; and that with the exception of a narrow region close to the wall towards the collector exit, the flow is smooth. It is remarked that although the main bulk of the collector area is axi-symmetric, the flow arriving from the symmetry plane and that arriving from the near wall follow very different paths as they approach the exit of the collector; the former follows a straight-through path, whereas the latter goes through a 900 bend. More careful design of the wall profile should eliminate this problem; at present the wall profile is simply taken to be an arc of a circle.
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442 Advanced Computational Methods in Heat Transfer IX It is also noted that the velocity magnitude is small throughout most of the collector region (~ 2 m/s) in order to minimize losses, but then accelerates rapidly as it approaches the turbine tunnel.
Figure 3:
Velocity vectors at mid-plane of collector region.
Figure 4 reveals the corresponding velocity vectors in the far downstream region where the duct width expands linearly. The arrow scale now displays a velocity of 75 m/s, reflecting the much larger velocity magnitudes leaving the turbine shaft. The area ratio for the case displayed was three.
Figure 4:
Velocity vectors in the far downstream section.
Figure 5 reveals the temperature distribution at ground level throughout the solar chimney plant. The ground level is horizontal up to the end of the turbine shaft, after which it rises with a slope of 45o to follow the mountain side profile. A colour code is employed to differentiate between regions of same temperature WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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range; the key on the right reveals their magnitudes. It is seen that with the exception of a narrow segment close to the side wall, the temperature distribution is nearly symmetrical. This is desired since it leads to more uniform temperature distributions and hence flow distribution at the turbine section. Near the walls temperatures are seen to be lower, influenced by heat loss from the side walls. The temperature of the fluid is seen to rise as it flows towards the exit of the collector due to the continuous heating from the ground and the absence of re-circulatory motions. Good mixing results in near uniform temperature distribution at the wind turbine section (W.T.)
Figure 5:
Figure 6:
Temperature distribution at ground level.
Temperature distribution half-way above ground.
Figure 6 reveals the corresponding temperature distribution at a solution surface half way up the duct cross-section; as expected the profiles reveal lower maximum temperatures and results of good mixing at the turbine plane. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
444 Advanced Computational Methods in Heat Transfer IX Figure 7 displays the “reduced-pressure” distribution over the vertical plane passing through the symmetry axis. The “reduced-pressure” refers to the difference between the local pressure and the external atmospheric pressure at the same horizontal height; which is the quantity actually solved for in order to simplify boundary conditions. A colour code is employed to reveal the reduced pressure levels. A large pressure drop is noticed at the turbine section (W.T.), and kinetic energy recovery is apparent in the diffuser section.
Figure 7:
Reduced pressure distribution in vertical symmetry plane.
MW Ouput
The effect of stack area-ratio on output of turbine is displayed in Fig. 8. Power output is estimated as the product of mean axial velocity at turbine section and total turbine thrust (drag) force. It is apparent that a 20% gain in output can be obtained by introducing an area ratio of three; however, further increase leads only to small gains.
100 90 80 70 60 50 40 30 20 10 0 0
2
4
6
Area Ratio
Figure 8:
Effect of exit area-ratio on output.
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Advantages of proposed design
When an appropriate terrain is encountered, the proposed design offers the following advantages over the conventional solar chimney design: 5.1 Cost savings It is expected that the exhaust duct, which is supported directly on the ground will be much cheaper to build than a towering chimney stack, over 500 m high. Moreover the wind turbine will be located much closer to the ground, thus cutting down the cost of its support structure. 5.2 Potential for higher heads Due to structural concerns, particularly those related to wind loads and seismic loads, the current technology limits feasible chimney stack heights to under 1000 m. With the present design considerably higher heights may be reached, the only limitation being posed by the height of the mountain peak. Indeed, this could also be extended partially, by building a chimney stack at the top of the mountain peak. 5.3 Less need for sophisticated technology Solar chimney power plants have been presented as an ideal energy source for many of the developing countries lying in the solar belt (Schlaich and Schiel [10]). Except for the towering chimney stack, they require only primitive technology to build. The new design dispenses with the technologically challenging stack, and therefore local expertise may be all that is needed. 5.4 Potential for co-generation and combined cycle One of the major disadvantages of solar chimney plants is their very low efficiency. Most of the solar energy used to heat the air is wasted in the form of hot air exhausting from the stack to the atmosphere. With the present design, the hot air may be employed at the top of the mountain for cogeneration or combined cycle purposes. 5.5 Kinetic energy recovery Another source of energy waste in conventional solar chimney plants is the kinetic energy of the exhausting air. For economic reasons the exit stack diameter is kept to a minimum and thus exhaust air leaves at substantial velocities. However, with the present design it is fairly economic and very feasible to build a diffuser section in the exhaust duct to reduce the kinetic energy loss.
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6 Summary and conclusions The paper proposes exploiting the height of appropriate nearby mountains to replace the costly chimney stack with a simple ground laid duct. This offers many advantages in terms of economy as well as potential for cogeneration and combined cycle applications. A demonstration case is presented and the concept is validated employing a given computational model. Although each design will necessarily be site specific, the design may be readily optimized with the aid of the presented computational model.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Padki, M.M. & Sherif, S.A., On a simple analytical model for solar chimneys,” Int. J. Energy Research, 23(4), pp. 345-349, 1999. VonBackstrom, T.W.& Gannon, A.J., Compressible flow through solar power plant chimneys,” ASME J. Solar Energy Eng.,122(3), pp.138-145, 2000. Pasumarthi, N. & Sherif, S.A., Performance of a demonstration solar chimney model for power generation, Proc. 35th Heat Transfer and Fluid Mechs. Inst., ed. F.H. Reardon, Sacremento, California, 1997. Gannon, A.J. & Von Backstrom, T.W., Solar chimney cycle analysis with system loss and solar collector performance, J. Solar Energy Eng., 122, pp.133-137, 2000. Serag-Eldin, M.A., Computing flow in a solar chimney plant subject to atmospheric winds”, Proc. ASME Heat Transfer/Fluids Engineering Summer Conference, Charlotte, NC, 2004. Serag-Eldin, M.A., Analysis of effect of geometric parameters on performance of solar chimney plants, Proc. ASME 2005 Summer Heat Transfer Conference, San Francisco, CA, HT2005-72340, 2005. Launder, B.E. and Spalding, D.B., The numerical computation of turbulent flows, J. Computer Methods in Applied Mechanics and Eng., 3(1), pp. 269-289, 1974. Hedberg, P.K., Rosten, H.I. and Spalding, D.B., The PHOENICS Equations, Report TR/99, CHAM , U.K., 1986. Serag-Eldin, M.A., Analysis of effect of turbine characteristics on performance of solar chimney plants, Proc. ASME 2005 Summer HT Conf., San Francisco, CA, Paper No. HT2005-72835, 2005. Schlaich, J. and Schiel,W., Solar Chimneys, Encyclopaedia of Physical Science and Technology, pp1-10, 3rd, edition, 2000.
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Section 7 Micro and nano scale heat and mass transfer
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Sinusoidal regime analysis of heat transfer in microelectronic systems B. Vermeersch & G. De Mey Department of Electronics and Information Systems, Ghent University, Belgium
Abstract The boundary element method has been used to evaluate the thermal impedances of electronic components in the micrometer range. All simulations gave rise to impedance plots composed of a few circular arcs. This conclusion has also been confirmed by experimental measurements and semi analytical calculations. Keywords: thermal impedance, thermal conduction, AC, numerical calculation, boundary element method.
1 Introduction All books devoted to heat transfer have mainly three chapters bearing the respective titles “conduction”, “convection” and “radiation”. The convective part is usually the most elaborated one for the obvious reason that this is the most important one in many industrial applications. If one is dealing with microelectronics, conduction turns out to be dominant. Indeed, the heat dissipated in the silicon semiconductor can only reach the cooling fins by thermal conduction through the silicon and the packaging materials. It often happens that the largest temperature drop occurs between the chip and the base of the cooling fin. The reason is obvious: the heat sources are so small that the immediate neighbourhood turns out to be the largest and hence most dominant thermal resistance. In most books on heat transfer, the last part of “conduction” is devoted to time dependent problems (Bejan [1], Incopera and De Witt [2]). A section related to AC thermal conduction (phasor notation) is non-existent. Whereas in electricity and electronics the use of phasor notation ( jω instead of ∂/∂t) is quite common, the application in the field of thermal analysis seems to be rather exceptional. In microelectronics however, an AC thermal analysis has a lot of importance. The WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060431
450 Advanced Computational Methods in Heat Transfer IX electronic signal and hence the corresponding heat production are usually periodic functions of time. In order to account for the thermal feedback on the electrical behaviour a thermal AC analysis for both electric and thermal phenomena becomes more and more useful. A limited number of papers has been devoted to the AC thermal behaviour (De Mey [3–5]). In this paper the emphasis will be on the calculation of the thermal impedance, i.e. the temperature difference between the heat source and the ambient divided by the power dissipation: Zth = Tsource P , in K/W. Note that all quantities like temperature, power,etc., are in phasor notation, thus complex functions of the spatial coordinates x, y and z. Some words should be said about the abbreviation AC. Strictly speaking it means Alternating Current. During the years this abbreviation has been used to denote any quantity which has a sinusoidal variation with respect to time. Even expressions like AC volts are used. In this contribution AC temperature simply means a temperature distribution varying sinusoidally with respect to time and certainly not Alternating Current Temperature.
2 Basic equations The heat transfer equation for the temperature in an IC is given by: k∇2 T − jωCv T = 0
(1)
√ where k is the thermal conductivity in W/mK, j = −1, ω = 2πf the angular frequency in rad/s and Cv the thermal capacity per volume unit in J/m3 K.
Figure 1: Cross section of a typical microelectronic structure.
The boundary conditions are explained with the help of a cross section being displayed in Fig. 1. On the bottom FA the ambient temperature T = 0 is applied. On the top a heat source dissipating P Watts is situated between the points D and C. Due to the small dimensions in microelectronic structures, the convective or WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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radiative heat transfer through the upper part is negligible. Hence the power can only be transferred by conduction into the substrate material or: ∂T P =k S ∂z
(on CD)
(2)
where S is the area of the heat source. A uniform power density inside the heat source has been assumed here. Along the other sides, the adiabatic boundary condition is taken into account: ∂T ∂T ∂x = 0 along AB and EF and ∂z = 0 along ED and CB. Remark again that this paper deals with 3D problems and Fig. 1 is just an examplary 2D cross section of a 3D structure. If two materials are in contact, the continuity of the temperature T and the normal heat flux k ∂T ∂n have to be taken into account. In order to set up an equivalent boundary integral equation for (1), the Green’s function of (1) has to be used: jωC 1 v exp − |r − r | (3) G(r|r ) = 4πk|r − r | k where r is the so called field point and r the so called source point. Further details about the integral equation, the discretisation procedure,etc., will be omitted here. For the impedance calculations, one cannot take the heat source temperature, because this is generally a non uniform function. Therefore the average temperature over the heat source has been used to evaluate the impedances.
3 Results First of all some experimental results will be shown (Kawka [6]). A typical result is displayed in Fig. 2.
Figure 2: Experimentally measured impedance plot.
Because a pure alternating heat source does not exist, measurements have been done using a step input power. The temperature has been recorded a function of WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
452 Advanced Computational Methods in Heat Transfer IX time, and by suitable Fourier techniques these data could be converted into an impedance plot as shown in Fig. 2. One observes very clearly that the plot is composed of two distinct circular arcs. The question now is whether these almost perfect circular shapes are just a pure coincidence or do they have a more general validity. The only way to answer this question is to investigate several structures analytically or numerically. For a rectangular shaped heat source on the surface of a half infinite thermal conducting medium, the impedance can be calculated directly by a numerical integration as shown by Vermeersch and De Mey [7]. The results for a square shaped heat source (200 µm × 200 µm) on a half infinite silicon substrate (k = 160 W/mK, Cv = 1.784 × 106 J/m3 K) are presented in Fig. 3.
Figure 3: Semi-analytically calculated impedance plot.
For the sake of comparison, the impedance has also been evaluated using the maximum instead of the average temperature inside the heat source. For both plots the so called central frequency fc , for which Im[Zth ] reaches a minimum, is indicated. Using the BEM, more general 3-D structures can be analyzed. A first example is shown in Fig. 4. The top of the cube has a uniform heat source whereas the bottom plate is at T = 0 reference temperature. The four other sides are assumed adiabatic. For this particular geometry, an analytical solution can be easily found because the temperature only depends on the vertical coordinate z. The agreement between the numerical and the analytical results is extremely good even for various frequencies ranging from f = 10 Hz up to f = 100 kHz. A second example is shown in Fig. 5. For clarity, the values for the DC impedance Z0 = Rth are indicated in the graphs. Two media with different thermal conductivities are involved here. The impedance plots clearly shows two circular arcs. By changing the thermal conductivity k2 of the bottom layer, the relative magnitude of the high frequency arc with respect to the low frequency arc is modified. One should however keep in mind that the WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Figure 4: Example 1 – cubic shaped silicon substrate completely covered by a uniform heat source: (a) geometry, (b) calculated impedance plot.
impedance curves are plotted on different scales. A closer look in Fig. 5 reveals that only the lower frequency arc is influenced, while the other remains unchanged throughout the graphs. This can easily be explained by taking into account that for AC thermal conduction, the size of the influenced region in the material depends on the source frequency. For high frequencies only the upper part of the silicon is heated, hence k2 does not play a role in the behaviour of the thermal impedance.
4 Conclusion It has been proved, with the help of the boundary element method, that the thermal AC behaviour of a microelectronic structure can be modelled. Comparisons have been made with experimental and analytical results. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Figure 5: Example 2 – silicon substrate mounted on packing material: (a) geometry, (b) calculated impedance plots for various thermal conductivities of the bottom layer.
References [1] Bejan, A., Heat transfer, Wiley: New York, 1993. [2] Incopera, F. & De Witt, D., Introduction to heat transfer, Wiley: New York, 1985. [3] De Mey, G., Integral equation approach to AC diffusion. International Journal of Heat and Mass Transfer, 19, pp. 702-704, 1976. [4] De Mey, G., Thermal conduction in phasor notation. Proc. of the Boundary WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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Element Technology Conference (BETECH’95), ed. C.A. Brebbia, Computational Mechanics Publications: Southampton, pp. 1-9, 1995. [5] De Mey, G., Heat transfer in electronics: a BEM approach. Proc. of the Conference on Boundary Element Research in Europe, ed. C.A. Brebbia, Computational Mechanics Publications: Southampton, pp. 203-209, 1998. [6] Kawka, P., Thermal impedance measurements and dynamic modelling of electronic packages (PhD thesis), Ghent University: Gent, 2005. [7] Vermeersch, B. & De Mey, G., Thermal impedance plots of micro-scaled devices. Microelectronics Reliability, 46(1), pp. 174-177, 2006.
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Viscous dissipation and temperature dependent viscosity effects in simultaneously developing flows in flat microchannels with convective boundary conditions S. Del Giudice, S. Savino & C. Nonino Dipartimento di Energetica e Macchine, Universit`a degli Studi di Udine, Udine, Italy
Abstract The effects of viscous dissipation and temperature dependent viscosity in simultaneously developing laminar flows of liquids in straight microchannels of arbitrary but constant cross-section are studied with reference to convective thermal boundary conditions. Viscosity is assumed to vary linearly with temperature, in order to allow a parametric investigation, while the other fluid properties are held constant. A finite element procedure, based on a projection algorithm, is employed for the step-by-step solution of the parabolized momentum and energy equations. Axial distributions of the local overall Nusselt number and of the apparent Fanning friction factor in flat microchannels are presented with reference to both heating and cooling conditions for two different values of the Biot number. Examples of temperature profiles at different axial locations are also shown. Keywords: laminar forced convection, microchannels, entrance region, temperature dependent viscosity, viscous dissipation, convective boundary conditions.
1 Introduction In several heat transfer problems concerning microchannel flows, the thermal resistance of the channel wall must be taken into account in order to obtain accurate solutions. Moreover, in many situations of practical interest, fluid velocity and temperature fields develop simultaneously, resulting in overlapping hydrodynamic and thermal entrance regions. This occurs when fluid heating or cooling begins at the microchannel inlet, where also the velocity boundary layer WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/HT060441
458 Advanced Computational Methods in Heat Transfer IX starts developing. In such a situation, entrance effects on fluid flow and forced convection heat transfer cannot be neglected if, as it happens very often in laminar flows, the total length of the microchannel is comparable with that of the entrance region. Temperature dependence of fluid properties can also play an important role in the development of velocity and temperature fields. If, as it is assumed in this paper, the fluid is a liquid, viscosity is the property which exhibits the most relevant variations with respect to temperature. Therefore, the main effects of temperature dependent fluid properties can be retained even if only viscosity is allowed to vary with temperature, while the other properties are assumed constant. Finally, viscous dissipation effects cannot often be neglected in ducts with very small hydraulic diameters, like microchannels, even for ordinary liquids, characterised by moderate values of viscosity. It is worth noting that, with temperature dependent viscosity, viscous dissipation affects both temperature and velocity distributions along the microchannel. Even if all these aspects have already been considered in the past, to the authors’ knowledge no systematic studies are reported in the literature taking into account the combination of wall thermal resistance, entrance, temperature dependent viscosity and viscous dissipation effects. Only recently, the authors have presented a couple of papers where the influences of the last three factors are analyzed, but the wall thermal resistance is neglected [1, 2]. It must be noticed, however, that a simplified way to account for it and, possibly, also for the external convection resistance, is represented by the specification of convective boundary conditions at the wall, with an appropriately defined heat transfer coefficient, when solving the energy equation. This approach is adopted in the present paper which, thus, represents an extension of the previous work. In the past decades, many authors have investigated, both analytically and numerically, simultaneously developing flows in straight ducts of constant crosssection. Comprehensive reviews of these theoretical studies, referring to ducts of different cross-sectional geometries, can be found in [3, 4]. However, since a basic assumption made in almost all such studies is that fluid properties are constant, the corresponding solutions are adequate only for problems involving small temperature differences. In fact, experimental results for problems involving large temperature differences substantially deviate from constant property solutions [3, 5]. In this paper, we present the results of a parametric study on the simultaneously developing laminar flow of a liquid in flat microchannels. The effects of wall thermal resistance, temperature dependent viscosity and viscous dissipation on heat transfer and pressure drop are investigated, while the other liquid properties are considered constant. A finite element procedure [6], based on a SIMPLE-like algorithm [7], is employed for the step-by-step solution of the parabolized momentum and energy equations [7, 8] in a two-dimensional domain corresponding to the cross-section of the duct. Due to the high value of the ratio between the total length and the hydraulic diameter in microchannels, such an approach is very advantageous with respect to that based on the steady-state solution of the set of governing elliptic equations in a three-dimensional domain corresponding to the whole microchannel. The above procedure has already been used, disregarding viscous dissipation effects, in the simulation of simultaneously developing flows of liquids WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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with temperature dependent viscosity in straight macro- and microchannels [9, 10] and, on the additional assumptions of non-negligible viscous dissipation effects, in the study of thermally and simultaneously developing liquid flows in microchannels when prescribed constant temperature boundary conditions are applied to the rigid walls [1, 2]. Here the same procedure is used to study a similar problem with reference to flows in flat microchannels with convective thermal boundary conditions. In order to allow a parametric investigation, viscosity is assumed to vary linearly with temperature in the range considered.
2 Mathematical model When the effects of axial diffusion can be neglected and there is no recirculation in the longitudinal direction, steady-state flow and heat transfer in straight microchannels of constant cross-section are governed by the continuity and the parabolized Navier-Stokes and energy equations. With reference to incompressible fluids with temperature dependent thermophysical properties, in the hypotheses of negligible body forces and significant effects due to viscous dissipation, these equations can be written in the following form ∂ ∂ ∂ (ρu) + (ρv) + (ρw) = 0 ∂x ∂y ∂z ∂u dp ∂u ∂ ∂u ∂u − ρw − + µ − ρv ∂y ∂z ∂z ∂y ∂z dx ∂v ∂ 2 ∂w ∂v ∂v ρu = − +µ 2µ µ ∂x ∂y ∂y 3 ∂y ∂z ∂v ∂v ∂p ∂ ∂w ∂v +µ − ρw − + µ − ρv ∂z ∂y ∂z ∂y ∂z ∂y ∂w ∂ ∂w ∂w 2 ∂v ρu = − +µ 2µ µ ∂x ∂z ∂z 3 ∂y ∂z ∂v ∂w ∂p ∂ ∂w ∂w +µ − ρw − + µ − ρv ∂y ∂y ∂z ∂y ∂z ∂z ∂t ∂ ∂t ∂t ∂ ∂t ∂t ρcu = − ρcw + µΦv k + k − ρcv ∂x ∂y ∂y ∂z ∂z ∂y ∂z ρu
∂ ∂u = ∂x ∂y
(1)
µ
(2)
(3)
(4)
(5)
where 2 2 ∂w ∂w ∂w Φv =2 + + + ∂z ∂y ∂z 2 2 2 ∂u ∂u ∂w 2 ∂v + + + − ∂y ∂z 3 ∂y ∂z
∂v ∂y
2
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
(6)
460 Advanced Computational Methods in Heat Transfer IX According to the assumption of parabolic flow, all the derivatives in the axial direction are neglected in the diffusive terms of eqns. (2) to (6) [8]. In the above equations, x and y, z are the axial and the transverse coordinates, respectively, while u and v, w represent the axial and the transverse components of velocity. Finally, t is the temperature, p is the deviation from the hydrostatic pressure, p is its average value over the cross-section, while ρ, µ, c and k represent density, dynamic viscosity, specific heat and thermal conductivity of the fluid, respectively. The solution domain can be bounded by rigid walls or by symmetry axes. On rigid boundaries the usual no-slip conditions, that is, u = v = w = 0, are imposed together with the convective boundary condition q = −k∂t/∂n = ha (t − ta ), where q is the local wall heat flux, n is the outward normal to the boundary, ha is the external convective heat transfer coefficient and ta is the ambient fluid temperature. Instead, symmetry conditions are ∂u/∂y = ∂w/∂y = 0, v = 0 and ∂t/∂y = 0 on boundaries perpendicular to the y axis and ∂u/∂z = ∂v/∂z = 0, w = 0 and ∂t/∂z = 0 on boundaries perpendicular to the z axis. The model equations are solved using a finite element procedure which represents an extended version of one previously developed for the analysis of the forced convection of constant property fluids in the entrance region of straight ducts [6]. The added new features mainly consist in the possibility of taking into account the effects of temperature dependent properties and of viscous dissipation. The adopted procedure is based on a segregated approach which implies the sequential solution of momentum and energy equations on a two-dimensional domain corresponding to the cross-section of the channel. A marching method is then used to move forward in the axial direction. The pressure-velocity coupling is dealt with using an improved projection algorithm already employed by one of the authors (C.N.) for the solution of the Navier-Stokes equations in their elliptic form [11].
3 Numerical results As stated above, the laminar forced convection in the entrance region of flat microchannels with convective thermal boundary conditions is studied. The hypotheses made here are that viscous dissipation effects are not negligible and that liquid heating/cooling begins at the microchannel inlet, where also the velocity boundary layer starts developing. Therefore, at the entrance of the microchannel, uniform values of the velocity ue and of the temperature te can be specified as the appropriate inlet conditions. At the microchannel wall the convective boundary condition = ha (tw − ta ) is applied, where qw and tw are the local wall heat flux and qw temperature, respectively. The dynamic viscosity is assumed to vary linearly in the temperature range considered and µe and µa are its values at te and ta , respectively. The temperature dependence of the dynamic viscosity in the range between te and ta is represented by the ratio µe /µa = Pre /Pra = Rea /Ree , where Pre = µe c/k and Pra = µa c/k are local Prandtl numbers, while Ree = ρue Dh /µe and Rea = ρue Dh /µa are local Reynolds numbers based on the hydraulic diameter Dh . Note that, while the values of local Reynolds and Prandtl numbers depend on temperWIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
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ature, the local P´eclet number Pe = RePr = Ree Pre = Rea Pra always has the same value. Since the viscosity of liquids decreases with increasing temperature, Pre /Pra > 1 corresponds to fluid heating (te < ta ) and Pre /Pra < 1 to fluid cooling (te > ta ), while Pre /Pra = 1 refers to isothermal flows (te = ta ) or to constant viscosity fluids. Moreover, the reference Brinkman number Brm = µm u2e /[k(te − ta )], evaluated at the reference fluid temperature tm = (te + ta )/2, is negative for fluid heating and positive for fluid cooling. In all the computations, the same values Rem = ρue Dh /µm = 500 and Prm = µm c/k = 5 of the Reynolds and Prandtl numbers have been assumed. Therefore, for the values of the ratio Pre /Pra = 1/2, 2/3, 1, 3/2 and 2 considered here, the local Reynolds number varies between 375 and 750 in the temperatures range between te and ta . In addition to Brm = 0, corresponding to negligible viscous dissipation, two reasonable non-zero values of the reference Brinkman number have been chosen, namely, Brm = ±0.01 and ±0.1. Finally, two values of the Biot number Bi = ha Dh /k = 5 and 20 have been selected as representative of situations where convective boundary conditions should be specified. In the following, numerical results concerning axial distributions of fapp Rem and of the local overall Nusselt number Nuo = U Dh /k are presented. In these expressions fapp is the apparent Fanning friction factor defined as [3] (pe − p)Dh 2ρu2e x
fapp = while
U=
qw tb − ta
(7)
(8)
is the local overall heat transfer coefficient, where tb is the bulk temperature [3, 4]. The local overall heat transfer coefficient U can be expressed as [3, 4] U=
1 1 + h ha
−1 (9)
with the local convective heat transfer coefficient h defined as h=
qw tb − tw
(10)
On the basis of eqn. (9), the local overall Nusselt number Nuo can be expressed as Nuo =
1 1 + Nu Bi
−1 (11)
where Nu = hDh /k is the local Nusselt number. The computational domain has been defined taking into account existing symmetries. Therefore, the cross-section of the flat channel considered here, having the half-spacing between the plates equal to a, corresponds to a rectangle of unit base and height a, which has been discretised by means of 9-node Lagrangian WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
462 Advanced Computational Methods in Heat Transfer IX 9
11 0.01 0.1
0.01 0.1 7
9
(Nuo )c
(Nuo )c 5
Brm 0.01 0.1 7
Brm 0.01 0.1
0
3
5
0
0.01
0.01
0.1
0.1 1 3 10 (a)
10 2
10 1
3 3 10
1
10 2
10 1
1
(b)
Figure 1: Axial distributions of the local overall Nusselt number (Nuo )c for simultaneously developing flows of constant property fluids with Prm = 5 in flat microchannels with different Brm : (a) Bi = 5 and (b) Bi = 20. parabolic elements. A total of 50 elements, whose sizes gradually increase with increasing distance from the walls, and 303 nodal points have been used in the domain discretisation. Of course, it has first been verified that this discretisation is fine enough to give mesh-independent results. In all the computations, the axial step has gradually been increased from the starting value ∆x/Dh = 0.0001 to the maximum value ∆x/Dh = 0.05. The procedure outlined in the previous section and employed for the numerical simulations had already been validated, on the assumptions of constant property fluid and negligible viscous dissipation, by comparing heat transfer and pressure drop results with existing literature data for laminar simultaneously developing flows in straight channels, both three-dimensional and axial-symmetric [6, 9, 10]. In order to assess the accuracy of the present computations, additional validation tests have been carried out. Asymptotic values of the Nusselt number (Nu∞ )c = h∞ Dh /k and fully developed values of the Poiseuille number (f Re)c for a constant property fluid are compared here with available literature data for flat microchannels. The computed values (Nu∞ )c = 7.54075 and 17.50000 for Brm = 0 and Brm = 0, respectively, are in excellent agreement with the corresponding literature values (Nu∞ )c = 7.54070 and 17.5 [3]. Moreover, the computed fully developed value of the Poiseuille number (f Re)c = 24.0000 coincides with the corresponding literature value (f Re)c = 24 [3]. The influence of viscous dissipation on the local overall Nusselt number is illustrated in fig. 1, where numerical results concerning axial distributions of (Nuo )c for constant property flows (Pre /Pra = 1) with different Biot and Brinkman numbers are presented. As expected, for a given value of the Biot number Bi the same asymptotic value of (Nuo )c is reached for fully developed conditions with any value of Brm , while Brm strongly affects the Nusselt number in the intermediate range of X ∗ = x/(Dh P e). The influence of Brm is also significant for low values of X ∗ , where (Nuo )c values are lower than those corresponding to Brm = 0 if Brm < 0 (heating), and higher if Brm > 0 (cooling). It is also apparent in fig. 1 WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX 1.050
1.050 Pr e Pr a
1.025 Nu o (Nuo )c
Pr e Pr a
2
3 2
Nu o (Nuo )c 1.000 1 2
0.975
Brm
(a)
10 2
2
1.025
1.000
0.950 3 10
463
3 2
2 3
2 3
0.975 1 2
0.01 0.1 10 1
1
0.950 3 10
10 2
Brm
0.01 0.1 10 1
1
(b)
Figure 2: Axial distributions of the ratio Nuo /(Nuo )c for flat microchannels with Prm = 5 and different Brm : (a) Bi = 5 and (b) Bi = 20.
that entrance effects on the local overall Nusselt number (Nuo )c are marginal for low values of Bi, that is, when the wall thermal resistance prevails. The influence of temperature dependent viscosity on the local overall Nusselt number is shown in fig. 2, where axial distributions of the ratio Nuo /(Nuo )c for flat microchannels with different Biot and Brinkman numbers are presented. As can be noticed, the ratio Pre /Pra significantly affects the Nusselt number as long as the flow develops, while its influence is rather small when fully developed conditions are reached. It must be observed that the very high values of the ratio Nuo /(Nuo )c found at intermediate X ∗ for heating (Pre /Pra > 1) are not very significant, since they are simply due to a moderate axial shifting of curves representing axial distributions of Nu with respect to the constant property ones reported in fig. 1. By comparing axial distributions of Nuo /(Nuo )c for different Brinkman numbers Brm and the same Biot number Bi, it can be seen that, in simultaneously developing flows, the influence of temperature dependent viscosity is more evident than that of viscous dissipation. Moreover, for a given |Brm |, liquid heating (Brm < 0 and Pre /Pra > 1) and liquid cooling (Brm > 0 and Pre /Pra < 1) lead to the same asymptotic value of the ratio Nuo /(Nuo )c provided that the corresponding values of the ratio Pre /Pra are reciprocal to each other, i.e., the liquid exhibits the same variation of viscosity in the temperature range between te and ta . On the basis of eqn. (10), this implies that fully developed profiles of the absolute values = (t∞ − tw )/(tb − tw ) are the same for both of the dimensionless temperature T∞ heating and cooling. This also implies that fully-developed axial velocity profiles are equal for liquid heating and liquid cooling, so that the term Φv , defined in eqn. (6), has the same distribution in both cases. The combined effects of temperature dependent viscosity and viscous dissipation on pressure drop are illustrated with reference to axial distributions of fapp Rem for flat microchannels with different Biot and Brinkman numbers. To facilitate the analysis of the results, the axial distribution of (fapp Rem )c for simultaneously developing flows of constant property fluids in flat channels is presented in fig. 3, while axial distributions of the ratio fapp Rem /(fapp Rem )c for WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
464 Advanced Computational Methods in Heat Transfer IX 120 Ref. 3
100 80 ( fapp Re)c 60 40 20 0 3 10
10 2
10 1
1
10
Figure 3: Axial distribution of (fapp Rem )c for simultaneously developing flows of constant property fluids in flat ducts: numerical results (solid line) and literature data (squares).
different Biot and Brinkman numbers are reported in fig. 4. The dimensionless axial coordinate appearing in figs. 3 and 4 is defined as X + = x/(Dh Rem ). Available literature data from [3] are also reported in fig. 3 for comparison. As can be seen, near the entrance the values of fapp Rem /(fapp Rem )c are more influenced by temperature dependence of viscosity than by viscous dissipation, while both effects increase and become comparable when fully developed conditions are approached. As expected, fully developed values of fapp Rem /(fapp Rem )c are larger than 1 for cooling and smaller than 1 for heating, while the opposite occurs near the entrance. Curves for heating and cooling cross each other at axial positions that get closer to the entrance as the Biot number increases. By inspection of figs. 2 and 4, it is evident that the influence of temperature dependent viscosity is greater on pressure drop than on heat transfer rate. The differences between the local values of Nuo /(Nuo )c and fapp Rem /(fapp Rem )c found for different values of the ratio Pre /Pra can be explained taking into 1.4
1.4 Brm
1.2 fapp Rem
Brm
0.01 0.1 Pr e Pr a
2
1.2 fapp Rem
3 2
( fapp Rem )c
( fapp Rem )c
1.0
1.0
0.8 1 2 0.6 2 10 (a)
2 3
10 1
0.01 0.1
0.8
1
10
1 2
Pr e Pr a
0.6 2 10
2
10 1
2 3
3 2
1
10
(b)
Figure 4: Axial distributions of the ratio fapp Rem /(fapp Rem )c for flat microchannels with Prm = 5 and different Brm : (a) Bi = 5 and (b) Bi = 20. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX 0.25
465
0.25 Pr e Pr a
0.20
2 0.20
Pr e Pr a
1 2 1
1 1 0.15
0.15 0.1
0.10
0.1
0.10
0.01 1
0.01 0.05
0.00 0.4
0.2
0
0.2
(a)
T
0.4
0.6
0.8
0.001
0.05
0.001
1
0.00 0.4
0.2
0
(b)
0.2
T
0.4
0.6
0.8
1
Figure 5: Transverse profiles of dimensionless temperature T at different axial locations for flat microchannels with Prm = 5, |Brm | = 0.1 and Bi = 5: (a) fluid heating and (b) fluid cooling. account the temperature distributions over the cross-sections, which, in turn, affect the velocity profiles. As an example, to show the effects of temperature dependent viscosity, transverse profiles of the dimensionless temperature T = (t − ta )/(te − ta ) at selected axial locations are compared in fig. 5 for flows with a temperature dependent viscosity fluid (Pre /Pra = 2 for heating and 1/2 for cooling) and a constant property fluid in the case of Bi = 5 and |Brm | = 0.1. The dimensionless transverse coordinate appearing in fig. 5 is defined as Z = z/Dh = z/(4a). It is worth noting that in the case of fluid heating, temperature profiles for intermediate axial positions exhibit a point of inflection because of viscous dissipation effects, while this does not occur in the case of fluid cooling. Moreover, it is apparent that the effect of temperature dependent viscosity increases with the distance from the entrance and is more relevant for fluid heating (Brm = −0.1) than for fluid cooling (Brm = 0.1).
4 Conclusions The effects of viscous dissipation and temperature dependent viscosity in simultaneously developing laminar flows of liquids in flat microchannels have been studied. Reference has been made to convective thermal boundary conditions at the wall of the microchannels. In order to allow a parametric investigation, viscosity has been assumed to vary linearly with temperature, while the other fluid properties have been held constant. Numerical results confirm that, in the laminar forced convection in the entrance region of straight microchannels, both temperature dependence of viscosity and viscous dissipation effects cannot be neglected in a wide range of operative conditions.
Acknowledgement This work was funded by MIUR (PRIN/COFIN 2005 project). WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
466 Advanced Computational Methods in Heat Transfer IX
References [1] Del Giudice, S., Nonino, C. & Savino, S., Thermally developing laminar flow in microchannels with temperature dependent viscosity and viscous dissipation. Proc. of the ECI ICHTFFM, Castelvecchio Pascoli, Italy, 2005. [2] Nonino, C., Del Giudice, S. & Savino, S., Effects of viscous dissipation and temperature dependent viscosity in simultaneously developing laminar flow in microchannels. Proc. of the ECI ICHTFFM, Castelvecchio Pascoli, Italy, 2005. [3] Shah, R.K & London, A.L., Laminar Flow Forced Convection in Ducts, Academic Press: New York, 1978. [4] Shah, R.K. & Bhatti, M.S., Laminar convective heat transfer in ducts (Chapter 3). Handbook of Single-Phase Convective Heat Transfer, eds. S. Kakac¸, R.K. Shah & W. Aung, Wiley: New York, 1987. [5] Kakac¸, S., The effect of temperature-dependent fluid properties on convective heat transfer (Chapter 18). Handbook of Single-Phase Convective Heat Transfer, eds. S. Kakac¸, R.K. Shah & W. Aung, Wiley: New York, 1987. [6] Nonino, C., Del Giudice, S. & Comini, G., Laminar forced convection in three-dimensional duct flows. Numer. Heat Transfer, 13, pp. 451-466, 1988. [7] Patankar, S.V. & Spalding, D.B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer, 15, pp. 1787-1806, 1972. [8] Hirsh, C., Numerical Computation of Internal and External Flows, Wiley: New York, vol. 1, p. 70, 1988. [9] Nonino, C., Del Giudice, S. & Savino, S., Influence of temperature dependent viscosity in laminar forced convection in axial-symmetric straight ducts (in Italian). Proc. of the 23rd UIT National Conference, Parma, Italy, 2005. [10] Nonino, C., Del Giudice, S. & Savino, S., Influence of temperature dependent viscosity in laminar forced convection in microchannels (in Italian). Proc. of the 23rd UIT National Conference, Parma, Italy, 2005. [11] Nonino, C., A Simple pressure stabilization for a SIMPLE-like equal-order FEM algorithm. Numer. Heat Transfer, Part B, 44, pp. 61-81, 2003.
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Experimental study of water evaporation from nanoporous cylinder surface in natural convective airflow S. Hara Department of Mechanical Engineering, Toyo University, Japan
Abstract Unglazed porous cylinders were used to study experimentally the evaporation features of water from the cylinder surfaces in natural convective airflows. Seto semi-porcelain clay No.6 was employed to make the unglazed porous cylinders. The firing temperature was changed from 773 to 1523 K to regulate the pore size of the porous. The diameter was from 80 to 85 mm, the thickness 7 mm, and the span 290 mm. The experiment was made at constant air temperature and the relative humidity. The air temperature ranged from 284 to 297 K and the relative humidity from 40 to 72%. The firing temperature had little effects on the evaporation characteristics. The evaporation rates were always 2.5 to 4 times as high as those of non-porous cylinders. The Grashof number, which shows macroscopic natural convection, did not have any appreciable effects on the evaporation. The evaporation rates were affected by the ambient temperature and relative humidity. Therefore, the following two conclusions were acquired: the evaporation on the nanoporous walls should be largely taken into account with the interaction between water and surface molecules which determines the evaporation energy of molecules, and the nanoporous does not consist of the pores among particles of the semi-porcelain, but of the nanoscale pores of a particle. Keywords: water evaporation, natural convection, nanoporous surface, unglazed semi-porcelain cylinder, molecular interaction, firing temperature, nanoscale pore, experiments, air temperature, relative humidity.
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468 Advanced Computational Methods in Heat Transfer IX
1
Introduction
Water evaporation phenomenon has been observed for water contained in an unglazed semi-porcelain cylinder. The mechanism of the evaporation from the nanoporous surface is unknown. If the porous passage is large enough for water molecules moving to surface, the evaporation on the porous surface goes on in the same manner as on the surface of water film, showing the same evaporation rate of the later. If the passage is resistive for the water molecules, the molecular interaction must be appreciable between the passage and the passer-by molecules. The evaporation latent heat is macroscopically defined as the energy difference between gaseous and liquid states of the molecular assembles at the same temperature and pressure. This means that it is the energy required for molecules to leave away the liquid-state cluster of the same homogeneous molecules. If the third molecules are related to the process of evaporation as in the solid-state cluster, the evaporating energy of the molecules must be different from that of the homogeneous molecules as shown in references [1 ~ 6]. Whether the change is positive or negative depends on the third-body interaction with the evaporating molecules. Concerning the water evaporation on the nanoporous surface of the unglazed pottery cylinder in forced convective airflow as shown in the references [7, 8], the evaporation rates were 2.5 times as high as those of non-porous cylinders at first. However, these evaporation rates decreased to the equilibrium value of 1.5 after 30 minutes. This is why the surface dry causes the water supply rate from the inside of the cylinder to be less than the evaporation rate from the surface of the cylinder. Therefore, the water evaporation on the surface of the nanoporous cylinders was not able to be detailedly analyzed. But in the natural convective airflow the cylinder surface does not get dry because the evaporation rate is low, and the evaporation rate remains steady as the time passes. It was, therefore, experimentally investigated how the water evaporation will be affected by the nanoscale pore size of the cylinder surface. The experiment was carried out in natural convective evaporation on the ground-based nanoporous cylinder with water inside.
2
Experimental apparatus and procedure
Figure 1 shows the experiment apparatus used. The nanoporous cylinder of 80 mm in diameter and 290 mm in length is surrounded by 700×700 mm cross-sectional area and 1000 mm high walls for getting stable natural convection. The walls are 700 mm away from the ground. The cylinder is located on the supporting cylinder of 80 mm in diameter and 1000 mm in height from the ground. Distilled water was absorbed up into the hollow cylinder of 7 mm in thickness from the storage tank, where the water surface level is 100 mm below the underneath end of the cylinder. Seto semi-porcelain clay No.6 is used to make the cylinder and the firing temperature was changed between 773 and 1523 K. The firing temperature causes the size of porous pores to change between 5 nm and 100 nm. The experiment was carried out under the stable
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conditions of the room temperature and relative humidity and with possible reduction of global natural convections in the room. The experiment started on the wetted surface of the cylinder. The evaporation rate was measured by the water flow supplied into the porous cylinder with a flow meter of Peltier element. The air temperature was measured with Pt 100 thermometer. The relative humidity was measured with HUMICAP of Vaisala (thin-film polymer sensor). The experimental air temperature T was 284 to 297 K and the relative humidity RH 40 to 72%. Unit: mm Natural Convective Air Flow
700
700
Nanoporous Cylinder
Nanoporous Cylinder
Closures
1000
Stop Valve
700
1000
Mass Meter
Cylinder Support Figure 1:
3
Flow
Distilled Water Ground
Experimental apparatus.
Experimental results
The experimental results obtained are summed up using two non-dimensional parameters, Sh; the Sherwood number for evaporation rate and Gr; the Grashof number for natural convection. The Sherwood number shows the evaporation rate, compared with the molecular diffusion rate under the same molecular-density conditions at the boundary. The evaporation rate can be expressed as hD ( ρ ∞ - ρ w ) S , (1) and the molecular diffusion rate is given by ρ -ρ D ∞ w S, (2) d WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
470 Advanced Computational Methods in Heat Transfer IX where ρ w and ρ ∞ are the water concentrations or air densities at the cylinder-wall surface and its infinity, hD the total mass transfer rate at the cylinder surface, D the molecular diffusion coefficient, d the cylinder diameter, and S the total surface area causing the evaporation. Then the Sherwood number is shown as h ⋅d Sh d ≡ D . (3) D The Grashof number is an inertial force compared with the viscous force like as the Reynolds number, although the inertial force without any forced flows is given by the natural convection force as u2 ρ∞ = g ( ρ∞ - ρw ) , (4) d and the viscous force is given by u (5) ρ∞ ⋅ ν 2 , d where u is the characteristic velocity and ν the kinematic viscosity. These give the non-dimensional number, Grd ; Grashof number as 2
ρ∞ - ρ w d 3 u2 u u2 ⋅ d 2 g , (6) Grd ≡ ρ ∞ ρ ∞ ⋅ ν 2 = = d ρ∞ ν2 d ν2 The heat transfer handbook [9] shows the Nusselt number for the cylindrical surface as
( Nu ) ( Nu )
56
*
l
c
*
l
p
2l / d = 1 + 0.428 * Nu l p
(
)
,
(7)
where (Nu l )c is the Nusselt number for the circular cylinder based on the cylinder length l and (Nu l ) p is the Nusselt number for the flat plate and with the
plate length l . Since the latter is given as * 14 Nu l p = 0.515 Ra l ,
(
(Nu l )c or eqn. (7) gives
(Nu ) *
d
c
)
= 0.515 Ra l
14
(8)
+ 0.683 (l / d ) Ra l 56
= 0.515 (Ra d ⋅ d l )
14
1 24
,
(9)
+ 0.683 (Ra d ⋅ d l )
1 24
,
(10)
where (Nu d )c and Ra d are the Nusselt number and Rayleigh number based on the cylinder diameter d . Since Ra d is given as Ra d = Grd ⋅ Pr ,
(Nu ) *
d
c
= 0.515 (Grd ⋅ Pr ⋅ d l )
14
(11)
+ 0.683 (Grd ⋅ Pr ⋅ d l )
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
1 24
.
(12)
Advanced Computational Methods in Heat Transfer IX
471
Substituting Nu with Sh, and Pr with Sc to eqn. (12) by the analogy between heat and mass transfer, then the Sherwood number for the non-porous cylinder fully wetted by water is given by * 1/ 4 1 / 24 Sh d c = 0.515 (Grd ⋅ Sc ⋅ d l ) + 0.683 (Grd ⋅ Sc ⋅ d l ) , (13)
(
)
where Sc is the Schmidt number. It is the purpose of the present experiment how different evaporation rates Sh will be resulted in the water evaporation from the nanoporous surfaces, under the same natural convective conditions, Gr. The evaporation rates resulted are all given in the term of the ratio compared with the Sherwood number for the fully wetted cylindrical surfaces at the same Grashof * number, Sh d / Sh d vs. Grd . 3.1 Effects of the firing temperature on the cylinder porous 3.1.1 Effects of the firing temperature on the cylinder diameter Figure 2 shows the effect of the firing temperature on the cylinder diameter. The shape, size and porosity distribution will be affected by the firing temperature to make the unglazed porous cylinder from the clay. The diameter of the semi-porcelain cylinder remained stable by 1273 K, and then the diameter decreased rapidly as the firing temperature increase.
87
Cylinder Diameter [mm]
86 85 84 83 82 81 80 79 750
950
1150
1350
1550
Firing Temperature [K] Figure 2:
Effects of the firing temperature on the cylinder diameter.
3.1.2 Effects of the firing temperature on the water leakage mass rate Figure 3 shows the effect of the firing temperature on the water leakage mass rate at the loaded pressure of 500 mm Aq. The water leakage mass rate raised as the firing temperature increased by 1373 K, and then the water leakage mass rate fell rapidly as the temperature increased. At 1523 K, there was no leakage. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
472 Advanced Computational Methods in Heat Transfer IX
2
Leakage mass rate [kg/(m · h )]
Specifically, the size of the pore among the semi-porcelain particles came to be smaller as the temperature increased by 1373 K, then the size of the pore rapidly decreased, and at the temperature of 1523 K there was no leakage. This shows that the pore size among the semi-porcelain particles changed due to the firing temperature. 0.5
T = 291 K ,
RH = 70 %
0.4 0.3 0.2 0.1 0 750
950
1150
1350
1550
Firing temperature [K] Figure 3:
Effects of firing temperature on the water leakage mass rate.
3.2 Effects of the firing temperature on the evaporation rate Figure 4 shows the effect of the firing temperature on the water evaporation rates. It is clear from the figure that the firing temperature has little effect on the water evaporation from the porous cylinders at the firing temperature of 973 ~ 1473 K. At 1523 K there was no evaporation because of no water supply to the cylinder surface. The size of the pore among the particles of semiporcelain then affects no evaporation rate, and the nanoporous with evaporation characteristics does not consist of the pores among particles of the semiporcelain, but of the nanoscale pores of a particle. 3.3 Effects of air temperature and humidity Figures 5 and 6 show the effects of the temperature and relative humidity of the ambient air surrounding the cylinder. The air temperature and relative humidity must be included in the terms of non-dimensional parameters, but they still have some effects on the evaporation rates. It is not known whether this comes from the insufficient estimation of physical properties such as the diffusion coefficients, viscosity and density or fundamentally from the evaporation from the porous cylinder surface. WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
4
Sh/Sh*
3
2
T=294 K, RH=55～60 %
1
T=291 K, RH=45～50 % T=284 K, RH=45～55 %
0 950
1050
1150
1250
1350
1450
1550
Firing Temparature [K] Figure 4:
Effects of firing temperature on the Sherwood number ratio.
4
Sh/Sh*
3
2 RH=70 ～75 % RH=50 ～55 % RH=40 ～45 %
1
Gr=(1.8～2.3)×10
4
Gr=(2.8～4.1)×10
4
Gr=(3.9～5.2)×10
4
0 280
285
290
295
300
T [K] Figure 5:
Effects of air temperature on the Sherwood number ratio.
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
473
474 Advanced Computational Methods in Heat Transfer IX 4
Sh/Sh*
3
2 4 T=295 ～ 297 K Gr=(1.8～5.2)×10
1
4 T=290 ～ 292 K Gr=(2.1～4.5)×10 4 T=283 ～ 285 K Gr=(2.9～4.4)×10
0 30
40
50
60
70
80
RH [%] Figure 6:
Effects of relative humidity on the Sherwood number ratio.
4
Sh/Sh
*
3
2
1
T=293 ～ 298 K, RH=64 ～72%
T=293 ～ 298 K, RH=50 ～59 %
T=293 ～ 298 K, RH=45 ～49 %
T=289 ～ 292 K, RH=70 ～ 71 %
T=289 ～ 292 K, RH=60 ～62 %
T=289 ～ 292 K, RH=42 ～ 55 %
T=284 ～ 289 K, RH=60 ～ 62 %
T=284 ～ 289 K, RH=50 ～ 58 %
T=284 ～ 289 K, RH=40 ～ 49 %
0 0
1
2
3 Gr×10
Figure 7:
4
5
-4
Effects of Grashof number on the Sherwood number ratio.
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
Advanced Computational Methods in Heat Transfer IX
475
3.4 Effects of Grashof number As shown in figure 7, the evaporation rate increases 2.5 to 4 times compared with the fully wetted cylinder surfaces in the range of the Grashof number (1~5) ×104. In this range, the Grashof number does not show an appreciable effect on the evaporation characteristics. Specifically the macroscopic flow due to the natural convection is the same as that of the evaporation from the nonporous cylinder fully wetted by water, but those microscopic features of evaporation must be appreciably affected by the interaction of water molecules with porous material of nanoscale configuration, which gives constants 2.5 to 4 times as large as those in eqn. (13).
4
Concluding remarks
(1) Following figures 5, 6 and 7, the evaporation features depend on the ambient temperature and relative humidity, almost independent of the Grashof number. Due to this independency, the same convective flows will appear if induced by the buoyancy force by the difference of density distribution. As far as the macroscopic flows, the same flows will appear even when the evaporation mechanism itself is different from each other. If the nanoporous surface affects only the microscopic mechanism of evaporation, that is, the interaction between water and porous-surface molecules, the evaporation features must be influenced by the ambient temperature and relative humidity, independent of the Grashof number which gives the macroscopic flow field. The particular evaporation mechanism on the porous surface is not straightforwardly deducible because the molecular interaction includes the quantum effects which cannot be expected by the molecular dynamics with a constant potential of interfacial molecules. (2) Following figure 4, the nanoporous related to the evaporation characteristics does not consist of the pores among particles of the semi-porcelain, but of the nanoscale pores of a particle of the semi porcelain.
Nomenclature d D g Gr hD l RH S Sh Sh*
: diameter of cylinder, m : diffusion coefficient, m2/s : gravitational acceleration, m/s2 : Grashof number, Gr = d 3 ⋅ g (ρ ∞ - ρ w ) / ρ ∞ ⋅ ν 2 : mass transfer rate, m/s : length of cylinder, m : relative humidity, % : total surface area causing evaporation, m2 : Sherwood number, Sh = hD ⋅ d / D . : Sherwood number of fully wetted surface cylinder in the handbook,
(
)
WIT Transactions on Engineering Sciences, Vol 53, © 2006 WIT Press www.witpress.com, ISSN 1743-3533 (on-line)
476 Advanced Computational Methods in Heat Transfer IX Sh* = 0.515 (Gr ⋅Sc ⋅ d / l ) + 0.683 (Gr ⋅ Sc ⋅ d / l ) Sc : Schmidt number, Sc = ν / D . T : temperature, K u : characteristic velocity, m/s ν : kinematic viscosity, m2/s ρ w : density at wall surface, kg/m3 3 ρ ∞ : density of main flow, kg/m 14
1 24
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Krim, J., Coulomb, J.P. & Bouzidi, J., Triple-Point Wetting and Surface Melting of Oxygen Films Adsorbed on Graphite, J. Phys. Review Letters , 58 (6), pp. 583-586, 1987. Sokol, P.E., Ma, W.J., Herwig. K.W., Snow, W.M., Wang, Y., Koplic, J. & Banavar, J.R., Freezing in Confined Geometries, Appl. Phys. Lett. 61 (7), pp. 777-779, 1992. Molz, E., Wong, A.P.Y., Chan, M.H.W. & Beamish, J.R., Freezing and Melting of Fluids in Porous Grlasses, Phys. Review B, 48 (9), pp. 57415750, 1993. Unruh, K.M., Huber, T.E. & Huber, C.A., Melting & Freezing Behaviour of Indium Meta in Porous Glasses, Phys. Review B, 48 (12), pp. 90219027, 1993. Hansen, E.W., Stöcker, M. & Schmidt, R., Low-Temperature Phase Transition of Water Confined in Mesopores Probed by NMR. Influence on Pore Size Distribution, J. Chem. Phys., 100 (6), pp. 2195-2200, 1996. Maddox, M.W. & Gubbins, K.E., A Molecular Simulation Study of Freezing/Melting Phenomena for Lennard-Jones Methane in Cylindrical Nanoscale Pores, J. Chem. Phys. 107 (22), pp. 9659-9667, 1997. Hara, S. & Suzuki, T., Water Evaporation on a Nanoporous Cylinder in Forced Airflows, Thermal Science & Engineering, 6(4), pp. 33-38, 1998. Hara, S., Water Evaporation on a Nanoporous Cylinder in Forced Airflows, Thermal Science & Engineering, 8(2), pp. 21-27, 2000. JSME, Heat Transfer Handbook, Japan Society of Mechanical Engineer, pp. 144-154, 1997.
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Advanced Computational Methods in Heat Transfer IX
477
Author Index Alane A.................................... 337 Altaç Z. .................................... 109 Amano R. S................................ 65 Andrews M. J........................... 259 Anzai K.................................... 249 Bailey M. ................................. 401 Beckers P. ................................ 121 Bhatti S. K. .............................. 323 Boudebous S. ............................. 33 Boukhanouf R.......................... 269 Bourquin F............................... 131 Brener A. M............................. 181 Buffone C. ............................... 269 Buikis A................................... 311 Ceci S....................................... 143 Černý R.................................... 153 Chaube A. .................................. 43 Chen W.-K................................... 3 Chiba R. ..................................... 23 Ciambelli P. ............................. 299 Correia E. L. ............................ 289 Covill D. G. ............................. 401 De Biase L. .............................. 143 De Mey G. ............................... 449 Deb R....................................... 423 Del Giudice S........................... 457 Domke K.................................. 221 Döner N. .................................. 109 Du X. ....................................... 241 Elamin G.................................... 97 Fedorovich E............................ 199 Ferguson F. ................................ 97 Fossati G. ................................. 143 Gabrielaitiene I. ....................... 381 Giedraitis V.............................. 213 Goriletsky V. I. ........................ 191 Goshayshi H. ........................... 173
Grinyov B. V. .......................... 191 Großmann S............................. 391 Guan Z. W. .............................. 401 Gylys J. ............................ 213, 231 Haddad A................................. 269 Hara S. ..................................... 467 Heggs P. J. ....................... 337, 371 Izumi M. .................................... 23 Jebrail F. F. .............................. 259 Jones W. P. .............................. 413 Kadja M. .................................... 53 Kholai O. ................................... 53 Kiela A..................................... 231 Krishna Ch. M. ........................ 323 Lakner M. ................................ 391 Liu D........................................ 241 Löbl H...................................... 391 Luo X........................................... 3 Maděra J. ................................. 153 Magier T. ................................. 391 Mai T. H. ................................... 53 Mazzaro M............................... 279 Meo M. G. ............................... 299 Mirzaee I.................................. 361 Musabekova L. M.................... 181 Nassiopoulos A........................ 131 Neelapu M. L........................... 323 Nemouchi Z. .............................. 33 Neves M. M. ............................ 289 Niranjan Kumar I. N................ 323 Nonino C. ................................ 457 Oosthuizen P. H......................... 13 Pahlevani F. ............................. 249 Pandey K. M............................ 423
478 Advanced Computational Methods in Heat Transfer IX Paul M. C................................. 413 Pletnev A. ................................ 199 Rahmani R. .............................. 361 Ramezanpour A. ...................... 361 Raval H. ................................... 401 Russo P. ................................... 299 Sacramento Rivero J. C. .......... 371 Sahoo P. K. ................................ 43 Savino S................................... 457 Schoenemann T. ...................... 391 Scholz R................................... 163 Serag-Eldin M. A..................... 437 Shang F. ................................... 241 Shirvani H................................ 361 Sidletskiy O. Ts. ...................... 191 Sinkunas S. ...................... 213, 231 Solanki S. C. .............................. 43 Spitzer K.-H............................. 163 Sugano Y. .................................. 23 Sumin V. I................................ 191 Sun D. ........................................ 65 Sundén B.......................... 351, 381 Taigbenu A. E............................ 77 Talalov V. ................................ 199
Teixeira S. F. C. F.................... 289 Tesárek P. ................................ 153 Tinaburri A. ............................. 279 Tymoshenko M. M. ................. 191 Vaccaro S................................. 299 Vasilyev V. V. ......................... 191 Vermeersch B. ......................... 449 Viscorova R. ............................ 163 Vueghs P.................................. 121 Vundru Ch. .............................. 323 Wendelstorf J........................... 163 Wilhelmsson C. ....................... 351 Wollerstrand J.......................... 381 Xian H. .................................... 241 Yan Y. Y.................................... 87 Yang Y..................................... 241 Yaokawa J. .............................. 249 Yuan J...................................... 351 Zdankus T................................ 213 Zu Y. Q...................................... 87
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