INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order.
University Microfilms International A Bell & Howell Information C om pany 300 North Z eeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600
Order Number 9325512
Investigation and modeling of complex interfacial effects for porous/non-porous configurations
Huang, Po-Chuan, Ph.D.
The Ohio State University, 1993
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
INVESTIGATION AND MODELING OF COMPLEX INTERFACIAL EFFECTS FOR POROUS/NON-POROUS CONFIGURATIONS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of the Ohio State University
By
Po-Chuan Huang, B.S., M.S.
The Ohio State University
1993
Dissertation Committee: Approved by Prof. K. Vafai
Prof. T. E. Blue Adv Prof. Robert H. Essenhigh Department of Mechanibal E eenng To My Parents, Wife and My Beloved Daughter ACKNOWLEDGMENTS
I would like to express my deepest appreciation to Prof. K. Vafai for his guidance
and insight throughout the research and for his infinite patience. I also thank my
committee members, Prof. T. E. Blue, Robert H. Essenhigh and S. Roklin for the
time they took to make critical comments.
I am grateful to Ohio Supercomputer Center for providing me with the computer
time for the numerical computation. In addition, I am grateful for many helpful discussions related to this study from James Wang, Hsing-Sen Hsiao, Wen-Tong
Wang and my colleagues.
Finally, I want to express my indebtedness to my wife, Meir-Chyun Tzou for her understanding and everlasting support, and my daughter Rouh-Yau Huang since I was frequently not available at home while they really needed me. Finally, I would like to dedicate this dissertation to my parents, brothers, and sisters for their love and support during the course of study. VITA
January 20, 1955 ...... Born-Tainan, Taiwan, R. O. C.
1978 ...... B.S., National Chiao Tung University, Hsichu, Taiwan, R. O. C.
1983 ...... M.S., Tatung Institute of Technology, Taipei, Taiwan R. O. C.
1984-1989 ...... Instructor, Tung-Nan College and China Maritime College, Taipei, Taiwan
1990-1993 ...... Graduate Student The Ohio State University Columbus, Ohio
FIELD OF STUDY
Major Field: Mechanical Engineering
Studied in Heat transfer and Fluid Mechanics
Prof. K. Vafai TABLE OF CONTENTS
ACKNOWLEDGEMENTS...... iii
VITA ...... iv
LIST OF TABLES...... vii
LIST OF FIGURES...... viii
NOMENCLATURE...... xviii
CHAPTER PAGE
I . GENERAL INTRODUCTION...... 1
II. ANALYSIS OF FLOW AND HEAT TRANSFER OVER AN EXTERNAL BOUNDARY COVERED WITH A POROUS SUBSTRATE
2.1 Statement of the problem ...... 7 2.2 Theory of the problem ...... 10 2.3 Analysis and solution ...... 14 2.4 Results and discussion ...... 20
III. FORCED CONVECTION OVER INTERMITTENTLY EMPLACED POROUS CAVITIES
3.1 Statement of the problem ...... 24 3.2 Analysis and formulation ...... 27 3.3 Numerical scheme ...... 34 3.4 Results and discussion ...... 38
IV. FORCED CONVECTION OVER POROUS BLOCK ARRAY
4.1 Statement of the problem ...... 56 4.2 Theory ...... 60 4.3 Numerical method and procedure ...... 66 4.4 Results and discussion ...... 71
V. AN INVESTIGATION OF FORCED CONVECTION THROUGH ALTERNATE POROUS CAVITY-BLOCK OBSTACLES
5.1 Statement of the problem ...... 97 5.2 Analysis ...... 101 5.3 Numerical scheme ...... 108 5.4 Results and discussion ...... 115
VI. ANALYSIS OF FORCED CONVECTION ENHANCEMENT IN A CHANNEL USING POROUS BLOCKS
6.1 Statement of the problem ...... 134 6.2 Mathematical formulation ...... 138 6.3 Computational details ...... 144 6.4 Results and discussion ...... 150
VII. INTERNAL HEAT TRANSFER AUGMENTATION IN A PARALLEL PLATE CHANNEL USING AN ALTERNATE SET OF POROUS CAVITY-BLOCK OBSTACLES
7.1 Statement of the problem ...... 174 7.2 Theory ...... 178 7.3 Boundary conditions ...... 181 7.4 Numerical solution method ...... 187 7.5 Results and discussion ...... 194
VIII. CONCLUSIONS AND RECOMMENDATIONS
8.1 Conclusions ...... 216 8.2 Recommendations for further research ...... 220
LIST OF REFERENCES...... 221 LIST OF TABLES
TABLES PAGE
3.1 Input data of governing parameters for intermittently emplaced c a v itie s ...... 39
5.1 Coefficients of equation (5.24) ...... 110 LIST OF FIGURES
FIGURES PAGE
2.1 Schematic diagram of the flow over an external boundary with anattached porous substrate ...... 11
2.2 Control volume for integral momentum analysis of 1) porous boundary layer, 2) interfacial region between the porous and fluid layers and 3) and the fluid boundary layer ...... 15
2.3 (a) Velocity and (b) temperature distribution along the flat plate at three different locations, *=0.2, 0.5 and 0.8, for Re^ = 3 x105> Al =0.35, DaL = 8xl0"\ Pr=0.7, kf/r/ k f = 1 .0 ,H/L=0.02...... 21
2.4 Comparison of the integral solutions with numerical solutions for x-compoment interfacial velocity along the flat plate for AL = 0.35 , ReL = 3 x l 0 5, DaL = 8 x 10-6, / / /L=0.02 ...... 23
3.1 Schematic diagram of flow and heat transfer over intermittently emplaced porous cavities ...... 28
3.2 (a) The non-uniform grid system for the whole computational domain, (b) Local integration cell in the computational domain ...... 35
3.3 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s) DaL = 8 x 1CT6, AL = 0.35, Pr=0.7, k ^ /k , = 1.0, A=6, 5 = 1 ...... ’...... 40
3.4 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 2 x 105, DaL = 8 x 10"6, AL = 0.35, P r=0.7,kcj!r / k f = 1.0, A=6, 5=1 ...... 41
viii 3.5 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, DaL = 8 x 1CT6, Pr = 0.7, k ^ /k , = 1.0, Al =0.35,4=6, 5=1 ...... ’...... 43
3.6 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, DaL = 8 x 10-6, AL = 0.35, Pr=7, k ^ /k , =1.0, A=6,5=l...... 45
3.7 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for Ret = 3 x 10s, Da£ = 8 x 10"6, At = 0.35, Pr=100, keff /k f = 1.0, A=6, 5=1 ...... 46
3.8 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, DaL = 8 x 10"6, AL = 1.05, Pr=0.7, keJf/ k f =1.0, A=6,5=l...... 48
3.9 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for RcL = 3 x 10s, DaL = 8 x 10"6, A^ = 0.35, Pr=0.7, keJf/ k f = l-°, A =3, 5 = 1 ...... 49
3.10 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, D aL = 8 x 10-6, At =0.35, Pr=0.7, kejr / k f = 1.0, A=6, 5 = 0 .8 ...... 51
3.11 Vorticity contour for four porous cavities at different spacings (a) A=6, 5=0.8 and (b) A=3, 5 = 2 ...... 52
3.12 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, DaL = 8 x 10-6, AL =0.35, Pr=0.7, K ff /K = 1-0, A=3, 5 = 2 ...... 53
3.13 (a) Streamlines and (b) isotherms for flow over two obstructing porous cavities for ReL = 3x 10s, DaL=8xlO -6, AL=0.35, Pr=0.7, kejr/ k f = 1.0, A=12.5, 5=1 ...... 54
4.1 Schematic diagram of flow and heat transfer through a multiple porous block system ...... 61
ix 4.2 (a) Grid system for the computational domain, (b) Local integration cell ...... 67 4.3 (a) Velocity distribution, (b) streamlines, and (c) frictional coefficient for flow through four obstructing porous blocks for Ret =3xl05, Da^SxKT*, AL =0.35,4=6, B=l, H*=0.02 ...... 73
4.4 Effects of the Darcy number on streamlines for flow through four obstructing porous blocks for Ret = 3 x 10s, AL = 0.35, <4=6, 5=1, //*=0.02 ...... 76-77
4.5 Effects of the Darcy number on isotherms for flow through four obstructing porous blocks for ReL = 3 x10s, AL = 0.35, Pr=0.7, k ejr / k f = 1.0, A=6, 5=1, //*=0.02 ...... 78-79
4.6 Effects of the Darcy number on the Nusselt number for flow through four obstructing porous blocks for Ret = 3 x 10s, A l =0.35, Pr=0.7, / k f =1.0, <4=6, 5=1, //*=0.02 ...... 80
4.7 Effects of the inertial parameter on streamlines for flow through four obstructing porous blocks for ReL = 3 x 10s, DaL = 8 x 10-6, A=6, 5=1, //*=0.02 ...... 83
4.8 Effects of the inertial parameter on isotherms for flow through four obstructing porous blocks for Ret = 3 x 10s, D at = 8 x 10"*, A=6, 5= 1, H*=0.02 ...... 84
4.9 Effects of the inertial parameter on the Nusselt number for flow through four obstructing porous blocks for ReL = 3 x 10s, D at = 8 x 10"*, Pr=0.7, ktff / kf = 1 0 .M 5=1, H*=0.02...... 85
4.10 Effects of the Reynolds number on streamlines for flow through four obstructing porous blocks for Da^ = 8 x 10"6, AL = 0.35, <4=6, 5=1,//*=0.02 ...... 87
x 4.11 Effects of the Reynolds number on isotherms for flow through four obstructing porous blocks for DaL = 8 x 1CT6, AL = 0.35, Pr=0.7, k eff / kf = 1.0, A=6, 5=1, H*=0.02 ...... 88
4.12 Effects of the Reynolds number on the Nusselt number for flow through four obstructing porous blocks for Dat = 8 x 10-6, Al =0.35, Pr=0.7, k eJf/ k f =l.0,A=6, 5=1, H*=0.02...... 89
4.13 Effects of the Prandtl number on isotherms for flow through four obstructing porous blocks for Re^ = 3 x 10s, Dat = 8 x 10"6, Al = 0.35, keff/ k f = 1.0,A=6,5=l,//*=0.02 ...... 91
4.14 Effects of the Prandtl number on the Nusselt number for flow through four obstructing porous blocks for, Re^ = 3 x 105, DaL = 8 xlO"6, At =0.35, keff / k f = 1.0, A=6, 5=1, H*=0.02...... 92
4.15 Streamlines for flow through four obstructing porous blocks for Re^ = 3 x 10s, DaL = 2 x 10-*, AL = 0.35, k tJf /k f = 1.0, Pr=0.7, H*=0.02...... 94
4.16 Isotherms for flow through four obstructing porous blocks for Ret = 3 x 10s, Da^ = 2 x 10-6, AL =0.35, kejr/ k f = 1.0, Pr=0.7, H*=0.02...... 95
4.17 (a) Streamlines and (b) isotherms for flow through five obstructing porous blocks for ReL = 3 x 105, Dat = 2 x 10"6, AL = 0.35, Pr=0.7, keff/ k f =1.0, A=6,5=l, //*=0.02 ...... 96
5.1 Schematic diagram of flow and heat transfer through alternate porous cavity-block obstacles ...... 102
5.2 (a)Grid system for the computational domain, (b) Local integration cell ...... 109
xi 5.3 (a)Streamlines, (b) velocity distribution, and (c) isotherms for flow through alternate porous cavity-block obstacles for Ret = 3xl04, DaL = 8x10“®, AL =0.35, A=3, 5=1, H*=0.02 ...... 116
5.4 Effects of the Darcy number on streamlines for flow throughaltemate porous cavity-block obstacle for ReL = 3 x 105, A l = 0.35, A=3, 5=1, H*=0.02...... 119
5.5 Effects of the Darcy number on isotherms for flow through alternate porous cavity-block obstacles for Re£ = 3 x 105, A l =0.35, Pr=0.7, keff / k, = 1.0. 4=3, 5=1, H*=0.Q2...... 121
5.6 Effects of the Reynolds number on streamlines for flow through alternate porous cavity-block obstacles for DaL = 8 x 10-6, AL =0.35, A=3, 5= 1,//*=0.02 ...... 123
5.7 Effects of the Reynolds number on isotherms for flow through alternate porous cavity-block obstacles for DaL = 8 x 10-6, A l =0.35, Pr=0.7, k eff /k f = 1.0. 4=3, 5=1, H*=0.02 ...... 124
5.8 The influence of the inertial parameter on streamlines for flow through alternate porous cavity-block obstacles for ReL = 3 x 10s, Dat = 8 x 10"*, A=3, 5=1, #*= 0.02 ...... 125
5.9 The influence of the inertial parameter on isotherms for flow through alternate porous cavity-block obstacles for ReL = 3 x 10s, DaA = 8 x 10"6, Pr=0.7, keff / k f = 10.4=3, 5=1, H*=0.02 ...... 126
5.10 Prandtl number effects on streamlines and isotherms for flow through alternate porous cavity-block obstacles for Ret = 3 x 10s,
DaL = 8x10^, Al =0.35, keff/ k f = 1.0,4=3, 5=1, //*=0.02 ....128
5.11 The influence of the geometrical layout on streamlines for flow through alternate porous cavity-block obstacles for ReL = 3 x 105,
Dat = 2 x 10"7, Al = 0.35, k eff / k f = 1.0, Pr=0.7, H*=0.02 at (a)./4=3 and 5=1, (b) A=4 and 5=1, and (c) A=3 and 5 = 2 ...... 130
xii 5.12 The influence of the geometrical layout on isotherms for flow through alternate porous cavity-block obstacles for ReL = 3 x 10s, Dat = 2 x 10-7, Al = 0.35, k ^ / k, = 1.0, Pr=0.7, #*=0.02 at (a).A=3 and 5=1, (b) A= 4 and 5=1, and (c) A=3 and 5= 2 ...... 131
5.13 Effects of larger set of porous cavity block configurations, (a) Streamlines and (b) isotherms for Ret = 3 x 10s, DaL = 2 x 10”7, Al = 0.35, Pr=0.7, ktff / k f = 1.0, 4=3, 5=1, H*=0.02, N=3...... 133
6.1 Schematic diagram of force convection in a parallel plate channel with porous block obstacles ...... 139
6.2 (a) Grid system for the computational domain, (b) Local integration cell ...... 145
6.3 (a) Streamlines, (b) velocity distribution, (c) isotherms, and (d) local Nusselt number distributeion for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10“5, A=0.35, Pr=0.7, ktff /k f = 1.0, A =4, 5 =1, #*=0.25...... 151-152
6.4 Effects of the Darcy number on streamlines for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, A =4, 5=1, #*=0.25...... 155
6.5 Effects of the Darcy number on isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, Pr=0.7, keff / k, = 1.0, A =4, 5 =1, #*=0.25...... 156
6.6 Effects of the Darcy number on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, Pr=0.7,k^ / kf = 1.0, A =4, 5 =1, #*=0.25...... 157
6.7 Effects of the Reynolds number on streamlines for flow in a parallel plate channel with porous block obstacles for Da=l x 10-5, A=0.35, A =4, 5=1, #*=0.25...... 159
X lll 6.8 Effects of the Reynolds number on isotherms for flow in a parallel plate channel with porous block obstacles for Da=l x 10"5, A=0.35, Pr=0.7, k tJ[ / k f = 1.0, A =4, 5=1, H*=Q.25...... 160
6.9 Effects of the Reynolds number on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Da=l x 10'5, A=0.35, Pr=0.7, ktff / kf = 1.0, A =4, B =1, H*=0.25...... 161
6.10 The influence of the inertial parameter on streamlines for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l x 10'5, A =4, B =1, H*=0.25...... 163
6.11 The influence of the inertial parameter on isotherms for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l xlO '5, Pr=0.7,k^ / k f = 1.0, A =4, B =1, H*=0.25...... 164
6.12 The influence of the inertial parameter on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l x 10'5, Pr=0.7, k,^ / k f = 1.0, A = 4 ,5 = 1 , H*=0.25...... 165
6.13 Prandtl number effects on streamlines and isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, ktff / k f = 1.0, A =4, B =1, H*=0.25...... 167
6.14 Prandtl number effects on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, k ^ / kf = 1.0, A =4, B =1, H*=0.25...... 168
6.15 The influence of the geometric parameter A on streamline and isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, Pr=0.7, kt]T /k f =1.0,5 =1, H*=0.25...... 169-170
xiv 6.16 The influence of the geometric parameter 5 on streamline for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, //*=0.25 at (a) A=4 and B =3, (b) A=4 and B =2, and(c) A=4 and B =1...... 172
6.17 The influence of the geometric parameter B on isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, Pr=0.7, kt/f / kf = 1.0, //*=0.25 at (a) A=4 and B =3, (b) A=4 and B =2, and(c) A=4 and B =1...... 173
7.1 Schematic diagram of force convection in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate ...... 179
7.2 (a) Grid system for the computational domain, (b) Local integration cell ...... 188
7.3 (a) Streamlines, (b) isotherms, and (c) local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10'5, A=0.35, Pr=0.7, / k f = 1.0 A =4, B =1, H*=0.25...... 195-196
7.4 Effects of the Reynolds number on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for D a=3xl0"5, A=0.35, A=4, 5= 1, H*=0.25...... 199
7.5 Effects of the Reynolds number on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Da=3 x 10"5, A=0.35, Pr=0.7,k<# / kf = 1.0, A=4, 5=1, H*=0.25...... 200
7.6 Effects of the Reynolds number on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Da=3 x 10“5, A=0.35, Pr=0.7, k ejr /k { =1.0 A =4, 5 =1, H*=0.25...... 201
7.7 Effects of the Darcy number on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the
xv bottom plate for Re=750, A=0.35, A=4, B=l, H -0 .2 5 ...... 203
7.8 Effects of the Darcy number on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, A=0.35, Pr=0.7, Kff / k / = 1.0, A =4, B =1, H*=0.25...... 204
7.9 Effects of the Darcy number on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Re=750, A=0.35, Pr=0.7, kejr / k f = 1.0 A =4, B =1, H*=0.25...... 205
7.10 The influence of the inertial parameter on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=1500, Da=3 x 10"5, A =4, B =1, H*=0.25...... 207
7.11 The influence of the inertial parameter on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=1500, Da=3 x 10”5, Pr=0.7, ktff/ k f =1.0, A =4, B =1, H*=0.25 ...... 208
7.12 The influence of the inertial parameter on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=1500> Da=3 x 10"5, Pr=0.7, / k f = 1.0, A =4, B =1, H*=0.25...... 209
7.13 Prandtl number effects on streamlines and isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10"5, A=0.35, kejr/ k f = 1.0, A =4, B =1, H*=0.25...... 210
7.14 Prandtl number effects on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10”5, A=0.35, k eff/ k f = 1.0 A =4, B =1, H*=0.25...... 211
xvi 7.15 The influence of the geometric parameters A and B on streamline for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Re=750, Da=3 x 10"5, A=0.35, Pr=0.7, keJJ- / = 1.0, £ =1, #*=0.25...... 213
7.16 The influence of the geometric parameters A and B on isotherms for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Re=750, Da=3 x 10"5, A=0.35, Pr=0.7, =1.0,2? =1, //*=0.25 ...... 214 NOMENCLATURE
A dimensionless geometric parameter, equations (3.23), (5.23), (6.12), and (7.10)
B dimensionless geometric parameter, equations (3.23), (5.23), (6.12), and (7.10)
C coefficient of the discretization equations (3.24), (4.17), (5.24),(6.15), and (7.14) Cf friction coefficient, equation (4.14)
D spacing between cavities or blocks, [n f1 ]
Da Darcy number, equations (2.14), (3.16b), (4.7b), (5.16b), (6.9), and (7.4)
F a function used in expressing inertia terms, which depends on the Reynolds number and the microstructure of the porous medium
h convective heat transfer coefficient, [ Wm-2K_1 ]
H height of the porous slab, block, or cavity, [m]
g(x,y) curve defining the porous/fluid interface
k thermal conductivity, [Wm'1K"‘]
K permeability of the porous medium, [m2] tx length of plate upstream from the cavities, blocks, or cavity-blocks, [m]
i 2 length of plate downstream from the cavities, blocks, or cavity-blocks, [m]
L length of the external boundary or channel, [m] n coordinate normal to the porous/fluid interface
xviii N number of cavities, blocks, or cavity-blocks
Nu Nusselt number, equations (4.15), (6.13), and (7.11) p pressure, [Pa]
Pe Peclet number, equations (3.16), (4.7), (5.16), (6.8), and (7.4) r ratio of x-component velocity in the interfacial region to free-stream
velocity, equation (2.14)
Pr Prandtl number
R height of channel, [m]
Re Reynolds number, equations (2.14), (3.16), (4.7a), (5.16a), (6.8), and (7.4a)
5° source term, equations (5.24), (6.15), and (7.13) t coordinate tangential to the porous/fluid interface
T temperature, [K] u x-component velocity, [ms-1]
y-component velocity, [ ms_1] v velocity vector, [ms-1]
V Volume, [m3]
W width of porous cavity or block, [m] x horizontal coordinate, [m] y vertical coordinate, [m]
Greek symbols a thermal diffusivity, [m2s-1] a ejr effective thermal diffusivity (=keff / p fcpJ)t [m 2s-1 ]
T general diffusive coefficient, used in equations (4.16) and (7.13)
xix Ax x-direction width of the control volume
Ay y-direction width of the control volume
Bx x-direction distance between two adjacent grid points
By y-direction distance between two adjacent grid points
5 boundary-layer thickness, [m]
e porosity of the porous medium
0 dimensionless temperature, equations (2.24), (3.22b), (4.12b), (6.9),and (7.8)
A inertial parameter, equations (2.14), (3.16b), (4.7b), (5.16b), (6.9), and (7.5)
^ dynamic viscosity, [ kgm"ls_1 ] v kinematic viscosity, [ m V ] vorticity p fluid density, [ kgirf3] xw wall shear stress, [kgm'’s“2]
O transported property
^ stream function
Superscripts
* dimensionless quantity
Subscripts
0 condition at the inlet av average xx C main grid point
E east cff effective
/ fluid
/ interface m bulk mean
N north p porous
S south
t thermal
W west
w condition at the wall x local
oo free-stream
Other
< > locall volume average o f a quantity [a, b] larger o f a and b
xxi CHAPTER I
GENERAL INTRODUCTION
The flow of viscous flow through permeable materials has been a problem of
long-standing interest for fluid dynamicists. Historically, the study of such flows
begins with Darcy's law (1856)
u = - — VP.
This states that the average fluid velocity is proportional to the average pressure
gradient. The coefficient k is a physical property of the porous material and has units
of length squared. Although Darcy's law was originally proposed on empirical
grounds, it may be justified theoretically for statistically homogeneous, isotropic
materials and low-Reynolds-number fluid flow. The Darcy's law has been applied to
a vast array of problems involving flow through porous media and has proved to be a reliable model for flow in the interior of such material.
However, despite its success for interior flows, Darcy's law is not a complete model for porous media. When a fluid flows past a porous body of finite size, the interior flow must be matched with the exterior pure fluid flow at the boundary surface. Under normal circumstances, the continuity of velocity and surface stress
1 across this boundary is required; however, this is not possible owing to the reduced
order of Darcy's law as compared with the Navier-Stokes equations. Moreover, the
Darcy's law neglects the effects of a solid boundary, inertia forces, and variable
porosity on fluid flow and heat transfer through porous media. Therefore, when the
fluid velocity is high, the porous medium is bounded, and the porosity is variable,
thus rendering the Darcy's flow model ineffective.
The main objective of the present work is to gain a full understanding of the
interaction phenomena occurring between a porous medium and another medium. In
general, the other medium could be a fluid, a solid or another porous medium. A
specific example of the interface region can be cited from petroleum reservoirs
wherein the oil flow encounters different layers of sand, rock, shale, limestone, etc.
Similar situations are encountered in many other cases of practical interest such as
solid matrix heat exchangers, iron blast furnaces, metal processing, geothermal
operations, nuclear waste repositories, underground coal gasification, ground water
hydrology, etc. Among the various configurations, six fundamental cases are
analyzed in depth in the present work. Four of them are external convective flow
problems and the rest are internal convective flow problems.
In chapter 2, laminar forced convection over an external boundary covered with a porous substrate is analyzed using the integral method. The primary objective of this study is to employ an integral analysis for investigating forced convection boundary layer flow and heat transfer through a composite porous/fluid system and thereby provide a comprehensive yet extremely fast alternative as well as a comprehensive base for numerical solutions addressing these types of interfacial 3 transport. A general flow model that accounts for the effects of the impermeable boundary and inertial effects is used to investigate the changes in the flow pattern and heat transfer performance due to the existence of a composite medium consisting of a long thin porous substrate attached to the surface of a flat plate. Several important characteristics of the flow and temperature fields in the composite layer are investigated. The present analysis provides a rather accurate approximate simulation of the interfacial transport while reducing previously reported computational times for these type of simulations by three orders of magnitude.
In chapter 3, external forced convection over a flat plate with intermittently emplaced porous cavities is considered based on the general model, whereas in chapter 4 the same situation with multiple porous blocks attached is investigated. The primary objective of these studies is to analyze the changes in the flow pattern and heat transfer characteristics due to the existence of the multiple porous cavity or block structure. The effects of inertia as well as the boundary surfaces on the fluid- saturated porous region are taken into account. Solutions of the problem have been carried out using a finite-difference method through the use of a stream function- vorticity transformation. Various interesting characteristics of the flow and temperature fields in the composite layer are analyzed and discussed in detail. The effects of several governing dimensionless parameters, such as the Darcy number,
Reynolds number, Prandtl number, the inertia parameter as well as the effects of pertinent geometric parameters are thoroughly explored.
In chapter 5, external forced convective flows through alternate porous cavity- block obstacles are studied numerically. The objective of this study is to analyze the 4 changes in the flow pattern and heat transfer characteristics due to the existence of a composite system made of alternately emplaced porous cavity-block obstacle. The
Brinkman-Forchheimer-extended Darcy Model, which accounts for the effects of impermeable boundary and inertia, is used to characterize the flow field inside the porous region. The formulation of the problem shows that flow and heat transfer characteristics depend on seven dimensionless parameters, namely, the Reynolds number, Darcy number, the Prandtl number, inertial parameter, two pertinent geometric parameters, and the number of porous cavity-block obstacles. Solution of the governing equations are carried out using the stream function-vorticity formulation and an in-depth discussion of the results for various physical interactions between the recirculating flows inside of the cavity and the external flow is presented.
Several interesting phenomena such as the interactions between the blowing and displacement effects from the porous blocks and the vortices penetrating into the porous cavities are presented and discussed, and it is shown that altering some parametric values can have significant effects on the external momentum and thermal boundary layer characteristics.
In chapter 6, internal forced convection in a parallel plate channel with porous blocks attached to the bottom plate is considered. The objective of this study is to present an numerical investigation for fully developed flow and forced convection within a constant-temperature parallel plate channel on which multiple porous blocks are emplaced. The Brinkman-Forchheimer-extended Darcy Model was used to characterize the flow field inside the porous regions in order to account for the inertia effects as well as the viscous effects. Solutions of the coupled governing equations for the porous/fluid composite system are obtained through a stream function- vorticity analysis with a finite difference scheme. Important results of engineering
interest were obtained and reported in this chapter. These results thoroughly
document the dependence of the streamline, isotherm and local Nusselt number distributions on the governing parameters of the problem, such as the Reynolds
number, Darcy number, Prandtl number, inertial parameter, and two pertinent geometric parameters. An in-depth discussion of the formation and variation for recirculation caused by porous medium is presented. It is shown that altering some parametric values can have significant effects on both flow pattern and heat transfer characteristics.
In chapter 7, internal forced convection in a parallel plate channel with porous cavity and block alternately emplaced on the bottom plate is investigated numerically.
The present study aims at investigating numerically forced convection in a isothermal parallel plate channel with porous cavity and block alternately emplaced on the bottom plate. The Brinkman-Forchheimer-extended Darcy Model, which accounts for the effects of impermeable boundary and inertia, is used to characterize the flow field inside the porous region. Solutions of the coupled governing equations are carried out through the stream function-vorticity analysis. An insight into the behavior of fluid flow and forced convection heat transfer has been obtained by the examinations of various governing parameters, such as the Reynolds number, Darcy number, inertial parameter, Prandtl number, and two geometric parameters. Several interesting phenomena such as the heat transfer augmentation in the thermal entrance region of channel were presented and discussed. The results of this investigation indicate that the size of recirculation caused by porous block will affect the flow and heat transfer characters inside the inter-block porous cavity. 6
In the last chapter general discussions on the physical aspects of the problems are made and the relevance of the present work is summarized. In addition, several aspects associated with the above-mentioned problems are identified for the future research. Throughout this work a conscious effort has been made in order to deliver a compact idea to bare essentials. To this end, unnecessary peripheral information has been minimized in each chapter. CH APTER II
ANALYSIS OF FLOW AND HEAT TRANSFER OVER AN EXTERNAL
BOUNDARY COVERED WITH A POROUS SUBSTRATE
2.1 STATEMENT OF THE PROBLEM
During the past decade there has been a renewed research interest in fluid flow and heat transfer through porous medium due to its relevance in various applications such as drying processes, thermal insulation, direct contact heat exchangers, heat pipes, filtration, etc. Comprehensive reviews of the existing studies on these topics can be found in Combarnous and Bories (1975), Cheng (1978), and Tien and Vafai (1989).
An important problem related to convection through porous media is flow and heat transfer through composite porous systems. The convection phenomenon in these systems is usually affected by the temperature and flow field interactions in the porous space and the open space. This type of composite system is encountered in many applications, such as some solidification problems, crude oil extraction, thermal insulation, and some geophysical systems. Due to the mathematical difficulties in simultaneously solving the coupled momentum equations for both porous and fluid
7 regions, it is usually assumed that there is only one fluid-saturated porous region (i.e.,
there is no fluid region and interfacial surface ) and the flow is through this infinitely
extended uniform medium. Thus the interaction between the porous-saturated region
and the fluid region did not form a part of most of these studies. In addition, most of
the existing studies deal primarily with the mathematical formulations in the porous
medium based on the use of Darcy's law, which neglect the effects of a solid boundary and inertial forces. These assumptions will easily break down since in most applications the porous medium is bounded and the fluid velocity is high.
Inertial and boundary effects on forced convection along a flat plate embedded in a porous medium were studied by Vafai and Tien (1981, 1982), and Vafai et al.
(1985). Among these studies Vafai and Tien (1981) treated a fluid-saturated porous medium as a continuum, integrated the momentum equation over a local control volume, and derived a volume averaged momentum equation, which included the flow inertia as well as the boundary effects. They found that these boundary effects can significantly alter the heat transfer from the plate, especially at high Prandtl numbers.
There have been few investigations related to porous/fluid composite systems.
Poulikakos (1986) presented a detailed numerical study of the buoyancy-driven flow instability for a fluid layer extending over a porous substrate in a cavity heated from the bottom. Another related problem is that of Poulikakos and Kazmierczak (1987). In that work a fully-developed forced convection in a channel that is partially filled with a porous matrix was investigated and the existence of a critical thickness of the porous layer at which the value of Nusselt number reaches a minimum was demonstrated.
Kaviany (1987), Beckermann and Viskanta (1987), and Nakayama et al. (1990) evoked the boundary layer approximations and solved the generalized momentum 9 equation presented in Vafai and Tien (1981) to investigate the same flow configuration.
Vafai and Kim (1990) performed a numerical analysis of forced convection over a porous/fluid composite system, which consisted of a thin porous substrate attached to the surface of the flat plate. They showed that both heat transfer retardation and enhancement of an external boundary can be achieved through the attachment of a porous medium.
The main focus of this research is to analyze laminar forced convection over a composite porous/fluid system using the integral method and to form an extremely fast alternative as well as a comparative base with numerical solutions for this type of interfacial transport problems. In this study, a thin porous slab attached to the surface of the plate will be discussed. The details of the interaction phenomena occurring in the porous medium and the fluid layer are systematically analyzed, revealing the effects of various parameters governing the physics of the problem under consideration. The present analysis reduces typical CPU times for the interfacial simulations presented in
Vafai and Kim (1990), which take about an hour on CRAY YMP, to only few seconds on a VAX 8550. 10
2.2 THEORY OF THE PROBLEM
The flow configuration and the coordinate system for this problem are shown in
Fig. 2.1. The thickness of porous medium is H, the free stream velocity u„, the length of the external boundary L, and the free stream temperature is T„, and the wall is maintained at a constant temperature Tw. In this study, we are assuming that
(i) the flow is steady, laminar, incompressible, and two-dimensional;
(ii) the properties are constant for both fluid and porous matrix;
(iii) the fluid-saturated porous medium is considered homogeneous, isotropic, and
in local thermodynamic equilibrium with fluid; and
(iv) the boundary layer approximations hold.
Based on these assumptions, the conservation equations, which include the boundary and inertial effects, in the porous region can be written as (Vafai and Tien
(1981)&(1982), Vafai (1984), Vafai (1986))
(2 .1)
dup _ 1 dPp (2.2) dyP Pf dxp
(2.3)
4 BSSS§ porous medium
Fig. 2.1 Schematic diagram of the flow over an external boundary with an attached porous substrate. 12 where subscript p refers to the porous medium, K and g are the permeability and
porosity of the medium, respectively, F is a function used in expressing inertia term and oC'jf = keff / PfCP f . Note that variables u, v, and T are volume-averaged
quantities. Since at a sufficiently large distance from the wall the flow field is uniform,
the free-stream axial pressure gradient in porous region required for maintaining the x- component interfacial velocity u, can be expressed as
= P .4) p dxp K VK dxP
Inserting equation (2.4) into (2.2), the momentum equation becomes
dll? du,p s F g . 2 2\ d 2u P d ll/ c. Up!hp pl h ~ K (U,~ Up) + jK (U,~Up) *~dri+U,d ^
In the fluid region, the conservation equations for mass, momentum and energy are
d ll, d \)r - L + — L = Q (2.6) dxj dyf
du, du, 1 dp, d2uf _ m/dXj ^ + u /3 dyfT ^ = - 3 pf +vf dxf T T 1 dyf ( 2 J )
dr. dr, d% u'*;+ v'l£ ma'l £ (28)
The boundary conditions are x, y < 0 : u = u„, P = P„ (2.9) 13 x > 0, y = 0 : u - v = 0 (2.10)
x>0,y-^°°:u = u„,P = P„ (2.11)
In addition, the matching conditions at the interface of the porous/ fluid system are
u y=It — u y-it , v yslt ... = t> Jy=// (2.12a)
- pi in\ ~ in. (2.12b) >=//- ~ t \y=ir’ P'ff dy\y=ir dy y=H+
(du dv'] _ (du dv] (2.12c) ^tir\dy d x/=,r ^\dy dx)y°H*
_rr k 3 7 - k 3 7 T >=//" T y=ir > K‘ff dy y=II- y=H* (2.12d) 14
2.3 ANALYSIS AND SOLUTION
An integral analysis is applied to three different regions: the porous boundary
layer region, fluid boundary layer within the porous substrate, and the fluid boundary
layer outside of the porous substrate as shown in Fig. 2.2.
Integral momentum equation for the vorous boundary layer Following the
Karman- Pohlhausen integral method the parabolic velocity distribution is described as
(2.13)
where 5P is the thickness of the momentum boundary layer in the porous region. After
a lengthy analysis the integral momentum equation in dimensionless form for the
porous region is derived as
--o 8 „ r r- - = (2.14) 5 dxp 15 dxp
where
This equation is subject to the following initial conditions:
8 ,(0 ) = 0, r(0) = 0 (2.15) ►
►
►
Fig. 2.2 Control volume for integral momentum analysis of 1) porous boundary layer, 2) interfacial region between the porous and fluid layers and 3) and the fluid boundary layer 16
Integral momentum equation for the fluid boundary layer within the porous
substrate A lengthy integral analysis for the control volume 2 shown in Fig. 2.2 leads
to the momentum integral equation for that region
p ' dxf dxf 0
1 d8F (2.16) -(1 ~r)ul —r — £ 3 dx„ dx
where x, is the shear stress in the porous/fluid interface and 5p and S/ are the thicknesses of porous and fluid momentum boundary layers, respectively. Assuming the following parabolic velocity distribution for the fluid boundaiy layer
— = r + (2-2r)^- —(1-r) h . (2.17) f KdfJ where subscript f refers to fluid and r = u, and substituting Eq. (2.17) into Eq.
(2.16) and replacing x, by Pf(duf / dyf)yf= 0> the approximate integral momentum equation in dimensionless form becomes
15 15 5 ) dxf 'U 5 5 k
dr — r - — H (2.18) (2- 2r)R ^ +(1- r) 3 dx7 \dxr where This equation is subject to the following initial conditions:
8,(0) = 0, 5/(0) = 0, r(0) = 0 (2.20)
Integral momentum equation for the fluid boundary layer outside of the porous substrate A similar procedure for the control volume 3 in Fig. 2.2, which incorporates the interfacial boundary conditions, leads to the integral momentum equation for the interfacial region
\2 V f~ - + veJru,(x)lH-8p(x)] +-j=- “ J {H — 8p) = 0 (2.21)
Substituting the velocity distributions for both porous and fluid boundary layers into
Eq. (2.21) gives the dimensionless form of the integral momentum equation
+ (^L~ + A r2(/7 - 8 ) = 0 (2.22) R eLSf ReLDaL 1 p)
This equation is subject to the following initial conditions: 18
Integral enp.rpv p.nuation Since only one thermal boundary layer was observed in the forced convection through the porous/fluid composite system under consideration
(Vafai and Kim (1991)), in this study the control volume 1 in Fig. 2.2 is used to derive the integral energy equation, which depends on the relative values of bp and 5/ .
Performing an energy balance over control volume 1 and using the following parabolic temperature distribution
f V Tw _ 2 2 p_ _ y P (2.24) T „ -T w 5, 8, the integral energy equation in dimensionless form is obtained after a very lengthy analysis for three different possible conditions as follows:
f s 8 82 S3 ) dr r '{ s , db, _ f 8 3 8 0 dSp ut p | p p 1 + — —, 1 —=—1---- = r { 3 3 68, 308,2 d§ p _ 8, > 8, (2.25a) d xp ~Preff ReL 8t ■=■ dr db. ^ 8.— + r — 8, = 6 , (2.25b) dx dx and ■' ~ c2 ^ f 8? Sf ] dr 1 , 8t St d8fc dbP — + r {6SP 308^dXp ^ 38P 108?^dxp 68? 158? V. t V) $ I 19 5, < 5P (2.25c) P re#R cl 8t where Re, = Pr . = — and <5, is the thermal boundary layer thickness. This V a «r equation is subject to the following initial conditions: 8,(0) = 0, 5/(0) = 0,r(0) = 0 (2.26) 2 0 2.4 RESULTS AND DISCUSSION The combinations of equations (2.16), (2.18), (2.22), and (2.25) form a set of four nonlinear simultaneous ordinary differential equations for the four unknowns 8p, 5 S,, and r. The fourth-order-Runge-Kutta method described in Gerald (1978) is applied to solve the above initial condition problem. As discussed earlier, the present analysis is extremely effective and expedient in showing the physics of the interfacial transport. In what follows, the solutions obtained by integral analysis are examined and compared with the velocity and temperature distributions obtained by Vafai and Kim (1990). Figure 2.3 shows how the boundary layer thickness, the velocity and temperature distributions are affected by the presence of a porous matrix. The results in Fig. 2.3 are presented for Reynolds number of ReL = 3 x10s, Darcy number Dat = 8 x 10"6, inertial number A L = 0.35, Prandtl number Pr=0.7, effective conductivity ratio ktjr I kf = 1 and the dimensionless thickness of the porous slab of H*= 0.02. As expected, there are two distinct momentum boundary layers: one in the porous region and the other one in the fluid region. Inside the porous region as the transverse coordinate increases the velocity profile is shown to increase from zero to a constant value, which is maintained until the outer boundary layer appears. Once it crosses the porous/fluid interface, it goes through a smooth transition and approaches a free-stream value in the fluid region. As expected, the momentum boundary layers in the porous medium as well as in the fluid region grow in the streamwise direction. Consequently, the magnitude of the 0.05 Numerical tnelhod Inlegral method 0.04 2: x=0.5 0.03 3: x=0.8 0.02 0.01 0.00 — 0.0 0.2 0.4 0.6 0.8 1.0 Ut I 0.020 Numerical melhod Inlegral method 0.016 0.012 2: x=0.S 3: x=0.H 0.008 0.004 0.000 0.0 0.2 0.4 0.6 0.8 1.0 0 = ±— K-Tm Fig. 2.3 (a) Velocity and (b) temperature distribution along (lie flat plate at three different locations, x=0.2, 0.5 and 0.8, for ReL = 3 x105» = 0.35, Da* = 8 x IQ’6, Pi-0.7, k lff /k f = 1.0, HIL=0 .0 2 . 22 interfacial velocity decreases to adapt this growth. Figure 2.3(b) shows the temperature distribution along the flat plate at three different locations. The velocity and temperature distributions obtained by integral method are in good agreement but somewhat different from those obtained by numerical method. This is due to the approximate expressions which were used for the velocity and temperature profiles in the integral analysis. Figure 2.4 compares the results of the numerical method and the integral method in the streamwise direction for the interfacial velocity. The results show a remarkably good agreement between the integral analysis and the full numerical solution considering the complexity and the enormously larger CPU times requirements for the numerical simulations. It should be noted that the type of agreements found in Figs. 2.3(a), 2.3(b) and 2.4 are typical for a wide range of pertinent parameters and are not presented here for the sake of brevity. 23 Numerical method 0.9 Inlegral method 0.8 0.6 0.5 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fig. 2.4 Comparison of the integral solutions with numerical solutions for x-compoment interfacial velocity along the flat plate for At = 0.35, Ret = 3xl05, Da^SxlO**, H/L=0.02. CHAPTER III FORCED CONVECTION OVER INTERMITTENTLY EMPLACED POROUS CAVITIES 3.1 STATEMENT OF THE PROBLEM Force convection heat transfer through porous media has been a major topic for various studies during the past decades due to many engineering applications such as thermal insulation engineering, water movement in geothermal reservoirs, underground spreading of chemical waste, nuclear waste repository, grain storage, and enhanced recovery of petroleum reservoirs. Majority of the previous investigations include flow through a semi-infinite porous medium for external boundaries, and flow through structures which are fully filled with the porous medium for internal flows. However, consideration of the interaction between the porous-saturated region and the fluid region did not form a part of most of these studies due to the difficulties in simultaneously solving the coupled momentum equations for both porous and fluid regions. There has been very little work done on these type of interactions which can occur in various practical applications. Furthermore, the majority of existing studies on convective heat transfer in porous media are based on the Darcy's law, which is found to be inadequate for the formulation of fluid flow and heat transfer problems in porous media when there 24 25 is an impermeable boundary and/or the Reynolds number based on the pore size is greater than unity. A general theory and numerical calculation techniques for flow field and heat transfer in recirculating flow without including the porous medium has been developed by Gosman (1976), and was successfully used by various investigators such as Gooray (1982), and Gooray et al. (1981, 1982). They have shown that these methods can be applied to give reasonably accurate quantitative heat transfer results for the separated forced convection behind a backstep. A few studies have been reported on forced convection in a rectangular cavity. Yamamoto et al. (1979) experimentally studied the forced convection on a heated bottom surface of a cavity situated on a duct wall. They have shown that reattachment of separated flow for shallow cavities and vortex flow for deeper ones had a large effect on the heat transfer behavior on the heated bottom surface. Sinha et al. (1981) reported the experimental results for laminar separating flow over backsteps and cavities. They found that cavities can be classified as closed, shallow open, and open depending on the range of the value of the aspect (depth-width) ratio. Aung (1983) performed an experimental investigation of separated forced convection laminar flow past two-dimensional rectangular cavities where the walls are kept at a constant temperature. He found that the temperature distribution outside of the cavity had little influence on the flow in the cavity and the local heat transfer distribution on the cavity floor attains a maximum value that is located between the midpoint of the cavity floor and the downstream wall. Bhatti and Aung (1984) numerically examined the laminar separated forced convection in cavities. They found that the average Nusselt number in open cavity i 26 flow is related to the Reynolds number raised to a power, which depends on the aspect ratio of the cavity, and the influence of the upstream boundary layer thickness on the heat transfer in the cavity is negligible. Most of the existing related studies on convection in composite systems were focused on the problem of natural convection in an enclosure (Beckermann et al. (1988); Bejan (1984) and Sathe et al. (1988); Cheng (1978), Poulikakos and Bejan (1985)) or forced convection in a duct (Poulikakos and Kazmierczak (1987); Bejan (1984)), or external and internal boundaries (Vafai and Kim (1990); Poulikakos (1986); Bejan (1984)). However, to the best of authors' knowledge, there has not been any investigations on the forced convection over multiple porous cavities. The present work constitutes one of the first analyses of the laminar separated forced convection through porous cavities. The present investigation provides a valuable fundamental framework for predicting heat transfer and fluid flow characteristics for other composite systems. The results and fundamental information presented here can be extended to examine various applications such as in electronic cooling and in heat exchanger design, reduction of skin friction and heat transfer enhancement or augmentation, some of the manufacturing processes, geothermal reservoirs and oil extraction. 27 3.2 ANALYSIS AND FORMULATION The configuration for this problem is shown in Fig. 3.1. The height and width of the porous cavities are H and W, respectively, the distance between two cavities is D, the length of the wall is L, the free stream velocity is and the free stream temperature is T„. The wall is maintained at a constant temperature Tw. It is assumed that the flow is steady, laminar, incompressible, and two-dimensional. In addition, the thermophysical properties of the fluid and the porous matrix are assumed to be constant and the fluid-saturated porous medium is considered homogeneous, isotropic and in local thermodynamic equilibrium with the fluid. For the fluid region the conservation equations for mass, momentum and energy are V -v = 0 ( 3 . 1 ) v-Vv = — — VF + VyV^ (3.2) Pf v-VT = afV2T (3-3) Based on the Brinkman-Forchheimer-extended Darcy model which accounts for the effects of the inertial and impermeable boundary, the mass, momentum and energy equations in the porous matrix (Vafai and Tien (1981)&(1982), Vafai (1984), Vafai (1986)) can be expressed as V- v = 0 (3.4) porous medium j W77777Z Fig. 3.1 Schematic diagram of flow and heat transfer over intermittently emplaced porous cavities. to 00 where K , and e are the porous medium's permeability and porosity, and veJf and a cjr are the effective kinematic viscosity and thermal conductivity of the porous medium. It should be noted that velocity v and temperature T in the porous region are both volume- averaged quantities as described by Vafai and Tien (1981). Also it should be noted that the effects of local non-thermal equilibrium and dispersion are neglected at this time based on the work of Vafai and Sozen (1990) and Sozen and Vafai (1990,1993). The boundary conditions necessary to complete the problem formulation are u = u„, "0 = 0, P = P„, T = T„ at x=0 (3.7) u = 0, v = 0 ,T = Tw on all solid walls (3.8) u = um,P = P„,T = T„ as y _>oo (3.9) In addition to these the two sets of conservation equations are coupled by the following matching conditions at the porous/fluid interface: Vorticity-Stream Function Formulation The above governing equations are cast in terms of the vorticity-stream formulation. Introducing stream function and vorticity as 0X1/ 3\ir (3.11) ~Bx e _ dv du (3.12) the governing equations for the whole region can be expressed in dimensionless form as 3y* 3£* By* 3%‘ 1 2,. . (3.13) 0y dx Bx By Re£ * V V* = -K (3.14) B y * 30 3\jr* 30 _ -V9 (3.15) ~By~Bx^ ~ B /B /~rcL Pe where in the fluid region 31 Re£ = , PeL = - ^ , 5* = 0 (3.16a) V/ « / and in the porous region u L _ K F L e Pe£ — _ , Da£ — 2 , Al — 1/2 (3.16b) “ L1' “ K1 1 e* a I *Ib* * f *3IV*I * 3IV1 =-p- ^ -5 "AJV S -Al v ^ s Re£Da£ 1 1 dx dy u d 1 ) v ‘ d (3.17) Re£ 9y Da£J Re£ dx* VDar . +KK^(aJ-Iv'K ^ ( a 0 the dimensionless boundary conditions thus become ¥ * = /. V = 9 = 0, at x* = 0 (3.18) dx a V . K + (N-i)(H' + d ) \|/ = 0, ^ = - —-^-,0 = 1, at* =•! . 0>y >-H (3.19) dx [^; + NVP +(N-1)D 32 . j.. a > 0 ¥ =0 , £ =“T^"»0 = 1’ 3y i\ + (N - 1)(W* + D*) 0 { \ - £ 2) < x ' < i 3\|/* j.. 3V * Q n 2^ = U = - ^ S r , e = 0 , as? (3.21) where N (=1, 2, 3, 4) is the number of porous cavities. Note that the variables in the above equations are defined as follows: x . v . u . 1) . 2 *2" * =T’L y =T' L u = — u ' v -— u ’ V + u p .zza; (3-22b) W* - ^ /* _ A _ ^2 r)* _ & irr* _ W H -T'e'-Te'~T'D ~T" T ' 3 -2 2 o > From the above equations, boundary conditions, and geometric arrangement of cavities it is seen that the present problem is governed by seven dimensionless parameters. These are Darcy, Reynolds, and Prandtl numbers, the inertia parameter, the number of cavities N, and the geometric parameters A and B, where 34 3.3 NUMERICAL SCHEME Employing a non-uniform rectangular grid system, the finite-difference form of the vorticity transport, stream function and energy equations were derived using control-volume integration of these differential equations over discrete cells surrounding the grid points, as shown in Fig. 3.2. In the above discretization scheme the upwind and central-differencing formats are also introduced for the convective and diffusive terms, respectively. This results in a system of equations of the following form: Cc0 c = CN0 N + CS0 S + CE0 E + CW0W + S* (3-24) where stands for the transported variables, C's are coefficients combining convective and diffusive terms, and S®is the appropriate source term. The subscripts on C denote the main grid points surrounded by the four neighboring points denoted as N, S, E and W. To ensure the continuity of the diffusive and convective fluxes across the interface without requiring the use of an excessively fine grid structure, the harmonic mean formulation suggested by Patankar (1980) was used to handle abrupt changes in thermophysical properties, such as the permeability and the thermal conductivity, across the interface. Moreover, the source terms incorporated with the boundary and inertia effects were linearized as described by Patankar (1980). The vorticity at sharp corners requires special consideration. Seven different methods of handling this comer 35 (a) N 1 i 1 i 1 i 1 i w 1 i E i I c i 1 i 1 i 1 .11 i s (b) Fig. 3.2 (a) The non-uniform grid system for the whole computational domain, (b) Local integration cell in the computational domain. 36 vorticity are discussed in Roache (1976). Here average treatment for the evaluation of vorticity suggested by Greenspan (1969) is used to model the mathematical limit of a sharp comer as appropriately as possible. The finite difference equations thus obtained were solved by the extrapolated- Jacobi scheme. This iterative scheme is based on a double cyclic routine, which translates into a sweep of only half of the grid points at each iteration step (Adams and Ortega (1982)). In this work convergence was considered to have been achieved when the absolute value of relative error on each grid point between two successive iterations was found to be less than 10“6. The iterative procedure was then terminated. To examine the independence of the results on the chosen Ax and Ay, many numerical runs with different combinations of Ax and Ay were performed. This was done by a systematic decrease in the grid size until further refinement of the grid size resulted less than 1 percent difference in the converged results. A grid size of 162x 188 was finally found to model accurately the flow field described in the results for all of the considered cases. The application of the boundary condition at infinity at a finite distance from the wall was given careful consideration. This was done through the following procedure. The length of the computational domain in the vertical direction was systematically increased until the maximum vorticity changes for two consecutive runs would become less than 1%. Therefore, in our investigation the computational domain is chosen to be larger than the physical domain. Along the x direction, the computational domain starts at a distance of one-fifth of total length upstream of the physical domain. This procedure eliminates the errors 37 associated with the singular point at the leading edge of the composite system. On the other side, the computational domain is extended over a distance of two-fifths of the total length downstream from the trailing edge of the physical domain. Since the present problem has a significant parabolic character, the downstream boundary condition on the computational domain does not have much influence on the physical domain. In the y-direction the computational domain is extended up to a distance sufficient enough to ensure that even for the smallest value of the Reynolds number the upper boundary lies well outside the boundary layer through the entire domain. In the present study locating the upper boundary at a distance of 8 times the depth of the cavity has been found to be sufficient. Extensions beyond 8 times the depth of the cavity had no effect on the solution. To validate the numerical scheme used in the present study, initial calculations were performed for laminar flow over a flat plate (i.e., //*= 0 , for no porous substrate) and that over a flat plate embedded in a porous medium (i.e., and ^*—> 00, representing the full porous medium case). The results for H*= 0 agree to better than 1 percent with boundary layer similarity solutions for velocity and temperature fields. The results for H*—>°° and W* —»<*> agree extremely well with data reported by Vafai and Thiyagaraja (1987) and Beckermann et al. (1987). 38 3.4 RESULTS AND DISCUSSION The effects of the geometric arrangements of the porous cavities as well as the effects of different values of RaL, DaL, Pr, and AL on the flow and temperature fields were investigated. Table 3.1 displays various input parameter sets considered in this analysis. The parameter sets presented in Table 3.1 were only a subset of much larger set which was investigated in this work. The parameter sets presented in Table 3.1 were found to be most important in revealing pertinent aspects of the geometric arrangements of the porous cavities and variations in the thermophysical properties. Figures 3.3-3.12 show streamlines and isotherms over multiple porous cavities for the corresponding cases listed in Table 3.1. To illustrate the results of flow and temperature fields inside the porous cavity, only the portion which concentrates on the porous/fluid region and its close vicinity is presented. However, it should be noted that the computational domain included a significantly larger region than what is displayed in the subsequent figures. Effect of the Reynolds number Figure 3.3 shows streamlines and isotherms over four porous cavities with A=6 and 5=1 for the case where Pr=0.7, AL = 0.35, DaL = 8 x 10 -6 and ReL = 3 x 105. It can be seen that a laminar vortex resides within each of these cavities. The strength of the eddies within each cavity decreases further along the flow direction. These recirculating flows are formed as the primary flow impinges on the downstream cavity wall and then flows toward the bottom surface. Due to an increase in the thickness of external boundary layer along the plate, there is a reduction in the mass flow rate that Table 3.1 Input data of governing parameters for intermittently emplaced cavities Case# ReL Da L Pr A z. A B N 1 3 x l0 5 8 x 10^ 0.7 0.35 6 1 4 oo o 2 X 2 x l 0 5 0.7 0.35 6 1 4 j 3 x l0 5 8 x 10"* 0.7 0.35 6 1 4 4 3 x10s 8 x 10"* 7 0.35 6 1 4 T oo o D 3 x l0 5 X 100 0.35 6 1 4 6 0.7 1.05 6 1 4 X o 8 x 10^ 7 3 x 10s 8 x 10"5 0.7 0.35 3 1 4 8 3 x 10 5 8 x 10^“ 0.7 0.35 6 0 .8 4 9 3 x l0 5 8 x 10"* 0.7 0.35 3 2 4 10 3 x 10s 8 x 10-6 0.7 0.35 12.5 1 2 0.08 3x10' 0.06 1x10' 0.04 9x10* 0.02 •SxlO' -6x10' 0.0 •2x10- 0.0 0.20 0.60 O.SO 1.00 * X (a) 0.06 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 (b) Fig. 3.3 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities " 1 nS T"\rt o .. 1 n-6 A _ A n c n n n 1. /I, — i n A —C Q — 1 0.08 4x10' 0.06 3x10' 2x10' y* 0.04 9xio: 0.02 0.0 0.0 0.20 ' 0.40 0.60 0.80 1.00 X (a) 0.08 0.06 9*10* y* 0.04 0.02 0.0 0.20 0.40 0.60 0.80 1.00 (b) Fig. 3.4 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for Re^ = 2x10s, D a ^ S x lO "6, AL = 0.35, Pr=0.7 ,Kjr / k/ = 10, ^=6, 5=1. 42 penetrates into each subsequent cavity. This in turn causes a reduction in the strength of vortices within the cavities as the flow moves further downstream. Figure 3.3(a) also shows that within each cavity the streamlines between the vortex center and the downstream wall of that cavity are denser than those between the vortex center and the upstream wall of the cavity. This is because the magnitude of the downstream vertical velocity is larger than the upstream one. The small fluctuations of the porous/fluid interfacial streamline are due to the macroscopic shear frictional resistance at the interface. As the Reynolds number decreases from 3 x 10 5 to 2 x 105, the center of the vortex for each cavity moves further to the left as seen in Fig. 3.4. The reason for this trend is that the lower the Reynolds number, the lower the flow inertia, which in turn results in lower bulk frictional resistance for the flow. As a result a stronger vorticity in the cavity is formed, which makes the streamlines bulge more towards the upstream cavity wall. Comparison of the isotherms in Figs. 3.3(b) and 3.4(b) indicates that at a smaller Reynolds number the isotherms penetrate further inside the cavity. This is again due to lower bulk frictional resistance for the lower Reynolds number, which results in higher velocities thus increasing the convective energy transport within the cavity. Effect of the Darcv number The Darcy number Da£ = K / L 2 is directly related to the permeability of the porous medium. Figures 3.3 and 3.5 show the streamlines and isotherms for Re/- = 3xl05, AL=0.35, Pr = 0 .7 >A=6 , and 5 = 1 but with DaL= 8 xlO -6 and 8 x 1 0 '8> respectively. Comparison of the streamlines in Figs. 3.3(a) and 3.5(a) shows 0.08 0.06 4x10' 3x10' 1x10- 9x10: 1x10' 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 (a) 0.06 o-< »«io-‘ 0.04 0.02 0.0 0.20 0.40 0.60 0.80 1.00 (b) Fig. 3.5 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, DaL = 8 x 10"8, Pr = 0.7, k,# / k / = 1.0, AL = 0.35,A=6, 5=1. 44 that as the Darcy number decreases, the size of the vortices is reduced. This is because smaller values of DaL translate into larger bulk frictional resistance for the flow in the porous medium. This in turn reduces the extent of penetration of the primary flow into the cavity. Comparison of the isotherms in Figs. 3.3(b) and 3.5(b) depicts that for the lower Darcy number case the isotherms penetrate deeper inside the cavity, especially in the left half section of cavity. The reason for this interesting effect is that for a lower Darcy number, heat diffusion is more significant than heat convection in the porous region, which in turn increases the energy transfer. This aspect is similar to some of the previous investigations (Bejan (1981)). Note that in this study the conductivity of the porous medium is taken to be equal to that of the fluid, to concentrate on the effects of the geometrical and thermophysical variations. Effect of the Prandtl number To study the effects of the Prandtl number on the flow and temperature fields, three different Prandtl numbers were chosen such that they will cover a wide range of thermophysical fluid properties. The numerical results are presented in Figs. 3..3, 3.6 and 3.7 for Ret = 3xl05, At =0.35, DaL = 8 x10"6, A =6 , and B=1 for three different fluids with Pr=0.7 (air), Pr=7 (water), and Pr=100 (some oil), respectively. Obviously, the Prandtl number variations have no effect on the flow field. However, as expected, the isotherms penetrate further inside the porous cavities as Prandtl number decreases. o .o a 3x10' 0.06 0.04- 9xlo; 1x10' 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 X (a) o .o a 0.06 0.04 IxlO' 1x10* 5x10- 9xl( 0.02 .0.993 ).995 0.0 0.0 0.20 0.400.60 0.80 1.00 X Fig. 3.6 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x l0 5, DaL = 8 xlO-6, AL = 0.35, Pr=7, ktff/kf = 1.0, A=6, 5=1. 4^ 0.00 •5x10' 0 .0 0 0.00 1 x 10' 0 . 0 3 0 .0 2 0 .0 1 0 . 0 .000 .2 0 0 .0 0 0 .0 0 1.00.0 X (a ) 0 .0 0 0 .0 0 0.00 1x10* 5x10* 9x10* 0 . 0 3 0.02 0.0 1 QS9Tj 0 .0 0.999- 0 .0 0.20 0.00 0 .0 0 1 .a .X (b ) Fig. 3.7 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3x10s, DaL = 8xlO-6, AL = 0.35, Pr=100, keJT / k f = 1.0, A=6, 5=1. •t*. ON 47 Inertial Effects The inertial effects become noticeable when the Reynolds number based on the pore diameter becomes large. The effect of the inertial parameter is shown in Figs. 3.3 and 3.8 for Ret = 3 x10s, Dat = 8 x 10"6, Pr=0.7, A = 6 and 5=1 but AL = 0.35 and 1.05, respectively. Comparison of the streamlines in Figs. 3.3(a) and 3.8(a) shows that as the inertial parameter increases, the strength of the vortices is reduced. This is due to the larger bulk frictional resistance for the flow which the flow experiences at a larger inertial parameter. This in turn reduces the extent of penetration of the primary flow into the cavity. Comparison of the isotherms in Figs. 3.3(b) and 3.8(b) shows that for the smaller inertial parameter case the isotherms penetrate deeper into the cavity. This is owing to higher velocities for the smaller inertial parameter, which increases the transport of the convective energy. Effect of the First Geometric Parameter A. The effects of the aspect ratio (the first geometric parameter) A on the flow and temperature fields were studied for the general case of Ret = 3x10s, AL =0.35, Da*, = 8 x 10-6, Pr=0.7 and 5=1. The streamlines and isotherms for aspect ratios of A=3 and 6 are presented in Figs. 3.9 and 3.3, respectively. As seen in Fig. 3.3 for A=6 the flow in the cavity consists of a single laminar vortex that occupies the entire cavity. For A=3 the flow is still characterized by a single vortex, but the vortex center is displaced towards the upstream cavity wall. This kind of flow situation may persist up to a certain value of A where the center of vortex just coincides with the center of cavity. o . o a o.oa 4x10* 0 .0 6 3x10- 2x10' 0.02 • 1x10" 0.0 1 ■6x10* 0 . 0 >2x10* 0.0 0.20 0.^0 0.60 0.60 1.0 .X (a) 1x10** 5xt0** (b) Fig. 3.8 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3x 10s, D a^SxK T*, AL = 1.05, Pr=0.7, / k f = 1.0, A=6, 5=1. 00 0.08 0.06 •4x10 1 0.04, •2x10- 9xio; 1x10' 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 X (b) Fig. 3.9 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3x 10s, DaL =8x10^, AL = 0.35, Pr=0.7, ktJf / k f = 1.0, A=3, 5=1. 50 Effect of the second geometric number 5 . The second geometric number 5 = D* IW* reflects the influence of cavity array arrangement on the flow. There are two configurations considered in this analysis. The numerical runs were carried out for the general case of Ret = 3x10s, DaL = 8 x 10-6, A l = 0.35, Pr=0.35, and A=6. The streamlines and isotherms for the cases of 5=1 and 5=0.8 are presented in Figs. 3.3 and 3.10. It can be seen from Fig. 3.10 that when A = 6 and 5<1 the flow separates from the upstream top left corner of the first cavity, reattaches on the bottom surface, and then reaches the downstream cavity wall. After passing through the first cavity the flow separates from the plate and an eddy region appears behind the separation point. This is due to a very steep velocity gradient (vorticity) in the fluid as it turns around the top right comer of the first cavity as shown in Fig. 3.11(a). Under a special condition where both parameters A and 5 were changed a few interesting results were found. As can be seen in Fig. 3.12, when the value of A was decreased from 6 to 3 and 5 increased from 0.8 to 2, a secondary boundary layer starts from the top right comer of the first cavity and is maintained over the external boundary between the first two cavities. This secondary boundary layer results in vortex flow inside the other three cavities. The reason for the appearance of the secondary boundary layer is that as the fluid turns around the top right comer of the first cavity, it experiences a milder velocity gradient as seen in Fig. 3.11(b). The velocity gradient is so small that the longitudinal pressure gradient is enough to reattach the flow to the wall. Comparison of the isotherms in Figs. 3.3(b) and 3.10(b) or Figs. 3.9(b) and 0.06 4x10' 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 X (a) o .p s 0.06 0.04 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 X (b) Fig. 3.10 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x l0 5, Dat = 8 x lO ^, AL = 0.35, Pr=0.7, ktJf / k f = 1.0, A=6, 5=0.8. 0.03 0.05 -10 0.06 o.ot 0.04 - 10. 0.02 0.7 0.0 0.0 0.20 0.60 0.80 1 .0 0 X (a) 0.08 .0.03. 0.06 .0 .0 1 . 0.04 -100' 0.02 - 1 0 0.0 0.0 0.20 0.40 0.60 o.eo 1.00 X (b) Fig. 3.11 Vorticity contour for four porous cavities at different spacings (a) A=6 , 5=0.8 and (b) A=3, 5=2. U\ to 0.08 0.06 3x10' 0.04 9x10- 0.02 •3 xlO- 0.0 0.0 0.20 0.60 0.80 1.00 X (a) 0.08 1x10' 1x 10* 1x10* 3x10* 0.06 0.02 0.99 0.0 0.0 0.20 0.40 0.60 0.80 1.00 « X Fig. 3.12 (a) Streamlines and (b) isotherms for flow over four obstructing porous cavities for ReL = 3 x 10s, Dat = 8xl0"6, AL = 0.35, Pr=0.7, kejr / k f = 1.0, A=3, 5=2. U\ 0.08 4x 10' 0.06 3X10- y ■ 0.04 9 x 10' 0.02 0.0 0.0 0.20 0.40 0.60 0.80 1.00 (a) 0.08 0.06 0.04 .1x 10- 1x 10' 1x 10' 3x 10- 1x 10' 0.02 •0.93 3.96 0.0 *0.97 0.0 0.20 0.40 0.60 0.80 1.00 (b) Fig. 3.13 (a) Streamlines and (b) isotherms for flow over two obstructing porous cavities for Rei = 3x10s, DaL = 8 xlO -6, AL = 0.35, Pr=0.7, kejr/k f = 1.0, A=12.5, 5= 1. 55 3.12(b) indicates that a reduction in the second geometric parameter B creates a significant distortion of the isotherms. This is due to the separated and reattached flow regions which were described previously. Figure 3.13 illustrates the results for two porous cavity array with ,4=12.5 and 5=1 for Rei =3xl05j DaL= 8 xlO -6j A l =0.35, and Pr=0.7. The streamlines and isotherms for the two porous cavity array are very similar to those for the four cavity array. CHAPTER IV FORCED CONVECTION OVER A POROUS BLOCK ARRAY 4.1 STATEMENT OF THE PROBLEM Convective heat transfer in fluid-saturated porous media has gained considerable attention in recent decades due to its relevance in a wide range of applications such as thermal insulation engineering, water movements in geothermal reservoirs, heat pipes, underground spreading of chemical waste, nuclear waste repository, geothermal engineering, grain storage, and enhanced recovery of petroleum reservoirs. Most of the previous studies related to transport through porous media are based on investigating flow through a semi-infinite porous medium or flow through structures which are fully filled with the porous medium. But, due to the difficulties in simultaneously solving the coupled momentum equations for both porous and fluid regions, analysis of the porous/fluid composite structures which occur in wide variety of practical applications have been mostly ignored. A general theory and numerical calculation techniques for flow field and heat transfer in recirculating flow without including the porous medium has been developed 56 57 by Gosman (1976), and was successfully used by various investigators such as Gooray (1982), and Gooray et al. (1981, 1982). They have shown that these methods can be applied to give reasonably accurate quantitative heat transfer results for the separated forced convection behind a backstep. However, to the best of authors' knowledge, there has not been any investigations on the forced convection over multiple porous blocks. For an external boundary exposed to semi-infinite porous medium, Cheng (1977) and Bejan (1984) have presented closed form results for the local Nusselt number as function of the Peclet number based on Dracy's law. Kaviany (1987) has solved a similar problem and obtained Karman-Pohlhausen solutions based on the Brinkman- Forshheimer-extended equation which accounts for the boundary and inertia effects. Vafai and Thiyagaraja (1987) presented a theoretical analysis for a general class of problems involving interface interactions on flow and heat transfer for three different types of interface zones. They obtained analytical solutions for both velocity and temperature distributions as well as the analytical expressions for the Nusselt number for all of these interface conditions. Vafai and Thiyagaraja (1987) also presented a detailed theoretical solution for the velocity and temperature fields as well as the Nusselt number distribution for flow over an external boundary embedded in a porous medium. Bejan (1982) obtained the temperature distribution for a horizontal line source and a point source placed in a uniform flow through a porous medium, and Cheng and Zheng (1985) reported the inertia and thermal dispersion effects on flow and temperature fields for a horizontal line heat source in a porous medium. 58 While many studies have been performed in these areas, little attention has been focused on porous/fluid composite system. Vafai and Kim (1990) have studied numerically the flow and heat transfer over a flat plate with an attached porous substrate. They carried out a fundamental study of the effects of Darcy and Prandtl numbers, inertia parameter, and the ratio of the conductivity of the porous material to that of the fluid. They found that the porous substrate caused the flow to deflect and resulted a decrease in the frictional drag. They also found that, depending on the value of the thermal conductivity of the porous wafer, the substrate can either augment or decrease the heat transfer from the external boundary. However, there is very little work on forced convective flow in more complicated configurations. The present work addresses a fundamental investigation of one such complicated configuration, namely a system composed of intermittent porous blocks. The main focus of this research is to analyze laminar forced convection over a composite porous/fluid system composed of multiple porous blocks. Since very little work has been done on external forced convective flow and heat transfer in composite systems, the objective of the present work is to study the interaction phenomena occurring in the porous medium and the fluid layer. In addition, the effects of various parameters governing the physics of the problem under consideration are also analyzed. The results presented in this work will provide a valuable and fundamental framework for predicting heat transfer and fluid flow characteristics for other composite systems such as in electronic cooling and in heat exchanger design, heat transfer enhancement or augmentation, drag reduction (as compared to the presence of solid blocks), some of the manufacturing processes, geothermal reservoirs and oil extraction. The present 59 work constitutes one of the first analyses of the laminar separated forced convection over porous blocks. 60 4.2 THEORY A schematic representation of the system under investigation is shown in Fig. 4.1. It consists of flow over a flat plate with multiple porous blocks attached on the external boundary. The height and width of the porous blocks are H and W, respectively, the distance between two blocks is D, the length of the wall is L, the free stream velocity is , and the free stream temperature is TM. The wall is maintained at constant temperature Tw. It is assumed that the flow is steady, laminar, incompressible, and two-dimensional. In addition, the thermophysical properties of the fluid and the porous matrix are assumed to be constant and the fluid-saturated porous media are considered homogeneous, isotropic and in local thermodynamic equilibrium with the fluid. An efficient alternative method for combining the two sets of conservation equations for the fluid region and porous regions into one set of conservation equations is to model the porous substrate and the flow regions as a single domain governed by one set of conservation equations, the solution of which satisfies the continuity of the longitudinal and transverse velocities, normal and shear stresses, temperature, and the heat flux across the porous/fluid interface as described by the following equations (Sathe et al. (1988); Vafai and Kim (1990)): Mp |« (x .y )=0 M/|*(*.y)= 0 ’ ^P\g(x,y)=0 V / g(x,y)=0 (4.1a) 9 u ,| g(*.y)=o *(*.y)= 0 » V-eJJ | s (x,y )=0 “ M y ^ (4.1b) 61 porous medium Tc^U,60 m%8m MSS Fig. 4.1 Schematic diagram of flow and heat transfer through a multiple porous block system. where g(x, y)=0 is the curve defining the porous/fluid interface. The derivative with respect to n and t represents the normal and tangential gradients, respectively, to the curve g(x, y)=0 at any point on the interface. In dimensionless form the above- mentioned equations governing the whole porous/fluid composite system can be written as (Vafai and Tien (1981)&(1982), Vafai (1984), Vafai (1986)): (4.2) dy* dx' dx' 3y* Ret V V* = -K (4.3) 3\|/* 30 3\)/* 30 _ f 1 _ " ^ “ V • vC7 (4.4) 3y* dx' dx* 3y* KPeL y where (x*, y*) are dimensionless rectangular Cartesian coordinates and V* is the stream function which is related to the fluid velocity components u and V by u = • (4.5 ) and ^ is the vorticity defined by 63 y _ 9l) dll (4.6) * ~ dx dy the non-dimensional parameters in the fluid region are ReL = i ^ , Pql = ^ , S' = 0 (4.7a) *7 a7 and in the porous region the non-dimensional parameters are Pe = !^ k Da = A = — (4.7b) L a eJf' L I} ' L K 1/2 . a|v*| . a|v*| S*= ------1) —' — U —! L Re^Da^ ^ * dx dy u d ( 1 1 u* d ( + • (4.8) l DaJ ReL dx' vDa, + Iv 1 " ‘^ ( a 0 - K K s - ( a . ) Where the source terms 5* can be considered as those contributing to the vorticity generation due to the presence of the porous medium. It should be noted that the effects of local non-thermal equilibrium and dispersion are neglected at this time based on the work of Vafai and Sozen (1990) and Sozen and Vafai (1990, 1993). The dimensionless boundary conditions are v* = y \ K = e = 0. at ** = 0 (4.9) dx 64 ay (4.10) (4.11) (4.12) All of the above variables have been nondimensionalized based on the following definitions: (4.13a) (4.13b) It should be noted that these conservation equations for forced convection in porous media are developed here using the local volume-averaging technique (Vafai and Tien (1981)). This is done by associating with every point in the porous media a small volume V bounded by a closed surface A. Let Vp be that portion of V containing the fluid. The local volume average of a quantity ®, such as u, "u.or T, associated with the fluid is then defined as in Vafai and Tien (1981): (4.14) 65 In addition, the conservation equations in porous media are also based on the generalized flow model, which take into account the effects of flow inertia as well as friction caused by macroscopic shear. In this study, the problem is governed by seven parameters, i.e., the geometric parameters A = W* / H* and B = D* /W *, Reynolds number, u„L / v, Darcy number, K/L2, inertia parameter, FLe/VK , the Prandtl number, v /a , and the number of blocks N. To evaluate the effects of the porous material on the shear stress and heat transfer rate at the wall, additional calculations were carried out. For the shear stress the results were cast in dimensionless form by means of the local friction coefficient as /=o (4.15) pul / 2 ReL dy" and for the heat transfer rate the results were represented in dimensionless form in terms of a dimensionless Nusselt number (4.16) kf kf dy /=o Note that the conductivity of the fluid was chosen in the formulation of the Nusselt number. This results in more meaningful comparisons for the heat flux at the external boundary between the composite system and the case where there was no porous substrate. 66 4.3 NUMERICAL METHOD AND PROCEDURE The fluid flow and heat transfer characteristics in the configuration of interest will be revealed after solving numerically the mathematical model outlined in the previous section. The following is a general formulation for the diffusion-convection equation, which can be applied to vorticity, stream function, and temperature equations. _(,*)+_ w .-^ r-j+ 5 4 r _ ' + (4.17) Here d> represents any one of the pertinent variables. Figures 4.2(a) and (b) show the grid system for the computational system and the local integration cell. Based on the non-uniform rectangular grid system of Fig. 4.2(a), the finite-difference form of equation (4.17) is derived by volume integration of the differential equation over discrete cells surrounding the grid points. Application of central differencing for the diffusive term and the upwind differencing for the convective term gives Q c£ = r, |^(l + [-r,&tA,°]) (4.19a) (4.19b) 67 N i Sy i i W w |( ij) Ay \e ~C A] ' 0 + 1 -j) Sx 8x w (b) Fig. 4.2 (a) Grid system for the computational domain, (b) Local integration cell. 68 cN = rn^-(\+ [-rnSynvn,o)) (4.19c) Ca« r ,|i( l + [r.8y.i>.,0]) (4.19d) b = S*AxAy (4.19e) and Cc — cE + cw + CN + c, (4.19f) Applying the above discretization procedure for vorticity, stream function, and temperature equations gives C £i.j = CE^ i+u + Cy£i- + CW$V i +C £ ij-1 + S' AxAy (4.20) (4.21) (4.22) Equations (4.12)-(4.22) give a system of linear equations for \|/\ and 0. These finite difference equations were solved by the extrapolated-Jacobi scheme - an iterative scheme based on a double cyclic routine -- which translates into a sweep of only half of the grid points at each iteration step (Adams and Ortega (1982)). It was necessary to use underrelaxation to prevent instability and divergence due to nonlinearity in these 69 finite difference equations. The numerical integration was performed until the following convergence criterion was satisfied: max <10- (4.23) K * * where Figure 4.2(a) depicts the non-uniform grid system for the computational domain in this problem. This non-uniform grid system possesses a very fine grid structure through the porous block array as well as its immediate surroundings, and gradually becomes coarser towards the far field. A grid independence test for ReL = 3 x 10s, Dat = 8 x KT6, AL = 0.35, Pr=0.7, A=6 , 5=1, and N=4 showed that there is only a very small difference (less than 1%) in the streamlines and isotherms among the solutions for 82 x 82, 144 x 162, and 172 x 202 grid distributions. As this difference is small, all of our computations were based on the 144x162 grid system. It was found that any further increase in the grid distribution resulted less than 1% difference for cases studied in this work. To accommodate the simultaneous solutions of the transport equation in both fluid and porous regions, the effective viscosity of the fluid-saturated porous medium is set to be equal to the viscosity of the fluid. It has been found that this approximation provides good agreement with experimental data (Lundgren (1972); Neale and Nader (1974)). Moreover, the dimensionless groups, AL and DaL at the interface of a control volume are computed by the harmonic mean. 70 The interface between the porous medium and fluid space requires special consideration. This is due to the sharp change of thermophysical properties, such as the permeability, porosity, and the thermal conductivity, across the interface. All of these effects on the porous/fluid interface are summarized in the dimensional parameters DaL, A l , and Pr. The harmonic mean formulation suggested by Patankar (1980) was used to handle these discontinuous characteristics in the porous/fluid interface. In our investigation the computational domain is chosen to be larger than the physical domain. Along the x direction, the computational domain starts at a distance of one-fifth of total length upstream of the physical domain. This procedure eliminates the errors associated with the singular point at the leading edge of the composite system. On the other side, the computational domain is extended over a distance of two-fifths of the total length downstream from the trailing edge of the physical domain. Since the present problem has a significant parabolic character, the downstream boundary condition on the computational domain does not have much influence on the physical domain. In the y-direction the length of the computational domain was systematically increased until the maximum vorticity changes for two consecutive runs became less than 1 percent. In order to examine the validity of the present numerical model, comparisons with more classical results were made. They were performed for laminar flow over a flat plate (i.e., //*= 0 , no porous substrate) and that over a flat plate embedded in a porous medium (i.e., //* —»°° and W‘— full porous medium). The results for //*=0 agree to better than 1 percent with boundary layer similarity solutions for velocity and temperature fields. The results for //*—>°° and W '—>°° agree extremely well with data 71 reported by Vafai and Thiyagaraja (1987) and Beckermann et al. (1987). These comparisons were found to be similar to those presented in Vafai and Kim (1990). 72 4.4 RESULTS AND DISCUSSION The effects of governing physical parameters, such as the Reynolds number, Darcy number, Prandtl number, and inertial parameter, as well as the goemetric arrangements of the porous block array on the flow and temperature fields were explored. To illustrate the results of the flow and temperature fields, only the portion which concentrates on the porous block array and its close vicinity is presented. However, always, the much larger domain was used for numerical calculations and interpretation of the results. The influences of the porous block array on the velocity field is depicted in Fig. 4.3(a) for a case where the Reynolds number is 3 x 10s, the Darcy number is 8 x 1CT6, the inertial number is 0.35, the dimensionless height and width of the porous block are 0.02 and 0.12, respectively, and the spacing between the porous blocks is 0.12. It can be seen that there are two distinct momentum boundary layers in the flow field. One is along the impermeable wall and the other is along the top of porous obstacles. Inside the porous media as the normal coordinate increases the velocity distribution increases from zero to a constant value, which is maintained until the outer boundary layer appears. The velocity distribution goes through a smooth transition once it crosses the porous/fluid interface, and finally approaches a free-stream value. As expected, both momentum boundary layers grow in the streamwise coordinate. Consequently, in the porous region the magnitude of the interfacial velocity between these two boundary layers decreases to adapt this growth, resulting in further increase in the thickness of the boundary layer near the wall. Inside of the inter-block spacing, the flow is 0.03 0.0*4. lL 33 0.03 :± 7 0.02 0.0 1 8845 0.0 x* (a) 0.03 0.03 •0.03* 0.02 0.15' ■0.003. 0.0 1 •0.006 .009' .0.0003. 0.0 0.0 o . s 1 . o X* (b) Fig. 4.3 (a) Velocity distribution, (b) streamlines, and (c) frictional coefficient for flow through four obstructing porous blocks for Re^ = 3 x 10s, DaL = 8 x 10-6, AL = 0.35, A=6, 5=1, H*= 0.02. U> 74 0.0030 ■without porous media •with porous media 0.0026 0.0021 0.0017 0.0012 0.0008 0.0003 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 (c) Fig. 4.3 (a) Velocity distribution, (b) streamlines, and (c) frictional coefficient for flow through four obstructing porous blocks for Rez. = 3xl05, Dajr =8x10^, At = 0.35, A=6 , 5=1, H*=0.02. 75 decelerated and is eventually forced to reverse its direction near the wall resulting in a boundary layer separation. This is due to the adverse pressure caused by the porous obstacle. The above-described flow field can also be observed by streamlines shown in Fig. 4.3(b). It can be observed that the streamlines move upward due to the presence of the porous blocks. This indicates that the flow is directed upwards periodically while encountering the porous blocks. This effect is more pronounced for smaller Darcy numbers or larger inertial parameters. This phenomena is due to the relatively larger resistance that the flow encounters inside the porous blocks, which in turn displaces the fluid by blowing it from the porous region into the fluid region. These results were also found in the work of Vafai and Kim (1990). The flow patterns, including the shape of boundary layer and the blowing effect from porous block, play an important role in affecting the temperature distribution. Figure 4.3(c) displays the periodical streamwise variation of the local friction coefficient at the wall. It can be seen that the friction coefficient decreases with an increase of the boundary layer thickness and increases with a decrease in the boundary layer thickness. This is the reason for the fluctuation of the local friction coefficient along the external boundary. It can be seen that inside porous medium the friction at the wall decreases. This decrease is a direct result of the blowing effect, as discussed previously. However, in the space between the porous blocks the adverse pressure caused by porous obstacle reduces the fluid momentum resulting in a rapid increase in the boundary layer thickness. Consequently, the friction coefficient significandy decreases in the inter-block spacings. 0.02 -0.1* 0.01 •0.003' >0.006 3.009- o . o a T T .0.03. 0.02 •0.1J- 0.01 .0.003- -0.006. >.009* T Dat = 4 x 10' 0.03 -0.1J- 0.01 •0.003' .0.006 .0.0003. 0.0 0.0 0.2 0.0 o . a 1 . o Fig. 4.4 Effects of the Darcy number on streamlines for flow through four obstructing porous blocks for Ret = 3 x10s, AL = 0.35, A=6, 5=1, #*=0.02. CT\ Da, =2x10' 0.03 0.03* 0.0 0.03 DaL =1.75x10' 0.03 .0.03* .0.1! 0.0 1 0.0 0.03 0.03 0.03- .0.003' 0.0 “X 0.0 0 .2 0.0 o . a i . o X* Fig. 4.4 Effects of the Darcy number on streamlines for flow through four obstructing porous blocks for Rql = 3 x 105, AL = 0.35, 4=6, 5=1, H*=0.Q2. -j o . o a Da, =2x10* 0.03 0.01 0.0 0.03 T TT DaL = 1.75x10' 0.03 0.02 0.01 '0.95- Da, =1.5x10' 0.03 0.0 1 •0.9J- 0 .0 0 .0 o . e o . a 1 .o X* Fig. 4.5 Effects of the Darcy number on isotherms for flow through four obstructing porous blocks for ReL = 3xlOs, AL =0.35, Pr=0.7, kejr/ k , = 1.0, A =6, B = 1, H*=0.02. 00 Da, xlQ' 0.02 0.01 .0.01. 0 .0 0.02 0 .0 -*. Da, =8x10" 0.02 0.02 0.01 0.0 0 .0 2 DaL = 4x10* .0.02 0.02 0.0 1 .0.01. 0 .0 0.0 0 .2 1 .o Fig. 4.5 Effects of the Darcy number on isotherms for flow through four obstructing porous blocks for Re^ = 3x10s, AL = 0.35, Pr=0.7, keff / k f = 1.0. A=6, 5 = 1, H*= 0.02. VO 80 200.000 0 :Daj (without porous media) 180.000 l:D aj,= 8 xlOM 2: Dat = 8 x 10 "4 3: Dat = 4 x 10"* 160.000 4:DaJrj = 2 x l0 '< 5 :Dat = 1.75x10' 6 : Da, =1.5x10^ 140.000 120.000 — —1 jvju 100.000 80.000 60.000 40.000 20.000 0.000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 X* Fig. 4.6 Effects of the Darcy number on the Nusselt number for flow through four obstructing porous blocks for Ret = 3xl05, AL=0.35, Pr=0.7, k eff / k f = 1.0, A=6, 5=1, H*=0.02. 81 Effect of the Darcy Number The Darcy number is directly related to the permeability of the porous medium. To investigate the effect of Darcy number on the flow and temperature fields, computations were carried out at Dat = 8 x KT4, 8 x KT6, 4 x KT6, 2 x KT6, 1.75x10"*, and 1.5 x 10"* for HIL=0.02, A =6 , B = 1, Re*, = 3 x 105and Al =0.35. Results of the computations for streamlines, isotherms, and local Nusselt numbers are presented in Figs. 4.4-4.6. It can be seen, from Fig. 4.4, that by decreasing the Darcy number up to a certain value (less than 4 x 10-6) the boundary layer starts to separate from a certain location at the wall, and then forms a separation region along the streamwise direction near the wall. The size of separation region increases in the inter-block spacing due to the adverse pressure gradient caused by porous obstacles, and decreases inside of the porous media due to the bulk frictional resistance offered by porous matrix. The appearance of vortex zone depends on whether the flow reattaches the wall or not. Outside of the boundary layer flow, the blowing effect caused by porous media increases as the Darcy number decreases, which leads to the larger upward moving of the streamlines and the reducing of mass flow rate passing through the porous media. The closely spaced streamlines indicate that there are larger volume flow rates. The temperature fields shown in Fig. 4.5 correspond to a case where the Prandtl number is 0.7 and the conductivity of the porous media is equal to that of the fluid. Smaller value of Da L translates into larger blowing effects which diverts the flow through the porous medium. Therefore, as the Darcy number decreases the thickness of thermal boundary layer increases and the distortion of isotherm near the wall becomes more noticeable. The increase in the distortion of the isotherms at lower values of DaL is due to the presence of a larger 82 fluctuating separation zone near the wall. The fluctuations in the momentum and thermal boundary layers also create the oscillations in the local Nusselt number distribution as depicted in Fig. 4.6. It is seen that the Darcy number has a significant impact on the local Nusselt number distribution. The local Nusselt number fluctuates periodically as the streamwise coordinate increases due to the presence of the porous blocks, with an increasing mean at the higher Darcy numbers (higher than 2x 10-6) and a decreasing mean at the lower Darcy numbers. Moreover, the extent of the fluctuation increases for lower values of Darcy numbers. The trough in each cycle occurs at an x * value corresponding to the left face of each porous block. This is due to the slow circulating motion inside the separating zones in front of the porous obstacles, resulting in low heat transfer rates in these regions. In particular, for the lower Darcy number heat transfer in circulating regions decreases markedly due to the larger size of separation zone. Inertial Effects When the Reynolds number based on the pore diameter becomes large the inertial effects become significant. The effects of an increase or decrease in the inertial parameter are shown in Figs. 4.7-4.9 for Ret = 3x 10s, Da£ = 8 x 10-6, Pr=0.7, A = 6 , and B =1 at AL=0.35, 3.5, 5, and 7, respectively. Comparison of the streamlines in Fig. 4.7 shows that as the inertial parameter increases, the distortion of streamlines becomes more significant and the size of the vortices near the wall increases. This is because of the larger bulk frictional resistance that the flow experiences for larger inertial parameters. Therefore, larger values of AL would lead to a larger blowing effect thus reducing the mass flow rate through the porous blocks. In At = 0.35 •0.15* •0X0* 0 .0 3 At =3.5 0.0-4. .0.03. •0.13- o . o n 0.0CCO9*~ y+ 0 . 0 3 TTT A, = 5 .0.00* ‘ 0.02 .0.0003- o . o s T A t = 7 0 .0 1 — 0.2 0.00.2 0 .0 0.0 1 .o Fig. 4.7 Effects of the inertial parameter on streamlines for flow through four obstructing porous blocks for ReL = 3 xlO5, DaL = 8x 10”6, A=6, B=\, H =0.02. 00 .0.01. 0.0 o . o s 0.02 0.01 0.0 o . o s TT TT 0.02 0.01 o r o s o . o s 1 >-0.2 0.0 0.2 O.*- 0.0 0.0 1.0 Fig. 4.8 Effects of the inertial parameter on isotherms for flow through four obstructing porous blocks for ReL = 3 x l0 5, DaL = 8xl(T6, Pr=0.7, ktjr/ k f = 1 0 . A =6, 5= 1, H*=0.02. 00 200.0 i i i i i i i i 0: Al = 0 (without porous media) 1: = 0.35 180.0 2: At = 3.5 3: At = 5 160.0 4:At =7 o-' 140.0 120.0 Nu 100.0 / 80.0 / s ' 60.0 // . . . . /S / ' - S U/-///• - -v s \ / \ 40.0 i f x s C/ \ - ^ 3-. 20.0 { ^ 0.0 I I I I I I I I L . 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 X* Fig. 4.9 Effects of the inertial parameter on the Nusselt number for flow through four obstructing porous blocks for R tL = 3 x 10s, DaL = 8 x 1CT6, Pr=0.7, k e]f / k f = 1 .0 ,4 =6, 5=1, H*= 0.02. 86 addition, the larger bulk frictional resistance caused by the porous matrix results in a larger adverse pressure gradient in front of the porous obstacle. As a result, the flow near the wall is decelerated more quickly and eventually forced to reverse its direction forming a larger separation zone. When entering the porous block, the laminar eddies in the separation zone are weakened by the frictional resistance of the porous matrix. This results in the reattachment of the boundary layer to the wall, and formation of a closed vortex region. Comparison of the isotherms in Fig. 4.8 shows that the larger the value of A L, the larger the thickness of the thermal boundary layer, and the more noticeable the distortion of the isotherms. The reason for this behavior is similar to the effect of Darcy number on the temperature field. As in the case of Darcy number variations, a periodic variation of local Nusselt number is observed. As can be seen in Fig. 4.9 for smaller inertial parameters the mean value of the Nusselt number increases while for larger inertial parameters the mean value of the Nusselt number decreases with the flow direction. This shows that the inertial parameter has a significant effect on the formation and the size of the separation vortex and the convective energy transport. Effect of Reynolds Number Figures 4.10-4.12 show streamlines, isotherms, and Nusselt number distributions for Da£ = 8xl0"6, AL=0.35, Pr=0.7, A=6, and B = 1 with Re£ = 3 x 105, 2 x 105, 1.5 x 10s, and 1 x 10 55 respectively. As expected, distortions in the streamlines and boundary layer thickness become evident as the Reynolds number decreases (see Fig. 4.10). The reason for this trend is that the lower the Reynolds number, the lower the flow inertia, thus reducing the extent of penetration of the flow Ret = 3xlOs .... 0.03— - -O .l! ------0.003 — | — o.oo< ------j ------0.009 ------i OQ003 Re, =2x10 0.0003 Re, =1. o.± Fig. 4.10 Effects of the Reynolds number on streamlines for flow through four obstructing porous blocks for DaL = 8 x 10^, AL = 0.35, A=6, 5=1, H =0.02. OO R e^SxlO3 Re, =2x10 ReL = 1.5x10 O.OS Ret = 1 x 10; O.OS 0.02 •0.1 ' 0.0 0.2 0.0 0.2 O.S O.S 1 Fig. 4.11 Effects of the Reynolds number on isotherms for flow through four obstructing porous blocks for Dai = 8 x l0 -6, Ai =0.35, Pr=0.7, /k y = 1.0. A =6, 5 = 1, H*=0.02. 89 200.0 — without porous media — with porous media 180.0 1: Ret = 3 x 10s 2:ReL=2xl0s 160.0 3 :Ret = 1.5 x10s 4:R et = lxlOs 140.0 120.0 Nu 100.0 80.0 60.0 40.0 20.0 0.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2* Fig. 4.12 Effects of the Reynolds number on the Nusselt number for flow through four obstructing porous blocks for DaL = 8xl0“*, AL = 0.35, Pr=0.7, keJf / k , = 1.0, A=6, 5=1, H*=0.02. 90 into the porous blocks. This results in a larger blowing effect in the flow field, and a larger thickness of boundary layer near the wall. Comparison of the temperature fields corresponding to different Reynolds numbers (Fig. 4.11) clearly shows that for the lower Reynolds number the heat is transfered further out into the flow field. The variation of the local Nusselt number corresponding to Fig. 4.11 is shown in Fig. 4.12. with a different increasing mean for different Reynolds number. As expected, the heat transfer rate from the wall decreases with a decrease in the Reynolds number. This obviously is due to lower velocities near the wall for smaller Reynolds numbers. It is this decrease in the transfer of convective energy that causes a lower temperature gradient and heat flux at the wall. As with the other two cases a periodic variation of local Nusselt number is again observed. Prandtl Number Effects To investigate the effect of the Prandtl number on the flow and temperature fields, three different Prandtl numbers were chosen such that they will cover a wide range of thermophysical fluid properties. The numerical results are presented in Figs. 4.13 and 4.14 at ReL = 3 x l0 5, DaL = 8x KT6, A^O.35, A=6, andB=l for three different fluids with Pr=0.7 (air), Pr=7 (water), and Pr=100 (a representative oil), respectively. Obviously, the Prandtl number variations have no effect on the flow field and the velocity distribution will be exactly the same as that shown in Fig. 4.3(a). As seen in Fig. 4.13, due to the lower value of the thermal diffusivity the temperature gradient is larger for larger Prandtl numbers. As expected, the local Nusselt number and its fluctuations increases with an increase in the Prandtl number (Fig. 4.14). o.os Pr=0.7 0.02 0.0 1 .0 .01. 0.0 o.os T Pr=7 0.02 0.0 1 .0 .01, T Pr=100 0.03 0.02 0.01 0.01 0.0 —0.2 0.0 0.2 0.4. 0.6 O.S 1 .O X* Fig. 4.13 Effects of the Prandtl number on isotherms for flow through four obstructing porous blocks for ReL = 3 x l0 5, DaL = 8xlO-6, AL = 0.35, / k f = 1.0* ^=6* 5=1, #*=0.02. VO 900.0 — without porous media — with porous media 800.0 Pr=100 700.0 600.0 500.0 Nu 400.0 300.0 200.0 .Pr=0.7 100.0 0.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Fig. 4.14 Effects of the Prandtl number on the Nusselt number for flow through four obstructing porous blocks for, ReL = 3 x 10s, DaL = 8 x 10-6, At =0.35, kt]f /kf- 1.0, A=6, B=1, //*=0.02. 93 Effect of the First Geometric Parameter A The effect of the aspect ratio (the first geometric parameter) A=W*IH* on the flow and temperature fields were studied for Ret = 3 x10s, DaL = 2 x 1CT6, At =0.35, Pr=0.7, A-6 , and 5=1. The streamlines and isotherms for aspect ratios of A=3 and 6 are presented in Figs. 4.15 and 4.16, respectively. As seen in Figs. 4.15 and 4.16, when the value of A was decreased from 6 and 3, the distortions of streamlines and isotherms becomes less pronounced, and there are no closed vortex zones along the wall. This is the direct result of reduction in the aspect ratio which lessens the blowing action and subsequent deceleration of the flow field. Effect of the second Geometric Parameter B. The second geometric parameter B=D*IW* reflects the influence of the inter-block spacing. There are two configurations investigated in this analysis. Figures. 4.15 and 4.16 show the streamlines and isotherms for ReL = 3xl05, Da^, = 2 x 10"*, Al =0.35, Pr=0.7, A=6 for cases 5=1 and 0.5. Comparison of Figs. 4.15 and 4.16 shows that as the inter-block spacing decreases, streamline and isotherm distortions become more pronounced. However, for lower values of 5, the interblock separation zones along the wall is reduced while a larger separation zone forms behind the last block. This is because the smaller inter-block spacing weakens deceleration action on the boundary layer thus reducing the extent of the separation zone within that space. Figure 4.17 illustrates the results for the flow over five porous block array with A=6 and 5=0.5 for ReL = 3 xlO5, DaL = 2 x 10"*, AL=0.35, Pr=0.7. As can be seen the increase in the number of blocks does not alter the main features of the flow and temperature fields. (a) A=6 and 5=1 0.02 ■0.1S- 1.006* 0.0 1 0.0 o . o s T 43.03 y* -0.13 .0.003- 0 .0 1 >.009- .0.0003. 0.0 o . o s (c) A=6 and 5=0.5 0.03 ——0.03' 0.02 .0.15. 1.003' 0.0 1 .0.0003. •0J0006* 0.0 — 0.2 0.0 0.2 o . e o .s 1 .o X* Fig. 4.15 Streamlines for flow through four obstructing porous blocks for ReL = 3 xlO5, Dat = 2 xKT6, Al = 0.35, / k , = 1.0, Pr=0.7, tf*=0.02. VO (a) A=6 and B=l o.oo 0.01 0.0 0.05 0.02 0.0 1 •0 .6* • 0.0 0.05 I TT (c) A=6 and £=0.5 0.0 1 0.0 0.2 0.0 o . a 1 .o 4.16 Isotherms for flow through four obstructing porous blocks for ReL = 3 x = 2xl(T 6, Al =0.35, k^/kr = 1.0, Pr=0.7, H*=0.02. o . o s o . o s .0.03- .0.003' 0.0 1 0.0 0.0 0.2 . 0.6 • O .S 41 .o o . o s 0.03 0.02 0.0 1 0.0 0.0 0.2 0.6 O.S 1 .O X* Fig. 4.17 (a) Streamlines and (b) isotherms for flow through five obstructing porous blocks for ReL = 3x10s, Dai = 2xlO -6, At = 0.35, Pr=0.7, k tjr / k f = 1.0, A=6, 5=1, H*= 0.02. vo Ov CHAPTER V AN INVESTIGATION OF FORCED CONVECTION THROUGH ALTERNATE POROUS CAVITY-BLOCK OBSTACLES 5.1 STATEMENT OF THE PROBLEM Forced convection over external boundaries in the presence of a porous medium has constituted an important area of research for the past several decades. This is due to the very fundamental and generic nature of this type of problem which makes it pertinent to a wide variety of applications, including drying processes, heat pipes, filtration, direct contact heat exchangers, electronic cooling, thermal insulation, etc. Therefore, it is important, from both applied and basic points of view, to investigate the transport processes in the porous medium. Combamous and Bories (1975) and Cheng (1987) provide some well-thought views on heat transfer in fluid-saturated porous medium. Based on Darcy's law, Cheng (1977) and Bejan (1984) documented the local Nusselt number for forced convection over a semi-infinite flat plate embedded in a porous medium with constant temperature and heat flux. 97 98 Vafai and Tien (1981) analyzed the effects of a solid boundary and inertial force on flow over an external boundary after establishing the governing equations by a local volume-averaging technique. They showed that for the flow field the boundary effect is confined within a thin momentum boundary layer which often plays an insignificant role in the overall flow consideration, but when the thermal boundary layer's thickness is less than or of the same order as that of the momentum boundary layer, the effect of boundary on the heat transfer is more pronounced. Kaviany (1987) obtained Karman-Pohlhausen solutions for the same flow configuration on the basis of the Brinkman-Forschheimer-extended equation. An important problem related to forced convection through a porous medium is flow and heat transfer in composite systems. This involves the study of fluid flow above and through a porous medium. The flow over the fluid region is governed by the Navier-Stokes equation and the flow through the porous medium is governed by the generalized momentum equation which includes the effects of flow inertia as well as friction caused by macroscopic shear stress (Vafai and Tien (1981)). These two flows are coupled through the interface boundary conditions at the porous/fluid interface. The interactions of flow and temperature fields between the porous- saturated region and the fluid region have a significant influence on the convection phenomenon in these systems. This type of basic composite system is of importance in various applications such as crude oil extraction, solidification of castings, geothermal operations, nuclear waste repositories, thermal insulation, etc. Several investigations were devoted to the problem of finding the proper set of boundary conditions at the interface between a 99 fluid flow in a porous medium and the adjacent region without a porous medium. Beavers and Joseph (1967) experimentally reported the mass efflux of a poiseuille flow over a naturally permeable surface based on Darcy's law. They found that when a viscous fluid passes over a porous solid, tangential stress entrains the fluid below the interface with a velocity which is slightly greater than that of the fluid in the bulk of the porous medium. Levy and Sanchez-Palencia (1975) found that when the typical length scale of the external flow is large compared with the microscopic scale, the velocity field transition at the interface from the porous media to the free fluid region occurs over a thin region of the order of the pore scale. They also showed that depending on the direction of the pressure gradient in the porous medium two different kinds of phenomena may appear at the interface. More relevant to the present study is the work of Vafai and Thiyagaraja (1987) which was based on the Brinkman-Forschheimer-extended Darcy model. They performed a theoretical analysis for a general class of problems involving interface interactions on the flow and temperature fields for three basic types of interface composites. Vafai and Kim (1990) studied forced convection over an external boundary with a porous substrate. They found that the porous substrate causes a blowing effect on the flow field. They also found that the porous substrate significantly reduces the frictional drag and it can either enhance or reduce the heat transfer at the wall. Another related problem is the recirculating flow created by boundary layer separation due to an abrupt change in body geometry. Heat transfer in such flows without any porous medium has been reviewed by Fletcher et al. (1974) and Aung and Watkins (1979). 100 Analysis of external forced convection in a porous/fluid composite system is significantly more complicated due to the complex geometric configuration of these types of systems. This work presents a numerical study of forced convection over a composite system, which is composed of alternating porous cavity and block regions separated by the exposed areas of the plate. Since little attention has been focused on external forced convection fluid flow and heat transfer in the porous/fluid composite system, the major goal of the present study is to investigate fundamental changes in the flow and temperature fields owing to the existence of cavity-block porous obstacles and the interactions between the blowing and displacement effects from the porous blocks and the vortices penetrating into the porous cavities. Futhermore, the influence of the governing physical parameters is also thoroughly analyzed and it is shown that altering some parametric values can have significant effects on the external momentum and thermal boundary layer characteristics. 101 5.2 ANALYSIS The configuration for the problem under investigation is depicted in Fig. 5.1. The width and height of the rectangular porous cavities and blocks are H and W, respectively, the distance between any given cavity and a block is D, the length of the external boundary L, the stream velocity and the free stream temperature is TM. The external boundary is maintained at a constant temperature Tw. It is assumed that the flow is steady, laminar, incompressible, and two-dimensional. In addition, the thermophysical properties of the fluid and the matrix are assumed to be constant and the porous medium is considered homogeneous, isotropic, nondeformable, and in local thermodynamic equilibrium with the fluid. The governing conservation equations for the present problem will be separately written for the porous and fluid regions. Treating the solid matrix and the fluid as a continuum, the local volume averages of the conservation equations for mass, momentum, and energy in the porous region, which account for the effects of the inertial and impermeable boundary, are (Vafai and Tien (1981)&(1982), Vafai (1984), Vafai (1986)) V•(v)= 0 (5.1) ({v) • v(r)) = «,j v 2(7') (5.3) where { } denotes the local volume average of a quantity, v represents the velocity vector, P/ the fluid density, s the porosity, K the permeability, J a unit vector 5888 Porous medium ^77777777, %wXw5 Fig. 5.1 Schematic diagram of flow and heat transfer through alternate porous cavity-block obstacles. oriented along the velocity vector, (P)f the intrinsic phase average of pressure, P-f the fluid viscosity, and F is an empirical function which depends primarily on the microstructure of the porous medium. It should be noted that the effects of local non- thermal equilibrium and dispersion are neglected at this time based on the work of Vafai and Sozen (1990) and Sozen and Vafai (1990, 1993). The conservation equations for mass, momentum, and energy in the fluid region are V- v = 0 (5-4) v • Vv = — — VP + v, V2v (5.5) Pf v-Vr = a / V2r (5.6) The associated boundary conditions necessary to complete the formulation of the problem are a = v = 0, P = Pm, T = 7L at *=0 (5.7) it = (u) = 0, v = (v) = 0, T - (T) = Tw on the solid wall (5.8) u = u ^ ,P = P ^ T = Tm asy —> oo (5.9) In addition these two sets of conservation equations are coupled by the following matching conditions at the porous/fluid interface, which satisfy the continuity of the velocity, pressure, stress, temperature, and heat flux across the interface, where g(x, y)=0 is the curve defining the porous/fluid interface. The derivative with respect to n and t represents the normal and tangential gradients, respectively, to the curve g(x, y)=0 at any point on the interface. To accommodate the solution of the transport equation in both fluid and porous regions, the effective viscosity of the fluid-saturated porous medium is set equal to the fluid viscosity. It has been found that this approximation provides a good agreement with experimental data (Lundgren (1972); Neale and Nader (1984)). Introducing the stream function and vorticity as " = = W = (5-ID £ = = 3(}>)_3(m) (5 12) dx dy dx dy these two sets of conservation equations are transformed into one set of dimensionless stream function-vorticity formulation, which is valid throughout the composite system, 105 3\)/* 39 3vj/* 39 _ V 7 -V9 (5.15) 3 / 3 7 “ a T 3 7 " vPet the non-dimensional parameters in the fluid region are ReL= ^ , PeL= ^ , S* =0 (5.16a) a , and in the porous region the non-dimensional parameters are umL K F L e - : > Dat ~ j2< ^ L ~ j^l/2 (5.16b) a_’"tjf and 1 . 3|v‘| , 3|v* S* = v — u (5.17) Re^Da^ 3 / 3 / Here 5* can be considered as the term which contributes to the vorticity generation due to the presence of the porous medium. In addition, the dimensionless boundary conditions are * * £* 3 \J/ f\ * n V = y - S = — 0 = 0, at * = 0 (5.18) dx V * =0, A t*S = - 3 - \j/^ r , 9 = 11, dx . K + 2(N-1)(W* + D‘) . at* =1 . . . . . 0>y >-// (5.19) + 2 (N -1 )(W + D ) and 106 dy 'i\ + 2(N - 1)(W‘ + D*) < x' < t\ + W* + 2(N - 1)(W* + D*), y* = -H‘ 0 < x ‘ < i\ i\ + W* + 2(N - 1)(W* + £>*) (1-£'2) (5.21) where N (=1, 2,...) is the number of the porous cavity-block composites. The dimensionless variables in the above equations are defined as follows: (5.22a) (5.22b) (5.22c) Based on the above equations, boundary conditions, and geometric arrangement of porous cavities and blocks, it is seen that the present problem is governed by seven dimensionless parameters. These are the Darcy number, Reynolds number, two Prandtl numbers, inertia parameter, geometric parameters A and B, and the number of cavities N, where 107 (5.23) H W 108 5.3 NUMERICAL SCHEME To obtain the solution of the foregoing system of equations, the region of interest is overlaid with a variable grid system as shown in Fig. 5.2(a). Applying the central differencing for the diffusion terms and the second upwind differencing for the convective terms, the finite-difference form of the vorticity transport, stream function, and energy equations were derived by control-volume integration of these differential equations over discrete cells surrounding grid points, as shown in Fig. 5.2(b). This results a system of equations of the following form: Cc Oc = CnO n + Cs Os + Ce O e + CWQ>W + S* (5.24) where O stands for the transport variables, C's are coefficients combining convective and diffusive terms, and S 0 is the appropriate source term, which is listed in Table5.1. The subscripts on C denote the main grid points surrounded by the four neighboring points denoted as N, S, E and W. The finite difference equations for £*, \j/*, and 0 obtained in this manner were solved by the extrapolated-Jacobi scheme. This iterative scheme is based on a double cyclic routine, which translates into a sweep of only half of the grid points at each iteration step (Adams and Ortega (1984)). The numerical procedure for solving the finite-difference equations is as follows: 1. Overlay the computational domain with a variable mesh. 2. Assign values of Rai? Dai; AL,A,B, N, and initial values for £*, \(/*, u ,v , and 0 in Eqs. (5.13)-(5.15), and boundary conditions. (a) N i i (ij+D n 5y, i , ni w W j(’ij) Ay 1 0 E i 1 C I i\, i-1-j) | (i+1 j) c ul 1 5y s (U-1) ' S ■M.------—- ► 5x Sx w (b) Fig. 5.2 (a) Grid system for the computational domain, (b) Local integration cell. Table 5.1 Coefficients of Equation (5.24) V Ay Sx, Ay i s | s -(u [-(P. j . 5i.„.,oJ) 5x„ Ax (P^r(i+HP'O.Sy.«..o]) sy. Ax jP^ ( i +[-(p,1),6y...„o]) Sy, Cc CE + Cw + CN + Cs + ^ ^ + AL|vc|AxAy CE + CW+CN + CS CE + CW+CN +CS f O Ill 3 Calculate the new values of vorticity at each node by using the finite difference set of equations for as given by Eq. (5.24). 4. Calculate the new values of stream function \y* at each node from Eq. (5.24) for y * by using the values of E,* found from step 3. 5. Calculate the new values of the velocity from w=\(/* and u=-\|/*. 6. Update new boundary values using the new nodal values for \)/* and £*. 7. Repeat steps 3 to 6, until the following convergence criterion is satisfied C ' - C i max <10-* (5.25) where 9 stands for £,*, orQ and n denotes the iteration number. 8. Calculate the temperature 0 by using the finite-difference set of equations for 0 as given by Eq. (5.24) with the assigned values of Pr and the values of \|/* obtained from step 7, until the convergence criterion for 0 is satisfied. The treatment of the. vorticity at sharp corners requires careful consideration. Several methods of handling this corner vorticity are discussed in Roache (1976). Here to model properly the mathematical limit of a sharp corner, the method of average vorticity values is used (Greenspan (1969)). The bifurcation of the vorticity at the comers is essentially handled through the introduction of two different vorticity values. These are and £*, where both are evaluated by using the no-slip wall equation, but is evaluated by considering the comer being part of the horizontal 112 wall while is evaluated by considering it to be part of the vertical section of the cavity. Then a single comer vorticity equal to the average of two wall values is obtained. The interfacial properties play important roles on the porous/fluid system. This is due to the discontinuity of thermophysical properties, such as the permeability, porosity, and thermal conductivity, across the interface. All of these effects on the porous/fluid interface are summarized in the nondimensional parameters Da,, A, and Pr. In order to ensure the continuity of the convective and diffusive fluxes across the interface, the harmonic mean formulation (Patankar (1980)) was used to deal with these discontinuities. For the present case Da,, AL and Pr at the interface of a control volume are as follows: 2Da, Da, 2A, A, 2Pr„Pr, Da = ____ =2 a = Pr - eff f *L> D a, +Da, L‘ A, +A, ^ " f t . + Pr, (5>26) Lir Lt . Ar Lf . <# I where the subscripts eff, f and / stand for effective, fluid, and interfacial, respectively. Therefore, instead of the source terms in Eqs. (5.16) and (5.17), the following source terms were used across the interface: * u d u a f 5*=- f 1 ^ Re, dy l DaJ Re, 113 5 u d f 1 ] v* d ( 1 ' + - — t t t ------— t t - — ReL dy I^Da^ J Ret dx l^DaL , (5.28) where equation (5.27) was used for the fluid and equation (5.28) was used for the porous region. Note that constant values of DaL, and AL were used for a specified porous substrate. A nonuniform grid system with a large concentration of nodes in regions of steep gradients, such as the wall, corners and blocks, was employed. Figure 5.2(a) depicts the nonuniform grid system for the computational domain. A very careful analysis was made to ensure grid-independence and the upper boundary was systematically increased until it would have no detectable effect on the results. Three sets of grid systems, 162x136, 162x195, and 202x 292 were investigated in this work. It was found that for the most extreme cases there was only less than 1% difference in the values of the streamlines and isotherms between the 162x195, and 202 x 292 grid systems. Therefore, a 162x195 grid system was adopted for the present work. 114 In this study the computational domain was always chosen to be larger than the physical domain. Along the x direction, the computational domain starts at a distance one-fifth the total length upstream of the physical domain. This procedure eliminates the errors associated with the singular point at the leading edge of the composite system. On the other side, the computational domain is extended over a distance two- fifths the total length downstream from the trailing edge of the physical domain. Since the present problem has a significant parabolic character, the downstream boundary condition on the computational domain does not have much influence on the physical domain. In the y-direction the computational domain is extended up to a distance sufficient enough to ensure that even for the smallest value of the Reynolds number the upper boundary lies well outside the boundary layer through the entire domain. In the present study, locating the upper boundary at a distance of eight times the depth of the cavity has been found to be sufficient. Extensions beyond eight times the depth of the cavity had no effect on the solution. To further validate the numerical scheme used in the present study, initial calculations were performed for laminar flow over a flat plate (i.e., H*= 0, for no porous substrate) and that over a flat plate embedded in a porous medium (i.e., H*—>°° and W '— representing the full porous medium case). The results for H =0 agree to better than one percent with boundary layer similarity solutions for velocity and temperature fields. The results for and agree extremely well with data reported by Vafai and Thiyagaraja (1987). These comparisons were found to be similar to those presented in Vafai and Kim (1990). 115 5.4 RESULTS AND DISCUSSION The dimensionless parameters that need to be specified for this system are ReL, Dat , A l , Pr, A, B, and N. Since these seven basic dimensionless parameters are required to characterize the system, a comprehensive analysis of various combinations of these parameters were done. The results given in this work present only a small fraction of the cases which were investigated. The displayed results were chosen to represent the most pertinent effects of these parameters. In addition, to better illustrate the flow and temperature fields, only the portion which concentrates on the porous/fluid region and its close vicinity is presented. However, the much larger domain was always used for numerical calculations and interpretation of the results. The influence of the cavity-block structure on the flow field is depicted in Fig. 5.3(a) for a case where the Reynolds number is 3 x 10s, the Darcy number is 8 x 1CF6 > the inertial number is 0.35, the dimensionless height and width of the porous cavity or the block are 0.02 and 0.06, respectively, and the spacing between the porous cavities and the blocks is 0.06. It can be seen that the streamlines are considerably distorted due to the presence of the porous cavity-block structure. The streamlines move upward while piercing into the porous block, and become sparser after passing through it. Physically, this is due to the relatively larger resistance that the flow encounters in the porous block, which in turn displaces the fluid by blowing it from the porous region into the fluid region and reduces mass flow rate through the porous -0 x 10 ■ - 2*10 - 1x 10 Li iii i I i 0.20 0.40 0.60 0.80 1 .O Fig. 5.3 (a) Streamlines, (b) velocity distribution, and (c) isotherms for flow through alternate porous cavity-block obstacles for Ret = 3 x 104, Da^ = 8 x 10-6, AL = 0.35, A=3, B=l, H*-0.02. 117 blocks. This effect is more pronounced for the cases with smaller Darcy number or larger inertial number. Figure 5.3(a) also displays a laminar vortex contained within each cavity. The intensity of the eddies within each cavity decreases along the flow direction. These vortices are formed as a result of the entraped flow striking on the downstream cavity wall and then flowing toward the bottom surface. Due to an increase in the thickness of the external boundary layer along the plate, the mass flow rate penetrating into the subsequent cavity decreases, which in turn reduces the intensity of vortex in the cavities along the flow direction. The small fluctuations of streamlines close to the inlet of porous cavities is due to the macroscopic shear frictional resistance at the porous/fluid interface. The above-described flow field can also be observed by the velocity field shown in Fig. 5.3(b). It can be seen that after the primary flow field encounters the porous blocks, two relatively distinct momentum boundary layers are formed. One layer is along the impermeable wall and the other is on the top of the porous obstacles. Inside of the porous media, as the normal coordinate increases the velocity distribution increases from zero to a constant value, which is maintained until the outer boundary layer appears. The velocity distribution goes through a smooth transition once it crosses the porous/fluid interface, and finally approaches a free-stream value. As expected, both momentum boundary layers grow in the streamwise coordinate. Consequently, in the porous region the magnitude of the interfacial velocity decreases to adapt this growth, resulting in a further increase in the thickness of the boundary layer near the wall. These results are consistent with that found in the work of Vafai 118 and Kim (1990). The flow patterns, including the shape of boundary layer, the blowing effect, and the interaction between the recirculating flow inside the cavity and the external flow play a significant role in affecting the temperature field. Figure 5.3(c) shows isotherms corresponding to the flow field shown in Figs. 5.3(a) and 5.3(b). There is only one thermal boundary layer and there is little distortion of the temperature field over the flat wall. Inside of the cavities, spacing between isotherms close to the left wall is larger than that close to the right wall. This is due to the larger convective energy transport over the downstream wall of the cavity as compared to the one upstream. Effect of the Darcy Number The Darcy number is directly related to the permeability of the porous medium. To investigate the effect of Darcy number on flow and temperature fields, computations were carried out at Da^ = 8 x 10“*> 4 x 10~7> 2 x 10-7> and 1 x 10"7 for H/L=0.02, A=3, 5 = 1 ,Re^ = 3 x 105and A L=0.35. Results of the computations for streamlines and isotherms are presented in Figs. 5.4 and 5.5. The flow fields displayed in Fig. 5.4(a)-(d) reveal that as the Darcy number decreases, the interaction between the vortex inside the cavity and the external flow becomes more pronounced. This is due to the larger bulk frictional resistance to the flow within the porous block for the smaller Darcy number, which in turn results in a larger adverse pressure gradient for the porous obstacle. Consequently, the flow rate near the wall is decreased due to the viscous shear stress and the adverse pressure gradient. This results in the formation of a separation zone before the porous block. 0.08 0.07 -0.03- 0.01 0.01 *DaL = 8x10^ 0.0 -4*10 -T«ClO -*■» 0.20 0.40 0.60 0.80 1.0 Fig. 5.4 Effects of the Darcy number on streamlines for flow through alternate porous cavity-block obstacle for Re, = 3 x 10s, A, = 0.35, A=3, 5=1, H*= 0.02. ~ H-A VO 120 For the case of lower Darcy numbers, the larger bulk frictional resistance reduces the mass flow rate perforating into the cavity, causing the uplifting of the streamlines from the base of the cavity. Consequently, an intricate eddy region is formed at the inlet of the porous/fluid interface. For Darcy numbers below 2 x 10~7, the vortex region in front of the porous blocks gradually becomes part of the recirculating flow inside of the cavities. For low Darcy numbers, the strength of the laminar eddies in the separation zone before the porous blocks is damped out by the porous matrix frictional resistance, which allows the boundary layer to reattach itself to the wall, and form a closed vortex region. Furthermore, for low Darcy numbers, the boundary layer, separation zone, or vortex region covering the inlet of cavities creates a complicated recirculating flow inside the cavities, as seen in Fig. 5.4. The temperature distribution corresponding to the flow field shown in Fig. 5.4 is displayed in Fig. 5.5. The temperature fields shown in Fig. 5.5 correspond to a case where the Prandtl number is 0.7 and the conductivity of porous media is equal to that of the fluid. For smaller Darcy numbers, the larger separation zone after the porous cavities increases the distortion of the isotherms near the heated wall. It can be seen that as the Darcy number decreases the thickness of the external thermal boundary layer increases. Initially, as the Darcy number decreases, (up to DaL=2 x 10-7) the thermal penetration into the cavities increases, especially within the left half section of the cavity. This is the result of the relative increase in the thermal diffusion compared to convection within the porous medium. The thermal penetration ceases to propagate further into the cavity for Darcy numbers less than 2 x 10"7 due to the aforementioned formation of the intricate eddy region at the inlet of the porous/fluid 0.08 0.07 0.06 0.05 0.04- 0.03 0.02 0.01 0.0 • 0.08 0.07 0.06 0.05 0.04- 0.03 0.02 0.01 0.0 Da, =4x10' y * 0 .0 8 0.07 0.06 0.05 •0:04- 0.03 0.02 0.01 0.0 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.0 0.0 0.20 0.40 0.60 0.80 1.0 x* Fig. 5.5 Effects of the Darcy number on isotherms for flow through alternate porous cavity-block obstacles for ReL = 3 x 10s, AL = 0.35, Pr=0.7, / kf = 1.0. A=3, B= 1, H*=0.02. 122 interface. It should be noted that as the Darcy number is reduced below 1 x 10'7, an almost stagnant region is formed at the inlet of the cavities. Effects of the Reynolds number The effect of an increase or decrease in the Reynolds number is shown in Figs. 5.6 and 5.7 for Da^ = 8x 10^, AL=0.35, Pr=0.7, A= 3, and 5=1 for Re£ = 3x 10s, 3 x 10"7» 1 x 104 > and 5 x 103> respectively. Comparison of the streamlines in Fig. 5.6 shows that as the Reynolds number decreases, the distortion of streamlines and boundary layer thickness along the wall becomes more significant. In addition, the center of the vortex for each cavity moves further to the center. This is caused by a reduction in the magnitude of the inertial forces at a lower Reynolds number thus reducing the penetration extent of the flow into the porous blocks and cavities. This results in a larger blowing effect over the porous block resulting in a larger thickness of boundary layer near the wall. For larger Reynolds numbers, the external flow basically skims past the cavity without a strong interaction with the flow inside of the cavities. However, there is a significant effect on the porous/fluid interface structure at larger Reynolds number. The distortion of the isotherms at lower Reynolds numbers shown in Fig. 5.7 is obviously a direct result of the described flow field. Inertial Effects The inertial effects become significant for the higher permeability and the lower fluid viscosity (Vafai and Tien (1981)). Figures 5.8 and 5.9 show the effect of the inertial parameter on the flow and temperature fields for ReL = 3 x 10s, DaL = 8 x 10"6, Pr=0.7, A=3, and 5=1, for AL=0.35, 35, 70, and 210, respectively. . o , , , ------. 0 * 7 . o e - .o s “ . ©-*. - “ .03 r ~ 1 0.01 I i .02 ------~i~ —^ ■ I .ixtO~3 ~~ . o n -•xlO'i- - Rct = 3 x 10s t -ixio’L JL -1*10“* L -1x 10“* . o A -ixio“i. ! -5x I0 ~ J -Sxio'2 -3x10 Re, =3x10 -1x10 -5x10 = 1x10 - 3 x 0 -1x10 -1x10 0.0 0.20 0.4-0 0.60 0.80 1.0 Fig. 5.6 Effects of the Reynolds number on streamlines for flow through alternate porous cavity-block obstacles for DaL = 8 x 10-6, AL = 0.35, A=3, B=l, H*= 0.02 _ to o.oa 0.07 0.06 o.os .0.04 0.03 0.02 0.01 - Ret =3xl05 o.oa 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.08 0.07 0.06 0.05 .0.04 0.03 0.02 0.01 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.0 0.20 0.40 0.60 0.80 1 .O Fig. 5.7 Effects of the Reynolds number on isotherms for flow through alternate porous cavity-block obstacles for DaL = 8x10"*, Ai = 0.35, Pr=0.7, / k 7 = 1.0> ^ = 3, 5=1, #*=0.02. - J x lO '3 -3x 1 0 A, =0.35 -1x10 -IXIO 0.0 0.20 0.40 0.60 o.ao 1 .o Fig. 5.8 The influence of the inertial parameter on streamlines for flow through alternate porous cavity-block obstacles for ReL = 3 x10s, DaL = 8 x 10"6, ,4=3, 5=1, H*= 0.02. _ Lnto A l = 0.35 t 0.999- -0-95 0.999- 0.98 ------1------1------,______- 1______» ... > y* 0.999— A, =210 0.20 0.4-0 0.60 0.80 1 .0 Fig. 5.9 The influence of the inertial parameter on isotherms for flow through alternate porous cavity-block obstacles for Ret = 3xl05, DaL = 8 x 10"6, Pr = 0. 7 , keJT /k , = 1.0, A=3, 5=1, H*=0.02. 127 As expected, the distortion of the streamlines and isotherms and the size of the vortices near the wall increase with an increase in the inertial parameter. This is the result of the larger bulk frictional resistance that the flow encounters at larger values of the inertial parameter. This in turn leads to a larger blowing effect, which reduces the mass flow rate through the porous media, and results in a larger adverse pressure gradient in front of the porous obstacles, creating a larger separation zone. Prandtl Number Effects To investigate the effect of the Prandtl number on the flow and temperature fields, three different Prandtl numbers were chosen such that they will cover a wide range of thermophysical properties. The numerical results are presented in Fig. 5.10 for Re^ = 3 x 105, Da^ = 8 x 10"^, AL=0.35, A=3, and B =1 for three different fluids with Pr=0.7 (air), Pr=7 (water), and Pr=100 (a representative oil), respectively. Obviously, the variations °f Prandtl number have no effect on the flow field and therefore the flow field is the same for all Prandtl numbers. This flow field is shown in Fig. 5.10(a). As seen in Fig. 5.10(b)-(d), due to the lower value of the thermal diffusivity relative to the momentum diffusivity the extent of the thermal penetration over the external boundary as well as into the cavity becomes significantly confined for larger values of Prandtl numbers. Effect of the First Geometric Parameter A The first geometric parameter A=W*/H* represents the influence of the geometry of the porous obstacles on the flow. For this purpose, two configurations o . OS O.OT 'Tfl ' ' ' ~ o .o e — 0.03 0.0 — 0.03 i i 001 i ...i 0.02 —■imo'i-- 0.0 1 - - 0.0 s r " » 0.08 0.07 0.06 0.05 0.04- 0.03 0.02 0.01 Pr=0.7 y* 0.08 0.07 0.06 0.05 0.04- 0.03 0.01 0.08 0.06 0.05 0.04- 0.03 -1x10 0.02 Pr=100 0.20 0.4-0 0.60 0.80 1.0 Fig. 5.10 Prandtl number effects on streamlines and isotherms for flow through alternate porous cavity-block obstacles for ReL=3xlOs, DaL = 8xlO"6, AL=0.35, Kjf / k , = 1.0, 4=3, 5=1, H*=0.02 to 00 have been investigated- Figures 5.11 and 5.12 show the streamlines and isotherms for Rez_ = 3xl05, Dai = 2xl(T7, AL=0.35, Pr=0.7, and 5=1 for these two cases corresponding to A =3 and 4, respectively. It should be noted that Figs. 5.11 and 5.12 also represent the effects of the variations of the second geometric parameter 5 which is to be discussed later. Comparison of Figs. 5.11 and 5.12 shows that as the value of A increases, the distortion in streamlines and isotherms becomes more pronounced. Moreover, for larger values of A, the boundary layer separation zone in front of the porous block increases which in turn affects the flow and temperature fields inside of the cavities. This is a direct result of the relative increase in the length of the porous block, which extends the blowing action, and subsequent deceleration of the flow field. Effect of the Second Geometric Parameter B The effect of interspacing between a porous cavity and a porous block (B=D*/W *) were studied for two cases. These cases corresponded to Ret = 3 x 105, Da^ = 2 x l0 -7, At =0.35, Pr=0.7, and A =3 for 5=1 and 2. The streamlines and isotherms for these cases are also presented in Figs. 5.11 and 5.12, respectively. As can be seen, when the value of B increases from 1 to 2, the distortions of streamlines and isotherms as well as the effect of boundary layer separation zone on the recirculating flow inside the cavity become less pronounced. This is due to the larger interspacing between porous cavities and blocks which delays the blowing and deceleration actions. -jx.o-5^ la) A=3 and 5=1 , -ixio'2 __ ------1------1------—==------1 _ —■ * i __- -5X 10 (b) A =4 and 5=1 . -•*«> -5 x 1 0 -5 x 1 0 -9x10 (c) A=3 and 5=2 -1X10 0.0 0.20 0.40 0.60 0.80 1 .O Fig. 5.11 The influence of the geometrical layout on streamlines for flow through alternate porous cavity-block obstacles for Ret = 3xl05, Dat = 2xl0-7, AL = 0.35, k eff / k f =1.0, Pr=0.7, H*= 0.02 at (a).A=3 and 5=1, (b) A=4 and 5=1, and (c) A=3 and 5=2. LO o 0.08 0.07 0.06 0.05 0.04 0.05 0.02 0.01 (a) A=3 and 5=1 0.999 0.08 0.07 0.05 0.04 0.03 0.02 0.01 (b) A=4 and 5=1 0.08 0.07 0.06 0.05 0.04 0.03 0.02 (c) 4=3 and 5=2 0.20 0.40 0.80 1.0 x* Fig. 5.12 The influence of the geometrical layout on isotherms for flow through alternate porous cavity-block obstacles for Re£- = 3xl05, Dat =2xl0“7, AL=0.35, k eff / kr = 1.0. Pr=0.7, H*= 0.02 at (a).A=3 and 5=1, (b) A= 4 and 5=1, and (c) A = 3 and 5=2. 132 Effects of Larger Set of Porous Cavity Block Configurations The effects of larger set of porous cavity block configurations are shown in Fig. 5.13. This figure displays the results for the external flow over three porous cavity- block structures with A =3 and 5=1 for ReL=3xl05, Dai = 2xl0~7, At =0.35, Pr=0.7. As can be seen in this figure the increase in the number of cavity-blocks obstacles does not alter the main features of the flow and temperature fields. -5 * 1 0 -5 * 1 0 -i*io-J “ 1 * 1 0 0.07 0.06 0.05 0.04- 0.02 0.01 0.0 0.20 0.4-0 0.60 0.80 1 .O Fig. 5.13 Effects of larger set of porous cavity block configurations, (a) Streamlines and (b) isotherms for Ret = 3xlO5, Dat = 2xlO~7, AL =0.35, Pr=0.7, k ^ /k / = 1.0, A=3, 5=1, H*= 0.02, N=3- CHAPTER VI ANALYSIS OF FORCED CONVECTION ENHANCEMENT IN A CHANNEL USING POROUS BLOCKS 6.1 STATEMENT OF THE PROBLEM Forced convection heat transfer in a channel or duct fully or partially packed with a porous material is of considerable technological interest. This is due to the wide range of applications such as direct contact heat exchangers, electronic cooling, heat pipes, etc. It has been demonstrated that insertion of a high-conductivity porous material in a cooling passage can have a positive effect on convective cooling. Koh and Colony (1974) performed a numerical analysis of the cooling effectiveness of a heat exchanger containing a conductive porous medium, while Koh and Stevens (1975) conducted an experimental investigation for the same problem. It was shown that for the case of a fixed wall temperature the heat flux at the channel wall can be increased by over three times by using a porous material in the channel. Rohsenow and Hartnett (1973) presented a constant Nusselt number for the fully developed region in a porous medium bounded by two parallel plates, based on the Darcy flow model. To account for the effect of a solid boundary, Kaviany (1985) performed a numerical study of laminar flow through a porous channel bounded by isothermal 134 135 plates based on the Brinkman-extended Darcy model for constant porosity media. Poulikakos and Renken (1987) have investigated the effect of flow inertia, variable porosity, and a solid boundary on the fluid flow and heat transfer through porous media bounded by constant-temperature parallel plates and a circular pipe. They found that boundary and inertial effects decrease the Nusselt number where variable porosity effects increase the Nusselt number. The above-referenced investigations were based on filling the entire channel with a porous medium. This method, while beneficial in augmenting the heat transfer rate, can significantly increase the pressure drop inside the channel. Furthermore, the results of those investigations cannot be extended to parts of other applications such as electronic cooling, fin configurations, solidification of castings and geothermal applications. An important and a fundamental problem in heat transfer augmentation in a channel is related to forced convection through a channel with multiple porous emplaced blocks. The flow over the fluid region is governed by the Navier-Stokes equation and the flow through the fluid-saturated porous medium is governed by a volume-averaged momentum equation based on the Brinkman-Forchheimer-extended Darcy model (Vafai and Tien (1981)). These two flows are coupled through the interface boundary conditions at the porous/fluid interface. The interactions of flow and temperature fields between the porous-saturated region and the fluid region have a significant influence on the convection phenomenon in these systems. Several investigations were devoted to the problem of finding the proper set of boundary conditions at the interface between a fluid flow in a porous medium and the adjacent region without a porous medium. Beavers and Joseph (1967) experimentally reported the mass efflux of a poiseuille flow over a naturally permeable medium based on 136 Darcy's law. They found that when a viscous fluid passes through a porous solid, tangential stress entrains the fluid below the interface with a velocity which is slightly greater than that of the fluid in the bulk of the porous medium. Levy and Sanchez- Palencia (1975) found that when the typical length scale of the external flow is large compared with the microscopic scale, the velocity field transition at the interface from the porous media to the free fluid region occurs over a thin region of the order of the pore scale. They also showed that depending on the direction of the pressure gradient in the porous medium two different kinds of phenomena may appear at the interface. Due to the mathematical difficulties in simultaneously solving the coupled momentum equations for both porous and fluid regions, very little work has been done on internal forced convection on the porous/fluid composite system. Vafai and Thiyagaraja (1987) have performed an analytical investigation of a fully developed forced convection for three basic types of interface composites. They obtained analytical solutions for the velocity and temperature distributions as well as analytical expression for the Nusselt numbers for all three classes of interface composites investigated in their work. Poulikakos and Kazmierczak (1987) have presented a theoretical study of forced convection in a channel with a porous region attached at its wall, based on the Brinkman-extended Darcy model. In the present study, a numerical investigation of forced convection in a parallel plate channel with porous blocks emplaced at the bottom wall is presented. The analysis is based on the use of Brinkman-Forchheimer-extended Darcy Model in the porous media and the Navier-Stokes equation in the fluid region. The use of the porous medium generally enhances the mixing within the fluid region resulting in a 137 higher heat transfer than that obtained in the corresponding smooth channel. In the present investigation the basic interaction phenomena between the porous substrate and the fluid region for these types of composite systems as well as the methodology for enhancing the heat transfer rate within the channel have been analyzed. Furthermore, the effects of various parameters governing the hydrodynamic and thermal characteristics of the problem are analyzed. 138 6.2 MATHEMATICAL FORMULATION A schematic diagram of forced convection enhancement in a channel using porous blocks is displayed in Fig. 6.1. The fluid enters at ambient temperature T0, with a parabolic velocity profile. The plate walls are maintained at constant temperature Tw, the channel width and total length are R and L, the width and height of the rectangular porous blocks are H and W, respectively, the distance between the blocks is D, the lengths of plate upstream and downstream from the blocks are and respectively. The flow is assumed to be steady, incompressible, and two dimensional. In addition, the thermophysical properties of the fluid and the porous matrix are assumed to be constant and the porous medium is considered homogeneous, isotropic, nondeformable, and in local thermodynamic equilibrium with the fluid. In this study the Brinkman-Forchheimer-extended Darcy model, which accounts for the effects of the inertia as well as friction caused by macroscopic shear (Vafai and Tien (1981); Kaviany (1987)), is used to demonstrate the flow inside the porous region. The equations governing momentum and energy conservation for the present problem will be separately written for the porous and fluid regions in dimensionless forms. For the porous region (Vafai and Tien (1981)&(1982), Vafai (1984), Vafai (1986)) \ 1 V72C* 1 e* A| .|e* * L * 3KI * 9KI (6 . 1) Up dx p dy' Re,ff•ff ^ R e^D a ^ p dx Up dy (6.2) Porous Medium 1 I Parabolic Velocity Profile Fig. 6.1 Schematic diagram of force convection in a parallel plate channel with porous block obstacles. OJ VO and for the fluid region (6.4) (6.5) (6.6) where subscripts p and/ refer to the porous and fluid regions. The operator V2 is the Laplacian and all the other symbols are defined in the nomenclature. It should be noted that the effects of local non-thermal equilibrium and dispersion are neglected at this time based on the work of Vafai and Sozen (1990) and Sozen and Vafai (1990, 1993). The following dimensionless variables used in Eqs. 6.1-6.6 are defined as (6.7) (6.8) (6.9) Note that the field variables in the porous region are volume-averaged quantities as described in Vafai and Tien [6]. The stream function and vorticity are defined in the 141 * £ 3V (6.10) dy dx E = d v „ d u (6.H ) dx dy The appropriate boundary conditions for the present problem are: (1) atx*=0, 0 u*= 6y*(l-y*)> v*=0 f *2 *3 \ (2) at x*=L*, 0 Jo u d y ' = 1, D* = 0 *-N L* = o V) ^ -V l* = o ^ - = o ox dx dx (3) at 0< x* u = 0, u* = 0, \\r' = 1 ? = - f £ . e = i 3y (4) at 0< x* u = 0, t>* = 0, \|/‘ = 0 142 V ~ | £ . e - o dy The above boundary conditions correspond to a fluid entering the domain with a fully developed profile along with the application of the no-slip condition on the two parallel walls. At the channel exit, axial diffusion is set equal to zero to satisfy the closure for the elliptic problem, and the x-component velocity u is calculated to satisfy the conservation of mass. The exit boundary conditions were evaluated very carefuly. This was done through choosing two different domains corresponding to the physical and computational domains. The location of the exit boundary condition was systematically moved further downstream until it was ensured that the exit boundary condition has no detectable effect on the physical domain. The upper and bottom plates of the channel are maintained at a constant temperature. At the porous/fluid interface, the following quantities evaluated in both porous and fluid regions are matched: horizontal and vertical velocities, normal and shear stresses, temperature, pressure, and heat flux. The matching conditions for the present governing equations can be expressed as v*P =V/> C =^/> a n d e ;= e ; where the subscripts p and/stand for porous and fluid, respectively. Based on the above coupled governing equations, boundary conditions, and the shape of porous/fluid interface, it is seen that the present problem is governed by six 143 dimensionless parameters. These are the Darcy, Reynolds, two Prandtl numbers, inertia parameter, and two geometrical parameters A and B, where W D a „* H D W (fi 12) A = ——, B = —— , and H = — , D = —, W = — H W ' R R R To evaluate the effects of the porous material on the heat transfer rate at the wall, the local Nusselt number is evaluated as follows: x‘ir 00 Nu IlR (6.13) k, k,(Tw-Tm)d/ y*0 where 0m = (Tm-T 0)/(TW-T0) is the dimensionless form of the bulk mean temperature Tm defined by (6.14) Here the absolute value of the velocity is used as in Kelkar and Patankar (1987), so that the regions of recirculating flow are properly represented. Note that the definition of Nusselt number, based on the conductivity of the fluid, permits a direct comparison between the smooth and blocked channels. 144 6.3 COMPUTATIONAL DETAILS To obtain the solution of the foregoing system of equations, the region of interest is overlaid with a variable grid system as shown in Fig. 6.2(a). Applying the central differencing for the diffusion terms and a second upwind differencing for the convective terms, the finite-difference forms of the vorticity transport, stream function, and energy equations were derived by a control-volume integration of these differential equations over discrete cells surrounding any given grid point, as shown in Fig. 6.2(b). This results in a system of equations of the following form: CcO c = CnO n + CS0 S + Ce O e + C„&w + 5° (6-15) where E and W. The finite difference equations for I;*, \\r*, and 0 obtained in this manner were solved by the extrapolated-Jacobi scheme. This iterative scheme is based on a double cyclic routine, which translates into a sweep of only half of the grid points at each iteration step (Adams and Ortega (1982)). The numerical procedure for solving the finite-difference equations is as follows: 1. Overlay the computational domain with a variable mesh. 2. Assign values of Re, Da, A , A, B, and initial values for !;*, \j/\ ,u, n, and 0 in Eqs. (6.1)-(6.6), and the corresponding boundary conditions. 145 0102020201000100010000000200000002010201029002000088 (a) OJ+1) (U) Ac (b) Fig. 6.2 (a) Grid system for the computational domain, (b) Local integration cell. 146 3 Calculate the new values of vorticity at each node by using the finite difference set of equations for if as given by Eq. (6.15). 4. Calculate the new values of stream function \|/’ at each node from Eq. (6.15) for \|/‘ by using the values of found from step 3. * 5. Calculate the new values of the velocity from M=\|/y and • 6. Update new boundary values using the new nodal values for \jr* and £,*. 7. Repeat steps 3 to 6, until the following convergence criterion is satisfied Ln+1 xn max C where <{> stands for % , y , or© and n denotes the iteration number. 8. Calculate the temperature 0 by using the finite-difference set of equations for 0 as given by Eq. (6.15) with the assigned values of Pr and the values of \j/* obtained from step 7, until the criterion of convergence for 0 is satisfied. The interfacial properties play very important roles in the porous/fluid composite system. This is due to the abrupt change of thermophysical properties, such as the viscosity, permeability, porosity, and the thermal conductivity, across the interface. These effects on the porous/fluid interface are represented by the nondimensional parameters Re, Da, A , and Pr. The harmonic mean formulation recommended by Patankar (1980) was used to treat these discontinuous characteristics at the porous/fluid surface. This ensured the continuity of the convective and diffusive fluxes across the interface without requiring the use of an excessively fine grid structure. For the present case Re, Da, A, and Pr at the interface of a control volume were found as: 2 R e - R e, 2Da.ffD a, 2A~A, 2Pr,ffPrf Re, = - f Da, = —- A, = - ,ff / Pr, = ------^ - ( 6 . 1 7 ) Re,j5r+ Re/ "*■ ^ a/ if / where the subscripts and I stand for effective, fluid, and interfacial, respectively. Therefore, instead of the momentum equations in Eqs. (6.1) and (6.4), the following were used across the interface: M* W ^ — V2r + S* (6.18a) p dx p dy Re, ^ p where 148 where |v^ ^ - ( A , ) (6.19b) f Re, dy* ^Da, J Re, dx’ l^Da, where Eqs. (6.18a) and (6.18b) are for the porous side and Eqs. (6.19a) and (6.19b) are for the fluid side of the interface. Note that constant values of Da, and A were used for a specified porous substrate. In the present work due to lack of other information, the effective viscosity of the fluid-saturated porous medium is set equal to the fluid viscosity. It has been found that this approximation provides a good agreement with experimental data (Lundgren (1972); Neale and Nader (1974)). The vorticity at the wall is evaluated using the linear Taylor series approximation, that is: r3(<-Vp),C 1 (6.20) where the subscript np denotes the first neighboring point next to the boundary and Avv represents the normal distance from the wall to the point np. In this study, the computational domain was chosen to be larger than the physical domain to eliminate the entrance and exit effects and to satisfy continuity at the exit. A systematic set of numerical experiments was performed to ensure that the use of a fully developed velocity profile for the outflow boundary condition has no detectable effect on the flow solution within the physical domain. That is, the downstream length beyond the physical domain was determined by trial and error to 149 ensure that the effects of the outflow boundary condition were well outside of the physical domain. A nonuniform grid system with a large concentration of nodes in regions of steep gradients, such as those close to the walls and porous blocks, was employed, as shown in Fig. 6-2(b). A grid independence test showed that there is only a very small difference (less than 1%) in the streamlines and isotherms among the solutions for 218x69, 258x158, and 300x69 grid distributions. As this difference is small, our computations in this work were based on a 218x69 grid system. These computations, performed on a CRAY/YMP, took about 300-500 CPU seconds depending upon the different governing parameters. To validate the numerical scheme used in the present study, comparisons with two relevant results were made. These comparisons were carried out for the problem of hydrodynamically fully developed forced convection in a channel with a porous medium partially covering the external boundary (Poulikakos and Kazmierczak (1987)), and external forced convection over a flat plate embedded in a porous medium (i.e. //*—><» and W*—>°°), representing the full porous medium case (Vafai and Thiyagaraja (1987) and Beckermann et al. (1987)). The result of these comparisons (being similar to those presented in Vafai and Kim (1990)) showed that the numerical model predicts very accurately the velocity and temperature fields in a porous/fluid composite system. 150 6.4 RESULTS AND DISCUSSION As discussed earlier, the present problem is governed by six dimensionless parameters. These are the Darcy, Reynolds, two Prandtl numbers, inertia parameter, and two geometrical parameters A and B. In this section, the effects of these parameters on the flow field, temperature field and local Nusselt number distribution will be examined. The fixed input parameters that were used for all cases were R=l, .f ,=6, and k eff / k f =1.0. Note that for illustrating the flow and temperature fields clearly, only part of the figures were presented. However, at all times, the much larger domain was used for numerical calculations and interpretation of the results. Furthermore, in this study the conductivity of the porous media is taken to be equal to that of fluid in order to concentrate on the effects of geometric and thermophysical variations. Figure 6.3 displays the effects of rectangular porous blocks on the fluid flow and convection heat transfer for a case where the Reynolds number is 750, Darcy number is lxlO"5, inertia parameter is 0.35, Prandtl number is 0.7, the dimensionless height and width of the porous blocks are 0.25 and 1.0 respectively, and the spacing between the porous block is 1. Several interesting features are observed from these plots. The streamlines are considerably distorted in the channel due to the presence of the porous block array (see Fig. 6.3(a)). The velocity distribution is parabolic at both the entrance and exit of the two plates. However, this distribution changes rapidly as the fluid encounters the porous block array, especially at the comers of block. As seen in Fig. 6.3 the blocks have a more prominent effect on the flow conditions downstream compared to the conditions upstream of the blocks. Another interesting feature is the formation of a relatively large vortices (a) o 0.8 6 (b) 0 .4 0.2 7 0.0 o 0.8 ■OJS 0.6 (C) 0.2 0.0 o 24 6 8 10 11 4 1 6 20 22 24 26 Fig. 6.3 (a) Streamlines, (b) velocity distribution, (c) isotherms, and (d) local Nusselt number distributeion for flow in a parallel plate channel with porous block obstacles for Re=750, Da=lxlO"5, A=0.35, Pr=0.7, k eJf / k f = 1.0, A =4, B =1, H*= 0.25. 40 — without porous blocks — with porous blocks 35 30 25 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 * X Fig. 6.3 (a) Streamlines, (b) velocity distribution, (c) isotherms, and (d) local Nusselt number distributeion for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10"5, A=0.35, Pr=0.7, k ejr / k f = 1.0, A =4, B =1, H*= 0.25. 153 behind each porous block separated by a small recirculation region rotating in a direction opposite to that of the larger vortices. The height of these recirculation regions is about twice the height of the porous blocks. A weak eddy is generated on the smooth upper plate surface corresponding to the reattached region on the bottom plate. The complicated flow field within the channel is the result of four interrelated effects: a penetrating effect pertaining to the porous medium, a blowing effect caused by porous media displacing the fluid from the porous region into the fluid region, a suction effect caused by the pressure drop behind the porous blocks resulting in a downward flow, and the effect of boundary layer separation. Therefore, the characteristic of the porous substrate plays a significant role on the flow and temperature fields. The temperature field in the channel is displayed in Fig. 6.3(c). As expected, the thickness of both upper and lower thermal boundary layers increases along the length of heated plates, and becomes significantly distorted within and around the porous- block region. As expected, the symmetric character of the temperature field re establishes itself far enough downstream of the porous block region. The variation of local Nusselt number corresponding to the above temperature fields is illustrated in Fig. 6.3(d). A periodic variation of Nusselt number on the bottom plate is observed starting before the leading edge of porous block array (from x* =6). The peak in each cycle occurs at an x value corresponding to the center of the larger vortex behind the porous block, while the minimum in each cycle occurs at around the location where the larger and the smaller vortices behind each block meet. The heat transfer in the rear part of each porous block is higher due to increased convection aided by higher velocities in the recirculation eddy. Whereas the heat transfer at around the location, 154 where the larger and the smaller vortices meet, is lower due to an almost stagnant flow field within that region. Comparison of local Nusselt number distributions for a channel with and without porous blocks shows that the recirculation flow caused by porous blocks can augment significantly the heat transfer rate. Effect of the Darcy Number The Darcy number is directly related to the permeability of the porous medium. The effect of variations in the Darcy number is depicted in Figs. 6.4-6.6 for Re=750, A=0.35, Pr=0.7, A=4, and 5=1, for Da=lx 10'5, 5x 10'5 and 9x 10"5. Comparison of the streamlines in Fig. 6.4 shows that the distortion of streamlines and the size of recirculation zones behind the porous blocks becomes less pronounced as the Darcy number increases. This in turn accelerates the core flow to satisfy the mass continuity and confines the development of recirculation zones in the transverse direction. For smaller Darcy numbers, the recirculation cell occupies only the space between the porous blocks and the flow penetration into the porous block array is significantly reduced. In the limit, if the Darcy number is reduced to a value approaching zero, there will be no streamlines penetrating the porous block and the flow passes over the solid block array. Comparison of isotherms in Fig. 6.5 shows that as the value of the Darcy number is reduced the distortion of the isotherms becomes less pronounced. This is the direct results of the discussed flow field. The variation of local Nusselt number for various Darcy numbers is displayed in Fig. 6.6. It is seen that the Darcy number has a significant impact on the local Nusselt number distribution. Here there is an interesting phenomenon for the overall trend in the Nusselt number distribution. There exists an optimum Darcy number corresponding 1.0 ------1-----'1----- ■' 1 ...... ' " ' ‘ V > J '.... o.a (a)Da = lxlCTs / 0.6 — 0.4 y yjlr\ /#a%—//Ir\ /)!,AW — — 0.2 ------fjlrTtra ^ ------0.01------Jv* Air/ill nr/I IW^/llllUr/liv— ------1------ro.oi— ;— ------;------» » 0.8 •a7- . 0.6 y 0.4 0.2 .0.1 ■O.OL 0.0 •0.01* J. 1.0 (c)Da=9xl0‘ 0.8 *0.7' 0.6 •OJ 0.4 0.2 .0.1. .0.01. >0.01. 0.0 O 2 4 6 8 10 12 14 16 18 20 22 24 26 4. X Fig. 6.4 Effects of the Darcy number on streamlines for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, A=4,5=1, #*=0.25. Or,U\ 0.8 0.6 0.4 0.2 0.0 1.0 0.8 . 0.6 y 0.4 0.2 0.0 0.8 ■0A- 0.6 (c)Da = 9 x 10' 0.4 0.2 0.0 0 24 6 8 10 T2 14 16 18 20 22 24 26 X Fig. 6.5 Effects of the Darcy number on isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, Pr=0.7, keff/kf = 1.0, A =4, B =1, H*=0.25. - o\ 157 40 1: Da = °o (without porous blocks) 2: Da = 1 x 10-2 35 3: Da = 3xl0~5 4: Da = 5 x 10~5 5: Da = 9 x 10'5 30 25 '* 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 * X Fig. 6.6 Effects of the Darcy number on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=750, A=0.35, Pr=0.7,kt„ /k , =1.0, A =4,5 =l,//*=0,25. 158 to the largest values of the Nusselt number distribution. Below and above this optimum value the peak values of the Nusselt number drop off. This is the result of three competing effects. These are the the flow through the porous blocks, the blowing effect and the suction effect caused by the porous blocks. Effect of Reynolds Number Figures 6.7-6.9 show the streamlines, isotherms, and Nusselt number distribution for Da=l x 10’5, A=0.35, Pr=0.7, A=4, and B=1, with Re=750, 1200 and 1500, respectively. It can be seen from Fig. 6.7 that increasing the Reynolds number From 750 to 1200 increases the distortion level in the core flow streamlines. As the Reynolds number increases the larger vortex behind each block diminishes and ultimately it vanishes. At the same time the size of the smaller vortex ahead of each block grows occupying most of the porous block. For larger Reynolds numbers (>1200), a large recirculation region is formed behind the last porous block. The reason for this is that at these larger Reynolds numbers a very sharp velocity gradient occurs at the right top comer of the last porous block, which delays the reattachment of the core flow to the bottom plate. As expected, the temperature fields corresponding to different Reynolds numbers (Fig. 6.8) clearly show that at higher Reynolds number the extent of distortion for isotherms increases. The effect of Reynolds number on the local Nusselt number distribution is depicted in Fig. 6.9. It shows that both peak and trough values of Nux increase with an increase in the Reynolds number. The peak values of the Nusselt numbers correspond to the re attachment regions of the core flow to the external boundary behind each porous block, which increases the heat transfer by convection. It should be noted that a high 1.0 (a) Re=750 0.8 0.6 0.4 ■OJ' 0.2 0.0 •O.ai. 1 .0 TTT T (b) Re=1200 0.8 y,* 0.6 0.4 0.2 ■o.i- 0.0 ■0.01' 0.8 (c)Re=1500 • 0.6 0.4 •OJ' 0.2 .0.1 0.0 -o.or 0 2 4 6 8 10 12 14 16 18 20 22 24 26 X Fig. 6.7 Effects of the Reynolds number on streamlines for flow in a parallel plate channel with porous block obstacles for Da=lxlO"5, A=0.35, A=4, 5=1, H*=0.25. 0.8 0.6 (a) Re=750 0.4- 0.2 0.0 ± 1.0 0.8 (b) Re=1200 0.4 0.2 .04. 0.0 0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Fig. 6.8 Effects of the Reynolds number on isotherms for flow in a parallel plate channel with porous block obstacles for Da=l x 10"5, A=0.35, Pr=0.7, kljr / kf = 1.0, A=4, 5=1, H*=0.25. CT\ o 50 — without porous blocks — with porous blocks 45 1: Re=750 2: Re=1200 3: Re=1500 40 35 30 Nujj 25 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 * X Fig. 6.9 Effects of the Reynolds number on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles forD a= lxl0‘5, A=0.35, Pr=0.7,k^ / k , = 1.0, A =4, B =1, H*=0.25. 162 rate of increase in the overall rate of heat transfer from the channel into the flow can be attained by using the porous blocks. Inertial Effects. The effect of an increase or decrease in the inertial parameter is shown in Figs. 6.10-6.12 for Re=1500, D a=lx 10'5, Pr=0.7, A=4, and B=1, for A=0.35, 21 and 35. The flow fields displayed in Fig. 6.10 reveal that as the inertial parameter increases, the vortex behind each block gradually grows inside the inter-block spacing. This is due to the larger bulk frictional resistance that the flow encounters at larger values of the inertial parameter. This in turn causes a larger blowing effect through porous blocks, which displaces the fluid deeper into the core flow, and creates a larger recirculation in the right top comer of porous blocks after encountering the primary flow field. As expected, the distortion in the isotherms becomes pronounced as the inertial parameter increases due to the increase in the size of the vortices behind the porous blocks (Fig. 6.11). Again this is the direct result of larger blowing effect for a larger inertial parameter. Figure 6.12 shows the variation of Nux with inertial parameter. In general, as the inertial parameter increases, the peak value of Nux increases. This is due to the larger fluid mixing caused by a larger recirculation zone for larger values of the inertial parameter. Prandtl Number Effects In order to determine the effect of the Prandtl number on the flow and temperature fields, three different Prandtl numbers were compared such that they will cover a wide range of thermophysical properties. These comparisons, shown in Figs (a) A=0.35 0.8 0.6 0 .4 0.2 0.0 .0.01. 1.0 TTTTTT (b) A=21 0.8 0.4 0.2 .0.1- 0.0 •0.01- (c) A=35 0.8 0.6 0.4 0.2 0.0 4 6 10 12 14 16 18 20 22 24 26 X Fig. 6.10 Influence of the inertial parameter on streamlines for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l x 10‘5, A =4,5 =1, H*=0.25. 0.8 - 0.6 (a) A=0.35 0.4 0.2 0.0 1.0 0.8 0.6 a" (b) A=21 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 ■0.7-----— 0.0 2 4 6 80 10 12 14 16 18 20 22 24 26 4 X Fig. 6.11 The influence of the inertial parameter on isotherms for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l x 10'5, Pr= 0.7, k eff / k f -1.0/4 =4,5 =l,7/*=0.25. 55 1: A=0 50 2: A=0.35 3: A=21 4: A=35 45 40 35 30 25 20 15 10 5 0 6 8 10 12 14 16 18 20 22 24 26 * X Fig. 6.12 The influence of the inertial parameter on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=1500, Da=l x 10'5, Pr=0.7, kejr / kf = 1.0, A =4, B =1, H*=0.25. 166 6.13 and 6.14, were performed for Re=750, Da=l x 10"5, A=0.35, A -4 and 5=1, for three different fluids with Pr=0.7 (air), Pr=7 (water), and Pr=100 (typical value for oil), respectively. Since Re, Da and A are fixed, the variation of Prandtl number has no effect on the flow field and therefore the flow field is the same for all Prandtl numbers. This flow field is shown in Fig. 6.13(a). As expected and as seen in Figs. 6.13(b)-(d), increasing the Prandtl number decreases the thickness of the thermal boundary layer. As expected, the local Nusselt number and its fluctuations increase with an increase in the Prandtl number (Fig. 6.14). Note that utilizing a combination of Re and Pr alone (i.e., Pe=Re Pr) is insufficient for depicting the temperature field or describing Nusselt number correlations. Effect of Geometric Parameters A and B The geometric parameters, A and B, are related to the porous-block's aspect ratio and pitch. The effect of aspect ratio on the flow and temperature fields were studied for Re=750, Da=l x 10~5, A=0.35, Pr=0.7, and 5=1. The streamlines and isotherms for A=4 and .4=8 are represented on Fig. 6.15. As can be seen in Fig. 6.15, when the value of A is increased from 4 to 8, the distortions for streamlines and isotherms become less pronounced. In addition, the size of recirculation and interaction between successive porous blocks reduce. This is due to the relative decrease in the height of the porous blocks, which in turn offers a lower degree of obstruction to the flow for larger values of A. The effect of pitch on flow and temperature fields is shown in Figs. 6.16 and 6.17 for a case where Re=750, Da=l x 10'5, A=0.35, Pr=0.7, and A=4, for 5=3,2, and 1. It 0.8 0.6 (d)Pr=100 0.4 0.2 0.1 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Fig. 6.13 Prandtl number effects on streamlines and isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, / k f = 1.0, A =4, B =1, #*=0.25. 160 — without porous blocks_ 150 — with porous blocks 140 1:P p =0.7 2: Pjp=7 130 3: Pp=100 120 110 100 Nux 80 70 50 40 30 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 X Fig. 6.14 Prandtl number effects on local Nusselt number distribution for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, ktJf /k f = 1.0, A =4, B =1, H*=0.25. 0.8 0.6 0.4 0.2 .0.1. * 0.0 y 1 .o TTT 0.8 0.6 •0.5. •0.7 0.4 0.7 0.2 0.0 •0.01- O 2 4 6 8 10 12 14 16 18 20 22 24 26 X (a) Fig. 6.15 The influence of the geometric parameter A on streamline and isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, D a=lxlO '5, A=0.35, Pr=0.7, k f// / k f = 1.0, B =1, H*=0.25. 1 .0 o.s 0.6 0.4 .0.4- 0.2 ■0.7— 0.0 A=8 and B=\ X 0» Fig. 6.15 The influence of the geometric parameter A on streamline and isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, D a=l x 10"5, A=0.35, Pr=0.7, / k, = 1.0, B =1, H*= 0.25. -a o can be seen that for larger values of the pitch parameter (6=3 in Fig. 6.16(a)), the recirculation zones caused by the porous blocks are relatively independent of each other. At the same time, several eddies are generated on the upper smooth plate due to the core flow attaching to the bottom plate between the blocks. With decreasing pitch up to 5=2 (Fig. 6.16(b)), recirculation zones behind the first and second porous blocks vanish. Comparison of the temperature fields in Fig. 6.17 shows that as the pitch decreases from 5=3 to 6=2, the isotherms are more distorted. However, as 6 further decreases to 1, the distortion of the isotherms is less pronounced. 1.0 0.8 0.6 .90' 0.4 .0.7- 0.2 .0.1. 0.0 0.8 y ■os- 0.4 0.2 05. .0.01. 0.0 0.8 0.6 0.4 0.2 ■o.v ■0.01. 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 * X Fig. 6.16 The influence of the geometric parameter B on streamline for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10'5, A=0.35, H*=0.25 at (a) A=4 and B =3, (b) A=4 and B =2, and(c) A=4 and B =1. 0.8 .0.8 ' 0.6 0.4 0.2 ■ 0.8------0.0 0.8 .0.8 - , • 0.6 0.4 0.2 0.0 ■0.8 — 0.8 0.6 0.4 0.2 ).s— 0.0 o 2 4 6 8 10 12 14 16 18 20 22 24 26 X Fig. 6.17 The Influence of the geometric parameter B on isotherms for flow in a parallel plate channel with porous block obstacles for Re=750, Da=l x 10’5, A=0.35, Pr=0.7, / k f = 1.0, H*=0.25 at (a) A=4 and B =3, (b) A= 4 and B =2, and(c) A=4 and B =1. ^ CHAPTER VII INTERNAL HEAT TRANSFER AUGMENTATION IN A PARALLEL PLATE CHANNEL USING AN ALTERNATE SET OF POROUS CAVITY-BLOCK OBSTACLES 7.1 STATEMENT OF THE PROBLEM The problem of convective heat transfer and fluid flow in horizontal ducts with fins and ribs has been well studied and documented because of the augmentation effect on the heat transfer process. The similar problem with porous constructure has also gained extensive attention due to the wide range of applications which include, but are not restricted to, areas such as thermal insulation, crude oil extraction, solidification of castings, nuclear waste repositories, solid matrix heat exchangers. Shah and London (1987) provided a comprehensive survey of literature pertinent to the heat transfer performance studies within a channel without porous medium while Koh and Colony (1974) numerically investigated the cooling effectiveness for a porous material in a coolant passage. Koh and Stevens (1975) performed an experimental study for the same problem. They showed for the case with fixed allowable wall temperature that the heat flux at the channel wall can be increased by over three times by using high-conductivity porous material in the 174 175 channel. Rohsenow and Hartnett (1973) presented the constant Nusselt number for the fully developed region in a porous medium bounded by two parallel plates, based on the Darcy's flow model. To account for the effect of a solid boundary, Kaviany (1985) performed a numerical study of laminar flow through a porous medium bounded by isothermal plates based on the Brinkman-Extended Darcy model for constant porosity model. Poulikakos and Renken (1987) have examined the effect of flow inertia, variable porosity, and a solid boundary on forced convection in a duct filled with porous media. An important problem related to forced convection through a porous medium is flow and heat transfer over a porous /fluid composite system. This involves the study of fluid flow above and through a porous medium. Under this circumstance, the porous/fluid interfacial region represents a zone of discontinuity of material properties. It has a direct influence on the fluid flow and heat transfer, especially when there is a large gradient of physical properties such as permeability, porosity, and thermal conductivity, across the interface. This type of composite systems is encountered in many cases of practical interest such as nuclear waste repositories, crude oil extraction, iron blast furnaces etc. There exists an extensive set of literature (Joseph and Tao (1966)), which describes coupled fluid motions satisfying the Navier-Stokes equations in the free fluid, empirical or semiempirical set of equations (typically Darcy's law) in the permeable material, and matching conditions at the interfacial boundaries. Beavers and Joseph (1967) experimentally reported the mass efflux of a Poiseuille flow over a naturally permeable boundary based on Darcy's law. They found that when a viscous fluid passes a porous solid, tangential stress entrains the 176 fluid below the interface with a velocity which is slightly greater than that of the fluid in the bulk of the porous medium. Both experimental and theoretical investigations for validating such slip-flow interface condition were done by Taylar (1971) and Richardson (1971). Levy and Sanvhez-Palencia (1975) found that when the typical length scale of the external flow is larger compared with the microscopic scale, the velocity field transition at the interface from the porous media to the free fluid region occurs over a thin region of the order of the pore scale. They also showed that depending on the direction of the pressure gradient in the porous medium two different kinds of phenomena may appear at the interface. Recently, Vafai and Thiyagaraja (1987) analytically studied a general class of problems involving interactions on flow and heat transfer for three basic types of interface zones. They obtained analytical solutions for the velocity and temperature distributions as well as analytical expression for the Nusselt numbers for all three classes of interface composites investigated in their work. More revelant to the present study is the work of Poulikakos and Kazmierczak (1987). They analyzed fully developed forced convection in a channel partially filled with a porous matrix and showed that a critical thickness exists at which the value of Nusselt number reaches a minimum, based on the Brinkman-extended Darcy model. Analysis of internal forced convection in a porous/fluid composite system is significantly more complicated due to the complex geometric configuration of these type of systems. In this study, a numerical investigation for a channel, within which multiple porous cavity and block structures are alternately emplaced has been performed. The analysis is based on the use of Brinkman-Forchheimer-extended Darcy Model in the porous media and the Navier-Stokes equation in the fluid region. 177 The porous media provides a penetrating random structure, which augments the mixing in the fluid and profoundly changes the heat transfer characteristics within the channel. Therefore, the present study is aimed at a fundamental investigation of changes in the flow pattern and heat transfer performance due to the existence of cavity-block porous obstacles. Effects of various governing physical parameters are also considered in order to investigate their influence on the flow and thermal characteristics within the channel. 178 7.2 FORMULATION The problem consists of flow between parallel plates with a multiple porous cavity-block structure on the bottom boundary, as depicted in Fig. 1. The fluid enters the channel at ambient temperature T0 . It is assumed that the hydrodynamic entry length is small resulting in an entry parabolic velocity profile into the channel. Alternatively, this parabolic velocity profile can also occur for a regular channel entrance region prior to the multiple porous cavity-block structure region. The plate walls are maintained at constant temperature Tw, the channel width and total length are R and L, and the width and height of the rectangular porous cavities and blocks are H and W, respectively. The distance between cavity and block is designated as D, and the length of the plate upstream and downstream from the porous cavity-blocks are i x and i 2, respectively. The flow is assumed to be steady, incompressible, and two dimensional. In addition, the thermophysical properties of the fluid and the porous matrix are assumed to be constant and the porous medium is considered as homogeneous, isotropic, nondeformable, and in local thermodynamic equilibrium with the fluid. In this study, the Brinkman-Forchheimer-extended Darcy model, which accounts for the effects of flow inertia as well as friction caused by macroscopic shear (Vafai and Tien (1981); Kaviany (1987)), is used to describe the flow inside the porous regions. This work is based on the application of an efficient method combining the two sets of governing equations for the fluid and the porous regions into one set of conservation equations satisfying the matching conditions at the porous/fluid interface. The resulting equations for governing momentum and energy conservations in terms of dimensionless variables are as follows (Vafai and Tien (1981)&(1982), Vafai (1984), Vafai (1986)): profile Fig. 7.1 Schematic diagram of force convection in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate. 180 3y* 95* _ 9vl 9|1 = J_V2t* . C* a* a/ a/ Re 5 (7.1) v y = < (7.2) ^_vae__ n > (7.3) 9y * dx" Bx By* lP e J the non-dimensional parameters in the fluid region are R e = llsJL^ p e = if2Z ) s* =0 (7.4a) v, ec r and in the porous region the non-dimensional parameters are Pe = ^ i , Da = A = —7^ (7.4b) a . R 2 K 9 H _ a H 5 * = -----— 2;*-A|v*|i;*-A (7.5) ReDa 1 ^ at* a/ where the nondimensional quantities are . x . y . u . l) I .1 71 = —, y = —, u = — , -u = — , v =V« +u (7.6) A? It 1/ 1 • ¥ (7.7) um.R T„-Tn 181 The source term S* in the porous region, is composed of those terms contributing to the vorticity generation due to the presence of the porous medium. In addition, the above stream function and vorticity are defined as follows (7.8) Oy Ox 3\) du ox Oy c7-9) It should be noted that the variables v and T in the porous regions are both volume- average quantities, as described by Vafai and Tien (1981). 7.3 BOUNDARY CONDITIONS Due to the elliptic nature of the conservation equations the boundary conditions for all the field variables have to be specified along the entire boundary enclosing the solution domain. At the inlet of the channel, the stream function distribution is calculated from the specified inlet fluid velocity profile that is a fully developed parabolic profile. At the outlet, the gradient of the stream function in the axial direction is assumed to be zero; i.e., the streamlines are assumed to be peipendicular to the exit plane of the channel. This boundary condition frequently appears in the literature (Roache (1982)) and implies that the flow is almost fully developed at the exit. Even though the fully developed flow may not be achieved at the exit of the channel, this zero-gradient boundary condition offers sufficient flexibility for the flow distribution. Furthermore, by choosing an extended computational domain it was 182 The source term 5* in the porous region, is composed of those terms contributing to the vorticity generation due to the presence of the porous medium. In addition, the above stream function and vorticity are defined as follows (7.8) Oy aX ox ay (7-9> It should be noted that the variables v and T in the porous regions are both volume- average quantities, as described by Vafai and Tien (1981). It should also be noted that the effects of local non-thermal equilibrium and dispersion are neglected at this time based on the work of Vafai and Sozen (1990) and Sozen and Vafai (1990, 1993). 7.3 BOUNDARY CONDITIONS Due to the elliptic nature of the conservation equations the boundary conditions for all the field variables have to be specified along the entire boundary enclosing the solution domain. At the inlet of the channel, the stream function distribution is calculated from the specified inlet fluid velocity profile that is a fully developed parabolic profile. At the outlet, the gradient of the stream function in the axial direction is assumed to be zero; i.e., the streamlines are assumed to be perpendicular to the exit plane of the channel. This boundary condition frequently appears in the literature (Roache (1982)) and implies that the flow is almost fully developed at the exit. Even though the fully developed flow may not be achieved at the exit of the channel, this zero-gradient boundary condition offers sufficient flexibility for the flow distribution. Furthermore, by choosing an extended computational domain it was 183 ensured that the outflow boundary conditions had no detectable effect on the solution within the physical domain. This process is explained in more detail later on. The vorticity boundary conditions are derived from velocity distribution. For the thermal boundary conditions, the fluid is assumed to have a uniform temperature distribution at the inlet while at the outlet the temperature gradient along the flow direction is taken to be negligible, indicating that the convective effects are taken to be more dominant than the diffusion of heat. Again, by choosing an extended computational domain it was ensured that the thermal boundary conditions at the exit had no significant effect on the solution. For closure, the matching conditions, which satisfy the continuity of longitudinal and transverse velocities, normal and shear stresses, temperature, pressure, and heat fluxes, are applied across the porous/fluid interfaces (Sathe et al. (1990); Vafai and Kim (1990)). In summary, the boundary conditions can be described in the following dimensionless form (1) at the entrance (x*=0, 0 < /< l): u*= 6y*(l-y*) o*=0 / .2 . 3 \ $*=6(1- 2/) 0 = 0 (2) at the exit ( x*=L*, 0 (3) along the bottom plate 0 < x* < i\ + N W‘ + 2(N - 1)(W‘ + D ‘) < x* < i\ + NW* + N(W* + 2£>*) ‘ = 0 ( l - 4 ) < x ‘ i\ + 2(N - 1)(W* + D*) u = 0, v* = 0, \}r‘ = 0 $•=“ £ .0 = 1 ay (4) along the upper plate (0 u = 0, v* = 0, \|/‘ = 1 S‘ = “ !sr. 8 = 1 dy ® aIO"S [he side »aUS df the cavi,ies ( j :' = K +2 u = 0, u* = 0, \j/‘ = 0 185 r = - f ^ , e = i ox (6) along the porous/fluid interface: uf =up, vf = vp ♦ \ dvf 9u‘„ —9«:L + ,— a uJ- n 9 m! 9o 4 + — £ - ^ a/ ~ ]XeJT 9 / ’ dy* 9x* J ' 9;c* 99, 99 O — 9 1/ L= b- E. where subscripts / and p stand for fluid and porous, respectively. The parameter N (=1,2,...) is the number of cavity-block mixed porous obstacles. Note that the variables in the above equations are defined as follows: L: =—, 4=4-, 4=4., d*=—, w*=— RRR RR From the above equations, boundary conditions, and geometry arrangement of porous cavities and blocks, it is seen that the present problem is governed by six dimensionless parameters. These are the Darcy, Reynolds, and Prandtl numbers, inertia parameter, and geometric parameters A and B, where . W* D D* . H A = —r > B = —r and H - — (7.10) H W R 186 Further insight into the porous cavity block interactions on the fluid flow and heat transfer processes can be obtained by observing the variation of the local heat transfer rates on the channel wall. The local Nusselt number along the bottom plate may be defined from the local heat transfer coefficient as Nn _ ^ _ Kff(T»~ To) d 9 1 _ Kff 1 dQ j /71n “ ■ k, k,(T.-T.)dy'’-° k, ( l - e „ ) 3 / l ’ -° ( ' where keJf reverts to kf over regions with no porous substrate and 9m = (Tm- T 0)/(TW- T 0) is the dimensionless form of the bulk mean temperature Tm defined by \\u\Tdy T ^ b iC — (7-12) \y \d y The absolute value of the velocity, proposed by Kelkar and Patankar (1987), is used here, so as to properly account for regions of recirculating flow. It should be noted that conductivity of the fluid was chosen in the formulation of the Nusselt number. This choice gives a more meaningful comparison for the heat flux at the channel between the composite system and the case where there was no porous substrate. Therefore, the heat transfer augmentation will be even larger for a porous medium which has a larger thermal conductivity than that of the fluid's. 187 7.4 NUMERICAL SOLUTION METHOD The following is a general formulation for the diffusion-convection equation, which can be applied to vorticity, and temperature equations. r ^ W 0 - 13) Here d> represents either the temperature or vorticity function. The stream function equation (7.2) is solved using the SOR method. By using finite difference approximations, the governing equations can be reduced to a set of nonlinear algebraic equations that can be solved by an iteration scheme. Based on the nonuniform rectangular grid system in Fig. 7.2(a), the finite difference form of the vorticity and temperature equations is derived by volume integration over discrete cells surrounding the grid points (see Fig. 7.2(b)). Calculations were performed using the second upwind-differencing scheme for the convective terms with central- difference for the diffusive terms. This integration process results in a discretized equation that can be put into the form ^C^i.j = + + Q/^i.j+l + Q^i.j-1 + b (7.14) where Q = r,|Z(io]) (7.15a) (7.15b) C£ = r- f ^ 1+[“r'5*A ’°J) 188 ( 1 I|I 1 n |{ !{!! llllflliil II H UtlUiiil Sy. } } 0+1J) 8x i > s s OJ-1) N c (a) (b) A C (i.j) w w O.j+1) w (i-1 (i-1 j) lllliliillliilii IUIUIIU Fig 7.2(a) Grid system for the computational domain, (b) Local integration 189 c„ =r.^-(i+[-r.Sj-„«.,°]) (7.15c) cs = r,^(i+[r,5yA,°]) (7.15d) b = S° Ax&y (7.15e) and Cc — CE + Cw + CN + C5 ( 15f) The finite difference equations for £,* and 8 obtained in this manner were solved by the extrapolated-Jacobi scheme. This iterative scheme is based on double cyclic routine, which translates into a sweep of only half of the grid points at each iteration step (Adams and Ortega (1969)). The numerical procedure for solving the finite- difference equations is: 1. Overlay the computational domain with a finite difference mesh. 2. Assign values of Re, Da, A ,A,B, and N, and initialize values for^*, \jr* u, t), and 0 in Eqs. (7.1)-(7.6), and set boundary conditions. 3. Calculate the new values of vorticity at each node by using Eq. (7.14) for 4. Calculate the new values of stream function t|/* at each node using the SOR method and utilizing the obtained values of q* from step 3. 5. Calculate the new values of the velocity from u=\|ry* and 6. Update new boundary values using the new nodal values of \)/‘ and 7. Repeat step 3 to 6 , until the criterion of convergence for t," and \j/* is satisfied. 190 8. Calculate the temperature 9 from the Eq. (7.14) for 9 the Y* values obtained from step 7 until the criterion of convergence for 9 is satisfied. Here the following convergence criterion is used in these iterative procedures: max < 1 0 ' (7.16) In the present numerical calculation of coupled elliptic governing equations with an extended computational boundary condition downstream of channel, as explained previously, it is necessary to artificially specify the exit boundary location. The suitable location was chosen by trial and error to ensure that recirculation zone was inside of the computational domain. Therefore, by choosing an extended computational domain it was ensured that the computational outflow boundary condition had no effect on the physical domain solution. A nonuniform mesh system with a very fine grid spacing in regions of steep gradients, such as those close to the wall, corners and blocks, was selected to obtain accurate vorticity, streamline, and isotherm distributions. Fig. 2(a) depicts the nonuniform grid system for the computational domain. We employed a proper combination of A* and Ay to assure stability. This was done by a systematic decrease in the grid size until further refinement of the grid size showed no more than a 1 percent difference in the convergent result. The choice of 105 grid points in the y direction and 250 points in the x direction were found to provide grid independence for most of our result. 191 In order to obtain the vorticity at the wall, the assumption of the linear variation of vorticity from the wall to the neighboring point was used (Roache (1982)), that is, e. r30lC-Y*,) , Cm where i denotes the boundary node and An is the spatial interval in the direction normal to the boundary. The vorticity at sharp corners requires special consideration. Here, average treatment for the evaluation of vorticity suggested by greenspan (1969) was applied to model the mathematical limit of a sharp comer as appropriately as possible. To ensure the continuity of the diffusive and convective fluxes across the porous/fluid interface, the harmonic mean formulation suggested by Patankar (1980) was employed to handle the abrupt changes in thermophysical properties, such as the permeability and the thermal conductivity, across the interface. All of these effects on the interface are summarized in the nondimensional parameters Da, A, and Pr. For the present case Da, A, and Pr at the interface of a control volume are as follows: 2Dae/rD a, 2A e/rA , 2Pr„Pr, Da, = ------— f- , A . = Pr, = &-JL (7.17) Da„+Da, A„ + A/ 7 P^+Pr, } where the subscripts and I refer to effective, fluid, and interfacial, respectively. Therefore, instead of the source terms in Eqs. (7.4) and (7.6), the following were used across the interface: 192 in the fluid side of the interface (7.18a) Re dy and in the porous side of the interface ReDa (7.18b) u d + — — Re dy In addition, to accommodate the solution of the transport equations in both fluid and porous regions, the effective viscosity of the fluid-saturated porous medium is set to be equal to the viscosity of fluid. It has been found that this approximation provides good agreement with experimental data (Lundgren (1972) and Neale and Nader (1984)). Note that at any time constant values of Da and A for a specified porous substrate were used. The mathematical model and the numerical scheme were checked by comparing the results obtained from the present numerical results with other relevant limiting cases available in the literature. The relevant studies for our case correspond to the problem of hydrodynamically fully developed forced convection in a channel partially filled with a porous medium on the wall (Poulikakos and Kazmierczak (1987)), and external forced convection over a flat plate embedded in a porous medium (i.e. H*~>oo and IV*—>oo, representing the full porous medium case (Vafai and Thiyagaraja (1987)). The result of these comparisons (being similar to those 193 presented in Vafai and Kim (1990)) showed that the numerical model predicts quite accurately the velocity and temperature fields in a porous/fluid composite system. 194 7.5 RESULTS AND DISCUSSION The fixed input parameters that were used in all the simulations were R=1, ^,—13, and k e/r / k f = 1.0. To demonstrate the flow and temperature fields, only the portion concentrating on the porous/fluid region and its close vicinity are presented. However, it should be noted that the computational domain includes a larger region than what is displayed in the subsequent figures. The effects of the cavity-block porous obstacles on the flow and temperature fields are illustrated in Fig. 7.3 for a typical case. For the case shown in Fig. 7.3, the Reynolds number is 750, the Darcy number is 3x 10'5, the inertia parameter is 0.7, the dimensionless height and width of the porous cavity and block are 0.25 and 1.0 respectively, and the spacing between the porous cavities and the blocks is 1. It can be seen from Fig. 7.3(a) that the presence of cavity-block porous obstacles causes the flow to bend significantly and to detach from the wall surface forming a recirculation region behind each porous block. Small eddies are generated on the smooth upper plate surface corresponding to the reattached region on the bottom plate. However, even though the local behavior of flow adjacent to the cavity-block porous obstacles is affected by the existence of porous obstacles, flow somewhat downstream of the porous block obstacles is not influenced at all. Here the mechanism for the formation of recirculations in the rear part of porous block is due to the relatively larger resistance that the flow encounters inside the porous block, which in turn displaces the flow by inducing a blowing effect from the porous region into the fluid region. Shortly after the porous block, the blowing effect disappears. Instead, the longitudinal pressure gradient, caused by the pressure drop behind the porous block, 1.0 0.75 0.50 0.25 .0.1. 0. •0 .01- •-0.00I 7*—0.25 1.0 0.75 ■0.6 0.50 .0.1 0.25 0. 0.9- -04 0.9 .0.95 -0 .2 5 , 0.95- . 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 X Fig. 7.3 (a) Streamlines, (b) isotherms, and (c) local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10'5’ A=0.35, Pr=0.7, kf// / k/ = 1.0 A =4, B =1, H*= 0.25. vO U\ 50 without porous medium but with the empty cavities with porous media 45 40 35 30 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 * X Fig. 7.3 (a) Streamlines, (b) isotherms, and (c) local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Re=750, Da=3xl0"5, A=0.35, Pr=0.7, k tff / k f = 1.0 A =4, B =1, H*= 0.25. 197 creates a suction effect which moves the flow downwards to the space between the porous blocks. The flow patterns, including the shape of recirculation zone, and the interactions between vortex flow inside the cavities and the external circulating flow play a significant role in affecting the temperature field. Figure 7.3(b) shows the isotherms corresponding to the above flow field. It can be seen that the thickness of both upper and lower thermal boundary layers increases along the length of heated plates. These thermal boundary layers become considerably distorted around the porous obstacle regions. Shortly downstream of the porous cavity-block obstacles, the symmetrical character of temperature field recurs. It should be noted that compared to the case without porous blocks in the channel, both upper and lower thermal boundary layers meet earlier due to the presence of porous blocks, which as explained earlier pushes the flow near the bottom plate upward. The variation of local Nusselt number corresponding to the flow field shown in Fig. 7.3(a) is depicted in Fig. 7.3(c). It should be noted that the case without the porous medium is for the channel which still does include the empty cavities. This way the effects of the porous medium in enhancing the heat transfer from empty cavities is also illustrated in this work. In general, the porous cavity creates a close vortex region and reduces the heat transfer rate from the heated wall (the trough a in Fig. 7.3(c)), while the porous block produces recirculating flow, which improves fluid mixing, resulting in a heat transfer augmentation (the peak f). In addition, the interaction between the circulation behind the porous blocks and the vortex inside the porous cavities, can result another heat transfer augmentation (the peak g). Similarly other fluctuations in the Nusselt number distribution are the result of various separations and reattachments occurring around the porous cavity-block region. 198 Effects o f the Reynolds Number Figures 7.4-7.6 show the effect of the Reynolds number on the flow and temperature fields for Da=3 x 10'5, A=0.35, Pr=0.7, A=4,5=1, and N=3, for Re=750, 1000, and 1500, respectively. Comparison of the streamlines in Fig. 7.4 shows that as the Reynolds number increases, the relative strength of the recirculation zone decreases while its lateral size increases. The recirculation zone thus occupies the whole inter-block spacing, which reduces the interaction between the vortex inside the cavity and the closest recirculation zone. The reason for this trend is that increasing the Reynolds number, increases the fluid's momentum, resulting in a larger penetration into the porous blocks. This in turn increases the required length before reattachment occurs. Comparison of the isotherms in Fig. 7.5 indicates that as Reynolds number increases the isotherms in channel region become less distorted. Also as a result of the above-described flow field the thermal penetration into each of the cavities which are followed by a porous block is reduced. The variation of local Nusselt number for various Reynolds numbers are displayed in Fig. 7.6. Again, it should be noted that the case without the porous medium is for the channel which still does include the empty cavities. As expected, when Reynolds number grows from 750 to 1000, both peak and trough values of Nux increase. However, as Reynolds number increases further to 1500, the results for both peak and trough values of Nu^ are significantly reduced. The reason for this is that as the Reynolds number increases to 1500 the circulation zone occupies the whole inter block spacing, which completely separates the core flow from the heated wall, thus, reducing the convective heat transfer rate from the wall. 0.75 0.50 0.25 -o.i- -0.01 -0.01 .0.01. (c)Re=1500 -0.25 O 2 4- 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 X Fig. 7.4 Effects of the Reynolds number on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Da=3 x 10’5, A=0.35, A =4, 5=1, H*= 0.25. I—* vO VO 1 .0 (c)Re=1500 Fig. 7.5 Effects of the Reynolds number on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Da=3 x 10~5, A=0.35, Pr=0.7, f k f =■ 1.0, A=4, 5=1, H*=0.25. o o 60 — without porous medium but with the empty cavities — — with porous media 1: Re=750 S 2 \ 50 - 2:Re=1000 j \ 3: Re=1500 ! \ 45 40 35 Nu* 30 20 X Fig. 7.6 Effects of the Reynolds number on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Da=3xl0"5, A=0.35, Pr=0.7, kejr/kf = 1.0 A =4, B =1, H*=0.25. 202 Effects of the Darcy Number The Darcy number, Da=K /R 2, is directly related to the permeability of the porous medium. The effect of Darcy number is shown in Figs. 7.7-7.9, for Re=750, A =0.35, Pr=0.7, A =4, 5 = 1, and N=3, for Da=3xl0"5, 5x1 O’5 and 9xl0"5, respectively. It can be seen from Fig. 7.7 that as Darcy number increases, the distortion of streamlines becomes less significant and the height of recirculation behind porous blocks becomes smaller. The interaction between the vortex inside the cavity and the external flow is found to depend on whether the recirculation zone occupies the whole inter-block spacing or not. Note that the vortex inside the first cavity is relatively unaffected by the flow outside the cavity. As expected, the distortion of isotherms in the channel region corresponding to the flow field becomes less noticeable with an increase in the Darcy number (Fig. 7.8). The effect of Darcy number on the local Nusselt number distribution is shown in Fig. 7.9. It can be seen that the Darcy number has a significant impact on the variation of Nu*. The peak of each cycle in the local Nusselt number distribution lowers and moves to the right as Darcy number increases. This is because for the larger Darcy number the accelerated fluid penetrates deeper into the porous block. Comparison of local Nusselt number distribution between the channel with and without porous media shows that for the range of Darcy numbers investigated (3 x 10" 5 to 9x 10'5) the heat transfer augmentation increases as the Darcy number decreases. Inertial Effects When the Reynolds number based on the pore diameter of porous medium is large, the inertial effects become significant. Figures 7.10-7.11 illustrate the effect of o 0.75 •0.9' 0.50 0.25 .0.1' ■0.1 ■ 0.01. -0.0001^ ^ --O.001 -0 .2 5 •0.9- 0.75 ■0.7. y* 0.50 0.25 •0. 1' •0.01 •0.0! ■0.01. •0.005' -0.001 -0 .0001- -0 .2 5 0.75 •0.9- 0.50 0.25 •0 .01- ^ --o.oor -0.005 — 0.25 -0 .0001- 0 2 4 6 8 10 12 14 16 18 20 Fig. 7.7 Effects of the Darcy number on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, A=0.35, to A=4, 5=1, //*=0.25. o 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 X Fig. 7.8 Effects of the Darcy number on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, A=0.35, Pr=0.7, / k f = 1.0, A =4, B =1, H*= 0.25. 204 50 1: Da = o® (without porous medium but with the empty cavities) 45 2: Da = 3 x 10"5 3: Da = 5xl0"5 4: Da = 9 x 10"5 40 35 30 Nux 20 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 * X Fig. 7.9 Effects of the Darcy number on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, A=0.35, Pr=0.7, k ejr / k^- = 1.0 A =4, B =1, H*=0.25. 206 inertial parameter on streamlines and isotherms for Re=1500, Da=3xlO'5, Pr=0.7, A=4, and 5=1 for A=0.35, 21 and 35, respectively. It can be seen that the strength of the of the recirculation zone increases as the inertial number increases. Furthermore, for larger inertial numbers the interaction between the vortex inside the inter-block cavity and the core flow increases. This is due to the larger bulk frictional resistance that the flow inside the porous block will experience for larger inertial numbers. Therefore, larger values of A would lead to a larger blowing effect which increases the distortion in the streamlines and reduces the penetrating extent of the flow into the porous block. As a direct result of the discussed flow field (Fig. 7.11), the larger the value of A , the more noticeable the distortion of the isotherms. As can be seen in Fig. 7.12, with the increase of the value of A, the augmentation of heat transfer rate caused by porous blocks increases. Prandtl Number Effects The Prandtl number effects are shown in Fig. 7.13 for three different Prandtl numbers for fixed values of Re=750, D a=3xl0'5, A=0.35, A=4, and 5=1. These values for the Prandtl number - 0.7 (air), 7.0 (water) and 100 (typical value for oil) - are chosen such that they will cover a wide range of thermophysical properties. Obviously, the variations of Prandtl number have no effect on the flow field since the values of Re, Da and A are fixed. Therefore, the flow field shown in Fig. 7.13(a) is the same for all Prandtl numbers. As can be seen in Fig. 7.13(b)-(d), increasing the Prandtl number in the same flow field decreases the thickness of thermal boundary layer in the core flow and increases the extent of thermal penetration into the cavity. Due to the lower value of the thermal diffusivity, the temperature gradient is larger for 0.75 •0.9- 0.50 0.25 -0.1- -0.01 •-0.01 .0.01, (a) A=0.35 - 0.0 0 1 -0.005 —0.25 0.75 .0.9 0.50 0.25 ■o.oi .0.1- .0.1. (b) A=21 -0.0001 \ v - 0.001 -0 .2 5 0.75 .0.9. 0.50 0.25 .0 .1, .0.1. .0 .01 . —0.001» (c) A=35 .-o.oooi* —0.25 O 2 4 6 8 10 12 '14 16 18 20 22 24 26 28 30 32 34 36 X Fig. 7.10 Influence of the inertial parameter on streamlines for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate ror Re=l 500, Da=3xl0"5, A =4, B =1, H*=0.25. -0.2_ -°*9 Ib^h0-9 (a) A=0.35 N , M0.95 p^j-0.95 - ...... ------0 .7 5 y 0 .5 0 0 .2 5 0 .2 5 0 .7 5 0 .5 0 0.2 5 0 .2 5 Fig. 7.11 The influence of the inertial parameter on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=1500, D a = 3 x l0 '5, Pr=0.7, / k f = 1.0, A =4, B =1, H*=0.25. ^ o 00 209 65 1: A=l=0 (without . porous medium but with the empty cavities) 60 - 2: A=0.35 3: A=21 55 - 4: A=35 50 45 40 35 30 25 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 X Fig. 7.12 The influence of the inertial parameter on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity- block obstacles on the bottom plate for Re=1500, D a=3xl0 5, Pr=0.7, kejr / k, = 1.0, A=4,B =1, //*=0.25. 0 0.75 .0 9 0.50 0.25 ■o.i- 4.00! -0 .2 5 0.75 ■or 0.50 0.25 or -0 3 0 3 -0.25 , 0.93- 0.75 0 .5 0 0.25 .0 3 . (c)Pr=7 0 3 - -0.25 0.95- , 0.93. “W 0.75 0.50 0.25 (d)Pr=100 .0.9 —0.25 O 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 52 34 36 X Fig. 7.13 Prandtl number effects on streamlines and isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10”5, A=0.35, k ejr / k , = 1.0, A =4, B =1, H*= 0.25. to o “i— r—i— i— i i i r i — i— r — without porous media i , but with the empty cavities i ,■ — witli porous media 1: Pr=0.7 2: Pr=7 3: Pr=100 H ,' i J n, 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Fig. 7.14 Prandtl number effects on local Nusselt number distribution for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3xl0 , A=0.35, k tff / k f = 1.0 A =4, B =1, H*=0.25 212 larger Prandtl number. As expected, both peak and trough values of Nu^ increase with an increase of the Prandtl number as seen in Fig. 7.14. Effect of the Geometry of the Porous Cavity-Blocks The geometric parameters A and B reflect the influence of the porous cavity's or block's aspect ratio and interspace between porous cavity and block. The effect of changing the height of the porous block on the flow and temperature fields is depicted in Figs. 7.15 and 7.16. The streamlines and isotherms for Re=750, Da=3xl0~5, A=0.35, Pr=0.7, and B=1 for A=4 to 8 are shown in Fig. 7.15(a)-(b) and Fig. 7.16(a)- (b), respectively. It can be seen that the streamlines and isotherms get less distorted as the height of porous block is decreased . In addition, the strength and size of recirculation regions behind the porous block is lessened significantly for smaller block heights. Also, for smaller porous block heights the influence of the external core flow on the vortex inside each cavity is diminished, which reduces the thermal penetration into the porous cavity. This is due to the considerable decrease of blowing effect caused by shorter porous blocks (see Fig. 7.16(a)-(b)). The effect of porous-block interspace is shown in Fig. 7.15(a)-(c) and Fig. 7.16(a)-(c), respectively. Comparison of streamlines in Fig. 7.15(a)-(c) shows that the porous block-cavity interaction decreases as the spacing between the porous block and cavity increases from 5=1 to 5=2, in which each porous cavity is occupied by a closed vortex. This is a direct result of the increase in the space between the porous block and cavity. The closed vortex inside the cavities leads to a decreased thermal penetration into the porous cavity, as shown in Figs. 7.16(a)-(c). Finally it was also 1 .0 0.75 0.50 0.25 0. (a) A = 4 aiid5=l -0 .2 5 1.0 0.75 y* 0.50 0.25 0. -0 .2 5 1.0 0.75 0.50 0.25 0. ■•0.001 -0.001 •0 .0001 . -0 .2 5 0 2 4 5 8 10 12 14 16 18 20 22 24 26 28 30 32 54 36 X Fig. 7.15 The influence of the geometric parameters A and B on streamline for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10"5, A=0.35, Pr=0.7, / kf = 1.0, B =1, H*=0.25. 1 .0 0 .7 5 •0 .6 ' 0 .5 0 •0 .1- 0 .2 5 .0.6 . 0. 0.3- 0.3 - 0 . 2 5 1.0 0 .7 5 •02' y 0 .5 0 0 .2 5 •0.6* 0. •0.9 (b) 4=8 for the block only, B = 1 °*®‘ « » t - » « t 0.95* 0 .7 5 0 .5 0 0 .2 5 .0.6 - 0. 0.3 •0.9 04' 0.9 0.9 - 0 .2 5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 . • * Fig. 7.16 The influence of the geometric parameters A and B on isotherms for flow in a parallel plate channel with alternate porous cavity-block obstacles on the bottom plate for Re=750, Da=3 x 10"5, A=0.35, Pr=0.7,k^ /kf = 1.0, B =1, //*=0.25. £ 215 shown that increasing the number of porous cavity-block obstacles had no effect on the flow and heat transfer characteristics which were presented in this work. CHAPTER VIII CONCLUSIONS AND RECOMMENDATIONS 8.1 CONCLUSIONS Throughout this work the effects of boundary friction and inertia on convective heat transfer and fluid flow at the porous/fluid composite system have been studied by both integral and numerical methods. Among various configurations, six fundamental ones are studied in depth: these are external forced convection over a flat plate emplaced with porous slabs, blocks, cavities or alternate cavity-blocks, and internal forced convection in a parallel flat plate channel with porous block or cavity-block structure on the bottom plate. In chapter 2 the problem of external forced convection over a boundary covered with a porous substrate is investigated by integral method. Under the assumption that two momentum boundary layers and a thermal boundary layer exist in the flow fluid of porous/fluid composite system, three integral momentum boundary layer equations for three different regions — porous boundary layer region, fluid boundary layer region, and the interface region between these two regions— and an integral energy equation were described. The fourth-order-Runge-Kutta method is applied to solve this nonlinear simultaneous ordinary differential equation for the four unknowns 216 217 5P, 8f , 5, and r. The solutions obtained by integral analysis are examined and compared with the velocity and temperature distributions obtained by numerical method. The results indicate a qualitative agreement between integral and numerical methods. However, the presented integral method dose not have the mathematical complications and therefore reduced tremendously the computational time. In chapter 3 the problem of forced convection in an external boundary with intermittently emplaced porous cavities is analyzed numerically. Characteristics of the flow and temperature fields in the composite layer and the effects of various governing dimensionless parameters, such as the Darcy number, Reynolds number, Prandtl number, the inertia parameter as well as two pertinent geometric parameters were thoroughly analyzed and discussed. The fundamental information presented here can be extended to examine various applications such as in electronic cooling and in heat exchanger design, reduction of skin friction and heat transfer enhancement or augmentation, some of the manufacturing processes, geothermal reservoirs and oil extraction. The present work constitutes one of the first analyses of the laminar separated forced convection through porous cavities. In chapter 4 the problem of external forced convection over a porous block array is investigated numerically. The flow field and thermal characteristics of external laminar forced convection flow over a porous block array are investigated numerically. Computations for flow and temperature fields have been performed to study the effects of various governing parameters, such as Reynolds number, Darcy number, inertial parameter, Prandtl number and two geometric parameters. Comparisons of the local Nusselt number at the wall with and without porous blocks 218 were also made. Two distinct boundary layers were shown to exist for the velocity field, while only one boundary layer was observed for the temperature field. It was shown that the porous block array significantly reduces the heat transfer rate at the wall. The extent of the reduction depends on the values of the governing parameters. The variation of local Nusselt number distribution with an increasing or decreasing mean is determined by whether or not there is the isolated vortex flow near the wall. Therefore, this configuration can be used as a thermal insulator for flows which have strong parabolic characters. Overall it is shown that the presence of a porous block array near an impermeable boundary significantly changes the convection characteristics. In chapter 5 the problem of external forced convection through alternate porous cavity-block obstacles is examined numerically. The effects of all of the governing parameters such as the Darcy number, Reynolds number, inertia parameter, Prandtl number, the two geometric parameters, and the number of cavity-block structure on the flow and temperature fields were explored in detail. Throughout the study a reasonably wide range of the independent parameters were covered. An in-depth discussion of the results for various physical interactions between the recirculating flows inside of the cavity and the external flow was presented. Several interesting phenomena such as the interactions between the blowing and displacement effects from the porous blocks and the vortices penetrating into the porous cavities were presented and discussed. These results indicate that altering some parametric values can have significant effects on the external momentum and thermal boundary layer characteristics. 219 In chapter 6 the problem of forced convection in a parallel flat channel using porous blocks on the bottom plate is numerically studied based on Brinkman- Forchheimer-extended Darcy Model. The rectangular porous blocks change the incoming parabolic velocity field considerably, resulting in the formation of recirculating flow penetrating these porous blocks. These vortices which can be controlled by altering some governing parametric values have significant effects on the transfer characteristics. The effects of the Reynolds numbers, Darcy numbers, inertial parameters, Prandtl numbers, and geometric parameters on forced convection enhancement in a channel using multiple emplaced porous blocks have been analyzed in detail and the existence of an optimum porous matrix is demonstrated. Comparison of the local Nusselt number distributions between the channel with and without porous blocks clearly shows that significant heat transfer augmentation can be achieved through the emplacement of porous blocks. In chapter 7 the problem of forced convection in a parallel plate channel with alternate porous cavity-block obstacles is solved numerically based on Biinkman- Forchheimer-extended Darcy Model. The rectangular porous cavities and blocks change the incoming parabolic velocity field significantly, resulting in the formation of recirculating zone between the porous blocks and cavities. It is shown that the porous media provides a penetrating random structure, which augments the mixing in the fluid and profoundly changes the heat transfer characteristics within the channel. The dependence of flow and temperature characteristics on the governing parameters, such as the Darcy number, the Reynolds number, inertia parameter, the Prandtl number and two geometric parameters, is documented. The results of this investigation show that the interaction between the vortices residing inside the 220 cavities and the vortices after the porous blocks have a significant effect on the flow and thermal characteristics of the channel. 8.2 RECOMMENDATIONS FOR FURTHER RESEARCH Several possible modifications for the present study are recommended. These are: 1. Additional effects such as variable porosity and thermal dispersion near the wall can be incorporated into the present model when the porous cavity or block has large aspect ratio. 2. The accumulated knowledge gained through this investigation will provide a fundamental framework for predicting heat transfer and fluid flow characteristics for more complicated configurations. 3. It would be of interest and importance to study the relationship between the heat transfer augmentation or retardation caused by the porous obstacles and the associated pressure-drop penalty. LIST OF REFERENCES Adams, J. and Ortega, J., 1982, “A Multicolor SOR Method for Parallel Computation,” Proceedings of Int. Conf. on Parallel Procession, pp. 53-56. Aung, W., 1983, “An Interferometric Investigation of Separated Forced Convection in Laminar Flow Past Cavities,” ASME J. Heat Transfer. Vol. 105, pp. 505-512. Aung, W. and Watkins, C. B., 1979, “Heat Transfer Mechanisms in Separated Forced Convection,” Proceedings of the NATO institute on Turbulent Forced Convection in Channels and Rod Bundles. Beavers, G. I. and Joseph, D. D., 1967, “Boundary conditions at a naturally permeable wall,” J. Fluid Mech.. Vol. 30, pp. 197-207. Beckermann, C. and Viskanta, R., 1987, “Forced Convection Boundary Layer Flow and Heat Transfer Along a Flat Plate Embedded in a Porous Medium,” Int. J. Heat Mass Transfer. Vol. 30, pp. 1547-1551. Beckermann, C., Ramadhyani, S., and Viskanta, R., 1987, “Natural Convection Flow and Heat Transfer Between a Fluid Layer and a Porous Layer Inside a Rectangular Enclosure.” ASME J. Heat Transfer. Vol. 109, pp. 363-370. Beckermann, C., Viskanta, R., and Ramadhyani, S., 1988, “Natural Convection in Vertical Enclosures Containing Simultaneously Fluid and Porous Lavers.” J. Fluid Mech.. Vol. 186, pp. 257-284. Bejan, A., 1984, Convection Heat Transfer, John Wiley and Sons, New york. Bhatti, A., and Aung, W., 1984, “Finite Difference Analysis of Laminar Separated Forced Convection in Cavities,” ASME J. Heat Transfer. Vol. 106, pp. 49-54. Cheng, P., 1977, “Combined Free and Forced Convection Flow about Inclined Surfaces in Porous Media.” Int. J. Heat Transfer. Vol. 20, pp. 807-814. Cheng, P., 1978, “Heat Transfer in Geothermal System,” Adv. Heat Transfer, Vol. 14, pp. 1-105. Cheng, P., 1982, “Mixed Convection about a Horizontal Cylinder and a Sphere in a Saturated Porous Medium.” Int. J. Heat Mass Transfer. Vol. 25, pp. 1245-1247. 221 222 Cheng, P. and Zheng, T. M., 1985. “Thermal Dispersion Effects in Forced Convection Plume above a Horizontal Line Source of Heat in a Packed Bed,” Int. J. Comm. Heat Mass Transfer. Combamous, M. A. and Bories, S. A, 1975, “Hydrothermal Convection in Saturated Porous Media,” Adv. Hydroscience, Vol. 10, pp. 231-307. Fletcher L. S., Briggs, D. G. and Page, R. H., 1974, “Heat transfer in separated and reattached flow: an annotated review.” Israel J. Tech. Vol. 12, pp. 236-261. Gerald, C. F., 1978, Applied Numerical Analysis, Chap. 5, 2nd ed., Addison- Wesley, New York. Gooray, A. M., Watkins, C. B., and Aung, W., 1981, “Numerical Calculations of Turbulent Heat Transfer Downstream of a Rearward-Facing Step,” Proceedings of the 2nd International Conference on Numerical Method in Laminar and Turbulent Flow, Venice, Italy, pp. 639-651. Gooray, A. M., Watkins, C. B., and Aung, W., 1982, “K-e Calculations of Heat Transfer in Redeveloping Turbulent Boundary Layers Downstream of Reattachment,” Presented at AIAA/ASME Thermophysics Conference, St. Louis, Mo, June 6-9. Gosman, A. D., 1976, The TEACH-T Computer Program Structure, Flow, Heat and Mass Transfer in Turbulent Recirculating Flows-Prediction and Measurement, Lecture Notes from McGill University, Canada. Greenspan, D., 1969, “Numerical Studies of Steady, Viscous, Incompressible Flow in a Channel with a Step,” J. Eng. Mathematics. Vol. 3, No. 1, pp. 21-28. Joseph, D. D. and Tao, L. N., 1966, “Lubrication of a porous bearing-stokes' Solution.”J. app. Mech. pp. 753-760. Kaviany, M., 1985, “Laminar Flow Through a Porous Channel Bounded by Isothermal Parallel Plates.” Int. J. Heat Mass Transfer. Vol. 28, pp. 851-858. Kaviany, M., 1987, “Boundary Layer Treatment of Forced Convection Heat Transfer from a Semi-Infinite Flat Plate Embedded in Porous Media,” ASME J. Heat Transfer. Vol. 109, pp. 345-349. Kelkard, K. M. and Patankar, S. V., 1987, “Numerical Prediction of Flow and Heat Transfer in a Parallel Plate Channel with Staggered Fins,” J. Heat Transfer. Vol. 109, pp. 25-30. Koh, J. C. Y. and Colony, R., 1974, “Analysis of Cooling Effectiveness for Porous Material in a Coolant Passage,” ASME J. Heat Transfer. Vol. 96, pp. 324-330. 223 Koh, J. C. Y. and Stevens, R. L., 1975, “Enhancement of Cooling Effectiveness by Porous Material in Coolant Passage,” ASME J. Heat Transfer. Vol. 97, pp. 309- 311. Levy, T. and Sanchez-Palencia, E., 1975, “On Boundary Conditions for Fluid Flow in Porous Media,” Int. J. Engne. Sci.. Vol. 13, pp.923-940. Lundgren, T. S., 1972, “Slow Flow Through Stationary Random Beds and Suspensions of Spheres,” J. Fluid Mech.. Vol. 51, pp. 273-299. Nakayama, A., Kokudai, T., and Koyama, H., 1990, “Non-Darcian Boundary Layer Flow and Forced Convective Heat Transfer Over a Flat Plate in a Fluid-Saturated Porous Medium,” ASME J. Heat Transfer. Vol. 112, pp. 157-162. Neale, G. and Nader, W., 1984, "Practical Significance of Brinkman's Extension of Darcy's Law Coupled Parallel Flows within a Channel and a Bounding Porous Medium," Can. J. Chem. Eng.. Vol. 52, pp. 475-478. Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D. C. Poulikakos, D., and Bejan, A., 1985, "The Departure from Darcy Flow in Natural Convection in a Vertical Porous Layer," Phvs Fluids. Vol. 28, pp. 3477-3484. Poulikakos, D., 1986, “Buoyancy-Driven Convection in a Horizontal Fluid Layer Extending Over a Porous Substrate,” Phvs. Fluids. Vol. 29, pp. 3949-3957. Poulikakos, D. and Kazmierczak, M. , 1987, “Forced Convection in a Duct Partially Filled with a Porous Material,” ASME J. Heat Transfer. Vol. 109, pp. 653-662. Poulikakos, D. and Renken, K., 1987, “Analysis of Forced Convection in a Duct Filled with Porous Media in Heat Transfer,” in Geophysical and Geothermal System, ASME HTD, Vol. 76, pp. 9-20. Richardson, S., 1971, “A Model for the Boundary Condition of a Porous Material Part II,” J. Fluid Mech. Vol. 49, PP327-336, 1971. Roache, P. J., 1976, Computational Fluid Dynamics, Hermosa, Albuquerque, NM. Rohsenow, W. M. and Hartnett, J. P., 1973, Handbook of Heat Transfer, Mcgraw- Hill, New York. Shah, R. K. and London, A. L., 1987, Laminar Flow Forced Convection in Duct, Suppl. 1, in Advances in Heat Ttansfer, ed, T. F. Irvine and J. P. Hartnett, Academic press, New York. 224 Sathe, S. B., Lin, W. Q., and Tong, T. W., 1988, “Natural Convection in Enclosures Containing an Insulation with a Permeable Fluid-Porous Interface,” Int. J. Heat and Fluid Flow. Vol. 9, pp. 389-395. Sinha, S. N., Gupta, A. K., and Oberai, M. M., 1981, “Laminar Separating Flow Over Backsteps and Cavities-Part II: Cavities,” AIAA J. Vol. 20, No. 3, pp. 370-375. Sozen, M. and Vafai, K., 1990, "Analysis of the Non-Thermal Equilibrium Condensing Flow of a Gas Through a Packed Bed" Int. J. Heat and Fluid Flow. Vol. 33, pp.1247-1261 Sozen, M. and Vafai, K., 1993, "Longitudinal Heat Dispersion in Packed Beds with Real Gas Flow," AIAA J. of Thermophvsics Heat Transfer. Vol. 7, pp. 153-157 Taylar, G. I., 1971, “A Model for the Boundary Condition of a Porous Material Part I.”J. Fluid Mech. Vol. 49, pp. 319-326. Tien, C. L. and Vafai, K., 1989, “Convective and Radiative Heat Transfer in Porous Media,” Adv. Appl. Mech., Vol. 27, pp. 225-281. Vafai, K., 1984, "Convective Flow and Heat Transfer in Variable Porosity Media," L Fluid Mech.. Vol. 147, 233-259 Vafai, K., 1986, "Analysis of the Channeling Effect in Variable Porosity Media," ASME J. Energy Resources Technology. Vol. 108, pp. 131-139 Vafai, K. and Kim, S. J., 1990, “Analysis of Surface Enhancement by a Porous Substrate.” ASME J. Heat Transfer. Vol. 112, pp. 700-705. Vafai, K., Alkire, R. L. and Tien, C. L., 1985, “An Experimental Investigation Heat Transfer in Variable Porosity Media,” ASME J. Heat Transfer. Vol. 107, pp. 642- 647. Vafai, K. and Sozen, M., 1990, "Analysis of Energy and Momentum Transport for Fluid Flow Through a Porous Bed," ASME J. Heat Transfer. Vol. 112, pp. 690- 699 Vafai, K. and Thiyagaraja, R., 1987, “Analysis of Flow and Heat Transfer at the Interface Region of a Porous Medium,” Int. J. Heat Mass Transfer. Vol. 30, pp. 1391-1405. Vafai, K. and Tien, C. L., 1981, “Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media,” Int. J. Heat Mass Transfer. Vol. 24, pp. 195-203. Vafai, K. and Tien, C. L., 1982, “Boundary and Inertia Effects on Convective Mass Transfer in Porous Media,” Int. J. Heat Mass Transfer. Vol. 25, pp. 1183-1190. 225 Yamamoto, H., Seki, N. and Fukusako, S., 1979, “Forced Convection Heat Transfer on Heated Bottom Surface of a Cavity,” ASME J. Heat Transfer. Vol. 101, pp 475-479.