Embed Size (px)
Citation preview
NUMERICAL INVESTIGATION OF INCOMPRESSIBLE FLOW IN
GROOVED CHANNELS-HEAT TRANSFER ENHANCEMENT
BY SELF SUSTAINED OSCILLATIONS
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
THE MIDDLE EAST TECHNICAL UNIVERSITY
BY
TÜRKER GÜRER
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
OF
DOCTOR OF PHILOSOPHY
IN
THE DEPARTMENT OF MECHANICAL ENGINEERING
MARCH 2004
ii
Approval of Graduate School of Natural and Applied Sciences
Prof. Dr. Canan Özgen
Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of Doctor
of Philosophy
Prof. Dr. Kemal der
Head of Department
This is to certify that we read this thesis and that in our opinion it is fully adequate, in
scope and quality, as a thesis for the degree of Doctor of Philosophy
Prof. Dr. Hafit Yüncü
Supervisor
Examining Commitee Members
Prof. Dr. Faruk Arınç (Chairman) Prof. Dr. Nevzat Onur Prof. Dr. Tülay Özbelge Assoc. Prof. Dr. lker Tarı Prof. Dr. Hafit Yüncü
iii
ABSTRACT
NUMERICAL INVESTIGATION OF INCOMPRESSIBLE FLOW IN
GROOVED CHANNELS-HEAT TRANSFER ENHANCEMENT
BY SELF SUSTAINED OSCILLATIONS
GÜRER, A. Türker
Ph. D., Department of Mechanical Engineering
Supervisor: Prof. Dr. Hafit Yüncü
March 2004, 133 pages
In this study, forced convection cooling of package of 2-D parallel boards with
heat generating chips is investigated. The main objective of this study is to determine
the optimal board-to-board spacing to maintain the temperature of the components
below the allowable temperature limit and maximize the rate of heat transfer from
parallel heat generating boards cooled by forced convection under constant pressure
drop across the package. Constant heat flux and constant wall temperature boundary
conditions on the chips are applied for laminar and turbulent flows.
Finite elements method is used to solve the governing continuity, momentum
and energy equations. Ansys-Flotran computational fluid dynamics solver is utilized to
iv
obtain the numerical results. The solution approach and results are compared with the
experimental, numerical and theoretical results in the literature [1].
The results are presented for both the laminar and turbulent flows. Laminar flow
results improve existing relations in the literature. It introduces the effect of chip
spacing on the optimum board spacing and corresponding maximum heat transfer.
Turbulent flow results are original in the sense that a complete solution of turbulent
flow through the boards with discrete heat sources with constant temperature and
constant heat flux boundary conditions are obtained for the first time. Moreover,
optimization of board-to-board spacing and maximum heat transfer rate is introduced,
including the effects of chip spacing.
Keywords: Forced convection, self-sustained oscillations, grooved channels,
parallel plates
v
ÖZ
OYUKLU KANALLARDA SIKITIRILAMAZ AKIIN NÜMERK OLARAK
NCELENMES -KEND KENDN DEVAM ETTREN SALINIMLARLA ISI
TRANSFERNN ARTTIRILMASI
GÜRER, A. Türker
Doktora Tezi, Makina Mühendislii Bölmü
Tez Yöneticisi: Prof. Dr. Hafit Yüncü
Mart 2004, 133 sayfa
Bu çalımada, üzerlerinde ısı üreten çipler bulunan iki boyutlu bir kart
paketinin zorlanmı konveksiyon yoluyla soutulması incelenmitir. Çalımanın esas
amacı, sabit basınç kaybı altında, belli bir hacime yerletirilmi kartların tolere
edilebilen en yüksek çalıma sıcaklıını salayan optimum kart uzaklıının ve buna
karılık gelen maksimum ısı transfer hızının bulunmasıdır. Laminer ve türbülans akılar
için sabit ısı akısı ve sabit yüzey sıcaklıı sınır artları incelenmitir.
Denklemler sonlu elemanlar yöntemi kullanılarak çözülmütür. Çözüm
sırasında Ansys-Flotran Hesaplamalı Akıkanlar Dinamii kodundan faydalanılmıtır.
vi
Çözüm yaklaımı ve sonuçlar, literatürdeki deneysel ve nümerik çözümlerle
karılatırılmıtır [1].
Sonuçlar laminer ve türbülans akı için ikiye ayrılmıtır. Laminer akı çözümleri
literatürdeki denklemlere çip aralıının etkisini de ekleyerek, mevcut denklemleri
iyiletirmitir. Türbülans akı sonuçları üzerlerinde münferit ısı kaynakları olan plakalar
arasındaki türbülans akı için komple bir çözüm sunması açısından bu konuda
literatürde ilktir. Bunun yanı sıra, plaka aralıkları ve maksimum ısı transfer hızı çipler
arası mesafenin etkisi de göz önünde bulundurularak optimize edilmitir.
Anahtar Kelimeler: Zorlanmı Konveksiyon, kendi kendini devam ettiren
salınımlar, oyuk kanallar, paralel plakalar.
vii
ACKNOWLEDGEMENTS
I would like to thank and express my sincere appreciation to my supervisor
Prof. Dr. Hafit Yüncü, for his guidance and support throughout my thesis and in my
graduate study.
A great deal of gratitude goes to my wife, Banu Bayazıt Gürer, for her
understanding, patience, and support in all aspects.
I would like to thank Mr. Fuat Sava, my director at Aselsan, for his support
during the last two years.
My special thanks goes to my family Berin, brahim and Nilüfer Gürer for
their encouragement and support during my education.
viii
TABLE OF CONTENTS
ABSTRACT ...................................................................................................... iii
ÖZ ..................................................................................................................... v
ACKNOWLEDGMENTS..................................................................................vii
TABLE OF CONTENTS...................................................................................viii
LIST OF TABLES.............................................................................................xii
LIST OF FIGURES ...........................................................................................xiii
LIST OF SYMBOLS .........................................................................................xix
CHAPTER
1. INTRODUCTION ...................................................................................1
2. REVIEW OF THE PREVIOUS WORK ..................................................8
3. DESCRIPTION OF MODEL AND GOVERNING EQUATIONS...........13
3.1. Flow Geometry and Assumptions ..............................................13
3.2 Governing Equations ..................................................................15
3.2.1.Laminar Flow.....................................................................15
3.2.2.Turbulent Flow ..................................................................17
3.2.2.1 Zero Equation Models ............................................23
3.2.2.2 One Equation Models .............................................24
ix
3.2.2.3. Two Equation Models............................................25
4. DESCRIPTION OF THE SOLUTION METHOD....................................31
4.1. Solution Strategies .....................................................................31
4.2 Details of the Numerical Solution ...............................................33
4.2.1. Discretization Equations....................................................33
4.2.1.1. Advection Term.....................................................35
4.2.1.1.1. Monotone Streamline Upwind Approach .....37
4.2.1.1.2. Streamline Upwind/Petro-Galerkin ...............41
Approach
4.2.1.1.3. Collocated Galerkin Approach .....................42
4.2.1.2. Diffusion Terms ....................................................42
4.2.1.3. Source Terms ........................................................43
5. NUMERICAL SOLUTION .....................................................................45
5.1 Segregated Solution Algorithm ...................................................45
5.2 Matrix Solvers ............................................................................50
5.3 Overall Convergence and Stability..............................................52
5.3.1 Convergence ......................................................................52
5.3.2 Stability .............................................................................54
5.3.2.1. Relaxation .............................................................55
5.3.2.2. Inertial Relaxation .................................................55
5.3.2.3. Artificial Viscosity ................................................56
5.3.2.4. Residual File..........................................................57
5.4. Numerical Modeling ..................................................................57
x
5.4.1. Grid Configuration............................................................57
5.4.2. Grid Independency ...........................................................60
5.4.3 Numerical Data ..................................................................67
5.5. Comparison of the Results with Experimental and
Numerical Results in the Literature.............................................68
5.5.1 Comparison of Flat Plate Results with the Literature ..........70
5.5.1.1 Laminar Flow.........................................................70
5.5.1.1.1. Developing Velocity .....................................70
5.5.1.1.2. Pressure Drop ...............................................71
5.5.1.1.3. Nusselt Number ............................................73
5.5.1.2 Turbulent Flow.......................................................76
5.5.1.2.1. Developing Velocity .....................................76
5.5.1.2.2. Fully Developed Velocity .............................76
5.5.1.2.3. Pressure Drop ...............................................78
5.5.1.2.4. Nusselt Number ............................................79
5.5.2 Comparison of Grooved Plate Results with the Literature...79
5.5.2.1 Flow and Temperature Fields..................................82
5.5.2.2 Local Nusselt Number ............................................85
5.6. Typical CFD Data......................................................................86
5.6.1 Laminar Flow.....................................................................86
5.6.2 Turbulent Flow ..................................................................91
6. RESULTS AND DISCUSSION ...............................................................95
6.1. Method of Optimization.............................................................95
6.1.1. Chips with Constant Temperature .....................................95
xi
6.1.2. Chips with Constant Heat Flux..........................................97
6.1.3. Dimensionless Pressure.....................................................99
6.2 Laminar Flow .............................................................................99
6.2.1. Chips with Constant Temperature .....................................99
6.2.2. Chips with Constant Heat Flux..........................................105
6.3. Turbulent Flow ..........................................................................111
6.3.1. Chips with Constant Temperature .....................................111
6.3.2 Chips with Constant Heat Flux...........................................116
7. CONCLUSION........................................................................................123
REFERENCES ............................................................................................129
CURRICULUM VITAE ..............................................................................133
xii
LIST OF TABLES
Table ........................................................................................................ page
3.1 Default Values of Constants in the Basic k-ε Equation ...................................27
4.1 Transport Equation Representation for laminar flow ......................................34
4.2 Transport equation representation for turbulent flow ......................................35
5.1 Axial velocity in the hydrodynamic entrance region of a flat duct ..................70
5.2. Experimental velocity distribution of the turbulent developing flow between parallel plates for Re=200000 by Dean [32] .........................................................76
xiii
LIST OF FIGURES
Figure....................................................................................................... page
1.1 Geometrical representation of the problem............................................ 5
1.2 Scheme of the physical situation in grooved channel ........................................ 6
3.1 Stack of heat generating boards cooled by forced convection ........................... 14
3.2 Computational domain ..................................................................................... 14
4.1 Streamline Upwind Approach .......................................................................... 38
4.2 Downwind node definition ............................................................................... 39
4.3 Possible downwind nodes ................................................................................ 40
4.4 Downwind node identification ......................................................................... 40
5.1 Typical convergence monitor of the variables .................................................. 54
5.2 Different Meshing techniques of the solution domain....................................... 58
5.3 Detailed mesh configuration around the chips for laminar flow ........................ 59
5.4 Detailed mesh configuration around the chips for turbulent flow...................... 59
5.5. Variation of non-dimensional exit temperature with number of elements for turbulent flow over 4 chips per board configuration ............................................... 61
5.6. Variation of non-dimensional exit temperature with number of elements for turbulent flow over 6 chips per board configuration ............................................... 61
5.7. Variation of non-dimensional exit temperature with number of elements for turbulent flow over 8 chips per board configuration ............................................... 62
5.8. Variation of non-dimensional exit temperature with number of elements for turbulent flow over 10 chips per board configuration ............................................. 62
xiv
5.9. Variation of non-dimensional exit temperature with number of elements for turbulent flow over flat plate .................................................................................. 63
5.10. Variation of non-dimensional exit temperature with number of elements for laminar flow over 4 chips per board configuration ................................................. 63
5.11. Variation of non-dimensional exit temperature with number of elements for laminar flow 6 chips per board configuration ......................................................... 64
5.12. Variation of non-dimensional exit temperature with number of elements for laminar flow over 8 chips per board configuration ................................................. 64
5.13. Variation of non-dimensional exit temperature with number of elements for laminar flow over 10 chips per board configuration ............................................... 65
5.14. Variation of non-dimensional exit temperature with number of elements for laminar flow over flat plate .................................................................................... 65
5.15. Variation of non-dimensional exit temperature with number of elements for laminar flow over flat plate, q=const boundary condition ....................................... 66
5.16. Variation of non-dimensional exit temperature with number of elements for turbulent flow over flat plate, q=const boundary condition..................................... 66
5.17. Variation of viscosity ratio with Reynolds number for flat plate .................... 69
5.18. Dev. axial velocity in the entrance region of a flat duct for laminar flow ....... 71
5.19. Dimensionless pressure drop for the laminar developing flow in a flat duct... 73
5.20. Local Nusselt number for the simultaneously developing laminar flow with constant temperature boundary condition ............................................................... 75
5.21. Local Nusselt number for the simultaneously developing laminar flow with constant heat flux boundary condition.................................................................... 75
5.22. Velocity distribution for developing turbulent flow in a parallel plate channel for the Reynolds number 200000................................................................................. 77
5.23 Fully developed velocity distribution for turbulent flow in a parallel plate channel for Reynolds number 9370 and 17100.................................................................... 78
5.24. Turbulent flow apparent friction factor in the hydrodynamic entrance region of a flat duct with uniform inlet velocity ....................................................................... 79
5.25. Thermally developing turbulent flow in a parallel plate channel with constant wall temperature for Reynolds number 9370 and 17100 ................................................ 80
xv
5.26. Thermally developing turbulent flow in a parallel plate channel with constant heat flux boundary conditions for Reynolds number 9370 and 17100 ............................ 80
5.27. Schematic of experimental setup ................................................................... 82
5.28. Comparison of streamline maps [9] and present study for Re=100................. 83
5.29. Comparison of streamline maps [9] and present study for Re=620................. 83
5.30. Comparison of streamline maps [9] and present study for Re=1076............... 83
5.31. Comparison of temperature field of the present study with experimental and numerical results of [9] for Re=354........................................................................ 84
5.32. Comparison of temperature field of the present study with experimental and numerical results of [9] for Re=1760...................................................................... 85
5.33. Comparison of experimental and local Nusselt numbers for Re=1481 ........... 86
5.34. Comparison of experimental and local Nusselt numbers for Re=620 ............. 86
5.35. Axial velocity distribution between the boards for laminar flow .................... 87
5.36. Temperature distribution between the boards for laminar flow ...................... 88
5.37. Streamlines between the boards for laminar flow........................................... 89
5.38. Pressure drop across the boards for laminar flow........................................... 90
5.39. Axial velocity distribution between the boards for turbulent flow.................. 91
5.40. Temperature distribution between the boards for turbulent flow .................... 92
5.41. Streamlines between the boards for turbulent flow ........................................ 93
5.42. Pressure drop across the boards for turbulent flow......................................... 94
6.1. Dimensionless heat transfer (Ω) versus spacing of four chips per board configuration for laminar flow, constant wall temperature boundary condition....... 100
6.2. Dimensionless heat transfer (Ω) versus spacing of six chips per board configuration for laminar flow, constant wall temperature boundary condition ............................ 100
6.3. Dimensionless heat transfer (Ω) versus spacing of eight chips per board configuration for laminar flow, constant wall temperature boundary condition....... 101
xvi
6.4. Dimensionless heat transfer (Ω) versus spacing of ten chips per board configuration for laminar flow, constant wall temperature boundary condition ............................ 101
6.5. Dimensionless heat transfer (Ω) versus spacing of flat plates for laminar flow, constant wall temperature boundary condition ....................................................... 102
6.6. Optimum spacing versus non-dimensional pressure drop (Π) for laminar flow constant wall temperature boundary condition ....................................................... 103
6.7. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for laminar flow, constant wall temperature boundary condition............................................................................. 103
6.8. Maximum heat transfer (Ω) versus pressure drop (Π) for laminar flow, constant wall temperature boundary condition ..................................................................... 104
6.9. Coefficients (a) of Ωmax. versus chip spacing (b/L) for laminar flow, constant wall temperature boundary condition............................................................................. 104
6.10. Dimensionless heat transfer (Ω) versus spacing of four chips per board configuration for laminar flow, constant heat flux boundary condition ................... 106
6.11. Dimensionless heat transfer (Ω) versus spacing of six chips per board configuration for laminar flow, constant heat flux boundary condition ................... 106
6.12. Dimensionless heat transfer (Ω) versus spacing of eight chips per board configuration for laminar flow, constant heat flux boundary condition ................... 107
6.13. Dimensionless heat transfer (Ω) versus spacing of ten chips per board configuration for laminar flow, constant heat flux boundary condition ................... 107
6.14. Dimensionless heat transfer (Ω) versus spacing of flat plates for laminar flow, constant heat flux boundary condition.................................................................... 108
6.15. Optimum spacing versus non-dimensional pressure drop (Π) for laminar flow constant heat flux boundary condition.................................................................... 109
6.16. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for laminar flow, constant heat flux boundary condition.................................................................................. 109
6.17. Maximum heat transfer (Ω) versus pressure drop (Π) for laminar flow, constant heat flux boundary condition.................................................................................. 110
6.18. Coefficients (a) of Ωmax. versus chip spacing (b/L) for laminar flow, constant heat flux boundary condition ......................................................................................... 110
xvii
6.19. Dimensionless heat transfer (Ω) versus spacing of four chips per board configuration for turbulent flow, constant wall temperature boundary condition..... 112
6.20. Dimensionless heat transfer (Ω) versus spacing of six chips per board configuration for turbulent flow, constant wall temperature boundary condition..... 112
6.21. Dimensionless heat transfer (Ω) versus spacing of eight chips per board configuration for turbulent flow, constant wall temperature boundary condition..... 113
6.22. Dimensionless heat transfer (Ω) versus spacing of ten chips per board configuration for turbulent flow, constant wall temperature boundary condition..... 113
6.23. Dimensionless heat transfer (Ω) versus spacing of flat plates for turbulent flow, constant wall temperature boundary condition ....................................................... 114
6.24. Optimum spacing versus non-dimensional pressure drop (Π) for turbulent flow constant wall temperature boundary condition ....................................................... 114
6.25. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for turbulent flow, constant wall temperature boundary condition ..................................................................... 115
6.26. Maximum heat transfer (Ω) versus pressure drop (Π) for turbulent flow, constant wall temperature boundary condition ..................................................................... 115
6.27. Coefficients (a) of Ωmax. versus chip spacing (b/L) for turbulent flow, constant wall temperature boundary condition ..................................................................... 116
6.28. Dimensionless heat transfer (Ω) versus spacing of four chips per board configuration for laminar flow, constant heat flux boundary condition ................... 117
6.29. Dimensionless heat transfer (Ω) versus spacing of six chips per board configuration for turbulent flow, constant heat flux boundary condition................. 117
6.30. Dimensionless heat transfer (Ω) versus spacing of eight chips per board configuration for turbulent flow, constant heat flux boundary condition................. 118
6.31. Dimensionless heat transfer (Ω) versus spacing of ten chips per board configuration for turbulent flow, constant heat flux boundary condition................. 118
6.32. Dimensionless heat transfer (Ω) versus spacing of flat plates for turbulent flow, constant heat flux boundary condition.................................................................... 119
6.33. Optimum spacing versus non-dimensional pressure drop (Π) for turbulent flow constant heat flux boundary condition.................................................................... 120
xviii
6.34. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for turbulent flow, constant heat flux boundary condition.................................................................................. 120
6.35. Maximum heat transfer (Ω) versus pressure drop (Π) for turbulent flow, constant heat flux boundary condition.................................................................................. 121
6.36. Coefficients (a) of Ωmax. versus chip spacing (b/L) for turbulent flow, constant heat flux boundary condition ......................................................................................... 121
xix
LIST OF SYMBOLS
Ae Coefficient of Transport Equation
b Gap Between Two Successive Chips
bi Modified Source Term
Brf Inertial Relaxation Factor
cp Specific Heat
Cφ Convection Coefficient
d Board-to-board spacing
dopt Optimal board-to-board spacing
D Fixed Volume Electronic Package Height
Dh Hydraulic Diameter
DOF Degree of Freedom
f Fanning Friction Factor
fapp Apparent Friction Factor
Gε Production of Turbulent Viscosity
h Chip Height
H Package Height
J Momentum Flux
k Thermal Conductivity
xx
ke Effective Conductivity
L Board Length
'm Mass Flow Rate per Unit Width from a Single Channel
Mφ Convergence Monitor for Degree of Freedom
N Number of the Chips
Nu Nusselt Number
qchip Chip Heat Flux
Pi Inlet Pressure
Po Outlet Pressure
P Mean Pressure
'P Fluctuating Component of pressure
Pe Peclet Number
Pr Prandtl Number
'Q Heat Transfer Rate from a Single Channel
tQ Heat Transfer Rate from the Package
r Relaxation Factor
Re Reynolds Number
Sφ Source Term
t Time
Tchip Maximum Chip Temperature
T Free Stream Temperature
Tme Mean Exit Temperature
Tmi Mean Inlet Temperature
xxi
U Free Stream Velocity
Vx Velocity in x-direction
Vy Velocity in y-direction
V Mean Velocity
V’ Fluctuating Component of Velocity
w Distance Between Two Successive Chips
We Weighting (Shape) Function
x+ Dimensionless Axial Coordinate
Yε destruction of Turbulent Viscosity
Greek Symbols:
α Thermal Diffusivity
αe Effective Thermal Diffusivity
∆P Pressure Drop
∆P* Dimensionless Pressure Drop
ε Kinetic Energy Dissipation Rate
φ General Variable in Description of Transport Equation
φd Downstream Value of the General Variable
φu Upstream Value of the General Variable
Γφ Diffusion Coefficient
κ Turbulent Kinetic Energy
µa Artificial Viscosity
µe Effective Viscosity
µt Turbulent Viscosity
xxii
ν Kinematic Viscosity
νe Effective Kinematic Viscosity
νt Eddy diffusivity of Momentum
νh Eddy diffusivity of Heat
Π Non-dimensional Pressure Drop
θ Non-dimensional temperature
ρ Density
τt Shear Stress
Ω Non-dimensional Heat Transfer
σR Reynolds Stress Term
σt Turbulent Prandtl Number
1
CHAPTER 1
INTRODUCTION
In recent years, electronics has developed and become a part of our lives. As
the number of applications that electronics is involved in our lives increases, the
successful operation of electronic systems becomes a major consideration. Prevention of
failure of an electronic system can be crucial in many areas such as health and defense
applications.
Reliability of the components in an electronic system depends on many criteria
such as construction and density of the components on the printed circuit boards,
operating conditions, architecture of the electronic system, and type of applications the
system is used for. Among them, one of the most important parameter that satisfies the
successful operation of the device is the correct thermal management of the devices,
which has become a major problem due to increased chip intensity at the module level.
Silicon chips are required to be maintained at temperature between 65oC-125oC
depending on the application [2]. In many applications, thermal design of the systems
constitutes the most critical part of the whole design process. Thus experimental and
numerical studies analyzing heat transfer phenomenon especially in chip-on-board and
multi-chip modules where high heat dissipation may occur, have drastically increased
for the last 20 years.
2
The purpose of the thermal design is to provide equipment to remove the heat
from heat sources to one or more heat sinks in the environment while keeping the
temperature of the individual elements within their operational limits. If the operation
temperature is exceeded, performance of the device decreases and failure of the system
is likely to occur.
In order to enhance the overall heat transfer from the components, both the
internal and the external thermal resistances should be reduced. Internal resistance
largely depends on material properties, geometric configuration, and assembly
processes affecting contact resistance between layers of elements. The external
resistance on the other hand depends on major mode of heat transfer, geometry, size of
heat transfer area, and coolant. Good thermal design can be achieved by performing an
optimum thermal management while considering performance, manufacturability,
maintainability, compatibility, and cost of the system.
Due to the wide power dissipation range of electronic systems, there are
various cooling methods employing different fluids:
1. Air cooling
a. Natural convection
b. Forced convection
2. Liquid cooling (direct or indirect)
a. Natural Convection
b. Forced convection
3. Phase-change cooling
Although most of the recent studies is focused on water cooling and direct
liquid immersion cooling to support high chip heat fluxes and high packaging densities,
3
air cooling is by far the most widely used cooling technique in the computer industry
due to the availability of the coolant, simple low cost designs, ease of maintenance and
high reliability. It is the preferred method for small to medium scale computers. Even in
large-scale computers where water-cooling is widely used, still many components are
cooled by air.
Depending on the power levels to be dissipated, either natural convection or
forced convection can be employed. Natural convection is utilized in low levels of
power dissipation, due to low or no power requirements and low noise levels. On the
other hand, high levels of power dissipation are handled by forced convection methods.
Achievement of an efficient cooling relies on the full understanding of convection
phenomenon. Much of the work today is devoted to this field [2].
Although the size and configuration of the electronic equipment varies greatly,
it is still possible to identify some generic cooling problems and related flow
configurations in order to derive some useful correlations from the numerical and
experimental research [1]. The most well known generic cooling problem is the forced
convection of air between the arrays of vertically or horizontally stacked printed circuit
boards carrying electronic components.
Instead of cooling electronic components serially, that is instead of using the
heated air for cooling successive chips, baffles are provided to separate the modules and
the fins, enabling fresh air to move perpendicular to chip area. The method is called
impinging flow. Heat transfer coefficient increases compared to serial configuration,
enabling increased chip and module cooling capability. However, in the impinging
cooling technique, the density of the chips in the volume is decreased due to channels
required to supply fresh air.
4
Because of the high heat transfer coefficients of liquids, especially in case of a
direct contact with inert or dielectric liquids, liquid cooling is getting to be a preferable
cooling technique. Single-phase liquid forced convection, single-phase liquid jet
impingement, and pool boiling liquid cooling are the available techniques.
In liquid forced convection the liquid is forced to flow over the components
resulting in a heat transfer coefficient over an order of magnitude higher than that of air.
Forced convection liquid cooling has advantages over boiling such that the temperature
of the components is more accurately controlled and there is no need to condense the
vapor.
Utilizing liquid phase-change phenomenon can further increase the heat
transfer rate from the components. Dielectric liquids are used in these applications. One
of the problems associated with pool boiling in general is the thermal hysteresis
problem [3]. This behavior can be characterized by a delay in the inception of nucleate
boiling such that the heated surface continues to be cooled by natural convection, with
high superheats until boiling finally does occur. Dielectric liquids are particularly prone
to this behavior and occurrence of hysteresis is almost inevitable with smooth surfaces
such as silicon chips.
Another alternative to achieve higher heat transfer rates is to allow liquid to
flow over the modules while evaporating. It is shown that due to the absence of thermal
hysteresis, higher critical heat fluxes than that of pool boiling can be achieved.
One of the best methods to achieve highest heat transfer rates with a minimum
coolant is jet impingement. It is possible to remove heat fluxes as high as 100 W/m2
from the system using jets [3].
5
The purpose of this work is to determine the optimal board-to-board spacing to
maintain the components temperature below the allowable temperature limit and
maximize the rate of heat transfer from parallel heat generating boards cooled by forced
convection. The geometry of electronic package under consideration is illustrated in
Fig. 1.1. This geometry is a very popular one, chip-on-board configuration with multi-
layers of printed circuit boards. A sufficiently large number of parallel electronic boards
cooled by forced convection are installed in a fixed package volume. The coolant enters
the package through the left opening of the package, flow through the board-to-board
grooved channels and exits through the right opening. The pressure difference across
the package is a known constant, and maintained by fan or pump. Pump or fan is
located either upstream or downstream of the package. The electronics boards are
sufficiently wide in the direction perpendicular to the flow. The heat generating
electronic chips are mounted on one side of the electronic boards.
Printed Circuit Boards
Microelectronic chip
Figure 1.1 Geometrical representation of the problem
The flow in grooved channel is a complex one with separated flow in which the
complex interactions of separated vortices, free shear layers and driving wall-bounded
Coolant in
Coolant out
6
shear flows can be observed. The self-sustaining oscillations are observed in the flow
due to the instability of the free shear layer in conjunction with the disturbance
feedback. Fig 1.2 represents the physical situation in a single channel of fixed volume
electronic package. The geometry of the channel is specified by the chip width, chip
height and the hydraulic diameter of the channel.
In this study, optimal spacing between parallel boards to maximize the total
heat transfer rate with chips cooled by forced convection is investigated numerically.
The continuity, momentum and energy equations are solved using finite elements
method for constant pressure drop across the boards. For this purpose, Ansys Flotran
finite element code is used. In Chapter 2, previous work on the problem along with
related subjects is reviewed. In Chapter 3, physical model, governing equations, and
boundary conditions are presented. Description of the solution method used in Ansys
Flotran is presented in Chapter 4. Details of numerical solution, convergence and
Redeveloping thermal boundary layer
FLOW
Separation Reattachment
Free shear layer Recirculation Electronic chip
Figure 1.2 Scheme of the physical situation in grooved channel
stability parameters, boundary conditions and numerical grid, and comparison of
present results with the results in literature are given in Chapter 5. Optimization
7
approach for a fixed volume and the results of the optimization are represented in
Chapter 6.
Chapter 7 represents the findings of the present work in two parts. In the first
part, optimum spacing and total maximum heat transfer in terms of pressure drop and
chip spacing for laminar flow are given. Although there are similar solutions in the
literature, laminar results improve existing approximate fully developed relations by
introducing the effect of chip spacing on the optimum board-to board spacing and
corresponding maximum heat transfer including the entrance region. In the second part,
correlations for optimum spacing and maximum heat transfer rate are presented. This
part of the study is original in the sense that a complete solution of turbulent flow
through the boards with discrete heat sources at constant temperature and constant heat
flux boundary conditions are obtained for the first time. Moreover, optimization of
board-to-board spacing and maximum heat transfer is introduced, including the effects
of chip spacing. Comparison of correlations of the present study and the literature is
also included in Chapter 7.
8
CHAPTER 2
REVIEW OF THE PREVIOUS WORK
Forced convection cooling with air is the most traditional and preferred cooling
method in electronics applications, due to simple design and easy maintenance of
cooling systems and availability of air in desired amounts. Moreover, air-cooling
provides economical and reliable solutions. Thus, in recent years, especially after
1980’s, the subject has become very popular and many studies have been performed.
Among these studies, those related particularly to the present study are reviewed in this
chapter.
The studies on the physics of the problem go back to the 1960’s. Mehta and
Lavan carried out one of the first studies investigating the flow in two-dimensional
channels [4]. The work formed the basis of all the other studies in that a single cavity
located in the lower wall of the two dimensional channel was examined. In this work, to
minimize the number of parameters, length of the channel was taken to be infinite and
the upper wall was moved with constant velocity. The flow was laminar, and the fluid
was incompressible and Newtonian. The nature of the shear driven vortex was
examined in terms of Reynolds number and aspect ratio of the channel. The results
pointed out that as Re number increased;
9
- The strength of the vortex increased
- The vortex center shifted downstream and upward
- The streamlines in the free shear layer clustered together
- The streamlines at the interface turned out to be convex from concave
Although the work performed by Mehta and Lavan provided a good idea to
understand the physical situation in the groove (as cavity), it was not representing the
real problems in today’s applications. Rockwell and Naudascher [5] presented a
general study showing the possible geometrical configurations where free shear
layers and self sustained oscillations were likely to occur. Their work focused on the
physics of the flow and was rather a literature survey with experimental illustrations.
The free shear layers were classified as planar, axisymmetric and both. The
oscillations from the noise and undesirable structural loading in acoustics and aircraft
applications were also examined.
Among the numerical and experimental studies in the ducts, the book by Shah
and Bhatti [6] and the paper by Shah and London [7] described the fundamentals of the
problem in flat ducts and form a basis for all the related studies. However, a more
specific work addressing to a real problem i.e. maximizing heat transfer from a bundle
of flat plates in a control volume, was first performed analytically by Bejan and Lee [8].
The work explained the main idea of maximizing heat transfer from heat generating
boards within a fixed volume, which was the idea behind the present study as well. Both
natural and forced convection cooling were considered in the work where two limiting
cases were examined: the spacing between the boards was small (small D-limit) and the
10
spacing between the boards was large (large D-limit). In the first case, the spacing
between two boards, D, was assumed to be sufficiently small and heat transfer rate
extracted by the coolant from the finite space occupied by the package was calculated
using channel flow correlations. It was shown that heat transfer rate is proportional with
D2. In large D-limit, the spacing between the boards D was large enough such that it
exceeds the thickness of the thermal layer that forms on each surface of the boards. In
this limit, the flow could be considered as boundary layer flow over a flat plate with the
center region of the flow being at inlet temperature T. The total heat transfer rate from
the package volume was proportional to D-1. These two limiting cases were plotted on a
graph, y-axis being heat transfer rate and x-axis being the spacing between the boards.
Intersection of D2 and D-1 asymptotes gave the location of optimum D.
The experimental studies by Farhanieh et.al., [9], by Herman [10], and by Kakaç
and Cotta [11] are all closely related to the present work. These experiments not only
enable to test the real situations but also provide data to verify the validity of many
numerical studies, including the present work.
Farhanieh and colleagues presented the numerical and experimental analysis of
laminar fluid flow and forced convection heat transfer in a grooved duct [9]. The work
consisted of four grooves kept at constant temperature. Before the test section, there was
a long entrance section in order to achieve fully developed flow. The plane walls of the
duct were kept cold, whereas the grooves were heated and kept at a uniform temperature
in the test section. During the experiments holographic interferometry technique was
used. The visualized temperature fields were used to predict heat transfer coefficients.
The governing equations were also solved using finite volume method. The results were
obtained for different Reynolds numbers and compared with experimental findings. The
11
results showed that local heat transfer coefficient could be 2.4 times higher than that of
channel flow without grooves.
Experimental work, performed by Herman [10], investigated the laminar flow in
a two-dimensional grooved channel using holographic interferometry technique.
Grooved walls were heated and kept at uniform temperature. Heat transfer enhancement
by passive modulation, in other words, introduction of hydrodynamic instabilities, was
examined. Heat transfer and pressure drop data were presented for a wide range of
Reynolds number.
The experimental work performed by Kakaç and Cotta [11] focused on two-
dimensional flow in grooved channels. The study consisted of two parts: The theoretical
approach using generalized integral transform technique to solve the problem and the
experimental findings in order to validate the theoretical results.
The works by Eryurt [12] and by Ekici [13] have special importance for the
present work due to the similar physical considerations and the approach to the
mathematical modeling of the physical problem. In both studies, flash mounted boards
were investigated, for natural convection in [12] and for forced convection in [13]. The
optimization in general could be performed for constant mass flow rate, constant
pressure drop or constant power consumption. For the forced convection studies
including the present work, the driving force was taken as constant pressure drop, which
could be perceived as the idealization of a fan. In [12] and [13], channel spacing was
changed to find an optimum spacing corresponding to maximum heat transfer. It was
also shown that, optimum spacing of the channels was independent of the type of
thermal boundary conditions.
12
Ghaddar et.al. [14] investigated incompressible moderate-Reynolds number
flows in periodically grooved channels by direct numerical simulation using spectral
element method. It was shown that for Reynolds numbers less than a critical value the
flow approached steady state, consisting of an outer channel flow, a shear layer at the
groove lip and a weak recirculating vortex in the groove. The study also included a
detailed stability analysis and a frequency analysis of the self-sustaining oscillations. In
[15], heat transfer enhancement in grooved channels was investigated using spectral
element method on the energy equation by Ghaddar et.al. It was shown that oscillatory
perturbations of the flow results in heat-transfer enhancement as the critical Reynolds
number of the flow approaches. Finally, for a single groove, it was shown that resonant
oscillatory forcing results in doubling of heat transfer rate.
Majumdar and Amon [16] performed a similar work as [14,15] but for a
different geometry, the flow being symmetrical streamwise. It was shown that above a
critical Reynolds number, the flows bifurcated to a time periodic, self-sustained
oscillatory state. Traveling waves were observed even at moderately low Reynolds
numbers inducing self-sustained oscillations that result in very well-mixed flows,
which, in turn, leaded to convective heat transfer augmentation. Results were presented
for laminar and transitional incompressible flows in grooved channels.
Poulikakos and Wietrzak [17] worked on turbulent flow in grooved channels,
performing the analysis for a single groove. Standard k- method was utilized to model
the turbulent flow. Effects of height and location of block with respect to channel was
examined. The results showed that recirculation adversely affects the heat flux from the
surfaces nearby.
13
CHAPTER 3
DESCRIPTION OF MODEL AND GOVERNING EQUATIONS
3.1. Flow Geometry and Assumptions
The objective of this study is to determine the optimum spacing of the heat
generating boards, modeled as parallel plates with discrete sources, which are cooled by
single-phase forced convection (effect of buoyancy force is neglected). The aim is to
maximize heat transfer from a specified package volume for the fixed pressure drop
(∆P) across the package. Fixed pressure drop assumption is a representative model for
installations where pressure difference is maintained by fan or pump, which is located
either upstream or downstream of the package [8].
The geometry of electronic package under consideration is illustrated in Fig.
3.1. The fixed volume of electronic package has height H, and length L. A sufficiently
large number of parallel electronic circuit boards, cooled by forced convection are
installed in the package. The thickness of the boards is neglected throughout the
calculations. The coolant (air) at temperature T enters the package through the left
opening of the package with uniform velocity U, flow through the board-to-board
channels and exists through the right opening. The heat generating electronic chips are
14
mounted on the upper side of the electronic boards. Chip surfaces are modeled as either
constant temperature or constant heat flux.
Figure 3.1. Stack of heat generating boards cooled by forced convection
One channel of fixed volume of electronic package is illustrated in Fig. 3.2.
This channel is the computational domain for this study. In Fig. 3.2, d represents board-
to-board spacing, h stands for chip height, w stands for distance between two successive
chips, b represents the distance between two successive chips and Tchip designates chip
temperature.
Figure 3.2 Computational domain
L
H
Pi Po
L
U∞, T∞ y x
h Tchip b
w
d
15
Before introducing the governing equations for laminar and turbulent
incompressible flows, it is important to list the assumptions made during modeling.
Since the electronic boards are sufficiently wide in the direction perpendicular to the
flow, the flow is taken as two-dimensional. In addition to that, the flow is steady. The
fluid is Newtonian and incompressible, and thermo-physical properties are constant.
The duct walls are considered to be smooth, non-porous, and rigid with negligible
thickness compared to the electronic package height.
3.2 Governing Equations
3.2.1 Laminar Flow
The simplest class of flows, in which viscous phenomena are important, occurs
when the streamlines form an orderly parallel pattern. The fluid in the viscous region
may be thought of as proceeding along in a series of layers with smoothly varying
velocity and temperature from layer to layer. Viscous flows of this class are called
laminar.
Conservation of mass, momentum and energy equations for two-
dimensional laminar flow can be expressed as
Continuity: 0=∂
∂+
∂∂
yyV
xxV
(3.1)
x-momentum:
∂
∂+
∂
∂⋅+
∂∂−=
∂∂
+∂
∂2
2
2
21
y
xV
x
xVxP
yxV
yVxxV
xV νρ
(3.2)
y-momentum:
∂
∂+
∂
∂⋅+
∂∂−=
∂
∂+
∂
∂2
2
2
21
y
yV
x
yV
yP
yyV
yVxyV
xV νρ
(3.3)
Energy:
∂
∂+∂
∂=∂∂+
∂∂
2
2
2
2
y
T
x
TyT
yVxT
xV α (3.4)
16
where Vx and Vy stand for velocity component in x and y directions respectively.
The governing equations are elliptic requiring boundary conditions to be
prescribed around all boundaries. There are inlet plane, outlet plane, board surfaces and
chip surfaces. Air at uniform temperature T enters the computational domain through
the inlet plane with uniform velocity U with inlet pressure Pi. Thus, the inlet boundary
conditions applied are:
∞==∞=<<= TTyVUxVdyx 000 (3.5)
If the length of the computational domain is sufficiently larger than the board
spacing d, it is common practice to assume that the flow is perpendicular to the outlet
plane and heat transfer rate on the outlet plane is purely by convection rather than by
conduction. Outlet pressure at the exit of the computational domain is Po. The outlet
boundary conditions can be written as:
0002/2/ =∂∂=
∂
∂=
∂∂
<<−=yT
xyV
xxV
dydLx (3.6)
Since the board surface is smooth impermeable and with no slip, the velocity
components are zero. Thermal boundary conditions applied are those of either uniform
temperature or uniform heat flux at chip surfaces. The lower board boundary conditions
can be written as
surfaceschipForchipqqorchipTT
wallsplaneallonyT
yVxV
Lxy
==
=∂∂==
<<=
0,0
0,0
(3.7)
Unlike lower board, there are no chips on the upper board. Therefore, boundary
conditions for the upper board is
17
000 =∂∂==<<=
yT
yVxVLxdy (3.8)
3.2.2 Turbulent Flow
Very often the flows of the real fluids differ from the laminar flows considered
in the preceding section. As the velocity of the fluid is increased, boundary layers
formed on solid bodies undergo a transition from laminar to turbulent regime. The
incidence of turbulence was first recognized in relation to flows through straight pipes
and channels and illustrated by Reynolds [18], by feeding into the flow a thin thread of
liquid dye. As long as the flow is laminar, the dye maintains sharply defined boundaries
along the stream. As soon as the flow becomes turbulent, the dye diffuses into the
stream. In this case there is a superimposed subsidiary motion right angle to the main
motion, which causes mixing. In laminar flow, according to Hagen-Poiseuille solution,
velocity distribution in a pipe over the cross section is parabolic, but in the turbulent
flow, owing to the transfer of momentum in the transverse direction, it becomes
considerably more uniform [19]. With a closer investigation, it appears that at a given
point in the flow, the velocity and pressure are not constant in time but exhibit very
irregular, high frequency fluctuations. The velocities at a given point can only be
considered constant on the average and over a longer period of time.
Reynolds conducted the first systematic investigation. He discovered the law of
similarity which now bears his name, and which states that transition from laminar to
turbulent flow always occur at the same Reynolds number. The numerical value of the
Re number at which transition occurs was established as being approximately 2300. The
numerical value of the critical Re number depends very strongly on the conditions,
which prevail in the initial pipe length as well as inlet conditions. This fact was
18
experimentally confirmed by other researchers [20]. Barnes and Coker and later Schiller
reached values up to 20000 in maintaining laminar flow. Ekman reached up to Re
number of 40000 without any disturbances. There are also numerous experiments
showing that there exists turbulence even below critical Re number [20]. Therefore, it
may not be always possible to use Re number as the criteria to determine whether the
flow is laminar or turbulent during numerical modelling.
Transition from laminar to turbulent flow is accompanied by a noticeable
change in pressure drop. In laminar flow, axial pressure gradient, which maintain the
motion, is proportional to the first power of the velocity. On the other hand, in turbulent
flow, the pressure gradient becomes nearly proportional to the square of the mean
velocity [20].
The fluctuations imposed on the principal flow is so complex that it seems to
be impossible to solve by analytical methods, but it must be realized that the resulting
mixing motion is very important for the course of the flow and for the equilibrium of
the forces. The effects caused by this mixing are as if the viscosity were increased by
factors of hundred, thousand or even more. At large Re numbers, there exists a
continuous transport of energy from the main motion into the eddies. The goal of the
turbulent analysis is to provide a description of the mean flow, in other words time
averages of the turbulent motion.
Upon close investigation of the flow, the most striking feature of turbulent
motion consists in the fact that the velocity and pressure at a fixed point in space do not
remain constant with time but perform irregular fluctuations of high frequency. In
describing the turbulent flow in mathematical terms, it is convenient to separate the
flow into a mean motion and a fluctuation or eddy motion. Denoting the time average of
19
the Vx-component of velocity by xV and its velocity of fluctuation by 'xV , we can write
down the following relations for the velocity components and pressure
'xVxVxV += '
yVyVyV += 'PPP += (3.9)
The time averages are formed at a fixed point in space and are given by
dttt
txV
txV +
=10
0
1
1 (3.10)
At this point, it must be made clear that the mean values are to be calculated
over a sufficiently long interval of time, t1, for them to be completely independent of
time. Thus by definition, the time averages of fluctuating components are equal to zero.
0'0'0' === PyVxV (3.11)
Before introducing the relations between the mean motion and the apparent
stresses caused by fluctuations, it is wise to give a physical explanation illustrating their
existence. The arguments are based on the momentum transfer.
Let us consider an elementary area dA in a turbulent stream whose velocity
components are Vx and Vy. Normal to the area is imagined as x-axis and the direction y
is in the plane of dA. The mass of the fluid passing through the area per unit time is
given by (ρVxdAdt), and thus the flux of momentum in the x-direction is
dJx=(ρVx2dAdt). Similarly the flux in the y direction is dJy=(ρVxVydAdt). Assuming the
density is constant, time averages of the momentum per unit time can be calculated as
20
2xVdAxdJ ρ= xVyVdAydJ ρ= (3.12)
'2'222'2
xVxVxVxVxVxVxV ++=
+= (3.13)
' 22' 2'222xVxVxVxVxVxVxV +=++=
(3.14)
Similarly
''yVxVyVxVyVxV += (3.15)
Therefore the expressions for momentum fluxes per unit time becomes
+= ' 22
xVxVdAxdJ ρ (3.16)
+= ''yVxVyVxVdAydJ ρ (3.17)
When examined closely, the quantities above have the dimension of forces and
upon dividing by area, stresses are obtained. Therefore it can be concluded that the area
under consideration, which is normal to the x-axis is acted upon by stresses
+− ' 22
xVxVρ in the x-direction, and
+− ''yVxVyVxVρ in the y-direction. The first
of the two is the normal stress whereas the latter is the shear stress. It is seen that the
superposition of fluctuations on the mean motion result in two additional stresses. They
are termed as Reynolds Stresses of turbulent flow and must be added to the stresses
caused by the steady flow [20].
21
It is apparent that the time averages of the mixed products of velocity
fluctuations such as ''yVxV differ from zero. The stress component
− ''yVxVρ can be
interpreted as the transport of x-momentum through a surface normal to the y-axis.
Upon introducing the velocities given by Eq. (3.9) into Navier Stokes and
energy equations, the following expressions are obtained:
∂
∂+
∂
∂−∇+
∂∂−=
∂∂
+∂
∂y
yVxV
xxV
xVxP
yxV
yVxxV
xV''' 2
2 ρµρ (3.18)
∂
∂+
∂
∂−∇+
∂∂−=
∂
∂+
∂
∂
xyVxV
yyV
yVyP
yyV
yVxyV
xV''' 2
2 ρµρ (3.19)
∂∂+
∂∂−∇=
∂∂+
∂∂ ''''2 TyV
yTxV
xT
yT
yVxT
xV α (3.20)
The left hand sides of the Eqs. (3.18), (3.19) and (3.20) are formally identical
with the steady-state Navier-Stokes and energy equations, if the velocity components Vx
and Vy and temperature are replaced by their time-averages and the same is true for the
pressure, friction and diffusion terms on the right hand side. In addition, the equations
contain terms, which depend on the turbulent fluctuations of the stream. As explained
before, they are called Reynolds stresses. The method of calculation of turbulent flow
and temperature mostly depends on empirical or numerical hypothesis, which establish
a relationship between the Reynolds stresses produced and the mean values of velocity
components together with a suitable relation for the heat transfer [20].
22
Various different models are used to find unknown Reynolds stress term, and
turbulent heat diffusion ranging from simple algebraic to second order closure models.
In current study, - model is used as a turbulence model since the flow has separation,
re-attachment and circulation.
Boussinesq was the first scientist to work on the problem [20]. In a similar
analogy with the Stoke’s law, he suggested a mixing coefficient for the Re stress in the
turbulent incompressible flow in the form of
∂∂
=−yxV
tyVxV ν'' (3.21)
∂∂=−
yT
hTyV'
'' ν (3.22)
where νt and νh are called eddy diffusivity of momentum and eddy diffusivity of heat
respectively. Introducing Eqs. (3.21) and (3.22), into Eqs. (3.18), (3.19), and (3.20), and
neglecting small terms, the Navier-Stokes and energy equations become:
( )
∂
∂+
∂
∂++
∂∂−=
∂∂
+∂
∂2
2
2
21
y
xV
x
xVtx
PyxV
yVxxV
xV ννρ
(3.23)
( )
∂
∂+
∂
∂++
∂∂−=
∂
∂+
∂
∂2
2
2
21
y
yV
x
yVty
PyyV
yVxyV
xV ννρ
(3.24)
( )
∂
∂+∂
∂+=∂∂+
∂∂
2
2
2
2
y
T
x
Thy
TyV
xT
xV να (3.25)
Actual problems cannot be solved by these equations unless the dependence of
νt and νh on velocity is known. Therefore it is necessary to find empirical or numerical
methods in order to suggest a relation between ε and mean velocity.
23
There are different methods used in analysis in order to relate turbulent
kinematic viscosity to mean velocity [21]. The techniques can be classified as zero
equation models, one-equation models and two-equation models.
3.2.2.1 Zero Equation Models
The first zero equation model based on the Boussinesq eddy viscosity is the
Prandtl’s mixing length formulation. Prandtl [20] is one of the first scientists making an
important advance in the direction of dependence of eddy viscosity on mean velocity.
With Prandtl’s simplified mechanism of motion, the flow can be visualized such that as
the fluid passes along the wall in turbulent motion, fluid particles move bodily for a
given length, both in the longitudinal and transverse direction, retaining their
momentum parallel to x. It will be assumed that such a lump of fluid, which comes from
a layer at (y1-L) and has a velocity, )1( LyxV − is displaced over a distance L in the
transverse direction. This distance L is known as Prandtl’s mixing length. As the lump
of fluid retains its original momentum, its velocity in the new layer is smaller than the
previous layer. The difference can be given by
≈∆
dyxVd
LxV (3.26)
Velocity differences caused by transverse motion can be regarded as the
turbulent velocity components. Hence, the time average of the absolute value of the
fluctuations can be calculated by
( ) ( )( )ydy
xVdLLyxVLyxVxV y
=+∆+−∆=21' (3.27)
Referring to above equation, the following physical interpretation of the
mixing length can be made. The mixing length is the distance in the transverse direction
24
which must be covered by a lump of fluid particles travelling with its original mean
speed in order to make the difference between its velocity and the velocity in the new
layer equal to the mean transverse fluctuation in turbulent flow. The argument implies
that the transverse component 'yV is of the same order and can be written as
dyxVd
cLxcVyV == '' (3.28)
22''''
⋅−=−=
dyxVd
LconstyVxVcyVxV (3.29)
Including c into unknown mixing length
22''
−=
dyxVd
LyVxV (3.30)
Recalling Eq. (3.21) above expression can be introduced into shear stress
definition as
dyxVd
tdyxVd
dyxVd
Ldy
xVdLt µρρτ =
=
= 22
2 (3.31)
Above equation is known as Prandtl’s mixing length hypothesis. Kinematic
viscosity, therefore, can be defined as
dyxVd
Ltt
2==ρµν (3.32)
3.2.2.2 One Equation Models
Standard one-equation approach calculates a length scale related to local shear
layer thickness. The relation is given by Eq. (3.33),
25
021
LKt =ν (3.33)
where xVxVK 21=
New one-equation models suggest a modeled transport equation for the eddy
viscosity νt. Baldwin and Barth and Spalarat and Allmaras proposed one-equation
models for νt [22].
In its original form, the Spalart-Allmaras model is a low-Reynolds-number
model, requiring the near-wall region of the boundary layer to be properly resolved. The
transported variable in the Spalart-Allmaras model, ε , is identical to the turbulent
kinematic viscosity except in the near-wall (viscous-affected) region. The transport
equation for ε is
( ) ( ) ( )
( ) εεερερεµ
εσε
ρερερε
SYybc
yyG
yVyxV
xt
+−
∂∂+
∂∂+
∂∂+
=∂∂+
∂∂+
∂∂
21 (3.34)
where Gε is the production of turbulent viscosity and Yε is the destruction of turbulent
viscosity that occurs in the near-wall region due to wall blocking and viscous damping.
σε and Cb are constants. Yε is a user-defined source term [22].
3.2.2.3 Two Equation Models
The simplest "complete models'' of turbulence are two-equation models in
which the solution of two separate transport equations allows the turbulent velocity and
length scales to be independently determined. The standard -ε model falls within this
26
class of turbulence model and has become the workhorse of practical engineering flow
calculations in the time since it was proposed by Launder and Spalding [23].
Robustness, economy, and reasonable accuracy for a wide range of turbulent flows
explain its popularity in industrial flow and heat transfer simulations.
In the -ε model, stands for kinetic energy and ε stands for its dissipation
rate, and can be defined as
''21
xVxV=κ (3.35)
yxV
yxV
∂∂
∂∂
=''
νε (3.36)
where ν is the kinematic viscosity. l can be defined as a length scale representing the
macro scale of turbulence [24], which is expressed in terms of κ, ε and a constant CD as
εκ 5.1DCl = (3.37)
Two equation models require the solution of partial differential equations for
turbulent kinetic energy () and its dissipation rate (ε). The turbulent kinetic energy and
dissipation rate equations given by Spalding and Launder [17] are as follows:
The turbulent kinetic energy equation:
∂∂+
∂∂−−+
∂∂
+
∂∂+
∂∂
+
∂∂=
∂
∂+
∂∂
22
122
1
2
)()(
yxGt
ykt
yxkt
xyyV
xxV
κκνεν
κσννκ
σνν
κκ
(3.38)
27
The Dissipation rate equation
∂
∂+
∂
∂+
∂
∂+
∂
∂+−
+
∂∂
+
∂∂+
∂∂
+
∂∂=
∂
∂+
∂∂
2
2
22
2
22
2
22
2
22
22
1)()(
y
yV
x
yV
y
xV
x
xVC
GtCy
tyx
txy
yV
xxV
νκ
ε
κενε
εεσ
ννε
εσν
νεε
(3.39)
where 222
2
∂
∂+
∂∂
+
∂
∂+
∂∂
=xyV
yxV
yyV
xxV
G
These two equations enable the turbulent viscosity to be found from
εκµκµν 221
ClCt == (3.40)
Extensive investigations of the turbulent flows by Launder [24] have led to the
determination of constants in Eqs. (3.38) and (3.39). Slightly different values may be
used for the flows near the wall but it has been proved that the values given in Table 3.1
have led to as satisfactory predictions as obtained with those originally employed.
Default values of constants in the basic -ε equations are:
Table 3.1. Default Values of Constants in the Basic κ-ε equation. Value C1, C1εεεε
C2 Cµ σk σε σt Default 1.44 1.92 0.09 1.0 1.3 0.9
The solution of the turbulence equations is used to calculate effective viscosity
and the effective thermal diffusivity.
εκ
µνν2
Ce += (3.41)
28
tt
e σναα += (3.42)
where νe is the effective viscosity, and t is the turbulent Prandtl number given by
Table 3.1.
To summarize the turbulent flow formulation, it is useful to recall the turbulent
equations together with the boundary conditions.
Continuity: 0=∂
∂+
∂∂
yyV
xxV
(3.43)
x-momentum: ( )
∂
∂+
∂
∂++
∂∂−=
∂∂
+∂
∂2
2
2
21
y
xV
x
xVtx
PxxV
yVxxV
xV ννρ
(3.44)
y-momentum: ( )
∂
∂+
∂
∂++
∂∂−=
∂
∂+
∂
∂2
2
2
21
y
yV
x
yVty
PxyV
yVxyV
xV ννρ
(3.45)
Energy: ( )
∂
∂+∂
∂+=∂∂+
∂∂
2
2
2
2
y
T
x
Thy
TyV
xT
xV να (3.46)
κ-equation
∂∂+
∂∂−−+
∂∂
+
∂∂+
∂∂
+
∂∂=
∂∂
+∂
∂
22
122
1
2
)()(
yxGt
ykt
yxkt
xyyV
xxV
κκνεν
κσννκ
σνν
κκ
(3.47)
ε-equation
∂
∂+
∂
∂+
∂
∂+
∂
∂+−
+
∂∂
+
∂∂+
∂∂
+
∂∂=
∂∂
+∂
∂
2
2
22
2
22
2
22
2
22
22
1)()(
y
yV
x
yV
y
xV
x
xVC
GtCy
tyx
txy
yV
xxV
νκ
ε
κενε
εεσ
ννεεσ
ννεε
(3.48)
When compared with laminar flow equations, the continuity, momentum and
energy equations have the similar form, velocities being replaced by their time average
29
values with effective viscosity and thermal diffusivity. Two more equations for
turbulent kinetic energy and its dissipation rate , are included in the formulation to
evaluate turbulent viscosity and diffusivity. The boundary conditions Eq. 3.5 to 3.8 for
laminar flow are valid for turbulent flow as long as variables are replaced by their time
average values, but additional boundary conditions are required to solve and
equations. For the inlet of the channel, the boundary conditions are as follows [17]:
dUdyx
005.0
23
20275.000κεκ =∞=<<= (3.49)
On the board and chip surfaces, and are zero and can be written as
0,,0 ==<< εκboundariessolidallonLx (3.50)
For the exit of the channel, following boundary conditions are valid:
000 =∂∂=
∂∂<<=
xxdyLx
εκ (3.51)
If the inertial effects are great enough with respect to viscous effect, the flow
can be assumed to be turbulent. The classical method is to check Re number in order to
determine whether the flow is turbulent or not. It works quite well when the geometry is
simple. However, for the complex geometries as in the present work, the flow becomes
turbulent much earlier then the Re number estimates. For this reason it is more effective
to check the ratio of effective viscosity to dynamic viscosity.
νν eRatioityVis =cos (3.52)
During the calculations, for each pressure drop the analysis in Ansys Flotran
start with performing a trial run corresponding the smallest channel spacing. The flow
equations are solved assuming the flow is turbulent. Then, the viscosity ratio is checked;
30
if the viscosity ratio is greater than five the flow can be assumed to be turbulent [25]
and analysis for other spacing values for the pressure drop under consideration can be
performed. It should be noted that, as the board spacing increases for the prescribed
pressure drop, velocity increases between the boards, and effects of turbulence become
more severe. If viscosity ratio is smaller than 5, in that case the flow is assumed to be
laminar and laminar formulation is used. It is also possible to use turbulent formulation
for the laminar flow since the equations become identical as the viscosity ratio
approaches to 1, however two more equations for and are involved during iterations,
which increase computation time.
31
CHAPTER 4
DESCRIPTION OF THE SOLUTION METHOD
4.1. Solution Strategies
In any thermo-fluid problem, prediction of the flow, temperature and the
mechanisms behind the problem can be based on two methods, namely experimental
investigations, and theoretical approach. Before giving the details of the method used in
the present work, it is useful to express capabilities, limitations, advantages, and
disadvantages of each method.
Experimental investigations are by far the most reliable way of finding a
correct solution to any physical problem. However, it can be time consuming and
expensive in some cases such as aerodynamic tests. Due to financial limitations, in
many applications small-scale prototypes are used to simulate the behavior of the real
systems, but it may not reflect the true system under operating conditions as well. On
the other hand, the need for increased computing power for complex systems as well as
for complex physical phenomenon makes experimental investigations advantages in
certain cases.
In contrary to experimental solutions, theoretical approach is fast, cheap and
can be performed at any time for any boundary conditions. It can be defined as solution
of mathematical model representing the physical phenomenon. The theoretical approach
gives flexibility in other aspects such as simulating limiting cases or ideal conditions
32
which cannot be achieved in real life. But, most of the time, theoretical approach should
be verified with the experiments to prove its correctness.
Theoretical solutions can be divided into two categories: Those who have an
explicit analytical solution and those who have an approximate numerical solution.
Problems having explicit solutions are very limited in real life and most of the time they
are simplified versions of more complicated problems. Most of the problems faced
today, has no analytical solutions. Therefore numerical methods are widely utilized to
find answers for complicated problems. At this point it is also possible to classify these
complex problems as well: Problems that have adequate mathematical description (heat
conduction, laminar flow, and simple turbulent boundary layers.), and problems without
any adequate mathematical description (complex turbulent flows, some two-phase
flows.) Thus, there exist two sources of errors in numerical solutions: The one due to
nature of numerical solution and the other due to insufficient mathematical model of the
real problem.
Numerical methods can be defined as the discretization of the mathematical
models. The simplification is the use of algebraic equations instead of the differential
ones, which makes the numerical prediction powerful and widely applicable. As the
discretization can be done in many ways, the methods take different names for different
discretization approaches. However, the numerical methods can be divided into three
general categories:
i. Finite Element Method (FEM) formulation: It is a global formulation which
is most suitable for the irregular geometries and it gives more accurate
results than the finite difference formulation for the given discretization, but
with the expense of more algebra requirement.
33
ii. Finite difference formulation: It is a local formulation and requires relatively
little algebra to setup for problems having regular boundaries.
iii. Spectral Element Method: A high order finite element method that
combines the generality of the finite element method with the accuracy of
spectral techniques.
4.2. Details of the Numerical Solution
Finite element method is a valuable tool in the solution of many engineering
problems [26]. In situations where the governing equations are known, but complicated
geometry or boundary conditions make analytical solutions difficult or impossible to
obtain, the finite element method is often employed. The finite element method makes
use of a spatial discretization and a weighted residual formulation to arrive at a system
of matrix equations. Solution of the matrix equations yields an approximate solution to
the original boundary value problem. The following sections give the details of
numerical solution as used in Ansys Flotran.
4.2.1 Discretization Equations
For the discretization of the governing equations a segregated, sequential
solution algorithm is used. In other words, element matrices are formed, assembled and
the resulting system solved for each degree of freedom separately. Development of the
matrices proceeds in two parts. In the first part, the form of the equations is achieved,
and each term in these equations is evaluated. Next, the segregated solution algorithm is
outlined and the element matrices are developed from the equations.
Since the given flow field must satisfy the continuity equation and the
governing differential equations (momentum and energy equations) for the laminar
34
flow, and time average governing equations for the turbulent flow, they can be
expressed in the form of a single elliptic differential expression given by Eq. (4.1)
Momentum, energy, and turbulence equations all have the form of a scalar transport
equation consisting of the three identical terms, namely convection, diffusion, and
source terms [27]. The pressure equation is derived using the continuity equation and it
will be discussed in the section of the segregated solver. The general variable φ is used
in the description of discretization method. The form of the general scalar transport
equation is:
φφ
φφ
φφφφφ Syyxx
CyVy
CxVx
+∂∂Γ
∂∂+
∂∂Γ
∂∂=
∂∂+
∂∂
)()()()( (4.1)
where represents a generic dependent variable which is Vx, Vy and T for laminar flow
and TyVxV ,, for turbulent flow. The terms on the left-hand side of the equation
represent convective terms. The first two terms on the right-hand side of the equation
represent diffusion term. Cφ and Γφ are the coefficients of convection and diffusion terms
respectively. Sφ represents the source term. The values to generate the governing
equations (x-momentum, y-momentum and energy) for laminar flow and time average
governing equations (x-momentum, y-momentum, energy, dissipation rate and kinetic
energy) for turbulent flow are given in Table 4.1 and Table 4.2 respectively [27].
Table 4.1. Transport equation representation for laminar flow
Transport Equation
φφφφ Cφφφφ ΓΓΓΓφφφφ Sφφφφ
x-momentum Vx 1 ν xp ∂∂− /1ρ
y-momentum Vy 1 ν yp ∂∂− /1ρ
Energy T Cp k/ρ
35
Table 4.2. Transport equation representation for turbulent flow
Transport Equation
φφφφ Cφφφφ ΓΓΓΓφφφφ Sφφφφ
x-momentum xV 1 ( )tνν + xp ∂∂− /1ρ
y-momentum yV 1 ( )tνν + yp ∂∂− /1ρ
Energy T cp k/ρ
Kinetic energy 1 kt
σνν + MGt νεν 2−−
Dissipation rate ε 1 εσ
νν t+ NCGtC νκ
εκενε 2
221 +−
where G, M and N in the source terms represent some functions of velocity and its
derivatives, and are given in Eqs. (3.38) and (3.39). The constants C1, C2, and k are
represented in Table 3.1.
Since the approach is the same for each equation, the generic transport
equation will be treated only. Each term in the transport equation will be explained in
turn. In the discretization process, the element matrices are derived and put together to
form the general matrix equation:
[ ] [ ]( ) φφ eediffussion
econvection
e SAA =+ (4.2)
Galerkin's method of weighted residuals is applied to form the element
integrals. Each term in the transport equation is multiplied with a weighting function,
which is also the shape function, denoted by We and then integrated over the solution
domain.
36
4.2.1.1. Convective Term
The most common weighted residual formulation is the Galerkin method, in
which weighting and interpolation functions are from the same class. The Galerkin
method when applied to heat conduction problems leads to symmetric stiffness matrix.
In this case, it can be shown that the solution possesses the best approximation property
of the weighting function. However, for the fluid flows or convective heat transfer, the
matrix associated with the convection term is non-symmetric and as a result the best
approximation property is lost [26]. The works [26] and [28] focus on the problem
associated with the use of Galerkin method and suggest improvements in the solution by
the use of streamline upwind technique. As in the present work, for fluid flows over
grooves in narrow channels, wiggles are most likely to appear when downstream
boundary conditions force a rapid change in the solution. The only way to eliminate
oscillations is to severely refine the mesh such that convection no longer dominates on
an element level. However, often only the global solution features such as total heat
transfer and pressure drop across the solution domain, are desired and in this case mesh
refinement is required to prevent oscillations. This has led an alternative formulation
included in the works [26,28] and explained in this section.
It is well known that Galerkin method gives rise to central-difference type
approximations and may yield to oscillations in the solution. It has been discovered that
more accurate and stable solutions could be obtained using upwind differencing of the
convective terms [26]. The drawback is that upwind differencing is only first order
accurate. It is also possible to construct upwinded convective terms by adding artificial
diffusion to a central difference treatment or by employing modified weighting
37
functions to achieve the upwind effect. In essence, upstream element of a node is
weighted more heavily, than a downstream element of the same node.
But it has also been discovered that upwind finite element formulations lead to
the same system of matrix equations and give exact solutions for one-dimensional
problems. When generalized to more complicated situations, they are far from the
reality. In multidimensional problems, upwind solutions often exhibit excessive
diffusion in the direction perpendicular to flow. Therefore it has become apparent that a
combination of central and upwind differences based on Reynolds number is better than
either upwind or central differences alone. The shortcomings of the upwind method can
be overcome by using monotone streamline upwind approach. The basic idea of this
method is to add diffusion (or viscosity) only in the flow direction.
Three approaches to discretize the convection term in this new approach can be
utilized [27]. The monotone streamline upwind (MSU) approach is first order accurate
and tends to produce smooth and monotone solutions. The streamline upwind/Petro-
Galerkin (SUPG) approach and the collocated Galerkin (COLG) approach are second
order accurate and tend to produce oscillatory solutions.
4.2.1.1.1.Monotone Streamline Upwind Approach (MSU)
In the monotone streamline approach it is assumed that pure convection
transport occurs along a characteristic line. Fig. 4.1 illustrates the idea behind this
approach.In the figure φD and φU indicate downstream and upstream values of the
general variable respectively. The velocity field is considered as a set of streamlines
tangent to the velocity vectors everywhere. The convection terms can therefore be
38
Figure 4.1. Streamline Upwind Approach
expressed in terms of the streamline velocities. In pure convective transport, it can be
assumed that all transport occurs along the characteristic lines. In that case, it can be
written:
s
VC
y
VC
x
VC syx
∂∂
=∂
∂+
∂∂ )()()( φφφ φφφ (4.3)
When expressed along a streamline, this expression is constant throughout an element:
[ ] dAWds
VCdA esconvection
e =)( φφ (4.4)
The derivative is calculated using a simple difference:
s
VCVC
ds
VCd DsUss
∆−
=)()()( φφφ φφφ (4.5)
The value at the upstream location is unknown but can be expressed in terms of
the unknown nodal values it is between. The process consists of cycling through all the
elements and identifying the downwind nodes. A calculation is made based on the
Streamline
M K
N
∆S
L φU
φD
39
velocities to see where the streamline through the downwind node came from.
Weighting factors are calculated based on the proximity of the upwind location to the
neighboring nodes [26].
The evaluation of the integral given by Eq. (4.4) is performed considering the
element illustrated in Fig. 4.2 with the streamlines included. Since the velocity variation
on the element is two-dimensional, the streamlines may curve as illustrated. Node 3 on
this element is a downwind node. The term downwind defines a node for which the
negative of the velocity vector at that node points back into the element. Note that an
individual element may have a number of possible configurations as illustrated by Fig.
4.3. Note further that a node not lying on the boundary will be a downwind node on at
least one element.
The determination of whether or not a node is a downwind node on a given
element is straightforward. Referring to Fig. 4.4, a downwind node is one for which
Figure 4.2. Downwind node definition
2
3
4
1
40
(a) (b)
(c) (d)
Figure 4.3. Possible downwind nodes: a) One downwind node; b) Two downwind nodes
c) Interior corner-no downwind node; d) No downwind node
Figure 4.4. Downwind node identification
the velocity vector has a non-negative outward normal component on both of the
element sides adjacent to the node [26]. Once the downwind node has been identified in
this manner, the convection term approximation requires the determination of the
upstream location illustrated in Fig. 4.3, denoted by x’ and y’. Once x’ and y’ are
located, the convection terms for the node is approximated as
−∆
WdAsissV
)( φφρ (4.6)
where
4 3
2
1
x’ y’ φ’
xi yi φi
41
( ) ( )2'2' yiyxixs −+−=∆ (4.7)
2,
2, iyVixVsV += (4.8)
The location of x’ and y’ coordinates as well as the calculation of φ’ itself is
based on the interpolation factors. The interpolation factors are calculated based on the
mass flow rates on each element side [26].
4.2.1.1.2. Streamline Upwind/Petro-Galerkin Approach (SUPG)
The SUPG approach consists of a Galerkin discretization of the convection
term and an additional diffusion-like perturbation term, which acts only in the
convection direction.
dAy
CyV
x
CxV
y
eWyV
x
eWxV
magUzh
C
dAy
yVC
xxVCeWconvection
eA
∂
∂+
∂
∂
∂
∂+
∂∂
+
∂
∂+
∂
∂=
φφφφτ
φφφφ
22
)()(
(4.9)
where C2τ is global coefficient set to 1. h is the element length along convection
direction. The other variables are:
22yVxVmagU += (4.10)
≥<≤
=33/301
PeifPe
Peifz (4.11)
NumberPeclethmagUC
Pe =Γ
=φ
φρ2
(4.12)
42
It is clear from the SUPG approach that as the mesh is refined, the perturbation
terms goes to zero and the Galerkin formulation approaches second order accuracy. The
perturbation term provides the necessary stability, which is missing in the pure Galerkin
discretization [28].
4.2.1.1.3. Collocated Galerkin Approach (COLG)
The COLG approach uses the same discretization scheme with the SUPG
approach with a collocated concept. In this scheme, a second set of velocities, namely,
the element-based nodal velocities are introduced. The element-based nodal velocities
are made to satisfy the continuity equation, whereas the traditional velocities are made
to satisfy the momentum equations.
dAy
CeyV
x
CexV
y
eWeyV
x
eWexV
magUzh
C
dAy
eyVC
x
exVCeWconvection
eA
∂
∂+
∂
∂
∂
∂+
∂∂
+
∂
∂+
∂
∂=
φφφφτ
φφφφ
22
)()(
(4.13)
where all the parameters are defined similar to those in the SUPG approach.
In this approach, the pressure equation is derived from the element-based nodal
velocities, and it is generally asymmetric even for incompressible flow problems. The
collocated Galerkin approach is formulated in such a way that, for steady-state
incompressible flows, exact conservation is preserved even on coarse meshes upon the
convergence of the overall system.
43
During the analysis, for the discretization of velocities and temperature
streamline upwind/Petro-Galerkin (SUPG) approach is used whereas for κ, ε, and
pressure, Monotone Streamline Upwind Approach (MSU) is used.
4.2.1.2. Diffusion Terms
Diffusion terms can be obtained by integrating the expression given in Eq.
(4.1.) over the solution domain using the weighting functions [27].
dAyy
eWdAxx
eWonContributiDiffusion
∂∂Γ
∂∂
+
∂∂Γ
∂∂
= φφ
φφ (4.14)
Integration by parts yields to:
∂∂Γ
∂∂=
∂∂Γ
∂∂
dAxx
eWdA
xxeW
φφ
φφ (4.15)
Once the derivative is replaced by the nodal values and the derivatives of the weighting
function, the nodal values will be removed from the integrals
xW
WwhereWx
ee
xe
x ∂∂==
∂∂ φφ
(4.16)
The diffusion matrix may now be expressed as:
dAeyWe
yWexWe
xWdiffusioneA
Γ+Γ=
φφ (4.17)
4.2.1.3. Source Terms
Using the weighting functions and integrating over the volume, source terms
can be evaluated as:
= dASeWeS φφ (4.18)
44
The momentum equations obtained by employing the formulation given above
can be solved only when the pressure field is given or somewhat estimated. Unless the
correct pressure field is employed, the resulting velocity field from the momentum
equations will not satisfy the continuity equation. Such an imperfect velocity field
(based on guessed pressure field) can be improved and will progressively get closer to
satisfying the continuity equation by applying velocity and pressure corrections.
45
CHAPTER 5
NUMERICAL SOLUTION
5.1. Segregated Solution Algorithm
The preceding section outlines the discretization of each term in the
momentum, energy, kinematic energy (κ) and dissipation (ε) rate equations. In this
section, solution of these coupled equations will be explained. Each equation is solved
in a sequential manner, using an intermediate value for the other degrees of freedom
[29]. After solving all the equations, the values are updated and next iteration is
performed until the convergence criteria are satisfied.
The preceding section outlined the approach for every equation except the
pressure equation, which comes from the segregated velocity-pressure solution
algorithm. In this approach, the momentum equation is used to generate an expression
for the velocity in terms of the pressure gradient. This expression is used in the
continuity equation after integrating by parts. This nonlinear solution procedure belongs
to a general class of Semi-Implicit Method for Pressure Linked Equations (SIMPLE)
[29]. The numerical solution is achieved using (SIMPLEN) algorithm; an enhanced
(SIMPLE) algoritm [29]. During derivation, 2-D incompressible flow equations will be
considered.
46
Application of Galerkin's method to the continuity equation yields to:
( ) dAyW
yVxW
xVdAyVxVWdAyyV
xxV
W
∂∂+
∂∂−+=
∂
∂+
∂∂
(5.1)
The next step is to find an expression for the velocities in terms of the pressure
gradient. When the momentum equations are solved, it is with a previous value of
pressure. Algebraic expressions of the momentum equations can be written assuming
that the coefficient matrices consist of the convection and diffusion contributions as
before, and all the source terms are evaluated except the pressure gradient term.
dAeE
e xP
WSxAV =
∂∂−=
1φ (5.2)
dAeE
e yP
WSyAV =
∂∂−=
1φ (5.3)
Each of these sets represents a system of N algebraic equations for N unknown
velocities. After the summation of all the element quantities, it is possible to show an
expression for each velocity component at each node in terms of the neighboring
velocities, the source terms, and the pressure drop. Using the subscript ‘i’ to denote the
nodal equation, for i = 1 to N, where N is the number of fluid nodes and subscript “j” to
denote its neighboring node:
dAxP
Wij
j
xija
xr
xiia
xiVxiV Ω
∂∂
≠
+
−= 1ˆ (5.4)
47
dAyP
Wij
j
yija
yr
yiia
yiVyiV Ω
∂∂
≠
+
−= 1ˆ (5.5)
where
( )
≠
+
≠
+−−
=ij
j
xija
xr
xiia
ij
j
xibxVxVx
ija
xiVij
ˆ (5.6)
( )
≠
+
≠
+−−
=ij
j
yija
yr
yiia
ij
j
yibyVyVy
ija
yiVij
ˆ (5.7)
Here aij represent the values in the x and y coefficient matrices for the two
momentum equations, r is the relaxation factor, and bi is the modified source term
taking into effect the relaxation factors.
For the purposes of this expression, the neighboring velocities for each node
are considered as being known from the momentum equation solution. At this point, the
assumption is made that the pressure gradient is constant over the element, allowing it
to be removed from the integral. This means that only the weighting function is left in
the integral, allowing a pressure coefficient to be defined in terms of the main diagonal
of the momentum equations and the integral of the weighting functions:
48
=
⋅
≠
+
=N
edAW
ij
j
xija
xr
xiia
xM1
1 (5.8)
=
⋅
≠
+
=N
edAW
ij
j
yija
yr
yiia
yM1
1 (5.9)
Therefore, expressions for unknown nodal velocities have been obtained in
terms of the pressure drop and a pressure coefficient.
∂∂−=
xP
MVV xxixiˆ (5.10)
∂∂−=
yP
MVV yyiyiˆ (5.11)
These expressions are used to replace the unknown velocities in the continuity
equation to convert it into a pressure equation. The terms coming from the unknown
velocities (replaced with the pressure gradient term) and from the unknown density
(expressed in terms of the pressure) contribute to the coefficient matrix of the pressure
equation while all the remaining terms will contribute to the forcing function.
The entire pressure equation can be written on an element basis, replacing the
pressure gradient by the nodal pressures and the derivatives of the weighting function,
putting all the pressure terms on the left hand side and the remaining terms on the right
hand side
49
[ ]
[ ] [ ] dss
yVWdss
xVWedAyW
yVyW
xVxW
edAyW
yMyW
xW
xMxWeP
−−
∂∂
∂∂+
∂∂
=
∂∂
∂∂+
∂∂
∂∂
ˆˆ (5.12)
The final step is the velocity update. After the solution for pressure equation,
the known pressures are used to evaluate the pressure gradients. In order to ensure that a
velocity field exists which conserves mass, the pressure term is added back into the
“hat”(previous) velocities:
∂∂
≠
+
−= ePedAx
WW
ij
j
xija
xr
xiia
xVxV1ˆ (5.13)
∂∂
≠
+
−= ePedAyW
Wij
j
yija
yr
yiia
yVyV1ˆ (5.14)
The global iterative procedure in SIMPLEN algorithm can be summarized as
follows [29]:
• Start assuming turbulent flow
• Solve velocities (V ) approximately (Eqs. 5.6-5.7)
• Solve pressure equation for P using Eq. (5.12)
• Solve turbulence equations for and ε
• Update effective properties based on turbulence solution
50
• Calculate viscosity ratio and decide laminar or turbulent flow
• Update velocities based on (V ) and P
• Formulate and solve energy equation for T
• Check rate of change of the solution (convergence monitors)
• End of global iteration
During the solution algorithm it should be noted that after the calculation of
viscosity ratio, if the flow turns out to be laminar, then governing equations (continuity,
x-momentum, y-momentum and energy) for laminar flow are solved using the
coefficients given in Table 4.1. Otherwise, the iterations continue with turbulent
formulation until the convergence criteria are satisfied. With such an approach,
computation time is reduced by discarding and equations from the solution of
laminar flow.
5.2. Matrix Solvers
The algorithm requires repeated solutions to the matrix equations during each
set of iterations. It is common practice to use fast approximate algorithms for the
momentum equations since it is time saving even if slightly slower convergence rates
are achieved compared to exact solvers [29]. In the case of the pressure equation, more
accurate results are required to ensure conservation of mass. Accuracy of pressure
equation also directly affects the convergence of the momentum equations in the next
global iteration. In a thermal problem with constant properties, there is no need to solve
the energy equation at all steps until the flow problem has been converged.
51
To accommodate the varying accuracy requirements, there are two major types
of matrix solvers in Ansys, Flotran. These are iterative and exact solvers [27]. The first
solver is an iterative one called Tri-Diagonal Matrix Algorithm (TDMA). The other
solvers are exact solvers namely the conjugate residual method for non-symmetric
matrix equations and preconditioned conjugate gradient method for incompressible
pressure equation.
TDMA is an approximate solver performing a user specified number of
iterations through the problem domain. The method consists of breaking the problem
into a series of tri-diagonal problems where any entries outside the tri-diagonal portion
are treated as source terms using the previous values. For a completely unstructured
mesh, or an arbitrarily numbered system, the method reduces to the Gauss-Seidel
iterative method. Since it is considered an approximate method, TDMA is not executed
to convergence. The iteration technique is based on improving the initial guessed values
by systematic repetitions until the solution is sufficiently close to the correct solution of
the algebraic equations. The round-off errors that create a problem in direct methods are
no longer important, since the user can control the level of error. Besides this, the
problem of using large amount of computer memory is eliminated because only one set
of variables is held in computer storage. The major disadvantage of the iterative
methods is convergence is too slow, especially when the number of grid points is large.
The reason for the slowness is easy to understand; the method transmits the boundary
condition information at a rate of one grid interval per iteration [29].
Exact solvers are semi-direct methods that iterate to a specified convergence
criterion. These are iterative methods used to attempt an exact solution to the equation
52
of interest. The performance of the semi-direct solvers is monitored by the behavior of
the inner product of residuals, which should be reduced to a fraction typically 1x10-7 of
its initial value. The conjugate gradient method is used only for the pressure equation in
incompressible flows. The sequential solution algorithm must allow space for a non-
symmetric coefficient matrix for the momentum and energy equations. Only half of this
storage is required for the symmetric matrix and the other half is used to store the
decomposition. The conjugate residual method can be used with or without
preconditioning, the latter approach requiring significantly less computer memory [27].
Semi-direct methods combine the advantages of direct solvers that is solving algebraic
equations to a specified convergence requirement in a short time with the iterative
solvers that is stability and better convergence of the solution.. Therefore, during the
solutions, TDMA method is used for the evaluation of the velocity and temperature
values. Pressure equation is solved using a semi-direct solver, namely pre-conditioned
conjugate gradient method.
5.3. Overall Convergence and Stability
5.3.1. Convergence
The fluid problems are difficult to solve due to their non-linear nature and
convergence may not be achieved always. Instabilities can result from a number of
factors: the matrices may have poor condition numbers because of the finite element
mesh or very large gradients of variables in the actual solution. The fluid phenomena
being observed could also be unstable in nature.
53
Overall convergence of the solution is controlled through the convergence
monitoring parameters. A convergence value is calculated for each variable at each
global iteration. It can be defined as normalized rate of change of the solution from one
global iteration to the next and is calculated for any variable as follows [27]:
=
=
−−=
N
i
ki
N
i
ki
ki
M
1
1
1
φ
φφφ (5.15)
where:
Mφ= convergence monitor for degree of freedom φ
N= total number of finite element nodes
φ= Degree of freedom (Vx, Vy, T, ε etc.)
k=current global iteration number
It is thus the sum of the absolute value of the changes over the sum of the
absolute values of the degree of freedom. User sets the convergence criteria. But it is
not guaranteed to reach to these criteria at the end of the global iterations, which can be
due to several factors. The nature of the problem may not let the solution converge
below the preset criteria (oscillations in the flow, instabilities etc.). Another reason can
be the coarse solution mesh. Wrong matrix solvers for the equations may prevent
convergence. Wrong under-relaxation, over relaxation and artificial viscosity values
will cause divergence in the solution, or sometimes will slow down the convergence. In
order to satisfy convergence of variables, the convergence criteria given by Eq. (5.15)
are observed through the solution monitor until the prescribed convergence criteria are
met by all the variables. A typical solution monitor is
54
0 50 100 150 200 250 300 350 400 450 5001.10
11
1.1010
1 .109
1 .108
1 .10 7
1 .10 6
1 .10 5
1 .10 4
1 .103
0.01
0.1
1
VxVyPressureENKEENDSTEMP
# of iterations
resi
dual
An 1,
An 2,
An 4,
An 5,
An 6,
An 7,
n
Fi
gure 5.1. Typical convergence monitor of the variables
shown in Fig. 5.1. The convergence criteria set for all variables is 10-6. The iterations
continue until either the prescribed convergence criteria or the number of global
iterations is reached. During the runs, it is observed that the convergence is achieved at
400 to 800 global iterations for laminar flow and 800-1200 iterations for turbulent flow.
Besides, flat plate solutions converge much faster than that of boards with chips.
5.3.2.Stability
Three techniques are available in Flotran to slow down and stabilize a solution.
These are relaxation, inertial relaxation, and artificial viscosity [27].
55
5.3.2.1. Relaxation
Relaxation is simply taking as the answer some fraction of the difference
between the previous global iteration result and the newly calculated values. In addition
to the degrees of freedom, relaxation can be applied to effective viscosity and effective
conductivity calculated through the turbulence equations. Denoting by φi, the nodal
value of interest, the expression for relaxation is as follows:
( ) calci
oldi
newi rr φφφ φφ +−= 1 (5.16)
where rφ is called relaxation factor.
5.3.2.2. Inertial Relaxation
Inertial relaxation is used to make a system of equations more diagonally
dominant. It is similar to a transient solution. It is most commonly used in the solution
of the compressible pressure equation and in the turbulence equations. It is only applied
to the DOF.
The algebraic system of equations to be solved may be represented as, for i=1
to the number of nodes:
≠
=+ij
ijijiii faa φφ (5.17)
With inertial relaxation, the system of equations becomes:
( ) oldi
dii
ijijiji
diiii AfaAa φφφ +=++
≠
(5.18)
where =rf
dii B
areaWdA
)(ρ (5.19)
56
Brf is called as inertial relaxation factor. When the solution converges, Aii.φi
and Aii.φiold on both sides of the equations are equal to each other and cancel each other
out. This form of relaxation is always applied to the equations, and in order to reduce
the dominancy of inertial relaxation, the factor is set to a very large number (the default
value of Brf = 1.0 x 1015) so that the term including inertial relaxation becomes
practically zero.
5.3.2.3. Artificial Viscosity
Artificial viscosity is a stabilization technique that has been found useful in
compressible problems and incompressible problems involving distributed resistance.
The technique serves to increase the diagonal dominance of the equations where the
gradients in the momentum solution are the highest. Artificial viscosity enters the
equations in the same fashion as the fluid viscosity. The additional terms are:
∂∂
+∂
∂∂∂=
y
V
xV
xR yx
ax µ (5.20)
∂∂
+∂
∂∂∂=
y
V
xV
yR yx
ay µ (5.21)
where µa is the artificial viscosity.
In each of the momentum equations, the terms resulting from the discretization
of the derivative of the velocity in the direction of interest are additions to the main
diagonal, while the terms resulting from the other gradients are added as source terms.
Note that since the artificial viscosity is multiplied by the divergence of the velocity,
57
(zero for an incompressible fluid), it should not impact the final solution. For
compressible flows, the divergence of the velocity is not zero and artificial viscosity
must be regarded as a temporary convergence tool, to be removed for the final solution.
5.3.2.4. Residual File
One measure of how well the solution is converged is the magnitude of the
nodal residuals throughout the solution domain. The residuals are calculated based on
the “old” solution and the “new” coefficient matrices and forcing functions. Residuals
are calculated for each degree of freedom (VX, VY, VZ, PRES, TEMP, ENKE, ENDS).
The residuals provide information about where a solution may be oscillating. The values
at each node are normalized by the main diagonal value for that node in the coefficient
matrix. This enables direct comparison between the value of the residual and value of
the degree of freedom at the node.
5.4. Numerical Modeling
5.4.1. Grid Configuration
Grid generation plays a crucial role in the solution of any numerical problem.
A well-established computation grid is essential not only to capture the physics of the
problem, but also to achieve convergence, stability and a correct solution.
It is possible to mesh the computation domain with unstructured tetrahedral
and hexahedral and structured hexahedral elements in Flotran. Each meshing technique
has advantages and disadvantages. Tetrahedral mesh has the ability to represent any
physical model accurately i.e. cylinders, complex shapes, curved surfaces etc. On the
contrary, number of elements with tetrahedral mesh becomes very large compared to
58
hexahedral elements, which results in a drastic increase in the computation time.
Compared to tetrahedral mesh, hexahedral mesh will generate less elements yielding to
decrease computation time, while sacrificing from accuracy of the results. The best
method of meshing is mapped meshing using hexahedral elements. It will not only yield
minimum number of elements for the same solution domain, but also yield the most
correct answers compared to the previous techniques. However, mapped meshing has
very limited application areas, because the solution domain should be consisting of all
rectangles.
For the problem under consideration, it is wise to use mapped mesh with
rectangular elements since the geometry is 2-D and rectangular in shape. There are
many ways to mesh the computation domain. Three major meshing techniques are given
in Fig. 5.2. The first configuration consists of same size elements. The advantage of
such a configuration is the ease of grid generation. The disadvantage is that,
unnecessary numerical computation is performed at those locations where
Figure 5.2. Different Meshing techniques of the solution domain
59
accuracy is not vital. Besides, if finer grid configuration is required due to stability and
convergence problems, computation time enormously increases. Second grid
configuration is created to eliminate the disadvantage of first configuration by focusing
the finer grids at critical locations where the geometry changes abruptly. This
configuration consists of two different size elements, namely a coarse and a fine mesh.
Using such grid configuration, computation time relatively decreased with respect to the
first one. But the penalty is the additional computation time at the interface of different
size grid sections. The final configuration focuses primarily on the locations where
numerical error due to grid size can be an important criterion in the stability of the
solution. The grid configuration enables to increase number of grid points at desired
locations only. Thus it is the configuration used in this study. When the channel spacing
is increased for the optimization calculations, number of grids in the y direction is
increased in order to keep the grid size in the desired range. A more detailed
representation of numerical grid used in the calculations is shown in Fig. 5.3 and 5.4.
The first figure belongs to the laminar flow with four chips per board. The second figure
is for turbulent flow with six chips per board.
Figure 5.3. Detailed mesh configuration around the chips for laminar flow
Figure 5.4. Detailed mesh configuration around the chips for turbulent flow
60
5.4.2. Grid Independency
In order to verify that the solutions are independent of grid size, i.e. grid
independency of the solution, a series of calculations is performed at the beginning of
each case. The worst case i.e. the largest board spacing, is examined for a series of
solution at a given pressure drop. For this purpose, it is logical to observe total heat
transfer rate from a single channel, which is also interest of the present study. The heat
transfer rate from a single channel can be written as:
( )miTmeTpcmQ −= '' (5.22)
The inlet mean temperature is equal to free stream temperature and heat transfer rate
depends on exit mean temperature. Non-dimensional mean exit temperature for constant
chip temperature and heat flux boundary conditions can be written as:
∞−∞−
=TchipT
TmeTmeθ for constant chip temperature (5.23)
kdwqTmeT
me /′′∞−
=θ for constant chip heat flux (5.24)
The variation of mean exit temperature with number of elements is shown through Fig.
5.5 to 5.16. There are five different chip configurations for two different flows namely
laminar and turbulent, and for two thermal boundary conditions, resulting in 20
different cases. Each case is solved at least for 7 different pressure drops and each
pressure drop includes 8 different board-to-board spacing, which makes a total of
minimum 1000 runs. Thus, it is not possible to illustrate grid independency of each
case. Therefore for each chip configuration, one laminar and one turbulent flow cases
are chosen for constant chip temperature boundary condition. Two additional figures are
included for constant chip heat flux boundary condition as well.
61
Turbulent Flow, T=const.# of chips=4, spacing =2.8 mm
0,63
0,64
0,65
0,66
0,67
0 5000 10000 15000 20000 25000 30000 35000
# OF ELEMENTS
θθθθme
Figure 5.5. Variation of non-dimensional mean exit temperature with number of
elements for turbulent flow over 4 chips per board configuration
Turbulent Flow, T=const.# of chips=6, spacing =2.6 mm
0,89
0,9
0,91
0,92
0,93
0,94
0 10000 20000 30000 40000 50000 60000 70000# OF ELEMENTS
θθθθme
Figure 5.6. Variation of non-dimensional mean exit temperature with number of
elements for turbulent flow over 6 chips per board configuration
62
Turbulent Flow, T=const.# of chips=8, spacing =2.6 mm
0,978
0,98
0,982
0,984
0,986
0,988
0,99
0,992
0 10000 20000 30000 40000 50000 60000 70000 80000 90000# OF ELEMENTS
θθθθme
Figure 5.7. Variation of non-dimensional exit temperature with number of elements for
turbulent flow over 8 chips per board configuration
Turbulent Flow, T=const.# of chips=10, spacing =2.6 mm
0,985
0,99
0,995
1
0 10000 20000 30000 40000 50000 60000 70000 80000 90000# OF ELEMENTS
θθθθme
Figure 5.8. Variation of non-dimensional mean exit temperature with number of
elements for turbulent flow over 10 chips per board configuration
63
Turbulent Flow, T=const.# of chips=flat, spacing =3 mm
0,3
0,33
0,36
0,39
0,42
0,45
0 10000 20000 30000 40000 50000 60000 70000 80000 90000# OF ELEMENTS
θθθθme
Figure 5.9. Variation of non-dimensional mean exit temperature with number of
elements for turbulent flow over flat plate
Laminar Flow, T=const.# of chips=4, spacing =25 mm
0,373
0,374
0,375
0,376
0,377
0,378
0,379
0 2000 4000 6000 8000 10000 12000 14000 16000
# OF ELEMENTS
θθθθme
Figure 5.10. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow over 4 chips per board configuration
64
Laminar Flow, T=const.# of chips=6, spacing =20 mm
0,626
0,627
0,628
0,629
0,63
0,631
0,632
0,633
0 5000 10000 15000 20000 25000# OF ELEMENTS
θθθθme
Figure 5.11. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow 6 chips per board configuration
Laminar Flow, T=const.# of chips=8, spacing =30 mm
0,355
0,356
0,357
0,358
0,359
0,36
0 5000 10000 15000 20000 25000 30000# OF ELEMENTS
θθθθme
F
igure 5.12. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow over 8 chips per board configuration
65
Laminar Flow, T=const.# of chips=10, spacing =18 mm
0,958
0,9584
0,9588
0,9592
0,9596
0,96
0 5000 10000 15000 20000 25000 30000 35000# OF ELEMENTS
θθθθme
Figure 5.13. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow over 10 chips per board configuration
Laminar Flow, T=const.# of chips=flat, spacing =20 mm
0,924
0,926
0,928
0,93
0 1000 2000 3000 4000 5000 6000 7000# OF ELEMENTS
θθθθme
Fi
gure 5.14. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow over flat plate
66
Laminar Flow, q=const.# of chips=10, spacing =18 mm
0,768
0,77
0,772
0,774
0,776
0,778
0 5000 10000 15000 20000 25000 30000 35000# OF ELEMENTS
θθθθme
Figure 5.15. Variation of non-dimensional mean exit temperature with number of
elements for laminar flow over flat plate, constant heat flux boundary condition
Turbulent Flow, q=const.# of chips=4, spacing =2.8 mm
0,39
0,4
0,41
0,42
0,43
0,44
0 5000 10000 15000 20000 25000 30000 35000
# OF ELEMENTS
θθθθme
Figure 5.16. Variation of non-dimensional exit mean temperature with number of
elements for turbulent flow over 4 chips per board configuration, constant heat flux
boundary condition
67
The Figs. 5.5 to 5.16 show the dependence of the exit mean temperature on the
solution grid. The dependence is not linear as expected. As the number of elements
increases, the change in non-dimensional exit temperature decreases. Therefore, it is not
logical to increase number of elements beyond a certain point, since it increases
computation time drastically, with a minor change in results. This point can be regarded
as third data point in the Fig. 5.5 to 5.16. During the calculations, element numbers
corresponding to third data point are chosen and satisfactory results have been obtained.
5.4.3. Numerical Data
The geometrical parameters, boundary conditions, and computational data
related to numerical solution are presented in this section. Although the results
presented in this work are all dimensionless, they are modeled as dimensional terms in
Flotran. The data is chosen such that it represents physical dimensions in real world.
During the construction of geometry, chip positions are chosen such that the
board length is equally divided i.e. spacing between the chips and the distance between
the trailing edge and first chip and the distance between the leading edge and last chip
are all equal.
Geometrical Parameters:
Length of the boards: 300 mm.
Chip width: 15 mm
Chip height: 1.5 mm
Number of chips per board: 4, 6, 8, 10, flat
Board spacing: 2.5mm-50 mm
68
Boundary Conditions:
Pressure Drop: (0.16x10-2-0.1224x10-1)Pa for laminar flow
(81.6-816)Pa for turbulent flow
Inlet temperature (To): 20 oC
Chip wall temperature (Tchip): 100 oC
Computational Data:
Number of elements: 5000-20000 for laminar flow
15000-60000 for turbulent flow
Average Computation time: 200 CPU seconds for laminar flow
1000 CPU seconds for turbulent flow
Total Runs performed: 600 runs for laminar flow
600 runs for turbulent flow
Computer Used: HP X2100 Workstation, Pentium 4,
2.4GHz. Processor, 1.5 GB RD RAM
Convergence criteria: 1x10-6 for all DOF
5.5. Comparison of the Results with Experimental and Numerical Results
in the Literature
In this section, laminar and turbulent flow results of the current study will be
compared with the ones in literature to verify the analysis. There are numerous
experimental and numerical studies available for two-dimensional flow between parallel
plates [5]. In the first part, axial velocity distributions, pressure drop and Nusselt
69
numbers will be considered for the flat plate. In the second part of this section, the
grooved channel results are compared with the experimental work [9].
Before introducing the comparisons of laminar and turbulent results with the
results in literature, it is useful to emphasize the role of viscosity ratio in the
determination of flow regime. For that purpose, change of viscosity ratio with Reynolds
number is plotted in Fig. 5.17 for smooth channel results of the present work.
Theoretically, for channel flow, transition from laminar to turbulent occurs at Re of
2300 [20]. The results of the present study show that corresponding viscosity ratio for
Re 2300 is 5 for smooth channels. Therefore, as an alternative to Reynolds number,
viscosity ratio of 5 can be used to determine whether the flow is laminar or turbulent
[25]. Use of viscosity ratio eliminates the difficulty of Reynolds number definition for
channels with chips since the height of channel varies. Moreover, during the solution of
grooved channels, it is observed that the fluctuations occur earlier than the predicted
Reynolds number. Therefore, the use of viscosity ratio gives better estimation of the
flow field than the use of Reynolds number.
Change of Re with Viscosity Ratio
0
5
10
15
20
25
1000 10000 100000Re
Vis
cosi
ty R
atio
Retheo=2300
turbulentlaminar
Figure 5.17. Variation of viscosity ratio with Reynolds number for smooth duct
70
5.5.1. Comparison of Smooth Duct Results with the Literature
In this part, validity of numerical results of the current study for the laminar
flat plate is determined by comparing the results with various other numerical,
analytical and experimental results available in literature [6].
5.5.1.1. Laminar Flow
5.5.1.1.1. Developing Velocity
The problem of laminar flow development in a flat duct has been studied in
detail by employing both boundary layer theory idealizations and a complete set of
Navier-Stokes equations. Among the boundary-layer type of solutions, the numerical
work of Bodoia and Osterle [30], on laminar hydrodynamically developing flow is
regarded as one of the most accurate ones [6].
Dimensionless axial velocity distribution from the numerical results of [30] is
presented in Table 5.1. The axial velocity distribution of this study is plotted in Fig.
5.18 in order to compare the results with [30], where the dimensionless axial coordinate
x+ for the hydrodynamic entrance region is defined as follows:
Re2Re dx
Dx
xh
==+ (5.25)
Table 5.1 Axial velocity in the hydrodynamic entrance region of a flat duct.
Axial Velocity, u/um
103x+ Y/d=1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.25 1.1013 1.1013 1.1013 1.1013 1.1012 1.1010 1.0993 1.0863 1.0132 0.7194 0
0.5 1.1443 1.1443 1.1443 1.1442 1.1438 1.1414 1.1290 1.0788 0.9204 0.5567 0
2.0 1.2882 1.2811 1.2778 1.2684 1.2447 1.1923 1.0918 0.9246 0.6825 0.3708 0
5.0 1.4111 1.4039 1.3803 1.3357 1.2635 1.1565 1.0095 0.8917 0.5870 0.3132 0
71
0
1
2
3
4
5
6
7
8
0 0,2 0,4 0,6 0,8 1
y/d
Vx/Vx,m
Numerical Results, [30]
Present study
Figure 5.18. Developing axial velocity in the entrance region of a flat duct for laminar
flow
As can be seen from the figure, there is a good agreement between the
developing axial velocity distribution of the present study and [30].
5.5.1.1.2. Pressure Drop
The fluid flow characteristics of all the ducts are expressed in terms of certain
hydrodynamic parameters. The dimensionless distance for the hydrodynamically
developing flow is already defined by Eq. (5.25) where the hydraulic diameter is
defined as 4 times the duct cross-section area divided by the wetted parameter. The
103x+=0.2
103x+=0.5
103x+=2.0
103x+=5.0
72
hydraulic diameter is consistently used as the characteristic dimension in the definition
of Reynolds and Nusselt numbers especially for the simple geometries, in other words
geometries without groove.
The fanning friction factor, f, is defined as the ratio of the wall shear stress τw
to the flow kinetic energy:
2)2/1( mu
wfρ
τ= (5.26)
In the entrance region of the ducts, the pressure drop is due to the combined
effect of the wall shear and the change in momentum flow rate due to the developing
velocity profile. Therefore it is wise to define a combined friction factor, which is called
apparent friction factor. Shah and London [6], developed a correlation, Eq. (5.27), that
predicts the apparent friction factor for laminar developing flow
( ) 2
2/1
2/1 )(000029.01)/(44.3)4/(674.02444.3
Re −+
++
+ +−++=
xxx
xf app (5.27)
The fanning friction factor and axial pressure drop are related as:
+∆=
x
Pappf
4
*Re (5.28)
Then the dimensionless pressure drop:
dL
appfP =∆ * (5.29)
In the Fig. 5.19, dimensionless pressure drop results of the current study can be
compared with Shah and London’s [6]. Although there is a little deviation from the
theoretical results, the results of the present study are in the acceptable range of ±5%.
73
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10x+
∆∆∆∆P*
Shah and London [6], correlationPresent study
Figure 5.19. Dimensionless pressure drop for the laminar developing flow in a flat duct
5.5.1.1.3. Nusselt Number:
The solution with equal and uniform temperatures on both duct walls is of
special importance, as it constitutes the limiting case of rectangular ducts with constant
temperature boundary condition. The solution was first developed by Nusselt in 1923,
in the form of Taylor series expansion [6]. Other scientists investigated the topic
furthermore focusing on the effect of axial conductance on the flat duct solution with
the uniform and equal temperatures at both duct walls. They showed that the effect of
axial conduction is negligible for Pe>50 [6].
For the simultaneously developing flow in a flat duct, the results are available
for uniform temperature and uniform heat flux on both walls. The Nusselt number can
be defined as:
( )mTmwTkxq
khdxh
xNu−
==,
" (5.30)
74
Different scientists have studied the problem of simultaneously developing
flow in a flat duct. The most accurate results are those suggested by Hwang and Fan
[31], referring to a numerical work. The results for the constant wall temperature
boundary condition are correlated in the following form, which is valid for
0.1<Pr<1000:
64.017.0
14.1
, )(Pr0358.01)(024.0
55.7 −++
−++
++=
xx
Nu Tm (5.31)
Then Shah and Bhatti [6], developed the following formula for the local
Nusselt numbers by differentiating the correlation suggested by Hwang and Fan:
[ ][ ]264.017.0
64.017.014.1
,)(Pr0358.01
14.0)(Pr0179.0)(024.055.7
−++
−++−++
+
−+=x
xxNu Tx (5.32)
In Eq. (5.32), dimensionless axial coordinate for the thermal entrance region
x++ was defined by,
PrRe2dx
PeDx
xh
==++ (5.33)
In Fig. 5.20, comparison of the local Nusselt numbers of the current study with
the one suggested by Eq. (5.32) is presented. Both results converge to the same fully
developed Nusselt number although there is a small variation at the developing region.
The problem of simultaneously developing laminar flow with constant heat flux at both
walls was again studied by Hwang and Fan [31], and the results are presented in the Fig.
5.21. The results of the constant heat flux boundary perfectly are in agreement with the
experimental results.
75
6
7
8
9
10
11
12
13
14
15
0 0.005 0.01 0.015 0.02 0.025 0.03
x++
Nux,T
Hwang and Fan [31],numericalPresent study
Figure 5.20. Local Nusselt number for the simultaneously developing laminar flow
with constant temperature boundary condition
7
8
9
10
11
12
13
14
15
0 0.01 0.02 0.03 0.04 0.05 0.06
x++
Nux,q
Present study
Hwang and Fan [31], numerical
Figure 5.21. Local Nusselt number for the simultaneously developing laminar flow
with constant heat flux boundary condition
76
5.5.1.2. Turbulent Flow
In this section, results of turbulent flow between parallel plates are compared
with the ones in the literature. The results presented consist of developing and fully
developed velocity distributions, friction factor and local Nusselt number variation
through the entrance region for Tw-constant and qw-constant cases.
5.5.1.2.1. Developing Velocity
In the experimental study, conducted by Dean [32], developing velocity
profiles for Reynolds number of 200000 are investigated. The results are presented in
Table 5.2 and compared in Fig. 5.22 with the results of the current study. In general,
they are in good agreement with the experimental results of [32].
Table 5.2. Experimental velocity distribution of the turbulent developing flow between
parallel plates for Re=200000 by Dean [32]
x/d 2y/d
Vx/Vx,mean
0.05 0.1 0.22 0.31 0.4 0.5 0.6 0.7 0.8 0.9 1 8.3
0.73 0.82 0.986 1.029 1.063 1.063 1.063 1.063 1.063 1.063 1.063
0.05 0.1 0.2 0.32 0.4 0.55 0.6 0.67 0.8 0.9 1 20.4
0.73 0.786 0.89 0.979 1.02 1.07 1.1 1.114 1.114 1.114 1.114
0.05 0.11 0.2 0.31 0.4 0.49 0.61 0.7 0.8 0.9 1 38.7
0.72 0.828 0.895 0.964 1 1.028 1.085 1.112 1.134 1.153 1.156
0.05 0.11 0.2 0.31 0.4 0.5 0.6 0.71 0.8 0.9 1 75.3
0.75 0.856 0.915 0.97 1.005 1.039 1.071 1.096 1.107 1.123 1.13
5.5.1.2.2. Fully Developed Velocity
There are many studies on the determination of fully developed velocity profile in
turbulent flow. Among them, Laufer [33] is one of the first scientists, who performed
experiments to obtain the fully developed velocity profile in turbulent flow. His results
are used by other scientists to develop mathematical expressions for the velocity profile
77
in a smooth-walled flat duct. For the fully developed velocity distribution in turbulent
flow, experimental results [33] are used to determine the validity of the results of the
current study. The results are in good agreement with the experiments as shown in Fig.
5.23.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2y/d
Vx/
Vx,
mea
n x/d=8.3
x/d=20.4
x/d=38.7
x/d=75.3
0(8.3)
0(20.4
0(38.7
0(75.3
1.2 (8.3)
1.0 (8.3)
1.2
1.0 (20.4)
1.2
1.0 (38.7)
1.2
1.0
Experimental [19] Present Study
Figure 5.22. Velocity distribution for developing turbulent flow in a parallel plate
channel for the Reynolds number 200000
78
Re=9370
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y/d
Vx/
Vx,
mea
n
Present Study
Experimental [33]
Re=17100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y/d
Vx/
Vx,
mea
n
Present Study
Experimental [33]
F
igure 5.23 Fully developed velocity distribution for turbulent flow in a parallel plate
channel for Reynolds number 9370 and 17100
5.5.1.2.3. Pressure Drop
Deissler investigates the apparent friction factor, fapp, in the hydrodynamic
entrance region of a smooth flat duct by an integral method [34]. Fig. 5.24 shows the
79
comparison between numerical results of this study and Deissler’s, which are in very
good agreement except a deviation in the entrance region.
Re=10000
0
0.005
0.01
0.015
0.02
0.025
0 5 10 15 20 25
x/Dh
fapp
Numerical, [34]Present study
Figure 5.24. Turbulent flow apparent friction factor in the hydrodynamic entrance
region of a flat duct with uniform inlet velocity
5.5.1.2.4. Nusselt Number
The thermally developing flow in a flat duct with uniform and equal
temperatures or heat fluxes at both walls has been examined by Paykoç [35]. The results
of the present study are presented in Figs. 5.25 and 5.26 in comparison with the
Paykoç’s results [35]. The numerical results of turbulent Nusselt numbers show very
good agreement except a deviation in the entrance region.
5.5.2. Comparison of Grooved Duct Results with the Literature
The results of the present work are considered in two categories: Turbulent
flow and laminar flow. The grooved channel studies in focus on specific geometries.
Such a systematic approach to the problem under consideration is performed for the first
80
time by the present study. To verify the results of laminar flow, experimental setup of
[9] is modeled in Ansys. In case of turbulent flow, the situation is worse. There are only
a few examples focusing on the single grooves. Thus, turbulent solutions of the present
study are going to help to fill the gap in this field.
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10
x/Dh
Loca
l Nu T
Numerical, [35]
Present Study
Re=9370
Re=17100
Figure 5.25. Thermally developing turbulent flow in a parallel plate channel with
constant wall temperature for Reynolds number 9370 and 17100
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6 7 8 9 10
x/Dh
Loca
l Nu q
Numerical, [35]Present Study
Re=9370
Re=17100
Figure 5.26. Thermally developing turbulent flow in a parallel plate channel with
constant heat flux boundary conditions for Reynolds number 9370 and 17100
81
The work performed by Farhanieh, Herman, and Sunden [9] constitutes an
extensive study on laminar fluid flow in grooved channel with both numerical and
experimental approaches. The results of [9] are compared with the present study in
order to validate the findings.
During the experiments of [9], holographic interferometry technique is used to
obtain temperature distribution in the computation domain. Holographic interferometry
was introduced to heat transfer measurement some 30 years ago. The technique uses
light as information carrier to obtain qualitative and quantitative data on the investigated
phenomenon; for this case, the temperature fields, in the grooved and communicating
channels. The measurement method is based on the comparison of wave fronts, where
at least one of the wave fronts is reconstructed holographically. The light source is a
laser. The laser beam is divided into two; a reference beam and a measuring beam by
means of a beam splitter. Both beams are then expanded into parallel ray bundles by a
beam expander. The object wave passes through the test section with the phase object,
which is the temperature field for this case and falls on the holographic plate. The
reference beam falls directly onto the plate recording the reference state. Then the
photographic plate is developed and exactly repositioned with a precision plate holder.
In the second stage, reference state is reconstructed by illuminating the holographic
plate with the reference beam. Then, the test section is heated. The object wave
experiences a phase change on its way through the test section. The difference between
the reference state and the measured state is measured in form of a fringe pattern. This
very high-speed phenomenon can be investigated by using a high-speed camera.
82
Figure 5.27. Schematic of experimental setup
Experimental arrangement used in the visualization of temperature fields are
modeled in Ansys and shown in Fig. 5.27. The experimental channel consists of an
entry section, test section, and the exit section. The length, height and depth of the entry
section are 0.9 m, 0.01 m, and 0.22 m respectively. The selected length of the entry
section provides the fully developed velocity profiles at the entrance of the test
section. The depth to height ratio of 22:1 is needed to obtain the two-dimensional
temperature distribution in the test section. The test section is 0.22 m long, and is kept at
constant wall temperature by circulating water from the reservoir. The grooves are
heated to 50 oC whereas the rest of the channel is kept at ambient temperature of 20 oC.
The length and the depth of the grooves are 0.02 m and 0.005 m respectively. The exit
section is 0.3 m. long, so that it removes possible downstream effects from the test
section.
5.5.2.1. Flow and Temperature Fields
The effects of Reynolds number on the flow field in the duct for three different
Re numbers are presented in Fig. 5.28 to 5.30. Although the limits of the streamline
patterns are not presented in [9], the figures give a qualitative understanding of the fluid
flow pattern in the duct and allow comparison in terms of location of circulations. At
low Re number, the flow occupies the front part of the groove and separation can be
observed at the upstream part of the groove as the streamlines are deflected into the
83
grooves. As the Re number increases, the separation bubbles grow larger and the
streamlines of the main stream become straight, unchanged due to groove, indicating
the establishment of fully developed periodic flow.
Figure 5.28. Comparison of streamline of [9] with present study (m2/s)for Re=100
Figure 5.29. Comparison of streamline of [9] with present study (m2/s) for Re=620
Figure 5.30. Comparison of streamline of [9] with present study (m2/s) for Re=1076
84
Fig. 5.31 and 5.32 show comparison of temperature results of the present study
with interferogram recordings at two different flow velocities. Besides, the numerical
results of [9] are also presented in the figures. Although the numerical values of
temperature distribution are not available in [9], the figures give a qualitative
understanding of the temperature pattern in the grooves. .As the flow velocity is
increased, the temperature gradient becomes steeper at the straight narrow parts of the
duct, while in the grooves the effect is minor.
5.5.2.2. Local Nusselt Number
Distributions of the experimental and numerical local Nusselt numbers along
the heated plate are presented for two different Reynolds numbers in Fig. 5.33 and 5.34.
The agreement between the numerical and experimental results is quite
Figure 5.31. Comparison of temperature field (oC) of the present study with experimental and numerical results of [9] for Re=354
Experiment
Numerical
Present
85
Figure 5.32. Comparison of temperature field (oC) of the present study with
experimental and numerical results of [9] for Re=1760
good. The local Nusselt number shows the same characteristics for different Reynolds
numbers. The distribution shows that immediately at the beginning of the first groove
Nusselt number decreases sharply. This is due to the low flow velocities in the
recirculating zone. Within the wall it increases gradually and reaches a maximum at the
second half of the groove. After the maximum, Nusselt number decreases again due to
the increased recirculation effects towards the end of the groove. Immediately
downstream of the groove, Nusselt number reaches a maximum, due to sudden increase
in the velocity, and gradually decreases along the straight section towards the next
groove. This behavior is called re-development of thermal boundary layers.
In conclusion, the flat plate results and grooved channel results of the present
study show good agreement with the results in literature. Thus, the results of the present
study are verified with the available results in the literature [9].
Numerical
Present
Experiment
86
Re=1481
0
5
10
15
20
25
30
4 6 8 10 12 14 16 18 20
Duct Length
Loca
l Nus
selt
Num
ber
Present studyHerman [*],ExperimentalHerman[*],Numerical
Figure 5.33. Comparison of experimental and local Nusselt numbers for Re=1481
Re=620
0
5
10
15
20
25
4 6 8 10 12 14 16 18 20
Duct Length
Loca
l Nus
selt
Num
ber
Present studyHerman[*], numericalHerman[*], experimental
Figure 5.34. Comparison of experimental and local Nusselt numbers for Re=620
87
5.6. Typical CFD Data
5.6.1. Laminar Flow
Below are the sample results for velocity, temperature, pressure distributions
and stream functions in the flow field sorted with respect to chip numbers for the
laminar case.
Figure 5.35. Axial velocity(m/s) distribution between the boards for laminar flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
88
Figure 5.36. Temperature distribution (oC) between the boards for laminar flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
89
Figure 5.37. Streamlines (m2/s) between the boards for laminar flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
90
Figure 5.38. Pressure drop (Pa) across the boards for laminar flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
91
5.6.2. Turbulent Flow
Similar to laminar flow results, the sample results for velocity, temperature,
pressure distributions and stream functions in the flow field sorted with respect to chip
numbers for the laminar case are presented through Fig.5.39 to 5.42. Effective viscosity
is larger than 5 for turbulent flow.
Figure 5.39. Axial velocity distribution (m/s) between the boards for turbulent flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
92
Figure 5.40. Temperature distribution (oC) between the boards for turbulent flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
93
Figure 5.41. Streamlines (m2/s) between the boards for turbulent flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
94
Figure 5.42. Pressure drop (Pa) across the boards for turbulent flow
# of chips=4
# of chips=6
# of chips=8
# of chips=10
# of chips=flat
95
CHAPTER 6
THE OPTIMAL BOARD-TO-BOARD SPACING
The main objective of this study is to maximize the total rate of heat transfer
from the finite space occupied by the package to the air flowing through the package.
The maximum heat transfer from the package can be achieved only when the plates are
optimally spaced. Before expressing the results, parameters used in the optimization
approach are explained in detail.
6.1. Method of Optimization
6.1.1. Chips with Constant Temperature
In the first configuration, the chip temperature is assumed to be constant. The
optimum spacing is the one corresponding to the maximum heat transfer from the stack
of boards in a fixed volume undergoing a specified pressure drop. Thus, in the
optimization procedure, for a given number of chips per board and a fixed height (H),
the total heat transfer rate is calculated corresponding to different board spacing but for
a specified and fixed pressure drop. Calculations are repeated for different pressure drop
values, and heat transfer rate versus (d/L) values are plotted for each pressure drop in
order to obtain optimum spacing for the given geometry. Dimensionless heat transfer
rate, and dimensionless pressure drop can be formulated as follows:
96
From the 1st law of thermodynamics, total heat transfer rate per unit width from
the package:
dH
miTmeTpcmTQ )( −′= (6.1)
where the inlet mean temperature Tmi is, ∞= TmiT , and the mass flow rate per unit
width from a single channel is defined as,
dmxVdUm ,ρρ =∞=′ (6.2)
Since the properties of air does not depend on temperature, the mean
temperature can be calculated as
=
AdAxV
AdAxTV
mT (6.3)
By using dimensionless temperature θ for the constant wall temperature
boundary condition, dimensionless mean temperature can be defined as,
∞−∞−
=TchipT
TmTmθ (6.4)
Substituting Eq. (6.4) into Eq. (6.1) and rearranging the terms, total heat
transfer rate becomes
meTchipTHmxVpcTQ θρ )(, ∞−= (6.5)
From the definition of Reynolds number, mean velocity is defined as,
LLddhDmxV1
)/2(Re
2ReRe
,ννν === (6.6)
Replacing the mean velocity, Vx,m in Eq. (6.5) gives,
97
meTchipTHLLdpcTQ θνρ )(1
)/2(Re
∞−= (6.7)
In order to group constant or given parameters in one side, Eq. (6.7) is
rearranged as,
meLdTchipTHpcLTQ θ
νρ
=∞− /2
Re)(
(6.8)
Substitute Prandtl number and thermal diffusivity into Eq (6.8), the
dimensionless heat transfer rate can be defined as:
meLdHL
TchipTTQ
kθ
=∞−
=Ω/2
Re)(Pr
1
(6.9)
where αν=Pr ,
pck
ρα =
In Eq. (6.9), Pr, and k are the constants while W, H, Tchip, T∞ are the given
parameters, it is evident that,
Ω∝TQ
Plotting dimensionless heat transfer rate (Ω) versus (d/L) graphs for constant
pressure drop yields to optimal spacing corresponding to maximum value of the Ω for
constant chip temperature boundary condition.
6.1.2. Chips with Constant Heat Flux
In the second configuration, for which the heat flux dissipated from chip
surfaces is constant, the objective is to maximize total heat transfer rate per width while
keeping maximum wall temperature as low as possible. In the formulation T denotes
the local wall temperature at the downstream, and it takes its maximum value, Tchip, at
the last chip. The aim is to maximize the total heat transfer from a fixed volume keeping
98
the Tmax at a specified value, yielding to the optimal spacing. The formulation can be
done as follows:
Dimensionless temperature θ for the constant wall heat flux case:
kdwqTT/′′∞−
=θ (6.10)
The total heat transfer rate from an LxH space filled by a stack of N=H/d
number of parallel boards with chips of total length Lchip of uniform heat flux wq ′′ can
be written as
chipLdH
wqTQ "' = (6.11)
where ( )[ ]hblNchipL 2+−= . Substituting Eq. (6.10) into (6.11) and rearranging the
terms
( )
∞−
=max2/
1'θ
TchipT
LdLchipL
LH
kTQ (6.12)
Recalling that Pr
νρ pck = , total dimensionless heat transfer rate can be calculated by
( ) ( )
=∞−
=Ω2/max
1Pr1'
LdLchipL
HL
TchipTpcTQ
θνρ (6.17)
where L, Lchip, H, ν, cp and ρ are fixed. Therefore, the objective is to maximize the
quantity ( )
2/max
1Pr1
Ldθ on the right hand side of Eq. (6.17). As in the case of
chips with constant wall temperature, the optimization constraint is the constant
pressure drop in the x-direction.
99
6.1.3. Dimensionless Pressure
The optimization constraint is the constant pressure drop (∆P=const.) and
dimensionless pressure drop can be defined as:
2
22*
Re2
∆=∆=∆ dPuP
Pm ρνρ
(6.10)
Rearranging the terms, one can define the dimensionless number Π,
2Re*
2
24
∆=∆=Π
Ld
PL
Pρν
(6.11)
Importance of this new dimensionless number Π is that optimization is done
by taking pressure drop constant for different d/L values, accordingly Π is constant.
P∆Πα =constant for the varying d/L values
The objective is to find a relation between (dopt/L) and Π, if exists, using the
single channel results and optimization parameters defined in this chapter.
6.2. Laminar Flow
6.2.1 Chips with Constant Temperature
The non-dimensional heat transfer rates are plotted against d/L values for
different pressure drops. The results are shown in Fig. 6.1 to 6.5 for different number of
chips per board. Locus of maximums is obtained by curve fitting and differentiation.
Then, equation of the optimum spacing vs. maximum heat transfer is derived for
different chip numbers and plotted. Non-dimensional pressure drop changes from 2x106
to 15x106.
100
Laminar, T=constant# of chips=4
0.0
500.0
1000.0
1500.0
2000.0
0.000 0.050 0.100 0.150 0.200
d/L
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
Figure 6.1. Dimensionless heat transfer (Ω) versus spacing of four chips per board
configuration for laminar flow, constant wall temperature boundary condition
Laminar, T=constant# of chips=6
0.0
500.0
1000.0
1500.0
2000.0
2500.0
0.000 0.050 0.100 0.150
d/L
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
Figure 6.2. Dimensionless heat transfer (Ω) versus spacing of six chips per board
configuration for laminar flow, constant wall temperature boundary condition
101
Laminar, T=constant# of chips=8
0.0
500.0
1000.0
1500.0
2000.0
2500.0
0.000 0.050 0.100 0.150
d/L
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
Figure 6.3. Dimensionless heat transfer (Ω) versus spacing of eight chips per board
configuration for laminar flow, constant wall temperature boundary condition
Laminar, T=constant# of chips=10
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
0.000 0.050 0.100 0.150d/L
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
=5x10e6
F
igure 6.4. Dimensionless heat transfer (Ω) versus spacing of ten chips per board
configuration for laminar flow, constant wall temperature boundary condition
102
Laminar, T=constant# of chips=Flat
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
3500.0
0.000 0.050 0.100 0.150
d/L
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
=5x10e6
Figure 6.5. Dimensionless heat transfer (Ω) versus spacing of flat plates for laminar
flow, constant wall temperature boundary condition
In order to find maximum heat transfer and corresponding optimum spacing for
laminar flow constant chip wall temperature, maximum Ω and corresponding (d/L)
values are evaluated from Fig. 6.1 to 6.5 and plotted with respect to pressure drop. The
results are shown through Fig. 6.6 and 6.8. It can be seen from the figures that the
optimum spacing and maximum heat transfer is of the form
naxy = (6.18)
In Eq. (6.18), y represents optimum spacing or heat transfer, x represents
dimensionless pressure drop, and (a) is a coefficient, which is function of chip spacing
(b). Variations of the coefficient (a) with chip spacing (b) for optimum board spacing
and maximum heat transfer are given in Fig. 6.7 and 6.9.
103
Laminar, T=constant
y = 0,34x0,25
y = 0,3x0,25
y = 0,295x0,25
y = 0.32x0.25
y = 0.325x0.25
8.0
10.0
12.0
14.0
16.0
18.0
20.0
1.0E+06 1.0E+07 1.0E+08
ΠΠΠΠ
L/d
chip4chip6chip8chip10flat
Figure 6.6. Optimum spacing versus non-dimensional pressure drop (Π) for laminar
flow constant wall temperature boundary condition
Laminar, T=constant
y = -0,33x + 0,33
0
0.1
0.2
0.3
0.4
0.001 0.01 0.1 1
b/L
a
Figure 6.7. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for laminar flow,
constant wall temperature boundary condition
104
Laminar, T=constant
y = 0,21x0,5
y = 0,25x0,5y = 0,26x0,5
y = 0,27x0,5
y = 0.285x0.52
100.0
1000.0
1.0E+06 1.0E+07 1.0E+08
ΠΠΠΠ
ΩΩΩΩ
duz
chip4
chip6
chip8
chip10
Figure 6.8. Maximum heat transfer (Ω) versus pressure drop (Π) for laminar flow,
constant wall temperature boundary condition
Laminar, T=constant
y = -0.28x + 0.28
0
0.05
0.1
0.15
0.2
0.25
0.3
0.001 0.01 0.1 1
b/L
a
Figure 6.9. Coefficients (a) of Ωmax. versus chip spacing (b/L) for laminar flow,
constant wall temperature boundary condition
41
2
2133.0
∆
−=ρν
LP
Lb
optdL
(6.19)
105
The relations for optimum spacing and maximum heat transfer are obtained by
curve fitting to the above graphs. Equation (6.19) shows the relation between optimum
spacing and pressure drop. It is found out that L/dopt changes with one quarter of the
pressure drop. Besides, number of chips is linearly related to optimum board spacing for
laminar flow. Although there are a few specific examples in literature on the flows in
grooved channels, smooth channel results for laminar flow can be used to compare the
findings. By setting b/L to 0, Eq. (6.10) yields to flat plate results and the comparison is
given in the next chapter.
( )2
1
2
21285.0
max
'max
∆
−=∞− ρννρ
LP
Lb
HL
TTpcQ
(6.20)
The maximum heat transfer rate from the package is the value of heat transfer
rate that corresponds to optimal spacing value at a given pressure drop.The relation
between the maximum heat transfer and pressure drop is expressed in Eq. (6.20) with
the effect of number of chips is included. As for optimal board-to-board spacing,
maximum heat transfer also changes linearly with number of chips, yielding to smooth
channel results for b=0. Maximum heat transfer, on the other hand, changes with square
root of pressure drop.
6.2.2. Chips with Constant Heat Flux
The results of constant heat flux boundary condition are shown in Fig. 6.10 to
6.14 for different number of chips per board. Maximums of the curves and
corresponding spacing values are evaluated as explained section 6.2.1. Non-dimensional
pressure drop changes from 2x106 to 15x106.
106
Laminar, q=constant# of chips=4
0.0
100.0
200.0
300.0
400.0
500.0
600.0
0.000 0.050 0.100 0.150 0.200 0.250d/L
ΩΩΩΩ
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
Figure 6.10. Dimensionless heat transfer (Ω) versus spacing of four chips per board
configuration for laminar flow, constant heat flux boundary condition
Laminar, q=constant# of chips=6
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
0.000 0.050 0.100 0.150d/L
ΩΩΩΩ
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
Figure 6.11. Dimensionless heat transfer (Ω) versus spacing of six chips per board
configuration for laminar flow, constant heat flux boundary condition
107
Laminar, q=constant# of chips=8
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
0.000 0.050 0.100 0.150
d/L
ΩΩΩΩ
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
Figure 6.12. Dimensionless heat transfer (Ω) versus spacing of eight chips per board
configuration for laminar flow, constant heat flux boundary condition
Laminar, q=constant# of chips=10
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0.020 0.070 0.120d/L
ΩΩΩΩ
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
=5x10e6
Figure 6.13. Dimensionless heat transfer (Ω) versus spacing of ten chips per board
configuration for laminar flow, constant heat flux boundary condition
108
Laminar, q=constant# of chips=Flat
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
0.020 0.070 0.120d/L
ΩΩΩΩ
=2x10e6
=3x10e6
=4x10e6
=6x10e6
=8x10e6
=10x10e6
=12x10e6
=15x10e6
=5x10e6
Figure 6.14. Dimensionless heat transfer (Ω) versus spacing of flat plates for laminar
flow, constant heat flux boundary condition
The results for optimum chip spacing and corresponding maximum total heat
transfer with respect to pressure drop are shown in Fig. 6.15 and 6.17. Relations for
optimum board-to-board spacing and maximum total heat transfer in terms of pressure
drop are given in Eqs. (6.21) and (6.22).
41
2
2133.0
∆
−=ρν
LP
Lb
optdL
(6.21)
( )2
1
2
2121.0
max
'max
∆
−=∞− ρννρ
LP
Lb
HL
TTpcQ
(6.22)
109
Figure 6.15. Optimum spacing versus non-dimensional pressure drop (Π) for laminar
flow constant heat flux boundary condition
Laminar, q=constant
y = -0,33x + 0,33
0
0.1
0.2
0.3
0.4
0.001 0.01 0.1 1
b/L
a
Figure 6.16. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for laminar flow,
constant heat flux boundary condition
Laminar,q=constant
y = 0,29x0,25
y = 0,3x0,25
y = 0,32x0,25
y = 0,325x0,25
y = 0,33x0,25
10.0
12.0
14.0
16.0
18.0
20.0
1.0E+06 1.0E+07 1.0E+08
Π
L/dopt
chip4
chip6
chip8
chip10
düz
110
Laminar, q=constant
y = 0.175x0.5
y = 0.215x0,5
y = 0.19x0,5
y = 0.2x0,5
y = 0.205x0,5
100.0
1000.0
1.0E+06 1.0E+07 1.0E+08
ΠΠΠΠ
ΩΩΩΩ
düzchip4chip6chip8chip10
Figure 6.17. Maximum heat transfer (Ω) versus pressure drop (Π) for laminar flow,
constant heat flux boundary condition
Laminar, q=constant
y = -0.21x + 0.21
0
0.05
0.1
0.15
0.2
0.25
0.001 0.01 0.1 1
b/L
a
Fi
gure 6.18. Coefficients (a) of Ωmax. versus chip spacing (b/L) for laminar flow, constant
heat flux boundary condition
111
Chips with constant heat flux boundary condition showed that optimal board-
to-board spacing is insensitive to the thermal boundary condition. However, total
maximum heat transfer rate is 35% less than that of chips with constant temperature
boundary condition. This result is to be expected, because the temperature of the
isothermal chips is equal to allowable surface temperature for all the chips in the
channel, while the allowable temperature for the chips with constant heat flux boundary
condition occurs at the last chip.
6.3. Turbulent Flow
6.3.1 Chips with Constant Temperature
Similar to laminar flow, for turbulent flow constant wall temperature case
dimensionless heat transfer versus d/L behavior for different number of chips are shown
through Fig. 6.19 to 6.23. Non-dimensional pressure drop changes from 1x1011 to
12x1011. However, unlike the laminar flow results, turbulent flow results are not
smooth, however still showing a maximum at some d/L values as expected.
The maximums of each data set corresponding to Π=constant is obtained by
curve fitting, differentiation of the equation of curve fit, and finding the roots of
differentiation. Then, maximums are plotted with respect to pressure drop in order to
derive a relation if possible.
Optimum spacing and maximum heat transfer for turbulent flow change with
pressure drop in the form of power series solution, given by Eq. (6.23) and (6.24). The
results are presented through Fig. 6.24 to 6.27.
112
Turbulent, T=constant# of chips=4
0.0
50000.0
100000.0
150000.0
200000.0
250000.0
300000.0
350000.0
0.005 0.010 0.015 0.020 0.025 0.030
d/L
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.19. Dimensionless heat transfer (Ω) versus spacing of four chips per board
configuration for turbulent flow, constant wall temperature boundary condition
Turbulent, Tconstant# of chips=6
0
50000
100000
150000
200000
250000
300000
350000
400000
0.005 0.010 0.015 0.020 0.025
d/L
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.20. Dimensionless heat transfer (Ω) versus spacing of six chips per board
configuration for turbulent flow, constant wall temperature boundary condition
113
Turbulent, T=constant# of chips=8
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
0.005 0.007 0.009 0.011 0.013 0.015 0.017 0.019 0.021 0.023 0.025
d/L
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.21. Dimensionless heat transfer (Ω) versus spacing of eight chips per board
configuration for turbulent flow, constant wall temperature boundary condition
Turbulent, T=constant# of chips=10
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
0.005 0.010 0.015 0.020 0.025 0.030
d/L
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.22. Dimensionless heat transfer (Ω) versus spacing of ten chips per board
configuration for turbulent flow, constant wall temperature boundary condition
114
Turbulent, T constant# of chips=Flat
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
0,004 0,009 0,014 0,019
d/L
P=1E11
P=2E11
P=3E11
P=4E11
P=6E11
P=8E11
P=10E11
Figure 6.23. Dimensionless heat transfer (Ω) versus spacing of flat plates for turbulent
flow, constant wall temperature boundary condition
Turbulent, T=constant
y = 0,56x0,19
y = 0,538x0,19
y = 0,51x0,195
y = 0,47x0,21
y = 0,6x0,2
10.0
30.0
50.0
70.0
90.0
110.0
130.0
150.0
1.00E+11 1.00E+12ΠΠΠΠ
L/d
chip4chip6chip8chip10duz
Figure 6.24. Optimum spacing versus non-dimensional pressure drop (Π) for turbulent
flow constant wall temperature boundary condition
115
Turbulent T=constant
y = -0.59x + 0.59
00.20.40.60.8
0.0001 0.001 0.01 0.1 1
b/L
a
Fi
gure 6.25. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for turbulent flow,
constant wall temperature boundary condition
Figure 6.26. Maximum heat transfer (Ω) versus pressure drop (Π) for turbulent flow,
constant wall temperature boundary condition
Turbulent, T=constant
y = 2,9x0,45
y = 3,5x0,45
y = 3,9x0,45
y = 4,05x0,45
y = 4,1x0,49
10000
100000
1000000
1.00E+11 1.00E+12
ΠΠΠΠ
ΩΩΩΩ
duzchip4chip6chip8chip10
116
Turbulent, T=constant
y = -2.1x + 2.1
0
0.5
1
1.5
2
2.5
0.001 0.01 0.1 1
b/L
a
Figure 6.27. Coefficients (a) of Ωmax. versus chip spacing (b/L) for turbulent flow,
constant wall temperature boundary condition
51
2
2159.0
∆
−=ρν
LP
Lb
optdL
(6.23)
( )
45.0
2
211.2
max
'max
∆
−=∞− ρννρ
LP
Lb
HL
TTpcQ
(6.24)
The Eqs. (6.23) and (6.24) shows the optimum spacing and corresponding total
maximum heat transfer from the fixed volume package consisting of boards with
discrete heat sources on them. L/dopt changes with ∆P0.2. Again, chip spacing is found to
be linearly related to the L/dopt. Total heat transfer is found to be proportional with
∆P0.45 and linearly proportional with chip spacing (b).
6.2.2. Chips with Constant Heat Flux
The results for constant heat flux boundary condition are shown through Fig.
6.28 to 6.32 for different number of chips per board. Non-dimensional pressure drop
changes from 1x1011 to 10x1011.
117
Turbulent, q=constant# of chips=4
0
10000
20000
30000
40000
50000
60000
0.008 0.010 0.012 0.014 0.016 0.018
d/L
ΩΩΩΩ
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=7E11
P=8E11
P=10E11
Fig
ure 6.28. Dimensionless heat transfer (Ω) versus spacing of four chips per board
configuration for laminar flow, constant heat flux boundary condition
Turbulent, q=constant# of chips=6
0
10000
20000
30000
40000
50000
60000
70000
0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022
d/L
ΩΩΩΩ
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.29. Dimensionless heat transfer (Ω) versus spacing of six chips per board
configuration for turbulent flow, constant heat flux boundary condition
118
Turbulent, q=constant# of chips=8
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024
d/L
ΩΩΩΩ
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.30. Dimensionless heat transfer (Ω) versus spacing of eight chips per board
configuration for turbulent flow, constant heat flux boundary condition
Turbulent, q constant# of chips=10
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
0,005 0,010 0,015 0,020 0,025 0,030
d/L
ΩΩΩΩ
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.31. Dimensionless heat transfer (Ω) versus spacing of ten chips per board
configuration for turbulent flow, constant heat flux boundary condition
119
Turbulent, q=constant# of chips=Flat
0
50000
100000
150000
200000
250000
300000
350000
0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018d/L
ΩΩΩΩ
P=1E11
P=2E11
P=3E11
P=4E11
P=5E11
P=6E11
P=8E11
P=10E11
Figure 6.32. Dimensionless heat transfer (Ω) versus spacing of flat plates for turbulent
flow, constant heat flux boundary condition
Similar plots are presented for optimum board-to-board spacing and
corresponding total maximum heat transfer rate for chips with constant heat flux
boundary condition through Fig. 6.33 to 6.36. The relations for optimal spacing and
maximum heat transfer are given by Eq. (6.25) and (6.26).
51
2
2159.0
∆
−=ρν
LP
Lb
optdL
(6.25)
( )
45.0
2
2115.1
max
'max
∆
−=∞− ρννρ
LP
Lb
HL
TTpcQ
(6.26)
The Fig. 6.25 and 6.34 show that the optimum board-to-bard spacing is
insensitive to the type of the thermal boundary condition. Curve fitting shows that
optimal spacing is proportional to L0.6 and inversely proportional to ∆P1/5 . The
120
Turbulent, q=constant
y = 0,56x0,19
y = 0,54x0,195
y = 0,6x0,2
y = 0,47x0,21
y = 0.515x0.2
10
30
50
70
90
110
130
150
1.00E+11 1.00E+12
ΠΠΠΠ
L/dopt
flatchip8chip10chip4chip6
Figure 6.33. Optimum spacing versus non-dimensional pressure drop (Π) for turbulent
flow constant heat flux boundary condition
Turbulent q=constant
y = -0.5916x + 0.5921
00.10.20.30.40.50.60.7
0.0001 0.001 0.01 0.1 1
b/L
a
Figure 6.34. Coefficients (a) of (L/d)opt versus chip spacing (b/L) for turbulent flow,
constant heat flux boundary condition
121
Turbulent q=constant
y = 1,5x0,4
y = 2x0,41
y = 2,2x0,42
y = 2,25x0,43
y = 2,3x0,45
10000
100000
1000000
1.00E+10 1.00E+11 1.00E+12
ΠΠΠΠ
ΩΩΩΩ
chip4
chip6
chip8
chip10
duz
Figure 6.35. Maximum heat transfer (Ω) versus pressure drop (Π) for turbulent flow,
constant heat flux boundary condition
Turbulent, q=constant
y = -1.14x + 1.15
00.20.40.60.811.21.4
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
b/L
a
Figure 6.36. Coefficients (a) of Ωmax. versus chip spacing (b/L) for turbulent flow,
constant heat flux boundary condition
122
properties of air affect the optimum board-to-board spacing through the group of
properties (ρν2). The maximum heat transfer rate from a space filled by a stack of
parallel boards given by Eq. (6.26) is the value of heat transfer rate that corresponds to
the optimal spacing at a given pressure drop.
Despite the same optimum spacing for both thermal boundary conditions,
maximum heat transfer values for constant heat flux are different than that of constant
wall temperature and given by equations (6.26). Although the dependence of chip
spacing and pressure drop to the heat transfer are same, the total heat transfer from the
package for chips with constant heat flux is 40% less than that of chips with constant
heat flux. As explained earlier, this is natural, since chips with constant temperature
dissipate heat to the coolant at the maximum allowable temperature whereas chips with
constant heat flux dissipate heat such that allowable maximum temperature occurs at the
last chip only.
123
CHAPTER 7
CONCLUSION
In this study, parallel boards with discreet heat sources in a fixed volume of
electronic package cooled by forced convection are investigated using Ansys, Flotran.
The main objective is to find optimum spacing between the boards corresponding to
maximum heat transfer. Since the electronic boards are sufficiently wide in the direction
perpendicular to the plane, the flow is taken as two-dimensional. The optimization
constrain is the pressure difference which is constant across the package. This is a good
representative model for installations in which pressure difference is maintained by fan
or pump. The solutions are obtained both for laminar and turbulent flow in developing
region for chips with constant temperature and with constant heat flux thermal boundary
conditions.
The computations are performed for a single channel using a finite element
program Ansys, Flotran. Single channel solutions are utilized to verify the results of the
present study with the available results in the literature. It has been proved that the
results are in very good agreement with that of in the literature. Furthermore, single
channel results are combined with the optimization procedure described in Chapter 6 in
order to derive a relation for optimum spacing and maximum heat transfer in terms of
pressure drop, chip spacing, board length and fluid properties. The results are classified
as laminar and turbulent flow for chips with constant wall temperature and chips with
124
constant heat flux boundary conditions. It is worth to emphasize once more that the
study is an original one in the sense that they not only induces effect of chip spacing on
the optimal board-to-board spacing and maximum heat transfer, but also gives a
complete solution for turbulent developing flow between the boards with discrete heat
sources.
The total heat transfer from the array strongly depends on the spacing of the
boards. The total heat transfer starts to increase with board spacing reaching a maximum
corresponding to optimum board spacing. If board spacing is further increased, the heat
transfer from the package starts to decrease. When the physical phenomenon in the
package is closely examined, it can be seen that as the board spacing decreases, the
number of boards in the package increases, resulting in an increase in total heat transfer.
On the other hand, the decrease in spacing causes an increase in the maximum
temperature on the boards. In order to keep the maximum temperature within the
desired limits, the heat transfer from a single chip should be decreased, resulting in a
drop in total heat transfer from the package. Consequently, for the optimum design, the
board spacing should be kept at optimum value. The optimum value is a function of
board length, chip spacing, and pressure drop across the package. The results of the
present work are examined in two categories.
Laminar Flow
The optimal board-to-board spacing is independent of the type of thermal
boundary condition and is given by the correlation Eq. (7.1). Similar result is available
in the literature for flash mounted boards and presented in Eq (7.2). Setting chip spacing
(b) to zero in Eq. (7.1) yields to flash mounted configuration.
125
Present Study 4
1
2
2133.0
∆
−=ρν
LP
Lb
optdL
(7.1)
Ekici [13] 4
1
2
231.0
∆=
ρν
LP
optdL
(7.2)
Bejan and Sciubba 4
1
2
233.0
∆=
ρν
LP
optdL
(7.3)
The optimal board-to board spacing is proportional to square root of the board
length, the property group 4
12
ρν and inversely proportional to ∆P1/4.
The maximum heat transfer rate per unit volume from a space filled by stack of
parallel boards that corresponds the optimal spacing at a given pressure drop is given by
Eq. (7.4) and (7.5). Note that, since the results in the literature are for boards with same
boundary conditions on both sides, number of boards in the control volume increases by
2. Thus, the solutions of the present study is represented in a similar fashion in order
make the comparison easier:
( ) 21
PL
TchipTpc
Lb
157.0LH
maxQ∆ρ
∞−
−=×
(7.4)
( ) 21
PL
TchipTpc
Lb
143.0LH
maxQ∆ρ
∞−
−=×
(7.5)
for chips with uniform temperature and uniform heat flux respectively. Eqs. (7.4) and
(7.5) are the results of intersection of asymptotes and is admittedly approximate. In
126
convection literature the dimensionless pressure difference group
∆
µν
2PL is termed as
the Bejan number [37]. For air Pr=0.72, as in this study the optimal spacing must be:
41
358.0 BeoptdL = (7.6)
Flash mounted board results of [8] and [13] are given in Eq. [7.7] to [7.10] for
different thermal boundary conditions:
Constant Temperature:
Ekici[13]: ( ) 21
PL
TchipTpc57.0
LHmaxQ
∆ρ
∞−=
×
(7.7)
Bejan and Lee[8]: ( ) 21
PL
TchipTpc6.0
LHmaxQ
∆ρ
∞−≤
×
(7.8)
Constant heat flux:
Ekici [13]: ( ) 21
PL
TchipTpc43.0
LHmaxQ
∆ρ
∞−=
×
(7.9)
Bejan and Lee[8]: ( ) 21
PL
TchipTpc45.0
LHmaxQ
∆ρ
∞−≤
×
(7.10)
The inequality signs in Eqs. (7.8) and (7.10) is a remainder that if Q is plotted
on the ordinate and d on the abscissa, the peak of actual Q versus d curves is located
under the intersection of asymptotes. The right hand sides of Eqs. (7.8) and (7.10)
represents the correct order of magnitude of maximum heat transfer rates and can be
expected to anticipate within 30%.
127
Turbulent Flow
As in the case of laminar flow, the optimal board-to-board spacing is found out
to be independent of the type of thermal boundary condition. The relation between the
optimum spacing, pressure drop, board length, air properties and chip spacing is given
by Eq. (7.11) for turbulent flow. A similar result is available in the literature for flash
mounted boards and presented in Eq (7.12). Setting chip spacing (b) to zero in Eq.
(7.11) yields to flash mounted configuration. As predicted from Eq. (7.11), optimal
board-to-board spacing increases with L0.6 and decreases with ∆P0.2. The properties of
air affect the optimum board-to-board spacing through the group of properties ( ) 2.02ρν .
Present Study: 5
1
2
2159.0
∆
−=ρν
LP
Lb
optdL
(7.11)
Ekici (13): 87.4
1
2
249.0
∆=
ρν
LP
optdL
(7.12)
The maximum heat transfer rate per unit volume from a space filled by stack of
parallel boards that corresponds the optimal spacing at a given pressure drop for
turbulent flow is given by Eq. (7.13) and (7.14).
( ) 45.010.2max PL
TchipTpc
Lb
LHQ
∆
∞−
−=×
ρ
(7.13)
( ) 45.0115.1max PL
TchipTpc
Lb
LHQ
∆
∞−
−=×
ρ
(7.14)
for chips with uniform temperature and uniform heat flux respectively.
128
When examined closely, independent of the flow regime, maximum heat
transfer corresponding to chips with constant heat flux boundary condition is 35-50%
less than that of chips with constant surface temperature boundary condition. This result
is to be expected, because the temperature of the isothermal chips is equal to allowable
surface temperature for all the chips in the channel, while the allowable temperature for
the chips with constant heat flux boundary condition occurs at the last chip.
A fundamental question in the design of in the design of finned heat exchanger
surfaces of electronic packages is how to determine the spacing between heat generating
plates in a stack of fixed volume. When the stack peak temperature (hot spot) is fixed;
the optimal board-to-board spacing is the one corresponding to maximum heat transfer
rate from the entire package to the ambient fluid flow.
The reviews of the literature on cooling of electronic equipment show that the
optimal spacing and corresponding volumetric heat generation rate have determined for
packages cooled by natural convection. Stacks cooled by convection were optimized
only in cases where the flow is laminar. This work fills this void and develops concrete
means for calculating optimal board-to-board spacing with discrete heat sources cooled
by turbulent forced convection. Besides, laminar flow results are improved by the
introduction of chip spacing.
129
REFERENCES
1. Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John
Wiley & Sons, USA, 1987. 2. Bar-Cohen, A., “Trends in the Packaging of Computer Systems”, Cooling of
Electronic Systems, Kakaç, S. et al. (eds.), Kluwer Academic Publishers, Netherlands, pp. 17-45, 1994.
3. Chu, R.C., and Simons, R.E. “Cooling Technology for High Performance
Computers: IBM Sponsored University Research”, Cooling of Electronic Systems, Kakaç, S. et al. (eds.), NATO ASI Series, Kluwer Academic Publishers, Netherlands, pp. 97-123, 1994.
4. Mehta, U.B., and Lavan, Z., “Flow in a two dimensional channel with a
rectangular cavity”, ASME J. Applied Mechanics, pp 897-901, December 1969. 5. Rockwell, D., and Naudascher, E., , “Self-sustained oscillations of impinging free
shear layers”, Ann. Rev. Fluid Mech, vol. 11, pp. 67-94, 1979. 6. Shah, R.K. and Bhatti, M.S., “Laminar Convective Heat Transfer in Ducts”,
Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John Wiley & Sons, USA, 1987.
7. Shah, R.K. and London, A.L., “Laminar Flow Forced Convection in Ducts”, (cited
in) Shah, R. K. and Bhatti, M. S., “Laminar Convective Heat Transfer in Ducts”, Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John Wiley & Sons, USA, 1987.
8. Bejan, A. and Lee, S.W., “Optimal Geometry of Convection Cooled Electronic
Packages”, Cooling of Electronic Systems, Kakaç, S. et al. (eds.), Kluwer Academic Publishers, Netherlands, pp.277-292, 1994.
9. Farhanieh, B., Herman, C., and Sunden, B., “Numerical and experimental analysis
of laminar fluid flow and forced convective heat transfer in a grooved duct”, Int. J. Heat and Mass Transfer, vol. 36, No. 6, pp.1609-1617, 1993.
130
10. Herman C., “Experimental visualization of temperature fields and measurement of heat transfer enhancement in electronic system models”, Cooling of Electronic Systems, Kakaç, S. et al. (eds.), NATO ASI Series, Kluwer Academic Publishers, Netherlands, pp.313-337, 1994.
11. Kakaç, S., and Cotta, R.M., “Unsteady forced convection in a duct with and
without arrays of block-like electronic components”, Cooling of Electronic Systems, Kakaç, S. et al. (eds.), NATO ASI Series, Kluwer Academic Publishers, Netherlands pp. 239-275, 1994.
12. Eryurt, E., “The Optimum Spacing Between Parallel Heat Generating Boards
Cooled by Natural Convection” MS. Thesis, METU, December 2001. 13. Ekici, Ö., “The Optimal Spacing Between Parallel Heat Generating Boards
Cooled by Turbulent Forced Convection”, MS. Thesis, METU, July 2001. 14. Ghaddar, N.K., Korzcak, K.Z., Mikic, B.B., Patera, A.T., “Numerical
investigation of incompressible flow in grooved channels; Part 1. Stability and self-sustained oscillations”, J. Fluid Mech., vol. 163, pp. 99-127, 1986.
15. Ghaddar, N.K., Magen, M., Mikic, B.B., Patera, A.T., “Numerical investigation of
incompressible flow in grooved channels; Part 2. Resonant and Oscillatory Heat Transfer Enhancment”, J. Fluid Mech., vol. 168, pp. 541.567, 1986.
16. Majumdar, D., and Amon, C.H., “Heat and Momentum Transport in Self-
Sustained Oscillatory Viscous Flow”, ASME J. Heat Transfer, vol. 114, pp. 866-873, 1992.
17. Poulikakos, D., and Wietrzak, A., “Cooling of micro electronic sensor by
turbulent forced convection”, Cooling of Electronic Systems, Kakaç, S. et al. (eds.), NATO ASI Series, Kluwer Academic Publishers, Netherlands, pp. 203-224, 1994.
18. Schetz, J.A., Boundary Layer Analysis, Prentice Hall, 1993 19. Deissler, R.G., Turbulent Fluid Motion, Taylor and Francis, 1998 20. Schlichting, H., Boundary Layer Theory, trans. By Kestin, J., McGraw Hill, 1968. 21. Gatski, T., Hussaine, M.Y., and Lumley J.L., Simulation of Turbulent Flows,
ICASE /LaRC Series in Computational Science and Enginnering, Oxford University Press, 1996.
22. Icepak Theory Reference Manual, Fluent Inc, 2003. 23. Mohamadi B., and Pironneau O., Analysis of the K-Epsilon Turbulence Model,
John Wiley and Sons, 1994.
131
24. Launder, B.E., and Spalding, D.B., “The Numerical Computation of Turbulent
Flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 3, pp. 269-289, 1974.
25. Ansys CFD Flotran Analysis Guide, Ansys Inc., 2000. 26. Rice, J.G., and Schnipke, R.J., “ A Monotone Streamline Upwind Finite Element
Method for Convection Dominated Flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 48, pp 313-327, 1985.
27. Ansys Theory Reference Manual, Ansys Inc., 2000. 28. Brooks, A., and Hughes, T.J.R., “Streamline Upwind/Petrov-Galerkin
Formulations for Convection Dominated Flows with Particular Emphasis on Incompressible Navier-Stokes Equations”, Computer Methods in Applied Mechanics and Engineering, Vol. 32, pp. 199-259, 1982.
29. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Series in Computational
Methods in Mechanics and Thermal Sciences, ed. By Sparrow E.M., and Minkowycz, W.J., Mcgraw Hill 1980.
30. Bodoia, J.R. and Osterle, J.F., “Finite Difference Analysis of Plane Poiseuille and
Couette Flow Developments”, (cited in) Shah, R. K. and Bhatti, M. S., “Laminar Convective Heat Transfer in Ducts”, Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John Wiley & Sons, USA, 1987.
31. Hwang, C.L. and Fan, L.T., “Finite Difference Analysis of Forced Convection
Heat Transfer in Entrance Region of a Flat Rectangular Duct”, (cited in) Shah, R.K. and Bhatti, M.S., “Laminar Convective Heat Transfer in Ducts”, Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John Wiley & Sons, USA, 1987.
32. Dean, R.B., “An Investigation of Shear Layer Interaction in Ducts and Diffusers”,
PhD Thesis, (cited in) Emery, A. F. and Gessner, F. B., “The Numerical Prediction of Turbulent Flow and Heat Transfer in the Entrance Region of a Parallel Plate Duct”, Transactions of the ASME, Journal of Heat Transfer, pp. 594-600, 1976.
33. Laufer, J., “Investigation of a Turbulent Flow in a Two-Dimensional Channel”,
National Advisory Committee for Aeronautics, NACA TN 2123, Washington, 1950.
34. Deissler, R.G., Analysis of Turbulent Heat Transfer and Flow in the Entrance
Region of Smooth Passages, (cited in) Shah, R.K. and Bhatti, M.S., “Turbulent and Transition Flow Convective Heat Transfer”, Handbook of Single-Phase Convective Heat Transfer, Kakaç, S. et al. (eds.), John Wiley & Sons, USA, 1987.
132
35. Paykoç, E., An Investigation of Steady State Forced Convection Heat Transfer for
Air Flow Between Parallel Plates and Pipes, Ms Thesis, METU, 1966. 36. Bejan A., and Sciubba, A., “The Optimal Spacing of Parallel Plates Cooled by
Forced Convection”, Int. J. of Heat and Mass Transfer, Vol.35, No=12, pp3259-3264,1992.
37. Bhattacharjee, S., and Grosshandler, W.L., “The Formulation of a Wall Jet Near a
High Temperature Wall Under Microgravity Environment”, ASME HTD, Vol 96, pp. 711-716 1988.
133
CURRICULUM VITAE
A. Türker Gürer was born in Helsinki, Finlandiya on July 20,1972. He received his
B.S. degree in Mechanical Engineering from the Middle East Technical University in
June 1994. Then, he worked for the Department of Mechanical Engineering of
METU as teaching assistant in thermodynamics and heat transfer courses between
1994 and 1999. He completed his M.S. degree in 1997. He worked as instructor in
the Mechanical Engineering department from 2000 to 2001. Since then, he has been
working as a senior mechanical design engineer in Aselsan. His main area of
interests are heat transfer, numerical methods in heat transfer, electronics cooling,
and thermodynamics.
![An adaptive discretization of incompressible ow using a multitude …physbam.stanford.edu/~fedkiw/papers/stanford2013-02.pdf · e.g. [54, 53, 17, 64, 68]), and adaptive mesh re nement](https://img.dokumen.tips/doc/110x75/5f9f751ded22241cb346b778/an-adaptive-discretization-of-incompressible-ow-using-a-multitude-fedkiwpapersstanford2013-02pdf.jpg)
