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Multiphase Science and Technology, 25 (2–4): 311–338 (2013)

EFFICIENT HYDRAULIC AND THERMAL

ANALYSIS OF HEAT SINKS USING VOLUME

AVERAGING THEORY AND GALERKINMETHODS

Krsto Sbutega∗ & Ivan Catton

University of California, Los Angeles, UCLA MAE, Box 951597, 48-121E4, Los Angeles, California 90095-1597, USA

∗Address all correspondence to Krsto Sbutega E-mail: [email protected]

Air- and water-cooled heat sinks are still the most common heat rejection devices in electronics,making their geometric optimization a key issue in thermal management. Because of the complex

geometry, the use of finite-difference, finite-volume, or finite-element methods for the solution of the governing equations becomes computationally expensive. In this work, volume averaging theory isapplied to a general heat sink with periodic geometry to obtain a physically accurate, but geometri-cally simplified, system model. The governing energy and momentum equations are averaged overa representative elementary volume, and the result is a set of integro-partial differential equations.Closure coefficients are introduced, and their values are obtained from data available in the literature.The result of this process is a system of closed partial differential equations, defined on a simple ge-ometry, which can be solved to obtain average velocities and temperatures in the system. The intrinsicsmoothness of the solution and the simplified geometry allow the use of a modified Fourier–Galerkin Method for efficient solutions to the set of differential equations. Modified Fourier series are chosen asthe basis functions because they satisfy the boundary conditions a priori and lead to a sparse systemof linear equations for the coefficients. The validity of the method is tested by applying it to modelthe hydraulic and thermal behavior of an air-cooled pin-fin and a water-cooled micro-channel heatsink. The convergence was found to be O(N −3.443), while the runtime was ∼0.25 s for N = 56. The

numerical results were validated against the experimental results, and the agreement was excellentwith an average error of ∼4% and a maximum error of ∼5%.

KEY WORDS: Galerkin method, volume averaging theory, pin-fin heat sink,micro-channel heat sink

1. INTRODUCTION

The omnipresence of electronic equipment in today’s world, although at times daunting,

is a fact. Electronic components pervade our entertainment and communication systems,

existing in their most apparent forms as cellular phones, smart televisions, and comput-ers. However, their various roles in life support, military defense, economic prediction,

etc., make their reliability a vital concern for our society. Our dependency on electronic

0276–1459/13/$35.00 c 2013 by Begell House, Inc. 311


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312 Sbutega & Catton

NOMENCLATURE

a arbitrary constant

A1, A2 non-dimensional aspectratios

Afs fluid/solid interface area

AT heat input area

Av inlet area

B boundary operator

cd overall drag coefficient

cdp pressure drag coefficient

cf friction coefficient

c p specific heat

D pin fin diameter

dh volume averagingtheory–defined hydraulic

diameter

f f Fanning friction factor

f, g general forcing function

G non-dimensional parameters

H micro-channel height

h heat transfer coefficient

H c heat sink height

H,K,B,I boundary expansion

coefficients

k thermal conductivityk effective thermal conductivity

tensor

L heat sink length/linear

partial differential equation

operator

lmfp mean-free-path length

l p pore length scale

M non-dimensional parameter

N number of basis functions

n normal vector

p pressure, pitch (micro-channel

heat sink)

q heat flux

R residualRh porosity-weighted thermal

conductivity ratio

S,F,P,E expansion coefficients

S L longitudinal pitch (pin-fin

heat sink)

S T transversal pitch (pin-fin

heat sink)

S w area per unit volume in

representative elementary

volume

S wp representative elementaryvolume frontal area

T temperature

u x-component of velocity

U avg average velocity

V volume

v velocity

w fin thickness (micro-channel

heat sink)

x position vector

y vector with respect to the

representative elementaryvolume centroid

Greek Symbols

α thermal diffusivity

γ,φ constants inside trial

functions

δ Kronecker delta

ε porosity

ν dynamic viscosity

σ filter function

ϕ trial function

ψ arbitrary function of interest

Multiphase Science and Technology


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Efficient Hydraulic and Thermal Analysis of Heat Sinks 313

NOMENCLATURE (Continued)

Subscripts

dh with respect to length

scale dhf fluid phase

in inlet

n, m coefficient numbers

s solid phase

w wall

Superscripts and Symbols

disp dispersion

stag stagnation

ˆ dimensional quantity

∼ fluctuation/deviation fromintrinsic average

superficial average

f intrinsic average

devices continues to increase exponentially, which also amplifies the potential risks as-

sociated with the degradation or failure of these devices. Inter-diffusion, corrosion, and

electro-migration are the leading causes of reliability degradation in electronic compo-

nents, and each has a thermal-activation component. It is widely understood that reliabil-

ity increases exponentially with a decrease in temperature (Schmidt, 2004). Furthermore,

computational speed increases with a decrease in operating temperature. Therefore, an

improved heat removal system leads to an improvement in both performance and relia-

bility. However, the contemporary reduction in size of the devices leads to a strong need

for more efficient heat rejection devices. Because of the ease of fabrication and applica-

tion, reliability, and low cost (Chu, 2004), air-cooled heat sinks are still the most com-

mon cooling solutions for electronics. First proposed by Tuckerman and Pease (1981),

micro-channel heat sinks have gained popularity in the recent decade as an alternative to

air-cooled heat sinks. Because of their high volume-to-surface area, they seem to be the

most viable solution for the next generation of compact cooling devices.

Heat sinks are complex, multi-scale, heterogeneous geometrical structures, which

make them difficult to analyze. The most commonly used numerical methods, such as

the finite-difference (FD), finite-element (FE), and finite-volume (FV) methods, require

domain discretization. The complex and multi-scale nature of the geometry makes such

discretization challenging and requires a large number of elements. The recent advances

in computational speed and memory storage now allow for obtaining solutions to the

governing equations on such large meshes. Large computational fluid dynamics (CFD)

software packages have now become the standard for analysis of heat sinks (Chein and

Chen, 2009; Park et al., 2004). Solutions using CFD are less expensive than obtaining

experimental data, allow great flexibility, and provide a very large amount of informa-

tion about the flow and heat transfer. Nevertheless, the computational time required to

evaluate the performance of each heat sink is on the order of hours.The ultimate goal in most applications is to design a heat sink that can dissipate the

given heat load with the minimum amount of pumping power, while keeping the heat

Volume 25, Number 2–4, 2013


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generating device under a certain temperature. This is a multi-parameter optimization

problem. The description of the complex geometrical structure of heat sinks requires a

large number of parameters (about 10); therefore, the optimization study would require

a very large number of system evaluations (on the order of 103). Because of the com-

putational time required, CFD cannot be used as a tool in such studies. Furthermore, if only overall temperature differences, pressure drop, and heat flux are of interest, it is

inefficient to first carry out detailed local flow and energy calculations, and then average

them in post-processing. Thus, it would be highly desirable to trade the detailed local

information of the solution for a significant decrease in computational time.

In this study, the volume averaging theory (VAT) is applied to a general heat sink to

obtain a set of partial differential equations (PDEs) in the average temperatures, fluxes,

and velocity. The result is an accurate model that can be used to evaluate heat sink per-

formance very quickly, therefore enabling a multi-parameter optimization study. The

VAT was developed for analysis of transport phenomena in porous media, which are

inherently multi-scale and complex geometrically. The VAT is a rigorous mathematical

tool that spatially smooths the governing conservation equations and produces a set of

PDEs that are defined at every point in the heat sink, essentially transforming the com-

plex structure of the heat sink into a homogenous medium. The basics of the VAT are

discussed in a short review in the next section. Every averaging process involves the

loss of some information at the lower scale. This loss of information requires a closure

scheme to correlate the lower-scale phenomena to the averaged quantities. In this work,

the schemes developed by Kuwahara et al. (2001) and Travkin and Catton (2001) will

be used to obtain a set of closed PDEs.

The application of the VAT leads to a set of PDEs, which is defined in the entire

domain and whose solution is inherently smooth because of the averaging process. These

features suggest that a Galerkin method (GM) solution, with a modified Fourier series

as the basis, can be used to efficiently obtain a solution. The GM is a subset of spectral

methods. Spectral methods have become increasingly popular recently (especially for

direct numerical simulations of turbulence) because of better convergence and memory

management. The most popular spectral method is the Chebyshev collocation method.

Horvat and Catton (2002, 2003) used a Fourier–GM as a basis in one direction to reduce

non-dimensionalized VAT equations to a system of ordinary differential equations that

was solved analytically with a matrix exponential. However, this is possible only for

constant temperature boundary conditions because in this case the system reduces to an

eigenvalue problem. In this work, the GM will be used in both directions to solve the

VAT equation for any temperature or heat flux boundary condition.

2. THEORETICAL FUNDAMENTALS

The VAT is a rigorous mathematical tool that allows the study of heat transfer and fluid

flow in hierarchical, geometrically complex systems. It was developed in the 1960s by


https://www.researchgate.net/publication/267584310_Modeling_of_Forced_Convection_in_an_Electronic_Device_Heat_Sink_as_Porous_Media_Flow?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==https://www.researchgate.net/publication/222646713_A_Numerical_Study_of_Interfacial_Convective_Heat_Transfer_Coefficient_in_Two-Energy_Equation_Model_of_Porous_Media?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==


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Whitaker (1967, 1969), Gray (1975), and Gray and O’Neill (1976) to deal with the em-

piricism and lack of rigor involved in modeling of flow through porous media. The fun-

damentals can be found in Whitaker (1999). In its goal and mathematical approach, VAT

is analogous to the continuum approach in mechanics. Theoretically, to model transport

phenomena in any system, Newton’s second law could be applied to each molecule toobtain its motion, and from it any physical quantity of interest. However, this is impos-

sible computationally because of the prohibitive amount of molecules contained in most

systems of interest. Thus, after averaging the discrete space over a loosely defined parti-

cle, a continuum is defined in which quantities of interest can be defined at every point

of the domain, independently of whether a molecule is present or not at the location

of interest. Mathematically, the governing equations at the molecular scale, defined by

the mean-free path, lmfp, are averaged over an intermediate length scale, defined by the

length scale of the vague concept of a particle. This process leads to continuum equa-

tions at the length of scale of interest, i.e., the system length scale, L. The validity of the

procedure is dictated by the Knudsen number, Kn = lmfp/L. If the Knudsen number issmall, the averaging is statistically valid and the continuum approach can be used. In the

averaging process, some of the information at the scale of lmfp is lost and some closure

coefficients are required. The effect of lower-scale information on the higher scale is

incorporated in the closure coefficients, which can include viscosity, thermal conductiv-

ity, diffusivity, etc. These closure coefficients have to be determined and provided either

analytically (i.e., kinetic theory of gases), experimentally, or numerically (i.e., molecu-

lar dynamics). In a very similar manner, the VAT replaces a complex discrete geometric

structure (e.g., porous medium) with a fictitious continuum. Theoretically, the point-wise

governing equations can be solved over each domain (e.g., solid and fluid) to obtain any

of the quantities of interest. However, this can be very computationally expensive or

impossible because of the complicated geometry. Therefore, the point-wise governing

equations are averaged and the discrete complex structure is substituted with a fictitious

medium in which the quantities of interest are defined at every point, independently of whether it is in the solid or fluid phase. Mathematically, the governing equations at the

pore length scale, l p (e.g., Navier–Stokes) are averaged over a representative elementary

volume (REV) to obtain a set of governing equations at the length scale of interest, i.e.,

system length scale L. The validity of the approach is, once again, given by the length

scale disparity, l p/L. If this number is small, the averaging is statistically valid and theapproach is justified. Again, in the averaging process, some information of phenomena

at the lower scale (l p) is lost and some closure is required to model the effect of these

phenomena on the quantities of interest. In our case, these closure coefficients will be

the REV heat transfer coefficient, thermal conductivity, and friction factor. These closure

coefficients can be determined analytically, experimentally, or numerically (CFD).

The derivation of the VAT energy, mass, and momentum conservation equationsstarts from the incompressible, laminar, steady, constant property; point-wise mass; mo-

mentum; and energy conservation equations. Furthermore, it is assumed that the walls


https://www.researchgate.net/publication/229599842_Diffusion_and_Dispersion_in_Porous_Media?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==https://www.researchgate.net/publication/239154863_A_Derivation_of_the_Equations_for_Multi-Phase_Transport?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==


7/29


are impermeable, the effects of gravity are negligible, and no viscous dissipation or heat

generation is present. With these assumptions, the point-wise governing equations are

∇ · v = 0 (1)

∇ · (vv) = − 1ρ

∇ p + νf ∇2v (2)

∇ · (vT f ) = α f ∇2T f (3)

0 = α s∇2T s (4)

The appropriate boundary conditions can be divided in the system and interface bound-

ary conditions. The interface boundary conditions are internal; they determine the energy

exchange within the system and are applied on surfaces of scale l p. The system bound-

ary conditions are applied on the external boundaries, determine energy inputs into the

system, and are applied to surfaces of scale L. The interface boundary conditions are

given by

v = 0T f = T s, on Afs

−kf ∂T f

∂ x

· n =

−ks

∂T s

∂ x

· n

(5)

where Afs is the fluid/solid interface area inside the system and is usually very hard to

characterize. The system boundary conditions are usually more dependent on the prob-

lem, but some examples are

v = vin, on Av

T f = T w or

−kf

∂T f

∂ x

· n = q w on AT

(6)

where Av and AT are the inlet area and heat input area, respectively. As discussed pre-

viously, Eqs. (1)–(4) are defined only within their domains. The mass and momentum

equations are defined only in the fluid domain, while the solid energy equation is de-

fined in the solid domain. Thus, the challenge in these equations comes mostly from

the determination of the interfaces in the intricate geometry. However, after the VAT is

applied, the internal boundary conditions are absorbed by the governing equations and

the difficulties related to internal boundaries are bypassed.

The averaging of the governing equations starts by associating to every point x a

REV V , of which x is the centroid (see Fig. 1). The superficial and intrinsic averaging

operators are defined, respectively, as

ψ|x = 1

V (x)

V f (x)

ψ (x + yf ) dV (7)



8/29


FIG. 1: Definition of the vectors in REV (Whitaker, 1999).

ψf

x= 1V f (x)

V f (x)

ψ (x + yf ) dV (8)

where ψ is a function of interest (tensor of any rank). The two averages are related by

ψ = V f

V ψf = εf ψ

f (9)

where εf is the fluid volume fraction. In order to be able to average Eqs. (1)–(4), a

relationship between the average of a gradient and the gradient of the average is required.

An extension of

Leibniz’s rule, known as the spatial averaging theorem (SAT), gives the necessary

relationship:∇ ψ = ∇ ψ +

1

V

Afs

n · ψdS (10)

where Afs is the interface area and n is the unit vector normal to Afs pointing from

the fluid toward the solid. Slattery (1972) gives a detailed derivation of the theorem.

With averaging operators (7) and (8) and the SAT, it is possible now to move on to the

development of the VAT equations.

The superficial averaging operator is applied to Eqs. (1)–(4) and they are averaged

over the REV to obtain

∇ · v = 0 (11)

∇ · (vv) = − 1ρ ∇ p + ν

∇2v

(12)

∇ · (vT f ) = α f

∇2T f

(13)



9/29


0 = α s

∇2T s

(14)

The SAT can be applied to Eq. (11), along with the no-penetration boundary condition,

to obtain

∇ · v = 0 (15)

which shows that the superficial velocity is solenoidal. This is an expression of the VAT

continuity equation. An alternative form can be found using the relationship given in

Eq. (9):

∇ · vf = −∇ε

ε

vf (16)

This equation shows that for constant porosity, the intrinsic velocity is also solenoidal;

however, this is not the case when the porosity is changing. In the validation process, the

results will be calculated for a constant porosity heat sink and the homogeneous form of

Eq. (16) will be used. Next, the SAT is applied to the pressure term of Eq. (12):

∇ p = ∇ p + 1

V Afs

pndS (17)

Applying the SAT to the diffusive terms, they can be rewritten as

∇ · ∇v = ∇ · ∇v + 1

V

Afs

n · ∇vdS (18)

∇ · ∇T i = ∇ · ∇T i + 1

V

Afs

n · ∇T idS (19)

where i = {f , s}. Using the SAT again, and the no-penetration boundary condition to the

first term in Eq. (18), the momentum diffusion term reduces to

∇ · ∇v = ∇2 v + 1

V

Afs

n · ∇vdS (20)

Similarly, Eq. (19) can be rewritten as

∇ · ∇T i = ∇2 T i + ∇ ·

1

V i

Afs

nT idS

+ 1

V i

Afs

n · ∇T idS (21)

The averaging process of the convective terms is more elaborated. First, the SAT andno-flow-through conditions are used to reduce the terms to

∇ · (vT f ) = ∇ · vT f (22)



10/29


∇ · (vv) = ∇ · vv (23)

These equations are still incomplete because the terms are expressed as the average of the

product while a closed form of the VAT equations requires the product of the averages.

The two terms are separated by decomposing the function into the average and a spatial

deviation: ψ = ψi + ˜ ψ (24)

A similar decomposition is performed in time averaging of turbulent Navier–Stokes

equations. However, in Eq. (24) function ψ is averaged spatially and the fluctuations

are the differences from the spatial average. If the averaged function is smooth enough

(for a detailed length scale analysis, see Carbonell and Whitaker, 1984), the right-hand

sides in Eqs. (22) and (23) can be decomposed to obtain

∇ · vv = ∇ ·εf vf v

f + εf ṽṽf

(25)

∇ · vT f = ∇ ·εf v

f

T f

f

+ εf

ṽ˜T f

(26)

Following this analysis and with some manipulation of the equation, a first form of the

VAT mass, momentum, and solid and fluid energy equations can be written as follows:

∇ · vf = 0 (27)

εf vf · ∇ vf = −

εf

ρf ∇ pf + εf νf ∇

2 vf + νf

V

Afs

n · ∇vdA + 1

ρf V

Afs

pndA

(28)

− εf ∇ · ṽṽf

εf vf · ∇ T f

f = εf α f ∇2 T f

f + α f

V

Afs

n · ∇T f dS + ∇ ·

Afs

T f ndS

(29)

− εf ∇ ·

ṽT̃ f

f

0 = εsα s∇2 T s

s + ∇ ·

α s

V

Afs

nT sdS

+ 1

V

Afs

n · α s∇T idS (30)

The goal is to obtain a set of equations for the average velocity and temperatures. The set

given above still contains certain point-wise terms in the integrals and some fluctuation



11/29


terms. Therefore, this is not a closed form of the governing equations because it contains

information about the lower-scale phenomena that is not known a priori. In order to close

these equations, it is necessary to develop a scheme that will relate these fluctuation and

integral terms to averaged quantities. Travkin and Catton (2001) defined the following

closure coefficients for the VAT momentum equation:

cd = cdpS wp

S w+ cf −

∇ ·εβ ṽβṽβ

β

1

2ρβ v̄β

2 (31)

cdp = 2

Afs

n pdA

ρf v2 S wp

(32)

cf = 2

Af

nτ wdA

ρf v2S w

(33)

where S w is the interface surface per unit volume in the REV:

S w = Afs

V (34)

When substituted into Eq. (28), a deceivingly simple closed form of the VAT momentum

equation for constant porosity is obtained

εf vf · ∇ vf = −

εf

ρf ∇ pf + εf νf ∇

2 vf − 1

2cdS w

vf

2(35)

Similarly, Kuwahara et al. (2001) defined a VAT heat transfer coefficient and an effective

thermal conductivity as

h =

(kf /V ) Afs n · ∇T f dAS w

T s

s − T f f

(36)

ki,eff · ∇ T ii = ki,stag · ∇ T i

i + δf iki,disp · ∇ T ii (37)

where i = {s, f } and the Kronecker delta is used simply to point out that dispersion iszero in the solid phase. Dispersion thermal conductivity is a tensor and it is defined as

ki,disp · ∇ T ii = −ρic pi

εi

T̃ iṽi

i (38)

while the stagnation thermal conductivity is the sum of the diffusive term and the tortu-

osity

ki,stag · ∇ T ii = εiki∇ T i

i + 1V

Afs

nT idA (39)



12/29


As it can be inferred from the name, the stagnation thermal conductivity is defined as

the effective thermal conductivity that the system would have if the velocity was zero.

As discussed previously, these closure coefficients are the equivalent of transport coef-

ficients such as thermal conductivity, viscosity, etc., and need to be determined before

moving on to the solution of the VAT equations.Using these closure coefficients, Eqs. (29) and (30) can be rewritten as

εf ρf c pf vf · ∇ T f

f = ∇

kf,eff · ∇ T f

f + hS w

T s

s − T f f

(40)

∇

ks,stag · ∇ T ss = hS w

T s

s − T f f

(41)

Now, a full set of closed equations, given by Eqs. (16), (35), (40), and (41) has been

developed for the averaged quantities of interest, namely vf f , T s

s, and T f f . A key

point to emphasize is that these equations, and therefore their solutions, are defined at

every point in the domain. Therefore, an average velocity and solid and fluid temper-

atures are assigned to each point, without consideration of whether the point is actuallywithin the solid or fluid phase. The discrete heat sink geometry has now been turned

into a continuous medium. The determination of the closure coefficients is obviously a

key component in the accuracy of the solution. Vadnjal (2009) showed that the Fanning

friction factor, f f , closely approximates the VAT-defined drag coefficient, cd; therefore,

the Fanning friction factor will be used in the governing equations. The heat transfer

coefficient and the friction factor can then be taken from available experimental data for

common geometries.

This study analyzes the behavior of a periodic heat sink in two dimensions. The

extension to a three-dimensional domain is not difficult. However, the implementation

can be quite tricky and will be more computationally expensive, thus it is not consid-

ered here. Because of the periodicity of the geometry, the fluctuation terms will die out

quickly away from the boundaries and as a result can be ignored. Furthermore, because

of the small local Péclet number and the high thermal conductivity ratio, the tortuosity

and dispersion effects can be ignored. It is further assumed that the flow is fully devel-

oped; however, axial conduction is not assumed to be negligible because of the small

length scale considered. A graphical description of the domain and boundary condition

is given in Fig. 2. With the given assumptions, the VAT governing equations, Eqs. (35),

(40), and (41), become

0 ≤ ẑ ≤ Ĥ c

−εf ̂ νf ∂ 2 ûf

f

∂z2 +

1

2

f f Ŝ w ûf f 2

= −εf

ρ̂f

∂ ˆ pf f

∂ ̂x

(42)

0 ≤ x̂ ≤ L̂, 0 ≤ ẑ ≤ Ĥ c


https://www.researchgate.net/publication/234368820_Modeling_of_a_heat_sink_and_high_heat_flux_vapor_chamber?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==


13/29


FIG. 2: Schematic of the geometry and boundary conditions.

ĉ pf ρ̂f εf ̂uf f ∂ T̂ f

f ∂ ̂x

= εf k̂f

∂ 2T̂ f

f ∂ ̂z2

+ εf k̂f

∂ 2T̂ f

f ∂ ̂x2

+ ĥŜ w

T̂ s

s−T̂ f

f (43)

0 ≤ x̂ ≤ L̂, 0 ≤ ẑ ≤ Ĥ c

εsk̂s

∂ 2T̂ s

s∂ ̂x2

+ εsk̂s

∂ 2T̂ s

s∂ ̂z2

= ĥŜ w

T̂ s

s−T̂ f

f (44)

where ˆ denotes dimensional quantities.

As discussed previously, the internal boundary conditions have been absorbed in the

equations. Also, the continuity equation [Eq. (16)] was not listed but has been used todetermine that the z-velocity component is zero. The system boundary conditions for

the momentum equation are given by the no-slip condition (assuming no by-pass) as

follows:

ûf f z=0

= ûf f z= Ĥ c

= 0 (45)

The energy equation system inputs are given by

T̂ f

f x=0

= T̂ in

∂ T̂ f f

∂ ̂x

x=L̂

=

∂ T̂ f f

∂ ̂z

z= Ĥ c

= 0

(46)



14/29


∂ T̂ s

s

∂ ̂x

x=0

=∂ T̂ s

s

∂ ̂x

x=L̂

=∂ T̂ s

s

∂ ̂z

z= Ĥ c

= 0 (47)

Because there is no heat source or sink at the outlet, it is assumed that any heat transport

at this location is simply convected away by the fluid, therefore resulting in an adiabaticboundary condition at the outlet. The boundary condition at the bottom of the channel

can vary depending on whether temperature or heat flux is specified. For the case of

specified temperature T̂ w, the boundary condition is straightforward:

T̂ f

f z=0

=T̂ s

sz=0

= T̂ w (48)

For the case of a heat flux boundary condition, the situation is more complicated. Be-

cause of a scale discontinuity, the determination of the heat transfer from the base to

the porous channel is still an active area of research (Imani et al., 2012; Ouyang et al.,

2013; Yang and Vafai, 2011). In this work, the base is assumed to be thin relative to its

length, and conjugate effects are ignored. It is also assumed that the input heat flux willbe distributed to the two phases as follows (Imani et al., 2012):

−εf ̂kf

∂ T̂ f

f

∂ ̂z

z=0

= −εsk̂s∂ T̂ s

s

∂ ̂z

z=0

= q̂ w (49)

In the next section, a method will be developed to efficiently solve Eqs. (42)–(44) with

the boundary conditions given by Eqs. (45)–(49). The result will be a fast running code

for modeling of thermal and hydraulic behavior of heat sinks.

3. SOLUTION METHOD

A solution to the VAT energy equations will be obtained using a GM with a modi-

fied Fourier series as the basis functions. The GM gets its name from Boris Galerkin

(Galerkin, 1915), a Russian engineer that developed and used the method to solve dif-

ferential equations resulting from problems in statics. The semi-analytical nature of the

method and its rapid convergence made it very popular when computers were not widely

available (Finlayson, 1972). The GM, in its global formulation, is advantageous for sim-

ple geometries and smooth solutions. In recent years, the wide availability of computers

has made the FD and FV methods more popular because of the ease of implementation

and versatility. Local GMs have been used quite often in FE methods. The application

of the VAT to the governing equation ensures that the solution is smooth and simplifies

the geometry, effectively bypassing the shortcomings of the GM.

The GM is a subset of the larger class of spectral methods (Canuto et al., 2006) anda brief review is given here. A linear PDE can be formulated as

L (u) = f on Ω (50)


https://www.researchgate.net/publication/276982720_Estimation_of_heat_flux_bifurcation_at_the_heated_boundary_of_a_porous_medium_using_a_pore-scale_numerical_simulation?el=1_x_8&enrichId=rgreq-49d533c5-df74-4139-bd6e-e9c887efddec&enrichSource=Y292ZXJQYWdlOzI2NDM4Mjc4MDtBUzoyMTEyOTE1NTQ3NTA0NjVAMTQyNzM4NzI0MDM5OA==


15/29


where L is a linear operator; Ω is the domain of the solution; and f is a forcing function.The extension to a non-linear equation is very similar but will not be addressed in detail

here. The boundary conditions can be expressed as

B (u) = g on ∂ Ω (51)

Spectral methods assume that the solution can be expanded in a truncated series:

uN (x) =N n=0

anϕn (52)

with undetermined coefficients an. The substitution of this assumed solution in Eq. (50)

will not satisfy the solution exactly, and it will result in a residual R:

L (uN ) − f = R on Ω (53)

The idea behind spectral methods is to multiply both sides of Eq. (53) with weight func-

tions wm (x). The resulting expression is integrated over the domain, and the coefficients

are chosen such that they drive the residual to zero Ω

L (uN )wmdΩ −

Ω

fwmdΩ = 0 (54)

Various spectral methods differ in the choice of the weight functions. Pseudo-spectral

methods, such as the collocation methods, force the residual to zero at given points in

the domain. The GM uses trial functions that are equal to the basis functions, ϕ = w.The result is a set of algebraic equations for coefficientsan. The boundary conditions can

be treated in several ways; however, choosing basis functions that satisfy the boundary

conditions a priori often gives the best convergence (Finlayson, 1972). The method can

be expanded to two-dimensional problems by using a tensor product of basis functions:

uN (x, y) =N n=0

N m=0

anmϕn (x)φm (y) (55)

Following the same procedure described for the one-dimensional cases, a set of N 2

algebraic equations for coefficients anm is obtained. In this work, a modified Fourier

series will be used as the basis functions for the problem described in Fig. 2 because of

its ability to satisfy the boundary conditions a priori.

It is always good practice to non-dimensionalize the governing equation. The domain

and the quantities of interest are non-dimensionalized as follows:

T ii =

T̂ ii

− T̂ in

T̂ in, uf

f = ûf

f

Û avg, Û avg =

dh

2ρf f f

∆ p

L



16/29


x = x̂

L̂,

... z = ẑ

Ĥ c, dh =

4εf

S w(56)

where i = {s, f }, and the superscript ˆ denotes dimensional quantities. By substitutingthese parameters into the governing equations [Eqs. (42)–(44)], the following equations

are obtained:0 ≤ z ≤ 1

−M 1∂ 2 uf

∂z2 +

uf

2

= 1

(57)

0 ≤ x ≤ 1, 0 ≤ z ≤ 1

G1 uf f ∂ T f

f

∂x − G2

∂ 2 T f f

∂x2 −

∂ 2 T f f

∂z2 − G3

T s

s − T f f

= 0

(58)

0 ≤ x ≤ 1, 0 ≤ z ≤ 1

G2

∂ 2 T ss

∂x2 +

∂ 2 T ss

∂z2 − G3Rh

T ss

− T sf

= 0

(59)

The non-dimensional parameters are defined as

M 1 = ˆ νf

Ĥ 2c

ρ̂f L̂d̂h

2f f ∆ˆ p, G1 = Prf RedhA1A2, G2 = A

2

2, G3 = 4NudhA

2

1

A1 =Ĥ c

d̂h, A2 =

Ĥ c

L̂, Rh =

εf k̂f

εsk̂s

(60)

The non-dimensional number M 1 represents the ratio of diffusive effects to friction and

pressure forces. Parameter G1 is a Péclet number multiplied by two geometrical non-

dimensional ratios; A1 is the ratio of the z-direction diffusion length scale ˆH c to the

convection length scale d̂h, while A2 is the ratio of z - and x-direction diffusion length

scales. The boundary conditions [Eqs. (45)–(49)] can be rewritten in non-dimensional

form as

uf f z=0

= uf f z=1

= 0 (61)

T f f x=0

= 0

∂ T f f

∂x

x=1

= 0, ∂ T f

f

∂z

z=1

= 0 (62)

∂ T ss

∂x

x=0

= ∂ T ss

∂x

x=1

, ∂ T ss

∂z

z=1

= 0 (63)



17/29


∂ T f f

∂z

z=1

= − q̂ w Ĥ c

ε̂f ̂kf T̂ in= q w,

∂ T ss

∂z

z=1

= − q̂ w Ĥ c

ε̂sk̂s T̂ in= q wRh (64)

T f f = T w =

T̂ w − T̂ in

T̂ in

= T ss (65)

The non-linearity of the momentum equation makes a FD solution advantageous. The

diffusive term is approximated using a second-order centered difference on a non-uniform

grid. Chebyshev nodes are chosen for the grid points to capture the steep gradients at the

boundaries. The non-linear term is linearized around point i. The discretized form of the

momentum equation is given by

− 2M 1

(zi+1 − zi) (zi+1 − zi−1) uf

k+1

i+1+

2M 1(zi+1 − zi) (zi − zi−1)

uf

k+1

i(66)

− 2M 1

(zi − zi−1) (zi+1 − zi−1) uf

k+1

i−1+ uf

k+1

iuf

k

i= 1

The resulting system is solved using a Gauss–Seidel iterative method. Further iteration

is required because the friction factor, which is hidden in M 1, depends on the Reynolds

number. The tolerance was set to be ε = 10−6 for the relative error. The grid was refined

until the solution reached the tolerance condition, and it was found that 100 points were

enough to capture the gradients at the boundaries. Following this procedure, the VAT

momentum solution can be obtained in about 0.04 s.

The solution to the VAT energy equations will obviously depend on the boundary

conditions. A general expansion of the temperature in the modified Fourier series can be

given for both boundary conditions given in Eqs. (64) and (65):

T f f = I n sin ( γnx) + H n sin ( γnx)sin( γ0z) (67)

+ sin ( γnx) [F nm cos (φmz) + E nm sin ( γnz)]

T ss = Bn cos (φnx) + K n cos (φnx)sin( γ0z) (68)

+ cos (φnx) [S nm cos (φmz) + P nm sin ( γmz)]

where γn = (2n + 1)π /2 and φm = mπ . The coefficients K n and Bn are the Fourierseries coefficients of the input functions. For analytical input functions these can be

obtained analytically (such as in the constant case analyzed in the Results section). For

any other input, these coefficients can be determined in O (N log N ) operations using adiscrete Fourier transform. The coefficients H n and I n are the modified Fourier series

coefficients of the input functions and can also be calculated analytically for given func-

tions. For general discrete functions they can be determined using a Filon-type quadra-ture in O(N ) operations (Adcock, 2011). For specified temperature boundary conditionsthe coefficients K n, H n, F nm, and S nm will be zero. For the case of a specified heat flux,



18/29


Bn, I n, P nm, and E nm will be zero. Thus, given the boundary conditions, there will be

2N 2 + 2N coefficients to be determined. The number of basis functions needed for

a converged solution will be controlled by either the number of coefficients required to

correctly represent the input functions or the number of coefficients required to represent

the solution function. The convergence of these Fourier coefficients is determined by thesmoothness of the function; therefore, the overall convergence will be determined by the

smoothness of either the input functions (which is known a priori) or the smoothness

of the solution function (which is not known a priori). It is well known that Fourier ap-

proximation of discontinuous functions leads to Gibbs phenomena and they have a slow

rate of convergence; therefore, they are the worst-case scenario. However, filtering can

be used to reduce Gibbs phenomena and recover exponential conversion away from the

discontinuity. The validation procedure will require inputting constant input functions

(constant heat flux), which are discontinuous in a sine Fourier representation; hence, a

raised cosine filter will be employed:

σn =

1 + cos (2nπ /N )

2 (69)

The series expansions of the coefficients given in Eqs. (67) and (68) are differentiated

and substituted in the governing equations. The result is then multiplied by weight func-

tions that are the same as the basis functions and the equations are integrated over the

domain. The resulting integrals can all be solved analytically, except for the integral in-

volving the velocity, which would have to be solved numerically. However, in both cases

considered, the non-dimensional parameter M 1 in Eq. (57) is at least three orders of

magnitude smaller than the other parameters. Therefore, the velocity profile will be very

uniform and a constant average velocity can be used in the energy equation without much

loss in accuracy. After this process, all the integrals can be solved analytically and the

system of linear equations for the coefficients can be obtained without any computation.

The resulting linear system of equations will obviously depend on the boundary condi-

tions selected. For the case of a heat flux input, the system is given in tensor notation as

follows:

G1

γnJ 1,pn

δc,qm

2

+ G2

γ2

n

δs,np

2

δc,qm

2

+

φ2m

δs,np

2

δc,qm

2

(70)

+G3

δs,np

2

δc,qm

2

F nm +

−G3

J 2,pn

δc,qm

2

S nm

=

−G1 ( γnJ 1,pnJ 2,0q )−G2

γ2

n

δs,np

2 J 2,0q

−

γ20

δs,np

2 J 2,0q

−G3

δs,np

2 J 2,0q

H n + G3 (J 2,pnJ 2,0q )K n



19/29


G3Rh

J 2,np

δc,qm

2

F nm +

−

φ2m

δc,np

2

δc,qm

2

− G2

φ2n

δc,np

2

δc,qm

2

(71)

− G3Rhδc,np

2

δc,qm

2 S nm =

γ20

δc,np

2 J 2,0q

+ G2φ2n

δc,np

2 J 2,0q

+ G3Rh

δc,np

2 J 2,0q

K n + [−G3Rh (J 2,npJ 2,0q)] H n

where the tensors J 1,np, J 2,np and δs,np, δc,qm are defined as

J 1,pn =

1 0

cos( γnx)sin( γ px) dx

=

[ p + (1/2)] − (−1)n (−1) p [n + (1/2)]

π ( p2 − n2 + p − n) if n = p

1π (2n + 1) if n = p

(72)

J 2,pn =

1 0

sin( γ px)cos(φnx) dx = 4 p + 2

π (4 p2 − 4n + 4 p + 1) (73)

δc,np =

2 if n = p = 0

1 if n = p = 0

0 otherwise

, δs,np =

1 if n = p = 0

0 otherwise

(74)

The tensors J 1 and J 2 are full; however, their outer product with a diagonal matrix will

produce a block diagonal matrix. The resulting linear system can be cast in a matrixform:

A1 A2A3 A4

F

S

=

b1b2

(75)

Matrices A1, A2, and A3 are block diagonal (because they result from an outer product of

full matrices with the identity), while matrix A4 is diagonal. Figure 3 shows the structure

of the matrix for N = 56. The number of non-zero elements is 3N 3 + N 2 out of thetotal 4N 4 elements, and for N > 6 the matrix will be sparse and the system can be

solved very efficiently even for large N . The entire code was developed in MATLAB

(The MathWorks, Inc.) and the built-in sparse solver was used. All calculations were

performed using double-digit precision on a quad core Intel i7 2700k processor running

Windows 7 with 16GB of RAM. For N = 64, which is used during the validation process,the solution time is about 0.15 s. After the coefficients are obtained, the temperature and

heat flux can be obtained by recombining the solution at any point or grid.



20/29


0 1000 2000 3000 4000 5000 6000

0

1000

2000

3000

4000

5000

6000

nz = 529984

FIG. 3: Matrix structure.

4. RESULTS AND DISCUSSION

The accuracy of the theory, closure, validity of assumptions, and solution method are

determined by validating the numerical results with experimental results available in theliterature. In both experimental cases, the boundary condition at the base is a constant

heat flux condition. As discussed previously, coefficients H n and K n can be calculated

analytically:

H n = σn2q w

γn γ0, K n =

q wRh

γ0n = 0

0 otherwise

(76)

The raised cosine filter [Eq. (69)] is applied to the modified sine Fourier series because

it is the approximation of a step function. The filter will reduce Gibbs phenomena and

improve the convergence rate away from the discontinuity point. However, a significant

number of terms will still be necessary to obtain a good approximation of the constant

function; therefore, the overall convergence will be controlled by the H n coefficients. Itis important to note that this is a worst-case scenario, and for the case of a smooth heat

input, the solution would converge within much fewer terms.



21/29


The code will be validated by modeling two different geometries with different work-

ing fluids: an air-cooled staggered pin-fin heat sink and a water-cooled micro-channel

heat sink. The Nusselt and Reynolds numbers are defined as

Nudh =ĥd̂

hk̂f

(77)

Redh =Û avg d̂h

ˆ ν(78)

where dh is the VAT-derived hydraulic diameter and is defined in Eq. (56). The geometry,

pressure drop, and heat input for the staggered pin-fin heat sink were taken from Rizzi

(2002) and are shown in Table 1.

The Rizzi (2002) experiments were conducted by placing an aluminum heat sink

in a wind tunnel and recording its thermal response for four different inlet velocities,

which modeled four different Reynolds numbers in the 900–2400 range. The Nusselt

and Reynolds numbers were defined using the same length scale used in Eqs. (77) and(78), and the heat transfer coefficient and friction factor were defined as

ĥ = q̂

T̂ s,max − T̂ in(79)

f = 1

2

∆ P̂

L̂ d̂h

ρ̂f Û avg

(80)

where T̂ s,max is the maximum temperature in the heat sink. The same definition of the

heat transfer coefficient and friction factor is used when comparing the numerical results.

Because the heat flux is constant, the thermocouple closest to the exit, which was placed

TABLE 1: Pin-fin heat sink inputs

Parameter Value

L 113.75 mm

W 113.75 mm

H c 38.10 mm

tb 8.25 mm

D 3.18 mm

S T 4.76 mm

S L 4.76 mmT in 298.0 K

Q 50.0 W



22/29


about 15 mm from the outlet, recorded the highest temperature. Hence, the numerical

T s,max was calculated at the location of the last thermocouple. The geometry and def-

inition of the REV are shown in Fig. 4. In the experimental studies, a wooden block

was used to eliminate flow bypass but is omitted in Fig. 4 to better illustrate the internal

geometry. The closure coefficients h and f f for the pin-fin heat sink were taken from theZukauskas (1987) experimental correlations. The formulas for porosity and the specific

surface can be obtained geometrically, and are given by

εf = 1− π D2

8S T S L, S w =

π D

2S T S L(81)

The geometry, pressure drop, and heat input for the micro-channel heat sink were taken

from the Qu and Mudawar (2002) experimental study and are shown in Table 2. The ex-

periments were conducted for single-phase deionized water flowing over an oxygen-free

copper heat sink fitted with a polycarbonate plastic cover plate. The Reynolds number

was varied by changing the volumetric flow rate through the heat sink, and the tem-

peratures were recorded. Qu and Mudawar (2002) defined the Reynolds number usingthe traditional hydraulic diameter as the ratio of four times the cross-sectional area to

the wetted perimeter. However, it can be shown that the VAT-defined hydraulic diameter

FIG. 4: Schematic of the pin-fin heat and REV definition (without the cover plate) (Zhou

et al., 2011).



23/29


TABLE 2: Geometry and inputs

for micro-channel heat sink

Parameter Value

L 44.8 mm

W 10.0 mmH c 0.713 mm

tb 3.00 mm

p 0.467 mm

t 0.236 mm

T in 288.0 K

Q 448.0 W

used in our definition of the Reynolds number reduces to the same expression for rect-

angular channels; therefore, no conversion factors are necessary.

The closure coefficients h and f f

were obtained from the Copeland (2000) numerical

results for plane fins. A schematic of the geometry is given in Fig. 5 and the formulas

for porosity and specific surface for this geometry are given by

εf = p− w

p , S w =

2H c + 2 ( p−w)

Hp (82)

FIG. 5: Schematic of the micro-channel geometry.



24/29


First, a basis independence study is conducted to determine the number of basis func-

tions N necessary to obtain a converged solution. The solution was refined until the

relative difference in the interface fluid midpoint temperature was less than 10−3. The

fluid temperature was chosen because it is known that its convergence will be slower due

to the slower convergence of coefficients H n. The results are shown in Fig. 6. The errorwas interpolated using a power fit, and the interpolation was almost perfect with R2 =

0.9994. It was determined that the convergence rate is of ON −3.858

, and N = 64 will

give a relative error below the required tolerance. For further proof of convergence, the

energy imbalance was found to be only 0.01% for N = 64. Therefore, this value will be

used in the rest of the validation process.

To compare the numerical and experimental results, the input pressure drop for the

staggered pin-fin heat sink was varied to obtain a relationship for Nudh and f versus

Redh . The numerical and experimental results are compared in Fig. 7. The agreement

for the Nusselt number is excellent. The average error is only 3.2% while the maximum

error is 5.1%. The agreement for the friction factor is also excellent with an error of

less than 2.6% everywhere except for the last point, where it is 4.5%. This is to be

expected because the last point is at a Reynolds number of 2560, which is already in

the transition zone. The error bars for the experimental values are not provided, but

the numerical results are expected to be well within experimental uncertainties for all

Reynolds numbers.

30 40 50 60 70 80 90 100 11010

−4

10−3

10−2

10−1

N

R e l a t i v e E r r o r

FIG. 6: Relative error in the midpoint fluid temperature as a function of the number of

basis N .



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800 1000 1200 1400 1600 1800 2000 2200 2400 26000

20

40

60

80

100

120

140

160

180

Red

h

N u d

h

800 1000 1200 1400 1600 1800 2000 2200 2400 26000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f

f experimental

f predicted

Nu experimental

Nu predicted

FIG. 7: Nusselt number and friction factor versus the Reynolds comparison for a pin-fin

heat sink.

To demonstrate the geometric flexibility of the code and to further validate the theory

and numerical method, the code is used to model a water-cooled micro-channel heat sink.

It is important to notice that the equations to be solved and their domain do not change.

The only differences are in the calculation of the porosity, specific surface, the closure

parameters h and f f , and the viscosity of the fluid. The comparison of the bottom wall

temperature is shown in Fig. 8 and the results are once again excellent. At a Reynolds

number of 890, the average and maximum errors are 3.8 and 7.3%, respectively. At a

Reynolds number of 1454, the errors are 2.4 and 1.2%. The higher error at the lower

Reynolds number can be explained by the fact that at lower Reynolds numbers the effect

of the base will be more significant. The results for the pressure drop are shown in Fig. 9.

Once again, the agreement between the numerical and experimental data is excellent.

The average error is 4.1% while the maximum error is 5.2%. Both are expected to be

within experimental uncertainties.

Hence, it has been shown that the code can accurately model the hydraulic and ther-

mal behavior of heat sinks with considerably different geometries and different working

fluids. The ability to quickly compare different geometries makes the code particularlysuitable for population-based optimization algorithms such as genetic algorithms and

particle swarm optimization. In such algorithms, the confidence in the optimality of the



26/29


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

x/L

T s

[ 0 C

]

Numerical at Re=890

Numerical at Re=1454

Experimental at Re=890

Experimental at Re=1454

FIG. 8: Comparison of the numerical and experimental bottom wall temperatures at

different x locations for the water-cooled micro-channel heat sink (q = 100 W/cm2).

400 600 800 1000 1200 1400 16000

10

20

30

40

50

60

70

Red

h

∆ P [

k P a ]

FIG. 9: Comparison of the numerical and experimental pressure drop versus the

Reynolds number for the water-cooled micro-channel heat sink.



27/29


global maximum increases with the number of heat sinks evaluated. One of the lim-

its of the numerical method is that the number of basis functions required will depend

on the smoothness of the solution, which often is not known a priori, and that of the

boundary conditions. In the validation process considered, the overall convergence of

the solution is controlled by coefficients H

n. It is important to note that the solutionfor variable heat flux is already built into the numerical solution and can be obtained

by using different coefficients H n and K n. Reduced convergence is also expected if the

solution itself has very steep gradients (e.g., low Prandtl fluids). In contrast, for smooth

input heat fluxes (or temperatures), the convergence is expected to improve significantly;

as a consequence, this implies that the number of basis functions required to meet the

tolerance would be significantly lower, and the solution time would be further reduced

considerably.

5. CONCLUSIONS

The governing equations for mass, momentum, and energy in a heat sink were derivedby averaging the corresponding point-wise equations over a REV. The equations were

closed using transport coefficients that model micro-scale behavior. A code was devel-

oped to solve the resulting VAT-derived system of PDEs. The momentum equation is

solved using a FD scheme because of the non-linearity. The solid and fluid tempera-

tures are expanded in a modified Fourier series, and the GM is used to obtain a sparse

linear system in the coefficients. The basis functions are chosen such that they satisfy

the boundary conditions a priori. The system is solved using a sparse solver to model

different heat sinks with constant heat flux. The method was applied to model thermal

behavior of an air-cooled pin-fin and a water-cooled micro-channel heat sink for a given

heat load and pressure drop. It was found that about 64 basis functions in each direc-

tion were necessary to obtain convergence, and the overall runtime was about ∼0.25

s. The convergence is known to improve as the smoothness of the boundary condition

increases. The thermal behavior of the heat sink obtained numerically is compared to

experimental values, and the agreement is excellent with an average error of less than

4% for both cases. The code also predicted the average velocity for the given pressure

drop with very good accuracy for both geometries. For the analyzed air-cooled pin-fin

heat sink, the agreement with experimental data is excellent with an error of less than

2.6% for Re < 2300. For the water-cooled micro-channel heat sink, the average error is

4.1% and the maximum error is 5.2%. Although these errors are slightly higher than in

the pin-fin case, the prediction is still excellent and the error is expected to be within the

experimental uncertainties. Overall, the agreement with experimental data and the com-

putational efficiency show the validity, accuracy, and advantage of using a GM solution

to solve VAT-based conservation equations. The code will be extended to include casesof constant temperature, non-negligible base conduction, and changing geometry in the

flow direction.



28/29


6. ACKNOWLEDGMENTS

We would like to sincerely thank the late Dr. Novak Zuber and the Kerze-Cheyovich

endowment, which made this research possible.

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