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Cooling Technologies Research Center
Research Publications
Purdue Libraries Year
Analysis and optimization of the thermal
performance of microchannel heat sinks
Dong Liulowast S V Garimella dagger
lowast
daggerPurdue Univ sureshgpurdueedu
This paper is posted at Purdue e-Pubs
httpdocslibpurdueeducoolingpubs9
Analysis and optimizationof the thermal performance of
microchannel heat sinksDong Liu and Suresh V Garimella
Cooling Technologies Research Center School of Mechanical EngineeringPurdue University West Lafayette Indiana USA
Abstract
Purpose ndash To provide modeling approaches of increasing levels of complexity for the analysis ofconvective heat transfer in microchannels which offer adequate descriptions of the thermalperformance while allowing easier manipulation of microchannel geometries for the purpose of designoptimization of microchannel heat sinks
Designmethodologyapproach ndash A detailed computational fluid dynamics model is first used toobtain baseline results against which five approximate analytical approaches are compared Theseapproaches include a 1D resistance model a fin approach two fin-liquid coupled models and a porousmedium approach A modified thermal boundary condition is proposed to correctly characterize theheat flux distribution
Findings ndash The results obtained demonstrate that the models developed offer sufficiently accuratepredictions for practical designs while at the same time being quite straightforward to use
Research limitationsimplications ndash The analysis is based on a single microchannel while in apractical microchannel heat sink multiple channels are employed in parallel Therefore theoptimization should take into account the impact of inletoutlet headers Also a prescribed pumpingpower may be used as the design constraint instead of pressure head
Practical implications ndash Very useful design methodologies for practical design of microchannelheat sinks
Originalityvalue ndash Closed-form solutions from five analytical models are derived in a format thatcan be easily implemented in optimization procedures for minimizing the thermal resistance ofmicrochannel heat sinks
Keywords Optimization techniques Heat transfer Convection
Paper type Technical paper
The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at
wwwemeraldinsightcomresearchregister wwwemeraldinsightcom0961-5539htm
NomenclatureAc frac14 microchannel cross-sectional areaAf frac14 fin cross-sectional areaAs frac14 area of heat sinkCp frac14 specific heatDh frac14 hydraulic diameterf Re frac14 friction constant
h frac14 heat transfer coefficientH frac14 height of heat sinkHc frac14 microchannel depthk frac14 thermal conductivityL frac14 length of heat sink_m frac14 mass flow rate
The authors acknowledge the financial support from members of the Cooling TechnologiesResearch Center (available at httpwidgetecnpurdueedu CTRC) a National ScienceFoundation IndustryUniversity Cooperative Research Center at Purdue University
Analysis andoptimization
7
Received February 2003Revised November 2003
Accepted May 2004
International Journal for NumericalMethods in Heat amp Fluid Flow
Vol 15 No 1 2005pp 7-26
q Emerald Group Publishing Limited0961-5539
DOI 10110809615530510571921
IntroductionThe potential for handling ultra-high heat fluxes has spurred intensive research intomicrochannel heat sinks (Tuckerman and Pease 1981 Weisberg and Bau 1992Sobhan and Garimella 2001) For implementation in practical designs the convectiveheat transfer in microchannels must be analyzed in conjunction with the choice andoptimization of the heat sink dimensions to ensure the required thermal performanceDesign procedures are also needed to minimize the overall thermal resistance
The focus of this paper is to present a comprehensive discussion and comparison offive different (approximate) analytical models of increasing sophistication which offerclosed-form solutions for single phase convective heat transfer in microchannelsA general computational fluid dynamics (CFD) model is first set-up to obtain anldquoexactrdquo solution Details of the approximate models and the assumptions involved arethen presented along with a comparison of the thermal resistance predictions fromthese models Optimization of the thermal performance of microchannel heat sinks isthen discussed
Description of the problemThe microchannel heat sink under consideration is shown in Figure 1 Materials forfabrication may include conductive materials such as copper and aluminumfor modular heat sinks or silicon if the microchannels are to be integrated into thechip For conservative estimates of thermal performance the lid (top plate) may be
n frac14 number of microchannelsNu frac14 Nusselt numberP frac14 pressureq00 frac14 heat fluxq frac14 heat removal rateQ frac14 volume flow rateR frac14 thermal resistanceRe frac14 Reynolds numbert frac14 substrate thicknessT frac14 fin temperatureTb frac14 temperature at the base of the finTf frac14 fluid temperatureum frac14 mean flow velocityW frac14 width of heat sinkwc frac14 microchannel widthww frac14 fin thickness
Greek symbolsa frac14 aspect ratio of microchannelshf frac14 fin efficiencym frac14 dynamic viscosityu frac14 thermal resistancer frac14 density of fluidDP frac14 pressure dropDT frac14 temperature difference
Subscripts and superscriptsc frac14 channel fluidf frac14 fluidi frac14 inlets frac14 solid finw frac14 wall
Figure 1Schematic of amicrochannel heat sink
HFF151
8
to be insulated The width of individual microchannels and intervening fins (wcthornww)is typically small compared to the overall heat sink dimension W and numerouschannels are accommodated in parallel flow paths
Continuum equations for conservation of mass momentum and energyrespectively for the convective heat transfer in microchannel heat sinks can bewritten as (Fedorov and Viskanta 2000 Toh et al 2002)
7 middot ethr kVTHORN frac14 0 eth1THORN
kV middot7ethr kVTHORN frac14 27P thorn 7 middot ethm7 kVTHORN eth2THORN
kV middot7ethrCpTTHORN frac14 7 middot ethkf7TTHORN for the fluid eth3THORN
7 middot ethks7TsTHORN frac14 0 for the fin eth4THORN
This set of equations assumes steady-state conditions for incompressible laminarflow with radiation heat transfer neglected With an appropriate set of boundaryconditions these equations provide a complete description of the conjugate heattransfer problem in microchannels
CFD modelA numerical model was formulated to solve the three-dimensional heat transfer inmicrochannels using the commercial CFD software package FLUENT (Fluent Inc1998) The simulation was performed for three different sets of dimensions as listed inTable I These three cases are chosen to simulate experiments in the literature(Tuckerman and Pease 1981) that have often been used for validating numericalstudies (Weisberg and Bau 1992 Toh et al 2002 Ryu et al 2002)
The computational domain chosen from symmetry considerations is shown inFigure 2 The top surface is adiabatic and the left and right sides are designatedsymmetric boundary conditions A uniform heat flux is applied at the bottomsurface In the present work water is used as the working fluid (r frac14 997 kg=m3Cp frac14 4 179 J=kg K m frac14 0000855 kg=ms and kf frac14 0613 W=mK evaluated at 278C)and silicon is used as the heat sink substrate material with ks frac14 148 W=mK
In the numerical solution the convective terms were discretized using afirst-order upwind scheme for all equations The entire computational domain wasdiscretized using a 500 pound 60 pound 14 (x-y-z) grid To verify the grid independence of
Case1 2 3
wc (mm) 56 55 50ww (mm) 44 45 50Hc (mm) 320 287 302H (mm) 533 430 458DP (kPa) 10342 11721 21373q00 (Wcm2) 181 277 790Rexp (8CW) (Tuckerman and Pease 1981) 0110 0113 0090Rnum (8CW) 0115 0114 0093
Note L frac14 W frac14 1 cm
Table IComparison of thermal
resistances
Analysis andoptimization
9
the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work
The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork
The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered
Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)
2ksrsaquoT
rsaquoy
yfrac140
frac14 q00 eth5THORN
or
2ksdT
dy
yfrac140
frac14wc thorn ww
wwq00 eth6THORN
in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of
Figure 2Computational domain
HFF151
10
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
Analysis and optimizationof the thermal performance of
microchannel heat sinksDong Liu and Suresh V Garimella
Cooling Technologies Research Center School of Mechanical EngineeringPurdue University West Lafayette Indiana USA
Abstract
Purpose ndash To provide modeling approaches of increasing levels of complexity for the analysis ofconvective heat transfer in microchannels which offer adequate descriptions of the thermalperformance while allowing easier manipulation of microchannel geometries for the purpose of designoptimization of microchannel heat sinks
Designmethodologyapproach ndash A detailed computational fluid dynamics model is first used toobtain baseline results against which five approximate analytical approaches are compared Theseapproaches include a 1D resistance model a fin approach two fin-liquid coupled models and a porousmedium approach A modified thermal boundary condition is proposed to correctly characterize theheat flux distribution
Findings ndash The results obtained demonstrate that the models developed offer sufficiently accuratepredictions for practical designs while at the same time being quite straightforward to use
Research limitationsimplications ndash The analysis is based on a single microchannel while in apractical microchannel heat sink multiple channels are employed in parallel Therefore theoptimization should take into account the impact of inletoutlet headers Also a prescribed pumpingpower may be used as the design constraint instead of pressure head
Practical implications ndash Very useful design methodologies for practical design of microchannelheat sinks
Originalityvalue ndash Closed-form solutions from five analytical models are derived in a format thatcan be easily implemented in optimization procedures for minimizing the thermal resistance ofmicrochannel heat sinks
Keywords Optimization techniques Heat transfer Convection
Paper type Technical paper
The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at
wwwemeraldinsightcomresearchregister wwwemeraldinsightcom0961-5539htm
NomenclatureAc frac14 microchannel cross-sectional areaAf frac14 fin cross-sectional areaAs frac14 area of heat sinkCp frac14 specific heatDh frac14 hydraulic diameterf Re frac14 friction constant
h frac14 heat transfer coefficientH frac14 height of heat sinkHc frac14 microchannel depthk frac14 thermal conductivityL frac14 length of heat sink_m frac14 mass flow rate
The authors acknowledge the financial support from members of the Cooling TechnologiesResearch Center (available at httpwidgetecnpurdueedu CTRC) a National ScienceFoundation IndustryUniversity Cooperative Research Center at Purdue University
Analysis andoptimization
7
Received February 2003Revised November 2003
Accepted May 2004
International Journal for NumericalMethods in Heat amp Fluid Flow
Vol 15 No 1 2005pp 7-26
q Emerald Group Publishing Limited0961-5539
DOI 10110809615530510571921
IntroductionThe potential for handling ultra-high heat fluxes has spurred intensive research intomicrochannel heat sinks (Tuckerman and Pease 1981 Weisberg and Bau 1992Sobhan and Garimella 2001) For implementation in practical designs the convectiveheat transfer in microchannels must be analyzed in conjunction with the choice andoptimization of the heat sink dimensions to ensure the required thermal performanceDesign procedures are also needed to minimize the overall thermal resistance
The focus of this paper is to present a comprehensive discussion and comparison offive different (approximate) analytical models of increasing sophistication which offerclosed-form solutions for single phase convective heat transfer in microchannelsA general computational fluid dynamics (CFD) model is first set-up to obtain anldquoexactrdquo solution Details of the approximate models and the assumptions involved arethen presented along with a comparison of the thermal resistance predictions fromthese models Optimization of the thermal performance of microchannel heat sinks isthen discussed
Description of the problemThe microchannel heat sink under consideration is shown in Figure 1 Materials forfabrication may include conductive materials such as copper and aluminumfor modular heat sinks or silicon if the microchannels are to be integrated into thechip For conservative estimates of thermal performance the lid (top plate) may be
n frac14 number of microchannelsNu frac14 Nusselt numberP frac14 pressureq00 frac14 heat fluxq frac14 heat removal rateQ frac14 volume flow rateR frac14 thermal resistanceRe frac14 Reynolds numbert frac14 substrate thicknessT frac14 fin temperatureTb frac14 temperature at the base of the finTf frac14 fluid temperatureum frac14 mean flow velocityW frac14 width of heat sinkwc frac14 microchannel widthww frac14 fin thickness
Greek symbolsa frac14 aspect ratio of microchannelshf frac14 fin efficiencym frac14 dynamic viscosityu frac14 thermal resistancer frac14 density of fluidDP frac14 pressure dropDT frac14 temperature difference
Subscripts and superscriptsc frac14 channel fluidf frac14 fluidi frac14 inlets frac14 solid finw frac14 wall
Figure 1Schematic of amicrochannel heat sink
HFF151
8
to be insulated The width of individual microchannels and intervening fins (wcthornww)is typically small compared to the overall heat sink dimension W and numerouschannels are accommodated in parallel flow paths
Continuum equations for conservation of mass momentum and energyrespectively for the convective heat transfer in microchannel heat sinks can bewritten as (Fedorov and Viskanta 2000 Toh et al 2002)
7 middot ethr kVTHORN frac14 0 eth1THORN
kV middot7ethr kVTHORN frac14 27P thorn 7 middot ethm7 kVTHORN eth2THORN
kV middot7ethrCpTTHORN frac14 7 middot ethkf7TTHORN for the fluid eth3THORN
7 middot ethks7TsTHORN frac14 0 for the fin eth4THORN
This set of equations assumes steady-state conditions for incompressible laminarflow with radiation heat transfer neglected With an appropriate set of boundaryconditions these equations provide a complete description of the conjugate heattransfer problem in microchannels
CFD modelA numerical model was formulated to solve the three-dimensional heat transfer inmicrochannels using the commercial CFD software package FLUENT (Fluent Inc1998) The simulation was performed for three different sets of dimensions as listed inTable I These three cases are chosen to simulate experiments in the literature(Tuckerman and Pease 1981) that have often been used for validating numericalstudies (Weisberg and Bau 1992 Toh et al 2002 Ryu et al 2002)
The computational domain chosen from symmetry considerations is shown inFigure 2 The top surface is adiabatic and the left and right sides are designatedsymmetric boundary conditions A uniform heat flux is applied at the bottomsurface In the present work water is used as the working fluid (r frac14 997 kg=m3Cp frac14 4 179 J=kg K m frac14 0000855 kg=ms and kf frac14 0613 W=mK evaluated at 278C)and silicon is used as the heat sink substrate material with ks frac14 148 W=mK
In the numerical solution the convective terms were discretized using afirst-order upwind scheme for all equations The entire computational domain wasdiscretized using a 500 pound 60 pound 14 (x-y-z) grid To verify the grid independence of
Case1 2 3
wc (mm) 56 55 50ww (mm) 44 45 50Hc (mm) 320 287 302H (mm) 533 430 458DP (kPa) 10342 11721 21373q00 (Wcm2) 181 277 790Rexp (8CW) (Tuckerman and Pease 1981) 0110 0113 0090Rnum (8CW) 0115 0114 0093
Note L frac14 W frac14 1 cm
Table IComparison of thermal
resistances
Analysis andoptimization
9
the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work
The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork
The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered
Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)
2ksrsaquoT
rsaquoy
yfrac140
frac14 q00 eth5THORN
or
2ksdT
dy
yfrac140
frac14wc thorn ww
wwq00 eth6THORN
in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of
Figure 2Computational domain
HFF151
10
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
IntroductionThe potential for handling ultra-high heat fluxes has spurred intensive research intomicrochannel heat sinks (Tuckerman and Pease 1981 Weisberg and Bau 1992Sobhan and Garimella 2001) For implementation in practical designs the convectiveheat transfer in microchannels must be analyzed in conjunction with the choice andoptimization of the heat sink dimensions to ensure the required thermal performanceDesign procedures are also needed to minimize the overall thermal resistance
The focus of this paper is to present a comprehensive discussion and comparison offive different (approximate) analytical models of increasing sophistication which offerclosed-form solutions for single phase convective heat transfer in microchannelsA general computational fluid dynamics (CFD) model is first set-up to obtain anldquoexactrdquo solution Details of the approximate models and the assumptions involved arethen presented along with a comparison of the thermal resistance predictions fromthese models Optimization of the thermal performance of microchannel heat sinks isthen discussed
Description of the problemThe microchannel heat sink under consideration is shown in Figure 1 Materials forfabrication may include conductive materials such as copper and aluminumfor modular heat sinks or silicon if the microchannels are to be integrated into thechip For conservative estimates of thermal performance the lid (top plate) may be
n frac14 number of microchannelsNu frac14 Nusselt numberP frac14 pressureq00 frac14 heat fluxq frac14 heat removal rateQ frac14 volume flow rateR frac14 thermal resistanceRe frac14 Reynolds numbert frac14 substrate thicknessT frac14 fin temperatureTb frac14 temperature at the base of the finTf frac14 fluid temperatureum frac14 mean flow velocityW frac14 width of heat sinkwc frac14 microchannel widthww frac14 fin thickness
Greek symbolsa frac14 aspect ratio of microchannelshf frac14 fin efficiencym frac14 dynamic viscosityu frac14 thermal resistancer frac14 density of fluidDP frac14 pressure dropDT frac14 temperature difference
Subscripts and superscriptsc frac14 channel fluidf frac14 fluidi frac14 inlets frac14 solid finw frac14 wall
Figure 1Schematic of amicrochannel heat sink
HFF151
8
to be insulated The width of individual microchannels and intervening fins (wcthornww)is typically small compared to the overall heat sink dimension W and numerouschannels are accommodated in parallel flow paths
Continuum equations for conservation of mass momentum and energyrespectively for the convective heat transfer in microchannel heat sinks can bewritten as (Fedorov and Viskanta 2000 Toh et al 2002)
7 middot ethr kVTHORN frac14 0 eth1THORN
kV middot7ethr kVTHORN frac14 27P thorn 7 middot ethm7 kVTHORN eth2THORN
kV middot7ethrCpTTHORN frac14 7 middot ethkf7TTHORN for the fluid eth3THORN
7 middot ethks7TsTHORN frac14 0 for the fin eth4THORN
This set of equations assumes steady-state conditions for incompressible laminarflow with radiation heat transfer neglected With an appropriate set of boundaryconditions these equations provide a complete description of the conjugate heattransfer problem in microchannels
CFD modelA numerical model was formulated to solve the three-dimensional heat transfer inmicrochannels using the commercial CFD software package FLUENT (Fluent Inc1998) The simulation was performed for three different sets of dimensions as listed inTable I These three cases are chosen to simulate experiments in the literature(Tuckerman and Pease 1981) that have often been used for validating numericalstudies (Weisberg and Bau 1992 Toh et al 2002 Ryu et al 2002)
The computational domain chosen from symmetry considerations is shown inFigure 2 The top surface is adiabatic and the left and right sides are designatedsymmetric boundary conditions A uniform heat flux is applied at the bottomsurface In the present work water is used as the working fluid (r frac14 997 kg=m3Cp frac14 4 179 J=kg K m frac14 0000855 kg=ms and kf frac14 0613 W=mK evaluated at 278C)and silicon is used as the heat sink substrate material with ks frac14 148 W=mK
In the numerical solution the convective terms were discretized using afirst-order upwind scheme for all equations The entire computational domain wasdiscretized using a 500 pound 60 pound 14 (x-y-z) grid To verify the grid independence of
Case1 2 3
wc (mm) 56 55 50ww (mm) 44 45 50Hc (mm) 320 287 302H (mm) 533 430 458DP (kPa) 10342 11721 21373q00 (Wcm2) 181 277 790Rexp (8CW) (Tuckerman and Pease 1981) 0110 0113 0090Rnum (8CW) 0115 0114 0093
Note L frac14 W frac14 1 cm
Table IComparison of thermal
resistances
Analysis andoptimization
9
the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work
The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork
The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered
Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)
2ksrsaquoT
rsaquoy
yfrac140
frac14 q00 eth5THORN
or
2ksdT
dy
yfrac140
frac14wc thorn ww
wwq00 eth6THORN
in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of
Figure 2Computational domain
HFF151
10
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
to be insulated The width of individual microchannels and intervening fins (wcthornww)is typically small compared to the overall heat sink dimension W and numerouschannels are accommodated in parallel flow paths
Continuum equations for conservation of mass momentum and energyrespectively for the convective heat transfer in microchannel heat sinks can bewritten as (Fedorov and Viskanta 2000 Toh et al 2002)
7 middot ethr kVTHORN frac14 0 eth1THORN
kV middot7ethr kVTHORN frac14 27P thorn 7 middot ethm7 kVTHORN eth2THORN
kV middot7ethrCpTTHORN frac14 7 middot ethkf7TTHORN for the fluid eth3THORN
7 middot ethks7TsTHORN frac14 0 for the fin eth4THORN
This set of equations assumes steady-state conditions for incompressible laminarflow with radiation heat transfer neglected With an appropriate set of boundaryconditions these equations provide a complete description of the conjugate heattransfer problem in microchannels
CFD modelA numerical model was formulated to solve the three-dimensional heat transfer inmicrochannels using the commercial CFD software package FLUENT (Fluent Inc1998) The simulation was performed for three different sets of dimensions as listed inTable I These three cases are chosen to simulate experiments in the literature(Tuckerman and Pease 1981) that have often been used for validating numericalstudies (Weisberg and Bau 1992 Toh et al 2002 Ryu et al 2002)
The computational domain chosen from symmetry considerations is shown inFigure 2 The top surface is adiabatic and the left and right sides are designatedsymmetric boundary conditions A uniform heat flux is applied at the bottomsurface In the present work water is used as the working fluid (r frac14 997 kg=m3Cp frac14 4 179 J=kg K m frac14 0000855 kg=ms and kf frac14 0613 W=mK evaluated at 278C)and silicon is used as the heat sink substrate material with ks frac14 148 W=mK
In the numerical solution the convective terms were discretized using afirst-order upwind scheme for all equations The entire computational domain wasdiscretized using a 500 pound 60 pound 14 (x-y-z) grid To verify the grid independence of
Case1 2 3
wc (mm) 56 55 50ww (mm) 44 45 50Hc (mm) 320 287 302H (mm) 533 430 458DP (kPa) 10342 11721 21373q00 (Wcm2) 181 277 790Rexp (8CW) (Tuckerman and Pease 1981) 0110 0113 0090Rnum (8CW) 0115 0114 0093
Note L frac14 W frac14 1 cm
Table IComparison of thermal
resistances
Analysis andoptimization
9
the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work
The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork
The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered
Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)
2ksrsaquoT
rsaquoy
yfrac140
frac14 q00 eth5THORN
or
2ksdT
dy
yfrac140
frac14wc thorn ww
wwq00 eth6THORN
in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of
Figure 2Computational domain
HFF151
10
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
the convective heat transfer results three different meshes were used in the fluidpart of the domain 20 pound 5 30 pound 7 and 50 pound 15 The thermal resistance changed by34 percent from the first to the second mesh and only by 03 percent upon furtherrefinement to the third grid Hence 30 pound 7 grids were used in the fluid domain forthe results in this work
The agreement between the experimental and predicted values of thermalresistance in Table I validates the use of the numerical predictions as abaseline against which to compare the approximate approaches considered in thiswork
The numerical results may also be used to shed light on the appropriate boundaryconditions for the problem under consideration For instance it is often assumed inmicrochannel heat sink analyses that the axial conduction in both the solid fin andfluid may be neglected Using the numerical results for case 1 as an example the axialconduction through the fin and fluid were found to account for 03 and 02 percent ofthe total heat input at the base of the heat sink respectively Thus the assumption ofnegligible axial conduction appears valid for heat transfer in the silicon microchannelsconsidered
Two alternative boundary conditions have been commonly used at the base of thefin in microchannel analyses (Zhao and Lu 2002 Samalam 1989 Sabry 2001)
2ksrsaquoT
rsaquoy
yfrac140
frac14 q00 eth5THORN
or
2ksdT
dy
yfrac140
frac14wc thorn ww
wwq00 eth6THORN
in which equation (5) implies that the imposed heat flows evenly into the fluid via thebottom of the microchannel and into the fin via the base of the fin while equation (6)implies that all the heat from the base travels up the base of the fin Clearly neither of
Figure 2Computational domain
HFF151
10
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
these two extreme cases represent the actual situation correctly The computed heatflux in the substrate in the immediate vicinity of the fin base is shown in Figure 3 forcase 1 The heat fluxes into the fluid and the fin are 555 and 333 Wcm2 respectivelyHence the error associated with employing equations (5) and (6) as the boundarycondition at the base of the fin would be 50 and 24 percent respectively A reasonablyaccurate alternative for the boundary condition could be developed as follows
q frac14 hwc
2L
ethTb 2 T fTHORN thorn hethH cLTHORNhfethTb 2 T fTHORN eth7THORN
Hence the ratio of the heat dissipated through the vertical sides of the fin to thatflowing through the bottom surface of the microchannel into the fluid is 2hfHcwc or
2hfa This leads to a more reasonable boundary condition at the base of the fin
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth8THORN
This condition results in a heat flux of 366 Wcm2 through the base of the fin which iswithin 10 percent of the computed exact value of 333 Wcm2
In light of this discussion equation (8) is imposed as the thermal boundarycondition at the base of the fin for all the five approximate models developed in thiswork
Approximate analytical modelsIn view of the complexity and computational expense of a full CFD approach forpredicting convective heat transfer in microchannel heat sinks especially in searchingfor optimal configurations under practical design constraints simplified modelingapproaches are sought The goal is to account for the important physics even if some
Figure 3Heat flux distribution at
the base of the fin
Analysis andoptimization
11
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
of the details may need to be sacrificed Five approximate analytical models (Zhao andLu 2002 Samalam 1989 Sabry 2001 Kim and Kim 1999) are discussed along withthe associated optimization procedures needed to minimize the thermal resistanceThe focus in this discussion is on the development of a set of thermal resistanceformulae that can be used for comparison between models as well as for optimizationof microchannel heat sinks
As shown in Figure 1 for the problem under consideration the fluid flows parallelto the x-axis The bottom surface of the heat sink is exposed to a constant heat fluxThe top surface remains adiabatic
The overall thermal resistance is defined as
Ro frac14DTmax
q00Aseth9THORN
where DTmax frac14 ethTwo 2 T fiTHORN is the maximum temperature rise in the heat sink ie thetemperature difference between the peak temperature in the heat sink at the outlet(Two) and the fluid inlet temperature (Tfi) Since the thermal resistance due tosubstrate conduction is simply
Rcond frac14t
ksethLW THORNeth10THORN
the thermal resistance R calculated in following models will not include this term
R frac14 Ro 2 Rcond eth11THORN
The following assumptions are made for the most simplified analysis
(1) steady-state flow and heat transfer
(2) incompressible laminar flow
(3) negligible radiation heat transfer
(4) constant fluid properties
(5) fully developed conditions (hydrodynamic and thermal)
(6) negligible axial heat conduction in the substrate and the fluid and
(7) averaged convective heat transfer coefficient h for the cross section
In the approximate analyses considered this set of assumptions is progressivelyrelaxed
Model 1 ndash 1D resistance analysisIn addition to making assumptions 1 ndash 7 above the temperature is assumed uniformover any cross section in the simplest of the models
For fully developed flow under a constant heat flux the temperature profile withinthe microchannel in the axial direction is shown in Figure 4 The three components ofthe heat transfer process are
qcond frac14 ksAsTwo 2 Tbo
teth12THORN
HFF151
12
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
qconv frac14 hAfethTb 2 T fTHORN eth13THORN
qcal frac14 rQCpethT fo 2 T fiTHORN eth14THORN
The overall thermal resistance can thus be divided into three components
Ro frac14DTmax
q00ethLW THORNfrac14
1
q00ethLW THORNfrac12ethTwo 2 T fiTHORN frac14 Rcond thorn Rconv thorn Rcal eth15THORN
in which the three resistances may be determined as follows
(1) Conductive thermal resistance
Rcond frac14t
ksethLW THORNeth16THORN
(2) Convective thermal resistance
Rconv frac141
nhLeth2hfH c thorn wcTHORNeth17THORN
with fin efficiency hf frac14 tanhethmH cTHORN=mH c
(3) Caloric thermal resistance
Rcal frac141
rfOCpeth18THORN
Model 2 ndash fin analysisIn this model assumptions 1-7 mentioned above are adopted and the fluid temperatureprofile is considered one-dimensional (averaged over y-z cross section) T f frac14 T fethxTHORNThe temperature distribution in the solid fin is then
Figure 4Temperature profile in a
microchannel heat sink
Analysis andoptimization
13
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
d2T
dy 2frac14
2h
kswwethT 2 T fethxTHORNTHORN eth19THORN
with boundary conditions
2ksdT
dy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 eth20THORN
dT
dy
yfrac14H c
frac14 0 eth21THORN
It follows that
Tethx yTHORN frac14 T fethxTHORN thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNeth22THORN
where m frac14 eth2h=kswwTHORN1=2
The fluid temperature Tf(x) can be obtained from an energy balance
_mCpdT fethxTHORN
dxfrac14 q00ethwc thorn wwTHORN eth23THORN
with T fethx frac14 0THORN frac14 T0 The bulk fluid temperature is then
T fethxTHORN frac14 T0 thornq00ethwc thorn wwTHORN
rfCpumH cwcx eth24THORN
and equation (22) can be rewritten as
Tethx yTHORN frac14 T0 thorn1
m
q00
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshmethH c 2 yTHORN
sinhethmH cTHORNthorn
q00ethwc thorn wwTHORN
rfCpumH cwcx eth25THORN
The thermal resistance is thus
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORN
frac141
m
1
ks
2hfa
2hfathorn 1
wc thorn ww
ww
coshethmH cTHORN
sinhethmH cTHORN
1
ethLW THORNthorn
ethwc thorn wwTHORN
rfCpumH cwc
1
Weth26THORN
Model 3 ndash fin-fluid coupled approach IFollowing the same line of reasoning as in the fin analysis (model 2) and adoptingassumptions 1-7 mentioned above but averaging the fluid temperature only in the zdirection (Samalam 1989) the energy equation in the fin can be written as
rsaquo2T
rsaquoy 2frac14
2h
kswwethT 2 T fethx yTHORNTHORN eth27THORN
with
HFF151
14
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
2ksrsaquoT
rsaquoy
yfrac140
frac142hfa
2hfathorn 1
wc thorn ww
wwq00 frac14 J eth28THORN
rsaquoT
rsaquoy
yfrac14H c
frac14 0 eth29THORN
The energy balance in the fluid is represented by
rfCpumwcrsaquoT f
rsaquoxfrac14 2hethT 2 T fethx yTHORNTHORN eth30THORN
and it is assumed that T fethx frac14 0 yTHORN frac14 0 Substituting h frac14 Nukf=Dh into equation (30)yields
1
2rfCpumwcDh
rsaquoT f
rsaquoxthorn kfNuT f frac14 kfNuT eth31THORN
Defining X frac14 x=a and Y frac14 y=a where a frac14 rfCpumwcDh=2kfNu the solution toequation (31) can be written as
T fethX Y THORN frac14
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0 eth32THORN
Hence equation (27) can then be transformed to
rsaquo2TethX Y THORN
rsaquoY 2frac14 b TethX Y THORN2
Z X
0
TethX 0Y THORNe2ethX2X 0THORN dX 0
eth33THORN
in which
b frac14a2
l 2eth34THORN
l 2 frac14kswwDh
2kfNueth35THORN
Solving equation (33) by Laplace transforms
rsaquo2f ethY sTHORN
rsaquoY 2frac14 gf g frac14
bs
s thorn 1
eth36THORN
The boundary conditions in equations (28) and (29) become
2ksrsaquof
rsaquoY
Yfrac140
frac14Ja
seth37THORN
2ksrsaquof
rsaquoY
Yfrac14 ~Hc
frac14 0 eth38THORN
where ~Hc frac14 H c=a and J is defined in equation (28) The solution to this system ofequations is
Analysis andoptimization
15
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
f ethsTHORN frac14Ja
kssffiffiffig
pcosh
ffiffiffig
pethY 2 ~HcTHORN
sinh
ffiffiffig
p ~Hc
eth39THORN
The inverse Laplace transform yields the temperature
TethX Y THORN frac14 L21frac12f ethsY THORN
frac14Ja
ks~Hcb
eth1 thorn XTHORN thornb
2ethY 2 ~HcTHORN
2 2b ~H
2
c
6
(
thorn 2X1nfrac141
eth21THORNnethsn thorn 1THORN2
sncos
npethY 2 ~HcTHORN
~Hc
esnX
)eth40THORN
in which
sn frac142n 2p 2= ~H
2
c
bthorn n 2p 2= ~H2
c
This is a rapidly converging infinite series for which the first three terms adequatelyrepresent the thermal resistance
R frac14DT
q00ethLW THORNfrac14
TethL 0THORN2 T0
q00ethLW THORNfrac14
Ja
ks~Hcb
eth1 thorn L=aTHORN thorn1
3b ~H
2
c
1
ethLW THORNeth41THORN
Model 4 ndash fin-fluid coupled approach IIIn this model assumptions 1-7 mentioned above are again adopted except that axialconduction in the fin is not neglected (Sabry 2001) The governing equations in thesolid fin and liquid respectively are therefore
72Tethx y zTHORN frac14 0 eth42THORN
7 middot ethrfCp kVT fethx y zTHORNTHORN frac14 kf72T fethx y zTHORN eth43THORN
At the fin-fluid interface the condition is
2ksrsaquoTs
rsaquozfrac14 2kf
rsaquoT f
rsaquozfrac14 hethT i 2 T fTHORN eth44THORN
in which the averaged local fluid temperature is
T fethxTHORN frac14
Z wc=2
0
vT f dz=ethumwc=2THORN eth45THORN
with
HFF151
16
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
um frac14
Z wc=2
0
v dz=wc=2
Along with equation (28) the following boundary conditions apply
rsaquoT
rsaquoy
yfrac14H c
frac14rsaquoT
rsaquox
xfrac140
frac14rsaquoT
rsaquox
xfrac14L
frac14rsaquoT
rsaquoz
zfrac142ww
2
frac14rsaquoT f
rsaquoz
zfrac14wc
2
frac14 0 eth46THORN
Integrating equation (42) over z from 2ww2 to 0 the fin temperature varies as
ww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T thorn
rsaquoT
rsaquoz
0
2ww2
frac14 0 eth47THORN
in which
T frac14
Z 0
2ww2
T dz=ethww=2THORN
Combining equations (44) and (46)
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethTi 2 T fTHORN frac14 0 eth48THORN
Assuming Ti frac14 T since Bi frac14 hethww=2THORN=ks 1 equation (48) becomes
ksww
2
rsaquo2
rsaquox 2thorn
rsaquo2
rsaquoy 2
T 2 hethT 2 T fTHORN frac14 0 eth49THORN
If the axial conduction term rsaquo2T=rsaquox 2 is neglected equation (49) would reduce toequation (27)
Since fully developed conditions are assumed and axial conduction in the fluid isneglected equation (43) may be integrated over z from 0 to wc2 to yield
rsaquo
rsaquox
Z wc=2
0
uT f dz frac14kf
rfCp
rsaquoT f
rsaquoz
wc2
0
eth50THORN
Using the boundary condition in equation (44) this reduces to
umwc
2
rsaquo
rsaquoxT f thorn
1
rfCphethT f 2 TTHORN frac14 0 eth51THORN
The following dimensionless variables are introduced
X frac14 x=L Y frac14 y=H c and T frac14T 2 T0
DTceth52THORN
in which
Analysis andoptimization
17
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
DTc frac142hfa
2hfathorn 1
ethwc thorn wwTHORN=2
hH cq00 eth53THORN
The system of equations above can be cast in dimensionless terms
A 2 rsaquo2
rsaquoX 2thorn
rsaquo2
rsaquoY 2
T 2 ethmH cTHORN
2ethT 2 T fTHORN frac14 0 eth54THORN
rsaquoT f
rsaquoXthorn SethT f 2 TTHORN frac14 0 eth55THORN
rsaquoT
rsaquoY
Yfrac140
frac14 2ethmH cTHORN2 eth56THORN
rsaquoTs
rsaquoX
Xfrac141
frac14rsaquoTs
rsaquoY
Yfrac140
frac14rsaquoTs
rsaquoY
Yfrac141
frac14 0 eth57THORN
T fjXfrac140 frac14 0 eth58THORN
where A frac14 H c=L and the modified Stanton number (Sabry 2001) is given by
S frac14 hL
rfCpumwc
2
Employing similar techniques as adopted for model 3 the fin temperature isobtained as
TethX Y THORN frac14 mH ccoshethmH ceth1 2 Y THORNTHORN
sinhethmH cTHORNthornX1nfrac140
cosethnpY THORNf nethXTHORN eth59THORN
in which the first term in the infinite series provides results of acceptable accuracy(5 percent deviation from the complete series)
f 0ethXTHORN frac14 SX thornX2
ifrac141
C0i
w0i
ew0iX thorn C03 eth60THORN
with
w01 frac14 2S
2thorn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
w02 frac14 2S
22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS
2
2
thornmH c
A
2s
C01 frac14 2Sethew02 2 1THORN
ethew02 2 ew01 THORN
HFF151
18
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
C02 frac14 2Sethew01 2 1THORN
ethew02 2 ew01 THORN
C03 frac14SA
mH c
2
The thermal resistance is thus obtained as
R frac14DT
q00ethLW THORNfrac14
Teth1 0THORN2 T0
q00ethLW THORNDTc
frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
X2
ifrac141
C0i
w0i
ew0i thorn C03
1
ethLW THORN
eth61THORN
In most practical cases ethmH c=ATHORN S=2 and equation (61) reduces to
R frac142hfa
2hfathorn 1
wc thorn ww
2hH cmH c
coshethmH cTHORN
sinhethmH cTHORNthorn S thorn
SA
mH c
2
1
ethLW THORNeth62THORN
Model 5 ndash porous medium approachThe convective heat transfer process in microchannels can also be treated as beingsimilar to that in a fluid-saturated porous medium with the extended Darcy equationused for fluid flow and a volume-averaged two-equation model used for heat transferas demonstrated in Vafai and Tien (1981)
Following the analysis of Kim and Kim (1999) a two-equation model can beemployed to obtain the volume-averaged properties over a representative elementaryvolume for the solid region and the fluid region separately The momentum equationand boundary conditions are
2d
dxkplf thorn mf
d2
dy 2kulf 2
mf
K1kulf frac14 0 eth63THORN
kulf frac14 0 at y frac14 0H c eth64THORN
where kulf is the volume-averaged velocity 1 frac14 wc=ethwc thorn wwTHORN is the porosity andK frac14 1w2
c=12 is the permeability Equations (63) and (64) may be written as
U frac14 Dad2U
dY 22 P eth65THORN
U frac14 0 at Y frac14 0 1 eth66THORN
using the dimensionless parameters
U frac14kulf
um Da frac14
K
1H 2frac14
1
12a2s
Y frac14y
H P frac14
K
1mfum
dkplf
dx
The solution to the momentum equation is then
Analysis andoptimization
19
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
U frac14 P cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 2 1
8gtltgt
9gt=gt eth67THORN
The volume-averaged energy equations for the fin and fluid respectively are
ksersaquo2kTlrsaquoy 2
frac14 haethkTl2 kTlfTHORN eth68THORN
1rfCpkulf
rsaquokTlf
rsaquoxfrac14 haethkTl2 kTlfTHORN thorn kfe
rsaquo2kTlf
rsaquoy 2eth69THORN
with boundary conditions
kTl frac14 kTlf frac14 Tw at y frac14 0 eth70THORN
rsaquokTlrsaquoy
frac14rsaquokTlf
rsaquoyfrac14 0 at y frac14 H c eth71THORN
where a is the wetted area per unit volume h the local heat transfer coefficient definedas the ratio of the interfacial heat flux to the solid-fluid temperature difference and kse
and kfe the effective conductivities of the solid and fluid defined as kse frac14eth1 2 1THORNks kfe frac14 1kf
For fully developed flow under constant heat flux it is known that
rsaquokTlf
rsaquoxfrac14
rsaquokTlrsaquox
frac14dTw
dxfrac14 constant eth72THORN
and
q00 frac14 1rfCpumHrsaquokTlf
rsaquoxeth73THORN
The energy equations (68) and (69) and boundary conditions can thus be written indimensionless form as
d2u
dY 2frac14 Dethu2 ufTHORN eth74THORN
U frac14 Dethu2 ufTHORN thorn Cd2uf
dY 2eth75THORN
with
u frac14 uf frac14 0 at Y frac14 0 eth76THORN
du
dYfrac14
duf
dYfrac14 0 at Y frac14 1 eth77THORN
in which
HFF151
20
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
C frac141kf
eth1 2 1THORNks D frac14
haH 2
eth1 2 1THORNks u frac14
kTl2 Tw
q00Heth121THORNks
uf frac14kTlf 2 Tw
q00Heth121THORNks
Substituting the solution obtained for velocity equations (74) and (75) can be solvedto give
uf frac14P
1 thorn C
242
1
2Y 2 thorn C1Y thorn C2 2 C3 cosh
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth1 thorn CTHORN
CY
r 1A
2 C4 sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
CY
r
thorn C5 cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt375
eth78THORN
u frac14 P
24Da cosh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r thorn
1 2 coshffiffiffiffi1
Da
q sinh
ffiffiffiffi1
Da
q sinh
ffiffiffiffiffiffiffiffiffiffiffi1
DaY
r 8gtltgt
9gt=gt
21
2Y 2 thorn C1Y 2 Da
352 Cuf
eth79THORN
where
N 1 frac14 Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Da1 2 cosh
ffiffiffiffiffiffi1
Da
r ( )vuut
N 2 frac14C
Da
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
rsinh
ffiffiffiffiffiffi1
Da
r sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1 thorn CTHORN
C
r
C1 frac14 1 2
ffiffiffiffiffiffiDa
pcosh
ffiffiffiffi1
Da
q 2 1
sinh
ffiffiffiffi1
Da
q
C2 frac14 2Da thorn1
Deth1 thorn CTHORN
C3 frac14C
DaDeth1 thorn CTHORND1
Analysis andoptimization
21
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
C4 frac14N 1 thorn N 2
Deth1 thorn CTHORN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
Ccosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDeth1thornCTHORN
CY
q sinh
ffiffiffiffi1
Da
q D1
s
C5 frac14 Da 21
D1
Finally the thermal resistance can be obtained as
R frac1412K
rfCpw3caW1 2um
2ufbH c
eth1 2 1THORNksLWeth80THORN
in which ufb is the bulk mean fluid temperature defined as
uf b frac14
Z 1
0
Uuf dY
13Z 1
0
U dy
Key features of the five approximate models discussed above including theassumptions governing equations and resistance formulae developed are summarizedin Table II
Assessment of the approximate modelsFor the microchannel parameters listed in Table I thermal resistances were computedwith Fluent as well as from the five approximate models The results are shown inTable III It can be seen that all the approximate models would provide acceptablepredictions for the thermal resistance of the microchannel heat sink with the maximumdeviation being 78 percent Models 2-5 are more complex to apply than model 1 andinvolve the solution of the differential governing equations In spite of its simplicity model1 appears to adequately represent the physics of the heat transfer problem and isrecommended for use in the design and optimization of practical microchannel heat sinks
It may be noted that in model 2 the fluid temperature is considered to be only afunction of the x-coordinate and the fin temperature is solved in a truly 1D mannerThe thermal resistance expression from model 2 is therefore identical to that frommodel 1 Also in models 3 and 4 since 2D temperature fields are considered in both thefin and fluid the new terms H c=eth3nkswwLTHORN and eth1=ethrfCpQTHORN2THORNethnksH cww=LTHORN appearin addition to the other terms in the simpler models 1 and 2 The difference betweenmodels 3 and 4 is that the axial conduction term appears explicitly in the fin equationof model 4 while it is neglected in model 3
In the calculations above expressions for Nusselt number Nu and the frictionconstant f Re are needed for computing the convective heat transfer coefficient h andthe average velocity um in the microchannel In all the five approximate modelsdiscussed above the flow is assumed to be thermally and hydrodynamically fullydeveloped Hence the following relations are used in terms of microchannel aspectratios (Incropera and DeWitt 1996 Shah and London 1978)
Nufd frac14 8235eth1 2 1883athorn 3767a 2 2 5814a 3 thorn 5361a 4 2 2a 5THORN eth81THORN
ethf ReTHORNfd frac14 96eth1213553=athorn19467=a 2 217012=a3 thorn09564=a 4 202537=a 5THORN eth82THORN
HFF151
22
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
Tem
per
atu
reA
xia
lco
nd
uct
ion
Mod
elF
inF
luid
Fin
Flu
idG
over
nin
geq
uat
ion
sT
her
mal
resi
stan
ce(R
)
11D
1Dpound
poundN
otu
sed
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfQ
Cp
(88)
22D
1Dpound
pound
d2T
dy
22
hP
k sA
cethT
2T
fethxTHORNTHORNfrac14
0
_ mC
pdT
fethxTHORN
dx
frac14q
middotethw
cthorn
wwTHORN
1
nhLeth2h
fHcthorn
wcTHORNthorn
1
rfC
pQ
(89)
32D
2Dpound
poundk srsaquo
2T
rsaquoy
2frac14
2h wwethT
2T
fethx
yTHORNTHORN
rfC
pu
mw
crsaquoT
f
rsaquox
frac142hethT
2T
fethx
yTHORNTHORN
Ja
k s~ H
cb
eth1thorn
L=aTHORNthorn
1 3b~ H
2 c
1
ethLW
THORN(9
0)
42D
2Dp
pound7
2TethxyzTHORNfrac14
0
7middotethk VT
fethx
yzTHORNTHORNfrac14
af7
2T
fethx
yzTHORN
2hfa
2hfa
thorn1
1
2nhH
ch
fLthorn
1
rfC
pQthorn
1
ethrfC
pQTHORN2
nk s
Hcw
w
L
(91)
51D
1Dpound
pound
2d dxkpl fthorn
mf
d2
dy
2kul f2
mf k1kul ffrac14
0
k sersaquo
2kT
lrsaquoy
2frac14
haethk
Tl2
kTl fTHORN
1r
fCpkul frsaquokT
l frsaquox
frac14haethk
Tl2
kTl fTHORNthorn
k fersaquo
2kT
l frsaquoy
2
12K
rfC
pw
3 ca
W1
2u
m2
uf
hH
c
eth12
1THORNk
sL
W(9
2)
Notespound
ndashn
otco
nsi
der
ed
andp
ndashco
nsi
der
ed
Table IISummary of approximate
analytical models
Analysis andoptimization
23
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
However the fully developed assumption is not always valid especially for microchannelswith the larger hydraulic diameters and short lengths With hydrodynamic and thermallengths defined as Lthorn frac14 L=ethDhReTHORN and L frac14 L=ethDhRePrTHORN the following relations(Samalam 1989 Harms et al 1999) could be employed instead of equations (81) and (82)
Nu frac14 335ethL THORN2013a 012Pr20038 0013 L 01 eth83THORN
Nu frac14 187ethL THORN2030a 0056Pr20036 00005 L 0013 eth84THORN
f appRe frac1432
ethLthornTHORN057
2
thornethf ReTHORN2fd
1=2
Lthorn 005 eth85THORN
In the present calculations L=Dh lt 100 with moderate Reynolds numbers so that thehydrodynamically fully developed condition is satisfied For the working fluid in thisstudy (water Prandtl number 58) Nusselt numbers calculated from equations (81) and(83) are listed in Table IV The deviation between the two sets of results is within 6 percentand therefore the assumption of thermally fully developed conditions is acceptableIn general developing thermal effects should be carefully considered before fullydeveloped conditions are assumed
OptimizationThe optimization of microchannel heat sink design can be motivated using the thermalresistance approach in model 1 As indicated in equation (18) Rcal is inverselyproportional to the mass flow rate When the pressure head along the microchannellength is prescribed as the constraint Rcal will decrease as wc increases when Hc
reaches the maximum allowable value However the convective heat transfercoefficient h will increase when Dh decreases leading to a reduction in Rconv asshown by equation (17) The heat transfer from the substrate through the fins willalso be enhanced if the fin efficiency increases which requires a larger fin
Case
Thermal resistance (8CW) 1 2 3
Ronum 0115 0114 0093Romodel 1 0112 0112 0091Romodel 2 0112 0112 0091Romodel 3 0106 0106 0087Romodel 4 0106 0106 0087Romodel 5 0115 0106 0089
Table IIIOverall thermalresistances
Case1 2 3
Nufd 597 581 606Nu 560 555 585
Table IVNusselt numbers
HFF151
24
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
thickness ww However the increase in ww will reduce the number of microchannelfinpairs in a heat sink for a prescribed heat sink size Due to these competing factors thereexists an optimal microchannel dimension that minimizes the overall thermalresistance
In order to optimize the thermal performance of a microchannel heat sink thefollowing variables must be specified from implementation constraints
(1) thermal conductivity of the bulk material used to construct the heat sink (ks)
(2) overall dimension of the heat sink (L and W from the size of the chip Hc and tfrom fabrication and structural considerations)
(3) properties of the coolant (rf m kf Cp) and
(4) allowable pressure head (DP)
To illustrate the procedure the example considered uses water as the working fluid tocool a chip with L frac14 W frac14 1 cm and a given pressure head of DP frac14 60 kPa The heatload is 100 Wcm2 The microchannel heat sink is to be made of silicon witht frac14 100mm and H c frac14 400mm The fluid properties are evaluated at 27 8C Theoptimization process involves finding the optimal microchannel geometry (channelwidth wc fin thickness ww and aspect ratio a frac14 H c=wc) that will minimize thermalresistance
Solutions to the following equations would yield the optimum
rsaquoR
rsaquowcfrac14 0 eth86THORN
rsaquoR
rsaquowwfrac14 0 eth87THORN
In this work the optimization computations were performed using the commercialsolver MATLAB (The Math Works Inc 2001) The optimized results derived from thefive approximate models are listed in Table V The optimal thermal resistance valuesreported from the five models agree to the within 10 percent It may also be noted thatthe minimum thermal resistance is always attained at the largest allowable aspectratio In practical designs the aspect ratio would be determined by the limits on themicrochannel depth and the substrate thickness
ConclusionsFive approximate analytical models for predicting the convective heat transfer inmicrochannel heat sinks are presented and compared Closed-form solutions from thesemodels are compared to full CFD simulation and experimental results and the efficacy
Model wc (mm) ww (mm) a Ro (8CW)
1 64 18 625 009652 65 19 615 009653 65 24 615 009734 61 16 656 009075 64 27 625 01072
Table VOptimal dimensions
Analysis andoptimization
25
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26
of the different models assessed Optimization procedures are discussed for minimizingthe thermal resistance of the heat sinks The results obtained demonstrate that themodels developed offer sufficiently accurate predictions for practical designs while atthe same time being quite straightforward to use
References
Fedorov AG and Viskanta R (2000) ldquoThree-dimensional conjugate heat transfer in themicrochannel heat sink for electronic packagingrdquo International Journal of Heat and MassTransfer Vol 43 pp 399-415
Fluent Inc (1998) Fluent Userrsquos Guide Lebanon New Hampshire
Harms TM Kazmierczak MJ and Gerner FM (1999) ldquoDeveloping convective heat transfer indeep rectangular microchannelsrdquo International Journal of Heat and Fluid Flow Vol 20pp 149-57
Incropera FP and DeWitt DP (1996) Fundamentals of Heat and Mass Transfer WileyNew York NY
Kim SJ and Kim D (1999) ldquoForced convection in microstructure for electronic equipmentcoolingrdquo Journal of Heat Transfer Vol 121 pp 639-45
The MathWorks Inc (2001) Matlab Version 61 Natwick MA
Ryu JH Choi DH and Kim SJ (2002) ldquoNumerical optimization of the thermal performance ofa microchannel heat sinkrdquo International Journal of Heat and Mass Transfer Vol 45pp 2823-7
Sabry M-N (2001) ldquoTransverse temperature gradient effect on fin efficiency for microchanneldesignrdquo Journal of Electronic Packaging Vol 123 pp 344-50
Samalam VK (1989) ldquoConvective heat transfer in microchannelsrdquo Journal of ElectronicMaterials Vol 18 pp 6111-7
Shah RK and London AL (1978) ldquoLaminar flow forced convection in ductsrdquo Advances in HeatTransfer Supplement 1 Academic Press London
Sobhan CB and Garimella SV (2001) ldquoA comparative analysis of studies on heat transfer andfluid flow in microchannelsrdquo Microscale Thermophysical Engineering Vol 5 pp 293-311
Toh KC Chen XY and Chai JC (2002) ldquoNumerical computation of fluid flow and heattransfer in microchannelsrdquo International Journal of Heat and Mass Transfer Vol 45pp 5133-41
Tuckerman DB and Pease RF (1981) ldquoHigh-performance heat sinking for VLSIrdquo IEEEElectronic Devices Letters EDL-2 pp 126-9
Vafai K and Tien CL (1981) ldquoBoundary and inertia effects on flow and heat transfer in porousmediardquo International Journal of Heat and Mass Transfer Vol 24 pp 195-203
Weisberg A and Bau HH (1992) ldquoAnalysis of microchannels for integrated coolingrdquoInternational Journal of Heat and Mass Transfer Vol 35 pp 2465-74
Zhao CY and Lu TJ (2002) ldquoAnalysis of microchannel heat sinks for electronics coolingrdquoInternational Journal of Heat and Mass Transfer Vol 45 pp 4857-69
Further reading
Tien CL and Kuo SM (1987) ldquoAnalysis of forced convection in microstructures for electronicsystem coolingrdquo Proc Int Symp Cooling Tech for Electronic Equipment Honolulu HIpp 217-26
HFF151
26