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International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
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International Journal of Heat and Mass Transfer
journal homepage: www.elsevier .com/locate / i jhmt
Analytical heat diffusion models for different micro-channel heat sinkcross-sectional geometries
Sung-Min Kim, Issam Mudawar *
Boiling and Two-Phase Flow Laboratory (BTPFL), Purdue University International Electronic Cooling Alliance (PUIECA), Mechanical Engineering Building,585 Purdue Mall West Lafayette, IN 47907-2088, USA
a r t i c l e i n f o
Article history:Received 20 November 2009Received in revised form 10 April 2010Accepted 10 April 2010Available online 10 June 2010
Keywords:Analytical heat diffusion modelMicro-channel heat sinkFlow boiling
0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.05.019
* Corresponding author. Tel.: +1 765 494 5705; faxE-mail address: mudawar@ecn.purdue.edu (I. Mud
a b s t r a c t
This study explores heat diffusion effects in micro-channel heat sinks intended for electronic cooling appli-cations. Detailed analytical models are constructed for heat sinks having micro-channels with rectangular,inverse trapezoidal, triangular, trapezoidal, and diamond-shaped cross sections. Solutions are presented forboth monolithic heat sinks and heat sinks with perfectly insulating cover plates. The analytical results arecompared to detailed two-dimensional numerical models of the same cross-sections over a broad range ofcover plate thermal conductivities for different micro-channel aspect ratios, fin spacings and Biot numbers.These comparisons show the analytical models provide accurate predictions for Biot numbers of practicalinterest. This study proves the analytical models are very effective tools for the design and thermal resis-tance prediction of micro-channel heat sinks found in electronic cooling applications.
� 2010 Elsevier Ltd. All rights reserved.
1. Introduction temperature, respectively. However, as direct temperature mea-
The past two decades have witnessed intense interest amongresearchers in the use of micro-channel heat sinks, which has beenspurred by such unique attributes as compactness, high power dissi-pation to volume ratio, and small coolant inventory. Because of therelative simplicity of fabricating rectangular micro-channels, the vastmajority of these studies involve rectangular cross-sections. How-ever, an increasing number of single-phase and two-phase heat trans-fer studies is focusing on sinks having non-rectangular cross-sectionssuch as triangular [1–3], trapezoidal [4,5], and diamond-shaped [6],citing thermal benefits to these cross-sections related to heat diffu-sion effects and bubble nucleation in sharp corners. Since the heatsink is typically designed to conduct heat away from a high-heat-fluxelectronic or power device that is attached directly to the heat sink,heat diffusion within the heat sink is responsible for an importantthermal resistance between the device’s surface and the coolant.Unfortunately, published work describing heat diffusion effects inheat sinks having non-rectangular micro-channels is quite sparse.
In case of a monolithic micro-channel heat sink, the local heattransfer coefficient at an axial location along the micro-channelcan be determined from the relation
h ¼ q00
Tw � Tf; ð1Þ
where q00, Tw, and Tf are the average heat flux along the micro-chan-nel wall, the average channel wall temperature, and the bulk fluid
ll rights reserved.
: +1 765 494 0539.awar).
surement of the micro-channel perimeter is not practical, the finanalysis method is often used in experimental studies (see [7,8])to evaluate the local heat transfer coefficient. For example, in caseof a heat sink with rectangular micro-channels and three-sidedheating (i.e., with a perfectly insulating top cover plate), applyingthe fin analysis method to the system illustrated in Fig. 1a yields [7]
h ¼q00eff ð2Hch þWchÞ
ðTw;b � Tf Þð2gHch þWchÞ¼ q00baseðWch þWsÞðTw;b � Tf Þð2gHch þWchÞ
; ð2Þ
where q00base is the device heat flux, and the fin efficiency and finparameter are defined as [9]
g ¼ tanhðmHchÞmHch
and m ¼
ffiffiffiffiffiffiffiffiffiffi2h
kWs
s; ð3Þ
respectively. In Eq. (2), Tw,b represents the temperature correspond-ing to the entire bottom plane of the micro-channels, which is as-sumed uniform in Refs. [7,8]. This temperature can be determinedfrom the temperature Tbase of the device by assuming one-dimen-sional heat conduction between the plane of the device and the bot-tom plane of the micro-channels.
Tw;b ¼ Tbase �q00baseHb
k: ð4Þ
The one-dimensional fin analysis method is a convenient andaccurate means to determining the heat transfer coefficient in amicro-channel heat sink provided the Biot number, based onhalf-width of the solid wall separating channels for the rectangularcross-section is sufficiently small [10,11]. As will be shown later in
Nomenclature
a coefficient in Eq. (19)Ac cross-sectional area of finAs surface area of fin exposed to convectionAR aspect ratio of micro-channel, Hch/Wch
b fin lengthBi biot number, hWs/2kC1, C2 coefficients in fin equationd fin lengthh heat transfer coefficientHb distance from bottom of heat sink to bottom of micro-
channelHc height of cover plateHch micro-channel heightIn Nth-order Bessel function of first kindk thermal conductivity of heat sink solidkc thermal conductivity of cover plateKn Nth-order Bessel function of second kindm fin parameterQ heat transfer rate per unit lengthq0 heat transfer rate per unit lengthq00 heat fluxq00base heat flux based on base area of heat sink; device heat
fluxq00eff heat flux based on heated perimeter of micro-channelT temperatureTbase temperature of heat sink base; device temperatureTw,b mean temperature of bottom plane of micro-channels
Tw,t mean temperature of top plane of micro-channelsWch micro-channel widthWs width of solid wall separating micro-channelsWs,e width of endwallx coordinate
Greek symbolsdb fin thickness at basen1, n2, n3, n4 parameters in temperature functionsg fin efficiencyh temperature difference in fin equation
Subscripts1 micro-channel left wall2 fin tip; tip of solid wall separating micro-channels2a micro-channel top wallA analyticalb micro-channel bottom wallbase heat sink basec cover platech micro-channelf bulk fluidfin fin base; base of solid wall separating micro-channelsN numericalt micro-channel top walltip adiabatic fin tip for diamond channelw micro-channel wall
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4003
this study, the one-dimensional approach is valid for most micro-channel experiments with rectangular micro-channels becauseBi << 0.1. However, it should be emphasized that Eq. (2) is validonly for three-sided heated rectangular micro-channels, i.e., whenthe cover plate is perfectly insulating.
The objective of this study is to derive one-dimensional analyt-ical solutions for heat diffusion in micro-channel heat sinks havingrectangular, inverse trapezoidal, triangular, trapezoidal, and dia-mond-shaped micro-channel cross-sections as illustrated inFig. 1. Unlike earlier published studies, the effect of the thermalconductivity of the cover plate will be taken into consideration.Excepting the diamond-shaped cross-section, two sets of exactsolutions are presented for each cross section: monolithic heatsinks (kc = k), and heat sinks with perfectly insulating cover plates(kc = 0). To validate the analytical solutions, a parametric study ofchannel aspect ratio, fin spacing and Biot number is performed,and the analytical predictions are compared to numerical solutionsof the two-dimensional heat diffusion equation for the differentcross-sectional geometries.
2. Longitudinal fin analyses
Fig. 2 illustrates the schematic diagrams for three basic longitu-dinal fins with rectangular, trapezoidal, and inverse trapezoidalprofiles that are used to derive the diffusion models for the heatsink micro-channel geometries illustrated in Fig. 1. It should beemphasized that, while certain solutions have been published forsome of these fin profiles under boundary conditions of interestto micro-channel heat sink geometries, solutions are not availablefor others and are therefore derived in detail in the present paper.In the following, solutions to all relevant boundary conditions willbe presented, with each solution obtained from prior sources care-fully indicated as such.
The following conventional assumptions are made in the one-dimensional fin analysis [12,13]:
1. The heat conduction in the fin is steady and one-dimensionalalong the x-direction.
2. The fin material is homogeneous and isotropic.3. The thermal conductivity of the fin is constant and uniform.4. The heat exchange between the fin and the surrounding fluid is
solely by convection along the fin surface.5. There is no heat generation within the fin itself.6. The heat transfer coefficient and the temperature of the sur-
rounding fluid are constant and uniform.
Applying energy conservation to the differential elementsshown in Fig. 2, the governing second-order ordinary differentialequation for one-dimensional steady-state temperature distribu-tion can be derived as follows [9]:
d2TðxÞdx2 þ 1
Ac
dAcðxÞdx
� �dTðxÞ
dx� 1
Ac
hk
dAsðxÞdx
� �ðTðxÞ � Tf Þ ¼ 0; ð5Þ
where Ac and As are the cross-sectional area and the surface area ofthe fin exposed to convection, respectively.
2.1. Rectangular fin
For the rectangular fin, Ac is constant and Eq. (5) can be writtenas
d2h
dx2 �m2h ¼ 0: ð6Þ
Eq. (6) has the solution
hðxÞ ¼ C1expðmxÞ þ C2expð�mxÞ: ð7Þ
Solid
Hch
Hb
Fluid
q”eff
q”base
Ws,e
Ws Wch
HcWs,e
Cover Plate
Solid
Hch
Hbq”eff
q”base
Ws Wch,t
Hc
Wch,b
Fluid
Cover Plate
Ws,e Ws,e
Solid
Hch
Hb
q”eff
q”base
Ws Wch
Hc
Fluid
Cover Plate
Ws,e Ws,e
Solid
Hch
Hbq”eff
q”base
Ws Wch,b
HcWch,t
Fluid
Cover Plate
Ws,e Ws,e
Solid Hbq”eff
q”base
Ws Wch
Hc
Fluid Hch
Ws,e Ws,e
a b
e
c d
Fig. 1. Schematic diagrams of micro-channel heat sinks with (a) rectangular, (b) inverse-trapezoidal, (c) triangular, (d) trapezoidal, and (e) diamond-shaped cross-sections.
b
fin
x
dx
qx qx+dx
b
fin
x
d
dx
qx+dx qx
b
fin
x
d
dx
qx qx+dx
a b c
Fig. 2. Schematic diagrams of (a) rectangular, (b) trapezoidal and (c) inverse trapezoidal fins.
4004 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4005
where
hðxÞ ¼ TðxÞ � Tf and m ¼ffiffiffiffiffiffiffiffiffi2h
kdfin
s: ð8Þ
The above equation is subjected to the following boundary condi-tions that are of interest to micro-channel heat-sink modeling.
2.1.1. Constant base temperature and adiabatic tip
hð0Þ ¼ hfin ð9Þ
and
dhdx
����x¼b
¼ 0: ð10Þ
The exact solution of Eq. (6) with boundary conditions (9) and(10) gives the following temperature distribution along the fin,
hðxÞhfin¼ coshðmðb� xÞÞ
coshðmbÞ ; ð11Þ
from which the heat transfer rate at the fin base can be obtained as
q0fin ¼ �kA0cdhdx
����x¼0¼ kdfinhfinm tanhðmbÞ: ð12Þ
2.1.2. Constant base temperature and prescribed tip temperature
hð0Þ ¼ hfin ð13Þ
and
hðbÞ ¼ h2: ð14ÞThe solution to Eq. (6) with boundary conditions (13) and (14) is gi-ven by
hðxÞhfin¼ ðh2=hfinÞ sinhðmxÞ þ sinhðmðb� xÞÞ
sinhðmbÞ ð15Þ
and the heat transfer rates at the fin base and at the fin tip are given,respectively, as
q0fin ¼ �kA0cdhdx
����x¼0¼ kdfinm
coshðmbÞhfin � h2
sinhðmbÞ ð16Þ
and
q02 ¼ �kA0cdhdx
����x¼b
¼ kdfinmhfin � h2 coshðmbÞ
sinhðmbÞ : ð17Þ
The same results of Eqs. (11), (12) and (15)–(17) are repre-sented in [14] as elements of linear transformations.
2.2. Trapezoidal fin
Similar to the solution procedure for a rectangular fin, the gov-erning Eq. (5) may be expressed as
d2h
dx2 þ1x
dhdx�m2a
hx¼ 0; ð18Þ
hðxÞ ¼K0 2m
ffiffiffiffiffiffiadp�
hfin � K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� h2
h iI0 2m
ffiffiffiffiffiaxp� �
þ I0 2mffia
p�hI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffia
p�
q0fin ¼ kdfinmffiffiffiffiffiffiffiffiffiffiffiffi
abþ d
r I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
þ K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p�hI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffia
p�
where
m ¼ffiffiffiffiffiffiffiffiffi2h
kdfin
sand a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2
fin
4þ ðbþ dÞ2
s: ð19Þ
The exact solution for the above second-order ordinary differentialequation can be written in the form of modified Bessel functions [14].
hðxÞ ¼ C1I0 2mffiffiffiffiffiaxp� �
þ C2K0 2mffiffiffiffiffiaxp� �
; ð20Þ
where I0 and K0 are modified, zero-order Bessel functions of the firstand second kinds, respectively. Eq. (20) is subjected to the followingboundary conditions.
2.2.1. Constant base temperature and adiabatic tip
hðbþ dÞ ¼ hfin; ð21Þdhdx
����x¼d
¼ 0: ð22Þ
The exact solution of Eq. (18) with boundary conditions (21)and (22) gives the following temperature distribution along the fin.
hðxÞhfin¼
I0 2mffiffiffiffiffiaxp� �
K1 2mffiffiffiffiffiffiadp�
þK0 2mffiffiffiffiffiaxp� �
I1 2mffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� K1 2m
ffiffiffiffiffiffiadp�
þK0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� I1 2m
ffiffiffiffiffiffiadp� ; ð23Þ
where In and Kn are modified, nth-order Bessel functions of the firstand second kinds, respectively. The heat transfer rate at the fin baseis given by
q0fin ¼ kdfinhfinmffiffiffiffiffiffiffiffiffiffiffi
abþd
r
�I1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� K1 2m
ffiffiffiffiffiffiadp�
�K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� I1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� K1 2m
ffiffiffiffiffiffiadp�
þK0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþdÞ
p� I1 2m
ffiffiffiffiffiffiadp� :
ð24ÞIt should be noted that similar expressions to Eqs. (23) and (24)
are found in Bejan and Kraus [15], in which dAs/dx in Eq. (5) is as-sumed equal to 2L, where the fin thickness is small compared to itsheight. Although it is mentioned that this assumption is valid formost practical thin fins [13], the following general expression from[14,16] is used for more accurate results.
dAs
dx¼ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dy
dx
� �2s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2
fin þ 4ðbþ dÞ2q
bþ d: ð25Þ
2.2.2. Constant base temperature and prescribed tip temperature
hðbþ dÞ ¼ hfin ð26Þand
hðdÞ ¼ h2: ð27Þ
The temperature distribution and the heat transfer rates at thefin base and the fin tip with boundary conditions (26) and (27) aregiven, respectively, as follows:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ dÞ
h2 � I0 2m
ffiffiffiffiffiffiadp�
hfin
iK0 2m
ffiffiffiffiffiaxp� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ dÞ
I0 2m
ffiffiffiffiffiffiadp� ; ð28Þ
I0 2m
ffiffiffiffiffiffiadp� i
hfin � h2
2mffiffiffiffiffiffiffiffiffiffiaðbþdÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðbþ dÞ
I0 2mffiffiffiffiffiffiadp� ð29Þ
4006 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
and
q02 ¼ kdfinm
ffiffiffiffiffiffiadp
bþ d
hfin
2mffiffiffiffiadp � I1 2m
ffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� h ih2
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� : ð30Þ
The same results of Eqs. (23), (24) and (28)–(30) are repre-sented in [14] as elements of linear transformations.
2.3. Inverse trapezoidal fin
The governing Eq. (5) can be written as follows:
d2h
dx2 þ1x
dhdx�m2a
hx¼ 0; ð31Þ
where
m ¼ffiffiffiffiffiffiffiffiffi2h
kdfin
sand a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2
fin
4þ d2
s: ð32Þ
The general solution to Eq. (31) is given by [14]
hðxÞ ¼ C1I0 2mffiffiffiffiffiaxp� �
þ C2K0 2mffiffiffiffiffiaxp� �
: ð33Þ
The above equation is subjected to the following boundaryconditions.
2.3.1. Constant base temperature and adiabatic tip
hðdÞ ¼ hfin ð34Þ
and
dhdx
����x¼bþd
¼ 0 ð35Þ
hðxÞ ¼K0 2m
ffiffiffiffiffiffiadp�
h2 � K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� hfin
h iI0 2m
ffiffiffiffiffiaxp� �
þ I0 2mffia
p�hI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffia
p�
q0fin ¼ kdfinm
ffiffiffiad
r I1 2mffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffia
p�hI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ d
p�
q02 ¼ kdfinm
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
pd
hfin
2mffiffiffiffiffiffiffiffiffiffiaðbþdÞp � I1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
þ K1ðh
I0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
pÞK0 2m
ffiffiffiffiffiffiadp�
� K0 2mp�
The temperature distribution, and the heat transfer rate at the finbase with boundary conditions (34) and (35) are given, respectively,by
hðxÞhfin¼
I0 2mffiffiffiffiffiaxp� �
K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� þ K0 2m
ffiffiffiffiffiaxp� �
I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� I0 2m
ffiffiffiffiffiffiadp�
K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� þ K0 2m
ffiffiffiffiffiffiadp�
I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� ð36Þ
and
q0fin ¼ kdfinhfinm
ffiffiffiad
r
�I1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� K1 2m
ffiffiffiffiffiffiadp�
� K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� I1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiadp�
K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� þ K0 2m
ffiffiffiffiffiffiadp�
I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
p� :ð37Þ
2.3.2. Constant base temperature and prescribed tip temperature
hðdÞ ¼ hfin ð38Þ
and
hðbþ dÞ ¼ h2: ð39Þ
The temperature distribution, and the heat transfer rates at thefin base and the fin tip with boundary conditions (38) and (39) aregiven, respectively, as
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ dÞ
hfin � I0 2m
ffiffiffiffiffiffiadp�
h2
iK0 2m
ffiffiffiffiffiaxp� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ dÞ
I0 2m
ffiffiffiffiffiffiadp� ; ð40Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ dÞ
ihfin � h2
2mffiffiffiffiadpffiffiffi
Þ
I0 2mffiffiffiffiffiffiadp� ð41Þ
and
2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
pÞI0 2m
ffiffiffiffiffiffiadp� i
h2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ dÞ
I0 2m
ffiffiffiffiffiffiadp� : ð42Þ
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4007
3. Micro-channel heat sinks
To obtain exact solutions for heat diffusion in the micro-channelheat sinks, the following additional assumptions are made basedon the unit cells shown in Fig. 3 (see also Fig. 1):
1. The top surface of the cover plate is perfectly insulated.2. In case of rectangular, inverse trapezoidal, and trapezoidal
channels, the temperature of the fin base is the same as thatof the micro-channel bottom wall, i.e., Tfin = Tb.
3. In case of kc = k (monolithic heat sink), the temperature of thefin tip is the same as that of the micro-channel top wall, i.e.,T2 = T2a.
4. The effective heat flux, q00eff , is averaged over the heated perim-eter of the micro-channel. The heated perimeter consists ofthe entire wetted perimeter of the micro-channel, or, for thecase of a perfectly insulating cover plate, the micro-channelperimeter minus the portion in contact with the cover plate.
5. The endwall width of the micro-channel heat sink is equal tothe solid sidewall’s half-width, i.e., Ws,e = 0.5Ws.
3.1. Rectangular cross-section
The heat transfer rate at fin base can be obtained from the fol-lowing energy balance for the micro-channel unit cell shown inFig. 3a,
Q base ¼ Qb þ Qfin ¼ Qb þ Q1 þ Q2 ½W=m�: ð43Þ
The heat transfer rate at the micro-channel bottom wall is givenby
Q b ¼ hWchðTw;b � Tf Þ: ð44Þ
Introducing Eqs. (16) and (17) for the rectangular fin, the heat trans-fer rates at the fin base and the fin tip for the micro-channel unitcell can be written, respectively, as
Wch/2
Qb/2
Ws/2
Fluid
Cover Plate
Q1/2
Q2/2
Q2a/2
Solid
Q2=Q2a
Qfin=Q1+Q2
Qbase/2
Qfin/2
Qbase=Qfin+Qb
Hc
Hch
Hb
T2a
Tb
T1
Tfin
T2
Tbase
Wch,b/2
Qb/2
Cover Plate
Q1/2
Q2/2
Q2a/2
Solid
Fluid
Wch,t/2Ws/2
Qfin/2
Q2=Q2a
Qfin=Q1+Q2
Qbase=Qfin+Qb
Qbase/2
Hc
Hch
Hb
T2a
Tb
T1
Tfin
T2
Tbase
Cover Pla
Q1/2
Q2/2
Q2a/2
Solid
Flu
Wch/2Ws/2
Qfin/2
Q2=Q2a
Qfin=Q1+Q
Qbase=Qfin
Qbase/2
T
Tfin
T2
Tbase
Cover Pla
Q1/2
Q2/2
Q2a/2
Solid
Flu
Wch/2Ws/2
Qfin/2
Q2=Q2a
Qfin=Q1+Q
Qbase=Qfin
Qbase/2
T
Tfin
T2
Tbase
a b c
Fig. 3. Heat sink unit cells with: (a) rectangular, (b) inverse trapezoidal, (c) triang
Qfin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hkWs
p ðTw;b � Tf Þ coshðmHchÞ � ðTw;t � Tf ÞsinhðmHchÞ
ð45Þ
and
Q2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hkWs
p ðTw;b � Tf Þ � ðTw;t � Tf Þ coshðmHchÞsinhðmHchÞ
; ð46Þ
where
m ¼
ffiffiffiffiffiffiffiffiffiffi2h
kWs
s: ð47Þ
By neglecting heat loss from the top of the cover plate, applyingan energy balance to the cover plate gives
Q2a ¼ hWchðTw;t � Tf Þ ¼ Q 2: ð48Þ
This procedure yields the following expression for tip temperature,
Tw;t ¼ Tf þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hkWs
pðTw;b � Tf Þ
hWch sinhðmHchÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hkWs
pcoshðmHchÞ
: ð49Þ
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1a yields
q00eff ð2Hch þ 2WchÞ ¼ q00baseðWs þWchÞ ¼ Q fin þ Qb: ð50Þ
It may be noted that the heat influx to the micro-channel can beexpressed in terms of the effective heat flux or the base heat flux,as originally defined by Qu and Mudawar [7]. In the present study,the effective heat flux, q00eff , and the base heat flux, q00base, are definedas the mean heat flux based on the channel perimeter subjected toheating, and the heat flux based on heat sink base area, respec-tively, as illustrated in Fig. 3. In case of a perfectly insulating coverplate (kc = 0), q00eff has a zero value at the surface adjoining the coverplate, and is uniform over the other surfaces. For example, in thecase of the rectangular micro-channel with insulating cover plate,q00eff is assigned a value that is averaged over the three heatingwalls. If the base heat flux, q00base, and the base temperature, Tw,b,
te
id
2
Hc
Hch
Hb
2a
T1
te
id
2
Hc
Hch
Hb
2a
T1
Wch,b/2
Qb/2
Ws/2
Cover Plate
Q1/2
Q2/2
Q2a/2
Solid
Fluid
Wch,t/2
Qfin/2
Q2=Q2a
Qfin=Q1+Q2
Qbase=Qfin+Qb
Qbase/2
Hc
Hch
Hb
T2a
Tb
T1
Tfin
T2
Tbase
Hc
Hch
Hb
Q1/2
Q2/2
Q2a/2
Solid
Wch/2Ws/2
Fluid
Qfin/2
Q2=Q2a
Qfin=Q1+Q2
Qbase=Qfin
Adiabatic fin tip
Qbase/2
T2a
T1
Tfin
T2
Tbase
Ttip
d e
ular, (d) trapezoidal, and (e) diamond-shaped micro-channel cross-sections.
4008 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
are available from the actual device or measured from experiment,the mean heat transfer coefficient of the working fluid can be cal-culated from Eqs. (4), (44), (45), (49) and (50).
3.1.1. Insulating cover plateIn case the cover plate has a very small thermal conductivity
compared to that of the heat sink solid, the cover plate will behaveas a perfect insulator, i.e., Q2 ¼ Q 2a � 0. Criteria where thisassumption is valid will be discussed in more detail in the next sec-tion. For an insulating cover plate,
Q base ¼ Qb þ Qfin ¼ Qb þ Q1 ½W=m�: ð51Þ
Using Eq. (12) for a rectangular fin, the heat transfer rate at the finbase of the micro-channel unit cell can be written as
Q fin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hkWs
pðTw;b � Tf Þ tanhðmHchÞ: ð52Þ
Applying the energy balance to the micro-channel heat sink shownin Fig. 1a yields
q00eff ð2Hch þWchÞ ¼ q00baseðWs þWchÞ ¼ Q fin þ Q b: ð53Þ
3.2. Inverse trapezoidal cross-section
Similar to the solution procedure for the rectangular micro-channel unit cell, an energy balance for the inverse trapezoidalcross-section shown in Fig. 3b yields
Q base ¼ Qb þ Qfin ¼ Qb þ Q1 þ Q2 ½W=m�; ð54Þ
where
Q b ¼ hWch;bðTw;b � Tf Þ: ð55Þ
Using Eqs. (29) and (30) for a trapezoidal fin, the heat transferrates at the fin base and the fin tip of the unit cell can be written,respectively, as
Q fin ¼ kðWs þWch;t �Wch;bÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aHch þ d
r I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
þ K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� h i
ðTw;b � Tf Þ �ðTw;t�Tf Þ
2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞp
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp�
ð56Þ
and
Q 2 ¼ kWsm
ffiffiffiad
r ðTw;b�Tf Þ2mffiffiffiffiadp � I1 2m
ffiffiffiffiffiffiadp�
K0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
pÞ þ K1 2m
ffiffiffiffiffiffiadp�
I0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
pÞ
h iðTw;t � Tf Þ
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� ; ð57Þ
where
m ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h
kðWs þWch;t �Wch;bÞ
s;d ¼ WsHch
Wch;t �Wch;band
a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14ðWs þWch;t �Wch;bÞ2 þ Hch þ
WsHch
Wch;t �Wch;b
� �s 2
: ð58Þ
An energy balance for the cover plate yields
Q2a ¼ hWch;tðTw;t � Tf Þ ¼ Q 2: ð59Þ
This procedure yields the following expression for fin tiptemperature,
Tw;t ¼ Tf þðTw;b � Tf Þ
2mffiffiffiffiffiffiadp 1
n1; ð60Þ
where
n1 ¼ I1 2mffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �þ K1ð2m
ffiffiffiffiffiffiadpÞI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �þ hWch;t
kWsm
ffiffiffida
rI0ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
qÞK0 2m
ffiffiffiffiffiffiadp�
�K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �I0 2m
ffiffiffiffiffiffiadp� �
: ð61Þ
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1 yields
q00eff Wch;t þWch;b þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þ ðWch;t �Wch;bÞ2q �
¼ q00baseðWs þWch;tÞ ¼ Q fin þ Q b: ð62Þ
3.2.1. Insulating cover plateFor an insulating cover plate, the following energy balance is
applied,
Qbase ¼ Q b þ Q fin ¼ Q b þ Q 1 ½W=m�: ð63Þ
Using Eq. (24) for a trapezoidal fin, the heat transfer rate at thefin base of the micro-channel unit cell can be written as
Qfin ¼ kðWsþWch;t�Wch;bÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aHchþd
r
�I1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� K1 2m
ffiffiffiffiffiffiadp�
�K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� I1 2m
ffiffiffiffiffiffiadp�
I0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
pÞK1 2m
ffiffiffiffiffiffiadp�
þK0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� I1 2m
ffiffiffiffiffiffiadp�
�ðTw;b�Tf Þ:ð64Þ
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4009
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1b yields
q00eff ½Wch;b þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þ ðWch;t �Wch;bÞ2q
� ¼ q00baseðWs þWch;tÞ ¼ Q fin þ Q b:
ð65Þ
3.3. Triangular cross-section
As shown n Fig. 3c, energy conservation for a triangular cross-section can be expressed as
Q base ¼ Qfin ¼ Q1 þ Q2 ½W=m�: ð66Þ
Using Eqs. (29) and (30) for the trapezoidal fin, the heat transferrates at fin base and fin tip of triangular micro-channel unit cellshown in Fig. 3c can be written, respectively, as
Q fin ¼ kðWs þWchÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aHch þ d
r I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
þ K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� h i
ðTw;b � Tf Þ �ðTw;t�Tf Þ
2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞp
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
pÞI0 2m
ffiffiffiffiffiffiadp� ð67Þ
and
Q 2 ¼ kWsm
ffiffiffiad
r ðTw;b�Tf Þ2mffiffiffiffiadp � I1 2m
ffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ d
pÞ
� h iðTw;t � Tf Þ
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� ; ð68Þ
where
m ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h
kðWs þWchÞ
s; d ¼WsHch
Wchand
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14ðWs þWchÞ2 þ Hch þ
WsHch
Wch:
� �2s
: ð69Þ
Q fin ¼ kðWs þWchÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aHch þ d
r I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K1 2m
ffiffiffiffiffiffiadp�
� K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K1 2m
ffiffiffiffiffiffiadp�
þ K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I1 2m
ffiffiffiffiffiffiadp�
ðTw;b � Tf Þ: ð75Þ
Applying an energy balance to the cover plate gives
Q 2a ¼ hWchðTw;t � Tf Þ ¼ Q 2: ð70Þ
These relations yield the following expression for the fin tiptemperature,
Tw;t ¼ Tf þðTw;b � Tf Þ
2mffiffiffiffiffiffiadp 1
n2; ð71Þ
where
n2 ¼ I1 2mffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �þ hWch
kWsm
ffiffiffida
rI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �K0 2m
ffiffiffiffiffiffiadp�
�K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �I0 2m
ffiffiffiffiffiffiadp� �
: ð72Þ
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1c yields
q00eff Wch þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þW2ch
q �¼ q00baseðWs þWchÞ ¼ Q fin: ð73Þ
3.3.1. Insulating cover plateFor an insulating cover plate,
Qbase ¼ Q fin ¼ Q 1 ½W=m�: ð74Þ
Using Eq. (24) for a trapezoidal fin, the heat transfer rates at thefin base of the unit cell can be written as
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1c yields
q00eff
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þW2ch
q¼ q00baseðWs þWchÞ ¼ Q fin: ð76Þ
3.4. Trapezoidal cross-section
Following a solution procedure similar to that employed withthe inverse trapezoidal cross-section, energy conservation for thetrapezoidal micro-channel unit cell shown in Fig. 3d yields
q00eff Wch;t þWch;b þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þ ðWch;b �Wch;tÞ2q �
¼ q00baseðWs þWch;bÞ ¼ Q fin þ Q b ¼ kWsmffiffiffiad
r I1 2mffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� þ K1 2m
ffiffiffiffiffiffiadp�
I0ð2mffiffiffiapðHch þ dÞÞ
h iðTw;b � Tf Þ �
ðTw;t�Tf Þ2mffiffiffiffiadp
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� þ hWch;bðTw;b � Tf Þ; ð77Þ
4010 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
where
m ¼
ffiffiffiffiffiffiffiffiffiffi2h
kWs
s; d ¼ WsHch
Wch;b �Wch;tand
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2
s
4þ WsHch
Wch;b �Wch;t
� �2s
: ð78Þ
This procedure yields the following expression for the fin tiptemperature,
Tw;t ¼ Tf þðTw;b � Tf Þ
2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
p 1n3; ð79Þ
Table 1Dimensions of micro-channels tested in the study of aspect ratio and fin spacing (total 30
Rectangular Inverse Trapezoidal Tr
Hch 715 (AR = Hch/Wch = 4.3), 168 (AR = Hch/Wch = 1.0)Ws 42 (Ws/Wch = 0.25, Bi = 0.0027)Wch Wch,t 168 16
Wch,b 62
Hch 528 (AR = Hch/Wch = 4.3), 124 (AR = Hch/Wch = 1.0)Ws 86 (Ws/Wch = 0.69, Bi = 0.0055)Wch Wch,t 124 12
Wch,b 46
Hch 341 (AR = Hch/Wch = 4.3), 80 (AR = Hch/Wch = 1.0)Ws 130 (Ws/Wch = 1.63, Bi = 0.0084)Wch Wch,t 80 80
Wch,b 30
Fig. 4. Sample computational domain (shown turned 90�) of monolithic heat sink withcross-sections for all five cross-sections with same channel height.
where
n3 ¼ I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �K0 2m
ffiffiffiffiffiffiadp�
þ K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �I0 2m
ffiffiffiffiffiffiadp�
þ hWch;t
kðWs þWch;b �Wch;tÞm
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHch þ d
a
rI0 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞ
q� �K0 2m
ffiffiffiffiffiffiadp�
�K0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch þ dÞÞ
qI0 2m
ffiffiffiffiffiffiadp� �
: ð80Þ
cases, all dimensions are in lm).
iangular Trapezoidal Diamond
8 168 62 168168
4 124 46 124124
80 30 8080
rectangular cross-section and Hch = 124 lm, with magnification of micro-channel
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4011
3.4.1. Insulating cover plateUsing Eq. (37) for an inverse trapezoidal fin, the heat transfer
rate at the fin base of the unit cell can be written as
Q fin ¼ kðWs þWchÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aHch=2þ d
r I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
þ K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� h i
ðTw;b � Tf Þ �ðT2�Tf Þ
2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þdÞp
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� K0 2m
ffiffiffiffiffiffiadp�
� K0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞÞI0 2m
ffiffiffiffiffiffiadp� r
ð83Þ
Q 2 ¼ kWsm
ffiffiffiad
r ðTw;b�Tf Þ2mffiffiffiffiadp � I1 2m
ffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� h iðT2 � Tf Þ
I0ð2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
pÞK0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
p� I0 2m
ffiffiffiffiffiffiadp� ; ð84Þ
Qfin ¼ kWsm
ffiffiffiad
r
�I1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþd
pÞ
� K1 2m
ffiffiffiffiffiffiadp�
�K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� I1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiadp�
K1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� þK0 2m
ffiffiffiffiffiffiadp�
I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHchþdÞ
p� �ðTw;b�Tf Þ:
ð81Þ
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1d yields
bQ 2 ¼ kWsðT2 � Tf Þm
ffiffiffia
d
sI1ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
qÞK1ð2m
ffiffiffiffiffiffiad
pÞ � K1ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
qÞI1ð2m
ffiffiffiffiffiffiad
pÞ
I0ð2mffiffiffiffiffiffiad
pÞK1ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
qÞ þ K0ð2m
ffiffiffiffiffiffiad
pÞI1ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
qÞ; ð86Þ
q00eff Wch;b þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þ ðWch;b �Wch;tÞ2q �
¼ q00baseðWs þWch;bÞ
¼ Q fin þ Q b: ð82Þ
3.5. Diamond-shaped cross-section
As shown in Fig. 3e, two separated trapezoidal regions may beconsidered for the analysis of a diamond-shaped micro-channel.For the lower trapezoidal region, the equations for the trapezoidalfin with boundary conditions of constant base temperature andprescribed tip temperature are applied. For the upper trapezoidalregion, the equations for the inverse trapezoidal fin with boundaryconditions of constant base temperature and adiabatic tip areapplied, where Q2 is used as the heat transfer rate at the fin base.
For the lower trapezoidal region, the heat transfer rate at fin baseand fin tip are given, respectively, as
and
where
m ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h
kðWs þWchÞ
s; d ¼WsHch
2Wchand
a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14ðWs þWchÞ2 þ
Hch
2þWsHch
2Wch
� �2s
: ð85Þ
For the upper trapezoidal region, the heat transfer rate at the finbase is given by
where
m ¼
ffiffiffiffiffiffiffiffiffiffi2h
kWs
s; d ¼WsHch
2Wchand a ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2
s
4þ WsHch
2Wch
� �2s
: ð87Þ
An energy balance at the interface between the upper and lowertrapezoidal regions gives
Q2 ¼ bQ 2: ð88Þ
This procedure yields the following expression for temperatureat the interface between the two trapezoidal regions,
T2 ¼ Tf þðTw;b � Tf Þ
2mffiffiffiffiffiffiadp 1
n4; ð89Þ
T [K
]a b
Q [
W/m
]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
Qfin
Qb
Q2 &Q2a
Q1
Qbase
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
Tfin
Tb
T2 T2a
Tbase
Hch = 715 m, AR = 4.3, Ws/Wch = 0.25Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Hch = 715 m, AR = 4.3, Ws/Wch = 0.25Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Rectangular
Rectangular
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
Tfin Tb
T2
T2a
Tbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb Q2
&Q2a
Q1
Qbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Rectangular
Rectangular
c d
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
Tfin
Tb
T2
T2a
Tbase
Hch = 341 m, AR = 4.3, Ws/Wch = 1.63Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb Q2 &Q2a
Q1
Qbase
Hch = 341 m, AR = 4.3, Ws/Wch = 1.63Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Rectangular
Rectangular
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
Tfin
Tb
T2
T2a
Tbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb Q2 &Q2a
Q1
Qbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Rectangular
Rectangular
Fig. 5. Effects of thermal conductivity of cover plate for rectangular channel with (a) Hch = 715 lm, (b) Hch = 528 lm, (c) Hch = 341 lm and (d) Hch = 124 lm.
4012 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370a b
Tfin
Tb
T2
T2a
Tbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb
Q2 &Q2a
Q1
Qbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Inverse Trapezoidal
Inverse Trapezoidal
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
Tfin
Tb
T2
T2a
Tbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb
Q2 &Q2a
Q1
Qbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Inverse Trapezoidal
Inverse Trapezoidal
Fig. 6. Effects of thermal conductivity of cover plate for inverse trapezoidal channel with (a) Hch = 528 lm and (b) Hch = 124 lm.
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4013
where
n4 ¼ I1 2mffiffiffiffiffiffiadp�
K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �þ K1 2m
ffiffiffiffiffiffiadp�
I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �
þ mm
ffiffiffiffiffiffiad
ad
s I1 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �K1 2m
ffiffiffiffiffiffiad
p� � K1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �I1 2m
ffiffiffiffiffiffiad
p� I0ð2m
ffiffiffiffiffiffiffiffiadÞ
qK1ð2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
qÞ þ K0ð2m
ffiffiffiffiffiffiad
pÞI1 2m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia Hch=2þ d� r� �
� I0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �K0 2m
ffiffiffiffiffiffiadp�
� K0 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðHch=2þ dÞ
q� �I0 2m
ffiffiffiffiffiffiadp� �
: ð90Þ
Applying an energy balance to the micro-channel heat sinkshown in Fig. 1e yields
q00eff
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4H2
ch þ 4W2ch
q¼ q00baseðWs þWchÞ ¼ Q fin: ð91Þ
4. Results
Table 1 shows the dimensions examined in the present study forthe five micro-channel geometries. These dimensions are based onvalues from a parametric experimental study by Lee and Mudawar[17] involving rectangular micro-channels. Table 1 includes a total
of 30 different cases that include channel aspect ratios of AR = Hch/Wch = 4.3 and 1.0 and dimensionless fin spacings of Ws/Wch = 0.25,
0.69, and 1.63. For validation of the analytical results, FLUENT 6.3[18] was used to numerically solve the two-dimensional heat diffu-sion equation for each of the 30 cases. All simulations were con-ducted with copper as conducting solid using k = 387.6 W/m K,q00base ¼ 300 W=cm2, h = 50,000 W/m2 K, Tf = 300 K, Hb = 2000 lm,and Hc = 300 lm [17]. A convergence criterion of 10�11 was appliedto the residuals of the energy equation. The two-dimensionalmeshes were created using GAMBIT 2.2 software [19]. Fig. 4 showssample computational domains of the monolithic heat sink forHch = 124 lm. Grid independence was examined using two gridsystems with 29,000 and 86,000 cells for rectangular channel withHch = 124 lm, which showed nearly identical temperature
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370a b
Tfin T2
T2a
Tbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
&Qfin
Q2 &Q2a
Q1
Qbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Triangular
Triangular
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
Tfin
T2
T2a
Tbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
&Qfin
Q2 &Q2a
Q1
Qbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Triangular
Triangular
Fig. 7. Effects of thermal conductivity of cover plate for triangular channel with (a) Hch = 528 lm and (b) Hch = 124 lm.
4014 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
distributions with differences in the area-averaged temperature andheat transfer rate at the fin base of less than 0.01%. For calculationefficiency, the grid system with 29,000 cells was used for all the sim-ulations. For non-rectangular channels, meshes were generatedwith the same grid aspect ratio of the rectangular channel.
4.1. Effects of micro-channel aspect ratio and spacing
In order to examine the effects of thermal conductivity of coverplate, as well as various geometries, the area-averaged tempera-tures and heat transfer rates for Hch = 528, 124, 715, and 341 lmwere obtained from numerical simulations as shown in Figs. 5–8.The same input values of q”base = 300 W/cm2, h = 50,000 W/m2 K,and Tf = 300 K were used for all numerical simulations, and thetemperatures and heat transfer rates represented are area-aver-aged values across the corresponding planes indicated in Fig. 3.In the case of kc = k = 387.6 (monolithic heat sink), Figs. 5–8 showthe rectangular micro-channel produces lower T2 and Tfinvaluescompared to the other channel geometries. Conversely, thediamond channel produces higher T2 and Ttipvalues, as indicatedin Table 2. Moreover, comparing Figs. 5b and d, 6a and b, 7a andb and 8a and b, show T2 and Tfin values are much lower for highAR channels than those for lower AR. For kc = 387.6 and AR = 4.3,Fig. 5a–c show channels with the smallest fin spacing of Ws/Wch = 0.25 produce much lower T2 and T2a values compared tothose with larger spacings, Ws/Wch = 0.69 and 1.63. Although rela-tively limited in parametric range, the present two-dimensionalsimulations suggest a rectangular micro-channel with a high as-pect ratio and small fin spacing provides the best overall thermal
performance compared to the other cross-sectional geometriesbased on the assumptions provided in Sections 2 and 3.
4.2. Effects of thermal conductivity of cover plate
For all micro-channel geometries, the two-dimensional simula-tions yield Tfin values that are nearly identical to those of Tb, whichvalidates the second assumption adopted in the present analyticalheat sink models. Figs. 5–8 show that T2a approaches T2 withincreasing thermal conductivity of the cover plate, which validatesthe third assumption of the present analytical heat sink models forfairly to highly conducting cover plates. On the other hand, in caseof a cover plate with very low thermal conductivity, Q2 (which isequal to Q2a) becomes negligible and Q1 � Qfin; i.e., the cover platebehaves as perfectly insulating.
4.3. Effects of biot number
Fig. 9 shows the percent differences in heat transfer rate at thefin base, Qfin, between the two-dimensional numerical results andpresent analytical solutions for AR = 4.3 and 1.0, Ws/Wch = 0.69,and Bi ranging from 0.0055 to 0.2. Fig. 9 shows that for Bi < 0.2,the rectangular and trapezoidal channels for both aspect ratiosyield higher differences between numerical and analytical resultscompared to the other channel geometries. It should be noted thatthe temperature difference between the fin base and the micro-channel bottom wall increases with increasing Biot number. There-fore, the second assumption adopted in the present micro-channelanalytical heat sink models is not valid for high Bi values. Fig. 9
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370a b
Tfin
Tb
T2
T2a
Tbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb
Q2 &Q2a
Q1
Qbase
Hch = 528 m, AR = 4.3, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Trapezoidal
Trapezoidal
T [K
]Q
[W
/m]
290
700
kc [W/m.K]
kc [W/m.K]
0.01 0.1 1 10 100 1000
300
310
320
330
340
350
360
370
0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
Tfin
Tb
T2
T2a
Tbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Qfin
Qb
Q2 &Q2a
Q1
Qbase
Hch = 124 m, AR = 1.0, Ws/Wch = 0.69Hb = 2000 m, Hc = 300 m
q"base = 300 W/cm2, Tf = 300 Kh = 50,000 W/m2, k = 387.6 W/m.K
Trapezoidal
Trapezoidal
Fig. 8. Effects of thermal conductivity of cover plate for trapezoidal channel with (a) Hch = 528 lm and (b) Hch = 124 lm.
Table 2Comparison of 2D numerical (area-averaged) and 1D analytical results for AR = 4.3, Ws/Wch = 0.69 and kc = k.
Units Rectangular Inverse trapezoidal Triangular Trapezoidal Diamond-shaped
Num. Anal. %Diff. Num. Anal. %Diff. Num. Anal. %Diff. Num. Anal. %Diff. Num. Anal. %Diff.
Tbase K 328.10 – 327.77 – 327.90 – 328.12 – 328.99 –Tfin K 312.40 312.62 0.07 312.26 312.29 0.01 312.42 312.42 0.00 312.42 312.64 0.07 313.52 313.51 0.00Tb K 312.78 �0.05 312.41 �0.04 – 312.80 �0.05 –T2 K 307.95 308.09 0.05 309.05 309.06 0.00 309.60 309.57 �0.01 309.12 309.19 0.02 311.40 311.39 0.00T2a, orTtip for
diamondK 307.89 0.06 308.98 0.03 309.51 0.02 308.98 0.07 310.55 310.57 0.01
Qb W/m 79.28 78.24 �1.31 28.54 28.27 �0.95 – 79.38 78.37 �1.27 –Q2 W/m 48.88 50.14 2.58 55.66 56.17 0.92 58.90 59.33 0.73 20.56 21.13 2.77 291.40 292.97 0.54Qfin W/m 550.62 561.63 2.00 601.36 603.76 0.40 630.00 630.76 0.12 550.48 562.59 2.20 630.00 631.12 0.18
%Diff. = (TA � TN)/TN � 100 [%] or (QA � QN)/QN � 100 [%].
S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016 4015
shows that differences between the analytical and numerical re-sults become appreciable only for fairly high Bi values that are be-yond the range of interest for practical electronic cooling heatapplications.
4.4. Assessment of overall accuracy of analytical models
To assess the accuracy of the analytical models for realistic heatsink geometries, values of the percent differences in the tempera-tures and the heat transfer rates for Ws/Wch = 0.69, AR = 4.3 andBi = 0.0055 are provided in Table 2. Very good agreement is real-ized between the present analytical model predictions and thetwo-dimensional numerical simulations, with maximum differ-
ences in temperature and heat transfer rate of 0.07% and 2.77%,respectively. The analytical models for the inverse trapezoidal, tri-angular, and diamond-shaped micro-channels show better predic-tions than for the rectangular and trapezoidal micro-channels.
Table 3 summarizes percent differences in Qfin between analyt-ical and numerical predictions for all 60 cases (30 cases indicatedin Table 1 repeated for kc = 0 and kc = k), which feature the follow-ing Bi values: Bi = 0.0027 for Hch = 715 and 168 lm, Bi = 0.0055 forHch = 528 and 124 lm, and Bi = 0.0084 for Hch = 341 and 80 lm.The percent difference increases with decreasing fin spacing (Ws/Wch) and increasing aspect ratio. Overall, the highest percent dif-ference in heat transfer rate is 4.65%, which corresponds to thetrapezoidal case with Hch = 715 lm and kc = k.
0.001
Bi
(Qfin
,A -
Qfin
,N)
/ Qfin
, N x
100
[%
]
Hch = 528 m, AR = 4.3, RectangularHch = 528 m, AR = 4.3, Inv. Trapez.Hch = 528 m, AR = 4.3, TriangularHch = 528 m, AR = 4.3, TrapezoidalHch = 528 m, AR = 4.3, DiamondHch = 124 m, AR = 1.0, RectangularHch = 124 m, AR = 1.0, Inv. Trapez.Hch = 124 m, AR = 1.0, TriangularHch = 124 m, AR = 1.0, TrapezoidalHch = 124 m, AR = 1.0, Diamond
Hb = 2000 mHc = 300 m
0.01 0.1 1
0
5
10
15
20
25
30
q"base = 300 W/cm2 Tf = 300Kh = 50,000 W/m2 k = kc= 387.6 W/m.K
Fig. 9. Variation of percent difference between analytical and two-dimensionalnumerical results with Biot number for Ws/Wch = 0.69.
Table 3Percent difference in heat transfer rate between 2D numerical (area-averaged) and 1Danalytical results.
Hch
(lm)(Qfin,A � Qfin,N)/Qfin,N � 100 (%)
Rectangular Inversetrapezoidal
Triangular Trapezoidal Diamond
715 kc = 0 3.72 0.53 �0.01 4.55 –kc = k 3.78 0.58 0.05 4.65 0.25
168 kc = 0 1.46 0.15 0.00 2.00 –kc = k 2.53 0.55 0.37 2.36 0.57
528 kc = 0 1.80 0.32 0.07 2.14 –kc = k 2.00 0.39 0.15 2.21 0.18
124 kc = 0 0.50 0.12 0.10 0.81 -kc = k 1.01 0.36 0.24 0.96 0.30
341 kc = 0 0.70 0.24 0.17 0.84 –kc = k 0.76 0.25 0.23 0.81 0.26
80 kc = 0 0.25 0.17 0.18 0.44 –kc = k 0.42 0.27 0.23 0.45 0.25
4016 S.-M. Kim, I. Mudawar / International Journal of Heat and Mass Transfer 53 (2010) 4002–4016
5. Conclusions
This study examined heat diffusion effects in micro-channel heatsinks found in electronic cooling applications. Analytical modelswere constructed for heat sinks with rectangular, inverse trapezoi-dal, triangular, trapezoidal, and diamond-shaped micro-channels.Solutions were presented for both monolithic heat sinks as well asheat sinks with perfectly insulating cover plates. The analytical re-sults were compared to two-dimensional numerical simulationsfor different micro-channel aspect ratios, fin spacings and Biot num-bers. Key findings from the study can be summarized as follows:
(1) A systematic analytical methodology was developed for heatdiffusion in the heat sinks which enables the determinationof effective heat flux along the heated portion of the micro-channel and the mean wall temperature.
(2) Overall, a rectangular micro-channel with a high aspect ratioand small fin spacing provides the best overall thermal per-formance compared to the other cross-sectional geometriesbased on the assumptions provided in Sections 2 and 3.
(3) The analytical models for the inverse trapezoidal, triangular,and diamond-shaped micro-channels show better predic-tions than for the rectangular and trapezoidal micro-chan-nels. Nonetheless, a maximum percent difference in heattransfer rate between the analytical and numerical resultsfor 60 cases of practical interest of only 4.65% proves theanalytical models are very accurate and effective tools forthe design and thermal resistance prediction of micro-chan-nel heat sinks found in electronic cooling applications.
Acknowledgement
The authors are grateful for the support of the Office of NavalResearch (ONR) for this study.
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