Description

Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Analytical Modeling of Fluid Flow and Heat Transferin Microchannel/Nanochannel Heat Sinks
W. A. Khan
∗
National University of Science and Technology, PNS Jauhar, Karachi 75350, Pakistan
andM. M. Yovanovich
†
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
DOI:10.2514/1.35621
Laminar forced convection in two-dimensional rectangular microchannels and nanochannels underhydrodynamically and thermally fully developed conditions is investigated analytically in the slip-
ﬂ
ow regime.Closed-formsolutionsfor
ﬂ
uidfrictionandNusseltnumbersareobtainedbysolvingthecontinuummomentumandenergy equations with the
ﬁ
rst-order velocity slip and temperature jump boundary conditions at the channel walls.An iso
ﬂ
ux thermal boundary condition is applied on the heat sink base. The results of the present analysis arepresented in terms of the channel aspect ratio, hydraulic diameter, momentum and thermal accommodationcoef
ﬁ
cients, Knudsen number, slip velocity, Reynolds number, and Prandtl number. It is found that
ﬂ
uid frictiondecreases and heat transfer increases compared with no-slip
ﬂ
ow conditions, depending on the aspect ratios andKnudsen numbers that include the effects of the channel size or rarefaction and the
ﬂ
uid/wall interaction.
Nomenclature
A
= total heating surface area,
m
2
, or constant,
1
=
2
=
P=L
A
c
= cross-sectional area of a single
ﬁ
n,
m
2
D
h
= hydraulic diameter, m
f
= skin-friction coef
ﬁ
cient or friction factor
G
= volume
ﬂ
ow rate,
cm
3
=
s
H
c
= channel height, m
h
= average heat transfer coef
ﬁ
cient,
W
=
m
2
K
K
= coef
ﬁ
cient of pressure loss,
P=
12
U
2av
Kn
= Knudsen number,
=D
h
k
= thermal conductivity of solid,
W
=
m
K
k
ce
= sum of contraction and expansion losses in the channel
k
f
= thermal conductivity of
ﬂ
uid,
W
=
m
K
L
= length of the channel in the
ﬂ
ow direction, m
L
= characteristic length, usually taken as the hydraulicdiameter of the channel,
m
m
=
ﬁ
n parameter,
m
1
,
h
av
=k
2
w
w
p
_
m
= total mass
ﬂ
ow rate,
m
1
,
kg
=
s
N
= total number of microchannels/nanochannels
Nu
D
h
= Nusselt number based on hydraulic diameter
Pe
D
h
= Peclet number based on hydraulic diameter
Pr
= Prandtl number
Q
b
= heat transfer rate from heat sink base, W
Q
fin
= heat transfer rate from
ﬁ
n, W
q
= heat
ﬂ
ux,
W
=
m
2
Re
D
h
= Reynolds number based on hydraulic diameter,
U
av
D
h
=T
= absolute temperature, K
t
= thickness, m
U
av
= average velocity in the channels,
m
=
s
W
= width of the heat sink, m
w
c
= half of the channel width, m
w
w
= half of the
ﬁ
n thickness, m
= thermal diffusivity,
m
2
=
s
or constant,
2
u
=
1
c
c
= channel aspect ratio,
2
w
c
=H
c
hs
= heat sink aspect ratio,
L=
2
w
c
=
ﬁ
n spacing ratio,
w
c
=w
w
= ratio of speci
ﬁ
c heats,
c
p
=c
v
P
= pressure drop across the microchannel/nanochannel, Pa
fin
= ef
ﬁ
ciency of the
ﬁ
n,
tanh
mH
c
=mH
c
= mean free path, m
= absolute viscosity of
ﬂ
uid,
kg
=
m
s
= kinematic viscosity of
ﬂ
uid,
m
2
=
s
u
= slip velocity coef
ﬁ
cient,
2
=
Kn
t
= temperature jump coef
ﬁ
cient,
2
t
=
t
2
=
1
Kn=Pr
=
ﬂ
uid density,
kg
=
m
3
= tangential momentum accommodation coef
ﬁ
cient
t
= energy accommodation coef
ﬁ
cient
w
= shear stress at channel wall,
N
=
m
2
,
d
u
d
y
j
y
0
= reduction of the friction factor due to the rarefactioneffect,
fRe
Dh
j
Kn
=
fRe
Dh
j
Kn
0
Subscripts
a
= ambient av = average
b
= base surface
f
=
ﬂ
uid
ﬁ
n = single
ﬁ
n
g
= gashs = heat sink
s
= slipth = thermal
w
= wall
Introduction
C
ONVENTIONAL heat sinks and heat pipes are unable tohandle high heat removal rates. After the pioneering work of Tuckerman and Pease [1], microchannel heat sinks have receivedconsiderable attention, especially in microelectronics. Most of thesemicrochannel heat sinks were water cooled. They could dissipate anextremelyhigh-powerdensitywithaheat
ﬂ
uxashighas
790 W
=
cm
2
[1]. These heat sinks do not include slip effect.
Received 12 November 2007; revision received 11 February 2008;accepted for publication 12 February 2008. Copyright © 2008 by theAmerican Institute of Aeronautics and Astronautics, Inc. All rights reserved.Copies of this paper may be made for personal or internal use, on conditionthatthecopierpaythe$10.00per-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923;includethecode0887-8722/ 08 $10.00 in correspondence with the CCC.
∗
Associate Professor, Department of Mechanical Engineering, PNEngineering College.
†
Distinguished Professor Emeritus, Department of MechanicalEngineering.
J
OURNAL OF
T
HERMOPHYSICS AND
H
EAT
T
RANSFER
Vol. 22, No. 3, July
–
September 2008
352
In microchannels/nanochannels, the gas
ﬂ
ow can be modeledusingNavier
–
Stokesandenergyequationsusing
ﬁ
rst-orderslip-
ﬂ
owand temperature jump boundary conditions. Microscopic effectsbecome more important when the molecular mean free path of thecoolant gas has the same order of magnitude as the channel size.Because of these effects, the friction factors decrease and the heat transfer coef
ﬁ
cients increase with increase in Knudsen number.Inthisstudy,thefullydevelopedlaminar
ﬂ
owisanalyzedthroughrectangular microchannels/nanochannels in the slip-
ﬂ
ow regime(
0
:
001
< Kn <
0
:
1
), and closed-form solutions are obtained for friction factors and heat transfer coef
ﬁ
cients in terms of the channelaspect ratio, Knudsen number, Reynolds number, and Prandtlnumber.There is currently no model for predicting corresponding velocitypro
ﬁ
lesorpressuredistributionintheslipregime.Moreover,thereisnosuchmodelthatcanbeusedinothergeometries,forexample,two-dimensional channels and rectangular ducts with different aspect ratios. The objective of the current investigation is to develop a uni
ﬁ
ed, physics-based model appropriate for the slip-
ﬂ
ow regimeand for two-dimensional channels and ducts.
Literature Review
Inthelasttwodecades,the
ﬂ
owandheattransferinmicrochannel/ nanochannel heat sinks have become subjects of growing researchattentioninmicroelectronics.Althoughtheworkonthesechannelsisnot new, microchannel heat sinks have received considerableattention after Tuckerman and Pease [1], especially in micro-electronics. Following this work, several experimental, numerical,and theoretical studies on rare
ﬁ
ed gas
ﬂ
ows in microchannels havebeen carried out in a wide range of Knudsen numbers with theobjectiveofdevelopingsimple,physics-basedmodels.Thesestudiesare reviewed in this section.Harley et al. [2] and, later, Morini et al. [3
–
5] presented analyticaland experimental studies. In these studies, they investigated therarefaction effects on the pressure drop through silicon micro-channels having rectangular, trapezoidal, or double-trapezoidalcrosssections.TheypointedouttherolesoftheKnudsennumberandthe cross-sectional aspect ratio in the friction factor reduction due tothe rarefaction and obtained solutions for velocity pro
ﬁ
les, frictionfactors,shearstresses,momentum
ﬂ
ux,andkineticenergycorrectionfactors.Ebert and Sparrow [6] formulated an analytical slip-
ﬂ
ow solutionfor a rectangular channel. They found that the effect of slip is to
ﬂ
atten the velocity distribution relative to that of a continuum
ﬂ
owand that the compressibility increases the pressure drop through anincrease in the viscous shear rather than through an increase in themomentum
ﬂ
ux.Inman [7
–
9] presented theoretical analyses of
ﬂ
uid
ﬂ
ow and heat transferforlaminarslip
ﬂ
owinaparallelplatechannelwithdifferent thermal boundary conditions. The solutions contain a seriesexpansion and analytical expressions for the complete set of eigenvalues and eigenfunctions for the problems. They obtainedexpressions for the temperature of the gas adjacent to the wall, thewall heat
ﬂ
ux, and the Nusselt numbers for the conduits for variousvalues of the rarefaction parameters. The results indicated that thethermal entrance length is decreased with increasing gas rarefactionand also that for a given mean free path the thermal entrance lengthis greater for unsymmetrical heating than for a symmetrical wallheat
ﬂ
ux.Colin et al. [10] proposed an analytical slip-
ﬂ
ow model based onsecond-order boundary conditions for gaseous
ﬂ
ow in rectangular microchannels. They designed an experimental setup for themeasurement of gaseous micro
ﬂ
ow rates under controlled tempera-ture and pressure conditions. It was shown that, in rectangular microchannels, the proposed second-order model is valid for Knudsen numbers up to about 0.25, whereas the
ﬁ
rst-order model isno longer accurate for values higher than 0.05. The best
ﬁ
t is foundfor a tangential momentum accommodation coef
ﬁ
cient
0
:
93
,both with helium and nitrogen. Yu and Ameel [11,12] analytically
investigated the laminar forced convection in thermally developingslip
ﬂ
ow through iso
ﬂ
ux rectangular microchannels. They obtainedlocal and fully developed Nusselt numbers,
ﬂ
uid temperatures, andwall temperatures by solving the continuum energy equation for hydrodynamically fully developed slip
ﬂ
ow with the velocity slipandtemperaturejumpconditionatthewalls.Theyfoundthattheheat transfer may increase, decrease, or remain unchanged, comparedwith the no-slip
ﬂ
ow conditions, depending on the aspect ratios andtwo-dimensionless variables that include the effects of the micro-channel size or rarefaction and the
ﬂ
uid/wall interaction.Zhao and Lu [13] presented an analytical and numerical study onthe heat transfer characteristics of forced convection across a micro-channelheatsink.Theyusedporousmediumand
ﬁ
napproachesandinvestigated the effects of channel aspect ratio and effective thermalconductivityratioontheoverallNusseltnumber.Theyfoundthattheoverall Nusselt number increases as the channel aspect ratio isincreasedanddecreaseswithanincreasingeffectivethermalconduc-tivityratio.Theyproposedanewconceptofmicrochannelcoolingincombinationwithmicroheatpipesandestimatedtheenhancementinthe heat transfer. They conducted two-dimensional numericalcalculationsforbothconstantheat
ﬂ
uxandconstantwalltemperatureconditions to check the accuracy of the analytical solutions and toexamine the effect of different boundary conditions on the overallheat transfer.Quarmby [14] and Gampert [15] used
ﬁ
nite-differencesimulations to investigate developing slip
ﬂ
ow in circular pipesand parallel plates.Barber and Emerson [16,17] examined the role of Reynolds and
Knudsen numbers on the hydrodynamic development length at theentrance to parallel plate microchannels. They carried out numericalsimulations over a range of Knudsen numbers covering thecontinuum and slip-
ﬂ
ow regimes. Their results demonstrate that, at the upper limit of the slip-
ﬂ
ow regime, the entrance development region is almost 25% longer than that predicted using continuum
ﬂ
ow theory.Andrei and Raymond [18] developed a three-dimensional modelto investigate the
ﬂ
ow and conjugate heat transfer in themicrochannel-based heat sink for electronic packaging applications.They solved the Navier
–
Stokes equations of motion numericallyusing the generalized single-equation framework. They alsodeveloped and validated the theoretical model by comparing thepredictions of the thermal resistance and the friction coef
ﬁ
cient withthe available experimental data for a wide range of Reynoldsnumbers. Their analysis provides a unique fundamental insight intothe complex heat
ﬂ
ow pattern established in the channel due tocombined convection
–
conduction effects in the three-dimensionalsetting.Arkilic et al. [19,20] performed analytic and experimental
investigations into gaseous
ﬂ
owwith slight rarefaction through longmicrochannels.Theyusedatwo-dimensionalanalysisoftheNavier
–
Stokes equations with a
ﬁ
rst-order slip-velocity boundary conditionto demonstrate that both compressibility and rare
ﬁ
ed effects arepresent in long microchannels. They reported the tangentialmomentum accommodation coef
ﬁ
cients (TMAC) for nitrogen,argon, and carbon dioxide gases in contact with single-crystalsilicon. For all three gases, the TMAC is found to be lower than one,ranging from 0.75 to 0.85.Beskok and Karniadakis [21] developed simple, physics-basedmodels for
ﬂ
ows in channels, pipes, and ducts at microscales for a wide range of Knudsen numbers at low Mach numbers. Theyproposed a new general boundary condition that accounts for thereduced momentum and heat exchange with the wall surfaces andinvestigateditsvalidity.Theyfoundthat,asthevalueoftheKnudsennumber increases, the rarefaction effects become more important and, thus, the pressure drop, shear stress, heat
ﬂ
ux, and correspond-ing mass
ﬂ
ow rate cannot be predicted from standard
ﬂ
ow and heat transfer models based on the continuum hypothesis. They alsodetermined that simple models based on kinetic gas theory conceptsare not appropriate either, except in the very high Knudsen number regime corresponding to near-vacuum conditions.Bower et al. [22] presented experimental results on the heat transferand
ﬂ
owinsmall SiCheat exchangerswith multiple rowsof
KHAN AND YOVANOVICH
353
parallel channels oriented in the
ﬂ
ow direction. They analyzed theoverall heat transfer and pressure drop coef
ﬁ
cients in single-phase
ﬂ
ow regimes and found that liquid-cooled SiC heat sinks easilyoutperform air-cooled heat sinks.Harms et al. [23] obtained experimental results for a single-phaseforced convection in deep rectangular microchannels. They testedsingle-andmultiple-channelsystems.Allofthetestswereperformedwithdeionizedwaterastheworking
ﬂ
uid,withtheReynoldsnumber ranging from 173 to 12,900. The experimentally obtained localNusselt number agreed reasonably well with classical developingchannel
ﬂ
ow theory. Furthermore, their results show that a multiple-channel system designed for developing laminar
ﬂ
ow outperformsthe comparable single-channel system designed for turbulent
ﬂ
ow.Hetsroni et al. [24,25] performed experimental and theoretical
investigations on single-phase
ﬂ
uid
ﬂ
ow and heat transfer inmicrochannels. They considered both problems in the frame of a continuum model, corresponding to small Knudsen number. Theyanalyzed the data of the pressure drop and heat transfer in circular,triangular,rectangular,andtrapezoidalmicrochannels.Theeffectsof geometry and the axial heat
ﬂ
ux due to thermal conduction throughtheworking
ﬂ
uidandchannelwalls,aswellastheenergydissipation,werediscussed.Theycomparedtheexperimentaldata,obtainedbya number of investigators, to the conventional theory on heat transfer.Hsiehetal.[26,27]presentedexperimentalandtheoreticalstudies
of incompressible and compressible
ﬂ
ows in a microchannel. Theyused nitrogen and deionized water as working media in their experiments. The results were found to be in good agreement withthose predicted by analytical solutions in which a 2-D continuous
ﬂ
ow model with
ﬁ
rst slip boundary conditions is employed andsolved by a perturbation method with a proposed new completemomentum accommodation coef
ﬁ
cient.
Analysis
ThegeometryofamicrochannelheatsinkisshowninFig.1a .Thelength of the heat sink is
L
and the width is
W
. The top surface isinsulated, and the bottom surface is uniformly heated. The surfacesofthechannelsareassumedtobesmooth.Acoolantpassesthrougha number of microchannels along the
x
axis and takes heat away from the heat dissipating electronic component attached below. The
ﬂ
owin the channels is steady, laminar, and fully developed. There are
N
channels, and each channel has a height
H
c
and width
2
w
c
. Thethickness of each
ﬁ
n is
2
w
w
, whereas the thickness of the base is
t
b
.The
ﬁ
n tips are assumed to be adiabatic. The temperature of thechannel walls is assumed to be
T
w
with an inletwater temperature of
T
a
. At the channel wall, the slip-
ﬂ
ow velocity and temperature jumpboundary conditions were applied to calculate the friction and heat transfer coef
ﬁ
cients. Taking advantage of symmetry, a controlvolume (CV) is selected, as shown in Fig.1b. The length of thecontrol volume is taken as unity for convenience, and the width andheight are taken as
w
w
w
c
and
H
c
t
b
, respectively. This controlvolumeincludeshalfofthe
ﬁ
nandhalfofthechannel alongwiththebase.ThesidesurfacesABandCDandthetopsurfaceACofthisCVcan be regarded as impermeable, adiabatic, and shear free (i.e., nomass transfer and shear work transfer across these surfaces). Theuniform heat
ﬂ
ux over the bottom surface BD of the CV is
q
.
Governing Equations
The continuum equations for the conservation of mass,momentum, and energy can be used with slip-
ﬂ
ow and temperature jump boundary conditions. Using scale analysis, the axialmomentum and energy equations for the control volume shown inFig.1breduces to
d
2
u
d
y
2
1
d
p
d
x
(1)and
u
d
T
d
x
d
2
T
d
y
2
(2)
Hydrodynamic Boundary Conditions
1) At the channel surface:
u
u
s
u
L
d
u
d
y
j
y
0
2) At the symmetry plane:
y
w
c
d
u
d
y
0
Thermal Boundary Conditions
1) Following Liu and Garimella [28], the thermal boundarycondition at the base of the
ﬁ
n can be determined from an energybalance:
k@T @y
y
0
q
w
w
w
c
w
w
2
fin
2
fin
c
q
w
2) At the wall:
T
g
T
w
t
L
d
T
d
y
j
y
0
3) At the symmetry plane:
y
w
c
d
T
d
y
0
Fluid Flow
IntegratingEq.(1)twicewithrespectto
y
andusinghydrodynamicboundaryconditions,thevelocitydistributionindimensionlessform can be written as
u
Aw
2
c
2
2
4
(3)The average velocity in the channel is de
ﬁ
ned as
U
av
1
w
c
Z
w
c
0
u
y
d
y
23
Aw
c
1
6
(4)The normalized velocity distribution and slip-
ﬂ
ow velocity can bewritten as
u
U
av
32
2
2
4
1
6
U
s
U
av
6
1
6
9>>=>>;
(5)Themomentumtransfertothechannelwallcanbeexpressedintermsof the skin-friction coef
ﬁ
cient or friction factor, de
ﬁ
ned as
f
w
12
U
2av
24
Re
D
h
11
c
11
6
(6)
c
H
qW L
b
Q
w
w
Q
q
z
b
t
c
w
y
c
H
c
w
2
w
w
2
A D BC
a) b)
Fig. 1 Geometry of the microchannel/nanochannel heat sink.
354
KHAN AND YOVANOVICH
which gives the Poiseuille number
fRe
D
h
for rectangular microchannels/nanochannels in terms of aspect ratio
c
and theslip-velocity coef
ﬁ
cient.
fRe
D
h
24
=
1
c
1
=
1
6
(7)ThevaluesofthePoiseuillenumberarecomparedwiththeanalyticalvalues quoted by Shah and London [29] and the numerical valuesgivenbyMorinietal.[4]inTable1forthecontinuum
ﬂ
ow(
Kn
0
).Morini et al. [4]de
ﬁ
nedthe reduction of thefriction factor due to therarefaction effect as follows:
fRe
Dh
Kn
fRe
Dh
Kn
0
11
121
c
2
Kn
(8)For a
ﬁ
xed cross section, the friction factor reduction
has beencalculatedbycomparingthePoiseuillenumberforanassignedvalueof the Knudsen number with the value that the Poiseuille number assumes for
Kn
0
(i.e., the continuum
ﬂ
ow). The friction factor reduction
depends on the channel aspect ratio and on the Knudsennumber.Table2shows the comparison of the present values of the frictionfactor reduction
with the numerical values presented by Moriniet al. [4] for some values of the Knudsen number between 0.001 and0.1.Itshowsthat
decreasesas
Kn
goesfrom0.001to0.1;thisresult con
ﬁ
rmsthatgasrarefactionreducesthefrictionbetweenthegasandthe microchannel/nanochannel walls. The reduction of the frictionfactor is stronger for rectangular microchannels/nanochannels withsmallchannelaspectratios.For
Kn
0
,thefrictionfactorreduction
reachestheminimumvalueof45.5;thevalueof
becomes56.5%for a square microchannel/nanochannel (
c
1
).The rectangular microchannels/nanochannels with smaller channel aspect ratios have higher values of
c
; hence, for thesemicrochannels/nanochannels, thedecrease ofthefriction factor withthe Knudsen number is larger. In other words, the rarefaction effectsappear to be higher in microchannels/nanochannels with smaller aspect ratios. This is due to the de
ﬁ
nition of the Knudsen number based on the hydraulic diameter of the channel.It shows that the present values are in good agreement for smaller aspectratioswiththepreviousresults.Thiscanbeconsideredagoodvalidation of the assumptions made in the present work.The coef
ﬁ
cient of pressure loss can be determined from
K
P=
12
U
2av
k
ce
f
L=D
h
(9)where
k
ce
is the sum of the contraction and expansion losses in thechannel. Kleiner et al. [30] used experimental data from Kays andLondon [31] and derived the following empirical correlation for theentrance and exit losses
k
ce
in terms of the channel width and
ﬁ
nthickness:
k
ce
1
:
79
2
:
32
w
c
=
w
c
w
w
0
:
53
w
c
=
w
c
w
w
2
(10)
Heat Transfer
Theenergyequation,Eq.(2),indimensionlessformcanbewrittenas
1
2
c
f
@
@
4
Pe
D
h
@
2
@
2
(11)where
x=w
c
;
y=w
c
; f
32
2
2
4
1
6
T
T
a
D
h
q
w
=k
f
; U
av
u=f
; Pe
D
h
Re
D
h
Pr
From an energy balance on a
ﬂ
uid element in the channel,
d
T
d
x
q
w
=k
f
w
c
U
av
(12)In dimensionless form, it can be written as
@
=@
1
=Pe
D
h
(13)Combining Eqs. (11) and (13), we get
@
2
@
2
1
c
4
f
1
c
4
3
32
2
1
3
32
2
U
s
U
av
(14)In dimensionless form,
ﬁ
rst thermal boundary condition can bewritten as
Table 1 Comparison of Poiseuille numbers
fRe
D
h
for micro-channels/nanochannels
Poiseuille number
fRe
D
h
c
Shah and London [29] Morini et al. [4] Present
0 24 24 240.2 19.07 19.07 20.000.4 16.37 16.37 17.140.6 14.98 14.98 15.000.8 14.37 14.37 13.331.0 14.22 14.22 12.00
Table 2 Comparison of friction factor reduction (
) for microchannels/nanochannels
Friction factor reduction (
)
Kn
0
:
001
Kn
0
:
01
Kn
0
:
1
c
Present Morini et al. [4] Present Morini et al. [4] Present Morini et al. [4]
0.0 0.988 0.988 0.893 0.893 0.455 0.4550.1 0.989 0.989 0.902 0.901 0.478 0.4770.2 0.990 0.990 0.909 0.907 0.500 0.4960.3 0.991 0.990 0.916 0.912 0.520 0.5140.4 0.991 0.991 0.921 0.917 0.539 0.5290.5 0.992 0.991 0.926 0.920 0.556 0.5410.6 0.992 0.992 0.930 0.923 0.571 0.5510.7 0.993 0.992 0.934 0.924 0.586 0.5570.8 0.993 0.992 0.937 0.925 0.600 0.5620.9 0.994 0.992 0.941 0.925 0.613 0.5641.0 0.994 0.992 0.943 0.926 0.625 0.565
KHAN AND YOVANOVICH
355
k@
s
@
0
k
f
w
c
D
h
(15)where
s
T
T
w
D
h
q
w
=k
f
isthedimensionlesstemperatureforthesolidsurface.Also,fromthecontinuity of the temperature and heat
ﬂ
ux at the solid
–
ﬂ
uidinterface,
k@
s
@
0
k
f
@
@
0
(16)Combining Eqs. (15) and (16), we get
@
@
0
w
c
D
h
(17)Using this boundary condition and integrating Eq. (14) with respect to
, we get
@
@
1
c
4
32
2
12
3
1
32
2
12
3
U
s
U
av
(18)From an overall energy balance on the
ﬂ
uid element, we get thefollowing additional condition
Z
10
f
d
0
(19)Integrating Eq. (18) and applying Eq. (19), we get
1
c
18
3
132
4
14
17140
1210
U
s
U
av
2
18
2
18
3
132
4
370
U
s
U
av
(20)Integrating Eq. (13) and applying the condition
T
g
T
a
D
h
q
w
=k
f
at
0
, we get
T
g
T
a
D
h
q
w
=k
f
Pe
D
h
0
(21)where
0
can be determined from Eq. (18). From the secondthermal boundary condition, we get
T
g
T
w
D
h
q
w
=k
f
t
(22)Combining Eqs. (21) and (22), we get
T
w
T
a
D
h
q
w
=k
f
Pe
D
h
0
t
(23)By de
ﬁ
nition, the bulk temperature is given by
T
b
T
a
D
h
q
w
=k
f
Pe
D
h
(24)which gives
T
b
T
a
D
h
q
w
=k
f
Pe
D
h
(25)Combining Eqs. (23) and (25), we get
T
w
T
b
D
h
q
w
=k
f
0
t
1
c
17140
370
U
s
U
av
1210
U
s
U
av
2
t
(26)Foruniformwall
ﬂ
ux(UWF),theaverageheattransfercoef
ﬁ
cientfor the
ﬁ
n is de
ﬁ
ned as
h
fin
q
w
=
T
w
T
b
(27)In dimensionless form it can be written as
Nu
D
h
h
fin
D
h
k
f
(28)
Overall Heat Transfer Coef
ﬁ
cient for the Heat Sink
The heat balance for the whole CV can be written as
Q
NQ
fin
Q
b
(29)where
Q
hA
hs
b
Q
fin
hA
fin
b
Q
b
hA
b
b
9=;
(30)which gives the overall average heat transfer coef
ﬁ
cient for a microchannel/nanochannel heat sink:
h
hs
N
1
Ah
fin
hA
b
A
hs
(31)with
N
W
2
w
w
=
2
w
c
w
w
A
hs
NA
fin
A
b
hs
1
NA
fin
=A
hs
1
fin
A
fin
2
H
c
2
w
w
L
A
b
LW
N
1
2
w
w
L
9>>>>=>>>>;
(32)The average heat transfer coef
ﬁ
cient for the
ﬁ
n can be determinedfrom Eq. (29), whereas
h
b
for the UWF boundary condition wasdetermined by Khan et al. [32] and could be written as
h
b
0
:
912
k
f
=L
Re
1
=
2
L
Pr
1
=
3
(33)where
Re
L
is the Reynolds number based on the length of the baseplate and is de
ﬁ
ned as
Re
L
U
av
L
=
(34)
Case Studies and Discussion
The slip-
ﬂ
ow range (
0
:
001
< Kn <
0
:
1
) dictates the channelwidth for the
ﬂ
ow of any gas through microchannels/nanochannels.For air (
69
:
2 nm
), Fig.2shows that the channel width rangesfrom
35
m
to
350 nm
. Qin and Li [33] have shown a noveltechniquein creatingmicrochannels/nanochannels usingaNd:YAGlaser in a dry process.In these channels, the friction losses are reduced, as shown inFig.3. It is demonstrated that the friction losses are highest in thecontinuum
ﬂ
ow (
Kn
0
). As the
Kn
number increases, the frictionlosses decrease with an increase in the aspect ratio. Arkilic et al.[19,20] demonstrated experimentally that, for nitrogen, argon, and
carbondioxide,theTMACisfoundtobelowerthan1,rangingfrom 0.75 to 0.85. The effect of these TMAC on the friction factors areshown in Fig.4in the slip region. It shows that the friction factorsdecrease monotonically as TMAC decreases and the channel aspect ratioincreases.Theeffectsoftheaspectratiosonthepressuredropinthe slip-
ﬂ
ow region are investigated in Fig.5. It is obvious that the
356
KHAN AND YOVANOVICH

Search

Similar documents

Related Search

Fluid flow and heat transfer in MicrochannelsMicroscale Fluid Flow and Heat TransferNumerical Modeling of Multiphase Flow and ConNumerical simulation of blood flow and fluid-Experimental Fluid Dynamics and Heat Transfer-\tFluid flow, heat and mass transfer in poroFlow and Heat Transfer in Porous MediaFluid Mechanics and Heat TransferSimulation of Fluid Flow Using Lattice BoltzmComputational Fluid Dynamics and Heat Transfe

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x