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

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Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks
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  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   hA  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
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