Effects Of Entrance Region Transport Processes On Slip Flow Regime In A Wavy Wall Microchannel With Isothermally Heated Walls

Transcription

1 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. Effects Of Entrance Regon Transport Processes On Slp Flo Regme In A Wavy Wall Mcrochannel Wth Isothermally Heated Walls H. Shokouhmand, S. Bgham Abstract In ths study, thermal and hydrodynamc characters of a hydrodynamcally and thermally developng flo n a avy mcrochannel are analyzed. A numercal smulaton has been carred out to solve the contnuum, momentum and energy equatons n curvlnear coordnate. The equatons are dscretzed usng the fnte-volume method on a staggered mesh and solved by SIMPLE algorthm. A fully mplct scheme s used for the temporal terms and the hybrd dfferencng s appled for the approxmaton of the convectve terms. Maxell slp condton and Von-Smoluchosk's temperature jump boundary condton are mposed. The effect of rarefacton on thermal and hydrodynamc characters of flo n avy mcrochannels s explored. Also the effect of creep flo s assumed. The results sho that Knudsen number has an mportant effect on both the f.re and Nusselt number on the undeveloped flud flo. To verfy the code a comparson s carred out th avalable results and good agreement s acheved. Index Terms Wavy Wall Mcrochannel, Slp ondtons, Enterance Regon, FVM Method I. INTRODUTION In the past decade, nvestgaton of mcro-scale flos has become more and more mportant due to ther de applcaton n electronc equpments, heat exchangers, sensors and flo controls, reactors, poer systems and mcroelectromechancal systems (MEMS. One of the mportant goals related to mcro thermal devces s the removal of a large amount of dspersed heat from a small space. So, the exstences of accurate and effcent analyss are very mportant to acheve ths goal. Flo n avy channels has had de applcaton because of ncreasng the rate of heat transfer. Ths advantage mght become more mportant n mcro-scale devces. In spte of ths mportant advantage, accordng to the knoledge of authors, there are a fe nvestgatons nto avy channels n mcro-scale. The hydrodynamc and thermal characterstcs of flo n mcrochannels n the smple geometres have been studed by many researchers though [1] [4]. Based on the magntude of Knudsen number (Kn=λ/L, Manuscrpt receved November 25, The 2010 Internatonal onference of Mechancal Engneerng (IME'10 H. Shokouhmand, dstngushed professor of Mechancal Engneerng, ollege of Engneerng, Unversty of Tehran, Tehran, Iran (correspondng author to provde phone: ; Fax: ; e-mal: hshokoh@ut.ac.r.8 S. Bgham, M.Sc. student of Mechancal Engneerng, ollege of Engneerng, Unversty of Tehran, Tehran, Iran (e-mal: sajjadbgham@gmal.com. ISSN: (Prnt; ISSN: (Onlne the rato beteen the gas mean free path and the length scale flud flos exhbt dfferent behavors. Therefore, some classfcatons have been done. Accordng to the magntude of Knudsen number flos are dvded nto four regmes: contnuum, slp, transton and free molecular flos. In mcrochannels due to the mnute length scale, the Knudsen number gets a sgnfcant value and t determnes the behavor of flud flo. In ths ork, the slp regme s studed and the Knudsen number s n the range from to 0.1. Dfferent nvestgatons on the mcrochannel heat transfer n ths regme have been carred out [5]-[7]. In ths regme, velocty slp and temperature jump occur at the all surface and thereby, flo characterstcs such as Nusselt number and f.re are nfluenced. Theoretcal and numercal studes thn the slp flo regme typcally use the Naver Stokes approach model and energy equaton along th approprate slp all and temperature jump models. In the past decade, consderable efforts have been devoted to analyze flud flo and heat transfer n avy channels and mcrochannels. heng [8] studed a famly of locally constrcted channels, and n each case, the shear stress at the all as found to be sharply ncreased at and near the regon of constrcton. Vrads et al. [9] studed the steady, to-dmensonal case n channel n curvlnear orthogonal and non-orthogonal coordnate systems. They llustrated that the vortcty remans constant n the straght secton of the channel close to the nlet, ncreasng rapdly as the all starts convergng. Also t as shon that the vortcty peaks at the pont here the cross-sectonal area becomes mnmum and drops rapdly as the flo enters the dvergng part of the channel. Wang et al. [10] numercally studed forced convecton n a symmetrc avy all macro channel. Ther results shoed that the ampltudes of the Nusselt number and the skn-frcton coeffcent ncrease th an ncrease n the Reynolds number and the ampltude avelength rato. The heat transfer enhancement s not sgnfcant at smaller ampltude avelength rato; hoever, at a suffcently larger value of ampltude avelength rato the corrugated channel ll be seen to be an effectve heat transfer devce, especally at hgher Reynolds numbers. Arklc et al. [11] nvestgated helum flo through mcrochannels. It s found that the pressure drop over the channel length as less than the contnuum flo results. The frcton coeffcent as only about 40% of the theoretcal values. Beskok et al. [] studed the rarefacton and compressblty effects n gas mcroflos n the slp flo regme and for the Knudsen number belo 0.3. Ther formulaton s based on the classcal Maxell/Smoluchosk boundary condtons that allo WE 2010

2 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. partal slp at the all. It as shon that rarefacton negates compressblty. They also suggested specfc pressure dstrbuton and mass flo rate measurements n mcrochannels of varous cross sectons. Although the hydrodynamc and thermal aspects of flud flo n normal mcrochannels have been extensvely studed, to the best of the author's knoledge, the flud/thermal analyss of avy mcrochannel has never reported. The present ork s an attempt to fll the lterature gap n ths regard. Y-momentum: ( vu + ( vv + ( q + ( x 22 η v + ( q p ( x ξ = p 1 Re { ( q 11 v + ( q v v } (4 II. FORMULATION OF THE PROBLEM To begn th, Fg. 1 shos the geometry of nterest hch s seen to be a to-dmensonal symmetrc avy channel. The channel alls are assumed to extend to nfnty n the z-drecton (.e., perpendcular to the plane. The mathematcal non-dmensonal expresson of avy all s gven as y ( = (1 sn(2π ( x x a 0.5 (1 λ Fg 1 Physcal doman of avy mcrochannel Steady lamnar flo th constant propertes s consdered. The present ork s concerned th both thermally and hydrodynamcally developng flo cases. In ths study the usual contnuum approach s coupled th to man characterstcs of the mcro-scale phenomena, the velocty slp and the temperature jump. A general non-orthogonal curvlnear coordnate frameork th (ξ,η as ndependent varables s used to formulate the problem. The non-dmensonal governng equatons can be rtten as: ontnuty: U V + = 0 X-momentum: 1 u ( uu + ( uv = { ( q11 Re u u u + ( q 22 + ( q + ( q } ( y η p + ( y ξ p (3 (2 U Energy: ( θu + ( q here : 22 + ( θv θ + ( q 1 = Pe { ( q 11 θ + ( q = uy η vx η, V = uy ξ + vxξ θ θ } J = xξ yη xη yξ, q 11 = ( y η + xη J q = ( xξ xη + yξ yη, q 22 = ( x ξ + yξ J J u, v are the physcal velocty components and U c and V c are the veloctes n ξ,η coordnates, respectvely. Here, θ represents non-dmensonal temperature. The employed dmensonless varables are defned as follos: x x =, L v v = θ = u y y =, L L, ρ u Re =, µ T T T T p = p, ρ u 2 u u = Pe = Re Pr = u u L α Here, u and T are the nlet velocty and nlet temperature, respectvely. III. SLIP FLOW EFFETS AND BOUNDARY ONDITIONS In order to estmate the slp effect at all under rarfed condton, the Maxell slp condton has been dely used hch s based on the frst-order approxmaton of all-gas nteracton from knetc theory of gases [13, 14]. The magntude of tangental accommodaton coeffcent expresses the degree of non-elastc dffusve reflecton beteen gas molecules and all molecules [15]. Usng Von-Smoluchosk's model e have the follong boundary condtons at all n curvlnear coordnate form: U S 2 σ v U s = Kn σ n v 2 σt 2γ Kn θ θs = 1 ( ( σ γ + 1 Pr n T 2 3 (1 γ Kn Re θ + 2π γ Ec s (5 (6 here γ and σ represent the specfc heat rato and accommodaton coeffcent, respectvely. The second term n the slp velocty assocates th the thermal creep. ISSN: (Prnt; ISSN: (Onlne WE 2010

3 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. Here, Ec means the Eckert number hch s defned as p 2 u Ec = (7 c ( T T here, Pr and Kn mean the Prandtl number and Knudsen number, respectvely. The Knudsen number shos the effect of rarefacton on flo propertes. A nonzero Knudsen number means a slp flo th nonzero flo velocty at the boundares and nonzero temperature dfference beteen the boundares and adjacent flo. In the present ork, the slp flo regme th the Knudsen number rangng from 0.01 to 0.1 s consdered. Also n ths ork, the study s lmted to ncompressble flo. The flo can be consdered ncompressble for Mach number loer than 0.3 [16]. Moreover, the other boundary condtons should be defned. A unform nlet velocty and temperature are specfed as u = 1, v = 0, θ = 0 (8 In the outlet, fully developed boundary condtons are assumed as u v θ = = = 0 (9 x x x Also the frcton coeffcent and Nusselt number for a hydrodynamcally-thermally developng flo n the avy mcro channel are calculated by, V. GRID INDEPENDENY The resultng numercal velocty and temperature felds may be used to calculate f.re and Nu along the length of the mcrochannel. The accuracy of the numercal solutons and the tme requred to reach a soluton are dependent on the grd resoluton. In ths paper, all computatons are performed on three grds comprsng , and control volumes respectvely. The obtaned results shoed suffcent accuracy on these range of grd resolutons. For nstance, ths accuracy s ndcated for Nu along the mcrochannel th the condtons specfed n Fg. 3. Grd dependence studes have been completed th smlar results for each numercal soluton presented n the results secton. Hence, for smplcty all subsequent results presented n result secton are obtaned usng grd. f tang 4( y ( x u ( x Re = 2 ( u( x, y dy n Nu = θ ave 2 1 θ ( x ( x 1 n (10 (11 Fg. 3 Numercal results of local Nusselt number along the avy mcrochannel th KN=0.075 at Re=2 and a=0.2 IV. VALIDATION OF NUMERIAL ODE In Fg. 2, a comparson th the prevously publshed result of Wang and hen [10] s carred out to valdate the numercal code and non-orthogonal grd dscretzaton scheme of the present study. Ther model s analogous to the present model but th the ater as orkng flud and macro scale channel. Also there s no slp effect th fxng Kn number at zero. VI. SOLUTION PROEDURE The governng equatons th approprate boundary condtons are solved by employng the SIMPLE algorthm [17], a fnte volume method, n non-orthogonal curvlnear coordnate frameork. A fully mplct scheme s used for the temporal terms and the HYBRID dfferencng [18] s appled for the approxmaton of the convectve terms. The Posson equatons s solved for (x, y to fnd grd ponts [19] and are dstrbuted n a non-unform manner th hgher concentraton of grds close to the curvy alls and normal to all alls, as shon n Fg. 1. In ths ork, a full-staggered grd s used. The dscrete form of the momentum and energy equatons and all the boundary condtons are obtaned by applyng a second order central dfference scheme. Whle for boundary nodal ponts, the one-ay dfference scheme s appled (.e., forard for loer and nlet boundares and backard for upper and outlet boundares. One convergence crtera s a mass flux resdual less than 10-8 for each control volume. Another crtera that s establshed for the steady state flo s ( +1 - / here represents any dependent varable, namely u, v and, and s the number of teraton. Fg. 2 Valdaton of the numercal code th avalable results ISSN: (Prnt; ISSN: (Onlne WE 2010

4 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. VII. RESULTS AND DISUSSION In ths secton, the results of computer program based on the mathematcal model developed n the prevous sectons are presented. The velocty feld, local temperature feld, frcton factor and local Nusselt number through the avy mcrochannel are exhbted. To reduce the computaton ork, only one half of mcrochannel, shon n Fg. 1, s consdered due to the symmetrcal condtons. Hoever, the results presented n the results secton are shon for the hole mcrochannel. The boundares are mantaned at temperature T =70 o and the unform nlet temperature s consdered T =25 o. The tangental momentum accommodaton coeffcent σ v and the thermal accommodaton coeffcent σ T are set at 0.9. The results are obtaned for the specfc heat rato γ=1.4 and Pr=0.7. Also geometry parameters s taken λ=2. A. The flo feld The effect of Knudsen number on slp velocty n the avy mcrochannel s shon at Fg. 4. By ncreasng the Knudsen number, the channel dmensons decrease and approaches to molecular dmensons and the effect of slp velocty ould become more and more mportant. Moreover, n the convergent regons, the average velocty ncreases that contrbutes to a rapd rase n the slp velocty n ths regon. Fg. 5 Schematc llustraton of Knudsen number effect on velocty profle at Re=2 and a=0.2 The varatons of f.re along the mcrochannel for varous Knudsen numbers n the hydrodynamcally/thermally developng regon are depcted at Fg. 6. It s evdent that there s hgh frcton at the entrance regon due to presence of hgh velocty gradents. Hoever, t rapdly decreases as the flo develops. Moreover, rarefacton has a decreasng effect on the frcton factor. Ths effect can be nterpreted mathematcally. Eq. (10 shos that f.re depends on the average velocty and the gradent of tangental velocty. As Knudsen ncreases, because of a fxed Re e have hgher Mach number that results n greater average velocty. In addton, larger Knudsen number decreases the slope of velocty near the all and ths means havng lesser tangental velocty gradent. Therefore, n accordng to the prevous explanaton, the larger Knudsen number causes the lesser f.re. Fg. 4 Varaton of slp velocty along the avy mcrochannel th Knudsen number Re=2 and a=0.2 Fg. 5 compares the velocty profle n dfferent Knudsen numbers and n dfferent cross sectons. It schematcally shos hen rarefacton ncreases, the slp velocty values become greater. In addton, n each Knudsen number as the flud approached throttle regons, the slp velocty becomes more consderable because of ncreasng the average velocty. Fg. 6 Varaton of f.re along the avy mcrochannel th Knudsen number at Re=2 and a=0.2 As t can be observed n Fg. 6, hen the flud flos n the dvergent regon, f.re experences a rapd decrease n the mcrochannel because of decreasng the average velocty. ISSN: (Prnt; ISSN: (Onlne WE 2010

5 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. mcrochannel s presented n Fg. 9. As expected, very hgh heat transfer rate s experenced n the entrance regon of the mcrochannel due to hgh temperature gradent. As expected also, hgh heat transfer rate dmnshes rapdly as the thermally developng flo approaches the fully developed flo. Because of ncreasng of the average velocty and especally slp velocty n the convergent regons, there s a jump n the local Nusselt n these regons. Besdes, Nusselt number n the mcrochannel loer as the rarefacton ncreases. As already sated, hen rarefacton ncreases the temperature jump s ntensfed. The temperature jump means the absolute dfference beteen the average temperature and all temperature. So the temperature jump decreases the Nusselt number by ncreasng the absolute dfference of the all temperature and mean gas temperature. Fg. 7 Varaton of f.re along the avy mcrochannel th geometry at Re=2 and Kn=0.075 Fg. 7 shos the varaton of f.re as a functon of ampltude of the ave hle keepng the Reynolds number and Knudsen number constant. By ncreasng ampltude of the ave, the flud flo senses the varaton of cross secton more. As t can be seen by ncreasng ampltude of the ave and subsequently the more decrease n the average gradent of tangental velocty, f.re experences more ntense decrease n the dvergent regon as explaned for Fg. 6. B. The temperature feld The varaton of flud temperature near the all along the mcrochannel for varous Knudsen numbers s depcted n Fg. 8. As shon, the larger Knudsen numbers, the hgher temperature jumps. By decreasng of the channel dmensons, the thckness of the Knudsen number layer ncreases that brng about further temperature jump. In addton, t s found that ths effect s gradually dmnshng as the flud flo approaches the developed regon. Fg. 9 Varaton of local Nusselt number along the avy mcrochannel th Knudsen at Re=2 and a=0.2 Fg. 10 shos the varaton of Nusselt number as a functon of ampltude of the ave hle keepng the Reynolds number and Knudsen number constant. By ncreasng ampltude of the ave, the flud flo senses the varaton of cross secton more. As t can be seen by ncreasng ampltude of the ave, Nusselt number experences much larger fall n the dvergent regon due to decrease n the average velocty. Fg. 8 Varaton of slp temperature th Knudsen number along the avy mcrochannel at Re=2 and a=0.2 The effect of Knudsen number on local Nusselt number for hydrodynamcally/thermally developng flo n the avy Fg. 10 Varaton of local Nusselt number along the avy mcrochannel th Knudsen at Kn=0.075 and Re=2 ISSN: (Prnt; ISSN: (Onlne WE 2010

6 Proceedngs of the World ongress on Engneerng 2010 Vol II WE 2010, June 30 - July 2, 2010, London, U.K. VIII. ONLUSION In ths study, the contnuum approach th the velocty slp and temperature jump condton at the sold alls s appled to develop the mathematcal model of flo phenomenon n the avy mcrochannel. These equatons are solved by SIMPLE algorthm n curvlnear coordnate. The flo and heat transfer characterstcs of lamnar ncompressble gaseous flo n a avy mcrochannel are analyzed. The effects of Knudsen number and geometry on thermal and hydrodynamc characterstcs of flo n the avy mcrochannel at constant Reynolds number are llustrated n ths ork. It s found that the Nusselt number and f.re decrease th Knudsen number. It s also found that f.re and Nusselt numbers experence a rapd jump n the convergent part and ths jump s more consderable for loer Knudsen numbers. In addton, the model successfully predcts the groth of temperature jump and slp velocty th Knudsen number at the sold alls. Moreover, by decreasng ampltude of the ave, the varaton of Nusselt number and f.re n the avy regon become more ntense. NOMENLATURE a ampltude of the ave (m k thermal conductvty of ar (W/m.K h local heat transfer coeffcent (W/m 2.K J jacoban of the coordnate transformaton p dmensonless pressure Re Reynolds number (Re= u L / Pr Prandtl number (Pr = / Nu local Nusselt number Nu fully developed Nusselt number Kn Knudsen number Ma Mach number Pe Peclet number Ec Eckert number f skn-frcton coeffcent c p specfc heat (J/kg K n dmensonless normal drecton to the all s dmensonless tangental drecton to the all R gas constant (J/kg.K T temperature (K q heat flux u dmensonless velocty component n x-drecton v dmensonless velocty component n y-drecton L channel nlet dth x dmensonless horzontal coordnate y dmensonless vertcal coordnate Greek Symbols a thermal dffusvty(m 2 /s surface avelength (m densty of flud (kg/m 3 dynamc vscosty (kg/m.s rato of specfc heats (c p /c v molecular mean free path (m knematc vscosty(m 2 /s T energy accommodaton coeffcent momentum accommodaton coeffcent dmensonless temperature curvlnear horzontal coordnate curvlnear vertcal coordnate τ Subscrpts ave s shear stress mean value surface condtons nlet condtons flud property near the all Superscrpt contravarant veloctes tang tangental drecton returns to dmensonal parameters REFERENES [1] Jennfer van Rj, Todd Harman, Tmothy Ameel, The effect of creep flo on to-dmensonal soflux mcrochannels, Internatonal Journal of Thermal Scences 46 ( [2] Lütfullah Kuddus, Edvn Çetegen, Thermal and hydrodynamc analyss of gaseous flo n trapezodal slcon mcrochannels, Internatonal Journal of Thermal Scences 48 ( [3] Graur IA, Me olans JG, Zetoun DE Analytcal and numercal descrpton for sothermal gas flos n mcrochannels, Mcroflud Nanoflud 2 ( [4] Jj LM, Effect of Rarefacton, Dsspaton, and Accommodaton oeffcents on Heat Transfer n Mcro cylndrcal ouette Flo, ASME J. Heat Transfer 130 ( [5] hen S, Kuo WJ, Heat transfer characterstcs of gaseous flo n long mn- and mcrotubes, Numercal Heat Transfer; Part A: Applcatons 46 ( [6] Larrode FE, Housadas, Dreossnos Y, Slp-flo heat transfer n crcular tubes, Int J. Heat and Mass Transfer, 43 ( [7] Kavehpour HP, Faghr M, Asako Y, Effects of compressblty and rarefacton on gaseous flos n mcrochannels, Numercal Heat Transfer; Part A: Applcatons 32 ( [8] heng RT-S, Numercal Soluton of the Naver-Stokes Equatons by the Fnte Element Method, Phys. Fluds 15 ( [9] Vrads G, Zalak V, Bentson J, Smultaneous, varable solutons of the ncompressble steady Naver-Stokes equatons n general curvlnear coordnate systems, Trans. ASME I: J. Fluds Engng. 114 ( [10] Wang, hen K, Forced convecton n a avy-all channel, Internatonal Journal of Heat and Mass Transfer 47 ( [11] E.B. Arklc, K.S. Breuer, M.A. Schmdt, Gaseous Flo n Mcrochannels, ASME Applcaton of Mcrofabrcaton to Flud Mechancs 197 ( [] A. Beskok, G.E. Karnadaks, W. Trmmer, Rarefacton and compressblty effects n gas mcroflos, Journal of Fluds Engneerng, Transactons of the ASME, 118 (3 (1996 pp [13] Kennard EH, Knetc theory of gasses, Ne York: McGra-Hll (1938. [14] Gombos TI, Gas knetc theory, Ne York: ambrdge Unversty Press (1994. [15] Gad-el-Hak M, The MEMS Handbook, R Press LL, Boca Raton (2002. [16] Kandlkar S, Garmella S, L D, oln S, Kng MR (2006, Heat Transfer And Flud Flo In Mnchannels and Mcrochannels, Elsever, Brtan. [17] Patankar SV, A calculaton procedure for heat, mass and momentum transfer n three-dmensonal parabolc flos, Int J Heat Mass Transf 15 ( [18] Spaldng DB, A novel fnte dfference formulaton for dfferental expressons nvolvng both frst and second dervatves, Int J Numer Methods Eng 4 ( [19] Hoffman KA, omputatonal flud dynamcs for engneers, Engneerng Educaton System, Austn (1989. ISSN: (Prnt; ISSN: (Onlne WE 2010