Developing of laminar fluid flow in rectangular microchannels

Transcription

1 Developng of lamnar flud flow n rectangular mcrochannels A. AKBARINIA 1,*, R.LAUR 1, A. BUNSE-GERSTNER 1 Insttute for Electromagnetc Theory and Mcroelectroncs (ITEM) Center of Industral Mathematcs (ZeTeM) Unversty of Bremen GERMANY * a.akbarna@tem.un-bremen.de Abstract: -Two dmensonal ellptc dfferental equatons are solved numercally to nvestgate lamnar flud flow n rectangular mcrochannels. The model employs the Naver-Stokes equatons wth velocty slp at the boundary condton to smulate the flow behavor n mcrochannels. The numercal soluton s obtaned by dscretzng the governng equatons usng the fnte-volume technque. A numercal code was developed. The developed code s used to evaluate the effects of velocty slp, the flow rate and sze of mcrochannel on the flud flow n rectangular mcrochannels. The numercal smulatons are done on a wde range of the Reynolds number (Re), the Knudsen number (Kn) for three dfferent values of the wdth of mcrochannel. The results shows good agreement wth prevous publshed expermental data. The effects of rarefacton and flow rate on the flow behavor and the hydrodynamc developng flud feld are presented and dscussed. The developng profle of velocty and pressure due to dfferent Re and Kn, are shown. It s found that at a gven Re, ncreasng the Knudsen number causes to decrease the dmensonless pressure along the mcrochannel. An ncreasng n the Knudsen number reduces the maxmum velocty n the mcrochannels whle the entrance length ncreases at any Re. Key-words: mcrofludcs, lamnar flow, MEMS, mcrochannel, two dmensonal. 1 Introducton Throughout the last 0 years, the fronter of mcrofludcs has expanded rapdly. It has been shown that the technology of mcrofludcs has many ndustral applcatons such as lab-on-a-chp devces, heat exchangers, pumps, combustors, gas absorbers, solvent extractors, and fuel processors [1]. Mcrochannels, however, are the basc structures n these systems. Based on Kn number, Beskok [] classfed the flow n mcrochannel nto four flow regmes: contnuum flow regme (Kn 01), slp flow regme (01 < Kn 0.1), transton flow regme (0.1 < Kn 10) and free molecular flow regme (Kn>10). The flow n most applcaton of these systems, such as Mcro Gyroscope, Accelerometer, Flow Sensors, Mcro Nozzles, Mcro Valves, s n slp flow regme, whch s characterzed by slp flow at. Tradtonally, the no-slp condton at the s enforced n the momentum equaton and an analogous no-temperature-ump condton s appled n the energy equaton. Strctly speakng, no-slp/no-ump boundary condtons are vald only f the flud flow adacent to the surface s n thermodynamc equlbrum. Ths requres an nfntely hgh frequency of collsons between the flud and the sold surface [1]. In practce, the no-slp/no-ump condton leads to farly accurate predctons as long as Kn < 01. Beyond that, flow n devces shows sgnfcant slp snce characterstc length s on the order of the mean free path of the gas molecules. It means that the collson frequency s smply not hgh enough to ensure equlbrum and a certan degree of tangental velocty slp and temperature ump must be appeared. Slp at s the most mportant feature n mcro-scale that dffers from conventonal nternal flow. So, slp flow characterstcs are very mportant for desgnng and optmzng systems wth mcrochannel. Slp-flow heat transfer n parallel plate mcrochannels have been studed by numerous authors [14 17] for dfferent boundary condtons. Hadconstantnou and Smek [17] showed that for the slp-flow regon, the contnuum approach predcton s n good agreement wth the drect smulaton Monte Carlo (DSMC) results. The extended Graetz problem wth velocty slp and temperature ump at the has been studed analytcally and numercally for the cases of constant temperature and constant heat flux boundary condtons by several researchers [18 3]. Axal conducton effects were neglected n all solutons and the smplfed energy equaton was solved assumng hydrodynamcally fully developed velocty profle. Slpflow heat transfer n rectangular mcrochannels has been studed by several researches [4 9]. Yu and Ameel [4,5] studed lamnar slp-flow forced convecton under thermally developng flow for constant temperature and soflux boundary condtons. The ISSN: ISBN:

2 energy equaton was solved analytcally usng ntegral transform technque neglectng axal conducton and the heat transfer augmentaton due to rarefacton was studed. Renkszbulut et al. [6] solved slp-flow and heat transfer n the entrance regon of rectangular mcrochannels for cases where Prandtl number s equal to unty. Hettarachchet al. [1] studed threedmensonal slp-flow and heat transfer n rectangular mcrochannels wth velocty slp and temperature umps, they proposed a correlaton for the fully developed frcton factor. Wang et al. [3] nvestgated the flud flow n square mcrochannels expermentally and they measured the velocty of water n mcrochannels wth - mm wdth by usng a mcro- PIV technque. The obectve of the present paper s to study the effects of rarefacton on the hydrodynamc parameters of lamnar flud flow n rectangular mcrochannels at dfferent Re and Kn. Smultaneous effect of rarefacton, nertal force and vscous force on the entrance length, the developng of velocty and pressure are presented and dscussed. Governng equaton The schematc of the mcrochannel and coordnate system are shown n Fg. 1. L and W are the length and the wdth of the mcrochannel, respectvely. The flow s consdered along the X-axs and the channel length s chosen so that hydrodynamcally developed flow s reached at the ext. u n y x Fg.1. Schematc dagram of two dmenson rectangular mcrochannel Flud flow n rectangular mcrochannels has been consdered. The flow s lamnar, steady and ncompressble. The flud s water wth constant propertes. Dsspaton, pressure work and body forces are neglected. Rarefacton effects set velocty slp at the flud- nterface. Therefore, the steady state governng equatons descrbng flud flow n the mcrochannel n the Cartesan coordnate and n the tensors form are as follows: Contnuty: u = 0 (1) x The momentum equaton: L W u p u u ( ρ ) = μ( ) x x x () x Where ρ s densty, μ s dynamc vscosty, =1,, 3 and u =(u x, u y ) are velocty vectors. The momentum equatons are non-dmensonalzed usng followng dmensonless parameters:,, and (3) Where A s the cross sectonal area, s s the wetted permeter of the cross-secton and p s pressure. Thus: and also So: The non-dmensonalzed governng equaton becomes as follow: Contnuty equaton: U = 0 X Momentum equaton: 1 U P ( U U ) = ( ) X Re X X (4) (5) Where the Reynolds number s..1 Boundary condtons Ths set of nonlnear ellptcal governng equatons has been solved subect to followng boundary condtons: At the mcrochannel nlet, flud velocty profles are assumed unform and constant (u n ). At the flud-sold nterface: The velocty n y drecton s zero whle the flud flow satsfes velocty slp at the s. At the mcrochannel outlet (x=l) the dffuson flux n the drecton normal to the ext s assumed to be zero for velocty whle a zero pressure s assgned at the flow ext... Slp velocty boundary condton In a tradtonal (.e., contnuum) flow analyss, velocty contnuty s enforced along all flud nterfaces. In the no-slp-flow regon, a dscontnuty of the velocty and temperature at the flud nterface arses due to breakdown of the local thermodynamc equlbrum between the and the adacent flud. The velocty slp of the flud adacent to the s proportonal to normal velocty gradents at the flud nterface. The non-dmensonalzed tangental velocty slp at the flud nterface s expressed as [1,] σ V U 3 ( γ 1) Kn Re θ U = Kn + (6) σ V n π γ Ec X ISSN: ISBN:

3 s the transverse velocty gradent and s the tangental temperature gradent n the flud at the flud nterface. The frst term of ths equaton represents the slp-flow nduced by the transverse velocty gradent, whle the second term accounts the slp-flow nduced by the thermal creep. The second term becomes neglgble snce at the for moderate temperature gradent slp-flows and also due to the fact that t s second order n Knudsen number. The tangental momentum accommodaton coeffcent,, descrbes the nteracton of the flud molecules wth the. Generally the values of these coeffcents depend on the surface fnsh, velocty at the flud nterface, and are determned expermentally. It vares from near zero to unty for specular and dffuse reflectons, respectvely. For most engneerng applcatons, values of accommodaton coeffcents are near unty and consderng approxmate nature of the slp-flow they are taken as unty n ths present study. The non-dmensonal form of the smplfed slp-flow velocty boundary condtons can be expressed as: U U U = Kn (7) n.3 Implementaton of the slp-velocty boundary condton The followng process s presented at [1]. A control volume adacent to the boundary s shown n Fg.. A staggered-mesh arrangement s consdered n ths analyss. Slp velocty at the s gven by Eq. (7). By usng a frst order approxmaton for the normal velocty gradent n Eq. (7), the slp velocty can be expressed as a functon of the neghborng velocty, U e, as: Kn U = ΔY U Kn e (8) 1 + ΔY whch s then ncorporated n to the numercal soluton as a coupled boundary condton. A dscretzed momentum equaton for a control volume adacent to the can be wrtten as [19]: 0 (9) where the coeffcents a s present the convecton and dffuson nfluence at the sx faces of the boundary volume as shown n Fg.. The terms U and U N are the veloctes at the and lqud nterface, respectvely, whch could be assumed nfntesmal dstance apart. Flud ΔY U W U =0 U N U P U s Fg.. Control volume adacent to the. The velocty slp at the flud nterface can be wrtten as: Kn U U N U =, (10) n By usng a frst order approxmaton for the normal velocty gradent,,n Eq. (10) and rearrangng, the term (U P - U N ) can be wrtten as:, (11) By substtutng Eq. (11) nto Eq. (10), the dscretzed momentum equaton can be expressed for a boundary control volume as (1) 0 where U s zero and a modfed boundary convecton dffuson coeffcent,, can be defned as: 1 (13) Thus, the dscretzed equatons for boundary control volumes can be wrtten smlar to nternal control volumes wth a smple adustment to the boundary convecton dffuson coeffcent, a N, as shown n Eq. (13). 3 Numercal method and valdaton The sets of coupled non-lnear dfferental equatons were dscretzed usng the fnte volume technque as descrbed by Patankar [19]. A power law scheme was used for the convectve and dffusve terms whle the SIMPLER procedure was ntroduced to couple the velocty-pressure. A structured non unform grd dstrbuton has been used to dscretze the computatonal doman. It s fner near the mcrochannel entrance and near the where the velocty gradents are hgh. Several dfferent grd dstrbutons have been tested to ensure that the calculated results are grd ndependent. The selected grd for the present calculatons conssted of 00 and 40 nodes n the x and y drectons respectvely. As t shows n Fg.3 ncreasng the grd numbers does not change sgnfcantly the U e ISSN: ISBN:

4 x/x X=408 X= X=0.317 X=357 X=70, w=100 μm y/y Fg.3 Grd ndependent test y/w Expermental results [Wang et al., 009] Smulaton results w= mm X=408 X=601 X=1.336 X=.4646 X= , w=100 μm Fg.5 Developng of the velocty profle at and 0 y/w Expermental results [Wang et al.] Smulaton results w= mm velocty at the centerlne regon. The resultng set of algebrac equatons s solved usng lne-by-lne method. The soluton s assumed converged when 10 s satsfed for all ndependent varables. In order to demonstrate the valdty and precson of the model assumptons and the numercal analyss, the calculated velocty at developed regon s compared wth the correspondng expermental results carred out by Wang et al. [3] for water at n the two dfferent square mcrochannels. As t s shown n Fg.4 a sgnfcant agreement between the numercal and expermental results are observed at dfferent wdth of the square mcrochannels.therefore, ths smulaton s relable to use n rectangular mcrochannel. u/u n Fg4. Comparson wth the prevous expermental results at W= mm and W= mm 4 Results The developng process of velocty profles at dfferent Re for and W=100μm are shown n Fg.5. At ISSN: ISBN:

5 1.3 Re=5 0 w=100 μm 1 Kn=5 Re=5, w=100 μm Kn=5, W=100 μm 1 Kn=5 0, w=100 μm Fg.6 Developng of the velocty along the mcrochannel at and begnnng of mcrochannel, the velocty profles are very sharp near the and they are unform n the center. When the flud flows n the mcrochannel, velocty gradent near the becomes low whle the maxmum velocty appears at the centerlne. The fully developed condton takes place at X=70 for, but t takes place at X= for 0. (See also Fg.6. At a gven Kn and wdth of mcrochannel, ncreasng the mass flow (.e. ncreasng Re) postpones the fully developed condton. Increasng Re does not have any sgnfcant effect on the fully developed nondmensonal velocty. Ths means that although the mass flux ncrease but the fully developed non-dmensonal velocty does not change. The effects of the slp-velocty on the velocty profle at the developng regon are shown n Fg.7. Increasng Kn causes to decrease maxmum velocty at the mddle of mcrochannel, but the velocty near the augments due to an ncrease n the slp-velocty. at a gven flow rate, the velocty profles n a mcrochannel for hgh Kn are more unform than low Kn. Fg.7 Varaton of the velocty profles wth Kn at Re=5 and 0 The developng of pressure along the mcrochannel s shown n Fg.8. At the begnnng of the mcrochannel, the pressure profles are not unform and constant, but the pressure profles become unform and constant n streamwse drecton. At a gven Kn, an ncrease n the velocty (.e. an ncrease n Re) amplfes the mean pressure along the mcrochannel. After passng the entrance length of mcrochannel, the velocty becomes fully developed; therefore the gradent of pressure n x drecton (the pressure drop) becomes lnear. Increasng the Knudsen number from 0 to 0.1 decreases the pressure along the mcrochannel as s shown n Fg.9. The effects of the slp-velocty, wdth of the mcrochannel and Re on the varatons of the entrance length are shown n Fg.10. The dmensonless entrance length, X en, s defned as the dstance where the maxmum velocty reaches 99% tmes the correspondng fully developed value over hydraulc dameter. Increasng wdth of the mcrochannel ncreases the entrance length, because a bgger wdth of the mcrochannel causes to postpone the developng of the velocty. (See fgures 10a and 10. ISSN: ISBN:

6 p/(ρu n ) X=408 X= X=0.317 X=357 X=70, w=100 μm p [kpa] Re=5 0 w=100 μm p/(ρu n ) X=408 X=601 X=1.336 X=.4646 X= , w=100 μm Fg.8. Developng of the pressure profle at and 0 It s llustrate n Fg.10a and Fg.10b that the Knudsen number has an ncreasng effect on the entrance length for all wdths of the mcrochannel, but ts effect s more sgnfcant at hgher Re. Fg.9c shows that an ncrease n Re mplements the entrance length. When the Re s hgh, the flud can flow far from the nlet untl reaches the fully developed. 5 Concluson The two-dmensonal Naver-Stokes equatons wth the velocty slp at the boundary condton have been solved numercally to nvestgate the lamnar flud flow n mcrochannels. It has been shown that ncreasng the Knudsen number causes to decrease the maxmum velocty whle the velocty close to the augments. An ncrease n Kn reduces the pressure and the fully developed velocty along the mcrochannel. Increasng the Reynolds number postpones the fully developed condtons whle t does not have any sgnfcant effects p [kpa] Kn=5 W=100 μm Fg.9 Developng of the pressure along the mcrochannel at and on the non-dmensonal velocty. The Knudsen number has an ncreasng effect on the entrance length at any Re and any wdth of the mcrochannel, but ths effect s more sgnfcant at hgh Re. References: [1] G.H. Mohamed, J. Flud Eng. 11 (5) (1999) 5. [] A. Beskok, G.E. Karnadaks, J. Thermophys. Heat Transfer 8 (4) (1994) 355. [3] O. Aydn, M. Avc, Thermally developng flow n mcrochannels, J. Thermophys. Heat Transf. 3 (0) (006) [4] J.V. R, T. Harman, T. Ameel, The effect of creep flow on two-dmensonal soflux mcrochannels, Int. J. Therm. Sc. 11 (46) (007) [5] L. Bswal, S.K. Som, S. Chakraborty, Effects of entrance regon transport processes on free convecton slp flow n vertcal mcrochannels wth sothermally heated s, Int. J. Heat Mass Transf. 50 (7 8) (007) [6] N.G. Hadconstantnou, O. Smek, Constant-temperature Nusselt number n mcro and nanochannels, J. Heat Transf. 14 (00) ISSN: ISBN:

7 X en X en X en W[μm] 1 Kn= Re=5 0 W=100 μm W [μm] 1 Kn= c) Kn Fg.10 Varaton of the entrance length wth the wdth of mcrochannel, Re and Kn at 0 and c) W=100μm [7] Z.P. Duan, Y.S. Muzychka, Slp flow n ellptc mcrochannels, Int. J. Therm. Sc. 11 (46) (007) [8] R.F. Barron, X. Wang, T.A. Ameel, R.O. Warrngton, The Graetz problem extended to slp-flow, Int. J. Heat Mass Transf. 40 (8) (1997) [9] T.A. Ameel, R.F. Barron, X. Wang, R.O. Warrngton, Lamnar forced convecton n a crcular tube wth constant heat flux and slp flow, Mcrosc. Thermophys. Eng. 1 (4) (1977) [10] F.E. Larrodé, C. Housadas, Y. Drossnos, Slpflow heat transfer n crcular tubes, Int. J. Heat Mass Transf. 43 (000) [11] W. Sun, S. Kakac, A.G. Yazcoglu, A numercal study of sngle-phase convectve heat transfer n mcrotubes for slp flow, Int. J. Therm. Sc. 50 (17) (007) [1] M. Barkhordar, S.G. Etemad, Numercal study of slp flow heat transfer of non- Newtonan fluds n crcular mcrochannels, Int. J. Heat Flud Flow 5 (8) (007) [13] S. Yu, T.A. Ameel, Slp flow heat transfer n rectangular mcrochannels, Int. J. Heat Mass Transf. 44 (001) [14] S. Yu, T.A. Ameel, Slp-flow convecton n soflux rectangular mcrochannels, J. Heat Transf. 14 (00) [15] M. Renkszbulut, H. Nazmand, G. Tercan, Slpflow and heat transfer n rectangular mcrochannels wth constant temperature, Int. J. Therm. Sc. 45 (9) (006) [16] L. Kuddus, Predcton of temperature dstrbuton and Nusselt number n rectangular mcrochannels at slp condton for all versons of constant temperature, Int. J. Therm. Sc. 10 (46) (007) [17] L. Kuddus, E. Ce_tegen, Predcton of temperature dstrbuton and Nusselt number n rectangular mcrochannels at slp condton for all versons of constant heat flux, Int. J. Heat Flud Flow 4 (8) (007) [18] L. Ghodooss, N. Eg_rcan, Predcton of heat transfer characterstcs n rectangular mcrochannels for slp flow regme and H1 boundary condton, Int. J. Therm. Sc. 6 (44) (005) [19] S.V. Patankar, Numercal Heat Transfer and Flud Flow, Hemsphere, Washngton, [0] W.A. Ebert, E.M. Sparrow, Slp flow n rectangular and annular ducts, J. Basc Eng., Trans. ASME (1965) [1] H.D. M. Hettarachch, M. Golubovc, W. M. Worek, W.J. Mnkowycz, Three-dmensonal lamnar slp-flow and heat transfer n a rectangular mcrochannel wth constant temperature, Int. J. of Heat and Mass Transfer 51 (008) [] Mohamed Gad-el-Hak, The MEMS Handbook, MEMS: Introducton and Fundamentals, Second Edton, Taylor & Francs Group, 006. [3] Haol Wang, Yuan Wang, Measurement of water flow rate n mcrochannels based on the mcrofludc partcle mage velocmetry, J. of Measurement 4 (009) ISSN: ISBN: