Pseudo-particle modeling for gas flow in microchannels

Chinese science Bulletin⊙2007◆ Science in China PressPseudo-particle modeling for gas flow in microchannelsWANG LIMin 1,2 GE Weit CHEN FeiGuo1 21 State Key Laboratory of Multi-phase Complex System, Institute of Process Engineering, Chinese Academy of Sciences, Beijing00080,Chna;The velocity profiles and temperature distributions of gas flow in microchannels, for Knudsen numbersranging from 0.01 to 0. 20, are investigated with pseudo-particle modeling(PPM). It has been found thatthe velocity profiles are mainly affected by Knudsen number and the external force fields applied. WhenKnudsen number was increased, the slip velocities on the walls increased at the beginning, and thendecreased. The temperature distributions were also significantly affected by the external force. TheDarcy friction factor increased with increasing Knudsen number, and its variation with Mach numberunder increased Knudsen number was similar to the so-called premature laminar-turbulent transitionobserved in experiments.pseudo-particle modeling, microchannel, Knudsen number, Mach numberResearches on micro flows can be traced back to the factor is under-predicted by classical correlations. Li etdesigning of high-performance compact micro heat ex- al. have also found that the Fanning friction factors forchangers. It is found that miniaturized fluidic circuits gas flow in micro circular channels are larger than theare insensitive to electromagnetic interference and may theoretical values for laminar flow. But Pfahler et al. 81find applications in precise control and also in medicine found that for nitrogen gas and silicone oil liquids infor implanted drug-delivery devices. In addition, mi- microchannels, the friction factors were consistently lesscro-reactor system can enhance the mixing, mass and than the theoretical laminar values. Choi et al. 9 alsoheat transfer in reactive processes, resulting in great reported a friction factor being 17% below theoreticalelevation in the output rate. The micro-reactor tech- predictions for nitrogen flow in micro circular channelnique has been applied to catalytic combustion of hy. In addition, Wu and Little 5. b found that the criticaldrogen/air mixture, direct fluorination of aromatic com- Reynolds number for transition to turbulent flow obpounds and so onservably reduces for gas flow in miniature conduits, formicro flows, however, forces and phenomena newhich the relatively high roughness in microchannels isligible in macroscale may turn to be dominant. For in- generally believed to be the cause ostance, the flow appears to be granular for liquids and Essentially, the number of influential factors forrarefied for gases, and the walls"move"4. Therefore, highly non-equilibrium micro flows is far more than thatraditional hydrodynamics and transport theories are not for macro flow and quasi-equilibrium flow. If a model isreadily applicable to such flow. Theoretical research in compared with experiments or other models based onthis field is not on a par with the technological devel- only a few criteria, neglecting relevant molecular detailsopment, and is becoming a hinder of the latterwhich still lack quantification at present, confusionExperimental research into micro flows also encounters many difficulties because of its complexity. Fordoi;10.1007/s11434-007-0075-6example, the friction factors of gas flow in microchanSupported by the National Natural Science Foundation of China(Grant Nosnels are in dispute. The measurements of Wu and Lit- 20336040, 20490201 and 20221603), and the Knowledge Innovation Program of the,b for gas flow in miniature conduits showed that the Chinese Academy of Sciences(Grant No. KJCX-Sw-L08)www.scichina.comwww.springerlink.comcome yH中国煤化工:1m0.41450455CNMHGbuild a strict relationship between the simulation results spatial homogeneity and isotrope that can guaranteealmost inevitable. In other words, it is, so far, hard to processed in a predefined sequencuOand a highly non-equilibrium micro flows in realiTherefore. it should be beneficial to investigate microflows on a simplified model so as to understand someaspects of their behavior firstFor this purpose, we note that particle methods are0o∞o0flexible in building up simple and ideal models that arenot restricted by experimental conditions and are phcally reliable in terms of this discrete nature ll. To dateslip flow and tranflow are direct simulation Monte-Carlo (DSMC) 2. 13.(LBM)4, but they still890000OOface difficulties in dealing with dense and highlynon-equilibrium gas flow. Whereas more rigorous mo-lecular dynamics (MD) tools may complicate theQGnGg ooproblem by introducing many molecular and interfacialdetails. In this paper, we employ microscopic pseudoFigure 1 Simulation layout. 1, Pseudo-particle(PP); 2, velocity of thepseudo-particle; 3, side walls; 4, periodic boundary in the axial directionlingl6-18 to simulate gas flow in microchannels with hard sphere directly. The Knudsen num-In this work, periodic boundary conditions are appliedbers(Kn) range from 0.01 to 0.20. We wish that the ba- in the flow direction. The tangential momentum aosic transport characteristics obtained in such flow can commodation coefficient 9 o on the fluid-solid interprovide benchmarks for more sophisticated experiments face is taken as 0.75 that is. 75% molecules are re-and simulationsflected diffusely and 25% molecules are reflectedspecularly. The landscape of surface roughness and the1 Simulation methoddetails of the reflection process are not considered. ThePPM is a particle method proposed as a combination of luid is driven by a constant external force, which avoidsAD and DSMC. As shown in Figure 1, PPM discretizes the non-uniformity of mean free path (a)and localKnudson numbers along the flow direction in preslave fourproperties: mass(m), Sure-driven flow. This enables us to separate in the studyradius(r), position(P)and velocity (v), among which mof flow and transportation behavior the effect of geand r are held constant in the simulation. All particles metrical size and boundary conditions from other fac-move synchronously in uniform time steps(4t). In eachtorsThroughout this work, the simulation parameters aretime stepparticlesunder some external forces. At the end of each step, ifcast into dimensionless forms by taking r=l, m=l andthe distance between two particles PI-P2l is less thanAt=l. The thermal velocity of the pps is taken to be 0.02the sum of their radii (2r), and the inner product of their packing fraction /=0.087, and their mean free path zPi-P2 and vI-v2 is negative, they will collide as two 1=5.503. The height of the microchannel H=2400,rigid and smooth particles, i.e. they do not move but its width W=AKn=30-600. The number of PPs in the yacquire new velocities as followssimulation is therefore 2000-40000. The sizes of the2m2(v1-v2)(B1-P2)channels are about 5.6-111x222 nm if the PPs are com-sV1=V1+P1-P2)2P2-P1)parable to Argon under 273 K and I atm, which is still inuthe nanometer range. Therefore, the phrase"microIn the next time step, the particles move to new posi- flows"in this paper actually refers to micitions with new velocities, and so on. Collisions are rather than micrometer flow in particular oscopic flowWANG LiMin et al. Chinese Science Bulletin February 2007 I veH中国煤化工CNMHG2 Results and discussion016F(a)g=1.00.014Figure 2 shows the velocity profile in microchannel flow▲-g-30×100.012under different Knudsen numbers for g=10. As showng4.0×10slip on the walls, the velocity0.010profiles in the center region of the microchannels are0.008still parabolic. When the Knudson number is in th0.006range of 0.05-0.20, the absolute slip velocity decreaseswith the increasing Knudsen number while the relativeslip velocity increase-0.10.00.102030405060708091.01.11200124(a)00100.00.006g=1.0×10°g=2.0×10▲-g=30×106g-40×1060.002-g=5.0×100.10.0010.20.304050.6070.8091.01.1(b)13F(c-g7.0×10=9.0×0.6g=1.0×100.4-8=1.2×100.10.00.020.3040.50.60.70.80.910L.I0.1000.1020304050.60.70.809xMFigure 2 Velocity profile in microchannel flow under different Knudsen Figure 3 Velocity profile in microchannel flow under different externalnumbers for 8=10.(a)Absolute velocity;(b)relative velocity. . Kn= force fields for Kn=0. 1834.(a) Absolute velocity in weak external force0.1843;,Kn=0.1152;★,Kn=0.0922;◆,Kn=0.0709;,Knfield;(b)relative velocity in weak external force field; (c)relative velocity0.0614:■,Kn=0.0512in strong external force fieldFigure 3 shows the velocity profile in microchannelflow under different external force fields for Kn=0. 1834n(x)3(x/W)(1-x/w)+31KAs shown in Figure 3(a), the absolute slip velocity in2-3Kncreases with the increasing intensity of external forcewhere u is the cross-sectional average velocity. Com-fields. But as shown in Figure 3(b)and(c), the relative pared with the empirical results 20l in the slip flow re-slip velocity does not change accordingly. Thus, a cor-relation for the distribution of relative velocities in the()21-(/R)2]+8Knrange of Kn=0.01 to 0.20 is found based on the simulation data in figure 3. thatalthough the constants in our correlation is dependent on452WANG LiMin et al. Chinese Science Bulletin February 2007 vO中国煤化工CNMHGthe boundary conditions(e.g. Ov)we choose, it can beuObeen verified by more precise event-driven hard sphere dfound that the general features are consistent in that the simulations, suggesting that it may be related to therelative velocity distribution is only dependent on the second-order effects of Burnett equation21)Knudsen number and takes a parabolic form in the cenFigure 5 shows the wall slip velocities in microchan-tral regionnel flow under different g and Kn. They are found teThe heat generated by viscous dissipation is absorbed increase with Kn at first, and then decrease. with thein the isothermal walls, which leads to temperature gra- same Knudsen number, stronger external force field re-dients in the radial direction as will be discussed in the sults in higher slip velocity. a qualitative analysis is asfollowing. In general, the temperature variation is not followspronounced, the relative difference between the highestand the lowest temperatures is under 10%00.030Figure 4 shows the temperature distributions in mi-025cochannel flow under different external force fields forKn=0. 1834. As shown in Figure 4(a), the temperature0.020distributions in the center of microchannels is nearly flatunder low intensity of the external force field, and thehighest temperature presents on the centerline. But when0.010highest temperature location shifts to the near wall region and the temperature on the centerline decreasesaccordingly. This seemingly strange phenomenon has0000.020.040.060080.100.120.140.160.18020Figure 5 Wall slip velocity in microchannel flow under different g andKI1.0×10:·,g=50×10;▲,g=10×10;,g=2.0×10°g=50×106;,g=1.0×103;·,g=15×10-3;★,g=2.0×10Firstly, by neglecting thermal effects, the generalsecond-order slip condition for isothermal gas flow in0.96g=20×10°microchannels 4 has the non-dimensional form0.94-g-4.0×105.0×10°Lt1000.1020.30405060.70.809101.1where B=/Ou1.041.02小imnn2×2, n which n is the di-s coordinate perpendicular to the wall whichnormalized by the channel width, and w indicates thatthe variables are taken on the wall. For continuum flow(3)redell-known no-slipg-7.0×108-=8.0×106On the other hand the force balance for the gas flowin microchannelsgl.I×l0pgH,0.1000.l020.3040.50.60.70.8091.01.1where p donates gas density. If the gas viscosity u can 5external force fields for Kn=0. 1834.(a) Weak external force field; (b) be considered as a constant,will increase withstrong external force fieldWANG LiMin et al. Chinese Science Bulletin February 2007 vOTYH史453stronger external force field with the same Knudsen that is fo Ma. with the increase of Ma, the decrease ofnumber, and slip increases too. From eq. (1), we have f slows down gradually and finally tends to a constantalue. The mach number at thisincreases with Kn for3Knwith increasing Kn. Such trends are similar to the pre-is constant while the b decreases with the in- Reynolds numbers at the turning points here are abouttens to hundreds. The physical implications of this transe of Kn. In the low Kn range, it may incisition in such a microscopic flow are, therefore, stillthan B, so the slip velocity increases with Knsubjected to further researcha certain value, the increase of Kn may be slower, andthe slip velocity decreases with increasing Kn3 ConclusionFigure 6 shows the Darcy friction factor (f) depictedas a function of the Mach number in microchannel flow Scaling-down from traditional flow to micro flows, justunder different Knudsen numbers. Because of the scale like widely practiced scaling-up processes, involveseffect on fluid viscosity in micro flows, rigorous defini- typical nonlinear phenomena and multi-scale structurestion and precise measurement of this property are diffi- Multi-scale modeling is, therefore, necessary and particult under highly non-equilibrium conditions, and hence cle methods are promising candidates!l as they natuthe Reynolds number, which is closely related to viscos- rally deal with intermediate scales between the atomisticity, cannot readily characterize the micro flows. Howand continuum levels. In this work, we use pseudo-par-ever, Kn and Ma are less influenced by the scale effect, ticle modeling for the numerical investigation of microas they are well defined in equilibrium. Thereforeflows. which can avoid the errors and uncertainties inshould be more meaningful to use Ma and Kn as primary experiments and investigate the influence of Mach andcriteria in partitioning the flow regimes in micro flowsKnudsen numbers on transport properties independentlywn in Figure 6, f increases with the increase ofThe result indicates that the velocity profiles are mainlyKn in microchannels For low mach numbers aaffected by the Knudsen number. when Knudsen num-power-law dependence can be found between and Ma,ber was increased the slip velocities on the walls increased at the beginning and then decreased. The tem-■一0.1834perature distributions were also significantly affected by▲-0.09◆-0.0459the external force. In weak external force field. the-0.0229highest temperature presents in the center. But inv-0.0153SGforce field. the highest temperature+-0.0092presents in the near wall region. In additionfriction factor increased with increasing Knudsen num▲▲▲ber. and its variation with mach number and with in-creased Knudsen number was similar to the so-calledpremature laminar-turbulent transition observed in experimentsMach number ( Ma)Figure 6 The Darcy friction factor depicted as a function of Ma in mi- The authors wish to thank Prof. Li Jinghai, D: Li Ying and Prof. Luocochannel flow under different KLingai for illuminative discussion.I Tuckerman D B, Pease R F w. High performance heat sinking forDissertation for the Doctoral Degree. Dalian: Dalian Institute ofVLSL. IEEE Electron Device Lett. 1988. 2: 126-129Chemical Physics, Chinese Academy of Sciences, 2005. 106-1172 Groisma A, Enzelberger M, Quake S R, Microfluidic memory and 4 Karniadakis G E, Beskok A Micro Flows: Fundamentals and Simu-control devices. Science. 2003. 300: 955--958lation. Berlin: Springer-Verlag, 20023 Cao B. Transport and reaction in microchannel reactor(in Chines5 Wu P Y Little wA. Measurement of friction factors for the flow of454WANG LiMin et al. Chinese Science Bulletin February 2007 I ve中国煤化工CNMHGgases in very fine channels used for microminiature Joule-Thomson 14 Chen S Y, Doolen G D. Lattice Boltzmann method for fluid flowsuOrefrigerators. Cryogenics, 1983, 23: 273-27Annu Rey Fluid mech. 1998. 30: 329--3646 Wu P Y, Little WA. Measurement of the heat transfer characteristics 15 Ge W, Li J H. Pseudo-particle approach to hydrodynamics of partivery fine channels used for microminiature Joulecle-fluid systems In: Kwauk M, Li J, eds. Proceedings of the 5thThomson refrigerators. Cryogenics, 1984, 24: 415-420International Conference on Circulating Fluidized Bed. Beijing7 LiZX, Du DX, Guo Z Y. Experimental study on flow characteristicsScience press. 1996. 260-265of liquid in circular microtubes. Microscale Thermophys Eng, 200316 Ge W. Multi-scale simulation of fluidization. Dissertation for the7(3):253-265Doctoral Degree. Harbin: Harbin Institute of Technology, 19988 Pfahler J, Harley J, Bau HH, et al. Liquid and gas transport in smallchannels. ASMC Proc DCS. 1990. 19: 149-15417 Ge w, Li J H. Macro-scale phenomena reproduced in microscopic9 Choi S B, Barron R F, Warrington R O. Fluid flow and heat transfer insystems- pseudo-particle modeling of fluidization Chem Eng Sci,microtubes. ASMC Proc DCS. 1991. 322003,58(8):1565-158510 Pfund D, Rector D, Shekarriz A Pressure drop measurements in a 18 Rapatort D C. The Art of Molecular Dynamics Simulation. Cam-microchannel, AIChE J. 2000. 46: 1496-1507bridge: Cambridge University Press, 200411 Ge W, Ma J S, Zhang j Y. et al. Particle methods for multi-scale19 Wang L M. Ge W,Li J H. A new boundary condition in particleimulation of complex flows. Chin Sci Bull, 2005, 50(l1)methods. Comput Phys Comm, 2006, 174: 386-3901057-106920 Barron R F, Wang X M, Ameel T A, et al. The Graetz problem ex-12 Bird G A. Molecular Gas Dynamics. New York: Oxford Universittended to slip-flow. Int J Heat Mass Transfer, 1997, 40(8)1817-182313 Bird G A Molecular Gas Dynamics and the Direct Simulation of Gas 21 Ge w. Personal communication. Institute of Process Engineering,Flows. New York: Oxford University Press, 1994Chinese academy of Sciences, 2006Science in China Series B: ChemistryEDITORXU Guangxian(Hsu, Kwang-Hsien)College of Chemistry and Molecular EngineeringAIMS AND SCOPEcience in China Series B: Chemistry, an academic journal cosponsored by the Chinese Academy of Sciences and the National NaturalScience Foundation of China, and published by Science in China Press and Springer, is committed to publishing high-quality, original re-sults in both basic and applied researchScience in China Series B: Chemistry is published bimonthly in both print and electronic forms. 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