Simulation of rarefied gas flow and heat transfer in microchannels

Vol. 45 No. 3SCIENCE IN CHINA( Series E)June 2002Simulation of rarefied gas flow and heat transfer inmicrochannelsWANG Xian(王娴y, WANG Qiuwang(王秋旺y,TAO Wenquan(陶文铨y& ZHENG Ping(郑平School of Energy Power Engineering Xi an Jiaotong University XI an 710049, China2. Department of Mechanical Engineering, Hong Kong University of Science and Technology Hong Kong, ChinaCorrespondenceshouldbeaddressedtoWangQiuwangemail:wangqw@xjtu.edu.cnReceived june 18 2001Abstract Analysis and simulation of rarefied nitrogen gas flow and heat transfer were performedwith the Knusden number ranging from 0. 05 to 1.0, using the direct simulation of Monte Carlo(DSMC )method. The influences of the Kn number and the aspect ratio on the gas temperature andwall heat flux in the microchannels were studied parametrically. The total and local heat fluxes of themicrochannel walls varying with the channel inlet velocities were also investigated in detail. It wasfound that the Kn number and the aspect ratio greatly influence the heat transfer performance of microchannels, and both the channel inlet and outlet have higher heat fluxes while the heat flux in themiddle part of channels is very low. It is also found that the inlet free stream flow velocity has smallaffect on the wall total heat flux while it changes the distribution of local heat fluxKeywords: microchannel, Kn number DSMC, fluid flow and heat transferFluid flow and heat transfer can be studied at either macroscopic or microcosmic levels. Therapid development of mems has enabled us to investigate the fluid flow and heat transfer in mcro-dimensional devices 1. However the Navier-Stokes equation used for continuum flow is oftennot suitable for the fluid flow described by Boltzmann equation in micro-dimensional device. Thegas flow can be divided into four domains based on the Kn number which gives the rarity of thegases. It is defined asKn=入/L(1)where A is the gas mean free path i L is the characteristic length. Generally the Navier-Stokesequation loses its validity when Kn>0. 1, so the gas macroscopic properties should be obtainedby the statistical method at the microcosmic level. But based on the current knowledge it is veryhard to obtain the analytical solution of boltzmann equation For these reasons to satisfy theneeds for numerical simulations of rarefied gas Bird et al. proposed a new method to simulategas flow and heat transfer in the 1960s--direct simulation of Monte Carlo( DSMC )method 3It proves a powerful tool to solve such problems w中国煤化工In this paper, with the aid of DSMC methodCNMH Gation were made carefully aiming at the subsonic flow in a microchanel Wuurue m lumber langing from 0.05 to 1.0In order to investigate the influence factors on the heat transfer performance in microchannels,afew cases with different aspect ratios were simulated and compared for every Kn numberSCIENCE IN CHINA Series E)1 Numerical methodsDSMC method was derived from molecular gas dynamics i it does not solve the Boltzmann e-quation directly but imitates the real physical process described by this equation. In DSMCmethod, the real gas was modeled by a large number of simulated molecules. This probabilisticprocedure depends on the dilute gas assumption. The essential DSMC approximation is the uncoupling over a small time interval of the molecular motion and intermolecular collisions so wecan follow the trajectories of the very large number of simulated molecules simultaneously using acomputer and record their microcosmic properties. Then the macroscopic properties may be identified with average values of the appropriate molecular quantities at any location in the flow. Themolecule movement routines are trivial and a simple indexing of the molecules to sub-cells andcells causes all collisions to be between near neighbors. The procedures for the establishment ofthe correct collision rate are based on the cells while individual collision pairs. In order to makethe typical simulated molecules remain in a cell at least one time step the cell s dimension is ofthe order of the local mean free path. On the other hand to implement the uncoupling of molecular movement and collision the time step should be smaller than the mean collision timeNitrogen was used as the working fluid in this work Because of the differences among thestructures of monatomic diatomic and polyatomic molecules the solving methods involved in theDSMC procedure are also different. Monatomic molecules only have translational energy so theintermolecular collision should be regarded as elastic. However, diatomic and polyatomicmolecules also incluIde rotational and vibrational energy hence the collisions are inelastic. Butthe molecular characteristic temperature related to vibration is so high( for example the temperature for nitrogen is 3337 K )that the temperatures of most working conditions are not high enoughto excite the vibrational mode So in the common cases, it is reasonable not considering vibrationenergy in simulation[ 5]. If a collision is regarded as inelastic although the total energy does notange it may be reassigned between the translational and internal modes. Whether the total en-ergy is reassigned or not depends on the rotational and vibrational relaxation collision numbersZrot and Zib which are functions of the temperature. A general Larsen-Borgnakke distribution 3function for division of energy between the translational and internal modes was adopted in thispaper and the overall kinetic gas temperature may be defined as the weighted mean of the transla-tional and internal temperature based on the degrees of freedom3Tr+ srot Trot 5vibTV(2)where 3 is the number of translational degree of freedom. For DSMC method in spite of thespace dimension the velocity must be treated as three-dimensional Srot and swib stand for thenumber of rotational and vibrational degrees of freedom respectively i Ttr, Trot and Tvib stand forthe temperatures related to translation rotation and vibration respectivelyThe heat flux through a surface is also the result of molecule-surface interaction which can中国煤化工ECNMHG(3)where the subscripts i and r refer to the incident and reflect molecules respectively, E is the totalenergy including translational and internal energy m is the molecule mass while At and AareaThetfaperaure of the reflected molecules can be adjusted by the thermal accommodationMULATION OF RAREFIED GAS FLOW HEAt TRaNSFER IN microchannelscoefficientqia。(4)away by the reflected molecules when T,=Tw. It is obvious that ae ranges from 0 to/carriedwhere q; and qr are the incident and reflected energy fluxes while q w is the energy flux2 Physical problem and simulation results2.1 Physical problerThe microchannel model is shown in fig. 1. The gas in the channel is nitrogen a freestream at atmosphere with a temperature of 273K and velocity of 100 m/s flows into the mi-cochannel. The temperature of the two wall surfaces of the microchannel was kept at 500 KComputations were conducted for five Kn numbers: 0. 05,0. 08,0.25,0 75, and 1. 0. Foreach of them comparison was made among three aspect ratios that is, Ar=5, 15, 25The time step was 10 s and the stream boundary conditionTswas used at the inlet and outlet. Variable hard sphere( VHS )e.T.model was adopted as the molecular model 3].For the surfaceboundary diffuse reflection model was used in which the velocityof each molecule after refection is independent of its incident velocity. The velocities of the reflected molecules as a whole are dis-Fig.1. Microchannel sketchtributed in accordance with the Maxwellian distribution 3 1fc)(5)where cis the molecule thermal velocity m is the molecule mass K is the boltzmann constantand Tr is the temperature of reflected molecule. In this paper the thermal accommodation coefficlentunity that is, Tr= TwIn addition the number of the simulated molecule must be reasonable for both computationalfficiency and physical reliability. If the number is too small, the results will lose their statisticmeanings and if it is too large the computational time would increases greatly. According toBird 6], there must be 20-30 simulated molecules in a cell. During the simulation, the numbersof the simulated molecules in cells are usually not constant but change slightly in the subsonicflow. Therefore in our work the same weighted factor the number of actual molecules represented by one simulated molecule )was chosen in all cells and the average number of simulatedmolecules per cell is 25--352.2 Results and analysisThe variation of heat flux through microchannel s surface with the Kn number when Ar= 5is shown in fig. 2. By eq. (3), the negative heat flux indicates that heat is transferred from sur-faces to gas. Fig.2 shows that in the range of kn=0-10, which includes the whole transitionalregion the surface heat flux increases with an increase in Kn number. When Kn 1 the heatflux increases rapidly and when kn> l the increa中国煤化工 mainly because of thefact that the higher the kn number the more rareCN MH Gncrease in Kn numberthe molecule-surface collision frequency becomes much higher, while the collision of intermolecular decreases. In the free molecule region( Kn>10 ), the intermolecular collision frequency canbe ignored compared with that between molecule and surface. So in this circumstance the sur-face heat flux is mainly dependent on the molecule-surface collision. When the Kn number islow, the intemtfecular collision is dominant especially in continuum flow( Kn<10-3),and324SCIENCE IN CHINA Series E)10only those molecules close to the surface can collidewith the surface resulting in a decrease in the surface heat flux. That is why the surface heat flux of amicrochannel(10-10 W/m2)is much higher thanthat of the macroscopic channel usually less than10 W/m2). When the Kn number increases to a certain degree although the intermolecular collision decreases the number of molecules colliding with sur-face does not increase appreciably resulting in onlya moderate increase in the surface heat flux in the68 10 high Kn number regionFig. 3 shows the variation of local heat flux inFig.2. Variation of heat flux with the Kn number the flow direction under different Kn number whenhen ar=5AR=5, and fig. 4 shows the gas temperature varia-tion accordingly. From these two figures it can be found that the varying trend of heat flux is al-most the same for different Kn numbers the heat fluxes at inlet and outlet are much higher thanthose at the middle part of the channels. This phenomenon was also reported by mavriplis etal.[70.0E-1.0长kn005Kn=O-2.0360Kn=0.75"=0750.00.20.40.608Fig. 3. Variations of local heat flux in the flow direFig. 4. Variations of local temperature in the flowtion at different Kn numbers with AR= 5direction at different Kn numbers with ar= 5Figs. 5 and 6 show the variations of local heat flux and gas temperature in the flow directionat different AR when Kn =0. 25. Fig. 5 indicates that the higher the AR is the lower the localheat flux will be. While with the decrease in the aspect ratio the heat flux in the middle part in-creases. When AR is increased the gas entering the microchannel is heated rapidly the heatluxes increase in the inlet and outlet but are close to zero in the middle part because the gas temperature is nearer to that of the surface( fig. 6). Most part of the channel keeps almost adiabaticand the flow is almost isothermal. However the dimensional starting point of the adiabatic regionis nearly the same for different AR, in the order凵中国煤化工The phenomena shown in figs. 3-6 are quitCNMHGnuum flow. The maireason is that there are molecules entering into and escaping from microchannel both in its inland outlet and the number of which can be obtained by the theory of molecular gas dynamics 33Once the molecules enter into the microchannel the molecules move arbitrarily with their thermalvelocities in every direction with equal opportunity. Therefore there are molecules escaping fronboth inlerafFtlet of the channel. The energy of entering molecules in the channel inlet andMULATION OF RAREFIED GAS FLOW HEAt TRaNSFER IN microchannels325100.0500440-1.5AR=SAR=250.000.8Fig.5 variallocal heat flux in the fleFig. 6. Variations of local temperature in the nlow dition under different ar with Kn=0. 25rection under different AR with Kn =0. 25outlet is much different from that of the molecules in the channel. After the gas enters into chan-fied, and almost all entering molecules would collide with theface so there is a large amount of energy exchange at the channel inlet and outlet. On the otherthe inlet and outlet, which causes a decrease in the mean molecular temperature. The nergy inhand the heat transfer is somewhat enhanced due to the escaping molecules with hightion of the above two factors with the first one dominating )results in a rapid increase influx in inlet and outlet. As long as the free stream velocity is larger than zero the number of entering molecules in the inlet will be larger than that in the outlet so the heat flux in inlet is muchhigher. In the middle part of the channel although there is a large amount of molecules collidingwith the surface the energy of these molecules is close to that of the surface through colliding frequently with the surface so that the molecule-surface interaction will exchange very small enerFor the smaller AR( such as AR=5 in fig. 6), the gas molecules have not enough chancesto collide with the surface before they escape from the channel which results in large energy dif-ference between the molecules and wall surface therefore heat exchange will happen all over thechannelFig. 7 shows the temperature distribution in 496the direction perpendicular to the gas flow at thecross section x/L=0. 3 when Ar=5. It can be 492seen that on the surface(y/H=0.0 and 1.0there exists a temperature slip region( that is the 488\知的3to the given wall temperature 500 K). With the in- E44crease in Kn number the slipping temperature dif-ference increasesFigs. 8 and 9 represent the variations in theiB.中国煤化工tal and local heat flux with subsonic stream velocCNMHGu oo with Kn=1.0 and AR= 5. Here the analysis.81.0is only focused on the cases with small AR. Fig. 8shows that the total heat flux varies with u oo verylightly but the distribution of local heat flux is Fig. 7. Temperature distribution in the direction perpd疚方鹳拇ig,9). When u o is lower,theA=5dicular to the gas flow at the cross section x/L=0. 3 withange326SCIENCE IN CHINA Series E)heat flux in the inlet and outlet is almost the same. With the increasing u oo the heat fluxcreases in the inlet and decreases in the outlet but in the middle part the heat flux hardlchanges with u oo. This is because when u oo increases the number of entering molecules in theinlet will increase and those in the outlet will decrease In our work the ratios between the enterng molecules in inlet and outlet for u o=10 m/s, 100 m/s, 300 m/s are approximately 1: 1, 21, 15: I respectively. So the increasing u enlarges the number of entering molecules whoseenergy level is quite different from the surface and the heat exchange due to the collision in-creases accordingly. When the outer macroscopic flow velocity is supersonic or even larger, thevacuum boundary condition can be used on the outlet of the channel i that is there are nomolecules entering in the outlet. Under this circumstance, the heat transfer trend in the outlet isalmost the same as that in the middle part of the channel and the heat flux will approach zeroThis point of view was also confirmed by Mavriplis et al. [71106-[ =10 m/sU =200 m/st =400 m/s50100150200250300040.60.81.0U,/m·s-Fig. 8. Variation of the total heat flux with the freeFig. 9. Variation of the local heat flux with the freem velocity with Kn =1. 0 and AR=5stream velocity with Kn =1. 0 and Ar=53 Conclusion(1)As far as the heat transferrochannel is concerned the heat flux is mainly locatedin the inlet and outlet. When AR is small the heat exchange will happen all over the channelWith the increasing AR, the middle region of the channel in which the heat flux approaches zerobecome larger and the gas flow is much closer to isothermal flow whose temperature is almost thesame as that of the surface. The total heat flux increases with the increasing Kn number and de-creases with increasing Ar1(2)The gas temperature slipping happens on the surfaces of the channel and the higher thenumber the greater the slipping temperature differencea microchannel, but it influences the local hew city has no effect on the total heat transfer rate of(3 )In subsonic flow the free stream vel中国煤化工Acknowledgments This work was supported by the NatG2000026303)and the National Natural Science Foundation of ChCNMHGRD of China grant NoRefe1. Liu,J., Micro/Nano-scale Heat Transfer( in Chinese ), Beijing: Science Press,2001,1-6mulation OF RAREFIED GAS FLOW HEat transFer IN microchannels3273. Bird, G, A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, New York Oxford University Pres4, Wu,Q. F, Chen , W. F,, DSMC Method in Thermochemistry and Nonequilibrium Flow of Rarefied Gas with High Tature( in Chinese ) Changsha: National University of Defence Technology Press,1999,61-715. Cheng, C. H., Liao, F. L., DSMC analysis of rarefied gas flow over a rectangular cylinder at all Knudsen numbers , J. Fluids Engineering 2000, 122: 720--7296. Bird, G.A., Molecular Gas Dynamics, Oxford: Claredon Press, 19767. Mavriplis C., Ahn ,J. C,, Goulard R. Heat transfer and flow fields in short microchannels using direct simulation MonteCarlo,J. Thermophysics and Heat Transfer, 1997, 11(4): 489--49TH中国煤化工CNMHG