THERMALLY DRIVEN GAS FLOWS IN CAVITIES FAR FROM LOCAL EQUILIBRIUM

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1 8 th GRACM International Congress on Computational Mechanics Volos, 1 Jul 15 Jul 15 TERMALLY DRIVEN GAS FLOWS IN CAVITIES FAR FROM LOCAL EQUILIBRIUM Giorgos Tatsios 1, Dimitris Valougeorgis 1 and Stefan K. Stefanov 1 Department of Mechanical Engineering Universit of Thessal Volos, GR-38334, Greece tatsios@mie.uth.gr; diva@mie.uth.gr Institute of Mechanics Bulgarian Academ of Sciences Sofia, Bulgaria stefanov@imbm.bas.bg Keords: Kinetic Theor, Rarefied Gas Dnamics, DSMC, Knudsen Number, GASMEMS. Abstract. The flo of a rarefied gas confined in a to dimensional cavit due to non-isothermal alls is investigated. To set-ups of boundar conditions are considered. In the first case the top and bottom alls are kept at constant temperatures and a linear temperature distribution is applied along the lateral alls and in the second case three alls are kept at constant lo temperatures and the fourth one is heated. Since the flo is far from local equilibrium in a ide range of the Knudsen number, a kinetic approach based on the Bolltzmann equation is required. The flo is simulated b the deterministic solution of the Shakhov kinetic model and stochasticall b implementing the DSMC method. Although the investigated flo configuration is simple, it is rich in non-equilibrium phenomena. 1 INTRODUCTION Rarefied gas flos are characterized b conditions far from local equilibrium. The measure of the departure from equilibrium is the Knudsen number, denoted as Kn, hich is defined as the ratio of the mean free path over a characteristic length of the problem. For Kn.1, i.e., hen the flo is in the so-called transition and free molecular regimes, the traditional Navier-Stokes-Fourier approach is no longer valid, and a more fundamental kinetic tpe approach should be applied. Such flos have latel received considerable attention due to their theoretical interest and their implementation in several emerging technological fields including vacuum packed MEMS [1], micropumps/microactuators [,3] and vacuum sstems [4]. The are also commonl applied in benchmarking of novel numerical schemes [5,6,7]. In the present ork a revie of some recent developments in temperature driven rarefied gas flos in cavities subject to to different boundar condition cases is presented. The first case concerns a cavit here the top and bottom alls are kept at constant temperatures, hile a linear temperature distribution beteen those temperatures is applied at the lateral alls [8]. In the second case the three alls are kept at constant lo temperatures hile the fourth is heated b a given heat flu distribution [9]. Modeling is based on the non-linear Shakhov model [1], hile the DSMC method [11] is used for benchmarking purposes. FLOW CONFIGURATION A rarefied monatomic gas is confined in a D enclosure of side length W. The phsical space variables are denoted b ', ' ith W / ' W / and ' W. Due to the temperature difference of the bounding alls a stead-state heat flo through the enclosure is developed defined b the heat flu vector Q Q ', ', Q ', ' and due to non-equilibrium (rarefied) phenomena a flo field is also developed, defined b the velocit vector U U ', ', U ', '. It is noted that in rarefied flo configurations, as the present ones, gravitational forces are commonl neglected. The gas temperature and number densit are denoted as T ', ' and N ', ' respectivel, hile the pressure is given b the equation of state P NkBT, here B k is the Boltzmann constant. At this point it is convenient to introduce the folloing dimensionless quantities:

2 N T P U U Q Q,, n,, p, u, u, q, q (1) W W N T P P P The dimensionless phsical space variables are 1/,1/,,1 and the quantities n,, p, u, u and q, q are the distributions of the dimensionless number densit, temperature, pressure, velocit and heat flu respectivel. The reference quantities are denoted b the subscript zero. The reference pressure is P N k T and the most probable molecular speed is k T / m, here m is the molecular mass. The B hard sphere model is used to model intermolecular collisions, providing for the viscosit the epression here is the viscosit at reference temperature T. The flo parameter characterizing the flo is the reference Knudsen number, defined as Kn B () PW The to boundar condition cases are distinguished: Case 1: the top and bottom alls are kept at constant temperatures T 1 and T respectivel ith T1 T and a T T T T W is applied at the lateral alls. So, apart from linear temperature distribution of the form S 1 '/ the reference Knudsen number, the other parameter characterizing the flo is the temperature ratio T1/ T. In this case the temperature of the hot all is used as the reference temperature ( T T ). The average number densit in the enclosure is N and P is taken from the equation of state. Case : three alls are kept at a constant temperature T C and the forth is heated b a constant heat flu of magnitude Q. In this case the second parameter characterizing the flo is the dimensionless applied heat flu q Q / P. The reference temperature in this case is the average temperature of the heated all, hile the reference number densit and pressure are taken as in case 1. 3 DETERMINISTIC AND STOCASTIC MODELING 3.1 Deterministic modeling The deterministic modeling is based on the direct solution of the dimensionless Shakhov model equation. Also, the ell-knon projection procedure can be ritten as [8,9] 1 S n Kn 1 Kn S, n ith, denoting the to components of the molecular velocit vector and S 4 1. (3) M 1 q / u q u u u 15 n. (4) S 4 1 hile the reduced local Maellians are M 1 q / 1 u q u u u 15 n. (5) M n ep u u / M n ep u u /. (6), The macroscopic quantities of practical interest are then taken as moments of the to reduced distribution functions and :, u, d d,, n d d, 1 n 1 u d d n. (7)

3 , d d u u 3n. (8) 3 q, q, q u u ud d. (9) The purel diffuse Maell boundar conditions have been implemented and the outgoing distributions at the boundaries denoted b and, are epressed as n ep / n ep /. (1), here n is a parameter calculated in terms of the ingoing distribution, satisfing the impermeabilit conditions and is the imposed dimensionless all temperature. In Case, at the heated all, is part of the solution q, q. ere, ithout loss of generalit, the bottom and the problem closes b appling the constraint all is considered as the heated all and these parameters are defined b the epressions [9] here 1 B q A, n A 1, (11), B u u u d d A d d (1) ith and denoting the ingoing distribution. The above set of integro-differential equations (3-6) coupled ith the epressions (7-9) subject to boundar conditions (1-1) are solved numericall discretizing the phsical space b a control volume approach and the molecular velocit space b the so-called discrete velocit method. The iterative process of the algorithm is terminated hen a convergence criteria of the from ma n n u u u u 1. (13) ( k) ( k) ( k 1) ( k) ( k 1) ( k) ( k 1) ( k) ( k 1) 1 i, j i, j i, j i, j i, j i, j i, j i, j i, j is fulfilled. In the above epression k denotes the iteration inde and 3. Stochastic modeling k the error after k iterations. Stochastic modeling is based on the DSMC method, described in [11]. In this method the real process of particle motion is divided into to steps, the ballistic motion of the particles over a distance proportional to their velocities, hich is purel deterministic and the collisions beteen particles that is done in a stochastic manner, folloing the traditional No Time Counter (NTC) scheme [11] together ith the S molecular interaction model. The implementation of the given heat flu boundar condition for case is described in [1]. The interaction of the particles ith the solid boundaries is purel diffuse. The phsical domain is divided into 1 1 square 5 6 cells ith a size smaller than the mean free path, hile 1-1 model particles are used and the time step is chosen to be about 1/ 3 of the cell transversal time. The sampling of the macroscopic properties starts after the 5 stead state has been achieved and carried out b volume base averaging for over time steps giving a sample size of approimatel scatter. 4 RESULTS AND DISCUSION samples per cell, hich is sufficientl large to reduce the statistical Results are presented for both cases, starting ith Case 1. Simulations have been conducted in a ide range of Knudsen numbers and temperature ratios. More specificall, these parameters var as.1 Kn 1 and.1 T1/ T.9. The effect of the gas rarefaction and the temperature ratio on the flo field can be seen in Fig. 1, here the streamlines and temperature contours are shon for values of the Knudsen number covering the transition to free molecular regimes and for small and large temperature differences. We observe that a stratified

4 temperature field is developed in all cases. In Figs. 1 (a) and (c), in the earl transition regime, the hole flo domain is covered b vortices that near the lateral alls have a velocit heading from cold-to-hot regions. This behavior is epected, as the leading mechanism for this flo is thermal transpiration or creep, hich leads to motion from cold to hot. What is ver interesting is that for larger degrees of rarefaction Figs. 1 (b) and (d) those vortices have been squeezed and to more vortices appear, counter-rotating to them having velocities in the vicinit of the lateral alls from hot-to-cold regions, contrar to thermal creep flo. The formation of these vortices as ell as the mechanism of their creation, have been etensivel studied in [8]. It is concluded that this hot-to-cold flo along the lateral alls are due to the interpla beteen the ballistic and collision motion of the particles. Moreover, as the temperature difference is increased those unepected flo structures epand covering larger parts of the enclosure, hile for small temperature differences the are restricted close to the lateral alls. Figure 1. Streamlines and temperature contours in a square enclosure for T1/ T.1,.9 and Kn.1,1 (case 1). More information regarding the flo field near the lateral alls can be taken from Fig., here the tangential velocit u along the lateral alls is given for small and large temperature differences and various values of the Knudsen number, ranging from the slip up to the free molecular regimes. For both temperature ratios hen the Knudsen number is small Kn.1 the direction of the velocit on the hole lateral alls is from cold-tohot, indicated b u and as the Knudsen number is increased some regions start having positive tangential velocities. For Kn 1 and 1 (transitional and free molecular regimes respectivel) along the lateral all there are onl positive tangential velocities. Of course as Kn all motion in the enclosure vanishes [13]. The flo field structure for Case has qualitative similarities ith that of Case 1. Again to kinds of vortices

5 Figure. Tangential velocit u along the lateral alls of a square enclosure for T1/ T.1,.9 and various values of the Knudsen number (case 1). appear along the boundaries depending on the Knudsen number and the temperature ratios (here the temperature ratio is defined in terms of the mean temperature of the heated all). Some quantitative differences hoever eist, mainl in the temperature distributions due to the different boundar conditions. Case configuration is close to the set-up of cooling devices in packed MEMS. Therefore, the most important results are regarding the heat flu departing from the bottom plate. As the heat flu departing from the bottom plate is given, the temperature distribution along the heated all is part of the solution. In order to investigate the results it is W / convenient to introduce the average temperature of the heated all as T T W / d. In Fig. 3, the temperature ratio TC / T is given in terms of the applied heat flu for various values of the reference Knudsen number. A ver interesting result is that for all values of the Knudsen number simulated, covering the slip, transition and free molecular regimes, for an given value of q, to values of the temperature ratio TC / T ma be obtained. More specificall as TC / T is initiall increased q is also increased, until a maimum value around TC / T.3 and then if e further increase the temperature ratio q is decreased. So the heat flu has a non-monotonous behavior ith respect to the temperature ratio. This is in agreement ith previousl reported results and a detailed eplanation of this phenomenon is provided in [9]. Figure 3. Temperature ratio T / T in terms of the dimensionless heat flu q for various values of the reference Knudsen number. C Figure 4. Departing heat flu Q W / m from the bottom plate in terms of reference pressure for various temperature ratios.

6 Some indicative dimensional results are presented in Fig. 4, here the dimensional heat flu given at the bottom plate Q W / m is plotted in terms of the reference pressure P Pa for various values of the temperature ratio T / T. The results are for a square cavit of side W 5m and the average temperature of the heated all is C T 3 1 K. The orking gas is Argon m Kg / kmol. We see that for an given temperature ratio as the pressure is increased the heat flu is also increased. Moreover at an given pressure, as the temperature difference increases the heat flu also increases until it reaches a maimum value for TC / T.3, and then it decreases if the temperature difference is increased. This result is important in the stabilit and efficienc optimization of cooling devices operating under vacuum conditions 5 CONCLUSIONS The thermall driven flo of a monatomic rarefied gas confined in a D enclosure far from local equilibrium has been investigated using both deterministic and stochastic modeling. Ver good agreement beteen the to approaches is taken. To boundar condition scenarios ere investigated, the first having mainl scientific importance, hile the second is of more of practical importance as it corresponds to cooling applications. A nonepected flo near the lateral alls contrar to the epected thermal creep flo is observed. A non-monotonic behavior of the heat flu departing from the bottom plate has been observed ith the heat flu attaining a maimum value at an intermediate temperature ratio. This result is important for the optimization of cooling devices operating under rarefied conditions. It is believed that this ork is of both scientific and technological interest. REFERENCES [1] Liu,., Wang, M., Wang, J. and Zhang, X. (7), Monte Carlo simulations of gas flo and heat transfer in vacuum packed MEMS devices, Applied Thermal Engineering, Vol. 7, [] Aleeenko, A., Gimelshein, S., Muntz, E. and Ketsdever, A. (6), Kinetic modeling of temperature driven flos in short microchannels, International Journal of Thermal Sciences, Vol. 45, [3] Ketsdever, A., Gimelshein, N., Gimelshein, S. and Selden, N. (1), Radiometric phenomena: From the 19th to the 1st centur, Vacuum, Vol. 86, [4] Vargas, M., Wüest, M. and Stefanov, S. (1), Monte Carlo analsis of thermal transpiration effects in capacitance diaphragm gauges ith helicoidal baffle sstem, Journal of Phsics: Conference 36, 113. [5] Masters, N., D. and Ye, W. (7), Octant flu splitting information preservation DSMC method for thermall driven flos, J. Comput. Phs., Vol. 6, [6] Rana, A., Torrilhon, M. and Struchtrup,. (1), eat transfer in micro devices packaged in partial vacuum, J. Phs.: Conf. Ser. 36, 134. [7] uang, J. C., Xu, K. and Yu, P. (13), A unified gas-kinetic scheme for continuum and rarefied flos III: Microflo simulations, Commun. Comput. Phs., Vol. 14(5), [8] Vargas, M., Tatsios, G., Valougeorgis, D. and Stefanov, S. (14), Rarefied gas flo in a rectangular enclosure induced b non-isothermal alls, Phsics of Fluids, Vol. 6, [9] Tatsios, G., Vargas, M., Valougeorgis, D. and Stefanov, S. (15), Non-equilibrium gas flo and heat transfer in a heated square microcavit, eat Transfer Engineering, in press. [1] Shakhov, E. M. (1968), Generalization of the Krook kinetic relaation equation, Fluid Dnamics, Vol. 3 (5), [11] Bird, G. A. (1994), Molecular Gas Dnamics and the Direct Simulation of Gas Flos, Oford: Clarendon Press. [1] Akhlaghi,., Roohi, E. and Stefanov, S. (1), A ne iterative all heat flu specifing technique in DSMC for heating/cooling simulations of MEMS/NEMS, International Journal of Thermal Sciences, Vol. 59, [13] Sone, Y. (9), Comment on eat transfer in vacuum packed microelectromechanical sstem devices [Phs. Fluids., 1713 (8)], Phsics of Fluids, Vol. 1,