Vacuum. Pressure driven rarefied gas flow through a slit and an orifice. S. Misdanitis *, S. Pantazis 1, D. Valougeorgis. abstract

Transcription

1 Vacuum 86 (212) 171e178 Contents lists available at SciVerse ScienceDirect Vacuum journal homeage: Pressure driven rarefied gas flow through a slit and an orifice S. Misdanitis *, S. Pantazis 1, D. Valougeorgis University of Thessaly, Deartment of Mechanical Engineering, Volos 38334, Greece article info abstract Article history: Received 3 October 211 Received in revised form 1 January 212 Acceted 9 February 212 Keywords: Rarefied gas dynamics Kinetic theory Vacuum flows Knudsen number The flow of a monatomic gas through a slit and an orifice due to an arbitrarily large ressure difference is examined on the basis of the nonlinear BhatnagareGrosseKrook (BGK) model equation, subject to Maxwell diffuse boundary conditions. The governing kinetic equation is discretized by a second-order control volume scheme in the hysical sace and the discrete velocity method in the molecular velocity sace. The nonlinear fully deterministic algorithm is otimized to reduce the comutational effort by introducing memory usage otimization, grid refinement and arallelization in the molecular velocity sace. Results for the flow rates and the macroscoic distributions of the flow field are resented in a wide range of the Knudsen number for several ressure ratios. The effect of the various geometric and hysical arameters on the flow field is examined. Comarison with reviously reorted corresonding Direct Simulation Monte Carlo (DSMC) results indicates a very good agreement, which clearly demonstrates the accuracy of the kinetic algorithm and furthermore the reliability of the BGK model for simulating ressure driven flows. Ó 212 Elsevier Ltd. All rights reserved. 1. Introduction Non-equilibrium flows may aear in several technological fields including microfluidics, vacuum technology and high altitude aerodynamics. The degree of dearture from local equilibrium is characterized by the Knudsen number, which is defined as the ratio of the mean free ath over a characteristic macroscoic length of the roblem. It is well known that when the Knudsen number is larger than.1 the tyical NaviereStokeseFourier formulation is not valid and alternative mesoscale aroaches based on kinetic theory, as described by the Boltzmann equation or reliable kinetic model equations, must be imlemented. Over the years, linearized kinetic equations have been extensively alied with great success to solve internal fully develoed gas flow through long channels of various cross sections in the whole range of the Knudsen number [1e3]. The numerical solution is fully deterministic based on the discretization of the hysical sace by finite difference schemes and of the molecular velocity sace by discrete velocity methods. Then, the discretized equations are solved in an iterative manner, while synthetic acceleration algorithms may be alied to accelerate the slow convergence of * Corresonding author. address: semisdan@mie.uth.gr (S. Misdanitis). 1 Current address: Physikalisch-Technische Bundesanstalt, Berlin 1587, Germany. the iteration ma at the sli and hydrodynamic regimes [4]. Overall, reliable results for linear fully develoed flows through long channels of any cross section may be obtained with moderate comutational effort. However, when the flow becomes nonlinear, i.e. in the case of fast flows through channels of finite length, including flows through slits and orifices, the comutational effort is significantly increased. The incoming distribution functions at the entrance and the exit of the channel are not Maxwellians and therefore adequately large comutational domains must be included before and after the channel to roerly imose the boundary conditions. It is noted that in fully develoed flows through long channels the flow is simulated only in one cross section of the channel and then, using a standard methodology [1,2], the mass flow rate is estimated, while in short channels kinetic modeling is required in the whole flow field. This is the main reason for the large comutational effort required in the latter case, comared to the one for fully develoed flows. The most common aroach for solving nonlinear flows is the DSMC method [5], which is a owerful stochastic method for high seed flows, circumventing the solution of a kinetic equation. The DSMC method has been recently imlemented to solve flow through slits, orifices and short tubes [6e1]. However, in general this method is associated with large comutational effort, certain difficulties in the arallelization of the code and statistical noise for low seed flows. Therefore, it is imortant to develo alternative aroaches based on comutationally efficient deterministic solutions of model kinetic equations. Several 42-27X/$ e see front matter Ó 212 Elsevier Ltd. All rights reserved. doi:1.116/j.vacuum

2 172 S. Misdanitis et al. / Vacuum 86 (212) 171e178 researchers have develoed such algorithms [11e13]. However, additional research work in this field is needed. In the resent work, the gas flow through a slit and an orifice has been simulated by introducing a comutationally efficient nonlinear fully deterministic algorithm based on the nonlinear BGK kinetic model equation subject to Maxwell diffuse boundary conditions. The kinetic equations have been solved by alying in the hysical sace a second-order control volume scheme and in the molecular velocity sace the discrete velocity method. Gradual grid refinement has been introduced to seed-u the slow convergence of the iterative ma in the sli and hydrodynamic regimes [14,15]. Moreover, the code is arallelized in the molecular velocity sace, while memory demands are reduced by roer handling of the allocated arrays. Results for the flow rates and the macroscoic quantities are resented in the whole range of the rarefaction for several values of the ressure ratio between the two reservoirs. Comarison with reviously reorted corresonding DSMC results indicates very good agreement. 2. Flow configuration and comutational domain The flow of a monatomic gas through a slit of height H and through an orifice of radius R contained in an infinitely thin artition that searates two containers is considered. The gas in the containers far from the oening is in equilibrium at ressures P 1 and P 2, with P 1 > P 2, and temeratures T, which is also the temerature of the artition. The imlemented coordinate system is Cartesian for the slit flow and Cylindrical for the orifice flow. Both flow configurations are two-dimensional in the hysical sace, with the ^x axis denoting the flow direction, while the slit and the orifice are located at ^x ¼. The second axis is denoted by ^y in the slit flow and by ^r in the orifice flow. The comutational domain consists of the two large comutational areas, before and after the artition, defined by ðx L ^x ; ^y; ^r Y L Þ and ð ^x X R ; ^y; ^r Y R Þ, which corresond to the ustream and downstream reservoirs resectively including the infinitely thin artition which contains the slit ð ^y H=2Þ or the orifice ð ^r RÞ. The flow configuration and the comutational domain are shown in Fig. 1. The reference quantities are H, R, T, P 1. In addition, N 1 is the reference number density defined as P 1 ¼ N 1 k B T, with k B denoting the Boltzmann constant, while the most robable molecular velocity, defined as y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k B T =m with m denoting the molecular mass, is the reference velocity. The flow is characterized by the reference rarefaction arameter for the slit and the orifice resectively defined as d slit ¼ P 1H m y and d orifice ¼ P 1R m y (1) where m is the gas viscosity at reference temerature T. The reference rarefaction is roortional to the inverse of the reference Knudsen number. The objective is to estimate for each flow the flow rates and all macroscoic distributions. 3. Formulation The flow is modeled by the nonlinear BGK model equation subject to urely diffuse boundary conditions. The choice of the BGK model for simulating the flow is justified by the fact that this is a ressure driven flow and the exected variation of temerature in the flow field is small. The main unknown is the distribution function f ¼ f ðs; xþ, while s ¼ð^x; ^yþ or s ¼ð^x; ^rþ is the osition vector and x ¼ðx x ; x y ; x z Þ is the molecular velocity vector. It is seen that the distribution function deends on five indeendent variables. The local ressure and temerature are defined by P and T resectively. It is convenient to introduce the following non-dimensional quantities: x ¼ ^x H or x ¼ ^x R ; y ¼ ^y H ; r ¼ ^r R ; c ¼ x y ; g ¼ f y3 N 1 ; n ¼ N N 1 ; u ¼ U y ; s ¼ T T ; ¼ P P 1 (2) In subsections 3.1 and 3.2, the formulation of the flow through a slit and an orifice resectively is resented. In both cases, the hard shere model is imlemented and therefore the variation of the viscosity is roortional to the square root of the temerature Flow through a slit The non-dimensional quantities are introduced into the governing equation to yield [16,17] c cos q vg vx þ c sin q vg vy ¼ d n s ffiffi g M g where g M ¼ n=ðsþ 3=2 ex½ðc uþ 2 =sš. Moreover, for comutational uroses it is convenient to exress the comonents (c x,c y,c z ) of the articleq velocity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in terms of cylindrical coordinates (c,q,c z ), where c ¼ c 2 x þ c2 y and q ¼ tan 1 ðc y =c x Þ. Also, by taking advantage of the two-dimensionality of the flow, it is ossible to eliminate the c z comonent of the distribution function by introducing aroriate rojections, exressed by the moments F x; y; c x ; c y ¼ J x; y; c x ; c y ¼ Z N Z N gðx; y; cþdc z and (3) c 2 z gðx; y; cþdc z (4) Then, oerating accordingly on Eq. (3), after some routine maniulation the following couled set of integro-differential equations is reduced: c cos q vf vx þ c sin q vf vy ¼ d n s ffiffi F M F (5) c cos q vj vx þ c sin q vj vy ¼ d n s ffiffi J M J (6) Fig. 1. Uer half of flow configuration and comutational domain. The corresonding reduced Maxwellians are:

3 F M ¼ n h c s ex cos q 2þ. i u x c sin q 2 u y s J M ¼ n h c 2 ex cos q 2þ. i u x c sin q 2 u y s The macroscoic quantities are exressed in terms of F and J as S. Misdanitis et al. / Vacuum 86 (212) 171e (7) (8) J þ X L =H; y; c ; q ¼ J þ x; Y L =H; c ; q ¼ 1 2 ec2 (17) Downstream reservoir F þ X R =H; y; c ; q ¼ F þ x; Y R =H; c ; q ¼ P 2 P 1 1 ec2 (18) Z 2 nðx; yþ ¼ u x ðx; yþ ¼ 1 n u y ðx; yþ ¼ 1 n sðx; yþ ¼ 2 3n Z N Z 2 Z N Fc dc dq (9) Z 2 Z N Z 2 Z N c cos q Fc dc dq (1) c sin q Fc dc dq (11) c 2 F þ J c dc dq 2 3 u 2 x þ u2 y (12) Based on the above, the slit flow configuration has been reduced to a 4D roblem gaining significantly in CPU time without losing any of the hysical findings of the flow. The mass flow rate through the slit is calculated by _M slit ¼ m Z H=2 H=2 Nð; ^yþu x ð; ^yþd^y (13) and is non-dimensionalized by the corresonding mass flow rate for free molecular flow into vacuum d ¼, P 2 /P 1 ¼ given by ffiffiffi _M fm;slit ¼ðHP 1 Þ=ðy Þ. Then, the dimensionless flow rate is defined as W slit ¼ where G slit ¼ M _ slit ¼ 4 ffiffiffi Gslit (14) _M fm;slit Z 1=2 nð; yþu x ð; yþdy (15) Turning to the issue of the boundary conditions, it is assumed that molecules are fully accommodated at both sides of the artition and therefore the outgoing distributions are described by Maxwellians having the temerature of the artition. Also, the gas at the fictitious boundaries of the comutational domain is assumed in equilibrium and therefore the incoming distributions are aroximated by Maxwellians defined by the local ressure P 1 or P 2, temerature T and zero bulk velocity. Symmetry boundary conditions are imosed at y ¼. Based on the above, the deduced boundary conditions in dimensionless form for the reduced distributions for the slit flow are: Ustream reservoir F þ X L =H; y; c ; q ¼ F þ x; Y L =H; c ; q ¼ 1 ec2 (16) J þ X R =H; y; c ; q ¼ J þ x; Y R =H; c ; q ¼ P 2 P ec2 (19) Partition wall F þ x/ H ; y; c ; q J þ x/ H ; y; c ; q ¼ rslit H ec2 (2) ¼ rslit H 2 ec2 (21) In all cases the suerscrits (þ) denotes outgoing distributions, while the notation ðhþ denotes the left () and right (þ) sides of the artition wall. Finally, in Eqs. (2) and (21) the quantities r H are secified by the no enetration condition at each side ¼ 2 ffiffiffi r slit r slit þ ffiffiffi ¼2 Z=2 Z N =2 Z3=2 Z N = Flow through an orifice c cos q F ; y; c ; q c dc dq (22) c cos q F þ ; y; c ; q c dc dq (23) The non-dimensional quantities for the cylindrical geometry are introduced into the governing equation to yield [14,18] vg c x vx þ c cos q vg vr c sin q vg r vq ¼ d n s ffiffi g M g (24) where g M ¼ n=ðsþ 3=2 ex½ðc uþ 2 =sš is the Maxwellian. However, unlike the flow through a slit, the rojection rocedure can not be imlemented in the orifice flow which remains a 5D roblem. The macroscoic quantities are given by nðx; rþ ¼2 u x ðx; rþ ¼ 2 n u r ðx; rþ ¼ 2 n sðx; rþ ¼ 4 3n Z N Z Z N Z N Z Z N Z N Z Z N Z N Z Z N gc dc dqdc x (25) c x gc dc dqdc x (26) c cos q gc dc dqdc x (27) h ðc x u x Þ 2 þ c cos q 2 u r þ c sin q 2 i gc dc dqdc x The mass flow rate through an orifice is calculated by ð28þ

4 174 S. Misdanitis et al. / Vacuum 86 (212) 171e178 _M orifice ¼ 2m Z R Nð; ^rþu x ð; ^rþ^rd^r (29) and it is non-dimensionalized by the corresonding mass flow rate for free molecular flow into vacuum d ¼, P 2 /P 1 ¼ given by _M fm;orifice ¼ðR 2 ffiffiffi P 1 Þ=ðy Þ. Then, the dimensionless flow rate is defined as W orifice ¼ where G orifice ¼ M _ orifice ¼ 4 ffiffiffi Gorifice (3) _M fm;orifice Z 1 nð; rþu x ð; rþrdr (31) Next, we move to the descrition of the boundary conditions regarding the flow through an orifice. Similar assumtions as is the case of the flow through a slit are taken into account. As a result, Maxwellians having the hysical characteristics of the artition are the outgoing distributions from the walls searating the two containers, while at the fictitious boundaries of the domain, Maxwellians defined by the local ressure P 1 or P 2, temerature T and zero bulk velocity are secified as the incoming distributions. Symmetry boundary conditions are imosed at r ¼. Based on the above, the boundary conditions in dimensionless form for the orifice flow are: Ustream reservoir g þ X L =R;r;c x ;c ;q ¼ g þ x;y L =R;c x ;c ;q ¼ 1 (32) 3=2ec2 Downstream reservoir g þ X R =R;r;c x ;c ;q ¼ g þ x;y R =R;c x ;c ;q ¼ P 2 P 1 1 3=2ec2 (33) Partition wall g þ x/ H ; r; c x ; c ; q ¼ rorifice H 3=2 ec2 (34) In all cases the suerscrits (þ) denotes outgoing distributions, while the notation ðhþ denotes the left () and right (þ) sides of the artition wall. Finally, in Eq. (34) the quantities r H are secified by the no enetration condition at each side ¼ 4 ffiffiffi r orifice r orifice ZN Z Z N þ ¼4 ffiffiffi Z c x g ; r; c x ; c ; q c dc dqdc x (35) Z Z N c x g þ ; r; c x ; c ; q c dc dqdc x (36) Summarizing the formulation it is noted that the flow through a slit is described by the kinetic equations (5,6) couled by the moments (9e12) and subject to boundary conditions (16e21), while the arameters r slit H are given by (22,23). The flow through an orifice is described by kinetic equation (24) couled by the moments (25e28) and subject to boundary conditions (32e34), while the arameters r orifice H are given by (35,36). The solution of the two roblems deends on two dimensionless arameters, namely the reference rarefaction arameter d and the ressure ratio P 2 /P 1. Table 1 Dimensionless flow rate through a slit W vs. rarefaction arameter and ressure ratio. W slit P 2 /P 1 ¼ P 2 /P 1 ¼.5 P 2 /P 1 ¼.9 d BGK DSMC [6] BGK DSMC [7] BGK 1. e.5 e e Numerical scheme The kinetic equations for both roblems under consideration are discretized in the molecular velocity and hysical saces. In articular, the continuum sectrum of the comonents c x and c of the molecular velocity vector is relaced by a set of discrete magnitudes c m x ½; cmax x Š and c m ½; cmax Š, with m ¼ 1,2,.,M, which are taken to be the roots of the Legendre olynomial of order M accordingly maed from [1,1] to ½; c max x; Š. Also, the olar angle sace is discretized in q k, k ¼ 1,2,.,K angles, uniformly distributed in [,2] for the Cartesian geometry and by making use of the axisymmetry in [,] in the Cylindrical geometry. In the hysical sace the comutational domain is discretized using (I J) square elements of side h. At each square element (i,j), i ¼ 1,2,.,I and j ¼ 1,2,.,J a central second-order scheme in sace is alied. In addition, the macroscoic moments of the reduced distributions are estimated by double or trile numerical integration consisting of one or two Gauss-Legendre quadratures and the traezoidal rule. A similar aroach has been imlemented in Ref. [14,15]. Then, the discretized equations are solved in an iterative manner consisting of two stes. In the first ste, the kinetic equations are solved for the unknown distributions assuming that the macroscoic quantities at the right hand side of the kinetic equations are known. In the second ste, udated estimates of the macroscoic quantities are comuted based on the moments of the distribution functions. The iterative rocedure is ended when the termination criterion is satisfied. This is the tyical iteration scheme used in discrete velocity kinetic solvers. However, due to the fact that the regions at the inlet and at the outlet of the comutational domain must be adequately large, the comutational cost rises significantly. Therefore, here this tyical algorithm has been ugraded by making use of a) message assing interface (MPICH2), b) memory handling techniques and c) automated grid refinement. The arallelization of the kinetic algorithm is imlemented in the Table 2 Dimensionless flow rate through an orifice W vs. rarefaction arameter and ressure ratio. W orifice P 2 /P 1 ¼ P 2 /P 1 ¼.5 P 2 /P 1 ¼.9 d BGK DSMC [8] BGK DSMC [8] BGK DSMC [8] 1. e.5 e.1 e e.52 e.11 e e 1.29 e.28 e

5 S. Misdanitis et al. / Vacuum 86 (212) 171e molecular velocity sace. This is a simle and natural way of arallelization inherent in the structure of the algorithm reducing significantly the required comutational effort. The estimates of the distribution functions at each rocessor are summed to estimate the udated macroscoic quantities. Before starting a new iteration, the macroscoic quantities and the imermeability constants are synchronized and re-transmitted to each rocessor. This rocedure takes lace in the host comuter of the arallel scheme. The comutational effort during the first ste of the iteration, which is the most comutationally demanding, can be slit u to M K rocessors, which corresond to the total number of discrete velocities, for the case of the slit and M 2 rocessors for the case of the orifice. The scaling characteristics of the algorithm are quite good, considering the number of variables that must be exchanged at each iteration. An average efficiency of about 94% for 64 rocessors and 75% for 256 rocessors is calculated. It is exected that extending the arallelization in the hysical sace will further reduce the comutational cost. Furthermore, memory handling techniques have been used to reduce storage requirements due to the 4D and 5D of the distribution function for the slit and orifice flows resectively. Due to the velocity magnitude indeendency, a temorary array can be allocated and overwritten after treating each magnitude. The dimensionality of this array can be further reduced by storing the distribution only in the arts of the domain required by the marching scheme of the discretized governing equation. For Fig. 2. Distributions of number density (to), temerature (middle) and axial velocity (bottom) along the symmetry axis for flow through a slit (left) and an orifice (right) with P 2 / P 1 ¼.5 and d ¼.1, 1 and 1.

6 176 S. Misdanitis et al. / Vacuum 86 (212) 171e178 examle, for the marching rocedure toward the ositive x -direction, the distribution is stored only at ositions x and x Dx. These techniques ermit having a two-dimensional array for the distribution function and ractically remove memory limitations. In this manner, channels of any length can be considered, since the size of this array is only determined by the height of the entrance/ exit regions and the number of angles in the case of the orifice. Finally, grid refinement techniques have been imlemented in order to accelerate the convergence of the numerical scheme. In this context, the simulations are initially erformed with a smaller amount of nodes. After convergence has been reached, the simulation rocedure is reeated in a refined mesh, where the hysical sace arameters have been doubled, using the revious solution as an initial condition. Linear interolation has been used here as a first aroximation. This rocedure is reeated until the final number of nodes has been reached. Following this multile grid rocedure, the gain in the number of iterations for large values of d is significant. It is noted that the number of iterations in dense grids is about two orders of magnitude smaller than the corresonding one in sarse grids and therefore the overall CPU time is greatly reduced. 5. Results and discussion Numerical results are resented in dimensionless form for the flow rates and the macroscoic distributions for the slit and orifice flows in a wide range of the rarefaction arameter d [,1] and three ressure ratios P 2 /P 1 ¼,.5 and.9. The resented results have been obtained with the ustream and downstream comutational domains varying from 2 8 u to 4 15 deending on the ressure ratio and rarefaction arameter. The iteration rocess is terminated when a relative convergence criterion of 1 6 imosed on the average residual of each comuted macroscoic quantity er node is fulfilled. Regarding the accuracy of the resented results, it is noted that all numerical arameters (number of nodes in the hysical sace, magnitudes and angles in the molecular velocity sace, size of inlet and outlet regions, convergence criterion) have been modified in order to ensure that the resented results are grid indeendent u to at least two significant figures. This accuracy requires CPU time of hours (d 1), days (d ¼ 1) and weeks (d ¼ 1). The dimensionless flow rates W through a slit and an orifice are tabulated in Tables 1 and 2, resectively. In both cases the flow rate is monotonically increased as the rarefaction arameter d is increased. It is seen that the increase of W is small in the free molecular and continuum regimes and more raid in the transition regime. Also, in both cases W is decreased as the ressure ratio P 2 / P 1 is increased (i.e., the ressure difference between the two reservoirs becomes smaller). It is also seen that as P 2 /P 1 is increased the deendency of W on d becomes weaker. For comarison uroses, corresonding results available in the literature for the slit [6,7] and the orifice [8] flow roblems obtained by the DSMC method are included in Tables 1 and 2 resectively. It is seen that for P 2 /P 1 ¼ and.5 the BGK and DSMC results agree u to at least two significant figures. This very good agreement clearly demonstrates the accuracy of the kinetic algorithm and furthermore the reliability of the BGK model for simulating ressure driven flows. For P 2 /P 1 ¼.9 available DSMC results exist only for the orifice flow. In this case the agreement remains very good u to d ¼ 1 and then at d ¼ 1 and 1 is reduced but always remains within 7%. It is imortant to note that, for both the BGK and DSMC algorithms, the required comutational effort is increased as d is increased. However, as the ressure ratio P 2 /P 1 is increased the required comutational effort for the DSMC algorithm is increased, while this is not the case for the kinetic algorithm, the comutational effort of which is slightly reduced as the ressure ratio is increased. This is a significant advantage of the kinetic comared to the DSMC aroach. Fig. 3. Flow through a slit for P 2 /P 1 ¼.5 and d ¼.1 (left) and 1 (right); streamlines (to) and isolines of number density (middle) and axial velocity (bottom).

7 S. Misdanitis et al. / Vacuum 86 (212) 171e Fig. 4. Flow through an orifice for P 2 /P 1 ¼.5 and d ¼.1 (left) and 1 (right); streamlines (to) and isolines of number density (middle) and axial velocity (bottom). Macroscoic distributions and tyical ictures of the flow field are shown in Figs. 2e4.InFig. 2, the dimensionless number density, temerature and axial velocity, for both a slit and an orifice flow, along the symmetry axis is shown for three characteristic values of the rarefaction arameter and P 2 /P 1 ¼.5. It is seen that the variation of the corresonding macroscoic quantities in the slit and orifice flow is qualitatively similar. Starting with the density variation, it is seen that far ustream is equal to one, it is raidly decreased in the region one unit length before and after the wall artition and then it gradually aroaches the far downstream conditions. The temerature equals unity in most of the domain, while close to the artition it is decreased. The dro is related to the rarefaction arameter and ranges from around 4% for small values of d u to about 16% for d ¼ 1. The axial velocity far ustream is almost zero and is increased in the region just before the artition, while after the artition it is decreased to almost zero far downstream. The maximum value of the velocity is increased as d is increased. The above described behavior is tyical for the corresonding macroscoic quantities when P 2 /P 1 s. In Figs. 3 and 4, a more comlete icture of the flow fields is rovided for the slit and orifice geometries resectively. Flow streamlines along with isolines of number density and axial velocity are lotted for P 2 /P 1 ¼.5 and d ¼.1 and 1. Again, there is a qualitative resemblance between the corresonding slit and the orifice flows. It is seen that the structure of the flow field between rarefied and dense atmosheres is different. At d ¼.1 the streamlines are almost symmetric with regard to the y axis, while at d ¼ 1 there is no symmetry and a vortex aears next to the artition at the downstream container. Also, as the atmoshere becomes more dense, the flow accelerates faster and the maximum axial velocity is increased. The saw-like shae of the isolines aearing for the small values of the rarefaction arameter has only numerical character and it is not related to the hysical roerties of the flow. This is a disadvantage of the discrete velocity method which is not easily circumvented. 6. Concluding remarks An efficient fully deterministic kinetic algorithm has been alied for the solution of the ressure driven slit and orifice flows. By solving the BGK kinetic model subject to Maxwell boundary conditions the flow rates and the macroscoic quantities of the flows have been obtained. The comutational algorithm includes arallelization in the molecular velocity sace, drastic reduction in memory demands and acceleration of the iteration ma by multile grid methods. The results are in very good agreement with reviously reorted DSMC results, clearly indicating the accuracy of the roosed algorithm and the reliability of the BGK model for solving ressure driven flows. Acknowledgments This work has been suorted by the Euroean Communities under the contract of Association EURATOM/Hellenic Reublic. The views and oinions exressed herein do not necessarily reflect those of the Euroean Commission. References [1] Shariov F. Rarefied gas flow through a long rectangular channel. J Vac Sci Technol A 1999;17:362e6. [2] Naris S, Valougeorgis D. Rarefied gas flow in a triangular duct based on a boundary fitted lattice. Eur J Mech B/Fluids 28;27:81e22. [3] Varoutis S, Naris S, Hauer V, Day C, Valougeorgis D. Comutational and exerimental study of gas flows through long channels of various cross sections in the whole range of the Knudsen number. J Vac Sci Technol A 29; 27:89e1. [4] Naris S, Valougeorgis D, Kalema D, Shariov F. Flow of gaseous mixtures through rectangular microchannels driven by ressure, temerature, and concentration gradients. Phys Fluids 25;17:167.1e [5] Bird GA. Molecular gas dynamics and the direct simulation of gas flows. Oxford: Oxford University Press; 1994.

8 178 S. Misdanitis et al. / Vacuum 86 (212) 171e178 [6] Shariov F, Kozak D. Rarefied gas flow through a thin slit into vacuum simulated by the Monte Carlo method over the whole range of the Knudsen number. J Vac Sci Technol A 29;27:479e84. [7] Shariov F, Kozak D. Rarefied gas flow through a thin slit at an arbitrary ressure ratio. Eur J Mech B/Fluids 211;3:543e9. [8] Shariov F. Numerical simulation of rarefied gas flow through a thin orifice. J Fluid Mech 29;518:35e6. [9] Varoutis S, Valougeorgis D, Sazhin O, Shariov F. Rarefied gas flow through short tubes into vacuum. J Vac Sci Technol A 28;26:228e38. [1] Varoutis S, Valougeorgis D, Shariov F. Simulation of gas flow through tubes of finite length over the whole range of rarefaction for various ressure dro ratios. J Vac Sci Technol A 29;27:1377e91. [11] Chigullaalli S, Venkattraman A, Ivanov MS, Alexeenko AA. Entroy considerations in numerical simulations of non-equilibrium rarefied flows. J Com Phys 21;229:2139e58. [12] Gimelshein N, Gimelshein S, Selden N, Ketsdever A. Modeling of low-seed rarefied gas flows using a combined ES-BGK/DSMC aroach. Vacuum 21; 85:115e9. [13] Titarev V, Shakhov E. High-order accurate conservative method for comuting the oiseuille rarefied gas flow in a channel of arbitrary cross section. Com Math Math Phys 21;5:537e48. [14] Pantazis S, Valougeorgis D. Heat transfer through rarefied gases between coaxial cylindrical surfaces with arbitrary temerature difference. Eur J Mechanics B/Fluids 21;29:494e59. [15] Pantazis S. Simulation of transort henomena in conditions far from thermodynamic equilibrium via kinetic theory with alications in vacuum technology and MEMS. Ph.D. thesis: University of Thessaly. Volos, Greece; 211. [16] Graur IA, Polykarov A Ph,. Shariov F. Numerical modelling of rarefied gas flow through a slit into vacuum based on the kinetic equation. Comut Fluids 211;49:87e92. [17] Graur A, Polykarov A Ph, Shariov F. Numerical modelling of rarefied gas flow through a slit at arbitrary gas ressure ratio based on the kinetic equation. Z Angew Math Phys 211;49:1e18. [18] Shariov F, Kremer GM. On the frame deendence of constitutive equations. i. Heat transfer through a rarefied gas between two rotating cylinders. Continuum Mech Thermodyn 1995;7:57e71.