VACUUM chambers have wide applications for a variety of

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1 JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER Vol. 2, No., January March 27 Free Molecular Flows Between Two Plates Equied with Pus Chunei Cai ZONA Technology, Inc., Scottsdale, Arizona Iain D. Boyd University of Michigan, Ann Arbor, Michigan 489 and Quanhua Sun ESI US R&D, Inc., a Meber of ESI Grou, Huntsville, Alabaa 358 DOI:.254/.2589 This study describes a theoretical analysis of free-olecular flows between two lates, both equied with vacuu us, with an ai to investigate the facility effects of large chabers equied with distributed cyrogenic vacuu us. Three free-olecular odels are roosed to study the unsteady flow behavior, and to calculate the average nuber density and average velocity of the rarefied background gas olecules between the two lates. The first odel is based on ass conservation and redicts that the unsteady evolution is a ultile tie scale rocess. The second odel is a classical steady kinetic odel, whereas the third odel is the ost advanced kinetic odel that includes chaber wall effects. In the last two odels, a detailed balance of flux along different flow directions and reflection relations lead to exact solutions. All three odels include the chaber facility effects, esecially u size and u sticking coefficient. Siulations with the direct siulation Monte Carlo ethod are erfored and the results are coared with the analytical results. The analytical results fro these odels can be used to estiate the average background ressure and velocity in large vacuu chabers equied with ultile us for exerients and siulations. Noenclature D = chaber diaeter d = atoic diaeter fc = velocity distribution function H, L = chaber length Kn = Knudsen nuber k = Boltzann constant _ = ass flow rate into vacuu chaber = atoic ass n = nuber density n X = nuber density for articles assing through station X fro one side n X = nuber density for articles assing through station X fro the other side n in = nuber density for flux into the vacuu chaber n = surface noral P b = chaber backressure R = gas constant, or chaber end section radius r = distance vector between two oints r = distance between two oints S = chaber cross-sectional area or u size s = area ratio, S =S c T = teerature U, V = ean velocity V = theral velocity characterized by u teerature V w = theral velocity characterized by wall teerature w i = article weight = u sticking coefficient = secific heat ratio = article conductance ercentage fro chaber wall to one chaber end = ean free ath = background density = initial background density d = seidecaying eriod = article conductance ercentage fro one chaber end to the other end = chaber volue, or solid angle in velocity hase sace or hysical sace subscrit b = background c = chaber in = inflow w = wall, 2 = u or 2 A, B = chaber end A, B Presented as Paer 382 at the 9th AIAA/ASME Joint Therohysics and Heat Transfer Conference, San Francisco, California; received 4 June 2; revision received 29 August 2; acceted for ublication 29 August 2. Coyright 2 by the Aerican Institute of Aeronautics and Astronautics, Inc. All rights reserved. Coies of this aer ay be ade for ersonal or internal use, on condition that the coier ay the $. er-coy fee to the Coyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 923; include the code $. in corresondence with the CCC. CFD Secialist, 8489 E. Ironwood Square Drive; reviously graduate student research assistant, Deartent of Aerosace Engineering, University of Michigan, Ann Arbor, Michigan, AIAA Meber. Professor, Deartent of Aerosace Engineering, 32 Beal Avenue, AIAA Associate Fellow. Research Scientist, 77 Old Madison Pike, AIAA Senior Meber. 95 I. Introduction VACUUM chabers have wide alications for a variety of uroses such as aterials rocessing and testing of sacecraft electric roulsion systes. The ain goal of vacuu chabers is to rovide and aintain a low ressure environent. For exale, in exerients of testing a cluster of high-ower electric roulsion (EP) devices, such as lasa thrusters, vacuu chabers are usually equied with cryous [ 3], and the backressure in the vacuu chaber is aintained at about 3 4 Pa with the thrusters in oeration. At such a low ressure, the background flow inside the vacuu chaber is usually free-olecular.

2 9 CAI, BOYD, AND SUN One fundaental concern about the vacuu chaber is to understand the effects of the chaber characteristics on the chaber erforance, such as the average background ressure and average velocity in the chaber. These facility characteristics include the u size, the u sticking coefficient, the sidewall length, the chaber wall teerature which is usually 3 K, and the cyrogenic u teerature which is usually aintained at 2 K. These facility characteristics can have significant iacts on the vacuu chaber background flow and it is iortant to roerly quantify these facility effects. For exale, in EP exerients with large vacuu chabers generally equied with ultile cyrogenic us, the real cyrogenic u sticking coefficient can be uch lower than its noinal value, because there will generally be a roellant frost built u on the cryogenic us [4]. For this situation, to accurately estiate the actual sticking coefficient, it is crucial to accurately redict the background ressure and background velocity in the chaber. More generally seaking, a study of facility effects can hel to answer the following questions: ) For a given set of thruster oerating conditions and given facility characteristics, what values can be exected for the average background ressure and average background velocity in a chaber? It is crucial to have a clear understanding of this question because the background ressure and background velocity are the two ajor roerties indicating the erforance disarity for EP test devices between exeriental conditions and real sace oeration conditions. In sace, the abient ressure and velocity is zero, whereas in a large vacuu chaber, there is always a finite background ressure, and a background average velocity towards the us. 2) What are the exact effects of the facility on the background gas? 3) What is the otial condition that a facility can achieve by adjusting facility roerties? Answering these three questions can rovide guidelines to irove existing chabers or to design new chabers. A study of the background flow can have significant influence on nuerical siulations as well. For exale, in article siulations of lasa lue flows in large vacuu chabers, usually a large full-scale 3-D siulation is relaced by a sile two-diensional or axis-syetric siulation. In such a silified siulation, traditionally the background flow is relaced with static articles whose velocities are assigned fro a zero-centered Maxwellian distribution. However, as ointed out by our revious study [5 7], in chabers equied with one-sided us at one chaber end, the real background flow usually does not follow a zero-centered Maxwellian distribution, and under extree situations, the average background velocity of the background articles can reach over =s. Another interesting result fro our revious study is the seidecaying eriod forula for the unsteady background flow. For large vacuu chabers, such as the large vacuu test facility at the University of Michigan, several seconds are required for the background flow to reach a steady-state condition, hence it is a challenge to erfor three-diensional article siulations of lue flows in the chaber with a background flow [,7]. The results of this study will be helful for nuerical siulations of lue flows in vacuu chabers. Recently, we erfored several studies on the facility of large chabers [5 7]. These studies investigated the background flow using sile odels of free-olecular flows between two lates equied with one-sided or two-sided cyrogenic us on or near one chaber end. The background flow is searated fro the ain lue flow inside the chaber. Through the to late, a net gas flow rate is assued to enter the chaber, which reresents the reflected lue flow at one chaber end, and the theral seed of the gas flow is characterized by the wall teerature. On the other late equied with us which are aintained at a low u teerature, when articles hit the u surface, they will either stick to the u and thus be reoved with a robability of,orreflect into the chaber with a robability of having a theral seed characterized by the u teerature. Analyzing the fluxes along two directions and the nuber density relations at different locations leads to colete Fig. Illustration of the flow roble for odels and 2. solutions of the background flow, including exressions for the velocity distribution function and the nuber density. This article reorts our recent rogress in a continuing investigation of facility effects for large vacuu chabers equied with distributed cyrogenic us. The roble studied is the freeolecular flow between two lates both equied with us, which is a fair aroxiation for large vacuu chabers with distributed us. For exale, the 2V chaber at Arnold Engineering Develoent Center [8] is a large chaber with distributed us: there are large cryogenic u lates located at the botto of the chaber working as the riary us and saller us working as the secondary us on the to of the chaber. The roble is illustrated by Fig. : through the to late of area S c, there is a diffusion flow characterized by the wall teerature and a ass flow rate into the sace between the two lates; both lates are equied with us; the us are characterized by individual u areas S and S 2, the sae u teerature, and the sae u sticking coefficient. Particles ove between the two lates freely. When the articles strike a wall, they reflect with a theral velocity characterized by the wall teerature. If articles strike a u surface, then by a robability of the articles stick to the u, or by a robability of they reflect back towards the other late with a theral seed characterized by the u teerature. Note for the reflections on us and walls, colete accoodations with the wall or u surface are assued. Sidewall effects are included in this study as well. To silify the study and without lose generality, we assue the to late is sall and the size is set to a fixed value S =S c :2. For this study, we still use the assution that the lue flow does not directly hit the u. In real alications such as chabers for EP device exerients, the chaber is usually quite long, L=R >. The chaber length is one of the facility characteristics that can have a significant effect on the background flow, and we classify the chaber to be a short chaber or a long chaber based on the ratio of chaber length to chaber radius for a cylindrical chaber [5,7]. The long chaber wall effects will be discussed with two aroaches illustrated in our revious study [5 7]. In the first aroach, sidewall length effect is suarized with a secific araeter reresenting the ercentage of articles traveling fro one chaber end to the other chaber end without any collisions with the chaber side walls. The other treatent is to sily set the ratio between the wall and u late teeratures, T w =T, to unity. This second aroach is based on the fact that in long wall situations, alost all articles reflected fro the us on one chaber end will collide with the chaber wall before they arrive at the other chaber end; hence when they arrive at the other chaber end, their theral velocities are characterized by the wall teerature instead of the u teerature. Details of these two aroaches can be found in our revious study [5 7], and both of the will be alied in this study as well. The following sections resent the results of several analytical odels and nuerical siulations to validate the analytical solutions. It is worthy to ention that this is a well-defined classical free-olecular flow roble with realistic alication background. Even though it will be shown that the solutions to this roble are rather colex, these solutions are ore general than those for the roble of one-sided us which we investigated reviously [5,7]. The analytical results fro this study are alicable to estiate the

3 CAI, BOYD, AND SUN 97 average background nuber density and velocities, which is significant for the evaluation of vacuu chaber erforance and article siulations of lue flows in a vacuu chaber. To our knowledge, there are siilar analytical results in the literature. II. Model : Fro Mass Conservation Relation Siilar to our revious work [5,7], this study assues a constant density distribution between the two lates equied with one-sided us. Using the ass conservation law for gas in the chaber, the following relation ust hold: R d d dv _ dt dt 4 S V 4 S 2V 2 () where V and V 2 are the average theral velocities towards us and 2 with u surface areas S and S 2, resectively. This ordinary differential equation is solved using the following two boundary conditions: ) At tie t, gas begins to effuse into the chaber through the to late, the us begin to oerate, and the average density between the ga is, hence t. 2) When the tie is sufficiently large, a steady flow is established in the chaber: dt!=dt. The solution for this equation is exressed as Eq. (2), which consists of one unsteady ter and one steady ter: 4 _ t ex S V S 2 V 2 t S V S 2 V _ (2) S V S 2 V 2 Pressure in a vacuu chaber is generally easured with an ionization gauge. The ressure corresonding to such exeriental easureents [] is: P b ttrt w RT w 4 _ S V S 2 V 2 RT w ex S V S 2 V 2 t 4 4 _ S V S 2 V 2 RT w (3) At steady state, for fixed chaber roerties, the ressure is roortional to the ass flow rate: P b S c = _ 2RT w 2 2RT w S c =S V S 2 V 2 (4) If the backressure and the ass flow rate are known, for exale, through exeriental easureents, then the u sticking coefficient can be calculated using the following relation: 4 _RT w =P b S V S 2 V 2 (5) Unfortunately, by including the to u, this sile odel cannot rovide an exression for the average velocity between the two lates, though it can be reasoned that the net flow direction is towards the botto late as long as the botto u has a nonzero size and a nonzero sticking coefficient. A. Deterining the Theral Seeds V and V 2. Short Chaber Situation For a short chaber case, the exressions for V and V 2 are colex. With the existence of the sall u on the to late, along both directions, a fraction of articles have theral velocities characterized by the wall teerature V w 8RT w =, and a fraction have theral velocities characterized by the u teerature V 8RT =. A reasonable exression for V can be constructed by careful consideration of the following facts: ) When S 2, all articles oving towards u have a theral velocity V V w. 2) When S 2 S c and, all articles oving towards u have a theral velocity V V. 3) When S 2 S c and, all articles hitting u 2 are absorbed and no articles ove towards u. Hence the average velocity is V. With consideration of the above three facts, the silest linear construction for V is V S S 2 =S 2 =S c 8kT w = c q S 2 =S c 8kT = () Constructing V 2 can follow a siilar aroach. There are three grous of articles oving towards u 2: one grou consists of articles that entered directly through the to late; one grou is resulted fro reflections on the to late wall; and the last grou is due to reflections on the to u. ) When S, there is no u on the to late, and all articles oving towards u 2 have a theral velocity V 2 V w. 2) When S S c and, all articles reflected fro the to late towards u 2 have a theral velocity V, while there is another grou of articles with a ass flow rate that oves with a theral seed V w. 3) When S S c and, all articles hitting the to us are absorbed and no articles are reflected towards the botto u. Hence, the theral velocity for this situation is V 2 V w. With a consideration of the above three effects, a roer construction for V 2 is V 2 AS =S c q A S =S c 8kT w = 8kT = where A is a araeter to be deterined. 2. Long Chaber Situation Under the long chaber situation, the exressions for V and V 2 are greatly silified as V V 2 8RT w =. Obviously, though the V and V 2 exressions in the short chaber situation are ore accurate, they are ore colex for ractical uroses, esecially with the need to deterine the araeter A for V 2. Hence, for silicity, in this study, we adot the velocity exressions for long chaber wall situations. This treatent greatly silifies the final results with a decreased accuracy. However, odel is a relatively crude odel fro which only aroxiate trends are exected. B. Discussion This crude odel, esecially Eq. (3), relates several facility roerties such as the u size, the ass flow rate, and the roellant roerty R; only the u teerature is not included. There are three conclusions fro odel A: ) It is evident fro Eq. (3) that there exists a seidecaying eriod for the flow evolution: or (7) d 4=S V S 2 V 2 (8) d 4=S V ; d2 4=S 2 V 2 (9) Obviously, the density evolution involves ultile tie-scales, esecially if different, detailed exressions for V and V 2 are considered. 2) No atter how efficiently the us work, there is a certain aount of finite backressure between the lates. This backressure is reresented by the second ter of Eq. (3). The exression also

4 98 CAI, BOYD, AND SUN indicates that with fixed u araeters, at the final steady state, the background ressure is roortional to the ass flow rate. 3) When S, the analytical results for both short and long chabers degenerate consistently back to the results obtained in our revious work [5,7]. The ressure can be calculated fro Eq. (4) with known facility roerties, including cryogenic u sticking coefficient, u sizes, roer estiations of V and V 2, and a given ass flow rate. This odel illustrates that the steady background ressure is inversely roortional to the sticking coefficient when the ass flow rate is fixed. At sall values of, a % difference in the coefficient ay result in a significant backressure variation, whereas for large values of the noralized ressure is not very sensitive to this araeter. III. : Steady Detailed Kinetic Model This odel is a steady, detailed kinetic odel illustrated by Fig. as well. This odel considers the detailed flux balance in the flowfield, hence it can redict ore accurate results than the first odel. When the gas flow between the two lates reaches a final steady state, fluxes along the two directions ust be exactly balanced everywhere in the chaber. This roerty requires several nuber density relations. Analysis of these flux and nuber density relations is the key to the forulation of this steady-flow odel. One iortant relation for this study is the nuber density for a grou of articles reflected fro a late with a different teerature. To reserve the equal ass flux, the following relation ust hold: n T n2 T 2 () where the subscrits, 2 reresent the incoing and reflected grous of articles, resectively [9]. With the general short chaber assution, there are four nuber density relations in the flowfield, and they reresent different grous of articles traveling along two directions with theral seed characterized by the wall teerature and the u teerature, as illustrated in Fig., and they for a colete velocity distribution for the free-olecular flow between the two lates: fc 8 >< >: n n q ex q ex C2 C2 q n 2 2kT ex q n 2 ex 2kT C2 2kT ; C> C2 2kT ; C< () where n reresents the nuber density of articles oving towards the botto late, with theral velocities characterized by the wall teerature; n 2 reresents the nuber density of articles oving towards the botto late, with theral velocities characterized by the u teerature. Siilar definitions are used for n and n 2 to reresent the nuber densities of articles oving uwards. Fro the receding velocity distribution function, it is evident that the flow between the two lates cannot be described using a Maxwellian distribution, and the average velocity is generally not zero. Following the reflection relation, Eq. (), the velocity distribution function, Eq. () and F ig., there are four nuber density relations existing in the flowfield as a result of reflections on both lates, and a set of solutions is obtained fro these relations. q n n in n S =S c n 2 S =S c T =T w (2) q n 2 n S =S c T w =T n 2 S =S (3) q n n S 2 =S c n 2 S 2 =S c T =T w (4) n 2 n S 2 =S c q T w =T n 2 S 2 =S (5) where n in _=S c 2RT w =, and it is the nuber density of inflow gas through the to end. Further denote M S =S c S 2 =S c, N S =S c S 2 =S c, and after straightforward derivations, the final solutions for the steady free-olecular flow between the two lates are n n in N= N M () q n 2 n in N T w =T = N M (7) n n in S 2 =S c = N M (8) q n 2 n in S 2 =S c T w =T = N M (9) The ratios aong these nuber densities are n :n 2 :n :n 2 N:N T w =T : S 2 =S c :S 2 =S c T w =T, and any of these four grous of articles can be the doinating grou. The noralized background ean velocity and the ressure corresonding to exeriental easureents are as follows: U S 2 =S c 2RT w f2 N S2 =S c N S 2 =S c T w =T P g (2) P b S c _ 2RT w f2 N S2 =S c N S 2 =S c T w =T P g (2) 4 N M Note the backressures are calculated with P b nkt w because an ionization gauge is generally used in exerients to easure the background ressure. The receding results are exressions for general short chaber situations. By sily setting T =T w, the results for long wall situations are obtained: n n in N= N M (22) n 2 n in N= N M (23) n n in S 2 =S c = N M (24) n 2 n in S 2 =S c = N M (25) We want to oint out that the incoing flux is ilicitly included in Eq. (2), whereas the outflux relation is ilicitly satisfied by the four nuber relations. Hence, even though we do not exlicitly list the flux balance equation, the flux into the chaber via the to late and out of the chaber via two us is ilicitly balanced. The validity of this odel will be illustrated via coarisons with nuerical siulations. It is interesting and necessary to verify the coatibility of these new results with our revious results of free-olecular flows

5 CAI, BOYD, AND SUN 99 between two lates equied with one-side us on the botto late only [5,7]. When S, M S 2 =S c, N, n 2, n n in =S 2 =S c, n S 2 =S c n in =s, and n 2 = T w =T n in. Hence, the new analytical results degenerate consistently to our revious results as exected and a erfect coatibility is obtained. IV. Model 3: Steady Detailed Kinetic Model with Wall Effect This odel is an advanced steady kinetic odel illustrated by Fig. 2. Different fro odel 2, this odel considers the sidewall effect using a geoetry araeter. With a side wall, the incoing nuber density at one chaber end can be attributed to two effects: a fraction of articles travels fro side wall to the chaber end with a transortation ercentage of, and a fraction of articles travels fro the other chaber end without any collisions with the chaber wall with a transort ercentage of [5,]. It is not necessary to evaluate the first araeter because it will be canceled, whereas the evaluation of the other key araeter involves two stes: first, coute the solid angle at one secific oint on one chaber end subtended by the other chaber end, with the following solid angle forula []: Z Z r n ds 2 Z R 2 d r 3 H d (2) r 2 H 2 2r cos 2 3=2 s The final forat of this integral involves the Heuan s Labda function and the colete ellitic integral of the first kind []. The second ste is to average the solid angle over all oints on this chaber end. Though there is an analytical result for the first ste, it is extreely difficult to obtain an analytical exression for the second ste, which is required to evaluate the analytical results fro this odel. Hence, the transortation coefficient is evaluated nuerically. For the two nuerical siulation cases in this study, equals.5 and.3 for cylindrical chaber L=R 9=3 and :9=3, resectively [5,]. In this odel, we assue at chaber ends A and B, there are two grous of incoing articles and two grous of reflected articles, as illustrated in Fig. 2. Hence, the velocity distribution functions can be exressed as follows: 8 >< f A C >: 8 >< f B C >: n A n A n B n B q ex q ex q ex q ex C2 C2 C2 C2 q n A2 2kT ex C2 2kT ; C> q n A2 ex C2 2kT ; C< 2kT q n B2 2kT ex C2 2kT ; C< q n B2 ex C2 2kT ; C> At both chaber ends, the incoing nuber densities for articles with a theral seed characterized by the wall teerature consist of two arts; one fro the other chaber end and one fro the sidewall: n A n B n w ; n B n A n w (29) The nuber densities of articles with a theral seed characterized by the u teerature are n A2 n B2 ; n B2 n A2 (3) There are four reflection relations at both chaber ends: q n A S =S c n A n A2 S =S c T =T w n in (3) 2kT q n A2 S =S c T w =T n A n A2 S =S c (32) (27) (28) q n B S 2 =S c n B n B2 S 2 =S c T =T w (33) q n B2 S 2 =S c n B T w =T n B2 S 2 =S c (34) Finally, the ass flux into the chaber is balanced by the ass flux absorbed by the two us: _ n A S V w n A2 S V n B S 2 V w n B2 S 2 V (35) The final solutions for the above nine equations are quite colex. Denote M S =S c, N S =S c T =T w, P S = S c T w =T, Q S =S c, R S 2 =S c, S S 2 =S c T =T w, K S 2 =S c T w =T, T S 2 = S c, C S V w, D S V, E S 2 V w, F S 2 V, M SP TQSQ NTP, TQ RSQ NK, then the solutions can be exressed as _=TQ=C FP DFQTP= TQn n B in =C FP DFQTP= TQ D FQK= TQE (3)

6 CAI, BOYD, AND SUN χ.4.2 Fig. 2 Illustration of odel 3 (not to scale). n A n b n in TQ (37) n B2 K TQ n B TP TQ n A (38) n A2 Pn A Qn B Fig. 4 Contours of velocity distribution U av = 2RT w (S =S c :2, S 2 =S c :4). P b S c _ n 2RT w 8n A n A2 n A n A2 n B in n B2 n B n B2 (42) n A Mn A Nn B2 n in ; n B Rn B Sn A2 (39) n A2 n B2 ; n B2 n A2 (4) The noralized average background velocity and ressure are evaluated by averaging the values at two chaber ends: At the end of this section, we want to oint out that odel 3 is a relatively advanced odel because it is caable of analytically estiating the chaber sidewall effects. Figures 3 and 4 show contours of average background ressure and average velocity in the chaber with different sidewall length reresented by and absortion coefficient reresented by. The u sizes are fixed at S =S c :2 and S 2 =S c :4. We can conclude fro these two ictures that when <:5, the value which is the U av n A n B n A n B n A2 n B2 n A2 n B2 T =T w 2RT w na n A2 n A n A2 n B n B2 n B n B2 (4) χ Fig. 3 Contours of ressure distribution P b S c = _ 2RT w (S =S c :2, S 2 =S c :4) indicator of wall length effect on the background ressure, is greater than zero; when < ; whereas in the range of.5 and.9 can take ositive or negative values. This fact indicates that the chaber wall has a colex influence on the final flow state. This fact will be illustrated later with nuerical siulations. V. Nuerical Siulations For vacuu chabers equied with distributed us, there are no exeriental easureents available, hence the validity of these odels relies on coarisons with nuerical siulations. Because the flow between the two lates is free-olecular, it is ideal to validate the receding analytical results with the direct siulation Monte Carlo (DSMC) ethod []. This study uses a secific DSMC ackage naed MONACO [2] to erfor siulations. DSMC siulations are hysically accurate by including collisions, even though collisions occur very infrequently. The DSMC ethod siulates collision effects in rarefied gas flow by collecting grous of articles into cells. Pairs of articles inside a cell are selected at rando and a collision robability is evaluated that is

7 CAI, BOYD, AND SUN roortional to the roduct of the relative velocity and the collision cross section for each air. The robability is coared with a rando nuber to deterine if that collision occurs. If so, soe for of collision dynaics is erfored to alter the roerties of the colliding articles. The no tie counter ethod [] is adoted to deterine the collision rate. In this study, oentu collisions, or elastic collisions, involving exchange of oentu between the articiating articles are included in the DSMC ethod. The variable hard shere [] odel is eloyed and the collision cross section of xenon is: el Xe; Xe2:2 8 g 2! 2 (43) y/l.8..4 Syetric line 7.5E+ 7E+.75E+.5E+.25E+ E+ Diffuse wall 5.75E+ where! is the relative velocity and! :2 is related to the viscosity teerature exonent for xenon. In all elastic interactions, the collision dynaics is odeled using isotroic scattering together with conservation of linear oentu and energy to deterine the ostcollision velocities of the colliding articles. In total, over 4 axisyetric siulations of background xenon gas flows in a cylindrical chaber are erfored. In half of these siulations, the interolecular collision odule in the ackage is deactivated, whereas for the rest of siulations, the interolecular collision odule is activated. Hence, these siulation results rovide both real hysical flow results and ideal free-olecular flow results. In this study, a cylindrical siulation doain is adoted and artially illustrated by Fig. 5. The to and botto sides are the chaber ends, the left side is the syetric axis, and the right side is the outer surface. The cylindrical end section radii are set to 3, and the ga between the two lates is set to either 9 or.9, reresenting long and short chaber situations, resectively. To silify the situation, we fix the to u size by setting S =S c :2. The ass flow rate into the chaber through the to late is set to _ :7 7 kg=s. The u teerature is set to 5 K, because cyrogenic us are usually aintained at low teerature of 2 K in oeration. The wall teerature is set to roo teerature 3 K. At the initial tie ste, an abient density or vacuu condition is assued. A esh size of 5 3 cells is adoted in the siulation to achieve relatively high resolution. The no concrete us treatent [5] is adoted on both lates. In this aroach, no secific locations on the two lates are secified as us. When a article hits a late, a rando nuber is selected to coare with the articular area ratio S =S c or S 2 =S c. If the rando nuber is saller than the area ratio, then this article hits the u, otherwise it hits the chaber end. By following this aroach, it is not necessary to generate a new esh for each new case, and it is an effective way to siulate sall u areas. Once a article hits a u, then another rando nuber is generated, if this nuber is saller than the u sticking coefficient then this article sticks to.2.75e+ 8E+.25E+ E r/r Fig. Contours of nuber density, 3, with collisions ( :4, S 2 =S c :, L=R 9=3, T w 3 K, T 5 K). the u and is reoved fro the siulation, otherwise the article is reflected fro the u with a theral velocity characterized by the low u teerature. In the DSMC siulations, articles are introduced into the siulation doain through one chaber end with given ass flow rate and given theral seeds characterized by the wall teerature. The tie ste and article weight, which reresent the aount of real olecules with one article, vary in different cases. Most siulation details are the sae as the siulations in our revious study [5,7] excet there is an extra sall u on the to late. A. Evaluation of Average Quantities Walker s exeriental results [] show that the background ressure deends on u locations, but to include the u location effect will need a two-diensional odel, and it is not the goal of this study. The riary concern of this study is to rovide forulas to estiate the average background ressure and average background velocity in the chaber. Hence, the detailed background flowfield structures are neglected. At each tie ste, an average density and an average velocity for the whole doain are couted by counting all articles in the siulation doain and averaging their velocities: Xwi t N i = (44).5 7 X Ut Uj =N (45) y/l Syetric line r/r Fig. 5 Contours of ean free ath,, with collisions ( :3, S 2 =S c :3, L=R 9=3, T w 3 K, T 5 K) Diffuse wall where N i is the total nuber of articles used in ith cell in the DSMC siulation; U j and N are the jth article velocity and total nuber of articles in the doain. B. Coent on Accuracy The ajor urose of this study is to rovide several analytical exressions to evaluate the average background ressure and average background flow velocities. Because the flow is freeolecular, the DSMC is an accurate siulation ethod to rovide accurate flow solutions. By counting all articles in the chaber and averaging their velocities, the receding two equations rovide accurate average results in the whole chaber. By including collisions, these DSMC siulations are exected to rovide ore accurate results than the analytical results. By neglecting collisions, other DSMC siulations are exected to rovide exact freeolecular flow results at extree conditions which are closer to analytical results. Hence, these article siulation results will be

8 2 CAI, BOYD, AND SUN y/l.8..4 Syetric line 8E+ 7.5E+ 7.25E+ 7E+.75E+.5E+ Diffuse wall.25e+ Noralized ressure Model /T =) Model 3(L/R=3) Model 3(L/R=.3) w/ Collision(L/R=3) w/ Collision(L/R=.3) w/o Collision(L/R=3) w/o Collision(L/R=.3) E+ 8E+ 7.5E+ 7E r/r Fig. 7 Contours of nuber density, 3, without collisions ( :4, S 2 =S c :, L=R 9=3, T w 3 K, T 5 K). Fig S 2 /S Average density (noralized by _=S c 2=RT w, :4). used to coare the effectiveness of the analytical odels. The detailed distributions in these two-diensional siulations ay be affected by subtle changes in boundary treatents, but the average nuber density and average velocity in the chaber are not sensitive to these changes in boundary treatent. This schee was successfully alied in our revious studies [5,7]. Figure 5 shows the contours of ean free ath for the case of :4, S 2 =S c :3 and with article collisions. It is obvious that in the flowfield, the ean free ath is larger than, which is the diaeter of the cylindrical siulation doain. Hence, a freeolecular flow will be achieved. Most of the siulations in this study have a higher degree of rarefaction than this case by a larger value of or S 2 =S c. Figures and 7 show two nuber density distributions fro DSMC siulations for a long chaber. Gas flows into the chaber fro the to late, and with a consideration of reflections, both the to and the botto lates are essentially sources where articles effuse into the chaber. Though this study is not focused on flowfield details, these two tyical figures are helful to understand the results. The siulation araeters for these two figures are the sae excet that the interolecular collision odule is turned off for Fig. 7. Hence, Fig. 7 is a free-olecular flow result. It is obvious that these two figures are quite siilar, but the first figure has a lower average density. The siilarity is a result of the sae chaber geoetry, whereas the difference arises by including or not including olecular collisions. By including collisions, articles are ieded in crossing the dense layers close to both chaber ends, they are reflected back, and absorbed by the us on the chaber ends. Whereas for free-olecular flow, in the second figure, articles are free to ass through the dense layers close to both ends and enter the iddle of the chaber. Hence, for free-olecular flows, with less absortion effects, a higher average nuber density is anticiated under soe secific araeter cobinations, and these results will be further illustrated later. Figure 8 shows several average density evolution results fro odel and DSMC siulations. Fro this figure, it is evident that both the analytical results and the DSMC siulation results redict several seidecaying eriods and final steady densities. The freeolecular DSMC siulation yields a higher final average nuber density than that with olecular collisions. With collisions, the seidecaying eriod is shorter because collisions aid the rocess of foring an equilibriu state. The analytical results overredict the seidecaying eriod, and the difference can be attributed to the silified velocity V V 2 V w. In general, odel is the crudest odel without any detailed flux balance at all. Figures 9 and show two series of average backressure with varying S 2 =S c, and :4,.8, resectively. Analytical results fro the three odels and DSMC siulations are included. It is evident that all of the odels redict the correct trends, and for :8 the gas in both analytical and nuerical siulations Backressure, Pa X DSMC, w/collision, ρ = DSMC, w/collision, ρ > DSMC, w/o Collision, ρ = DSMC, w/o Collision, ρ > Model A, ρ > Model A, ρ = Noralized ressure Model /T =) Model 3(L/R=3) Model 3(L/R=.3) w/ Collision(L/R=3) w/ Collision(L/R=.3) w/o Collision(L/R=3) w/o Collision(L/R=.3) 2 4 Tie, s Fig. 8 Coarison of evolution history ( :4, S 2 =S c :5) S 2 /S Fig. Average density (noralized by _=S c 2=RT w, :8).

9 CAI, BOYD, AND SUN 3 Noralized ressure 5 5 Model /T =) Model 3(L /R=3) Model 3(L/R=.3) w/ Collision(L/R=3) w/ Collision(L/R=.3) w/o Collision(L/R=3) w/o Collision(L/R=.3) U/(2RT w ) / /T =) Model 3(L /R=3) Model 3(L/R=.3) w/ Collision(L /R=3) w/ Collision(L /R=.3) w/o Collision(L /R=3) w/o Collision(L /R=.3) Fig. Average density (noralized by _=S c 2=RT w, S 2 =S c :4) S 2 /S Fig. 3 Average velocity ratio U= 2RT w, :4. decrease. Figure 9 indicates that the free-olecular DSMC siulations yield higher average densities than those siulations without olecular collisions, whereas in Fig., all siulation results are quite close. Models and 3 have iroved erforance when S 2 =S increases. Another iortant difference between these two figures is that in the first figure with :4 < :5, no intersection occurs in both the siulation data and odel 3 results; whereas in the second icture with :8, the analytical long chaber wall and short chaber wall curves intersect, and the siulation data intersect as well. These results are consistent with our revious observation fro Fig. 3. Hence, though odel 3 underredicts the average background ressure, it is caable of quantifying the effects of chaber wall length. Figures and 2 show two series of average nuber density results with fixed S 2 =S c :4,.8, and varying. It is evident that all of the odels redict the correct trends and for S 2 =S c :8, both analytical and nuber siulation results are in good agreeent. In general, odel 2 still has the best erforance. It is interesting to note that odel 3 redicts that at : :9 the short chaber curve and the long chaber curve intersect, and nuerical siulations confir these intersections, ore obviously shown in Fig. 2. These values at the intersections are different fro the results obtained in our revious study of single u situations due to the secondary sall Noralized ressure Model /T =) Model 3(L/R=3) Model 3(L/R=.3) w/ Collision(L /R=3) w/ Collision(L /R=.3) w/o Collision(L /R=3) w/o Collision(L /R=.3) Fig. 2 Average density (noralized by _=S c 2=RT w, S 2 =S c :8). U/(2RT w ) / /T =) Model 3(L/R=3) Model 3(L/R=.3) w/ Collision(L /R=3) w/ Collision(L /R=.3) w/o Collision(L /R=3) w/o Collision(L/R=.3) S 2 /S Fig. 4 Average velocity ratio U= 2RT w, :8. u. It is difficult to deterine this value analytically with the colex results fro odel 3. Figures 3 and 4 show two series of average background velocity results with fixed :4,.8, but varying S 2 =S c. It is evident that all of the odels redict the sae trends as the nuerical siulations. For :8, the free-olecular siulations have lower average velocity values than those siulations with olecular collisions. The reason for this result is, by neglecting the collisions, ore articles are reflected back fro the botto u with negative velocities. Hence, a lower average background velocity is exected. For these velocity results, odel 2 with T =T w and odel 3 have better erforance than odel 2 when coared with nuerical siulation results. Figures 5 and show two series of average background velocity results with fixed S 2 =S c :4,.8, and varying. All analytical odels redict the correct trends. For ost cobinations of araeters, the siulations without considering collisions have a lower average background velocity than those siulations with considering olecular collisions. The reason is sile: because the large u is located at the far chaber end, without considering collisions, all articles reflected at the u will ove to the other end or wall with negative velocities. With considering collisions, a ortion of those reflected articles will ove back to the riary u and be absorbed again, due to the dense collisions around the riary u, hence, less negative velocities will be included in the coutation for the average velocity.

10 4 CAI, BOYD, AND SUN U/(2RT w ) / /T =) Model 3(L /R=3) Model 3(L/R=.3) w/ Collision(L /R=3) w/ Collision(L /R=.3) w/o Collision(L /R=3) w/o Collision(L /R=.3) interolecular collisions. Generally, the analytical and nuerical results have the sae trends, whereas odels 2 and 3 offer quite good agreeent to the DSMC siulations. Siilar to our revious study, several conclusions can be drawn for the background free-olecular flows: the background flow is not static with zero average velocity and cannot be described as a full Maxwellian distribution, and the long seidecaying eriod rohibits a full 3-D siulation of lue flows in the whole chaber. Hence, extra attention is necessary for the background flow construction when siulating lue flows in large vacuu chabers with a article aroach such as the DSMC ethod. U/(2RT w ) / Fig. 5 Average velocity ratio U= 2RT w, S 2 =S c : /T =) Model 3(L /R=3) Model 3(L/R=.3) w/ Collision(L /R=3) w/ Collision(L /R=.3) w/o Collision(L /R=3) w/o Collision(L /R=.3) Fig. Average velocity ratio U= 2RT w, S 2 =S c :8. VII. Conclusions In this study, we roosed three odels to study the average background ressure and the average velocity between two lates both equied with cyrogenic us. The results indicate that the analytical results are ore colicated than the situations in our revious study. However, the results fro this study are ore general and can degenerate consistently to our revious study. To validate the analytical results fro the three odels, we erfored about 4 DSMC siulations, with and without Acknowledgents The authors gratefully acknowledge funding for this work fro the U.S. Air Force Office of Scientific Research under grant FA This work was carried out and artially coleted at the University of Michigan. References [] Walker, M. L., Effects of Facility Backressure on the Perforance and Plue of a Hall Thruster, Ph.D. Thesis, Ph.D. Dissertation, Aerosace Engineering Det., Univ. of Michigan, Ann Arbor, MI, 25. [2] Hargus, W. A., and Reed, G., Air Force Clustered Hall Thruster Progra, AIAA Paer , July 22. [3] Beal, B. E., Clustering of Hall Effect Thrusters for High-Power Electric Proulsion, Ph.D. Dissertation, Aerosace Engineering Det., Univ. of Michigan, Ann Arbor, MI, 24. [4] Biagioni, L., Boccaletto, L., Marcuccio, L., Nicolini, S., Servi, D., and Andrenucci, M., Large Sace Siulator for Electric Proulsion Testing: Design Requireents and Engineering Analysis, AIAA Paer 2-375, July 2. [5] Cai, C., Theoretical and Nuerical Studies of Plue Flows in Vacuu Chabers, Ph.D. Dissertation, Aerosace Engineering Det., Univ. of Michigan, Ann Arbor, MI, 25. [] Cai, C., Boyd, I. D., and Sun, Q., Free Molecular Background Flow in a Vacuu Chaber Equied with Two-Sided Pus, Journal of Vacuu Science and Technology A (Vacuu, Surfaces, and Fils), Vol. 24, No., 2, [7] Cai, C., Boyd, I. D., and Sun, Q., Rarefied Background Flow in a Vacuu Chaber Equied with One-Sided Pus, Journal of Therohysics and Heat Transfer, Vol. 2, No. 3, July Set. 2, [8] Meyer, J. W., Loane, J. T., Hallock, G. A., Wiley, J. C., and Dawbarn, R. A., Hall Thruster Plue Signal Transission Testing in the AEDC 2V Vacuu Chaber, AIAA Paer , July 25. [9] Gobosi, T. I., Gaskinetic Theory, Cabridge Univ. Press, New York, 994. [] Narasiha, R., Orifice Flow of High Knudsen Nuber, Journal of Fluid Mechanics, Vol., Oct. 9, [] Bird, G. A., Molecular Gas Dynaics and the Direct Siulation of Gas Flows, Oxford Univ. Press, New York, 994. [2] Dietrich, S., and Boyd, I. D., Scalar and Parallel Otiized Ileentation of the Direct Siulation Monte Carlo Method, Journal of Coutational Physics, Vol. 2, No. 2, July 99,