Investgaton of a New Monte Carlo Method for the Transtonal Gas Flow X. Luo and Chr. Day Karlsruhe Insttute of Technology(KIT) Insttute for Techncal Physcs 7602 Karlsruhe Germany Abstract. The Drect Smulaton Monte Carlo method (DSMC) s well developed foarefed gas flow n transton flow regme when 0.0<Kn<. However such a smulaton for a complex 3D vacuum system s stll a challengng task because of the huge demand on the memory and long computatonal tme. On the other hand f Kn>0 the gas flow s free molecular and can be smulated by the Test Partcle Monte Carlo method (TPMC) wthout any problem even for a complex 3D vacuum system. In ths paper we wll nvestgate the approach to extend the TPMC to transton flow regme by consderng the collson between gas molecules as an nteracton between a probe molecule and the gas background. Recently ths collson mechansm has been mplemented nto ProVac3D a new TPMC smulaton program developed by KIT. The prelmnary smulaton result shows a correct nonlnear ncreasng of the gas flow. However there s stll a quanttatve dscrepancy wth the expermental data whch means further mprovement s needed. Keywords: Test partcle Monte Carlo smulaton molecular collson transtonal gas flow PACS: 47.45.-n 05.0.Ln INTRODUCTION The gas flow n a vacuum system s characterzed by the Knudsen number Kn=λ/d wth λ beng the mean free path of the molecules and d the sze of the vacuum chamber. For an ultra hgh vacuum or a hgh vacuum system the mean free path s much greater than the sze of the vacuum chamber so Kn>0. In ths case the gas molecules seldom collde wth each other and the gas flow s called as free molecular flow whch can be smulated by the Test Partcle Monte Carlo method (TPMC). If Kn<0.0 the gas can be usually consdered as contnuous flud and the flow s called as vscous flow whch can be smulated by Computatonal Flud Dynamcs (CFD). The gas flow regme when 0.0<Kn< s called as transton flow regme whch can be smulated by knetc theory or the Drect Smulaton Monte Carlo method (DSMC). However the smulaton of a complex 3D vacuum system wth DSMC or knetc theory s stll a challenge task because of the huge demand on the memory and long computatonal tme. Unfortunately the gas flows n some ITER applcatons for example the gas flow nsde the neutralzer of the Neutral Beam Inecton system (NBI) the gas flow before the cryogenc pump n pellet necton scenaro s wthn the transton flow regme. So that t s reasonable to explore a new smulaton Monte Carlo method for the transtonal gas flow. DEVELOPMENT OF PROVAC3D INCLUDING MOLECULAR COLLISIONS ProVac3D standng for 3D densty PROfle n the VACuum system s a TPMC smulaton program developed by the Karlsruhe Insttute of Technology (KIT). In our prevous works [-3] we have well descrbed ts man deas and successfully cross checked and used t n dfferent applcatons n free molecular flow regme. By recordng the tme of flght of molecules we can calculate the gas densty dstrbuton. The natural dea to extend ProVac3D nto transton flow regme s to nclude the collson between gas molecules by consderng approprate nteracton between the probe molecule and the gas background whch s descrbed n Fgure. Suppose P(x) r s the probablty that the probe molecule meets a target molecule of the gas background before t has passed a dstance x r t reads as r P( x) = exp( σ n( x) dx) ()
FIGURE. The collson between the probe molecule and target molecule. where n(x r ) s the densty of the background gas molecules and σ the collson cross secton. The ntegral s along the path of the probe molecule. From ths relaton one can derve Equaton (2) to determne the collson tme t of the probe molecule wth one of the molecule n background. t log( rnd) ( )( ) (2) n x y z v probe ubulk dt = σ 0 where v probe s the speed of the probe molecule u bulk the bulk speed of the background gas rnd a random number. Suppose τ s the possble collson tme wth the wall f no collson wth the molecule of the background s taken nto account. When t < τ the collson wth the target molecule wll happen at the poston In such a case the velocty hard sphere collson model. v wth ( ) 2 relatve bulk velocty u r v v (3) r of the probe molecule after the collson can be calculated for the frst attempt by the v = and r v = 2 x = x0 + v r t. r [ v e + ( v + v )] relatve x) = sn( θ ) cos( ϕ ) y) = sn( θ ) sn( ϕ ) z) = cos( θ ) v r beng the velocty of the randomly chosen target molecule superposed by v ( x) = v sn( θ ) cos( ϕ v ( y) = v sn( θ v ( z) = v cos( θ In Equatons (4) and (5) the azmuth angles of the target molecule θ ϕ ) are randomly chosen n full space ( however the azmuth angles of the probe molecule after the collson ( θ ϕ ) are randomly chosen n the half-space determned by theelatve velocty. On the other hand when t > τ the collson wth the wall wll happen. In such a case Cercgnan-Lamps boundary condtons have been ncluded [45]. )sn( ϕ z. x y (4) (5) 2
NORMALIZATION PROCEDURE AND CONVERGENCE Above mentoned collson mechansm has been recently mplemented nto ProVac3D. Naturally t can be used to the applcaton n whch a dlute gas speces s movng nsde the gas background of another gas speces [6]. However the emphass of the present work s to smulate the transton flow of sngle gas speces through a crcular tube connectng wth a dosng dome wth gas densty n and pumpng dome wth neglgble gas densty n2<<n as llustrated n Fgure 2. FIGURE 2. ProVac3D smulaton model. The gas flow s smulated as the functon of the gas densty n n the dosng dome by an teraton process n = n0 + k n( k = 023 L). (6) In order to start from a free molecular background n 0 wth k=0 must be small enough. For our smulatons n s chosen at least one magntude smaller than n 0 so that the collsons between the probe molecules can be always neglgble. For each teraton step k a hgh number of probe partcles representng n has been smulated. As sad the smulaton doman s dvded nto cells n ProVac3D smulaton [-3]. If s the cell ndex we can calculate the average tme of flght of all test partcles () ; here subscrpt k means the k th teraton step. Because even there are collsons the densty s stll proportonal to fts k () and nversely proportonal to the cell volume V () we can normalze to obtan the densty dstrbuton untl the k th teraton step s fnshed fts k n k ( ) = C ( fts0 ( ) n0 + fts ( ) n + L + ftsk ( ) n) / V ( ) (7) wth C beng the global normalzaton constant. Please note that the densty dstrbuton () s then used n Equaton (2) to determne the new collson tme n n k the next (k+) th teraton step. We have checked ths normalzaton procedure wth dfferent cell shapes cell szes molecular collson cross sectons and ncrease step n and proven that t s convergent. SIMULATION RESULTS AND COMPARISON WITH EXPERIMENTAL DATA The smulaton s carred out for a crcular tube of L=0.570 m and D=0.06 m. For ths crcular tube of L/D=9.75 t s not only hard to calculate wth knetc theory because the length-to-dameteato s too small but also hard to calculate wth DSMC because the length-to-dameteato s too large. The gas smulated s ntrogen at 5 C. The collson cross secton s calculated from the expermental data of the vscosty [7]. The experment was carred out n the TRANSFLOW test rg [8]. The smulaton doman s only the tube tself f we neglect the collsons nsde the dosng dome. In such a case C = 4 va wth v beng the thermal molecular speed and A the area of the tube cross secton. Consequently the gas flow rate Q can be determned by the transmsson probabltes w k n each teraton step 3
Q = C L k ( w0 n0 + w n + + w n Other macroscopc parameters such as bulk velocty and temperature etc. can be obtaned n the smlar way. In the smulaton n 0 s chosen as 0 8 correspondng to an ntal P=3.98 0-3 Pa and an ntal Kn=00. respectvely. n s chosen as 0 7. After k=9990 teraton steps the fnal n s 0 2 whch s 000 tmes bgger than the ntal one correspondng to a fnal P=3.98 Pa and a fnal Kn=0. respectvely. So the gas flow evolves from the free molecular flow to the begnnng of the transtonal flow. For each teraton step 0 5 test partcles are used whch means that one test partcle represents about 2.37 0 0 ntrogen molecules. There are 5 meshes along the radal drecton and 40 meshes along the axal drecton of the tube. In total the smulaton doman s dvded nto 200 cells. The smulaton s fnshed roughly n 2 hours by a desktop PC wth a CPU at 2.67 GHz. ). (8) FIGURE 3. Comparson of expermental result and ProVac3D smulaton for a crcular tube of L/D=9.75. The ProVac3D smulaton result n Fgure 3 shows a correct nonlnear ncreasng of the gas flow as P ncreases but there s stll a quanttatve dscrepancy wth the expermental data. Frst of all we have neglected the pressure n the pumpng dome. From the experment we know that the pressure n the pumpng dome though always beng at least one order of magntude lower than that n the dosng dome for ths partcular tube wll ntroduce a small back streamng. Secondly the entrance effect or the collsons n the dosng dome may play a role. We wll mprove our model to nclude these effects. Addtonally the smulated relatve axal densty and the axal bulk velocty nsde the tube are shown n Fgure 4 and Fgure 5 respectvely. We can see that the results for n=0 8 and n=0 2 are dfferent whch s qualtatvely reasonable. Unfortunately we have nether the expermental data nor other numercal results to compare wth. FIGURE 4. Relatve densty nsde the tube for a crcular tube of L/D=9.75. 4
FIGURE 5. Bulk velocty n the axal drecton for a crcular tube of L/D=9.75. CONCLUSIONS We have expanded our ProVac3D Monte Carlo smulaton program to explore a new approach for smulatng the transtonal gas flow. In ths new approach the collsons between molecules are represented by the collsons of the probe partcle wth the target partcle n the background. One convergent normalzaton scheme s suggested. Wth ths procedure the gas flow through a crcular tube s smulated as a functon of the ncreasng molecular densty n the dosng dome. The prelmnary smulaton result shows the correct nonlnear ncreasng of the gas flow rate. However there s stll a quanttatve dscrepancy wth the expermental data whch means further mprovement s needed. Obvously ths new smulaton approach has several advantages. Frst of all lke usual TPMC the test partcle s smulated one by one. So we wll not need the huge memory lke DSMC whch makes the smulaton of a 3D problem possble. Actually ProVac3D has mplemented the enttes to model a complcated 3D vacuum system. The present work shows the computaton tme s much faster than DSMC. Secondly as well known DSMC s very hard to be parallelzed. However the parallelzaton of TPMC s straghtforward. Ths work for ProVac3D s n progress. Thrdly the gas flow s smulated by an teraton process n the suggested normalzaton scheme. Ths means that we can even nterrupt the smulaton process and later on contnue the smulaton of the further evoluton of the gas flow by usng the obtaned result as the background. Nevertheless the drawback of ths normalzaton scheme s also clear snce we always need to start from the free molecular flow and there are many smulaton steps needed to reach the transton flow regme. REFERENCES. X. Luo M. Dremel and Ch. Day ProVac3D and Applcaton to the Neutral Beam Inecton System of ITER n 26 th Internatonal Symposum on Rarfed Gas Dynamcs edted by T. Abe AIP Conference Proceedngs 084 Amercan Insttute of Physcs Melvlle NY 2009 pp. 099-04. 2. M. Dremel Chr. Day S. Hanke and X. Luo Cryopump desgn development for the ITER Neutral Beam Inectors Fuson Engneerng and Desgn 84 689-693(2009). 3. X. Luo Chr. Day 3D Monte Carlo vacuum modelng of the neutral beam necton system of ITER Fuson Engneerng and Desgn n Press. 2. C. Cercgnan and M. Lamps Knetc models for gas-surface nteractons Trans. Theory Stat. Phys. 0-04(97) 5. F. Sharpov Applcaton of the Cercgnan-Lamps Scatterng Kernel to Channel Gas Flows n 22 nd Internatonal Symposum on Rarfed Gas Dynamcs edted by T. J. Bartel and M. A. Galls AIP Conference Proceedngs 585 Amercan Insttute of Physcs Melvlle NY 200 pp. 347-353. 6. S. Longo and P. Domede A Monte Carlo model for seeded atomc flows n the transton regme J. Comput. Phys. 228 385-3857(2009). 7. W. A. Cole and W. A. Wakeham The vscosty of ntrogen oxygen and ther bnary mxtures n the lmt of zero densty J. Phys. Chem. Ref. Data 4 209-226(985). 8. S. Varouts S. Nars V. Hauer Chr. Day and D. Valougeorgs Computatonal and expermental study of gas flows through long channels of varous cross sectons n the whole range of the Knudsen number J. Vac. Sc. Technol. A27 89-00(2009). 5