Theoretical Analysis of Vacuum Evacuation in Viscous Flow and Its Applications

Transcription

1 ECIAL Theoreical Analysis of acuum Evacuaion in iscous Flow and Is Applicaions Yasuhiko ENDA The vacuum is classified ino hree caegories depending on he sae of gas: viscous, inermediae, and molecular s. The viscous evacuaion is generally regarded as a basic echnique, wih which pressure decreases along wih he exponenial curve of ime. However, his is no always he case when he conducance of he pipe is considered. To begin wih, he auhor heoreically solved he evacuaion equaion in he viscous. In addiion, he inroduced he noion of ransferring pressure, calculaed as /CC; refers o he pumping speed, while CC refers o he raio of he conducance of he pipe o he pressure. By examining he wo exreme cases, he auhor has revealed a new fac: (a) when pressure >>, he evacuaion curve is exponenial as generally saed; whereas (b) when pressure <<, he curve is no exponenial bu inversely proporional o ime. In his paper, he auhor describes he deails of his sudy and he applicaions of case (b). Keywords: vacuum, viscous, Knudsen number, conducance, evacuaion ime 1. Inroducion The vacuum is defined as he sae of he space filled wih he gas, he pressure of which is lower han ha of he amosphere in JI Z 816 (acuum echnology -ocabulary). The vacuum echnologies have found a wide variey of indusrial applicaions depending on he pressure level (Table 1). resisance dominaes he phenomena. The auhor also inroduced he noion of ransferring pressure and showed ha he general evacuaion curve, which includes case 1) and ), was explained oally. The resul of he analysis of case ), which is no menioned usually, is very simple and also ineresing. He also menioned is applicaions and aenions a he las. Table 1. acuum echnologies classified by he pressure Technology ressure(a) Uilized physical phenomena acuum sucion ressure difference wih he amosphere acuum disillaion acuum gas exchange Boiling poin drop Lower residual impuriies acuum evaporaion Molecular beam epiaxy Longer mean free pah of he gas Lower impingemen rae of residual molecules, for beer grown films. Caegories of acuum -1 Caegories by he pressure JI Z 816 defines he pressure ranges of vacuum as Table. acuum caegories Low vacuum Medium vacuum High vacuum Ulra high vacuum Table. ressure ranges of vacuum (1) ressure ranges (a) Amosphere Less han 1 5 Table 1 shows ha he pressure of vacuum ranges over 1 o he 14 h power, and ha he echnologies, equipmen srucure and specificaions, asks o be solved, and knowhow needed for he specific range differs from one anoher. In his paper, he auhor firs conduced a heoreical analysis of vacuum evacuaion in he viscous, which is usually regarded as a basic echnique, and so no much is menioned in general vacuum exs. Nex he deduced he exac soluion of he equaions, and found answers in wo exreme cases. The wo cases are: 1) in he case when he pipe s resisance can be negleced, ) in he case when he pipe s The auhor mus noe ha hese caegories are easy o observe and also easy o undersand inuiively, bu do no represen acual physical phenomenon (kineic sae of gas molecules) and so are inadequae o physical analyses. - Caegories by he kineic sae of gas In order o analyze vacuum phenomena, caegories by he kineic ( s) sae of gas are necessary. Is boundaries are acually inexac, bu i is usually classified ino 3 caegories in Table 3. 4 Theoreical Analysis of acuum Evacuaion in iscous Flow and Is Applicaions

2 iscous Inermediae Molecular Table 3. acuum caegories by he kineic sae of gas Caegories ub class ae of gas molecules Crieria (3) Turbulen Laminar Collision rae beween molecules are much higher han ha of molecules and wall Transien region of viscous and molecular Collision rae beween molecules are much lower han ha of molecules and wall K : Knudsen number: Index of viscous/molecular. K = λ/d, λ: mean free pah (m), D: diameer (m) Re : Reynolds number: Index of urbulen/laminar. Re = Dvρ/η, D: diameer (m), v: speed (m/s), ρ: densiy (kg/m 3 ), η: viscosiy (a s) K<.1, Re> K<.1, Re<1.1<K<.3 K>.3 collisions beween molecules are less han hose of molecules and he wall (ee Fig. 1 (b)). Here, we disinguish hose cases as follows: (a) viscous (b) molecular (a) viscous (b) molecular Fig. 1. iscous vs. molecular s Diameer D The reason why hese caegories are imporan is ha conducance, which is an imporan index of vacuum evacuaion, changes drasically depending on hem. In his paper, he auhor analyzes he viscous (laminar) region. Bu before ha, basics of Table 3 is explained firs in he nex secion. 3. Basics of acuum 3-1 Mean free pah The gas is a group of many molecules. Each of he molecules moves randomly and wih various speeds, he average of which is defined by he molecules ype and emperaure (a room emperaure, ca. 5 ~ 15 m/s). ince he number of molecules are usually very large (.7E pcs/l, a room emperaure), hey repea muual collisions a a high rae. The average disance beween one collision o he nex is called he mean free pah. I is known ha he mean free pah λ [m] is described as follows (3). kt T λ = (1) πd d Here, [a] is he pressure, T [K] is he emperaure, d [m] is he diameer of he molecule, and k [J/K] is he Bolzmann s consan. If one assumes ha T = 3 [K] and he gas is nirogen, hen he molecule s diameer d [m] is.37 [nm] (5),(6). 3-3 Index of : Knudsen number and D In Fig. 1, when D [m] is defined as he diameer of he pipe (or chamber), one can see ha he is viscous when λ << D, and is molecular when λ >> D. o if non-dimensional number K = λ/d is inroduced, hose condiions are K >> 1 and K << 1, respecively. This K is called Knudsen number. In pracical use, i is ofen he case when K <.1 is regarded as viscous, and K >.3 as molecular. And when we pu hese relaions ino Equaion (), we ge iscous : D >.68 [a m] Molecular : D <.. In pracical design of he vacuum sysem, hese represenaions are easier o grasp. 3-4 Conducance When gas s in a pipe, resisance caused by i is called evacuaion resisance, and is inverse is called conducance. In a schemaic diagram as Fig., le us call he pressures a boh ends 1 and [a], and rae Q [am 3 /s]. Then he conducance C [m 3 /s] is expressed as Q = C ( 1 ) (3) ipe s conducance boh in viscous and in molecular has been calculaed heoreically, as in Table 4. One can see he poins: C λ = / () (The value is almos he same, if he gas is amosphere.) 3- iscous and molecular As Fig. 1 (a) shows, when he pressure is high, collisions beween molecules are dominan. As one can see in Equaion (), when he pressure decreases, he mean free pah ges longer, and evenually here is a case when he Q 1 Fig.. chemaic diagram of conducance EI TECHNICAL REIEW NUMBER 71 OCTOBER 1 5

3 Table 4. Conducance () of (long) pipe iscous Molecular Conducance 1349 D4 11 D3 C [m 3 /s] L L = (1+)/ [a], D: Diameer [m], L: Lengh [m] iscous : is proporional o average pressure and 4 h power of D. Molecular : is independen from he pressure, bu is proporional o he 3 rd power of D. As for he conducance of he inermediae, aken he molecular conducance as 1, an approximae equaion (3) 1+1 D (D) 1+36 D is advocaed. (ee Fig. 3) According o Fig. 3, i is enough o hink ha he lower limi of viscous is D.3 ~.4. Δ = e Δ (4) When i is inegraed, we ge ln() = e + cons. (5) Afer solving Equaion (5) wih he iniial condiion =, =, we will ge { = exp( e / ) (6) =.33 log ( ) (7) e These wo Equaions(6),(7) are simple and clear enough, bu hey have a problem wih he prerequisie. Tha is, as described in secion 3-4, he pipe s conducance C is proporional o he average pressure. In he above calculaion, an assumpion is made ha he C is large enough, or jus is effec is 8%. Bu wha will happen, if i can no be negleced? Or is here any exac answer? In his paper, he auhor will consider his general and exac case. 1 1 Relaive value of conducance; as a funcion of D Inermediae Molecular Flow D [a m] iscous Fig. 3. Relaive conducance, as a funcion of D 4. General Answer o he Evacuaion Calculaion in iscous Flow 4-1 Inroducion of basic equaion For inroducing he basic equaion, le us redefine he physical parameers as in Fig. 5. ressure () [a] ressure () umping peed [m 3 /s] 3-5 Evacuaion speed & ime (simple calculaion) Le us hink a simple evacuaion sysem as in Fig. 4. We will evacuae he chamber (he volume [m 3 ]) by a pump (pumping speed = [m 3 /s]), conneced wih a pipe (he conducance C [m 3 /s]). In elemenary exs, he answer is o solve he nex Equaion (4), by simply omiing he pipe s conducance (since i is large enough), or by hinking ha he effecive pumping speed e is abou.8. olume [m 3 ] ipe s Conducance C [m 3 /s] Fig. 4. imple evacuaion sysem umping peed [m 3 /s] pump olume [m 3 ] ipe s Conducance C [m 3 /s] C = CC {() + ()}/, CC: Conducance Coefficien Fig. 5. Evacuaion sysem in viscous ump As described in secion 3-4, in a viscous he pipe s conducance C is proporional o he average pressure. o le us call he raio of he conducance o he pressure conducance coefficien Cc [m 3 /sa]. From Table 4, in a room emperaure & amosphere, = 1349 D4,where D [m] is he pipe s diameer, L [m] L is he lengh of i. As in Fig. 5, le us define ha is he pumping 6 Theoreical Analysis of acuum Evacuaion in iscous Flow and Is Applicaions

4 speed, () is he pressure a he pump inle, () is he pressure of he chamber and is a funcion of ime. We will se he assumpions below. The ulimae pressure of he pump is small enough, and can be negleced. The chamber is large enough, so ha he pressure in i is uniform, and he volume of he pipe can be negleced. Leak and ou-gassing in he sysem can be negligible. The is viscous, a leas in he region of our concern. Le us ry o ge (), saring () = a =, on he condiions above. When we name he rae in he pipe as Q() [am 3 /s], from Equaion (3) we ge, Q () = C ( () ()) C = c ( () () ) () () = Q () And from he definiion of he pumping speed, we ge And, from he maerial balance ha he pressure drop per uni hour in he chamber is equal o he rae, we ge he nex equaion. d = Q () (1) d When we pu Equaion (9) o (8) and (1), we ge { () () = () (11) d = () (1) d From Equaion (11), ( ) Q () = () (9) () = + + () And puing Equaion (13) o (1), we ge ) Our firs aim is o solve his differenial Equaion (14) on he iniial condiion of =, () =. 4- Deducion of he general soluion Luckily we can solve Equaion (14) analyically. Here, we pu We pu Equaion (15) ino (14) and ge (13) d = + + () (14) d ( (8) () = x () (15) dx = x () (16) d Here, we pu Afer inegraing Equaion (18), we ge Here, from a formulary (7), From Equaions () and (1), we ge Here, () = x () Le us solve Equaion (), on he iniial condiion = and () =. When we pu = (ime consan),we ge (17) dx = x () (18) d dx d = (19) 1+x 1 1+x +1 Lef hand side of (19) dx 1+x 1 + ln x + 1+x = A x x ( ) = x Noe ha hese are he general soluion of he Equaion (). 4-3 Two exreme cases ince = has he uni of [s] and appears by he form of /, we can safely assume ha i is a ime consan. In he same way, has he uni of [a] and so we can assume ha i is some x 1+x 1 = + ln x + 1+x x x () = x (), = x, = () 1+ 1+x A = ln x + 1+x (3) x ( ) 1+ 1+x = ln x + 1+x A x ( ) ( ) () Righ hand side of (19)= d = A(consan) (1) (4) EI TECHNICAL REIEW NUMBER 71 OCTOBER 1 7

5 kind of ypical pressure. From now on, we pu =, and call his as ransferring pressure. Wih his, Equaions (3) and (4) are expressed: (/ ) = ln + 1+ (/ ) A (/ ) ( ) ( / ) A = ln + 1+ ( / ) ( / ) ( ) (Noe ha is a funcion of ime.) We can say ha he pressure is normalized by he ransferring pressure Π. In general, since is amospheric pressure, >> Π can be concluded in many cases. o A 1 ln ( ) Here we examine wo exreme cases. Case 1 In he case of >> Π (Equaion (6) holds rue auomaically.) { } { } (5) ( ) 1 ln 1 ln = ln ( ) ( ) Namely, (6) is valid in general. ( ) ( ) exp = exp (7) (8) normal (or moderae). In his case, when one sars evacuaion from he amosphere, firs he pressure decreases exponenially o ime (as in case 1). When he pressure decreases enough lower han he ransferring pressure, he pressure decreases inversely proporional o ime, independenly o he iniial pressure (as in case ). The pressure when hese wo cases come across virually is he ransferring pressure Π. ) In he case when he pipe s diameer is large enough, and so he ransferring pressure is small and almos lower han he limi of he viscous. In his case, here is no acual case sae, bu almos all he evacuaion ime he case 1 happens. Tha is, he pressure decreases exponenially o ime. This is he sae ha he auhor described in secion 3-5, and also ha usual vacuum exs describe in hem. 3) On he oher hand, here is a case when he pipe s diameer is oo small and so he ransferring pressure is oo high o almos reach he amosphere. In his case he case 1 does no happen, bu all he ime of evacuaion he pressure decreases inversely proporional o ime (case ). This is he case when he pumping speed is masked by he resisance of he pipe. The evacuaion ime is he funcion of only he volume and he conducance coefficien Cc, and is independen of he iniial pressure or pumping speed (Cf. Eq. 3). In he general evacuaion of viscous, all hree saes (1 o 3) can be appeared. We mus noe ha he sae ), as described in exs, is no he only one; in he sae 1) and 3), he fac ha he region when he pressure decreases inversely proporional o ime exiss, is very noe-worhy. In Fig.6 hese saes are demonsraed graphically. Here, L = 3 [m], = 1 [L], = [L/min], = 1e5 [a]. Three ypical curves (1 o 3) are ploed on he graph. The lower limi of viscous is D =.3. Case In he case of << Π ln(1) A (9) / { In he case of In he case of A 1 + ln (1+ ) 1.53 >> >> A ln ( ) 1 ressure [a] 1.E+5 1.E+4 1.E+3 Transferring pressure Evacuaion ime [sec] 3) D = 4mm Inversely proporional o ime 1) D = 1cm 1.E+ ) D =.5cm Mix of ) & 3) Exponenially decrease iscous limi 1.E , 1, Fig. 6. Calculaion of hree ypes of evacuaion Namely, 1 = (3) 4-4 Inerpreaion of he general soluion Le us ry o inerpre he wo cases in he previous secion. 1) In he case when he ransferring pressure Π = /CC is 4-5 Measured example Figure 7 shows a measured example. = 8.6 [L], = 18 [L/min], L = 7.5 [m], D = 7.53 [mm] (3/8 ube, hickness 1 mm), = 1e5 [a] (amosphere) 8 Theoreical Analysis of acuum Evacuaion in iscous Flow and Is Applicaions

6 1 Measured Calculaed 5- Aenion Figure 9 is a redrawn graph of Fig. 6, wih he linear horizonal axis and he logarihmic verical axis. One may noice ha he curve 1) is no linear bu asympoic o he x-axis. If one believes elemenary exs descripion ha he roughing evacuaion curve should be linear in a semi-log graph, i is a serious misake. One may ry o search leakage which acually does no exis. Of course, i is no necessary. In his case, i is simply ha he pipe s diameer is oo small, if he slow evacuaion is no your inenion. ressure [a] 1 1.E Evacuaion ime [sec] Fig. 7. Measured example ressure [a] 1.E+4 1.E+3 1.E+ The reason his curve becomes fla is ha he characerisics change from exponenial o inversely proporional. NOT he leak. 1.E Applicaions and Aenions 5-1 Applicaion One may hink ha he case when he pressure decreases inversely proporional o ime is meaningless, for he evacuaion akes ime only. Bu i is no always so. When one evacuaes a chamber wih a large pipe from he amosphere, a urbulen occurs a he sar. This causes roubles by scaering paricles in he chamber in many cases. To preven his, i is a sandard echnique o sar evacuaion slowly wih a small pipe (Fig. 8). A his ime, one can answer he quesion How long does i ake o he pressure when he comes o laminar, wih wha size of he pipe is suiable for his?, using equaions described above. If one uses oo small pipes, i may ake oo much ime. omeimes here is a case ha wo channels are necessary for he slow piping. acuum Chamber low Evacuaion ipe Main ipe pump Evacuaion ime [sec] Fig. 9. Evacuaion curve wih semi-log scale 6. Conclusion In he world of vacuum indusry, descripions in exs abou viscous evacuaion is limied, probably because i is hough o be raher rivial. In his paper, he auhor analyzed i and found ha wha is hough o be simple is acually no so simple. He inroduced he noion of ransferring pressure and showed ha he evacuaion curve in a viscous is firs exponenial, and hen inversely proporional o ime. In paricular, when he pipe s resisance dominaes, he fac ha he pressure decreases inversely proporional o ime is simple and imporan. I is srange ha here is virually no ex which clearly poins ou his fac. The soluion deduced mahemaically his ime has a raher complicaed form, and is no so easy o grasp. Bu nowadays when compuer use is ordinary, i can be a useful ool when used wih a spread shee sofware for making graphs or simulaions. I is he auhor s pleasure ha his paper be some help for he people concerned wih vacuum indusries. Fig. 8. Example of slow evacuaion References (1) JI Z 816 acuum echnology-ocabulary p (199) () acuum handbook (rev. ed.), pp39-41, Nippon-hinku-Gijusu Co., Ld. (198) (3) Yoshiaka Hayashi, Inroducion o acuum Technology, pp18-19, p13, p1, Nikkan-Kogyo hinbun Co., Ld. (1987) (4) URL hp:// app-a/conduc.hml (5) URL hp:// lennard.hml (6) URL hp:// app-a/mfp.hml (7) Yoshihiko Oosuki(Translaor), Mahemaical Formulary, pp86-88, Maruzen (1983) EI TECHNICAL REIEW NUMBER 71 OCTOBER 1 9

7 (8) John F. Ohanlon, acuum Technology Manual, angyo Tosho Co., Ld. (1985) (9) Kasuya Nakagawa, acuum Technologies, engineers manual, Ohm-sha Co., Ld. (1985) (1) Hiroshi Nakagawa, Wha o do agains vacuum roubles; learning from failures, Trend Books (198) (11) Kazuo Tanida, acuum sysem engineering Basics and applicaions, Yokendo (1977) Conribuor Y. ENDA enior pecialis enior Assisan General Manager lan & roducion ysems Engineering Division He has engaged in design and developmen of producion equipmen, especially in he semiconducor indusries. 1 Theoreical Analysis of acuum Evacuaion in iscous Flow and Is Applicaions