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Calculation of capture coefficients for a cylindrical cryopump
Vacuum/volume 42/numbers 8/9/pages 555 to 560/1991
0042-207X/91S3.00+.00 © 1991 PergamonPressplc
Printed in Great Britain
Calculation of c a p t u r e c o e f f i c i e n t s for a cylindrical c r y o p u m p J W Lee a n d Y K Lee, Department of Mechanical Engineering, Pohang Institute of Science and Technology,
Pohang PO 125, Kyungbuk, Korea, 790-600 received for publication 14 October 1990
A new efficient method has been developed for calculating the pumping efficiency of a cryopump for different types of gases, which uses the geometric view factors between the surface elements of a cryopump, This method is as accurate as the Monte Carlo method, but much faster and more efficient when a systematic change in the system parameters is investigated. The method is applied for investigating the effect of cryopanel geometry on the pumping characteristics of cryopump arrays for different types of gases. One major result from such calculations is that the diameter of the cryopanel should be about 0.6 times that of the pumpbody diameter in order to give maximum pumping of He and H2 by the cryosorption element attached on the reverse side of the pane/. 1. Introduction
When a cryopump is used for pumping gases in the hv or uhv range, different types of gases are pumped on different parts of the pumping array, which means that the performance of a cryopanel array of fixed design changes depending on the composition of gas load. From the design point of view, optimum values exist for design parameters, examples of which are the diameter and height of the panel, and the gap between the baffle and the panel face, adopting the general geometry as cylindrical o1 conical, as in most of the commercial cryopumps. When the gas load is mainly of type II gases such as N> 02 Ar, and CO2 ~, the diameter of cryopanel should be as hugc as possible so that most of the gas molecules entering the pump through a baffle or a louver can be intercepted by and condensed on the front surface of the panel. However, when the gas load of H2 and He is appreciable, there should be an optimum size tbr the cryopanel. Too large a panel will prevent the gas molecules from going through the gap between the panel and the pump body and reaching the cryosorption element, which is usually put on the reverse side of the cryopanel. Too small a cryopanel also means a smaller capture efficiency since the gas molecules which passed through the gap and reached the bottom surface of the shield can easily return to the frontside of the panel without making collisions on the sorption element. When a cryopump is used in the medium vacuum range, the rapid growth of cryo deposits on the front (top) surface of the panel may sometimes cause an earlier regeneration which means an inefficient use of the surface area of the cryopanel. In order to determine the optimum shape and size of a cryopanel, estimations of the pumping rate at different positions of the panel for each type of gas are also necessary. Up to now, no work has been published on the regional pumping rate, and only a few values are available for the total capture coefficients-', for which two methods have been used. The Monte Carlo method is a kind of exact solution technique, but it generally takes a very long computation time because a totally new run is needed even when there is only a change in the operating conditions. The other method is to break the cryopump into many sections for
which conductance values can easily be calculated and to reassemble them using Oatley's method, in order to obtain the total capture coefficient. However, the molecular flux through the interface between the decomposed sections should be random (obeying the cosine law), in order for Oatley's method to give accurate results 3, which is not true for most cryopump conditions. In this paper, a new technique is suggested for calculating the pumping efficiency with the same accuracy as the Monte Carlo method but with much less computing time. This efficient technique makes it possible to analyse in detail the effect on the pumping characteristics for the several types of gases, of the design parameters such as shape, size, and sticking coefficients on different parts of the cryopanel array. 2. Theoretical basis
2.1. Formulation. Consider an enclosure consisting of N elements such that the temperature, sticking coefficient, condition, and molecular flux are all uniform on each element (Figure 1). The molecular flux balance for the kth
surface surface surface surface
entrance •
I
I
3 2
Jk I
exit Figure t. Enclosure for analysis, composed of N discrete surface elements. 555
J W Lee and Y K Lee. Calculation of capture coefficients for a cylindrical cryopump
each punlping stirfacc can be calculated if only tile vie\~ factors. sticking coetticien/s, a n d g a s s o n r c e l]LIX d i s t r i b u t i o n (dne to evaporation or source-gas introduction) are known. The transnlission probability Pr of a x,aCLILInl systenl with /, entrance surl'accs and n# exit surfaces can also be dclined as,
Ni, k = ni, k A k No, k - Re,k A k -- [ne, k + ( l - f k )Itl,k] a k
nl
II,/,~/ // i
Pr=
,
Iqgure 2. Molecular lluxcs on a typical SUl-Iilccclement.
element can be written as (Figure 2), N/, = hi,.41, = (11,./, --#li.i,)Ai, => 111, = #L,.I, --Ili.l,,
(I)
where n~ and n,, arc the rate o1" incoming and outgoing molecules per unit area, respectively, and &, and A~, arc t i l e net molecular flux per unit area and the surface area o[" the kih surface clcrnent. The second e q u a t i o n comes from tile fact that the outgoing molecules are composed of directly entitled pills reflected moleCtl]eS :
I;,)H,j,,
(2)
where n ~ is the sell emitting molecular flux due to ewlporation or source introduction, a n d / ) , the sticking coefficient for the kth surfacc clement. The incident molecular flux n,.j, can be considered as the sum Of fluxes coming directly from the other snrl'ace elements without making any collision on the way to the /,th surface. Since the o u t g o i n g flux l'rom any surface element can be assumed to be diffusely distributed, the fraction of outgoing molecules from t h e / t h surface that arrive directly on the kth surface is the so-called geometric view factor/://. Then n,,¢, can be rcwrittcn in terms of n,,j, and F~j.
Ai, ni. t =
ic;i, Ain<,., = I
~ /
Vl, iAI, n, i ~> I
HI./, =
~" /D,,.H.., j
(3)
I
where reciprocity relation between view factors 4 is used. Combining equations (1) and (2) gives an e q u a t i o n for n~, in terms of emission flux, It.j,
H/, = H, /,-
l
tic. k
I
.1;, ~Hj,
= l
.1;, (H,j,
l)n,,.L).
\
i
I
~ /
((J/~,
Fi, i)H,,,.
(5)
I
Equations (4) and (5) give the final equation for the kth surface : \
[5/,i /=:
(1
/j,)Fi,,]n,,.i
n,.~.
(6)
I
Writing N such equations for /, = I - N. we have a m a l r i x equation which gives n,,.~in terms of ~#<.z,and F~,,. n~, can be found 1"I"O111e q u a t i o n (4) and n,, r So the differential p u m p i n g action OI1 556
~1) = 27rR, 0 = arccosine
\
y" F/,in,,.i~nl,
2.2. View factor calculalion. A l t h o u g h there is a huge hod5 of view Iilctors obtained from analytical solutions, t h e ) arc all for simple geometries', and numerical calculation is unavoidable as the n u m b e r of snrfaces becomes hugc a n d ' o r the geometr~ bccolncs more complex. F o r typical crxopunaps, the shield and the cryopanel array have a cylindrical a n d o r conical shape a i t h a central blockage, l\~r which analytical evaluation of view factors is quite formidable. In this paper the view lactors are calculated by' the M o n t e Carlo m e t h o d for this reason. The M o n t e Carlo method for calculating the view factors consists of evaluating the fraction of the particles emitted l ' r o l ] ] one surface element that will directly hit the surface elelnenl concerned. Particles are ejected one by one from a r a n d o m position on a given surface elemenL into a direction of an azimLlth angle (/) and a direction angle 0. Particle trajectory is calculated in order to determine the surface on wllich it will land. After this process has been continued l \ a a large e n o u g h n u m b e r of cjcctcd particles, tile view thctor b , is calculated bv the fraclion of lhe particles, k), - N~ L"¢,. where N, is the n u m b e r of particles \~hlch hit snrl'ace i :/nd N; is the total n u m b e r of particles emitted from surface i. Thc method of choice IBr the angle of emission is important. Since the emission of particles from a diffuse surface should IBIlow the cosinc law. thc particle flux density per unit solid angle should be p r o p o r t i o n a l to cos 0. It follows that the total nunlber of particles emitted into a direction between 0 and ()-F d0 fronl the normal direclion should be p r o p o r l i o n a l to cos 0 sin 0 dO, because the solid angle stiblcnded by the stlrl]leC clernent between 0 and 0 + d 0 is p r o p o r t i o n a l to sin 0 dO. The "difl'use' condition is satisfied if the azimuth anglc 0 aind tile anglc to the skirl'ace nOlllla] 0 are taken :lccording to tile Iollox~ing proced u re", (8) R~.
(4)
and another equation for n~, comes from equations (3) and (I). hi, =#l<,j,
I
The fnrmuhllion as presented here is theoretically idenlical to tile thcrnlal radiation exchan,.ze, between difl'use surlaccs 4.
N k = nk A k
H<,.a = H<,L + (1
(7)
where R~ and R, are r a n d o m nulnbers. For a simple example of coaxial disks, 3000 molecules arc e n o u g h to give illa excellent agreenlent with exact results (Figtirc 3). 2.3. Comparison with the full Monte Carh) methud. The present m e t h o d is physically similar to tile M o n t e Carlo method, and gives identical results if only tile subdivision is line enough to ensure uniform molecuhlr flux on each surface element. Tile M o n t e Carlo m e t h o d tracks tile trajectory el'each molecule from the source surlhce until it is finally captured. On the other hand, the present m e t h o d calculates the capturc efficiency by a single inversion m a n i p u h l t i o n o f a nlalrix whose elements arc expressed ill terms of view factors. The view factors between c',cry pair of
J W Lee and Y K lee . Calculation of capture coefficients for a cylindrical cryopump t.O-
/ "\
1.0
[ G"1 ,r
0.8-
Pr ~. o.4 ~
R
o.0-
o
Exact ......... D.H. Dovis[7]
~
---
0.4
This study
-
(R,/Ro)==2.25
0.2.
0.2
ado
This work
// o.o
.~
~
'
f'z
'
;e
'
0.0
to
A/RI
L/Ro
Figure 3. Geometric view factor between coaxial disks.
Figure 5. Transmission probability for a baffle with a circular blocking plate and two restricted ends•
surface elements are computed by the Monte Carlo technique, where particle tracking is required only for a single collision, no matter whether it is captured or not due to that collision. For the complex geometries and for cases where multiple collisions to the capture are common, the present method is very efficient. A simple example to test the effectiveness of this method against the full Monte Carlo method is the molecular conductance of a cylindrical duct in the free molecular flow. The inner surface of the cylinder is divided into N identical rings of surface element, and the sticking coefficients are set to 1.0 for the entrance and the exit surface, and 0 to the others on the wall. If the amount of self emission n,. is set to zero, except on the entrance, the transmission probability can be obtained by calculating ni using the method described previously. The computation time with a full Monte Carlo simulation increases very rapidly with L/D (length/diameter) because the number of collisions with the cylinder wall increases very rapidly with L/D. But our method can obtain a nearly exact solution by solving a system of N algebraic equations once the view factors are known. As the number of dividing surfaces increases, the transmission probability obtained by the method approaches the analytical result. When LID is 2, our method with seven surface divisions gives quite a good agreement with the Monte Carlo and experimental results (Figure 4). For the symmetric plate baffle geometry (Figure 5), the number of surface divisions required for an excellent agreement with the experimental result and the Monte Carlo simulation is more than 30.
1.0
-1.0
,, Pr
0.6
~o ~ ,
0.4
- This work ooooo Experiment [8] AA*AA Monte Carlo [8] u=$
~
o
~
3.1. Model description. The model cryopump for analysis is schematically shown in Figure 6 : surface 1 is a baffle or a louver which is the only entrance and exit: surface 2 is a radiation shield without any hole for gas transmission ; and surface 3 is a cryopanel with cryosorption element, such as charcoal, bonded on the reverse side. The baffle and the radiation shield are usually maintained at a temperature 50 100 K, and the cryopanel at 1020 K. Since the condensation efficiency on a cryopanel with specified geometry and temperature differs much between different gas species, the gas species are grouped into three types based on the cryopump temperature required for maintaining uhv ~. Water vapour (H20) and carbon monoxide (CO) are examples of type I gases whose equilibrium vapour pressure at 70 K is in the uhv range, and are condensed quite easily on the baffle surface. Type II gases, such as N2, O_~, Ar and CH4 do not condense on the baffle but are pumped easily on the cryopanel and also on the cryosorption element. Type IlI gases, such as H> He and Ne, do not condense on the cryopanel but are pumped by the cryosorption element only. Since the aim of this study is to investigate the pumping characteristics o f a cryopanel, only types II and llI gases are considered, then the sticking coefficient on the radiation shield can be
!H c
-0.6
,~
~
~
Ho
" -0.4 " -0.2
N=20 0.0
3. Pumping characteristics of a model cryopanel
,i
~ _ _ ~
0.2
method [B]
Do
-
0.0
1o
(L/D) Figure 4. Transmission probability obtained with different number of surface elements, N, for a circular tube of length L and diameter D.
Figure 6. Schematic of a model cryopump for analysis : I baffle : 2 radiation shield : 3 cryopanel : .L, sticking coefficient of the front(top) cryosurface : f sticking coefficient of the bottom cryosurface (cryosorption element). 557
J W Lee and Y K Lee. Calculation
of capture coefficients for a cylindrical cryopump 0.20 •
l(1 be zero. It is also assumed that tile gas molecules of types 11 or 111 entering the cryopump through a bal]]e (serrate I) are distributed unirormly over the baffle surfacc, and their distribution is directionally difruse, rollowing the cosine law. Gas molecules can bc pumped out upon collision with the cryopanel. and thc probability (/1" capture per collision is 1], on the I'ront stlrlacc end .Ii on the back surface (sorption element). Then the [ISSLImcd
TYPE III GASES Hc/Ho=0.15
Pr o.z5 ~
capturc p r o b a b i l i t y o r the model cryopanel is defined as thc Iraction o r gas molecules captured on surracc 3 to gas molecules introduced rrom surface I. Punlping efficiency o r the whole
0,10
pump body can be obtained by an appropriute rch_ttiorlsMp such as Oallcy's law between the transmission probabilitics o1" thc baffle and the cryopanel. The whole pump surl'aec is di\.idcct into 50 surl'ace elements, and the view ractor bctwcen the Sull'qcc elements is calculated using the Monte Carlo method with 5000 particles emitted rrom each surl';.tce element into i-andom directions m order to satisfy the diffuse condition described previously. Any more increase in the ntunber of particles did not make any noticeable improven]er, t in accuracy.
/ Do : 20 cm fo = 0 . 0 0.05 + o.5 o.'~ ~77 o.'.
3.2. Effect o f c r y o p a n e l g e o m e t r y . There HIC four gcomctric puramctcrs rot a cylindrical/conical cryopancl : diarnetcr Di, length oz height or the vertical skirt I l L conical angle 0, and the gap t h belwecn the panel and the baMe. Variation of capture probabilit? Pr with the fcmr parameters is shown m Figures 7 II, l\/r both types 11 ;rod Ill gases. For type 11 gases, b o t h / ' i and 1, are set to 1.0, but l\)r type II1 gases Ii - 1.0 and/], - 0.0. The diamcter or
0.9
TYPE
II
GASES
Hc/Ho=O,15~.
J
/
0.8
0.7
0.6 HI = Do = Ho =
°'%.o
o.~
7 20 20
o.'v
cm cm cm
o.'8
0.9
I
i Ho =
20
cm
'~
0.9
D~/Do Figure 9. \.arialion olcaptLu'c probal',ilit~ l'r,,,,ith lh and t1+ Ior t?pc Ill gakc>;.
tile radiation shield I ) o and height 1t. arc lixed at 20 cm. and the o.,nic,ul angle 0 is zero unless mentioned ,.',therv,;ise. F o r type II gases, capture p r o b a b i l i t ) i n c r e a s e s m o n o t o n i c a l h ,,,,ill] I ) L giving about 25 30'!. increase when / ) i D , changed from 0.5 to 0.9 (Figure 7). Increasing the gap betv,.een the barite und the cryopanel rcsuhs in a reduced capturc prc, babilit~ becuuse a lurgcr gap rneans a higher probabilil_v ,.)fcolliding ,aith a shield wall and returning to the baltle. When Ihe gap is tripled
0.15 to 0.45. 1he capttlrc probability is redtlced b,, 22 25'!i, over the range of diameter ratio in l:igurc 7. The length of the skirt bus a relativcl',, small effect on the c:.tpttn-c coefficient, and e\cn less I'or the larger diameter (Figul-c N). For type Ill gases, the capture probability does not beha\c monotonicully with the gconletric parumctcrs. First of all, thcrc exists an optitnum cryopancl diameter which makes the capture probability nlaximum. Whcn tile diameter is lnuch larger thu]l this oplimuna \.ahtc. it is ,,cry. hard I\rr the molecules to ~,,o thromdl tile gap between thc panel and the shield, which is a pre-rcquisite for reaching the cryosorption element. When the diameter is tllLich smaller, molecules can easih go through tile gup. but at the same time they can easily return to the Iront side o r t h e panel. v,.Mch results in a smaller capture cllicicncv. Over a \cry wide range (/1"other parameters, the optimum diarneter ratio D I D ( , is about 0.6 (FigLtre 9). Incrcascd Hc results in a reduced capture probability again, bccuusc un increase in 11~ reduces the probfrom thllo-
DI/Do I'igure 7. Variation of cup[ui-c probability l'r v, ith I)i and th' for t)r,c II gases. 0.20
. . . . . . . . . . . . . . . . . . . . . . .
TYPE III GASES
i.ol
TYPE II GASES
I
0.15
--
0.6
1.0
0.8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
- --___2_-~--.......
Pr
~-=-c:-...... .... o.4
o.~o ~k
- - - 22"_--2.:+
O.4 I
0,4, ~
Hc = Do = HO =
0.8
5 cm 20 cm 20 cm
Di/Do=O. 2
0.05
0.2
i
-----
-
~
--
-
----
~
t
Dr/Do=0.2 Hc = 5 cm Do = 2 0 c m No = 2 0 c m
0.0 |
o.2
o.'s
o.',. H~/Ho
o.'~
0,00
0.6
Figure g. Varialion ,.',l'caphm: prohabili/', lb" with Hi for t>pc 11 gas,.'s 558
i !
0.2
o.s
----
o.~ H~/Ho
I
o:5
T.
I+igurc I0. Vuriati(m ,,fl capturL' prc, babitit_~ /'r ~ ith 1//for t~ pc I I 1 gases. Top cur,,,.' for I = 1.0. and the I,.w,,._'rcur',c-, l'or /[ - O.,"L 0.6. 0.4. and 0.2 dov,.nv,:trd.
J W Lee and Y K Lee .
Calculationof capturecoefficientsfor acylindricalcryopump
o0
0.20 •
T Y P E llI GASES ~ / ~ - - ' ~ ' ~ -
TYPE
GASES
II
-
o.0
H, = Hc =
Pr
He Di
o.,1.
= =
Do
Pr o.lo
cm
5
5 cm 20 cm 10 cm
0.10
I
20 c m
0.o0
°"
HI
=
7
Ho = Di = Do =
20 14 20
cm
-
o.o
Ib ~b sb ~
8'o~o
~ 6b qo (degree)
I. Variation of capture probability Pr with conical angle 0 for types I1 and Ill gases. ability of an entering molecule passing through the gap between the panel and the radiation shield. Increased skirt length reduces the capture efficiency, but its effect is much less than that caused by changes in diameter. A cylindrical cryopanel of optimum diameter has the highest capture efficiency, irrespective of the skirt length (Figure 10). If the cryopanel has a truncated conical shape, the conical angle 0 is thought to have a strong effect on the pumping characteristics. The capture probability for type II gases increases as the skirt becomes flat (0 --* 90 ), but for type III gases there exists an optimum angle, which can be explained in the same way as optimum diameter. The maximum horizontal reach of a conical cryopanel with an optimum angle is really close to the optimum diameter (Figure 11). 3.3. Effect of sticking coefficients. As pumping proceeds, surface conditions of the cryopanel change, so changing the sticking coefficient. The capture coefficient for type lI gases varies almost linearly with £,, the sticking coefficient on the front surface, but changes very little with f , the sticking coefficient of the cryosorption element (Figure 12). This is due to the small amount of type II gases pumped on the sorption clement. For type IIl gases (J;, = 0), however, the sticking coefficient /i has a profound effect, especially for f less than 0.5 (Figure 13). 3.4. Regional pumping speed. For the design of optimum array
shape, information about the distribution of local pumping rate
0.8-
T Y P E II GASES
3,,
=~
o.oPr 0.5-
0.4-
/
0.2
0.0
,~r'/
J
0.2
=
Ft e
l// O.S-
,
,
0.4
,
,
0.0
Hi
=
He
=
,
5
i
0.2
'
O.',t
'
0.'6
'
0.'
'
1.0
,
Figure 13. Variation of capture probability Pr with sticking coefficiem f
for type Ill gases.
t.o
~ C ~ TYPE II GASES
Dp,,,ao.4°'e~°'8 0.2 o.o
~
HH'c=
8cm = 5 crn He = 20 c m
)~/
~,o= 212 cm 0 cm ~
~
~
~
fc
l~
A
X (cm) Figure 14. Relative pumping rate at different positions of the cryopanel surface, x is the distance from the centre.
is very important. For type I1 gases, the local pumping rate shows a maximum at the centre of the front surface, and decreases away from the centre (Figure 14). There is a sudden drop in the pumping rate at the joint between the front disk and the skirt (at x = 6 for example). The values are normalized with respect to the maximum value at the centre, and the pumping rate is quite uniform on the front disk. The amount of type II gases pumped on the reverse side is quite small, being in the order of less than 10% of that pumped on the front surface. It follows that of the type II gases captured, about 40% is pumped on the front disk and 60% on the side skirt, for the model cryopump shown in Figure 14. For type Ill gases, the maximum pumping rate occurs at the lip of the skirt (x = 14), and the minimum at the joint. The ratio of the maximum to minimum is about 5 both for types II and Ill gases. 4. Conclusions
cm
7 cm 20 cm
0.8
'
f,
Figure I
0.7-
0.0
cm cm cm
t.O
fo Figure 12. Variation of capture probability Pr with sticking coefficients 11 and/;, for t~cpe 11 gases.
1. A new method of calculating transmission probability has been devised and used for analysing the pumping characteristics of a cylindrical/conical cryopanel. The method makes use of view factors and gives results as accurate as the Monte Carlo method, but is very efficient and easy to use, especially when the geometry is fixed and only the sticking coefficient is changed.
559
J W l e e and Y K Lee: Calculation of capture coefficients for a cylindrical c r v o p u m p
2. The capture probability l\)r type ll gases increases m o n o tonically with the diameter and the skirt length of the cryopancl but I\w type III gases there exists ~i11 o p t i m u m diameter which is a b o u t Di,'Do- 0.6 over a wide rallgc of other geometric p~.ll'LInl e tCFS. 3. Thc capture probability of type 1I gases increases almost linearl 5 \~ith the sticking cocfticicnt of the cryopancl upper surl'ace, but is quite inscnsitivc to the sticking coefficient of thc cryosorption clement. 4. For a conical cryopanel, there exists an o p t i m u m conical angle for type Ill gases, such that the nlLlxilllunl horizontal reach at the o p t i m u m angle is equal to the o p t i m u m diameter. 5. The local pumpirlg ralc J\)l- type I1 gases is quite u n i l \ m n on the Iront stir[ace, and decreases away troll] the centre, l:or type llI gases the p u m p i n g ralc occurs tit lhc lip of the skirl, and
560
the m i n i m u m ;it the .joint. The ratio of thc Ill;.tXlIllLIlII [o Ihe Illinillltllll is ~lbOtl[ 5. References i \ .I BartlcH and P ..X l.c~is. ('t:l'oqcm~ I ' r . ¢ c ~ ~m,/I:qml~/mvll~ 15ML. p T I { Ig~4). : R .\ l tacl'cr. (rropumpil~d 7 h c . r v am/l'raclicc, p 1:,7. ('k~rcndon Prc~. ()\l'¢~rd (19~9). ~[) J ~LIIIIC]Cr, .1 l a c 5ci 7cchm~/. A4(3). 3~S (I9~n). P. Sicecl and J R ttowci]. 771ermM RaJialioll fh'al 7m~l~/~'r. 2nd I d n .
p INS. Mc(h-a\~-14ill. Ncx~ York (IOSI). ".1 R tto\~cll. RtlJit/li~m (.fff~tfm'ali~m I'}l~mr~. Mc(h-;m tlill. Nc~ 5 ~rk (19~2). " M M Wcincr..1 W Tindall and 1. M C;indcll. P~q~cr No 6 > W & I f I - s l . 15'./1/:'. No~cmbcr I 1960}. I) tl l ) a x J s . . / . I F p I Phr~. 31. 1169 (1960). I. I. l.cvcl>on. N tfillc,'on and l) 11 l)~l\is, l ' r w > ?,'tmz la~' 5)m/~. 7.
372 ( I 9(,0).
0042-207X/91S3.00+.00 © 1991 PergamonPressplc
Printed in Great Britain
Calculation of c a p t u r e c o e f f i c i e n t s for a cylindrical c r y o p u m p J W Lee a n d Y K Lee, Department of Mechanical Engineering, Pohang Institute of Science and Technology,
Pohang PO 125, Kyungbuk, Korea, 790-600 received for publication 14 October 1990
A new efficient method has been developed for calculating the pumping efficiency of a cryopump for different types of gases, which uses the geometric view factors between the surface elements of a cryopump, This method is as accurate as the Monte Carlo method, but much faster and more efficient when a systematic change in the system parameters is investigated. The method is applied for investigating the effect of cryopanel geometry on the pumping characteristics of cryopump arrays for different types of gases. One major result from such calculations is that the diameter of the cryopanel should be about 0.6 times that of the pumpbody diameter in order to give maximum pumping of He and H2 by the cryosorption element attached on the reverse side of the pane/. 1. Introduction
When a cryopump is used for pumping gases in the hv or uhv range, different types of gases are pumped on different parts of the pumping array, which means that the performance of a cryopanel array of fixed design changes depending on the composition of gas load. From the design point of view, optimum values exist for design parameters, examples of which are the diameter and height of the panel, and the gap between the baffle and the panel face, adopting the general geometry as cylindrical o1 conical, as in most of the commercial cryopumps. When the gas load is mainly of type II gases such as N> 02 Ar, and CO2 ~, the diameter of cryopanel should be as hugc as possible so that most of the gas molecules entering the pump through a baffle or a louver can be intercepted by and condensed on the front surface of the panel. However, when the gas load of H2 and He is appreciable, there should be an optimum size tbr the cryopanel. Too large a panel will prevent the gas molecules from going through the gap between the panel and the pump body and reaching the cryosorption element, which is usually put on the reverse side of the cryopanel. Too small a cryopanel also means a smaller capture efficiency since the gas molecules which passed through the gap and reached the bottom surface of the shield can easily return to the frontside of the panel without making collisions on the sorption element. When a cryopump is used in the medium vacuum range, the rapid growth of cryo deposits on the front (top) surface of the panel may sometimes cause an earlier regeneration which means an inefficient use of the surface area of the cryopanel. In order to determine the optimum shape and size of a cryopanel, estimations of the pumping rate at different positions of the panel for each type of gas are also necessary. Up to now, no work has been published on the regional pumping rate, and only a few values are available for the total capture coefficients-', for which two methods have been used. The Monte Carlo method is a kind of exact solution technique, but it generally takes a very long computation time because a totally new run is needed even when there is only a change in the operating conditions. The other method is to break the cryopump into many sections for
which conductance values can easily be calculated and to reassemble them using Oatley's method, in order to obtain the total capture coefficient. However, the molecular flux through the interface between the decomposed sections should be random (obeying the cosine law), in order for Oatley's method to give accurate results 3, which is not true for most cryopump conditions. In this paper, a new technique is suggested for calculating the pumping efficiency with the same accuracy as the Monte Carlo method but with much less computing time. This efficient technique makes it possible to analyse in detail the effect on the pumping characteristics for the several types of gases, of the design parameters such as shape, size, and sticking coefficients on different parts of the cryopanel array. 2. Theoretical basis
2.1. Formulation. Consider an enclosure consisting of N elements such that the temperature, sticking coefficient, condition, and molecular flux are all uniform on each element (Figure 1). The molecular flux balance for the kth
surface surface surface surface
entrance •
I
I
3 2
Jk I
exit Figure t. Enclosure for analysis, composed of N discrete surface elements. 555
J W Lee and Y K Lee. Calculation of capture coefficients for a cylindrical cryopump
each punlping stirfacc can be calculated if only tile vie\~ factors. sticking coetticien/s, a n d g a s s o n r c e l]LIX d i s t r i b u t i o n (dne to evaporation or source-gas introduction) are known. The transnlission probability Pr of a x,aCLILInl systenl with /, entrance surl'accs and n# exit surfaces can also be dclined as,
Ni, k = ni, k A k No, k - Re,k A k -- [ne, k + ( l - f k )Itl,k] a k
nl
II,/,~/ // i
Pr=
,
Iqgure 2. Molecular lluxcs on a typical SUl-Iilccclement.
element can be written as (Figure 2), N/, = hi,.41, = (11,./, --#li.i,)Ai, => 111, = #L,.I, --Ili.l,,
(I)
where n~ and n,, arc the rate o1" incoming and outgoing molecules per unit area, respectively, and &, and A~, arc t i l e net molecular flux per unit area and the surface area o[" the kih surface clcrnent. The second e q u a t i o n comes from tile fact that the outgoing molecules are composed of directly entitled pills reflected moleCtl]eS :
I;,)H,j,,
(2)
where n ~ is the sell emitting molecular flux due to ewlporation or source introduction, a n d / ) , the sticking coefficient for the kth surfacc clement. The incident molecular flux n,.j, can be considered as the sum Of fluxes coming directly from the other snrl'ace elements without making any collision on the way to the /,th surface. Since the o u t g o i n g flux l'rom any surface element can be assumed to be diffusely distributed, the fraction of outgoing molecules from t h e / t h surface that arrive directly on the kth surface is the so-called geometric view factor/://. Then n,,¢, can be rcwrittcn in terms of n,,j, and F~j.
Ai, ni. t =
ic;i, Ain<,., = I
~ /
Vl, iAI, n, i ~> I
HI./, =
~" /D,,.H.., j
(3)
I
where reciprocity relation between view factors 4 is used. Combining equations (1) and (2) gives an e q u a t i o n for n~, in terms of emission flux, It.j,
H/, = H, /,-
l
tic. k
I
.1;, ~Hj,
= l
.1;, (H,j,
l)n,,.L).
\
i
I
~ /
((J/~,
Fi, i)H,,,.
(5)
I
Equations (4) and (5) give the final equation for the kth surface : \
[5/,i /=:
(1
/j,)Fi,,]n,,.i
n,.~.
(6)
I
Writing N such equations for /, = I - N. we have a m a l r i x equation which gives n,,.~in terms of ~#<.z,and F~,,. n~, can be found 1"I"O111e q u a t i o n (4) and n,, r So the differential p u m p i n g action OI1 556
~1) = 27rR, 0 = arccosine
\
y" F/,in,,.i~nl,
2.2. View factor calculalion. A l t h o u g h there is a huge hod5 of view Iilctors obtained from analytical solutions, t h e ) arc all for simple geometries', and numerical calculation is unavoidable as the n u m b e r of snrfaces becomes hugc a n d ' o r the geometr~ bccolncs more complex. F o r typical crxopunaps, the shield and the cryopanel array have a cylindrical a n d o r conical shape a i t h a central blockage, l\~r which analytical evaluation of view factors is quite formidable. In this paper the view lactors are calculated by' the M o n t e Carlo m e t h o d for this reason. The M o n t e Carlo method for calculating the view factors consists of evaluating the fraction of the particles emitted l ' r o l ] ] one surface element that will directly hit the surface elelnenl concerned. Particles are ejected one by one from a r a n d o m position on a given surface elemenL into a direction of an azimLlth angle (/) and a direction angle 0. Particle trajectory is calculated in order to determine the surface on wllich it will land. After this process has been continued l \ a a large e n o u g h n u m b e r of cjcctcd particles, tile view thctor b , is calculated bv the fraclion of lhe particles, k), - N~ L"¢,. where N, is the n u m b e r of particles \~hlch hit snrl'ace i :/nd N; is the total n u m b e r of particles emitted from surface i. Thc method of choice IBr the angle of emission is important. Since the emission of particles from a diffuse surface should IBIlow the cosinc law. thc particle flux density per unit solid angle should be p r o p o r t i o n a l to cos 0. It follows that the total nunlber of particles emitted into a direction between 0 and ()-F d0 fronl the normal direclion should be p r o p o r l i o n a l to cos 0 sin 0 dO, because the solid angle stiblcnded by the stlrl]leC clernent between 0 and 0 + d 0 is p r o p o r t i o n a l to sin 0 dO. The "difl'use' condition is satisfied if the azimuth anglc 0 aind tile anglc to the skirl'ace nOlllla] 0 are taken :lccording to tile Iollox~ing proced u re", (8) R~.
(4)
and another equation for n~, comes from equations (3) and (I). hi, =#l<,j,
I
The fnrmuhllion as presented here is theoretically idenlical to tile thcrnlal radiation exchan,.ze, between difl'use surlaccs 4.
N k = nk A k
H<,.a = H<,L + (1
(7)
where R~ and R, are r a n d o m nulnbers. For a simple example of coaxial disks, 3000 molecules arc e n o u g h to give illa excellent agreenlent with exact results (Figtirc 3). 2.3. Comparison with the full Monte Carh) methud. The present m e t h o d is physically similar to tile M o n t e Carlo method, and gives identical results if only tile subdivision is line enough to ensure uniform molecuhlr flux on each surface element. Tile M o n t e Carlo m e t h o d tracks tile trajectory el'each molecule from the source surlhce until it is finally captured. On the other hand, the present m e t h o d calculates the capturc efficiency by a single inversion m a n i p u h l t i o n o f a nlalrix whose elements arc expressed ill terms of view factors. The view factors between c',cry pair of
J W Lee and Y K lee . Calculation of capture coefficients for a cylindrical cryopump t.O-
/ "\
1.0
[ G"1 ,r
0.8-
Pr ~. o.4 ~
R
o.0-
o
Exact ......... D.H. Dovis[7]
~
---
0.4
This study
-
(R,/Ro)==2.25
0.2.
0.2
ado
This work
// o.o
.~
~
'
f'z
'
;e
'
0.0
to
A/RI
L/Ro
Figure 3. Geometric view factor between coaxial disks.
Figure 5. Transmission probability for a baffle with a circular blocking plate and two restricted ends•
surface elements are computed by the Monte Carlo technique, where particle tracking is required only for a single collision, no matter whether it is captured or not due to that collision. For the complex geometries and for cases where multiple collisions to the capture are common, the present method is very efficient. A simple example to test the effectiveness of this method against the full Monte Carlo method is the molecular conductance of a cylindrical duct in the free molecular flow. The inner surface of the cylinder is divided into N identical rings of surface element, and the sticking coefficients are set to 1.0 for the entrance and the exit surface, and 0 to the others on the wall. If the amount of self emission n,. is set to zero, except on the entrance, the transmission probability can be obtained by calculating ni using the method described previously. The computation time with a full Monte Carlo simulation increases very rapidly with L/D (length/diameter) because the number of collisions with the cylinder wall increases very rapidly with L/D. But our method can obtain a nearly exact solution by solving a system of N algebraic equations once the view factors are known. As the number of dividing surfaces increases, the transmission probability obtained by the method approaches the analytical result. When LID is 2, our method with seven surface divisions gives quite a good agreement with the Monte Carlo and experimental results (Figure 4). For the symmetric plate baffle geometry (Figure 5), the number of surface divisions required for an excellent agreement with the experimental result and the Monte Carlo simulation is more than 30.
1.0
-1.0
,, Pr
0.6
~o ~ ,
0.4
- This work ooooo Experiment [8] AA*AA Monte Carlo [8] u=$
~
o
~
3.1. Model description. The model cryopump for analysis is schematically shown in Figure 6 : surface 1 is a baffle or a louver which is the only entrance and exit: surface 2 is a radiation shield without any hole for gas transmission ; and surface 3 is a cryopanel with cryosorption element, such as charcoal, bonded on the reverse side. The baffle and the radiation shield are usually maintained at a temperature 50 100 K, and the cryopanel at 1020 K. Since the condensation efficiency on a cryopanel with specified geometry and temperature differs much between different gas species, the gas species are grouped into three types based on the cryopump temperature required for maintaining uhv ~. Water vapour (H20) and carbon monoxide (CO) are examples of type I gases whose equilibrium vapour pressure at 70 K is in the uhv range, and are condensed quite easily on the baffle surface. Type II gases, such as N2, O_~, Ar and CH4 do not condense on the baffle but are pumped easily on the cryopanel and also on the cryosorption element. Type IlI gases, such as H> He and Ne, do not condense on the cryopanel but are pumped by the cryosorption element only. Since the aim of this study is to investigate the pumping characteristics o f a cryopanel, only types II and llI gases are considered, then the sticking coefficient on the radiation shield can be
!H c
-0.6
,~
~
~
Ho
" -0.4 " -0.2
N=20 0.0
3. Pumping characteristics of a model cryopanel
,i
~ _ _ ~
0.2
method [B]
Do
-
0.0
1o
(L/D) Figure 4. Transmission probability obtained with different number of surface elements, N, for a circular tube of length L and diameter D.
Figure 6. Schematic of a model cryopump for analysis : I baffle : 2 radiation shield : 3 cryopanel : .L, sticking coefficient of the front(top) cryosurface : f sticking coefficient of the bottom cryosurface (cryosorption element). 557
J W Lee and Y K Lee. Calculation
of capture coefficients for a cylindrical cryopump 0.20 •
l(1 be zero. It is also assumed that tile gas molecules of types 11 or 111 entering the cryopump through a bal]]e (serrate I) are distributed unirormly over the baffle surfacc, and their distribution is directionally difruse, rollowing the cosine law. Gas molecules can bc pumped out upon collision with the cryopanel. and thc probability (/1" capture per collision is 1], on the I'ront stlrlacc end .Ii on the back surface (sorption element). Then the [ISSLImcd
TYPE III GASES Hc/Ho=0.15
Pr o.z5 ~
capturc p r o b a b i l i t y o r the model cryopanel is defined as thc Iraction o r gas molecules captured on surracc 3 to gas molecules introduced rrom surface I. Punlping efficiency o r the whole
0,10
pump body can be obtained by an appropriute rch_ttiorlsMp such as Oallcy's law between the transmission probabilitics o1" thc baffle and the cryopanel. The whole pump surl'aec is di\.idcct into 50 surl'ace elements, and the view ractor bctwcen the Sull'qcc elements is calculated using the Monte Carlo method with 5000 particles emitted rrom each surl';.tce element into i-andom directions m order to satisfy the diffuse condition described previously. Any more increase in the ntunber of particles did not make any noticeable improven]er, t in accuracy.
/ Do : 20 cm fo = 0 . 0 0.05 + o.5 o.'~ ~77 o.'.
3.2. Effect o f c r y o p a n e l g e o m e t r y . There HIC four gcomctric puramctcrs rot a cylindrical/conical cryopancl : diarnetcr Di, length oz height or the vertical skirt I l L conical angle 0, and the gap t h belwecn the panel and the baMe. Variation of capture probabilit? Pr with the fcmr parameters is shown m Figures 7 II, l\/r both types 11 ;rod Ill gases. For type 11 gases, b o t h / ' i and 1, are set to 1.0, but l\)r type II1 gases Ii - 1.0 and/], - 0.0. The diamcter or
0.9
TYPE
II
GASES
Hc/Ho=O,15~.
J
/
0.8
0.7
0.6 HI = Do = Ho =
°'%.o
o.~
7 20 20
o.'v
cm cm cm
o.'8
0.9
I
i Ho =
20
cm
'~
0.9
D~/Do Figure 9. \.arialion olcaptLu'c probal',ilit~ l'r,,,,ith lh and t1+ Ior t?pc Ill gakc>;.
tile radiation shield I ) o and height 1t. arc lixed at 20 cm. and the o.,nic,ul angle 0 is zero unless mentioned ,.',therv,;ise. F o r type II gases, capture p r o b a b i l i t ) i n c r e a s e s m o n o t o n i c a l h ,,,,ill] I ) L giving about 25 30'!. increase when / ) i D , changed from 0.5 to 0.9 (Figure 7). Increasing the gap betv,.een the barite und the cryopanel rcsuhs in a reduced capturc prc, babilit~ becuuse a lurgcr gap rneans a higher probabilil_v ,.)fcolliding ,aith a shield wall and returning to the baltle. When Ihe gap is tripled
0.15 to 0.45. 1he capttlrc probability is redtlced b,, 22 25'!i, over the range of diameter ratio in l:igurc 7. The length of the skirt bus a relativcl',, small effect on the c:.tpttn-c coefficient, and e\cn less I'or the larger diameter (Figul-c N). For type Ill gases, the capture probability does not beha\c monotonicully with the gconletric parumctcrs. First of all, thcrc exists an optitnum cryopancl diameter which makes the capture probability nlaximum. Whcn tile diameter is lnuch larger thu]l this oplimuna \.ahtc. it is ,,cry. hard I\rr the molecules to ~,,o thromdl tile gap between thc panel and the shield, which is a pre-rcquisite for reaching the cryosorption element. When the diameter is tllLich smaller, molecules can easih go through tile gup. but at the same time they can easily return to the Iront side o r t h e panel. v,.Mch results in a smaller capture cllicicncv. Over a \cry wide range (/1"other parameters, the optimum diarneter ratio D I D ( , is about 0.6 (FigLtre 9). Incrcascd Hc results in a reduced capture probability again, bccuusc un increase in 11~ reduces the probfrom thllo-
DI/Do I'igure 7. Variation of cup[ui-c probability l'r v, ith I)i and th' for t)r,c II gases. 0.20
. . . . . . . . . . . . . . . . . . . . . . .
TYPE III GASES
i.ol
TYPE II GASES
I
0.15
--
0.6
1.0
0.8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
- --___2_-~--.......
Pr
~-=-c:-...... .... o.4
o.~o ~k
- - - 22"_--2.:+
O.4 I
0,4, ~
Hc = Do = HO =
0.8
5 cm 20 cm 20 cm
Di/Do=O. 2
0.05
0.2
i
-----
-
~
--
-
----
~
t
Dr/Do=0.2 Hc = 5 cm Do = 2 0 c m No = 2 0 c m
0.0 |
o.2
o.'s
o.',. H~/Ho
o.'~
0,00
0.6
Figure g. Varialion ,.',l'caphm: prohabili/', lb" with Hi for t>pc 11 gas,.'s 558
i !
0.2
o.s
----
o.~ H~/Ho
I
o:5
T.
I+igurc I0. Vuriati(m ,,fl capturL' prc, babitit_~ /'r ~ ith 1//for t~ pc I I 1 gases. Top cur,,,.' for I = 1.0. and the I,.w,,._'rcur',c-, l'or /[ - O.,"L 0.6. 0.4. and 0.2 dov,.nv,:trd.
J W Lee and Y K Lee .
Calculationof capturecoefficientsfor acylindricalcryopump
o0
0.20 •
T Y P E llI GASES ~ / ~ - - ' ~ ' ~ -
TYPE
GASES
II
-
o.0
H, = Hc =
Pr
He Di
o.,1.
= =
Do
Pr o.lo
cm
5
5 cm 20 cm 10 cm
0.10
I
20 c m
0.o0
°"
HI
=
7
Ho = Di = Do =
20 14 20
cm
-
o.o
Ib ~b sb ~
8'o~o
~ 6b qo (degree)
I. Variation of capture probability Pr with conical angle 0 for types I1 and Ill gases. ability of an entering molecule passing through the gap between the panel and the radiation shield. Increased skirt length reduces the capture efficiency, but its effect is much less than that caused by changes in diameter. A cylindrical cryopanel of optimum diameter has the highest capture efficiency, irrespective of the skirt length (Figure 10). If the cryopanel has a truncated conical shape, the conical angle 0 is thought to have a strong effect on the pumping characteristics. The capture probability for type II gases increases as the skirt becomes flat (0 --* 90 ), but for type III gases there exists an optimum angle, which can be explained in the same way as optimum diameter. The maximum horizontal reach of a conical cryopanel with an optimum angle is really close to the optimum diameter (Figure 11). 3.3. Effect of sticking coefficients. As pumping proceeds, surface conditions of the cryopanel change, so changing the sticking coefficient. The capture coefficient for type lI gases varies almost linearly with £,, the sticking coefficient on the front surface, but changes very little with f , the sticking coefficient of the cryosorption element (Figure 12). This is due to the small amount of type II gases pumped on the sorption clement. For type IIl gases (J;, = 0), however, the sticking coefficient /i has a profound effect, especially for f less than 0.5 (Figure 13). 3.4. Regional pumping speed. For the design of optimum array
shape, information about the distribution of local pumping rate
0.8-
T Y P E II GASES
3,,
=~
o.oPr 0.5-
0.4-
/
0.2
0.0
,~r'/
J
0.2
=
Ft e
l// O.S-
,
,
0.4
,
,
0.0
Hi
=
He
=
,
5
i
0.2
'
O.',t
'
0.'6
'
0.'
'
1.0
,
Figure 13. Variation of capture probability Pr with sticking coefficiem f
for type Ill gases.
t.o
~ C ~ TYPE II GASES
Dp,,,ao.4°'e~°'8 0.2 o.o
~
HH'c=
8cm = 5 crn He = 20 c m
)~/
~,o= 212 cm 0 cm ~
~
~
~
fc
l~
A
X (cm) Figure 14. Relative pumping rate at different positions of the cryopanel surface, x is the distance from the centre.
is very important. For type I1 gases, the local pumping rate shows a maximum at the centre of the front surface, and decreases away from the centre (Figure 14). There is a sudden drop in the pumping rate at the joint between the front disk and the skirt (at x = 6 for example). The values are normalized with respect to the maximum value at the centre, and the pumping rate is quite uniform on the front disk. The amount of type II gases pumped on the reverse side is quite small, being in the order of less than 10% of that pumped on the front surface. It follows that of the type II gases captured, about 40% is pumped on the front disk and 60% on the side skirt, for the model cryopump shown in Figure 14. For type Ill gases, the maximum pumping rate occurs at the lip of the skirt (x = 14), and the minimum at the joint. The ratio of the maximum to minimum is about 5 both for types II and Ill gases. 4. Conclusions
cm
7 cm 20 cm
0.8
'
f,
Figure I
0.7-
0.0
cm cm cm
t.O
fo Figure 12. Variation of capture probability Pr with sticking coefficients 11 and/;, for t~cpe 11 gases.
1. A new method of calculating transmission probability has been devised and used for analysing the pumping characteristics of a cylindrical/conical cryopanel. The method makes use of view factors and gives results as accurate as the Monte Carlo method, but is very efficient and easy to use, especially when the geometry is fixed and only the sticking coefficient is changed.
559
J W l e e and Y K Lee: Calculation of capture coefficients for a cylindrical c r v o p u m p
2. The capture probability l\)r type ll gases increases m o n o tonically with the diameter and the skirt length of the cryopancl but I\w type III gases there exists ~i11 o p t i m u m diameter which is a b o u t Di,'Do- 0.6 over a wide rallgc of other geometric p~.ll'LInl e tCFS. 3. Thc capture probability of type 1I gases increases almost linearl 5 \~ith the sticking cocfticicnt of the cryopancl upper surl'ace, but is quite inscnsitivc to the sticking coefficient of thc cryosorption clement. 4. For a conical cryopanel, there exists an o p t i m u m conical angle for type Ill gases, such that the nlLlxilllunl horizontal reach at the o p t i m u m angle is equal to the o p t i m u m diameter. 5. The local pumpirlg ralc J\)l- type I1 gases is quite u n i l \ m n on the Iront stir[ace, and decreases away troll] the centre, l:or type llI gases the p u m p i n g ralc occurs tit lhc lip of the skirl, and
560
the m i n i m u m ;it the .joint. The ratio of thc Ill;.tXlIllLIlII [o Ihe Illinillltllll is ~lbOtl[ 5. References i \ .I BartlcH and P ..X l.c~is. ('t:l'oqcm~ I ' r . ¢ c ~ ~m,/I:qml~/mvll~ 15ML. p T I { Ig~4). : R .\ l tacl'cr. (rropumpil~d 7 h c . r v am/l'raclicc, p 1:,7. ('k~rcndon Prc~. ()\l'¢~rd (19~9). ~[) J ~LIIIIC]Cr, .1 l a c 5ci 7cchm~/. A4(3). 3~S (I9~n). P. Sicecl and J R ttowci]. 771ermM RaJialioll fh'al 7m~l~/~'r. 2nd I d n .
p INS. Mc(h-a\~-14ill. Ncx~ York (IOSI). ".1 R tto\~cll. RtlJit/li~m (.fff~tfm'ali~m I'}l~mr~. Mc(h-;m tlill. Nc~ 5 ~rk (19~2). " M M Wcincr..1 W Tindall and 1. M C;indcll. P~q~cr No 6 > W & I f I - s l . 15'./1/:'. No~cmbcr I 1960}. I) tl l ) a x J s . . / . I F p I Phr~. 31. 1169 (1960). I. I. l.cvcl>on. N tfillc,'on and l) 11 l)~l\is, l ' r w > ?,'tmz la~' 5)m/~. 7.
372 ( I 9(,0).