The relation between the conductance of an elbow and the angle between the tubes

Vacuum/volume 32/number 10/11/pages 655 to 659/1982

0042-207X/82/110655-05503.00/0 Pergamon Press Ltd

Printed in Great Britain

The relation b e t w e e n the conductance of an e l b o w and the angle b e t w e e n the tubes Xu Tingwei* and Wang Kaiping, Radio Electronics Department, Sichuan University, Chengdu, China

By the Monte Carlo method it has been shown that the molecular conductance of an elbow, the correction factor and the transmission probability are all functions of the supplementary angle between the two tubes. These three quantities at first decrease slowly with increasing angle, pass through a minimum between 49" and 73° and then again increase. Our calculations differ from the results of Roth who predicted a monotonous decrease in the transmission probabilities with angle and of Davis who stated that the transmission probabilities at 90 ° and (7 were equal. Using a non-linear regression technique, an empirical expression for the correction factor K of the elbow as a function of the angle 0 between the two tubes has been fitted to the calculated results. The expression for K has the form

K=A+BeC°+D sin E (0+F) where A, B, C, D, E and F are the regression coefficients.

1. Introduction The molecular conductance Co of a cylinder which is elbowed and whose axial length L is kept constant at L - - a + b (Figure 1) (L is the length of the cylinder before it is elbowed), is conventionally calculated as follows: the molecular conductance C of the straight cylinder is multiplied by a correction factor K, K being a function of the ratio of the cylinder's length to its radius L/R and the supplementary angle 0 between them. It is widely believed that Co monotonously decreases with the angle 0 (Figure 1) and some books published in recent years still perpetuate this belief. The conductance C of a straight cylinder is given on page 85 in Roth t by

C = 3.81~/TIM (D3/L)K'

(1) f

fr

o/~e" I

|

~--Outtet

Figure 1. Diagram of the cylindrical elbow.

* Unpaid Scientific Associate at CERN, Geneva, Switzerland.

where T is the temperature of the gas molecules, M is the mass number, D is the diameter of the cylinder and K' is a correction factor taking account of end effects and is given on the same page by 15 L/D + 12(L/D) 2 K ' = 20 + 38 L/D + 12(L/D)'"

(2)

The conductance of a circular orifice Co is given on page 79 in Roth t by

Co= 2.86~/T/M 0 2.

(3)

One can write the equation for the transmission probability p of a straight cylinder as 1.33 ( 1 5 + 1 2 L/D) P = C/C° = 20 + 38 L/D + 12(L/D) 2"

(4)

When the transmission probability ofan elbowed cylinder with an angle 0 is calculated, the 'equivalent axial length' L e should be used instead of L, where Le is defined on page 89 in Roth ~:

0

L.=L + l.33]-~6 D.

(5)

The correction factor K, i.e. the ratio of the conductance C s (or the transmission probability Pe) of the elbowed cylinder to the conductance C (or p) of the straight cylinder, can easily be derived from equations (1)-(5):

15 + 12 L JD 20 + 38L/D + 12(LID )2 K= 2 0 + 38 LJD+ 12(Lo/D)2 x 15+ 12 L/D . (6) 655

Xu Tingweiand Wang Kaiping:C o n d u c t a n c e

of an e l b o w and the angle b e t w e e n t h e tubes

Considering L/D a s a parameter, K is a monotonously decreasing function of 0 provided that L/D >_I. Early in 1960, however, using the Monte Carlo method, Davis et al z'3 obtained the following result: in comparison with the transmission probability for a straight cylinder, that of a rightangle elbow (0 = 90 °) is larger by about 5 ~o at L/R = 2 and smaller by about 7 ~ at L / R = I0. From the result of their calculations they concluded that both transmission probabilities are equal within the errors of the calculations, i.¢. both conductances are equal and hence the correction factor K = I. The experimental measurements subsequently carried out showed that, provided that L/R = 2--6, these two transmission probabilities were equal within the experimental errors (__.6~). From their work it was suggested that the conductance remains constant when a cylinder 1,30

(o)

is elbowed, namely K = 1, independent of 0. The above results have been also quoted by Roth 1 on page 103. These two conceptions, which are obviously contradictory, appear in Roth t without comment. In order to shed some light on this, a Monte Carlo method has been used to calculate the conductance of an elbow as a function of the angle. The empirical expression for the correction factor K of an elbow as a function of the angle 0 has been obtained for L/R = 2-10. The results show that neither view is quite correct. 2. Results and conclusions of the Monte Carlo simulation

The Monte Carlo simulation presented in this paper was performed on the Chinese computer TQ-16 using ALGOL. The 1.30

"

1.25

1.25

1,20

120

1.15

I .15

1.10

0.95

L/R = 3

- -

J

K 1.05

+---+...+_.+~'+

1.00,

A A

0.95

_\

0.90

_

\ \

\

A 0.85

-

A I

0.80 0

\

0.85

\

I

t

I

15 30 45

L3o 1.25

I

/

O

T_\

0,90 -

(b)

I .10

L/R-2

1.05

1,00-

-

60 75 9 0 8, deg

I

i

I

I

0.90

105 120 135 150

(c)

\

I

I

I

I

I\N.I

t

15 30 45 6 0 75 9 0 8, deg

/

?

+

(d)

1.30

I

105 120 135150

125

- -

1.20

I 20

/

1.15 - 1.10

L/R

= 4

K 1.05

-I-

1.15 I I0 K

L/R = 5

1.05 +

,oo

1.00

0.95-

\

0.90 - -

\

0.95

\

\

0.90

0.85 _ 0.80

o

0.85

I Is

I I 30 ,*5

I

8,

Figure 2 (a--d~ 656

I

8o 7s

I "~?'~.1 9 0 ,o5

de.o

I

I

12o=35mo

i

0.80 0

t

15 30

l

I

I

J

J 7-,I

I

45 6 0 75 9 0 105 120 135 150 8, deg

Xu Tingwei and Wang Kaiping: C o n d u c t a n c e of an e l b o w and the angle b e t w e e n the tubes formulae and steps of the calculation were similar to those adopted in previous publications *-~. The computer plotting of Figure 2 was performed on the CDC 7600 computer at CERN. To check the calculation, the transmission probabilities of the straight cylinder (0 = 0 °) and the right-angle elbow (0 = 90 °) were first calculated using the programme for calculating the transmission probability of an elbow with arbitrary angle 0. The trajectories of 104 molecules were traced and all the standard deviations were less than 0.5 %. Our results, those of Davis et al z'3 and Clausing v and those from equation (4) derived from Roth 1, have been listed together in Table 1 for comparison. The statistical tests based on the data listed in Table 1 show that the two estimates of P from our calculations and Davis' come from two

1.30

-

(e)

//

1.20 1.15 1.10

--

L/R

• 6

K 1.05

,oo÷~--~-+....

1.20

I .[0

1.so

-

I

I

I

I

f v't"

0.95

I

I

[

I

0.80

15 30 45 60 75 90 ~05 120 135 150 8. d e g

(g)

-

/

1.00

0.85

I

/

~~,,,.i.

;/

0.85 --

I

L/R = 7

1.05

0,90

0

+

1.15

0.90 --

0.80

/

1.25

~ . _+~-~

0.95 -

(f)

1.30

÷

1.25

normal populations with the same expectation and standard deviation for both straight cylinders and right-angle elbows. Also, the estimates agree with the values given by Clausing and those for straight cylinders obtained from equation (4). Hence, equation (4) may be used with confidence to calculate the transmission probability of a straight cylinder. However, for right-angle elbows the values obtained from equation (4) using L e instead of L are obviously lower than those from Monte Carlo simulations, except L / R = 10. Therefore, equation (4) is not valid for calculating the transmission probability of short elbows. After this test of the programme, the correction factors K of elbows with various L / R and 0 were calculated. The number of molecules traced was always 104 and all the standard deviations

0

I

I

I

I

I

15

30

45

60

75

1.3o -

125

J

I

I

i

I

90 105 120 135 150

O, dell

(h)

1.25

I 20

--

1.20

I 15

--

1.15

I ,10

--

L/R=

i.io --

8

K 1.05 --

L/R=IO

1.05 --

/

/

/ I O0 - I - ~ - I - ~ 0.95 --

1 . 0 0 ~

~

~+ ~"~'~"~'~

0.90

0.90 --

0.85

0.85

0.80

I

,÷

I

I

I

I.

I

I

I

I

I

O 15 30 45 60 75 90 105 120 155 150 8, deg

o.so

t 0

~

I

[

I

I

I

" I

~

I

I

15 30 45 60 75 90 105 120

8,

I

I

135

150

deg

Figure 2. The correction factor K ofan elbow as a function of its angle 0 and the ratio L/R. +, Values calculated by the authors; calculated by Davis; - - - curves calculated from equation (6).

, regression curves;

• values

657

Xu Tingwei and Wang Kaiping: Conductance of an elbow and the angle between the tubes Table 1. Values of P for straight cylinders and right-angle elbows. P are values calculated by the authors; Pc are values calculated by Clausing; P, are values calculated from equation (4); P~ are values calculated b~y Davis. L is the length of the cylinder; R is the radius; 0 is the angle.

0=90 ° e

0=00

L/R 2

4 5 6 7 8 10

e

e~

P,

P~

0.5130 0.4200 0.3546 0.3109 0.2748 0.2498 0.2248 0.1961

0.5136 0.4205 0.3589 0.3146 0.2807 0.2537 0.2316 0.1973

0.5130 0.4220 0.3602 0.3150 0.2802 0.2527 0.2301 0.1955

0.522 0.425 0.361 0.315

0.5506 0.4306 0.3549 0.3047 0.2670 0.2395 0.2212 0.1839

were less t h a n 0.5%. T h e results are listed in Table 2 a n d plotted in Figure 2. T h e solid curves in Figure 2 were o b t a i n e d by fitting the data points in T a b l e 2 by n o n - l i n e a r regression to the following empirical expression

P,

P,

0.541

0.3992 0.3438 0.3026 0.2705 0.2448 0.2236 0.2059 0.1779

0.357 0.272 0.221 0.183

120 h

I00

K = A + BeC° + D s i n E(O + F)

(7)

where A, B, C, D, E a n d F are regression coefficients (related to the p a r a m e t e r L/R) whose values are listed in Table 3. F o r c o m p a r i s o n , the dashed lines from e q u a t i o n (6) derived from R o t h I are s h o w n a n d also Davis' values. Figures 2 a n d 3 show that, with increasing 0, the correction factors (also the c o n d u c t a n c e s a n d transmission probabilities) decrease slowly at first, pass t h r o u g h a m i n i m u m between 49 ° a n d 73 ° and then again increase a n d reach their initial value a r o u n d 62°-105 ° (provided t h a t L / R = 2 - 8 ) , a n d then rapidly increase. The c o n d u c t a n c e of a n elbow neither decreases m o n o t o n o u s l y with angle as is usually stated, n o r remains constant. It is a complicated function of the angle.

80~

X 60-X

4c

I

I

I

4

2

I

6

8

I

I0

L/R

Figure 3. The angle 0 at which the conductance of the elbow is minimal ( x ) and that at which the conductance of the elbow equals that of the straight cylinder (O) as functions of the ratio L/R.

Table2. Valuesofthe correction f a c t o r K f o r elbows. L is the length ofthe cylinder; R is the radius; 0 is the angle

L/R

K 0=0"

O= 15°

0=30 °

0=45 °

0=60 °

0=75 °

0=90 °

O= 105°

O= 120°

O= 135°

2 3 4 5 6 7 8 I0

1.13000 1.0000 1.0000 1.00130 1.01300 1.0000 1.00~ 1.0000

1.0057 1.0036 0.9963 1.0035 1.0004 1~112 1.0027 0.9801

0.9988 1.0010 0.9853 0.9875 0.9931 0.9868 0.9969 0.9547

0.9988 0.9893 0.9755 0.9833 0.9807 0.9640 0.9662 0.9546

0.9996 0.9929 0.9752 0.9595 0.9625 0.9396 0.9786 0.9587

1.0150 0.9883 0.9915 0.9801 0.9643 0.9452 0.9560 0.9352

1.0733 1.0252 1.0008 0.9801 0.9716 0.9588 0.9840 0.9373

1.0286 1.0310 1.0187 1.0080 1.0008 0.9893 0.9587

1.1500 1.0827 1.0699 1.0256 1.0458 0.9939

1.3027 1.2700 1.1725 1.1472 1.0658

Table 3. Values of the regression coefficients in the empirical expression (7) for the correction factor K of elbows

658

L/R

A

B

2 3 4 5 6 7 8 10

0.9955 0.9938 0.9873 0.9818 0.9768 0.9704 0.9814 0.9683

0.2736 0.3613 0.7143 0.5145 0.2068 0.2251 0.5531 0.1601

x 10- 3 x I0- 3 x 10 -e x 10- 4 x 10 -4 x 10- ~ × 10- s x 10 -s

C

D

E

F

0.062 0.040 0.102 0.064 0.070 O.100 0.075 0.130

0.008 0.011 0.013 0.020 0.024 0.035 0.023 0.035

4.0 4.0 3.2 2.5 2.5 3.2 3.0 2.3

8 8 32 26 30 18 18 50

Xu Tingwei and Wang Kaiping: Conductance of an elbow and the angle between the tubes It is only coincidence that Davis et al concluded that the transmission probabilities for both a straight cylinder and a rightangle elbow are equal because the differences between the two kinds of cylinder are within _ 5 % and are quite by chance just covered by the experimental error of _ 6 ~ . The results given by the authors and Davis show that the transmission probability of a right-angle elbow 5 - 7 ~ larger than that of a straight cylinder when L/R-- 2, and 7 % less when L/R = 10. The value of K of a right-angle elbow varies from slightly more than unity to slightly less with increasing L/R. This tendency can be clearly observed from the original curve given by Davis and cannot simply be attributed to statistical fluctuations. We conclude that, when a straight cylinder is elbowed, with increasing angle, its conductance and also its transmission probability and correction factor at first decrease slowly, pass through a minimum and then again increase.

3. Discussions (1) When a straight cylinder is elbowed, its conductance varies relatively little despite quite a wide range of its angle. For example, the range of the angle 0 for which the conductance of the elbow varies within + 7 ~ is 0 ° _<0_< 90 ° when L/R = 2 and 0 ° _<0_< 135° when L/R = 10. The upper limits are between 90 ° and 135 ° for other L/R ratios. In view of vacuum engineering practice, one may consider that (as a rough estimate) the conductance is essentially constant when 0_<90 °. (2) According to the conventional method using the equivalent axial length L, instead of the physical length L, the calculated values of the conductance of an elbow are always less than the true values (within the studied range of 2_< L/R_<10), except when

L/R= 10 and 0 < 7 5 °. The lower the ratio L/R, the larger the deviation. The larger the angle 0, the larger the deviation. (3) For several L/R ratios, the conductance of an elbow with 0 around 15° was found to be somewhat larger than that of a straight cylinder, no matter what random number sequence was adopted. This unphysical phenomenon needs further investigation, however, it is thought to be due to a statistical fluctuation from the Monte Carlo simulation. (4) Calculations in the report were limited to those elbows with a = b. It needs further investigation to see how the conductance, the transmission probability and the correction factor vary when a cylinder is elbowed provided a:~ b and whether both conductances calculated for various inlets are the same or not.

Acknowledgements The authors would like to express their thanks to Dr Feng Yuguo for suggesting the investigation, Dr J-M Laurent for his help in the computer plotting at CERN, Dr A G Mathewson and Mrs X Sift for critical reading of the manuscript and Dr R S Calder for helpful discussions and Mrs L Xu for her help in the regressions.

References ~A Roth, Vacuum Technology, North-Holland Publishing Company (1976). 2 D H Davis, J Appl Phys 31, 1169 (1960). 3 L L Levenson, N Milleron and D H Davis, 1960 Vac Symp Tram, p 372. "Xu Tingwei and Xu Li, Vacuum Technique (Chinese) 6, 50 (1977). s Xu Tingwei and Feng Yuguo, Vacuum(Chinese) 3, 1 (1978). Feng Yuguo and Xu Tingwei, Vacuum 30, 377 (1980). 7 p Clausing, Ann Phys 12, 961 (1932).

659