The measurement of the speed of pumps

The measurement of the speed of pumps received24March 1965;accepted30March 1965 W Steekelmaeher, CentralResearchLaboratory,EdwardsHigh VacuumInternational Limited, Crawley, Sussex

It is proposed to specify an "intrinsic" speed of a pump as well as an "intrinsic" conductance of components as obtained when these are connected to a large chamber or between large chambers. It is shown that by separating out an aperture conductance, both for the case of a component and a pump, the reciprocals of conductances and of pump speeds in composite systems are additive in the usual way, provided that the number of aperture conductances have been correctly accounted for. On the other hand Dayton suggested that the pump speed should be specified as measured with a test dome of the same diameter as the pump. In this case, the conductance of a tube of the same diameter as the pump needs to be modified to allow for the beaming effect of the gas into the pump mouth. It is shown that the alternative procedure based on the use of intrinsic speed and conductance is preferable in many ways and gives identical results for the net speed of a composite system, as obtained by the method of Dayton. Finally, it is shown that by moving the position of a tubulated gauge along a test dome of the same diameter as the pump, a position can be found where the measured pump speed is approximately equivalent to the intrinsic speed. For a test dome of effective length equal to the diameter, the gauge tube should be at a position on the wall, half a diameter away from the pump mouth. There has been some controversy in recent years regarding the definition and measurement of pump speeds. This has arisen because the measured speed is influenced by the shape of the test dome. It has become customary to measure the pump speed with a test dome of the same diameter as the mouth of the pump, but then due to the beaming effect of this dome, the speed may be considerably enhanced compared with a speed measured with a very much larger test dome. In his latest contribution on this subject in the February issue of Vacuum, B B Daytonl gives an excellent review of the whole question particularly as regards the use of a test dome and the associated pressure measurement. In his analysis of the problem he makes a strong case for specifying the pump speed as that measured with a test dome of diameter equal to the pump inlet and static pressure being measured at the plane of the pump inlet. This hinges on his analysis of the problem of computing a net speed Sn to be expected in the interior of a large vacuum chamber connected by a pipe of diameter equal to that of the pump inlet to a pump of speed S (defined with the small test dome). In the following it will be shown that the alternative approach of defining the pump speed as that which would be obtained when the pump is connected to a large dome gives equally valid answers when calculating the net speed of a composite system. Dayton 1,7 has shown that the usual equation 1

1

1

S. = S + v

(1)

is not valid when the speed measured with a large dome is substituted for S and the conductance U is computed in various ways. He indicates, however, that (1) should give approximately correct results for a connecting pipe of radius R equal to that of the pump inlet, if S corresponds to the speed measured with a dome of diameter equal to the pump inlet and effective height

equal to, or greater than 2R, with static pressure measured at the plane of the pump inlet, and the conductance (for air at 25 °C) of the pipe between the chamber and the pump inlet given by U = 11.7 ~.R2m/(1 --JLW/2) (2) in which W is the Clausing probability of passage through the pipe, and JL is a "beaming coefficient" for the gas flow across the entrance of a hypothetical tubulated ionization gauge, located exactly in the plane of the pump inlet. For values of W and JL reference should be made to the paper by Stickney and Dayton. 2 It should be noted that although a special case was made out for the particular definition of S as noted above, the net speed Sn arrived at in this calculation corresponds to the speed as measured when using a large chamber. Hence, the problem has merely been transferred from the mouth of the pump to the end of the tube of conductance U. It may be asked what one should do, if one had to consider the net pumping speed of a combination with a further tube connected to the inlet of the tube on the pump? Furthermore it appears to be unsatisfactory for the speed of a pump to be defined in a different way from that obtained by working out the net speed arising from a combination of a pump and a tube. As the "measured" conductance of a duct or the speed of a pump will depend on the way the gas is led into it from any ducts or components attached to it, it is a useful concept to consider separately an "intrinsic conductance" or "intrinsic speed" as that conductance or speed, which would be measured, if the duct was connected between two very large chambers, or the pump was connected to a large chamber. This in fact is the condition generally assumed for calculations on the conductance of practical configurations, and was the basis of the earlier calculation on the conductance and beaming effect under molecular flow conditions by Clausing,3 and the Monte Carlo calculations by Davis.4

Vacuum / volume 15/number 5. PergamonPress Lid I Printed in GreatBritain

249

W Steckelmacher: The measurement of the speed of pumps The question then arises, how to deal with the flow of gas through composite systems at low pressures, when the intrinsic conductance of each part is known. A method of dealing with this was indicated in the paper by Oatley, 5 commented upon by Steckelmacher.6 Although this may be criticized as giving results which are still only approximations, it will be shown below that the equations derived by Dayton give in fact identical results. In any case for many purposes, the approximations involved are good enough. Consider a tube connecting two large chambers, then the conductance may be split up into an aperture and tube conductance, UA and UTrespectively. 1

V

=

1

1

- .

(3)

VA + vr

With an aperture of area A within a tube of area A0, the conductance of the aperture is modified to UA/Ao given by: UA/Ao = UA Ao/(Ao --A). (4) For a pipe of radius R, the Clausing formula gives as an "intrinsic" conductance

(5)

u = UA W,

where UA = 11.7 A = 11.7 ~ R 2 (for air at 25 °C). From (3) it then follows that the corresponding tube conductance is given by

UT = UA W/(l --W)

(6)

so that one could define a modified probability factor

WT = W/ (1 - - W )

(7)

valid for calculations involving tubes without aperture effects. The applications of these results to composite systems was given by Oatley (1957) and Steckelmacher (1957). Similar ideas can be applied to pumps, if the "intrinsic" speed of a pump is S, then S = ~rUA

(8)

where ~r is a "sticking coefficient" for the pump, so that for a "perfect" pump ~r = 1, whose speed becomes equal to the aperture conductance UA. Analogous to the case of composite pipe systems, it becomes necessary to consider the "intrinsic" pump speed as made up of an aperture effect and a "basic" speed SB, where 1

S

--

1

UA

+

l

(9)

SB

from (9), in terms of the "intrinsic" speed S: 1

1

1

Sn = S q- ~JT

(13)

where UT is given by (6). The same procedure would be used if the net speed was required of composite systems involving additional pipes, baffles, etc. Thus if the pipe to be connected, had a larger radius R~, so that the aperture area is An = ~R~ then the net speed is given by 1

1

l

1

Sn = Sl @ UA/A1 "@ S B '

(14)

from (3), (4), (7) and (9) this becomes 1

1

1

1

S£ = S-f- U1

A

UA Ai

(15)

Probably enough has been said to show how the concepts of "intrinsic" pump speeds and conductances of components can be used to compute effective net speeds and conductances of composite systems. It now remains to show that the results obtained are in fact not in conflict with the procedure advocated in the paper by Dayton. Write his equation (2) above for U as Ut Ut = UA W/(1 --JL W/2) (21) Also using the results (100 a) and (10D b) in Stickney and Dayton 2 one obtains

1 UA JL [ ( 2 : W ) Jo J2lL--x = 1+ . . . . . (16) ~r Stx 2 L 2w L where Stx indicates the speed as meausred at position x with a test dome of the same diameter as the pump mouth. For the case x = L as required in Dayton's analysis (16) becomes

1

UA Jr -StL 2 " ie in terms of the intrinsic speed S, from (8),: = 1 -~-

uA

UA

(17)

JC + 2 "

stL = T - - 1

(18)

Hence, following Dayton1 the net speed of a system consisting of a pipe with conductance given by (21) and pump of speed given by (18) is 1

1

1

1

1

JL

1

JL

(19)

S = StL + Ut ~ S -- UA -t- 2UA + U A W - - 2UA ' ie

and, hence,

(10)

s B -~ UA ~/(1 --o)

For a "perfect" pump (equivalent to merely an aperture connected to an infinitely large evacuated space) it is clear that S B ~ (3o.

With these concepts it is a simple matter to consider composite systems of pumps and ducts. For example consider the net speed Sn of a pipe of radius R connected to a pump of the same diameter. The only aperture to be considered in this case, is that at the inlet to the pipe and hence 1

Sn

1

1

U + SB'

(11)

from (3), 1

1

1

250

x

1

Sn -- UA -[- -UT -}- SB

sn = 3 + UA

'

(12)

-t

which is identical to (13), with (6) as given above. Hence, there is nothing to be gained by considering the speed of a pump in terms of StL and the corresponding tube conductance as Ut in preference to the alternative system, involving the concepts "intrinsic" speeds and conductances. If this is accepted, the only question which then remains, is to determine the "intrinsic" speed of a pump, ie the speed which would be obtained with measurements involving a very large test dome, without necessarily having to use a large dome for testing. One way out would be to consider (16) and make measurements at such a position x along the test dome that Stx becomes equal to S. From (16) such a position is given by

L

(

--

1

(2-- W) (

2W

Jo--

•

(20)

W Steckelmacher: The measurement of the speed of pumps C o n s i d e r the special case o f a test d o m e o f effective l e n g t h L e q u a l to 2R, t h e n f r o m Stickney a n d DaytonZ W = 0.5, OL = 0.52 a n d 00 = 0.87 a n d t h e n (20) gives a p p r o x i m a t e l y x / R = 1,

(21)

ie in this case a particularly simple test d o m e a r r a n g e m e n t s h o u l d give a g o o d a p p r o x i m a t i o n to the intrinsic speed o f a pump.

References 1 B B Dayton, Vacuum, 15 (2), Feb 1965, 53. 2 W W Stickney and B B Dayton, Trans 10 A VS Nat Vac Symp, MacMillan, New York, 105-116.

3 p Clausing, Ann dPhys, 12, 1932, 961-989. 4 D H Davis, J A p p l P h y s , 31, 1960, 1169-1176. 5 C W Oatley, Brit JApplPhys, 8, 1957, 15-19. 6 W Steckelmacher, Brit J Appl Phys, 8, 1957, 494-496. 7 B B Dayton, Industr Eng Chem, 40, 1948, 795-803.

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