CHAPTER 3

Gas Flow

In this chapter we discuss the flow of gas at reduced pressures, as it is encountered in a vacuum system. Gas flow is complex and the nature of the solution depends on the flow rate and gas properties as well as the geometry and surface properties of the duct. We begin by defining the flow regimes and introducing the concepts of throughput, mass flow and conductance. We describe the gas throughput and conductance for several kinds of flow. We show how approximation techniques and probability methods are used to solve complex problems, such as flow in ducts containing entrance and exit orifices, aperture plates, or other irregular shapes.

3.1 FLOW REGIMES

Gas flow regimes are characterized by the nature of the gas and by the relative quantity of gas flowing in a pipe. The nature of the gas is determined by examining Knudsen’s number, whereas Reynolds’ number describes the relative flow. In the viscous gas region (high pressures) the flow is called continuum flow. The flow can be further described as turbulent or viscous. Turbulent flow is chaotic, like the flow behind a moving vehicle or the rising smoke some distance from a cigarette. Laminar or stream flow occurs when the velocity and surface irregularities are small enough for the gas to flow gently past obstructions in laminar streamlines. In the molecular gas region, the mean free path is so long in comparison to the pipe size that the flow is entirely determined by gas- wall collisions. The flow in this region is called molecular flow. Between the continuum flow region and the molecular flow region is the transition region. In this region gas molecules collide with each other and with walls.

A viscous gas is characterized by a Knudsen number of < 0.01. Knudsen’s number Kn, is a dimensionless ratio of the mean free path to a characteristic dimension of the system, say, the diameter of a pipe:

25

A User’s Guide to Vacuum Technology, 3rd Edition. John F. O’Hanlon Copyright 0 2003 John Wiley & Sons, Inc.

ISBN: 0-471-27052-0

26

h Kn=- d

GAS FLOW

b(3.1)

In continuum flow the diameter of the pipe is much greater than the mean free path and the character of the gas flow is determined by gas-gas collisions. The flow has a maximum velocity in the center of the channel and zero velocity at the wall. Continuum flow can be either turbulent or laminar viscous. The boundary between turbulent and viscous flow can be expressed in terms of Reynolds’ dimensionless number R for round pipes:

(3 .2) R=- UPd

Y

where p is the mass density (kg/m3), of the gas of viscosity q flowing with stream velocity U in a pipe of diameter d. Reynolds’ number is used to characterize the relative quantity of gas flow. It is a ratio of the shear stress due to turbulence to the shear stress due to viscosity. Alternatively, it tells something about the forces necessary to drive a gas system in relation to the forces of dissipation due to viscosity. Reynolds [l] found two flow situations dynamically similar when this dimensionless number was the same. When R > 2200, the flow was always turbulent and when R < 1200 the flow was always viscous [2]. In the region 1200 < R < 2200 the flow was viscous or turbulent, depending on the geometry of the inlet and outlet and on the nature of the piping irregularities.

Laminar viscous flow, the ordered flow of a gas in streamlines, occurs in the region bounded by a Reynolds’ number lower than 1200 and a Knudsen number less than 0.01.

When the mean free path is equal to or greater than the pipe diameter, say Kn > 1, and when R < 1200, the gas is said to be a molecular gas, and the flow is called molecular flow. To be precise, Reynolds’ number does not have any meaning for a gas in the free-molecular regime, because classical viscosity cannot be defined. The nature of molecular flow is very different from laminar viscous flow. Gas-wall collisions predominate and the concept of viscosity is meaningless. For most surfaces, diffuse reflection at the wall is a good approximation; that is, each particle arrives, sticks, rattles around in a surface imperfection, and is re-emitted in a direction independent of its incident velocity. Thus there is a chance that a particle entering a pipe in which h >> d will not be transmitted, but will be returned to the entrance. In molecular flow, gas molecules do not collide with one another, and gases can flow in opposite directions without interaction.

In the region 1 > Kn > 0.01 the gas is neither viscous nor molecular. Flow in the transition region is difficult to treat theoretically. In this range,

3.1 FLOW REGIMES 27

called the transition, or slip flow range, where the pipe is several mean free paths wide, the velocity at the wall is not zero, as in viscous flow and the reflection is not diffuse, as in free molecular flow. Now let us define throughput, mass flow and conductance and develop some practical gas flow formulas.

3.2 THROUGHPUT, MASS FLOW, AND CONDUCTANCE

Throughput is the quantity of gas (the volume of gas at a known pressure) that passes a lane in a known time; ddt(PV) = Q. In SI throughput has units of Pa-m /s. Because 1 Pa = 1 N/m2, and 1 J = 1 N-my the units could be expressed as J/s or watts (1 Pa-m3/s = 1 W). Throughput is the energy per unit time crossing a plane. The energy in question is not the kinetic and potential energy contained in the gas molecules, but rather the energy required to transport the molecules across a plane. Expressing gas flow in units of watts is awkward and not used, but it helps to explain the concept that throughput is energy flow. Throughput is a volumetric dimension (volume of gadunit time). Throughput cannot be converted to mass flow unless the temperature is specified. It is in many ways unfortunate that vacuum technologists have chosen to use a volumetric unit, which conveys incomplete information. Volumetric flow does not conserve mass.

Mass flow, molar flow, or molecular flow are, respectively, the quantity of substance in units of kg, kg-moles, or molecules that passes a plane in a known time. Equation (3.3) describes the relationship between molar flow and throughput.

P

Q - Q N'(kg - mole/s) = - - - N,kT RT

(3.3)

In a similar fashion, mass flow is related to throughput by N'(kg/s) = MQ/N&T. Throughput can be related to molar or mass flow, only if the temperature is constant and known. A spatial change in the temperature can alter the throughput without altering the mass flow. We discuss applications of mass flow in Chapter 6 (flow meters) and in Chapter 15 where we describe cryogenically pumped systems.

The flow of gas in a duct or pipe is dependent on the pressure drop across the object as well as its cross-sectional geometry. Division of the throughput by the pressure drop across a duct held at constant temperature yields a property known as the conductance of the duct.

C = - Q 4 -4

b (3.4)

2s GAS FLOW

In SI the unit of throughput is the Pa-m3/s and the unit of conductance or pumping speed is the m3/s; however, related throughput units of Pa-L/s and conductance units of L/s are widely used. Unless explicitly stated, all formulas in this chapter use the cubic meter as the volumetric unit.

The pressures Pland P2 in (3.4) refer to the pressures measured in large volumes connected to each end of the channel or component. According to (3.4) conductance is the property of the object between the points at which the two pressures are measured. For those whose first introduction to flow was with electricity (3.4) is analogous to an electrical current divided by a potential drop. As with electrical charge flow, there are situations (transition, viscous and choked flow) in which the gas conductance is nonlinear, that is, a hc t ion of the pressure in the tube. Unlike electrical charge flow, there are cases in which the molecular conductance depends not only on the object, but also on the nature of adjacent objects and how they allow particles to be diffusely scattered from their surfaces. We will explore this last issue in detail when we describe methods for combining conductances in the molecular flow regime.

3.3 CONTINUUM FLOW

A gas is called a viscous gas when Kn < 0.01. The flow in a viscous gas can be either turbulent R > 2200, or viscous R < 1200. Equation (3.2) can be put in a more useful form by replacing the stream velocity with

u=- Q AP

If we replace the mass density, using the ideal gas law, (3.2) becomes

For air at 22OC, this reduces to

Q(Pa - L/s) R = 8.41 x 10-

(3.5)

b(3.6)

(3 -7) d

In ordinary vacuum practice turbulent flow occurs infrequently. Reynolds' number can reach high values in the piping of a large roughing pump during the initial pumping phase. For a pipe 250 mm in diameter connected to a 47-L/s pump, R at atmospheric pressure is 16,000. Turbulent flow will exist whenever the pressure is greater than 1 . 5 ~ 1 0 ~ Pa (100 Torr). In practice, roughing lines are often throttled during the initial portion of the roughing cycle to prevent the sudden out-rush of gas fiom scattering

3.3 CONTINUUM FLOW 29

process debris that may reside on the chamber floor. The flow in the throttling orifice is turbulent at high pressures.

In the high flow limit of the turbulent flow region the velocity of the gas may reach the velocity of sound in the gas. Further reduction of the downstream pressure cannot be sensed at the high-pressure side so that the flow is choked or limited to a maximum or critical value of flow. The value of critical flow depends on the geometry of the element, for example, orifice, short tube, or long tube, and the shape of the entrance. A detailed discussion of critical flow has been given by Shapiro [3].

Rather than divide the discussion of continuum flow into viscous, turbulent and critical, it is easier to discuss the flow in terms of the geometry of the pipe. We divide this discussion into orifice flow, long tube flow, and short tube flow, and we give equations for each region.

3.3.1 Orifices

For tubes of zero length (an extremely thin orifice) the flow versus pressure is a rather complicated function of the pressure. Consider a fixed high pressure, say atmospheric pressure, on one side of the orifice with a variable pressure on the downstream side. As the downstream pressure is reduced, the gas flowing through the orifice will increase until it reaches a maximum. At this ratio of inlet to outlet pressure (the critical pressure ratio), the gas is flowing at the speed of sound in the gas. The gas flow through the orifice is given by

for 1 > p2 /& 2 (2 / y + l))Y'(Y-')

The factor C accounts for the reduced cross-sectional area as the high- speed gas stream continues to decrease in diameter, after it passes through the orifice. This phenomenon is called the vena contracta. For thin, circular orifices, C is -0.85. If the downstream pressure P2 is further reduced, the gas flow will not increase, because the gas in the orifice is traveling at the speed of sound and cannot communicate with the high-pressure side of the orifice to tell it that the pressure has changed. In this region P2 cannot influence the flow so long as P2/PI < ( 2/(h+l) )U(h-'). The ratio of specific heats is h whose values are given in Appendix B.4. The flow is given by

b(3.9)

30 GAS FLOW

This value is called critical, or choked, flow. See Fig. 3.1. This limit is important in describing flow restrictors (devices that control gas flow and the rate of pumping or venting in a vacuum system), choked flow in air-to- air load locks, and flow through small leaks ftom atmosphere. In any of these relationships, the conductance can be found from C = Q/(P1-P2). For air at 22"C, h = 1.4 and P2/PI = 0.525; the choked-flow limit is

Q(Pa - m3 /s) = 2004AC'

for air at 22" C, when P2 / 4 50.52 b (3.10)

Q(Pa-Ws)= 2x1054(Pa)A(m2)C' b(3.11)

for air at 22"C, when Pz 14 20.52

3.3.2 Long Round Tubes

A general mathematical treatment of viscous flow results in the Navier- Stokes equations, which are most complex to solve. The simplest and most familiar solution for long straight tubes is the equation due to independently to Poiseuille and Hagen, and called the Hagen-Poiseuille equation:

nd4 ( 4 + P 2 ) Q=G 2 (4 -P2)

The gas flow for air at room temperature becomes

Q(Pa-m3/s)=718.5 d4(4 + P ~ ) ( ~ -p2) I

b (3.12)

(3.13)

1 1

0 0.5 1.0 P2'Pl

Fig. 3.1 Throughput versus pressure ratio in a circular orifice.

3.3 CONTINUUM FLOW 31

This specific solution is valid when four assumptions are met: (1) fully developed flow (the velocity profile is not position-dependent), (2) laminar flow, (3) zero wall velocity, and (4) incompressible gas. Assumption 1 holds for long tubes in which the flow lines are fully developed. The criterion for fully developed flow was determined by Langhaar [4] who showed that a distance of Z, = 0.0568dR was required before the flow streamlines developed into their parallel, steady-state profile. For air at 22°C this reduces to Z, (meters) = 0.0503Q when Q is given in units of Pa- m3/s. Assumptions 2 and 3 are satisfied if R < 1200 and if Kn < 0.01. The assumption of incompressibility holds true, provided that the Mach number U, the ratio of gas-to-sound velocity, is < 0.3.

(3.14)

For the special case of air at 22°C we have

Q(Pa - L/s) < 9.0 x 1 O5 d2P b (3.1 5 )

This is a value of flow that may be exceeded in many cases and would render the results of the Poiseuille equation incorrect.

Relationships for viscous flow between long, coaxial cylinders and long tubes of elliptical, triangular and rectangular cross section have been tabulated by Holland et al. [5 ] . Williams et al. [6] give the relation for flow in a long rectangular duct for air at 20°C

where the duct cross-section dimensions b and h and the length Z are given in cm. The function Y(h/b) is obtained from the following table:

Wb Y wb Y Wb Y

1 .o 0.4217 0.4 0.30 0.05 0.0484 0.8 0.41 0.2 0.175 0.02 0.0197 0.6 0.3 1 0.1 0.0937 0.01 0.0099

- -

In the limit h << b, the air flow reduces to the one-dimensional solution of Sasaki and Yasunaga [7]

32 GAS FLOW

Again h, b, and I are given in cm. The flow in (3.16) and (3.17), like (3.13), is inversely proportional to viscosity and may be accordingly scaled for other gases. These relations for long tubes are of limited use. They are of use in components such as mass flow meter tubes, controlled leaks, and piping that connects chambers with remotely located pumps and gas tanks. In most practical cases we connect chambers with as short a duct as possible to reduce unwanted pressure drops, and we need to know relationships which are valid for these cases.

3.3.3 Short Round Tubes

As we noted above, the flow in short tubes does not obey the Poiseuille equation. The flow may switch fiom viscous to to critical flow without there being any pressure region in which the Poiseuille equation is valid. This problem has been treated in several ways. Dushman [8] gives a non- linear relation for flow in short round tubes. It is valid only for unchoked flow. Santeler [9] devised a technique in which he models the short tube as an aperture in series with a short tube of length l‘. The problem is formulated by assuming an unknown pressure Pk between the “tube” and the “aperture.” This is the pressure that would be measured by a gauge just inside the end of the tube that was pointing upstream. Figure 3.2 illustrates an application of this technique-calculating the pressure drop and airflow through a 100-pm-diameter leak in a 1-cm-thick vacuum wall. Santeler’s model uses (3.13) with PI replaced by P,. The flow through the aperture was modeled using (3.10) with P, as its inlet pressure, and “high vacuum” (P = 0) as its outlet pressure. Since the two flows are in series, they are equal; the solution is P, = 44,560 Pa. The answer can be checked to ensure that the assumption of choked flow in the aperture is valid; if not, then (3.8) must be used in place of (3.10). This model predicts Poiseuille leak flow with significant gas expansion at the vacuum side.

3.4 MOLECULAR FLOW

A gas is called a molecular gas when Kn > 1 .O. This is equivalent to stating that Pd < 6.6 Pa-mm (4.95 Torr-mm) for air at 22°C. In this region the flow is called molecular flow. For completeness we could say that R < 1200; however, we cannot define a Reynolds number in the region where viscosity cannot be defined. The molecular flow region is theoretically the best understood of any flow type. This discussion focuses on orifices, infinite tubes, finite tubes, and other shapes, including combinations of components in molecular flow.

3.4 MOLECULAR FLOW

A

g 104-

103- v)

1 0 2 -

10’ - 1

e

33

e- Px = 44,560 Pa

1 100

Distance( pm) 0

Fig. 3.2 Pressure profile through a fine leak in a vacuum wall, as calculated with Santeler’s model. This model assumes Poiseuille flow through the tube with a precipitous drop in pressure immediately within the vacuum vessel caused by choked flow at the exit.

3.4.1 Orifices

If two large vessels are connected by an orifice of area A and the diameter of the orifice is such that Kn > 1, then the gas flow fiom one vessel (PI, nl) to the second vessel (P2, n2) is given by

and the conductance of the orifice is

V C = Q = - A P, - P, 4

(3.18)

b (3.1 9)

which for air at 22°C has the value

C(m3 /s) = 1 16A(m2) (3.20)

or

C(L/s) = 11 .6A(cm2) b(3.21)

From (3.18) we note an interesting property of the molecular flow regime. Gas can flow fiom vessel 2 to vessel 1; at the same time gas is flowing fiom vessel 1 to vessel 2 without either of the gases colliding with gas that originated in the other vessel.

GAS FLOW 34

3.4.2 Long Round Tubes

The diffusion method of Smoluchowski [lo] and the momentum transfer method of Knudsen [ 1 11 and Loeb [ 121 were the first used to describe gas flow through very long tubes in the free molecular flow region. For circular tubes both derivations yield conductances of

7[: d3 12 1

c,, = -v-

For air at 22°C this becomes 3 d3

Cmbe(m /s) = 1211

(3.22)

(3.23) 1

Conductance relations for long noncircular tubes have been derived [ 131.

3.4.3 Short Round Tubes

The flow equation for long tubes (3.22) indicates the conductance becomes infinite as the length tends toward zero, whereas in Section 3.4.1 we showed the conductance actually becomes vA/4. Dushman [ 141 developed a solution to the problem of short tubes by considering the total conductance to be the sum of the reciprocal conductances of an aperture and a section of tube of length 1.

(3.24)

As Ud + 0, (3.24) reduces to (3.19), and as l/d -+ 00 it reduces to (3.22). Although this equation gives the correct solution for the extreme cases, it is not correct for the intermediate. It can be in error by as much as 12-1 5%.

The difficulty in performing calculations for short tubes lies in the nature of the gas-wall interaction. Lorentz [15] assumed the walls of a pipe are molecularly rough; that is, molecules are scattered according to the cosine law (diffuse reflection). Molecules hit a wall, oscillate in potential wells, and recoil in a direction that is independent of their arrival angle. In diffuse reflection, scattered molecules have the greatest probability of recoiling at an angle of 90" from the surface. Particles not scattering at 90" have as much likelihood of going forward through the tube as going backward toward the source. See Fig. 3.3. Clausing [16] solved this problem by calculating the probability that a molecule entering the pipe at one end will escape at the other end after making diffuse collisions with the walls. Clawing's solution is in the form of an integral that is difficult to evaluate. For simple cases such as round pipes, Clausing and others have generated approximate solutions. The solution has been tabulated in many standard

3.4 MOLECULAR FLOW 35

Fig. 3.3 A molecule making diffuse collisions with a wall is scattered in a direction independent of its original path. A molecule forgets its original direction and is emitted with a probability proportional to cos cp fiom the normal. The most probable angle is cp = 0’. A molecule has an equal probability of going forward as it does backward.

texts, and it is usually given in the form of a transmission probability a that a molecule entering the pipe will leave the pipe at the other end. The conductance of a pipe is then found from (3.25)’ where A is the cross- sectional area of the pipe and v is the thermal velocity of the gas.

V C = a - A

4 b(3.25)

For the special case of very long round tubes (I >> d), the expression given in (3.22) can be written in the form of (3.25) to reveal that the transmission probability a of a long tube is (4d/31) as shown in Table 3.1.

For air at 22°C the conductance can be simplified to read

C (L/s) = 1.16x105aA(m2)

(3.26)

C(L/s) = 1.16aA(cm2) b(3.27)

Equation (3.27) describes the molecular conductance per unit area of any structure in molecular flow has a maximum value [l 1.6 L/(s-cm2) for air at 22OC] and that any structure (thicker than a thin aperture) will have a conductance less than this value.

DeMarcus [ 171 used a variational principle to solve the Clausing integral with improved accuracy. Berman [ 181 made a polynomial fit to DeMarcus’ solution, and extended it to larger Z/d values. Values of the transmission

36 GAS FLOW

Table 3.1 Transmission Probability a for Round Pipes

l/d a Ud a

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1 .o 1.1 1.2 1.3 1.4 1.5

1 .ooooo 0.95240 0.90922 0.86993 0.83408 0.80127 0.771 15 0.74341 0.71779 0.69404 0.67198 0.65143 0.63223 0.61425 0.59737 0.58148 0.56655 0.55236 0.53898 0.52625 0.51423 0.49185 0.47 149 0.4 5 2 8 9 0.43581 0.42006

1.6 1.7 1.8 1.9 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0

10.0 15.0 20.0 25.0 30.0 35.0 40.0 50.0

500.0 5000.0

Q)

0.40548 0.39195 0.37935 0.36759 0.35658 0.31054 0.27546 0.24776 0.22530 0.20669 0.19099 0.16596 0.14684 0.13175 0.11951 0.10938 0.07699 0.05949 0.04851 0.04097 0.03546 0.03127 0.02529

0.26643~0-~ 4d131

0.26479~

probability for round pipes obtained fiom his equations are given in Table 3.1 for a range of Zld values. The DeMarcus-Berman results agree with very precise calculations done by Cole [19] to between 4 and 5 decimal places. These values of a can be used in (3.25) and (3.27). “Exit effects”, which we shall discuss shortly, are included in these tabulations of probability values.

3.4.4 Other Short Structure Solutions

The calculation of molecular conductance in an arbitrarily complex short tube structure is not possible in closed form. The molecular conductance has been solved analytically for noncircular cross sections in only a few cases. Where exact solutions are not possible, probabilistic methods have been used. Tanigouchi et al. [20] have simulated the effects of varying accommodation coefficient and sticking coefficient on probability.

3.4 MOLECULAR FLOW 37

Table 3.2 Transmission Probability II for Thin, Rectangular, Slit-like Tubes

I / h a l/h a

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10

1 .ooooo 0.95245 0.90958 0.87097 0.83617 0.80473 0.77620 0.75021 0.72643 0.70457 0.68438 0.54206 0.45716 0.39919 0.35648 0.32339 0.29684 0.27496 0.25655 0.24080

15 20 30 40 50 60 70 80 90

100 200 500

1000 2000 5000

10000 20000 50000

100000 200000

0.18664 0.15425 0.1 1648 0.09471 0.08035 0.07008 0.06234 0.05627 0.05136 0.0473 1 0.02722 0.01276

0.70829~ 1 O 2 0.38914~ 1 0-2 0.17409~ 1 O 2 0.94000~ 10” 0.50472~10” 0.22023~10” 0.1 1705x lo3 0.6 1994x lo4

Analytical Solutions

A geometry that is not circular in cross section is the thin, rectangular, slit- like pipe. This geometry is often encountered in differentially pumped feedthroughs and joints between differentially pumped chambers. It consists of a thin gap of thickness h, length Z, and width b, with the condition h << b, where b, h, and Z are defined in Fig. 3.8. Berman [18] developed a polynomial fit to solutions for the transmission coefficient and values calculated with the use of his formula are given in Table 3.2. His results agree with those of Neudachin et al. [21]. The conductance of a slit can be calculated from (3.25) using the transmission probability from Table 3.2 and an inlet area of bh. “Exit conductance” drops are also included in these transmission probabilities. The transmission probability has been experimentally shown to decrease with increased surface roughness [22].

In addition to the short round pipe and the slit-like tube discussed above, solutions exist for the annular cylindrical pipe [23], the rectangular pipe [24], the elliptical tube, and the triangular tube [5] . Other short tube cross

38 GAS FLOW

sections and complex structures have been treated with statistical techniques such as the Monte Carlo technique described here.

Monte Carlo Technique

The Monte Carlo statistical methods developed for the calculation of molecular flow conductance by Davis [25] and by Levenson et al. [26] were a major breakthrough in the calculation of complex, but practical, vacuum system elements such as elbows, traps, and baffles. The Monte Carlo technique uses a computer to simulate the individual trajectories of a large number of randomly chosen molecules. Figure 3.4 is a computer graphical model of the trajectories of 15 random molecules entering an elbow. It yielded a transmission probability of 0.222. When a large number of particles was used, the transmission probability of 0.31 was calculated. This points to one difficulty of the Monte Carlo technique; its accuracy depends on the number of molecular trajectories used in the calculation. A great deal of computational time is required for accurate solutions to complex problems. Figures 3.5 through 3.12 contain examples of the Monte Carlo technique for some structures of interest [25-271. The molecular conductance is the product of the probability and conductance of

I / -

C D

Fig. 3.4 A computer graphical display of the trajectories of 15 molecules entering an elbow in free molecular flow. Courtesy of A. Appel, IBM T. J. Watson Research Center.

3.4 MOLECULAR FLOW 39

an aperture identical in shape to the entrance of the structure under consideration. End effects are included in these formulas as well. The great computational time required to perform a transmission probability calculation by the Monte Carlo technique, has driven others to approximate complex systems by combining cylindrical tubes, orifices, and baffle plates. See, for example, Fustoss and TOth [28], Harries [29], Steckelmacher [30], Oatley [31], Haefer [32], and Ballance [33]. However, molecular conductances must be combined in series with great care. The increased capability of desktop computing has made possible direct simulation with Monte Carlo methods [34].

3.4.5 Combining Molecular Conductances

Implicit in the definition of conductance (3.4) is the understanding that molecules will arrive at the entrance to the component distributed in a Maxwell-Boltzmann fashion, and depart into a void without colliding with another surface. This is possible only if there are no other walls in the vicinity of the entrance and exit of the component. It can be accomplished by connecting the component between two large reservoirs so that the pressures in the vessels will be unaffected by the flow through the component. In practice this condition is rarely met. Typically the shortest possible lengths of pipe are used to interconnect pumps, chambers, traps, and baffles or elbows (whose length is of the order of the pipe diameter).

Parallel Conductances

The conductance of tubes connected in parallel can be obtained from the simple sum and is independent of any end effects.

CT=C1+cZ+C3+... b (3.28)

Series Conductances

Series conductances of truly independent elements in molecular flow will yield a total conductance of

1 1 1 1 - = -+-+ - + ... ‘T ‘1 ‘2 ‘3

b(3.29)

Equation (3.29) gives the value of conductance we would measure if the elements were isolated from each other by large volumes. See Fig. 3.13~2. The large volume provides a place for the distribution of molecules exiting the prior conductance to completely randomize, or assume the distribution of a rarefied Maxwell-Boltzmann gas.

e

0

'0

1

2 3

4

5 6

7 8

VR

Fig.

3.5

M

olec

ular

tran

smiss

ion

prob

abili

ty o

f a

roun

d pi

pe. R

epri

nted

w

ith p

erm

issi

on fr

om L

e V

ide,

No.

10

3, p

. 42

, L

. L

. L

even

son

et a

l. C

opyr

ight

196

3, S

oci6

tte F

ranc

aise

des

Ing

enei

urs

et T

echn

icie

ns du

Vid

e.

1.0

I I

I I

I I

1 I

----

-- W

ith

Cen

tral

Pla

te

Wit

hout

Cen

tral

Pla

te

0.6

,

e,

0.4

0.2

----

----

----

1.5 -1

I I

I I

I I

I I

J 0

12

34

56

78

9

UR

0

Fig

. 3.

6 M

olec

ular

tra

nsm

issi

on p

roba

bilit

y of

a r

ound

pip

e w

ith

entr

ance

and

exi

t ap

ertu

res.

Rep

rint

ed w

ith p

erm

issi

on f

rom

J.

App

l. P

hys.

, 31,

p. 1

169,

D. H

. Dav

is. C

opyr

ight

196

0, T

he A

mer

ican

Inst

itute

of

Phy

sics

.

1.0 I

I I

I I

I 1

I I

0;

;

b ;

Ib 1;

If

1; U

R

lp

10

5 1

Fig

. 3.7

Mol

ecul

ar tr

ansm

issi

on p

roba

bilit

y of

an

annu

lar

cylin

dric

al

pipe

. R

epri

nted

with

per

mis

sion

fro

m L

e Vide, N

o. 1

03, p

. 42

, L.

L.

Lev

enso

n et

d.

Cop

yrig

ht

1963

, So

ci6t

6 Fr

anca

ise

des

Inge

neiu

rs e

t T

echn

icie

ns d

u V

ide.

du

Vid

e.

Fig

. 3.

8 M

olec

ular

tra

nsm

issi

on p

roba

bilit

y of

a r

ecta

ngul

ar d

uct.

Rep

rint

ed w

ith p

erm

issi

on fi

om L

e V

ide,

No.

103

, p. 4

2, L

. L. L

even

son

et a

l. C

opyr

ight

196

3, S

oci6

t6 F

ranc

aise

des

Ing

enei

urs

et T

echn

icie

ns

R

0.8

0.6

m t

.

4:

a 0.

4

0.2

0.51

I

I I

I I

I 1

1 I

I L

- - - - ?

0.4

0.3 -

- 8

. so"

0-

8.45"

8.30

"

-

0-

I I

I I

I I

I I

1

AIB

Fig

. 3.

9 M

olec

ular

tra

nsm

issio

n pr

obab

ility

of

a

chev

ron

baff

le.

Rep

rint

ed w

ith p

erm

issio

n fr

om L

e V

ide,

No.

103

, p. 4

2, L

. L. L

even

son

et a

l. C

opyr

ight

196

3, S

ocitS

ttS F

ranc

aise

des

Ing

enei

urs

et T

echn

icie

ns

du V

ide.

"O 1

5

0 0

12

34

56

78

W

R

Fig. 3

.10

Mol

ecul

ar t

rans

miss

ion

prob

abili

ty o

f an

elb

ow.

Rep

rint

ed

with

per

miss

ion

from

J. A

ppl.

Phy

s., 3

1, p

. 116

9, D

. H. D

avis

. Cop

yrig

ht

1960

, The

Am

eric

an In

stitu

te o

f Phy

sics

.

R.

0'

I

1 2

0 0.

241

0.96

6 2

2 0

0.12

0 0.

957

3 2

0 0.

078

0.92

8 4

2 0

0.060

0.95

5 5

2 0

0.

045

0.89

6 3

2 1

0.054

0.650

2 0.

041

0.494

3 2

3 2

3

0.034

0.407

3 2

4 0.029

0.353

5 1

0

0.06

5 0.

647

5 3

0

0.03

2 0.

966

2 1

0

0.18

6 0.

744

2 3

0

0.080

0.96

4

Fig.

3.1

1 M

olec

ular

tran

smis

sion

pro

babi

lity

of a

fru

stum

of

a co

ne.

Rep

rint

ed with

perm

issi

on f

rom

Tra

ns.

9th

Nut

l. V

uc. J

Lm

p., J

. D

. Pi

nson

and

A. W

. Pec

k. C

opyr

ight

196

2, M

acm

illan

, New

Yor

k, p

. 407

.

Fig.

3.1

2 M

olec

ular

tran

smis

sion

pro

babi

lity

of a

par

alle

l pla

te m

odel

. R

epri

nted

with

per

mis

sion

fro

m T

rans

. 9t

h N

utl.

Vuc

. Sy

mp.

, J.

D.

Pins

on a

nd A

. W. P

eck.

Cop

yrig

ht 1

962,

Mac

mill

an, N

ew Y

ork,

p. 4

07.

A

w

44 GAS FLOW

Fig. 3.13 Series conductance of two elements: (a) the pipes are isolated by a large volume, and (b) the pipes are connected directly together. The pressure readings are those measured by a gauge in the gas stream pointing upstream and parallel to the flow direction.

Exit and Entrance Effects

The simple reciprocal rule does not work where we combine two conductances directly. See Fig. 3.13b. Let us consider two tubes each of length-to-diameter ratio Zld = 1. From Table 3.1 we obtain the transmission probability of each tube as a = 0.51423. If we combine them according to the reciprocal rule in (3.29) we will obtain a net transmission probability of a = 0.25712. We know that the transmission of this structure (tube with l/d = 2) can be found from the data in Table 3.2 as a = 0.35685. The error in using the simple reciprocal rule is 27.9%. Why do we have this large error? The reason is the pressure distribution in the two tubes is not the same. For the case of the isolated conductances, the pressure in the (imaginary) large chamber between the two pipes can be defined. It is single-valued and could be measured with a gauge in the chamber. If we were to point a directional gauge in any direction in this chamber, we would measure the same pressure. In an analogous fashion, we could measure a voltage at the junction of two series resistors, which are carrying a current.

When we combine two pipes in series without the large volume in between, the situation changes drastically. The pressure at the exit of the first tube and the entrance to the second tube are now the same but not easy to define. If we were to place a directional gauge at the junction of the two pipes and point it upstream, it would read higher than downstream. Also, were it to face sideways, it would also depend on whether it were in the center of the tube or off-axis. The pressure is anisotropic. This is to be contrasted to the case of the two tubes connected by a chamber, which has a known pressure. In that case the only place where the pressure will differ is at the exit of the first tube as it enters the chamber as measured with a gauge looking upstream. That pressure will be higher than in the chamber.

3.4 MOLECULAR FLOW 45

Recall the definition of conductance given in (3.3). If we examine Fig. 3.13u, we see that the pressure difference in (3.3) is measured in the large chambers at each end of the tube. By measuring the pressure in this way we include the pressure drop schematically shown at the end of the tube. When we connect the two tubes directly, we have eliminated the pressure drop at the exit of the tube that would be measured by an upstream-directed gauge. For this reason we say the conductances tabulated here and in other sources include what Santeler [35] called the exit loss.

There is another effect, which is present to a small extent-gas beaming. Note that the gas entering the second tube in Fig. 3 . 1 3 ~ is randomly distributed, while the gas entering the second half of the tube in Fig. 3.13b is beamed. See for example the molecular exit angles depicted in Fig. 3.14. In normal cases this is a small correction-a few percent-which we will describe shortly. Beaming effects (entrance effects) in most real molecular flow situations do not introduce major errors, because real systems are made up of short tubes connected by elbows, traps, and so on. We observe in Fig. 3.14 that short tubes have near-cosine exit flux. Any component containing an elbow, baffle, chevron, or the like will also scatter molecules and shift the distribution toward cosine.

Series Calculations

Harries [29], Steckelmacher [30], Oatley [31] and Haefer [32] have each used the concept of a probability factor a, to calculate the conductance of a

A d k Fig. 3.14 Angular distribution of particles exiting tubes of various ratios of length to diameter. Reproduced with permission from Atom and Zon Sources, 1977, p. 86, C. Vhlyi. Copyright 1977, AkadCmiai Kiad6, Budapest.

46 GAS FLOW

series combination of vacuum elements in free molecular flow. We examine here the method developed by Oatley. Figure 3.15 illustrates the concept with a single component; r molecules per second enter at the left- hand side, J? molecules per second exit at the right-hand side, and (1 - a)r molecules per second are returned to the source vessel. The conductance is expressed by

V C = a - A

4 (3.30)

For two tubes in series, Oatley developed a technique for calculating a combined probability, the results of which are illustrated in Fig. 3.16. Among the r molecules per second that enter the first tube alJ? enter the second; r (1 - az)al of these are returned to the first tube and ralaz enter the second. From the group r(l - az)al molecules returned the first tube Tal( 1 - a,)( 1 - a]) are returned to the second, and so on, until an infinite series expression was developed that simplified to

(3.31) 1 1 1 -=-+--1 a a1 a2

The last term represents the exit pressure drop, which is subtracted in this formula. When generalized to several elements in series, this becomes

1-a-I-a, 1-a, 1-a,

a a1 a2 a3 b(3.32)

Now let us use (3.31) or (3.32) and calculate the series conductance of the two pipes of Zld = 1 described earlier. We obtain a value a = 0.3460. Note that this is closer to the Clausing value of a = 0.35685 obtained from Table 3.1. It is in error by 2.93%, because it cannot account for beaming effects.

Oatley’s formula, as given in (3.32), applies directly to elements of the same diameters, but it can be extended to elements of differing diameters. If the series components are of different diameters, or of increased complexity, the addition theorem developed by Haefer provides the easiest solution.

--- +-+- +...

=d-ar (l-a)r

Fig. 3.15 Model for calculating the transmission probability of a single element.

3.4 MOLECULAR FLOW 47

Tube 1 I Tube 2 I

I I

a2 ar- I r- a,

-=(-+--I) 1 1 1 a a, a2

Fig. 3.16 Model for calculating the transmission probability of two elements in series. Reprinted with permission from BY. J. Appl. Phys., 8, p. 15, C. W. Oatley. Copyright 1957, The Institute of Physics.

Haefer [32] developed a usehl addition theorem for elements in the molecular flow regime. It relates the total transmission probability of n elements a],, to the transmission probability a, and the inlet area Ai of each component. Extra terms are included in the equation whenever a cross- sectional area decreases upon entering the next element but not when the area increases. It is given here without proof.

where = 1 for < Ai, and = 0 for Ai+] 2 A,

The use of this formula is demonstrated with the example given in Fig. 3.17. Shown is a combination of three pipe sections of inlet areas A I , A2,

and A,. The pipes have corresponding transmission probabilities al, a2, and a3 . By use of (3.33), one obtains

after some simplification (3.34) becomes

From (3.35) it can be seen that this answer reduces to (3.32) when the pipe areas are all the same. Haefer’s method must be applied with consistency

48 GAS FLOW

I I u a3

a2

Fig. 3.17 Sample conductance to be evaluated with Haefer's addition theorem.

because the total transmission coefficient one calculates is a function of the end of the structure chosen as the origin. If the transmission probability were calculated from right to left, u,+ we would have to use an inlet area of A, in calculating the conductance. However, this does not affect the total conductance of a structure as

(3.36)

We can also see that the answer would have been different if the order of the second and third pipes were interchanged. Interchanging the order of components in a series configuration will affect the total transmission probability. As a rule, the conductance of the actual configuration should be calculated, because errors can creep in when components are mathematically rearranged to simplify calculation [36]. It is generally true that a complex structure made up of several series elements has the maximum conductance when the elements are arranged in increasing or decreasing order because the exit losses are the smallest and the beaming is the greatest. Arranging conductances in alternating large and small diameters introduces wall scattering, which makes the input to the following sections more like a Maxwell-Boltzmann gas, with a cosine distribution of scattered molecules.

Problems arise in conductance calculations because (3.29), which is valid for independently defined C's, is indiscriminately applied to series elements not isolated from one another, for example, in which the exit effect is not subtracted, and the inlet gas does not obey a cosine distribution. The most serious error is the introduction of the exit impedance of the tube multiple times. In all tabulated transmission coefficients, the exit term has been included. Therefore it is necessary to

3.4 MOLECULAR FLOW 49

remove it when combining conductances, and for this reason we use formulas like (3.32) or (3.33). The choice between an exact or approximate formula for the conductance of an individual pipe segment is usually less important than the correction for the exit effect. The Oatley and Haefer formulas remove the biggest error in calculating the conductance of combinations-the exit conductance drop at the end of each junction of equal diameter-but neither formula corrects for entrance effects, that is, non-cosine or beamed entrance flux.

Pinson and Peck [27] discuss beaming errors for pipe sections with and without baffles and show the difference between a calculation and Monte Carlo technique is 5 10% with the greatest differences seen when a baffle is not used. Beaming corrections for tube combinations have been developed by Santeler [35]. Components like chevron baffles and elbows tend to scatter the gas. For example, the conductance of a nondegenerate elbow can be calculated by using the conductances of the individual arms, obtained from Fig. 3.5, and summing these conductances with (3.31). Saksaganski [37] discusses efficient methods for analyzing complex systems and shows the angular coefficient and integral kinetic methods as alternatives to the Monte Carlo method.

The formulas developed by Haefer and Oatley may be used to calculate the pumping speed at the inlet of pipes connected to a pump. In this case the pump is characterized by its inlet area and Ho coefficient (defined in Chapter 7). The pump is simply considered to be a conductance of entrance area A and transmission probability up equal to its Ho coefficient.

3.5 THE TRANSITION REGION

The theory of gas flow in the transition region is not well developed. Thomson and Owens [38], and Loyalka et al. [39], have reviewed the state of the theory. DeMuth and Watson [40] have done additional work on the transition between molecular and isentropic flow in orifices. The simplest treatment of this region, due to Knudsen, discussed in many texts states

where for long circular tubes 2' is given by

b(3.37)

1 + 2.507( $) 1 + 3.095( $)

Z'= (3.38)

50 GAS FLOW

3.6 MODELS SPANNING SEVERAL PRESSURE REGIONS

Flow relations that span different pressure regions are difficult to construct. Two examples have already been given in this chapter. In addition to Dushman’s model for the transition region discussed above, we have discussed Santeler’s model for flow in a short tube in the viscous and choke regions. Santeler’s [9] model for a short tube separates the tube component fiom the exit effect. Therefore, it can be applied to a series of tubes by modeling the system as a series of tubes with one exit loss after the last tube section. Tison [41] developed an empirical fit with a form similar to Knudsen’s that described the flow through a metal capillary from the molecular to the viscous region. Other relations have been developed for specialized geometrical shapes. One is a relation developed by Kieser and Grundner [42] for the thin, slit-like tube. The thin, rectangular slit-like tube with one side in a rarefied gas and the other at atmospheric pressure is encountered in atmosphere-to-vacuum continuous feed systems and reel- to-reel coating systems known as web, or roll, coaters. This relation, which is valid in the molecular, transition and viscous flow regions, combines the ideas of Dushman and Knudsen. It is valid for any inlet pressure but only for low exhaust pressures (Po < 0.52Pi). Kieser and Grundner begin with Dushman’s relationship, which assumed a duct to be composed of a pipe and an entrance aperture. The conductance of the series combination is

(3.39)

or, for air at 20°C,

(3.40) 1

C,, (0.1 106eC + Z’)C, C, +- 1 - 1 --

CO and C, in (3.40) are given by

1 10 + 0.5(e / 10 + 0.3412(e/h)3’2

C, (L/s) = 1 1.6e

C, (L/s) = 1 1 . 6 e w ( k )

(3.41)

(3.42)

In the above relationships, e is the channel thickness, and w is the channel width. The form of (3.41) allows the (air) conductance of the aperture to vary from 11.6 L/s in molecular flow to 17 L h in the choked limit. Equation (3.41) is an experimental fit to the transition region for air [42].

The pipe conductance given in (3.39 is due to Knudsen and is a superposition of continuum and molecular flow. Since the aperture is

3.6 MODELS SPANNING SEVERAL REGIONS 51

included in the C, term, we have removed it from (3.42) [43]. The pipe conductance CM, calculated by Berman already contains an end effect Co. Also the premise on which (3.38) is based, that is, the representation of the total conductance as a series combination of a “pipe conductance” and an “aperture conductance”, is known to be in error by 10-15% in the region lle = 1-5. For longer pipes there is another small error. Equation (3.40) implicitly lets the average pressure in the ‘pipe” portion of the conductance be Pi /2. This forces the pressure across the viscous conductance to an incorrect value. However, the pipe conductance is in series with a choked orifice, so any error in the “pipe” conductance is greatly attenuated by the series combination. (It is like putting a large resistor in parallel with a very small one-the parallel combination isn’t greatly affected by the value of the larger resistor.)

There are few papers on flow relations, which are valid over several regions. For example, Schumacher [44] summarized the flow through small round tubes in graphical form, while Levina [45] developed nomographs for the same problem.

3.7 SUMMARY OF FLOW REGIMES

The values of gas flow, pressure, and pipe size discussed in the previous sections each extend over a wide range. We summarize this discussion by sketching a plot of flow divided by pipe size (Q/d) versus pressure times distance (Pd). Figure 3.18 depicts the various regions discussed in this section. Molecular flow occurs in the region R < 1200, and Kn > 1. The flow is proportional to the first power of the pressure (slope = 1). When Kn < 0.01 the gas is viscous, and the flow is either turbulent, fully developed (Poiseuille), undeveloped, or choked. Observe that fully developed flow is proportional to to the square of the pressure (slope = 2). The boundary between turbulent and laminar viscous is determined by the Reynolds’ number. Langhaar’s number and the Mach number determine the boundary between fully developed and undeveloped flow. We illustrate how a short tube can go from molecular to choked without ever having the Poiseuille equation apply. The region between molecular and viscous flow is the transition region. We see the transition from completely free molecular flow to completely viscous flow can take place over a two-decade pressure range.

In this chapter we have considered the equations of flow in each region. The equations developed here relate flow to pressure drop in several pressure regions and for several geometric cross sections. When we combine these with the dynamical equations of gas flow from a chamber, we can calculate the time required to reach a particular pressure.

52 GAS FLOW

10-2 100 102 I o4 Pd (Pam)

Fig. 3.18 Gas flow-pressure regimes.

REFERENCES

1. 0. Reynolds, Philos. Trans. R. SOC., London, 174 (1883). 2. A. Guthrie and R. K. Wakerling, Vacuum Equipment and Techniques, McGraw-Hill,

New York, 1949, p. 25. 3. A. H. Shapiro, Qvnamics and Thermodynamics of Compressible Fluid Flow, Ronald,

New York, 1953. 4. H. L. Langhaar, J. Appl. Mech., 9, A-55 (1 942). 5. L. Holland, W. Steckelmacher, and J. Yarwood, Vacuum Manual, E. & F. Spoon,

London, 1974, p. 26. B. J. Williams, B. Fletcher, and J. A. A. Emery, Proceedings of the 4th International Vacuum Congress, 1968, Institute of Physics and the physical Society, London, 1969. p. 753.

7. S. Sasaki and S. Yasunaga, J. Vuc. Soc. Japan, 25, 157 (1982). 8. S. Dushman, Scientific Foundations of Vacuum Technique, 2nd ed., J. M. Lafferty, Ed.,

Wiley, New York, 1962, p. 35. 9. I). J. Santeler, J. Vac. Sci. Technol. A, 4, 348 (1 986). 10. M. von Smoluchowski, Ann. Phys., 33,1559 (1910). 11. M. Knudsen, Ann. Physik, 28,75 (1909); 35, 389 (191 1). 12. L. B. Loeb, The Kinetic Theory of Gases, 2nd ed., McGraw-Hill, New York, 1934,

Chapter 7.

REFERENCES 53

13. W. Steckelmacher, J. Phys. D: Appl. Phys., 11,473 (1978). 14. Ref. 5, p. 91. 15. H. A. Lorentz, Lectures on Theoretical Physics, 1, Macmillan, London, 1927, Chapter 3. 16. P. Clausing, Ann. Phys., 12,961 (1932), English Translation in J. Vac. Sci. Technol., 8,

17. W. C. DeMarcus, Union Carbide Corp. Report K-1302, Part 3,1957. 18. A. S. Berman, J. Appl. Phys., 36,3365 (1965), and erratum, ibid, #37,4598 (1966). 19. R. J. Cole, Rarefied Gas Dynamics, 51, Part 1, of Progress in Astronautics and

Aeronautics, J . L. Potter, Ed., (loth International Symposium Rarefied Gas Dynamics), Am. Inst. of Aeronautics and Astronautics, 1976, p. 261.

636 (1 97 1).

20. H. Tanigouchi, M. Ota, and M. Aritomi, Vacuum, 47,787 (1996). 21. I. G. Neudachin, B. T. Porodnov and P. E. Suetin, Soviet Physics, Technical Physics,

22. W. Sugiyama, T. Sawada, and K. Nakamori, Vacuum, 47,791 (1996)/ 23. A. S. Berman, J. Appl. Phys., 40,4991 (1 969). 24. D. J. Santeler, and M. D. Boeckmann, J. Vac. Sci Technol. A, 9,2378 (1992). 25. D. H. Davis, J. Appl. Phys., 31, 1169 (1960). 26. L. L. Levenson, N. Milleron, and D. H. Davis, Le Vide, 103,42 (1963). 27. J. D. Pinson and A. W. Peck, Transactions of the 9th National Vacuum Symposium,

28. L. Fiistiiss and G. Teth, J. Vac. Sci. Technol., 9, 1214 (1972). 29. W. Harries, Z. Angew. Phys., 3,296 (1951). 30. W. Steckelmacher, Proc. 6th Int. Vacuum Cong., Kyoto, Jpn. J. Appl. Phys., Suppl. #2,

31. C. W. Oatley, Br. J. Appl. Phys., 8, 15 (1957). 32. R. Haefer, Vacuum, 30,217 (1980). 33. J. 0. Ballance, Transactions of the 3rd. International Vacuum Congress, 2, Pergamon,

34. A. Pace and A Poncet, Vacuum, 41,1910 (1990). 35. D. J. Santeler, J. Vac. Sci. Technol. A, 4,338 (1986). 36. D. J. Santeler et al., Vacuum TechnologV and Space Simulation, NASA SP-105,

37. G. L. Saksaganski, Molecular Flow in Complex Systems, Gordon and Breach, New

38. S. L. Thomson and W. R. Owens, Vacuum, 25, 151 (1975). 39. S. K. Loyalka, T. S. Storvick, and H. S. Park, J. Vac. Sci. Technol., 13, 1188 (1976). 40. S. F. DeMuth and J. S. Watson, J. Vac. Sci. Technol. A, 4, 344 (1 986). 41. S. A. Tison, Vacuum, 44,1171 (1993). 42. J. Kieser and M. Grundner, Proc. VIIIZntl. Vac. Congr., Suppl. Rev. Le Vide, 201, 376

43. J. F. O’Hanlon, J. Vac. Sci. Technol. A, 5,98 (1 987). 44. B. W. Schumacher, Proceedings of the gh National Vacuum Symposium 1961, Vol. 2,

45. L. E. Levina, Sov. J. Nondestruct. Test., 16,67 (1 980).

17, 1036 (1 972).

Macmillan, New York, 1962, p. 407.

Pt.1, 117(1974).

Oxford, 1967, p. 85.

National Aeronautics and Space Administration, Washington, DC, 1966, p. 1 15.

York, 1988.

(1978).

Pergamon, New York, 1962, p. 1192.

PROBLEMS

3. I 7 Describe the molecular, transition, laminar viscous and turbulent gas flow regimes.

3.2t Calculate Reynolds’ and Knudsen’s numbers for the following cases: (a) 80 Pa-m3/s of air through a 2-mm-diameter choked orifice at a

54 GAS FLOW

a pressure of 10 1,000 Pa, (b) an air flow of 5 x 1 O6 Pa-L/s ( 1 atm at 50 L/s) in a 5-cm-diameter mechanical pumping line at the time the pump is started, and (c) a flow of 0.1 Pa-L/s of water vapor in a 20- cm-diameter high vacuum pumping line. Assume room temperature. Characterize the flow in each case.

3.3 (a) Calculate the maximum gas flow in a 1-mm-diameter tube, which is 15 mm long, with atmospheric pressure on one end and 1 Pa on the other end using the Poiseuille equation. (b) Calculate the maximum choked flow through a 1-mm-diameter orifice. Explain the difference between two answers.

3.4 t What is the lowest average pressure necessary to keep the room temperature air flow predominantly viscous laminar in a 5-cm- diameter line?

3.5 t A 15-cm-diameter pipe, 15 cm long connects a high vacuum chamber and a pumping system. Which of the following modifications will give the greatest increase in conductance in the molecular flow region? (a) Reducing the length of the tube to 7.5 cm, (b) increasing the diameter to 17.5 cm from 15 cm.

3.6 A 1-cm-diameter tube, 10 cm long, interconnects two very large vessels. The vessel A contains nitrogen at lo-* Pa, and vessel B contains nitrogen at lo4 Pa. (a) What is the flow rate of the nitrogen molecules originally in vessel A from vessel A to vessel B? (b) Is there any flow from vessel B to A? If so, how much? (c) What is the net nitrogen flow from vessel A to vessel B?

3.7 The piston in Fig. 3.19 is moved to the right at a uniform, linear velocity U so that the nitrogen flow Q through the leak remains constant at 1x10" Pa-m3/s. (a) Assume that the pressure in the chamber remains constant at 10 Pa during this motion. Calculate the velocity of the piston. (b) Assume that the piston moves to the right at a velocity of U = 3 mm/s whereas the value of the leak remains unchanged. Calculate the rate of pressure rise of dP/dt in the chamber.

A = 10"

Fig. 3.19

PROBLEMS 55

3.8 tKurtz [Proc. 4th Int. Vuc. Cong. (1968), Inst. of Phys. and The Phys. SOC., London, 1969, p. 8171. published a clever, simplified Monte Car10 probability generator using six-sided dice. His method helps us to visualize the nature of molecular flow. It is easily applied to any two-dimensional structure. First, we divide the entrance to a structure in 6 equal areas and number them serially. We cast a (six-sided) die and randomly determine an entrance position. Second, we cast a pair of six-sided dice and determine the angle, or direction of particle motion (explanation below). We draw a ray fiom the starting point along that angle to a point where the particle “collides” with a wall. See Fig. 3.20~. We cast the dice again, determine a new angle, and repeat the procedure until the particle either leaves the structure at the exit or returns to the entrance. After a number of attempts, the transmission probability can be calculated. It is the ratio of particles successhlly navigating the structure divided by the total number of particles entering the structure.

7

Fig. 3.20

We can determine the angle of escape with the template depicted in Fig. 3.20b. Seven is the most probable sum obtained when casting two dies; it occurs 6/36 times. Six and 8 each will be obtained on average 5/36 times, or 5/6 of the probability of obtaining a 7. A sum of 7 is equated with an escape angle of 0” fiom the normal. A sum of 6 or 8 corresponds to cos I$ = 5/6, or Q = & 34”, and so on, as sketched in Fig. 3.20b. t (a) Determine the probabilities for each of the other possible combinations (2 through 12) and make a template of escape angle versus sum of the dies as sketched in Fig. 3.20b. t (b) Draw a two-dimensional “cold trap” like that sketched in Fig. 3.20~. Draw the length and diameter equal (L = 2R) and draw the

56 GAS FLOW

entrance and plate diameter as half the outer diameter (R = 2Ro). Determine the transmission probability by Kurtz’ technique for a gas that does not stick to the walls. This transmission probability is analogous to that of nitrogen or oxygen as it passes through a cold trap to the pump. t (c) Calculate the transmission probability of this structure for a vapor like water, that is fi-ozen on the interior plate. Do this by re- examining the individual paths of each molecule you traced, and only counting those that successfully traversed the trap without hitting the center (LNz-cooled) plate. This transmission probability is analogous to the trapping probability of water vapor in this kind of a cold trap.

(d) Why does the Monte Car10 technique require many attempts to calculate the transmission probability accurately?

3.9 t Under what conditions does the gas flux have a cosine distribution at the entrance to a vacuum component in the molecular flow regime?

3. I0 Using the addition method for molecular flow developed by Haefer, calculate the transmission probability of a 10-cm-diameter round pipe that is 20 cm long and terminated in a 5-cm-diameter diaphragm.