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Flow Calculations
CHAPTER 1.7
Flow Calculations The conductance of a duct is a measure of its ability to transport gas and is expressed in units of volume transported per unit time. The quantitative expressions used to calculate conductance of an element under different circumstances are fairly complex and depend on the type of flow as well as on geometrical and surface-related factors. Calculations of conductance and the corresponding gas flow rate for turbulent flow are difficult to treat analytically. Viscous flow is also somewhat difficult to treat quantitatively, because it depends not only on the shape of the duct but also on the gas pressure. Fortunately, at most pressures of interest to high-vacuum situations, the flow is molecular rather than viscous. Considerable effort has been expended for developing analytical techniques for determining flow under viscous and molecular regimes.
1.7.1 EQUATIONS FOR VISCOUS FLOW Generally, quantitative expressions that have been developed for calculating conductance and corresponding flow rate under viscous flow conditions are those for fairly simple geometrical configurations, such as circular tubes or rectangular ducts. These expressions are used, for example, to calculate the time required to evacuate a vessel or volume of some sort, through a pipe that is usually circular or rectangular in cross section.
1.7.1.1 Circular Tube The mass flow rate through a straight tube of circular cross section under viscous flow conditions, is determined by Poiseuille's equation, namely, ISBN 0-12-325065-7 •^^ ^ ^
Copyright © 1998 by Academic Press Ail rights of reproduction in any form reserved.
35
36
Chapter 1.7: Flow Calculations
Pi - Pi where
= K-— VL
(60)
d is the diameter of the tube L is the length of the tube 7] is the viscosity of the gas P is the average of Pj and Pj, the pressure at the opposite ends of the tube
For dry air at 20°C, this equation becomes Q=
750 J ^ P 1 iPi-Pi)
(61)
where Q is the mass flow rate in torr-L/sec, d is the tube diameter in inches, L is the tube length in centimeters, and P is the pressure in torn The expression for conductance for a circular pipe for air at 20°C is C=
17.12
2.94Prf^ ; L/sec
(62)
Rectangular Duct
For the rectangular duct, let a = long side and b = short side. The Poiseuille equation for the rectangular duct for air is at 20°C is C=
30a2b2KP
L/sec
(63)
JLi
where ^ is a shape factor whose value depends on b/a. As can be seen, the conductance of the rectangular slit increases rapidly as the cross section changes from slit to square. As in the case of round pipe, the expression for C leads to a relation for the volume flow in terms of the pressure drop along the duct. P^
/^ A\
AP where CP F =-^ AP
(64)
Thus K=
30a^b^K
^ AP L/sec
(65)
1.7.3
Knudsen's Formulation
37
1.7.2 EQUATIONS FOR MOLECULAR FLOW At low pressures, intermolecular collisions are less frequent than wall collisions, so the latter determine the gaseous flow characteristics through the channel. Specifically, two aspects determine the conductance of a duct during molecular flow: 1. 2.
Rate at which molecules enter the duct Probability that the molecules are transited through the system
The first depends on the entrance area of the system, while the latter is determined by the subsequent series of reflections from the walls, which result in the molecule eventually being transmitted through the duct or reflected back into original volume. Consider first the case of very thin aperture plate, for which the area A is more important in determining conductance than the wall area or wall conditions. The volume of gas traveling from one side of the aperture to the other side per unit time — the aperture conductance — is Q =
\AV,,
(66)
when the molecules have a Maxwell-Boltzmann velocity distribution. Conductance values depend on the molecular mass and kinetic temperature. The case where wall collisions are more important than the conductance of an aperture is considered next.
173 KNUDSEN'S FORMULATION The conductance Cj of a length of long tube of length L with uniform crosssectional area A and perimeter //, was calculated by Knudsen to be: C. = | ^ v „
(67,
The assumptions for obtaining the general result are 1. 2. 3.
Length of tube is much greater than the diameter. Direction of reflected molecules is independent of the incident direction. Reflected molecules are distributed equally per unit angle (cosine law for reflection from a Lambertian surface).
38
Chapter 1.7: Flow Calculations
Relationships derived from the general equation are given in Table 5 for simple geometries. Assumption 1 indicates that the effect of aperture is insignificant, and the conductance value is given by the preceding equation is for molecules that are well within the tube and are sufficiently removed from the aperture so that it is of no consequence. A rough attempt to correct this deficiency is to include a series conductance of the entrance aperture. Weissler and Carlson [5] gives a formula for a tube of perimeter //, area A, and length L:
-(-^•^r^.
(68)
1.7.4 CLAUSING FACTORS The conductance of a long tube is approximately related to the conductance of the entrance aperture through the factor [1 + (3/16)(L//M)]"' C^. This factor can be interpreted as the probability that a molecule incident on the aperature will be transmitted through the tube and leave at the other end. It is convenient to discuss conductance in terms of the aperture conductance and corresponding probability of passage, ^1^2'—^he Clausing factor—so that C=C,'
Pi^2 = (l/4)VavAiPi^2
(69)
Because conductance is independent of direction, A,P,^2 = A2P2^,
(70)
Examples: The throughput in pressure-volume units per unit time (torr • cm^ • s~^) through a long tube is given approximately by
e=i^y^'^ - ^^) where
^''^
d is the tube diameter in cm L is the tube length in cm /la is the average velocity of a molecule in cm s~^ Pi and P2 are the pressures (in torr) at the opposite ends of the tube
1.7.4 Clausing Factors
39
Approximate values for some probabilities of passage are accurate to within ±10%. A variety of techniques, which include analytical methods [6], Monte Carlo calculations, and variation methods [16]. Carlson lists different geometries that have been investigated and cites the corresponding references. Numerical examples can be found in Carlson [22].
Flow Calculations The conductance of a duct is a measure of its ability to transport gas and is expressed in units of volume transported per unit time. The quantitative expressions used to calculate conductance of an element under different circumstances are fairly complex and depend on the type of flow as well as on geometrical and surface-related factors. Calculations of conductance and the corresponding gas flow rate for turbulent flow are difficult to treat analytically. Viscous flow is also somewhat difficult to treat quantitatively, because it depends not only on the shape of the duct but also on the gas pressure. Fortunately, at most pressures of interest to high-vacuum situations, the flow is molecular rather than viscous. Considerable effort has been expended for developing analytical techniques for determining flow under viscous and molecular regimes.
1.7.1 EQUATIONS FOR VISCOUS FLOW Generally, quantitative expressions that have been developed for calculating conductance and corresponding flow rate under viscous flow conditions are those for fairly simple geometrical configurations, such as circular tubes or rectangular ducts. These expressions are used, for example, to calculate the time required to evacuate a vessel or volume of some sort, through a pipe that is usually circular or rectangular in cross section.
1.7.1.1 Circular Tube The mass flow rate through a straight tube of circular cross section under viscous flow conditions, is determined by Poiseuille's equation, namely, ISBN 0-12-325065-7 •^^ ^ ^
Copyright © 1998 by Academic Press Ail rights of reproduction in any form reserved.
35
36
Chapter 1.7: Flow Calculations
Pi - Pi where
= K-— VL
(60)
d is the diameter of the tube L is the length of the tube 7] is the viscosity of the gas P is the average of Pj and Pj, the pressure at the opposite ends of the tube
For dry air at 20°C, this equation becomes Q=
750 J ^ P 1 iPi-Pi)
(61)
where Q is the mass flow rate in torr-L/sec, d is the tube diameter in inches, L is the tube length in centimeters, and P is the pressure in torn The expression for conductance for a circular pipe for air at 20°C is C=
17.12
2.94Prf^ ; L/sec
(62)
Rectangular Duct
For the rectangular duct, let a = long side and b = short side. The Poiseuille equation for the rectangular duct for air is at 20°C is C=
30a2b2KP
L/sec
(63)
JLi
where ^ is a shape factor whose value depends on b/a. As can be seen, the conductance of the rectangular slit increases rapidly as the cross section changes from slit to square. As in the case of round pipe, the expression for C leads to a relation for the volume flow in terms of the pressure drop along the duct. P^
/^ A\
AP where CP F =-^ AP
(64)
Thus K=
30a^b^K
^ AP L/sec
(65)
1.7.3
Knudsen's Formulation
37
1.7.2 EQUATIONS FOR MOLECULAR FLOW At low pressures, intermolecular collisions are less frequent than wall collisions, so the latter determine the gaseous flow characteristics through the channel. Specifically, two aspects determine the conductance of a duct during molecular flow: 1. 2.
Rate at which molecules enter the duct Probability that the molecules are transited through the system
The first depends on the entrance area of the system, while the latter is determined by the subsequent series of reflections from the walls, which result in the molecule eventually being transmitted through the duct or reflected back into original volume. Consider first the case of very thin aperture plate, for which the area A is more important in determining conductance than the wall area or wall conditions. The volume of gas traveling from one side of the aperture to the other side per unit time — the aperture conductance — is Q =
\AV,,
(66)
when the molecules have a Maxwell-Boltzmann velocity distribution. Conductance values depend on the molecular mass and kinetic temperature. The case where wall collisions are more important than the conductance of an aperture is considered next.
173 KNUDSEN'S FORMULATION The conductance Cj of a length of long tube of length L with uniform crosssectional area A and perimeter //, was calculated by Knudsen to be: C. = | ^ v „
(67,
The assumptions for obtaining the general result are 1. 2. 3.
Length of tube is much greater than the diameter. Direction of reflected molecules is independent of the incident direction. Reflected molecules are distributed equally per unit angle (cosine law for reflection from a Lambertian surface).
38
Chapter 1.7: Flow Calculations
Relationships derived from the general equation are given in Table 5 for simple geometries. Assumption 1 indicates that the effect of aperture is insignificant, and the conductance value is given by the preceding equation is for molecules that are well within the tube and are sufficiently removed from the aperture so that it is of no consequence. A rough attempt to correct this deficiency is to include a series conductance of the entrance aperture. Weissler and Carlson [5] gives a formula for a tube of perimeter //, area A, and length L:
-(-^•^r^.
(68)
1.7.4 CLAUSING FACTORS The conductance of a long tube is approximately related to the conductance of the entrance aperture through the factor [1 + (3/16)(L//M)]"' C^. This factor can be interpreted as the probability that a molecule incident on the aperature will be transmitted through the tube and leave at the other end. It is convenient to discuss conductance in terms of the aperture conductance and corresponding probability of passage, ^1^2'—^he Clausing factor—so that C=C,'
Pi^2 = (l/4)VavAiPi^2
(69)
Because conductance is independent of direction, A,P,^2 = A2P2^,
(70)
Examples: The throughput in pressure-volume units per unit time (torr • cm^ • s~^) through a long tube is given approximately by
e=i^y^'^ - ^^) where
^''^
d is the tube diameter in cm L is the tube length in cm /la is the average velocity of a molecule in cm s~^ Pi and P2 are the pressures (in torr) at the opposite ends of the tube
1.7.4 Clausing Factors
39
Approximate values for some probabilities of passage are accurate to within ±10%. A variety of techniques, which include analytical methods [6], Monte Carlo calculations, and variation methods [16]. Carlson lists different geometries that have been investigated and cites the corresponding references. Numerical examples can be found in Carlson [22].