Hoffman D.M., Singh B., Thomas J.H. (Eds). Handbook of Vacuum Science and Technology

Подождите немного. Документ загружается.

24 Chapter 1.4: Throughput, Pumping Speed, and Rates

Evacuation rate: The pressure in a vacuum system falls as gas is removed,

and the rate of

gas

removal is the throughput of the vacuum pumps. Assuming

the pumping speed to remain constant, the throughput falls as the pressure falls,

and is in fact exponential, given by

P2 = P,exp(^-^j (41)

where P2

—

pressure (torr) at time t (seconds). Pi = pressure (torr) at time / = 0,

V = system volume (liters) and S = pumping speed (Ls

^).

Thus the pressure falls to half its original value in 0.69 VIS seconds, to quarter

that value in twice as long, to j that value in twice as long again, and so on. At

low pressures the evacuation rate falls off as a result of leakage and outgassing

from system walls. Pumpdown times to pressures below

10 ~-^

torr are seldom

computed but quoted as a result of experience with similar systems or extrapola-

tion of experimental data on smaller systems.

CHAPTER

1.5

Gas Flow

1.5.1

NATURE

GAS FLOW

The production and maintenance of vacuum involves flow of gas from the vac-

uum vessel through pipelines and pumps to the atmosphere. Hence mechanisms

that control and affect the nature of gas flow are of great importance. Typically,

gas flow in vacuum systems can be divided into three main categories or regimes:

• Turbulent flow

• Viscous or laminar flow

• Molecular flow

Many factors are responsible for determing the gas flow regime, including

• Magnitude of the flow rates

• The pressure drop at either ends of the pipe or duct

• The surface and geometrical properties of the pipe or duct

• The nature and characteristics of the gases being pumped

Quantitative expressions that relate factors such as duct geometry and nature of

gas to the type and quality are often complex. However, formulas and expressions

for number of conmionly encountered geometrical configurations have been de-

veloped, and these enable gas flow rate to be calculated with reasonable accuracy

and ease. Calculation of gas flow rates under for various geometries and different

conditions are presented in the following sections. Detailed derivations of many

of the formulas can be found in references. To simplify calculations, the three

just-listed flow categories are usually considered independently. All the three

regimes of gas flow are encountered when a vacuum system, connected by pipes

* ^ All rights of reproduction in any form reser\'ed.

26 Chapter 1.5: Gas Flow

to a vacuum pump, is pumped down from atmosphere to high-vacuum impact

system characteristics such as pumpdown time.

1.5.1.1

General Description

At high pressures and very high flow rates, the mean free path of the molecules is

very small compared with the dimensions of the pipe or vacuum vessel so that the

flow of the gas is limited by the viscosity of the gas. Gas flow under these condi-

tions is called viscous, and flow can be either turbulent or laminar. When the ve-

locity of the gas exceeds certain values, the flow is turbulent, the flowing gas lay-

ers are not parallel, and their direction is influenced by any obstacle in the way.

In spaces between the layers, spaces of lower pressures (cavities) appear. At lower

gas velocities, the viscous flow becomes laminar—that is, the layers of gas flow

are parallel, their velocity increasing from the walls toward the axis of the pipe.

As the pressure is reduced, the mean free path of the molecules becomes similar

to the dimensions of the vessel and flow is governed by a combination of the gas

viscosity and by molecular phenomena. These conditions give rise to a type of

flow known as intermediate flow. As the pressure continues to decrease, the mean

free path becomes larger than the vessel dimensions and flow depends only on

molecular collisions with the vessel walls; hence the flow under these conditions

is referred to as molecular. The molecular motion in the various flow regimes is

illustrated in Figure 2. The nature of gas flow (whether turbulent, laminar, inter-

mediate, or molecular) is determined by the values of two dimensionless parame-

ters,

namely, the Reynolds and Knudsen numbers. The limit between the turbulent

and laminar or viscous flow is defined by the Reynolds number, while those be-

tween laminar, intermediate, and molecular flow are determined by the Knudsen

number.

The Reynolds number is a dimensionless quantity expressed by

Re = pvD/r) (42)

where p is the density of the gas, v is the velocity, rj is the gas viscosity, and D is

the diameter of the tube.

For Re > 2100, flow is entirely turbulent.

For Re < 1100, flow is entirely laminar.

In the region 1200 < Re < 2200, the flow can be turbulent or viscous depend-

ing on the roughness of the surface of the tube and geometrical factors of the

tube.

As flow velocity increases, the Reynolds number increases at high pres-

sures,

and the gas, instead of flowing smoothly, develops eddies and oscillations.

Viscous flow occurs in the region bounded by a Reynolds number lower than

1200 and a Knudsen number lower than

0.001.

1.5.2 Turbulent Flow

The Knudsen number,

K„,

is also a dimensionless quantity, given by

*r. = i (43)

When the Knudsen number is equal to or greater than the pipe diameter, say

AT,,

> 1, the flow properties are determined by gas-wall collisions and the flow

is molecular. More detailed descriptions of the three main flow regimes are

given next.

1.5.2

TURBULENT FLOW

Turbulent flow of

gas

occurs at high pressure gradients, is characterized by eddies

and vortices in the gas stream, and is rarely encountered in most vacuum applica-

tions.

It does briefly occur, however, on the onset of pumping where the gas pres-

sure and velocity are sufficiently high and the gas flow in the vacuum vessel and

in the connecting pipes is very chaotic; the flow pattern lacks order and is charac-

terized by eddies that appear and disappear. The motion of gas molecules under-

going turbulent flow is complex and lacks order; the gas swirls and eddies, and

the individual particles of the gas may have velocities and directions that are quite

different from the average velocity and the overall flow direction. During turbu-

lent flow conditions the gas pressure and velocity of flow at any point in the sys-

tem fluctuates about a mean point. Except in very special cases (for example, in

very large vacuum systems), the duration of turbulent flow is short compared with

viscous flow and molecular flow. In most production thin film coating systems and

special "soft pumpdown" procedures are employed to reduce the initial pumping

speed, to reduce turbulence in the chamber and thus minimize dust or particle

generation and contamination of the substrates. Although it is irregular, the tur-

bulent flow can nonetheless be characterized by the laws of probabiUty [2]. The

existence of turbulent flow is determined by the value of the dimensionless Rey-

nolds number. If the Reynolds number is larger than 2100, the flow will always be

entirely turbulent; for example, for air at room temperature, flowing through a cir-

cular pipe, the flow is turbulent if

—- > 5 X 10^ (44)

where F is the flow rate through the pipe in L/sec; P is the mean air pressure in

millitorr, and d is pipe diameter in inches.

In many situations, however, turbulent flow can usually be neglected as it gen-

28 Chapter 1.5: Gas Flow

erally occurs only briefly (in the pipe between the mechanical pump and the vac-

uum system), during the initial stage of pumpdown from atmosphere.

1.5.3

VISCOUS, STREAMLINE, OR LAMINAR FLOW

This type of flow is much simpler than turbulent flow and occurs at moderate pres-

sure gradients when

< (J/100) (where

is the mean free path and d is the pipe

diameter). Viscous flow is smooth and orderly; every particle passing a point fol-

lows the same path as the preceding particle passing that point. Flow rates are pro-

portional to the pressure gradient, and the viscosity of the gas. Flow lines are

straight lines or gradual curves unlike the case with turbulent flow. That mean

free path of the molecules is small compared to the dimensions of the duct during

this type of flow, so that collisions are predominant in determining the character-

istics of flow and the flow rates will be affected by the viscosity of the gas. Vis-

cous laminar flow often occurs in backing lines to diffusion pumps. Viscous flow

conditions are reached as continued pumping reduces the pressure and the

Reynolds number decreases below 2200. Eddies cease to appear, and the energy

resulting from the pressure gradient is used to maintain a steady flow. The gas ve-

locity and pressure become uniform with time, and the flow becomes stream-

lined; that is, the lines of flow are smooth and continuous and curve gently in the

neighborhood of bends and other irregularities in the pipe. Near the wall the gas

is almost at rest, but progressing toward the center the layers of gas slide more

quickly over each other until the velocity reaches a maximum at the center. Vis-

cosity is important in determining the amount of gas passing per second through

the pipe under these conditions. This type of flow is referred to as either ''viscous''

flow or ''laminar" flow. It is governed by Poiseuille's law, which states that

Q d'P (45)

(P,

- P2) VL

where L = length of the tube of diameter d

Q = product of volume flow and corresponding pressure

rj = gas viscosity

K = numerical constant

The pressure gradient causes adjacent layers of the gas to exert a pressure on

each other in the direction of the negative pressure gradient, setting the gas in mo-

tion as a whole. Viscous flow can only occur when the mean free path is small

compared with the pipe diameter. Only those molecules near the pipe walls col-

lide with the walls, and since these will represent only a small fraction of the total

number molecules present, the nature of the walls does not have an important ef-

fect on the flow rate.

1.5.5 Flow Relationships

1.5.4

MOLECULAR FLOW

If the gas pressure is lowered even further (while still maintaining a pressure gra-

dient along the pipe), the mean free path of the gas molecules increases and ap-

proaches the pipe diameter and the nature of the flow changes. The laminar nature

the

flow disappears because molecules now collide with walls of

the

pipe rather

than with each other. When the pressure is low enough, the molecules move about

inside the pipe independently of each other and the flow is said to be "molecular."

The pressure along the tube no longer acts as a driving force pushing the gas along

the tube in a stream. At these low pressures the molecules move in random direc-

tions,

and there is net transfer of gas from the high- to the low-pressure region

simply because there are more molecules per unit volume.

Molecular flow is characterized by molecular collisions with the tube walls

rather than with other gas molecules. Flow rates are proportional to the difference

in pressures across the component and the reciprocal of the square root of the mo-

lecular weight of the gas. The dependence of flow rate on viscosity begins to de-

crease, because intermolecular collisions are less frequent. At pressures suffi-

ciently low for the mean free path to be several times greater than the vessel

diameter or the duct, the molecules migrate through a system freely and indepen-

dently of each other. This is called free-molecule flow or simply molecular flow,

and the gas flow rates are affected mainly by collisions of molecules with the tube

walls.

Molecular flow occurs at high vacuum when

(d/3).

As the mechanical

pump removes gas from the pipeline between the pump and vacuum chamber, the

gas is pushed toward the pump by means of collisions from its upstream neigh-

bors.

This describes the phenomenon of viscous flow and indicates its propor-

tionality to pressure gradient. When a high-vacuum pump moves gas directly

from a chamber, the pump waits for molecules to wander into the pump entrance

and once in, to prevent them returning: typical efficiencies in this respect for high-

vacuum pumps are about

40%.

During molecular flow, where intermolecular col-

lisions are rare, molecules can move away from the pump independently of mole-

cules moving toward the pump.

.5.5

FLOW RELATIONSHIPS

L5.5.1 Volume Flow

Rate,

S, and

Throughput, Q

The gas flow rate is an important aspect of the behavior of gases in a vacuum sys-

tem. It determines the time required to reach the operating pressure; it may also

30 Chapter 1.5: Gas Flow

determine the magnitude of gas leaks (gas leaking into the system from the out-

side) and outgassing (gas produced somewhere within the system) that can be tol-

erated without the pressure in the vessel rising above the desired operating pres-

sure.

The rate at which the gas flows through a system can be expressed in two

ways:

as a volumetric flow rate and as a mass flow rate.

1.5.5.2

Volumetric Flow Rate

Volumetric flow rate is usually given the designation S and is the volume of gas

flowing past a given point in a system per unit time. The volumetric rate measured

at the pump entrance is called the pump speed, Sp,

1.5.5.3

Throughput Q

Throughput Q is related to the mass flow rate and is given the designation Q. It is

proportional to the number of moles and hence to the mass of gas flowing past a

given point per unit time. The throughput and the volumetric flow rate are related

by the equation

Q = SP

where Q is the throughput and is the mass flow rate measured in torr liter/sec

(Ls~^) and S is the volumetric flow rate in liter/sec or Ls~^ measured at P, the

pressure in torr.

By definition, the throughput, Q, is constant for all points in a vacuum system

that is at equilibrium. Mass flow rate is measured in grams per second or the num-

ber of molecules per second. Since pressure (torr) is a measure of the molecular

concentration (number of molecules per liter) and the volumetric flow rate is

given as Ls~^ then the product torr-Ls~' is proportional to the number of mole-

cules passing any point in a vacuum system per second and hence the throughput.

The rate at which gas flows through the vacuum system depends on the pump

speed, the geometrical shape and the dimensions of the passages, and the type of

flow. Thus a volumetric flow rate of 1000 Ls~^ at a pressure of 10~^ torr gives a

throughput of 1.0 torr • Ls"^ This quantity 1.0 torr • Ls~^ = (1.0/760) atmo-

sphere Ls~^ For air, 1 mole (29 g) occupies 22.4 L, that is, 1 atmosphere liter =

29/22 g. Table 4 lists throughput conversion factors.

1.5.5 Flow Relationships

Table

Throughput Conversion Factors

Throughput Unit

1 lusec

1 clusec

lcm\NTP)s-^

\cm^iNTP)min-^

lft^(NTP)mm-^

1 meter3(A^rP)hour~'

lft3(A^rP)hour-'

1 watt^

Igs"'

at20°C

Equivalent torr-Ls ' Rate

0.001

0.01

0.76

0.0127

358

211

5.97

7.5

1.7 X

lO'*

molecular weight

*In the MKS system where the unit of force is the newton

and the unit of pressure is the newton per meter ^, the unit of

throughput is newton meter s

watt.

which equates to

joule

•

CHAPTER

1.6

Conductance

1.6.1

CONDUCTANCE

Any pipe or duct offers a certain resistance to gas flow of any type. This resis-

tance causes a pressure drop along the pipe. If F is the volume flow rate of gas

flowing per sec across any cross section of the pipe and P is the pressure at the

section, the quantity of gas passing per sec is

Q = FP (47)

The resistance R (or impedance) of a length of pipe is defined by

R = ^^ (48)

where

and

are the pressures at ends of the pipe.

Thus R has units of s/L. The reciprocal of R is called conductance and is des-

ignated by the symbol C.

C = — = quantity of flow/pressure difference (49)

Thus the conductance of an orifice, pipe or vacuum component is a measure

of the throughput for a given pressure drop across the pipe or component and is

therefore expressed as torr-Ls~Vtorr~^ which becomes Ls~^ The conductance

varies according to the mode of gas flow and the nature of the gas. Hence the net

transfer of gas through a component connected to a high-vacuum pump is propor-

tional to the pressure difference across the component. For a component with a

conductance C Ls

and pressures Pi and Pj torr existing at its ends, the net gas

$25.00 All rights of reproduction in any form reserved.

1.6.3 Conductance

Series

flow through the component is C (Pj - P2) torr-Ls "^ which is the difference in

mass flow rate from the high-pressure end (Cp, torr-Ls "^) and the backflow from

the low-pressure end

(Cp^

torr-Ls "

^).

1.6.2

CONDUCTANCES IN PARALLEL

Consider two pipes having conductances Ci and C2. The mass flow in each pipe

is given by

The total flow is

01 = CiP and 22 = ^^2^ (50)

Q^mdQ2

= (C,-^C2)P (51)

If the two pipes were replaced by a single pipe of conductance, Cj, which

would give the same flow, then

QT = CjP (52)

where Cj is the total conductance and the subscript j signifies total conductance

or flow for the element.

For series connection of

components,

pipes, or orifices, the total conductance is

given by

= Ci + C2 + C3 etc. (53)

1.6.3

CONDUCTANCES IN SERIES

Consider two conductances Cj and C2 in series; consisting, for example, of

pipe

joined at one end to another pipe of different diameter. Let

and

be the pres-

sure drops along pipes

and 2, respectively. Let Qi and

be the mass flow rates.

Let Pj be the total pressure drop in series, and Qj be the corresponding mean

flow

rate.

Thus

Q\ Qi Q\

P\=7r^ ^2 = ^, andAPT = -^ (54)

^1 C2 Cx