How to calculate flow rate and types of flow in vacuum physics

Types of flow

Three types of flow are mainly encountered in vacuum technology: viscous or continuous flow, molecular flow and, at the transition between these two, the Knudsen flow.

Viscous or continuum flow

This will be found almost exclusively in the rough vacuum range. The character of this type of flow is determined by the interaction of the molecules. Consequently internal friction, the viscosity of the flowing substance, is a major factor. If vortex motion appears in the streaming process, one speaks of turbulent flow. If various layers of the flowing medium slide one over the other, then the term laminar flow or layer flux may be applied.

Laminar flow in circular tubes with parabolic velocity distribution is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules’ mean free path is considerably shorter than the diameter of the pipe: λ « d.

A characteristic quantity describing the viscous flow state is the dimensionless Reynolds number Re. Re is the product of the pipe diameter, flow velocity, density and reciprocal value of the viscosity (internal friction) of the gas which is flowing. Flow is turbulent where Re > 2200, laminar where Re < 2200.

The phenomenon of choked flow may also be observed in the viscous flow situation. It plays a part when venting and evacuating a vacuum vessel and where there are leaks.

Gas will always flow where there is a difference in pressure

Δp = (p1 – p2) > 0. The intensity of the gas flow, i.e. the quantity of gas flowing over a period of time, rises with the pressure differential. In the case of viscous flow, however, this will be the case only until the flow velocity, which also rises, reaches the speed of sound. This is always the case at a certain pressure differential and this value may be characterized as “critical”:

(1.22)

A further rise in Δp > Δpcrit would not result in any further rise in gas flow; any increase is inhibited. For air at 68°F (20°C) the gas dynamics theory reveals a critical value of

(1.23)

The chart in Fig. 1.1 represents schematically the venting (or airing) of an evacuated container through an opening in the envelope (venting valve), allowing ambient air at p = 1000 mbar to enter. In accordance with the information given above, the resultant critical pressure is Δpcrit = 1000  ·  (1– 0.528) mbar ≈ 470 mbar; i.e. where Δp > 470 mbar the flow rate will be choked; where Δp < 470 mbar the gas flow will decline.

Fig 1.1 Schematic representation of venting an evacuated vessel.

1 – Gas flow rate qm choked = constant (maximum value)

2 – Gas flow not impeded, qm drops to Δp = 0

Molecular flow

Molecular flow prevails in the high and ultrahigh vacuum ranges. In these regimes the molecules can move freely, without any mutual interference. Molecular flow is present where the mean free path length for a particle is very much larger than the diameter of the pipe: λ >> d.

Knudsen flow

he transitional range between viscous flow and molecular flow is known as Knudsen flow. It is prevalent in the medium vacuum range: λ ≈ d.

The product of pressure p and pipe diameter d for a particular gas at a certain temperature can serve as a characterizing quantity for the various types of flow. Using the numerical values provided in Table III, the following equivalent relationships exist for air at 68°F (20 °C):

Table III Mean free path l Values of the product c* of the mean free path λ (and pressure p for various gases at 68°F (20°C.)

High and ultrahigh vacuum – Molecular flow

In the viscous flow range the preferred speed direction for all the gas molecules will be identical to the macroscopic direction of flow for the gas. This alignment is compelled by the fact that the gas particles are densely packed and will collide with one another far more often than with the boundary walls of the apparatus. The macroscopic speed of the gas is a “group velocity” and is not identical with the “thermal velocity” of the gas molecules.

In the molecular flow range, on the other hand, impact of the particles with the walls predominates. As a result of reflection (but also of desorption following a certain residence period on the container walls) a gas particle can move in any arbitrary direction in a high vacuum; it is no longer possible to speak of “flow” in the macroscopic sense.

It would make little sense to attempt to determine the vacuum pressure ranges as a function of the geometric operating situation in each case. The limits for the individual pressure regimes (see Table IX) were selected in such a way that when working with normal-sized laboratory equipment the collisions of the gas particles among each other will predominate in the rough vacuum range whereas in the high and ultrahigh vacuum ranges impact of the gas particles on the container walls will predominate.

Table IX Pressure ranges used in vacuum technology and their characteristics (numbers rounded off to whole power of ten)

In the high and ultrahigh vacuum ranges the properties of the vacuum container wall will be of decisive importance since below 10-3 mbar there will be more gas molecules on the surfaces than in the chamber itself. If one assumes a monomolecular adsorbed layer on the inside wall of an evacuated sphere with 1 l volume, then the ratio of the number of adsorbed particles to the number of free molecules in the space will be as follows:

at 1 mbar 10-2

at 10-6 mbar 10+4

at 10-11 mbar 10+9

For this reason the monolayer formation time τ is used to characterize ultrahigh vacuum and to distinguish this regime from the high vacuum range. The monolayer formation time τ is only a fraction of a second in the high vacuum range while in the ultrahigh vacuum range it extends over a period of minutes or hours. Surfaces free of gases can therefore be achieved (and maintained over longer periods of time) only under ultrahigh vacuum conditions.

Further physical properties change as pressure changes. For example, the thermal conductivity and the internal friction of gases in the medium vacuum range are highly sensitive to pressure. In the rough and high vacuum regimes, in contrast, these two properties are virtually independent of pressure. Thus, not only will the pumps needed to achieve these pressures in the various vacuum ranges differ, but also different vacuum gauges will be required. A clear arrangement of pumps and measurement instruments for the individual pressure ranges is shown in Figures 9.16 and 9.16a.

Fig 9.16 Common working ranges of vacuum pumps

Fig 9.16a Measurement ranges of common vacuum gauges

Units and definitions

Volume V (l, m3, cm3)

The term volume is used to designate

a) the purely geometric, usually predetermined, volumetric content of a vacuum chamber or a complete vacuum system including all the piping and connecting spaces (this volume can be calculated);

b) the pressure-dependent volume of a gas or vapor which, for example, is moved by a pump or absorbed by an adsorption agent.

Volumetric flow (flow volume) qv  (l/s, m3/h, cm3/s )

The term “flow volume” designates the volume of the gas which flows through a piping element within a unit of time, at the pressure and temperature prevailing at the particular moment. Here one must realize that, although volumetric flow may be identical, the number of molecules moved may differ, depending on the pressure and temperature.

Pumping speed S (l/s, m3/h, cm3/s )

The pumping speed is the volumetric flow through the pump’s intake port.

(1.8a)

If S remains constant during the pumping process, then one can use the difference quotient instead of the differential quotient:

(1.8b)

(A conversion table for the various units of measure used in conjunction with pumping speed is provided in Table VI).

Table VI Conversion of pumping speed (volume flow rate) units

Quantity of gas (pV value), (mbar ⋅ l)

The quantity of a gas can be indicated by way of its mass or its weight in the units of measure normally used for mass or weight. In practice, however, the product of p · V is often more interesting in vacuum technology than the mass or weight of a quantity of gas. The value embraces an energy dimension and is specified in millibar · liters (mbar · l) (Equation 1.7). Where the nature of the gas and its temperature are known, it is possible to use Equation 1.7b to calculate the mass m for the quantity of gas on the basis of the product of p · V:

(1.7)

(1.7b)

Although it is not absolutely correct, reference is often made in practice to the “quantity of gas” p · V for a certain gas. This specification is incomplete; the temperature of the gas T, usually room temperature (293 K), is normally implicitly assumed to be known.

Example:

The mass of 100 mbar · l of nitrogen (N2) at room temperature (approx. 300 K) is:

Analogous to this, at T = 300 K:

1 mbar · l O2 = 1.28 · 10-3 g O2

70 mbar · l Ar = 1.31 · 10-1 g Ar

The quantity of gas flowing through a piping element during a unit of time – in accordance with the two concepts for gas quantity described above – can be indicated in either of two ways, these being:

Mass flow qm (kg/h, g/s),

this is the quantity of a gas which flows through a piping element, referenced to time

or as

pV flow qpV (mbar · l · s–1).

pV flow is the product of the pressure and volume of a quantity of gas flowing through a piping element, divided by time, i.e.:

pV flow is a measure of the mass flow of the gas; the temperature to be indicated here.

Pump throughput qpV

The pumping capacity (throughput) for a pump is equal either to the mass flow through the pump intake port:

(1.9)

or to the pV flow through the pump’s intake port:

It is normally specified in mbar · l · s–1. Here p is the pressure on the intake side of the pump. If p and V are constant at the intake side of the pump, the throughput of this pump can be expressed with the simple equation

(1.10a)

where S is the pumping speed of the pump at intake pressure of p.

(The throughput of a pump is often indicated with Q, as well.)

The concept of pump throughput is of major significance in practice and should not be confused with the pumping speed! The pump throughput is the quantity of gas moved by the pump over a unit of time, expressed in mbar ≠ l/s; the pumping speed is the “transportation capacity” which the pump makes available within a specific unit of time, measured in m3/h or l/s.

The throughput value is important in determining the size of the backing pump in relationship to the size of a high vacuum pump with which it is connected in series in order to ensure that the backing pump will be able to “take off” the gas moved by the high vacuum pump.

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References

Vacuum symbols

Vacuum symbols

A glossary of symbols commonly used in vacuum technology diagrams as a visual representation of pump types and parts in pumping systems

Glossary of units

Glossary of units

An overview of measurement units used in vacuum technology and what the symbols stand for, as well as the modern equivalents of historical units

References and sources

References and sources

References, sources and further reading related to the fundamental knowledge of vacuum technology

Vacuum symbols

A glossary of symbols commonly used in vacuum technology diagrams as a visual representation of pump types and parts in pumping systems

Glossary of units

An overview of measurement units used in vacuum technology and what the symbols stand for, as well as the modern equivalents of historical units