Page 23 - Instrumentation Reference Book 3E

8 Measurement of flow

does not receive energy it must use its own heat The value of r is about 0.5 but it increases slightly
energy, and its temperature will fall. Thus the with increase of m and with decrease of specific
expansion that takes place owing to the fall in heat ratio. Values of r are tabulated in BS 1042
pressure does not obey Boyle's law, which applies Part 1 1964.
only to an expansion at constant temperature. The basic equation for critical flow is obtained
Instead it obeys the law for adiabatic expansion by substituting (1 - rc)p for Ap in equation (1.23),
of a gas: substituting rc for r in equation (1.31), and the
equation becomes
p1 . v; =p2 . v;I or p v? = constant
(1.30) W = 1.252 U . d2Jpp kglh (1.35)
where y is the ratio of the specific heats of the gas where
specific heat of a gas at constant pressure u = CJ@'/2)VC , (y - l)ly (1.36)
= specific heat of a gas at constant volume
The volume rate of flow (in m3/h) is obtained by
and has a value of 1.40 for dry air and other dividing the weight ratio of flow by the density (in
diatomic gases, 1.66 for monatomic gases such kglm3) of the fluid at the reference conditions.
as helium, and about 1.33 for triatomic gases such
as carbon dioxide.
If a metered fluid is not incompressible, 1.2.5.3 Departure from gas laws
another factor is introduced into the flow equa-
tions. This factor is necessary to correct for the At room temperature and at absolute pressures
change in volume due to the expansion of the less than 10 bar most common gases except car-
fluid while passing through the restriction. This bon dioxide behave sufficiently like an ideal gas
factor is called the expansibility factor E and has a that the error in flow calculations brought about
by departure from the ideal gas laws is less than 1
value of unity (1) for incompressible fluids. For
ideal compressible fluids expanding without any percent. In order to correct for departure from
change of state the value can be calculated from the ideal gas laws a deviation coefficient K (given
the equation in BS 1042 Part 1 1964) is used in the calculation
of densities of gases where the departure is sig-
nificant. For ideal gases K = 1.

1.2.5.4 Wet gases
where r is the ratio of the absolute pressures at the
upstream and downstream tappings (i.e., r = p1/p2) The above modification applies to dry gases. In
and y is the ratio of the specific heat of the fluid at practice many gases are wet, being a mixture of
constant pressure to that at constant volume. This gas and water vapor. Partial pressure due to satu-
is detailed in BS 1042 Part 1 1964. rated water vapor does not obey Boyle's law.
To apply working fluid flow equations to both Gas humidity is discussed in Chapter 6 of Part
liquids and gases the factor E is introduced and 2. If the temperature and absolute pressure at the
the equations become: upstream tapping and the state of humidity of the
gas are known, a correction factor can be worked
Q= 0.012 52CZ&Ed2fim3/h (1.32) out and applied to obtain the actual mass of gas
flowing.
W = 0.012 52 CZEEd2& kglh (1.33)
Gas density is given by the equation
E = 1 for liquids

1.2.5.2 Critical flow of compressible fluids
(1.37)
For flow through a convergent tube such as a
nozzle the value of r at the throat cannot be less where 6 is specific gravity of dry gas relative to
than a critical value r,. When the pressure at the air, T is temperature in Kelvin, p is pressure in
throat is equal to this critical fraction of the mbar at the upstream tapping, pv is partial pres-
upstream pressure, the rate of flow is a maximum sure in mbar of the water vapor, k is the gas law
and cannot be further increased except by raising deviation at temperature T, and p is gas density.
the upstream pressure. The critical pressure ratio For dry gas pv is zero and the equation becomes
is given by the equation
6P
2ry0h + - l)m2 . r;/l = - 1 (1.34) p = 6.196- kT kglm3 (1.38)