Page 22 - Instrumentation Reference Book 3E

Basic principles of flow measurement 7

Discharge coefficient when well away from their critical temperatures
and pressures) then the gas obeys several very
actual mass rate of flow
C= important gas laws. These laws will now be stated.
theoretical mass rate of flow
or if the conditions of temperature, density, etc., 1.2.5.1 DTJ gases
are the same at both sections it may be written in (a) Boyle’s law This states that the volume of
terms of volume.
any given mass of gas will be inversely propor-
c= actual volume flowing tional to its absolute pressure provided tempera-
theoretical volume flowing ture remains constant. Thus, if a certain mass of
gas occupies a volume vo at an absolute pressure
It is possible to determine C experimentally by po and a volume 1’1 at an absolute pressurep then
actual tests. It is a function of pipe size, type of
pressure tappings, and the Reynolds number.
Equation (1.21) is modified and becomes (1.27)

(1.24) (b) Charles’s law This states that if the volume
of a given mass of gas occupies a volume VI at a
This is true for flow systems where the Reynolds temperature TO Kelvin, then its volume v at T
number is above a certain value (20,000 or above Kelvin is given by
for orifice plates). For lower Reynolds numbers vllT0 = v1T or v = VI . TIT, (1.28)
and for very small or rough pipes the basic coeffi-
cient is multiplied by a correction factor 2 whose
value depends on the area ratio, the Reynolds (c) The ideal gas law In the general case p, v, and
number, and the size and roughness of the pipe. T change. Suppose a mass of gas at pressure po
Values for both C and Z are listed with other and temperature TO Kelvin has a volume vo and
relevant data in BS 1042 Part 1 1964. the mass of gas at pressure p and temperature T
We can use differential pressure to measure flow. has a volume v, and that the change from the first
Here’s a practical example: set of conditions to the second set of conditions
takes place in two stages.
Internal diameter of (a) Change the pressure from po to p at a con-
upstream pipe D mm stant temperature. Let the new volume be VI.
Orifice or throat From Boyle’s law:
diameter dmm
Pressure differential PO. vo =p. VI or VI = vo .po/p
produced hinm water gauge (b) Change the temperature from TO to T at
Density of fluid at constant pressure. From Charles’s law:
upstream tapping p kglm’
Absolute pressure at VllTo = dT
upstream tapping p bar Hence, equating the two values of v1

Then introducing the discharge coefficient C, the vo .palp = v . TOIT
correction factor and the numerical constant, the PO. vo/To =pvlT = constant (1.29)
equation for quantity rate of flow Qm31h becomes
If the quantity of gas considered is 1 mole, i.e., the
r
quantity of gas that contains as many molecules
(1.25) as there are atoms in 0.012 kg of carbon-12. this
constant is represented by R, the gas constant, and
and the weight or mass rate of the flow W kglh is equation (1 29) becomes:
given by
PV= Ro. T
W=0.01252C.Z.E.d2fi (1 26)
where Ro = 8.314 JlMol K andp is in N/m2 and Y
is in m3.
1.2.5 Modification of flow equations to apply to Adiabatic expansion When a gas is flowing
gases
through a primary element the change in pressure
Gases are compressible, while liquids, mostly, are takes place too rapidly for the gas to absorb heat
not. If the gas under consideration can be from its surroundings. When it expands owing to
regarded as an ideal gas (most gases are ideal the reduction in pressure it does work, so that if it