Consider an orifice plate placed in a pipe flow as shown in Fig.3.25 . We assume that the thickness of the plate is small in comparison to the pipe diameter. Let the orifice be sharp edged. The effect of a rounded plate is a matter of detail and will not be considered here.
We can deduce the flow rate through the pipe by measuring the pressure difference upstream of the nozzle and at the orifice.
We make a few assumptions about the flow as follows -
a) Continuity Equation.
b) Bernoulli Equation
The Bernoulli Equation gives,
Solving for V2 we have,
Consequently, the mass flow rate becomes,
The above equation gives the mass flow rate through the pipe in terms of the pressure drop and the areas. The equation gives only a theoretical value. In order to obtain a more realistic value one need to substitute the actual area at the minimum cross section or the Vena Contracta. This is not easy to measure. In addition losses may not be negligible as we have assumed. Extent of losses is a function of the Reynolds number. In practice, a Coefficient of Discharge is defined such that
Further if we define a ratio of diameters such that
Sometimes the ratio is referred to as Velocity of Approach Factor. Again it is usual to combine this and the Discharge Coefficient to define a Flow Coefficient given by
Consequently the mass flow rate is given by,
Thus the mass flow rate for a pipe can be calculated with the knowledge of pressure drop, the orifice diameter and the coefficient K. Extensive data exists in handbooks on the coefficient K.
Pressure drop is usually measured by using a manometer as shown in Fig. 3.25. Now the pressure drop is obtained as h, the height of a liquid column (which may be mercury). Accordingly the alternate form of Eqn.3.87 is
(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney