Orifice plate

An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a ""). 
Description.
An orifice plate is a thin plate with a hole in it, which is usually placed in a pipe. When a fluid (whether liquid or gaseous) passes through the orifice, its pressure builds up slightly upstream of the orifice but as the fluid is forced to converge to pass through the hole, the velocity increases and the fluid pressure decreases. A little downstream of the orifice the flow reaches its point of maximum convergence, the "vena contracta" (see drawing to the right) where the velocity reaches its maximum and the pressure reaches its minimum. Beyond that, the flow expands, the velocity falls and the pressure increases. By measuring the difference in fluid pressure across tappings upstream and downstream of the plate, the flow rate can be obtained from Bernoulli's equation using coefficients established from extensive research.
In general, the mass flow rate "formula_1" measured in kg/s across an orifice can be described as
Solving for formula_3:
formula_4
and:
formula_5
The above expression for formula_3 gives the theoretical volume flow rate. Introducing the beta factor formula_7 as well as the discharge coefficient formula_8:
formula_9
And finally introducing the meter coefficient formula_10 which is defined as formula_11 to obtain the final equation for the volumetric flow of the fluid through the orifice which accounts for irreversible losses:
formula_12
Multiplying by the density of the fluid to obtain the equation for the mass flow rate at any section in the pipe:
formula_13
Deriving the above equations used the cross-section of the orifice opening and is not as realistic as using the minimum cross-section at the vena contracta. In addition, frictional losses may not be negligible and viscosity and turbulence effects may be present. For that reason, the coefficient of discharge formula_8 is introduced. Methods exist for determining the coefficient of discharge as a function of the Reynolds number.
The parameter formula_15 is often referred to as the "velocity of approach factor" and multiplying the coefficient of discharge by that parameter (as was done above) produces the flow coefficient formula_10. Methods also exist for determining the flow coefficient as a function of the beta function formula_17 and the location of the downstream pressure sensing tap. For rough approximations, the flow coefficient may be assumed to be between 0.60 and 0.75. For a first approximation, a flow coefficient of 0.62 can be used as this approximates to fully developed flow.
An orifice only works well when supplied with a fully developed flow profile. This is achieved by a long upstream length (20 to 40 pipe diameters, depending on Reynolds number) or the use of a flow conditioner. Orifice plates are small and inexpensive but do not recover the pressure drop as well as a venturi, nozzle, or venturi-nozzle does. Venturis also require much less straight pipe upstream. A venturi meter is more efficient, but usually more expensive and less accurate (unless calibrated in a laboratory) than an orifice plate.
Compressible flow.
In general, equation (2) is applicable only for incompressible flows. It can be modified by introducing the expansibility factor, (also called the expansion factor) formula_18 to account for the compressibility of gasses.
formula_19
formula_18 is 1.0 for incompressible fluids and it can be calculated for compressible gases using empirically determined formulae as shown below in computation.
For smaller values of β (such as restriction plates with β less than 0.25 and discharge from tanks), if the fluid is compressible, the rate of flow depends on whether the flow has become choked. If it is, then the flow may be calculated as shown at choked flow (although the flow of real gases through thin-plate orifices never becomes fully choked By using a mechanical energy balance, compressible fluid flow in un-choked conditions may be calculated as:
formula_21
or
formula_22
Under choked flow conditions, the fluid flow rate becomes:
formula_23
or
formula_24
Computation according to ISO 5167.
Flow rates through an orifice plate can be calculated without specifically calibrating the individual flowmeter so long as the construction and installation of the device complies with the stipulations of the relevant standard or handbook. The calculation takes account of the fluid and fluid conditions, the pipe size, the orifice size and the measured differential pressure; it also takes account of the coefficient of discharge of the orifice plate, which depends upon the orifice type and the positions of the pressure tappings. With local pressure tappings (corner, flange and D+D/2), sharp-edged orifices have coefficients around 0.6 to 0.63, while the coefficients for conical entrance plates are in the range 0.73 to 0.734 and for quarter-circle plates 0.77 to 0.85. The coefficients of sharp-edged orifices vary more with fluids and flow rates than the coefficients of conical-entrance and quarter-circle plates, especially at low flows and high viscosities.
For compressible flows such as flows of gases or steam, an "expansibility factor" or "expansion factor" is also calculated. This factor is primarily a function of the ratio of the measured differential pressure to the fluid pressure and so can vary significantly as the flow rate varies, especially at high differential pressures and low static pressures.
The equations provided in American and European national and industry standards and the various coefficients used to differ from each other even to the extent of using different combinations of correction factors, but many are now closely aligned and give identical results; in particular, they use the same "Reader-Harris/Gallagher (1998)" equation for the coefficient of discharge for sharp-edged orifice plates. The equations below largely follow the notation of the international standard ISO 5167 and use SI units.
Volume flow rate:
Mass flow rate:
Coefficient of discharge.
Coefficient of discharge for sharp-edged orifice plates with corner, flange or D and D/2 tappings and no drain or vent hole (Reader-Harris/Gallagher equation):
Expansibility factor.
Expansibility factor, also called expansion factor, for sharp-edged orifice plates with corner, flange or D and D/2 tappings:
Overall pressure loss.
The overall pressure loss caused by an orifice plate is less than the differential pressure measured across tappings near the plate. For sharp-edged plates such as corner, flange or D and D/2 tappings, it can be approximated by the equation
or