Orifice Plate - Incompressible Flow Through An Orifice

Incompressible Flow Through An Orifice

By assuming steady-state, incompressible (constant fluid density), inviscid, laminar flow in a horizontal pipe (no change in elevation) with negligible frictional losses, Bernoulli's equation reduces to an equation relating the conservation of energy between two points on the same streamline:

or:

By continuity equation:

or and :

Solving for :

and:

The above expression for gives the theoretical volume flow rate. Introducing the beta factor as well as the coefficient of discharge :

And finally introducing the meter coefficient which is defined as to obtain the final equation for the volumetric flow of the fluid through the orifice:

Multiplying by the density of the fluid to obtain the equation for the mass flow rate at any section in the pipe:

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 is introduced. Methods exist for determining the coefficient of discharge as a function of the Reynolds number.

The parameter is often referred to as the velocity of approach factor and dividing the coefficient of discharge by that parameter (as was done above) produces the flow coefficient . Methods also exist for determining the flow coefficient as a function of the beta function 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 does. If space permits, a venturi meter is more efficient than an orifice plate.