volumetric flow of the fluid through the orifice which accounts for irreversible losses: 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 multiplying 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, 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) to account for the compressibility of gasses. 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,