**Discharge Coefficient Diagram and Conceptual Flow Lines for an Orifice in a Pipe**

**Discharge Coefficient Equation**

Valid for 0.4 ≤ C ≤ 0.8 and 0.2 ≤ β ≤ 0.7

**Derivation of Discharge Coefficient Equation**

Steady flow of an incompressible fluid through an orifice based on a discharge coefficient:

Pressure loss from location 1 to 3 in the pipe written in terms of discharge coefficient:

Pressure loss from 1 to 3 in the pipe due to the orifice written in terms of minor loss coefficient:

Also:

The goal is to solve the above equations for K. Begin by equating discharge coefficient equation (2) to (3):

Solve for K in terms of discharge coefficient:

Many pipe flow analysis programs do not readily implement orifice discharge coefficients because they use K's instead. Hence, having a value for K based on the orifice discharge coefficient is useful for flow calculators.

Symbols and units for discharge coefficient equations: A_{o}=orifice area (m^{2}), A_{p}=pipe area (m^{2}), d=orifice diameter (m), D=pipe inside diameter (m), P_{1}-P_{2}=differential pressure (N/m^{2}), P_{1}-P_{3}=non-recoverable pressure difference (N/m^{2}), Q=flow rate (m^{3}/s), V_{o}=velocity through orifice (m/s), V_{p}=velocity in pipe (m/s), ρ=fluid mass density (kg/m^{3}).

Reference: LMNO Engineering - Large diameter orifice flow meter calculation for liquids

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