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@wikipedia
Relates pressure drop
\Delta p = p_{in} - p_{out} on the choke with the flowrate through the choke
q arising from fluid friction with choke elements (ISO5167):
(1) |
\Delta p = p_{in} - p_{out} = \frac{ \rho \cdot (1- \beta^4)}{0.125 \, \pi^2 \, d^4 \, C_d^2 \, \epsilon^2} \cdot q^2 |
where
Derivation
Assume steady-state, incompressible, inviscid, laminar flow in a horizontal pipe (no change in elevation) with negligible frictional losses with incharge fluid velocity
v_{in} and discharge velocity
v_{out} at the orifice exit.
The mass conservation (equivalent to continuity equation):
(2) |
\rho \, q = \rho \, v_{in} \, A_{in} = \rho \, v_{out} \, A_{out} \Leftrightarrow v_{in} = \frac{q}{A_{in}}, \, v_{in} = \frac{q}{A_{out}} |
where
(3) |
A_{in} = \frac{\pi \, D^2}{4} \, A_{out} = \frac{\pi \, d^2}{4} |
Bernoulli's equation reduces to an equation relating the conservation of energy between two points on the same streamline:
(4) |
p_{in} + \frac{1}{2} \rho v^2_{in} = p_{out} + \frac{1}{2} \rho v^2_{out} |
(5) |
\Delta p = p_{in} - p_{out} = \frac{1}{2} \rho v^2_{out} - \frac{1}{2} \rho \, v^2_{in} = \frac{1}{2} \rho \, v^2_{out} \cdot \left[ 1 - \frac{v^2_{in}}{v^2_{out}} \right] |
(6) |
\Delta p = \frac{1}{2} \rho \, \left( \frac{q}{A_{in}} \right)^2 \cdot \left[ 1 - \frac{A^2_{out}}{A^2_{in}} \right] = \frac{\rho \, q^2}{2 \, A^2_{in}} \cdot \left[ 1 - \frac{d^4}{D^4} \right] = \frac{\rho \, q^2}{2 \, A^2_{in}} \cdot \left[ 1 - \beta^4 \right] = \frac{\rho \, q^2}{2 \, (\frac{\pi \, d^2}{4})^2} \cdot \left[ 1 - \beta^4 \right] = \frac{8 \, \rho \, q^2}{\pi^2 \, d^4 } \cdot \left[ 1 - \beta^4 \right] |
See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation (PFS)
[ Orifice Plate Discharge Coefficient ] [ Orifice Plate Expansion Factor @ model ]
Pipeline Engineering / Pipeline / Choke
[ Euler equation ] [ Water Pipe Flow @model ]