@wikipedia
Relates pressure drop
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| body | \Delta p = p_{in} - p_{out} |
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on the
choke with the flowrate through the choke
arising from fluid friction with
choke elements (
ISO5167):
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\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
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| Assume steady-state, incompressible, inviscid, laminar flow in a horizontal pipe (no change in elevation) with negligible frictional losses with incharge fluid velocity and discharge velocity at the orifice exit. The mass conservation (equivalent to continuity equation): | LaTeX Math Block |
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| \rho \, q = \rho \, v_{in} \, A_{in} = \rho \, v_{out} \, A_{out} \Leftrightarrow v_{in} = \frac{q}{A_{in}}, \, v_{out} = \frac{q}{A_{out}} |
where | LaTeX Math Block |
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| 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: | LaTeX Math Block |
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| p_{in} + \frac{1}{2} \rho v^2_{in} = p_{out} + \frac{1}{2} \rho v^2_{out} |
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| \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] |
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| \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] |
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| \Delta p = \frac{\rho \, q^2}{2 \, \left( \frac{\pi}{4} d^2 \right)^2} \cdot \left[ 1 - \beta^4 \right] = \frac{8 \, \rho \, q^2}{\pi^2 \, d^4 } \cdot \left[ 1 - \beta^4 \right] |
The above can rewritten as flowrate estimation with a given pressure drop :| LaTeX Math Block |
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| q = \frac{\pi \, d^2}{4} \, \sqrt{\frac{2 \, \Delta p}{\rho \, (1 - \beta^4)}} |
The actual rate through the orifice with account for the choke/orifice geometry, friction and viscous forces is corrected by introducing the discharge coefficient :| LaTeX Math Block |
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| q = C_d \frac{\pi \, d^2}{4} \, \sqrt{\frac{2 \, \Delta p}{\rho \, (1 - \beta^4)}} |
and correction for fluid compressibility is given by expansion factor :| LaTeX Math Block |
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| q = \epsilon \, C_d \frac{\pi \, d^2}{4} \, \sqrt{\frac{2 \, \Delta p}{\rho \, (1 - \beta^4)}} |
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| \Delta p = \frac{1-\beta^4}{С_d^2 \, \epsilon^2} \cdot \frac{\dot m^2}{2 \, \rho \, A_d^2} |
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| \Delta p = \frac{1-\beta^4}{С_d^2 \, \epsilon^2} \cdot \frac{j_m^2}{2 \, \rho} |
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| \dot m = \sqrt{ 2 \rho \ \Delta p } \cdot \frac{\epsilon \, C_d \, A_d}{\sqrt{1-\beta^4}} |
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where
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 ]
