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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

\rho

fluid  density

d

orifice diameter

D

pipe diameter 

\beta = \frac{d}{D}

orifice narrowing ratio

C_d

\epsilon

expansion factor



Assume steady-state, incompressibleinviscidlaminar 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_{out} = \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]
(7) \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 \Delta p:

(8) 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  C_d:

(9) 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 \epsilon:

(10) q = \epsilon \, C_d \frac{\pi \, d^2}{4} \, \sqrt{\frac{2 \, \Delta p}{\rho \, (1 - \beta^4)}}

Alternative forms


(11) \Delta p = \frac{1-\beta^4}{С_d^2 \, \epsilon^2} \cdot \frac{\dot m^2}{2 \, \rho \, A_d^2}
(12) \Delta p = \frac{1-\beta^4}{С_d^2 \, \epsilon^2} \cdot \frac{j_m^2}{2 \, \rho}
(13) \dot m = \sqrt{ 2 \rho \ \Delta p } \cdot \frac{\epsilon \, C_d \, A_d}{\sqrt{1-\beta^4}}

where

\dot m

fluid mass flowrate 

j_m\rho

orifice mass flux

A_d = 0.25 \, \pi \, d^2

orifice cross-section area

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 ]





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