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title Derivation

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Assume steady-state, incompressibleinviscidlaminar flow in a horizontal pipe (no change in elevation) with negligible frictional losses with incharge fluid velocity

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and discharge velocity
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body v_{out}
at the orifice exit.

The mass conservation (equivalent to continuity equation):

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\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

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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:

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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

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:

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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

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:

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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

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:

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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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