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\dot m  = M(p_{\rm out}, p_{\rm in})

It's often presented in terms of intake volumetric flowrate:

q = q_{in} = \frac{\dot m}{\rho(p_{in})}  = \frac{M(p_{\rm out}, p_{\rm in})}{\rho(p_{in})}

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In most practical cases  the pump model 

q = q(p_{\rm out} - p_{\rm in})

A popular popular pump proxy model is given by the quadratic equation with 3 inputs (

q = \frac{q_{\rm max}}{2 \cdot k_f} \cdot \left[ -1 + k_f +  \sqrt{ (1 + k_f)^2 - 4 \cdot k_f \cdot (p_{\rm out}- p_{\rm in})/\delta p_{\rm max}) \ } \, \right]

p_{\rm out} = p_{\rm in} +  \delta p_{\rm max} \cdot \left[ 1+ 
(k_f -1 ) \cdot \frac{q}{q_{\rm max}} - k_f \cdot \left( \frac{q}{q_{\rm max}} \right)^2
 \right ]

\eta(q) = 4 \, \eta_{\rm max} \cdot q/q_{\rm max} \cdot ( 1 -  q/q_{\rm max})

where

Real pumps have non-constant

Many The plunger pump and centrifugal pumps can be normally adjusted by the variation of the working frequency  frequency which affects the maximum the maximum pump flowrate and maximum pressure gain as:

q_{\rm max} = q^*_{\rm max}  \cdot \frac{f}{f^*}

\delta p_{\rm max} = \delta p^*_{\rm max}  \cdot \left( \frac{f}{f^*} \right)^2

where

See also

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Natural Science / Engineering / Device / Pump

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