The most general Pump model is given as a function of the mass flowrate on the intake $\begin{array}{l}p_{\rm in}\end{array}$ and discharge pressure $\begin{array}{l}p_{\rm out}\end{array}$ :

(1)	$\begin{array}{l}\displaystyle \dot m = M(p_{\rm out}, p_{\rm in})\end{array}$

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

(2)	$\begin{array}{l}\displaystyle q = q_{in} = \frac{\dot m}{\rho(p_{in})} = \frac{M(p_{\rm out}, p_{\rm in})}{\rho(p_{in})}\end{array}$

where

$\begin{array}{l}\rho(p)\end{array}$

fluid density as a function of fluid pressure $\begin{array}{l}p\end{array}$

The electrical power consumption $\begin{array}{l}\displaystyle W = \frac{dE}{dt}\end{array}$ is given by:

(3)	$\begin{array}{l}\displaystyle W = \eta(q) \cdot q \cdot (p_{\rm out}-p_{\rm in})\end{array}$

where

$\begin{array}{l}\eta\end{array}$

pump efficiency

In most practical cases the pump model (2) depends on the difference between intake and discharge pressure $\begin{array}{l}p_{\rm out} - p_{\rm in}\end{array}$ and called Pump Characteristic Curve (see Fig. 1):

(4)	$\begin{array}{l}\displaystyle q = q(p_{\rm out} - p_{\rm in})\end{array}$

Fig. 1. Example of Pump Characteristic Curve.

A popular pump proxy model is given by the quadratic equation with 3 inputs ( $\begin{array}{l}\{ q_{'rf max}, \delta p_{\rm max}, k_f \}\end{array}$ ):

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

(6)	$\begin{array}{l}\displaystyle p_{\rm out} = p_{\rm in} + \delta p_{\rm max} \cdot \left[ 1+ (k_f -1 ) \cdot \frac{q}{q_{\rm max}} - k_f \cdot \left( \frac{q}{q_{\rm max}} \right)^2 \right ]\end{array}$

(7)	$\begin{array}{l}\displaystyle \eta(q) = 4 \, \eta_{\rm max} \cdot q/q_{\rm max} \cdot ( 1 - q/q_{\rm max})\end{array}$

where

$\begin{array}{l}\delta p_{\rm max}\end{array}$	maximum pressure gain that pump can exert over the input pressure $\begin{array}{l}p_{\rm in}\end{array}$
$\begin{array}{l}q_{\rm max}\end{array}$	maximum flowrate that pump can produce
$\begin{array}{l}k_f \in [0,1]\end{array}$	total hydraulic pump friction (dimensionless)
$\begin{array}{l}\eta\end{array}$	pump efficiency
$\begin{array}{l}\eta_{\rm max}\end{array}$	maximum pump efficiency

Real pumps have non-constant $\begin{array}{l}k_f =k_f(q)\end{array}$ friction coefficient which often modelled as a 3^rd order polynomial and the overall real-pump model taking 6-inputs.

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

(8)	$\begin{array}{l}\displaystyle q_{\rm max} = q^_{\rm max} \cdot \frac{f}{f^}\end{array}$

(9)	$\begin{array}{l}\displaystyle \delta p_{\rm max} = \delta p^_{\rm max} \cdot \left( \frac{f}{f^} \right)^2\end{array}$

where

$\begin{array}{l}q_{\rm max}\end{array}$	maximum intake flowrate at the working frequency $\begin{array}{l}f\end{array}$	$\begin{array}{l}\delta p_{\rm max}\end{array}$	maximum pressure gain at the working frequency $\begin{array}{l}f\end{array}$	$\begin{array}{l}f\end{array}$	adjusted working frequency
$\begin{array}{l}q^*_{\rm max}\end{array}$	maximum intake flowrate at the nominal frequency $\begin{array}{l}f^*\end{array}$	$\begin{array}{l}\delta p^*_{\rm max}\end{array}$	maximum pressure gain at the nominal frequency $\begin{array}{l}f^*\end{array}$	$\begin{array}{l}f^*\end{array}$	nominal frequency

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References