Simple pump energy calculator

Calculations used

Navier–Stokes (viscous shear torque)

Idealized torque for viscous shear in a thin gap between radii \(r_1\) and \(r_2\):

$$ \tau \;=\; \pi\,\mu\,\frac{\omega}{h}\,\bigl(r_2^4 - r_1^4\bigr) $$

Here \(\mu\) is dynamic viscosity, \(\omega\) motor angular speed (rad/s), \(h\) the gap (m).

Flow & pump-housing volume

The housing is sized to hold one switching period worth of flow:

$$ V_{\text{cyl}} = Q \, t_{\text{sw}} $$

With a cylindrical chamber \( V = \pi r^{2} L = \frac{\pi}{4} D^{2} L \). If we choose \( L = D \) (stroke equals diameter):

$$ V = \frac{\pi}{4} D^{3} \quad\Rightarrow\quad D = \sqrt[3]{\frac{4V}{\pi}}, \qquad L = D, \qquad r = \frac{D}{2} = \sqrt[3]{\frac{V}{2\pi}} $$

Pipe sizing from flow

Choose a design velocity \(v_{\text{pipe}}\) (e.g. 0.6 m/s). From continuity \(Q=A\,v\):

$$ A_{\text{pipe}} = \frac{Q}{v_{\text{pipe}}}, \qquad d_{\text{req}} \;=\; \sqrt{\frac{4A}{\pi}} \;=\; \sqrt{\frac{4Q}{\pi\,v_{\text{pipe}}}} $$

Then round up to the nearest metric DN size:

$$ d_{\text{DN}} \;=\; \mathrm{ceil}_{\text{DN}}\bigl(d_{\text{req}}\bigr), \qquad v_{\text{corr}} \;=\; \frac{Q}{A(d_{\text{DN}})} $$

Backpressure to hydraulic head

Convert backpressure \(\Delta p\) to head \(H\) (water):

$$ H \;=\; \frac{\Delta p}{\rho g}, \qquad \Delta p[\mathrm{Pa}] = \text{mbar}\times 100 $$

Tip speed and RPM from head

Using the pump head coefficient \(\psi\) (0.6–1.0 typical):

$$ U_{\text{tip}} \;=\; \sqrt{\frac{g\,H}{\psi}}, \qquad \mathrm{RPM} \;=\; \frac{60\,U_{\text{tip}}}{\pi\,D_{\text{imp}}} $$

Velocity-head sanity check

Available head should at least cover exit velocity head:

$$ H_{\min} \;=\; \frac{v_{\text{pipe}}^{2}}{2g}, \qquad \text{passes} \;=\; \bigl(H \ge H_{\min}\bigr), \qquad \text{margin} \;=\; H - H_{\min} $$

Valve drive: torque, gearing, power

Viscous torque (above), geared by ratio \(G\) (e.g., 16 or 32):

$$ \tau_{\text{geared}} \;=\; \frac{\tau}{G} $$

Mechanical power and electrical input (efficiency \(\eta\)):

$$ P_{\text{mech}} \;=\; \tau_{\text{geared}}\,\omega, \qquad P_{\text{elec}} \;=\; \frac{P_{\text{mech}}}{\eta} $$

Duty-cycle multiplier (energy only during off-time)

If energy is used only during off-time:

$$ m \;=\; \frac{T_{\text{off}}}{T_{\text{on}} + T_{\text{off}}}, \qquad \overline{P}_{\text{valve}} \;=\; m \; P_{\text{elec}} $$

Pump hydraulic & electrical power

At operating point \((Q,H)\):

$$ P_{\text{hyd}} \;=\; \rho g Q H, \qquad P_{\text{elec,pump}} \;=\; \frac{P_{\text{hyd}}}{\eta_{\text{pump}}} $$

Notes

• Rounding up to DN reduces velocity \(v_{\text{corr}}\) and friction.
• The velocity-head check excludes friction and fittings since this pump's head calculations are solely based on friction and fittings: \( \Delta p = f \frac{L}{D}\frac{\rho v^2}{2} + \sum K \frac{\rho v^2}{2} \).
• Head coefficient \(\psi\) captures slip and geometry; refine with calibration or pump curves.