Centrifugal pump head curve

A centrifugal pump's developed pressure follows its quadratic head-vs-flow curve — the flow-dependent pressure rise the pressure-driven solver uses.

Governing equation

$Q$

Volumetric flow rate (m³ s⁻¹) — the unknown.

$\Delta P$

Pressure rise across the pump (Pa).

$\rho$

Fluid density (kg m⁻³).

$g$

Gravitational acceleration (m s⁻²).

$H_0$

Shut-off head (m).

$a$

Head-curve curvature (m per (m³ s⁻¹)²).

Derivation

1. 1
   $$H(Q) = H_0 - a\,Q^2$$

   A centrifugal pump's characteristic curve is well approximated near the operating point by a quadratic falling from the shut-off head.

2. 2
   $$\Delta P = \rho g\,H$$

   Convert developed head to a pressure rise through the fluid's specific weight.

3. 3
   $$F(Q) = \Delta P - \rho g\,(H_0 - a\,Q^2) = 0$$

   At the operating point the imposed pressure rise equals the curve value; flow is the unknown the surrounding network resistance closes.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Perry's Chemical Engineers' Handbook, 8th ed., §10 (transport & storage of fluids).
• Karassik et al., Pump Handbook, 4th ed., Ch. 2.