 Pump power affinity law

If pump power depends only on fluid mass density $\rho$, impeller diameter $D$ and angular velocity $\omega$, dimensional analysis fixes $P=C_P\rho {\omega }^3{D}^5$, where $C_P$ is dimensionless. Comparing pumps with the same $C_P$ gives the displayed scaling. Geometrical or flow-regime changes can change $C_P$; the law is not a universal comparison of arbitrary pumps.