Dimensionless parameters and their significance

Dimensional analysis of a turbomachine (variables: $D$, $N$, $\rho$, $\mu$, $Q$, $gH$, $P$) gives dimensionless groups that allow geometrically similar machines to be compared and model test results to be scaled up.

Use in practice

1. Similarity: two machines are dynamically similar if $\phi$ and $\psi$ (and $Re$, $Ma$) are equal, and their efficiencies are then equal.
2. Scaling laws: for the same fluid and constant $\phi$: $Q \propto N D^3$, $H \propto N^2 D^2$, $P \propto N^3 D^5$.
3. Selection of type: a high $N_s$ means large flow and low head, which needs an axial machine; a low $N_s$ means small flow and high head, which needs a radial one.
4. Performance prediction: a single non-dimensional curve of $\psi$ and $\eta$ against $\phi$ represents a whole family of machines.

For compressible flow, the mass flow parameter $\dot m \sqrt{T_{01}}/p_{01}$ and the speed parameter $N/\sqrt{T_{01}}$ are used in place of $\phi$ and $N$.