Energy coefficient for compound channel

Definition

The energy (kinetic energy) correction factor for a section of area $A$ and mean velocity $V=Q/A$ is

$$\alpha=\frac{\int v^3\,dA}{V^3A}$$

For a compound channel with subsections of area $A_i$ and mean velocity $V_i$:

$$\alpha=\frac{\sum V_i^3A_i}{V^3A}$$

Proof

1. $Q=\sum Q_i$, $A=\sum A_i$, $V=Q/A$.
2. The conveyance of each part is $K_i$, with $Q_i=K_i\sqrt{S_f}$ (all subsections share the same energy slope $S_f$). Then

$$Q=\sqrt{S_f}\sum K_i,\qquad V_i=\frac{Q_i}{A_i}=\frac{K_i\sqrt{S_f}}{A_i}$$

3. Numerator:

$$\sum V_i^3A_i=\sum\frac{K_i^3S_f^{3/2}}{A_i^3}A_i=S_f^{3/2}\sum\frac{K_i^3}{A_i^2}$$

4. Denominator:

$$V^3A=\frac{Q^3}{A^2}=\frac{S_f^{3/2}\left(\sum K_i\right)^3}{\left(\sum A_i\right)^2}$$

5. Dividing,

$$\boxed{\alpha=\frac{\sum\left(\dfrac{K_i^3}{A_i^2}\right)\left(\sum A_i\right)^2}{\left(\sum K_i\right)^3}}$$

Hence proved. For one uniform section $\alpha=1$; for a compound section $\alpha>1$ (typically 1.1 to 2 or more), and it must be used in $E=y+\alpha V^2/2g$.