Minimum hump height with energy loss

Given: $Q = 12\ \text{m}^3/\text{s}$, $b = 5\ \text{m}$, $y_1 = 1.5\ \text{m}$. Head loss over the hump $h_L = V_1^2/2g$. For the minimum hump height the flow is just critical on the crest.

Upstream:

$$V_1 = \frac{12}{5 \times 1.5} = 1.6\ \text{m/s},\quad \frac{V_1^2}{2g} = 0.1305\ \text{m},\quad E_1 = 1.5 + 0.1305 = 1.6305\ \text{m}$$

Critical conditions on the crest:

$$q = \frac{12}{5} = 2.4\ \text{m}^2/\text{s},\quad y_c = \left(\frac{2.4^2}{9.81}\right)^{1/3} = 0.837\ \text{m},\quad E_c = 1.5\,y_c = 1.256\ \text{m}$$

Energy equation between upstream and the crest:

$$E_1 = \Delta z + E_c + h_L$$ $$\begin{aligned} \Delta z &= E_1 - E_c - h_L \\ &= 1.6305 - 1.2561 - 0.1305 = 0.244\ \text{m} \end{aligned}$$

Answer: Minimum hump height $\approx 0.24\ \text{m}$.