Question

A 5.0 m wide rectangular channel carries a discharge of 10 m³/s at a depth of 1.5 m under uniform flow. To produce critical flow conditions without affecting the upstream conditions, the channel bottom elevation should be raised (in m) by ........ (rounded off to 2 decimal places).

Hint:

For critical flow conditions, use the critical depth formula to find the required depth, and then adjust the channel bottom elevation accordingly.

Answer 1

The discharge \( Q \) is given as 10 m³/s, and the depth \( y_1 \) is 1.5 m. The flow velocity \( v_1 \) is calculated using the discharge equation: \[ v_1 = \frac{Q}{B y_1} = \frac{10}{5 \times 1.5} = 1.33 \, {m/s}. \] For critical flow conditions, the critical depth \( y_c \) is calculated by: \[ y_c = \left( \frac{q^2}{g} \right)^{1/3} = \left( \frac{2^2}{9.81} \right)^{1/3} = 0.74 \, {m}. \] Now, to raise the bottom of the channel to produce critical flow conditions, we use the equation: \[ E_1 = E_2 + \Delta z_c \quad {and} \quad E_2 = E_c = \frac{3}{2} y_c. \] Substituting the values: \[ 1.5 + 1.33^2 / 2 \times 9.81 = \frac{3}{2} \times 0.74 + \Delta z \] \[ \Delta z = 0.48 \, {m}. \] Thus, the channel bottom elevation should be raised by \( \boxed{0.48} \) m (rounded to 2 decimal places).