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GATE 2023 CE (CE2) – Question 44

Hydraulics · Channel hydraulics: specific energy, critical flow, uniform flow, hydraulic jump · 2 marks · Multiple choice

The critical flow condition in a channel is given by ______. [Note: $\alpha$ – kinetic energy correction factor; Q – discharge; $A_c$ – cross-sectional area of flow at critical flow condition; $T_c$ – top width of flow at critical flow condition; g – acceleration due to gravity]

  1. $\alpha Q^2/g=A_c^3/T_c$
  2. $\alpha Q/g=A_c^3/T_c^2$
  3. $\alpha Q^2/g=A_c^3/T_c^2$
  4. $\alpha Q/g=A_c^3/T_c$

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Show answer and explanation

Correct answer: (A) $\alpha Q^2/g=A_c^3/T_c$

Explanation

**Step 1: specific energy** at a section with constant kinetic-energy correction factor $\alpha$:
$$E=y+\frac{\alpha Q^2}{2gA^2}.$$

**Step 2: critical flow means $E$ is a minimum**, so $dE/dy=0$. With $dA=T\,dy$:
$$\frac{dE}{dy}=1-\frac{\alpha Q^2}{gA^3}\frac{dA}{dy}=1-\frac{\alpha Q^2T}{gA^3}=0 .$$

**Step 3: rearrange** at the critical section:
$$\frac{\alpha Q^2}{g}=\frac{A_c^3}{T_c}.$$

Option **A**. (For $\alpha=1$ this is the familiar $Q^2/g=A^3/T$, or $Fr=1$.)