GATE 2023 CE (CE1) – Question 59
A 12-hour storm occurs over a catchment and results in a direct runoff depth of 100 mm. The time-distribution of the rainfall intensity is shown in the figure (not to scale). The $\phi$-index of the storm is (in mm, rounded off to two decimal places) ____________________________.
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Correct answer: 3.58 to 3.62
Explanation
The total rainfall is the area of the graph: $\frac{1}{2} \times 4 \times 20 + 2 \times 20 + \frac{1}{2} \times 6 \times 20 = 40 + 40 + 60 = 140$ mm, so the loss is $140 - 100 = 40$ mm. Let $u = 20 - \phi$. The part of the graph above the level $\phi$ is the area $\frac{u^2}{10}$ (rising part, slope 5 per hour) $+ 2u$ (flat part) $+ \frac{0.3u^2}{2}$ (falling part, slope $\frac{10}{3}$ per hour) $= 0.25u^2 + 2u$. Setting this equal to the runoff of 100: $u^2 + 8u - 400 = 0$, so $u = 16.40$ and $\phi = 3.60$ mm/h.