The relationship between outflow Q in m3 s−1 and storage S in m3 for an emergency spillway in a reservoir is Q = S/4000. Inflow, outflow and storage are assumed to be zero at time t = 0. If the inflow rate is 300 m3 s−1 at the end of t = 3 hours, the outflow rate in m3 s−1 is
GATE AG - 2012,
Answer: C
The keyed outflow is 172.34 m3 s−1.
The key follows the storage (continuity) equation applied over a single 3-hour routing interval, with the inflow rising linearly from 0 to 300 m3 s−1 and the outflow averaged over the interval. With \(\Delta t = 10800\) s and \(S = 4000\,Q\):
\[ \begin{aligned} \tfrac{I_{1}+I_{2}}{2}\Delta t - \tfrac{Q_{1}+Q_{2}}{2}\Delta t &= S_{2} - S_{1} \\ \tfrac{300}{2}(10800) - \tfrac{Q_{2}}{2}(10800) &= 4000\,Q_{2} \\ 1620000 &= 9400\,Q_{2} \\ Q_{2} &= 172.34 \end{aligned} \]
Splitting the 3 hours into shorter steps, or solving the differential equation exactly, gives a somewhat higher value of about 196 m3 s−1, which is not offered; the paper's options are built on the one-step calculation. The other values do not come from either form of the equation.
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