GATE 2025 CH – Question 35
The residence-time distribution (RTD) function of a reactor (in min$^{-1}$) is
$E(t) = \begin{cases} 1 - 2t, & t \le 0.5 \text{ min} \\ 0, & t > 0.5 \text{ min} \end{cases}$
The mean residence time of the reactor is ____ min (rounded off to 2 decimal places).
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Correct answer: 0.16 to 0.18
Explanation
The mean residence time is $\bar{t} = \frac{\int_0^{0.5} tE\,dt}{\int_0^{0.5} E\,dt}$. The integral $\int_0^{0.5}(1 - 2t)dt = 0.5 - 0.25 = 0.25$ and $\int_0^{0.5}t(1 - 2t)dt = \frac{0.25}{2} - \frac{2 \times 0.125}{3} = 0.125 - 0.0833 = 0.0417$. So $\bar{t} = \frac{0.0417}{0.25} = 0.17$ min.