GATE 2022 CH – Question 65

Chemical Reaction Engineering · Non-ideal reactors and residence time distribution · 2 marks · Numerical answer

An elementary irreversible liquid-phase reaction, $2P \xrightarrow{k} Q$, where the rate constant $k = 2$ L mol$^{-1}$ min$^{-1}$, takes place in an isothermal non-ideal reactor. The E-curve in a tracer experiment is shown in the figure. Pure $P$ (2 mol L$^{-1}$) is fed to the reactor. Using the segregated model, the percentage conversion of $P$ at the exit of the reactor is __________% (rounded off to the nearest integer).

$E_t$ in min⁻¹ against $t$ in min. The shaded region rises linearly from $E = 1$ at $t = 0$ to a higher value at $t = 0.5$ min, and the curve is zero afterwards.

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Correct answer: 50

Explanation

The curve is a straight line, $E = 1 + ct$ for $0 \le t \le 0.5$. The area must be 1: $0.5 + c\frac{0.25}{2} = 1$ gives $c = 4$, so $E = 1 + 4t$ (the height at 0.5 min is 3, as in the figure). The batch conversion of the second-order reaction is $X(t) = \frac{kC_0t}{1 + kC_0t} = \frac{4t}{1 + 4t}$. For the segregated model, $X = \int_0^{0.5}E(t)X(t)dt = \int_0^{0.5}(1 + 4t)\frac{4t}{1 + 4t}dt = \int_0^{0.5}4t\,dt = 0.5$, which is 50%.