GATE 2025 CH – Question 58

Chemical Reaction Engineering · Single and multiple reactions in ideal reactors · 2 marks · Numerical answer

The reaction $A \to products$ with reaction rate, $(-r_A) = kC_A^3$, occurs in an isothermal PFR operating at steady state. The conversion ($X$) at two axial locations (1 and 2) of the PFR is shown in the figure.

The value of $l_1/l_2$ is ____ (rounded off to 2 decimal places).

a horizontal PFR. Location 1 is at conversion $X_1 = 0.3$, at a distance $l_1$ from the inlet. Location 2 is at conversion $X_2 = 0.6$, a further distance $l_2$ along the reactor.

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Correct answer: 0.24 to 0.26

Explanation

For a constant density PFR, the length (volume) needed between two conversions is proportional to $\int\frac{dX}{(1 - X)^3}$ with the same constant, since $C_A = C_{A0}(1 - X)$ and the cross section is fixed. The integral is $\int_{X_a}^{X_b}\frac{dX}{(1 - X)^3} = \frac{1}{2}\left[\frac{1}{(1 - X_b)^2} - \frac{1}{(1 - X_a)^2}\right]$. From the inlet to $X = 0.3$: $\frac{1}{2}\left[\frac{1}{0.49} - 1\right] = 0.5204$. From 0.3 to 0.6: $\frac{1}{2}\left[\frac{1}{0.16} - \frac{1}{0.49}\right] = 2.1046$. So $\frac{l_1}{l_2} = \frac{0.5204}{2.1046} = 0.25$.