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GATE 2022 CH – Question 63

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

An elementary irreversible liquid-phase reaction, $2A \to B$, is carried out under isothermal conditions in a 1 m$^3$ ideal plug flow reactor (PFR) as shown in the figure. The volumetric flow rate of fresh $A$, $v_1 = 10$ m$^3$ h$^{-1}$, and its concentration $C_{A1} = 2$ kmol m$^{-3}$. For a recycle ratio $R = 0$, the conversion of $A$ at location 2 with respect to the fresh feed (location 1) is 50%. For $R \to \infty$, the corresponding conversion of $A$ is _________% (rounded off to one decimal place).

a PFR of 1 m³ with the fresh feed $v_1 = 10$ m³/h, $C_{A1} = 2$ kmol/m³ at location 1, the exit at location 2 and a recycle stream $v_R$ from the exit back to the inlet, with $R = v_R/v_2$.

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Correct answer: 38.01 to 38.39

Explanation

The space time is $\tau = \frac{1}{10} = 0.1$ h. For the second-order reaction, with $R = 0$ the PFR gives $\frac{1}{C_A} - \frac{1}{C_{A0}} = k\tau$. At 50% conversion $C_A = 1$, so $1 - 0.5 = 0.1k$ and $k = 5$ (m³ kmol⁻¹ h⁻¹, for the rate $kC_A^2$). As $R \to \infty$ the PFR behaves like a CSTR (it is perfectly mixed), with the same $\tau = 0.1$ h on the fresh feed: $C_{A0} - C_A = k\tau C_A^2$, that is $2 - C_A = 0.5C_A^2$, so $C_A^2 + 2C_A - 4 = 0$ and $C_A = -1 + \sqrt{5} = 1.236$. The conversion is $\frac{2 - 1.236}{2} = 38.2\%$.