GATE 2021 CH – Question 61
The following homogeneous, irreversible reaction involving ideal gases, $\mathrm{A\to B+C}$, $(-r_A)=0.5C_A$ (mol L⁻¹ s⁻¹), is carried out in a steady state ideal plug flow reactor (PFR) operating at isothermal and isobaric conditions. The feed stream consists of pure A, entering at 2 m/s. In order to achieve 50% conversion of A, the required length of the PFR is _____ meter (round off to 2 decimal places).
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Correct answer: 3.53 to 3.57
Explanation
The reaction increases total gas moles, so at constant temperature and pressure with pure-A feed, $v/v_0=1+X$ and $C_A=C_{A0}(1-X)/(1+X)$.
For a constant-cross-section PFR, $dL=(u_0/k)(1+X)/(1-X)\,dX$. Integrating to $X=0.5$ gives
$$L=\frac{2}{0.5}\left[-X-2\ln(1-X)\right]_0^{0.5}=4(2\ln2-0.5)=3.5451774\ \mathrm m.$$
Rounded, **L = 3.55 m**. Assuming constant gas velocity would omit the expansion and underestimate the required length.