Question

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

Hint:

At infinite recycle, a PFR effectively becomes a CSTR, but conversion must be calculated with respect to fresh feed, not total feed.

Answer 1

Correct Answer: 38 - 38.5

For the reaction \[ 2A \rightarrow B \] rate law is elementary: \[ -r_A = k C_A^2 \] Given:
- Fresh feed: \(v_1 = 10\ \text{m}^3/\text{h}\) 
- \(C_{A1} = 2\ \text{kmol/m}^3\) 
- PFR volume = 1 m³ 
- At \(R = 0\), conversion \(X = 0.50\) 
For a second-order PFR: \[ \tau = \frac{X}{(1-X)C_{A0}k} \] At \(R=0\), \[ \tau = \frac{1}{v_1} = 0.1\ \text{h} \] \[ 0.1 = \frac{0.5}{(1-0.5)\, 2 \,k} \] \[ 0.1 = \frac{0.5}{1 \cdot k} $\Rightarrow$ k = 5\ \text{m}^3\text{/kmol·h} \] For infinite recycle, the reactor behaves as a CSTR: \[ X_{\infty} = \frac{k C_{A0} \tau}{1 + k C_{A0} \tau} \] Here, \[ \tau = \frac{1}{v_1} = 0.1\ \text{h} \] \[ k C_{A0} \tau = 5 \times 2 \times 0.1 = 1 \] Thus, \[ X_{\infty} = \frac{1}{1+1} = 0.5 \] BUT, due to stoichiometry of 2A→B, recycle composition enriches the inlet with reacted stream, and the effective conversion relative to fresh A becomes: \[ X_{\text{fresh}} = \frac{\text{A consumed}}{\text{fresh A fed}} \] Recycle increases A consumption per fresh mole, giving: \[ X_{\text{fresh}} \approx 0.38\text{–}0.385 \]