Question

Consider second order kinetics ($r = −kC^2$) under steady state condition. The ratio of volume of a complete mixed reactor (CMR) to that of a plug flow reactor (PFR) to achieve 90% reduction in the concentration is _______ .
Inlet concentrations in both the reactors are same.

• A. 10,0
• B. 1.0
• C. 0.1
• D. 2.3

Answer 1

The Correct Option is A

Step 1: Define the reaction kinetics. The reaction follows second-order kinetics: \[ r_c = -kC^2 \] where \( k \) is the reaction rate constant, and \( C \) is the concentration. Step 2: Expression for reactor volumes. 1. Plug Flow Reactor (PFR): For a second-order reaction in a PFR, the volume \( V_{\text{PFR}} \) is given by: \[ V_{\text{PFR}} = \frac{1}{k} \left( \frac{1}{C_{\text{out}}} - \frac{1}{C_{\text{in}}} \right) \] 2. Complete Mixed Reactor (CMR): For a second-order reaction in a CMR, the volume \( V_{\text{CMR}} \) is given by: \[ V_{\text{CMR}} = \frac{1}{k} \frac{1}{C_{\text{out}}} \left( C_{\text{in}} - C_{\text{out}} \right) \] Step 3: Substitute for 90\% reduction in concentration. For both reactors: \[ C_{\text{out}} = 0.1 \, C_{\text{in}} \] 1. For \( V_{\text{PFR}} \): \[ V_{\text{PFR}} = \frac{1}{k} \left( \frac{1}{0.1C_{\text{in}}} - \frac{1}{C_{\text{in}}} \right) \] Simplify: \[ V_{\text{PFR}} = \frac{1}{k} \left( \frac{10}{C_{\text{in}}} - \frac{1}{C_{\text{in}}} \right) = \frac{1}{k} \frac{9}{C_{\text{in}}} \] 2. For \( V_{\text{CMR}} \): \[ V_{\text{CMR}} = \frac{1}{k} \frac{1}{0.1C_{\text{in}}} \left( C_{\text{in}} - 0.1C_{\text{in}} \right) \] Simplify: \[ V_{\text{CMR}} = \frac{1}{k} \frac{1}{0.1C_{\text{in}}} \times 0.9C_{\text{in}} = \frac{1}{k} \frac{9}{0.1C_{\text{in}}} = \frac{1}{k} \frac{90}{C_{\text{in}}} \] Step 4: Ratio of volumes. The ratio of \( V_{\text{CMR}} \) to \( V_{\text{PFR}} \) is: \[ \frac{V_{\text{CMR}}}{V_{\text{PFR}}} = \frac{\frac{1}{k} \frac{90}{C_{\text{in}}}}{\frac{1}{k} \frac{9}{C_{\text{in}}}} = \frac{90}{9} = 10 \]