Question

The conversion for a first-order reaction in a PFR compared to CSTR (same volume) is:

• A. Same
• B. Lower in PFR
• C. Higher in PFR
• D. Dependent only on temperature

Hint:

For all reaction orders \( n \gt 0 \), a PFR is more efficient than a CSTR, meaning:
\[ V_{\text{PFR}} \lt V_{\text{CSTR}} \quad \text{for the same conversion} \]
This general rule is a core concept in reactor selection.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to compare the fractional conversion achieved in an ideal Plug Flow Reactor (PFR) with that in a Continuous Stirred Tank Reactor (CSTR) of the same volume, for a first-order reaction operating under identical feed conditions.

Step 2: Key Formula or Approach:
For a first-order reaction (\( -r_A = kC_A \)) in constant-volume systems:
1. PFR Performance Equation:
\[ \tau = -\frac{\ln(1 - X_A)}{k} \quad \implies \quad X_{A, \text{PFR}} = 1 - e^{-k\tau} \]
2. CSTR Performance Equation:
\[ \tau = \frac{X_A}{k(1 - X_A)} \quad \implies \quad X_{A, \text{CSTR}} = \frac{k\tau}{1 + k\tau} \]

Step 3: Detailed Explanation:

• Let's evaluate the conversion for a sample value of \( k\tau = 1 \):
For PFR:
\[ X_{A, \text{PFR}} = 1 - e^{-1} \approx 1 - 0.368 = 0.632 \quad (63.2\%) \]
For CSTR:
\[ X_{A, \text{CSTR}} = \frac{1}{1 + 1} = 0.50 \quad (50.0\%) \]

• Why the PFR achieves higher conversion: A CSTR is completely mixed, meaning the reactant concentration drops immediately to its low outlet value upon entering the reactor.
Because the reaction rate is proportional to concentration (\( -r_A = kC_A \)), a CSTR operates entirely at this lower rate.
In contrast, the reactant concentration in a PFR decreases gradually along the length of the reactor.
This means the average reactant concentration—and thus the average reaction rate—is higher in a PFR, resulting in a higher conversion for the same volume.


Step 4: Final Answer:
For any positive-order reaction (including first-order), the conversion is higher in a PFR than in a CSTR of the same volume.