Question

At high Thiele modulus ($\phi \gg 1$), the effectiveness factor ($\eta$) approaches:

• A. 1
• B. 0
• C. 3/$\phi$ (for sphere)
• D. $\phi$

Hint:

The general asymptotic relation for catalyst pellets of any geometry at high Thiele modulus is:
\[ \eta \approx \frac{1}{\phi_{\text{p}}} \]
where \( \phi_{\text{p}} \) is defined using a characteristic length of \( V_p/A_p \). For a sphere, \( V_p/A_p = R/3 \), which yields \( \eta \approx 3/\phi \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the limiting value of the internal effectiveness factor (\( \eta \)) for a spherical catalyst pellet at high values of the Thiele modulus (\( \phi \gg 1 \)), which represents the regime of strong pore diffusion resistance.

Step 2: Key Formula or Approach:
For a first-order reaction inside a spherical catalyst pellet, the analytical solution for the effectiveness factor is:
\[ \eta = \frac{3}{\phi} \cdot \left[ \frac{1}{\tanh \phi} - \frac{1}{\phi} \right] \]
where \( \phi \) is the Thiele modulus for a sphere.

Step 3: Detailed Explanation:

• Asymptotic Analysis for Large \( \phi \):
When the Thiele modulus is very large (\( \phi \gg 1 \)), the hyperbolic tangent function approaches unity:
\[ \tanh \phi \to 1 \]

• Substitute this limit into the analytical expression:
\[ \eta \approx \frac{3}{\phi} \cdot \left[ \frac{1}{1} - \frac{1}{\phi} \right] = \frac{3}{\phi} \cdot \left[ 1 - \frac{1}{\phi} \right] \]

• Since \( \phi \) is very large, the term \( \frac{1}{\phi} \) is small compared to 1 and can be neglected:
\[ \eta \approx \frac{3}{\phi} \]

• Physical Meaning: This limit confirms that under strong pore resistance, the reaction rate is restricted to a narrow zone near the outer surface of the sphere, reducing the overall catalyst utilization.


Step 4: Final Answer:
At high Thiele modulus, the effectiveness factor for a spherical catalyst pellet approaches \( 3/\phi \).