Question

The reaction \( A \rightarrow {products} \) with reaction rate, \( (-r_A) = k C_A^3 \), occurs in an isothermal PFR operating at steady state. The conversion (X) at two axial locations (1 and 2) of the PFR is shown in the figure.
\includegraphics[width=0.5\linewidth]{58.png}
The value of \( l_1/l_2 \) is __________ (rounded off to 2 decimal places).

Hint:

To solve for the length ratio in a PFR, use the relationship between the conversion and concentration in the reactor, considering the reaction rate.

Answer 1

Correct Answer: 0.22

The reaction rate is given by: \[ (-r_A) = k C_A^3 \] The conversion \( X \) is related to the concentration of \( A \) through the following relationship for a PFR: \[ \frac{dX}{dx} = k C_A^3 \] For a PFR operating at steady state, the relationship between the length of the reactor and the conversion can be expressed as: \[ \frac{X_2 - X_1}{l_2 - l_1} = \frac{k}{3} \left( X_1^3 - X_2^3 \right) \] Given: - \( X_1 = 0.3 \), - \( X_2 = 0.6 \). Substitute the known values and solve for \( l_1/l_2 \): \[ \frac{l_1}{l_2} = \frac{3}{k} \left( X_1^3 - X_2^3 \right) \] Substitute the values of \( X_1 \) and \( X_2 \): \[ \frac{l_1}{l_2} = \frac{3}{k} \left( 0.3^3 - 0.6^3 \right) \] \[ \frac{l_1}{l_2} = \frac{3}{k} \left( 0.027 - 0.216 \right) \] \[ \frac{l_1}{l_2} = \frac{3}{k} \times (-0.189) \] Therefore, the ratio \( \frac{l_1}{l_2} \) can be calculated, and the answer can be rounded to the nearest decimal place.