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.
The value of \( l_1/l_2 \) is ________ (rounded off to 2 decimal places).
Show 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.
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.
