Question

Consider the flowsheet in the figure for manufacturing C via the reaction \( A + B \rightarrow C \) in an isothermal CSTR. The split in the separator is perfect so that the recycle stream is free of C and the product stream is pure C. Let \( x_i \) denote the mole fraction of species \( i \) (where \( i = A, B, C \)) in the CSTR, which is operated in excess B with \( x_B/x_A = 4 \). The reaction is first-order in A with the reaction rate \( (-r_A) = k x_A \), where \( k_x = 5.0 \, {kmol}/({m}^3 \cdot {h}) \). The reactor volume \( V \) in \( {m}^3 \) is to be optimized to minimize the cost objective \( J = V + 0.25 R \), where \( R \) is the recycle rate in \( {kmol/h} \). For a product rate \( P = 100 \, {kmol/h} \), the optimum value of \( V \) is ____ \( {m}^3 \) (rounded off to the nearest integer). Given: \[ \frac{d}{dz} \left( \frac{z}{(1-z)^2} \right) = \frac{1}{(1 - 2z)^2} \]

Hint:

When optimizing reactor volumes, use the material balance and the rate of reaction to express the recycle rate in terms of the volume. This will allow you to minimize the cost objective.

Answer 1

Step 1: Write the rate equation for the CSTR.
We know the rate of reaction in the CSTR is given by: \[ r_A = k_x x_A \] The molar flow rate of \( A \) in the reactor inlet stream is: \[ F_A = F \cdot x_A \] The outlet flow rate of \( A \) is \( F_A - r_A \cdot V \), where \( F_A \) is the inlet flow rate of A, and \( r_A \) is the rate of consumption of A.
Step 2: Derive the optimization objective.
The cost objective \( J \) is given as: \[ J = V + 0.25 R \] Where \( R \) is the recycle rate. To minimize \( J \), we must express \( R \) in terms of \( V \). The relationship between \( R \) and \( V \) is derived from the material balance and reaction kinetics. By using the provided differential equation, we can calculate the optimal reactor volume. 
Step 3: Solve for the optimum volume.
After solving the equations and optimizing the objective, we find the optimum value of \( V \) to be: \[ \boxed{150 \, {m}^3} \] Final Answer: The optimum value of \( V \) is \( \boxed{150} \, {m}^3 \).