Question

The first-order irreversible liquid phase reaction \(A \to B\) occurs inside a constant volume \(V\) isothermal CSTR with the initial steady-state conditions shown in the figure. The gain, in kmol/m³·h, of the transfer function relating the reactor effluent \(A\) concentration \(c_A\) to the inlet flow rate \(F\) is:

 

• A. 1.2
• B. 0.4
• C. 0.6
• D. 0.8

Hint:

The gain in CSTR problems can be derived by understanding the steady-state concentration relationship and applying the flow rate changes accordingly.

Answer 1

The Correct Option is C

Step 1: General Relationship for CSTR.
For a first-order reaction in a CSTR, the general equation is: \[ \frac{dC_A}{dt} = \frac{F}{V}(C_{A0} - C_A) - k C_A \] At steady-state (\( \frac{dC_A}{dt} = 0 \)): \[ 0 = \frac{F}{V}(C_{A0} - C_A) - k C_A \] Rearranging: \[ k C_A = \frac{F}{V}(C_{A0} - C_A) \] Solving for \(C_A\): \[ C_A = \frac{C_{A0}}{1 + \frac{V k}{F}} \] Step 2: Deriving the Gain.
The gain is the change in \(C_A\) with respect to the change in flow rate \(F\): \[ \frac{d C_A}{dF} = \frac{d}{dF} \left( \frac{C_{A0}}{1 + \frac{V k}{F}} \right) \] Using the given values and solving for the gain, the final answer is determined.