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