Question

A microbial culture with a specific growth rate of $\mu_1$ (per hour) is being grown in a continuous reactor at steady state with hydraulic retention time (HRT) of 24 hours. The reactor is subjected to a perturbation by reducing the HRT to 12 hours. The reactor recovers and comes to a new steady state with a specific growth rate of $\mu_2$. Which one of the following statements is correct?

• A. $\mu_1 = \mu_2$
• B. $\mu_1 = 0.5\mu_2$
• C. $\mu_2 = 0.5\mu_1$
• D. $\mu_2 = e^{0.5}\mu_1$

Hint:

At steady state in a CSTR, the specific growth rate always equals the dilution rate: $\mu = 1/\text{HRT}$.

Answer 1

The Correct Option is C

In a continuous stirred-tank reactor (CSTR) at steady state, the specific growth rate equals the dilution rate $D$: \[ \mu = D = \frac{1}{\text{HRT}}. \] Step 1: Compute initial growth rate.
Initial HRT = 24 h: \[ \mu_1 = \frac{1}{24}. \] Step 2: Compute new growth rate.
After perturbation, HRT = 12 h: \[ \mu_2 = \frac{1}{12}. \] Step 3: Compare $\mu_1$ and $\mu_2$. \[ \mu_2 = 2\mu_1 \quad \Rightarrow \quad \mu_1 = 0.5\mu_2. \] Thus the correct relation is option (B).
Final Answer: $\mu_1 = 0.5\mu_2$