Chemical Bioreactor Dissolved Oxygen Depletion (Chemical & Environmental Engineering)

In municipal wastewater treatment plants and pharmaceutical fermentation processes, aerobic microorganisms consume dissolved oxygen (DO) to break down organic contaminants. If dissolved oxygen drops too low, aerobic digestion halts; if aeration blowers run excessively, operational power consumption spikes.

An environmental engineer models the dissolved oxygen concentration $C$ (in milligrams per liter, $\text{mg/L}$) inside a continuous stirred-tank bioreactor over time $t$ (in hours, with $t \ge 0$) following a sudden organic shock loading:

1. Using the Quotient Rule, determine the first derivative $\frac{dC}{dt}$ and find all critical numbers in the domain $[0, \infty)$.
2. Construct a 1D derivative sign chart to determine the intervals on which dissolved oxygen is increasing and decreasing.
3. Apply the First Derivative Test and the Single Critical Point Rule to determine the absolute maximum dissolved oxygen concentration and the exact time $t$ at which this peak occurs.
4. Determine the long-term asymptotic behavior $\lim_{t \to \infty} C(t)$ and explain why plant automation systems must restart supplemental aeration turbines once $t$ exceeds the peak window.