Chemical Reactor Thermal Runaway Safeguards (Chemical Engineering)

In chemical process engineering, continuous stirred-tank reactors (CSTR) carrying out exothermic autocatalytic polymerizations exhibit nonlinear temperature trajectories. As reactants convert, heat generation accelerates exponentially before flattening as reagents deplete. If the heat-generation rate exceeds cooling jacket dissipation capacity, thermal runaway occurs, causing vessel over-pressurization and explosion.

A process safety engineer is modeling the temperature curve $T(t)$ (in $^\circ\text{C}$) of a batch polymerization reactor over time $t$ (in hours, $t \ge 0$). The temperature response is modeled by the sigmoidal logistic function:

1. Determine the initial reactor core temperature at startup ($t = 0$).
2. Evaluate the horizontal asymptote of the reaction temperature by calculating $\lim_{t \to \infty} T(t)$. Interpret what this asymptotic temperature represents physically in terms of reagent exhaustion.
3. Compute the reaction heating rate $\frac{dT}{dt}$ as a function of time. Show that $\frac{dT}{dt} > 0$ for all $t \ge 0$, proving that core temperature rises monotonically.
4. Use the Second Derivative Test to locate the time $t_{\text{crit}}$ where the temperature curve has an inflection point ($\frac{d^2T}{dt^2} = 0$).
5. In chemical safety design, explain why the inflection point $t_{\text{crit}}$ marks the instant of maximum thermal hazard, dictating the required emergency coolant injection pump capacity.