Safe Venting Limits for Chemical Refineries (Chemical Engineering)

At a specialized chemical processing and fuel refining facility at Apex Industrial Park northeast of Las Vegas, volatile liquid hydrocarbon feedstock is transferred into a spherical pressurized storage dome with a fixed internal radius of $R = 12.0\text{ meters}$.

As fluid accumulates in the bottom spherical cap of the tank, the liquid volume $V$ is constrained by the fluid depth $h$ (measured vertically from the bottom pole of the sphere, $0 \le h \le 2R$) according to the spherical cap volume formula:

An automated high-pressure pipeline injects hydrocarbon feedstock into the vessel at a constant volumetric charging rate of:

1. Apply implicit differentiation with respect to time $t$ to the spherical cap volume equation, treating liquid depth $h$ as a dynamic function of time $h(t)$, and isolate the rate of liquid level rise $\frac{dh}{dt}$.
2. Determine the exact rate of liquid level rise $\frac{dh}{dt}$ in $\text{m/min}$ and $\text{cm/min}$ at the instant the liquid depth reaches $h = 4.0\text{ meters}$.
3. Compare the rate of liquid level rise when the tank is near the bottom ($h = 2.0\text{ m}$) versus when the tank is half-full ($h = 12.0\text{ m}$), and explain how plant safety control microcontrollers utilize this related rates calculation to prevent catastrophic vapor over-pressurization and emergency venting flare activation.