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:

V(h) = \pi R h^2 - \frac{1}{3}\pi h^3 = 12\pi h^2 - \frac{1}{3}\pi h^3

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

\frac{dV}{dt} = +15.0\text{ m}^3/\text{min}

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 $\frac{dh}{dt}$ .
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 $h = 4.0\text{ meters}$ .
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.