Cryogenic Propellant Tank Bulkhead Surface Area Minimization (Mechanical & Aerospace Engineering)

In launch vehicle propulsion design and aerospace structural engineering, orbital payload capacity is dictated by the Tsiolkovsky rocket equation. Every kilogram of inert dry mass eliminated from an upper-stage propellant tank translates directly into additional payload delivery capability to low Earth orbit (LEO). Propellant tanks are pressurized vessels, and minimizing the structural surface area of the tank directly minimizes the mass of the specialized aluminum-lithium (Al-Li) alloy skin and cryogenic spray-on foam insulation (SOFI).

An aerospace engineering team is designing a cylindrical liquid oxygen (LOX) propellant tank for an orbital insertion stage. Mission delta-$V$ requirements mandate that the tank hold an internal cryogenic propellant volume of exactly $54\pi\text{ m}^3$ (approximately $170\text{ m}^3$).

                         Top Bulkhead (Radius r)
                            +---------------+
                           /                 \
                          +-------------------+
                          |                   |
                          |                   |
                          |   Liquid Oxygen   |
                          |   Volume = 54*pi  |  Height h
                          |                   |
                          |                   |
                          +-------------------+
                           \                 /
                            +---------------+
                        Bottom Bulkhead (Radius r)

The tank is modeled as a closed right circular cylinder with radius $r$ and barrel section height $h$. The structural skin has an areal mass density of $\rho_A = 15\text{ kg/m}^2$.

1. State the objective function for total tank surface area $S(r, h)$ and the volume constraint equation.
2. Use the Applied Optimization Modeling Framework to eliminate $h$ and express surface area as a single-variable function $S(r)$.
3. Determine the Context-Driven Domain for radius $r$.
4. Calculate the optimal tank radius $r$ and barrel height $h$ that minimize total structural surface area.
5. Apply the Second Derivative Test to verify that your critical point produces an absolute minimum surface area, and demonstrate that the optimal height equals the tank diameter ($h = 2r$), confirming the general theorem from Optimal Aspect Ratios in Geometric Containers.
6. Compute the resulting minimum structural dry mass of the tank skin.