Aerospace Structural Thermal Isosurfaces

In aerospace and mechanical propulsion engineering, rocket combustion chambers and hypersonic re-entry heat shields must withstand severe thermal gradients exceeding thousands of degrees. Because thermal stress and material yielding are volumetric phenomena occurring inside solid metal structures, aerospace engineers construct 3D thermal isosurfaces (level surfaces of temperature) to map heat flow and optimize regenerative cooling channels.

An aerospace propulsion team at UNLV is testing a prototype liquid rocket engine nozzle injector. During a steady-state hot-fire simulation, the internal temperature field $T$ (measured in hundreds of degrees Celsius, $^{\circ}\text{C}$) within the solid copper-alloy injector flange is modeled by the three-variable quadratic function:

1. Determine the mathematical equations for the level surfaces (isothermal shells) at temperature thresholds $c = 1$ ($100^{\circ}\text{C}$) and $c = 4$ ($400^{\circ}\text{C}$). Describe their 3D geometric shape and state their radii.
2. Determine the mathematical equation for the 2D cross-sectional temperature slice passing through the principal horizontal plane $z = 0$. Describe the resulting 2D profile.
3. Meaning & Real-Life Application: Explain what these level surfaces and cross-sections tell propulsion engineers about thermal barrier coating boundaries and structural thermal cracking risks.