Problem $1.$ Temperature on skin surface with tumor

A cancerous tumor grows rapidly and the increased metabolism generates a large amount of heat. In a highly simplified analysis, consider the skin as a $2$ mm thick slab with one side $($deep skin$)$ having heat transfer by convection with blood at a temperature of $37C$ and heat transfer coefficient of $h=10\frac{W}{m^2}C.$ The surface of the skin is covered with an insulating material that has a temperature sensor$.($See Figure below.$)$ The skin is first allowed to reach steady state before reading its surface temperature. The average thermal conductivity of the $2-$mm skin layer is $0\mathrm{.}50\frac{W}{m}C$ and the rate of volumetric heat generation in it due to the cancer is $500\frac{W}{m^3},$ which is uniform throughout the skin layer.

Draw the Schematic $+$ Data for the problem.

Starting from the most general partial differential equation for heat transfer, simplify it based on the following assumptions:

A$.$ Steady$-$state

B$.$ Conduction heat transfer

C$.1-$D heat transfer $($x$-$direction only$)$

D$.$ Constant properties

E$.$ Uniform volumetric heat generation within the system

Write the two boundary conditions applicable to the system.

Obtain the equation giving the temperature at any distance $x$ from the skin surface. Also determine the expressions for the constants $($use symbols only, don't plug in values.$)$

Calculate the values of the constants $($don$\text{'}$t forget their units$)$

What is the temperature at the surface of the skin?