1. (20 pts) A plane wall of thickness L and thermal conductivity k undergoes a chemical reaction, causing volumetric internal heating at a rate proportional to temperature: q'" = a T.

The left side of the wall (at x = 0) is held at constant temperature To and the right-hand side (x=L) is externally cooled via convection with a fluid at temperature To and heat transfer coefficient h. Follow the next steps sequentially so you can successfully solve this problem: a. (3 pts) Write the FULL heat conduction equation in Cartesian coordinates and simplify it for 1-D, steady state, and constant thermal conductivity. Make sure to include the Include the q"" = a T term. b. (3 pts) Write the boundary conditions in mathematical form. Remember to make sure that your convection boundary condition makes physical and mathematical sense. c. (3 pts) Solve the simplified conduction equation from (a), but don't apply the boundary conditions. NOTE: Direct integration cannot be applied here. You can consult ODE tables if needed. d. (3 pts) Solve the constants in your temperature profile by applying the boundary conditions. e. (3 pts) Derive expressions for the heat flux at x = 0 and x=L.