Calculate heat transfer rates for a plane wall with variable thermal conductivity, given convection at two surfaces.

here are the solutions to the heat transfer problems.

3) Plane wall with variable thermal conductivity

Step 1: Define the heat transfer rates.
The heat transfer rate per unit area ($q = Q/A$) is constant through the wall and its surfaces.
For convection at surface 1:

$$q = h_1(T_{\infty 1} - T_1) = 250(130 - T_1) \quad (1)$$

For convection at surface 2:

$$q = h_2(T_2 - T_{\infty 2}) = 500(T_2 - 30) \quad (2)$$

For conduction through the