One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/(m·K). For these conditions, the temperature distribution has the form, T(x) = a + bx +cx^2. The surface at x = 0 has a temperature of T(0) = To = 120°C and experiences convection with a fluid for which T? = 20°C and h = 500 W/(m^2·K). The surface at x = L is well insulated.
(a) Applying an overall energy balance to the wall, calculate the internal energy generation rate, E?.
(b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution.