. A steel bar 10.0 cm long is welded end-to-end to a copper bar 20.0 cm long. Each bar has a square cross-section of 2.0 cm side. The free end of the steel bar is maintained at 100 ?C and the free end of the copper bar is maintained at 0 ?C. Find the temperature at the junction of the two bars and the total rate of heat flow. The thermal conductivities of steel and copper are 50.2 and 385.0 W/m.K respectively.
no you don't want to start a fire
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Assume 1D heat flow, then break the problem into two: each bar has a dirichlet boundary condition on one side and a matching neumann condition on the other. A simplifying trick is knowing/noticing that in the steady state, the temperature versus span must be two straight lines (with different slopes) that meet at the material boundary. Recognizing this form of the solution reduces the differential equation problem into a simple algebraic problem. q_steel q_copper k_steel (T_left - T_m) / L_steel k_copper (T_m - T_right) / L_copper solve for T_m
