Question

Two rods whose lengths are `l_1` and `l_2` and heat conductivity coefficient `x_1` and `x_2` are placed end to end. Find the heat conductivity coefficient of a uniform rod of length `l_1 + l_2` whose conductivity is the same as that of the system of these two rods. The lateral surfaces of the rods are assumed to be thermally insulated.

Answer 1

We have
`k_1 (T_1 - T)/(l_1) = k_2 (T - T_2)/(l_2) = k(T_1 - T_2)/(l_1 + l_2)`
or using the previous result
`(k_1)/(l_1) (T_1 -((k_1 T_1)/(l_1) + (k_2 T_2)/(l_2))/((k_1)/(l_1) + (k_1)/(l_2))) = k (T_1 - T_2)/(l_1 + l_2)`
or `(k_1)/(l_1) ((k_2)/(l_2)(T_1 - T_2))/((k_1)/(l_1) + (k_2)/(l_2)) = k (T_1 - T_2)/(l_2 + l_2) ` or `k = (l_1 + l_2)/((l_1)/(k_1) + (l_2)/(k_2))`.