Question

A lagged copper rod of uniform cross sectional area has lenght of 60cm the free ends of the rod are maintained at 20°C and 0°C respectively at a steady state.calculate the temperature at a point 20cm from the high temperature end

Answer 1

Let's consider the heat equation in a form:

$$u_{xx}=u_{t}$$

where $u(x,t)$ is the temperature that depends on position $x$ and time $t$. Subscripts denote partial derivatives. At the steady state the temperature $u = T(x)$ should not depend on the time, thus: $T_t = 0$. Solving the remaining equation, find:

$$T_{xx} = 0\\ T_x = C_1\\ T(x) = C_1x+C_2$$

where $C_1,C_2$ are some constants. Substituting the boundary value on the one end, obtain:

$$T(0) = C_1\cdot 0+ C_2 =0\\ C_2 =0$$

On another end:

$$T(60) =C_1\cdot 60+0 =20\\ C_1 = 1/3$$

Thus, the equation is:

$$T(x) = \dfrac{x}{3}$$

And the temperature at $x = 20cm$ is:

$$T(20) = \dfrac{20}{3} \approx 6.67\degree C$$

Answer. $6.67\degree C$.