Question

A laterally insulated bar of length 10cm and constant cross-sectional area lem, of density 10.6gm/cm, thermal conductivity 1.04cal/(cm Sec °C), and specific heat of 0.056cal/gmºC), has initial temperature 100°C. Then at some instance, say t=0, the temperature at x = L is suddenly changed to 0°C and kept at 0°C, where as the temperature at x = 0 is kept at 100°C, find the temperature at the middle of the bar

at t = 1,2,3, 10, 50 sec.

Answer 1

$Q=\frac{\lambda}{L}(t\degree-0\degree)St,$

$Q=cm(100\degree-t\degree)=c\rho\frac L2S(100\degree-t\degree),$

$\frac{\lambda t\degree t}{L}=\frac{c\rho L}{2}(100\degree-t\degree),$

$\frac{100\degree}{t\degree}-1=\frac{2\lambda t}{c\rho L^2},$

$t\degree=100\degree\cdot(\frac{2\lambda t}{c\rho L^2}+1),$

$\def\arraystretch{1.5} \begin{array}{c|c} t, s & t ,\degree C\\ \hline 1 & 97 \\ \hdashline 2 & 93 \\ \hdashline 3 & 90 \\ \hdashline10 & 74 \\ \hdashline50 & 36 \\ \end{array}$