A thin plate is initially at a uniform temperature of 200Ā°C. At a certain time t = 0, the temperature of the east side of the plate is suddenly reduced to 0Ā°C. The other surface is insulated. Use the explicit finite volume method in conjunction with a suitable time step size to calculate the transient temperature distribution of the slab and compare it with the analytical solution at time (i) t = 40 s, (ii) t = 80 s, and (iii) t = 120 s. Recalculate the numerical solution using a time step size equal to the limit given by (8.13) for t = 40 s and compare the results with the analytical solution. The data are: plate thickness L = 2 cm, thermal conductivity k = 10 W/mĀ·K, and Ļc = 10 Ć 10^6 J/m^3Ā·K.
Solve the problem using the Crank-Nicolson scheme (Īø = 1/2). Write the discretized equations for the interior nodes. Modify Eq. 8.9 and obtain the discretized equations for the boundary nodes: