The nondimensional form for the transient heat conduction in an insulated rod (Eq. 30.1) can be written as

∂²u/∂x² = ∂u/∂t

where nondimensional space, time, and temperature are defined as

x = x/L             l = T/(pCL²/k)              u = T – T_o/T_L - T_o

where L = the rod length, k = thermal conductivity of the rod material, p = density. C = specific heat. T_o = temperature at x = 0, and T_L = temperature at x = L. This makes for the following boundary and initial conditions:

Boundary conditions              u(0, l) = 0        u(1, l) = 0

Initial conditions                     u(x, 0) = 0        0 ≤ x ≤ 1

Solve this nondimensional equation for the temperature distribution using finite-difference methods and a second-order accurate Crank-Nicolson formulation to integrate-in time. Write a computer program to obtain the solution. Increase the value of Δt by 10% for each time step to more quickly obtain the steady-state solution, and select values of Δx and Δt for good accuracy. Plot the nondimensional temperature versus nondimensional length for various values of nondimensional times.