Chapter 30, Problem 13P

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

can be written as

$\frac{{\partial }^{2}u}{\partial {\bar{x}}^2}=\frac{\partial u}{\partial \bar{t}}$

wherenondimensional space, time, and temperature are defined as

$\bar{x}=\frac{x}{L}\bar{t}=\frac{T}{\left(\rho C{L}^2/k\right)}u=\frac{T-T_0}{T_L-T_0}$

where $L=$ the rod length $k=$ thermal conductivity of the rod material, $\rho =$ density, $C=$ specific heat, $T_0=$ temperature at $x=0$, and $T_L=$ temperature at $x=L$. This makes for the following boundary and initial conditions:

Boundary conditions	$u\left(0,t\right)=0$	$u\left(0,\bar{t}\right)=0$
Initial conditions	$u\left(x,0\right)=0$	$0\le \bar{x}\le 1$

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 program to obtain the solution. Increase the value of $\Delta \bar{t}$ by 10% for each time step to more quickly obtain the steady-state solution, and select values of $\Delta \bar{x}$ and $\Delta \bar{t}$ for good accuracy. Plot the nondimensional temperature versus nondimensional length for various values of nondimensional times.