Chapter 28, Problem 48P

The temperature distribution in a tapered conical cooling fin (Fig. P28.48) is described by the following differential equation, which has been nondimensionalized

$\frac{d^{2}u}{d{x}^2}+\left(\frac{2}{x}\right)\left(\frac{du}{dx}-pu\right)=0$

where $u=$ temperature $\left(0\le u\le 1\right),x=$ axial distance $\left(0\le x\le 1\right)$, and p is a nondimensional parameter that describes the heat transfer and geometry

$p=\frac{hL}{k}\sqrt{1+\frac{4}{2m^2}}$

where $h=$ a heat transfer coefficient, $k=$ thermal conductivity, $L=$ the length or height of the cone, and $m=$ the slope of the cone wall. The equation has the boundary conditions

$u\left(x=0\right)=0u\left(x=1\right)=1$

Solve this equation for the temperature distribution using finite difference methods. Use second-order accurate finite difference analogues for the derivatives. Write a computer program to obtain the solution and plot temperature versus axial distance for various values of $p=10,20,50,\text{ and }100$.

FIGURE P28.48