Question 10.5.1: Find the temperature u(x, t) at any time in a metal rod 50 c......

The temperature in the rod satisfies the heat conduction problem (1), (3), (4) with L = 50 and f(x) = 20 for 0 < x < 50. Thus, from equation (19),

u(x,t)=\sum_{n=1}^{\infty}c_{n}u_{n}(x,t)=\sum_{n=1}^{\infty}c_{n}e^{-n^{2}\pi^{2}\alpha^{2}t/L^{2}}\sin\Bigl({\frac{n\pi x}{L}}\Bigr), (19)

the solution is

u(x,t)=\sum_{n=1}^{\infty}c_{n}e^{-n^{2}\pi^{2}\alpha^{2}t/2500}\sin\left({\frac{n\pi x}{50}}\right), (22)

where, from equation (21),

c_{n}={\frac{2}{L}}\int_{0}^{L}f(x)\sin\Bigl({\frac{n\pi x}{L}}\Bigr)d x. (21)

c_{n}=\frac{4}{5}\int_{0}^{50}\sin\left(\frac{n\pi x}{50}\right)d x=\frac{40}{n\pi}(1-\cos n\pi)=\begin{cases}\frac{80}{n\pi}, & \text{n odd;} \\0, & \text{n even.}\end{cases} (23)

Finally, by substituting for c_{n} in equation (22), we obtain

u(x,t)=\frac{80}{\pi}\sum_{n=1,3,5,…}^{\infty}\frac{1}{n}e^{-n^{2}\pi^{2}\alpha^{2}t/2500}\sin\left(\frac{n\pi x}{50}\right). (24)

The expression (24) for the temperature is moderately complicated, but the negative exponential factor in each term of the series causes the series to converge quite rapidly, except for small values of t or α². Therefore, accurate results can usually be obtained by using only the first few terms of the series.

In order to display quantitative results, let us measure t in seconds; then α² has the units of cm²/s. If we choose α² = 1 for convenience, this corresponds to a rod of a material whose thermal properties are somewhere between copper and aluminum. The behavior of the solution can be seen from the graphs in Figures 10.5.3 through 10.5.5. In Figure 10.5.3 we show the temperature distribution in the bar at several different times. Observe that the temperature diminishes steadily as heat in the bar is lost through the end points. The way in which the temperature decays at a given point in the bar is indicated in Figure 10.5.4, where temperature is plotted against time for a few selected points in the bar. Finally, Figure 10.5.5 is a three-dimensional plot of u versus both x and t. Observe that we obtain the graphs in Figures 10.5.3 and 10.5.4 by intersecting the surface in Figure 10.5.5 by planes on which either t or x is constant. Note that the temperature is constant along each of the curves drawn on the surface in Figure 10.5.5; in this case, these level curves are known as isotherms. The slight waviness in Figure 10.5.5 at t = 0 results from using only five terms in the series for u(x, t) and from the slow convergence of the series for t = 0 (remember the Gibbs phenomenon introduced in Section 10.3).

A problem with possible practical implications is to determine the time \tau at which the entire bar has cooled to a specified temperature. For example, when is the temperature in the entire bar no greater than 1° C? Because of the symmetry of the initial temperature distribution and the boundary conditions, the warmest point in the bar is always the center. Thus \tau is found by solving u(25, t) = 1 for t. Using one term in the series expansion (24), we obtain

\tau={\frac{2500}{\pi^{2}}}\ln\left({\frac{80}{\pi}}\right)\cong820\,{\mathrm{s}}.