1. The plate shown in Figure 1 is initially at a temperature of $T_i = 483^\circ C$ and it is then immersed in a
W
L
Figure 1: Plate suddenly subjected to convective environment. $L = 101cm$. $W = 60cm$. $t_p = 6.8cm$.
convective environment with $h = 595 \frac{W}{m^2 \cdot K}$ and $T_\infty = 20^\circ C$. The plate has the following thermal prop-
erties: $k = 74 \frac{W}{m \cdot K}$, $\rho = 2735 \frac{kg}{m^3}$, $c = 970 \frac{J}{kg \cdot K}$. Near the center of the plate, apply the approximation
of one-dimensional heat transfer.
(a) Solve for the first four eigenvalues for the one-dimensional transient approximation.
(b) For the region near the center of the plate, use the four-term series to plot the temperature
variation through the thickness at 15 seconds, 1 minute and three minutes.
(c) How long will it take before the center of the plate drops to $30^\circ C$?
(d) Apply the Crank-Nicolson algorithm in one dimension using 48 grid points in space to solve the
transient problem. Plot the results over the analytic solution you developed at 15 seconds and at
three minutes. Indicate the time step you have chosen.
