Page 293 - Excel for Scientists and Engineers: Numerical Methods

270 EXCEL: NUMERICAL METHODS

minus sign is required because temperature gradients are negative (heat flows
from a higher temperature to a lower). The material of which the rod is made has
heat capacity c (cal g-' deg-') and density p (g ~m-~).
The heat flow (cal s-') out of the volume element, at point x + dk, is given by

- d( $ + s( g)dx) (1 2- 18)

The rate of increase of heat stored in the element Adx is given by
dT
cp( Adx) - (12-19)
dt
From equations 12- 17 and 12- 18, the rate of increase of heat stored in the
element Adx is Hi, - Hout, and this is equal to the expression in 12-19

which can be simplified to
(a,::) dT
K- =Cp- (12-21)
dt
or
d2T cp dT =O (12-21a)
ax= K dt
an example of a parabolic partial differential equation.
There are several methods for the solution of parabolic partial differential
equations. Two common methods are the explicit method and the Crank-
Nicholson method. In either method, we will replace partial derivatives by finite
differences, as we did in the example of the parabolic partial differential
equation.

Solving Parabolic Partial Differential Equations:
The Explicit Method
Using equation 12-2 1 as an example and writing it in the form
a2F dF
-+k-=O ( 12-22)
ax2 dy
we can replace derivatives by finite differences, using the central difference
formula for 8~/a~