Parabolic Partial Differential Equations

The previous example showed the steady-state distribution of temperature within a metal plate. We will now examine how temperature changes with time. This so-called heat equation is an example of a parabolic partial differential equation.

Consider the flow of heat within a metal rod of length L, one end of which is held at a known high temperature, the other end at a lower temperature. Heat will flow from the hot end to the cooler end. We want to calculate the temperature along the length of the rod as a function of time. We'll assume that the rod is perfectly insulated, so that heat loss through the sides can be neglected.

Consider a small element dx along the length of the rod. Heat is flowing from the hot end (x = 0) to the cooler end (x = L). The rate of heat flow into the element at the point x is given by

dx where K is the coefficient of thermal conductivity (cal s"1 cm"1 deg"1), A is the cross-sectional area of the rod (cm2) and dT/dx is the temperature gradient. The

Temperature Distribution in a Metal Plate

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"1 deg"') and density p (g cm"3).

The heat flow (cal s"1) out of the volume element, at point x + dx, is given by

dx dx dx dx

The rate of increase of heat stored in the element Adx is given by cp(Adx)

From equations 12-17 and 12-18, the rate of increase of heat stored in the element Adx is Hm - Houh and this is equal to the expression in 12-19

dx dx dx dx

i.L dt which can be simplified to k d2T dx2

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.

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