Page 403 - Bird R.B. Transport phenomena

§12.3 Steady Potential Flow of Heat in Solids 385

The temperature profile may then be obtained by integrating the heat flux:

\ 'сП=-\\* dy (12.2-22)
1
%
or, in dimensionless form,

в ( т Л ) 1
? ' = -^7Г = ^ \ ФЛ~Х (12.2-23)
q l</K J x
o
Then the expression for ф is inserted into the integral, and the order of integration in the dou-
ble integral can be reversed (see Problem 12D.7). The result is

3
Here Г(§) is the (complete) gamma function, and Г(|, х ) is an incomplete gamma function. 13
To compare this result with that in Example 12.2-1, we note that 17 = 1 - f and Л = \t,. The di-
mensionless temperature is defined identically in §10.8, in Example 12.2-1, and here.

§123 STEADY POTENTIAL FLOW OF HEAT IN SOLIDS

The steady flow of heat in solids of constant thermal conductivity is described by
Fourier's law q = -JfcVT (12.3-1)
2
Heat conduction equation V T = 0 (12.3-2)
These equations are exactly analogous to the expression for the velocity in terms of the
2
velocity potential (v = -V</>), and the Laplace equation for the velocity potential (V</> =
0), which we encountered in §4.3. Steady heat conduction problems can therefore be
solved by application of potential theory.
For two-dimensional heat conduction in solids with constant thermal conductivity,
the temperature satisfies the two-dimensional Laplace equation:

Щ + Щ = 0 (12.3-3)
dx 2 dy 1
We now use the fact that any analytic function w(z) = f(x, y) + ig(x, y) provides two scalar
functions/and g, which are solutions of Eq. 12.3-3. Curves of/ = constant may be interpreted
as lines of heat flow, and curves of g = constant are the corresponding isothermals for some
heat flow problems. These two sets of curves are orthogonal—that is, they intersect at right
angles. Furthermore, the components of the heat flux vector at any point are given by

ik^ = q -iq y (12.3-4)
x
Given an analytic function, it is easy to find heat flow problems that are described by it.
But the inverse process of finding an analytic function suitable for a given heat flow
problem is generally very difficult. Some methods for this are available, but they are out-
side the scope of this textbook. ' 1 2

15
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 9th
Printing (1973), pp. 255 et seq.
1
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
(1959), Chapter XVI.
M. D. Greenberg, Advanced Engineering Mathematics, Prentice-Hall, Upper Saddle River, N.J., 2nd
2
Edition (1998), Chapter 22.