Page 392 - Bird R.B. Transport phenomena

Chapter II

More Than One Independent

Variable

§12.1 Unsteady heat conduction in solids

§12.2° Steady heat conduction in laminar, incompressible flow
§12.3° Steady potential flow of heat in solids
§12.4° Boundary layer theory for nonisothermal flow

In Chapter 10 we saw how simple heat flow problems can be solved by means of shell
energy balances. In Chapter 11 we developed the energy equation for flow systems,
which describes the heat transport processes in more complex situations. To illustrate
the usefulness of the energy equation, we gave in §11.4 a series of examples, most of
which required no knowledge of solving partial differential equations.
In this chapter we turn to several classes of heat transport problems that involve
more than one dependent variable, either two spatial variables, or one space variable
and the time variable. The types of problems and the mathematical methods parallel
those given in Chapter 4.

§12Л UNSTEADY HEAT CONDUCTION IN SOLIDS
For solids, the energy equation of Eq. 11.2-5, when combined with Fourier's law of heat
conduction, becomes

pC ^ = (V • kVT) (12.1-1)
p
If the thermal conductivity can be assumed to be independent of the temperature and
position, then Eq. 12.1-1 becomes

2
^ = aV T (12.1-2)
at
in which a = k/pC p is the thermal diffusivity of the solid. Many solutions to this equa-
tion have been worked out. The treatise of Carslaw and Jaeger 1 contains a thorough dis-

1
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
(1959).

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