Page 306 - Bird R.B. Transport phenomena

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Chapter 0

Shell Enemy Balances and

Temperature Distributions in

Solids and Laminar Flow

§10.1 Shell energy balances; boundary conditions
§10.2 Heat conduction with an electrical heat source

§10.3 Heat conduction with a nuclear heat source
§10.4 Heat conduction with a viscous heat source
§10.5 Heat conduction with a chemical heat source
§10.6 Heat conduction through composite walls
§10.7 Heat conduction in a cooling fin
§10.8 Forced convection
§10.9 Free convection

In Chapter 2 we saw how certain simple viscous flow problems are solved by a two-step
procedure: (i) a momentum balance is made over a thin slab or shell perpendicular to the
direction of momentum transport, which leads to a first-order differential equation that
gives the momentum flux distribution; (ii) then into the expression for the momentum
flux we insert Newton's law of viscosity, which leads to a first-order differential equa-
tion for the fluid velocity as a function of position. The integration constants that appear
are evaluated by using the boundary conditions, which specify the velocity or momen-
tum flux at the bounding surfaces.
In this chapter we show how a number of heat conduction problems are solved by
an analogous procedure: (i) an energy balance is made over a thin slab or shell perpen-
dicular to the direction of the heat flow, and this balance leads to a first-order differential
equation from which the heat flux distribution is obtained; (ii) then into this expression
for the heat flux, we substitute Fourier's law of heat conduction, which gives a first-order
differential equation for the temperature as a function of position. The integration con-
stants are then determined by use of boundary conditions for the temperature or heat
flux at the bounding surfaces.
It should be clear from the similar wording of the preceding two paragraphs that the
mathematical methods used in this chapter are the same as those introduced in Chapter
2—only the notation and terminology are different. However, we will encounter here a
number of physical phenomena that have no counterpart in Chapter 2.
After a brief introduction to the shell energy balance in §10.1, we give an analysis of
the heat conduction in a series of uncomplicated systems. Although these examples are

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