Page 307 - Bird R.B. Transport phenomena

§10.1 Shell Energy Balances; Boundary Conditions 291

somewhat idealized, the results find application in numerous standard engineering cal-
culations. The problems were chosen to introduce the beginner to a number of important
physical concepts associated with the heat transfer field. In addition, they serve to show
how to use a variety of boundary conditions and to illustrate problem solving in Carte-
sian, cylindrical, and spherical coordinates. In §§10.2-10.5 we consider four kinds of heat
sources: electrical, nuclear, viscous, and chemical. In §§10.6 and 10.7 we cover two topics
with widespread applications—namely, heat flow through composite walls and heat
loss from fins. Finally, in §§10.8 and 10.9, we analyze two limiting cases of heat transfer
in moving fluids: forced convection and free convection. The study of these topics paves
the way for the general equations in Chapter 11.

§10.1 SHELL ENERGY BALANCES; BOUNDARY CONDITIONS
The problems discussed in this chapter are set up by means of shell energy balances. We
select a slab (or shell), the surfaces of which are normal to the direction of heat conduc-
tion, and then we write for this system a statement of the law of conservation of energy.
For steady-state (i.e., time-independent) systems, we write:
(rate of 1 rate of I rate of rate of
energy in energy out + j energy in energy out
1 by convective [ by convective 1 by molecular | by molecular
[transport J transport [transport J transport
rate of rate of rate of
work done work done work done rate of j
on system by system on system I energy > = 0 (10.1-1)
by molecular by molecular by external production!
transport .transport forces
The convective transport of energy was discussed in §9.7, and the molecular transport (heat
conduction) in §9.1. The molecular work terms were explained in §9.8. These three terms
can be added to give the "combined energy flux" e, as shown in Eq. 9.8-6. In setting up
problems here (and in the next chapter) we will use the e vector along with the expres-
sion for the enthalpy in Eq. 9.8-8. Note that in nonflow systems (for which v is zero) the e
vector simplifies to the q vector, which is given by Fourier's law.
The energy production term in Eq. 10.1-1 includes (i) the degradation of electrical en-
ergy into heat, (ii) the heat produced by slowing down of neutrons and nuclear frag-
ments liberated in the fission process, (iii) the heat produced by viscous dissipation, and
(iv) the heat produced in chemical reactions. The chemical reaction heat source will be
discussed further in Chapter 19. Equation 10.1-1 is a statement of the first law of thermo-
dynamics, written for an "open" system at steady-state conditions. In Chapter 11 this
same statement—extended to unsteady-state systems—will be written as an equation of
change.
After Eq. 10.1-1 has been written for a thin slab or shell of material, the thickness of
the slab or shell is allowed to approach zero. This procedure leads ultimately to an ex-
pression for the temperature distribution containing constants of integration, which we
evaluate by use of boundary conditions. The commonest types of boundary conditions
are:
a. The temperature may be specified at a surface.
b. The heat flux normal to a surface may be given (this is equivalent to specifying
the normal component of the temperature gradient).
c. At interfaces the continuity of temperature and of the heat flux normal to the in-
terface are required.