Page 312 - Bird R.B. Transport phenomena

296 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

SOLUTION II
Another method makes use of the result obtained previously in Eq. 10.2-13. Although T o is
not known in the present problem, we can nonetheless use the result. From Eqs. 10.1-2 and
10.2-16 we can get the temperature difference
2
irR LS c
- T air = (10.2-24)
H(2TTRL) 2/7
Substraction of Eq. 10.2-24 from Eq. 10.2-13 enables us to eliminate the unknown T o and gives
Eq. 10.2-23.

jlO.3 HEAT CONDUCTION WITH A NUCLEAR HEAT SOURCE
We consider a spherical nuclear fuel element as shown in Fig. 10.3-1. It consists of a
{F
sphere of fissionable material with radius R \ surrounded by a spherical shell of alu-
{C
minum "cladding" with outer radius R \ Inside the fuel element, fission fragments are
produced that have very high kinetic energies. Collisions between these fragments and
the atoms of the fissionable material provide the major source of thermal energy in the
reactor. Such a volume source of thermal energy resulting from nuclear fission we call S,,
3
(cal/cm • s). This source will not be uniform throughout the sphere of fissionable mater-
ial; it will be the smallest at the center of the sphere. For the purpose of this problem, we
assume that the source can be approximated by a simple parabolic function

s = s R (F) (10.3-1)
n
Here S n0 is the volume rate of heat production at the center of the sphere, and b is a di-
mensionless positive constant.
We select as the system a spherical shell of thickness Ar within the sphere of fission-
able material. Since the system is not in motion, the energy balance will consist only of
heat conduction terms and a source term. The various contributions to the energy bal-
ance are:
Rate of heat in
by conduction (10.3-2)
atr

Coolant
Aluminum
cladding
Sphere of
fissionable
material

Fig. 10.3-1. A spherical nuclear fuel assembly, showing
the temperature distribution within the system.