Page 314 - Bird R.B. Transport phenomena

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

(F)
These equations may be integrated for constant k and к ю to give
7^ = 4 ^ + 4 ^ + (10.3-16)

(10.3-17)

The integration constants can be determined from the boundary conditions
{F
B.C3: at r = R \ T (f) = T (C) (10.3-18)
(C)
B.C. 4: at r = K , 7 4 0 = T o (10.3-19)
where T is the known temperature at the outside of the cladding. The final expressions
o
for the temperature profiles are

(10.3-20)

(10.3-21)

To find the maximum temperature in the sphere of fissionable material, all we have to
do is set r equal to zero in Eq. 10.3-20. This is a quantity one might well want to know
when making estimates of thermal deterioration.
This problem has illustrated two points: (i) how to handle a position-dependent
source term, and (ii) the application of the continuity of temperature and normal heat
flux at the boundary between two solid materials.

§10.4 HEAT CONDUCTION WITH A VISCOUS HEAT SOURCE

Next we consider the flow of an incompressible Newtonian fluid between two coaxial
cylinders as shown in Fig. 10.4-1. The surfaces of the inner and outer cylinders are main-
tained at T = T and T = T , respectively. We can expect that T will be a function of r
o
b
alone.

Outer cylinder moves with
angular velocity Q.

Fig. 10.4-lo Flow between cylinders with viscous
heat generation. That part of the system enclosed
within the dotted lines is shown in modified form
in Fig. 10.4-2.