Page 313 - Bird R.B. Transport phenomena

§10.3 Heat Conduction with a Nuclear Heat Source 297

Rate of heat out
)
2
{F)
by conduction q \ Г+Дг • 4тг(г + Ar) 2 = (4тгг ^^) | Г+Дг (10.3-3)
at г + Ar
Rate of thermal
energy produced S, • 4тгг 2 Ar (10.3-4)
;
by nuclear fission
Substitution of these terms into the energy balance of Eq. 10.1-1 gives, after dividing by
4TT Ar and taking the limit as Ar —> 0

(10.3-5)

Taking the limit and introducing the expression in Eq. 10.3-1 leads to

(10.3-6)

The differential equation for the heat flux q in the cladding is of the same form as Eq.
( C)
r
10.3-6, except that there is no significant source term:
(10.3-7)
Integration of these two equations gives

(10.3-8)
—
с ( С ) (10.3-9)
г 2
in which C\ n and C are integration constants. These are evaluated by means of the
( C)
1
boundary conditions:
B.C. 1: at r = 0, q\ is not infinite (10.3-10)
F)
B.C. 2: at r = R , q = q\° (10.3-11)
{ F)
iF)
r
Evaluation of the constants then leads to
Ф ъ А
q i F ) = < (10.3-12)

R (F)3
(С) (10.3-13)
Л) г 2

These are the heat flux distributions in the fissionable sphere and in the spherical-shell
cladding.
Into these distributions we now substitute Fourier's law of heat conduction (Eq.
B.2-7):
dT {F)
UF) L. + _ — (10.3-14)
dr
(10.3-15)
dr