Page 375 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 355

For a spherical differential volume of thickness Ar as shown in Figure 9.8, the
inventory rate equation for energy, Eq. (9.2-1)) is expressed as
47r (r 2 q,.)lr - 47r (r2q,)lr+A, + 47rr2Ar R = 0 (9.2-50)
Dividing each term by 47rAr and taking the limit as Ar + 0 gives

(9.2-51)
or,
(9.2-52)
Substitution of h. (9.2-49) into Eq. (9.2-52) gives the governing equation for
temperature as
(9.2-53)

Integration of Eq. (9.2-53) gives

(9.2-54)

where u is the dummy variable of integration. Integration of Eq. (9.2-54) once
more leads to
R(u)
k(T) dT = - Jdr $ [lr u2 du] dr - CI + C2
(9.2-55)
Evaluation of the constants C1 and C2 requires the boundary conditions to be
specified.
Type I boundary condition
The solution of Eq. (9.2-55) subject to the boundary conditions

at r= R1 T=T1
(9.2-56)
at r= R2 T=T2
is given by
1 1

1
+ 1" -$ [hr R(u) u2 du dr (9.2-57)

Note that when W = 0, Eq. (9.2-57) reduces to Eq. (C) in Table 8.5. Further
simplification of Eq. (9.2-57) depends on the functional forms of k and R.