Page 376 - Modelling in Transport Phenomena A Conceptual Approach

356 CHAPTER 9. STEADY MIcRoscoprc BALANCES wrr~ GEN.

W Case (i) k = constant

In this case EQ. (9.257) reduces to
1 1

1
+ I”” $ [Jdr ?R(u) u2 du dr (9.2-58)

When ?R = 0, EQ. (9.258) reduces to Eq. (D) in Table 8.5.

W Case (ii) k = constant; % = constant

In this case Eq. (9.2-57) simplifies to

[)
%e!?
+ - - ($)2] (9.259)
6k

Type I1 boundary condition

The solution of EQ. (9.2-55) subject to the boundary conditions
dT
at r=R1
dz (9.2-60)
at r=R2 T=T!

is given by

Ra 1
k(T) dT = [lr R(u) u2 d] dr

Note that when 92 = 0, EQ. (9.2-61) reduces to EQ. (C) in Table 8.6. Further
simplification of Eq. (9.2-61) depends on the functional forms of k and 3.