Page 363 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 343

Since T = T(z), Table C.4 in Appendix C indicates that the only non-zero
energy flux component is e, and it is given by

(9.2-2)

For a rectangular volume element of thickness AZ as shown in Figure 9.5, Eq.
(9.2-1) is expressed as
qZl, A - qzIr+Ar A + AZ = 0 (9.2-3)

Dividing each term by AAz and taking the limit as AZ ---f 0 gives

(9.2-4)

or
-- - g (9.2-5)
dqz
dz
Substitution of Eq. (9.2-2) into &. (9.2-5) gives the governing equation for tem-
perature 8s
(9.2-6)

Integration of Eq. (9.2-6) gives

(9.2-7)

where u is a dummy variable of integration and C1 is an integration constant.
Integration of Eq. (9.2-7) once more leads to

l'k(T) dT = - 1' [I' X(u) d] dz + CI + C2 (9.2-8)
z

Evaluation of the constants Cl and C2 requires the boundary conditions to be
specified. The solution of h. (9.28) will be presented for two types of boundary
conditions, namely, Type I and Type 11. In the case of Type I boundary condi-
tion, the temperatures at both surfaces are specified. On the other hand, Type I1
boundary condition implies that while the temperature is specified at one of the
surfaces, the other surface is subjected to a constant wall heat flux.

Type I boundary condition
The solution of Eq. (9.2-8) subject to the boundary conditions

at z=O T=T,
(9.2-9)
at z=L T=TL