Page 501 - Modelling in Transport Phenomena A Conceptual Approach

11.2. UNSTEADY CONDUCTION WITH HEAT GENERATION 481

TI < To as shown in Figure 11.2. We are interested in obtaining the temperature
distribution within the slab.

Figure 11.2 Unsteady-state conduction with generation.

If LIH << 1 and L/W << 1, then it is possible to assume that the conduction
is onedimensional and postulate that T = T(t,z). In that case, Table C.4 in
Appendix C indicates that the only non-zero energy flux component is e, and it is
given by
aT
e, = qa = - k - (11.2-1)
az
The conservation statement for energy is expressed &S

Rate of energy ) = ( Rate of energy
Rate of
( energy in ) - ( Rate of ) + ( generation accumulation
energy in
(1 1.2-2)
For a differential volume element of thickness Az, as shown in Figure 11.2, Eq.
(11.2-2) is expressed as
a
q,l, A - qrlz+Alr A + A Az R = - [AAzpCp(T - T&)] (1 1.2-3)
at
Dividing Eq. (11.2-3) by AAz and taking the limit Az 4 0 gives

(11.2-4)

or,
(11.2-5)