Page 456 - Modelling in Transport Phenomena A Conceptual Approach

436 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.
10.2 ENERGY TRANSPORT

The conservation statement for energy reduces to

( Rate of ) - ( Rate of ) = ( Rate of energy ) (10.21)
energy in energy out accumulation

As in Section 8.2, our analysis will be restricted to the application of Eq. (10.2-1)
to conduction in solids and stationary liquids. The solutions of almost all imagin-
able conduction problems in different coordinate systems with various initial and
boundary conditions are given by Carslaw and Jaeger (1959). For this reason, only
some representative problems will be presented in this section.
The Biot number is given by Eq. (7.1-14) as

(Difference in driving force),,lid
(10.2-2)
Bi = (Difference in driving force)fluid

In the case of heat transfer, the temperature distribution is considered uniform
within the solid phase when BiH << 1. This obviously brings up the question,
‘What should the value of BiH be so that the condition BiH << 1 is satisfied?” In
the literature, it is generally assumed that the internal resistance to heat transfer
is negligible and the temperature distribution within the solid is almost uniform
when BiH < 0.1. Under these conditions, the so-called lumped-parameter analysis
is possible as can be seen in the solution of problems in Section 7.5. When
0.1 < BiH < 40, the internal and external resistances to heat transfer have al-
most the same order of magnitude. The external resistance to heat transfer is
considered negligible when BiH > 40.

10.2.1 Heating of a Rectangular Slab

Consider a rectangular slab of thickness 2L as shown in Figure 10.2. Initially the
slab temperature is uniform at a value of To. At t = 0, the temperatures of the
surfaces at z = f L are increased to 7’1. To calculate the amount of heat transferred
into the slab, it is first necessary to determine the temperature profile within the
slab as a function of position and time.
If 2L/H << 1 and 2L/W << 1, then it is possible to assume that the conduction
is one-dimensional 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, = q, = - k - (10.2-3)
az