Page 457 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 437

Figure 10.2 Unsteady-state conduction through a rectangular slab.

For a rectangular differential volume element of thickness Az, as shown in
Figure 10.2, Eq. (10.2-1) is expressed as

d
%Ix WH - QXlZ+AX WH = - [WHAZ P&P(T - T4-j (10.2-4)
at
Following the notation introduced by Bird et ab. (1960), “in” and “out” directions
are taken in the +z-direction. Dividing Eq. (10.2-4) by WHAz and letting
Az -+ 0 gives
(10.2-5)

or,
(10.2-6)

Substitution of Eq. (10.23) into J3q. (10.2-6) gives the governing equation for
temperature as
= k I

1 pep (10.2-7)

All physical properties are assumed to be independent of temperature in the devel-
opment of Eq. (10.2-7). The initial and boundary conditions associated with Eq.
(10.2-7) are
at t=O T = To for all z
at z=L T=T1 t>O (10.2-8)
at z=-L T=T1 t>O