Page 265 - Modelling in Transport Phenomena A Conceptual Approach

8.2. ENERGY TRANSPORT WITHOUT CONVECTION 245

8.2 ENERGY TRANSPORT WITHOUT
CONVECTION

The inventory rate equation for energy at the microscopic level is called the equation
of energy. For a steady transfer of energy without generation, the conservation
statement for energy reduces to

(Rate of energy in) = (Rate of energy out) (8.2-1)
The rate of energy entering and leaving the system is determined from the energy
flux. As stated in Chapter 2, the total energy flux is the sum of the molecular
and convective fluxes. In this case we will restrict our analysis to cases in which
convective energy flux is either zero or negligible compared with the molecular flux.
This implies transfer of energy by conduction in solids and stationary liquids.

8.2.1 Conduction in Rectangular Coordinates

Consider the transfer of energy by conduction through a slightly tapered slab as
shown in Figure 8.5. If the taper angle is small and the lateral surface is insu-
lated, energy transport can be considered one-dimensional in the z-direction3, i.e.,
T = T(z).

Figure 8.5 Conduction through a slightly tapered slab.

Table C.4 in Appendix C indicates that the only non-zero energy flux component
is e, and it is given by
dT
e, = q, = - k - (8.2-2)
dz
The negative sign in Eq. (8.2-2) implies that positive z-direction is in the direction
of decreasing temperature. If the answer turns out to be negative, this implies that
the flux is in the negative z-direction
For a differential volume element of thickness Az, as shown in Figure 8.5,
Q. (8.2-1) is expressed as

(AqJ, - (Aqx)Ir+Ae = 0 (8.2-3)
3The z-direction in the rectangular and cylindrical coordinate systems are equivalent to each
other.