Page 203 - Modelling in Transport Phenomena A Conceptual Approach

7.1. APPROXMATIONS USED IN UNSTEADY PROCESSES 183

Example 7.1 We are testing a 2cm thick insulating material. The density,
thermal conductivity, and heat capacity of the insulating material are 255kg/m3,
0.07 W/ m. K, and 1300 J/ kg. K, respectively. If our experiments take 10 min, is it
possible to ussume pseudo-steady-state behavior?

Solution

For the pseudo-steady-state approximation to be valid, Eq. (7.1-7) must be satisfied,
%.e.,
-
a tch
>>1
LZh
The thermal digwivity, a, of the insulating material is

k
a=-
PCP
- 0.07 = 2.11 x 10-7m2/s
-
(255)( 1300)
Hence,
at& (2.11 x io-') (10)(60)
-- - = 0.32 < 1
Lzh (2 x 10-2)2

which indicates that we have an unsteady-state problem at hand.

7.1.2 No Variation of Dependent Variable Within the
Phase of Interest

In engineering analysis it is customary to neglect spatial variations in either tem-
perature or concentration within the solid. Although this approximation simplifies
the mathematical problem, it is only possible under certain circumstances as will
be shown in the following development.
Let us consider the transport of a quantity cp from the solid phase to the fluid
phase through a solid-fluid interface. Under steady conditions without generation,
the inventory rate equation, Eq. (1.1-l), for the interface takes the form

Rate of transport of cp from
Rate of transport of cp from
( the solid to the interface >=( the interface to the fluid ) (7.1-8)
Since the molecular flux of cp is dominant within the solid phase, Eq. (7.1-8) reduces
to
Molecular flux of cp from Flux of cp from ) (7.1-9)
the solid to the interface ) = ( the interface to the fluid