Page 283 - Modelling in Transport Phenomena A Conceptual Approach

8.2. ENERGY TRANSPORT WITHOUT CONVECTION 263

At first, it looks as if the resistance expression for the rectangular and the cylindrical
coordinate systems are different from each other. However, the similarities between
these two expressions can be shown by the following analysis.
Note that the logarithmic-mean area, ALM, can be defined as

(8.2-31)

Substitution of Eq. (8.2-31) into Eq. (8.230) gives
I Resistance = l (8.2-32)

Note that Eqs. (8.2-13) and (8.2-32) have the same general form of
Thickness
Resistance = (8.2-33)
(Transport property) (Area)

The electrical circuit analog of the cylindrical wall can be represented as shown in
Figure 8.15.

R2 - Rl
k Am

e 0
TA -Q TB

Figure 8.15 Electrical circuit analog of the cylindrical wall.

Example 8.6 Heat flows through an annular wall of inside radius R1 = lOcm
and outside radius Rz = 15cm. The inside and outside surface temperatures are
60°C and 30"C, respectively. The thermal conductivity of the wall is dependent
on temperature as follows:
T = 30°C k = 42W/m.K
T = 60°C IC = 49W/m.K

Calculate the steady rate of heat transfer if the wall has a length of 2m.

Solution
Assumption

1. The thermal conductiwity varies linearly with temperature.