Page 299 - Modelling in Transport Phenomena A Conceptual Approach

8.2. ENERGY TRANSPORT WITHOUT CONVECTION 279

Integration of Eqs. (8.2-69) and (8.270) over the cross-sectional area of the fin
gives the boundary conditions associated with Eq. (8.2-77) as

at z=O (T)=Tw (8.2-78)
-
at r=L d(T) =O (8.2-79)
dz
It is important to note that Eqs. (8.2-64) and (8.2-77) are at two different scales.
Equation (8.2-77) is obtained by averaging Eq. (8.2-64) over the cross-sectional
area perpendicular to the direction of energy flux. In this way the boundary condi-
tion, i.e., the heat transfer coefficient, is incorporated into the governing equation.
Accuracy of the measurements dictates the equation to work with since the scale
of the measurements should be compatible with the scale of the equation.
The term 2/B in Q. (8.2-77) represents the heat transfer area per unit volume
of the fin, i.e.,
2
_- 2 LW Heat transfer area (8.2-80)
B-BLW= Fin volume
The physical significance and the order of magnitude5 of the terms in Eq. (8.2-77)
are given in Table 8.7.

Table 8.7 The physical significance and the order of magnitude of the terms in
Eq. (8.2-77).

Term Physical Significance Order of Magnitude
Rate of conduction

2(h) Rate of heat transfer from 2(~)(Tw - Tco)
B
- ((T) - Tco) the fin to the surroundings B

Therefore, the ratio of the rate of heat transfer from the fin surface to the rate of
conduction is given by

Rate of heat transfer - 2(h)(Tw - T,)/B - 2(h)L2 (8.2-81)
-
--
Rate of conduction k(Tw - T,)/L2 kB
5The order of magnitude or scale analysis is a powerful tool for those interested in mathe-
matical modelling. As stated by Astarita (1997), “Very often more than nine-tenths of what one
can ever hope to know about a problem can be obtained from this tool, without actually solving
the problem; the remaining one-tenth requires painstaking algebra and/or lots of computer time,
it adds very little to our understanding of the problem, and if we have not done the first part
right, all that the algebra and the computer will produce will be a lot of nonsense. Of course,
when nonsense comes out of a computer people have a lot of respect for it, and that is exactly
the problem.” For more details on the order of magnitude analysis, see Bejan (1984), Whitaker
(1976).