ABSTRACT
Consider the classic „n problem introduced in Chapter 8 and shown in Figure 9.1. To perform the „n analysis, we assumed the „n was thin so that at any point along its length, the „n had a constant temperature throughout its cross section. We derived the „n number, a dimensionless grouping that assessed the relative resistance of conduction along the length of the „n to convection from its surface:
f L kA hA
hP L kAx c s
convection =
= −
1 2 R R
(9.1)
Since the „n number is a kind of Biot number, we can also de„ne the „n number so that it measures the conduction to convection resistance in a direction perpendicular to the „n axis. Using the „n of Figure 9.1, we would have
f y kA hA
hy ky
o=
=
1 (9.2)
If fy is very small, as is the case in a properly designed „n, then the temperature throughout the cross section is uniform. When fy is large, then the resistance to conduction is high and there is a temperature gradient in the y-direction as well as in the x-direction. Under those circumstances, we cannot use a 1-D equation to describe the heat transfer but must resort to a 2-D formulation at least.
