Page 323 - Bird R.B. Transport phenomena

§10.7 Heat Conduction in a Cooling Fin 307

In (r 3 /r 2 )
Region 23: T 2 ~ T 3 = r q (10.6-26)
o o
At the two fluid-solid interfaces we can write Newton's law of cooling:
Surface 0: т.-т = £ (10.6-27)
о
Ь
Surface 3: 1з 1Ь ~Ы~Й: (10.6-28)
Addition of the preceding five equations gives an equation for T - T . Then the equation is
b
n
solved for q to give
0
2ттШ а - T b )
Q = (10.6-29)
o
In (r 2 /r } ) In (r 3 /r 2 )
% /C O 1 ^ 1 2 ^23 r 3
We now define an "overall heat transfer coefficient based on the inner surface" LZ by
0
Q o = 1ттЬщъ = U (27rLr )(T a - T ) (10.6-30)
0
0
b
Combination of the last two equations gives, on generalizing to a system with n annular
layers,
1
(10.6-31)
The subscript "0" on U indicates that the overall heat transfer coefficient is referred to the
o
radius r .
0
§10.7 HEAT CONDUCTION IN A COOLING FIN 1

Another simple, but practical application of heat conduction is the calculation of the effi-
ciency of a cooling fin. Fins are used to increase the area available for heat transfer be-
tween metal walls and poorly conducting fluids such as gases. A simple rectangular fin
is shown in Fig. 10.7-1. The wall temperature is T and the ambient air temperature is T .
w a

Heat out by transfer
to air stream

Heat x Heat
in by out by
conduction conduction
Wall temperature Fig. 10.7-1. A simple cooling fin with
known to be T w В L a n d В « W.
«

1 For further information on fins, see M. Jakob, Heat Transfer, Vol. I, Wiley, New York (1949),
Chapter 11; and H. D. Baehr and K. Stephan, Heat and Mass Transfer, Springer, Berlin (1998), §2.2.3.