Page 170 - Mechanical Engineers' Handbook (Volume 4)

1 Conduction Heat Transfer 159

Figure 2 Heat transfer by extended surfaces.

dT hC
2
(T T ) 0

dx 2 kA
The solution of this equation depends on the boundary conditions existing at the tip,
i.e., at x L. Table 11 shows the temperature distribution and heat-transfer rate for fins of
uniform cross section subjected to a number of different tip conditions, assuming a constant
value for the heat-transfer coefficient, h.
Two terms are used to evaluate fins and their usefulness. The first of these is the fin
effectiveness, defined as the ratio of the heat-transfer rate with the fin to the heat-transfer
rate that would exist if the fin were not used. For most practical applications, the use of a
fin is justified only when the fin effectiveness is significantly greater than 2. A second term
used to evaluate the usefulness of a fin is the fin efficiency, , This term represents the ratio
ƒ

Table 11 Temperature Distribution and Heat-Transfer Rate at the Fin Base (m
hc/kA)

T T Heat-Transfer Rate
Condition at x L T T q/mkA (T b T )
b
T ) k h h
dT
h(T x L cosh m(L x) sinh m(L x) sinh mL cosh mL
dx mk mk
x L
(convection) h h
cosh mL mk sinh mL cosh mL mk sinh mL
cosh m(L x) tanh mL
dT
dx x L 0 cosh mL
(insulated)
(T T )/(T T ) sinh mx sinh m(L x) cosh mL (T T )/(T T )
T x L T L L b L b
(prescribed temperature) sinh ml sinh ml
e mx 1
T x L T
(infinitely long fin, L → )