Page 325 - Bird R.B. Transport phenomena

§10.7 Heat Conduction in a Cooling Fin 309

Equation 10.7-9 may be integrated to give hyperbolic functions (see Eq. C.l-4 and §C5).
When the two integration constants have been determined, we get

в = cosh Щ - (tanh N) sinh Щ (10.7-12)
This may be rearranged to give
cosh Ml - Q
coshN (10.7-13)

This result is reasonable only if the heat lost at the end and at the edges is negligible.
The "effectiveness" of the fin surface is defined 3 by
actual rate of heat loss from the fin
(10.7-14)
rate of heat loss from an isothermal fin at T
w
For the problem being considered here 17 is then
fW [L r\
h(T-T )dzdy
a
Jo Jo (10.7-15)
V = • J o
h(T -T )dzdy
w
a
Jo Jo Jo
or
(10.7-16)
N
in which N is the dimensionless quantity defined in Eq. 10.7-8.

EXAMPLE 10.7-1 In Fig. 10.7-2 a thermocouple is shown in a cylindrical well inserted into a gas stream. Esti-
mate the true temperature of the gas stream if
Error in Thermocouple
Measurement T } - 500°F = temperature indicated by thermocouple
T w = 350°F = wall temperature
h = 120 Btu/hr • ft • F = heat transfer coefficient
2
к — 60 Btu/hr • ft • F = thermal conductivity of well wall
3
В = 0.08 in. = thickness of well wall
L = 0.2 ft = length of well

Thermocouple wires
to potentiometer
Pipe wall at T w

Well wall of
thickness В
Thermocouple Fig. 10.7-2. A thermocouple in a cylindrical
junction at T] well.

M. Jakob, Heat Transfer, Vol. I, Wiley, New York (1949), p. 235.
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