Page 298 - Modelling in Transport Phenomena A Conceptual Approach

278 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

If the measuring instrument, i.e., the temperature probe, is not sensitive enough
to detect temperature variations in the x-direction, then it is necessary to change
the scale of the problem to match that of the measuring device. In other words, it
is necessary to average the governing equation up to the scale of the temperature
measuring probe.
The area-averaged temperature is defined by

1
= - 1" lyll T dxdy (8.2-71)
Jd WB

Note that although the local temperature, T, is dependent on x, y and z, the
area-averaged temperature, (T), depends only on z.
Area averaging is performed by integrating Eq. (8.264) over the cross-sectional
area of the fin. The result is

or,

+ -!!? da2 (Iw -B/2 Tdxdy) = 0 (8.2-73)

The use of the boundary conditions defined by Eqs. (8.2-65)-(8.2-68) together with
the definition of the average temperature, R. (8.2-71) in Eq. (8.2-73) gives

Since TIx=B/2 = TIz=-B/2 as a result of symmetry, &. (8.2-74) takes the form

(8.2-75)

k-= 8(T) z(h) ((7') - T,) (8.2-77)
dz2 B