Page 266 - Modelling in Transport Phenomena A Conceptual Approach

246 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

Dividing each term by AZ and taking the limit as AZ -+ 0 gives

(8.2-4)

1
d(A9% (8.2-5)
-- -0
dz
Since flux times area gives the heat transfer rate, Q, it is possible to conclude from
Eq. (8.2-5) that
Aq, = constant = Q (8.2-6)

in which the area A is perpendicular to the direction of energy flux. Substitution
of Eq. (8.2-2) into Eq. (8.2-6) and integration gives

(8.2-7)

where C is an integration constant. The determination of Q and C requires two
boundary conditions.
If the surface temperatures are specified, i.e.,

at z=O T=To
(8.2-8)
at z=L T=TL

the heat transfer rate as well as the temperature distribution as a function of
position are given in Table 8.1.
On the other hand, if one surface is exposed to a constant heat flux while the
other one is maintained at a constant temperature, i.e.,

dT
at z=O -k--
dz -” (8.2-9)
at z=L T=TL

the resulting heat transfer rate and the temperature distribution as a function of
position are given in Table 8.2. It should be noted that the boundary conditions
given by Eqs. (8.28) and (8.2-9) are not the only boundary conditions available for
energy transport. For different boundary conditions, Eq. (8.2-7) should be used to
determine the constants.