Page 290 - Modelling in Transport Phenomena A Conceptual Approach

270 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

Figure 8.19 Conduction through a hollow sphere.

Dividing Eq. (8.245) by Ar and taking the limit as Ar -+ 0 gives

(8.2-47)
Since flux times area gives the heat transfer rate, Q, it is possible to conclude that

Aq, = constant = Q (8.248)
The area A in Eq. (8.2-48) is perpendicular to the direction of energy flux in the
?--direction and it is given by
A = 4XT2 (8.2-49)
Substitution of Eqs. (8.2-44) and (8.2-49) into Eq. (8.248) and integration gives

(8.2-50)

where C is an integration constant.
If the surface temperatures are specified, Le.,
at r= R1 T=Tl
(8.2-51)
at T= RZ T=T2
the heat transfer rate as well as the temperature distribution as a function of
position are given in Table 8.5. 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 T=R~ -IC-==]
dr (8.2-52)
at T=R~ T=Tz