Page 377 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 357

H Case (2) k = constant

In this case Eq. (9.2-61) reduces to
Ic(T - T2) = R2 ;;z [Jd' 8(u)u2 du] dr + [qlR: - 1"' 8(u)u2 du] (1 - ')
1
T R2
(9.2-62)
Note that when 8 = 0, Eq. (9.2-62) reduces to Eq. (D) in Table 8.6

Case (ii) k = constant; 8 = constant
In this case Eq. (9.2-61) simplifies to

Macroscopic equation

The integration of the governing equation, Eq. (9.2-53) over the volume of the
system gives the macroscopic energy balance as

r2
-I'"Jd"J,, Rz FZ( r2k - Z) sin8drd8dq3

Integration of Eq. (9.2-64) yields

Net rate of energy out Rate of energy
generation
Equation (9.2-65) is the macroscopic energy balance under steady conditions by
considering the hollow sphere as a system.
It is also possible to make use of the Newton's law of cooling to express the rate
of heat loss from the system. If heat is lost from both surfaces, Eq. (9.2-65) can
be written as

(9.266)

where TI and TZ are the surface temperatures at r = R1 and T = Rz, respectively.