Page 370 - Modelling in Transport Phenomena A Conceptual Approach

350 CHAPTER 9. STEADY MTCROSCOPIC BALANCES WITH GEN.

W Case (ii) k = constant; 8 = constant

In this case Eq. (9.2-35) simplifies to

Macroscopic equation

The integration of the governing equation, Eq. (9.2-26) over the volume of the
system gives the macroscopic energy balance as

d
- 1" 1'" L, ; (rk $) r drdedz = 1" 1"' L: 8r drdedz (9.2-38)
1
R2
Integration of Eq. (9.2-38) yields
(- k g) 2nR2L + (k g) Rrdr (9.2-39)

r=R2 r=R1
\ + 4-
Net rate of energy out Rate of energy
generation
Equation (9.2-39) is simply the macroscopic energy balance under steady condition
by considering the hollow cylinder as a system.
It is also possible to make use of Newton's law of cooling to express the rate of
heat loss from the system. If the heat is lost from both surfaces to the surroundings,
Eq. (9.2-39) can be written as

(9.2-40)

where TI and T2 are the surface temperatures at T = RI and r = R2, respectively.
Example 9.2 A catalytic reaction is being carried out in a packed bed in the
annular space between two concentric cylinders with inner radius R1 = 1.5 cm and
outer mdius R2 = 1.8cm. The entire surface of the inner cylinder is insulated.
The rate of generation of energy per unit volume as a result of a chemical reaction
is 5 x lo6 W/ m3 and it is uniform throughout the annular reactor. The effective
thermal conductivity of the bed is 0.5 W/ m. K. If the inner surface temperature is
measured as 280°C, calculate the temperature of the outer surface.
Solution

The temperature distribution is given by Eq. (9.2-37). Since q1 = 0, it reduces to
T=T2+-1-(k)2]+x 8G In ($-)
8%
4k