Page 366 - Modelling in Transport Phenomena A Conceptual Approach

346 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

9.2.1.1 Macroscopic equation
The integration of the governing equation, Eq. (9.2-6) over the volume of the
system gives

(k
- s," I" Jd" 2 g) dxdydz = Jd" Jd" 1" Rdxdydz (9.2-19)
Integration of Eq. (9.2-19) yields

Net rate of energy out Rate of energy
generation
Equation (9.2-20) is simply the macroscopic energy balance under steady conditions
by considering the plane wall as a system. Note that energy must leave the system
from at least one of the surfaces to maintain steady conditions. The "net rate of
energy out" in Eq. (9.2-20) implies the rate of energy leaving the system in excess
of the rate of energy entering into it.
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. (9.2-20) can be written as

(9.2-21)
where To and TL are the surface temperatures at z = 0 and z = L, respectively.

Example 9.1 Energy generation rate as a result of an exothermic reaction is
1 x 104W/m3 in a 50cm thick wall of thermal conductivity 20Wlm.K. The
left-face of the wall is insulated while the right-side is held at 45°C by a coolant.
Calculate the maximum temperature in the wall under steady conditions.

Solution
Let z be the distance measured from the left-face. The use of Eq. (9.2-18) with

)
'
I
:
T = TL + - (
qo = 0 gives the temperature distribution as
[1-
92 L2
(1 x 104)(0.5)2 [
2k
= 45+ 1-
2 (20)
Simplification of Eq. (1) leads to
T = 107.5 - 250z2
Note that dTldz = 0 at z = 0. Therefore, the mm'mum temperature occurs at the
insulated surface and its value is 107.5OC.