Page 301 - Modelling in Transport Phenomena A Conceptual Approach

8.2. ENERGY TRANSPORT WITHOUT CONVECTION 281

8.2.4.1 Macroscopic equation
Integration of the governing differential equation, Eq. (8.2-77), over the volume of
the system gives the macroscopic energy balance, i.e.,
dz2 dxdydz= 1
B/2 2
iL lw lBj2 k d2(T> L W 1B/2
-(h) ((3”) - T,) dxdydz
-B/2 B
(8.2-92)
Evaluation of the integrations yields

= 2 W(h) 1” ((T) - Tho) ds (8.2-93)
, . /
Rate of energy entering into the Rate of energy loss from the top and bottom
fin through the surface at x=O surfaces of the fin to the surroundings
Note that Eq. (8.2-93) is simply the macroscopic inventory rate equation for ther-
mal energy by considering the fin as a system. The use of Eq. (8.2-91) in Eq.
(8.2-93) gives the rate of heat loss from the fin as

BWk(Tw - T,)A tmh A
1 Qaoss = L (8.2-94)
8.2.4.2 Fin efficiency

The fin eficiency, q, is defined as the ratio of the apparent rate of heat dissipation
of a fin to the ideal rate of heat dissipation if the entire fin surface were at T,, i.e.,

2 W(h) lL((T) - T,) d.2 lL((T) - Tm) dr
-
rl= - (8.2-95)
2 W(h)(Tw - T,)L (Tw - T’)L
In terms of the dimensionless quantities, Eq. (8.2-95) becomes

q=pC (8.2-96)

Substitution of Eq. (8.2-91) into Eq. (8.2-96) gives the fin efficiency as

(8.2-97)

The variation of the fin efficiency as a function of A is shown in Figure 8.24. When
A 4 0, this means that the rate of conduction is much larger than the rate of heat
dissipation. The Taylor series expansion of q in terms of A gives
17
1
2
rl = 1 - - A2 + - A4 - - + ... (8.2-98)
A6
3 15 315