Page 394 - Modelling in Transport Phenomena A Conceptual Approach

374 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Table 9.1 The physical significance and the order of magnitude of the terms in
Eq. (9.3-79).
Term Physical Significance Order of Magnitude
&T (To - Tl)
k- dx2 Conduction B2
’
- Viscous dissipation @
v2
B2 B2

Before solving J3q. (9.3-79)) it is convenient to express the governing equation
and the boundary conditions in dimensionless form. Introduction of the dimen-
sionless quantities
(9.3-81)

(9.3-82)

reduces Eqs. (9.3-79), (9.3-73) and (9.3-74) to

Br (9.3-83)
dc2 -
at <=O 8=1 (9.3-84)

at <=l 8=0 (9.3-85)
Integration of Eq. (9.3-83) twice gives
Br
e = - - t2 + cl + c2 (9.3-86)
(
2
Application of the boundary conditions, Eqs. (9.3-84) and (9.3-85)) gives the solu-
tion as
(9.3-87)

Note that when Br = 0, Le., no viscous dissipation, EQ. (9.3-87) reduces to J3q.
(8.3-10). The variation of e as a function of ( with Br as a parameter is shown in
Figure 9.13.
In engineering calculations, it is more appropriate to express the solution in
terms of the Nusselt number. Calculation of the Nusselt number, on the other
hand, requires the evaluation of the bulk temperature defined by
€3
lwpzTdzdy 1 vz T dx
-
Tb = - (9.3-88)
1” la dXdY Jd” Vz dX
vz