Page 393 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WITH CONVECTION 373

The velocity distribution for this problem is given by Eq. (8.1-12) as

21% X
-=h (9.3-71)
V
The use of Eq. (9.3-71) in Eq. (9.3-70) gives the rate of energy generation per unit
volume as
(9.3-72)

The boundary conditions for the temperature, i.e.,
at x=O T=To (9.3-73)

at x=B T=Tl (9.3-74)
suggest that T = T(x). Therefore, Table C.4 in Appendix C indicates that the
only non-zero energy flux component is ex and it is given by
dT
e, = q, = - k - (9.3-75)
dx
For a rectangular volume element of thickness Ax, as shown in Figure 9.12, &.
(9.2-1) is expressed as

(9.3-76)

Dividing each term by WLAx and taking the limit as Ax ---t 0 gives

lim qxlx - 4z1x+*m pv2 (9.3-77)
+-=O
B2
Ax-0 Ax
or.
(9.3-78)

Substitution of Eq. (9.3-75) into Eq. (9.3-78) gives the governing equation for
- 1 (9.3-79)
temperature as

Note that in the development of Eq. (9.3-79) both viscosity and thermal conduc-
tivity are assumed independent of temperature. The physical significance and the
order of magnitude of the terms in Eq. (9.3-79) are given in Table 9.1. Therefore,
the ratio of the viscous dissipation to conduction, which is known as the Brinkman
number, is given by
Viscous dissipation - p V2/B2
-
Br = - - CL v2 (9.3-80)
Conduction IC (To - Tl)/B2 (To - Tl)