Page 304 - Modelling in Transport Phenomena A Conceptual Approach

284 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

The velocity distribution for this problem is given by Eq. (8.1-12) as
v.2
X
-- -1-B (8.3-1)
On the other hand, the boundary conditions for the temperature, i.e.,
at x=O T=T, (8.3-2)
at x=B T=Tl (8.3-3)
suggest that T = T(x). Therefore, Table C.4 in Appendix C indicates that the
only non-zero energy flux component is e, and it is given by
dT
e, = q, = -k- (8.3-4)
dx
For a rectangular volume element of thickness Ax, as shown in Figure 8.25, Eq.
(8.2-1) is expressed as
wL - %),+A, wL = (8.3-5)
Dividing each term by WLAx and taking the limit as Ax + 0 gives

(8.3-6)
or.
-=o
dq2
dx (8.3-7)
Substitution of Eq. (8.3-4) into Eq. (8.3-7) gives the governing equation for tem-
perature in the form
d2T
-- (8.3-8)
dx2 -O
The solution of Eq. (8.3-8) is
T = Ci -i- C2 x (8.3-9)

The use of boundary conditions defined by Eqs. (8.3-2) and (8.3-3) gives the linear
temperature distribution as

(8.3-10)

8.4 MASS TRANSPORT WITHOUT
CONVECTION

The inventory rate equation for transfer of species A at the microscopic level is
called the equation of continuity for species A. Under steady conditions without
generation, the conservation statement for the mass of species A is given by
(Rate of mass of A in) - (Rate of mass of A out) = 0 (8.41)