Page 260 - Modelling in Transport Phenomena A Conceptual Approach

240 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

total momentum flux are expressed as

dv,
rxz = 7,, + (pv,)vx =7,, = -p- (8.1-2)
dx
ryz = 7yz + (PVZ) q/ = 0 (8.1.3)
Xzz = Tzz + (PVz) vz = PV, (8.1-4)
2
For a rectangular differential volume element of thickness Ax, length Ax and width
W, as shown in Figure 8.1, Eq. (8.1-1) is expressed as
(~zz.1~ WAX + rxzIx WAZ) - ( rzzIz+Az WAX + rxzIs+Ax wax) = 0 (8.1-5)

Following the notation introduced by Bird et d. (1960), “in” and ‘‘out’’ directions
for the fluxes are taken in the direction of positive x- and z-axes. Dividing Eq.
(8.1-5) by WAX Az and taking the limit as Ax + 0 and AZ + 0 gives

(8.1-7)
Substitution of Eqs. (8.1-2) and (8.1-4) into Eq. (8.1-7) and noting that dv,/dz = 0
yields
d
dv,
z (z) (8.1-8)
=O
The solution of Eq. (8.1-8) is
vz = Cl 2 + cz (8.1-9)

where C1 and C2 are constants of integration. The use of the boundary conditions
at x=O vz=V (8.1-10)

at x=B vz=O (8.1-11)
gives the velocity distribution as

(8.1-12)

The use of the velocity distribution, Eq. (8.1-12), in Eq. (8.1-2) indicates that
the shear stress distribution is uniform across the cross-section of the plate, i.e.,

(8.1-13)