Page 348 - Modelling in Transport Phenomena A Conceptual Approach

328 CHAPTER 9. STEADY MCROSCOPIC BALANCES WITH GEN.

and
(9.1-15)
Combination of Eqs. (9.1-8) and (9.1-14) yields

(9.1-16)

which implies that P = P(z) only. Therefore, the use of Eq. (9.1-15) in Eq.
(9.1-12) gives
&v, dP
p-=- (9.1-17)
dx2
dz
-v
f(x) f(x)
Note that while the rightrside of Eq. (9.1-17) is a function of z only, the left-side
is dependent only on 5. This is possible if and only if both sides of Eq. (9.1-17)
are equal to a constant, say A. Hence,
dP j A=- P* - PL (9.1-18)
-=A
dz L
where Po and PL are the values of P at z = 0 and z = L, respectively. Substitution
of Eq. (9.1-18) into Eq. (9.1-17) gives the governing equation for velocity in the
form
(9.1-19)

Integration of Eq. (9.1-19) twice results in

v, = - - pL x2 + c12 + c, (9.1-20)
2 PL
where C1 and C2 are integration constants.
The use of the boundary conditions
at x=O v,=O (9.1-21)

at x=B v,=O (9.1-22)
gives the velocity distribution as

(9.1-23)

The use of the velocity distribution, Eq. (9.1-23), in Eq. (9.1-3) gives the shear
stress distribution as

(9.1-24)