Page 65 - Bird R.B. Transport phenomena

50 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

(i) because v r = 0, we can drop the term pv,v in Eq. 2.3-9a; (ii) because v. = v (r), the term
z
z
pv v will be the same at both ends of the tube; and (iii) because v~ = v (r), the term
z
z z
—2ixdv /dz will be the same at both ends of the tube. Hence Eq. 2.3-8 simplifies to
z
L
d_ ) _ ЛРо ~ PS°) ~ (P/. ~ PZ ^
dr rTyz _ (2.3-10)
in which <3> = p - pgz is a convenient abbreviation for the sum of the pressure and gravi-
1
tational terms. Equation 2.3-10 may be integrated to give
(2.3-11)
2L
The constant Q is evaluated by using the boundary condition
B.C.I: atr = 0, r rz = finite (2.3-12)
Consequently C] must be zero, for otherwise the momentum flux would be infinite at the
axis of the tube. Therefore the momentum flux distribution is

(2.3-13)

This distribution is shown in Fig. 2.3-2.
Newton's law of viscosity for this situation is obtained from Appendix B.2 as
follows:
dv.
(2.3-14)

Substitution of this expression into Eq. 2.3-13 then gives the following differential equa-
tion for the velocity:
dv z (2.3-15)
2ixL

Parabolic velocity
distribution vAr)

v z,

Linear momentum-
flux distribution
т,.Лг) Fig. 2.3-2 The momentum-flux
distribution and velocity distribu-
2L tion for the downward flow in a
circular tube.

1
The quantity designated by 9> is called the modified pressure. In general it is defined by 7? = p + pgh,
where h is the distance "upward"—that is, in the direction opposed to gravity from some preselected
reference plane. Hence in this problem h = —z.