Page 81 - Bird R.B. Transport phenomena

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

water at 20°C, and the manometer fluid is carbon tetrachloride (CC1) with density 1.594
4
3
g/cm . The capillary diameter is 0.010 in. Note: Measurements of H and L are sufficient to cal-
culate the flow rate; в need not be measured. Why?
2 3
2B.9 Low-density phenomena in compressible tube flow ' (Fig. 2B.9). As the pressure is de-
creased in the system studied in Example 2.3-2, deviations from Eqs. 2.3-28 and 2.3-29 arise.
2
The gas behaves as if it slips at the tube wall. It is conventional to replace the customary "no-
slip" boundary condition that v z = 0 at the tube wall by
dv
v z=-£-^, 2 atr = R (2B.9-1)
dr
in which I is the slip coefficient. Repeat the derivation in Example 2.3-2 using Eq. 2B.9-1 as the
boundary condition. Also make use of the experimental fact that the slip coefficient varies in-
versely with the pressure £ = Up, in which £ 0 is a constant. Show that the mass rate of flow is

1 + j ^ - ) (2B.9-2)

in which p avg = \(p 0 + p L).
When the pressure is decreased further, a flow regime is reached in which the mean free
path of the gas molecules is large with respect to the tube radius (Knudsen flow). In that
regime 3

w =
in which m is the molecular mass and к is the Boltzmann constant. In the derivation of this re-
sult it is assumed that all collisions of the molecules with the solid surfaces are diffuse and not
specular. The results in Eqs. 2.3-29,2B.9-2, and 2B.9-3 are summarized in Fig. 2B.9.

- Free molecule flow
or Knudsen flow

w
PO~PL Slip flow S ,,' Poiseuille flow

' • Fig. 2B.9 A comparison of the flow regimes
Pavg in gas flow hrough a tube.
t
2B.10 Incompressible flow in a slightly tapered tube. An incompressible fluid flows through a tube
of circular cross section, for which the tube radius changes linearly from R at the tube en-
o
trance to a slightly smaller value R at the tube exit. Assume that the Hagen-Poiseuille equa-
L
tion is approximately valid over a differential length, dz, of the tube so that the mass flow rate is
f) (2ВЛ0-1)
dz)
This is a differential equation for & as a function of z, but, when the explicit expression for
R(z) is inserted, it is not easily solved.

2 E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill, New York (1938), pp. 292-295, 300-306.
3 M. Knudsen, The Kinetic Theory of Gases, Methuen, London, 3rd edition (1950). See also R. J. Silbey
and R. A. Alberty, Physical Chemistry, Wiley, New York, 3rd edition (2001), §17.6.