Page 67 - Bird R.B. Transport phenomena

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

The results of this section are only as good as the postulates introduced at the begin-
ning of the section—namely, that v = v (r) and p = p(z). Experiments have shown that
z z
these postulates are in fact realized for Reynolds numbers up to about 2100; above that
value, the flow will be turbulent if there are any appreciable disturbances in the sys-
tem—that is, wall roughness or vibrations. 3 For circular tubes the Reynolds number is
defined by Re = D{v )p/ ц, where D = 2R is the tube diameter.
z
We now summarize all the assumptions that were made in obtaining the Hagen-
Poiseuille equation.
(a) The flow is laminar; that is, Re must be less than about 2100.
(b) The density is constant ("incompressible flow").
(c) The flow is "steady" (i.e., it does not change with time).
(d) The fluid is Newtonian (Eq. 2.3-14 is valid).
(e) End effects are neglected. Actually an "entrance length," after the tube entrance,
of the order of L = 0.035D Re, is needed for the buildup to the parabolic profile.
e
If the section of pipe of interest includes the entrance region, a correction must
be applied. 4 The fractional correction in the pressure difference or mass rate of
flow never exceeds LJLii L> L .
e
(f) The fluid behaves as a continuum—this assumption is valid, except for very di-
lute gases or very narrow capillary tubes, in which the molecular mean free path
is comparable to the tube diameter (the "slip flow region") or much greater than
the tube diameter (the "Knudsen flow" or "free molecule flow" regime). 5
(g) There is no slip at the wall, so that B.C. 2 is valid; this is an excellent assumption
for pure fluids under the conditions assumed in (f). See Problem 2B.9 for a dis-
cussion of wall slip.

EXAMPLE 2.3-1 Glycerine (CH OH • CHOH • CH OH) at 26.5°C is flowing through a horizontal tube 1 ft long
2
2
and with 0.1 in. inside diameter. For a pressure drop of 40 psi, the volume flow rate w/p is
Determination of 0.00398 ftVmin. The density of glycerine at 26.5°C is 1.261 g/cm . From the flow data, find the
3
Viscosity from viscosity of glycerine in centipoises and in Pa • s.
Capillary Flow Data
SOLUTION
From the Hagen-Poiseuille equation (Eq. 2.3-21), we find

S(w/p)L
2
/ 1ЬЛ/ dyn/cm \/ i V
4
тг 40 —- 6.8947 X 10 — 0.05 in. X -±- Л ft
V in.VV lty/in. 2 Д 12 m./
1 min \
8 0.00398 A ,. , 1(1 ft)
\ min 60 s /
= 4.92 g/cm • s = 492 cp = 0.492 Pa • s (2.3-23)

3 A. A. Draad [Doctoral Dissertation, Technical University of Delft (1996)] in a carefully controlled
experiment, attained laminar flow up to Re = 60,000. He also studied the nonparabolic velocity profile
induced by the earth's rotation (through the Coriolis effect). See also A. A. Draad and F. Т. М.
Nieuwstadt, /. Fluid. Mech., 361, 207-308 (1998).
4
J. H. Perry, Chemical Engineers Handbook, McGraw-Hill, New York, 3rd edition (1950), pp. 388-389;
W. M. Kays and A. L. London, Compact Heat Exchangers, McGraw-Hill, New York (1958), p. 49.
5
Martin Hans Christian Knudsen (1871-1949), professor of physics at the University of
Copenhagen, did key experiments on the behavior of very dilute gases. The lectures he gave at the
University of Glasgow were published as M. Knudsen, The Kinetic Theory of Gases, Methuen, London
(1934); G. N. Patterson, Molecular Flow of Gases, Wiley, New York (1956). See also J. H. Ferziger and H. G.
Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam (1972), Chapter 15.