Page 202 - Bird R.B. Transport phenomena

186 Chapter 6 Interphase Transport in Isothermal Systems

When these expressions are substituted into Eqs. 6.3-5 and 6, it is then evident that the
friction factor in Eq. 6.3-4 must have the form f(t) = /(Re, t), which, when time aver-
aged over the turbulent fluctuations, simplifies to
/ = /(Re) (6.3-13)
by using arguments similar to those in §6.2. Hence from the definition of the friction fac-
tor and the dimensionless form of the equations of change and the boundary conditions,
we find that / must be a function of Re alone.
Many experimental measurements of the drag force on spheres are available, and
when these are plotted in dimensionless form, Fig. 6.3-1 results. For this system there is
no sharp transition from an unstable laminar flow curve to a stable turbulent flow curve
as for long tubes at a Reynolds number of about 2100 (see Fig. 6.2-2). Instead, as the ap-
proach velocity increases, / varies smoothly and moderately up to Reynolds numbers of
5
5
the order of 10 . The kink in the curve at about Re = 2 X 10 is associated with the shift of
the boundary layer separation zone from in front of the equator to in back of the equator
of the sphere. 1
We have juxtaposed the discussions of tube flow and flow around a sphere to em-
phasize the fact that various flow systems behave quite differently. Several points of dif-
ference between the two systems are:
Flow in Tubes Flow Around Spheres
• Rather well defined laminar-turbulent • No well defined laminar-turbulent
transition at about Re = 2100 transition
• The only contribution to / is the friction • Contributions to / from both friction
drag and form drag
• No boundary layer separation • There is a kink in the/vs. Re curve
associated with a shift in the separation
zone
The general shape of the curves in Figs. 6.1-2 and 6.3-1 should be carefully remembered.
For the creeping flow region, we already know that the drag force is given by Stokes'
law, which is a consequence of solving the continuity equation and the Navier-Stokes
equation of motion without the pDv/Dt term. Stokes' law can be rearranged into the
form of Eq. 6.1-5 to get

2
2
F = T T K ) ( > J L 2 4 ) (6.3-14)
(
k 7
Hence for creeping flow around a sphere
/ = Ц for Re < 0.1 (6.3-15)
and this is the straight-line asymptote as Re —> 0 of the friction factor curve in Fig. 6.3-1.
For higher values of the Reynolds number, Eq. 4.2-21 can describe / accurately up to
about Re = 1. However, the empirical expression 2

/ = f / Ц + 0.5407 Г for Re < 6000 (6.3-16)

1
R. K. Adair, The Physics of Baseball, Harper and Row, New York (1990).
2
F. F. Abraham, Physics of Fluids, 13,2194 (1970); M. Van Dyke, Physics of Fluids, 14,1038-1039 (1971).