Page 194 - Bird R.B. Transport phenomena

178 Chapter 6 Interphase Transport in Isothermal Systems

We start in §6.1 by defining the "friction factor/ 7 and then we show in §§6.2 and 6.3
how to construct friction factor charts for flow in circular tubes and flow around spheres.
These are both systems we have already studied and, in fact, several results from earlier
chapters are included in these charts. Finally in §6.4 we examine the flow in packed
columns, to illustrate the treatment of a geometrically complicated system. The more
complex problem of fluidized beds is not included in this chapter. 1

§6.1 DEFINITION OF FRICTION FACTORS
We consider the steadily driven flow of a fluid of constant density in one of two systems:
(a) the fluid flows in a straight conduit of uniform cross section; (b) the fluid flows
around a submerged object that has an axis of symmetry (or two planes of symmetry)
parallel to the direction of the approaching fluid. There will be a force _^ exerted by the
F f
fluid on the solid surfaces. It is convenient to split this force into two parts: F s, the force
that would be exerted by the fluid even if it were stationary; and F^., the additional force
associated with the motion of the fluid (see §2.6 for the discussion of F s and ¥ k for flow
around spheres). In systems of type (a), ¥ k points in the same direction as the average ve-
locity (v) in the conduit, and in systems of type (b), ¥ k points in the same direction as the
approach velocity v^.
For both types of systems we state that the magnitude of the force F A . is proportional
to a characteristic area A and a characteristic kinetic energy К per unit volume; thus
F k = AKf (6.1-1) 1
in which the proportionality constant / is called the friction factor. Note that Eq. 6.1-1 is
not a law of fluid dynamics, but only a definition for/. This is a useful definition, because
the dimensionless quantity / can be given as a relatively simple function of the Reynolds
number and the system shape.
Clearly, for any given flow system,/is not defined until A and К are specified. Let us
now see what the customary definitions are:

(a) ¥ от flow in conduits, A is usually taken to be the wetted surface, and К is taken to
2
be \p{v) . Specifically, for circular tubes of radius R and length L we define/by
2
F = {2irRL)(\p{v) )f (6.1-2)
k
Generally, the quantity measured is not F , but rather the pressure difference p 0 — p and
L
k
the elevation difference h 0 — h . A force balance on the fluid between 0 and L in the direc-
L
tion of flow gives for fully developed flow
h = VPo ~ Pi) + Pg<h ~ h )]irR 2
0 L
= (9> - VJirR 2 (6.1-3)
0
Elimination of F between the last two equations then gives
k

R. Jackson, The Dynamics of Fluidized Beds, Cambridge University Press (2000).
1
For systems lacking symmetry, the fluid exerts both a force and a torque on the solid. For
1
discussions of such systems see J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Martinus
Nijhoff, The Hague (1983), Chapter 5; H. Brenner, in Adv. Chem. Engr., 6, 287-^138 (1966)'; S. Kim and
S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Boston
(1991), Chapter 5.