Page 195 - Bird R.B. Transport phenomena

§6.2 Friction Factors for Flow in Tubes 179

in which D = 2R is the tube diameter. Equation 6.1-4 shows how to calculate / from ex-
perimental data. The quantity/is sometimes called the Fanning friction factor. 2
(b) For flow around submerged objects, the characteristic area A is usually taken to be
the area obtained by projecting the solid onto a plane perpendicular to the velocity of the
approaching fluid; the quantity К is taken to be \pvl>, where v x is the approach velocity
of the fluid at a large distance from the object. For example, for flow around a sphere of
radius R, we define/by the equation
2
F = (7rR )(lpvi)f (6Л-5) 3
k
If it is not possible to measure F , then we can measure the terminal velocity of the
k
sphere when it falls through the fluid (in that case, У has to be interpreted as the termi-
Ж
nal velocity of the sphere). For the steady-state fall of a sphere in a fluid, the force F k is
just counterbalanced by the gravitational force on the sphere less the buoyant force (cf.
Eq. 2.6-14):

Elimination of F between Eqs. 6.1-5 and 6.1-6 then gives
k

This expression can be used to obtain / from terminal velocity data. The friction factor
used in Eqs. 6.1-5 and 7 is sometimes called the drag coefficient and given the symbol c .
D
We have seen that the "drag coefficient" for submerged objects and the "friction fac-
tor" for channel flow are defined in the same general way. For this reason we prefer to
use the same symbol and name for both of them.

§6.2 FRICTION FACTORS FOR FLOW IN TUBES
We now combine the definition of/in Eq. 6.1-2 with the dimensional analysis of §3.7 to
show what / must depend on in this kind of system. We consider a "test section" of inner
radius R and length L, shown in Fig. 6.2-1, carrying a fluid of constant density and vis-
cosity at a steady mass flow rate. The pressures 2P a n d ^L at the ends of the test section
0
are known.

2
This friction factor definition is due to J. T. Fanning, A Practical Treatise on Hydraulic and Water
Supply Engineering, Van Nostrand, New York, 1st edition (1877), 16th edition (1906); the name "Fanning"
is used to avoid confusion with the "Moody friction factor," which is larger by a factor of 4 than the /
used here [L. F. Moody, Trans. ASME, 66, 671-684 (1944)].
If we use the "friction velocity" v* = Vr /p = VW^&JK/lLp, introduced in §5.3, then Eq. 6.1-4
o
assumes the form
f=2(vj(v)) 2 (6.1-4a)
John Thomas Fanning (1837-1911) studied architectural and civil engineering, served as an officer in the
Civil War, and after the war became prominent in hydraulic engineering. The 14th edition of his book A
Practical Treatise on Hydraulic and Water-Supply Engineering appeared in 1899.
3 For the translational motion of a sphere in three dimensions, one can write approximately
n (6.1-5a)
where n is a unit vector in the direction of v . See Problem 6C.1.
M