Page 211 - Bird R.B. Transport phenomena

Problems 195

For turbulent flow, an approximate boundary layer treat- 6B.9 Friction factor for flow past an infinite cylinder. 4
ment based on the 1/7 power velocity distribution leads to The flow past a long cylinder is very different from the
5
2Г = 0.073pil R ^fi/R ilp (6B.4-3) flow past a sphere, and the method introduced in §4.2 can-
2
2
Express these results as relations between / and Re. not be used to describe this system. It is found that, when
the fluid approaches with a velocity v , the kinetic force
6B.5 Turbulent flow in horizontal pipes. A fluid is acting on a length L of the cylinder is x
flowing with a mass flow rate w in a smooth horizontal
pipe of length L and diameter D as the result of a pressure F,= (6B.9-1)
difference p 0 - p . The flow is known to be turbulent. In (7.4/Re)
L
The pipe is to be replaced by one of diameter D/2 but The Reynolds number is defined here as Re =
with the same length. The same fluid is to be pumped at Equation 6B.9-1 is valid only up to about Re = 1. In this
the same mass flow rate w. What pressure difference will range of Re, what is the formula for the friction factor as a
be needed? function of the Reynolds number?
(a) Use Eq. 6.2-12 as a suitable equation for the friction factor. 6C.1 Two-dimensional particle trajectories. A sphere of
(b) How can this problem be solved using Fig. 6.2-2 if Eq. radius R is fired horizontally (in the x direction) at high ve-
6.2-12 is not appropriate? locity in still air above level ground. As it leaves the pro-
Answer: (a) A pressure difference 27 times greater will be pelling device, an identical sphere is dropped from the
needed. same height above the ground (in the у direction).
6Б.6 Inadequacy of mean hydraulic radius for laminar (a) Develop differential equations from which the particle
flow. trajectories can be computed, and that will permit compar-
of the behavior
the two
effects
ison
(a) For laminar flow in an annulus with radii KR and R, of fluid friction, and of make the spheres. Include the steady-
that
assumption
use Eqs. 6.2-17 and 18 to get an expression for the average state friction factors may be used (this is a "quasi-steady-
velocity in terms of the pressure difference analogous to state assumption").
the exact expression given in Eq. 2.4-16.
(b) What is the percentage of error in the result in (a) for (b) Which sphere will reach the ground first?
(c) Would the answer to (b) have been the same if the
sphere Reynolds numbers had been in the Stokes' law
Answer: 477c region?
6B.7 Falling sphere in Newton's drag-law region. A
sphere initially at rest at z = 0 falls under the influence of Answers: (a)-=--j
gravity. Conditions are such that, after a negligible inter-
val, the sphere falls with a resisting force proportional to
the square of the velocity. dt
(a) Find the distance z that the sphere falls as a function of t. in which/ = /(Re) as given by Fig. 5.3-1, with
(b) What is the terminal velocity of the sphere? Assume
that the density of the fluid is much less than the density of
the sphere. Re =
2
Answer: (a) The distance is z = (l/c g) In cosh cgt, where 5 7
2
c = |(0.44)(p/p )(l/gR); (b) 1/c 6C.2 Wall effects for a sphere falling in a cylinder. "
sph (a) Experiments on friction factors of spheres are generally
6B.8 Design of an experiment to verify the /vs. Re chart performed in cylindrical tubes. Show by dimensional
for spheres. It is desired to design an experiment to test analysis that, for such an arrangement, the friction factor
the friction factor chart in Fig. 6.3-1 for flow around a for the sphere will have the following dependence:
sphere. Specifically, we want to test the plotted value / = 1
at Re = 100. This is to be done by dropping bronze spheres f = f(Re,R/R ) (6C.2-1)
cyl
2
3
(Psph = 8 g/cm ) in water (p = 1 g/cm , д = 10" g/cm • s). Here Re = IRv^p/1±, in which R is the sphere radius, v x is
3
What sphere diameter must be used? the terminal velocity of the sphere, and R cyl is the inside
(a) Derive a formula that gives the required diameter as
a function of /, Re, g, /x, p, and p sph for terminal velocity
conditions.
4 G. K. Batchelor, An Introduction to Fluid Dynamics,
(b) Insert numerical values and find the value of the
sphere diameter. Cambridge University Press (1967), pp. 244-246, 257-261. For
flow past finite cylinders, see J. Happel and H. Brenner, Low
3/ReV
Answers: (a) D = (b) D = 0.048 cm Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague
(1983), pp. 227-230.
4(p sp h -