On the Settling of Spheres
Roger F. Gans
March 1988
Abstract
Four different groups of students measured the settling speeds of a total of eighty
nominally three millimeter diameter and eighty nominally six millimeter diameter nominal spheres
in nominally pure glycerin. Analysis shows that the Stokes law prediction that the drag
coefficient is equal to twenty-four divided by the Reynolds number is obeyed. The precision of
the measurements is not high enough to detect discrepancies caused by wall effects.
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Introduction
Nineteen groups of students were asked to perform the following experiment in the junior
fluids lab, ME 241: Glass balls, approximately spherical, with nominal diameters of 3 mm and 6
mm, were allowed to fall through nominally pure glycerin in a 2000 ml graduated cylinder to
assess the accuracy of the Stokes law prediction that the drag coefficient CD is equal to 24/Re,
where Re is the Reynolds number, defined by
Re = Ud/ν
(1)
where U is the speed of fall, d the diameter of the sphere and ν the kinematic viscosity of the
glycerin. The drag coefficient is defined in the usual way, such that the drag force
FD = CD•ρ U2πd2/8
(2)
where ρ denotes the density of the glycerin.
Twenty balls of each size were allowed to fall the full height of the cylinder. The first ten of
these were timed for nearly the full fall, and the second ten for the bottom half only. Were the
time necessary to attain terminal velocity a significant fraction of the measured time, the first set
of balls would appear to fall more slowly. This was, in general, observed. However, the fall
appears to have obeyed the Stokes law anyway, the acceleration terms being very small, so that a
quasisteady Stokes description applies to all the data.
The data, properly analyzed, confirm the predictions of Stokes law within experimental
error. I will discuss the experiment in detail in this note, and illustrate my conclusions with data
from groups 1, 2, 11 and 12. There is nothing special about these groups. They were the groups
I was grading at the time I was preparing this note.
Procedure
The students were asked to weigh each sphere, using a scale that measured "weight" to the
nearest 10 mg. As the three millimeter spheres had masses in the range of 30 to 50 mg, this was
insufficiently precise to resolve the masses clearly. The errors to be expected for the small
spheres are quite large. The larger spheres had masses ranging from 270 to 300 mg, large
enough to be measured to within 3% or so.
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The students measured the diameters of the spheres using a micrometer. The precision of
the micrometer was nominally 10 µm, so the precision was adequate for the task. The difficulty
that arose here was that the "spheres" were not spherical. After wrestling with this problem, the
students appear to have chosen undefined averages for the sphere diameters. As will be seen, this
is probably not a large source of error.
Dropping and timing the spheres was straightforward. The distances (typically about 300
mm and 150 mm) were measured to within 1 mm, and the timing was by digital timer, manually
operated. The timer was read to the nearest 0.01 s, and I estimate the potential timing error to be
less than 0.25 s, about 2%.
Results
I took the students' raw data for mass, diameter and settling speed and calculated the
Reynolds number and drag coefficient for each sphere. The Reynolds number was calculated
from equation (1) and the drag coefficient from
CD = 8{m - ρ πd3/6}g/{πρ d2U2}
(3)
which is equation (2) with the drag force specified as the difference between the gravity force and
buoyancy force on the sphere. Here m and g denote the mass of the sphere and the acceleration
of gravity, respectively.
I did not have measured values for the physical properties of the glycerin. I used the
nominal values ρ = 1258.23 kg/m3 and ν = 0.00118 m2/s. As I needed the results for 160
spheres, I wrote a small Pascal program to do the work. This, together with the output, appears as
appendix A.1 (The program is not well commented, but it is simple enough that it ought to be
readable.)
As the output tables in the appendix make clear, the results for the small balls are difficult to
interpret. The Reynolds number is independent of the mass of the ball, and so does not vary
dramatically. The drag coefficient depends strongly on the mass and varies wildly. There would
be less scatter in the data if all the balls were assumed to have the same mass, that given by
1
Appendix deleted from this text.
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averaging the mass of many of the balls. One could perhaps do even better by finding a mean
density and adjusting the mean mass by the measured diameter. I have not tested these ideas.
The large ball data show that the measured drag coefficient is remarkably close to the
prediction of 24/Re. The balls that fell the shorter distance fell faster, suggesting that the other
balls spent some of their fall at less than terminal velocity. This does not affect the agreement
between theory and observation. Apparently the acceleration terms are negligible even during
acceleration. (Analysis of the transient problem is possible, but will not be given here. It is
beyond the scope of this note.)
Figures 1 and 2 show the large ball data and the small ball data, respectively. Both figures
have been drawn so that the filled in symbols refer to the short (fast) fall data and the open
symbols to the long (slow) fall data. The large ball data have some scatter, but appear to obey the
Stokes prediction. The small ball data have a large amount of scatter, too much to be sure which
is more significant: (1) that the data are equally placed on both sides of the line, suggesting that
the theory is correct; or (2) that the trend seems steeper than theory would predict, suggesting that
the theory does not apply. The scatter is presumably caused primarily by the scatter in measured
mass.
Group 1 (L)
Group 1 (s)
Group 2 (L)
Group 2 (s)
Group 11 (L)
Group 11 (s)
Group 12 (L)
Group 12 (s)
Theory
3000
2000
Drag Coefficient
4000
1000
Reynolds number
0
0.005
0.015
Figure 1 Small Ball Data
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0.025
Group 1 (L)
Group 2 (s)
Group 1 (s)
Group 2 (L)
Group 11 (s)
Group 11 (L)
Group 12 (s)
Group 12 (L)
Theory
300
200
Drag coefficient
400
Reynolds number
100
0.07
0.08
0.09
0.10
0.11
Figure 2 Large Ball Data
3000
2000
Drag coefficient
4000
large balls
small balls
Theory
1000
Reynolds number
0
1.36e-20
2.00e-2
4.00e-2
6.00e-2
8.00e-2
Figure 3. Data for Large and Small Balls on the same Graph
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1.00e-1
1.20e-
Figure 3 shows all of the data on the same graph, with the theoretical curve. The data are
clearly consistent with the theoretical prediction, and with each other. I would guess that the small
ball data are dominated by scatter to the extent that the apparently too steep trend is not
significant.
Discussion and conclusions
The basic question about this laboratory is whether one can detect the influence of the walls
on the drag coefficient. If the walls were within an Oseen length of the balls, one would expect
some influence. The Oseen length is the characteristic viscous length scale for a moving object,
n/U. (The question of corrections to Stokes law is a complicated one, beyond the scope of this
note. Proudman and Pearson (1957) gave the first clear exposition. A good exposition appears
in Van Dyke's book (1975).)
For the present experiment the typical velocity for the large balls is 17 mm/s, and for the
small balls 5 mm/s. Thus the Oseen lengths, assuming n = 0.00118 m2/s, are 69 mm and 236
mm, respectively. Note that the effects of the wall would be expected to be more important for the
small balls, contrary to a simple intuitive idea that the ratio of ball diameter to tube diameter is the
important parameter. In any event, it seems likely that some wall effects will exist in both cases.
The question becomes whether they can be detected, and whether they have been detected. To
answer this question it is necessary to consider the probable errors associated with the calculation
of Re and CD.
U is calculated from the length of fall, H, and the time of fall t. These are independent, so
the relative error in U is the square root of the sum of the squares of these relative errors. The
upper bound on these is 0.02, and it is reasonable to set ∆U/U = 0.02. That will be done.
The Reynolds number depends on U and d. The error in d is hard to estimate. The reports
do not give a result for the degree of out of round, the difference in d at various places on a single
ball. It is clearly larger than the nominal precision of the micrometer of 50 µm. A conservative
estimate for this error is 0.05, which will dominate the error in U. Adding the sums of the
squares and taking the square root gives ∆(Re)/Re = 0.05.
The expression for CD, equation (3), is more complicated. Using the rules for combining
independent errors and simplifying the various derivatives leads to an approximate expected
variation in the drag coefficient
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∆(CD)/CD = 2{(∆m/m)2 + (∆U/U)2 + (∆d/d)2}1/2
(4)
For the smallest sphere, with m = 30 mg and ∆d of the order of 1 mg, the error is CD can be of
the order of 60%. The data are scattered around the nominal theory by about this much, so that
no conclusions can be drawn about the influence of the walls.
References
Proudman, I. & Pearson, J. R. A. 1957 "Expansions at small Reynolds number for the flow past
a sphere and a cylinder" J. Fluid Mech. 2 237-262.
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics 2nd ed. Parabolic Press: Stanford
pp149-160.
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