Wall Effects on the
Sedimentation of
Spherical Particles
University of Pennsylvania
Department of Bioengineering
BE 310 – Dr. Scherer
Group T3
Natalie Georgakis
Elizabeth Khaykin
Derek Wong
Armaghan Farooq
Abstract
The goal of this experiment was to determine the wall effects on the sedimentation of
spheres dropped through a cylinder of viscous medium. This experiment served as a
model to understand the wall effects that affect the velocity of blood cells traveling
through different sized vessels. This was modeled by dropping Delran spheres of five
different radii, ro, through glycerin in three graduated cylinders of varying radii, R, and
measuring their terminal velocities. The Ladenburg equation was found to give a
relationship between ro/R and the measured terminal velocities. The data best fit this
theory in a ro/R range of 0.0466  0.001 to 0.186  0.011. Bounded terminal velocities
were then estimated for a blood cell in plasma with ro/R ratios in that range. Showing
that the theory worked best in that range, the bounded terminal velocity for a red blood
cell in plasma with a ro/R = 0.0008 was found to be 6.26 x 10-6 m/s. For ro/R = 0.0294
the bounded terminal velocity was found to be 4.45 x 10-6 m/s.
Background
The motion of red blood cells in the body is an important field of study in bioengineering
today. Because red blood cells travel within blood vessels, their velocity is affected by
the system in which they are contained. This study attempts to model the effects of blood
vessel walls in a simplified laboratory experiment using spheres and graduated cylinders.
The following graphs demonstrate how the velocity of blood varies within different
vessels of the circulatory system. It is clear from these graphs that blood velocity is
inversely proportional to the total cross-sectional area of the blood vessel, whether
arteries, capillaries, or veins.
Figure 1 - Blood Velocity across the Circulatory System
According to Figure 1, blood velocity is highest in the arteries and veins where the radii
are large, and blood velocity is lowest in the capillaries where the radii are smallest. In
this experiment, different ratios of sphere to cylinder radii will model how red blood cells
react within different sized vessels. It is their ratio that will determine the degree of
sensitivity to the vessel walls. The following mathematical model illustrates exactly what
relationship can be derived between the motion of an erythrocyte in a blood vessel as
opposed to an open and infinitely wide container.
Mathematical Model
It has been hypothesized and proven that an object falling in a bounded medium will
experience a larger drag force than if the medium was infinite (unbounded). The
additional drag results from the forces of the walls “binding” the fluid medium. As a
consequence of a higher drag, or an increased force in the opposite direction of
movement, the same particle will move with a slower terminal velocity than if the walls
were not present. It is essential to also consider the size and shape of the particle falling
because of the effect on drag. Spherical particles were observed due to their accessibility
and the resources of information available on their drag properties. An equation relating
the terminal velocities of both conditions was derived in order to understand this concept
in mathematical terms.
First, the drag on a sphere can be described by a force balance of the buoyancy,
gravitational, and drag forces. Figure 2 depicts how these forces interact and Equation 1
explains them mathematically.
Figure 2 – Forces Acting on Sphere During Sedimentation
Ds
r
Fbuoy
Fgrav
Equation 1 – Force Balance on Sphere
Ds  Fgrav  Fbuoy
 g (msphere  mliquid )
 gVsphere sphere  liquid 
Where the variables are:
Fgrav – the gravitational force (Newton)
Fbuoy – the buoyancy force (Newton)
g - the gravitational acceleration (9.81 meters/second2)
msphere – the mass of the spherical particle (kilogram)
mliquid – the mass of the liquid displaced by the spherical particle (kilogram)
Vsphere – the volume of the sphere and of the displaced liquid (cubic meters)
sphere – the density of the spherical particle (kilograms/ cubic meters)
liquid – the density of the liquid (kilogram/ cubic meters)
In addition, Stokes’ Law of Drag (Equation 2) describes the general relationship of
terminal velocity to drag force.
Equation 2 – Stokes’ Law of Drag on a Sphere
Ds  6 *  * U T * rsphere
Where the variables signify:
Ds – Stokes’ Drag force on a spherical particle (Newton)
Ut – terminal velocity of the sphere (meters/second)
rsphere – radius of the spherical particle, also as ro (meters)
In specific, Equation 2 describes the relationship for drag when the medium is
unbounded. Thus, a correction factor must be taken into account when the medium is
bounded. Ladenberg researched this relationship and found that the correction factor
depended on the ratio of the particle’s radius, ro, to the cylinder’s radius, R. His equation
is shown below (Equation 3) and proves that “binding” the medium causes an increase in
drag force.
Equation 3 – Ladenberg Equation
r
Db  Ds (1  2.1044 0 )
R
where: Db - bounded drag force (Newton)
R – cylinder radius (meter)
At this point, the equations are manipulated to be in terms of terminal velocity. We can
equate both bounded and unbounded drag to the force balance on the sphere (Equation 4
a, b) to derive one equation that relates unbounded (Uu) to bounded (Ub) terminal
velocities.
Equation 4a – Unbounded Drag Equals Force Balance
gVsphere  sphere   liquid   6U u ro
where: Uu – unbounded terminal velocity (meters/second)
Equation 4b – Bounded Drag equals Force Balance
r
gVsphere sphere  liquid   6U b ro (1  2.1044 0 )
R
where: Ub – bounded terminal velocity (meters/second)
To relate these two equations together only through terminal velocity, the bounded is
divided by the unbounded in order to get Equation 5.
Equation 5a – Ratio of Bounded Terminal Velocity to Unbounded Terminal Velocity
1
Ub 
r 
1  2.1044 o 
Uu 
R
Or manipulated to be in linear equation form:
Equation 5b – Linear form of Ratio of Bounded to Unbounded Terminal Velocity
Uu
r
 1  2.1044
Ub
R
Now, having formulated a mathematical approach, the experimentation phase began. The
Methods and Materials section explains how to repeat this experiment in the future (see
Appendix). It can be used for future reference in the BE310 course.
Results and Analysis1
The following data was calculated by measuring the length a sphere traveled down the
cylinder and dividing this value by the time it took in seconds:
Table 1 – Experimental Terminal Velocities for Three Cylinders
ro (m)
Ub (m/s)
Ub (m/s)
(for R = 0.05105m)
( for R = 0.01745m)
0.00635
0.007571
0.003437
0.00476
0.004716
0.003035
0.00397
0.003173
0.002362
0.00318
0.002354
0.002087
0.00238
0.001515
0.001321
Ub (m/s)
(for R = 0.0128m)
0.002363
0.002277
0.001828
0.001534
0.001127
These results are plotted on the same graph in Figure 3 to compare the terminal
velocities reached by different sized spheres dropped in the three cylinders.
Figure 3 - Experimental Terminal Velocities for Three Cylinders
To determine whether our flow is laminar the Reynolds numbers were computed using
the experimental bounded terminal velocities with Equation 6 below:
1
In the results the following notation is implemented:
Ub – unbounded terminal velocity R – radius of cylinder
Uu – bounded terminal velocity
ro – radius of sphere
Equation 6 – Particular Reynolds number
Re 
2 * U b * r0

Table 2 – Reynolds Numbers (unitless)
ro (m)
Re for R = 0.05105 m
0.00635
1.34E-04
0.00476
6.26E-05
0.00397
3.51E-05
0.00318
2.08E-05
0.00238
1.01E-05
Re for R = 0.01745 m
6.08E-05
4.03E-05
2.61E-05
1.85E-05
8.77E-06
Re for R = 0.0128 m
4.18E-05
3.02E-05
2.02E-05
1.36E-05
7.48E-06
According to BE310 Lab Maunual Spring 1998, for a Reynolds number much smaller
than 1 (i.e. Re<<1), the flow is laminar. Thus, all of the flow was determined to be
laminar, and the data found below was computed based on this assumption.
From the following known data, the defined drag force on the sphere was found for each
ro using Equation 1 and plugging in the known parameters.
Table 3 – Theoretical Unbounded Drag Force on Sphere
ro
Volume, V Density of
Density of Liquid
Sphere, sphere
Displaced, liquid
m
M3
Kg/m3
kg/m3
0.00635
1.07E-06
1400
1256.7
0.00476
4.52E-07
1400
1256.7
0.00397
2.62E-07
1400
1256.7
0.00318
1.34E-07
1400
1256.7
Theoretical Unbounded Drag Force
on Sphere, Ds
N
0.001507
0.000636
0.000368
0.000188
Using this defined drag force, the unbounded terminal velocity (Uu) was calculated with
Equation 4a for each value of ro. This theoretically found value for Uu was used in each
case to compare with the experimental Ub.
Table 4 – Unbounded Terminal Velocity vs. Experimental Bounded Terminal Velocity
ro (m)
R (m)
ro/R
Uu (m/s)
Ub (m/s)
Uu/Ub
0.05105
0.124388
0.017548
0.007571
2.317627
0.00635
0.01745
0.363897
0.017548
0.003437
5.105094
0.01280
0.496094
0.017548
0.002363
7.425944
0.05105
0.093291
0.009871
0.004716
2.092969
0.00476
0.01745
0.272923
0.009871
0.003035
3.252269
0.01280
0.372070
0.009871
0.002276
4.336113
0.05105
0.077742
0.006855
0.003173
2.160524
0.00397
0.01745
0.227436
0.006855
0.002362
2.901781
0.01280
0.310059
0.006855
0.001828
3.750604
0.05105
0.062194
0.004387
0.002354
1.863477
0.00318
0.01745
0.181948
0.004387
0.002087
2.102418
0.01280
0.248047
0.004387
0.001534
2.859833
0.05105
0.046645
0.002468
0.001515
1.628585
0.00238
0.01745
0.136461
0.002468
0.001321
1.867713
0.01280
0.186035
0.002468
0.001127
2.190654
The following figure depicts Uu/Ub vs. ro/R, which are the two unitless parameters that
contain all of the data, as defined in Equation 5b. The two equations represent the two
fit curves, one being linear and the other parabolic. The theoretical curve (from
Ladenburg’s Equation) is also plotted. (Error bars are calculated by differential methods
as described in Error Analysis section below.)
Figure 4 – Dimensionless Parameters
Uu/Ub
Uu/Ub vs. ro/R
9
8
7
6
5
4
3
2
1
0
y = 28.654x2 - 3.1987x + 1.9752
R2 = 0.9754
y = 11.128x + 0.6836
R2 = 0.8699
Theoretical
0
0.1
0.2
0.3
0.4
0.5
0.6
ro/R
The following table contains the experimentally found Ub sorted in ascending magnitude
and its associated theoretical Ub, found by Equation 5b. The percent error is calculated
between the two Ub values.
Table 5 – Experimental Ub vs. Theoretical Ub
Ub Experimental Sorted by
Ascending Magnitude(m/s)
0.001127
0.001321
0.001515
0.001534
0.001828
0.002087
0.002276
0.002354
0.002362
0.002363
0.003035
0.003173
0.003437
Ub Theoretical
(m/s)
0.001774
0.001917
0.002247
0.002882
0.004148
0.003172
0.005536
0.003879
0.004636
0.008585
0.00627
0.005891
0.009938
% Error
36.48
31.08
32.57
46.78
55.94
34.22
58.88
39.31
49.04
72.48
51.59
46.14
65.41
The five data points with the lowest percent error are in red color. These data points are
plotted in the following figure (Figure 5). (Error bars are calculated by differential
methods as described in Error Analysis section below.)
Figure 5 - Uu/Ub vs. ro/R found by using points with lowest % error
Uu/Ub vs ro/RS Selected for lowest %
Error
y = 3.0785x + 1.553
R2 = 0.8261
3
Uu/Ub
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
ro/R
The figure below includes only the five lowest values of ro/R: (Error bars are calculated
by differential methods as described in Error Analysis section below.)
Figure 6 - Uu/Ub vs. ro/R found by using smallest five points for ro/R
Uu/Ub vs. ro/R: first 5 points
y = 2.4224x + 1.6588
R2 = 0.4691
2.5
Uu/Ub
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
ro/R
In Figures 4, 5, and 6 the y-intercept was not set to any value.
Discussion
Table 1 gives the experimental bounded terminal velocities as measured with a ruler and
stopwatch. The data were plotted in Figure 3. It is very obvious from Figure 3 that for
the same value of R (radius of cylinder), the terminal velocity, Ub, increases with
increasing ro.
To properly determine the effects of the walls upon this velocity, we must compare this
Ub with the unbounded terminal velocity Uu. Unbounded velocity is found theoretically
using the definition of drag force, Ds from the force balance equation (Equation 4a).
Thus, our Uu value is always theoretical and only changes with the masses (volume times
density) of the spheres as shown in Table 3. However, the force balance equation only
applies when the fluid following around the sphere is laminar. By calculating the
Reynolds number for each case, it was found that they are all much less than 1 (Re<<1),
as shown in Table 2, and the flow was, indeed, laminar.
The values for ro/R and Uu/Ub are given in Table 4 and plotted in Figure 4. Using the
“trendline” function in Excel, a linear function and a second-order polynomial function
were found that best approximate all of the data. The values of residual squared (R2) for
each fit equation show that the second-order equation (R2 = 0.9754) compared to that of
the linear equation (R2 = 0.8699) is closer to one, which suggests that the polynomial
equation provides a better fit. Another way of interpreting this outcome is that as the
ratio ro/R increase, the ratio of Uu/Ub does not increase linearly. This suggests that the
factor, (2.1044 x ro/R), in the Ladenberg Equation (Equation 3) does not stay constant
for the entire range of ratios of ro/R. This finding will be furthered explained in the
analysis below.
In order to find the range of data points that best coincide with the theoretical values
calculate from Ladenburg’s equation (Equation 3), the percent error was calculated
between the experimental values of Ub and the theoretical ones as shown in Table 5. The
five best points (the bolded entries of Table 5) appeared in the lowest eight values (from
(Ub = 0.001m/s to 0.002m/s) of the experimentally measured Ub (column 1 of Table 5).
This shows that Ladenburg’s equation is best when applied to the cases with the lowest
bounded terminal velocities. These five data points are plotted in a Uu/Ub vs. ro/R graph
in Figure 5. Using the trendline function a linear approximation is obtained that gives a
more accurate slope of 3.078 (the theoretical slope being 2.1044, from Ladenburg’s
Equation), a y-intercept of 1.553, and an R2 value of 0.8261. However, because lower
terminal velocities correspond to smaller radii as shown in Figure 3, this also implies that
the Ladenburg’s equation works best for smaller ro/R ratios. This indicates that lower
values of ro/R (between 0.0466 and 0.186) are more accurate than higher values.
The reason that smaller ro/R values are closer to theory, as suggested by Ladenburg’s
Equation, was explained by Chang (1961). Initially, Equation 3 was derived from the
following:
Equation 6 – Chang equation involving Oseen Function
r
r
D
 1  2.1044 0  O( 0 ) 2
Ds
R
R
where the Oseen function is a second-order polynomial function that takes into account
the change in moment caused by frictional forces (Brenner and Happel, p. 210), which
explains the higher accuracy of the second-order fit equation as shown in Figure 4. To
transform the Oseen equation into Ladenburg’s Equation (Equation 3), the assumption
that ro/R is very small is made. This explains why the lower values of ro/R (between
0.0466 and 0.186) are closest to the theory.
To further test how much better the low values of ro/R approximate the theory, a Uu/Ub
vs. ro/R graph of the five points with the lowest values of ro/R are plotted in Figure 4.
The slope obtained, 2.4224 is much closer to the theoretically desired slope of 2.1044.
The y-intercept is 1.6588, which is not as close to 1 as taking the best 5 of 8 points.
However, the R2 value (0.4691) is much less which indicates that the data points had a
tendency to fluctuate more from the linear function obtained by Excel.
Error Analysis
Differential methods were applied to calculate the errors of the various parameters. The
uncertainty of the radii of the spheres, dro, and the radii of the cylinders, dR, were
estimated to be 5 x 10-5 m and 0.0005 m respectively. The error of the ratio d(ro/R) in
each case was found differentially using the following equation:
1
r  r
d  o   o2  dR   dro
R
R R
The uncertainty of the measured bounded terminal velocities, dUb, were found by the
standard deviation of the 5 trials. The error of the theoretical unbounded terminal
velocities, dUu, were found by rearranging Equation 4a, isolating Uu, and then applying
differential methods, resulting in the following equation:
dU u 
g
6
 (  sphere   liquid )  dV V  d liquid V  (  sphere   liquid )




 dro 
2
ro
ro
ro


where dV and dliquid, are the errors for the volume of the spheres and the density of the
fluid respectively. The error for density is estimated to be 0.1 Pa.s. The error for the
ratio Uu/Ub, was also calculated using differential methods using the following equation:
U  U
1
d  u   u2  dU b 
 dU u
Ub
Ub  Ub
Here is a summary of the results of the error calculations:
dro = 5 x 10-5 m (or 0.00005 m)
dR = 0.0005 m
dliquid = 0.1 Pa.s
Table 6 – Summary of Results from Error Calculations
ro (m)
R (m)
d(ro/R)
DV
0.00635
0.05105
0.00220
7.16829E-05
0.01745
0.01329
6.97985E-05
0.01280
0.02329
0.000134335
0.004763
0.05105
0.00189
5.57438E-05
0.01745
0.01069
8.01747E-05
0.01280
0.01844
0.000118575
0.003969
0.05105
0.00174
1.7722E-05
0.01745
0.00932
3.95237E-05
0.01280
0.01601
4.13098E-05
0.003175
0.05105
0.00158
2.85339E-05
0.01745
0.00807
9.80359E-05
0.01280
0.01359
5.32554E-05
0.002381
0.05105
0.00143
4.97555E-05
0.01745
0.00677
0.000145835
0.01280
0.01117
6.25825E-05
dUu
0.0010
0.0010
0.0010
0.0006
0.0006
0.0006
0.0004
0.0004
0.0004
0.0003
0.0003
0.0003
0.0002
0.0002
0.0002
dUb
7.16829E-05
6.97985E-05
0.000134335
5.57438E-05
8.01747E-05
0.000118575
1.7722E-05
3.95237E-05
4.13098E-05
2.85339E-05
9.80359E-05
5.32554E-05
4.97555E-05
0.000145835
6.25825E-05
d(Uu/Ub)
0.1547
0.3961
0.8475
0.1520
0.2836
0.4895
0.1495
0.2331
0.3233
0.1490
0.2414
0.2933
0.1754
0.3459
0.2857
Biological Example
The equations derived in this project can be applied to biological conditions to
understand how the walls of different blood vessels can affect blood velocity. The
following parameters were found about two different blood vessels in the circulatory
system. From these values we can use our theory for bounded velocity to compare the
two conditions.
The average radius of a red blood cell is
ro = 10 microns
The average radius of a retinal artery is
Rr = 51.5microns
The density of a red blood cell is
c = 1090 g/L
The average viscosity of blood is
 = 1.39 mPa*s
The average radius of the aorta is
Ra = 1.25cm
The density of blood plasma is
p = 1050 g/L
Equation 4b is used to calculate bounded velocity, Ub. For the retinal artery, bounded
velocity is 4.45 x 10-6 m/s, and for the aorta, is 6.26 x 10-6 m/s. Since the unbounded
velocity does not depend on the radius of the vessel, it is the same for both the retinal
artery and the aorta, Uu = 6.27 x 10-6. The following graph shows how the two relate to
each other (Figure 7) through the Ladenburg Equation (Equation 5b).
Figure 7 – Ladenburg Relationship shown for Retinal and Aortic arteries
Uu/Ub
Ladenburg Equation for Arteries
1.60E+00
1.40E+00
1.20E+00
1.00E+00
8.00E-01
6.00E-01
4.00E-01
2.00E-01
0.00E+00
0.00E+
00
y = 2.1067x + 0.9999
R2 = 1
5.00E02
1.00E01
1.50E01
2.00E01
2.50E01
ro/R
Future Investigation
Because the existing model of blood flow in blood vessels is simplified, there is much
room for further study of this problem. First, viscosity can also be varied in order to
model the finding that males have more viscous blood than females. This difference in
viscosity has also been linked to atherosclerosis by a study at the University of Edinburgh
Medical School (Reference #5). Interestingly, this study also showed that males have
higher blood velocity than women. Another modification of this experiment would be to
vary the shapes of the particles. In particular, biconcave disks would better model the
actual shape of a red blood cell. Our hope is that this project will serve as a basis for
numerous additional adaptations.
References
1. BE310 Laboratory Manual Spring 1998 (Laminar Viscous Flow: Very Slow
Motion)
2. Brenner, Effect of finite boundaries on the Stokes resistance of an arbitrary
particle, Chemical Engineering Science Volume 18, p.35, (1963)
3. Brenner & Happel, Slow viscous flow past a sphere in a cylindrical tube, Journal
of Fluid Mechanics Volume 4, p195, (1958).
4. Hinghofer-Szalkay H. Method of high-precision microsample blood and plasma
mass densitometry. Journal of Applied Physiology. 60(3): 1082-8, 1986 Mar.
5.
http://www.pathfinder.com/living/latest/RB/1998Apr20?726.ht
ml , “Sticky Blood Linked to Heart Disease”.
6.
http://ucrwcu.rwc.uc.edu/koehler/biophys/3e.html , “Blood
Velocity and Turbulence”.
7.
http://www.medic.mieu.ac.jp/2NDCNG/POSTERS/DG0407/RES.HTM “Demonstration
of human pulsetile retinal blood flow”.
Appendix A
Methods and Materials
Preparation of the Cylinders
Empty the pure glycerin solution from the marked storage bottle into three different
graduated cylinders of sizes: 4L, 250mL, and 100mL. Before filling each cylinder place
a rubberband around it near the top. After the cylinder is filled, carefully slide the rubber
band down until it goes around the cylinder about 3 in below the liquid surface and is
level with the tabletop.
Viscosity and Density Measurements
Pour a small amount of glycerin into a beaker and measure and record its viscosity using
the viscometer. Refer to the instruction manual for the viscometer operation procedures.
Other “helpful hints” on the viscometer are also available. The readout times factor is
tabulated for determining the viscosity of your sample. Use the chart for CP-40 with ½
ml sample. Since it is possible to use other solutions of glycerin-water, you may refer to
the solution viscosity chart for determining the percentage solution for obtaining the
desirable CPS for glycerol. Pour some of each solution into a densitometer flask and
weigh it to determine and record the solution density.
Preparation of the Delran Spheres
Pick out five spheres for at least five different diameters ranging from 0.002 – 0.007 m.
Starting with the smallest spheres and working up to the largest, gently place each sphere
in a spoon and submerge it in the liquid at the top of graduated cylinder to wet it. You
may need to use your finger or another spoon to make sure it is thoroughly wetted. Then,
place the sphere in the center of the graduated cylinder below the liquid surface, gently
slide the sphere off the spoon and let it sink to the bottom of the cylinder. Be extra
careful not to push the sphere, but to let it fall on its own.
Terminal Velocity Measurements
As the sphere sinks, record the time of its fall, starting as it passes the top rubber band
and stopping as it passes the lower rubber band. Be sure to record the length of the fall,
too. To prove to yourself that the sphere reaches terminal velocity, use the first trial to
record velocity in several segments at the top of the cylinder. Where the velocity
becomes stable is the point at which you should place the first rubber band for subsequent
trials. This must be done for all three cylinders.
Clean Up
Once the spheres have reached the bottom of these cylinders, do not try to remove them
until the glycerin solution experiments are over and you are cleaning up. Pour the pure
glycerin solution back into the storage bottle. Carefully collect the spheres with the
particle salvaging funnels and rinse them in a beaker of pure water. Rinse the graduated
cylinders thoroughly.