Experimental determination of the
terminal settling velocity of
a falling solid sphere
Practical 16 for 2nd year BSc-ME students
Course code WB2543
Photographs: 2 bowling balls with a diameter of 0.216 m entering water with an
impact velocity of 7.6 m/s. The left ball is smooth, the right ball has a patch of
sand roughness on its nose. Photographs taken from [P1].
Version 5
Last modified: November 16th, 2018
Author:
Dr.ir. Wim-Paul Breugem / dr. René Delfos
TU Delft, 3ME Process&Energy
w.p.breugem@tudelft.nl
Preface
This manual has been developed for a practical on the determination of the
terminal settling velocity of a falling solid sphere. It is intended for 2nd year BScME students at the Faculty of 3mE of TU Delft. The experimental setup was
bought from G.U.N.T. Gerätebau GmbH in 2014. The present manual is in part
based on the manual provided by G.U.N.T. for this setup (Report HM 135: Drag
Coefficient for Spheres). I’m open for suggestions to improve this manual; please
send your remarks to w.p.breugem@tudelft.nl .
Wim-Paul Breugem
Delft, October 29th, 2014
The manual has been revised based on the feedback from the students of last
year. Among others, an appendix was added in which the working principle and
use of a hydrometer is explained.
Wim-Paul Breugem
Delft, November 7th, 2015
I added a few remarks to clarify a few issues. The remarks are put in red.
Wim-Paul Breugem
Delft, November 30th, 2015
I added a paragraph on path instabilities of a freely falling sphere. Furthermore, a
few typos were removed.
Wim-Paul Breugem
Delft, November 6th, 2017
Small structural changes were applied.
René Delfos
Delft, November 16th, 2018
[ 2 / 28 ]
Contents
1. Motivation and learning goals ........................................................................................ 4 2. Theory ............................................................................................................................. 6 2.1 Drag coefficient ............................................................................................................ 6 2.2 Estimate of settling distance ......................................................................................... 8 2.3 Estimate of the terminal settling velocity ................................................................... 10 2.4 Path instabilities of freely falling spheres ................................................................... 11 3. Experimental setup........................................................................................................ 13 4. Carrying out the experiments ........................................................................................ 15 4.1 Material properties of the spheres ............................................................................... 15 4.2 Properties of the fluids ................................................................................................ 16 4.3 Experimental determination of the terminal settling velocity..................................... 17 4.4 Theoretical prediction of the terminal settling velocity .............................................. 19 5. Discussion ..................................................................................................................... 21 Literature ........................................................................................................................... 22 Photographs....................................................................................................................... 22 Appendix A. Determining the settling velocity solving a non-linear implicit equation. .. 23 Appendix B. Newton-Raphson algorithm......................................................................... 24 Appendix C. Dynamic viscosity of a water/glycerine solution ........................................ 25 Appendix D. Density of a water/glycerine solution.......................................................... 26 Appendix E. Principle of a hydrometer ............................................................................ 27 Appendix F. Principle of an Ostwald viscometer ............................................................. 28 [ 3 / 28 ]
1. Motivation and learning goals
This practical is concerned with the determination of the terminal settling velocity
of a falling solid particle. A precise prediction of the fall (or rise) velocity of a
particle is important for many applications. A few of these applications are
discussed below.
Figure 1a shows the launch of a weather balloon with a radiosonde at
KNMI in De Bilt. The radiosonde contains instruments to measure quantities such
as the temperature, pressure, wind speed, etc., during its rise of 1-2 hours
through the atmosphere. The balloon reaches typically an altitude of about 20 km
before it pops due to the low atmospheric pressure at this height. During its
journey the measured data is transmitted to a station on earth. It is important to
choose the right initial diameter of the weather balloon, since the rise velocity
depends on it and the sampling frequency should match with the rise velocity to
properly sample the atmosphere. Furthermore, the journey till the top of the
atmosphere should not take too long if otherwise the weather conditions will
change too much in the meantime.
Figure 1b shows the discharge of a dense sediment/water mixture or
slurry from a pipe, which is used for instance to transport sediment from the
hopper of a dredging vessel to land. The pipe flow is highly turbulent and need to
provide enough energy to keep the sediment in suspension in order to prevent
clogging of the pipe line. Roughly speaking, the higher the settling velocity of the
sediments and the sediment volume concentration, the more energy is needed
and thus a higher flow velocity and more pump power.
Figure 1c shows another application from the dredging industry. It
concerns the jetting of a water/sediment mixture (known as rainbowing) to create
the “Maasvlakte 2”, an extension of the port of Rotterdam with about 2000
hectares of reclaimed land from the North Sea. The settling of the sediments
causes elevation of the sea bed, which continues until the bed level is above sea
surface level and land is formed.
Finally, to mention a last example, figure 1d shows a Fluidised Bed
Reactor (FBR). FBRs are used for chemical reactions. Solid catalyst particles are
injected into the reactor to enhance the chemical reaction rate. To keep the
catalyst particles in suspension, the upward gas flow through the reactor should
be strong enough to prevent settling of the particles under gravity.
The learning goals of this practical are that after this practical you are able to:
a) explain the relevance of studying the settling of a particle under gravity for
applications in practice;
b) conduct an experiment to determine the terminal settling velocity of a solid
sphere in a tube filled with a stagnant fluid;
c) apply the particle force balance and appropriate correlations for the drag
coefficient for an a priori prediction of the terminal settling velocity;
d) explain possible causes for differences between the experimentally
determined and the theoretically predicted values for the terminal settling
velocity of solid spheres.
[ 4 / 28 ]
(a)
(c)
(b)
(d)
Figure 1. (a) Launch of a weather balloon at KNMI. Picture taken from [P2]. (b)
Discharge of slurry from a horizontal pipeline. Picture taken from [P3]. (c) Jetting
of a dense sediment/water mixture during creation of the “Maasvlakte 2”, an
extension of the port of Rotterdam (2008-2012). Picture taken from [P4]. (d)
Schematic of a fluidised bed reactor. Picture taken from [P5].
[ 5 / 28 ]
2. Theory
The theory presented below is primarily based on the textbook of White [1],
section 7.6.
2.1 Drag coefficient
The drag force, , on an obstacle falling under gravity in a gas or liquid is
defined as the total force1 that the flow exerts on it. The fall velocity of the
obstacle is usually referred to as the settling velocity, . When the settling
velocity has reached a constant value, it is called the terminal settling velocity. It
is customary to parameterise the drag force according to:
,
(1)
where is
is the so-called frontal surface area,
is the fluid ‘mass density’,
usually simply called density, and
is the so-called drag coefficient. The frontal
surface area is the projected area of the body in the direction of the flow; for a
. The drag coefficient is a
sphere with diameter
it is equal to
function of the shape of the obstacle and the Reynolds number Re. For a perfect
sphere the Reynolds number is defined as:
Re
,
(2)
is the dynamic viscosity of the fluid. The Reynolds number
where
characterises the importance of fluid inertial forces relative to fluid viscous forces;
fluid inertial forces dominate for Re ≫ 1.
If an obstacle is fully immersed in a fluid, then there are two contributions
to the drag force , namely the friction (or viscous) drag and the pressure (or
form) drag. Friction drag is the integral viscous shear stress that the fluid exerts
on the obstacle surface area when it flows parallel to the surface. Pressure drag
is the force that the fluid exerts due to the overall pressure difference between
the front and the rear surface of an obstacle (relative to the flow direction).
Friction drag is important at low Reynolds numbers and/or slender bodies (large
ratio of total surface area over frontal surface area, such as is the case for big
container ships). Conversely, pressure drag is important at high Reynolds
numbers and/or for bluff bodies (small ratio of frontal surface area over total
surface area, such as is the case for flow perpendicular to a thin circular disc).
The drag force on the spheres shown on the cover page of this manual is almost
entirely determined by pressure drag since in both cases the flow separation
causes a very strong pressure difference over the sphere; flow separation is
characteristic for high Reynolds numbers.
1
To be precise, the drag force is the total flow-induced force when the obstacle would fall at a
constant velocity (i.e., it is the steady-state contribution of the flow to the force on the obstacle).
When the obstacle accelerates or decelerates, other flow-induced forces exist, such as the
added-mass force and history forces (the added-mass force will be discussed in section 2.2).
[ 6 / 28 ]
Figure 2 shows the drag coefficient as function of the Reynolds number for
various three-dimensional obstacles. Focusing on the perfect sphere, we observe
the following behavior for the drag coefficient [1,2]:
Re ≲ 0.1,
(3a)
0.44
5 ∙ 10 ≲ Re ≲ 10 .
(3b)
Figure 2. Drag coefficient as function of obstacle Reynolds number for various
three-dimensional obstacles. Figure taken from [1].
Equation (3a) is known as Stokes’ drag law. In this so-called Stokes regime the
flow field is mirror-symmetric with respect to the midplane of the sphere.
Furthermore, the contribution from friction drag is a factor 2 larger than the
contribution from pressure drag on the sphere. Substitution of Eq. (3a) into Eq.
(1) yields
3
and thus the drag force increases linearly with the
settling velocity for very low Reynolds numbers.
Equation (3b) corresponds to the case of a turbulent wake past the
obstacle as depicted for a bowling ball on the left photograph on the cover page.
0.17
: the drag force
From Eq. (1) it follows that for this flow regime
scales quadratic with the settling velocity. Notice furthermore that the drag force
is now independent of the fluid dynamic viscosity, but scales linearly with the fluid
density. The latter explains why speed skating world records are never ridden in
the Netherlands since the altitude of the ice skating rink in Heerenveen is low
1.2 /
is relatively high compared to, say,
(sea level) and hence
Salt Lake City at an altitude of about 1300 and
1.1 / . Notice
also that the drag force scales now with the square of the diameter: speed
skaters and racing cyclists know this very well and try to minimise their frontal
surface area as much as they can in order to reduce the drag force acting on
them. For Reynolds number up to about 6000 the following parameterisation
appears to accurately capture the measured values of the drag coefficient as
shown in Fig. 3 [2,3]:
[ 7 / 28 ]
0.5407
0.1 ≾ Re ≾ 6000.
(3c)
Eq. (3c) is preferred over Eq. (3b) for Reynolds numbers up to 6000.
Figure 3. Drag coefficient of a sphere as function of Reynolds number up to a
Reynolds number of 6000. Symbols represent experimental data. The solid line
is given by Eq. (3). Figure taken from [3].
2.2 Estimate of settling distance
Let us consider a solid sphere falling under gravity in a fluid at rest. The force
balance (Newton’s 2nd law) for the sphere is approximately given by Morison’s
heuristic equation [10]:
,
(4)
is the sphere density,
is the volume of the sphere and is
where
the gravitational acceleration. The meaning of the terms at the right-hand side of
Eq. (4) is as follows:
- Term 1 represents the gravitational force, which is the driving force for the
settling of the sphere;
- Term 2 represents the Archimedes force, the upward force from the
displaced fluid;
- Term 3 represents the so-called added-mass force and takes into account
that when the sphere accelerates also the surrounding fluid of the sphere
has to accelerate along with it;
[ 8 / 28 ]
-
Term 4 represents the steady drag force when the sphere has reached its
terminal (final) settling velocity.
Equation (4) can be rewritten into the following form:
,
(5a)
is a constant. The
where and are constants provided that we assume that
coefficients are given by, respectively:
, and
where
,
(5b)
is the sphere-to-fluid density ratio. From Eq. (5a) the following
analytical solution can be derived (see exercise P7.66 in [1]):
tanh / ,
(6a)
where tanh is the hyperbolic tangent function. The coefficient
is the terminal
and
is a characteristic
settling velocity that the sphere reaches for ≫
settling time that the sphere needs to reach the terminal settling velocity (for
2.65 ,
0.99 ). The coefficients are given by, respectively:
1
,
.
(6b)
(6c)
at
For this practical it is of special interest to estimate the settling distance
which the sphere has reached 99% of its terminal velocity. The settling distance
can be estimated by integrating Eq. (6b) from
0 to
2.65 :
. .
ln cosh 2.65 ,
(7a)
where cosh is the hyperbolic cosine function. Assuming that
(3b), this can be worked out and written as:
5.94
.
0.44, see Eq.
(7b)
Note that Eq. (7b) will likely overestimate the settling distance, since the drag
coefficient is assumed here to be constant and equal to 0.44, while during the
early falling stage it is actually larger than 0.44 (see Figs. 2 and 3). Nevertheless,
Eq. (7b) is expected to yield a reasonable estimate for this practical as long as
the Reynolds number based on the terminal velocity is much larger than 500. As
2860 /
and
0.01
an example consider an aluminium sphere with
falling in water with
1000 / . Equation (7b) then predicts that the
0.20 . As a second example,
settling distance is approximately equal to
consider a polyoxymethylene (POM) sphere with
1430 /
and
0.005 falling in water. We get
0.057 , which is more than 3 times as
small as in the previous example.
[ 9 / 28 ]
2.3 Estimate of the terminal settling velocity
When the terminal settling velocity is reached, the velocity of the sphere remains
constant. Eq. (4) can then be rewritten in the following form:
Re ∙ Re
Ar 0
(8a)
where Ar is the so-called Archimedes number. It is defined as:
Ar
1
,
(8b)
⁄
the kinematic fluid viscosity. The Archimedes number is a
with
dimensionless number that characterises the importance of the net gravitational
force to the fluid viscous forces. Note that it only depends on the material
properties of the fluid and the sphere. For the aluminium and the POM sphere in
the previous section the Archimedes number is approximately equal to 1.82 ∙ 10
and 5.27 ∙ 10 (using
10 / and
9.81 / ), respectively, which
indicates that friction drag is small compared to the net gravitational force on the
spheres.
To calculate the terminal settling velocity from Eq. (8a), we need to specify
first the drag coefficient. In section 2.1 several correlations were given, see Eqs.
(3a)-(3c). The problem is that we do not know the terminal settling velocity and
hence the Reynolds number a priori. In Appendix A, a method using numerical
mathematical techniques is presented to solve such an ‘implicit equation’. You
will learn about those during the course Numerical Analysis in Q4. Here we follow
a graphical method to determine the Reynolds number:
First rewrite Eq. (8a) into an expression for the drag coefficient. This
expression for the drag coefficient should be equal to Eq. (3c) for Reynolds
numbers up to about 6000:
0.5407
.
(11a)
as
Plot the left and the right-hand side of Eq. (11) both in one figure of
function of Re. The Reynolds number for which force equilibrium exists,
corresponds to the intersection point (zoom in to determine this accurately).
Check whether the Reynolds number is in the valid range. Next, the terminal
settling velocity can be determined from the definition of the Reynolds number:
.
(11b)
The graphical procedure is illustrated in Fig. 4 for one of the POM spheres, i.e.
for one specific value of Ar. We find that Re 1.23 ∙ 10 and
0.246 / .
This new estimate for the Reynolds number is in the right range for which Eq.
(3c) is valid. Furthermore, the estimate is about 3% more accurate than the
0.44 (the lower the Reynolds number, the
previous one in step 2 based on
larger the difference with the estimate from step 2). You may check yourself that
the results found from the graphical procedure is the same as found with the
Newton-Raphson procedure as presented in Appendix B.
[ 10 / 28 ]
4Ar
3Re
24
Re
0.5407
1232 0.463
Figure 4. The graphical approach in which the Reynolds number is determined
from Eq. (11a). The drag coefficient is plotted as function of Reynolds number.
2.4 Path instabilities of freely falling spheres
Freely falling spheres may exhibit path instabilities. This originates from
instabilities of the particle’s wake. Whether a path instability will occur and which
kind of instability will occur, depends on 2 parameters: (1) the particle/fluid
/ , and (2) the Archimedes number (Ar . This is indicated in
density ratio
the figure below.
IV
I
VI
II
III
V
Regimes:
I. Steady vertical trajectory
II. Steady oblique trajectory
III. Oscillating oblique trajectory
(low frequency)
IV. Oscillating oblique trajectory
(high frequency)
V. Periodic zigzagging trajectory
VI. 3D chaotic trajectory
VII. Alternating 3D chaotic and
periodic zigzagging trajectory
VII
√
Figure 5. Path instability diagram for a freely falling sphere, from numerical
simulations by Jenny et al. [11].
[ 11 / 28 ]
For sufficiently small Ar (√Ar ≲ 150) the particle path is steady and vertical for all
density ratios (regime I). Interestingly, in a limited range of Ar a regime exists
where the particle path is still steady, but slightly inclined with respect to the
vertical at an angle of approximately 5º (regime II). Note that in this case the
sphere also rotates over a horizontal axis. At higher Ar the particle path
undergoes several different instability regimes till its path becomes a 3D chaotic
trajectory at large Ar. In the last regime the particle moves in the vertical direction
on average, while for heavy particles the horizontal excursions tend to be very
small.
[ 12 / 28 ]
3. Experimental setup
A schematic of the experimental setup is depicted in Fig. 6. It consists of 2
columns filled with different liquids: one is filled with tap water and the other one
with an aqueous glycerine solution with approximately 50% glycerine by weight
20° this has a dynamic viscosity of around 6∙ 10 / ∙
and a
(at
density of 1125 / , see Appendix C and D, respectively). The tube cover on
top (1) contains a hole to insert a sphere in the tube. The sphere will sink under
the action of gravity. The settling velocity will be determined from the time the
sphere takes to travel through the measurement section marked by 2 O-rings (4).
At the bottom of the tube there is a sluice to collect a sphere after an experiment
and to remove it without much loss of liquid (5 and 6).
measurement section
with height
In total 10 spheres are provided for the experiments:
- 2 x 2 aluminium spheres with
5 and 10
5 and 10
- 2 x 2 POM (Polyoxymethylene) spheres with
- 2 Nylon (Polyamide 6.6) spheres with
10
3
Figure 6. A schematic overview of the experimental setup. 1: tube cover with inlet
hole. 2: Pipe brackets. 3: O-rings. 4: measurement section marked by the off Orings. 5: upper chamber valve. 6: lower chamber valve. Picture from [4].
[ 13 / 28 ]
Important remark: release only 1 sphere per experiment and first collect
this sphere from the sluice before moving to the next experiment. This
avoids the risk that multiple spheres get stuck in the sluice!
For the experiments the following measurement instruments are provided:
- a micrometer screw gauge (Dutch: schroefmicrometer)
- a tape measure (Dutch: rolmaat)
- a scale (Dutch: weegschaal) to measure the mass of the spheres
- a thermometer
- a hydrometer to measure the density of the aqueous glycerine solution
(see Appendix E for explanation of the working principle and use)
- an Ostwald viscometer to measure the viscosity of the aqueous glycerine
solution (see Appendix F for explanation of the working principle and use)
- a stopwatch (alternatively, you could use your cell phone when it contains a
stopwatch)
Important remark 2: Most of these instruments are sensitive and/or fragile.
Please handle them with care, as if they cost 100 euro each (some do!).
[ 14 / 28 ]
5. Discussion
Task 11: The theory in section 2.3 was based on a single sphere falling in a fluid
in an infinite medium. In the experiment the sphere falls in a column.
Mass conservation requires that the fluid near the sphere has to move
upwards when the sphere is moving downwards. The averaged
upward fluid velocity at the sphere midplane is equal to:
,
,
(15a)
where
is the inner tube diameter and
is the absolute
,
terminal settling velocity in a fixed frame of reference.
The relative terminal settling velocity of the sphere, , , relative to
the surrounding fluid at the sphere mid-plane, is thus equal to:
,
(15b)
,
,
As a crude model to predict ,
(since this is the velocity you have
actually measured in the experiments), we could replace , by the
terminal settling velocity that a sphere would have when falling in a
:
quiescent fluid in an infinite medium, ,
,
1
,
,
(15c)
can be estimated from the procedure detailed in section
where ,
2.3. Experimental data for the fall velocity of drops showed that Eq.
(15c) is fairly accurate [6].
Measure the (inner) diameter of the tube. Based on Eq. (15c), do you
expect that the tube confinement effect did have a strong effect on the
terminal settling velocity in the experiments?
Task 12: Optional.
In applications where sedimentation occurs such as shown in Fig. 1c
where large amounts of sand deposit on the sea floor, we need to deal
with a similar effect as above. Due to the settling of the particles, a
return flow exists, which implies that the effective settling velocity of the
sediment is higher than the absolute settling velocity. This effect is
known as hindered settling. Define the sediment volume concentration
(or fraction) as
(it has a value between 0 and 1). In a similar way as
above, try to incorporate the effect of the return flow on the terminal
settling velocity of the spheres. Show that this model predicts that:
1
,
(16)
,
,
with
1 and where
can
be
estimated
from
the
,
procedure detailed in section 2.3. Experimental data shows that the
exponent is actually a function of the particle Reynolds number and
generally significantly larger than 1 (see the data of [7], which suggests
that
2.39 for
≳ 500). A more detailed discussion is beyond the
scope of the present practical.
[ 21 / 28 ]
Literature
[1] F.M. White. Fluid Mechanics. McGraw-Hill, Boston, 4th edition, 1999.
[2] R.B. Bird, W.E. Stewart and E.N. Lightfoot. Transport Phenomena. John
Wiley and Sons, New York, 2nd edition, 2002.
[3] F.F. Abraham. Functional dependence of drag coefficient on a sphere on
Reynolds number. Physics of Fluids, vol. 13, pp. 2194-2195, 1970.
[4] Experiment instructions: HM 135 Drag Coefficients for Spheres. Publicationno.: 917.000 00 A 135 02 (A), G.U.N.T. Gerätebau GmbH, Barsbüttel, Germany,
2013.
[5] N.-S. Cheng. Formula for the viscosity of a glycerol-water mixture. Ind. Eng.
Chem. Res., vol. 47, no. 9, pp. 3285-3288, 2008.
[6] J.R. Strom and R.C. Kintner. Wall effect for the fall of single drops. A.I.Ch.E.
Journal, vol. 4, no. 2, pp. 153-156, 1958.
[7] J.F. Richardson and W.N. Zaki. Sedimentation and fluidisation: part I. Trans.
Inst. Chem. Eng., vol. 32, pp. 35-53, 1954.
[8] J.B. Segur and H.E. Oberstar. Viscosity of glycerol and its aqueous solutions.
Industrial and Engineering Chemistry, vol. 43, no. 9, pp. 2117-2120, 1951.
[9] Taken from the website of Dow Chemical at www.dow.com.
[10] J.R. Morison, M.P. O’Brien, J.W. Johnson and S.A. Schaaf. The forces
exerted by surface waves on piles. AIME, Petroleum Transactions, vol. 189, pp.
149-154, 1950.
[11] M. Jenny, J. Dušek and G. Bouchet. Instabilities and transition of a sphere
falling or ascending freely in a Newtonian fluid. J. Fluid Mech., vol. 508, 201-239,
2004.
Photographs
[P1] F.M. White. Fluid Mechanics. McGraw-Hill, Boston, 4th edition, 1999.
[P2] http://www.knmi.nl/cms/content/32978/weerballon_of_radiosonde
[P3] http://www.engineeringtoolbox.com/slurry-transport-velocity-d_236.html
[P4] http://www.nationalgeographic.nl/fotografie/foto/rainbowing-1
[P5] http://en.wikipedia.org/wiki/Fluidized_bed_reactor
[P6] http://fr.wikipedia.org/wiki/Hydrometre
[ 22 / 28 ]
Appendix A. Determining the settling velocity solving a
non-linear implicit equation.
Below a fairly simple solution procedure is presented in 3 steps.
1. Assume
0.44
In this case the terminal settling velocity and Reynolds number are equal to,
respectively:
1.74
1
1.74√
,
(9a)
.
(9b)
It is remarked that Eq. (9a) is actually the same as Eq. (6b) in this case.
2. Check the Reynolds number
The constant value of the drag coefficient is only valid when the Reynolds
number is in the range of 5 ∙ 10 ≲
≲ 10 , see Eq. (3b). Furthermore for
≲ 6000, Eq. (3c) will yield a more accurate estimate than the constant of 0.44.
Let us check this for the aluminium and the POM sphere. From Eq. (9b) we
1.27 ∙ 10 for
calculate that
7.44 ∙ 10 for the aluminum sphere and
the POM sphere. We conclude that the Reynolds number is in the right regime
for the aluminium sphere and we estimate from Eq. (9a) the terminal settling
velocity by
0.74 / . On the contrary, the Reynolds number for the POM
sphere is such that we expect a more accurate value of the terminal settling
velocity when the drag coefficient is estimated by Eq. (3c). This is discussed in
the next step.
3. Redo the calculation for another correlation of the drag coefficient, if
necessary
For the POM sphere we expect a more accurate estimate of the terminal settling
velocity when the drag coefficient is estimated by Eq. (3c). Equation (8a) can
then be rewritten as:
18.12
82.09
4.56
0,
(10)
where
is a 4th-order polynomial function of
√ .
The polynomial roots of
can be found numerically by using an iterative
procedure such as for example the Newton-Raphson method. In the software
program MATLAB you could use the function “roots” to find the polynomial roots
of Eq. (10), though you may also want to program it yourself in whatever
computer language you like. A schematic overview of the Newton-Raphson
method is shown in Appendix B.
[ 23 / 28 ]
Appendix B. Newton-Raphson algorithm
Below a schematic overview is given of the Newton-Raphson algorithm for
solving the terminal settling velocity from equation (10).
0 (initialisation, is the iteration step)
1.74√ (estimate from Eq. (9b))
18.12
82.09
4.56
4
54.36
164.18 (derivative of to )
(new estimate based on tangent line to )
|
|
| |
1 (next iteration)
18.12
82.09
4.56
4
54.36
164.18
(new estimate)
It is recommended to put the threshold value smaller than10 (less than 0.1%
error in the old estimate compared to the new one). Don’t forget to check whether
the Reynolds number is the correct range for which Eq. (3c) is valid.
[ 24 / 28 ]
Appendix C. Dynamic viscosity of a water/glycerine
solution
The table below is taken from [8].1 centipoise
[ 25 / 28 ]
10 /
∙
.
Appendix D. Density of a water/glycerine solution
The table below is taken from [9].
[ 26 / 28 ]
Appendix E. Principle of a hydrometer
A picture of a hydrometer is shown below [P6]. The principle of a hydrometer is
to determine the density of a liquid from the balance between the downward
gravitational force and the upward Archimedes force. To measure the density of
a liquid, stick the hydrometer in a tube filled with the respective liquid. Due to the
mass in the bottom of the hydrometer the hydrometer will sink until the downward
gravitational force balances the upward Archimedes force from the displaced
liquid. The density can be read from the scale in the upper part of the hydrometer
and is equal to the level indicated at the meniscus.
scale
value of liquid mass
density = value indicated
on scale at meniscus
mass in
bottom part
[ 27 / 28 ]
Appendix F. Principle of an Ostwald viscometer
A picture of a so-called Ostwald viscometer is shown below. The liquid of which
the viscosity has to be determined, is poured in the left arm of the U-shaped
glass tube till the 2 liquid menisci are approximately located at positions 1 and 2.
Due to the hydrostatic weight of the liquid in the left arm, the meniscus at position
1 will gradually move downwards and the meniscus at 2 will gradually move
upwards.
The flow rate is set by the balance between the net hydrostatic weight in the Utube on the one hand and the friction drag in the tube on the other hand.
Assuming laminar flow inside the tube, the flow rate depends linearly on the
kinematic viscosity (that is the dynamic viscosity divided by the density of the
liquid). The kinematic viscosity, , can thus be determined by measuring the time,
∆ → , it takes for the meniscus in the right arm to travel from position 3 till
position 4:
(D.1)
∙∆ →
where is a constant (dependent only on the geometry of the viscometer), ∆ →
/ . For the Ostwald
is measured in seconds and is determined in units of
viscometer shown below the value of is engraved in the left arm and equal to
0.03023 ∙ 10 / .
Remark: cleaning of the viscometer is a delicate thing with the fragile
glasswork. Please do NOT try yourself but ask one of the assistants!
flow direciton
1
/
right arm
left arm
Constant C (in 10 4
3
2
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