Falling Cylinder Viscometer
Engineering 1281H
Autumn, 2013
Tom Barrett, Seat 33
Ryan Hutcheson, Seat 23
Caroline Leiser, Seat 10
Taylor Yeater, Seat 28
A. Theiss
Wednesday 3:00 PM
Date of Experiment: 10/16/13
Date of Submission: 10/23/13
1. Introduction
In many engineering fields, being able to determine the viscosity and flow of various
liquids is important. Viscosity of a liquid is its resistance of flow and is different for every liquid.
For example, water is far less viscous than the oil samples used in this experiment. The purpose
of this experiment was to analyze a sample of 350 cSt and 1000 cSt oil to determine the
experimental viscosity. With these values, the percent difference needed to be calculated using
the documented viscosities of the liquid. Lastly, the viscosity needed to be found for a third
unknown liquid that the Local Oil Lab had discovered.
The following section of the report describes the procedure and set up that was used to
complete this experiment. The next section shows how that calculations were made using
different equations and the results of the experiment. The third section is the discussion portion
of the report, which analyzes the results of the experiment in order to answer the purpose of the
experiment. The last section, the summary and conclusion discuss the end result of the
experiment and also give ways to improve the experiment.
2. Experimental Methodology
In order to determine the viscosities of the two known tubes and of the mystery tube, a
program called LabVIEW was used. In the program, the Vi used was called Viscosity.exe. The
camera, tube and the calibration grid were then set up. The proper set up is displayed in Figure 1
on the next page.
2
Figure 1: Setup for the testing Viscosity [1]
The box was held together using four black binder clips and the calibration grid was
taped to the back of the box. The camera was set up so that the camera had a view similar to
Figure 2, below.
Figure 2: Cameras view of Tube and Calibration Grid. [1]
3
The tube was placed in the holes in front of the calibration grid and the cylinder was held
up at top of the tube by using the magnet. All of the air bubbles had to be above the cylinder so
that they would not interfere with the results for viscosity. The magnet was removed and the
cylinder fell. Once the cylinder entered the grid, the record button was pressed on the LabVIEW
program. After all of the cylinder had exited the cameras view, the record button was pressed
again to stop the recording. On the program, the Calibrate button was pressed and a large section
of the calibration grid that did not have any of the tube in it was selected. After, the Start frame
button was pressed in order to start the first analysis frame. The frame that was selected was the
last frame before the cylinder entered the viewing area. Next, the End frame was pressed when
the last frame of the entire cylinder was present in the viewing area. After this, the region of
interest was selected after pressing the Set ROI button. The region of interest for this experiment
was the tube. To view the result, the Process button was pressed.
This process was repeated three times per liquid for all three liquids. To create the
graphs, the data was analyzed in excel. The initial point was the first point of the set and, for
every point after, the Pythagorean theorem was applied to find the displacement. The
displacement verses time was graphed for each trial of the three liquids. If the data was not
completely linear, the ends were cut off.
3. Results and Description
The data found from the three different trials for the liquids needed to be analyzed in
order to find the percent error in the known samples and to determine the viscosity of the
unknown liquid. All of the displacement values from this experiment were put into a graph of
displacement v. time in Figure A1 in Appendix A. These values in the graph are in inches/second
4
and have all nine trials on the one graph. The slope of each of the nine lines is the terminal
velocity for that trial.
All of the terminal velocities calculated from earlier experiments and from this
experiment are in Appendix A in Table A1. From all of these values, the mean and standard
deviation were calculated. Excel was used to calculate these values, which are in Table A2 in
Appendix A. Once the mean and standard deviation were found, the values were converted to
meters per second from inches per second by multiplying by 0.0254.
Also for this experiment, the theoretical terminal velocity had to be calculated. This value
was found for both of the known liquids using equations B1-B8, which are in Appendix B where
there are also sample calculations for these equations. To calculate the experimental viscosities
for all of the liquids, equation B9 which is found in Appendix B. To use this equation, the values
that were found after calculating equations B1 – B9 were used. The last thing that was calculated
for all of the liquids was the Reynolds number. The equation used to calculate Reynolds number
is in Appendix B and is equation B10. The experimental terminal velocities, viscosities and
Reynolds numbers, theoretical values for viscosity and terminal velocity and percent difference
for terminal velocity and viscosity are all in Table A4 in Appendix A.
4. Discussion
The net position and time had a linear relationship. If the derivative of the line was taken,
the velocity was found and since the relationship between position and time was linear, the
velocity was found to be constant.
The data that was found from this experiment is in Figure A1 in Appendix A, Group B.
The values that were found were all within one standard deviation of the mean for the earlier
trials. For the 350 cSt silicon, the mean for the earlier trials was 0.02874 m/s and the standard
5
deviation was 0.003125 m/s. The value from this experiment was 0.02862 m/s. The values found
during this experiment were within one standard deviation of the earlier trials. For the 1000 cSt
silicon, the mean of the previous trials was 0.009938 m/s and the standard deviation was
0.001183 m/s. The average from the data was 0.01079 m/s, which is within one standard
deviation from the previous trials. For the unknown solution, the average was 0.001874 m/s. This
average was within one standard deviation from the previous trials because the mean was
0.002161 m/s and the standard deviation was 0.0003224 m/s. Because all of the values from this
experiment were within one standard deviation from the previous trial’s mean, the two data sets
can be assumed to be similar.
When the calculations were done, there were some assumptions made in order for the
equations to work. Assumption number three, in particular, may have thrown off the results
because there is not an infinite amount of liquid between the walls and the cylinder. This
assumption and some other errors lead to the discrepancies between the theoretical values for
viscosity and terminal velocity and the values that were found during the experiment. The
percent difference between the theoretical and experimental values for viscosity and terminal
velocity are in Table A4 in Appendix A. There are also some other errors that were associated
with this experiment, which are discussed later in this section.
After completing this experiment, the Reynolds number was calculated for all three
liquids. The values for all of the calculated Reynolds numbers are in Table A4 in Appendix A.
The Reynolds values for the three liquids were all under one, which signifies that the liquids all
had a laminar flow regime. This was supported by the observations made in class. When the
cylinder was released and started falling, there were not any bubbles created like if the flow
6
regime was turbulent. For both of these reasons, it was determined that all three of the liquids
had a laminar flow regime.
In this experiment, there were definitely errors that threw off the results. When the
camera was recording the cylinder fall, the table could have been bumped or the camera could
have moved which would have thrown off the displacement of the cylinder. This error would
throw off the entire experiment because terminal velocity was calculated directly from the
displacement. Another error that could have occurred was that the cylinder could have rubbed
against the sides of the tube, which would have caused it to slow down and give a different
terminal velocity. To remove this error, larger tubes could have been used and the cylinder could
be dropped in the top instead of being trapped inside of the tube. This would make it so the
cylinder would not interfere with the sides of the tube which would lead to a much more accurate
measurement of terminal velocity and viscosity.
5. Summary and Conclusions
This experiment was done to calculate the experimental viscosities of two known liquids
and then to determine the viscosity of an unknown solution. To determine the viscosity, many
different equations had to be used. For all three liquids, the terminal velocities were found based
on their displacements when the cylinders were released in the tubes. Also, the Reynolds
numbers for all of the liquids were found using the terminal velocities and the experimental
viscosities. From the Reynolds numbers, which were all under one, the flow of all of the liquids
could be determined to be laminar flow. Also, the viscosity of the unknown liquid was calculated
to be 4.0118 kg/ms. The percent difference between the experimental terminal velocities and the
samples theoretical terminal velocities was 12.57% for the 350 cSt and 11.21% for the 1000 cSt
7
liquid. The percent difference between the documented and experimental viscosity for the 350
cSt liquid was 11.16% and for the 1000 cSt liquid was 10.07%.
To improve the experiment for future use, the experiment should be set up to minimize
errors. To stop the cylinder from sliding down the sides of the tube, a larger tube should be used
so that when the cylinder is dropped, it is more likely to drop straight down without rubbing
against the sides. Also, more trials could be done for the three liquids to insure more accurate
results.
8
References
[1]
Falling Cylinder Viscometer Write Up. 2013, October 16. www.carmen.osu.edu.
[2]
Experimental Data. 2013, October 16. A. Theiss.
9
APPENDIX A
Title of Appendix
1000
Unknown
Trial 1
1.0295
0.3290
0.0915
Trial 2
Trial 3
Trial 1
1.0349
1.1691
1.2196
0.3294
0.3195
0.4152
0.0927
0.1077
0.0752
Trial 2
1.0353
0.4334
0.0701
Trial 3
Trial 1
1.1264
1.2909
0.4254
0.3122
0.0760
0.0761
Trial 2
Trial 3
1.2705
1.2628
0.3440
0.3116
0.0844
0.0818
Group F Group E Group D
Trial 1
Trial 2
Trial 3
Trial 1
Trial 2
Trial 3
Trial 1
Trial 2
1.0000
1.1220
1.1724
1.1392
1.1200
0.9737
1.2651
1.3122
0.3837
0.4508
0.4149
0.4200
0.4650
0.4205
0.3898
0.3627
0.0768
0.0720
0.0572
0.1032
0.0907
0.1048
0.0970
0.1019
Trial 3
1.2124
0.4064
0.0967
Group G
Trial 1
Trial 2
Trial 3
0.8512
1.0000
0.9999
0.4003
0.3937
0.4297
0.0849
0.0706
0.0941
Trial 1
Trial 2
Trial 3
Trial 1
Trial 2
Trial 3
1.0825
1.1836
0.9560
1.2791
1.2726
1.1702
0.3397
0.3797
0.3767
0.469
0.4371
0.4043
0.0946
0.0717
0.0801
0.0911
0.0783
0.0757
Group C Group
GroupBB
Group A
350
Group I Group H
Fluid (cSt)
Table A1: Gives all of the experimental terminal velocities [2]
A2
TJB
RJF 3:00
Figure A1: The net position v. Time for all of the three liquids
Inches
Meters
Average for 350
1.131522222 0.028740664
Average for 1000
0.391248148 0.009937703
Average for Unknown
Standard Deviation 350
Standard Deviation 1000
Standard Deviation
Unknown
0.08507037 0.002160787
0.123044955 0.003125342
0.046578068 0.001183083
0.012692724 0.000322395
Table A2: Gives all of the Averages and Standard Deviations for the three liquids.
k
0.7625
G(k)
𝜑
Cw
1/ECF
6.456 * 10-3
6.2295
0.06504
1.0309
Table A3: Gives all of the values for the constants that had to be calculated
A3
350 cSt
1000 cSt
Unknown
liquid
Re
0.433383
0.05244808
0.002757495
μf experimental
μf doc
Vt, experimental
Vt, Theroritical
% Difference
for Terminal
Velocities
% Difference
for viscosities
0.301616166
0.3395
0.02874
0.025532018
0.8722993
0.97
0.0099377
0.008936206
4.01180166
12.57%
11.21%
11.16%
10.07%
0.0021607
Table A4: All of the values calculated during this experiment.
A4
APPENDIX B
Sample Calculations
k = rc/rt
(B1)
[𝑘 2 (1 − ln 𝑘) − (1 + ln 𝑘)]
𝐺(𝑘) =
(1 + 𝑘 2 )
(B2)
𝜑 = 𝐿𝑐 ⁄𝑟𝑐
(B3)
𝐶𝑤 = 1.003852 − 1.961019𝑘 + 0.9570952𝑘 2
(B4)
1
8𝑘
𝐺(𝑘)
=1+(
)(
)
𝐸𝐶𝐹
𝜋 ∙ 𝐶𝑤
𝜑
(B5)
𝑘𝑔
𝑘𝑔
𝜇𝑓 (
) = 𝜌𝑓 ( 3 ) ∙ 𝜐(𝑐𝑆𝑡) ∙ 10−6
𝑚∙𝑠
𝑚
(B6)
c=mc/vc
(B7)
𝑉𝑡,𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
[𝑔𝑟𝑐2 (𝜌𝑐 − 𝜌𝑓 )𝐺(𝑘)]
=
2𝜇𝑓
𝐸𝐶𝐹
𝜇𝑓,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 =
[𝑔𝑟𝑐2 (𝜌𝑐 − 𝜌𝑓 )𝐺(𝑘)]
2𝑉𝑡 ,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
𝐸𝐶𝐹
𝑘𝑔
𝑚
)𝑉
( ) 𝑟 2 (𝑚2 )
𝑚3 𝑡,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑠 𝑐
𝑘𝑔
𝜇𝑓,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 (𝑚𝑠) (𝑟𝑡 + 𝑟𝑐 )(𝑚)
2𝜌𝑓 (
𝑅𝑒 =
(B8)
(B9)
(B10)
Sample Calculation for Equation B1
k = rc/rt = 0.0061 / 0.08 = .7625
Sample Calculation for Equation B2
[𝑘 2 (1 − ln 𝑘) − (1 + ln 𝑘)] [(0.76252 )(1 − ln(0.7625)) − (1 + ln(0.7625))]
𝐺(𝑘) =
=
(1 + 𝑘 2 )
(1 + (0.76252 ))
= 6.456 ∗ 10−3
B2
Sample Calculation for Equation B3
𝜑 = 𝐿𝑐 ⁄𝑟𝑐 = 0.038 / 0.0061 = 6.2295
Sample Calculation for Equation B4
𝐶𝑤 = 1.003852 − 1.961019𝑘 + 0.9570952𝑘 2
= 1.003852 − 1.961019(. 7625) + 0.9570952(0.76252 ) = 0.06504
Sample Calculation for Equation B5
1
8𝑘
𝐺(𝑘)
8(0.7625)
6.456 ∗ 10−3
=1+(
)(
)= 1+(
)(
) = 1.0309
𝐸𝐶𝐹
𝜋 ∙ 𝐶𝑤
𝜑
𝜋 ∗ 0.06504
6.2295
Sample Calculation for Equation B6
𝑘𝑔
𝑘𝑔
𝜇𝑓 (
) = 𝜌𝑓 ( 3 ) ∙ 𝑣(𝑐𝑆𝑡) ∙ 10−6 = 970 ∗ 350 ∗ 10−6 = .3395 𝑘𝑔/𝑚𝑠
𝑚∙𝑠
𝑚
Sample Calculation for Equation B7
𝜌𝑐 =
𝑚𝑐
0.038
=
= 8554.4178
(π ∗ 0.00612 ∗ 0.038)
𝑣𝑐
Sample Calculation for Equation B8
𝑉𝑡,𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
(9.81
=
[𝑔𝑟𝑐2 (𝜌𝑐 − 𝜌𝑓 )𝐺(𝑘)]
=
2𝜇𝑓
𝐸𝐶𝐹
𝑘𝑔
𝑘𝑔
(0.0061𝑚)2 (8554.4177 − 970) 3 6.456 ∗ 10−3 )
2
𝑚𝑠
𝑚
𝑘𝑔
2 ∗ 350 ∗ 10−6 ∗ 970 𝑚𝑠
0.970
= 0.02553 𝑚/𝑠
B3
Sample Calculation for Equation B9
𝜇𝑓,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
[𝑔𝑟𝑐2 (𝜌𝑐 − 𝜌𝑓 )𝐺(𝑘)] [(9.81)(8554.4177 − 970)(6.456 ∗ 10−3 )]
=
=
2𝑉𝑡 ,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
2 ∗ 0.02874 ∗ 1.0309
𝐸𝐶𝐹
= 0.301616 𝑘𝑔/𝑚𝑠
Sample Calculation for Equation B10
𝑘𝑔
𝑚
2𝜌𝑓 ( 3 ) 𝑉𝑡,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ( 𝑠 ) 𝑟𝑐2 (𝑚2 )
2(970)(0.02874)(0.00612 )
𝑚
𝑅𝑒 =
=
= 0.43338
𝑘𝑔
(0.301616)(0.008 + 0.0061)
𝜇𝑓,𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 (𝑚𝑠) (𝑟𝑡 + 𝑟𝑐 )(𝑚)
B4
APPENDIX C
Symbols
Cw
ECF
g
G(k)
k
Lc
mc
rc
Re
rt
vc
Vt
Vt,theoretical
νf
νf
νf
f,experimental
c
f
φ
Disk wall correction factor
Eccentric Correction Factor
Acceleration due to gravity
Wall correction factor
Unitless radius ratio
Length of Solid Cylinder
Mass of Cylinder
radius of Solid Cylinder
Reynolds number
Radius of Cylindrical tube
Volume of Cylinder
Experimental terminal velocity
Theoretical terminal velocity
Kinematic viscosity of fluid 1
Kinematic viscosity of fluid 2
Kinematic viscosity of fluid 3
Experimental fluid viscosity (dynamic)
Density of Cylinder
Density of fluid (silicone oil)
Unitless cylinder aspect ratio
C2
--9.81 m/s2
-k = rc/rt
0.038
0.038
0.0061
-0.0080
𝑣𝑐 = 𝜋𝑟 2 𝐿𝑐
--350
1000
unknown
--
c=mc/vc
970
φ=Lc/rc
unitless
unitless
m/s2
unitless
unitless
m
kg
m
unitless
m
m3
m/s
m/s
cSt
cSt
cSt
kg/ms
kg/m3
kg/m3
unitless