Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 2, Issue 12, Dece mbe r-2011 1

ISSN 2229-5518

Terminal Velocity of Canola Oil, Hexane, and

Gasoline Drops Rising in Water due to Buoyancy

Benjamin Michael Cole Friedman, Cynthia Ross Friedman

Abs tractDrops, globules of a liquid in another liquid, are extremely important in many natural processes and industrial applications. The purpose of this study w as to devise a method to measure the terminal velocity of drops rising in w ater due to buoyancy, a nd to compare observed values w ith the theoretical. Tw o questions w ere explored: (1) Do these drops continue to accelerate upw ard from a depth of 6 cm; and, (2) Does the terminal velocity of these drops (modifying the experiment accordingly if not) match the theoretical

(calculated) values? A syringe w as used to inject 0.1 cm3 (0.1 mL) drops of three liquids (oil, hexane, and gasoline) into a vessel at a depth

of 6 cm, and the resulting motion w as video captured and imported into the sharew are kinematics prog ram Tracker® f or analysis and determination of terminal velocity. The experiments show ed that the drops reached terminal velocity bef ore reaching the surf ace (2.23 ±

0.10 cm, 1.48 ± 0.07 cm, and 1.35 ± 0.06 cm above the injection point, respectively). Secondly, in addition to the accepted term of πr2

normally used f or the projected area in the theoretical equation f or terminal velocity, a new term, 2π r2, w as also employed in order to account f or drop f lattening during ascension. As a result, the calculated value w ith the new term accurately predicted the observed, doing so better than the accepted term f or all three liquids, and might be used to improve the accepted theory.

Inde x TermsAcceleration, buoyancy, drag, drops, friction, projected area, spheres, terminal velocity.

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1 INTRODUCTION

rops, globules of a liquid in another liquid, are of funda- mental importance in many natural physical processes and in a host of industrial and man -related activities [1].
Rainfall, air pollution, boiling, flotation, fermentation, liquid -
liquid extraction, and spray drying are only a few of the ph e-
nomena and operations in which drops as well as solid pa r- ticles play a primary role. Meteorologists and geophysicists study the behavior of raindrops and hailstones. Applied ma- thematicians and physicists have long been concerned with fundamental aspects of fluid-particle interactions. Chemical and metallurgical engineers rely on drops for such operations as distillation, absorption, flotation, and spray drying. Me- chanical engineers have studied droplet behavior in connec- tion with combustion operations. In all these phenomena and processes, there is relative motion between bubbles, drops, or particles on the one hand, and surrounding fluid on the other [1].
While an understanding of drop behaviour is obviously valuable in real-world applications, there have been very few studies that explore the basics of drop motion, such as their response to the buoyant force. The theoretical principles have been documented [1], but actual experimental data are scant.
Therefore, the purpose of this work encompassed two ques-
tions relevant to drop behaviour: 1. Do drops of liquids less dense than water (canola oil, hexane, and gasoline) moving upward due to buoyancy continue to accelerate upward from a depth of 6 cm; and, 2. Will there be a difference between the observed and the theoretical values for the terminal velocity of the drops, and if so, can this be explained mathematically?

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Benjamin Friedman is a student at South Kamloops Secondary School, Canada,

PH-1-250-377-3459. E-mail: bfriedman@shaw.ca

Cynthia Ross Friedman is an Associate Professor at Thompson Rivers University,

Canada, PH-1-250-828-5424. E-mail: cfriedman@tru.ca

2 THEORETICAL B ACKGROUND

2.1 Buo yancy

An object submerged in a fluid displaces a volume of fluid equal to the volume of the object itself, with the buoyant force acting upon that object if it is less dense than the surround- ing fluid [2]. The buoyant force is equal in magnitude to the weight of the displaced fluid, but opposite in direction; i.e., upwards. Thus, expressing mass as the product of density and volume, the buoyant force can be expressed by the for-
mula:
, (1)
in which is the density of the surrounding fluid and is the volume of both the displaced fluid and the immersed object, in this case a drop of liquid less dense than water ( = accelera- tion due to gravity). If the surrounding fluid is frictionless, the drop’s acceleration due to is not proportional to the vo- lume, as

, (2)

where is the density of the drop.

2.2 Drag and Terminal Velocity

However, only superfluids such as supercooled helium-2 are truly frictionless [3], [4]; therefore, modeling drop behaviour must consider drag , a second force acting on a drop.

(3) In (3), describes the force of the drag acting on an object

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 2, Issue 12, Dece mbe r-2011 2

ISSN 2229-5518

(drop) in a liquid of some density at some velocity with a drag coefficient , and projected area [5]. For stable smooth spheres, the projected area is the median cross- sectional area , and the drag coefficient is 0.47 [5], [6]. Notably, an of assumes perfect spheres. However, we suspect that the drops might become flattened perpendicular to the direction of travel, becoming olblate spheroids. To a c- commodate for this, we propose that would be better ap- proximated by 2 , which would more accurately represent the cross-sectional area of a flattened drop. This treatment of assumes the sphere becomes fully flattened, such that the cross-sectional area becomes ½ of the surface area of the sphere (i.e., ½ of thus
If drag becomes large enough to exactly equal the drop weight plus the buoyant force, , the net force will be-

4 RES ULTS

4.1 Experimental

In all cases, drops became flattened perpendicular to the direc- tion of travel, becoming olblate spheroids. Table 1 shows the mean values for terminal velocity , height above injection site (position) when was reached, and time when was attained for the oil, hexane, and gasoline trials. Our novel ex- perimental set-up revelaed that oil drops had the lowest ter- minal velocity, gasoline had second lowest, and hexane had the fastest. Oil drops reached terminal velocity last, while hexane reached terminal velocity first.

TABLE 1

TERMINAL VELOCIT Y, POSITION, AND TIME FOR OIL, HEXANE, AND

GASOLINE

Oil Hexane Gasoline

come zero, and acceleration will cease (i.e., become zero as
well). The drop will move at a constant velocity to the surface of the water, having reached terminal velocity, :


(4)

Mean terminal velocity,

(cm/s)

Mean position when is reached (cm above injection)

Mean time when is reached

(s)

8.05 ± 0.05 13.15 ± 0.05 11.55 ± 0.05

2.23 ± 0.10 1.48 ± 0.07 1.35 ± 0.06

0.33 ± 0.01 0.13 ± 0.01 0.17 ± 0.01

3 METHODS

3.1 Experimental

All experiments were performed at room temperature. A 1 cm3 syringe with a 26-gauge needle was used to inject 0.1 cm3 drops of canola (rapeseed) oil (density 0.92 g/cm3 at room temperature [7]), hexane (density = 0.66 g/cm3 at room tem- perature [7]), or standard (vehicle) gasoline “A” (0.74 g/cm3 at room temperature [8]) into a straight-sided centimeter ruled plastic vessel filled with distilled water 6 cm below the sur-

Mean values for terminal velocity as well as position and time when terminal velocity

was reached for rising oil, hexane, and gasoline drops (volume = 0.1 cm 3, n = 10).

4.2 Calculation s and Compariso n s

Table 2 compares the observed and predicted terminal veloci- ties for rising drops of oil, hexane, and gasoline. The ob- served values for were much closer to the predicted values for all three liquids when was used for projected area
rather than the accepted .

TABLE 2

COMPARISON BET WEEN OBSERVED AND PREDICT ED TERMINAL

VELOCITIES

face. A digital video camera (Fuijifilm FinePix E900) operating
at 30 frames per second was used to capture each drop as it
rose in the water column. The experiment was repeated ten
times for each liquid. The video data was imported into the

Liquid Density

(g/cm3 )

Terminal

Velocity (cm/s) Observed

Terminal

Velocity (cm/s) Predicted with

=

Terminal

Velocity (cm/s) Predicted with

=

freeware program Tracker® (Open Source Physics), and the kinematics of each liquid (velocity, acceleration) were deter- mined. Simple statistics were used to evaluate the mean height above the injection point at which terminal velocity was reached for each liquid.

3.1 Calculations and Compari son s

Following the experimental determination of mean terminal velocity for each liquid, the theoretical values were calcu- lated from (4). As the volume of a drop is known (0.1 cm3),
the radius r of the sphere to use in (4) for ( ) was simply:


(5) However, the drops did become flattened. To account for this
phenomenon, a second value for was also used in (4); name- ly, 2 .

Oil 0.92 8.05 ± 0.05 11.32 8.00

Hexane 0.66 13.15 ± 0.05 23.33 16.50

Gasoline 0.72 11.55 ± 0.05 20.41 14.43

Drop volumes were 0.1 cm3, n = 10.

5 DISCUSSION

These results imply that there might be a better interpretation of the accepted formula used for predicting the terminal veloc- ity of a sphere, if that sphere is a fluid that experiences com- pression during movement. Using for projected area rather than the accepted has the effect of taking ½ of
the surface area of a sphere. Obviously, this assumes that the sphere is infinitely compressed, which of course is not the case. Nonetheless, is a better predictor of the experimen- tally-determined values of . Alternatively, instead of mani- pulating the term for , the drag coefficient for a sphere, with a value of 0.47 [5], [6], could be adjusted to ac- count for fluid sphere flattening, perhaps simply by multipl y- ing by a factor of 2: i.e., 2 x 0.47 = 0.94.

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CONCLUSION

Considering the importance of drops in many natural processes and industrial applications, our revised term for determining projected area of a fluid compressible sphere,
, or a modification of the drag coefficient for flat-
tening spheres should be of interest to many scientists and engineers who study drop behaviour.
ACKNOWLEDGMENT
The authors wish to thank Dr. Mark Paetkau (Physical Sciences, Thompson Rivers University) for his advice, ideas, and support, as well as the Department of Biological Science at Thompson Rivers University for financial and logistical su p- port. B. M. C. Friedman is especially grateful to Aura Carriere and André Laprade for guidance, as well as Janice Karpluk, Don Poelzer, Tracy Poelzer, and Rob Schoen for their encou- ragment and help. Both authors thank Dr. Thomas B. Fried- man of Thompson Rivers University for his generosity and Dr. Pierre Bérubé (Civil Engineering, The University of British Columbia) for pointing the authors in the right direction.
REFERENCES

[1] R. Clift, J. R. Grace, and M.E. Weber, Bubbles, Drops, and Particles.

New York, New York: Academic Press Inc., p. 1, 1978.

[2] C. Oxlade, C. Stockley, and J. Wertheim, The Usborne Illustrated Dic- tionary of Science. London, England: Usborne Publishing Ltd., p 25,

1999.

[3] P. Kapitza, “Viscosity of Liquid Helium Below the λ-point,” Nature, vol. 141, no. 3558, p. 74, Jan. 1938.

[4] J. F. Allen and A. D. Misener, “Flow of Liquid Helium II,” Nature, vol. 141, no. 3558, p. 75, Jan. 1938.

[5] National Aeronautics and Space Administration (NASA), The Drag

Equation, T. Benson, ed., Glenn Resarch Centre, http://www.grc.nasa.gov/WWW/K-12/airplane/drageq.html. 2010.

[6] Aerospace Web, Drag of Cylinders and Cones, http://www.aerospaceweb.org/question/aerodynamics/q0231.shtml. 2005.

[7] Simetric, Mass, Weight, Density, or Specific Gravity of Liquids,

http://www.simetric.co.uk/si_liquids.htm . 2011.

[8] The Engineering Toobox, Liquids and Fluids - Specific Gravities - SG, http://www.engineeringtoolbox.com/specific-gravity-liquid-fluids-

d_294.html. 2011.

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