An Important Property Of A Fluid Biology Essay

Viscosity is an of import belongings of a fluid which governs the many phenomena that appear when work is done on them. Fluids of different viscousnesss behave really otherwise and therefore their applications can change. For illustration, with lubricators, high viscousness lubricators are used in the slow moving parts cut downing clash whereas liquids with low viscousness are used in fast moving parts to administer heat. My extended essay is hence based on analyzing the consequence of different viscousnesss on the Ascension of bubbles.

Why Bubbles? These unusual formations of a liquid bed around air signifier in a assortment of substances such as liquids and gases. They are even used in several practical state of affairss including boiling, cavitations, crystal growing, chemical reactions between gas and liquid in bubble columns and stirred vass, to advert merely a few illustrations. Parameters such as rise speed are really of import in the graduated table up and design of many gas-liquid contactors, as the rise speed of bubbles determine the sum of clip the two stages are in contact for which the commixture of the fluids occur.

The research inquiry proposed “ An experimental survey on the effects of viscousness on bubble retarding forces and rise speed in dead glycerin solutions ” is to understand the effects of the liquids belongings, viscousness, on such features as their rise speed and drag coefficient, i.e. their motions through the liquid. In this probe, I will be mensurating the viscousness of five different concentration ratios of water-glycerine and the clip it takes the bubble to go between two line sections drawn on the seeable screen, measured utilizing a picture camera. Comparing the viscousnesss and the gestures of the bubbles in the fluid mixtures, I will discourse the consequences and set up a possible tendency.

Theory

( FBD ) Rise of a bubble in dead Newtonian fluids

The diagram below illustrates the development of the bubbles and all of the internal and external forces moving on the domain as it is formed in the fluid. Figure 1.1 shows a study of the full system ( ab initio, the bubble evolves into a sphere as air is injected from the syringe ) . The force organic structure diagram on the bubble as it flows upstream is the dotted cross-section that has been removed and exploded in the left part of this figure. Note that the bubble is non really spherical over the scope of viscousnesss.

There are three forces moving on the bubbles during its rise to the surface, they are the perkiness ( FB ) , the retarding force ( FD ) and the force of gravitation ( FG ) affected by its weight. Figure 1.0 illustrated the forces and it should be noted that the magnitudes are non to scale.

FB

FD + FG

a‰?D

,

Polonium

Pi

Roentgen

Calciferol

Fig. 1.0 Free Body Diagram of bubble travel. Fig. 1.1 Free Body Diagram of development.

After development, a bubble quickly begins to lift to the surface with a speed, Vb. The value of Vb is determined by the ratio between the floaty rise force, and the retarding force force, which is inherently dependent on the fluids belongingss such as viscousness and denseness. The larger the floaty force the greater the velocity of Ascension. As the bubble rises, a new interface is created at the upstream hemisphere whereas the old hemisphere flows down towards the bottom watercourse hemisphere where it finally disappears. Naturally, when the magnitude of FB & gt ; FD is big, the rise speed will be greater and changing, where the retarding force force is relative to the alteration in viscousness. Hence for higher viscousnesss the retarding force coefficient will be higher.

Experiment Detailss

Common controlled variables between both experiments

Surface Tension:

Among all other variables this would hold to be controlled and isolated, which is why five different fluid mixtures were created with H2O and glycerol. Glycerine will allow alterations in the viscousness of H2O without any important alteration in the surface tenseness. It will hold to be added to condense H2O as the Mg and Ca found in potted H2O could respond with certain chemical constituents and compromise the fluids public presentation in the trials.

Temperature:

It has a big consequence on fluid viscousnesss ; the temperature affects the mean velocity of the molecules and the clip they spend near their neighbouring molecules. Therefore, the temperature of the mixture was maintained at room temperature ( ~20oC ) and a thermometer ( -10oC to +110oC, A±0.5oC ) was used to mensurate the temperature of the mixture at different deepnesss. The lab was even unbroken free from any beginning of bill of exchange.

Density:

The denseness ( expressed as mass per unit volume ) of the fluid will act upon the retarding force force experienced by the bubble as it flows upstream. In add-on, it affects the efflux rate of a liquid. The denseness besides varies with temperature and is carefully measured utilizing an electronic graduated table ( A±0.1g ) . Since glycerol ‘s denseness is non equal ( but reasonably near )[ 1 ]to the value for the denseness of distilled H2O, some variableness is expected in the values measured for the aqueous glycerol.

Entire Volume:

The entire volume in each test and experiment ( viscousness and supplanting ) is kept the invariable.

Distance Travelled:

In both experiments and in every test, the distance was fixed. During the step of the efflux clip to find kinematic viscousness, the distance between the two points were maintained by cleansing, drying and re-using the same container. The same process was used for the different mixtures in the column for the displacement experiment.

Experiment ( 1 ) : Fluid word picture

Kinematic Viscosity:

The viscousness of a liquid is its opposition against shear or flow and is caused by the intermolecular clash exerted when beds of fluids effort to skid by each other. The kinematic viscousness relies on the force of gravitation and the fluid ‘s denseness as supplying the impulsive force for the motion of the fluid and is hence measured utilizing a simple instrument, where the clip it takes the liquid to flux out as it is being sheared by the opening is measured. The kinematic viscousness can so be derived by utilizing a simple standardization invariable, which is gained by using a liquid of known kinematic viscousness through the setup ( at the appropriate mention temperature ) and mensurating its outflow clip.

Apparatus:

Boss Clamp, Clamp Stand

2x 100 milliliter Measuring Cylinders ( A±0.5ml )

5x 400ml Beakers

1x 300ml Plastic Beaker

Electronic Stop Watch ( A±0.05s )

Pen Marker

Thermometer ( -10oC to +110oC, A±0.5oC )

Electronic Scale ( A±0.05g )

Method Variables:

Independent Variable: Concentration per centum.

Dependent Variable: Time.

Procedure:

Using the 100 milliliter mensurating cylinders prepare five 250ml mixtures from water/glycerine and put them in separate 400ml beakers, in the concentration per centums of:

Table 1: exposing the concentrations of the separate mixtures of H2O and glycerol.

Water

Glycerol

10 %

90 %

30 %

70 %

50 %

50 %

70 %

30 %

90 %

10 %

During the readying of the mixtures, bantam bubbles may organize due to the consumption of air while pouring the liquids. In any such instance the mixture will necessitate to be stirred until the bubbles are removed and left to settle. The temperature of the mixture will necessitate to be measured to avoid any disagreements between other mixtures after the agitation of the liquids.

After allowing the mixture settee, topographic point it on the electronic measurement graduated table ( A±0.05g ) and step its mass and so, accordingly its denseness. The variableness of the denseness can be expressed as an uncertainness mistake in the values of the kinematic viscousness.

Take the plastic beaker and bore a hole of diameter 0.5cm ( similar to the diameter of the air injection pipe used in making bubbles ) and attach it to the clinch base utilizing the foreman clinch.

Using the pen marker, chalk out two randomly placed lines on the plastic beaker ( sensible distance between them ) .

Using either a stopper or your ain finger, plug the hole at the underside of the plastic beaker while pouring the mixture into it.

Put the now emptied beaker under the plastic beaker. Remove the stopper and step the sum of clip it takes for the liquid to run out between the two lines.

Repeat the measurings at least three times and take an norm of the consequences.

Experiment ( 2 ) : Measuring the Ascension of bubbles

The experiment was filmed utilizing a Cannon picture camera with a 25 Federal Protective Service entering velocity.

Apparatus:

Plastic Drain Pipe ( 12.00cm x 12.00cm ten 30.00cm )

Metre swayer ( A±0.05cm )

Steel Wire, higher opposition the better

A~C Generator

Perspex Glass

Glue

Wooden Block

Video Camera

1x 1000ml Measuring Cylinder ( A±0.5ml )

2x 1000ml Beakers

Additional Controlled Variables

Volumetric Flow Rate: The rate of air flow affects the rise speed as the velocity at which air is pumped into the liquid governs the initial speed gained by the bubble as the air molecules inside are rushed. A known volume of atmospheric air is injected into the liquid through a syringe ( 20.0ml A±0.5ml ) with an approximative flow rate, Gair a‰? 0.87ml3s-1.

Independent Variable

Kinematic Viscosity: The viscousness of a liquid as under the force of gravitation and is the ratio of viscousness to denseness. ( Measurements explicated in process )

Dependent Variable

Supplanting and Time ( Measurements explicated in process )

Procedure:

First, the column to keep the liquid was built utilizing a plastic drain pipe cut longitudinally ( A steel wire was connected to the terminuss of an A.C generator, running a electromotive force across and heating it up in order to slit the pipe with easiness ) with the approximate measurings for the length, comprehensiveness and height being 12.00cm, 6.00cm and 30.00cm severally.

A hole of diameter 0.5cm ( A±0.05cm ) was drilled into the dorsum of the pipe. A tubing of length 7.4cm and diameter 0.46cm ( A±0.05cm ) , from where the gas will be injected into the liquid, was attached.

A Perspex glass of approximative tallness, 30cm, and length, 12cm, was attached to the unfastened side of the pipe ( with gum ) enabling the perceiver to see the Ascension. The pipe was so attached to a wooden block to maintain it levelled on all surfaces every bit good every bit moving as the base.

A figure of preliminary trials should be run with the column to guarantee that there are no beginnings of leaks. If found, the beginning was sealed farther by tape.

Distilled H2O and glycerin were measured utilizing a 1000ml measurement cylinder and stored in two separate 1000ml beakers. The mixtures started with a higher concentration of glycerin than distilled H2O and were diluted down into the appropriate concentrations for each experiment ( hence, conserving the sum of glycerin ) . The sum of the original mixture required to add distilled H2O to, was measured utilizing the expression:

M1V1 = M2V2, where M1 is the initial concentration of glycerin and V1 is the volume of the original solution required for dilution. M2 is the concluding concentration expected after the dilution and V2 is the entire volume of the solution. For illustration, Let us take the 2nd experiment where the concentration per centum of glycerin to condense H2O is 70: 30.

Initial Concentration, M1 = ,

Number of Moles = ,[ 2 ]

Mass = VGlycerol ( 90 % of 2000ml ) ten ( Acquired )

= 1800ml ten 1.83gml-1 = 3294g.

Number of Moles = =35.8mol,

Initial Concentration, M1 =molml-3.

Final Concentration, M2 = ,

Number of Moles = ,

Mass = Expected VGlycerol x ( Acquired ) =

= 2562g,

Number of Moles = = 27.8,

Final Concentration, M2 = .

The concluding concentration must ever be lower than the initial concentration. Therefore, volume of original solution needed,

V1 == 1553.1ml

And, Volume of Distilled H2O = 2000.0 – 1553.1 = 446.9ml

All Calculations were done in a similar mode.

Table 2: exposing the volume of the mixture added to condense H2O from the above computation method.

a?† [ concentration ] % of glycerin

Volume of original solution ( A±0.5ml ) /ml

Volume of distilled H2O ( A±0.5ml ) /ml

90 % – 70 %

1553.1

446.9

70 % – 50 %

1428.6

571.4

50 % – 30 %

1200.0

800.0

30 % – 10 %

666.7

1333.3

Having prepared the mixtures and column, the outer tubing taking to the handheld syringe was constricted ab initio to forestall an escape of the liquid. The mixture was poured in and left to settle for 2 – 3 proceedingss.

The atmospheric air was injected with an approximative flow rate and the bubbles motions were recorded utilizing a picture camera, which was kept in line with the face of the column. The captured movie was so digitised onto the computing machine and analysed through appropriate package ( explained in method of analysis ) .

The bubbles were formed in sequence of each other over different tests.

Illustration of process:

Video Camera

30 centimeter

12 centimeter

6 centimeter

Gas injector ( D: 0.5cm )

Column,

Able to keep 2.5 liters

Hardy base

Care should be taken in alining the picture camera perpendicular to the plane of the face of the column. The glass used was level to forestall every bit much deformation as possible from refraction.

Experimental ( 1 ) Results ( Raw ) :

The rheological belongingss for different concentration of the glycerin solutions tested are summarized in table 3.0.

Table 3.0: Experiment ( 1 ) consequences: The clip taken for a specific volume of the liquid to run out during each consecutive test.

Water: Glycerol mixture

( Percentage ratio % )

Trial 1:

Efflux Time /s ( A±0.05s )

Trial 2:

Efflux Time /s

( A±0.05s )

Trial 3:

Efflux Time /s

( A±0.05s )

90: 10

8.43

8.54

8.59

70: 30

9.18

9.31

9.21

50: 50

9.82

9.89

9.84

30: 70

12.91

13.94

13.65

10: 90

17.82

17.54

18.67

Experimental Results ( Numerical Data analysis ) :

The kinematic viscousness ( ) is relative to the efflux clip ( T ) of the liquid fluxing through the points.

, i.e,

Re-arranging the equation gives the agencies to cipher the standardization invariable ( degree Celsius ) . A liquid of known kinematic viscousness ( 648 Central Time, Reference temperature =20.3oC ) was run through the experimental apparatus and the mean efflux clip was found to be 48s.

,

Therefore utilizing equation ( 1 ) the kinematic viscousness can be determined.

Table 4.0: Experiment ( 1 ) analysis: The belongingss of the mixtures have been measured utilizing the above attack and are displayed below in tabular signifier. The kinematic viscousnesss uncertainness is represented as a per centum of the amount of the uncertainnesss in the efflux clip and denseness.

Water: Glycerol mixture ( per centum ratio % )

Average Efflux Time ( A±0.05s ) ( T ) /s

Kinematic viscousness ( A±3.30 % ) ( ) /cSt

( Tempref a‰?20oC )

Density

/gml-1

( A±0.05g )

90: 10

8.52

115.02

1.68

70: 30

9.23

124.61

1.78

50: 50

9.85

133.00

1.83

30: 70

13.56

183.06

1.83

10: 90

18.01

243.13

1.93

Average Density: 1.83gml-1

Experiment ( 2 ) Method of analysis:

A method of Video Analysis was used to roll up informations on the bubbles. First, the experiments were recorded on a digital picture camera. The two points marked on the column ( detached 30cm ) acted as a known measuring to scale the distance. The trouble being the angle of the camera, which had to be levelled and kept absolutely perpendicular to the face of the column to hold every bit small deformation as possible. For the same ground, I had to bring forth the bubbles as near to the face of the column so that the graduated table on the plane of the bubble was as near to the graduated table on the plane that holds the face of the column every bit good.

After the experimenting stage was done, and they have been decently recorded, the footage was digitised onto the computing machine. The long footage was trimmed down into shorter cartridge holders for the Ascension of the bubbles in each experiment ; stoping up with six different film cartridge holders and were converted from.mpeg-II to.mov with a reduced frame rate of 6 Federal Protective Service. The cartridge holders were so imported in to Logger pro 3.8 demo ( graphical analysis package with video analysis capablenesss ) . In Logger pro I located the bubble in each frame. The cartridge holders were on norm about 8 seconds long for the full experiment, each experiment contained three tests, on mean 2 seconds at 5 Federal Protective Service that is 10 frames per test, and at 25 frames per second are 50 frames per test. Although, non rather as accurate, does salvage a batch of clip and the loss of truth is non important. The cartridge holder is so scaled utilizing the separation between the two points on the forepart and running a marker along the side on the column. Logger pro so proceeded to roll up the clip and place co-ordinates and from at that place on place and speed ( among many other ) graphs can be extracted.

Graphic Analysis:

The graphs of the two 2nd most utmost concentrations of glycerin are displayed below, i.e. , 30 % glycerin and 70 % glycerin, to show the assorted tendencies in the graphs of supplanting and speed, when the concentration for glycerin is increased. All of the informations sets start at the beginning of the line section until their Ascension to the top[ 3 ].

Fig.1.0 represents the graph of the supplanting against clip for the bubble traveling in 30 % glycerol measured every frame, which was gathered from analyzing the footage.

Fig.2.0 represents the graph of the supplanting against clip for the bubble traveling in 70 % glycerol measured every frame, which was gathered from analyzing the footage.

Through point-by-point picture analysis I was able to generalize place informations sets for the bubbles. The bubbles approach what seems to be a additive tendency really shortly after get downing their Ascension. There were three tests for every concentration of glycerin and in all three, the graphs for each concentration arrives at a additive tendency, and when fitted with a line of best tantrum has a correlativity of at least 0.980[ 4 ]. This additive tendency in the place means a changeless speed, which so means no acceleration. This subdivision of changeless speed is said to be the bubble ‘s terminal speed. Zero acceleration besides means that the net Force is zero ( F=ma ) . The terminal speed as represented by the graph has besides been calculated, which will be subsequently used to understand it as a map of the Reynolds figure.

Fig. 3.0 illustrates the place versus clip with the speed measurings done by logger pro superimposed on the graph every bit good.

However, though the speed has been established to be changeless, the graph does non demo up changeless. This is caused by the method of informations aggregation. Point by point picture analysis fundamentally collects a series of times and co-ordinates, which makes it perfect for roll uping place informations. However, it does non cipher speed that good due to limited frame rate as it takes the a?†y and a?†t values between each frame and possible deformations makes this procedure less accurate and farther from the desired instantaneous value of Dy and dt.

Reynolds figure computation:

The Reynolds figure is dimensionless figure that is compared to the terminal speed as it is a better representation of the forces moving on the bubble as it flows through the liquid, in footings of a ratio of the syrupy to inertial forces and is computed by [ 1 ] :

tRe… … . ( 2 ) , where… … .. ( 3 ) ,

Where, ut = terminus speed ( acquired, cms-2 ) , 5 = kinematic viscousness ( acquired, cm2s-1 ) , Pliquid = denseness of the liquid ( acquired, gcm-3 ) , Aµ = dynamic viscousness.

Bubble Diameter Measurement

A bubble tantamount diameter ( deq ) was measured from the still frames obtained from the picture recording. The still images were so processed utilizing Pixcavator IA 3.2 demo and the bubble tallness ( dh ) and bubble breadth ( dw ) were measured in pels. The pel measurings were converted to millimeters by graduating it from the picture camera ‘s declaration and the measurings are merely considered to be comparative to each other. The bubble tantamount diameter, Deq was determined [ 2 ] as

Deq = … … ( 4 )

Where dw is the horizontal length and dh is the perpendicular length of the bubble. For this measuring it was assumed that the bubble was symmetric with regard to its perpendicular axis[ 5 ]. The mean tantamount diameter ( deq ) utilizing equation ( 4 ) was found to be,

deq a‰? 2.54mm ( A±0.05mm ) .

Rise speed results/analysis:

Fig. 4.0 the rise speed on a domain as a map of the Reynolds figure measured utilizing equation ( 2 ) & A ; ( 3 ) Tempref a‰? 20oC.

The bubble speed was measured at a tallness of 30cm above the point of air injection and the consequences obtained for the different kinematic viscousnesss is illustrated above in figure 4.0. It can be seen from figure 4.0 that the bubbles fluxing in the solutions of higher Reynolds figure have a big terminal speed. It can be seen from figure 4.0 that bubbles fluxing in solutions of lower Reynolds figure are greatly dominated by syrupy forces and alterations in viscousness have larger impacts on their terminal rise speed. When the concentration of glycerin is lower ( 10 % to 50 % ) compared to condense H2O, higher Reynolds figure, the values are really near together, proposing that the syrupy force seems less dominant. Therefore, addition in viscousness from 115 to 133 Cst has small consequence on the terminal speed.

Drag coefficient calculation/analysis:

The retarding force force, similar to clash, slows the upward motion of the bubble and the coefficient of retarding force is used as a relationship to find the opposition of an object in the fluid, which is an of import belongings to understand and in the false instance of non spherical bubbles the retarding force coefficient is calculated harmonizing to the given expression [ 3 ] :

Cd = aˆ¦aˆ¦aˆ¦.. ( 5 )

Where deq is the equivalent sphere diameter and dw is the diameter of the horizontal length or long axis length of the bubble and a?†I? is the denseness difference between the liquid and the atmospheric air.

Having calculated the mean denseness of the liquid in table 4.0, the denseness for the atmospheric air can be calculated by utilizing an extension of the gas jurisprudence in footings of denseness:

… … … … .. ( 6 )

, where I?air = denseness of the air, P = force per unit area, 1atm = 101325 Nm-2, R = gas invariable ( dry air ) = 286.9 J K-1 mol-1, T = temperature, at 20oC = 293.16 K ( A±0.5K ) ,

Therefore, the denseness of air is, I? = 1.205 kgm-3 ( A±0.17 % )

And, utilizing those values measured in experiment ( 2 ) , I can cipher the retarding force coefficient utilizing equation ( 5 ) .

Figure 5.0 Log – Log secret plan of the retarding force coefficients on a domain as a map of Reynolds figure, Tempref a‰? 20oC.

As the atom Reynolds figure additions from 106 to 106.5 ( A±6.3 % ) there is a big lessening in the retarding force coefficient ( A±7.7 % ) , Cd, from a value of approximately 104.5 to a value of about 104.0. As harmonizing to certification [ 3 ] , this happens due to the syrupy flows around the bubble making “ aftermaths ” behind the domain and the lessening in viscousness, higher Reynolds figure, corresponds to the formation of a disruptive boundary bed in forepart of the domain while accompanied by a narrower “ aftermath ” behind the domain. This narrower “ aftermath ” that forms behind the bubble causes the retarding force force to diminish. Besides, there is clearly an opposite relationship between the retarding force coefficient and Reynolds figure for the liquid.

Decision:

An experimental set-up was used to analyze the features of the bubbles lifting in different kinematic viscousnesss of glycerin and distilled H2O. The bubble rise features, viz. , bubble speed and drag coefficient produced sensible consequences. The bubble rise phenomena showed how the bubble speed varies with the addition in liquid viscousness as the bubble rises through the liquid column. The speed graph displays a tendency for different concentrations of glycerin solutions that the norm bubble rise speed decreases with the addition in kinematic viscousness. For smaller concentrations of glycerin ( 10 % and 30 % ) the syrupy forces are less dominant and a little lessening in bubble speed is observed for increasing solution viscousness. The graph exposing the relationship between the Cd and Re for the fluid illustrates that with diminishing kinematic viscousness, the opposition felt by the bubble lessenings due to the formation of a turbulent boundary which transcends into a narrower aftermath and can be inferred that with lower concentrations the syrupy flows are non the dominant forces. This research undertaking could be extended to include the effects of viscousness on the bubbles flight as it rises to the surface, which is besides an of import feature to see in gas-liquid contactors. Besides, the possibility of widening the correlativity for bubble rise features to Non-Newtonian fluids for their many applications and broad handiness in the universe and industries.

Evaluation:

After reexamining my experimental process I have noted a few restrictions that might do divergences in the ensuing informations and with the largest per centum mistake of 7.7 % . The most common mistake in the experiment would be a parallax mistake and could hold been made during the measurings of required glycerin and distilled H2O as the setup was held in manus during which the points were read, taking to a possible misread of the semilunar cartilage from leaning. Besides, the setup used to keep the liquid while mensurating the outflow clip could hold tilted every bit good when placed on the base. Second, due to the limited frame rate of the camera, the images were distorted as the motions were excessively speedy for it to capture, go forthing it blurred, doing it hard to turn up during the point by point analysis. Third, the volumetric rate of atmospheric air injected into the column would hold differed somewhat each clip due to human mistake. And as antecedently mentioned the limited frame rate caused disagreements in the information which made the speed non appear changeless. The method of still frame analysis to find the breadth and tallness did non take into history deformations from the picture camera and the existent diameter may change widely. Although, safeguard was taken to forestall escapes, the board that was attached as the base to maintain it sturdy was made from wood, which absorbed some of the solution as it is permeable. As the glycerin and distilled H2O were assorted, so excessively were air molecules trapped in the beds and the agitation experienced by each solution to take the air bubbles would be different and this would impact the consistence of the information. Besides, an premise was made on the humidness in the air due to the conditions conditions, which would impact the truth of the denseness measurings.

Possible solutions that I would propose is a high velocity picture camera with a big frame rate clearly capturing the objects motion. The equipment should be placed on a base and the perceivers oculus sight perpendicular to the semilunar cartilage or line section, or perchance utilize a swayer as a mention for a consecutive line. The base should be fitted with an impermeable surface such as aluminum ( light-weight and strong ) . A picture camera with a larger frame rate would besides cut down the deformation in the liquid and do diameter measurings more accurate. I would hold liked to compare my values with those available in literature to see the extent of the divergences and perchance get at a solution to optimize the experimental method further. A hygrometer would hold helped set up the sum of humidness nowadays in the room at the clip of carry oning the experiment.

( Word count: 3954 )

Bibliography:

Kulkarni, A. A. and Joshi, J. B. , Bubble Formation and Bubble Rise Velocity in Gas-Liquid Systems: A Review, Ind. Eng. Chem. Res. , 44, 2005, 5873-5931.

Miyahara, T. and Takahashi, T. , Drag Coefficient of a Single Bubble Rising through a Quiescent Liquid, Int. Chem. Eng. , 25 ( 1 ) , 1985.

Burris, A.W. factors impacting bubble size in H2O, Technical notes pg.54 -57

Ron Darby. “ Chemical Engineering Fluid Mechanics ” Marcel Dekker, Inc. 2001

Web Resources:

ReferenceA forA fluidA mechanicsA equationsA andA relevantA variables [ hypertext transfer protocol: //www.engineeringtoolbox.com/fluid-mechanics-t_21.html ]

Dr. Zhang, J. Particle Technology – Study Notes Chapter 2. Gesture of Particles through Fluids [ hypertext transfer protocol: //lorien.ncl.ac.uk/ming/particle/cpe124p2.html ]