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TABLE OP CONTENTS

Section Title Page No

1.0 Abstract... 2

2.0 Introduction... 3

3.0 Cyclone Theories... 5

4.0 What is Natural Length?... 10

5.0 Experimental Apparatus and Procedure... 13

6.0 Results and Discussion... 25

7.0 Limitations of this Study and Recommendations... 32

8.0 Conclusions... 33

9.0 References... 34

1.0 ABSTRACT

This study is part of a larger investigation attempting to

understand gas flow patterns within a cyclone. For the first time,

a flow visualization method, using a bubble generator, was utilized

to illuminate the gas flow patterns in a cyclone for a certain

range of gas flowrates and design configurations. The main

objective of this study is to investigate the presence of natural

length and the effect of selected cyclone parameters on it. In

addition, the effect of these parameters on the number of turns

formed in a cyclone was also studied. The flow visualization

experiments were recorded on a VHS video cassette.

Thus, this study hopes to contribute towards a better

2.0 INTRODUCTION

Cyclones are very simple separation devices that are used to

remove dust particles, mostly from gases and sometimes from

liquids. This study concentrates on gaseous fluids. Cyclones have

found extensive use in a wide range of engineering processes, for

more than a hundred years. Some common applications are in

coal-based power plants, grain handling in agricultural processes,

woodworking factories, and oil refineries.

The operation of cyclones can be extended to high temperature

and high pressure conditions by simply using appropriate materials

of construction. They can also handle high dust loadings (with an

increase in efficiency) and high throughputs. It is this

versatility of cyclones that makes them a widely used device for

gas cleaning. However, the separating capacity of cyclones is

limited to particles down to only about 5 microns in size.

Consequently, for particles of size below about 5 microns, they

have to be used in conjunction with other cleaning devices. In air

pollution control operations, they are often used as precleaners.

The various structural components of a conventional reverse

flow cyclone are (i) a gas inlet, (ii) a gas outlet, (iii)

cylindrical body section, (iv) conical section, (v) a dust outlet,

and (vi) a dust hopper. The most commonly used cyclone has a

tangential gas inlet. A schematic diagram of cyclone compenents

is shown in Figure 1. The relative proportions of all the above

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3.0 CYCLONE THEORIES

The separation in cyclones is largely brought about by the

centrifugal force imparted to the particles by the spinning motion

of the gas. The gas enters the cyclone via an entry duct that is

usually tangential to the body of the cyclone (Figure 2).

Sometimes a scroll entry is used (Figure 3) . It is generally

understood that the gas forms a downward moving outer vortex in the

annulus above the gas outlet duct. The vortex narrows as the gas

goes down the body of the cyclone. There is a second inner vortex

that flows upward and out of the gas exit duct.

As simple as it appears, the design of a cyclone is still

largely empirical. There is yet to be a "first principle" design

technique. The gas flow within a cyclone is extremely complex and

incompletely understood. However, the research works of ter Linden

(1949), Stairmand (1951), Lapple (1940) and Alexander (1949) in the

past, have contributed greatly to the understanding of cyclones.

In recent years, the works of Leith (1979, 1985) and others have

helped build on this earlier body of research.

The ultimate goal of a design technique for cyclones is to

predict accurately the collection or separation efficiency. The

collection efficiency "i^" is defined as that fraction of particles

of a certain size, which are collected by the cyclone. It is also

termed as "fractional efficiency." The fractional efficiency for

various particle sizes yields the fractional efficiency curve.

PlG.'^.C^CLOU^ CnTRY - HaLP OCHOLL

shape. One crucial estimator of efficiency is the "cut diameter,"

or "cut point" which is the particle size corresponding to a

cyclone efficiency of 50%.

The objective of any cyclone theory is to be able to predict

accurately the efficiency for particles of all sizes. A number of

relationships are available from the studies of Rosin et al.

(1932), Stairmand (1951), Lapple and Shepherd (1940), Davies

(1952), Barth (1956), Leith and Licht (1972), Dirgo and Leith

(1985), Theodore and de Paolo (1980), and lozia and Leith (1988).

The latter study uses a logistic model for cyclone efficiency to

show the relationship between various cyclone dimensions and

collection efficiency for particles of all sizes. This study has

been shown to predict djQ better than the preceeding theories. The

expression for dgg is:

1/

dso = { (9 U Q)/(Pp Z, Y,^^^) }2

where Q = gas flow rate

u = gas viscosity

^tmax ~ maximum tangential velocity

p = particle density

Zj, = cone length or natural length.

The term "Z^" that appears in the above equation is the topic of

this study. It is evident that accurate prediction of 6.^^ requires,

in turn, an accurate prediction of V^^^ and Z^. V^^^ is presently

evaluated from the following empirical relation (lozia and Leith,

1988), while Z^. is determined geometrically.

Vtmax = 6.1 V. (ab/D2)0-61 (D,/D) "^-^^ (H/D)'^-^^

V,. = inlet gas velocity a = height of gas inlet

b = width of gas inlet ^e ~ ^^^ exit duct diameter

D = cyclone diameter H = overall cyclone height. The length of the core is defined by Barth (1956) as the distance

from the bottom of the gas outlet to the base of the cyclone, H-S

(refer Figure 1) . Dirgo and Leith (1985) interpreted the core

length to be the distance from the bottom of the gas outlet to the

intersection of the cylindrical core with the cone wall, a distance

that can be less than (H-S) if the core diameter is greater than

the dust outlet diameter B. Alexander called it the "natural

length." A systematic investigation of natural length (Z^.) has

been lacking. A better understanding of natural length and

improved methods for its estimation are necessary to enhance the

design techniques for cyclones. Such an understanding will also

contribute towards the overall comprehension of flow patterns

within a cyclone. Hence, the investigation of natural length and

its dependence on some geometric and flow parameters of the cyclone

is the main objective of this study.

This study is divided into two parts. Part I deals with the

experimental investigation of natural length, while Part II

concerns the study of the number of turns made by a gas streamline

inside the cyclone and correlates it to gas flowrate and the inlet

4.0 KHAT IS NATX7RAI. LENGTH?

It is already known that the gas forms a vortex as it enters

the cyclone. The pressure is high throughout the cyclone, but

there exists a core of low pressure that extends the full height of the cyclone (ter Linden, 1949) . Below the gas outlet, the

spinning gas gradually migrates to this core of low pressure.

Within the core, the flow is upward. As the gas spirals down the

length of the cyclone, more and more of it is drawn off into this

central core and removed. At a certain point in the cyclone, all

the gas will be drawn into the core. This is the point of total

flow reversal, because there is no more gas flowing beyond this

point. The distance from the bottom of the exit duct to the point

of flow reversal is defined as the "natural length" of the cyclone.

This term was coined by Alexander (1949). He defined natural

length as the distance below the bottom of the exit pipe where the

vortices naturally turn (refer Figure 4).

Alexander states that the natural length is dependent only on

the proportions of cyclone inlet and outlet, irrespective of

gasflow over a wide range of values. He also proposed an empirical

relationship to estimate natural length (Z^) .

Z^ = 2.3 D^ (D^/ab)°-^^

where D^ = gas outlet diameter

D = barrel diameter

a = gas inlet height

b = gas inlet width

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It is evident from the definition of natural length that it

actually signifies the functional length of the cyclone. Beyond

the natural length, there will exist a stagnant zone where no

separation of particles from the gas takes place. Hence, an

optimal design requires that the overall length of the cyclone not

exceed the natural length significantly, to avoid the existence of

a stagnant zone. On the other hand, a cyclone that is shorter than

its natural length will not achieve full separation potential.

The experiments that led Alexander to the equation of natural

length are not clear from the literature. Furthermore, no

additional experimental work has been reported in the last 40 years

to substantiate or report the idea of natural length, or to

substantiate its dependence on inlet and outlet dimensions. Hence,

the purpose of this study was to investigate the dependence of

natural length on gas inlet and outlet dimensions and on gasflow.

5.0 EXPERIMENTAL APPARATUS AND PROCEDURE

The cyclone used in this study was constructed from clear

acrylic and polycarbonate plastic (see photograph in Figure 5) .

All its components were detachable and interchangeable. The overall height of the cyclone, and the diameter of the cyclone barrel were maintained constant throughout the experiments. The cyclone was fitted at the bottom with a dust hopper, also made of

clear acrylic.

Three sizes of outlet ducts and six kinds of inlets (all

tangential) were used in this study. The air flowrate in the

cyclone was controlled using a magnehelic pressure controller and

a fan (see photograph in Figure 6) . In general, the cyclone

dimensions conformed to the Stairmand high efficiency design except

for changes in inlet and outlet dimensions.

Part I - Method for Detection of Natural Length;

During the initial phase of this study, numerous attempts were

made to detect indirectly the point of flow reversal. The devices

that were used in these trials to track air flow included: (i)

smoke tubes, (ii) titanium tetrachloride fumes, introduced into

the cyclone through a perforated steel capillary, and (iii) a fine

perforated rod, threaded crosswise with numerous silk filaments along its length.

However, the above tests were not successful in reliably

recording the point of flow reversal. Hence, a better method had

to be developed to track air flow patterns and clearly reveal flow

FlC>. 6 Photo GrRAPH ^HOWtNG.

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reversal. Such a method was found in the use of neutrally bouyant

bubbles that follow air streams. Soap bubbles filled with helium

gas were introduced into the cyclone through the air inlet duct,

and the track of bubbles was illuminated using a strobe. The

illuminated flow patterns were filmed using a video camera and

recorder. A more detailed description of the apparatus is given

below. The experimental set-up is shown in Figure 7. Figure 1

shows the different dimensions of the cyclone.

The SAI Bubble Generator (model 3), manufactured by Sage

Action Inc., was used in these experiments (see photograph in

Figure 8) . This device generated helium-filled bubbles of sizes

between 1/16" and 1/4" diameter. The head from which the bubbles

were produced, consisted of three concentrically arranged tubes.

Helium passed through the inner tube, while the soap solution

flowed through the intermediate tube, thus forming helium-filled

bubbles at the tip. Air passing through the outermost tube

provided the thrust to blow the bubbles from the tip.

A continuous stream of bubbles could be formed and controlled

from the console of the bubble generator. The console (see

photograph in Figure 9) housed the valves that metered flow of

helium, air and bubble soap solution to the head. Micrometer

scales were provided on these valves to reproduce desired

flowrates.

A cylinder of compressed helium supplied the gas at a pressure

of about 22 psig. Compressed air was provided at a pressure of 26

psig. The bubble film solution was made using the liquid soap

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"Joy." The bubbles were introduced in the cyclone at an average

rate of about 150 bubbles per second. The bubble generator head

(nozzle) was supported on a long narrow stem (see photograph in

Figure 10). During a flow visualization experiment, this stem was placed inside the inlet duct of the cyclone so that the bubbles

were entrained into the air stream, just inside the inlet.

A strobe was placed at the bottom of the cyclone, just below the dust hopper (see photograph in Figure 11). Since the entire cyclone was made of clear acrylic, the strobe light illuminated the bubbles as they followed the air streamlines within the cyclone. The farthest point in the cyclone, where the outer vortex reversed

direction and turned upward, was measured as the natural length.

Six different inlets and three sizes of outlets were used in

this study. Each of these combinations was in turn studied at five

different air flows. The configurations and flows tested are listed in Table 1.

Part II - Method for Determination of Number of Turns and Velocity;

In this part of the study, the same set-up as that of Part I was used. This time, the number of turns made by the gas was observed and visually estimated from the recorded image on the

video cassette, for 7 different inlet and outlet duct combinations,

at a gas flow of 0.054 m^/s (114.5 cfm).

For each configuration used in this part, the cyclone was divided into four sections. The picture on the video screen was

paused, then the frame advance feature was used to track the number of tracks made by a chosen streak of bubbles. This process was

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Table 1. List of Configurations Tested and Their Natural Length

Config¬ a b _D. S ͣ ' .-Td..^^ 'Z,

uration Alexander's This Study

Equation

i ^1

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*

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49.1 >=101 Ii + 0? 12.5 5 7.5 12.5

37.1 >=101

h + O3 12.5 5 5.0 12.5

24.8 >=101

h + O1 18.75 5 10.0 18.75

42.7 >=95 I2 + 0? 18.75 5 7.5 18.75

32.4 >=95

I2 + O3 18.75 5 5.0 18.75

21.6 >=95

h + Ol 6.25 5 10.0 6.25

61.5 >=1D7

^3 + 02 6.25 5 7.5 6.25

46.6 >=107

h + 03 6.25 5 5.0 6.25

31.1 >=107

I4 + 01 12.50 2.5 10.0 12.50

61.5 >=101

^4 + 02 12.50 2.5 7.5 12.50

46.6 >=101

I4 + 02 12.50 2.5 5.0 12.50

31.6 >=101

I5 + O1 12.50 7.5 10.0 12.50

42.7 >=101

^5 + 0? 12.50 7.5 7.5 12.50

32.4 >=101 I5 + O3 12.50 7.5 5.0 12.50

21.6 >=101

^6 + O1 12.25 12.25 10.0 12.25

45.7 >=101 ^6 + 0? 12.29 12.25 7.5 12.25

34.6 >=101

l6 + O3 12.25 12.25 5.0 12.25

23.1 >=101

D = 25 cm, H = 94 cm, h = 35.5 cm, B = 10 cm.

** I = Inlet, 0 = Outlet, D = Barrel Diameter, H = Overall Height

B = Dust Outlet Diameter, S = Length of Gas Outlet Within

Cyclone, D^ = Exit Duct Diameter, a = Gas Inlet Height

b = Gas Inlet Width, h = Barrel Length, Z^. = Natural Length

*** Each of the above configurations was run at the following 5 air

flow rates:

Q, = 0.038 m^/s (81 cfm)

Q2 = 0.054 m^/s (114.5 cfm)

Q3 = 0.066 m^/s (140 cfm)

Q^ = 0.076 m^/s (162 cfm).

repeated for each section and the cumulative number of turns made

by the gas in the cyclone was estimated. Four counts of turns was

made for each each configuration. Analysis of variance was done

to estimate the statistical significance of the effect of

configuration on the number of turns.

It was evident from the video recording that the speed of

turns changed for different configurations. Though the film was

recorded in real-time, the high degree of parallax present in the

videotaped image made it impossible to make actual estimates of

angular velocity. Instead, the vertical speed (^) was estimated

for all configurations, i.e., the vertical distance traveled by a

bubble track in one second was measured.

This parameter was used to compare various test

configurations. To estimate this parameter, a section of the

cyclone was selected. At a flowrate of 0.054 m^/s (114.5 cfm) , the

video image of the flow was frozen on the T.V. screen. A track of

bubbles was chosen and advanced frame by frame on the screen. The

vertical distance travelled by the track was directly measured from

the T.V. screen. The video tape recorder had been previously

calibrated to find the time elapsed between two frames. Also, the

distance between two chosen points on the cyclone was directly

measured from the screen, and compared with the actual distance

between these two points, as measured during experimentation. This

comparison provides the magnification factor of the T.V. screen.

This magnification factor was then used to convert the vertical

distance traveled by the track on the screen to the actual distance

in the cyclone. An average of four readings was taken for each configuration.

Magnification Factor = Length Measured on T.V. Screen Actual Length Measured on Cyclone

The vertical velocity (j2f) was then estimated as:

= Actual Vertical Distance Traveled by a Bubble Track per __________________Frame (cm)_________________________

Elapsed Time Between Two Frames (sec)

6.0 RESULTS AND DISCUSSION

Part I - Natural Length:

The findings of this part of the study are shown in Table 1.

The flow visualization studies for the various inlet and outlet

configurations yielded the following results:

1. The bubbles clearly verified the presence of the outer

downward-flowing vortex.

2. A faster, upward-flowing, vortex was also observed at the core

of the cyclone.

3. The flow reversal occured all the way at the bottom of the

dust hopper. Z^ was estimated by taking the difference

between the entire length of the cyclone and the height of the gas inlet. These values are shown in column 7 of Table 1.

4. The point of flow reversal did not change with change in gas

flow.

5. The point of flow reversal remained the same for all inlets and outlets tested.

6. Comparison of the experimental results with Alexander's

empirical relationship for natural length (Table 1) shows that

the equation Z^ = 2.3 D^ (D^/ab)^^^ consistently

underestimates the natural length. Part II - Number of Turns:

The findings of Part II, i.e., the number of turns and

estimation of speed parameter for the seven configurations selected

are given in Table 2. The signifcant observations are:

Table 2. Number of Turns and Speed Parameter for Various

Configurations.

Conf urat

ig-ion _(cm)

a

(cm)

b (cm)

Area

(cm^)

S No. of

Turns _(cm/s)

i^ 1

^1 + O1 10 12.50 5.0 62.5 12.50 11 68

^6 + O1 10 8.75 8.75 76.6 8.75 9 118 1, + Oi 10 18.75 5.0 93.7 18.75 12

68

I5 + O1 10 12.50 7.5 93.7 12.50 10 62 I5 + O3 5 12.50 7.5 93.7 12.50 9 92

^1 ^ O3 5 12.50 5.0 62.5 12.50 8 72 I2 + O3 5 18.75 5.0 93.7 18.75 8 88

* D = 25 cm., H = 94 cm., h = 35,5 cm., B = 10 cm.

** Q = 0.054 m^/s (114.5 cfm)

1. Analysis of variance done for the various configurations

tested and number of turns obtained, shows that the

configuration has significant effect (p=0.00) for the inlets

and outlets studied (refer to Table 3).

2. Configuration also has a significant effect on the vertical

velocity (p=0.00), when statistical significance was tested

(refer to Table 4).

3. The vertical velocities were consistently lower for the bigger

exit duct (Dg = 10 cm) than for the smaller exit duct (D^ = 5

cm). For the inlet configuration I^, turns were estimated

only when used in conjunction with outlet O^. The comparable

value for the conbination I^ + Og was not obtained, because the

turns formed in this case were too fast to be counted.

4. The vertical velocities were found to be close for the

combinations (I5 + O3) and (Ij + O3) . Both these inlets, I5 and

I2 have the same cross-sectional area, 93.75 cm^.

Discussion;

The most significant finding of the natural length study was

that, for all the configurations used, the flow reversal was seen

well below the dust outlet, at the base of the hopper. This

finding raises two questions:

(i) Does that part of the core vortex within the dust hopper have

dust separation potential?

(ii) Would the point of flow reversal extend still further if the

length of the cyclone was increased?

It would now seem that the "natural length" observed in these

Table 3. Analysis of Variance for Number of Turns and

Configuration.

Dep Var: Turns; N: 27; Multiple R: 0.927; Squared Mult. R: 0.85 Source Sum-of-Squares Df _{Mean-Square F-ratio P}

Combinat 51.796 6 8.633 20.312 0.0

Error 8.500 20 0.425

Table 4. Analysis of Variance for Vertical Velocity and

Configuration.

Dep Var: Factor; N:29; Multiple R:0.847; Squared Mult. R: 0.72

experiments is actually an "artificial length" occuring due to the

obstruction posed by the base of the hopper. It may not have been the point where the vortices would "naturally" have reversed.

If part of the vortex within the hopper has separating

potential, then it also follows that the efficiency of the cyclone

could be increased by increasing its length, or changing the core

to barrel ratio. Otherwise, the turning vortex within the hopper

will lead to re-entrainment of dust from the hopper into the gas stream. It is necessary to corroborate the results of this study

by measuring the operational efficiency of these configurations

when dust-laden gas is processed through the cyclone.

The point of flow reversal did not change at different gas

flows. This is in keeping with Alexander's statements on natural

length. Since no change in the point of flow reversal was detected when the configurations or gasflow was changed, the conclusion drawn is that natural length, if it is different for different

cyclones, would depend on the total length of the cyclone. The total length of the cyclone was not changed in this study. The

ratio of barrel length to cone length, h::(H-h) (see Figure 6) may

also change the point of flow reversal. The effect of cyclone

total length and cone to barrel ratio should be investigated in

future studies.

In addition to these findings, the results of Part II suggest

that outlet and inlet dimensions affect the number of turns made

by the gas in the cyclone. Vertical velocities also changed with

change in outlet and inlet dimensions. An examination of the

values in Table 2 shows that the number of turns is higher for the larger outlet O^, while the vertical velocities are lower. The

bigger outlet duct has a larger cross-sectional area, hence, it

will lead to lower exit velocities of the gas, thus contributing

to the reduction in velocity.

However, it should be noted that vertical velocity is not the

only component, or even the major component of gas velocity in the

cyclone. Hence, the actual effect of reducing the outlet dimension

cannot be accurately known unless information is also available on

the radial and tangential components of the velocity. Ter Linden

(1949) and Alexander (1949) have also stated in different studies

that, smaller outlet ducts lead to faster vortices, and hence, a

higher pressure drop. In this study, it was difficult to detect a discernible trend in vertical velocities as a function of inlet cross-sectional area.

The values of natural length, as estimated from Alexander's

equation are significantly lower than the results of this study.

Alexander' s equation gives values of Z^. which decrease in

proportion with smaller outlet ducts and larger inlets. This

effect was not observed in this study. It is evident from Alexander's description of his experiments, that the method used to detect natural length was the introduction of water droplets

into a clear glass cyclone with an adjustable cone. The furthest

tracks made by the water droplets hitting the wall of the cyclone

was then arbitrarily defined as the natural length. At least one

problem, inherent in using this method, is that water drops being

much heavier than air, will be thrown against the cyclone wall long

before they follow the entire path of an air stream. This

introduces a "sling" factor for the drops/bubbles used to track the

air stream. The sling factor will be proportional to the density

difference between the drops/bubbles and air (or gas). In this

study, the use of neutrally buoyant helium filled soap bubbles

virtually eliminated the sling factor. This sling factor could

account for the lower values of Z^. predicted by Alexander. The

main conclusion of this study is that the effect of inlet and

outlet dimensions on natural length is not as significant as stated

by Alexander.

7.0 LIMITATIONS OP THIS STUDY AND RECOMMENDATIONS

Limitations;

Some of the limitations of this study are:

1. This study was restricted to testing the effects of gas outlet

and inlet dimensions on natural length. The basic design of

the cyclone was confined to the Stairmand high efficiency

configuration.

2. The range of air flowrates used was limited by the capacity of the fan.

3. At higher air flowrates, it was difficult to track the bubbles

visually. A light source more intense than the strobe, such as a laser beam, may be able to illuminate these tracks more

effectively.

4. The diameter of the barrel and the height of the cyclone were

invariant throughout the study.

Recommendations;

Some recommendations are:

1. The inlets and outlets used in this study should be further

tested with various core and barrel ratios.

2. The study should be extended to include efficiency tests on the configurations used.

3. Another parameter important to natural length estimation is

barrel diameter. Further studies on the effect of this

variable on Z^, are necessary.

8.0 CONCLUSIONS

This study concludes that the inlet and outlet dimensions do

not affect natural length, for a constant total lentgh of a

cyclone, as was proposed in the studies of Alexander (1949). Gas

flow does not affect natural length either. The number of turns

and the vertical velocity of the gas change with change in inlet

and outlet dimensions.

Flow visualization using naturally buoyant bubbles has great

potential for providing further insights into the complex flow

patterns within a cyclone. Further study, using a more

sophisticated version of the bubble generator device, and a laser

light source, is already underway. This study has set the

precedent for the use of a flow visualization method in the

characterization of cyclone performance.

9.0 REFERENCES

1. Abrahamson, J., Martin, C.G., and Wong, K.K. Trans. Inst.

Chem. Eng., 56, 168-176 (1978).

2. Alexander, R. McK., Proc. Australian I.M.M., (1949).

3. Barth, W., Brennstoff-Warme-Kraft, 8, 1-9 (1956).

4. Dirge, J.A., and Leith, D., Aerosol Sci. Technol., A, 401-405

(1985).

5. lozia, D.L., Ph.D. Thesis, University of North Carolina,

Chapel Hill (1988).

6. lozia, D.L., and Leith, D., Submitted to Aerosol Sci.

Technol., (1988).

7. Leith, D., and Licht, W., AIChE Symp. Series No. 68, p. 196

8. Leith, D. , in Handbook of Environmental Engineering, Humana

9. Rosin, P., Rammler, E., and Intelmann, W., Zeit. Ver. Deutsch.

Ing., p.433 (1932).

10. Shepherd, and Lapple., Ind. Eng. Chem., 972-984 (1939).

11. Stairmand, C.J., Engineering, p.356 (1951).

12. Stern, A.C., Caplan, K.J., and Bush, P.O., Cyclone Dust

Collectors (1955).

13. Swift, P. Filtration and Separation, 24-27 (1986). 14. Ter Linden, A.J., Engineering, 165-168 (1949).

10. ACKNOWLEDGEMENT

I would like to thank my thesis advisor. Dr. David Leith, for

his support and guidance in the research and preparation of this

technical report. I am also thankful to Drs. Michael Flynn and

Francis Digiano for their valuable suggestions during the

preparation of this report.

I am especially grateful to my husband, Venu Menon, who has

been a constant source of inspiration and assistance.

This project was supported by a research grant from the

Department of Energy.