FINAL REPORT

A Novel, Remote Cleaning,Electromagnetic (Quasi-Electrostatic)

Air Filter

OR

Prototype Electrostatic Air Filterfor

Pu Contaminated Soil Excavation

by

Robert A. Schill, Jr.

Department of Electrical and Computer EngineeringUniversity of Nevada Las Vegas

4505 Maryland ParkwayBox 454026

Las Vegas, Nevada 89154-4026

Submitted to:

Department Of Energy - Nevada Operations

1

February 1, 1999

2

TABLE of CONTENTS

I. Introduction 2

II. Purpose and Objectives 2

III. Prototype Electrostatic Air Filter: Description, Design, and Experiments 3

1. The Dust Maker 32. Faraday Tube 83. Water Charging System and Tests 124. Electrode Configuration & Dust Charging Tests 175. Prototype Electrostatics Air Filter Assembly and Experimental Tests 216. Design Embellishments 31

IV. Alternative Novel Quasi-Electrostatic Air Filter Paper Design 32 V. Deliverables and Student Participation List 34 VI. Conclusion 35

Appendix A. Heuristic Theory for Dust Maker - Instantaneous Number of Dust Particles 36

Appendix B: Total Charge on a Metallic Sphere Between a Parallel Plate Capacitor by Means of Direct Contact Charging 37

Appendix C: Simple Theory for Faraday Tube Calibration 40

Appendix D: ROUGH DRAFT General Theory for Faraday Tube Calibration 41

Appendix E: Mechanical Drawing Sketches of the Levitating Water Droplet Experiment 50

Appendix F: ROUGH DRAFT Theory for a Dust Charging System 53

Appendix G: Mechanical Drawing Sketches of the Corona Charging Electrostatic AirFilter Prototype (14 pages not numbered) 59

Appendix H: Computer Code for the Study of the Electric Field in a Two DimensionalCorrugated, Rectangular Electrode Configuration(21 pages not numbered) 60

Appendix I: Computer Code for the Study of the Faraday Tube Voltage Characteristics(26 pages not numbered) 61

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I. Introduction

Nuclear testing resulted in plutonium soil contamination at the Nevada Test Site. Soil restorationoperations are currently underway. Excavation equipment must disrupt the natural (or artificial) enamelsurface of the alluvium in the clean-up process. When the enamel like top layer of the desert soil iscompromised, natural and man-made forces result in airborne dust particles. This is of specific concern inPu contaminated soil excavation.

Conventional water spraying techniques are ineffective for dust particles on the order of a fewmicrons in diameter and smaller. Once the micron size dust particles are airborne, the viscous forces of airtend to dominate the gravitational forces. Consequently, the dust particles remain airborne for long periodsof time. If radio-nuclide contaminated dust particles are inhaled, they become trapped in the lungs and maybe a health risk.

Air precipitators typically employ filters or charging electrodes. In the latter, one electrode chargesthe dust particles while a second collects them. Once micron size dust particles are airborne, the viscousforces of air tend to dominate the gravitational and electrostatic forces acting on the particle. As a result,the fine dust particles tend to move in the direction of airflow. If fine porous filters or porous conductors areused, then backflow becomes a problem. This undesirably loads down the exhaust system. Further, largerdust particles will have a tendency to "clog" the filtering system which further aggravates the backflowproblem.

It is common for workers excavating radio-nuclide contaminated soil to wear special suits isolatingthe worker from the dust inhabited environment. In the field, these workers may be further isolated in airconditioned cabs of excavation equipment. The worker may be safely isolated from a possibly harshenvironment in the field. Breathing apparatus may be effective in reducing health risk. However, the workermay think it is a nuisance in the hot weather or when operating excavation equipment from within the cab.Very fine airborne dust particles embedded in clothing, skin, hair, etc., may still be inhaled into the lungslong after the breathing apparatus is removed whether on or off the site.

One means of extracting these fine radio-nuclide particles from the air is with an electrostatic airfilter which charges, traps, transports, and collects dust particles with the aid of electrostatic fields. Aprototype electrostatic air filter has been designed to use the viscous flow of air to guide dust particlesthrough a corona column (corona cloud) into a large chamber nearly flooded with atomized water droplets ofopposite charge. The oppositely charged water droplets are Coulomb attracted to the corona charged dustparticles. In turn, the dust particles become relatively heavy and precipitate from the air environment in thechamber exiting through a water drain. The air is forced through a grounded, loosely woven, fine wire meshinserted in the air exhaust tube for further dust filtration. Because the fine wire mesh in the exhaust tube isloosely woven, backflow loading effects are not as significant as other techniques. The wire mesh and wateratomizing chamber can be cleaned by flooding the exhaust tube with water. The dust charging electrodesmay be cleaned in a similar fashion. Consequently, the system is self cleaning. As a result, humanintervention is virtually eliminated.

II. Purpose and Objectives

This report is to conclude the research study of novel air precipitators with application to Pu (radio-nuclide) contaminated soil excavation. The research was conducted in three phases. The first phase was apaper study of a novel quasi-electrostatic air precipitator. Size, prohibited cost, use of hazardous gases(fluorine), power requirements, and poor efficiency led to the second phase of the research; down sizing.Efficiency, minimal power consumption, flexibility, and portability relative to the earlier design were to betaken in consideration. Results obtained in this study initiated the final phase of the supported research; aprototype electrostatic air filter. Each phase of the study will be presented with emphasis given to the

4

design, building, and testing of the prototype. The earlier phases of this research is documented in thethesis by Mr. Zhongyi Gu.

The prototype electrostatic air filter is to charge, trap, and transport small re-suspended dustparticles to a localized container without human intervention. It is designed to be installed in the cabs ofexcavation equipment occupied by workers. The prototype air filter with peripherals being reported are: adust maker, dust filter assembly [corona dust charging system, water droplet charging system, exhaustsystem], and diagnostics. In many cases, both theoretical and experimental studies were initiated.Experimental results dictate the level of completion of corresponding theories. In some cases,computational efforts based on developed theories were abandoned or postponed due to negativeexperimental results. Even so, these theories have been provided in the appendix for completeness of thisreport.

The report is organized in the following fashion. A description and design of the prototypeelectrostatic air filter is provided in Sect. III. Experimental studies leading to the design are discussed.Section IV provides an alternative novel quasi-electrostatic filter which may be of futuristic use as the stateof the art in laser technology advances. A list of participants and deliverables are provided in Sect. V. Thereport is concluded in Sect. VI.

III. PROTOTYPE ELECTROSTATIC AIR FILTER: DESCRIPTION, DESIGN, ANDEXPERIMENTS

The design and testing of the electrostatic air filter required an assembly of supporting diagnosticsand a dust maker. This section is subdivided as follows: 1. Dust maker, 2. Faraday tube, 3. Water ChargingSystem and Tests, 4. Electrode Configuration and Dust Charging Tests, 5. Prototype Electrostatic Air FilterAssembly and Experimental Tests. Each subsection contains a description of the device or diagnostic, itsutility, a theory if appropriate, and experimental results.

1. The Dust Maker

Purpose

A means was needed to simulate the precipitator's dust environment. The original objective was tofocus on dust particle sizes on the order of a few microns in diameter. Even so, this is not realistic sinceairborne dust particle sizes may range from submicron to 100 microns or larger in diameter in the fieldpending weather conditions and proximity. As a result, a crude dust maker was designed to target dustparticle diameters roughly on the order of less than 50 microns. This is achieved by first preparing thecollected desert soil. The soil is sifted with a sifter composed of four randomly overlapped layers ofconventional aluminum screening. The grid size of each layer is 1 mm x 2 mm. As a worst case scenario,dust particles with diameters of roughly 0.5 mm are allowed inside the dust maker. A forced airflow isdirected to the sifted dust base disturbing the surface layer (Refer to Fig. 1.1). The disturbed dust particlesmust then have enough kinetic energy to rise about 3" with the air stream exiting the dust maker through anoutlet. Those particles which are not light enough to pass out of the dust maker are deposited outside ofthe main airflow stream inside the dust maker. Consequently, the dust base must be stirred after limitedruns in order to maintain a nearly uniform particle count.

5

Dust Maker

CardboardAir FlowDirector

Dust Base

Fan

FlowAir

Outlet

Fig. 1.1

Detailed Description of Design

The dust maker is 15 cm x 20 cm x 30.5 cm metal box supporting a 120 V DC, 11 cm fan,cardboard baffle, an offset liquid funnel, and roughly a 9 cm sifted dust base. Refer to Figs. 1.2 and 1.3.

Fig. 1.2 Fig. 1.3

The cardboard airflow director (baffle) is sealed on the top and sides with tape. The baffle bottom rests onthe surface of the dust base. The top edge is 8" above the dust base taped to the back wall (wallsupporting the dust maker fan). The bottom edge extends about 4" away from the back wall. The outlet isan offset funnel with a 10 cm to 2.4 cm diameter transition. The funnel is mounted and centered over a 11cm by 11 cm square hole existing at the outlet side of the dust maker. The fan is screw mounted andcentered on the back wall about 3" from the top of the dust maker. The lid of the dust maker contains a 0.5"x 5" slit covered with a single layer (1 ply) of loosely woven grain paper such as Kimwipes EX-L (Productnumber 34256). The top edge of the dust maker is lined with duct tape so that there is a snug fit betweenthe lid and dust container.

The range of dust particle diameters generated by the dust maker is experimentally determinedusing a quadruple baffle filter system. This filter was designed to minimize the amount of backflow

6

experienced by the air stream passing though it. A quadruple baffle filter is designed using two standardsize dixie cups connected at the open end as shown in Fig. 1.4. Four sets of slits are cut beyond the

Dixie Cups

Filter Paper

QUADRUPLE BAFFLE FILTER SYSTEM

FilterPaper

Air FlowAroundFilterPaper

Slit inDixieCup

FilterPaper

Fig. 1.4

midway point into the cups. Whatman 1 Qualitative filter paper (11 µm retention) is inserted in the slits suchthat better than half of the cup cross sectional area is covered. The baffle was mounted on the dust makeroutlet. The dust maker fan voltage was then varied and the filter paper was examined with a metallurgicalmicroscope having a magnification factor of 500. Figure 1.5 displays what is typically

Fig. 1.5

observed with the microscope. The filter paper is exposed to the dust maker's air stream for five minutes.Table 1.1 yields dust particle diameters observed for various dust fan voltages.

Table 1.1Fan Voltage

(V)Dust Particle Sizes Observed

(µm)70 < 2075 < 2080 < 3090 < 30

7

An optical scattering technique known as Mie Scattering is used to quantify both the number ofdust particles being generated by the dust maker and the angular uniformity of the dust density at the dustmaker outlet. The theory is presented in Appendix A. Figure 1.3 displays the experimental setup. A lasermount was designed to support the laser and photo diode detector so to achieve good alignment whenexamining the uniformity of the dust density with azimuthal angle. Figure 1.6 is a mechanical drawing of thelaser mount.

1/4-20

4"

4 1/4"

1/4 Diameter

3"x3" foot plate

195 deg

210 deg

6"

6 1/

2"

7"

5"2"

255 deg

240 deg

14.993°

300 deg

285 deg

4.977°

39.983°

45.000°

30.090°

Fig. 1.6

To contain the dust during the experiment, a dust tunnel was designed. An approximately four andthree fourths inch cube Plexiglas container was built minus one wall. Directly opposite to this wall, a oneinch centered hole with mount (1 ¾" PVC tube, glued to Plexiglas wall, with inside shaved to fit acardboard/PVC rectangular to circular funnel) is located. A 1' diameter by 6' long plastic bag is attached tothat side of the cube with wall missing. The opposite side of the cube is attached to the dust maker outletby means of a homemade rectangular to circular funnel made out of cardboard, PVC pipe, and duct tape.Before each experiment is conducted, the plastic bag is completely deflated. The experiment is conductedwithin the time frame when the bag is not fully inflated. Steady state diode voltages are recorded bothbefore (base reading) and after the dust maker is turned on. About thirty runs were recorded at each of thefollowing angles relative to the horizontal: 0o, 45o, 90o, and 135o. Figure 1.3 shows the laser positioned inthe 90o position relative to the horizontal. The first inch or so of the dust base was mixed for the first 12 runsand a full dust base mix was made for the last 18 runs. Figure 1.7 displays the instantaneous dust count in

8

the beam when the dust base is thoroughly mixed (full dust base mix). The dust count is per experiment isreasonably consistent and appears to be angle independent.

Instant. Dust Count in BeamFull Mix Condition

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Exp. Run (no.+12)

Par

ticle

Co

un

t 0 deg.

45 deg.

135 deg.

90 deg.

Fig. 1.7

The average dust count for a fan voltage of 78V DC is roughly 458 dust particles in the beam path at any oneinstant in time. A procedure on how to operate the dust maker is established to yield reasonably consistentdust loads within the time frame of the experiment. A dust maker assembly parts list and a diagnosticassembly parts list are provided in Tables 1.2 and 1.3 respectively.

Table 1.2 Dust Maker Assembly Parts ListPart Description Purpose

Muffin Fan 115 V DC fan To generate the forced airflowneeded to generate airbornedust particles

Variable Autotransformer Staco, 120 V input, 0-120/140V output, 10 A, 1.4 KVA

To control the speed of themuffin fan

Dust maker frame Tin Box with lid Body of dust makerAir filter A single layer (1 ply) of loosely

woven grain paper such asKimwipes EX-L (Productnumber 34256) taped onrectangular opening cut in thelid

Helps decrease the outletairflow

Outlet Port Offset Funnel Airflow director Cardboard Directs the flow of air onto the

surface of the dust base

Table 1.3 Diagnostic AssemblyPart Description Purpose

HeNe laser & power supply Polarized, 5 mW, 633 nm,HeNe laser with TEMoo 0.81mm spot size

Source of photons needed todetect dust particles

Photo Diode detector EG&G Photon Devices SeriesFND-100Q

Detects those photons notscattered by the dust particles

9

and other objects between thelaser and diode

High voltage source Keithley Instuments240A High Voltage Supply

Needed to bias the photo diode

Oscilloscope Used to measure the outputvoltage of the diode

Dust tunnel 1' x 6' plastic bag Contain dustDust tunnel transition ~ 4 ¾" x 4 ¾" Plexiglas cube

with 1" inlet and wall missingoutlet

Contain dust and act as a stiffwalled transparent window forthe laser

Ruler mounts Mounted on the dust tunneltransition

Alignment purposes

Diagnostic mount Refer to Fig. 1.6 Frame supporting the laser anddetector

2. Faraday Tube

A diagnostic is needed to measure the amount of charge on a moving dust particle. For thispurpose, a Faraday tube diagnostic was designed, built, and tested. The Faraday tube works on the sameprinciple as the Faraday cup. Unlike the cup, the tube has a separate inlet and a separate outlet. It isanticipated that the tube should not significantly alter the dust particle's dynamics.

3.50

0cm

Exhaust EndNylon Cap withBeveled End

Input EndNylon Cap withTapered End

Nylon Cap Tightly FitsBetween the Inner andOuter Electrodes

Air Space Betweenthe Outer Grounded Electrodeand the Inner ElectricallyFloating Electrode

Electrically Floating Inside Cylindrical Electrode1/4" Stainless PipeMeasured Dimensions: 0.182" (4.62 mm) Inside Dia. 0.2515" (6.39 mm) Outside Dia.

Grounded Outer Cylindrical ElectrodeStandard 5/8" Copper PipeMeasured Dimensions: 0.572" (14.52 mm) Inside Dia. 0.6.26" (15.89 mm) Outside Dia.

Wire Soldered toFemale Coaxial Connectorand Inner Cylindrical Electrode

Female Coaxial ConnectorPress Fitted and Clamped onGround Electrode. Clamp NOTShown.

0.79

3cm

Out

side

Rad

.

19.7

00cm

Inne

r Pip

e Le

ngth

20.3

00cm

Out

er P

ipe

Leng

th

22.2

00cm

Ove

rall

Leng

th

0.793cmOutside Rad.

0.231cmInside Rad.

0.30

0cm

0.60

0cm

0.231cmInside Rad.

0.30

0cm 1.30

0cm

10

Fig. 2.1

The Faraday tube is composed of two concentrically oriented, cylindrical, tubes separated by nylonend caps. Except for the ends, an air medium exists between the tubes. Figure 2.1 provides a detailedsketch of the construction of the Faraday tube. The 6.39 mm outside diameter inner stainless steel tube(1/4" stainless steel pipe) is 20.3 cm long. The 14.52 mm inside diameter outer copper tube (standard 5/8"copper water pipe) is 19.7 cm long. The nylon separator on the input side of the Faraday tube contains a21.8o outside taper starting at a 2.31 mm radius (inside radius of inner stainless steel tube) at the tip of thetube to 7.95 mm radius (outside radius of outer copper pipe). This was designed to gradually sample thedust stream without creating substantial turbulence or back flow at the entrance of the tube. Near the backside of the Faraday tube is a 90o stub of 5/8" copper pipe welded on the copper ground electrode. A femalecoaxial connector is press fitted on the end of the stub and then clamped into place. The clamp is notshown in Fig. 2.1. The inner conductor of the coaxial connector is soldered to an insulated wire. The otherend of the wire is soldered onto the surface of the inner 1/4" stainless steel electrode. The inner electrode iselectrically floating while the outer tube is electrically grounded through the test equipment by way of thecoaxial cable connector.

Fig. 2.2 Fig. 2.3

The Faraday tube system consists of the Faraday tube, a male coaxial to female triaxial connector,triaxial cable with connectors, electrometer with analog output, and a fast oscilloscope as pictured in Figs.2.2 and 2.3. The loading effect of the Faraday tube was measured to be 1.1 TΩ. Few instruments haveinput impedances that come close to this value. Consequently, most measuring devices will significantlyinfluence the voltage measurements measured. A 6517A Keithley Electrometer was purchased for thisreason. Its optically coupled internal circuitry allows for an input resistance greater than 200 TΩ. Therefore,voltage measurements obtained from this device are not significantly influence by the meter. The internalcircuitry of the electrometer is slow compared to the time scale of the test experiments. At best, onevoltage reading could be taken as a charged test bead passes through the tube. This is unacceptable sincethe position of the test bead is not known at the time of the voltage reading. As a result, an oscilloscopewas directly coupled to the analog output of the electrometer. Real time measurements could be made withthe oscilloscope being optically isolated from the Faraday tube by way of the electrometer. As a note inpassing, a significant amount of effort was spent in developing isolated amplifiers and buffers in lieu of theelectrometer. The high input resistance of the Faraday tube significantly influenced the operation of all highquality conventional amplifier chips. The chips did not operate as designed. Attempts of building anamplifier circuit for this purpose was then abandoned and the electrometer was purchased.

11

Fig. 2.4

Figs. 2.2-2.4 illustrate the calibration setup. A 1.625 mm radius ball bearing is charged by a set ofparallel charging plates separated by 2.7 cm as shown at the top of Fig. 2.4. The ball bearing has a smallhole drilled through its center which is threaded with standard clear plastic fish line. The bearing movesfreely on the fish line aligned approximately along the axis of the Faraday tube. As shown in Fig. 2.4, awooden base supports the Faraday tube, charging plates and fish line with ball bearing. The fish line isanchored at the bottom of the wooden base with a screw and is tied to a plastic slide at the top of thewooden frame. Using the plastic slide, the bearing is pressed against the center of one of the metal plates.With suitable normal forces imposed by the fish line attached to the plastic slide, friction between thebearing and plate wall prevents the bearing from sliding. The voltage source connected to the plates isturned on thereby charging the ball bearing. The amount of charge deposited on the bead is developedtheoretically in Appendix B and can be deduced from the expression

( )( )( )q V R

D R R

D D R D RT o o oo o

o o

= −− +

− −

42

2

2 2

πε (2.1)

where Ro is the radius of the bead, D is the distance of separation between the charging plates, εo=8.85 x

10-12 F/m the permittivity of free space, and Vo is the potential difference across the plates. Once voltageand current readings are stabilized, the plastic slide is slightly tapped allowing the bearing to fall freely alongthe fish line through the Faraday tube. As the charged bearing passes through the inner electrically floatingtube, a voltage difference is induced between the inner and outer tubes. It is this time varying voltage that is

measured. Knowing the capacitance of the Faraday tube ( )C q V= , the charge may be determined. To

calibrate the tube, the measured voltage Vmeas is compared to the predicted voltage Vpred. Treating theFaraday tube as an infinite in extent coaxial capacitor, a simple theory predicts that the voltage should begiven by

VR VLD

ba

DR

DR

DR

predo o o

o o

= −

−

+

−

−

2

1 1

1 2

2

2

ln (2.2)

where a and b are respectively the outside radius of the electrically floating inner electrode and b is theinside radius of the grounded outer electrode, and L is the length of the inner copper tube. This expressionis developed in Appendix C. This simple theory does not take into consideration the finite geometry of thetube or the position of the charged ball relative to the Faraday tube. Even so, experimental results arecompared with predicted results. Figures 2.5 and 2.6 show typical

WoodenBase

FaradayTube

ParallelPlates

SlidingPlastic Plate

ElectrodeConnected toOne Plate

12

Fig. 2.5

Fig. 2.6

measured real time voltages. For the same conditions, a maximum calculated ± 2.77 V should have beenmeasured. The predicted voltage is about 7.25 to 8.13 times larger than the average experimental maximumvoltage values observed in the figures. A more complex theory employing Laplace's equation and a dualintegral boundary technique has also been developed which considers some of the finite geometry effects.Refer to Appendix D. Numerical efforts for this work have not been completed at this time and are notpresented in the appendix.

3. Water Charging System and Tests

13

Various water charging systems have been experimentally examined: injection pumping,electrostatic atomization, direct charging, and induced charging. The source requirement for direct chargingwith mechanical atomization appear to require the least voltage. Since one of the objectives of the prototypeis to minimize the source requirements, direct charging with mechanical atomization is pursued.

Fig. 3.1

Fig. 3.2

Referring to Figs. 3.1 and 3.2, the water charging system is composed of a motor, positive flowpump, a fiberglass key (not shown), conventional atomizing spray nozzle, charging electrode with voltagesource (an earlier electrode configuration is shown), pressure gauge with controller, and plastic tubing (waterline) with water container. Some of these elements are identified in Fig. 3.3. The water lines with watercontainer, the positive flow pump, the pressure gauge, and the atomizer are held at a positive potential. Toprevent the motor from being electrically shorted, a ¼" thick Fiberglass key was machine to mechanicallycouple the motor to the positive flow pump. The length of the key was determined to give a 3/8" clearancebetween any edge on the motor and any edge on the pump. This distance of separation is more than fivetimes the distance between the atomizer and the grounded electrode. As a result, shorting though themotor should not result. When the key was used, all shorting through the motor is completely eliminated forthe charging potentials employed. Figure 3.3 shows the motor and pump coupled with the fiberglass keymounted on a electrically insulated board. A safety shield shown in the figure just in front of the motor wasbuilt to cover the exposed area around the fiberglass key. It is of importance to note that the water lines andwater container must be electrically isolated from the worker when the water charging system is electricallyactive. This is a common requirement in the electrostatic spray painting business.

SprayNozzle

WaterGauge

14

Fig. 3.3

A flat spray atomizing nozzle with deflector is chosen as the mechanical atomizer and the chargingelectrode. A flat spray pattern distributes the liquid as a flat- or sheet- type spray. The flat spray pattern isformed by a round orifice tangential to a deflector surface. In the deflector design, the deflection surfacediverts the spray pattern away from the axis of the inlet pipe connection. Figure 3.4 shows the spray nozzlewith deflecting surface used. The nozzle is designed by Spray Systems Company. The sharp edges of thedeflector are ideal for charging since the electric field lines are highly concentrated at edges relative to flat orlarge radius metallic surfaces. The nozzle orifice was chosen based on three constraints: 1. minimize thewater pressure and the amount of water used per minute, 2. large area coverage by a single flat spraynozzle, and 3. a mid-ranged nominal water droplet diameter (i.e., to support large amounts of charge, to beaffected by gravity, and to provide broad coverage with as many droplets as possible). Spray System's flatspray nozzle with deflector was chosen with a 0.016" orifice diameter. Table 3.1 below displays the waterpressure, water capacity, and volume median diameter (VMD) droplet size for this nozzle. The volumemedian diameter measure drop size is a value of measure where 50% of the total volume of liquid sprayed ismade up of drops with diameters larger than the median value and 50% with smaller values.

Fig. 3.4

Table 3.1

Motor PositiveFlow Pump

WaterOutlet Line

PressureReleaseLine

FiberglassKey

SafetyShield

Water InletLine

15

Some Characteristics for the 0.016" Orifice Flat Spray Nozzle(FloodJet Spray Nozzle Wide Angle for Liquid: 1/8 K-SS .25 )

Pressure(psi)

Water Capacity(gpm)

1Volume Median Diameter(VMD) Droplet Size

(µm)

20* 0.035 ~24040 0.05 ~210

60** .06 ~200* Spray angle information provided: 83o

** Spray angle information provided: 117o

1 Reasonable estimates based on the following product information: 0.05 gpm at 10 psi yields 260 µm VMD;0.1 gpm at 40 psi yields 220 µm VMD; 0.16 gom at 100 psi yields 190 µm VMD

A plastic cap covered with aluminum tape was one of two electrode styles examined. Thealuminum was covered with GE RTV Silicone for insulation purposes. The plastic cap was about a 1/16" to1/8" thick. It was very difficult to cleanly cut the aluminum tape with a razor blade. The burrs on the tapeedge seemed to short out the system. A second more successful electrode was developed by wrappingstandard door bell wire around the spray nozzle and then gluing the wire to the nozzle with epoxy. Carewas taken not to over bend or twist the wire. Earlier tests demonstrated that the wire insulation becomeselectrically compromised at high voltages (above 2 kV) when the wire is twisted or bend beyond a certainangle. Note that door bell wire is not designed to support large voltages even though the current is verysmall. The small wire radius is needed to increase the field intensity near the edge of the electrode.Figures 3.5 and 3.6 show the charging electrode geometry used in the prototype. Figure 3.6 is a croppedenlargement of Fig. 3.5. The loose wire end is covered with GE RTV Silicone to prevent shorting as a resultof the charged water droplets

16

Fig. 3.5 Fig. 3.6

in the air environment. During experimental tests, this small stub of wire shorted near the electrode. It wasthought that the wire may have been bent beyond the critical angle thereby compromising the insulationelectrically. To fix this, the wire stub was inserted in a ¼" in diameter, clear, plastic tube slightly longerthan the wire stub filled with GE RTV Silicone. Water charging voltages never exceeded 6 kV from that timeonward and shorting never reoccurred.

Tests were conducted to determine the charge on the water droplets. The Millikan Oil DropExperiment was performed using the charged water droplets in place of oil. Here, charged water dropletsenter between two parallel plates with a potential difference across them. If the electrostatic forcecounteracts the gravitational force acting on the water droplet levitation results. Knowing the mass of thewater droplet, the potential difference between the plates, and the plate separation, the water droplet chargecan be determined. The task is to obtain a ball park measurement of the total charge supported by thedroplet. For this purpose, a 10" by 10" parallel plate capacitor housed in an acrylic container with a ½"centered side hole was designed as shown in Figs. 3.7 and 3.8 below. Refer to Appendix E for mechanicalsketches of the capacitor geometry with acrylic housing. The orifice allows water droplets impinging on theopening to enter the parallel plate environment. The aluminum plate electrodes rest on clear plastic fish line.The fish line is used to control the separation and orientation of the electrodes in the acrylic container. Asmall beveled hole was drilled in the aluminum plates for a beveled Phillips screw. The screw head fits flushto the plate surface. The screw is needed to electrically connect the external voltage source. The screwhead perturbation to the plate surface is small and is neglected in calculations for the ball park accuracy's ofinterest. Button levels were glued onto both plates and used as a means to level the plates such that thenormal to the plates is parallel to the gravitational force acting on the plates. It is well know that an 8o

angular displacement from this position will introduce a 1% error in calculations. Since the exact charge onthe water droplet is not of interest, a 1% or 2% error in calculation is acceptable. The water pressure at theatomizer is 50 psi.

Fig. 3.7 Fig. 3.8

The atomized water droplets travel a distance of 53 to 54 cm in air before they reach the levitating plates.Roughly, those droplets sampled traveled about 47.5 to 48 cm horizontally to the earth's surface and about24 cm against gravity. The center of the ½" diameter hole in the acrylic wall is 1.5 cm from the bottom ofthe top aluminum electrode. The distance of separation between the electrode plates is 4.5 cm. Theangular portion of the sheet of droplets sampled with relation to the atomizer nozzle was not of concern.Figures 3.9 and 3.10 are two views of the experimental setup. Five different tests were conducted todetermine the average plate voltage needed to cause levitation. Because the water droplets have both avertical and a horizontal motion relative to the plate surface, levitation was stated to occur when the water

17

droplets would traverse a major portion of the distance along the plates. Field fringe effects are expected toprevent the droplets from traversing the entire plate length. All observations were made without prejudice bythird parties not significantly familiar with the experiment. To visually observe the water droplets, Miescattering techniques were applied. A conventional flashlight was appropriately positioned so that theobserver could see the scattered light from the droplets. Fig. 3.11 is a plot of the parallel plate potentialrelative to the water charging potential. The stared black line yields an average value of the test runs.

Fig. 3.9 Fig. 3.10

2 3 4 5 6 7 8 92

3

4

5

6

7

8

9

10

Atomizer Voltage [kV]

Levi

tatin

g V

olta

ge [

kV]

Voltage Needed to Levitate Charged Water Droplets

Run 1 Run 2 Run 3 Run 4 Run 5 Average

Fig. 3.11

18

Based on the average of these test runs, the total charge on the dust particles are determined. Balancingthe gravitational force with the electrostatic force, the total charge on the water droplet can be determinedfrom

qdmgV

m R

Tplates

m

=

=

ρ π

43

3

(3.1)

where ρm=16.02 kg/m3 is the mass density of water at room temperature, R is the radius of the water droplet(spherical geometry assumed), d is the distance of separation between the levitating parallel plates (parallelplate capacitor), g=9.8 m/s2 is the acceleration of gravity constant, m is the mass of the water droplet, andVplates is the potential difference between the parallel plate capacitor. Using Eq. 3.1 the charge on the waterdroplet for various water charging potentials is plotted in Fig. 3.12. It is observed that the average charge isroughly between 4.5 and 11.25 pC for charging voltages between 2 and 8 kV.

2 3 4 5 6 7 84

5

6

7

8

9

10

11

12

Atomizer Voltage [kV]

Ave

rag

e D

rop

let

Ch

arg

e [

pC

]

Average Charge on Droplet for Various Atomizer Voltages

Atomizer Pressure: 50 psi Water Droplet Dia.: ~210 microns

Fig. 3.12

4. Electrode Configuration & Dust Charging Tests

Various corrugated charging electrode designs were examined. Two will be described here of whichone is used in the prototype device. The electrodes are constructed out of aluminum about 1 mm thick.The first set of corona charging electrodes examined is shown in Figs. 4.1 and 4.2. The top electrodecontains three ridges angle at 45o with all corners replaced by large radius fillets. The bottom electrode isflat with a 1.2 cm radius bend on the front and the back as observed in the side view of Fig. 4.1. Plexiglasside walls separate the negative and ground electrode such that the distance of separation between theedge of the razor blade and the bottom electrode is about ½". As shown in the figures, six razor blades areinserted in the negative electrode, three in each groove. The blades are secured to the electrode with

19

epoxy. The blades extend between 1/16" to 1/8" beyond the aluminum electrode surface into the airflowregion. For ten nonconsecutive hours, 18 kV was applied to the electrodes. The average current drawn fromthe source ranged between 1.1 mA and 1.25 mA. This is less than a 15% current fluctuation from theaverage. Intense purplish-blue cones of light were generated just below the sharp blades and appeared toslowly migrate in different directions along the blades in a random manner. Below this intense light, a cloudof bluish light exists as illustrated in Fig. 4.3. Figure 4.4 is nearly identical to Fig. 4.3 when the picture wastaken with a camera flash lamp. Random arcing also occurred occasionally. The generation of ozone couldbe detected. Undesired edge effects were virtually eliminated for this plate geometry as a result of the largeradius fillets and bends. Oxidation on the

Razor BladesNegative Electrode

Clear PlexiglasSide Walls

GroundElectrode

FRONT VIEW

Negative Electrode

SIDE VIEW

Air Flow

Ground Electrode

NegativeElectrode

Fig. 4.1

Fig. 4.2

aluminum ground plate was very evident by its whitish change in color (white rust). The razor bladescontained some form of ferromagnetic material. Oxidation of this material resulted in a noticeable red rustover the blade surface. Even so, the glow discharge was very consistent and the corona columns were notsignificantly affected. Further, it was observed that the plates did not degenerate monotonically with time.This form of oxidation could have been prevented if stainless steel blades with minimal amounts of iron wereused. Stainless steel metals such as Stainless Steel No. 316 or 17-4-Ph are recommended since theycontain minimal amounts of iron.

Clamp

PlexiglasSide wall

Razor Blades

20

Fig. 4.3 Fig. 4.4

A theory based on Laplace's equation employing various conformal mapping transformations(Schwartz-Christoffel and Inverse Cosine transformations) was developed to characterize the initialbreakdown region around the negative electrode surface. Refer to Appendix F. The model does not takeinto consideration the presence of the Plexiglas side walls. Instead, the side walls are modeled as groundplates as shown in Fig. 4.5. Using a random number generator, Fig. 4.6 shows the region where breakdownwill occur. This should be the region of the corona in which intense light is generated. The change in thenature of the electrical conduit as a result of the breakdown is not considered.

V CeramicInsulator

CeramicInsulator

First Electrode Design

Fig. 4.5 Fig. 4.6

The plate geometry illustrated in Fig. 4.1 has a limited charging region due to both the large spacebetween grooves and the direction of airflow along the length of the razor blade electrodes. Corona columnsor "corona sheets" are aligned parallel to the direction of airflow in this electrode configuration. Amodification to this geometry was developed to increase the charging region with a smaller potential.Figures 4.7 - 4.9 illustrate the modified electrode design. Here, air and dust particles are forced by someexternal source, such as a fan, to pass through corona columns or clouds aligned

Purplish-Blue Discharge Glow

Ext. Light Source Reflection

21

1.2 cm x 1.2cm x Plexiglas SideWall

Modified Electrode ConfigurationCross Sectional ViewOverall Size(2.6 cm x 13 cm x 14 cm)

Al Electrode (Bottom)Ends have a1.2 cm raduis bend

Stainless SteelRazor BladesExtends 2 mmbelow plateat 45 deg. angle;1.1 cm space betweenrazor blade rows

Al Electrode (Top)1.2 cm fillet corners(not shown)

Fig. 4.7

Fig. 4.8 Fig. 4.9

perpendicular to the direction of the airflow. These corona columns are generated by an electrodeconfiguration with overall size of 2.6 cm x 13 cm x 14 cm. The upper electrode is composed of a planarsheet of aluminum with 1.2 cm fillet corners. Stainless steel razor blade sharp electrodes extend out of theupper electrode about 2 mm below the plate at a 45o angle. The razor blade electrode edge is orientedperpendicular to the direction of the airflow. The razor blade electrodes are angled in the direction of airflowto help minimize turbulence. The lower electrode is also composed of aluminum (but stainless steel wouldprobably be better). The corners contain a 1.2 cm fillet. The inlet and outlet edge of the lower electrodealso have a 1.2 cm radius bend. The bends and fillets are required in order to prevent unwanted electricalarcing and undesired coronas outside of the charging region between the upper and lower plates. The sidewalls of the corona electrode are made out of Plexiglas about 1.2 cm thick by 1.2 cm wide. The potentialdifference needed to generate the corona in this case is about 16 kV. This is 2 kV smaller than the 18 kVneeded in the original electrode configuration. This is a consequence of the configuration of the electrodepair.

To determine the charging characteristics of the modified electrode configuration on the dust, thedusty airflow stream was directed towards the Faraday tube as shown in Figs. 4.10 and 4.11. The designedprototype frame to be described in the next section is connected to the dust maker. The back of theprototype frame is removed so that the Faraday tube supported by a sponge-like material can be

22

Fig. 4.10 Fig. 4.11

placed in the airflow stream. A handful of tests were conducted. In each case, the electrometer was nevertriggered resulting in no wave pattern on the oscilloscope. The scope may be observed in the backgroundon the desk in Fig. 4.10. As the tube was removed from the experiment, it was observed that a thick layerof dust was deposited on the electrically grounded outer tube. Visually examining the inside surface of theinner tube against a light background, few dust particles could be identified. This tends to indicate that thedust particle trajectories are influence by the presence of the inner tube with tube holder. Further tests werenot conducted with the Faraday tube as a consequence of time constraints. Figure 4.12 displays thegrounded electrode plate. A significant amount of dust was deposited over the surface of the plate. Thearrows in the figure point out the areas were the deposited dust layer is significantly thicker. This is theregion where the corona column is predominately drawn to

Fig. 4.13

the grounded electrode surface. Although information from these tests would have been used as a guide todetermine the water charging potentials needed, a different set of tests were devised to characterize theprecipitator as a whole unit with airflow velocity control. This will be discussed in Section 5.0.

5. Prototype Electrostatics Air Filter Assembly and Experimental Tests

Regions of significantdust build-up

23

In this section, the prototype electrostatic filter with experimental tests will be described. Theremay be some repetition of past sections in order to develop a full complete picture of the prototype.Experimental studies will be examined.

Description and Design

The corona charging electrostatic air filter is naturally divided into four main sections; air exhaustsystem, water drain, corona dust charging region, and water charging and precipitation region. Figure 5.1 isa side view of the device. Detailed mechanical drawings of the prototype frame exist in Appendix G. Theoverall dimension of the mainframe of the prototype is 12" x 16" x 6 ½". The body of the mainframe isconstructed out of Acrylic and assembled with nylon screws. The dust charging region is a 6" x 2 ¼" x 5¾" rectangular prism region at the front of the air filter housed on four sides with Acrylic walls. The front andback of this region are open. The two side walls contain two sets of parallel slits which support the negativeand ground dust charging electrodes. The plate closest to the bottom wall is the negative electrode. Thiselectrode contains the razor blades shown in Fig. 4.7. To prevent dust from entering below the lowerelectrode, the lower electrode is inserted into a slit contained in a 5 ¼" x 5/8" x ½ " rectangular Acrylicprism. A metal spring is compressed between the lower electrode and the bottom wall. The bottom mostend of the spring is straightened and extends out of the precipitatior through a notch in the right side wallfacing the front of the prototype. The back end of the bottom electrode is supported by a ½" x ½" x 6"Acrylic bar which also acts as a means to prevent the dust from accumulating in the region below theelectrode. The spring is used to make electrical contact between an external source and the electrode.The electrode closest to the top wall is the ground electrode. In order that this electrode fits into the slits, asmall portion of each side containing the bend (Refer to Fig. 4.7) had to be cut away. A spring iscompressed between the top wall and ground electrode. The top most section of the spring is straightenedand extends out of the dust charging region through a notch in the left side wall when facing the front of themachine. The water charging and precipitation region which will also be referred to as the precipitationchamber is a 6" x 16" x 6 ½" rectangular prism. The chamber supports the dust charging assembly, thewater atomizing nozzle, the air exhaust valve, and the water drain. The dust charging region is screwedonto the front of the precipitation chamber with nylon screws. A 5 ¼" x 5 ¾" rectangular hole exists in thefront panel of the chamber to allow the air to flow freely into the water charging and precipitation region. Thebottom plate supports a 1" outside diameter PVC pipe for water drainage. The PVC pipe is inserted in thebottom plate flush to the inside surface of the plate. Glue is used for a more permanent connection. A bentwire mesh covered with a stiff but flexible plastic sheet is taped onto the end of the drain pipe. A stiff plasticbar balanced on fulcrum forces the plastic sheet to cover the drain opening. There is enough space forwater to seep out as it collects in the precipitation chamber. The exhaust system consists of the 3"diameter trailer drainage hose, a grounded, loosely and randomly woven, copper mesh inserted near the airexhaust outlet of the precipitation chamber, and the exhaust fan at the opposite end of the exhaust hose.The three inch in diameter conventional trailer drainage hose is attached to the top plate of the precipitationchamber with a Valaterra bladex 3" diameter waste system gate valve. The Valaterra gate valve and mountis screwed onto the top plate of this region with metal bolts. This gate valve provides another means ofcontrolling the airflow characteristics. For diagnostic purposes, an aluminum wire grid may be inserted inthe hose at the exhaust fan side of the hose.

24

Main Stream Water DropletTrajectory

Corona Dust ChargingElectrostatic Precipitator Prototype

Exhaust FanAir Exhaust System

Clean Air Exhaust(Fan Located on theClean Air Exhaust Line;3" Dia. Hole Max. Var.)

Razor BladeElectrode (Neg.)

Corona DustCharging Region

Charging PlateAssembly

GroundElectrode

Water Charging andPrecipitation Region

1" Dia. DrainPVC Drain Pipe

Water Drain

Air Flow

Water Atomizerwith Ground Wire+

Fig. 5.1

A voltage controlled, variable speed exhaust fan (115 V AC muffin fan) in the air exhaust linecontrols the airflow in the air filter. A conventional 1-120 V output, 120 V input atutotransformer is used asthe controlling source. The exhaust fan draws the air and dust particles through the corona columns orclouds generated between the charging electrodes in the corona dust charging region. The inlet airflowvelocity is directly related to the fan voltage. Air speeds at the input to the electrostatic air filter weremeasured with a velometer [ALNOR Instruments Inc., Model No. with pitot probe (Type 6060-P)] which hasthe capability of measuring airflow speeds between 100 to 10,000 ft/min [0.5 - 50 m/s]. Only moderateairflow speeds are needed. Controlling voltages were determined for airflow speeds ranging between 190 to1300 ft/min [0.97 to 6.6 m/s]. At speeds substantially greater than 1300 ft/min [6.6 m/s] dust and waterdroplets will be undesirably sucked into the air outlet of the air filter. Figures 5.2 and 5.3 display average airflow speeds versus controlling voltages based on three different calibrating data sets taken every 2 V withinthe 40 to 80 V range when the copper wire mesh is not inserted in the air exhaust line. Figures 5.4 and 5.5yield similar data when the copper mesh partially blocks the airflow in the air exhaust line. As expected, anoverall decrease in airflow velocity results when the copper mesh is present. It is noted that the decrease invelocity is more noticeable at the higher fan voltages and less noticeable at the lower fan voltages.

25

40 45 50 55 60 65 70 75 800

1

2

3

4

5

6

7

Fan Voltage [V]

Air

Flo

w [

m/s

]Average Air Inlet Speed as Function of Fan Voltage When Water Outlet is Closed

40 45 50 55 60 65 70 75 800

200

400

600

800

1000

1200

1400

Fan Voltage [V]

Air

Flo

w [

ft/m

in]

Average Air Inlet Speed as Function of Fan Voltage When Water Outlet is Closed

Fig. 5.2 Fig. 5.3

45 50 55 60 65 70 75 801

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Fan Voltage [V]

Air

Flo

w [

m/s

]

Ave. Air Inlet Speed as Function of Fan Voltage: Water Outlet-Closed; Wire Mesh-Inserted

45 50 55 60 65 70 75 80200

300

400

500

600

700

800

900

1000

1100

1200

Fan Voltage [V]

Air

Flo

w

[ft/

min

]

Ave. Air Inlet Speed as Function of Fan Voltage: Water Outlet-Closed; Wire Mesh-Inserted

Fig. 5.4 Fig. 5.5

The unobstructed cross section which the air passes through at the inlet of the prototype isroughly ½" x 5 ¼". Unlike initial design studies of the electrode configurations provided in Sect. 4.0, theprototype electrode no longer contains a rectangular Plexiglas bar separators. Besides this change, themodified electrode design described in Sect. 4.0 is used. It is more convenient to machine slits in the sidewalls of the corona dust charging region to mechanically support the plates. The air with charged dustparticles leave the corona charging region to enter the water charging and precipitation region (chamber).The mechanical atomizer is located above the dust charging region. The grounding wire is threaded into themachine by a small opening at the top of the precipitator frame. The positive source lead is externallyconnected to the brass nipples outside of the precipitator. As water is pumped into the precipitationchamber it becomes atomized and electrically charged. The water pressure is typically operated at 40 psi

26

generating ~210 µm VMD droplets. Lower pressures do not provide a uniform stream as a result of the 4 to6 kV drop at the nozzle. The pressurized mist of charged water droplets seek out the oppositely chargeddust particles as a result of Coulomb attraction. The dust particles that are attracted to the water droplets inthe pressurized water sheet stream are forced out of the main airflow as a result of the water pressure.These particles are washed away and exit the precipitator by way of the water drain. Outside of the mainwater stream, the charged water droplets attracted to the dust particles weigh the dust particles so thatdominate gravitational forces precipitate them from the air. The airflow velocity in the precipitation region issmaller than the dust charging region as a result of an 88 to 7 differential in area which the air must passnormal through. This assumes that the water drain is closed. A weight sensitive drain valve would be idealfor this purpose. The clean or nearly clean air is exhausted from the precipitation chamber through anexhaust hose. The exhausted air passes through a loosely woven, random, thin copper wire mesh insertedin the exhaust hose as shown in Figs. 5.6 and 5.7. The wire mesh is grounded so to attract all chargedparticles which were not filtered from the air with the charged water droplets. The airflow then passesthrough the fan to the outside environment.

Fig. 5.6 Fig. 5.7

Figures 5.8 and 5.9 respectively display the front and the back of the air filter before tests areconducted. As indicated in Sect. 3.0, the ¼" diameter white colored tube just above the water atomizer isan insulation jacket for the ground wire wrapped around the spray nozzle. This was required due electricalfailure through the rubber insulation around the conductor in earlier experiments before the

CopperWireFilter

CopperWireFilter

27

Fig. 5.8 Fig. 5.9

prototype design. Figure 5.10 illustrates the precipitator connected to the water charging system. Thewater charging system is on and the precipitator inlet is exposed to the atmosphere. The motor, positiveflow pump, pressure gauge and brass plumbing exist in the background of the picture on the workbench. Itis of interest to note the precipitation build-up on the surface of the water charging and precipitation

Fig. 5.10

region. Figures 5.11 and 5.12 provide a better view. Above the dust charging plate region, the chamberwalls support the charged water mist. Below the dust charging region, the chamber walls are streaked as aresult of the forceful nature or the sheet of water generated by the atomizer.

Special InsulationJacket for GroundElectrode

28

Fig. 5.11 Fig. 5.12

Figures 5.13 and 5.14 display the barely visible dust maker silhouette in relation to the prototype. A smallerdust tunnel protrudes from the larger tunnel housing the dust maker. The smaller tunnel is directlyconnected to the air inlet of the precipitator. The airborne dust particles drawn to the precipitator must travelapproximately 2 ½' horizontally to the earth's surface and 1 ¼' against gravity to reach the air inlet of theprecipitator. The arrow in Fig. 5.13 points out a design modification that was added in the precipitatorregion. A flexible plastic ramp is taped to the chamber walls with electrical tape and an extended hardplastic shelf. The plastic shelf itself is screwed to the side walls of the air filter. This is necessary becausethe accumulated water droplets on the front wall where the nozzle is inserted tended to act as an electricalconduit between the atomizer nozzle and the dust charging plates even though the plates are positionedover a half inch away from the inside of the water charging region. It was anticipated that the airflow itselfwould have been strong enough to prevent the water from finding a path to the dust electrodes. As pointedout in Figs. 5.12 and 5.14, a plastic support exists under the brass nipple connected to the spray nozzle.This was needed to stabilize the nozzle from being twisted and/or slightly moved.

When the machine is operational a few unanticipated characteristics are observed. With the lightsoff, bluish-purple tracking along water paths on the plastic walls may be observed. Although this isundesired, it is not serious as long as the electricity does not find a closed conducting path. A noticeableamount of current is drawn from the source (on the order of couple of milliamps). Near the dust chargingplates, it is observed that the nylon screw heads external to the precipitator noticeable burns at times. It isanticipated that the sharp edges and the material make-up of the screw may have caused this. Also as apreventive measure, electrical tape was applied over the corner seams of adjacent walls to prevent waterdroplet conduits from seeping into corner cracks and bringing the electricity closer to the nylon screws.Refer to Fig. 5.13. Unusual tracking was also observed along the plastic side walls of the dust chargingregion. The nature of the tracking is not what one should expect if breakdown is to occur. This may be aconsequence of the type of plastic used in the precipitator. Acrylic (Plexiglas is another trade name for thesame material) is hydroscopic. That is, its surface absorbs moisture and any other fluids that exist on it.As a result, the electrical or insulating nature of the plastic may change when appropriate field strengths arepresent. Acrylic was chosen because it is relatively inexpensive and clear.

MechanicalSupport forAtomizerAssembly

29

Fig. 5.13 Fig. 5.14

Experimental Results

The properties of the dust precipitator are determine experimentally. Visual base line comparisonsare made. Conclusions are formulated based on these comparisons. Figure 5.15 is a picture of theexperimental setup. The picture displays the following: 1. Clean air exhaust hose, 2. Large dust tunnelhousing the dust maker, 3. Exhaust dust tunnel, 4. Exhaust fan mounted on the frame of the exhaust dusttunnel, 5. Silhouette of the dust maker, 6. Ground wire connected to the aluminum grid, 7. Ground wireconnected to the copper wire mesh, 8. Button compass sitting on Valaterra gate valve, 9. Connectionbetween the small dust tunnel and the inlet to the corona charging dust precipitator, 10. Ground wireconnected to the ground electrode in the dust charging region, 11. Small dust tunnel, 12. Autotransformer tocontrol the exhaust fan speed, 13. Water atomizing spray nozzle, 14. Opening from the dust charging regionto the water charging and precipitation region, 15. Drain, 16. Water supply line, 17. Autotransformer tocontrol the dust maker speed, 18. Crude mechanical water drain valve, 19. Water and dust collection pail,20. High voltage wire to the water atomizer, and 21. Ceiling support used to separate high and low voltagelines.

The baseline experiment is conducted in the following manner. A new supply of filtered dust isadded to the dust maker. The dust is stirred prior to turning the dust maker's fan on. The copper metallicfilter is removed from the air exhaust tube connected to the precipitator. Water is not supplied toprecipitator during this experiment. The voltage across the dust charging plates is set at 18 kV. This isroughly the voltage in which corona is well under way but electrical arcing is at a minimum. The exhaust fanvoltage is set to 60 V. Figures 5.3 and 5.4 provide the experimentally tested air inlet speed at the input to

Special WaterDirective Ramp

Dust MakerSilhouette

TapedSeams

Neg. HighVoltage Wire

Pos. HighVoltage Wire

SupportforAtomizer

Gnd.Wire

30

the precipitator. The water drain is better than 98% blocked with plastic. The air exhaust fan and the dustmaker fan are turned on for a period of 60 seconds. Prior to turning on the fans, an aluminum grid typicallyused in aluminum screening is placed just before the exhaust fan in the airflow as shown in Figs. 5.16 and5.17. The grid is grounded. The diameter of the threaded wires in the grid is 0.95 x 10-3 inches. Apermanent blue dye is placed over the approximate center of the grid (roughly 6 mm x 6 mm area). This dyeis used to coat all blemishes, small holes, burrs and rough surfaces on that section of the wire grid to beexamined. A high contrast between the dust particles and the dyed wire surface results. Also, undesiredglare and reflection is minimized from the silvery surface of the aluminum when examined under amicroscope. The airflow is only slightly affected by the presence of the grid. Between a 0 to 30 psidecrease in the airflow results. Once the experiment is concluded, the wire grid is magnified between 330to 500 times. A Pacific Western System Inc. semiconductor probe station is used to magnify the grid. Thestation contains a super fine color CCD camera with a Panasonic color video monitor. The magnified pictureis video recorded for visual comparison. Only one or two squares on the dyed grid is recorded.

Each experiment, excluding the baseline, is conducted in the following manner. The coppermetallic filter is positioned in the exhaust tube near the exhaust outlet of the precipitator box and notremoved. It is grounded. The metallic filter was visibly dirty on one side at the onset of the initialexperiment. It hasdalready been introduced to dust environments. The filter is not cleaned between setupssince it would be impractical in the field where dust environments may be fairly harsh over a long period oftime. An aluminum grid with dyed center is placed near the exhaust fan in the exhaust tube with planarsurface area perpendicular to the airflow in the tube. The dust is stirred in the dust maker. First the wateratomizing system is turned on. A 6 kV source is used to charge the water droplets. The atomizing nozzleis at a positive potential and the wire wrap around the nozzle is grounded. Next 18 kV is place across thedust charging plates. The plate with razor blades protruding from its surface is at a -18 kV potentialcompared to the upper grounded plate. Finally the dust maker fan and the air exhaust fan are turned on.The sequence of initializing the electrostatic precipitator is of importance to minimize the amount of dustbuildup on the copper filter in the exhaust tube.

31

1 2

3

4

5

6

7

8

9

10

11 12

13

14

15

16

17

18 19

20

21

32

Fig. 5.15

33

Fig 5.16 Fig. 5.17

The sequence of events in conducting the experiment is briefly outlined below.

Experimental Procedure:

1. Test grid prepared and inserted2. Mix Sand3. Turn water on4. Apply voltage at water atomizer5. Apply voltage at dust charging plates6. Turn on Dust Maker Fan (78 V)7. Turn on Exhaust Fan

Table 5.1 lists the experimental parameters of each run. The first run was a sample run. Thepurpose of this run is to demonstrate that dust particles were indeed charged and could be collected withthe aluminum grid at the exhaust fan. As observed in the video supplied with this report, the grid is highlypopulated with dust particles. The blue dyed wires are difficult to observe. Runs two through four establishthe fact that the precipitator dramatically cleans the dust particles out of the airflow. As the airflow speedincreases, the number of dust particles populating the dyed grid is observed to increase. Even so, theairflow appears very clean. During the fifth run, the dust charging electrodes shorted due to tracking alongone of the Acrylic side walls supporting the electrodes. It is observed that that the Aluminum grid wires arehighly populated with the dust even though the water charging system is properly operating. This providessubstantial proof that it is not enough to spray airborne dust particles with water to precipitate them from theair environment. Further, a dramatic difference is observed between the two dust covered grids of runs fourand five. Even though the exhaust fan speed in run five is higher than that of run four, one should not expectsuch a significant change in the dust collection. This is born out by comparing consecutive runs betweentwo and four. At the time of this report, no further testing was conducted. It is noted that during and afterexperiments, ozone could not be detected. This may be due to the short period of time in which theexperiments were conducted. It is also noted that once the system is stabilized after all components havebeen switched on, audible sounds of arcing across the dust charging electrodes could not be detected.

Al Grid

Al Grid& GrndWire

34

Table 5.1

No.# ExhaustFanVolt.[V]

WaterPres.[psi]

WaterDroplet

ChargingVolt. [kV]

DustChargingVoltage

[kV]

Exp.Duration

[s]

VideoCounter -

Start

VideoCounter -

Stop

CopperFilterInst.[Y,N]

1 60 0 0 18 120-180sample

00-00-00 00-00-47 N

2 50 40 5 18.5 45 00-00-49 00-03-29 Y

31 55 40 6 18.5 45 00-03-29 00-05-38 Y

4 60 40 6 18.5 45 00-05-39 00-09-28 Y

5* 65 40 6 18.5shorted

45 00-09-32 00-11-14 Y

* Tracking on back right bottom plate up along plastic surface across tiny air gap between plastic wall andcurved section of upper plate1 Sand not mixed

6. Design Embellishments

The following is a list of design embellishments which would improve the performance and capabilityof the design. It is not meant to be exhaustive nor is it to indicate that portions of the developed designshould or should not be manufactured in the same fashion as the prototype.

1. To clean the exhaust tube and precipitation chamber without human intervention, a means should bedeveloped to pump water into the exhaust tube when the voltage is off. Water will wash out the entiresystem excluding the dust charging electrodes. The degrading of the soiled electrodes may becompensated for by increasing the potential difference between the electrodes. Spray nozzles may belocated in or near the dust charging region to clean plates.

2. Only a single atomizer was used in the water charging system of the prototype. To obtain a moreuniform spray, it may be appropriate to use more than one atomizing nozzle. This will be especiallyadvantageous when one spray nozzle becomes obstructed due to dirt or corrosion in the water line.

3. A means to recycle the water used is important out in the field. The water with dust collected should bedirected to a gyrating container where the water and dust can be separated without human intervention.The clean water is to be siphoned and reused. Large dust particles may be filtered with a course filterbefore entering the gyrating container. The gyrating system need not be designed to separate theextremely fine dust particles from the water. Very fine particles which cannot be separated will passthrough the water lines, positive pump, and spray nozzle without obstructing the system.

4. A weight sensitive drain valve will find utility in draining a small pool of collected water in the precipitationchamber without compromising the desired airflow velocity at the input end of the corona chargingelectrostatic precipitator.

5. The prototype needs a broader base in order to support the weight of the water and dust chargingcomponents without falling over. Further, the base should allow room for the drain plumbing.

6. Acrylic plastic was used because it is inexpensive. Unfortunately, Acrylic (Plexiglas, and Luran) arehydroscopic. That is, this material absorbs fluids. This may compromise the electrical insulatingproperties of the material resulting in unwanted tracking. It is anticipated that this is the reason why thedevice shorted during experimentation. Further, nylon screws were used. The screw threads and headscontain sharp corners. Breakdown is prevalent at these locations. Either a different material or a

35

different machining of the edges of this material are recommended. The elimination of the screws in theregion where burning resulted may correct the problem.

7. Epoxy was used as a means to connect the razor blades to the aluminum plate electrodes. Epoxydoes not stick well to aluminum. The connections are brittle. A better means of manufacturing theplates with sharp edges needs to be examined.

8. The razor blades used contained iron. As a result of oxidation, the blades exhibited a red rust. Thismay degrade the property of the electrode. It is suggested that the electrodes be made out of StainlessSteel No. 316 or 17-4-Ph. It is understood that these conducting materials contain minimal amounts ofiron and are not as susceptible to rust.

9. Epoxy was used as a means to hold the ground wire in place on the atomizing nozzle. Althoughadequate, it is very difficult to work with and is brittle. Further, if epoxy is undesirable placed on theatomizer's deflecting surface, the fan shaped spray pattern and the droplet size are significantly altered.It is suggested that a special cap with ground electrodes built within be designed to fit the atomizernozzle. This will allow for a simple means to detach and attach the ground to the nozzle especially ifcleaning is required.

10. The entire unit with water system will need to be enclosed in a grounded case so to electrically isolatethe precipitator system from the worker.

IV. Alternative Novel Quasi-Electrostatic Air Filter Paper Design

One novel method of filtering the air is with a quasi-electrostatic air filter which charges, traps,transports, and collects the dust particles with the aid of electrostatic and quasi-electrostatic fields. Thedust particles are charged using a photoionization mechanism. The electrostatic and quasi-electrostaticfields not only trap and guide the charged dust particles to a collection region, they also suspend theparticles away from the walls and electrodes of the device. This is of importance when handling micron dustparticles with radionuclides attached to them especially when addressing health issues resulting fromresuspension.

A two dimensional quasi-electrostatic air filter is exhibited in Fig. IV.1. The air filter is divided intofour major regions: charging region (Region I), electrostatic region (Region II), transport region (Region III),and the collection region (Region IV). As the labeling of the individual regions implies, the task of the airfilter is to charge, trap, transport and collect the dust particles without human intervention.

Enclosure isolatesinterior region fromexterior region

h

Base Plate; VBP

x

Air OutletGrid; Vb

Air Flow

Wall of ElectrostaticRegion; Va

L

Va

Air InletGrid; Vb

I

IIl V1 V3

IIIV2 V2 V1

Average Trajectoryof Charged DustParticle

Electrode Plate

z

V3 V2V2d

ElectretFiber Bag

IV

36

Fig. IV.1

Dust and air are drawn into an inlet tube by means of a fan (not shown in Fig. IV.1) and directed intothe charging region of the air filter. The airflow passes through a metallic air inlet grid upon entering thecharging region. The air and untrapped dust exit through a metallic air outlet grid. These two grids are heldat the same potentials. Large airborne dust particles entering the charging region may be pulled by gravityto the bottom of the charging chamber. The micron particles are photoionized by a fan shaped, pulsed, ultraviolet laser beam. The fan-like laser beam geometry is illustrated in Fig. IV.2. The charging of micronparticles is limited by recombination effects, dust loading densities, and the beam's energy, pulse duration,and pulse frequency. A pulsed ultra-violet excimer laser is commercially available to perform such a task.By adjusting the airflow and laser beam properties, the dust particle can be charged up to its saturationvalue before air breakdown occurs. If breakdown results at the particle surface, the particle discharges.

Fiber Optic BundleCylindrical-to-FlatConverter

Fan-ShapedLaser Beam

W

Original Laser Beam

ExpandedLaserLight

Beam ExpanderLens

L B

α

Fig. IV.2

A complete heuristic argument excluding the effects of secondary ionization mechanisms has beendeveloped to describe the charging of a 1 µm diameter, aluminosilicate, dust particle with a pulsed 10 mJ,157 nm laser having a repetition frequency of 10 Hz by means of photoionization. The laser specificationsare based on the Compex 110, fluorine gas laser. Recombination effects dictate the amount of charge asingle micrometer dust particle will obtain if a 7.91 eV photon completely gives up all of its energy toremoving an electron from the dust particle in the collision process. Here, first ionization potentials of theatom and not the molecule were employed as a first approximation in determining the amount of energy thatis required in the photoionization process. Since the diameter of the dust particle is about an order ofmagnitude larger than the wavelength of the photon, absorption effects have been neglected. Based onthese considerations with the aid of ultraviolet mirrors or diffusers, the above calculations indicate that it ispossible to charge a significant volume of 1 µm diameter airborne dust for particle trapping and transport.

The electrostatic fields are configured by the wall potentials of the open ended charging andelectrostatic cavities and the potential distribution of the base plate in the charging region. The electrostaticregion is terminated by a set of plates that protrude and constrict the aperture opening to the transportregion. The large electrostatic field is tailored to extract the charged micron particles from the chargingregion and position them in a region midway between the first set of bi-planar electrodes in the transportregion. The viscous forces of air are in direct competition with the electrostatic forces the particlesencounter. In a zero airflow case, this affect aids in causing the charged particle to follow the electric fieldlines of force.

37

The transport region is composed of identical sets of bi-planar plates aligned with a common axis.The transport region is assumed to be two dimensional and semi-infinite in length. To aid in focusing thedust particles, the bi-planar electrodes are at the same potential. Adjacent sets of parallel plates differ inpotential. The plate potential sequence of any four consecutive sets of plates follows the pattern ... V1, V2,V3, V2, ... where ... represents the repeated nature of the potential pattern indicated. Consequently,excluding the region near the ends of the transport region, the spatial period of the quasi-electrostatic fieldsis equivalent to four times the length of any one plate. End effects such as the cavity structure of theelectrostatic region significantly alters the spatial periodic nature of the field configuration near the ends ofthe transport region. A four plate series is employed due to its generated field's ability to trap charges overa relatively large region in space but not necessarily too large for practical application. To transport the dustparticles long distances toward the collection region, the plate potentials are shifted in the direction of thedesired particle motion. The time between transitions is anticipated to be about a particle transit time alongone or two electrodes. This is large compared to the field's transient time constant developed by the voltagechange on the electrodes. Further, the transient time constant is anticipated to be very small compared tothe time it takes for the charge to overcome the large inertial and viscous forces and move a substantialdistance before the steady state electrostatic forces dominate. Therefore, as far as the charged dustparticle is concerned, it only experiences electrostatic forces and damping mechanical forces. As a result,an electrostatic modeling of the fields in the guide is employed.

The collection region can be any number of different geometries and materials. For the most part, itis isolated from the environment. One plausible configuration is a fiber web of polarized electret material.The dust particles floating in a relatively motionless air medium will be Coulomb attracted to the electretfibers. This provides for a simple means of cleaning and disposing of the material. Depending on the dustload, the attached dust particles will not dislodge when small forces are exerted on the fiber mesh. Thisallows for a relatively safe means of disposing radio-nuclides without further resuspension during machinemaintenance and cleanup procedures.

Details on the theory, a numerical study, and a quantitative prototype design with limitations maybe found in the thesis entitled A Novel, Remote Cleaning, Electromagnetic Air Filter by Zhongyi Gu. Acomplete listing of the codes developed and used are also listed in the thesis.

V. Deliverables and Student Participation List

Student Participation List

Student Name Capacity Approximate DatesZhongyi Gu MS Candidate (9/1/94 - 11/1/96)Vamsee Pamula Graduate Research Assistant (9/1/94 - 2/28/95)Srinivas Maheshwartha Graduate Research Assistant (9/1/94 - 12/31/94)Hans Yeager Undergraduate Computer Assistant (3/1/95 - 9/1/95)Elias Dagher Undergraduate Computer Assistant (Summer EPSCoR Grant)-(6/95 - 8/95)Robert Sedeghi Undergraduate Laboratory Assistant (5/20/96 - 8/20/96)Jiebing Wang Graduate Computer Assistant (3/1/96 - 9/1/96) and (7/15/97 - 8/25/97)Donald Bush Undergraduate Research Assistant (5/15/96 - 1/1/98) and (8/98 - 9/98)Sara Larsen* Laboratory Assistant 2/98 - 4/98* No financial support

Publication List

38

1. Z. Gu and R.A. Schill, Jr., The two-dimensional electrostatic field configuration for parallel plate dustparticle guide including end effects, Journal of Electrostatics 38 (1996) pp. 317-336.

2. Z. Gu and R.A. Schill, Jr., Novel quasi-electrostatic air filter: A single-particle study, Journal ofElectrostatics 39 (1997) pp. 203-230.

3. Z. Gu, R.A.Schill, Jr., and R. Venkat, A novel electromagnetic air filter for micro-diameter dust particles:First design, Nevada Environmental Research Park/DOE Experimental Program to StimulateCompetitve Research (EPSCoR) Proceedings, Las Vegas, Nevada¸ Nov. 30- Dec. 1, 1995.

4. E. Dagher, R. Venkat, and R.A.Schill, Jr., An early numerical study of an electromagnetic air filter,Nevada Environmental Research Park/DOE Experimental Program to Stimulate CompetitveResearch (EPSCoR) Proceedings, Las Vegas, Nevada¸ Nov. 30- Dec. 1, 1995

5. Z. Gu, A Novel, Remote Cleaning, Electromagnetic Air Filter, Thesis, Fall 1996.6. R.A. Schill, Jr. & D. Bush, Experimental and Theoretical Advances in the Design of a Prototype

Electrostatic Air Filter for Pu Contaminated Soil Excavation, Nevada Science and TechnologySymposium, Jan. 8-9, 1998.

Papers in Progress

1. R.A. Schill, Jr. and Jiebing Wang, Theoretical and experimental studies of the electrostatic breakdownregion in a two dimensional corrugated, rectangular electrode configuration, work in progress.

2. R.A. Schill, Jr., Theoretical and experimental studies of a Faraday tube used to measure charge on amoving object in a non-vacuum region without significantly altering the dynamics of the moving object,work in progress.

3. R.A. Schill, Jr., An electrostatic air filter designed for filtering miron size radionuclide contaminated dustparticles, work in progress.

Hardware

1. Electrostatic Air Filter (excluding voltage sources) - Water charging system, dust charging system, airexhaust system (upon request)

2. Dust maker (upon request)

Software

Appendix H: Computer Code for the Study of the Electric Field in a Two Dimensional Corrugated,Rectangular Electrode Configuration (Developed for the paper entitled: Theoretical and experimental studiesof the electrostatic breakdown region in a two dimensional corrugated, rectangular electrode configuration)

Appendix I: Computer Code for the Study of the Faraday Tube Voltage Characteristics (Developed for thepaper entitled: Theoretical and experimental studies of a Faraday tube used to measure charge on a movingobject in a non-vacuum region without significantly altering the dynamics of the moving object)

VI. Conclusion

A number of air filter designs were investigated. Two of the designs show promise. Cost, size,weight, power requirements, and efficiency prohibit the initial paper design studied. A second air filter wasdesigned, and a prototype was built and tested. Tests conducted indicate that the prototype provides arelatively clean air stream. These qualifications are based on visual observations conducted with a highpower microscope. These observations have been recorded on video tapes. After the fifth recorded run, thedust charging electrodes shorted as a result of tracking over the Acrylic side wall. At this point, no furthertesting was performed. Even with minimal visual data, it is definitely apparent that the prototype coronacharging air filter does significant precipitate airborne dust particles of interest.

39

Appendix A.

Heuristic Theory for Dust Maker - Instantaneous Number of Dust Particles

Let the maximum amount of time a single dust particle will spend in the laser beam to be

t d vL avemax =

where dL is the laser beam diameter and vave is the average dust particle velocity. In the time it takes asingle dust particle to travel the diameter of the laser beam, there will be Nd particles in the beam

N A Ltd d L= ψ max

where ψd is the number of dust particles generated by the dust maker per second passing through the laserbeam interaction region, L is the interaction length of the laser beam, and AL is the cross-sectional area ofthe laser beam. In a ∆z thickness of the cylindrical laser beam, it is assumed that there is no dust particleoverlap along z. The number of photons per second ℵm passing through the slab of dust particles ofthickness L without being scattered is

( )ℵ =ℵ −m Lo d ost Lexp maxψ σ

where ℵLo is the number of photons per second generated by the laser in the laser beam and σos is theoptical scattering cross section of the dust particles. The steady state number of dust particles in the laserbeam at any point in time is

( ) ( )N Ad L os oL m= ℵ ℵσ ln

The number of photons is directly related to the laser beam power and the photon energy. Consequently,

ℵ=PE

L

p

where PL is the laser beam power and EP (=hf=hc/λ; where h is Planck's constant, f is the frequency of thelaser, c is the speed of light, and λ is the wavelength of the laser) is the energy of the photon generated bythe laser. An EG&G Photon Device photo diode Model #FND 100Q is used. The laser power measured atthe diode is related to the diode current IL by

PI

LL=

Γ

where Γ is the photon diode absolute measurement obtained from the diode specs (Γ=4.25 A/W).Consequently, the steady state number of dust particles in the laser beam at ant point in time is related tothe natural log of the ratio of the diode current measure without dust particles present, IoL, to that with dustparticle present, Im(t),

( ) ( )N A I Id L os oL m= σ ln

40

Appendix B

Total Charge on a Metallic Sphere Between a Parallel Plate Capacitor by Means ofDirect Contact Charging

The total charge of a spherical bead in direct contact with a set of parallel plates needs to bepredicted. The theory leading to expression Eq. 2.2 is developed here.

Ro

Vo

zz1

xD

θ r

Fig. B.1

Since the radius of the bead is much smaller than the length of the plates, we may assume that the platesare of infinite extent both in the x and y directions. Further, symmetry in aximuthal angle in the x-y planeallows the three dimensional problem to be considered as a two dimensional problem.

In the cartesian coordinate system (x,y,z) shown in Fig. B.1, boundary conditions state that

( )V x y z z Vo, , = =1 (B.1)

( )V x y z Ro, , = − = 0 (B.2a)

The potential on the surface of the bead is also equal to zero since it touches the grounded plate.Employing a spherical coordinate system (r,θ,φ) with origin located at the center of the bead yields

( )V r Ro= =, ,θ φ 0 (B.2b)

Laplace's equation characterizes the potential at any point between the plates excluding the volumeof the bead. In a spherical coordinate system, the solution to Laplace's equation taking into account theazimuthal symmetry (symmetry in angle φ) of the problem is

( ) ( )[ ] ( )V r A r B r Pll

ll

ll, cosθ θ= + − +

=

∞

∑ 1

0 (B.3)

where Pl is the Legendre function of first kind order l.

Applying the boundary condition Eq. (B.2b) and applying the orthogonality condition for Legendrefunctions yields the following constraint

41

B R Al ol

l= − −2 1 . (B.4)

With the aid of Eqs. (B.3) and (B.4) the voltage and the electric field anywhere between the plates excludingthe volume of the sphere respectively are

( ) ( )V r A Rr

RRr

Pl ol

o

l

ol

ll, cosθ θ=

−

+

=

∞

∑1

0(B.5)

and

( ) ( ) ( ) ( )

( ) ( )

r rE r V r A

Rr

A R lr

Rl

Rr

P r

rA R

rR

Rr

l P lP

oo

l ol

o

l

ol

ll

l ol

o

l

ol

l

l l

, , , cos $

cos cos cossin

θ φ θ θ

θ θ θθ

= −∇ = − +

+ +

−

−

−

−

− +

=

∞

+

−

∞−

∑

∑

21

1 2

1

1

1

1

1

1

$θ

(B.6)Far from the metallic sphere in x and y dimension, the fields between the plates are approximately

the fields of a parallel plate capacitor. Consequently, for r and→ ∞ →θ π 2 the electric field

between the plates are given as

( )rE r

VD

zo, , $θ φ = − (B.7)

Comparing Eqs. (B.6) and (B.7) on a term by term manner yields the following constraints

A unknown

AVD

A for k

o

o

k

=

=

= =

1

0 23 4, , ,...

(B.8)

At r=z1 and θ=0, the voltage is forced to be equal to the source voltage V(r,θ)=Vo. Applying this conditionand using Eqs. (B.5), (B.6) and (B.8), the electric field is expressed as

( ) ( )( )( )

rE r

Rr

R D R R

D D R D RV r

Rr

VD

o o o o

o oo

o o

, , $

sin $

θ φ

θ θ

= −− +

− −

+ −

2

2 3

3

2

1

(B.9)

Applying boundary conditions over the surface of the sphere r=Ro, and integrating, the total chargethe sphere can support for the potential difference applied across the plates is

42

( )( )( )q V R

D R R

D D R D RT o o oo o

o o

= −− +

− −

42

2

2 2

πε (B.10)

43

Appendix C

Simple Theory for Faraday Tube Calibration

The Faraday tube is treated as a cylindrical capacitor of infinite extent in this simple theory. It isdesired to predict the maximum voltage measured using this approach. The definition of capacitance isgiven as

C q VT= / (C.1)

where C is the capacitance of the coaxial capacitor. For the coaxial geometry depicted in Fig. C.1 below,the capacitance is commonly known as

CL

ba

o=

2πε

ln(C.2)

where εo = 8.85x10-12 F/m the permittivity of free space.

ab

L

Fig. C.1

Solving for the predicted voltage in Eq. C.1 using Eqs. C.2 and Eqs. B.10 yields

VR VLD

ba

DR

DR

DR

predo o o

o o

= −

−

+

−

−

2

1 1

1 2

2

2

ln (C.3)

44

Appendix D

ROUGH DRAFTGeneral Theory for Faraday Tube Calibration

(Figures NOT provided)(Equation numbers are contain within therefore D labeling has not been used in this section)

Governing Equation - Inside Inner Cylinder

Two infinitesimally thin, perfectly conducting, concentrically oriented, cylindrical shells of radii a andb (where b>a) are separated by a homogeneous, linear, nondispersive, lossless dielectric of permittivity ε 1

and thickness (b-a). Refer to Fig. 1. The origin of the coordinate system is positioned at the center of theinner shell of finite length, L, with z axis located along the cylindrical axis of the cylinders. The origin of the(x,y,z) coordinate system is chosen at the center of the inner cylinder. The outer shell of infinite in extentis physically grounded. A charged, perfectly conducting, sphere (metallic bead) of radius Ro is arbitrarilypositioned on the z axis (Ro<a). Excluding the spaced occupied by the metallic bead, the r<a cylindricalregion is composed of a perfect dielectric of permittivity ε 2

. The bead supports a total charge QT. The

distribution of this charge over the surface of the bead is dictated by the beads relative position with respectto the cylindrical shells. It is of interest to determine the potential difference between the inner and outercylinders as a function of the position of the bead on the z-axis and the bead charge.

Consider a translated rectangular coordinate system ( )~, ~,~x y z and its spherical compliment

( )~,~

,~

r θ φ to exist at the center of the spherical bead. Symmetry of the problem dictates that the surface

charge density over the surface of the perfectly conducting sphere is a function of the conical angle ~θ and

independent of the azimuth angle ~φ . Consequently, the surface charge density is expressed as

( ) ( )ρ θ ρ θs so f~ ~

= (1)

where

( )ρ

π θ θ θπs

T

o

Q

R f d0

2

0

2

=

∫ ~sin

~ ~.

(2)

The function ( )f ~θ is determined by the proximity of the bead with respect to the cylindrical tubes.

Poisson's equation is applied making use of the cylindrical coordinate system (r,θ,z) central to theinner cylindrical shell. External to the bead with r<a, Poisson's equation simplifies to Laplace's equationyielding a solution of the form

( ) ( ) ( )[ ] ( )V r z A z A z I r, cos sin= +1 2 0ν ν ν (3)

Because the boundary condition in z is not periodic and since the voltage must vanish at the ends of theinfinite in extent cylinder, Eq. (3) must be re-expressed in terms of an integration over the parameter ν.Consequently, the form of solution for the voltage satisfying Laplace's equation for r<a may be written as

45

( ) ( ) ( ) ( ) ( )[ ] ( )( )V r z A z A z

I zI a

d, cos sin= +∞

∫ 1 20

00

ν ν ν ννν

ν(4)

Boundary conditions over the surface of the bead dictate that the surface charge density is directlyrelated to the normal component of the electric flux density external to the bead. The governing boundarycondition is

( ) ( )ρ θ ε

∂∂s

oV r Rr

~~

~= −=

2

(5a)

With the aid of Fig. 2, Eq. (4) is transformed from the defined cylindrical coordinates to sphericalcoordinates with origin at the center of the bead. Written in terms of the potential at the surface of the bead,the surface charge density is expressed as

( ) ( ) [ ]( ) ( ) [ ]( ) ( ) ( )( )

( ) [ ]( ) ( ) [ ]( ) ( ) ( )( )

ρ θ ε ν ν θ ν ν θ ν θν θ

ν

ν ν θ ν ν θ ν θν θ

νν

s o o o o

o

o o o o

o

A z R A z RI R

I a

A z R A z RI R

I ad

~ sin ~ cos~ cos ~ cos ~ cos ~~ sin~

cos ~ cos ~ sin ~ cos ~ sin ~~ sin ~

= + + − +

− + + +

∞

∫2 1 2

0

00

1 2

1

0

(5b)

Integrating over the surface of the spherical bead yields the total charge supported by the bead. With theaid of Eqs. (A9) and (A10) tabulated at the end of this Appendix, the expression for the total chargesupported by the bead is given as

( ) ( ) ( ) ( )[ ] ( )Q R A z A zd

I aT o o o= − +∞

∫23 2

31 2

2

00

πε ν ν ν νν ν

νcos sin

(6)

Governing Equations in Terms of Cylindrical Functions - Between Inner and Outer Cylinder

The voltage in the region between the inner and outer cylinders is desired. Laplace's equation isadequately describes this region since this region supports no charge. Employing a cylindrical coordinatesystem, the potential is expressed as

( ) ( ) ( )V r z V r z V r ze od, , ,= + (7)

where

( ) ( ) ( )[ ] ( )V r z D I r D K r ze e e, cos= +1 0 2 0α α α(8a)

( ) ( ) ( )[ ] ( )V r z D I r D K r zod o o, sin= +1 0 2 0α α α (8b)

Because the voltage at the r=a interface is aperiodic, the above relations and a discrete series of theserelations are not adequate to satisfy the boundary conditions. Therefore, α is treated as a continuousvariable and the constants of integration are treated as functions of α. As a result, Eqs. (8a) and (8b) are re-expressed as

( ) ( ) ( ) ( ) ( )[ ] ( )V r z D I r D K r z de e e, cos= +∞

∫ 1 0 2 00

α α α α α α(9a)

( ) ( ) ( ) ( ) ( )[ ] ( )V r z D I r D K r z dod o o, sin= +∞

∫ 1 0 2 00

α α α α α α(9b)

Equations (7) and (9a -b) are valid subject to the following boundary conditions

46

( ) ( )V r a z L V r a z L= > = = >+ −, ,2 2 (10a)

( ) ( )ε

∂∂

ε∂

∂1 2

2 2V r a z Lr

V r a z Lr

= >=

= >+ −, , (10b)

( ) ( ) ( )V r a z L V V z V zF Fe Fo= < = = +− , 2 (11a)

( ) ( ) ( )V r a z L V V z V zF Fe Fo= < = = ++ , 2 (11b)

( )V r b z= =− , 0 (12)

where VF is the constant, floating, equipotential over the inner and outer surface of the inner cylinder. It isthe constant VF that is to be determined. This equipotential is separated in terms of two functions which aredirectly associated with the even and odd symmetric solutions of Laplace's equation. It is assumed that theinner cylinder is infinitesimally thin. The outer cylinder is grounded. A simplifying relationship between A1

and D1e and between A2 and D1o is obtained using Eq. (10a) and the orthogonality condition. Applying theremaining boundary conditions, one can recast the boundary conditions above in terms of dual integralequations. The boundary conditions governing: the even symmetric solution of Laplace's equation

( ) ( ) ( )V z A z dFe =∞

∫ 10

ν ν νcos for zL

<2

(13a)

( ) ( )[ ] ( )0 110

1= +

∞−

∫ ν ν ν ν νA N z dcos for zL

>2

(13b)

and the odd symmetric solutions of Laplace's equation

( ) ( ) ( )V z A z dFo =∞

∫ 20

ν ν νsin for zL

<2

(14a)

( ) ( )[ ] ( )0 120

1= +

∞−

∫ ν ν ν ν νA N z dsin for zL

>2

(14b)

where

( )[ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]N

I b I a K a I a K a I a I a K b

I a I a K b I b K aν

ν ν νεε

ν νεε

ν ν ν

ν ν ν ν ν+ =

+

+ −

−−

1

11

0 0 12

11 0

2

10 1 0

0 0 0 0 0

(15a)

For the special case when the dielectric mediums enclosed by the outer conductor are the same, Eq. (15a)simplifies to

( )[ ]( ) ( ) ( ) ( )[ ] ( )

( )

Na I a K b I b K a

I aI b

νν ν ν ν ν

νν

+ =−

−1

1

2

1

0 0 0 00

0

(15b)

Fredholm Equation of Second Kind for Even Symmetric CaseThe even symmetric and odd symmetric portion of Laplace's equation are now treated separately.

The even symmetric potential is considered in this section. A form of solution constrained by the boundarycondition Eq. (13b) requires the use of the Weber-Schafheitlin type discontinuous integrals[ ]. The constantof integration A1(ν) is chosen as

47

( ) ( )[ ] ( ) ( )A N F JL

t t J t dt

L

1 1 0 10

211

2ν

νν ν ν ν= +

+

∫ ~Φ

(16a)

where

( ) ( ) ( )~Φ Φ Φt t t= + − (16b)

Here the approach suggested by ____________ [ ] is followed. As indicated in their works, the extraconstant F1 is introduced to avoid a singular solution. The function Φ(t) in Eq. (16b) is defined over the entireinner cylinder (r=a, -L/2<z<L/2) and is unknown. It is anticipated that the function is not symmetric about

the inner cylinder. The function ( )~Φ t is defined over the upper half of the cylinder. Using the integral forms

Eqs. (A1) and (A3) provided in Appendix A, the boundary condition Eq. (13b) is satisfied.

The function ( )~Φ t is determined by the remaining boundary condition, Eq. (13a). Equation (16a) is

substituted in Eq. (13a) yielding

( )[ ] ( ) ( )[ ] ( ) ( ) ( )V z F N JL z

d N t t J t dt z dFe

L

( )cos ~ cos= +

+ +

∞ ∞

∫ ∫∫1 00

10

2

0

12

1υ νν

νν ν ν ν νΦ

(17)

valid for zL

<2

. It is desired to re-express a portion of the resulting equation containing the unknown

function ( )~Φ t into the form

( )( ) ( )[ ]

( )( ) ( )[ ]

~ ~Φ Φt dt

h z h tor

t dt

h t h za

z

z

b

− −∫ ∫α α

where a<z<b, 0<α<1, and h(t) is strictly an increasing monotonic function in the interval [a,b]. Conventionaltransformations can then be applied to extract the unknown function from the integrand. Examining thedifferent Weber-Schafheitlin discontinuous integral forms, Eq. (A4) allows for the desired form sought.Consequently, the second term in Eq. (17) is modified by first interchanging the order of integration and then

adding and subtracting the term ( ) ( )J t z d10

ν ν νcos∞

∫ in the integrand of the outer integral associated to the

differential dt. With some manipulation and the aid of Eq. (A4), Eq. (17) is expressed as

( )[ ]

( )~Φ t

z tdt g z

z

2 20

12−

=∫ valid for zL

<2

(18)

where

( ) ( ) ( )[ ] ( ) ( ) ( ) ( )g zz

V z F N JL z

d t dt t t K t z dtFe

L L

= − − +

− −

∞

∫ ∫ ∫1

121 0

0 0

2

0

2

ν νν

νν

cos ~ ~ ,Φ Φ

(19a)

( ) ( ) ( ) ( )K t z N J t z d, cos=∞

∫ ν ν ν ν10

(19b)

Numerical and theoretical examination of the term N(ν) suggests that over the entire range of ν, the functionvaries between -1 and -2. Approximating N(ν) as a constant, the integral in Eq. (19b) is discontinuous

48

(Refer to Eq. (A4)). Consequently, the last integral in Eq. (19a) requires careful treatment. Theoreticalstudies in the literature do not appear to account for this. As a result, difficulties tend to arise in numericalsimulations. Therefore, with t replaced by y for convenience, g(z) is rewritten as

( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( )

g zz

V z F N JL z

d y dy

y y K y z dy y y K y z dy

Fe

L

z

z

L

= − − +

−

− −

∞

∫ ∫

∫ ∫

11

21 00 0

2

0

2

ν νν

νν

cos ~

~ , ~ ,

Φ

Φ Φ

(20)

Equation (18) is transformed by first replacing z by u, multiplying by

[ ]2

2 212

u

z udu

−, and

integrating from 0 to z . Upon interchanging the limits of integration, applying the integral form Eq. (A5)

that is encountered, and differentiating with respect to z the following resulting expression is obtained whenz is replaced by t

( )( )

[ ]~Φ t

ddt

ug u

t udu

t

=−

∫2

2 2120π

(21)

valid for 0<t<L/2.

Since g(u) is a function of ~Φ , a Fredholm form is sought. Equation (20) is substituted into Eq.

(21). The integral form Eq. (A6) allows for various simplifications. Appropriately interchanging the u and yintegrations in the integral operator leads to the result

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

~ lim ~

~

Φ Τ Φ

Φ

t tddt

y y N J y J t d dy

y y N J y J t d dy

t

t

L

= +

+

→

∞−

∞

+

∫∫

∫∫

ε

ε

ε

ν ν ν ν

ν ν ν ν

0 1 000

1 00

2

(22)

where

( ) ( )[ ] ( ) ( )

[ ]Τ t F N J

LJ t d

ddt

V u

t uduFe

t

= − +

−−

∞

∫ ∫1 0 10 2 2

120

12

2ν ν ν ν

π

(23)

In Eq. (22) care has been taken to exclude the y=t point in the last two terms. Even so, variations in t willcause variations in the limits of the integrands. Taking this into consideration, Eq. (22) is further evaluatedand placed into the form of a Fredholm equation of second kind as follows

( ) ( ) ( ) ( ) ( )~ ~ ,Φ Τ Ρ Φ Τt F t y y t dy t

L

= + ℵ +∫1 10

2

2

(24)

where

( ) ( ) ( ) ( ) ( )ℵ = −∞

∫y tyt

N J y J t d,Ε

ν ν ν ν ν0

1 1

(25a)

49

( ) ( ) ( )[ ] ( )ΤΕ1 0 1

0

11

2t

tN J

LJ t d= − +

∞

∫ ν ν ν ν(25b)

( ) ( )( )

[ ]Τ2

2 2120

2t

tddt

V u

t uduFe

t

= −−

∫πΕ

(25c)

( ) ( ) ( ) ( ) ( ) ( ) ( )Ε t t N J t J t d N J t J t d= − −

−

∞+

∞

∫ ∫1 1 00

1 00

ν ν ν ν ν ν ν ν(25d)

where Ρ in Eq. (24) symbolizes the principle part of the integral, t+ and t- represents respectively a valueinfinitesimally greater and less than t. Because the principle value of the integral excludes the points wherethe integrand is discontinuous, one may interchange the derivative and the inner integration operations.

The constant F1 is obtained directly from Eqs. (13a) and (16a). A degree of freedom exists inevaluating the constant at any point z within the constraints of the relations. For simplicity, it is assumedthat the potential on the inner cylinder for the even symmetric solution of Poisson's equation is known atz=0. Consequently,

( ) ( ) ( )F

V t t t dtFe o

L

o1

0

2

0=

− ℵ∫ ~Φ

Τ

(26)

where

( ) ( )[ ] ( )ℵ = +∞

∫o t N J t dν ν ν1 10

(27a)

( )[ ]Τo NJ

L

d= +

∞

∫ νν

νν1

20

0

(27b)

Equation (24) is rewritten with the aid of Eqs. (26-27) as

( ) ( ) ( ) ( ) ( ) ( )( )

( )~ ~ ,Φ Ρ ΦΤΤ

ΤΤ

Τt y y ty t

y dy tV

to

L

Fe= ℵ − ℵ

+ +∫ 1

00

2

20

1

0(28)

Fredholm Equation of Second Kind for Odd Symmetric CaseThe odd symmetric case is now considered. Unlike the even case, an added constant is not

required in the solution for the coefficient A2(ν). Let Ψ(t) be an unknown function defined over the boundaryof the inner cylinder. Based on Eq. (A2), the boundary condition given by Eq. (14b) is satisfied if thecoefficient A2 is chosen as

( ) ( )[ ] ( ) ( )A N t t J t dt

L

2 00

2

1ν ν ν= + ∫ ~Γ

(29a)

where

( ) ( ) ( )~Γ Γ Γt t t= − − . (29b)

A condition on ( )~Γ t is obtained by satisfying the remaining boundary condition, Eq. (14a) based on

the following procedure. Equation (29a) is substituted into Eq. (14a) and the ν and t integrations are

50

interchanged. The term J0,(νt)sin(νz) is added and subtracted in the ν integrand. The integral form Eq. (A2)

is applied to one of the terms. Replacing z by u, multiplying by

[ ]2

2 212

u

z udu

−, and integrating from 0 to

z, the following condition on ( )~Γ t is obtained

( ) ( )[ ]

~Γ tt

ddt

ug u

t udu

t

=−

∫2 1

2 20

12π

(30)

where

( ) ( ) ( ) ( )g u V u y y B y u dyFo

L

10

2

= − ∫ ~ ,Γ

(31a)

( ) ( ) ( ) ( )B y u N J y u d, sin=∞

∫0

0ν ν ν ν(31b)

These relations were obtained with the aid of Eq. (A8). Due to the discontinuous nature of Eq. (A2), thederivative operation at y=u may lead to an undesired singularity. Therefore, the same procedure is followedas developed for the even case which excludes the singular point. Making use of the Eq. (A6), the

expression characterizing ( )~Γ t is

( ) ( ) ( )~ ( ) ~ ,Γ Ρ Γt F t y G y t dy

L

= + ∫0

2(32)

where

( ) ( )( )

[ ]F t

U t tddt

uV u

t uduFo

t

=−

∫1 2

2 20

12π

(33a)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )G y tU t

yt

N J y J t d yddt

N J y J t d, = − +

∞ ∞

∫ ∫1

0 10

0 10

ν ν ν ν ν ν ν ν(33b)

( ) ( ) ( ) ( ) ( ) ( ) ( )U t t N J t J t d N J t J t d= + −

−

∞+

∞

∫ ∫1 0 10

0 10

ν ν ν ν ν ν ν ν(33c)

Although not explicitly written, the time derivative in Eq. (33b) may be interchanged with the integrationoperation since y will never equal t as a result of the Eq. (32).

Potential Difference Between the Inner and the Outer CylindersThe potential over the finite in length cylinder relative to the outer cylinder is a constant value; VF.

As indicated in Eq. (11b), it is defined to be equal to the sums of the even voltage function, VFe(z), and oddvoltage function, VFo(z) with respect to z over the surface of the inner cylinder. For |z|>L/2 at r=a, thefunctions and there defined symmetry are not defined. The viewpoint is now taken that the length of theinner cylinder, L, is a spatial period of a spatially periodic voltage function between − ∞ < < ∞z . Theperiodic voltage function is to defined to be equal to the voltage over the inner cylinder and repeats thisnature outside of the definition of domain. A Fourier series may now be written for this periodic functionwhich will give the desired values in the interval of interest. Although it also gives the values outside of this

51

interval, this is of no concern since the original even and odd symmetric functions are not evaluated off of thesurface of the inner cylinder. Therefore, let

( ) ( ) ( ) ( )[ ]V z V z E w z D w zFe Fo w ww

+ = +=

∞

∑ cos sinβ β0

(34)

The even and odd functions are easily identified with the proper sinusiodal function. Consequently,a form of solution for VFe and VFo is identified. These have been appropriately substituted into Eqs. (25c) and(33a). With the aid of Eqs. (A11) - (A14), Eqs.(25c) and (33a) are respectively placed in the following forms

( ) ( ) ( )ΤΕ2 1

0

1 2t

tw D H w tw

w= −

=

∞

∑ βπ

β(35)

( ) ( ) ( )F tU t

D wJ w tww

==

∞

∑ββ0

0

(36)

The constant Dw is determined by choosing a periodic function that describes the voltage over thedomain of the inner cylinder. Figures 3a-3c depict three different possible constraints on the voltage at r=a.If enough terms are retained in the series, each condition will approach the other over the domain of thecylinder. The simplest case is evaluated illustrated in Fig.3a is considered. With VF(z)=VF for all z, it iseasily observed that only the first term in the even function of the voltage will exist. Further, this implies thatA2(ν) equals zero since the voltage is an even function. With the aid of Eq. (A6), the function Τ2(t)=0 andVFe(0)=VF. It is observed here that if the extra constant was not included in Eq. (16a), the form of the Eq.(16a) would revert to that of the Fredholm equation of first kind.

To develop a set of relations that may be readily handled, the following normalization is introduced;

( ) ( ) ( ) ( )~ , .Φ Φt V t and A V AF F= =1 1ν ν This is not a unitless normalization but a means to extract the

unknown desired voltage from Eq. (6). Consequently, the potential difference between the inner and outercylinders may be expressed as

( )( ) ( )V

QR

zI a

A dFT

o

o= −

∞ −

∫32 0

3

2

01

0

1

ε πν ν

νν ν

cos(37)

Given knowledge of the charge QT the voltage at any point inside of the geometry can now be determine.

Useful Discontinuous Integral Forms of the Weber-Schafheitlin Type

( ) ( ) ( ) [ ]G z z J dfor z

z for zβ ν νβ ν

β

β β, cos= =

≤ <

− ≥ >

∞

−∫ 00

2 212

0 0

0

(A1)

( ) ( ) [ ]sin ν νβ ν β ββ

z J d z for zfor z

00

2 212 0

0 0

∞ −

∫ = − ≤ <≥ >

(A2)

( ) ( )f t G t z dtL

L

,−∫ =

2

2

0

(A3)

52

where f(t) is defined in the interval [-L/2,L/2] and g(t,z) is given by Eq. (A1).

( ) ( )

[ ]J t z d

tfor z t

tz

t z tfor z t

1

2 212

0

10

10

± = ±

≤ <

−−

> >

∞

∫ ν ν νcos

(A4)

( )( )[ ]dv

z v v tt

z

2 2122

2

− −=∫ π

(A5)

( )

[ ]( )cos ν π

νu

t udu J t

t

2 212

00

2−=∫

(A6)

( )

[ ]( )sin ν π

νu

u tdu J t

t 2 212

02−=

∞

∫(A7)

( )( )[ ]2

2 2 2 21

2

u

u t z udu

t

z

− −=∫ π

(A8)

[ ]( ) ( ) ( )x c a x I cx dxa c

nn

n

a n n+

+ +

− =+∫ 1 2 2

0

2 3 11

2

2 3sin

!!

(A9)

[ ]( )[ ]

( )( )

xc a x

a xI cx dx

a caν

ν

ν ν

ν

π

ν

+

+

+

−

−=

+

∫ 1

2 2

2 20

2 1

1

12

12

12

232

cos

Γ

(A10)

( ) ( )( )

( ) ( )( )( )a x

bx

bxdx

ab

H ab

J ab

a

b

2 2 1

0

12

12 1

2

12

22

−

=

−−

−

−∫

ββ

βπβ

sin

cosΓ

(A11)

( ) ( )( )

( ) ( )( )( )x a x

bx

bxdx a

ab

J ab

H abaa

2 2 1

0

212

12 1

2

12

22

110 2

−

= ±

+ −

−−

+

+∫

ββ

β

β

βπβ

βsin

cosΓ

(A12)

where Hν(z) is the Struve function.

′ = −H H0 1

2π

(A13)

( )[ ] ( )ddx

x J x x J xnn

nn= −1

(A14)

53

Appendix E

Mechanical Drawing Sketches of the Levitating Water Droplet Experiment

Side Plates (Holes 3/8" dia.; Slit 1/8" cut)Overall Dimension ~ 11 7/8" x 11 7/8" x 1/8"

R3/16"

1"

5"

4"

54

1/2 x 1/2 Support Bar - 4 RequiredLength May range between 6" to 10"

1/2"

10

"

5"

R1/2"

Front & Back Plate (Slit 1/8" cut)Back Plate without HoleOver All Dimension ~ 11 7/8" x 11 7/8" x 1/8"

55

R3/16"

10"

10"

Top & Bottom PlateHole Need Not Be Centered

56

Appendix F

ROUGH DRAFTTheory for a Dust Charging System

(Figures NOT provided)(Equation numbers are contain within therefore F labeling has not been used in this section)

Conformal mapping techniques are employed to examine the field pattern and field strengths in aclosed electrode configuration with a two dimensional field distribution. Refer to Fig. 1. The lower electrodeis geometrically configured with three sharp pointed edges. The angle each edge makes with the horizontal

is 45o. The upper electrode is U-shaped with 90o degree corners. The upper and lower electrodes areseparated with insulating material. It is of interest to study the field configuration in the central region of thedevice.

Due to geometrical and electrical symmetry, only the fields in the right half of the device areexamined. Referring to Fig. 2a, the right section of the device is fitted to an x-y coordinate system which isdenoted as the Z-plane where Z=x+iy. The corners are labelled with alpha-numeric symbols. Employing aSchwartz Christoffel transformation, the electrode geometry in the Z plane is transformed into two semiinfinite coplanar electrodes in the W plane. Refer to Fig. 2b. The W plane coordinates are u and v. Theelectrodes lie on the v=0 plane. The transformation from the Z plane to the W plane is provided in the tablebelow. For simplicity, point u1 is placed at negative infinity. The corresponding position in the Z plane isindicated by point z1 which is the junction between the two electrodes of different potential. The field in thislocalized region is not of significant interest. Further, u3 and u4 are respectively mapped to the points -1and 1. As a result of this mapping, a second conformal mapping transformation, the inverse cosinetransformation, allows for a direct solution for the fields in the W plane.

With the aid of Figs. 2a and 2b, the Schwartz Christoffel transformation is

( ) ( )

( ) ( ) ( ) ( )

dZdW

C W u W W

W u W u W u W u

= − + − ×

− − − −

− − −

− − −

( )2

5 6 7 8

12

12

14

14

14

12

14

1 1 (1)

Multiplying Eq. (1) by dW and integrating between 0 and W yields

( )Z CF W u u u u u C= +~~ ; , , , ,2 5 6 7 8 1(2)

where

( ) [ ]( )( )[ ] ( )( )( )( )[ ]

~; , , , ,F W u u u u u

W u dW

W u W W W u W u W u

W

2 5 6 7 87

2 5 6 80

12

12

141 1

=−

− + − − − −∫ (3)

The integrand of the integral in Eq. (3) contains a significant number of branch points. The branch

information must be carefully considered in evaluating the integral. Placing ( )~ ; , , , ,F W u u u u u2 5 6 7 8 in

polar form and separating the branch information yields

57

( )[ ] [ ]

~, ; , , , ,F a b u u u u u e

W u e dW

W u W W W u W u W u

i

i

a

b

2 5 6 7 87

2 2 4 4 4 2 4

2 5 6 8

12

2 3 4 5 6 7 8

12

141 1

=−

− + − − − −

−

− + + + + − +

∫Φ

φ φ φ φ φ φ φ

(4)

where the branch information is contained in the phase term

Φ = + + + + − +

2

2 2 4 4 4 2 42 3 4 5 6 7 8π

m m m m m m m (5)

where the mj terms are suitable integers corresponding to the appropriate branches. The phase terms

corresponding to the principle branch are

φ jj

vu u

=−

−tan 1 (6)

where uj equals -1 and +1 when j equals 3 and 4 respectively. Figure 2b illustrates that φ j for j=1 to 8 is

restricted to the range 0 ≤ ≤φ πj . The constant factor containing the branch information times ~C is

redefined as C. Once a branch is chosen, one must remain on the branch. To determine the constantsand the unknown transformed corner points in the W plane, the following set of boundary conditions must besatisfied.

( )ih CF u u u u u C= − +1 2 5 6 7 8 1; , , , , (7a)

( )id CF u u u u u C= +1 2 5 6 7 8 1; , , , , (7b)

( )L CF u u u u u u C1 5 2 5 6 7 8 1= +; , , , , (7c)

( )L CF u u u u u u C2 6 2 5 6 7 8 1= +; , , , , (7d)

( )L id CF u u u u u u C3 7 2 5 6 7 8 1+ = +; , , , , (7e)

( )L CF u u u u u u C4 8 2 5 6 7 8 1= +; , , , , (7f)

( )L ih CF u u u u u u C5 2 2 5 6 7 8 1+ = +; , , , , (7g)

58

Consider Eqs. (7a) and (7b). Referring to Figs. 2a and 2b W=0, it is observed that Z is imaginary. Further,Eq. (3) indicates that F(0,0;uj) equals zero. Therefore, C1 is imaginary. Now consider the integral within the

range -1<u<1 and v=0. The phase information in the integrand equals -i. As shown in Fig. 2b, this is aconsequence of φ j = 0 for j=2 and 3 and φ πj = for j= 4,5,6,7, and 8. As a result,

( ) ( )F u u u u u u iG u u u u u u0 02 5 6 7 8 2 5 6 7 8, ; , , , , , ; , , , ,= − (8)

where

( )[ ] [ ]

G u u u u u uW u dW

W u W W W u W u W u

u

01 1

2 5 6 7 87

2 5 6 80

12

12

14

, ; , , , , =−

− + − − − −∫ .

(9)

The function G will yield a real number since the integrand with limits are real. Since F is purely imaginaryfor -1<u<1, F(0,-1;u) and F(0,1;u) are imaginary. As a result, for consistency in argument, C must be real.

Singularities arise at the branch points. When passing past the branch points; W=u2, -1, +1, u5,

u6, and u8, the contour along the real u axis must be deformed into the upper half W plane around these

points. In the limit that the radius of the contour approaches the branch point, it can be shown that theintegral contribution along the semi-circular arc vanishes. Hence the boundary conditions which need to besolved simutaneously are simplified as

( ) ( )F u iG uj j0 1 0 1, ; , ;− = − − + (10a)

( ) ( )F u iG uj j0 1 0 1, ; , ;+ = − + − (10b)

( ) ( ) ( ) ( )F u u F u i G u uj j j0 012

21 15 5, ; , ; , ;= + − + −

(10c)

( ) ( ) ( )F u u F u u G u u uj j j0 06 5 5 6, ; , ; , ;= + + − (10d)

( ) ( ) ( ) ( )F u u F u u i G u u uj j j0 02

217 6 6 7, ; , ; , ;= + + + −

(10e)

( ) ( ) ( ) ( )F u u F u u i G u u uj j j0 02

218 7 7 8, ; , ; , ;= + − + −

(10f)

Equations (9)-(10f) are in a form suitable for numerical studies which will be examined in a later section.

Field Expressions

59

The plates with their respective voltage in the Z-plane have been transformed into a two-dimensionalcoplanar pair of simi-infinite parallel plates in the W-plane. The plate to the left of the (u,v) origin is at thesame potential as the U-shaped plate (upper plate) in the Z-plane where the plate to the right of the (u,v)origin is at the potential of the corrugated (lower plate) in the Z-plane. The edge of the left and right platesoccur at u=-1 and u=+1 respectively.

The fields generated by the two plates can be determined directly by a second conformaltransformation. The two plates are equipotential surfaces. Consequently, the field lines originate orterminate perpendicular to these surfaces. As a result, it is anticipated that the field lines and equipotentialsurfaces are respectively an orthogonal family of elliptical and hyperbolic curves. Typically, the inversecosine transformation lends itself useful to these geometries. The complex potential is therefore expressedas

( ) ( ) ( )Φ Ψu v V u v i u v K K, , ,= + = +1 2W (11a)

where

W U V= + = −i Wcos 1 (11b)

Field line plots may be determined directly from Eq. (11b). Upon rearrangement, it follows that the electricflux or electric field lines area family of equal constant confocal ellipses satisfying the relation

u vcosh sinhV V

+

=

2 2

1(12a)

and the equi-potential surfaces are a family of equal constant confocal hyperbolas orthogonal to theellipses characterized by

u vcos sinU U

+

=

2 2

1 .(12b)

Employing the transformation given by Eq. 2, field line plots can be constructed.

The plate at v=0 and 1<u< ∞ [v=0 and -∞<u<-1] in the second transformation lies in the =0 [ =π]plane. The constants K1 and K2 are determined by the boundary conditions that V( =0, )=V1 andV( =π, )=V0. Consequently, the complex potential is

( ) ( )Φ u vV V

W V, cos=−

+−0 1 11π

(13)

The gradient of the complex potential in the W space yields the complex fields in this space.Consequently, it can be shown that

( ) ( )[ ] [ ]

− + =−

−=

−

−

−E u v iE u v

V V

W

V V

Weu v

o o i, , 1

2

1

2

2

1 11

21

2π π

θ (14)

where

60

θ = −+ −

−tan 12 2

21

uvv u

(15)

In the Z-plane, the fields transform as

( ) ( ) [ ] [ ]

[ ][ ]

− + =− + − − − −

−×

−

−

− + + + + − +

E x y iE x yW u W W W u W u W u

W u

V V

We

x y

i

, ,2 5 6 8

7

1 0

2

2 2 4 4 4 2 4

1 1

1

12

14

12

12

2 3 4 5 6 7 8

π

φ φ φ φ φ φ φ

(16)

It is of interest to examine the fields in the W plane for special cases employing Eq. (14). It will beassumed that (V1-V0)>0. In the region where v=0 and |u|<1, the right hand side of Eq. (14) is positive real.This implies that the field components Eu(u,v) and Ev(u,v) are respectively negative and zero. This is inagreement with expectations of the field line being perpendicular to an equi-potential surface. For v=0 and

u>1, the argument of the square root is negative. As v approaches 0+, θ approaches −π as indicated by Eq.

(15). Consequently, [ ]1 21

2−W is replaced by [ ]− −i W 2 11

2. For v approaches 0+ and u<1, it is

observed that θ approaches π implying [ ]1 21

2−W is equivalently represented by [ ]+ −i W 2 11

2. In all

cases, there is agreement with physical considerations. That is, the fields in the upper half W plane on theplate located on the v=0 axis for u>1 is oriented in the + $v direction. Further, the fields are correctlydirected to terminate on the remaining plate in the upper half W-plane. It is also observed that Eu is alwaysnegative and Ev is positive when u=0 and negative when u<0. The same choice of branches follows in Eq.(16) as well.

Numerical Techniques

A variety of techniques were examined to obtain a numerical solution to Eqs. (7a)-(7g) or,equivalently, Eqs. (10a)-(10f). Some of the techniques examine are described below.

Numerical accuracy in evaluating the transformation integral Eqs. (3), (4), and (9) is crucial indetermining the both the spatial point to point mapping and the electric field strengths associated in themapping. Two numerical methods were examined: the open-end Newton Cotes method and an adaptivequadrature method. In the open-ended Newton Cotes method, the integral interval is equally spaced. Theintegrand is evaluated at all points except the end points and summed to give an approximate value for theintegral. The subroutine used, midpnt.f & dependencies, found in Numerical Recipes in Fortran () wasshown to be very inefficient. For the accuracy required, the algorithm was extremely slow. As analternative, an adaptive quadrature method was employed. This algorithm allocates many interval points inthe neighborhood of singular points of the integrand. Further, the algorithm was able to handle improperintegrals as well. This dramatically improved the efficiency of calculating the integral. The subroutineemployed, qaws.f & dependencies, found in Quadpack (a subroutine package for automatic integration)handles integrals of the form

61

( ) ( ) ( )B x x A f x dxA

B

− −∫ α β .

The integral interval [A,B] is bisected and a modified Clenshaw-Curtis integration is applied on both halfs. Inremaining steps, the Clenshaw-Curtis integration is used for those subintervals which have either A or B asone of their end-points. A Gaussian Kronrod integration is applied to all other subintervals. The ClenshawCurtis integration approximates the integrand by a truncated Chebyshev expansion which is then integratedexactly.

The qaws.f & dependencies program has been tested on the integral

( )( )π =

− +−

+

∫1

1 11

1

x xdx .

A relative error of about 1x10-5 % was observed. This is consistent with the relative error generated by theprogram.

Equations ()-() represent a system of nonlinear equations which need to be solved simultaneously.Two methods of solution were investigated: an optimization method and a matrix (Broyden's) method. Theoptimization method used prefers analytical derivatives. Supplying these is a formidable task. As a result,approximate derivatives were used. Convergence problems arose. An unacceptable 5% relative error wasestimated. When Broyden's matrix method (a secant method) was employed, relative errors on the order of

2x10-4% was achieved. This code was extracted from Numeric Recipes in Fortran. These methods wheretested against a closed rectangular geometry governed by the transformation given by

( )( )( )Z W C

dW

W u W WC

W

( ) =− + −

+∫20

11 1

.

Boundary conditions required to determine the constants C, C1, and u2 are

h CG u CCG u C

l CG u u

= − += +

= −

( , ; )( , ; )

( , ; )

0 10 01

1

2 2

2 2

5 2 2

where

( )( )( )G a b u abs

W u W WdW

a

b

( , ; )22

11 1

12

=− + −

∫

which can be transformed into a complete elliptical integral of the first kind. The program yield for d=0, u2=-321.99316. Solving for the ratio of h/l5 and employing u2 and a numerical table, yields the result 0.4000008

which is in excellent agreement with the exact result of 0.4. An error of 2x10-4% is concurred.

62

Appendix G

Mechanical Drawing Sketches of the Corona Charging Electrostatic Air FilterPrototype

(14 pages of drawings in this section are not numbered)

Parts List on Assembly Drawing

Number Drawing Title/Part Title1 Top Pate of Dust Charging Region2 Bottom Plate of Dust Charging Region3 Side Plate of Dust Charging Region (Right Side)4 Dust Block for Front of Dust Charging Region (Top Ground Electrode)5 Dust Block for Front of Dust Charging Region (Moveable, Slips Over Front Edge of

Negative Electrode)6 Back Bar for Lower Electrode Support & Dust Block in Dust Charging Region7 Mechanical Support for Spray Nozzle Assembly8 Front Plate; Water Charging & Precipitation Region9 Bottom Plate; Water Charging and Precipitation Region

10 Side Plates Water Charging & Precipitation Region11 Top Plate; Water Charging & Precipitation Region12 Back Plate Water Charging and Precipitation Region13 Acrylic Shelf Attached to Side Walls Inside the Water Charging & Precipitation Region14 Spray Nozzle Assembly

63

Appendix H

Computer Code for the Study of the Electric Field in a Two DimensionalCorrugated, Rectangular Electrode Configuration

(21 pages of code in this section not numbered)

64

Appendix I

Computer Code for the Study of the Faraday Tube Voltage Characteristics

(26 pages of code in this section not numbered)

65

Equipment List

1 Prototype Electrostatic Dust Filter: Custom made box composed ofAcrylic plastic assembled with nylon screws, metal electrodes with razorblade anode, 3" air exhaust outlet with gate valve to connect to fan (fan notincluded), ¾" water drain (no hose attachment), hole for water chargingsystem on front panel of device

$275.00

Part No. Purpose Type Other Comments

Airflow suction fanat air outlet

PAMORESystem PAP STModel 4600X115 V AC, 60 Hz., 20 WImpedance protected

Air speed controlled by varying fanvoltage. Air speeds of 190 ft/s [0.97m/s] to 1300 ft/s [6.6 m/s] areobtained with voltages ranging between40 and 80v.

Outlet air gatevalve

VALTERRA, Bladex WasteSystem # 1003; 3" dia. valve

Outlet hose Common 3" dia. Septic Tankhose for recreational vehicles 6ft. ?

Couplers 3" plastic couplersMounting Bracket Mounting bracket with 3" dia.

openingDust electrodesScrews¼" Acrylic sheets½" Acrylic sheets1" PVC pipe