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I, I: '-' fligfi‘g‘fhfi I :f“ v \ldf‘z‘ ,fi 3’51‘11‘1" 'cr;$ 1.11 151%} “L” A INTEL "-7. z.- 'I J m (213%,? Mini” S.” ‘E'cc‘mr' ..: :1- ‘3 7””: 3%, mg: 4} . j“: .w,,_fl . ”$15? Ax" . cw:— . I “' twang-Cw". N . , .M _m§:£uw' ~23“: :T-‘g -"'“°~.': I tr. mJEnv-wm‘gfi" ' u.- 1" ?' nwa " .317. a}; w“ ---‘ “3.5435: -~~ gm. ”7-;5§~q~“ t . This is to certify that the thesis entitled Electric and Inertial Forces in Pesticide Application presented by Henry D. Bowen has been accepted towards fulfillment of the requirements for ELD—_ degree in Wm. 51v GIVEIZMIG mm Major professor Date Nov. 2/ /7\5'3 ‘ . ' . 0—169 l1. . I fififl-uflwfl—n ---.-—a —-o——--—— - a—n-I- ----- ' '- ELECTRIC AND INERTIAL FORCES IN PESTICIDE APPLICATION By Henry Dittimus Bowen A“ A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of ‘HY ’Tl DOCTOR OF PHILOSO Department of Agricultural Engineering \0 U1 \Jl Year 1. ’I'HEsm 5/24' “I K? ii HENRY DITTIMUS BOWEN AN ABSTRACT The author presents the problems involved in work deal- ing with the deposition of small particles on plant surfaces. It is stated that the low recovery efficiency resulting from the use of current dusting and spraying machinery cannot be improved substantially without further information regarding the nature of the forces active in particle deposition. A theoretical approach is considered to be desirable because present evaluation techniques are considered inadequate to provide meaningful data in the light of the many physical phenomena that may effect both deposition and sticking after the particle strikes the surface. The author proposes to obtain the order of magnitude of forces active in particle deposition and specifically the relation between electric and inertial forces that may be active within the plant regions. It is believed that this quantifying of the magnitudes of the forces will then allow the develOpment of a greater understanding of the deposition and adherence problems. Also the theoretical results will allow more pointed investigations to be set up for the test- ing of ideas relative to improving machinery and techniques for particle deposition. Instruments now available for measuring electric fields or potentials have relatively large capacitances and inertias associated with them. The volume distributions of charge that results when charged dust is blown into the inner regions of iii HENRY DITTIMUS BOWEN AN ABSTRACT a plant contains such a small amount of charge and the dis- tributions change so rapidly that present day instruments disturb the fields being measured greatly, and are not able to follow the rapid changes of the field. Because of the above situation, a measureable field was calculated by two methods and also measured. The correlation being very good, it was concluded that calculation methods and assumptions used could be considered valid for calculations of those fields that could not be measured. A method of calculating the force produced on charged particles near small conduct- ing disks at zero potential (approximating plant leaves) is given. A study is made of the inertial forces produced and the resulting path of an uncharged particle that is described when an airstream carrying the particle is deflected by a surface. A comparison is made between inertia, electric, and gravity forces for several electric fields. The results in- dicate that electric forces are generally small compared to inertia forces and are comparable in magnitude to gravity forces for uniform charge distributions. However, under actual conditions the forces may be considerably greater than calculated due to redistribution of charges in the cloud so that the charge density is no longer uniform. The above results have shown that the electric forces de— veloped within a plant region depend both on the charge density iv HENRY DITTIMUS BOWEN AN ABSTRACT and the thickness of the cloud blanket near the depositing surface. This explains partially the inability of workers to transfer laboratory work on single surfaces and Spheres with thick blankets of cloud surrounding them to the field where there exists much closer spacing of deposit surfaces with resultant thinner cloud blankets and thus lower forces. Henry Dittimus Bowen candidate for the degree of Doctor of Philosophy Final examination, November 21, 1955, 9:00 A.M., Room 218, Agricultural Engineering Building Dissertation: Electric and Inertial Forces in Pesticide Application Outline of Studies Major subject: Agricultural Engineering Minor subjects: Mathematics, Physics Biographical Items Born, October 16, 1921, Bear Lake, Michigan Undergraduate Studies, Michigan State College, 1946—49 Graduate Studies, Michigan State College, 1950-52 Experience: Graduate Assistant, Michigan State College, 1950-52, Assistant Professor, North Carolina State College, Agricultural Engineering Department 1953 Member of Tau Beta Pi, Phi Kappa Phi, Society of the Sigma Xi, Farmhouse Fraternity and American Society of Agricultural Engineers TABLE OF CONTENTS INTRODUCTION... ..... .. ...... ......................... Nature of Study................................. Purpose of Study............. ..... .... ..... ..... History of the Study............................ Definition of the Problem... ...... .............. Limiting the Problem............................ ELECTRIC F ELDS IN CHARGED DUST CLOUDS...... ..... .... Discussion of Electric Fields in General........ Calculated Distributions of a Spherical Cloud of Charged Dust ...... ........ ............. Calculations of Potential Distribution of a Spherical Cloud of Charged Dust that Was Measured.......................... Measured Potential Distribution of a Spherical Cloud of Charged Dust .......... ....... Description of Apparatus........ ...... ..... Procedure for Making a Potential Traverse along a Vertical Diameter of a Charged Dust Cloud ............... .................. Comparison of Calculated and Measured Potential Distributions.............. ........ ... 1..) 004:er 22 28 43 vii Page Present Concept of Dust Deposition.............. 49 Deposit Test... ..... ..... ..... .................. 54 Discussion of Results........................... 60 The Electric Field and Potential Distribution of a Finite Disk and a Finite Cloud............. 61 INERTIAL FORCES IN A MOVING DUST CLOUD....... ....... . 74 Review of Literature ....... ......... ..... ....... 7Ll Discussion of Force............. ..... ........... 75 Statement of the Problem............. ..... ...... 76 Assumptions Necessary ....... . ...... ............. 76 Two Dimensional Problem...... ...... . ..... .. 77 Streamline Flow............................ 78 Parallel Streamlines..... ...... ............ 80 Spherical Particles. ........... . ........... 80 The Streamline and Particle Paths are Coincident............................. 81 Stoke's Law as an Approximation for the Resistance to Motion Through Air ..... ...... 81 Discussion of Results ......... ... ............... 86 Discussion of Turbulence in Field Dusting.. 92 COMPARISON OF ELECTRIC, INERTIAL AND GRAVITY FIELDS.. 95 Estimation of Charge Density...... ..... . ..... ... 96 Force on a Metal Sphere Tithin a Cloud for Case III ......... .......... ....... . ....... .. 97 Force on a Charged Particle in Finite Cloud Next to Disk......... ..................... 98 viii Page Inertial Force on Particle in Deflected Airstream........ .................. ........ ..... 99 Gravity Forces on Particle ...... . ........ . ...... 100 Discussion of Results...... ...... ........ ....... 102 APPENDIX I........ ................................... 105 Development of the Potential Equation of a Grounded Conducting Sphere in a Charged Dust Cloud by Method of Superposition.. ............. . 105 Boundary and Other Conditions .............. 105 Math Derivation ........ . ........... ... 106 Potential Equation of Homogeneous Cloud of Charged Dust of Radius a and Charge Density” = -J0. ........................ 108 Potential of Hollow Cloud .................. 108 Discussion of System of Grounded Con- ducting Sphere Inside of Hollow Charged Cloud .............................. 109 The Potential of the Sphere with Charge 0.. 109 Potential Equation of Hollow Cloud Containing Grounded Sphere ................. 110 APPENDIX II .......................................... 111 Development of the Potential Equation of a Grounded Conducting Sphere in a Charged Dust Cloud by Method of Inversion in a Sphere ..... ... 111 Boundary and Other Conditions.............. 111 ix Page Mathematical Derivation of Potential Equations.................................. 112 Summary of Potential Equations in Inside and Outside of Charged Cloud............... 115 Summary of Field Intensity Equations for Grounded Sphere Inside of Charged Dust Cloud.............. ..... . .......... ... 115 APPENDIX III......................................... 117 An Analysis of Particle Deviation from a Circular Streamline............................. 117 Assumptions and Other Conditions........... 118 Initial ConditionS.. ...... ............ 119 Mathematical Derivation of Particle Movement in x-direction.. .................. 119 Mathematical Derivation of Particle Movement in y—direction.. ..... . ......... ... 120 Evaluation of Constant for a Specific Particle................................... 121 Evaluation of Components of the Particle Path ....... .... ........... . ......... ....... 122 APPENDIX IV.. ..... .................... ........ ....... 125 Measuring Instruments for Charged Cloud Measurements..... ...... .................... ..... 125 LIST OF FIGURES Figure Page 1a Cross-section of Large Positive Cloud......... 19 1b Potential along Horizontal Diameter of Cloud........ ..... ......................... 19 2a Small Negative Cloud ................... . ...... 19 2b Potential Distribution along a Horizontal Diameter of Negative Cloud...... ...... ........ 19 5a Cross-section of Hollow Cloud ................. 19 5b Potential Distribution of Hollow Cloud ...... .. 19 4a Cross-section of Metal Sphere................. 20 4b Potential Distribution of Negatively Charged Sphere ................. . .............. 20 5a Cross-section of Negatively Charged Sphere Inside of a Hollow Cloud................. ..... 20 5b Potential Distribution of the Sum of Negative Charged Metal Sphere and Hollow Cloud along Extended Horizontal Diameter...... 20 6a Cross—section of a hollow Cloud Enclosed by a Conducting Shell at Zero Potential....... 21 6b Potential Distribution along Extended Horizontal Diameter of Hollow Cloud with Boundary of Cloud Earthed......... ......... ... 21 Figure 7a 7b 10 ll 12 13 1A 15 xi Page Cross-section of a Metal Sphere in a Hollow Cloud Enclosed by a Conducting Shell at Ground.. ..... . ......... .............. 21 Potential Distribution along an Extended Horizontal Diameter of Sum of Potentials of Grounded Metal Sphere, Hollow Cloud and a Grounded Metal Shell.... ...... ... ....... 21 Spherical Flask with Filling Cap and Charging Apparatus in Position................ 51 Charging Nozzle and Filling Cocks as Used in Figure 8.. ......................... 51 View of Apparatus used for Measuring Potential Distributions along a Verticle Diameter of a Charged Dust Cloud ...... . ..... .. 32 View of Measuring Chamber showing Spherical Cage, Reference Probe (black lead), Traverse Probe (glass rod insulated), and Metal Sphere. In the Background can be seen Perforated Bulkhead.......... ..... .... ..... .............. 32 Side View of Apparatus for Measuring Potential Distribution in a Charged Dust Cloud.. ........... . ..... ....... .......... 33 Potential Distribution for Case I ...... ....... 40 Potential Distribution for Case II............ 46 Potential Distribution for Case III........... A? Figure 17 18 2O 21 22 23 all 40 A1 Potential Distribution for Case IV ..... ....... Calculated Potential Distribution..... Measured Potential Distribution............ View of Metal Shells in Order of Run Of DepOSj-t TeSt 0000000000 ......OOOCOOO... ..... Carrying Box for Deposit Test............. Cloud Next to a Single Disk at Zero Potential ......... ...... Cloud Sandwiched Between Two Disks at Zero Potential..... ..... Formula for Force Inside of a Charged Cloud... Formula for Force Outside of a Charged Cloud.. Relative Position of Charged Cloud and Point at which Force Equation is 7""’fr,:j--‘t;lcerlo o o o o o o ooooooo o o 0 Development of the Force Unit Charge Placed Along Axis in a Positive Cloud Disk at Zero Potential .......... ... ...... ..... Development of the Force Unit Charge Placed along Axis in a Positive Cloud ducting Disks at Zero Potentia1.. ....... .... Field Intensity along Axis of Cloud Next to the Single Conducting Disk at Zero Potential of Figure 21. The Force Experienced by a the Perpendicular near a Conducting Experienced by a the Perpendicular between two Con- Direction is that xii Page 47 A8 A8 58 58 63 69 Figure 242 A3 AA 45 A6 A? A8 50 51 of One of Population of a Uni-signed Cloud. Charge Density'A9= 15.75 ................ . ..... Field Intensity along Axis of the Cloud Sandwiched between the Two Conducting Disks at Zero Potential of Fig. 22. Force Direction is that of One of the Population of a Uni-signed Cloud. Charge Density Potential Distribution along Axis of Cloud Shown in Fig. 21 .............................. Potential Distribution along Axis of Cloud Shown in Fig. 22.... ...... . ..... .. ..... ....... 2. Streamline of 5-dimensional Flow xy = C ....... Streamlines of 2-dimensional Flow xy = C ...... Streamlines for 2-dimensional potential Flow Around a Flat Plate ...................... Streamlines for 2-dimensional Viscous Flow Around a Flat Plate ...................... Relation of vs used in Formula x + Ki: m k I iii sz and Actual V5,. that the Particle Ziould Experience due to y—component of Inertia ...... Manner in which Path Deviates from Circular Streamline... ........ ................ 20 Micron Particle Deviation from Circular Streamline of 5 cm. Radius with Streamline Velocity vs Equal to A50 cm/sec and the xiii Page 70 7O 71 71 8A 8A 8A 8A Figure 52 55 5A 55 xiv Page Particle Density Equal to 2 ........ . .......... 85 Cross-section of Conducting Sphere and Cloud Showing Distribution of Charge Density is Uniform in Cloud and Non-uniform as Imaged into Conducting Sphere........ ...... 116 Co-ordinate System for Development of Equation on Particle Deposition by Inertial Force ......................................... 117 Instruments for Electrostatic Potential Measured ........... . ....... ... ................ 128 Probes and SuSpensions Used for the Potential Measurement and Deposit Tests ....... 128 Table II III IV VI VII VIII IX LIST OF TABLES Potential of a Uniformly Charged Cloud vs Distance from Center ...... .... Potential of a Hollow Cloud vs the Distance from Center...................... Potential of Hollow C Metal Sphere Inside v the Center ...... ..... loud with Grounded s Distance from 00000000000000 Potential of a Hollow Cloud with the Outer Boundary at Zero vs Distance from center ..... . Potential of a Hollow Cloud Having a Metal Sphere Within at Zero Outer Boundary at Zero vs from the Center...... Calculated Potential from Center for Case Calculated Potential from Center for Case Calculated Potential from Center for Case Calculated Potential from Center for Case Potential and an 0000 0000000 vs Distance I0000000000 vs Distance II ........ .. vs Distance :Iloooooooo vs Distance The Distance 00000000000000 Page 12 1A 18 23 25 27 Table XI XII XIV XV XVI XVII XVIII Measured Potential vs Distance from Center for Case I........................ Measured Potential vs Distance from Center for Case II....................... Measured Potential vs Distance from Center for Case III...................... Measured Potential vs Distance from Center for Case IV............. Summary of Potentials for Measured and Calculated Potential of a Spherical Cloud of Charged Dust ............. ... ............... Data Sheet for Deposit Test................... Comparison of the Forces of Inertia, Electric and Gravity Fields ........ ........... x-components of Particle Path ................ . y-components of Particle Path ................. xvi Page A0 A1 A2 45 101 12} 12A ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Professor A. W. Farrall, head, Agricultural Engineering De- partment, Michigan State College, for his continued faith in the work and the provision of a research assistantship pro- vided by the Rackham Foundation, also to the Rackham Founda- tion in granting the research assistantship for carrying out the work on ”Electric and Inertial Forces in Pesticide Application". Sincere thanks are extended to Professor 1. O. Ebert, Electrical Engineering Department, Aichigan State College, for his early interest in the project and help with equipment, to Professor Henry Parkus, iathematical Department, Michigan State College, for his help in understanding the electric fields, to Dr. R. C. Bullock, Mathematical Department, North Carolina State College, for his help in setting up the equa— tion of motion for the inertia force analysis, to Professor A. E. Mitchell, Horticulture Department, Michigan State College, for his interest, and to all other instructors of math and physics that have contributed to the background in making this study possible. The author wants to express his thanks to all of the men in the Research Department at Michigan State College, especially xviii Mr. James Cawood, for continued help in construction of equip- ment. The author is especially grateful to his major professor, Dr. w. M. Carleton, for his guidanCeand continued encourage- ment throughout the period in which this study was made. Special appreciation is felt toward Mr. M. E. Splinter for his unselfish help throughout the planning and execution of all of the work in regard to the electrical field measure- ments. Sincere thanks are due Professor G. N. Giles, head, Agricultural Engineering Department, North Carolina State College, for his interest in this project, and also to the other members of the Agricultural Engineering Department who helped on the drawing of pictures and assembling of this thesis. The author wishes to express his appreciation to his wife, Mrs. Jean Bowen, for her encouragement and the typing of this thesis. INTRODUCTION Nature of the Study The following work deals with the nature and magnitude of the several forces usually active to some degree in the de— positing of agricultural dusts and sprays on to crop plant surfaces. The forces studied in some detail are the electric and inertial. Gravity and thermal fields are mentioned but no thorough study has been made. Purpose of the Study The study was undertaken to develop a greater understanding of the nature of the several forces active in the deposition of agricultural crop sprays and dusts in order that more effic- ient machinery and techniques for application could be developed. Although spraying and dusting operations have been carried on for many years and the need for =est control has increased yearly, the equipment for applying pesticides is yet woefully inefficient. From work carried out by Bowen and others (4 ), it is believed that with present techniques and machines the dust recovery seldom exceeds 10 to 20 percent under day time operation (windy and no dew). Dust recovery refers to the a- mount of dust actually deposited on the plant surfaces as compared to the amount discharged by the dusting machine. According to Brooks (59 when dusting tomato vines with calcium arsenate by airplane, 50 to 60 percent of the dust drifts off the tract (200 ft. width) of dusting interest. This waste dust serves to contaminate pasture and forage fields on the downwind side of the tract. Only a fraction of the dust that remains on the tract is actually deposited on the plant, and often that is not uniformly Spread over the plant surfaces. History of the Study The agricultural engineer has been greatly hampered in the past in working with dust and spray machinery because of two major obstacles. The first is inadequate evaluation techniques. The few evaluation techniques now available to him have either been unreliable or very expensive and have always required too much time. For this reason perhaps more than any other, the profession has not made any major advances in pesticide machinery. The second obstacle has been an almost complete lack of information as to the relative importance and nature of the forces that act to deposit dust or spray particles on to plant or other surfaces. The first problem is being studied by workers in several public experimental agencies, but to the author's knowledge no one in the profession is working to increase the information regarding the forces acting to deposit particles onto plant surfaces. It is inconceivable to the author that significant advances will be made without more information about the forces acting. The author and co-workers spent some three years of con- tinuous effort directed toward improving dust recovery by utilizing the electrostatic forces produced when charged dust is blown into the plants. An increase in dust recovery of some 100 percent was encountered, but biological results have not shown a corresponding improvement in plant protection. It has not been determined why this is the case since evalu- ation techniques available have not indicated the relative amounts of all the various chemical constituents that are de- posited on the plants. The same problem of evaluation has plagued the entomologist and plant pathologist, and as a re- sult often all that is known about a pesticide is that if a certain quantity is mixed with a carrier and then blown or Sprayed onto the plants at a certain rate, the pests will sometimes be eliminated. Ultimately, however, an improved recovery efficiency will result in improved pest control or increased economy. To this end it is desirable to increase recovery . -u- The workers in electrostatic dusting soon realized that elementary physics explanation of induced charges on the plants attracting the charged dust cloud gave no indication whatever of the magnitudes and directions of the forces in a charged dust cloud. It was impossible to develop a clear concept of the physical processes being carried out because of the comolete lack of information as to the magnitudes of the inertial and electric forces occurring. Definition of the Problem The problem is to determine the magnitude and directions of the forces active in dust clouds. Limiting the Problem The problem is of considerable complexity and each of the two more important forces merit an individual treatment beyond what was possible by considering the two. However, since there was such a lack of information of the forces of either phase it was believed desirable to devote a portion of the time to each force in order to determine its relative importance. It is hoped that the material presented will stimulate other workers in the field to take up those portions that need further clarification and that in time the force systems will be fully -5 understood by all workers in the field. For the reasons stated above, it has been considered proper to determine the order of magnitudes of the forces and not try to deter- mine the exact forces of a given system. ELECTRIC FIELDS IN CHARGED DUST CLOUDS Discussion of Electric Fields in General Electric fields refer to those regions of space in which an electric charge placed therein would have a force exerted upon it. An electric field may be present in a region even though actual charges are not present. An electric field in the macroscopic sense need not be present in a region where charges of both signs are present in equal numbers and uni- formly diSpersed throughout the region. Local fields will be present in this case, however. In the case where there are charges of one sign only diSpersed throughout a region there will always be an electric field within the region. The magnitude of the field will depend upon the distribution of the charges and the proximity and magnitude of charges on the neighboring boundaries of the system. The field intensity is the negative gradient of the potential and, therefore, may be determined if the potential distribution can be found. The field intensity is the force that would be exerted upon a unit positive charged particle, if that particle were placed in the field. Thus the force experienced by any charged particle due to the electric field can be calculated by knowing the magnitude and sign of the charge and the field intensity. Calculated Distributions of a Spherical Cloud of Charged Dust Direct measurement of the electric fields occurring in regions surrounding plants in the field when a charged dust cloud is blown into the region is of great difficulty. The difficulty comes as the result of two conditions that are more or less incompatible. Available instruments have rela- tively large electrical capacities and inertias, and the electrical quantities being measured are generally very small and dissipate very quickly. In most cases the capacity of the instrument and its necessary leads is as large or is larger than the capacity of the system being measured. This, of course, violates the rule that a measuring system should not unduly disturb the values of the quantity being measured. Secondly, the electric fields formed by a dust cloud and any conducting boundaries are quickly dissipated both by the electric forces set up and by diffusion processes. The rela- tively large inertia of the instrument movement does not allow the needle to follow the fast changing fields. In attempts at direct measurement of charged smoke in a pyrex boiling flask, the potential distribution proved to be essentially constant all over the volume of the flask within the time (about one minute) required to measure the potential at two different positions. In order to be confident of the calculations for fields that are impossible to measure with today's equipment, it is -8- required that a measureable field be both calculated and measured. The measured field should show a reasonable corres- pondence to the calculated field for the same boundary con- ditions and charge distribution. The field selected for this purpose was that of a spherical cloud. The calculations for a uniform charge density are worked out in Appendices I and II. The calculations in Appendices I and II are for a spher- ical cloud surrounding a metal Sphere at ground or zero potential with a zero boundary at infinity. Other cases using several values for the potential on the Sphere and a zero potential at the outer boundary of the cloud have been worked out in the text. The ultimate goal is to determine the electric field forces acting on charged dust particles. These forces may be obtained from the product of the electric field intensity and the particle charge. Either the electric field intensity or the electric potential may be calculated using superposition methods. However, it was somewhat easier to measure the po- tential distribution (from which field intensity could be de- duced) than to measure the field intensity directly. It is also believed that the potential distribution concept of elec- trostatic deposition is a valuable contribution to the under- standing of the phenomena. For the above reasons the calcu- lations of the potential distributions of the several cases will be compared with the measured potential distributions. If these prove to be substantially in agreement, then the field intensity (negative of first derivative of potential) -9- will be derived from the calculated equations of potential distribution. The initial calculations of a homogeneous cloud enclos- ing a grounded sphere were made using a Kelvin transformation or inversion within a sphere. Because it was not certain at all in the investigator's mind that the field could be satis- factorily measured, a second calculation of the same problem using superposition methods was made. Exactly the same equations resulted from both methods and since the super- position method was considerably less complex and would be more easily understood by other workers in the profession, the superposition method is the only one presented in the text. For the special case of a uniform cloud surrounding a grounded conducting sphere and a zero boundary at infinity both the superposition and Kelvin transformation methods are given in Appendices II and III. Graphical additions of sev- eral fields are presented in the body and actual calculations are presented in Appendix I. The method of superposition has the advantage that it may be used for synthesis of any complex field when all of the simple components of the field can be calculated or measured. In fact, by knowing the boundary conditions, certain of the fields may be found quickly by a trial solution without re- sorting to the simultaneous solutions of several equations. It is hoped that the superposition approach will lead other workers to a better appreciation of the actual nature of the _ 10 - electric fields and will demonstrate the method in which geo- metric arrangement of boundaries may be applied to produce the desired fields with a dusting machine for field use. A graphical solution for the case of a cloud with a grounded Sphere within it will be made. The first calcula- tion is for a cloud with a zero potential at infinity of the type calculated in Appendices I and II. From Appendix I the potential of a charged spherical cloud with a uniform charge density x2 and a radius b is equal to UV = - ambpa/B + 27r¢9b2 .......... P23 b (1) where /° is the distance from the center of the cloud. U,= Awflb?5P .................. ...Fé- b (2) The following table gives the value of Ub/ww>from 0 5 P é 5, where b = 3 TABLE I POTENTIAL OF A UNIFORMLY CHARGED CLOUD VS DISTANCE FROM CENTER f0 Ub/v£> In Units In Units 0 18.00 1 17.33 2 15.33 5 12.00 A 9.00 5 7.20 -11.. Figure (la), page I?, Shows the cross—section of the uniform cloud of charge density +J9. Figure (lb), page Hi, Shows the potential distribution along an extended horizontal diameter of the cloud. If a metal sphere of radius a is to be placed in the center of the cloud, the metal sphere will displace some of the cloud, thus leaving a hole in the cloud. This will change the potential by a definite amount due to the absence of the charge originally in the space occupied now by the metal sphere. We must then fashion a cloud with a hole within it of a size that the metal sphere will Just fit into. This is most easily done by adding a small cloud of ~$9 charge den- sity (having a volume equal to the metal Sphere) to the large cloud. The small negative cloud superposed at the center of the positive cloud leaves a net charge of zero and thus makes a hole in the large cloud from the charge distribution stand- point. This negative cloud also subtracts from the potential throughout the exterior of the cloud. The potential of the small negative cloud is given by \N U... WAD/’73 - 2ND aa .......... Pé a ( 4:- U, = - Mimi/59 .............. M a ( Figure (2a) and Figure (2b), page/9 , give the cross- section of a small negative cloud and the potential distri- bution along a horizontal diameter of the small negative cloud respectively. - 12 _ The sum of the large positive cloud and the small nega- tive cloud is as follows U...= 2mm} - a2“) ........... . ........ . ..... P e a (5) 11...: - empf/a + 21rombe - Mamas/3P ..... a 4.- f’ .4. b (a) .3 3 Um; Ame - a )/3p.. ..... .. ............. . P?- b (7) The following table gives values of U/mvo for the hollow cloud. Figure (3a), page Hi, shows the cross-section of charge density of a hollow cloud. Figure (3b), page F1, shows the potential distribution along an extended horizontal diameter of a hollow cloud. TABLE II POTENTIAL OF A HOLLOW CLOUD VS THE DISTANCE FROM CENTER ’0 U /v£’ In Units In Units 1 16.00 2 14.67 3 11.56 A 8.67 5 6.93 In order to bring the potential to zero throughout the metal Sphere it is only necessary to place the metal Sphere in the hole of the cloud and add to the metal sphere a nega- tive charge of magnitude such as to produce a potential _ 13 - equal to but opposite in Sign to the potential of the hole. We know that the potential of a conducting sphere is constant throughout and an examination of the hole in the cloud shows that it too has constant potential throughout. The charge Q on a metal sphere to produce the potential necessary to cancel the potential of the hole is given by Q = V'C, where V is potential, and C is the capacitance of the sphere. The capacitance of a conducting Sphere is numerically equal to the radius, and thus we find that the Q on the sphere must be the negative voltage of the hole times the radius a or Q = 2713.9(b2 - a2)a. The potential V to depress the potential of hole to zero is given as follows — V'= Q/C = Q/a = - 2mn(b2 - aa)a/a = - 27rvh9(b2 - a2). . Pé a (8) - v = M" = - 2mm; - a2)a/¢° ......... F5 a (9) The addition of the charged Sphere to the hollow cloud is shown graphically in Figure (5b), page 20, and the values for U/wao are given in Table III on the following page. - 14 - TABLE III POTENTIAL OF HOLLOW CLOUD WITH GROUNDED METAL SPHERE INSIDE VS DISTANCE FROM THE CENTER p U.» + (-V) In Units In Units 1 0 2 5.67 5 6.23 t 4.67 5 4.00 The above addition shows a zero potential throughout the small sphere, a maximum potential about two-thirds of thick- ness of the cloud outward from the metal sphere and then a gradual dropping off until it reaches zero potential at in- finity. This type is approximated when a blanket of charged dust is laid down over a field by an airplane. The potential will be zero at the ground or plant surfaces, reach a maximum some- where within cloud and again approach zero somewhere out in space. The other case of interest is that of a cloud with boun- daries of known controllable boundary values. We may take as an example a spherical cloud with a metal Sphere in it, but this time with the outer boundary of the cloud at zero as well as the metal Sphere at zero. The method used will be essen- tially the same as before, namely, the addition of several _ 15 _ simple charge distributions. We may now start with the hollow cloud previously de— veIOped and add a negative charge at the outer boundary (a metal shell) until the potential of the shell is reduced to zero. The effect on the inside of the cloud due to this nega- tive charge is a constant reduction of potential all over the cloud. This is shown in Figure (6b), page.2(. The value of the constant to be subtracted is given by (U... ),,., = um/3b (10) The potential of the distribution is equal to U...- (U...).=b... ........................ 05 P 5. b (11) U...- (m... ),...,= 2mm“ - a“) - umow’ - 3 )/3b ............. P £- a (12) U‘“, — (114,0. )fi-‘b 2' - 21T$9F73 + 277-601): - MWAa3/3P - Amflb: - 21’ )/3b ........ a £- (0 9 b (13) UQ‘Hr ‘ (U414. )P=(r= 0. .......................... P 2 b (14) Table IV gives the values of an- (Uau, )P=I-/fp from 0 6.? £ 5,where a = l, b = 3. -16-— TABLE IV POTENTIAL OF A HOLLOW CLOUD WITH THE OUTER BOUNDARY AT ZERO VS DISTANCE FROM THE CENTER ‘0 UaHr - (Umo )nzo /TT+9 In Units In Units 1 14.1.14 2 5.11 3 0 A 0 5 0 The final case of interest is that of a hollow cloud con- taining both a metal sphere at zero potential and the boundary at zero potential. This is shown in Figure (7b), page2( , and can be calculated for the general case as follows: The potential of a hollow cloud is added to the potentials due to the negative charges on a grounded Sphere and a grounded metal shell. The value of the negative charges on the sphere and the shell can be, but need not be found. It is just as easy to find the value of the depressing potential necessary to bring the metal sphere and metal shell to zero Simultaneously. The following formula gives this potential throughout the cloud. 0 = UW - (U... )M _ v (1 — a/b) ..... (a = a (15) 1H:p = a the potential = 0 by definition. - 17 - We have taken the hollow cloud at zero potential on the outside boundary and added an amount aV/b to it to compen— sate for the negative value of the depressing potential - V at P = b from the consideration that Q (the charge on the metal sphere) = VC = Va, where C = a is the capacitance of the metal sphere. U4”, - (UM-L )P:b- v'= (1 - a/b) - (16) a 2 3 3 3 V _ - 2mm P/3 + 21mb - Amwa L/Br’ + Amo(a - b )/3b (1 - a/b) If v' = V/mw, then for a = 1, b = 3 -I = - _ _ _ O _ fr v g5 + 18 Ali/j Al (27 1M, z 4:111; _ 6.0% (17) l - .933 orb? Throughout the metal Sphere the depressing voltage = V, but since potential at a distance from the charged sphere due to the charge Q = VC = Va is given by the relation 0/9 = Va/p, then at p'= b the depressing potential = aV/b. In order to get the potential distribution throughout the cloud we now must subtract the depressing potential from, and add the con- stant aV/b to the potential of the hollow cloud whose outer boundary was at ground potential. U = UM — (U,.._ )M - v4. + aV/b C II A H (“I“) v - 27rloPI/3 + 27r4~9b‘z - AvNa3/3P + AmL9(a‘ - b3 )/3b - V/p + aV/b _ 18 - 3 U/m: - 2073 + 2 b" - u a’/3p + 4(a'R - b )/3b -W¢+awm um rora=1,b=3,V 26.66 U_/TrJO= - .667!” + 18 - 1.33/P - 11.56 -6.66/p +2.22..................a5Péb (19) Wm: OforOéPéaand forPab (20) Table V gives the values of U/voo for 0 £- /’ .4: 5. Figure (7b), page at , shows the potential distribution of this case. TABLE V POTENTIAL OF A HOLLOW CLOUD HAVING A METAL SPHERE WITHIN AT ZERO POTENTIAL AND AN OUTER BOUNDARY AT ZERO POTENTIAL VS THE DISTANCE FROM THE CENTER ’0 U/vl’ In Units In Units 1 0 1.5 1.83 2.0 2.00 2.5 1.30 3 0 A 0 5 0 - 19 - Superposition of Simple Electric Fields " :vvirvgfi-“w fv:1 1 {wide-io-ISJQS Fig. 2a. Small negative cloud. the lfrjtvrT'II—W ..(q-ys;~1013:q§ ig. 3a. Cross-section of hollow cloud. 5” U1 3 O 509x9A~1%W73 1 o‘ I Fig. la. Cross-section of large positive cloud. \ fi 9‘ J 1 § {—1 r 1 r 1 ‘ x r “7* I w 4-31-19 1 :3q 5 p :2 Distance from center in units. ig. 1b. Potential along horizontal diameter of cloud. ”flap Mr [/V/n' i Fig. 2b. Potential distri- bution along a horizontal diameter of negative cloud. Fig. 3b. Potential distri- bution of hollow cloud. - 20 _ Superposition of Simple Electric Fields Fig. Aa. Cross-section of metal sphere. Fig. 5a. Cross-section of negatively charged Sphere inside of hollow cloud. Potential distri- bution of negatively charged sphere. Fig. 5b. TYTFT—TTI t‘lOllJ‘ls- Potential distri- bution of the sum of negatively charged metal sphere and hollow cloud along extended horizontal diameter. Fig. 6a. Cross-section of a hollow cloud enclosed by a conducting shell at zero potential. Fig. 7a. Cross-section of a metal sphere in a hollow cloud en- closed by a conduct- ing shell at ground. ”/70 {IV Z/n/r f5 1' 0&3 lJ - 21 - .1 I Fig. 6b. Potential distri- bution along ex- tended horizontal diameter of hollow cloud with boundary of cloud earthed. d. ABove‘ CLOUD + V77 ..V-S- Du: ro OVA/P65 ON THE Spueu: Potential distribution along an extended hori- zontal diameter of sum of potentials of grounded metal sphere, hollow cloud, and a grounded metal shell. _ 22 _ Calculations of Potential Distribution of a Spherical Cloud of Charged Dust That was Measured. Since it is desirable to compare a measured field and calculated field, the calculated field must have the same boundaries and charge density.;9 as the measured fields. The charge density” is taken to be .6A5 units/cu. inch, which value is 1938 times the value ofcial in e.s.u./ cc. If Q_= 4WR%&993 _for total charge contained in a spher- ical cloud of radius R and charge density J0, then V’= Q/R = AvR3A973R = potential at outer boundary of this cloud. Thenfl': swims". Considering that 1 stat volt = 300 volts and 1 cm. = .39A inches, we may convert from c.g.s. units to volts and inches by using the following factor 1 ,(9’2 (at 17-7/3 = 1 ,2)" 300 1938 [Av(.3947‘]73 I (21) 4L9 (1938) =00 (1) The diameter of the cloud is 35.5 inches and the diameter of the metal Sphere to be used inside it is .90 inches. In terms of the distance 1° from the center this means the outer boundary of cloud is (b = 17.75 inches) and the radius of the metal sphere is (a = .A5 inches). The calculation for Case I is that of a cloud with no metal Sphere in it but depressed to zero potential at the outer boundary. former section is as follows withd~9 = The following table gives the values of U = 27mph)z - (’2')/3 .6u5: b -"-'- 17.75 U=ZL35<3H3'PZ) 23 - The equation as found by methods of the (22) (22) the potential for the various positions as calculated from the above for- mula to the nearest volt. CALCULATED POTENTIAL VS TABLE VI DISTANCE FROM CENTER FOR CASE 1 f’ j P1 (bi-f”) U Inches z 2 L from (315 - P ) 1.35 (b -P ) Center volts .45 202 515 425 2.25 5.06 510 418 4.25 . 18.1 297 I 401 0.25 39.1 276 ) 373 8.25 38. 2A7 ) 333 10.25 105 210 : 284 12.25 150 165 222 14.25 205. 112 151 15.25 267. 48 ’35 17.75 , 515. 0 C - 24 _ For Case II, where the metal Sphere is depressed to 200 volts and the outer boundary is at zero, the equation for the cloud depressed to zero potential at the outer boundary is given from a previous section as 27,.”(bz -p")/3 - Avd9a3/3I” + AnyoaVBb ..... .....a 5 F 9 b c: I! (13a) C! II 1.35 (515 - F2) - .246/p + .2A6/l7.75..a e p a b It can be seen that the second term on right can never be greater than .550 nor less than .013. The third term on right Side of equation equals .013. Thus the maximum error that can result from neglecting these two terms is .550 - .013 = .537 volt. This is an entirely negligible value when considering the magnitude of the first term on right side of equation having a maximum value of A25 volts. The second and third terms will be dropped from all subsequent calculations. The equation of hollow Cloud is now given as U = 1.55 (5’15 4") ............ aéf’é b (13b) The equation of hollow cloud with metal sphere at 200 volts and outer boundary at zero potential is given by U = 27r,lp(b2 -(=‘)/5 - av/(n + aV/b (25) V being the depressing potential and found by method of pre- vious section as follows U = 200 = 1.55 (515 - .2) - v (1 - a/b) ..... f’= a (24) V'= 231 volts aV/b = - volts U'= 1.55 (515 -p’) - .45 (251)4p + 6 volts a 4,0 b (25a) TABLE VII CALCULATED POTENTIAL VS DISTANCE FROM CENTER FOR CASE F 2mn(bz-PH/3 aV/fl aV/b (I Inches 2 from 1.35 (315 -(3) - .A5 (281)AF + 6 volts center .45 425 251 + 200 2.25 418 46 578 4.25 401 24 585 6.25 373 17 362 8.25 333 13 326 10.25 28A 10 280 12.25 222 8.5 220 1A.25 151 7.3 150 10.25 65 6.4 55 17.75 O 5.9 O For Case III where the metal Sphere is at ground, a negative charge is induced on the sphere sufficient to depress the original A25 volts down to zero at the surface of the metal sphere. The procedure for calculation is exactly the _ 26 - same as in the preceding case and a value for the depressing voltage V must be obtained. U = O = 2774.4)(b2‘ - a2)/3 - aV/a + aV/b....(‘J = a 0 = 425 - v (1 - a/b) V ” volts II p- \N G aV/b = 11 volts U = 1.55 (515 ..p‘) - .45 (456)/p + 11 volts....a .479 .4: b (25) TABLE VIII CALCULATED POTENTIAL VS DISTANCE FROM CENTER FOR CASE III ,0 27w (b1 - P')/5 aV/{J aV/b U Iggges 1.35 (315 —P1) - .A5 (A36)flp + 11 volts center .45 425 456 0 2.25 418 87 542 4.25 401 46 566 6.25 575 51 555 8.25 333 24 320 10.25 284 IQ 276 12.25 222 16 217 (14.25 151 14 148 16.25 65 12 64 17.75 0 11 0 _ 27 - Case IV has the metal sphere depressed to a negative 198 volts, and is handled exactly the same fashion. Solving for V we have. V = 639 volts aV/b = 16 volts U= 1.55 (515 ~p') - .45 (659)/p + 16 volts...a-éF 1: b (26) TABLE IX CALCULATED POTENTIAL VS DISTANCE FROM CENTER FOR CASE IV P 21r,'.9(bz -p’)/5 aV/p aV/b U 526288 1.35 (315 -/°L) - .AS (639)49 + 16 volts center .45 425 639 - 198 2.25 418 130 + 504 4.25 401 69 + 548 6.25 ' 575 47 + 542 8.25 333 35 + 314 10.25 284 28 + 272 12.25 222 24 + 214 14.25 _ 151 20 + 147 16.25 65 18 + 63 17.75 0 l6 0 _ 28 _ Measured Potential Distribution of a Spherical Cloud of Charged Dust The first real attempt to measure the potential distri- bution (from which the field intensity could be deduced) was a complete failure. A closed pyrex spherical container was filled with a cloud of charged smoke, by evacuating the con- tainer partially so that the smoke was drawn through an ion- ized field type charging nozzle. The filling apparatus was all mounted in a tight fitting stopper. Once the flask was filled with charged smoke, the stopper with the filling apparatus was replaced by another stopper containing a mov- able radioactive probe connected to an electrostatic volt- meter. However, it was found that the time required to re- place the filling stopper with the measuring stopper and bring the instrument up to potential was sufficient that an essentially constant potential occurred throughout the volume of the flask. This could only mean that the smoke particles had almost all precipitated onto the inside of the flask. Nothing whatever was learned of the potential distribution that would occur in a uniformly charged cloud. This appara- tus is shown in Figures (8) and (9). The second attempt utilized a continuous cloud of charged dust which was not so greatly affected by precipitation. The continuous cloud of dust was made available to the volume re- gion being measured to replenish the dust that precipitated out of the air. Also, conducting surfaces held at known _ 29 - potentials were used as boundaries to the cloud instead of insulated boundaries that changed potential with a precipi- tation of dust and charge. These conducting boundaries dissipated the charge brought to the boundaries by the pre- cipitated dust so there was no appreciable effect due to dust precipitation such as occurred in the first attempt. The apparatus to provide the charged dust and the measur- ing system are pictured in Figure (10) and (11), and a schematic diagram is shown in Figure (12). Description of Apparatus. The apparatus for producing a charged dust cloud and de- livering it to the measuring region consisted of the following items: A variable Speed motor operating a centrifugal fan for producing a dust cloud, an ionized field type charging nozzle mounted on the end of fan discharge pipe for charging the dust, a power supply for supplying power to the charging needle, a mixing chamber into which the dust was blown and mixed with air, a measuring chamber, a perforated bulkhead between mixing chamber and measuring chamber to provide a reasonably uniform discharge of charged dust into measuring chamber, an exhaust fan to draw the dust from mixing chamber into measuring chamber, and an axial flow fan to provide tur- bulence in the measuring chamber. Apparently due to the location of the exhaust outlet in the top of the rear end of the measuring chamber, the dust - 5o - cloud was prone to move in such manner as to allow a greater concentration of charge (and presumably dust, although this could not be visibly detected) nearer the upper part of the measuring chamber. This condition resulted in a somewhat lop-sided potential distribution. This condition was greatly improved by the use of an axial flow fan placed as shown in the diagram of Figure (12). The fan improved the distribution no matter whether directed along the floor, toward center of cage or toward ceiling, but gave best results when positioned along floor of measuring chamber as shown. The apparatus for measuring the potential of the charged cloud consisted of a chicken wire spherical cage of 55.5 inches diameter (which formed a cloud with a Spherical shape when the cage was placed at zero potential), a reference probe and electroscope type voltmeter, a movable probe (tra- verse probe) and electrostatic voltmeter, a cord and pulley arrangement for supporting the movable probe, and an indexed pulley for taking up or letting out cord which raised or lowered the traverse probe. Other equipment included a sling psychrometer for measur- ing the wet and dry bulb temperatures. 1-; 1 ., q 1‘ . - 31 _ Fig. 8. Spherical flask with filling cap and charging apparatus in position. , ('0 . * r ~ 64.... Fig. 0 Charging nozzle and filling cocks as used in Figure 8. Fig. _ 32 _ Fig. 10. View of apparatus used for measur- ing potential distributions along a vertical diameter of a charged dust cloud. . ‘1 _| -. _ , - . . . ... , 7 ....\ ’7, _. ,. , , , ., ... .5 A...» X: 1:5“. , . _,“,: t (- — ~ vi ”v 'w‘"'» .7 1 ‘ - / ‘q'a‘fll’nr.l;;¢ \rv— ‘ _ j ,. :7- 11. View of measuring chamber Showing spher- ical cage, reference probe (black lead), traverse probe (glass rod insulated), and metal sphere. In the background can be seen perforated bulkhead. -33.. .Usofio wasp .man. oomhano a :H cofipfindhumau HeapCOpoa wcfinsmwoe hon maumpmaom Mo Empmwfio .mfi ... I : ...... \ 0’ an Q‘K \\M\..\ HK \ // __ 933$ $.23 A \ I __ _ n A. _ __ .36 MS} qwusmwtmm. , ll .\ \ __ / \ :x// \\ __ mexxwxxw MmSSVQ i new I \ __- h.2x\ I thcfiskm 9 IQ _ MANNQE wlxwmvfixb ll mwakqfiw ImV ”kamdskv&; mammfl>xmh.nnw ka WEL we; wuzwwwurwwt I U onQQ went30§< I. Q wmswmux wukww mm mm. ... q WQ0&¢.Q&\.NUGHV 0 «wt». .SQ QUL («suwwnxl/ \\l// /J: \ / _ \ / I __ J a , / i. __ a... \ __ / xx __ 2w: 0Q -314- Procedure for Making a Potential Traverse along a vertical Diameter of a Charged Dust Cloud. The adjustable gate on the exhaust fan outlet was nearly closed. This reduced the pulse effect due to the uneven feeding of dust to the blower. An axial flow fan (of house- hold type) was operated to promote a mixing of dust and charge within the measuring chamber. The centrifugal blower was operated at 1500 r.p.m. for all traverse measurements. The charging nozzle was supplied with an ionizing voltage (approximately + 15000 volts D.C.). The dust (525 mesh atta- clay) was fed into the centrifugal blower by hand in discrete quantities of somewhat less than a gram by Shaking'through a sieve (Similar to a small flour Sifter). After an initial filling of the measuring chamber only one addition of dust was required for a reading. The reference probe was taped in place at one of the measuring stations and the reference dots on the glass faces of the meter were lined up at a voltage that would allow near full scale deflection on the traverse meter when at the sta- tion of maximum potential. The potentials on the metal sphere dictated to a large extent where the reference probe could be placed. The reference meter was operated in the range of 215 volts to A25 volts depending on the particular case being measured. The traverse probe had a three foot glass tube for an insulator at the measuring end, which was sufficiently o n . . 5" u u v . .- _... . n 3 u\' h . ..'-.|.-.._%1. -35.. heavy to keep the supporting cord under tension at all times. An indexed pulley (Figure 10) was calibrated so that by turning the pulley to a given mark, the probe was raised or lowered to one of the observations stations. The station positions were measured in two inch increments from the top of the spherical cage for convenience in calibration. This led to some inconvenience in all subsequent operations with the data since the sphere was only 35.5 inches in diameter. The cotton cord stretched the first couple of days but was recalibrated for each test, and checked after each test to be sure no change had taken place. After two or three days the cord did not stretch any observable amount. Three replications of readings were taken for each sta- tion. The order of observation of each reading for all sta- tions was determined by a complete randomization of the total readings to be taken. Thus the first reading of sta- tion one might be followed by the first reading of station twelve. Since the complete traverse required 51 observations not counting the two end points on the grounded cage, the last reading of some of the stations was taken as long as an hour and a half after the first reading. This randomization was considered necessary in order to confound any gradual changes in the measuring system due to changes in relative humidity, or other variables unknown to the experimenter over the two hour duration of the test. The reference probe was located in one position through- out test and the position checked by means of the traverse ‘1‘ 9 . ... ...\ -35.. probe several times during the test. Once the reference meter was calibrated for a traverse, all readings on the tra- verse meter were taken at the moment the tip of the gold leaf of the reference meter fell past the bottom of the index dots on the meter faces. A falling leaf on reference meter and falling needle on traverse meter was used for all observation of readings. This was done to insure that the bending stress in the gold leaf was always in the same direction. Charac- teristic of a radioactive probe such as was used for both reference and traverse probes is that it approaches the sur- rounding potential asymptotically and, therefore, there is necessarily a small amount of lag with the falling leaf. This means that a falling leaf or needle would read somewhat higher than the true potential and a rising leaf or needle somewhat less than the true potential. However, indications were that this lag was not of nearly as large magnitude as discrepancies that would arise due to variable bending stress in the gold leaf. The voltage difference on the same point for a falling as compared to rising leaf was somewhere in the neighborhood of fifteen volts, which variation could be ex- pected among different readings of the same point if no con- sideration were given to method of approach to the point. 0n the other hand, replications of a given reading using fast falling, slow falling and steady held readings approached from tOp down did not Show any observable differences in value of potential other than could be attributed to less -37.. accurate observing and announcing of arrival at point by the reference meter observer. A hovering leaf is harder to de- tect when it is at an exact position than is a moderately falling leaf. A fast falling leaf is often not called correctly because of inaccurate anticipation of the observer and inaccurate announcement of arrival. For example, an in- accurate announcement of one—tenth of a second on a fast falling leaf of perhaps 30 volts per second would amount to 3 volts error. Also a fast falling needle on traverse meter is subject to the Same type of error which may add another two or three volts to the error. For this reason, a slowly falling leaf of five to ten volts per second gave the most consistent results. Repeated observations of a given point Would usually be within two or three volts and often four or five readings could be taken in rapid succession that were within one volt of each other. The technique described re- quires some practice for consistency and naturally is subject to some errors, however, even unpracticed observers are able to read to within one or two volts of the reading of exper- ienced observers. There were four potential traverses carried out tracing the potential distributions due to four different boundary conditions placed on a spherical dust cloud of approximately uniform charge density. In all cases the outer boundary of the cloud (formed by a chicken wire cage) was held at zero potential. In the first case the traverse was of the potential of a vertical diameter of the cloud without a metal sphere in -38.. it. The second case was of approximately the same cloud with a metal Sphere of diameter .9 inches centered in it having a potential of + 200 volts impressed upon it. The third case was for a metal sphere centered in approximately the same dust cloud with zero potential impressed upon it. The fourth case was for approximately the same cloud with a - 198 volts impressed upon the sphere. In the second and fourth cases the potential on the sphere was obtained by connecting the metal Sphere to a 200 volt battery hook-up. In the third case the metal sphere was connected to ground. The lead of the metal sphere was inside of an 8,000 volt break down strength spaghetti tubing. The whole traverse from top of Sphere to bottom was carried out several times in preliminary traverses and also for the Case I. However, it was decided that since the two halves of the diameter from top to center, and bottom to center were quite similar (though not exactly so) it would be expedient to carry out only the bottom half of the tra- verse (from bottom to center). This decision came about be- cause the test was Shortened from two hours to one hour and thus would reduce variations due to changes in relative humid- ity, temperature and fatigue of operators. The bottom half was chosen because it most closely followed the theoretical calculations for Case I. The data for all four cases are given in Tables X, XI, XII and XIII. Discussion as to the results of the tests will -39- be reserved until a later chapter where the traverses are com- pared with the calculated traverses of the previous chapter. The original data reckoned the stations of observation as taken in two inch increments from the top of the sphere. However, in the data presented the positions of observation are given in inches from the center to correspond to the for- mulae for the calculated traverses which gives the potential as a function of the distance from the center. Case I represents the condition of a homogeneous cloud of charged dust with the outer boundary at zero potential. The data given in Table X are for the potentials existing along a vertical radius. TABLE X - MEASURED POTENTIAL VS DISTANCE FROM CENTER FOR CASE 1 =‘IEches from Potential in Volts center Rep. 1 Rep. 2 Rep.3 Ave. 0 405 402 405 405 2.25 393 393 394 393 4.25 371 367 373 370 6.25 540 541 550 544 8.25 510 506 516 511 10.25 258 256 257 257 12.25 215 220 21A 216 14.25 163 150 153 155 16.25 75 - 85 7O - 80 7O - 80 75 17.75 0 O O O - 40 _ Case II represents the conditions where the potential on a metal Sphere at the center of the cloud is approximately half way between zero and the potential it would assume if isolated. The potential was held on the sphere by means of a 200 volt battery hook up. The reference probe was placed at 12.25 inches from the center of the cloud and calibrated so that it read 216 volts at that position. This was done be- cause it was desired to have comparative densities for all cases, and it was noted that theoretically for all four cases the potential of the curves were close to a common value at any of the positions close to the outer boundary of the cloud. The data for Case II are given in Table XI. TABLE XI MEASURED POTENTIAL VS DISTANCE FROM CENTER FOR CASE II Inches from _FL Potential in Volts center Rep. 1 Rep. 2 Rep. 3 IRep. A Rep. 5 Rep. 6 Ave. .A5 + 200 + 200 + 200 + 200 2.25 356 375 37LL 386 385 384 385* 380 4.25 378 372 370 375 373 373 373 6 25 558 545 546 542 8.25 500 505 506 505 10.25 250 246 248 248 12.25 225 216 215 214* 218 fl4.25 160 159 153 157 A6.25 7O - 80 7O - 80 70 - 80 75 7.75 O O O O * These two readings taken after the other readings to as- certain whether or not the circled group was higher than previous readings as station because of a change in reference voltage. - A1 - Case III represents the conditions where both the metal sphere and the outer boundary are at zero potential. The data for Case III are given in Table XII. TABLE XII MEASURED POTENTIAL VS DISTANCE FROM CENTER FOR CASE III Inches from Potential in Volts center Rep. 1 Rep. 2 Rep, 3 Ave. .45 0 0 0 o 2.25 546 550 546 547 4.25 372 373 372 372 6.25 343 338 3A0 3A0 8.25 298 300 295 298 10.25 250 246 253 250 12.25 213 217 220 217 1A.25 155 160 165 160 16.25 70 - 8O 7O - 8O 7O — 8O 75 17.75 0 O O O Case IV represents the conditions where a potential of a negative 198 volts is on the metal sphere and all other conditions remain the same. The values of potentials at various distances from the center are given in Table XIII. - 42 _ TABLE XIII MEASURED POTENTIAL VS DISTANCE FROM CENTER FOR CASE IV IEChes' from Potential in Volts center Rep. 1 Rep, 2 Rep. 3 IRep, Rep. 5 Rep. 6 Ave. .45 — 198 - 198 - 198 - 198 2.25 + 296 + 300 + 313 e 300 + 310 + 305 + 30A 4.25 555 550 528 + 530 6.25 329 326 320 + 325 8.25 245 298 298 295 298 293 + 297 10.25 243 255 253 251 255 253 + 253 12.25 218 223 215 + 219 14.25 162 175 165 + 167 16.25 70 - 8O 7O - 80 7O - 80 + 75 17.75 0 O O O * The thirty readings included in the first three replica- tions of Table XIII were completely randomized as to order in which reading was taken. At those positions where some one value differed from the average of the closest two by more than ten volts, three more replications were made, one after the other at the end of the test. When only three replications were used the mean of the three was taken for an average. when six repli- cations were used, the highest and lowest were discarded and the mean of the other four was used as an average. a . - - n . . I n '9- . l a u r ' _ h s . .- .1 . . . I Comparison of Calculated and Measured Potential Distributions The purpose of obtaining both measured and calculated potential distributions was to verify the correctness of the calculated method by means of a comparison of the similarity of calculated and measured curves. There are only two details by which we can judge the similarity. They are (l) the correspondence of the general shape of each case between measured and calculated and (2) the order of ascendency of the curves of the four cases. The measured potential distribution has been adjusted in those cases where necessary to correspond to the calculated potential at the position of the reference probe using a charge density 19 of .6A5 units/cu. inch. This meant multi- plying the potentials except at boundaries for Case I by a factor 1.03, Case II by 1.01, Case III by 1.00, and Case IV by 0.978. It is recognized that there is some danger in making an adjustment of this kind because the standard deviation of this doint is quite large being 3.18 volts for Case I, A.36 volts for Case 11, A.96 volts for Case III and A.05 volts for Case IV. However, under the circumstances some sort of adjustment had to be made and all cases were treated in like fashion. It will be noted that the calculated potential is higher in the range 6 12 than the measured, but becomes less ,. L111- than the measured potential for the range 1345(94 17.75 for all four cases. This indicates that there is a low density of charge in the region 64 P< 12 and a high density of charge in 13 ((9417.75. The high and low density are with respect tOJJ = .6A5 units/cu. inch. There are two possible causes of this slight inhomogeneity, (l) the dust cloud pro- duced in the chamber was not uniform regardless of whether the dust was charged or not, and (2) the electric field forces set up by the charged dust cloud tended to gravitate dust nearer the conducting surfaces at the expense of the inner regions of the cloud. There is some indication, though not conclusive, that this was taking place both near the sphere and near the outside boundary. Probably both of the above factors contribute to the condition. There is little doubt as to the Similarity of shape of measured and of calculated potential distribution curves for each of the cases, as may be noted by observing Figures (13), (1A), (15) and (16) which compare measured and calculated po- tentials for Cases I, II, III and IV respectively. The Case IV, Figure (16), deviates the greatest amount, but there can be no mistaking the similarity of the general shape of Case IV for it is quite distinct from any of the other cases. The order of ascension of the curves starting with Case IV and going up is the same for both calculated and measured as can be seen by comparing Figures (17) and (18). - A5 - .£ocH .So\mpfics mzm. n Qszpfiz HmfipcmpOQ pmumHSono o .QSOLw ecu p50 ...-@653 awash on» omeonade o6. 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ITQQV .umumHSOHwo mampm> Smashups .H mmwo you cofiusn nfipumfip fisfiucmpoa mo comahmdfioo .nfi .wwm Q hNthwsx “Nix SNNV d q 5170/\ N/ “Nu/V1100! wwkkwb $00M mwt0$ 3 m %N§\.hN“v th..MN§v MNMw .v 7 1 . .uwwsom¢m§s\ .- “QM! - W .vougaoamo mamas». 00.5305 63332.60 mama? 623on .>H onto you coausnfiuunae .HHH 0666 you coausnappmae . Hwapcouoa no :omflnsqaoo .mH .me HafiucchQ no comfipmcsoo .m. .w.a anQ\.l Muck Rub gawk hwtb>< D.\ K 36k DEV stux $3.03 >\\ K hkfi MN! “Ni mug “Mex huh “we «NV ANN . bkx wwwx “NS. 8% 3.3 amp .86 MN.» EN 6% Q d H - 8 4 J a) J .- d u 4 q 4 d d d / / L.OQ\ / l / d O / z- / 9 N 1- 1 . cow w ... 7 o / / / N /O / 985... (NE QWUSthSV / m // / \\ 4..ng 7 l. I: / ,- QNAQV§UVRUIV // \\ 5 QkavauonvL/ \ III‘ I \ \ lr§ I: .20.: $33.25 9.0. .1. 02s .20Ca sc\mpacs mam. umfi .>H use HHH .HH .H sumac sou .>H can HHH .HH .H sense you . 0 soapsnaapsae Haaucouod amassed: .ma .maa coaesnaapnae asapcopod eoueassaso AH sad . . ..sswn j. as _ u. . _ TQQ\I “I. mswk>rww Sinkvx.mw$u ~60- Discussion of Results As can be noted from the preceding page, the sample mean for treatment B (the grounded Sphere) was 0.00127 gms. as compared to the sample mean of 0.00025 gms. for treatment A. On the basis of the theory the mean for treatment A Should have been zero but in all cases there was a small number of large clots of dust that dropped down from the wire of cage above during the run and in handling after the run. These clots appeared on all Shells to some degree. Although the cage was cleaned before each run, it collected dust in con- siderable amounts because of the high field intensities assoc- iated with the fine wires. The population mean at the 95 percent confidence level Should lie between m = 0.00127 f .00029 gms. for treatment B and m = 0.00023 t .00016 gms. for treatment A. The minimum fiducial limit of treatment B is more than twice that of the maximum fiducial limit of treatment A at the 95 percent confidence level. No actual dust deposit other than the clots due to gravity were detected on the shells of treatment A. The very high metallic lustre on the Shells of treatment A attests to the absence of more than a few small particles collected by electrical deposition. The treatment B Shells had a very uniform coating of very fine dust which greatly dulled the natural metallic lustre of the uncoated shells. -51- It is the author's belief that the deposition theory has been verified to the extent that deposition occurs as long as the collecting surface is kept at a lower potential than the surrounding cloud and that an isolated conductor will assume the potential of its surroundings and, therefore, collect no dust due to electrical forces. The Electric Field and Potential Distribution of a Finite Disk and a Finite Cloud The study of Spherical clouds enve10ping a metal sphere has served to verify the methods of electric field analysis but in themselves have little to merit as regards agricul- tural crop dusting. The electric fields of interest are those that are found next to the leaves and stems of agricultural crop plants when a cloud of charged dust is blown into the inner plant regions. A complete analysis of the field surrounding a plant leaf would be difficult indeed and probably would not be worth what it would cost in time and effort. However, an analysis of the forces that occur along a perpendicular axis of a small disk when a cylindrical cloud of dust is next to it would be of considerable help. This would give the mini- mum force that would be available to a leaf, as all other areas of the leaf would usually have as large or larger forces active. _ 62 _ By use of two formulas developed in potential theory for the force along the axis of a cylindrical charged cloud of uniform density, and the use of the principles of imaging and superposition, one may construct the field intensity distri- bution along a perpendicular axis of a conducting disk. The fields to be constructed are those of a four inch diameter conducting disk with a cylindrical cloud four inches in diameter and four inches high placed in contact with it, Figure (21), and a four inch high cylindrical cloud sand- wiched between two four inch diameter conducting disks at zero potential, Figure (22). The following pages Show the steps used in constructing these fields. Fig. 22. Cloud sandwiched be- tween 2 disks at zero potential. I. 8- (...—n... Fig. 24. - 53 - Cloud next to a single disk at zero potential. Fig. 21. Fig. 23. Diagram showing constants for formula. “" ticle with uni 26, 2 , 28 and 29 { ,Ih;+ a‘ 25, ' ‘“.- d": h'."+ a' , dz: + d. d9, Force on par- ht + charge for Figures Diagram showing constants for formula. F = 2v¢?(h + d.- d), Force on a particle with unit + charge for Figures 30, 31, 32 and 33. h = (c - b), d,= a‘+ b: d‘=.rE'I‘E‘ :2 >4 “0 Fig. 25. ,' A._ - ' ,w -. - v. H '42“- z a.” 5“ . J . .. bl...~....,m~.i- V/?,_-I__- "...“? . 3 z. . r 1; ’ y t'. , _ -‘ . - .-.‘. .. ht..." ..' (' : " , , u.“~N/4%~V -1:-—'\-"'i . . ".~ Fig. 26. Fig. 27. *IJ’IJ'IJ'IJ *13 'IJ’IJ'IJ ll 2w (hr d2+ a) 2w$9[u - J36 + 2] 2wd’[6 - 4.48] 9.5549 21w (hf h,+ d, - at) and} - 1 + 2.25 - 3.61) 21rJ~9(.64) 4.02.#> 21r40(h.- h,+ d,- (1,) 21w(2 - 2 +./2‘+ 2‘— I???) O h = h‘g d.= d a d,- .51;. J2‘+ 1‘ NR) 2‘+ 2‘ .\ Fig. Fig. 28. '13'11'12121 From symmetry with (85) the force is 2w49[h2— h‘+ d‘- dJ :anw[1.- 3 + If?-f?i] 2v19[1 - 3 + 3.61 - 2.25] — 4.02.L9 F = - 9.55 $3 Fig. 30. F: - 21r¢L9[h + d,- d‘] F‘= - 2wi9[4-+J3?-JE§1 F = — 21rl9[4 + 2.25 - 5.39] F = — 2WA9[6.25 - 5.39] F: - 27rA9[.86] #1.] = - 5,4 49 - 65 _ h a 3 ht: 1 d,= /2‘+ 3‘ d‘= I2‘+ 1‘ d,=./5‘+ 2’: 29 Fig. 3.1 '11 ’11 '31 II II Fig. '13 ll '13 ’13 ’13 ‘13 II II 31. —21rp[h + d. - d‘] -21M9[4 +m-fl0] ~2mw[4 + 2.83 - 6.34] -2wto[6.83 - 6.34] =-3.08J~9 . 32. -ZFW[h + d.- ck] ~21r.LD[4 + J31;- [53] -21rJ.0(4 + 3.61 - 7.29] ~21TJ~9[7.61 - 7.29] = - 2.0110 33. ~27r)o h + d,- d] I .. ~21er[4 + f2“ ~ RB] —21rJ0[4 + 4.48 - 8.25] -21r.U[8.48 - 8.25] 4719(2)] = — 1.44M -66-— h = ‘5 d;3 W‘. [—5- dza J6‘+ 2’:- J70 0.0- 3' J2‘+ A's-f5?) (2‘4 8‘: /6/8 -57- Fig. 34. F: 21rl~9(2+f4-f8) F = 21rU(4 — 2.82) F: 7.410 Fig. 35. F: 21mm +F- J33) F = 210901.24 - 3.60) F = 4.02%9 71g. 36. F== 2w(2+F-f§o) F ... 21rU(4.82 — 4.48) = 2.14M ’13 Fig. *13 N Fig. ’11 *1] ’11 ll 37. 2v.lfi(2 + [T3 'JE) 2mU(5.6 - 5.4) 1.26M 38. 21r+9(h + d. - C1,) 21rp2+J—2_0-N0) 21nd 1.1 ( ( 6.48 . 6.30) p -68— FORCE Due 71: Pas. CLOUD Fence DUE To N56. To {A L Font ...... 1?.5510 1AA“? 18.1!» _ .__. 14.0219 12.01» “mm + ... o .00 13.080 1mm —— 14w.» 1514049 19.120 4 —— 19.55» 1 9.55.13 mac» n- Fig. 39. Development of the force experienced by a unit charge placed along the perpendicular axis in a positive cloud near a conducting disk at zero potential. FORCE DUE FORCE pus Faces: DUE 131m. Force - To Top (136. re (’05. To Lowuueo. _. ..__.____..___.___...__ 1» 7.104: f 7.5519 1 1.13.9 1r 16.82.40 ... T AIM-Jo f m.» 4. ..u {-9 f 5. 78 .49 + .— f J-HJO o«00 t 2.14» w» w .... 11 1.2500 14.9”.) 4.4.02.1: .L @7820 ...... ...---_——— 1 1.13.19 19.55» t 7.10 p 1,5382 M Fig. 40. Development of the force experienced by a unit charge placed along the perpendicular axis in a positive cloud between two conducting disks at zero potential. .325 "Q3356 . empwno .psofio pocwfim 0 sac: m no sofipmfisaoa no one 7. no awn» mH cofiuoonau venom . age .Hm .mfia do Hwapcouoa ohms um xmfio MCHuosUGOo .mbfifi “Os 2&ch on» O» axon psofio .uo apfimcoe mmnmno .nsoao . mans macaw suamcopcw camaa .fis .wfia cecwamuans a ho ceapwasaoa 2.3 no 98 no page m.“ «mine: 2x ...wa teak MuttkflQ cofiuoepfip venom .mm .wfim v n. N a o no Hmfiucmuoq open us wxmfic wcduospcoo can on» amok» i use ponofizpcwm cacao one no a max» macaw muamcopcw names .ma .wfia .2 IQQ\I mutusN 3 SEC 4s mwxgmb tokau‘ .35..ka O A O ...,._./ _ I 02 r O O N 1 OQN ”My/5.179] N/ 3 AJJSNBANJ any F.-'$.})_ ..' _ "‘ . HJA/j/fJ 7% M ALIS/V471 N] 0 1223/ r ooh eon - 71 .mm .maa ca axons psofio 90 cans wcoam NI ‘7 VAL/Viiod 91744 cowpznawpmae Hmapcmuoa .3: .wwa hutt§N .3 3‘3.» 46 «been» tskk mussitQ m N ros\ room .Hm .wwm cw czonm Unofio mo mfiwm wcofim . coausnappmfip Hmfiucmpom .mn .Lwa MNIVRN \S taxQ {skux Noixknxua w .a m a a .. 0 W w w .v C a . _ i . w e ,m . w 4. CQN /./ . . .... ...z /. ...: .. 1 0Q. ”I {‘J~AJ I“. _ 72 - The force F as calculated is the force on a unit posi- tive charge and thus is equal to the field intensity. The direction of the force is the direction a positive charge would be urged if placed in the position under consideration. Thus at the top or bottom of a positive cloud a positive charge would be urged away from the cloud. At the center of a positive cloud no force would act upon the positive particle. A positive particle would be urged toward the center of a negative charge distribution. A view of Figure (41) Shows that the maximum force occurs near the surface of the disk, and acting to move a positive particle to the surface gradually diminishing until at a dis- tance approximately two-thirds of height of cylinder from disk. There the force is zero and for the rest of the dis- tance to the outside boundary of the cloud we find the field urging a positive particle outward and away from the surface. This considers a zero boundary at infinity. This condition is approximated by a cloud next to the outside leaves of a plant in field work. The potential distribution, Figure (43), may be constructed by the addition of the average intensities for each successive point. The potential and intensity distributions have the same general Shape as did those of a spherical cloud when the central Sphere is large and the cloud envelope thin. A study of either this or the Spherical case will show that the po- tential distribution and field intensity magnitude are very dependent upon the thickness of the dust cloud blanket for a .7}. given charge density. Both an increasing of charge density and an increase in the thickness of the blanket will increase the potential maximum and magnitude field intensity at the disk surface. Of course this will prove to be important in considerations of very close Spaced leaves. A second im- portant field is that of a cloud sandwiched between two para- llel leaves, as this condition is the rule rather than the exception in the inner regions of the plant. Figure (42) shows the intensity distribution and Figure (44) Shows the potential distribution for this case. The maximum intensity at the surface has been reduced by about twenty percent and now the position of zero force is midway between the disks, and the potential distribution is symmetrical as one would expect. The field forces available in this boundary configuration are very feeble compared to the forces available for the case of the Sphere surrounded by a thick cloud. No doubt this accounts for the inability of the workers to transfer to the field the almost unbelievably good results obtained when a charged cloud is blown around an apple. The thickness of the cloud envelope is in a large measure responsible for this. The general form of the field will remain the same regardless of the thickness of the cloud. A comparison of the forces available for the above cases as well as for the metal Sphere and Spherical cloud will be made with inertial and gravity forces in the next section. INERTIAL FORCES IN A MOVING DUST CLOUD Review of Literature A review of literature has not disclosed a solution to this problem except in some Special cases that do not apply directly to a clear understanding of particle deposition in pesticide application. According to Brooks (5), a development by Langmuir has Shown that the inertial or dynamical catch of aerosol drop— lets is directly related to the Size of the particle and velo- city of the airstream carrying and inversely related to the diameter of the collecting cylinder. His work was done in relation to the collection of water droplets on airplane wings passing through clouds. A method of plotting the trajectories of water droplets as air passes around a wing profile has been developed by Bergrun (3) using numerical integration. This method is valid when the streamlines are known. This latter method deals with free airstream velocities of the order of 300 miles per hour or greater in the region where incompressible fluid calculations are definitely invalid and the streamline and Reynolds numbers are such that Stoke's law is not even a close approximation. The method outlined in the above article -75.. will yield a more accurate solution than the one about to be executed, but requires a familiarity with the mathematics of numerical methods and a considerable amount of time. Discussion of Force According to Newton's second law, a body in motion con- tinues to move in a given direction without diminishing velo- city until acted upon by an external force. The resistance to change of motion in a straight line is known as inertia and the amount of force required to overcome this inertia is the time derivative of the change in linear momentum of a given body. One of the most important of forces operating in the deposition of particles out of the air stream con- veying them is the inertial force. An inertial force is pro- duced on a particle when an air stream is deflected by a sur- face and the linear momentum of the particle carries it to- ward the surface. The particles dealt with in dusting work are usually passed through a 325 mesh screen (44 microns opening) and usually vary in Size from 1/2 micron to 44 microns in their largest transverse crossection diameter. The active ingre- dients in dusts are generally in the range of 1/2 to 20 microns because of the methods of manufacture and or subse- quent treatment. Since a given amount of poison can be Spread over much greater area and many materials are greatly increased -75.. in activity, it is generally believed and has been demonstrated in laboratories that less material is required to control a pest when the material is relatively fine than when coarse. The active particles of interest are for the most part those in the 10 to 20 micron range. That is to say, this range is a compromise between the advantages of larger surface coverage of small particles and the inertia and resistance force advan- tages in deposition of larger particles. Statement of the Problem The problem is to determine the net force that is active in driving a particle toward a surface when the air stream conveying the particle is deflected by the surface. Assumptions Necessary In order for this problem to be solved with any reason- able amount of labor certain assumptions must be made to Simplify the problem. (a) The system is two dimensional. (b) The air is carried around the obstructing surface in streamline flow. (0) The streamlines are parallel within the regions of interest (that is region of particle path). -77... (d) The particles are spheres. (e) The streamline and particle path line are coincident Just before the deflection occurs. (f) That Stoke's law is valid over the range of action considered. All of the above assumptions lead to errors, but these can be kept small providing certain requirements are met as regards the Size of particles, velocity of particle movement with reSpect to air, velocity of air movement past the de- flecting surface and smoothness of the surface. Two Dimensional Problem. The actual streamlines formed around a finite surface are three dimensional and according to Prandtl ([0) are cubic hyberbolas of the form x ya: constant which are rotationally symmetrical about the z-axis as Shown in Figure (45). The pro- Jection of the streamlines on the x-y plane are of the form shown in Figure (45). This type of flow is that that would be expected if a large diameter Jet of non-compressible fluid impinged upon an infinite plane. The mathematics of this type of flow iS somewhat complicated and will not be carried out at this time. In two dimensional flow on the other hand, the streamlines all lie in the x-y plane and the object around which the fluid is flowing is long as compared to its cros— section. When the fluid flows in two dimensions the stream- lines will be as indicated in Figure (46) for impingment of - 78 - fluid against a flat plate. A comparison of the flow for a two dimension plate and a three dimension plate shows a difference in streamlines of quite some importance in particle deposition. However, limited time has dictated that the two dimensional flow be done first and later three dimensional work will be done since there are mathematical complications in the three dimensional form that must be eliminated before successful computation is possible. A great deal can be learned of the order of magnitude of the forces involved from the two dimensional case and the characteristics caused by changing values of mass and velocity of particle and air- stream will be the same for two dimensional and three dimen— sional flow. As a further simplification to get the method an approximation to the path, a circular path is substituted for the hyperbolic path of two dimension flow. This is justi— fiable on the basis that it will afford a great deal of in- formation for a fairly reasonable time expenditure. Streamline Flow. Streamline flow or potential flow means that the fluid does not go into vortices and break away from the obstacle past which it is flowing, Figure (47). Streamlines give a plot of the velocity direction at each point in the field. According to Prandtl (ll) gases may be treated as non-compress- ible as far as streamline form are concerned as long as the -79.. velocity of the gas is kept within reasonable limits. For air at atmospheric pressure and at a temperature of 15° C an error in the form of the streamline reaches 1.0 percent when the velocity reaches 160 ft./sec. Since the density de- creases as velocity increases in the following fashion from Prandtl (II). t (0 a/f (1 - 1/2 2313+ ....) where w = velocity of air and 0.: velocity of sound in the gas, the change in density is introduced by the second term in the above formula. It is a function of the velocity squared and the error at 1 percent iS found by setting the second term to .01. The velocity of sound in air at the above conditions is equal to 1120 ft./sec. so that by setting 1 CI. I w = ).02Cf = 95Wl2 = 160 ft./Sec. Studies of dusting and spray work will lie within the <‘ .01 = J R) l-—’ air velocities where the streamlines will not depart by more than 2 percent Since this will give a maximum velocity of 226 ft./sec. which is approximately 152 miles/hr. Except for those leaves directly in front of a duster nozzle where the velocity may be as high as 125 miles/hr., the velocities will be considerably less and will not exceed 20 miles/hr., except in a very narrow band, according to _ 80 _ measurements by Barnes (2). In a great majority of the plant regions the velocities directed perpendicular to the surfaces will not exceed 10 miles per hour and may be much less at which condition the compressibility of air will not change the form of the stream- lines more than a fraction of 1 percent. Except for the region near the boundary surface, the flow will be streamline and of a form considering the gas as non-compressible. Behind the surfaces there is a dead region formed, Prandtl (II), where the pressure is that of the undisturbed fluid as in Figure (48). Parallel Streamlines. Parallel streamlines are assumed in order for the calcu- lation to be carried out without a step by step numerical in- tegration. The streamlines that are crossed will be considered parallel. If the distance travelled by the particle away from a given streamline is relatively small, this will not intro- duce large errors; however, for very large distances across streamlines this can be large and must always be considered when analyzing the results. Spherical Particles. Spherical particles have been very thoroughly investigated as regards their movement through fluids because of the rela- tively Simple mathematics involved. However, other shaped -81- particles have been experimentally studied and can be related to an equivalent Spherical particle by means of suitable co- efficients. In general, the relationship for spheres holds fairly well (within 1 20%) for irregular particles for values of N of less than 50 where the particle shapes are not extreme and where the diameter as measured by screens, elutriation, microscope, or otherwise is taken as the diameter of an equivalent Sphere.1 The Streamline and Particle Paths are Coincident. The streamline and particle paths are coincident and the velocity of the particle is equal to the velocity of the stream at time Just before disturbance of free flow begins. This is almost strictly true for motion where the air and particles are not being accelerated. Here gravity forces have been considered small or acting perpendicular to the plane of the streamlines. Stokes Law as an Approximation for the Resistance to Motion Through Air. According to Lapple (8') Stokes law is a good approxi- mation in the range where RPiS less than 2, and the velocity does not exceed a certain value for a given particle diameter. The values of RP, the value of the drag coefficient C, and the velocity of relative movement of particle and air are all l Lapple. Fluid and Particle Mechanics. P. 288. - 82 - such that Stokes law E»: 3quFk¢ does not vary over the whole range of values of u encountered in the example from the true 5;: EEQ%EE by more than a few percent. Development of particle path derivation from the equa- tion of the path circular streamlines of a Spherical particle can be evaluated approximately provided the equation of the particle path is known in terms of the time t. According to Lapple and Shepherd (9), for streamline flow the motion is Newtonian and the motion in the ‘x direc- tion may be superposed on the motion in the y direction and the resultant gives the actual particle path. The equation developed in Appendix III gives the path of a Spherical par- ticle of density 2 and diameter 20 microns. This particle approaches a surface perpendicular to the flow as Shown in Figure (46), and the true hyperbolic streamline is approxi- mated by a quarter circle, Figure (49). The set-up is shown in Figure (49), and the path and manner in which the particle departs from it is Shown in Figure (50). The streamline path flowing around a flat plate would have a form as in Figure (47) if the fluid were perfect, that is had no viscosity. In a real fluid with viscosity there is a separation of the boundary layer behind the plate and there is a turbulence in that area with a slightly lower pressure than that of the free stream. In front of the plate, however, the streamlines approach those of the perfect fluid when the plate is smooth. This type of flow is pictured in Figure (48). -83- We are interested in what happens to a particle of dust as it is carried by the streamline as pictured in Figure (48). This part of the streamline is represented by xy = 6, Figure (46), a streamline that at 1 cm. from x-kaxiS is 6 cm. from y axis and further is almost perfectly represented by a circle of radius 5 cm. The density of our particle is taken to be 2, its diameter is 20 microns and it is a Sphere. The velocity of the free stream is taken to be 450 cm./sec. (approx- imately 10 MPH). The streamlines are taken to be parallel as in Figure (49). A better approximation would be concentric circular streamlines, but this would require step by step numerical integration. The parametric equations for the particle paths are equal to the following: ..f-t x(t) = - 6 + l + .964 (5 Sin wt - cos wt) - .036 e '“ gt 6 + .0225 - .964 (5 cos wt + sin wt) - .1925 e’ x(t) Values of x«)and yulare now plotted for values of t and yield the plot of Figure (51). 4'3 A3 lax ' aiflx Fig. 45. Streamline of Fig. 46. Streamlines of 3-dimensional 2-dimensional flow xy‘= C. flow xy = C. )) (is) J . V Fig. 47. Streamlines for Fig. 48. Streamlines for 2—dimensiona1 2-dimensional potential flow viscous flow around a flat around a flat plate. plate. (/ Fig. 49. Relation of.V@ Esed Fig. 50. Manner in which in formula 2 + —1F particle path m deviates from 1.1;. V5,, and actual V; circular stream- that the particle line' would eXperience due to y-component of inertia. -85- MS -----. 6 a: I “I L II 41 II E STREAMLINE .. 5 'é‘ \ I“ K! E -.| _ 4 w I \ ' 32 ' I )4 S q 'l x I _ 2 1:3, 2 PflfT/C'LE a PArH v; - Ics s I: -1 -1 .3 J 1"" X Dar/me: Ala/VG X‘AX/S m 5"Ervr/MEr6R: Fig. 51. 20 micron particle deviation from circular streamline of 5 cm. radius with streamline velocity vs equal to 450 cm/sec and the particle density equal to 2. Discussion of Results A study of Figure (49) will Show that for the circular case calculated in Appendix III, the V3 of the streamline as used in the analysis has a smaller x-component than does the V; of the particular circular streamline on which the particle finds itself due to the y-component of the inertia. Thus the x-component of the deviation from the original circular path would be in the neighborhood of 25 percent larger than the deviation as given. However, a look at Figure (46) will Show that as we approach the surface the streamlines them- selves have a smaller x-component of velocity for a given value of y and thus there is a compensation for the in- creased x-component of velocity of streamline. The calculation based on a circular streamline is thus more valid for the real case than for the circular streamlines. Another factor is that because of varying Reynolds numbers the particles are subjected to greater drags at times than is in- dicated by Stoke's laws. However, assuming the equations of motion are reasonably close, the drag forces do not vary from Stoke's law by more than a few percent at any time. It is the opinion of the writer that the equations of motion are a good approximation to the true deviation of particle from the streamlines and that the order of magnitude deviation in the x-direction for a 20 micron Spherical par- ticle under the conditions previously described, is one centi- meter. -87- The substance of the aforewritten is that the final position of the particle is very nearly correct, but the true path would deviate somewhat from that Shown. This indicates that a particle of these Specifications in the stream one centimeter from the perpendicular axis of a long plate twelve centimeters wide would just be caught by the surface. All particles of this Size within one centimeter of the x-axis Should be caught. All those more than one centimeter from the axis would pass by this surface without striking the surface. This means that at 10 miles per hour velocity only about 17 percent of the particles of 20 microns diameter and den- sity of 2 would strike the surface. The rest would go around the surface. A decrease in mass of the particle such as re- ducing the density would reduce the path deviation from the streamline in linear fashion. Half the mass would give half the deviation. An increase in velocity would increase the path deviation and the percentage of particles caught. Smaller particle Sizes with their increased drag and decreased inertia will take a decreased deviation that will vary approximately as the square the ratio of the Size. That is, a ten micron particle will deviate approximately one-quarter as much as the twenty micron particles. This means only about 4 percent of the particles in the projected area of the surface across the stream would strike the surface. The foregoing discussion would be reasonably valid for those cases in which a collecting surface was moved through -88- an essentially undisturbed suspension of particles, such as for an airplane moving through a cloud or a plate moved through a room full of a suspended aerosol. Brooks (5) has transferred material from the field of aeronautics, as has the author in this paper, and has used it to calculate the deposition efficiency and explain the phenomena that takes place in ordinary dusting and plant pesticide work. It iS believed by this author after considerable reflec- tion and a scrutiny of the laws governing the resistance forces acting, that the resistive forces to the passage of a particle through a fluid is very greatly modified (decreased) because of general turbulence within the fluid carrying the suspended particles. Turbulence within the fluid breaks down the shear forces in such a manner that the particle acts as though it were in a medium of lower viscosity. This does not necessarily invalidate all of the previous work, as the general streamline form and concept of the deposition method is yet the same, but the apparent viscosity of the fluid medium has been reduced in proportion to the amount of general turbulence present. According to Dallavalle's (b) dimensional analysis of a real fluid without a free boundary, we find a dimensionless constant made up of the ratio of inertial forces/frictional forces equal %V.z/4I!J.¥; :9?“ = R, the very familiar Reynolds number. Where/0 = the density of the fluid, L is a linear dimension, V is velocity, andgx is the coefficient of -89- viscosity. With a constant density of the fluid.f’for a given (LV) which is a function of the velocity of flow, Shape and size of vessel in contact with it the Reynolds number reflects the effect of viscosity and R xi}? where k is a constant = (LV). In studies of particle dynamics we have two distinct criteria for the Reynolds number R involved. One criteria is the R,p(Reynolds number of a particle moving through a viscous fluid) and the other is the R..(Reynolds number or index indicating apparent viscosity of the fluid medium of the particle). The Reynolds number reflects the amount of destruction of Shear forces in a fluid and thus effectively reduces the frictional forces in a fluid. At very high Reynolds numbers we have approximately the perfect fluid used in potential flow theory in which the viscosity is considered zero. When we consider the resistance to motion of a particle to passage through a fluid medium, we find there are three distinct types of resistance to motion. They are, streamline, intermediate and turbulent. All three types of resistance can be represented by one equation 2. FA: wt, 0 M78 [ A] where FA: total drag force acting on body, dynes _ 90 - diameter of Sphere, centimeters U I = overall drag coefficient, dimensionless density of dispension medium or fluid, grams/cubic " O H centimeter u = relative velocity between main body of fluid and particle or body, centimeter/second It: viscosity of dispersion medium or fluid, poises The drag coefficient for the streamline or Stoke's law region is given by cs: 2i . RP R for the Sphere = LVE= D u/’ F l‘ LA!- This reduces equation [A] to 5;: 3wDrKu, the familiar Stoke's equation. For all kinds of motion, an increase in R? causes a de- crease in drag coefficient. However, in a quiescent viscous fluid the RF is a function of Dpulkh, that is, the diameter of the particle, relative velocity of medium and particle, density of fluid medium and viscosity of fluid medium. Dallavalle (6) gives the ranges of the three types of motion and the correSponding range of the drag coefficient C. (a) streamline motion 10-411425 0‘: %% .439 (b) intermediate motion 2 (R (500, CL: RF 5 (c) turbulent motion 500 (R 410 , Cc: 0.44 Lapple (8') gives slightly different ranges. (a) streamline motion R less than . to R = 2, Cs: %5 - 91 _ .6 (b) intermediate motion 2 41141000, 0.0: 18.5/ R‘. (c) turbulent motion 10004R<2 x 10‘, Ct: .1411 Referring back to equation [A] we see that the force E. is a minimum for turbulent flow at RP)than 1000. We also note that at an Rf of 1000 the empirical value of Ce is .46 and the coefficient for streamline flow would be Cs: Tg00:= .024 which would yield a force greatly less from equation [A] an: wD;CP‘}"8, than does the turbulent flow coefficient q. The motion will always be turbulent when this apparent in- crease of efficiency due to a reduction of value of the co- efficient of drag C, because of the manner in which C, a function of RF is obtained. R1,being Dpufflh, it can be made large and thus 0 small by increasing particle size, rela- tive velocity, density of fluid, or by reducing the viscosity. In reducing the drag coefficient C in any manner except by reducing viscosity we see from equation [A] that we greatly increase the drag force F,. However, another concept may now be brought out that has not been emphasized formerly and is very important in understanding the resistive force of a particle as it moves through a fluid medium. This is brought out by Dallavalle (6) as follows: An interesting Situation arises with reference to the application of particle dynamics to fluids and may be stated as follows: Suppose a fluid moves with a velocity V in a duct; the motion of this fluid is de- termined in accordance with the variables stated in equation (2 - 4); now inject into this moving fluid a particle of diameter d having an initial velocity VF. What is the motion of the particle? _ 92 - We observe that Reynolds criteria for pipes which distinguishes turbulent from streamline flow has a value of 2500, where as the value of the criterion for particles is about 1.0, as may be seen in Figure 3. But we must not be led astray by such criteria. Tur- bulent motion of a particle in a still fluid is a localized condition, while turbulence in a fluid in motion is general. AS we have Shown, turbulence de- notes a destruction of parallel Shearing elements in the fluid so that motion becomes a function of fluid density only. Hence, a particle injected into a tur- bulent fluid behaves as though its Reynolds number, R , is greater than 1.0 even though its true value may actually be less. In general we may summarize the above in two state— ments as follows: (1) If a fluid is in streamline motion, the motion of a particle injected into it may be stream— line or turbulent, depending on whether the relative velocity between the fluid and the particle is greater or less than 1.0 thus....... ....................... (2) If the fluid is in turbulent motion, the motion of a particle injected into it will be turbulent, regard- less of the relative velocity between the particle and the fluid. Discussion of Turbulence in Field Dusting. Although the degree of turbulence, and thus breakdown in shear forces within the cloud produced by a conventional duster is not at this time known by the author, it is believed that at least under certain conditions it would be consider- able. The use of turbulence as a means of decreasing the apparent viscosity of the fluid medium and thus increasing the dynamical catch has not been explicitly stated as an ob- Jective in previous literature to the author's knowledge. Brooks (5) indicates that an increase in dynamic catch would be expected by forced turbulence in the foliage. The general l Dallavalle. Micrometrics. P. 39. -93- concept (not necessarily that of Brooks) has been that forced turbulence in the foliage would merely cause more particles to be brought to close proximity of the plant surfaces and thus the inertia would have more chance to carry particles to the surfaces. Turbulence in the sense above considered need not be as microscopic in size of eddies and need not con- tain the energy of the cloud as would be required to substan- tially reduce the apparent viscosity of the air. When the boiling in the air dies down the air returns to its former apparent viscosity and will again support particle transport as it would before energizing to turbulence. The amount of turbulence that is present in dusting practice is probably very low compared to that which could be generated with a tractor mounted duster and proper design of equipment to create this turbulence. Possibly an increase of efficiency could be gained by lower discharge velocities with a higher energy content in the air in the form of turbulence rather than velocity, so that the dust could remain within the plant region for a longer period of time. At present the dust enters the plant region and then leaves again in a matter of seconds. This type of apparent viscosity lowering could greatly enhance the electrical deposition capacities of a dust cloud if it turns out to be mechanically feasible to create this condition. It is believed that the author's treatment of particle deposition as related in the early part of the test will -911... serve to increase the understanding of the problem as pre- Sented. However, the particle path as calculated would appear to be incorrect by whatever influence the apparent viscosity decrease and the turbulent motion would have upon the path. In all probability there would be a considerable increase in the deviation of the particle from the streamline and thus a greater percentage of inertial catch than was in- dicated. The author does not believe that the introduction of a general high level of microsc0pic turbulence would invalidate the potential flow concept around the collecting surface but would on the contrary enhance it for the action of turbulence is to, in effect, lower the viscosity and thus make more nearly perfect the potential flow concept. COMPARISON OF ELECTRIC, INERTIAL, AND GRAVITY FIELDS In any electric field, static or dynamic, the force on the charge q is the vectorial summation F = qE qunB where F = force on a particle q = charge on particle E = field intensity V3: velocity normal to direction of magnetic flux B = magnetic flux density. Although the fields cannot be considered static in any sense, the force due to the dynamic part of the above equa- tion is almost entirely negligible because of the relatively small velocities and small flux densities. Thus the elec- tric forces available in a charged dust cloud almost entirely originate from the static part of the equation and are equal approximately to F = qE. The inertial forces available are due altogether to the dynamic phase of the moving dust cloud and are given by the equation F = ma. The gravity forces are given by F = mg where g = acceleration of gravity m = mass gms W = mg = F in dynes _ 95 - The electric forces available for a given geometry are entirely dependent on boundary potentials, charge density and charge level on the particle. The boundary conditions have already been prescribed, but we have yet to consider what charge density can be obtained easily with a field type machine and at what place the charge denSity (always changing) is to be considered. We are interested in the charge density that is available within the region of the plant surfaces. The question of what density of charge is available is not easily answered because heretofore no instrument has been available for indicating the charge density that is made avail— able by a field machine. The charge level on the particle can be calculated for given particle sizes and given charging nozzles when using the ionized field method of charging par- ticles. Estimation of Charge Density 15 lb/acre on 42 inch rows. 2 “3,26% gg/acre 12,500 ft, of row/acre. For cotton must fill a volume of foliage 2 x 3 x 1 ft. = 6 cu. ft./ft. of row. 75,000 cu. ft./acre. . 15 lb. x 454 gm/lb- Dust concentration in air 75,000 cu. ft. Total volume = 6 x 12,500 .1 9.06 x 10 ng/cu. ft. -97.. -4 = 5.25 x 10 gms/cu. in. -6 = 3.2 x 10 ng/cu. cm. Consider average particle at 15 microns diameter and density _ 3 5 = 2. There would be g-wrij: %1T(7.5 x 104) x 2 —‘i = 3.54 x 10 ng/particle -5 3.2 x 10 gms/cu. cm. 3.54 x lO”‘ng/particle The number of particles per cu. cm. = = 904 particles/cu. cm. Using the ionized field method for charging, the charge level per particle in a field machine will be approximately 2 3E,rz where E, is the charging field in the nozzle, §.is 3 the percentage of maximum charge that can be placed on the particle during the time it is in the charging field, the 3 is the a term that takes into account the material of the dust particle, and r is the radius of the dust particle. Charge per particle with E0: 7 stat volts/cm. All of the above conditions are attainable without difficulty in the field. ‘ —6 Q.= §-3E,rL= 2 x 7 x (7.5 x 104) = 7.87 x 10 esu/particle Total charge/cu. cm. of air==d02= 904 particles x 7.87 x 10‘6 esu/particle I ..3 40 = 7.1 x 10 esu/cu. cm. Force on a Metal Sphere Within a Cloud for Case 111 We calculate the depressing voltage by ~98- O=2/3 Ww'(bL ‘aL)--:—Y+E‘oooooooocp=a .3 O = 2.1 x 7.1 x 10 (45L- 1.1;) - V(l - g) -22-2 __ V'— 1 _ .025 — 31 stat volts. I L L av a Ufor grounded sphere: 2/3 7ND(b -/0) -?J—+‘3V 9U_ 4 . aV °U_ e.._i '3 2.1__ _ W- a_ 31r(7.1x10)1.1+ .1../J—a ..L -E= -3.3x10 +28.2.... ...... ......fl=a 28.2 stat volts/cm. I m n 28.2 stat volts/cm. \ m \ II For a 15 micron diameter particle the force is -6 —6 F = /E/ q = 28.2 x 7.87 x 10 = 222 x 10 dynes. Force on a Cloud in Finite Cloud Next to Disk Force on a particle of a uniform cloud of dust of dimen- sions 4 inches height and 4 inches diameter next to a con- ducting disk of 4 inches diameter and at zero potential. Using 19’1938 =9U , 7.1 x 10.3x 1.938 x 103 = 13.75 units/cu. in. Using maximum intensity as found in previous section as 19.149 E = 19.119 = 19.1 x 13.75 = 263 volts/inch 263 volts/inch Convertin to esu units this is g 300 volts/Stat volt x 2.54 cm/in. _ 99 - [E(= %g%-= .344 stat volts/om. Force on a 15 micron particle at surface of disk is 6 . -6 F = /E/q = .344 x 7.87 x 10 = 2.71 x 10 dynes. The force is toward the disk. Force on Particle in Cloud Between Two Disks Calculations for force on a particle along the axis and at the surface of a 4 inch conducting disk when the cloud is sandwiched between two disks as in Figure (22). Using the same charge density as in the previous cases A9 = 13.75 units/cu. inch. The intensity /E/ is given as 15.82.é9 /E/ = 15.82 x 13.75 = 217 volts/inch = .285 stat volts/cm. F = /E/ Q,= .285 x 7.87 x 10'6= 2.24 x 10'6dynes The force is toward the disk. Inertial Force on Particle in Deflected Airstream Calculation of inertial force on a 20 micron particle of density (5 = 2. From Appendix III 4t X“): x, + .964 (5 Sin wt - cos wt) - .036 e‘fi +( f = .904 (5 w cos wt + w Sin wt) + % (.030 e ) Q10: cr>< - 100 - %%‘= .964 (- 5 w’sin wt-v w‘cos wt) — %:(.036 e-£3 2 -n - -4 = .964 (- 5wt) - %8:4§2xxlégax 3.6 x lO'R 3.8 x 10 8 %%t= - 39,000 - 2 §g%1= acceleration The actual force of inertia of a particle is in the opposite direction to the force that causes the particle to decelerate. Maximum force of a 20 micron diameter particle having a density of 2 and carried by a circular streamline of 5 cm. radius with streamline velocity of 450 cm/sec. is -7 a -6 F = ma = 8.4 x 10 x 39 x 10 = 327 x 10 dynes. Gravity Forces on Particle Gravity force is given by F = mg, and for a 15 micron particle of density J = 2 is cm 4 a -9 F: 3-7r13rg-_- 3.54;: 10 gms x 980 sec. ‘6 F = 3.48 x 10 dynes 4).. O 101 .E mmmE so maofipsma co apfi>msm mo mosom n oosou hua>mpo .Ammv magmas as we oomMMSm knew as mxwflp 03p Cmmzpop psofio m CH oaofiusma co oopom u o mosow oaspoofim .xmfip esp or use: use Afimv ohswfim CH mm xmwp a mo mass QQHSOHpcoanQ map mcoHs oopom c m moaom oaspoofim .mpmop pcoEmsSmmmE esp mo HHH omwo .psoHo pmsp HmOHpmgdw m cfinpflz msonmm HHmEm a mo moweLSm pm mopom u < mosom oaspomfim .opm msflpms .Eo m m ome mocfiasmoapm esu page on pmpomfimep pcm oom\Eo om: um mcfi>oE Ewmspmafim cm Ca pew m u w mpfimcmp mo oaoflpsmm s so mopom HmeLmCH mQQmHm wBH>N mqmdfi mH.o mm.o m.o mm H.m .a. ncoaoas m i mo.H mm.o om.a mm a: .a. nsoaoas ca m:.m am.m as.m mmm me .u. eschews ma mm.w mm.m m®.: mom Smm mCOLOHE ON 0 m < fibH x moshp «rod x macaw o-oH x mocmp n-0H x mochp ang x mochp efioflpwmm mosom ooaom mosom moaom oosom .Emfla hpfi>wsw cappomam ofiapomfim ofispoodm mesocH mo mmomom Mme mo ZOmHmdmzoo 1.1a - 102 - Discussion of Results The comparison of forces in the chart of Rable XVI was one of the main goals of this thesis. The quantifying of forces was done to enable the research workers to clear up any misconceptions they may have had, to reveal the ways the forces might be varied and to better understand the mechanism of particle deposition in general. This has not been com— pletely accomplished, but it is felt by the author that con- siderable light has been brought to bear on the problem. Considering the inertial forces, it is apparent that when an air stream containing dust is directed perpendicular to a surface with sufficient velocity, it is a most powerful force for the deposition of particles of dust. When the dust stream is not directed perpendicular to the surface, the radius r of Appendix III becomes larger and w falls off very rapidly thus greatly lowering the depositing power of the air stream. The electric field forces of Case III of text using” = 7.1 x 10—3yields values comparable in magni- tude to the inertial forces. The results shown here are the author's best estimate of charge density available in a field duster being perhaps a little on the conservative side when using 15 microns as the average diameter to determine number of particles and in- directly charge density. If 7.5 had been used as the average diameter, the charge density would have been twice what it is, all other factors remaining constant. The electric force _ 103 _ available for deposition on a leaf considering a uniform charge density¢£9= 7.1 x 10'3esu/cc yields values that may be considered small when compared to even gravity forces. How then can we account for an increased deposition when using the charged versus uncharged even in those regions that have small leaf spacings that would not allow a sufficient sized cloud to form even the minute force Shown in the chart? The answer is not apparent to the writer, but it is to be expected that the charge densities arrange themselves in such fashion that the potentials are nearly constant within the spacings between the leaves. This means that all of the charged dust circulates as close to the boundaries as is possible consis- tent with the viscosity of the air and thus creates a higher density and larger forces near the surfaces than is indicated by a consideration of a uniform charge density. There is, of course, a possibility that the charge densityé9is unrealis- tically low, but it would take several times the densities believed reasonable to make the forces highly important. It may be possible that electric forces in conjunction with gravity and inertial forces make the particles adhere better when the foliage is dry due to polarization of waxes on the plant surfaces than would uncharged particles. There is no question but that the electric forces are very significant on the outer surfaces when the dust cloud blanket is of considerable thickness. In such crops as onions, the small radii of the stalks and relatively open growth -104- allows substantial electrical forces to develop. There are at least three ways by which electrical forces can be enhanced to the advantage of dust deposition. They are (l) dense clouds of dust can be blown into the plant re- gion and then recovered after a certain plant exposure, (2) other methods of charging may be used to obtain a higher charge level on each particle such as by contact charging, and (3) a very high voltage machine producing an ionized stream of air in all the plant region can be used to raise the charge density of the inner plant regions while at the same time charging the particles. The first of the above methods would require a hood and recovery system. The other two methods would require Special equipment to produce their higher charge densities. It is always desirable to obtain the highest level possible on each particle since a lesser number of high level particles can give as high an electric field as a large number of low level charged particles. However, the relative movement of any single particle is improved by the condition of high level of charge on the individual particles. Thermal repulsion, the force on a particle in a tempera— ture gradient may be Significant for very small particles when the leaf temperature is considerably higher than the surrounding air. This force acts to prevent deposition when the leaf is hot and aids deposition when the leaf is colder than surrounding air. APPENDIX I Development of the Potential Equation of a Grounded Conducting Sphere in a Charged Dust Cloud by Method of Superposition The method of solution is to calculate the potential throughout the charged spherical cloud then superpose a spherical cloud of smaller Size with a negative charge. This will give a hole with no charge in it. The next step is to insert in this hollow cloud a metal sphere of same radius as hole, carrying the charge just necessary to lower the poten- tial to zero at the boundary of the sphere. This charge happens to be equivalent to the potential of the hole before the Sphere was inserted times the capacitance of the Sphere. Potential of a Homogeneous Cloud of Charged Dust of Radius b and charge densityAD =19. Boundarygand other conditions: b = radius of cloud €==€, é_= dielectric constant of medium. ' a. J9 = 00, ”49 is charge density of cloud... .. ....... P4 b 4L9: 0 ........................ . .................. 10> b U£= Potential inside of cloud ................... 0‘ b U5: Potential outside of cloud.. ................ f’2 b -106- UL: U0 . ....... . ...... . .............. . ...... ”to: b DU .911. é'DF: €155“ .................... . ...... ....... (9: b Lim U0: 0 ,o 4.0 Lim PU = E ’0.~?«D 1 d ZDU l7U = [-33 a" (,0 573) = — 4mg ......... Poisson's Equation V‘U = *1/3z g]; (pig—g) = 0 .......... .....Laplace's Equation Math derivation: By starting with Poisson's and La- place's Equations we develop the potential within a homo- geneous dust cloud of charge densityOLQO . ‘ ..___ _1 <1- 130; = _ VU‘ d0 (Po—F) W82 7 ’- U; 4 3 pg‘P“=-37TOLQP+C‘ 3U' 4 c 95"‘314’0’D1‘fi (1) C H 1 04m) :1 .15 “b ' I. E}? + O p K? -107- V‘U.= %— 1,- (Hg?) - 9:, (Hg—{3}) 0 P3? = Cs gig. = g“; (3) U0: - %§. + C, (4) From Lim' U0: 0, 04: O fl—aw From LimFUoz E, 0,: - E fl—>°° : E = total charge = Vol x charge density = :— TT’ b We a _ 4 b U0- 1362/9- .................... fie b (48.) DU 4 .19 b3 -3: “...:J'" 0.. a DP 3P" .................. r) b (3a) ‘1 0U« = we ............. = b From ("DP evbp f C, 1477'”de - 108 - From U7: U0 ............. . ..... F== b 2 4 a: - 3Ww.b + CL: “5'71"”.b L Cs: 27rk9,b 2 L A p 4 UL: - 3- vp,f’ + 27rawob ........ 0 - - b (28) DU; _ _ 4 4 4 ’DP _ 3”“? .......... . ...0-f"—b (la) Potential Equation of Homogeneous Cloud of Charged Dust of Radius a and Charge density”: -U.. From above. U,= grow, PL- QTYUJ a7: ......... fl-é a (5) it}; 2 ngap ................ p9- a (6) {-41:38 . . ... ..Pea (7) $1; Aggogi ................. p9 a <8) Potential of Hollow Cloud. The hollow cloud is made up of sum of cloud of radius a and $9 = -49° , and cloud of radius b anquD=u€9o where a < b. By addition of equations (2a) and (5) - 109 - U=+21r49.(bL-a)... ............ Pia (9) By addition of equation (2a) and (7) Q. 3 U=~§7TJ30F+ 2vwobt-9—%§~...aétoéb (10) By addition of (4a) and (7) 3 __ uTTUo(b ' a) 5 U— 3,, ........... ,a b (11) Discussion of System of Grounded Conducting Sphere Inside of Hollow Charged Cloud. The potential equation of this system is developed by adding the potential of a charged conducting Sphere of radius a to the potential of hollow cloud. The negative of (9) must be put on the conducting sphere to depress the potential of the hole to zero. The charge Q to produce this negative potential is found by the relation 0,: VC, where Q is the charge, V is the potential and C a is the capacitance of the conducting sphere*. There- fore, the charge to depress the hole to zero is given by 0:: - 21rJ-9.(b2’ - a‘)a. \I The Potential of the Sphere with Charge Q. Q: - 2vw,(b" - a‘)a * The capacitance of a Sphere is equal to its radius in un- rationalized units. - 110 - U : §~= %=-27T&0.(b - a‘) ..... . ....pé a (12) U : Q. " ”‘32”; " a‘)a .............. 9- a (13) P /“ Potential Equation of Hollow Cloud Containing Grounded Sphere. By adding equations (9) and (12) U: 0 ................................... fléa (14) Adding (10) and (13) U: -§-1ri9,Pt+ ammo" W3 -27r#’.(bz -a1)a ............... ...aéPsb (15) /J By adding (11) and (13) U ___ 47r¢0c(bi- aa) _ 2'rn“’u(be - az)a””fl3 b (16) 3” /0 Field Intensity E in Charged Cloud with Grounded Conducting Sphere Inside. The field intensity is given as the negative of 63%.. 311 _ L - E - __. — O ........................... - 1 'DP P a ( 7) _ W _ 211 = - Ll'TTnf 1417'”. a: “‘ " an 3 I 5 10'- s L + TIME/’0 J a )3 ........... a e P e b (18) DU -4/a.(b3-a3) 2fifl‘-aa) _ E = __ = Tr I. + IT 0 a P2 b (19) '3? 3 F p” APPENDIX II Development of the Potential Equation of a Grounded Conducting Sphere in a Charged Dust Cloud by Method of Inversion in a Sphere By use of a Kelvin transformation or inversion in a sphere the surface formed by r = a can be made at zero po- tential, so that the surface charge on the conducting Sphere is replaced by a volume distribution within the conducting 2 Sphere. The inner boundary will be at c = a/b where a is the sphere radius and b is the radius of the dust cloud. Boundary and Other Conditions. U1: U2, 6.01]. /<)P = QDUz/JP” ............ .P = ayb U2: U3: EQUz/DP = {PIE/bf? ............... F = a U3: U4: 5,3U3/9P = “DUN/JP ............... [3: b 6,: 6,: 6,: 6‘ , where 6 = dielectric constants of mediums $2: 0, whereéf= charge density of region 6‘ - pa: " oa/F" 4%: 49° 19,: 4 a l . . . . ._. I. . e».- I - i .5 , I ' . . . m . . e . _ . ‘ . u - Limit U4: 0 p—r 00 Limit FUq-‘f— Q [0"90 0 VtU =/511 $7; ([3171]) = - 47129, Poisson's equation Mathematical Derivation of Potential Equations. U1: (C3, + Ca. from (1) U,= constant,.'. 0,: 0 U]: 02 9.2.- 0 DP V‘Uz=plx_ 37- ([3; 2%) = - 4 a9; ”“9335: from (1) dP p3 path- 2W9..a5. C op ' pi + 3 0.11 ... - siege” 9: a p7 P‘ -ll2- (l) (3) U4: g7+ 03 from integration of (1) Ce: 0, from consideration of lim U4: 0 Paco 07: Q, from consideration of lim PU4= Q fl—wo C7 Q U = _.= _. ‘1 P {3 92+- -2 W’ F" 9U DU: GIW'L. 6‘5; oooooooooooooooooooooooo oooop=a7b 2 b4 a 0 = - Jig—’3,— + C2? from above condition C3: 271'40.bza Q = total charge = vol times charge densityaw a A 2 = c 2 Q [HTML/o dP]o:_p°§__‘ + [41ra9f/3 dP]‘O=‘U. /’ a “a; 62 Q 2:: - 47rjo°f _‘d{’ + 47T «90f? dP s /’ . 3 Q = % moaaa+ grab - 21rU.ab" DU: 'DUq (3:97 - 6,75 ......... . .................. P: b - 47r§tb - %f- g, from above condition 3 05: Q — %. «mat : 3. was - 271-».ab‘ U = U )0: b - 113 - -114- 2 " 4/3 10b! - 3— Troq,b + Q ‘ b 7r ' + 0‘: g. , from above condition 2 C6: 27170,,b 3 U’= U10. ..... O ............... O. O. 0 Op: a/b C - - 4 Tram 103+ C from above e uation (10) 2” 3 'a‘ 4 ’ ‘1 U1: U3 ................................. .p= a 2 I. 1. =5 7r)~0,a — 27mm + 04: 0, from above condition (11) Solving (10) and (11) simultaneously u 3 CZ: - 3' TTd-ZE + Cq (10) 0 = gvma’+ 0,- 2mm)" (11) c—uwbse ‘2b‘ (1) L_-3-7r.g-3—1r12a+ FM. 2 By substitution of (12) into (10) L C = 27rrk9ub - g- raga" q 3 2 mP" . 3 . l. U3: - TT3 _ + 275up0a _ 2v; ab + 2'”sz (6a) 91.13: _ 413,8 27749.a3+ 2ww.ab‘ 3'0 3 3/" ft (78.) _. 9. _ 2790031 “Trombj_ 21rgg.ab" U _ ,0 — 3 {J + 3r: [2 (8b) _ 115 - — = _ Q 2ma3 Awab’ emboabz' DP 9‘ 5P1 ”3F7'+ “7;?"‘ (9a) Summary of Potential Equations in Inside and Outside of Charged Cloud. U = O by definition in a grounded conductor.../?5 a (13) U — - 2 2+ 2w¢9bl- EE§£§3 3" 3‘ ”W ' 3P 2. 1 - 2”“%(b ’ a )a inside of cloud ..... a é4P é'b (6a) 3 /O a L 2r U = “WM” " a ) - 2"”0”(b ' a )8 outside of cloudft‘: b (8b) 4 3 F p Summary of Field Intensity Eguations for Grounded Sphere Inside of Charged Dust Cloud. Q) - E = 3.3: 0 within conductor ..... . ..... f’9 a __ E: 2.9.3— M+ W3 DP 3 BP‘ 2, 2— + 2wxm(b/o: a )a inside of cloud...a él’é b (7a) - 3 3 z 1 _ E = 224: _ LLTrADO(b - a ) + 2W49o(b - a )a .../,2 b (98.) - 116 _ CHM‘W‘I” ‘ SPHams 1g. 52. Cross-section of conducting Sphere and cloud showing distribution of charge density as uniform in cloud and non-uniform as imaged into conducting sphere. APPENDIX III An Analysis of Particle Deviation from a Circular Streamline A first approximation to the path of a particle of mass m, having an air drag resistance ku (u 2 relative velocity particle and air), and subjected to the inertial forces pro- duced on a particle when the particle attempts to follow the streamlines of a deflected air stream may be obtained from the following derivation. A diagram of a circular streamline and relation of co-ordinate system to the streamline is shown in the following figure. Fig. 53. Co—ordinate System for development of equation on particle deposition by inertial force. - 118 - Assumptions and Other Conditions._ The circular streamline equation from the preceding figure is given by x = - C + r sin wt + c - r cos wt y The y-axis represents a surface which has deflected a stream- line of an impinging airstream into a circular form. The air- stream before deflection was directed parallel to the x-axis at a velocity V. Assume that for a given starting position (X0, yg) the particle path does not deviate from the stream— line circle enough to appreciably change the velocity of the airstream and that the latter velocity is constant, say'%. It is assumed that at the starting position (12,, yo) both the streamline and the particle have the same velocity and direc- tion. Let z = a vector giving the position of a particle P, and V5: the velocity of the air along a streamline at (X, y). M2 = k (\{g— z) = Drag = k (wind vel. - part. vel.) k ~ k Z+fiz=ffivs (l) The X and y components are given as 00 o k x+§x=fiwx (a 0. LC 0 — E y + fi-y - n1v33 (3) -119- g._(-c+rsinwt)=rwcoswt %€(+C-rcoswt)=rwsinwt Initial conditions: 0, X: X. (4) o, v”: X = v0 (5) 0, y= y. (6) 0, V}: 3} = o (7) Mathematical Derivation of Particle Movement in X-direction. k' k fix=+n7v9x EX=+Ewr cos wt m m . f t X e2== + KHEmjfeé cos wt m . at , at X e = 1913] e cos wt m . at k at - ..‘i2 8 .7 o . X e — m TWTII—EY) (a cos Lt + w Sin wt) + A ' kw -ai X = r 1 (a cos wt + w sin wt) + Ae (8) m(w‘ + a ) kwr a l 'M :: ~ —— . - - -- \ O X m(wi¥+ a‘) (w Sin wt cos wt) a Ae + 3 (,) - 120 - t=O,X=X i=v. (4)and(5) o, V: kWI' . ° m(w‘+ a‘) (a) + A kwra A V" - m(w'I + a‘) lcwr V0 kwr X0=W(Il) -a-+m(w‘+a‘)+B __ kwr v._ kwr B _ X°+ m(w“ + at) a m0? + a") V B = X°+ 'a—o t __ k'WI’ a . Va ’6» XCU— mTw"+ a‘) (w Sin wt cos wt) a e 0a + kwr e‘a’: X + y! (/ ) m(’vq1. + at)’ 0 Mathematical Derivation of Particle Movement in y-direction. .o k a _ E .. k . k _ = -_ w sin wt y + m y m I’ . gt t y e = 5%?! e7,6 sin wt dt ' “t urw [ ‘t( i t t) p (10) e:n easnw-wcosw +' y m(w‘ + a‘) 5 _ mi s. - l '“ y .. mfw‘ + a‘) ( - w cos wt - Sin wt) - E Ae + B (11) - 121 - t=O,y=O (7) _ kwr _ O‘Nw"+aT)( W)+A A = krwz m(W‘ + a‘) att: O,,,..y=yo (6) y = krw (_ é) - l . krwz z _ kra ‘ krw B — 370+ m(‘Ni + at)’ + am(Wt + a...) yw= y - krw ( E cos wt + sin wt) - .1. . km: ‘ 6;“ a m(w"+ 6‘” W a flw + a ) W + 3.) amth + at; (11.) Evaluation of Constant for a Specific Particle. V 450 cm/sec = velocity of particle and streamline at t = O r = 5 cm = radius of streamline arc W 90 radians/sec -? m = 8.u X 10 gm = mass of particle -6 k = 3.78 X 10 gm/sec = drag constant = 6muR -3 R = 10 cm = particle radius —6 ;(::200 X 10 poises = viscosity of air 2 a = %== 4.5 X 10 L 4 g) = 20.2 x 10 at: ( Fwd - 122 - -6 z kwr _3.78x10x.90x10x5 _ m(W' + a‘) _ 8.4 X 104(.81 X lO‘+ 20.2 X 10?) I '96“ cm 3r: &%9-= 5 (dimensionless) wr _ 450 _ ”5- [FE—5_ 1 cm. szr _ 3.78 X 10-; .81 X qux 5 _ 102 am(w‘+ a‘f " 450 x 8.4 x 10"): 21 x 10“ ‘ - 2 5 0m- k 450 103 3 78 10" 5 a r _ . X X . X X _ m(W‘ + at) _ 17.63 X 10'45 — 4.83 cm. Evaluation of Components of the Particle Path. The X and y components of the particle path may be found by substituting the constants of the preceding section into the formula for Xh)and yen This is done on the follow- ing two pages for four time increments, each corresponding to 22-1/2 degrees of the circular arc. The particle path is found by plotting the X and y components of the path. - 123 0H.0 u u H + smH0000. u Hm.: + m u n x H + Ammooc.v one. . Amy :00. + 0 - n x «pom m n p3 qoom msHo. n u so I new 900 "coprHSOHwo oHcEwm A, _ i _ 1!! H0. o.H .. .910 .. 0.0 - mmooo. usso 80.0 0004 _ ooo mic. . m ms. mm.H u no.0 u 0.m u smoo. u SYo mwm.0 :m©.0 owso HMHO. om. tin - saga - oo - So. ".....-o soso soso om: osmoo. so. 00.: .. 8.: - 0.0 - omH. netwo smmd mmmd ommm Race. 0 0.0 n 0.0 u 0.0 : 000.H "so H 000.0 0 0 .MMHU oHokHo Apvx u:mx : m 03 p3 as p [I— mo to: woo ch 3px _ H + :.o mmo. I A»: moo I p3 ch mv :00. + ox Hpvx vd meqm mquemdm m0 Bzmzomzooux HH>X MQMQB - 124 - mod u some. - mmmop u a mSprLm n as .msHo. u 0. 6 n o ”COHpmHSono oHcEwm :0. 00.0 mood H mmooo. u also 0006 ooo.H ooo. 3.8. :w. 00.: ems H smoo. .... ...?o 5m. smm. omso RS. so. sa.m ms.a a oao. u .swo sos. sos. om: mswoo. mm. mm; 91H H mmH. u En smm. man. ommm Reoo. 0 H H H H H 0 0 0 ..HMHU oHohHo Apt“ .H .. 0 Hon». ... o B... us as o. l mo 3% v4- moo ch o......o mmmH. - 32 can + oz moo 3 ago. - mmmo. + u “3K Mm megm MHQHBqu m0 ezmzomzoouh H H 52% mag ..H. APPENDIX IV Measuring Instruments for Charged Cloud Measurements The instruments used for electrostatic potential measure- ments are of two types. The gold leaf electroscope and those depending on the force developed between two charged plates. Both of these types have a variable capacitance, the electro- sc0pe having between 5 and 5 cm. and the other types having capacities depending on the design of the instrument. Elec- trostatic voltmeters do not require continuous current to measure. However, some considerable charge is required to bring these instruments up to potential and this is at the expense of the system being measured. Therefore, care must be exercised in taking results when small systems are being measured. The leads and their position with respect to the surrounding objects and walls of the room effect the capaci- tance of the total measuring system. The ordinary Cenco physics laboratory electroscope "A“ of Figure (5?) is an inexpensive and reliable instrument when a range of 50 to 500 volts is to be measured. They are difficult to use for anything but a reference meter. Cer- tain precautions should be taken if reasonably accurate - 126 - results are to be obtained. There is usually about ten to fifteen volts difference in the voltage of a given position of the tip of the gold leaf depending on whether the position is approached from above or below. This is largely due to the bending stress in the gold leaf itself. For more accur- ate results one Should approach a point from the same side as was used for calibrating the meter. The meters are easily calibrated using known sources of d.c. voltage. They read equally well positive or negative electricity and can be used to determine sign of charge by comparing deflection from un- known sign with that of a known sign, such as a comb which is negative after combing through ones hair. A meter that may be read fairly well from 1000 to 8500 volts "02 of Figure (54), can easily be constructed from an electros00pe, by replacing the gold leaf and suspension with an aluminum foil leaf of double thickness suspended from a pivot shaft and cup bearings from a Spring-wound alarm clock. The use of electrostatic voltmeters in charged cloud measure- ment requires probes and leads. The probes are best made by using a piece of copper or brass that has been coated with an alpha emmitting radioactive salt. Polonium nitrate is relatively easy to obtain and does a very effective job. The leads are easily made up by stringing a number 56 or 40 enameled COpper wire inside a length of spaghetti tubing obtained from a radio shop. The smaller the wire the less capacitance the lead will have. The spaghetti tubing should - 127 _ have a voltage breakdown rating of at least 8000 volts for use on the high range instrument. The purpose of the radioactive salt is to emit alpha particles which ionize the air sufficiently in the vicinity of the probe to supply whatever sign of charge is needed to raise the probe and leaf system to the potential of the posi- tion of the probe. The charge of sign that does not go on the probe is driven from the probe and tends to lower the po- tential of the system being measured by whatever amount of opposite sign of ion is used in raising the potential of the probe and instrument movement. Once the probe has reached the potential of the surround- ings the equal number of positive and negative ions formed by action of the radioactive salt will not effect the potential of the system as long as they stay mixed throughout the system in equal numbers. The greater the radioactivity, the faster the instrument can come to the potential of the surroundings. However, this type of sensing element is inherently slow especially with instruments of large electrical capacities. The small ball on the reference probe for the cloud measurement “A" of Figure (5'5”) and the flat disk on the end of the movable probe ”0” of Figure (53) were given four coats . each of polonium nitrate. 5 The suspension and grounding wire for the metal Sphere used in Case II, Case III and Case IV of the measured poten- tial test are shown in ”B" of Figure (55). The suSpension system for the grounded part of the deposit test is shown in - 128 - "D" of Figure (553. The instruments used in tests throughout this thesis are "A" and "B" of Figure (54). 5 Fig. 54. Instruments for electrostatic potential measured. gm ”offlmz‘uwn ‘J a 2'3. \fi‘V‘R‘N‘Hé ‘~\~ ' Fig. 5‘. Probes and suspensions used for the potential measurement and deposit.tests. O\ SELECTED BIBLIOGRAPHY Atwood, Stephen S. Electric and Magnetic Fields. 5rd ed. John Wiley and Sons, Inc. New York, 1949. 475 pages Barnes, Neal A. Crop Dusting Equipment. Unpublished Progress Report on Research Project. North Carolina State College, Agricultural Engineering Dept., 1951. 26 pages. Bergrun, Norman R. A Method for Numerically Calculating the Area and Distribution of Water Impingement on the Leading Edge of an Airfoil in a Cloud. National Advisory Commit- tee for Aeronautics, Washington, D.C., Tech. Note No. 1597. 1947. Bowen, Henry D., Hebblethwaite, Peter, and Carleton, W. M. Application of Electrostatic Charging to the Deposition of Insecticides and Fungicides on Plant Surfaces. Agricultural Engineering Journal, June 1952. pp. 347-550. Brooks, F. A. The Drifting of Poisdnous Dusts Applied by Airplanes and Land Rigs. Agricultural Engineering Journal, June 1947. pp. 255-239. Dallavalle, J. M. Micromeritics, The Technology of Fine Particles. 2nd ed. Pitman Publishing Corp. New York and London, 1948. 495 pages. Jeans, Sir James Hopwood The Mathematical Theory of Electricity and Magnetism. 5th ed. The University Press. Cambridge, 1955. 645 pages. Lapple, C. E. Fluid and Particle Mechanics. University of Delaware. Newark, Del., March 1951. 353 pages. _ 130 _ Lapple, C. E. and Shepherd, 0. B. Calculations of Particle Trajectories. Industrial and EngineeringChemistry. Vol. 52, No. 5, 1940. pp. ”07-617. Prandtl, L. and Tietjens, O. G. Translated by J. P. Den Hartog Applied Hydro and Aeromechanics. lst ed. McGraw— Hill Book Company, Inc. New York and London, 1954. 506 pages. Prandtl, L. and Tietjens, 0. G. Translated by L. Rosenhead Fundamentals of Hydro- and Aeromechanics. lst ed. McGraw-Hill Bock Company, Inc. New York and London, 1954. 265 pages. Nilson, E. Bright Jr. An Introduction to Scientific Research. McGraw-Hill Book Company, Inc.. New York, Toronto and London, 1952. 565 pages. r1 1,1, 1*” l _. R36??? USE ONLY Q . ; 339.35 US}: am}: it '3 '5‘ 0C 1 2 ’54 INTERUBRARY LOAN SEP 1 7 "55 lNTERJiBsiR-z WAN FEB 4 '56 .5; 5:5 ‘4’); IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII mu111119131ulmtumuuuilwgurlml