Airborne particles are present throughout our environment. ... important aspect of
aerosol technology is the quantitative description of ... sic variability of aerosol
properties and measurements, correction factors and errors .... WILLIAM C.
HINDS.
SecondEdition
William C. t;Jj({ds Departmentof EnvironmentalHealt~iences Centerfor Occupationaland EnvironmentalHealth UCLA School of Public Health Los Angeles,California
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JOHN WILEY & SONS,INC.
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Preface to the First Edition Preface to the Second Edition List of Principal Symbols Introduction
xi xiii YV
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I. I Definitions / 3 1.2 Particle Size, Shape,and Density / 8 1.3 Aerosol Concentration/ 10 Problems / 12 References/ 13 2 Properties of Gases
15
2.1 Kinetic Theory of Gases/ 15 2.2 Molecular Velocity / 18 2.3 Mean Free Path / 21 2.4 Other Properties / 23 2.5 ReynoldsNumber / 27 2.6 Measurementof Velocity, Flow Rate, and Pressure/ 31 Problems / 39 References/ 41 3
Uniform Particle Motion 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
42
Newton's ResistanceLaw / 42 Stokes'sLaw / 44 Settling Velocity and MechanicalMobility / 46 Slip Correction Factor / 48 NonsphericalParticles / 51 Aerodynamic Diameter / 53 Settling at High ReynoldsNumbers / 55 Stirred Settling / 62 v
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7 Brownian Motion and Diffusion 7.1 7.2 7.3 7.4
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Filtration
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9.1 MacroscopicPropertiesof Filters / 182 9.2 Single-Fiber Efficiency / 190 9.3 Deposition MechanismsI 191 9.4 Filter Efficiency / 196 9.5 PressureDrop I 200 9.6 MembraneFilters / 202 Problems / 204 References/ 204 10 Sampling and Measurement of Concentration 10.1 Isokinetic Sampling / 206 10.2 Sampling from Still Air / 213 10.3 Transport Losses / 216 10.4 Measurementof Mass Concentration / 217 10.5 Direct-ReadingInstruments / 222 10.6 Measurementof Number Concentration / 225 10.7 Sampling Pumps / 228 Problems / 230 References/ 231
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15.6 ChargeLimits / 333 15.7 Equilibrium ChargeDistribution / 335 15.8 ElectrostaticPrecipitators / 338 15.9 Electrical Measurementof Aerosols / 341 Problems / 346 References/ 347 16 Optical Properties
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16.1 Definitions / 350 16.2 Extinction / 352 16.3 Scattering / 358 16.4 Visibility / 364 16.5 Optical Measurementof Aerosols./ 370 Problems / 376 References/ 377
17 Bulk Motion of Aerosols Problems / 385 References/ 385 18 Dust Explosions Problems / 392 References/ 392 19 Bioaerosols
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19.1 Characteristics/ 394 19.2 Sampling / 396 Problems /400 References/ 400 20 Microscopic Measurement of Particle Size 20.1 Equivalent Sizesof Irregular Particles / 402 20.2 Fractal Dimensionof Particles / 408 20.3 Optical Microscopy / 413 20.4 Electron Microscopy / 416 20.5 AsbestosCounting / 422 20.6 Automatic Sizing Methods / 424 Problems /425 References/ 426
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Airborne particles are presentthroughout our environment. They come in many different forms, such as dust, fume, mist, smoke,smog,or fog. Theseaerosolsaffect visibility, climate, and our healthandquality of life. This book coversthe properties, behavior, and measurementof aerosols. This is a basic textbook for people engagedin industrial hygiene, air pollution control, radiationprotection,or environmentalsciencewho must, in the practiceof their profession,measure,evaluate,or control airborne particles. It is written at a level suitable for professionals,graduatestudents,or advancedundergraduates.It assumesthat the studenthas a good backgroundin chemistry and physicsand understandsthe conceptsof calculus.Although not written for aerosolscientists,it will be useful to them in their experimentalwork and will serveas an introduction to the field for studentsstartingsuchcareers.Decisionson what topics to includewere basedon their relevanceto the practical application of aerosolscience,which includes an understandingof the physical and chemical principles that underlie the behavior of aerosolsand the instrumentsusedto measurethem. Although this book emphasizesphysical rather than mathematicalanalysis,an important aspectof aerosol technology is the quantitative description of aerosol behavior. To this end I have included 150 problems, grouped at the end of each chapter.They are an important tool for learning how to apply the information presentedin the book. Becauseof the practical orientation of the book and the intrinsic variability of aerosolpropertiesand measurements, correctionfactorsanderrors of less than 5 percenthave generally been ignored and only two or three significant figures presentedin the tables. Aerosol scientistshavelong beenawareof the needfor a betterbasicunderstanding of the propertiesand behaviorof aerosolsamongappliedprofessionals.In writing this book, I have attemptedto fill this need,as well as the long-standingneed for a suitable text for studentsin thesedisciplines. The book evolved from class notes preparedduring nine years of teaching a required one-semestercourse on aerosoltechnologyfor graduatestudentsin the Departmentof EnvironmentalHealth Sciencesat Harvard University School of Public Health. Chaptersare arrangedin the order in which they are covered in class, starting with simple mechanicsand progressingto more complicatedsubjects.Particle statistics is delayeduntil the studenthasa preliminary understandingof aerosolproperties and can appreciate the need for the involved statistical characterization. Applications are discussedin eachchapterafter the principleshavebeenpresented. The more complicatedapplications,suchas filtration andrespiratorydeposition,are xi
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More than 16 years have passed since the first edition of Aerosol Technology was published in 1982. During this time the field of aerosol science and technology has expanded greatly, both in technology and in the number of scientists involved. When the first edition was published there were two national aerosol research associations, now there are 11 with regular national and international meetings. Growth areas include the use of aerosols in high-technology material processing and the administration of therapeutic drugs, and there is an increased awareness of bioaerosols, aerosol contamination in microelectronic manufacturing, and the effect of aerosols on global climate. While the first edition proved to be popular and useful, and became a standard textbook in the field, changes in technology and growth of the field have created the need to update and expand the book. The objective of the book has remained the same: to provide a clear, understandable, and useful introduction to the science and technology of aerosols for environmental professionals, graduate students, and advanced undergraduates. In keeping with changes in the field, this edition uses dual units, with SI units as the primary units and cgs units as secondary units. Besides updating and revising old material, I have added a new chapter on bioaerosols and new sections on resuspension, transport losses, respiratory deposition models, and fractal characterization of particles. The chapter on atmospheric aerosols has been expanded to include sections on background aerosols, urban aerosols, and global effects. There are 26 new examples and 30 new problems. The latest edition of Air Sampling Instruments remains an excellent companion book, as does Aerosol Measurement, by Willeke and Baron. Both provide greater depth and detail on measurement methods and instruments. Of the many people who have helped with this edition, I would like to particularly acknowledge Janet Macher, Robert Phelan, and John Valiulis for reviewing specific chapters; Rachel Kim and Vi Huynh for typing manuscript changes; doctoral student Nani Kadrichu for entering the equations; and finally, my wife Lynda for her continued support during this long process. WILLIAM
C. HINDS
Los Angeles. California
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LIST OF PRINCIPAL SYMBOLS
F F(a) F(x) ~dh FD FE Ff Fa Ff Fn Fth Fy Fr g G
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force cumulative frequencyat a, Eq. 4.8 cumulative fraction at x, Eq. 11.12 force of adhesion,Eqs. 6.1-6.4 drag force, Eqs. 3.4 and 3.8 electrical force, Eq. 15.8 frictional force on a fluid element,Eq. 2.36 force of gravity, Eq. 3.11 inertial force on a fluid element,Eq. 2.39 form componentof Stokesdrag, Eq. 3.6 thermal force, Eqs. 8.1 and 8.4 volume fraction of spheresin liquid, Eq. 21.6 frictional componentof Stokesdrag, Eq. 3.7 accelerationof gravity gravitational settling parameter,Eq. 9.29; ratio of cloud velocity to particle velocity, Eqs. 17.6and 17.7 geometricstandarddeviation, O"g,Eq. 4.40 height; velocity head,Eq. 2.43 height of chamber;thermophoreticcoefficient, Eq. 8.5; latent heatof evaporationof a liquid Mie intensity parameterfor perpendicularcomponentof scatteredlight, Eqs. 16.23and 16.24 Mie intensity parameterfor parallel componentof scatteredlight, Eqs. 16.23and 16.25 number of intervals for groupedsize data, Eq. 4.14; light intensity, Eq.16.7 incident light intensity, Eq. 16.7 intensity of scatteredlight at angle e, perpendicularpolarization,Eq. 16.24 intensity of scatteredlight at angle e, parallel polarization, Eq. 16.25 inhalable fraction, Eq. 11.7, 11.8 inhalable fraction for nosebreathing,Eq. 11.9 diffusion flux, Eqs. 2.30 and 7.1 Boltzmann's constant thermal conductivity of a gasor vapor a constant;correctedcoagulationcoefficient, Eq. 12.13 uncorrectedcoagulationcoefficient, Eq. 12.9 effective coagulationcoefficient for polydisperseaerosols,Eq. 12.17 electrostaticconstantof proportionality (51 units), Eq. 15.1 and Table 15.1 kinetic energy Knudsen number
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The microscopicparticles that float in the air are of many kinds: resuspendedsoil particles, smoke from power generation,photochemically formed particles, salt particles formed from oceanspray,and atmosphericclouds of water dropletsor ice particles. They vary greatly in their ability to affect not only visibility and climate, but also our health and quality of life. Theseairborneparticles are all examplesof aerosols.An aerosolis defined in its simplest form as a collection of. solid or liquid particles suspendedin a gas.Aerosolsare two-phasesystems,consistingof the particles and the gas in which they are suspended.They include a wide range of phenomenasuchas dust, fume, smoke,mist, fog, haze,clouds,and smog.The word aerosol was coined about 1920as an analog to the term hydrosol, a stableliquid suspensionof solid particles.Aerosolsare also referredto as suspendedparticulate matter, aerocolloidal systems,and dispersesystems.Although the word aerosol is popularly usedto refer to pressurizedspray-canproducts,it is the universally acceptedscientific term for particulatesuspensionsin a gaseousmedium and is used in that sensein this book. Aerosols are but one of the several types of particulate suspensionslisted in Table 1.1.All are two-componentsystemshaving specialpropertiesthat dependon size of the particles and their concentrationin the suspendingmedium. All have varying degreesof stability that also dependon particle size and concentration. An understandingof the propertiesof aerosolsis of great practical importance. It enablesus to comprehendthe processof cloud formation in the atmosphere,a key link in the hydrological cycle. Aerosol properties influence the production, transport,and ultimate fate of atmosphericparticulatepollutants.Measurementand control of particulate pollutants in the occupationaland generalenvironmentsrequire the applicationof this knowledge.Aerosol technologyhascommercialapplication in the manufactureof spray-driedproducts,fiber optics, and carbon black; the production of pigments;and the application of pesticides.Becausethe toxicity of inhaled particles dependson their physical as well as their chemicalproperties, an understandingof the propertiesof aerosolsis requiredto evaluateairborneparticulate hazards.The sameknowledgeis used in the administrationof therapeutic aerosolsfor the treatmentof respiratoryand other diseases. Aerosol technologyis the study of the properties,behavior, and physical principles of aerosolsand the application of this knowledgeto their measurementand control. The particulate phaseof an aerosolrepresentsonly a very small fraction of its total massand volume, lessthan 0.000I %. Bulk propertiesof aerosols,such as viscosity and density,differ imperceptiblyfrom thoseof pure air. Consequently, to study the propertiesof aerosols,onemust adopta microscopicpoint of view.This
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FIGURE 1.1 (a) Coal-burning power plant. (b) Scanning electron microscope (SEM) photographof coal fly ashparticles.
1.1 DEFINffIONS Aerosolscan be subdividedaccordingto the physical fonn of the particlesand their method of generation.There is no strict scientific classification of aerosols.The following definitions correspondroughly to commonusageand are preciseenough for most scientific description. Aerosol A suspensionof solid or liquid particles in a gas. Aerosols are usually stablefor at leasta few secondsand in somecasesmay last a year or more. The
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FIGURE 1.3 (a) Arc welding. (b) SEM photographof iron-oxide particles. Magnification 2300x.
Fume A solid-particleaerosolproducedby the condensationof vaporsor gaseous combustionproducts.Thesesubmicrometerparticlesare often clustersor chains of primary particles.The latter areusually lessthan 0.05 ~. Note that this definition differs from the popular use of the term to refer to any noxious contaminant in the atmosphere. Haze An atmosphericaerosolthat affects visibility. Mist and Fog Liquid-particle aerosolsformed by condensationor atomization. Particlesare sphericalwith sizesranging from submicrometerto about 200 ~m. Smog 1. A generalterm for visible atmosphericpollution in certain areas.The term was originally derived from the words smokeandfog. 2. Photochemical smog is a more preciseterm referring to an aerosol formed in the atmosphere by the action of sunlight on hydrocarbonsand oxides of nitrogen. Particlesare generally lessthan 1 or 2 ~m.
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FIGURE 1.5 (a) Volcanic eruptionof Mount St. Helens,May 1980.(b) Optical microscope photographof volcanic ash.Magnification 125x.USGSphotographby Austin Post.Reprinted from Mount St. Helens: Five YearsLater. Courtesyof EasternWashingtonUniversity Press and W. C. McCrone and J. G. Deily, The Particle Atlas. Reprinted by pennission from McCrone ResearchInstitute.
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having a diameterd and a density p = 1000 kg/m3 [1.0 g/cm3]. What is the surfaceareaof I g of O.l-l.1m-diameter particles? ANSWER: 60 m2.
1.9 Determinethe ratio of the surfaceareaof a sphereto that of a fiber with the
samevolume.Assumethat the fiber is a cylinder with a diameterequalto 20% of the diameterof the sphere.Assumethat the sphereandthe fiber havestandard density. ANSWER: 0.3.
REFERENCES Listed below are some general references on aerosol science and technology. A comprehensive list of books and journals is "given in Aerosol Science and Technology,14, 1-4 (1990). Cadle,R. D., TheMeasurementof Airborne Particles, Wiley, New York, 1975.(Emphasizes microscopicmeasurement.) Clift, R., Grace,J. R., andWeber,M. E.; Bubbles,Drops, andParticles,AcademicPress,New York, 1978.(Referencetextbook on fluid dynamics,heat transfer, and masstransfer of single bubbles,drops,and particles.) Davies, C. N. (Ed.), Aerosol Science,Academic Press,New York, 1966.(Thorough coverageon selectedtopics.) Friedlander,S. K.. Smoke,Dust and Haze,Wiley, New York, 1977.(Goodcoverage;oriented toward air pollution.) Fuchs,N. A., TheMechanicsof Aerosols.Pergamon,Oxford, UK., 1964.(This outstanding classichasbeenrepublishedin paperbackby Dover Publications,New York, 1989). Hidy, G. M., andBrock, J. R., TheDynamicsof Aeroco//oidal Systems.PergamonPress,New York, 1971.(A mathematicalapproachto the theory of aerosolscience.) Hinds, W. C., Aerosol Technology:Properties,Behaviorand Measurementsof Airborne Particles, Wiley, New York, 1982.(The predecessorto this edition.) Mercer, T. T., Aerosol Technologyin Hazard Evaluation, AcademicPress,New York, 1973. (Good coverage;orientedtoward occupationalhygiene.) Reist, P. C., Introduction to Aerosol Science,McGraw-Hili, New York, 1993.(An introductory textbook on aerosolscience.) Shaw,D. T. (Ed.), Fundamentalsof AerosolScience,Wiley, New York, 1978.(Detailedcoverageof selectedtopics.) Vincent, J. H., Aerosol Sciencefor Industrial Hygienists,Pergamon,Oxford, UK., 1995.(A textbook on the applicationof aerosolscienceand technologyto the field of occupational hygiene.) Vincent, J. H., Aerosol Sampling:Scienceand Practice, Wiley, New York, 1989.(Comprehensivecoverageof aerosolsampling,especiallywith blunt samplers.)
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4 The sizeof the particlesof a monodisperseaerosolis completelydefinedby a single parameter,the particle diameter.Most aerosols,however,arepolydisperseand may have particle sizesthat range over two or more orders of magnitude.Becauseof this wide size rangeand the fact that the physicalpropertiesof aerosolsare strongly dependenton particle size, it is necessaryto characterizethesesize distributionsby. statistical means.For the purposesof this chapter,we neglectthe effect of particle shapeand consideronly sphericalparticles. We begin by characterizingsize distributions in a generalway without specifying the type or shapeof the distribution used.Later we apply theseconceptsto the most common aerosolsize distribution, the lognormal distribution.
4.1 PROPERTIESOF SIZE DISTRIBUTIONS In this section,we characterizesize distributions and their propertiesby using examplesbasedon a specificsetof particle sizedata.The resultofa carefulsizeanalysis might be a list of 1000particle sizes.In somesituations,keepingthe datain this form may be desirable--for example,if the list is stored in a computer. In most situations, however.we would like to have a picture of how the particles are distributed amongthe various sizesand to be able to calculateseveraldifferent kinds of statistics that describethe propertiesof the aerosol.For that purpose,a list of 1000numbersis an awkward format, so it is necessaryto resort to descriptivestatistics to summarizethe information. The first step in such a summarizationis to divide the entire size range into a seriesof successiveparticle sizeintervalsanddeterminethe numberof particles(the count) in eachinterval. The int~rvalsmust be contiguousand cover the entire size range.so that no particlesare left out. If thereare 10 size intervals,our list of 1000 numbersis reducedto 20 numbers,the lower (or upper) size limits and the count for each interval, as shown in Table 4.1. The upper size limit of each interval coincides with the lower limit of the next-higher interval. If a particle size falls exactly on the interval limit, it is groupedin the higher interval. Thesegroupeddata are much easierto deal with and give the first glimpse of the shapeof the size distribution. Onegraphicalrepresentationof groupeddatais the histogram,shownin Fig. 4.1, where the width of eachrectanglerepresentsthe size interval and the height representsthe numberof particles in the interval. Unfortunately, the figure gives a dis75
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40
FIGURE 4.2 Frequency/~mversusparticle size.
number of particlesper unit of size interval (number/l.1m), and the heightsof intervals with different widths are comparable.Furthermore,the areaof eachrectangle is proportional to the number, or frequency, of particles in that size range. This relationship can be seenfrom the units of the graph: The height h; in number/j.1m times the width L1diin I.1mgives an areaequal to the numberof particles in the interval. The total area of all the rectanglesis the total number of particles in the sampleN.
The histogramis usually standardizedfor samplesize by dividing the heightsof the rectangleshj (in frequencY/Jlm)by the total numberof particlesobservedin the sample,giving heightsas fraction/Jlm.The areaof eachrectanglein the units of the graph, hjAdj, is equal to the fraction./; of particles in that size range,and the total area is equal to 1.0. This changeallows direct comparisonof histogramsobtained from samplesof different size. Ii
= ~n = (hiMi) N
As shown in Fig. 4.3, the shapeof the distribution is the sameas that shown in Fig. 4.2, but the ordinateis now the fraction of the total numberof particlesper unit of size interval.
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Becausethe frequencyfunction is such an important way of characterizingdistributed quantities, we review its properties here. Although we describe the frequency function in terms of the particle size distribution, we can extendthe analysis by analogy to any other distributed quantity, such as particle massor velocity. The dp + fraction ddp is of the total number of particles dfhaving diametersbetweendp and df= f(dp)ddp
(4.4)
wheref(dp) is the frequencyfunction and ddp is a differential interval of particle size.This function is the mathematicalrepresentationof the curve shownin Fig. 4.4. The areaunder the curve is always
which is equivalentto Eq. 4.3..In thinking about frequencyfunctions, always keep in mind which interval is under consideration.The interval may be from zero to infinity, as in Eq. 4.5, or it may be betweengiven sizesa and b, or it may even be the tiny interval ddp. The areaunder the frequencyfunction curve betweentwo sizesa and b equals the fraction of particles whosediametersfall within this interval. Mathematically, this is expressedas
which is equivalentto Eq. 4.2. Note that the fraction of particles having diameters exactly equal to diameterb is zero, becausethe interval width is zero:
Size distribution infonnation can also be presentedas a cumulative distribution function, F(dp), which is defined by
F(a) gives the fraction of the particles having diameterslessthan a. Note that the cumulative distribution function has the size interval built into its definition. Figure4.5 showsthe cumulativedistributionfunction for the datagiven in Table4.1. The fraction lessthan a particular size is obtainedfrom the location on the vertical axis associatedwith the point on the curve directly abovethat particular size. An alternativecumulativedistribution function can be constructedby considering the number of particles whose size is greaterthan, insteadof less than, a particular size. A plot of this function containsthe sameinformation as is in Fig. 4.5, but has the S-shapedcurve reversed.In this book we consideronly the more common variety, the "less than indicatedsize" cumulative distribution.
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PROPERTIES OF SIZE DISTRIBUTIONS
>:nj
where nj is the number of particles in group i, having a midpoint of size dj, and where N = Lnj, i.e., the total numberof particles,and the summationsare over all intervals. The midpoint size dj can be the geometricmidpoint-the squareroot of the product of the upper and lower limits of the interval-or the arithmetic midpoint-the mean of the limits. The former is often preferred,but causesa distortion when the first interval hasa lower limit of zero.The arithmeticmidpoint is used here for simplicity. The median is defined as the diameter for which one-half the total number of particles are smaller and one-half are larger. The median is also the diameterthat divides the frequencydistribution curve into equal areas,and the diameter correspondingto a cumulativefraction of 0.5. The modeis the most frequentsize, or the diameter associatedwith the highest point on the frequency function curve. The mode can be determinedby.setting the derivative of the frequencyfunction equal to zero and solving for d. For symmetricaldistributions such as the normal distribution, the mean,median,and mode will have the samevalue, which is the diameter of the axis of symmetry. For an asymmetricalor skeweddistribution, these quantities will have different values.The median is commonly usedwith skewed distributions,becauseextremevaluesin the tail havelesseffect on the medianthan on the mean.Most aerosolsizedistributionsare skewed,with a long tail to the right, as shown in Fig. 4.4. For sucha distribution, mode < median < mean The geometric mean,dg, is defined as the Nth root of the product of N values, dg
= (d1d2d3
dN)I/N
(4.13)
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dg = (d;ld;2d;3
where nl' n2' n3' . . . , n, are the numberof particles in intervals I through I and dl, d2,d3, . . . , d, are the midpoints, or someother characteristicdiameterfor intervals 1 through I. The geometric mean can also be expressedin terms of In(d) by converting Eq.4.14 to naturallogarithms: Indg = ~nj Indj N
(4.15)
(4.16)
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weighed on a balanceand the weights summedand divided by their number, the averagemasswould be obtained. -m=
Lmi N
(4.17)
Note that m could, of course,be obtainedmore easily by weighing the whole basket and counting the apples.In either case,this is the first stepof indirect sizemeasurement.Assuming a sphericalapple, the averagemassis related to the size (diameter)of the apple with averagemassdm by (4.18) Combining Eqs. 4.17 and 4.18 gives the diameterof averagemassfor a basketof apples,or, a sampleof particles,
d-m =
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6
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\
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S:)I.LSI.LV.LS3ZIS 3':)I.L~Vd
MOMENT DISTRIBUTIONS
particle has 1000times the massof a I-f.1nlparticle. The 10-llm particlescontribute 50,000 units of massto the total massof the particles,whereasthe l-llm particles contribute only 50 units of mass;thus, the massis distributed such that 99.9% is in the 10-llm particles and 0.1% in the l-llm particles.A similar analysisfor surface area attributes99% of the surfaceareato the 10-llm particles and 1% to the l-llm particles.The distinction betweencount and massdistributions can be put in
0
10
20 30 Particle diameter (I'm)
40
FIGURE 4.7 Cumulative massdistribution
50
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98
87
MOMENT DISTRIBUTIONS
The numeratorand denominatorof the rightmost side of Eq. 4.27 are moment sumsand are closely relatedto momentaverages.The numeratorcan be written in terms of the diameterof averagemass.From Eq. 4.22 for groupeddata, Ln.d3'
113
d-m = -!-L N Dt.d3
I I
( =
d-
N
m
)
3
Similarly, for the diameterof averagesurface, }::J-z.df = N
(d- )2
-
-
I I
S
Substituting into Eq. 4.27 gives dsm
N(d-)3 m
(d-)3 m
-M-;J;)2"-(d;)2
(4.31)
Thus, eachweightedmeandiametercan be written as the ratio of two momentaverages. Weighted mean diameterscan also be expressedin terms of aggregatequantities. If M and S are the total mass and surface area, respectively, of an aerosol sample,then
M= l'~6
) Ln.d3 I I
S = 7tLn.d? I I
(4.33)
and Eq. 4.27 becomes
(4.34) Equation 4.34 illustrates a characteristicof weightedmeandiameters:An average particle size (dsm)can be determinedif the particle density is known by simply weighing a sampleof particlesand measuringthe surfaceareaof the sample.There are gas adsorptionand radioactivecoating methodsto do this. In addition to mass,surface,and count distributions,there is thc lesscommonly used distribution of length. This distribution expressesthe fraction of the total length, if all particles are laid side by side, in any size range.In fact, there can be a moment distribution for any property that is proportional to a constantpower of diameter. The general form for the distribution function giving the fraction of a quantity proportional to d; in an interval betweendp and dp + ddp is
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S;)I.1SI.1V.1S3ZIS 31;)1.1'MVd
88
89
MOMENT DISTRIBUTIONS
If we desired the total weight to determine whether all 20 people could ride together on an elevator, we need only weigh that person whose height is given by the previous equation (or interpolate between the two nearest people) and multiply his or her weight by 20. The person with the median height will also have the median weight, because the order will not change by cubing heights.
2 An example of a moment distribution is that obtained from an analysis of the grain size of sand by sieving. The sand is passed through a stack of sieves arranged in order of decreasing mesh size. The mass of sand retained on each sieve is determined. From the characteristics of the screens, we can assign a diameter to each mass of sand retained. With these data, we can determine the distribution of mass and calculate the mass mean diameter of the sand particles as follows: Calculation of Mass Mean Diameter Sieve Opening (mm)
Midsize of Collected Sand, d.I (mm)
Mass Collected, m,
1.5 0.75 0.375 0.188 0.062 Total
8 34 40 12 3 97
1.0 0.50 0.25 0.125 Final Pan
(g)
~jdj dmm= --::-
Percentof Total Mass (%)
-~=O.57mm - 97
M
3 3 5 2
1.0 2.0 6.0 9.0
3 10 12 25
3 20 72 95
3 40 432 475
Arithmetic average,or count meandiameter= ~nd/N = 2.50 Median = 2.0
Mode= 2.0 Diameter of averagesurface= (~nd2/N)1/2= (9.5)1/2= 3.08 Length mean diameter
= (~nd3/N)1/3= (47.5)1/3 = 3.62 = ~nd2lLnd = 95/25 = 3.80
Surface mean diameter
= ~nd3/~nd2 = 475/95 = 5.00
Diameter of average volume
Volume meandiameter= ~nd4lLnd3=2675/475= 5.63
m,d; (g.mm) 12.0 25.5 15.0 2.26 0.19 54.9
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06
SJI.LSI.LVlS3ZIS 3'JI.L~Vd
THE LOGNORMAL
DISTRIBUTION
1
91
10 Particlediameter(pm)
100
FIGURE 4.8 Frequencydistribution curve (logarithmic size scale).
size data given in Table 4.1. Note that the logarithmic scale has no negativevalues, thus overcomingthe problem of negativeparticle size. The lognormal distribution hasbeenapplied to such diversedistributionsas incomes,populations(of both peopleand bacteria),pricesof stockslisted on the New York Stock Exchange,concentrationsof environmentalcontaminants,and particle sizesof aerosols.Thereis no fundamentaltheoreticalreasonwhy particle size data should approximatethe lognormal distribution, but it has been found to apply to most single-sourceaerosols.(SeeAppendix 2 at the end of the Chapter.)Mixtures of lognormal distributions will not be lognormal. The lognormal distribution is most useful in situations where the distributed quantity can have only positive valuesand coversa wide rangeof values-that is, where the ratio of the largestto the smallestvalue is greaterthan about 10. When this range is narrow, the lognormal distribution approximatesthe normal distribution. The lognormal distribution is used extensively for aerosol size distributions becauseit fits the observedsize distributionsreasonablywell and its mathematical form is convenientfor dealingwith the momentdistributionsand momentaverages describedin the precedingsections. Becausethe logarithm of dp is normally dis!Eibuted,the lognormal distribution frequencyfunction canbe formedby replacingdp and cr in Eq. 4.37 with their logarithmic counterparts.Thus, dp is replacedby the arithmetic mean of In d, defined by Eq. 4.15 as the geometricmeandiameterdg, In dg
- 1:n.lnd. I I -
N
(4.39)
Either the natural logarithm or the logarithm to the base 10 can be used, but the natural logarithm is more commonand is usedhere.
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g!d
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(
dpp-
(lv.v)
0
'., co -:; E 0.4
~ u
0.2
0 1
10 Particle diameter (I'm)
100
FIGURE 4.9 Cumulative distribution curve (logarithmic size scale).
bution of count and massplotted on the samelogarithmic diameterscale.For the massdistribution, we replacethe geometricmean diameterwith the massmedian diameterMMD, analogousto what we did for the count distribution. The massdistribution hasthe sameshapeasthe count distribution but is displacedalongthe size axis by a constantamountequalto MMD/CMD. The ratio MMD/CMD can be calculated knowing only the GSD. This calculation is coveredin Section4.6.
1
10 Particle diameter (I'ml
100
FIGURE 4.10 Count and massdistributions (logarithmic size scale).
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S:)I.LSI.LV.LS3ZIS 31:)1.L~Vd
.,6
LOG-PROBABILITY
95
GRAPHS
d
=~%
- d5(J'1o
(4.44)
For a lognonnal distribution, which is nonnal with respectto In d, thesedifferences becomeratios in 0" g
= in
dS4%
-
in dSO%
= in(dS4%/dsoo/o) GSD
= cr =!!J.lli-= ~ g
dSO%
d16%
=
( !!J.lli-) 1 d16%
The geometricstandarddeviation,beingthe ratio of two sizes,hasno units andmust always be greaterthan or equal to 1.0. The CMD (50% cumulative size) and the GSD can be determineddirectly from a log-probability plot and completelydefine a lognormal distribution. The first step in the graphical procedurefor determining a lognormal size distribution is to determinethe numberof particles in eachof a consecutiveseriesof size intervals. We then expressthesenumbersas a percentof the total and (starting with the smallestsize intelval), sequentiallyadd the percentfor eachinterval, to obtain the cumulativefrequencyin percent,as shownin the last column of Table 4.1. Next we plot the datum for eachinterval on log-probability paper as the percentageless than the upper size limit versusthe upper size limit for that interval. If a straight line fits the data, the distribution can be representedby a lognormal distribution andthe CMD andGSD canbe determinedfrom the 50th percentilesize and Eq. 4.46. While there will usually be somescatterabout the straight line, the
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THE HATCH-CHOATE
CONVERSION EQUATIONS
97
80 60 40
E 3 t 'Qj E ~ "-t; .. u 't ~ ~
20
10 81 6
2. -. 1 2
I
I I I . I I I
I
10 30 50 70 90 Percentlessthan indicatedsize
I. 9899
FIGURE 4.14 Cumulative count and cumulative mass distributions on log-probability graph.
representa bimodal distribution by two overlapping lognormal distributions. (See Knutson and Lioy (1995).) One great advantageof the lognormal distribution is that, for a given distribution, the geometricstandarddeviationremainsconstantfor all momentdistributions. If the GSD is the same,the ratio dS4%/dso% will be the same,for eachdistribution, and the cumulative distribution lines for other momentdistributions will be parallel to the cumulative count distribution line on a log-probability graph. The separation betweenthe lines (andbetweenmediandiameters)is a function only ofGSD. (SeeSection4.6.) Figure 4.14 showsboth count and massdistributions for the data given in Table 4.1, plotted as cumulativedistributions on log-probability graphpaper. The horizontal axis refersto cumulativepercentof massfor the massdistribution line and cumulative percentof count for the count distribution line.
4.6THE HATCJl;..cHOATECONVERSIONEQUATIONS The real power and utility (and perhapsthe popularity) of the lognonnal distribution comesfrom the fact that any type of averagediameterdiscussedin this chapter can easily be calculatedfor any known lognonnal distribution-that is, a distribution for which one averagediameter and the GSD are known. This is useful becauseit is frequently necessaryto measureone characteristicof the size distribution, suchasthe numberdistribution,when what is really neededis anothercharacteristic, such as the massdistribution or the diameterof averagemass.
(Z~'v)
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(l~.t)
(Z.ozUI~.£)dx~ow:)
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(O~'v)
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SJI.LSI.LV.LS 3Z1S 31JI.L'MVd
THE HATCH-CHOATE
CONVERSION EQUATIONS
99
Thus, the diameterof the particle with averagemass(that size particle whosemass multiplied by the total numberof particles gives the total mass)is difi
= CMD exp(l.5 In2(Jg;
(4.53)
The mode,d, is given by
J = CMD exp(-lln2{Jg)
(4.54)
The general conversion equation for any averagediameter described by Eq. 4.36--that is, thepth momentaverageof the qth momentdistribution-is given by (dqm)p = CMD ~Xp[(q +
f)
In.2(Jg]
(4.55)
For usewith the logarithm to the base10,the (q + p/2) tenD is multiplied by 2.303, and logJOand lOX are used. When q = 0, Eq. 4.55 becomesthe conversionequation for the diameterof averagedP (Eq. 4.52). Whenp = 0, Eq.4.55becomes the conversionequationfor the medianof the qth moment distribution (qMD). When p = I, we get the conversionequationfor the meanof the qth momentdistribution, dqm.Table 4.2 gives the value of b to be usedin Eq. 4.47 for the most commonconversions.Figure 4.15 showsthe relative positions of severalaveragediameterson an arithmetic plot of the count distribution data given in Table 4.1. The larger the coefficient in the conversionequation,the further that convertedaveragediameter is from the CMD. Also, the largerthe GSD, the greaterthe ratio betweenthe MMD and the CMD, as shown in Fig. 4.16. Table 4.3 summarizesinfonnation about the various types of averageswe have discussedand gives the coefficient b for the lognonnal conversionequation, Eq. 4.47. There is an obviouspatternto the valuesof b, and the table can be extended by analogyin any direction. However,as will be discussedin the next section,care must be taken to ensurethat the dataare accurateand properly representa lognormal distribution in the region of the convertedaveragediameter.
TABLE 4.2 Coefficients for Equation 4.47 for the Most Common Conversions To Convert from CMD to' Mode, d Count meandiameter,d Diameter of averagemass,d-m Mass median diameter,MMD Mass meandiameter,dmm "Diameters listed replace d,j in Eq. 4.47.
Use, for the value of bin Eq. 4.47, 0.5 1.5 3 3.5
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= ZE6"O;}~o£ = «rVzUl
w1i 608
L...
~'1)dx~OW;) = ~p pu~ '~
= q '.!1!pJOd
wl1 £Z = 9v"9 x ~"£ = ~981a ~"£ = (U9"O x £)dx~ ~"£ = aww
(Jzozuy)£)dx~~'£= (aD.UIq)dx~aw:) = aww :£
= q q~!M Lv"v
"b3 ~sn 'oww
"Z"Z = (a:.o) OSD pu~ wrl t;"£ = ow:)
~q~ Jo.:{
qt!M UO!tnq!Jt
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001
SJI.LSI.LV.LS 3ZIS 31JI.L~Vd
STATISTICAL
101
ACCURACY
FIGURE 4.16 Ratios of massmedian diameter and diameter of averagemass to count median diameterversusgeometricstand~rddeviation.
butions should be consideredestimatesand used cautiously. For example,a 10% error in the 84% size changesthe GSD from 2.0 to 2.2 and the calculatedMMD by 53%. The converseis also true: When deriving count sizesfrom massdata,care must be taken to ensurethat the lower end of the massdistribution is statistically accurate. Most of the particles in a samplewill be in a relatively few size intervals, and intervals in the tail of a distribution will contain few particles.To ensurestatistical reliability of the measurements at the tail, a commoncriterion is to observeat least 10 particles in every size interval of importanceto the distribution (either massor count) curve. An efficient procedurefor doing this for microscopic sizing is described in Section20.1. Confidenceintervalsfor the CMD andGSD can be constructedto provide a measure of the range in which the true CMD and GSD values are likely to lie. Thus, the 95% confidenceinterval for the CMD expressesthe likelihood (19 chancesout of 20) that the true CMD lies within the range defined by the confidenceinterval of a random sample. For a lognonnal distribution the distribution of In d is nonnal, and a confidence interval for In CMD (and CMD) can be constructedusing the standarddeviation of In CMD,
O"lnCMO' --!.-
O"ln CMD =
In2 0'
N
+
0' 2
1 1/2
EE
(4.56)
0.. tI1 of' :'" v. v.
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APPENDIX 3: DERIVATION OF THE HATCH-CHOATE
EQUATIONS
107
(4.73)
In dp = In CMD +
(4.74)
and dp, the diameterof averagedP, is
which is identical to Eq. 4.52. To obtain the conversion equation for the median (qMD) of the qth moment distribution, we start with the distribution function, Eq. 4.35. SubstitutingEq. 4.71, with p replacedby q, for the numeratorand usingthe procedurejust outlined in Eqs. 4.66--4.72to evaluatethe integral of Eq. 4.71 for the denominator,we get
=
df.
q
1
exp
(-
[In d
-
(In CMD
J2"ii d In O"g
+ q In20"g)p
21n20"g
(4.76)
This equation has the form of a lognormal distribution, Eq. 4.42, with a median diametergiven by In qMD qMD
= In
CMD + q In20"g
= CMD
exp(q In20"g)
(4.77)
which is identical to Eq. 4.48. We can use Eq. 4.71 to get dqm.the meanof the qth momentdistribution. This mean is the ratio of two momentsof diameterand has the generalform
f-
f(x) dx dqm = f~ 0 xq+t xqf(x)dx
Both the numeratorand denominatorof Eq. 4.78 are equivalentto the integralsof Eq. 4.71. EvaluatingEq. 4.78 using the procedurefollowed in Eqs.4.66--4.72gives
(4.79)
In dqrn = In CMD + [(q + 1)2
- q2] 2
In2(Jg
(4.80)
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PROBLEMS
109
4.5 An aerosolwith a lognonnal size distribution hasa count mediandiameterof 2.0 ~m and a geometricstandarddeviation of 2.2. If the massconcentration is 1.0 mg/m3,what is the numberconcentration?Assumesphericalparticles with Pp = 2500 kgim3 [2.5 gicm3].
ANSWER: 5.8 x 106/m3[5.8/cm3]. 4.6 Consideran aerosolwith a lognonnal size distribution (GSD = 1.8). The size distribution is in a rangewhere slip correction can be neglectedand Stokes's law holds. If the diameterof the particle of averagesettling velocity is determined to be 6.0 Ilm, what is the count median diameter? ANSWER: 4.2 Ilm. 4.7 An aerosolwith a lognonnal size distribution hasa count mediandiameterof 0.3 J.1mand a GSD of 1.5. If the number concentrationis 106particles/cm3, what is the massconcer:ttration? The particles are sphereswith a density of 4500 kgim3 [4.5 g/cm3]. . ANSWER: 130 mgim3. 4.8
An aerosolis known to have a lognonnal size distribution with a geometric standarddeviation ag' The averagetime required for the particles to settle a known distanceh is t. Derive an expressionfor the diameterof averagesettling time. Neglect slip correction.Also, give the equationfor convertingthis diameterto the count mediandiameter.Assumesphericalparticles. ANSWER:
4.9
di
= (~d-2/N)---O.5, b = + 1.
An aerosol is mixed with radon gas, resulting in a surfacecoating of radioactive radon decayproductson the particles.The aerosolis then divided into eight aerodynamicsizegroups,andthe radioactivityof eachsizegroup is measured.How can this information be usedto calculatethe count mediandiameter if the distribution is lognormal?All particles havethe samedensity and are geometrically similar. Assumethat log-probability graph paper is available.
4.10 For particles less than 0.05 ~m, light extinction efficiency is proportional to d4. Ifan aerosolis lognonnally distributedwith a CMD ofO.Ol~m and a GSD of 1.8, what is the diameterof averageextinction efficiency? ANSWER: 0.020 ~m. 4.11 Measurements of specific surface indicate that 16% of the total specific surface is contributed by particles less than 0.3 I.1mand 84% by particles less than 1.5 I.1m.Assuming that the particle size distribution is lognormal, calculate (without the aid of log-probability graph paper) the specific surface median diameter, the CMD, and the GSD. The specific surface is the surface area per gram of particles. ANSWER: 0.67 I.1m,1.28 I.1m,2.24.
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NATURAL BACKGROUNDAEROSOL
305
TABLE 14.1 Sourcesand Estimates of Global Emissions of Atmospheric Aerosols'
Source Natural Soil dust Seasalt Botanical debris Volcanic dust Forest fires Gas-to-particleconversionb Photochemicalc Total for natural sources Anthropogenic Direct emissions Gas-to-particleconversiond Photochemical" Total for anthropogenicsources
1000--3000 1000--10000 26-,S0 4-10000 3-150 100--260 40--200
1500 1300 50 30 20 180 60
2200-24000
3100
50-160 26~60 5-25
120 330 10
32~0
460
"Data primarily from Andreae (1995) and 8MIC (1971). bIncludes sulfate from 8°2 and H28. ammonium salts from NH3, and nitrate from NO.. cPrimarily photochemical particle formation from isoprene and monoterpenes vapors from trees. dIncludes sulfate from 8°2 and nitrate from NO.. .Primarily photochemical particle formation from anthropogenic volatile organic compounds.
the industrialized regions of the world. In theseareas,the humancontribution exceedsthat from natural sources. Once emitted into or formed in the atmosphere,particles can grow by vapor condensationor by coagulation with other particles. Particles are removed after growing to micrometeror larger sizesby settling or impaction on surfacessuchas tree leaves.Small particlesmay diffuse to surfacesor serveas nucleation sites for raindrops(rainout).Largerparticlesmay be sweptout by falling rain or snow(washout). Small particles have lifetimes of days in the troposphere; large particles (do> 20 11m)are removedin a matter of hours. Although most of the atmosphericparticulatemassis confinedto the troposphere (region below an altitude of II kID), the stratosphericaerosol can have significant effectson climate.This subjecthasbeenreviewedby Pueschel(1996). The primary sourceof particulate in the stratosphere(altitude from II to 50 kID) is the formation of sulfuric acid dropletsby gas-to-particleconversionof SOl injected into the stratosphereby major volcanic eruptions.Thesedroplets are formed by homogeneousnucleationinvolving photochemicalreactionsof SOl and water vapor. They spreadwidely over the hemisphere(north or south) in which they originated. The ozone layer in the lower stratosphereabsorbssolar radiation and heatsup relative to the surroundingair. This heatingcreatesa temperatureinversion and a region of atmosphericstability. The sulfuric acid dropletsformedin the stratosphere
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Ions are formed continuously in the atmosphereby the action of cosmic radiation and radioactive gasesemanating from soil. Over land, about 10 ion pairs (a positive and a negativeion) are formed in every cubic centimeterevery second. Theseions attachto aerosolparticles and recombinewith eachother at a rate that yields an equilibrium concentrationof small ions (chargedmolecular clusters)of lOG-SOOO/cm3 with an averagevalue of8 x IO8/m3[800/cm3].
14.2 URBAN AEROSOL The urban aerosol-that found in the lowest kilometer of the atmosphere over large cities--is dominated by anthropogenic sources. Aerosol mass concentrations in urban areas range from a few tens of micrograms per cubic meter to 1 mg/m3 during air pollution episodes in heavily polluted cities in developing countries. The horizontal distribution of aerosol concentration in urban areas varies greatly, depending on the proximity to natural and anthropogenic sources. Atmospheric stability and the thickness of the mixing layer (usually 0-2 km) also affect local concentrations. Table 14.2 gives historical particulate concentrations for different types of land areas. The particle size distribution of urban aerosol is complex, because it is a mixture of aerosols from various sources, each with a different size distribution, and is modified by size-dependent processes of growth, evaporation, and removal. The most common way to present particle size distribution data for the urban aerosol is in terms of the three modes, the nuclei, accumulation, and coarse particle modes.
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sus particle size on a log scale.Theseshow clearly the modal nature of the urban aerosol and the contribution of each mode to the distribution. Most information about atmosphericaerosolsize distributions is now presentedin the form of Fig. 14.2cor Fig. 14.2d.Thesefigures show that most of the particles are in the nuclei mode and most of their volume (and mass)is split betweenthe accumulationand coarse-particlemodes.The volume (or mass)distribution shown in Fig. 14.2d is typical of most urban aerosolscontaining photochemicalsmog. Suchdistributions are usually bimodal, with a saddlepoint in the 1-3-llm-diameterrange.Particlesurface areaand light scatteringare associatedprimarily with particlesin the accumulation mode. The nuclei modeconsistsprimarily of combustionparticlesemitted directly into the atmosphereand particles formed in the atmosphereby gas-to-particleconversion. This modeis not alwayspresent,but is usually found nearhighwaysand other sourcesof combustion.Becauseof their high numberconcentration,especiallynear their source,thesesmall particles coagulaterapidly with eachother and with particles in the accumulationmode.Consequently,nuclei particleshaverelatively short lifetimes in the atmosphereand end up in the accumulationmode.Nuclei particles may serveas sites for the formation of cloud dropletsand may subsequentlybe removed from the atmosphereas rain droplets (rainout). The accumulationmodeincludescombustionparticles,smogparticles, and nuclei-mode particles that have coagulatedwith accumulation-modeparticles. The smog particles are formed in the atmosphereby photochemicalreactionsof volatile organicsand oxidesof nitrogen in the presenceof strongsunlight. There is significant overlapbetweenthe nuclei and accumulationmodes.As the namesuggests, particlesaccumulatein this modebecauseremovalmechanismsare weak. Particles can be removedby rainout or washout,but they coagulatetoo slowly to reachthe coarse-particlemode.The nuclei and accumulationmodestogetherconstitute"fine" particles.The sizerangeof the accumulationmodeincludesthe wavelengthsof visible light, and theseparticlesaccountfor mostof the visibility effectsof atmospheric aerosols.(SeeSection 16.4.) Under conditions of high humidity, such as in a cloud or fog, the accumulation mode may itself have two submodes:a condensationmode with MMAD of 0.20.3 Ilm and a droplet mode with MMAD of 0.5-0.8 Ilm. The droplets are formed by the growth of hygroscopiccondensation-modeparticles. This processmay be facilitated by chemical reactionsin the droplet. The coarse-particlemodeconsistsof windblown dust, large salt particles from seaspray, and mechanicallygeneratedanthropogenicparticles suchas those from agriculture and surface mining. Becauseof their large size, the coarseparticles readily settle out or impact on surfaces,so their lifetime in the atmosphereis only a few hoursor days.The dividing line betweencoarseand fine particlesis the saddle point between I and 3 Ilm. The fine-particle mode containsfrom one-third to twothirds of the total mass,with the remainderin the coarseparticle mode. The ratio varies accordingto the region of the country. Fine and coarse-particleshave different chemical compositions,sources,and lifetimes in the atmosphere.There is comparativelylittle massexchangebetweenthe two modes.In somelocations,there
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REFERENCES
ANSWERS: 1.4 x IO2o/kg,8.7 x IO17/kg 8.7 x IO14/g,1.8 x lOll/g).
8 x lOl4/kg
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x IOI7/g
14.2 Estimate how long it would take for the diameter of average mass of the nuclei mode to equal the CMD of the accumulation mode if the nuclei mode particles were allowed to coagulate by themselves. Use modal data given in Table 14.3. Assume that the GSD is constant at its initial value and K is constant at 30 x IO--J6m3/s[30 x 10--10 cm3/s] (an interpolated average value from Table 12.4). ANSWERS: 3.3 x 104s(9.1 hrs).
REFERENCES Andreae, M. 0., "Climatic Effects of ChangiI)g Atmospheric Aerosol Levels," in Future Climates oJthe World: A Modelling Perspective,A. Henderson-Sellers(Ed.), Elsevier, Amsterdam,1995. t
as
Corn, M., "Aerosols andthe Primary Air Pollutants--Nonviable Particles,Their Occurrence, Properties,and Effects," in Stern,A. C. (Ed.), Air Pollution, AcademicPress,New York, 1976. Finlayson-Pitts,B. J., and Pitts, J. N., "Atmospheric Chemistry:Fundamentalsand Experimental Techniques,Wiley, New York, 1986. First, M. W., "Aerosols in Nature," Arch. Intern. Med., 131,24-32 (1973). Hidy, G., Mueller, P., Grosjean,D., Appel, B., andWesolowski,J.,TheCharacterand Origins oj SmogAerosols,Wiley, Ne~ York, 1980. Jaenicke,R., "Physical Characterizationof Aerosols," in Lee, S. D., Schneider,T., Grant, L. D., and Verkerk, P. J., (Eds.),Aerosols: Research,Risk Assessment,and Control Strategies, Lewis Publishers,Chelsea,Ml, 1986. Kiehl, J. T. and Rodhe,H., "Modeling Geographicaland SeasonalForcing due to Aerosols," in Charlson,R. J. and Heintzenberg,J., (Eds.),Aerosol Forcing oJ Climate, Wiley, New . York, 1995. National ResearchCouncil, Airborne Particles, University Park Press,Baltimore, 1979. Nyeki, S., Li, F., Rosser,D., Colbeck,1.,andBaltensperger,U., "The backgroundaerosolsize distribution at a high alpine site: An analysisof the seasonalcycle," J. Aerosol Sci., 28, 5211-5212 (1997). Pueschel,R. F., "Stratosphericaerosols:Formation,properties,effect," J. Aerosol Sci., 27, 383--402(1996). Schwartz, S. E., "The Whitehouse Effect--shortwave Radiative Forcing of Climate by AnthropogenicAerosols:an Overview," J. Aerosol Sci., 27, 359-382 (1996). SMIC, Inadvertent Climate Modification, Report oj the Study oj Man's Impact on Climate, MIT Press,Cambridge,MA, 1971. Sienfeld,J. H. andPandis,S.N., AtmosphericChemistryandPhysics,Wiley, New York, 1998. Whitby, K. J., "The Physical Characteristicsof Sulfur Aerosol," Atmos. Env., 12, 135-159 (1978).
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317
UNITS
where fa is the permittivity of a vacuum, 8.85 x 10-12C2/N . m Combining Eqs. 15.1 and 15.2gives
, FE=~R2 in cgs units
(15.4)
where FE is in .dyn, q and q' are in .stC, and R is in cm. Having defined charge in this way, we must define unit current and unit potential difference in consistent units. Unit current, the statampere (stA), is equal to a flow of charge of 1 stC/s along a , wire. Unit potential difference, the statvolt (stV), is defined as the voltage between two points that causes 1 erg of work to be done moving I stC of charge between the two points. The cgs system has the advantage of simplified equations, but the disadvantage of having to use "stat" units. The SI system has the advantage of using common electrical units, but requires including the constant KE in the equations. Table 15.1 gives conversion factors relating the two systems of units. From the definitions given and Coulomb's law, the following relationships can be derived:
A
= CIs
= N . rn/C N = KEC2/m2 V
V where KF
= 9.0
X
109 N
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= KEC/m
= stC2/cm2
stY
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= stC/cm
by Eq. 15.2.
TABLE 15.1 Conversion Factors and Constants for Sf and cgs Electrostatic Units Quantity
SI
cgs
Charge Current Potential difference KE Chargeon an electron, e Electronsper unit charge Electrical mobility Field strength
lC lA IV
3.0 x 109stC 3.0 x] l09 stA 0.00313stY
9.0 X 109N. m2/C2 1.60xl0-J9C 6.3 X 1018/C 1 m2N S
.
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ELECTRIC
319
FIELDS
When FE is expressedby Eq. 15.8and Eq. 15.9 is rearranged,an alternative defi. nition of field strengthis obtained:
E= ~W ~x
(15.10)
Thus, the field strengthin any direction at a point is equalto the potential gradient in that direction at the point. The detenninationof field strengthis primarily a problem of detennining the potential gradient near chargedsurfaces. The solution for an electric field (that is, an equationthat gives the field strength at any point), can be detenninedfor simple geometries.Three such casesare consideredhere.The field arounda singlepoint charge q can be detenninedeasily by placing an imaginary test chargeq' in the field and detennining the force on it by Coulomb's law and the field strengthby Eq. 15.6. . E = fA. q'
-- R2 KEq
(15.11)
The field strengthbetweentwo oppositely charged,closely spacedparallel plates is uniform in the region betweenthe plates (neglectingedge effects) and is given by
E= L\W x
(15.12)
where ilW is the algebraicdifferencein voltagebetweenthe two platesandx is the separationbetweenthe plates.In a unifonn field, the electrostaticforce on a charged particle is constanteverywherebetweenthe plates. For a positively chargedparticle, the direction of the force is toward the more negatively chargedplate; for a negativelychargedparticle,the force is directedtowardthe more positively charged plate. The field strengthinside a cylindrical tube with a wire along its axis is given by E=
~W R In(d,/dw)
15.13)
where ~W is the algebraic difference in voltage betweenthe wire and the tube, R is the radial position for which the field is being calculated,and dl and dware the diametersof the tube and the wire, respectively.Equation 15.13indicatesthat the field strengthgoesto infinity asR goesto zero.The finite diameterof the conducting wire precludesa field at R = O. The maximum field will be at the surfaceof the wire and will increaseas the wire diameterdecreases.Equation 15.13can also be usedfor the field strengthin the spacebetweentwo concentriccylinders.It becomes much more difficult to calculatethe field strengthfor complex geometriesor when there is significant spacecharge(ions or chargedparticles in the field region).
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To determine VTEwhen the particle motion is outside the Stokes region, it is necessary to follow a procedure analogous to that used to calculate VTSfor Re > I. (See Section 3.7.) A value of Re > I is more common for electrostatic motion than for settling, because the electrostatic force can be much greater than the force of gravity. Equating Eqs. 15.8 and 3.4, we obtain
l15.1T and solving for CDgives
ff ~
CD=
8neE 7tPvd2 ViE
(15.18)
Multiplyipg both sidesof Eq. 15.18by Re2gives CDRe2= 8neEpg -~
(15.19)
C~e2 can be calculatedwithout knowing the velocity or Reynoldsnumber at the terminal electrostaticvelocity. The terminal electrostaticvelocity can then be determined from the value of C~e2 by Eq. 3.33 with VTsreplacedby VTE'or by the graphical or tabular proceduresgiven in Section3.7. The Millikan oil-drop experiment,one of the classical experimentsof physics in the early 1900s,usedthe precedingelectrostaticprinciplesto measurethe charge of an individual electron. The apparatusused,called a Millikan cell, is shown in Fig. 15.1. A spherical,submicrometer-sizedparticle (an oil droplet) is introduced betweentwo horizontal parallel plates.The potential differencebetweenthe plates can be carefully controlled. The particle is illuminated by a beamof light between the platesand observedby a horizontal microscopepositionedperpendicularto the
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TABLE 15.2 Electrical Mobility of Electrons, Ions, and Aerosol Particles at Standard Conditions Particle Diameter (~m)
Electrical Mobility {m2N . S)8 Singly Charged
Electron Negative air ion Positive air ion 0.01 0.1 1.0 10 100
6.7 X
10--2
1.6 X 1.4 x 2.1 x 2.7 x 1.1 x
lQ-4 lQ-4 lQ--6 1~ 10-'1
Maximum Chargeb
7.3 x 1()'-4 9.3 x 1()'-4 (2.5 x 10-3)" (6.7 x 10-3)" (1.1 x 10-2)"
9.7 x 10--11 9.3 X 10--12
"For mobility in cm2/stV. s, multiply the value shownby 3 x 106. bBasedon the ion limit. (SeeSection 15.6.) "Velocity (m/s) in a unit electric field, but becauseRe > 1.0, Eq. 15.22doesnot hold.
The particle's mechanicalmobility can be obtainedfrom Table All
Z = neB = 40 x 1.6X 10-19x 1.23X 1010= 7.9 X 10-8 m2N.s [40 x 4.8 x 10-10X 1.23X 107 = 0.24 cm2/stY.s] "
= 6.6 stV/cm
[2000 ""10.0033 VTE
Re
= ZE = 7.9X 10-8X 200,000= 0.016 m/s
[0.24 x 6.6 = 1.6 cm/s]
66,000x 0.016x 0.6 x 10-6 = 6.3X 10-4 [6.6 X 1.6x 0.6 x 10-4 = 6.3X
10-4]
Note: VTE/~S = 0.016/1.37x 10-5 = 1170
15.4 CHARGING MECHANISMS The principal mechanismsby which aerosol particles acquire charge are flame charging, static electrification, diffusion charging,and field charging.The last two require the productionof unipolar ions, usually by coronadischarge(Section 15.5), and are used to producehighly chargedaerosols.It is often necessaryto estimate the chargeon a particle basedon the conditionsunderwhich it acquiresthe charge. Flame charging occurs when particles are fonned in or passthrough a flame. At the high temperatureof the flame, direct ionization of gasmoleculescreateshigh concentrationsof positive and negativeions and thennionic emissionsof electrons
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n(t)
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In[ 1 + 7tKEde~ 2kT
15.24)
where Ciis the mean thermal speedof the ions (c; = 240 m/s [2.4 x 104cm/s] at standardconditions) andNi is the concentrationof ions. At standardconditions,for Nit> 1012s/m3 [106 s/cm3], Eq. 15.24 is accurate to within a factor of two for particles from 0.07 to 1.5 ~ and, for Nit> 1013s/m3[107s/cm3],to within a factor of two for particles from 0.05 to 40 ~m. A more accurate,but less convenient, expressionrequiring numericalintegrationis given by Lawless(1996). Even in the presenceof an electrostaticfield, diffusion chargingis the predominantmechanism for charging particles less than 0.2 ~m in diameter.A processsimilar to diffusion charging, but utilizing bipolar ions, is used to dischargehighly chargedaerosols, 'see Section 15.7. Field charging is chargingby unipolar ions in th~ presenceof a strong electric field. This.mechanismis depicted in Figs. 15.2a-15.2c, in which the negatively chargedplate is at the left and negativeions are present.The rapid motion of ions in an electric field resultsin frequentcollisions betweenthe ions and the particles. When an unchargedsphericalparticle is placed in a uniform electric field, it distorts the field, as shownin Fig. 15.2a.The field lines shownrepresentthe trajectories of ions. The extent of the distortion of the field lines dependson the relative permittivity (dielectric constant)£ of the particle material and the charge on the particle. For an unchargedparticle, the greaterthe value of £, the greaterthe number of field lines that convergeon the particle. Ions in this electric field travel along the field lines and collide with the particle where the field lines intersectthe particle. All ions on intersectinglines to the left of the particle will collide with the particle and transfer their chargeto it. As the particle becomescharged,it will tend to repelthe like-chargedincoming ions. This situation is shownin Fig. 15.2b,in which the particle is partially charged.The presenceof the chargeon the particle reducesthe field strengthand the numberof field lines convergingon the particle from the left and increasesthesetwo quantitieson the right. Becauseof thesechanges,the rate of ions reachingthe particle decreases as the phrticle becomescharged. Ultimately, the chargebuilds up to the point where no incoming field lines converge on the particle (Fig. 15.2c)and no ions can reachthe particle. At this maximum-chargecondition, the particle is said to be at saturation charge. When diffusion charging can be neglected,the number of charges,n, acquired by a particle during a time t in an electric field E with an ion number concentration Ni is
n(t) = I 3e e+2
t I
Ed2
~~
'!!L (15.25)
where E is the relative pennittivity of the particle and Z; is the mobility of the ions, approximately0.00015m2N . s [450 cm2/stV. s]. In this equation,the first two fac-
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CHARGING MECHANISMS
tors representthe saturationchargen.,reachedafter sufficient time at a given charg. ing condition. - 3£
)( Ed2
,i:+2 4KE-; ,
(15.26)
The first factor on the right in Eqs. 15.25 and 15.26 depends only on the material of the particle and ranges from 1.0 for E = 1 to 3.0 for E = 00. The relative permittivity, or dielectric constant E, reflects the strength of the electrostatic field produced in different materials by a fixed potential relative to that produced in a vacuum under the same conditions. For most materials, 1 < E < 10; E is 1.0 for a vacuum, 1.00059 for air, 5.1 for DOP, 4.3 for quartz, 80 for pure water, and is infinite for conducting particles. The second factor in Eq. 15.25 indicates that the saturation charge is proportional to the surface area of the particle and to the electrostatic field strength. The final factor is a time-dependent term tJIat reaches a value of 1 when 1tKEeZ;N;t » 1.0. The rate of charging does not depend on the particle size or field strength, but only on the ion concentration. When particles are intentionally charged by field charging, the ion concentration is usually 1013/m3[107/cm3] or greater, so charging will be 95% complete in 3 s or less. The charge that is acquired is proportional to d; in field charging and to dp in diffusion charging, so field charging is the dominant mechanism for particles larger than 1.0 f.1m,and diffusion chargihg is the dominant mechanism for particles less than 0.1 f.1m,even in the presence of an electric field. Between these sizes, both mechanisms are operating and the situation is much more complicated. Equations 15.24 and 15.25, while explicit, involve restrictive assumptions and are somewhat oversimplified. More accurate expressions exist, but they are not explicit and usually require computer calculation. Lawless (1996) presents approximations for charging rates that are applicable to diffusion, field, and combined charging. The charging rate obtained must be numerically integrated to get the charge acquired by a particle in a given period. Table 15.3 gives the number of charges acquired in 1 s by particles of various sizes by the diffusion, field, and combined charging mechanisms at an ion concentration of 1013/m3[107/cm3]. For the conditions of the table calculations, the field-charging equation, Eq. 15.25, is accurate for particles larger than 5 f.1m,and Eq. 15.24 is useful from 0.2 to 2 ~m. Figure 15.3 shows the number of charges acquired by a particle having E = 5.1 for various charging conditions. Figure 15.4 shows particle mobility versus particle size for a typical field-charging condition. Although particle charge decreaseswith particle size, mechanical mobility increases rapidly with decreasing size; consequently, there is a size for minimum electrical mobility in the submicrometer range. Also, mobility under typical charging conditions is a relatively weak function of particle size in the size range 0.1-1 f.1m.
EXAMPLE . How many elementarychargesare acquiredby a 2.0-f.lmwater droplet in 1 s by (a) diffusion charging in an ion concentrationof lO13/m3 [lO7/cm3]and (b) field
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.ube--
FIGURE 15.7 Crosssection of a wire-and-tubeelectrostaticprecipitator.
ticles stick if they touch the wall and are not reentrained.The fraction of particles removed during this period, dN/N, is the negativeratio of the areaof the annulus, 2nRVTPdt,to the total cross-sectionalareaof the tube. nR2,and is given by
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EM grid
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Lucitecap FIGURE 15.9 Cross-sectionaldiagram of a point-to-plane electrostaticprecipitator used to obtain aerosol samples for electron microscopy.
a sampling bias that favors the collection of larger particles (Cheng et al., 1981). The use of a low sampling flow rate and a moderateion current reducesthis bias.
15.9 ELECTRICAL MEASUREMENT OF AEROSOLS A simple electrical mobility analyzer is shown in Fig. 15.10. Aerosol particles are introduced along the centerline between two oppositely charged parallel plates. Filter samples (or other aerosol measurements) are taken downstream of the device while a specific voltage is maintained on the plates. For a given voltage, all particles with mobility greater than a certain amount will migrate to the plates and be trapped. Those with lower mobility will get through, to be collected on the downstream fil-
Charged plate
Charged plate
FIGURE 15.10 Diagram of a simple electrical mobility analyzer.
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FIGURE 15.11 Schematicdiagramof an electrical aerosolanalyzer.
through a central tube. Only particles with a narrow range of mobilities can enter the gap. Particleswith greatermobility migrate to the central rod before reaching the gap, while those with lower mobility go beyond the gap and are filtered Oit. The exiting aerosolis nearly all singly chargedand nearly monodisperse.Its si:'~ is controlled by adjustingthe voltage on the central rod. The DMA was originally intendedto be usedasa monodisperse aerosolgenerator to producesubmicrometer-sizedaerosolsfor testing and calibration. However, it is used more commonly to measureparticle size distribution with high resolution in the submicrometersize range.By monitoring the exit aerosolstreamwith a CNC (see Section 13.6), the numberconcentrationin a narrow range of mobilities, and thus the particle size,can be determined.The entire submicrometerparticle sizedis-
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OF AEROSOLS ~
tions. A monodispersesubmicrometeraerosolof known sizeis producedby the first DMA. The aerosolis subjectedto a growth or shrinkageprocess,and then its size is measuredby the secondDMA. By careful operation and data analysis,particle size changesas small as 0.3% can be measured.This method works best for particles in the 0.01-to-0.2-~msize range. Although not an aerosol-measuringinstrument,the Coulter counter, an instrument manufacturedby Coulter Electronics,of Hialeah, Florida, usesthe electrical resistanceprinciple to measureparticle sizedistributionsin liquid suspensions.The instrumentis suitableonly for insolubleparticles,andthe particlesmustbe captured directly in a conductive liquid, such as with an impinger, or transferredto such a liquid before measurement.As shownin Fig. 15.13,the liquid flows from one section to the other through a small orifice 10-400 ~m in diameter.A current is established through the orifice by platinum electrodesin the liquid on either side of the orifice. Becausethe particle has an electrical resistancedifferent from that of the liquid, thereis a momentarychangein the current as the particle passesthrough the orifice due to the displacedelectrolyte. This change,which is directly proportional to the volume of the particle, is sensedelectronically and convertedto particle size, and thesedata are accumulatedto producea distribution of particle size weighted by volume. The instrument can be used for particle sizes ranging from 0.3 to 200 ~m, but for a given orifice, the rangeis limited to 2-40% of its diameter. Sizecalibration is donewith polystyrenespheresof known size. (SeeSection21.2.) The maximum counting rate is 3000/s.A samplesize of about I mg is requiredto detennine a size distribution. Wetting agentsand ultrasonic energy are used to en~urethat all narticles are fullv disoersedin the liquid.
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REFERENCES
347
15.9 Show that Rayleigh limit disintegration leads to the formation of chargestable.smaller droplets. 15.10 Determine the electrostatic velocity and the ratio of this velocity to the settling velocity for a 0.6-~m particle (e = 10) in a field strength of 1200 kV/m
[12 kV/cm]. Assumethat the particle is chargedto the averageBoltzmann equilibrium charge. ANSWER: 0.0043 mls [0.43 cmls], 320. 15.11 Detenninethe electrostaticmigration velocity for a 0.6-~m particle (£ =10) in a field strength of 1200 kV/m [12kV/cm]. Assume that the particle is chargedin the samefield for one secondat an ion concentrationof 1014/m3 [108/cm3]. ANSWER: 0.44 m/s [44 cm/s]. 15.12 What particle size receives equal numbersof chargesin 1 s by diffusion chargingand by field chargingat 2000kV/m [20 kV/cm]? Assumethat each mechanismoperatesindependentlyand that the ion concentrationis 1014/m3 [108/cm3] for both types of charging. 6 = 1. [Hint: A trial-and-error spread-
sheetapproachmay be the easiestway to do this problem.] ANSWER: 0.15 J.1m. 15.13 What voltage is required to just balance a singly charged (negative) 0.8-~mdiameter particle in a Millikan cell? The plate spacing is 5 mrn and particle density is 1000 kg/m3 [1.0 g/cm3]. ANSWER: 82 v.
REFERENCES Cheng,Y-S., Yeh, H-C., and Kanapilly, G. M., "Collection Efficiencies ofa Point-to-plane ElectrostaticPrecipitator,"Am. Ind. Hyg. Assoc.J., 42, 605-610 (1981). Flagan,R. C., "History of ElectricalAerosolMeasurements," AerosolSci. Tech.,28, 301-380 (1998). Hoppel, W. A., andFrick, G. M., "Ion-Aerosol AttachmentCoefficientsandthe Steady-State ChargeDistribution on Aerosols in a Bipolar Environment,"Aerosol Sci. Tech.,S, 1-21
I
( 1986). Lawless, P. A., "Particle ChargingBounds,SymmetryRelations,and an Analytic Charging Rate Model for the Continuum Regime,"J. Aerosol Sci., 27,191-215 (1996). Liu, B. Y. H., andKapadia,H., "CombinedField andDiffiIsion Chargingof Aerosol Particles in the ContinuumRegime,"J. Aerosol Sci., 9, 227-242 (1978). Liu, B. Y. H., and Pui, D. Y. H., "Electrical Neutralization of Aerosols," J. Aerosol Sci., S, 465-472 (1974). Liu, B. Y. H., Pui, D. Y. H., andKapadia,H., "Electrical AerosolAnalyzer:History, Principle,
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The optical propertiesof aerosolsare responsiblefor many spectacularatmospheric effects,suchas richly coloredsunsets,halosaroundthe sunor moon, and rainbows. They also causethe degradationof visibility associatedwith atmosphericpollution. The interactionof aerosolparticleswith light forms the basisfor an importantclass of instrumentsused for measuringaerosolparticle size and concentration.Optical measurementmethodshavethe advantagesof being extremelysensitiveand nearly instantaneousand of not requiring physical contactwith the particles. The optical phenomenacoveredin this chapterare all a direct result of the scattering and absorptionof light by aerosolparticles.Black smokelooks black because the particles absorbvisible light. Denserain clouds, on the other hand, look black eventhough the dropletshavenegligible absorption,becausethe droplet scattering is so completethat light cannot get through the clouds. Fair-weatherclouds look white becauseof the extensivescatteringof the incident light from the surfaceof the cloud. When we seean object, we are seeingprimarily the scatteredlight from that object. The scatteredlight reaching our eyes contains information about the color, surfacetexture, and form of the object. The scatteringof light by very small particles--less than 0.05 Ilm in diameteris describedin relatively simple termsby Rayleigh's theory, the theory ofmolecular scattering.Scatteringby largeparticles--greaterthan 100Ilm-!1
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DEFINITIONS
Material
Index of Refraction
Vacuum Water vapor Air Water Glycerine Benzene Ice (H2O) Glass Sodium chloride
1.0 1.00025 1.00028 1.3330 1.4730 1.501 1.305 1.52-1.88 1.544
Material Quartz (SiOl) Polystyrenelatex Diamond Urban aerosol(avg.) Methylene blue Soot Carbonb Iron Copper
'Data taken primarily from HandbookofChemisrry
Index of Refraction 1.544 1.590 2.417 l.56-0.087i 1.55-0.6i 1.96-0.66i 2.o-1.0i 2.8o-3.34i 0.47-2.81i
and Physics. 74th ed,. CRC, Boca Raton, FL (1993).
b", = 0,491 J.1m,
The intensity of electromagneticradiation arriving at any surface,such as the surfaceof a detector,is expressedin terms of the radiant power arriving per unit area.Whenpower (energyper unit time) is measuredin watts(W), intensityis stated in W/m2 [W/cm2]. Intensity of light emitted from a point source is expressedas radiant power emitted into a given solid angle, i.e., watts per steradian(W/sr); for visible light, this intensity can also be expressedin candela(cd), the SI baseunit for luminous intensity. The steradian(sr) is the dimensionlesssolid angle equalto the cone from the center of an imaginary spherethat subtends1/41tof the total spherical surface. Thus, there are 41t sr in a sphere.The light scatteredfrom an aerosolparticle can be consideredas coming from a point source,and its intensity in a given direction may be expressedin any of the units just mentioned. Although we are concernedprimarily with the scattering of visible light, the principles presentedherecan be applied to electromagneticradiation of any wavelength. Thus, the scatteringof radio wavesby artificial satellites,the scatteringof microwavesby raindrops,and the scatteringof light by aerosolparticlesareequivalent processeswith wavelengthsand object sizesthat are the sameorder of magnitude. The scatteringin eachcaseis governedby the ratio of the particle size to the wavelengthA of the radiation.This ratio, which is dimensionless,is called the size parameter and is given by 1td
a=T
(16.6)
The factor 7tis introducedto simplify certain light-scatteringequationsand hasthe effect of making a. equal to the ratio of the circumferenceof the particle to the wavelength.For visible light, the value of a. is approximatelyequalto six times the particle diameterexpressedin micrometers. Light can be considereda streamof photonsor a transversewave. To describe light scatteringby aerosolparticles, it is convenientto consider light as the elec-
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intensity per unit path length associatedwith an elementalthicknessdr. The units of O"eare (length)-I, and O"eand L must be expressedby the sameunit of length to makethe exponentofEq. 16.7dimensionless.The extinctioncoefficient O"e is analogous to y in Eq. 9.11 for penetrationthrough a filter. (Note that Eqs. 16.7and 9.11 have the sameform.) The apertureand lens arrangementshown in Fig. 16.1 is neededto ensurethat the particlesare illuminatedwith a parallel beamof light (i.e., to ensurethat all rays are parallel) and that only the attenuatedparallel light reachesthe detector.Failure to include the lens and apertureat the detectorallows forward-scatteredlight from the particles to reachthe detector,in which caseEq. 16.7does not hold. The extinction producedby a particle is a function of the particle extinction efficiency Qe.
~=
I I
radiant power scattered and absorbed by a particle
radiantpower geometricallyincidenton the particle
(16.8)
The geometricallyincidentpower is the amountof energyper secondthat intercepts the cross-sectionalareaAp of the particle. The particle extinction efficiency gives a relative measureof a particle's ability to removelight from a beamcomparedwith simple blocking or interception by the projected area of the particle. The extinction efficiency times the projected area of a particle is the cross-sectionalareaof light removedfrom a beamby the'particle. A Qeof 2.0 indicatesthat a particle removes twice as much light as it would by simple projected-areablocking. Values of Qe range from 0 to about 5. The particles in a cubic centimeterof a monodisperseaerosolwill eachremoveApQewatts from a beamof unit intensity. From the definition of the extinction coefficient, a monodisperseaerosolof N particles per unit volume has an extinction coefficient of
0"e = NAp~ n = ~~ 4
(16.9)
Qeis defined (Eq. 16.8) in the sameway as the combinedsingle-filter efficiency, given in Eq. 9.14. Also, Eqs. 16.7 and 16.9 are analogousto Eq. 9.19 for filters. Both relate macroscopicphenomena-light attenuationand filter penetration--to microscopic-scaleparticle properties,Qeand £1:,respectively. The extinction efficiency of a particle is the sum of its scatteringefficiency Qs and its absorptionefficiency QQ' ~=Qs+Qa
(16.10)
where Qsand Qa are defined by equationsequivalentto Eq. 16.8.It follows from Eq. 16.10that, for monodisperseparticles O"e = o"s + O"a
(16.11)
where O'sand O'aare defined by Eq. 16.9 with Qereplacedby Qs and Qa, respectively. For nonabsorbingparticles,Qe= Qsand O'e= O's.For polydisperseaerosols, Eq. 16.9 holds for eachparticle size, and the combinedeffect is given by the sum of the 0'8'Sfor all the particle sizes.
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,
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I 0
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I
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I
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1 2 Particle diameter for A = 0.52 /lm (/lm)
I
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FIGURE 16.2 Extinction efficiency versus particle size for spheres(after Hodkinson, 1966).
lations. The curves for nonabsorbingparticles in Fig. 16.2have been smoothedto remove numeroussmall bumpsand wiggles. Thesebumpsare greatly reducedfor absorbingparticles.Extinction efficiency decreases rapidly with decreasinga in the region below a = 2, asshownin Figs. 16.2and 16.3.
2
4
6 cx(In-1)
8
10
12
FIGURE 16.3 Extinction efficiency versus a(m - 1) for nonabsorbing sphereswith refractive indices between 1.33 and 1.50.Data from Kerker (1969).
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EXTINCTION
357
It is of interestto expressBouguer'slaw, Eq. 16.7,in tenDSof the particle mass concentrationCmfor monodisperseaerosols. c
m -
Np
p
7td3
(16.18)
6
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(16.19)
2ppd -
For large particles, for which Qe is a constant, extinction increases (scattering becomes greater) with decreasing particle size for a constant mass concentration. Figure 16.5 shows the relative scattering per unit mass for three materials as a function of particle size. Because Qedecreasesrapidly (Qe ocd4) for small particles, there is a particle size for maximum extinction, as shown in the figure, and a range in which a reduction in size gives an increase in extinction for a given mass concentration. This fact explains why the visual impact of atmospheric aerosols and smokestack plumes is governed by the concentration of particles in the size range from 0.1 to 2 Jlm and why a reduction in concentration outside this range has little effect on extinction or visibility. Table 16.2 gives the relative extinction for a given mass of water in the atmosphere in the form of molecules, aerosol particles, or raindrops. For a given mass concentration of water, aerosol particles in the range of 0.1 to 1.0 Jlm are a million times more effective than vapor molecules and a thousand times more effective than
16 1
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FIGURE 16.5 Relative scatteringper unit massof aerosolversusparticle size for light at a wavelengthof 0.52 ~m.
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FIGURE 16.6 Diagramshowingscatteringangle,scatteringplane,and the polarizedcomponentsof scatteredlight.
slightly from its incident direction has a small scatteringangle and is said to be forward-scattering light. Light that is reflectedor scatteredback to the source(i.e., e = 180°) is called back-scatteredlight. The incident and scatteredbeamscan each be resolvedinto two independentlypolarizedcomponents,one whoseelectric vector is perpendicularto the scatteringplane (designatedby the subscript 1) and one whose electric vector is parallel to the scatteringplane (subscript2). The general theory of light scatteringby aerosolswas developedin 1908 by Gustav Mie. The theory gives the intensity of light scatteredat any angle e by a sphericalparticle with known values of a and m illuminated by light of intensity 10.The resulting Mie equationsare very complicatedfor particleswith sizesgreater than the wavelengthof light. When particlesare much smallerthan the wavelengthof light (d < 0.05 ~), the much simpler Rayleigh scatteringtheory can be used.For theseparticles (and gas molecules),the instantaneouselectromagneticfield of the incident light is unifonn over the entire particle, creatinga dipole that oscillatesin synchronizationwith the incident electromagneticfield. This oscillating dipole reradiateselectromagnetic energyin all directions.The intensityandpatternof scatteringis known as Rayleigh scatteringand, for unpolarizedillumination, is given by for d < 0.05 ~m
(16.20)
where 1(8) is the total intensity, at a distance R from the particle, of the light scattered in the direction 8. Equation 16.20 indicates that the intensity of the scattered light at any angle will be proportional to (j6/1-.4. This is a strong function of diameter and a strong inverse function of wavelength. Rayleigh scattering depends on the particle volume squared and, as a result, is independent of the particle shape. The dependence of scattered light on wavelength (1-.-4)accounts for the blue color of the sky: Blue light is scat-
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When Eq. 16.20 is integrated over the surface of a sphere,the result for nonabsorbingparticles is the total radiant power of the light scatteredby the particle in all directions, which is equal to lo7td2~/4. For particles larger than about 0.05 1.1ffi, the Mie equationsmust be usedto determine the angulardistribution of scatteredlight. Mie scatteringtheory providesa complicatedbut very generalsolution to the problem of scatteringby spheres.The theory is valid for absorbingand nonabsorbingspheresfrom moleculardimensions to particles large enoughto be treatedby classicaloptics. The Mie theory is identical to the Rayleigh theory for particle sizesin the Rayleigh region. . The Mie solution is the completeformal solution of Maxwell's equationsfor the combinedvector wave equationsfor the incident wave,the wave insidethe particle, and the scatteredwave, subjectto a setof boundaryconditionsat the surfaceof the particle.A derivationof this theoryandan explanationof the equationsarepresented in van de Hulst (1957, 1981)and Kerker (1969). The Mie solution .of Maxwell's electromagneticequationsis a solution to one of the classicalproblemsof physics.Although the solution is exact, its utility was severelylimited beforethe arrival of digital computers,becauseof the extremecomplexity of the computations.With a mechanicalcalculator,a single solution for one value of m, a, and e could take from severalhoursto severalmonths.Efficient computer programsthat provide nearly instant solutions to the Mie equationsare now available. (SeeDave, 1969,and Wilson and Reist, 1994.) At a distanceR in the direction e from a sphericalparticle illuminated with unpolarized light of intensity 10(W1m2[W/cm2]), the scatteredintensity is 1(9) = loA2(4 + i2) 81t2R2
16.23)
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~ 47t2R2
(16.24)
for perpendicularpolarization and by
~
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more over a few degrees,for an a of 10 or more. As the particles get larger, the scatteringin the forward direction becomesmuch strongerthan at other angles.The segmentof the curve for a = 10 between0 and about 20° is known as theforward lobe and consistsprimarily of diffracted light. It is absentin Rayleigh scattering. Agglomeratedor very irregularparticlesand absorbingparticlestend to smoothout the irregularscatteringpatternsshown.The irregularscatteringpatternsshownapply only to single particles or monodisperseaerosols.Most aerosolsare polydisperse, and the patterns from many different sizes add together to give much smoother curves. Figure 16.9gives the combinedMie intensity parameterfor water droplets as a function of the particlesizeparameterfor scatteringat e = 30° and90°. Thesecurves
0 = 30°
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~ ~ QI .S QI ~ CI
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where Bois the luminanceof the object andB' is the luminanceof the background. Luminance describesthe brightnessof a surface and is defined as the luminous intensity per unit solid angle per unit area of surface.The units of luminanceare lumens/m2.sr, or candelasper squaremeter (cd/m2).The sky nearthe horizon has a luminanceof 1Q4cd/m2on a clear day and 1Q-4cd/m2on an overcast,moonless night. The luminanceof a pieceof white paperis 25,000cd/m2in sunlight and 0.03 cd/m2in moonlight. When the luminanceof an object and its surroundingbackgroundare the same, the contrast is zero, and the object cannotbe distinguishedfrom the background. If the object is less luminousthan its background,Cois negative,reachinga value of -1 for an ideal black object againsta white (or lighted) background.When the object is brighter than its background,Cocan haveany positive value.For example, a light at night representsa casein which Co has a large value. Contrastsgreater than about 10 seldomoccur during daytime. As we look at progressivelymore distantobjects,suchasthe hills shownin Fig. 16.10,each more distant hill is lighter in tone. At the limit of visibility, the hills havethe samebrightnessasthe surroundinghorizon. What lightensthe appearance of the distant hills and mountainsand reducestheir contrastrelative to the sky is the light scatteredto the observerby the interveningaerosol.As the distanceis increased,more aerosolis betweenth~ observerand the mountains,and they appear lighter. The inherent contrast-that is, the contrastwithout the intervening aerosol--of forest-coveredmountainsagainstthe sky is approximately-1. The apparentcontrast CRwhen the mountainsare observedthrough the intervening aerosolis less than the inherent contrastbecauseof the light scatteredby the aerosoland is defined as
-
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FIGURE 16.10 Photographshowing the increasein the apparentluminance of forestcoveredhills with increasingdistance.Photographcourtesyof W. L. Hinds.
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where Ba/crehas been replaced by BRaccording to Eq. 16.29. Substituting Eqs. 16.29 and 16.31 into 16.27 gives Koschmieder's equation, an expression for the apparent contrast of an object when viewed against the horizon as a function of the distance between the observer and the object:
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by Eq. 16.35 in the sameunits of length in which O"eis expressed.The equation relatesthe visual rangeto O"e and,consequently,to the aerosolpropertiesof number or massconcentrationand particle size by Eq. 16.9, 16.13,or 16.19.When the visual range is greaterthan 30 kIn, the effect of scatteringby air moleculesmust be included by summingthe extinction coefficients for air (Eq. 16.17)and aerosol. In 1899, Lord Rayleigh used the precedingvisibility principles and Eq. 16.17, a result of the Rayleigh theory, to determineAvogadro's numberby visual observation. One clear day, while vacationing in Darjeeling, India, he observedMount Everestat a distanceof about 150kIn from the terraceof his hotel. Through simple calculations,he arrived at a value of 3 x 1019for n (adjustedto sealevel), which correspondsto a value of No of 6.7 x 1023.This is remarkably accurateconsidering the simplicity of the measurementand illustratesthe power of light scattering as an atmosphericmeasuringtool.
EXAMPLE What is the daylight visual range in polluted air containing O.3-J:lm-diameter particles at a concentrationof2 x IO!o/m3[2 x 104particles/cm3]?Assumethat the particle refractive index is 1.5.
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4 3.91 = 1.97 x 105 cm = 2.0 km I 1.98x 1O~5
Equation 16.35 gives the true visual range, when the correct value of E is 0.02. The latter depends on the level of illumination. The meteorological range is defined as the distance that yields a contrast threshold of 0.02, regardless of the illumination level or the uniformity ofcre' Equation 16.35 is the definition of the meteorological range. In many situations the absorption ,by the atmospheric aerosol is negligible compared to scattering, and crecan be replaced by crs(Eq. 16.11). This assumption permits the meteorological range to be estimated by an instrument called an integrating nephelometer (see Section 16.5), which measures cr, (also called bscat)
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ond method relies on the color of the scattered light as a function of angle to determine particle diameters from 0.2 to 1.5 I.1m.Monodisperse aerosols in this size range produce colored bands in their angular scattering pattern called higher order Tyndall spectra, or HOTS. The number or the angular positions of red bands can be used to determine the particle size by comparing the observed values with calculated values. The sharpness of the colored bands increases with increasing monodispersity. Transmissometers, or light-attenuating photometers, are instruments that measure the attenuation (extinction) of light through a path length that may range from a few centimeters to a few kilometers. Stack transmissometers have a focused light source positioned on one side of a stack and directed across the stack to a detector. These instruments measure the extinction by the smoke in the stack when constructed according to Fig. 16.1, but often the detected beam is inadequately collimated, and such instruments detect significant amounts of forward-scattered light. Because of multiple scattering effects at high aerosol concentrations, the transmitted light intensity varies in an unpredictable way with concentration. This unpredictability is not a problem for pure extinction by aerosols. A gravimetric calibration can be used if the particle refractive index and size are constant. To eliminate forward scattering, the size of the pinhole before the detector must have an angular diameter, measured from the center of the preceding lens, of not more than 0.38/a radians. Three types of instruments rely on measurementsof light scattered at fixed angles from polydisperse aerosols: photometers, nephelometers, and optical-particle counters. Photometers are used to measure relative concentration by measuring the combined light scattered from many particles at once. The aerosol flows continuously through the instrument at several liters per minute. The illumination and collection optics are arranged so that the light scattered at a fixed range of angles reaches the detector. Depending on the design, the instrument may measure the scattered light in the region of 90°, 45°, or less than 30°. Forward-scattering photometers« 30°) are less sensitive to refractive index than 90° scattering instruments. Photometers are used with monodisperse test aerosols for filter penetration testing. Typical instruments can measure penetrations as low as 0.001 %. In the field of occupational hygiene, portable photometers are used as direct-reading instruments to measure mass concentration. The scattering angles and wavelengths used are selected to reduce the effect of particle size and refractive index on the response of the instrument. Electronic averaging is used to provide a digital readout of mass concentration over the range of 1 I.1g/m3to 100 mgim3. The output of such instruments is sensitive to aerosol material and particle size distribution, and errors in concentration of a factor of two or more are possible with variation in these parameters. Instruments should be calibrated with side-by-side filter samples if they are used for aerosols that differ appreciably from the manufacturer's calibration aerosol. These photometers provide continuous output and are well suited to monitoring concentration changes with time and location, provided that the particle size and chemical composition remain constant. Commercially available photometers and other light-scattering instruments are reviewed by Pui and Swift (1995).
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(16.38) where Q is the flow rate through the sensitivevolume and t is the time for a particle to traversethe beam,plus the signal-processingtime. Coincidenceerror can be minimized by reducingthe flow rate and the dimensionsof the beam;however, small beam dimensionsrequire a small aerosolstreamdiameterso that nearly all particles will passthrough the beam for a high counting efficiency. Someinstruments use aerodynamicfocusing of the aerosolstreamto achievea streamdiameter of lessthan 0.1 mm. Table 16.3gives maximum concentrationsfor a 10%error in count due to coincidence. Care must be taken in interpreting size distribution data from optical particle counters.The distribution obtainedis the size distribution by count or the number distribution, but the numberof particlesbelow the lower size limit is unknown and may be large. Plotting the results as counts per micrometer (number divided by channelwidth) gives an accuraterepresentationof the data.Plotting the fraction per micrometer may be seriously in error if there is uncertaintyin the total numberof particles. Another particle-sizing method,called photon correlation spectroscopy(PCS), dynamic light scattering(DLS) or quasi-electriclight scattering(QELS), relies on both the light scatteringand the Brownian motion of the particles.An ensembleof
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16.6 One sunny day, while measuring the atmospheric aerosol concentration (25 ~g/m3)on the top of the UCLA Centerfor Health Sciencesbuilding, you notice that you canjust seePalos Verdes Hill 22 mi away. If you assume that the atmospheric aerosol is monodisperse,what is its particle size? Assumethat Qe = 1.0 and Pp = 1000kg/m3 [1.0 g/cm3]and neglect extinction due to air molecules. ANSWER: 0.34 ~m. 16.7 Repeatproblem 16.6,but include the effect of air molecules.Assumethat
II.= 0.5 ~m. ANSWER:0.40~m.
16.8 You observethe visual rangein urban air. If you assumethat the rangedependssolely on the fine-particle mode and that it can be representedby an equivalent nonabsorbingmonodisperseaerosol, what is its particle size? Assumethat Qe= Qs = 1 and the particle density is 1500kg/m3[1.5 g/cm3]. [Hint: Usethe empiricalexpressionrelatingvisual rangeto fine-particlemass concentrationand Bouguer'slaw expressedin termsof massconcentration.] ANSWER: 0.3 ~m. 16.9 What is the meteorologicalrange in particle-free air at 20°C at sea level? Assume that A.= 0.5 ~m. ANSWER: 220 kIn. 16.10 (a) What is the maximum distance at which one can see a black automobile on a foggy, moonlit night? Assume that the fog consists of 5-~m droplets at 101O/m3 [104/cm3], that the background luminance is 10-3cd/m2, and that your eyes are completely dark adapted. (b) Repeat part (a) if the headlights are on and are directed toward you; assume that the inherent contrast is 104. ANSWER: (a) 6.1 m; (b) 30 m.
REFERENCES Blackwell, H.R., "Contrast Thresholdsof the Human Eye," JOSA,36, 624-643 (1946). Cadle, R. D., TheMeasurementof Airborne Particles, Wiley, New York, 1975. Cooke, D. D., and Kerker, M., "ResponseCalculationsfor Light ScatteringAerosol Particle Counters,"Appl. Optics, 14,734-739 (1975). Dave, J. V., "Scatteringof ElectromagneticRadiationby a Large, Absorbing Sphere,"IBM J. Res.Develop.,302-313 (May 1969). Dave, J. V., Subroutinesfor Computing the Parametersof the ElectromagneticRadiation Scatteredby a Sphere,Rept. No. 320-3237,IBM Scientific Center,Palo Alto, CA, May 1968. Fenn,R. W., "Optical Propertiesof Aerosols," in Dennis, R. (Ed.), Handbook on Aerosols, USERDA, 1976,TID-26608, Oak Ridge, TN, 1976.
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The analysisof aerosolmotion in the precedingchaptershas focusedexclusively on the motion of individual particles.Thereare,however,somesituationsin which individual particle motion is negligible comparedwith motion on a larger scale.In one such situation, the gas phaseof an aerosolhas a density that differs from the surroundinggas. For example,the buoyancyof a stack gasplume causesthe particles containedin it to rise many metersafter leaving the stack.The movementof gasesunder thesecircumstancesis a subject of fluid mechanicsand meteorology and is not coveredhere.Whenthereis no differencein gasdensitybetweenan aerosol cloud and its surroundings,there are still situations in which the cloud as an entity can move much faster than the individual particles that make up the cloud. For the purposesof this chapter,an aerosol cloud is defined as a region of high aerosolconcentrationhaving a definite boundaryin a much larger region of clean air. The mechanicsof clouds is far more complicatedthan that of individual particles, and a complete description of the motion of clouds does not exist. In this chapter,we define the order of magnitudeof the conditionsunderwhich cloud motion can occur. Sincecloud motion dependson the bulk propertiesof aerosols,we evaluatefirst the viscosity of an aerosolcloud relativeto that of pureair. Considera monodisperse aerosol of I-J,lmspheresof standarddensity at a massconcentrationof 100 g/m3. This is a very high massconcentrationand correspondsto the densestsmokesthat can be produced.The numberconcentrationof suchan aerosolis 2 x 108/cm3.The averagespacingbetweenparticles is 17 J,lm,or 17 times the diameterof the particles, so the aerosolis still mostly air, andthe particlesare far enoughapartto have little effect on eachother. The viscosity of such a two-phasesystemllc is Tlc= Tl(1 + 2.5Cv) where C" is the volume concentration-that is, the volume of particles per unit volume of aerosol. In this example, C" = 0.0001 and the aerosol has a viscosity 0.025% greater than that for pure air, a negligible difference. The density of a cloud exceeds that of particle-free air by an amount equal to the mass concentration of the particles. At a mass concentration of 100 gim3, an aerosol has a density 8.3% greater than that of clean air. It is this difference in density that causesbulk motion of aerosols. When such motion is caused by gravity, it is called clbud settling or mass subsidence. Cloud settling occurs when the aerosol concentration is sufficient to cause the entire cloud to move as an entity at a velocity significantly greater than the indi379
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EXAMPLE A sphericalcloud of water droplets 5 m in diameterhas a massconcentrationof g/m3.What is the terminal settling velocity of the cloud? Pc = 0.001 kg/m3 [lO~ g/cm3 4d~PcPgg 3'12 = CDRe: =
This value is beyond the range of values given in Table 3.4 and is in the region where CDhas a constantvalue of 0.44. Hence,
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whereC" is the slip correctionfactor of the particles.Equation17.7holds only when Re < 1 for the cloud, which usually implies a cloud size smaller than a few centimeters.Cloud settling in the Stokesregion is favoredby largecloud size,high mass concentration,and small particle size. A massconcentrationof only 9.8 mgim3 is required for a G value of unity for a cloud I cm in diameterwith particles 1 ~m in diameter. Table 17.1 gives the minimum massconcentrationsrequired for cloud settling (G = 1). The values were calculatedusing Eqs. 17.4 and 17.6 and the procedure given in Section3.7. The table showsthe importanceof the cloud size and particle size in determiningwhether cloud settling occurs.It is apparentthat most aerosol cloudswill, at leastinitially, havecloud motion, althougha valueof G > 100is probably required for sustainedmotion. Becauseof the dilution, breakup,and erosionmechanismdescribedearlier, the lifetime of clouds lessthan a few metersin diameteris brief. Furthermore,a value of G = 1 meansonly that the cloud settling velocity is equalto the particle settling velocity, a velocity that is very small,particularly for submicrometer-sized particles. Consequently,it is of greaterpractical significanceto identify the concentrationrequired for a cloud to have an appreciablesettling velocity, suchas 0.01 m/s [I cm/ s], ashasbeendonein Table 17.2.In this casethe requiredmassconcentrationdoes not dependon the particle size, althoughthe numberconcentrationdQes.Note that the massconcentrationvalues are microgramsper cubic meter in Table 17.1 and milligrams per cubic meter in Table 17.2. When the particle numberconcentrationin a cloud is high, coagulationwill rapidly changeboth the numberconcentrationand the particle size, althoughthe mass concentrationwill remain constant.As can be seenfrom Eq. 17.6, an increasein particle sizewith a constantcloud diameterand massconcentrationleadsto a rapid decreasein the value of G. Thus, coagulationinhibits cloud settling. In sum, the severaleffectsdescribedhere suggestthat aerosolclouds often move initially as an entity, but quickly revert to individual particle motion becauseof the combined effect of erosion,breakup,size distribution, dilution, and coagulation. Generally, the gas composition and temperatureinside a cloud and in the surroundingair will be different.This differencein densityis usuallymuch greaterthan the net cloud density p", due to the weight of the particles.For example,in natural atmosphericclouds, a temperaturedifference of 0.2°C producesa buoyant force TABLE 17.1 Minimum Concentration for Cloud Settling (G = 1) at Standard Conditions
0.04 0.1 0.4 1.0
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REFERENCES
385
cloud settling and only a very slight reduction of particle settling velocity, due to the increasein viscosity of the aerosolat high concentrationsand the slight upward velocity of the suspendinggascausedby the displacedparticle volume. Thesetwo
effectshaveonly a minor influence« 0.1%) on the settlingvelocityof particles in containedaerosols,even extremely denseaerosols.
PROBLEMS
7.1 A cigar smokerblows a smokering that falls with a noticeablevelocity. If the ring is consideredto be a sphere0.05 m [5 cm] in diameter,what is its cloud settling velocity? Assumethat the smokegas is air at room temperature and the smokeparticle concentrationis 30 g/m3.If the smokeparticles are 0.4 I.1n1 in diameterand of standarddensity, what is the value of G? ANSWER: 0.17 m/s [17 cm/s], 25,000. 17.2 Repeatthe first part of Problem 17. for a smokegas-phasedensity that is 2% greaterthan that of air. ANSWER: 0.24 m/s [24 cm/s].
REFERENCES Chen, B. T. and Yeh, H. C., "Effects of Cloud Behavior on a Cigarette SmokeAerosol," LovelaceFoundationReport LMF-129, Albuquerque,NM, 1990. Fuchs,N. A., TheMechanicsof Aerosols,Pergamon,Oxford, 1964. Kaiser, G. D., and Griffiths, R. F., "The Accidental Releaseof AnhydrousAmmonia to the Atmosphere:A SystematicStudy of FactorsInfluencing Cloud Density and Dispersion," J. Air Pol/. Control Assoc.,32, 66-71 (1982). Martonen,T. B., "Deposition Patternsof CigaretteSmokein HumanAirways," Am. Ind. Hyg. Assoc.J., 53, 6-18 (1992). Maude,A. D., andWhitmore, R. L., "A GeneralizedTheory of Sedimentation,"Brit. J. Appl. Phys.,9. 477-482 (1958). Phalen,R. F., Oldham, M. J.. Mannix, R. C., and Schum,G. M., "Cigarette SmokeDeposition in the TracheobronchialTree: Evidencefor Colligative Effect," Aerosol Sci Tech,20, 215-226 (1994). Plesset,M. S., and Whipple; C. G., "Viscous Effects in Rayleigh-Taylor Instability," Phys. Fluids, 17,1-7 (1974). Stober,W., Martonen,T. B., and Osborne,S., Jr., "On the Limitations of AerodynamicalSize Separationof DenseAerosolswith the Spiral Duct Centrifuge," in Shaw,D. T. (Ed.), Recent Developmentsin Aerosol Science,Wiley, New York, 1978. Wen, C. S., TheFundamentalsof Aerosol Dynamics,World Scientific, Singapore,1996.
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CHARACTERISTICS
TABLE 19.1 Particle Size and Natural Background Concentration of Bioaerosols'
Viruses Bacteria Fungal Spores Pollen
0.5-1000 0-10,000 0-1000
'Data primarily from Jacobson and Morris (1976).
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ria have been identified. Those that causehuman diseaseare describedas human pathogens.Someexamplesof infectiousdiseasesknown to be causedby aerosolized bacteriaare tuberculosis,legionellosis,and anthrax.Opportunisticpathogensarenot harmful to healthypeople,but can infect thosewith compromisedimmunesystems. Many environmentalbacteria colonize water or soil and are releasedas aerosols when the water or soil is disturbed.A gram of soil can contain 109bacteria.In indoor environments,bacteriamay colonize accumulationsof moistur! in ventilation systemsand becomeaerosolizedby air currentsor vibration. Somebacteriarelease sporescalled endospores.Theseare hardy, dormantversionsof the bacteria,0.5 to 3 f.1min size, that are easily carried by air currents.Included in this categoryare actinomycetesassociatedwith hypersensitivitypneumonitis. Fungi representa unique group of organismsthat are found everywherein the outdoor environmentand in most indoor environments.They may occur as singlecelled organismssuch as yeast or, more commonly, as microscopic,multicellular branchingstructurescalled hyphae.Massesof hyphaeare easily visible. Molds and mildew are visible fungal growths on surfaces.Only about70,000of the estimated 1.5 million fungi have been identified. The majority of fungi are saprophytes,organismsthat get their nutrientsfrom deadorganic materialand facilitate its decomposition. They are found primarily in soil, in damplocations.and on decayingvegetation. Most fungi disperseby releasingsporesinto the air. Thesefungal spores, particlesof 0.5 to 30 f.1min size,are resistantto environmentalstressesandadapted to airborne transport.They may be single celled or fragmentedhyphae,typically 2-4 JiIn, or fruiting bodiesthat form at the end of hyphae.They are aerosolizedby air currentsor mechanic,aldisturbances.Yeasts,by contrast,do not producespores, but becomeairborneby aerosolizationof the liquid in which they grow. Inhalation of fungi or sporescan causeinfectious disease,such as histoplasmosis,or opportunistic infection, suchas aspergillosis.Most fungi are associatedwith allergic diseases(e.g., asthma). Virusesare intracellularparasitesthat can reproduceonly insidea hostcell. They consistof encasedRNA or DNA and little else.Although nakedvirusesrangefrom 0.02 to 0.3 ~m, most airbornevirusesare found aspart of droplet nuclei or attached
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for bioaerosolparticles employsmany of the methodsusedfor nonbioaerosolparticles, but the analytical methodsimposecertain limitations on the sampling process.There are physical and biological aspectsto bioaerosolsampling.The physical aspectsare the samefor bioaerosolsand nonbioaerosols.The biological aspects may require aseptichandling to preventcontaminationand specialattention to the viability of the organismsduring the processof sampling and analysis.Viability, which dependson the organism,the environment,and the samplecollection and analysismethodsused,introducessignificantuncertaintyinto the quantitationof airborneconcentrationsofbioaerosols.Estimatesof the fraction of viable particlesthat survive the sampling and analysisprocess,grow, and are countedrange from 0.1 to 0.00I (Macnaughtonet al., 1997).Other importantfactorsin bioaerosolsampling are high variability in number concentration,wide particle size range of interest, analytical sensitivity to under- and overloading,and interferenceby otherparticles. Comprehensivereviews of bioaerosolsamplingmethodsare given by Nevalainen et al. (1993), Macher (1995), and Willeke and Macher (1998). There are three stagesof bioaerosolsampling.The first concernsthe inlet efficiency, which is the sameas that for nonbioaerosolparticles.This stageis covered in Sections 10.1and 10.2 on isokinetic and still air sampling.The secondstageis the collection or depositionof particles onto glass slides, a semisolidculture medium, or water. This is accomplished by adapting the basic methods used for nonbioaerosolsto bioaerosols.The third stageis biological analysisto identify and quantitatethe bioaerosolparticles present.This stageis unique to bioaerosolsampling. Most bioaerosolanalysisinvolves either microscopicexaminationor growth in culture medium followed by counting colonies.A discussionof the secondstage of bioaerosolsampling follows. The most common collection methodsrely on impaction, the sameprocessas describedin Section5.5 for nonbioaerosols.Slit impactorsimpactparticlesdirectly onto a culture medium. For bacteriaand fungal spores,the culture medium,called agar, is a semisolid material containing water and nutrients that foster the growth of the viable particles that are collected. For viruses, cell or tissue culture media are used.Typically, the agarfills a 100-mmor 150-mmdisposablepetri dish, called a culture plate,that is slowly rotatedunderthe slit to provide a history of bioaerosol concentration.Rotating slit impactors have flow rates of 28-50 L/min and cutoff diametersof about 0.5 J:lm. Multijet impactorshave 100-500jets impactingdirectly onto agarculture plates. This action spreadsthe collected particles to many locationsto prevent overloading. Single and multistage impactorsare available with cutoff diametersfrom 0.6 to 8 J:lm.Shearforces in the high-velocity jet and during impactionmay causeloss of viability. '
Other types of impactors are used to collect pollen. The pollen grains are impacteddirectly onto glassslidesor adhesive-coated transparenttapefor microscopic examinationand counting. Culturing is not used for pollen. One type of sampler (the RotorodOO sampler)impacts pollen grains directly onto adhesive-coatedpolystyrenerods.The rods are transparentfor direct microscopicparticle counting.The rods are squarein cross section,are 60 mm long and 0.5 to 1.6 mm wide, and are
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countedas a single colony. Second,if two different speciesare sufficiently close, one may inhibit the growth of the other. Finally, due to its chemical nature, a nonbioaerosolparticlemay inhibit the growth of a nearbyviable particle.For a given sampler,the samplingtime t to achievea desiredsurfacedensitys (coloniesor particles per unit area) is given by t=-=--
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whereA is the area onto which particles are deposited,CNis the averagenumber concentrationof bioaerosolparticles, and Q is the samplerflow rate. For microscopic counting, a surfacedensity of 108/m2[10,000fcm2]gives good results; for counting colonies,a surfacedensity of 104fm2[lfcm2] is desired. A different approachis taken formultijet impactors.There is a deposition site directly below eachjet. After culturing, one cannot tell whetherjust one or more than one organismhas been depositedat a particular site. Regardlessof the number of organismsinvolved, a site with a colony is called afilled site or a positive hole. Platesare analyzedby countingthe numberof filled sites.For a given multijet impactor, the more filled sites, the greaterthe proportion of those filled sites that havemultiple organisms.Willeke andMacher(1998)andMacher(1989)give tables with correction factors to estimatethe total numberof viable organismscollected, basedon the numberof filled sites.Also given is the statisticaluncertaintyof these estimates.As shown in Table 19.2,the correction is less than 5% when less than 10% of the sites are filled sites.The correction factor is 2.0 when 80% of the sites are filled sites.For an impactorwith Nj jets, the following empirical equationgives
TABLE 19.2 Correction Factors for Multiple Particle Collection in Multijet Impactors. Filled Fractionb 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Correction FractionC
Filled Fraction
Correction Factor
1.026 1.054 1.084 1.116 1.151 1.189 1.231 1.,277 1.329 1.386
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1.452 1.527 1.615 1.720 1.848 2.012 2.232 2.559 3.154 >5.878
'Calculated from Macher (1989). bfraction of deposition sites with colonies. "Total number of viable particles collected equals the number of filled sites times the correction factor.
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TABLE 20.3
Spreading Factor for Liquids on Glass Slides'
Liquid
Surface Coatingb
Particle Size Range (J:lm)
DOPC DOP DOP DOP DOP Oleic acid Oleic acid Oleic acid Sulfuric acid
None L-1429 L-1428 FC-72I L-IO83 none L-1428 FC-72I L-1429
2-50 2-5 2-50 10-20 50-150 2-50 2-50 3-29 0.02-0.07
Spreading Factor 0.30 0.58 0.69 0.74 0.69 0.42 0.70 0.75 0.71
"Data from Liu, Pui, and Wang (1982); Cheng, Chen, and Yeh (1986); OlanFigueroa, McFarland, and Ortiz (1982); and John and Wall (1983). bL-IO83, L-1428, L-1429, and FC-721 are fluorocarbon oelophobic surfactants manufactured by 3M, Inc., St. Paul, MN. CDi (2-ethylhexyl)
phthalate.
sion. Fractalsare structuresthat havegeometricallysimilar shapesat different levels of magnification. This characteristicis calledself-similarity or fractal morphology. Thefractal dimensionrelatesa property such as the perimeter or surfacearea of an object to the scaleof the measurement. For example,the length ora coastlinecan be representedby a continuousseries of equal straight-line segments.The size of eachsegmentrepresentsthe step size, or "ruler" length, from one contactpoint to the next. The total length dependson the step size. The smaller the step size, the finer is the detail one can measureand the longer the measuredcoastline.The relationshipbetweenstepsize and measured length is shownin Figure20.4 for the coastlineof GreatBritain. This log-log graph, called a Richardsonplot, of the measuredcoastlinelength L versusthe step size A approximatesa straight line with a negativeslopem. The equationfor this straight line is L
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(20.5)
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OPTICAL MICROSCOPY
413
longer, but retains its self-similarity. Other examplesof quantitiesthat can be describedas fractalsare the bordersof clouds and smokeplumes,turbulence,and the geometry of the airways of the lungs.
20.3 OPTICAL MICROSCOPY Becausethe optical microscopeis a commoninstrumentusedin many areasof science,only thosecharacteristicsof microscopesthat are important to particle sizing arecoveredhere.Optical microscopesaremostsuitablefor countingand sizing solid particles having diametersof 0.3-20 j.1m.While any professional-gradecompound microscopecan be usedto size particles, those with multiple objectives mounted in a nosepieceand a mechanicalstageare the most convenient.Figure 20.7 shows the major componentsof such a microscope.Other desirablefeatures,not shown, are a built-in substageilluminator and binocular eyepieces. The light path for an optical microscopeis shown in Fig. 20.8. Light from an external source,usually a microscopelamp, is reflected8P into the condenserby a mirror. The condenserconcentratesthe light onto the samplefrom below. The particles to be measuredare mountedon a glassslide held in the mechanicalstage.A magnified image of the particles is formed by the objective lens positionedjust abovethe slide. The image is locatedat the eyepiece,which further magnifies it to createa virtual image at a distancecomfortablefor viewing. In this situation, the
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FIGURE 20.7 Diagram of an optical microscope.
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nosepiece,shown in Fig. 20.7, permits 2-5 objectivesto be mountedon the microscopeand rotatedinto place as needed.The objectivesare positionedso that, when the microscopeis exactly in focus for one objective,the otherswill also be in focus. Particlescan be captureddirectly on a glassslide by sedimentation,electrostatic precipitation, or thermal precipitation and viewed directly in a microscope.Powders or bulk dust samplescan be spreadout on a slide by gentle smearingwith a toothpick. The procedurefor viewing particles capturedon a filter is to cut out a small pie-shapedpiece of the filter and place it on a clean glassmicroscopeslide. A drop of immersionoil is allowed to saturatethe filter to make it transparent.The filter is coveredwith a glass coverslip and examinedin the microscope.The immersion oil must have the samerefractive index as the filter material. The most common type of filter for optical microscopeexaminationof aerosolparticles is a cellulose estermembraneof 0.45-l.1mpore size. Thesefilters have a refractive index of 1.510.Unfortunately, particles having exactly this refractive index are also made transparentand cannotbe observedby this method. Although a total magnification of about 2500x can be obtained with standard objectives and eyepieces,the size of the smallestobservableparticle is governed not by magnification,but by the limit of resolution,which is controlledby the wavelength of light and the characteristicsof the objective lens.The limit of resolution, or, more simply, the resolution, is the smallesJscaleof detail that can be observed in the magnified image. Although greatermagnification producesa larger image, it doesnot increasethe observabledetail beyondthe limit of resolution.Resolution is usually defined as the minimum separationdistancerequired for two dots to be observedas two separatedots rather than as one elongateddot. Resolutionis characterized by a property of the objective lens called the numerical aperture, NA, defined as NA
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that producedin optical microscopy.The magnified image is viewed by projecting the electron image onto a fluorescentscreenor a photographicplate. The principles of resolutionand depth of field discussedin Section20.3 hold for electron microscopes,but the wavelengthof the radiation used in electron microscopesis much lessthan the wavelengthof light. The wavelengthof an electron is given by A e
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are cleared by the normal clearance mechanisms. Each of the asbestos-related diseases is associated with fibers in a specific size range. Fibers 0.15 to 3 J.1min diameter and 2-100 J.1mlong are associated with asbestosis, fibers less than 0.1 J.1m in diameter and 5-100 J.1mlong with mesothelioma, and fibers 0.15-3 J.1min diameter and 10 to 100 J.1Inlong with lung cancer (Lippmann, 1991). Because the risk of disease depends critically on the shape of the particles, microscopy must be used to evaluate the concentration of those asbestos particles whose size and shape make
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them hazardous. The U.S. occupational exposure standard for asbestos is 100,000 fibers/m3 [0.1 f/cm3] greater than 5 J.1Inin length and having an aspect ratio greater than 3: 1, as determined by the membrane filter method using phase contrast microscopy (PCM). The membrane filter method, or "NIOSH method," is an optical microscope technique used for the quantitative determination of the concentration of asbestos fibers in air. The method consistsof collecting breathing zone samplesat 0.5-16 L/min on 25-mm membrane filters (mixed cellulose ester membrane filters with a pore size of 0.45-1.2 J.1m). Open-facefilter holderswith a 50-mm protective cowl (inlet extension) are used. The sample volume is adjusted to produce a fiber surface density in the optimal range of 100-1300 f/mm2. A sample volume of 1.0 m3 [1000 L] is suitable for concentrations of 40,000-500,000 f/m3 [0.04-0.5 f/cm3]. The filter is "melted"to a solid, transparent film by hot acetone vapor and is sealed in triacetin. Fibers meeting the size and shape criteria are counted at 400--450x magnification (45x objective, NA = 0.65-0.75, and lOx eyepiece) by phase contrast optical microscopy. (Phase contrast microscopy is used to enhance the contrast of the fibers because the refractive index is near that of the mounting medium.) Any particle having a length greater than 5 J.1mand an aspect ratio of 3: I (length:diameter) or greater is considered a fiber, whether or not it is asbestos.Standard practice in the United States calls for using a Walton-Beckett graticule, shown in Figure 20.12. The Walton-Beckett graticule defines a 100-J.1m-diametercounting field with standard fiber shapes and sizes around the perimeter for comparison. Fibers meeting the size and shape criteria that are completely within the field are counted as one fiber, while those that are partially within the field are counted as one-half fiber. Sample fibers are also shown in Figure 20.12. Because of the limit of resolution of optical microscopes, this method cannot detect fibers less than 0.25 J.1min diameter. Consequently, a significant number of hazardous fibers are too small to be detected by PCM, so the count that is obtained should be considered an index of asbestos concentration, and not the true concentration. Because of the difficulty of sizing particles near the limit of resolution of the microscope, novice counters can underestimate fiber concentrations by a factor of two or more. More advanced methods of optical and electron microscopy may be used to determine whether a fiber that meets the size and shape requirements is in fact an asbestos fiber. This can be done most definitively by transmission electron microscopy (TEM), which allows observation and measurement of fiber size and shape for even the smallest asbestos fibers. TEM also provides the opportunity for positive identification through elemental analysis by energy-dispersive X-ray analysis
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PROBLEMS
operatorerror or bias. Automatedimage analysisattemptsto overcometheseproblems by using computeranalysisof particle images.First, the image of many particles is scanned,creatinga digital image in the form of an array of millions of picture elements(pixels). The image may be from an optical or electron microscope or a photograph.The computerinterpretsparticle size and shapefrom the arrangement of contiguousdark pixels on a light backgroundor vice versa.The computer can then calculatethe projectedareadiameter,maximum chord, Feret's diameter, Martin's diameter,perimeter,and various shapefactors. Mean and standarddeviations and momentsof thesequantitiesfor all particlesin the field can be calculated automatically. Measurementscan be made on all or selectedparticles in a field. Image analysiscan be combinedwith elementalanalysisin an electronmicroscope. Here,the microscopeis operatedundercomputercontrol to obtaininformationabout particle size, shape,and elementalcomposition of many particles automatically. Analysis of hundredsor thousandsof particles yields more elaborateinformation, suchas size distributions for selectedelementalcompositionor averageconcentrations of a specific elementversusparticle size. The accuracy,speed,versatility, and ability to make measurementswithout the possibility of operatorbias are the primary virtues of theseautomaticinstruments. They work best with a high-contrastimage in which the smallestparticles have an image size of severalmillimeters. When particle imageshave low contrast with fuzzy, gray edges,the size of the particles measuredis controlled by instrument settingsthat define what gray level constitutesthe edgeof the particle. In this situation, the entire sizing process,microscopy,photography,and imageanalysismust be calibratedto yield accurateresultsthat are free from bias. The procedurecan be further complicatedby overlappingparticles and a "noisy" background.Caremust be taken that the backgroundtexture, such as the holes of a capillary pore membrane(Nuclepore) filter, is not included in the analysis.Suchelementsare instinctively screenedout when we view an image, but the computerdoesnot have our instincts. PROBLEMS
.
20.1 Consideran aerosolcomposedentirely of sphericaldoublets(particlesthat consistof two spheresjoined at a point). Assumethat all spheresare identical with a diameterd. If the particles are viewed perpendicularto the line betweentheir centers: (a) What is the projectedareadiameterfor theseparticles? (b) What is their volume shapefactor? (c) What is the maximum and minimum ratio of Feret'sdiameterto the projected areadiameter? ANSWER: 1.41d,0.37, 1.41,0.71. 20.2 A straight fiber has an aspectratio of 5: 1. If the fiber is viewed perpendicular to its axis, what is the maximum ratio of Feret's diameterto projected areadiameter? ANSWER: 2.0.
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REFERENCES
Fletcher,R. A., and Small, J. A., "Analysis of Individual CollectedParticles,"in Willeke, K., and Baron, P. A. (Eds.),Aerosol Measurement,Van NostrandReinhold,New York, 1993. John, W., and Wall, S. M., "Aerosol Testing Techniques for Size-Selective Samplers," J. AerosoISci., 14,713-727 (1983). Kaye, B. H., A RandomWalkthrough Fractal Dimensions,Wiley-VCH Verlagsgesellschaft, Weinheim, Germany, 1989. Lippmann, M., "Industrial and EnvironmentalHygiene: ProfessionalGrowth in a Changing Environment,"Am. Ind. Hyg. Assoc..I:, 52, 341-348 (1991). Liu, B. Y. H., Pui, D. Y. H., and Wang X-Q., "Drop Size Measurementof Liquid Aerosols," Atm. Environ., 16,563-567 (1982). Mandelbrot, B. B., Fractals, Form, Chanceand Dimension,Freeman,SanFrancisco,1977, 1983. McCrone,W. C., andDeily, J. G., TheParticle Atlas, 2d ed.,Ann Arbor Science,Ann Arbor, MI,1973. Mercer, T. T.,Aerosol Technologyin Hazard Evaluation, AcademicPress,New York, 1973. NIOSH, "Asbestosand Other Fibers 7400," in NIOSH Manual of Analytical Methods,U.S. Gov. Printing Office, Washington,DC, 1989. Olan-Figueroa,E., McFarland,A. R., and Ortiz, C. A., "Flattening coefficients for DOP and oleic acid droplets depositedon treated glass slides," Am. Ind. Hyg. Assoc.J., 43, 395 (1982). Silverman,L., Billings, C. E., and First, M. W., Particle SizeAnalysis in Industrial Hygiene, Academic Press,New York, 1971. Sorenson,C. M., andFeke,G. D., "The Morphology of MacroscopicSoot,"AerosolSci. Tech, 25,328-337 (1996).
i
AppendixAI.
UsefulConstantsand ConversionFactors
Fundamental units 1 micrometer (~m) = 1 micron (~) = 10--6 m = IQ--4cm = 10-3mm == 103nm = 104A = 3.94 X 10-5 in 1 nm = 10-3~m = 10--6 mm == 10-7cm = IQ-9m 1 m = 39.4 in = 3.28 ft 1 kIn = 1000m = 3280 ft == 0.621 mi 1 in. =0.0254 m = 2.54 cm = 25.4mm 1 ft =0.305 m = 30.5 cm = 305 mm 1 newton(N) 1 kg. m/S2= 105dyn 105g. Cm/S2
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APPENDICES
449
Specific heat ratio
= K = cplcv= 1.40
Diffusion coefficient = 2.0 x 10-5 m2/s = [0.20 cm2/s] Composition by volume (dry air) N2
78.1%
02 Ar
20.9% 0.93%
CO2
0.031 %
Other
0.003%
Water at 293 Kf20°Cf and 101 kPa (1 atm) Viscosity = 0.00100Pa' s [0.0100 dyn. s/cm2] Surfacetension = 0.0727 N/m = [72.7 dyn/cm] Saturationvapor pressure= 2.34 kPa = [17.5 mm Hg] Water vapor at 293 Kf20°C} and 101 kPa (1 atm) Diffusion coefficient = 2.4 x 10-5m2/s[0.24 cm2/s] Density =0.75 kgim3 =0.75 x 10-3g/cm3 AppendixA2. SomeBasicPhysicalLaws The inertia of a body is that property of the body which tends to resist changein its stateof rest or of motion. Mass is a quantitativemeasureof inertia. Momentum= massx velocity = mV Newton's secondlaw: The rateof changeof momentumof a body is proportional to the net force on the body, LF =d(mV)/dt. For a body with constantmass, F
= m(%) = ma
where force is in N (newtons)for m in kg and a in m/S2,and in dyn for m in g and a in Cm/S2. Work
= force
x distance;
Kinetic energy Power
= work
= tmV2;
1N .m
= 1 l;
1 kg. m2/s2= 1 l; [1 g'
per unit time; 1 W
= 1 lis = 107erg/s = 0.00134Hp
Centripetalforce =
i
= 1 erg] cm2/s2 = 1 erg]
[1 dyn . cm
!!!.~ R
= mRro2
where Vr = tangentialvelocity, mls [cm/s]; co= angularvelocity, rad/s; and R = radiusof motion.
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APPENDICES
APPENDIX A3. Relative Density of Common Aerosol Materials (Multiply Values by 1000 for Density in kg/m3 and by 1.0 for Density in g/cm3) So/ids Aluminum Aluminum oxide Ammonium sulfate Asbestos Asbestos,chrysotile Carnaubawax Coal Fly ash Fly ash cenospheres Glass,common Granite Ice Iron Iron oxide Limestone Lead Marble Methylene blue dye Liquids Alcohol Dibutyl phthalate Dioctyl phthalate[DOP; di-2 (ethylhexyl) phthalate] Dioctyl sebacate Hydrochloric acid
APPENDIX A4.
2.7 4.0 1.8 2.0-2.8 2.4-2.6 1.0 1.2-1.8 0.7-2.6 0.7-1.0 2.4-2.8 2.6-2.8 0.92 7.9 5.2 2.7 11.3 2.6-2.8 1.26
Natural fibers Paraffin Plastics Pollens Polystyrene Polyvinyl toluene Portlandcement Quartz Sodium chloride Sulfur Starch Talc Titanium dioxide Uranine dye Wood (dry) Zinc Zinc oxide
1-1.6 0.9 1-1.6 0.45-1.05 1.05 1.03 3.2 2.6 2.2 2.1 1.5 2.6-2.8 4.3 1.53 0.4-1.0 6.9 5.6
0.79 1.045
Mercury Oils Oleic acid Polyethyleneglycol Sulfuric acid Water
13.6 0.88-0.94 0.894 1.13 1.84 1.00
0.983 0.915 1.19
Standard Sieve Sizes Nominal Wire Size (~m)
250
Designation ISO Std," Alternate (/lm) (No.)
Nominal Wire Size (~m)
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APPENDIX AS. Propertiesof Water Vapor
I
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781 71:i
70
pressure
Surface tension
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20
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68 66111,111"'\111'_'0 -10 0 10
20
30
40
50
Temperature (oC)
APPENDIX A9. Propertiesof Water
60
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APPENDICES
0.01
0.1
1.0
10
100 juoo
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Charging Kelvin effect
Siinificant
Light scattering
Rayleigh
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APPENDICES
APPENDIX All (b). Particle Diameter d (11m)
Slip Correction
Properties of Airborne Particles at Standard Conditions (cgs Units)' Diffusion
Relaxation
Factor Cc
Settling Velocity VTS (cm/s)
0.001 0.0015 0.002 0.003
224.332 149.752 112.463 75.174
6.75E-O7 I.OIE-O6 1.35E-O6 2.04E-O6
0.004 6.665 0.006 0.008
56.530 '({.:1"4i 37.888 28.568
2.72E-O6
2.78E-O9
!"A(t[-(\'(; 4.IIE-O6 5.5IE-O6
0.01 0.015 0.02 0.03
22.976 15.524 11.801 8.083
0.04 0.05 0.06 0.08 0.1 0.15 0.2
Time 't (5)
Mobility B (cm/dyn. s)
6.89E-IO 1.O3E-O9 1.38E-O9 2.08E-O9
Coefficient
Coagulation Coefficient
D (cm2/s)
K (cmJ/s)
1.32E+12 5.85E+ll 3.30E+ll 1.47E+ll
5.32E-O2 2.37E-O2 1.33E-O2 5.94E-O3
3.IIE-IO 3.8IE-IO 4.40E-IO 5.39E-IO
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