Processes Sources and Particle Size Distributions BY J. R.BROCK Dept. of Chemical Engineering University of Texas Austin Texas 78712 U.S.A. Received 28th March 1973 Rational control policies for particulate air pollutants ultimately must consider the particle size distribution. This point is illustrated by a simple example of the inadequacy of regulations based on limitation of total particulate mass emissions from primary sources. A general model for the evolu- tion of the atmospheric aerosol size distribution is discussed for the particular case of an urban area. Conservation equations for particulate matter and nucleating vapour are developed. An important element in the development of the general model is knowledge of the particle size distribution associ- ated with various primary sources of particulate matter.Such primary sources are classified according to the basic aerosol generation process of homogeneous and heterogeneous nucleation and com- minution. Aerosols from primary sources in which particle generation occurs by homogeneous nucleation are considered to be well aged aerosols which by virtue of coagulation have reached asymptotic limit or " self preserving " distributions. As a practical matter however randomization will alter the asymptotic limit distributions from primary sources. Inherent complications in attempts to characterize the particle size distributions of aerosols formed by heterogeneous nucleation are discussed. Aerosols generated by comminution are known to approach asymptotic limit distributions which in certain cases have the log normal form although randomization again will alter such asymptotic forms.1. INTRODUCTION The pollution of the atmosphere with aerosols is a topic of current concern. Adverse health effects have been traced to exposure of urban populations to high ambient levels of particulate matter and sulphur oxides. Reduction in visibility is an obvious consequence of an increase in the amount of atmospheric aerosol. Consider-able speculation continues as to the possible role of pollutant aerosol in inadvertent modification of weather and even the global energy balance. In many of these instances the unfavourable effects are related not solely to the total quantity of suspended particulate matter but in a detailed manner to the aerosol size distribution.Therefore it would appear that formulation of a rational policy for control of pollutant aerosol must consider the size distribution. As an illustration of this point consider a single elementary example. Controls on emissions of particulate matter from sources have been directed primarily toward reduction of total mass emissions. From a regional or international standpoint this may not be sufficient. Consider that the rate of change of total suspended particulate mass m owing to a primary source emission is a function of the primary source emission rate a and the residence time b-l of the aerosols from that source. As we shall discuss later these simple assumptions are not generally correct but they will serve for the purposes of this simple illustrative example.The functional relationship described above may be expressed in the form dm -= a-bm dt 198 J. R. BROCK 199 where b-l is a function of particle size which for a particular source category will be chosen as the residence time of the mass median diameter. For short time periods dm -0 = a-bm dt N so that rn = a/b gives the total mass of atmospheric aerosol at any time owing to a particular source category. Table 1 presents some comparisons for several source categories in the U.S.A. In these examples the largest tonnage sources of particulate emissions are not the largest contributors to the inventory of suspended particulate matter. There is of course no pretension that the data in table 1 are accurate but they do serve to illustrate the point that total mass emissions may not be a reliable guide to assessment of atmospheric degradation by primary sources.TABLE l.-M~ss OF AEROSOL IN ATMOSPHERE OWING TO INDICATED PRIMARY SOURCES primary source (uncontrolled) mass median diam./pm(uncontrolled)ref. (I) inverse particle residence time/yr-1 ref. (2)b source mass emission rate/ton yr-1 U.S.A. ref. (1) a mass of particulate matter of atmosphereU.S.A. alb 1. automobile 0.4 20 4~ 105 2~ 104 2. coal-fired 10 625 3x lo6 5x lo3 electric utility pulverized sand gravel FFC units 3. crushed stone 4. petroleum 20 0.5 1250 30 5x lo6 4.5~105 4~ 103 i.5~104 In general rational control policy for particulate matter requires detailed answers to such questions as What are the sources of aerosols in the atmosphere? What are the processes producing changes in the aerosol size distribution in the atmosphere? How are particles removed from the atmosphere ? Detailed answers to these questions would imply a knowledge of the evolution of the aerosol particle size distribution in the atmosphere.The purpose of this paper is to develop an analytical framework for the examina- tion of some of these questions. We begin with the formulation of a model for the evolution of the atmospheric aerosol size distribution. Consequences and approxi- mations of the model are discussed. As an important element of the general model examination is made of the particle size distributions arising from basic aerosol generation processes and associated with primary sources of particulate matter.2. MODEL OF THE ATMOSPHERIC AEROSOL In this section an analytical model for the atmospheric aerosol is discussed. While the general model is not necessarily limited the direction of the discussion is toward the evolution of the particle size distribution in the atmospheric boundary layer of urban areas. A major element in the evolution of the particle size distribu- tion and the essential feature of the hydromechanical description of the atmospheric boundary layer is the presence of turbulence. These facts make difficult a rigorous derivation of an evolution equation for the atmospheric aerosol. The nature of these difficulties in the development of aerosol dynamics in a turbulent medium has been outlined in ref.(3). In the development of this model a first assumption is that the 200 PROCESSES SOURCES AND PARTICLE SIZE DISTRIBUTIONS velocity of the suspending gas is not altered by the presence of particles. While this assumption may be valid for the ambient urban aerosol it clearly excludes detailed consideration of aerosol dynamics in such systems as dense or hygroscopic plumes fogs clouds etc. A second assumption is that aerosol particles and gaseous contami- nants diffuse similarly in the presence of atmospheric turbulence. Justification for this assumption is given in ref. (3) and (4). Brownian diffusive transport of individual particles is considered to be negligible compared with turbulent diffusive transport.With this discussion in view the following relation is assumed to describe the evolution of the density function n(p X t) for the atmospheric aerosol an 6(p-p' p') + ~(,d)n(p-p') dp' -1.1 b(p',p)n(p')dp' + at a"2 a __ [cc(,u)n]-[P(p)n]+ G(p) Vn+ ip+ vN (2.1) aP2 3P P N where n(p X t) dp is the number of aerosol particles haveing masses in the range p dp at a poiiit in space X at time t. co is the fluid velocity. The first two terms on the right hand side of the equation represent the change in n owing to coagulation. The third and fourth terms describe the change in n owing to condensation of trace gaseous substances. The change in n owing to gravitational settling is given by the fifth term in which G(p)is the gravitational settling velocity of a particle of mass p.Lp(p,X t) is the rate of input at X t of particles of mass p from primary source P,and the summation is extended over all primary sources which may be treated as point line or area sources. Similarly j2Ni(p X t)represents the rate of production of particles of mass p at X t by homogeneous nucleation of the ith chemical species. It should be noted that heterogeneous nucleation of condensable species is accounted for by the condensation terms three and four. It should be noted that the coagulation coefficient b(p p') and condensation coefficients a@) p(p) may be very complicated functions if refinements such as particle shape composition and other physico-chemical properties are included. Coupled to eqn (2.1) are conservation equations for those substance undergoing heterogeneous and homogeneous nucleation as co -+G VS = 1Sjr+cSjp-csjrl-s (s, p)n(p)dp at r P r' 0 Sj is the mass concentration of the jth chemical species undergoing heterogeneous nucleation.Sjr represents the rate of production of this species by the vth chemical reaction and sj the rate of production by the Pth primary source. Similarly Sjr' is the rate of removal ofj by the rth chemical reaction. The last term on the right hand side of eqn (2.2) is the rate of removal of speciesj by condensation onto the existing aerosol; the function y(Sj,p) will in general be a complex function of the physico- chemical properties of the particles. The first three terms on the right hand side of eqn (2.3) are analogous respectively to those in eqn (2.2).The last term is the rate of removal of i by homogeneous nucleation as defined in eqn (2.1). Eqn (2.1) (2.2) and (2.3) are of course coupled closely to the energy and momen- tum balance equations for the atmosphere.' We shall discuss this coupling shortly. J. R. BROCK 20 1 However consider now the first two moments of n 03 N =lon(P)dP N being the total number of particles per unit volume and M the total mass. Through performing the indicated integrations one obtains the following moment equations from eqn (2.1) The details of the integration of eqn (2.1) to obtain eqn (2.6) and (2.7) are well .. known and are therefore omitted here. Np,APlvand jlp,A?Nrepresent respectively total number and mass of particles contributed by both primary sources and homo- geneous nucleation.It is clear from eqn (2.6) and (2.7) that N is unaffected by condensation processes but is altered by coagulation whereas the reverse is the case for M. Hence the two moment eqn (2.6) and (2.7) provide complementary information on these two important processes shaping the size distribution. For a turbulent fluid eqn (2.1) (2.2) and (2.3) are not in a useful form. By analogy with the usual procedure for turbulent dispersion of non-reactive gases we can express quantities in these equations in terms of time averaged quantities with an overbar and fluctuating components. For example v =V+V’ sj=sj+s; n =E+n‘ hi=hi+hi where by definition V’ =0 etc. Unfortunately eqn (2.1) (2.2) and (2.3) are non-linear in n Sj and hi so that the usual procedure of introducing Prandtl’s mixing length hypothesis is not sufficient.This is a familiar problem occurring in the analysis of non-linear chemical reactions in turbulent fluids. It has recently been discussed for atmospheric chemical reactions.6 Unfortunately there is no simple resolution. For example if one seeks to ignore a product such as relative to E2 it is necessary that characteristic coagulation times l/bn and temporal variations in 5 be much greater than the Lagrangian time scale of the turbulence. In addition the spatial variation of ii must be large compared with the length scales of the turbulence. These conditions may not always be met ;and if they are not we do not have a convenient hypothesis such as Prandtl’s to relate quantities such as n-and ii2.More detailed study of these questions appears to be essential for further development of the present theory. We proceed then under the restrictions indicated above and neglect terms such as nT relative to EE etc. With the introduction of Prandtl’s mixing length hypothesis eqn (2.1),(2.2) and (2.3) become 202 PROCESSES SOURCES AND PARTICLE SIZE DISTRIBUTIONS cop -+V a hi Vhi = V KVgi+xzjy+xHjp-I gjq,-s &(p) dp (2.11) at 4 q‘ 0 where K is the so-called eddy diffusivity. The difficulties in the concept of eddy diffusivity applied to atmospheric dispersion are well known and extensively re~iewed.~ We restrict our discussion here to urban particulate pollution.In order to apply these equations in the descriptions in the description of the aerosol concentration in an urban area it is clear that detailed knowledge of chemical reaction rates nucleation rates and physico-chemical alterations of the particulate matter is necessary. In addition meterological data in the form of mean winds atmospheric stability conditions relative humidity tempera- ture and radiative flux are essential. The boundary and initial conditions will depend also in part on this meteorological information. A discussion of the solution of eqn (2.9) (2.10) and (2.11) is beyond the scope of this paper. It is possible however to indicate the relative importance of some of the terms in these equations which will serve to provide a framework for later investiga- tions.With the restriction of the discussion to pollution by aerosols of an urban area various characteristic time scales may be introduced as scale factors in eqn (2.9) (2.10) and (2.11). For an urban area a characteristic residence time tRES may be introduced as the ratio of the crosswind diameter D of the area to the mean wind velocity U-that is tREs -D/U. Obviously a problem occurs in this definition if winds are light and variable as occurs during inversion conditions. This point will be touched on later. There are also characteristic diffusion times for longitudinal ?LONG and vertical tVERT dispersion tLoNG N L2/KL and where L is the longitudinal distance perpendicular to a line in the mean wind direction through a point of observation His the vertical distance above the surface.K,and KH are the corresponding eddy diffusivities. For particulate systems the characteristic particle gravitational settling time tCRAV -H/ Vg(,u),must be considered along with ~VERT-There are also characteristic times for the aerosol growth processes. For coagulation 1 tCOAG -____ b(p p‘)’‘’’ J. R. BROCK 203 where for the collision pair n(,u) > n(,u'). For condensation where ACj is the difference between the number density of jin the gas and just at the surface of the particle. iij is the molecular mean speed of j. /z(r) is a function of particle radius r and Knudsen number Kn such that for Kn -+0 h(r) -r. In addition to these characteristic times there are times associated with the gas phase reactions homogeneous nucleation processes flux rates of particulate matter to boundary surfaces etc.If one is interested solely in the evolution of the aerosol distribution in urban areas it is clear that only those processes with characteristic times within the period tREs need be considered. The assumption here of course is that upwind outside the urban area the aerosol concentration is low and plays no important role in the evolution processes occuring in the urban area. By way of example for an urban area with a cross wind diameter of 30 km and a mean wind speed of 2 m/s tREs-4 h. Consequently for KL -8 x 106cm2fs primary sources located longitudinally more than approximately 3 km from the line in the direction of the mean wind speed through a reference or monitoring point need not be considered in interpreting observations at that monitoring station.In the same manner one can conclude that coagulation between two particles both with radii greater than 0.1 ,urn can be neglected in eqn (2.9) inasmuch as for such coagulating pairs of particles at known urban concentrations tCoAG230 h. Also in photo- chemical smog reactions the induction period for formation of ozone which appears to be one of the reactants producing particles by nucleation is of the order of 4 h so that for the conditions of this particular example such processes cannot be neglected in eqn (2.9). From this one can infer that it is not generally necessary to consider all the complication inherent in eqn (2.9) (2.10) and (2.11) in that some of the terms may be of negligible order for certain meteorological and boundary conditions.A difficulty arises with these arguments under inversion conditions where the mean wind speed is very small and highly variable in direction. In these circum- stances the ground surface and inversion base suggest the applicability for the urban raea of a so-called '' box model " in which unfortunately much detail is lost. It is assumed that the urban region closed at the top by the inversion base corresponds to a well-mixed container. If eqn (2.9) is integrated over this defined volume and the volume integrals are converted where applicable to surface integrals the following result is readily obtained where (n) is the volume-averaged density function.~(p)represents the inverse residence time of particles of mass p in the volume and is a function for the conditions specified above of the mean wind dispersion coefficients and surface removal processes. zifi(p) represents the rate of input of particles from primary or homo- geneous nucleation source i with density mi@) and characteristic time zt '. This model has been applied' in the discussion of the evolution of the size spectrum for particle radii greater than 0.1 ,urn. As noted above coagulation of pairs of particles found in this portion of the atmospheric size spectrum can be 204 PROCESSES? SOURCES AND PARTICLE SIZE DISTRIBUTIONS neglected over reasonably long time periods.Hence for radii greater than 0.1 pm eqn (2.12) becomes (2.13) where the small nuclei below 0.1 pm may be considered as participants in the conden- sation process on the large particles ; the new coefficients a' and p' include this con- sideration. If primary source inputs are neglected it is easy to show ** that as a result of condensation eqn (2.13) leads to certain characteristic forms for the particle size distribution for particles greater than 0.1 pm radius. Of course primary sources in many urban areas may be dominant. Therefore in such circumstances if generalizations concerning urban particle size distributions are to be found one must look at charac- teristics of the particle size distributions of typical dominant primary sources. Consideration is given to this topic in the next section.3. PROCESSES AND SIZE DISTRIBUTIONS The necessity for consideration of the particle size distribution of primary sources has been noted above. Primary sources as defined by the relations developed in Section 2 include all those sources injecting particles as such directly into the atmosphere. Particles from a given primary source may be generated by the processes of nucleation comminution or by combinations of these processes. Nucleation may be homogeneous or heterogeneous. The term homogeneous nuckation embodies all those processes in which vapour molecules interact physically or chemically to form particles ;the particle growth process begins from particle sizes of molecular order and may proceed by coagulation condensation or a combination of these.In heterogeneousnucleation,new particles are not formed ;vapour molecules condense physically or chemically onto existing particles and primarily one is dealing with a condensational growth process. Particle generation by comminution involves successive usually mechanical sub- divisions of liquids or solids to the fine particle state. Aerosol generation at the air- sea interface and dust rise by wind action at the air-land interface are important examples of natural primary sources of particles formed by comminution. These sources in fact are estimated lo to constitute the two largest contributors of aerosol mass on a world wide basis. With these definitions characteristics of the particle size distribution produced by the particle generation processes of nucleation and comminution will now be examined.HOMOGENEOUS NUCLEATION Automobile exhaust represents perhaps one of the important examples of an anthropogenic primary source in which particles are apparently generated principally by homogeneous nucleation as defined here. The residence time of the aerosol before injection and subsequent fairly rapid dilution in the atmosphere in this and other important industrial combustion sources is usually of the order 0.1 -1 s. Therefore the particulate emissions from such sources are comparatively well-aged aerosols for which the particle size distribution has had sufficient time to reach a "self-preserving "form by coagulation.' ' The term "self-preserving "refers to the tendency of aerosols coagulating with the same collision parameter b(p p') to achieve similar particle size distributions after sufficient time of coagulation.Also J. R. BROCK 205 a simple calculation is sufficient to show that usually aerosols formed by homogeneous nucleation will in the time period 0.1 -1 s have an average size which is in the sub- micrometre range. If A$’ is the total mass concentration of condensed material formed by nucleation the order of the mean radius of the coagulated aerosol should be r -[3 Jzl( 1+bNot)/47EpN0]+ (3.1) where No is the initial embryo concentration and p is the particle density. Eqn (3.1) becomes for No t % 1 r -(3Abt/4np)+ (3.2) and the order of the mean radius becomes independent of No.As an example for an automobile using leaded gasoline the undiluted exhaust has a total particulate mass concentration A%’,of -lO-’-lO-* g/cm3. Eqn (3.2) indicates as do measurements,12 that most of the aerosol is certainly in the submicrometre range. For anthropogenic primary sources of aerosol formed principally by homogeneous nucleation and in which subsequent particulate growth is by coagulation one might infer that particle size distributions from all such sources should have the same functional form and should differ only in the parameters of the “self preserving ” form. The inference is the same if simultaneous condensation occurs. Published studies l1 of the numerical solution of the coagulation equation in the free molecule and continuum regimes support the foregoing conclusion.However experimental measurements of coagulating aerosols reveal that generally the aerosol is more polydisperse than predicted by the self-preserving functional form. The explanation for this behaviour is a very familiar one to statistician^.'^ The aerosol measured in the coagulation experiments does not represent a single population but instead a mixture usually in random proportion of a heterogeneous population. In other words the aerosol actually measured is a composite of many different aerosol populations each with a different history. Thus the particle distribution function realized in an experiment G(X)is where the pi are random weights attached to the various members of the heterogeneous population each with distribution function Fi(X).For an infinitely composite population G(X) = 1F(X a) dU(a). (3.4) In experimental realizations of coagulation not only systematic spatial or time variations or random experimental error serve to create a composite population but for dilute systems unavoidable random fluctuations also contribute. As a result of the effect of heterogeneity of population the aerosol formed by homogeneous nucleation from a given primary source will always be more poly- disperse than predicted by the “self-preserving” theory. Unfortunately the extent of this increase in polydispersity for a given primary source will probably depend on the details of that source such as geometry flow dynamics etc. As a result it remains to be determined whether or not for various primary sources of this type generaliza- tions axe possible.Certainly the fact of the “ self-preserving ” form provides a useful base from which to proceed in the inquiry. S7-8 206 PROCESSES SOURCES AND PARTICLE SIZE DISTRIBUTIONS HETEROGENEOUS NUCLEATION The dense hygroscopic plumes emitted from various industrial processes are examples of aerosols formed by heterogeneous nucleation. In these instances water vapour has condensed on an existing hygroscopic aerosol which may itself in turn have been generated by homogeneous nucleation or comminution. For the simple process of condensation of a pure substance on an aerosol of some given initial density function no(r)dr of particle radius r it is a simple matter to examine the development in time of n(r t).In this case the evolution of n(r t) is dn a -+-[f(r)n]at ar =0 (3.5) wheref(r) is the growth law for a particle of radius Y. For example in the continuum region Kn +0 neglecting the Kelvin effect f(r) =a/r where a is a con~tant.~ Similarly in the free molecule region f(r) =a a constant if the Kelvin effect is neglected. Eqn (3.5) is a first order equation for which solutions may readily be found for arbitrary initial conditions. However perhaps the most interesting feature of the pure condensation process is the tendency of condensation to produce a less poly- disperse aerosol in the continuum region when the Kelvin effect can be neglected. In this case it is a simple matter to show that the ratio of the standard deviation CT to the mean radius yl approaches zero with increasing time d -60 Y1 (Yl.0 +24+ where the subscript designates initial conditions.Similarly in the free molecule regime (3.7) No being the initial total particle concentration. This characteristic of pure conden- sational growth has been utilized for the production of approximately monodisperse aerosols in variations of the original Sinclair-La Mer aerosol generat~r.~ If as in the previous examples the concentration of condensing vapour is held fixed but the Kelvin effect is included in the termf(r) it can be shown that a/yl -,0 as a result of condensation. However if the quantity of condensing vapour is limited one finds that in an initially polydisperse aerosol after condensation has proceeded the smaller particles will begin to evaporate while the larger ones continue to grow.Also additional complication beyond the scope of this discussion arises in con- sideration of a hygroscopic aerosol which grows at humidities below the critical supersaturation of some of the particles. In such cases n(r)can become bimodal and very polydisperse. More general condensation processes including stochastic effects have been examined elsewhere (see also Section 2). Such processes as well as randomization indicated in eqn (3.3) and (3.4),usually act to increase the polydispersity ofan aerosol. Additional complication can be introduced by considering as well simultaneous coagulation and condensation. When the deterministic condensational growth described by eqn (3.5) is the only process altering the aerosol size distribution the final distribution clearly will be J.R. BROCK 207 determined by the initial size distribution. This initial size distribution will be that owing either to homogeneous nucleation or comminution or both. When the condensation process is stochastic and/or randomization occurs the final particle size distribution resulting from condensational growth will become asymptotically independent of the initial distribution the equivalent of the “ self-preserving ” behaviour for a coagulating aerosol. However in general it is much more difficult to draw conclusions concerning the nature of the particle size distribution resulting from condensation in these cases than for coagulation.Unlike the coagulation equation the condensational growth equation is coupled to the conservation equation of the condensing vapour; the state of the suspending gas usually plays a secondary role in the coagulation of fine particles. Furthermore the ability of particles to grow by condensation depends in detail on particle composition or surface properties ;such characteristics are usually not considered to be of great importance in coagulation. Therefore for sources in which particle generation by heterogeneous nucleation plays an important role detailed examination of the process dynamics will be necessary to characterize the particle size distribution. COMMINUTION Important natural sources of aerosol particles generated by comminution have been cited at the beginning of this section.Anthropogenic sources of aerosol generated by comminution are also of common occurrence and include emissions from industrial operations such as mineral rock and gravel processing sand blasting cement manufacture etc. as well as inadvertent emissions resulting from farming operations etc. The process of comminution begins with a body of macroscopic size and by successive subdivisions or splittings liquid or solid particles capable of aerosolization are formed. It is therefore the inverse operation to homogeneous nucleation and subsequent coagulation. The evolution equation for comminution may be represented by the relation = Ja c(p/p’)n(p‘jdp’-c(p)n(p) at P where c(p/p’)dt is the probability that a particle of mass p’ will split in time dt to form 1,2,3...particles of mass p and c(p)dt is the probability that in the same time a particle of mass p undergoes splitting.The basic assumption of eqn (3.8) is that each particle splits with a probability independent of the presence of other particles. Clearly additional detail can be introduced. It is possible to show that the splitting process approaches asymptotically a limit distribution,l which for certain assumptions concerning the splitting probabilities can be approximated by the log normal distribution. Just as the coagulation process has for certain assumptions concerning collisions an asymptotic limit distribution so too does the process of comminution. A common assumption in the discussion of the splitting process l6 is that the probability of splitting is proportional to some power of the mass of a particle.Clearly if a comminution process is carried out so that a particle of say 1000pm is split with unit probability a particle of 1 pm radius will be split with a probability orders of magnitude less in fact if splitting is directly proportional to particle mass for particles of unit density). For this reason many large sources of particles produced by comminution such as those cited above will produce particles in the range of larger particle sizes. 208 PROCESSES SOURCES AND PARTICLE SIZE DISTRIBUTIONS Although asymptotic limit distributions may exist for a given comminution process randomization can be expected to be important owing to the comparatively small number of particles per unit volume in typical comminution processes.How-ever very large particles are not important in the consideration of sources of air pollution so that the distribution produced by a comminution process can be truncated at the order of 100pm radius. Therefore the particle size variation of interest will generally be over only one or two orders of magnitude of particle radius. As a result the range of polydispersity which might arise from randomization is restricted. 4. CONCLUSION While the model described in Section 2 for the evolution of the atmospheric aerosol appears to be very complex it is nevertheless suggested that owing to the important role of the particle distribution in many of the detrimental effects of air pollution this complexity eventually must be faced.Characterization of the primary sources of pollutant aerosol is one of the necessary first steps. It is recognized of course that primary sources of particulate matter will not generally fit into the separate categories discussed here. The aim here has been to examine the basic aerosol generation processes and to inquire into the characteristics of the resultant particle size distributions. Further progress along these lines will require adequate field data which are at present insufficient. This work was supported in part by a research grant from the Division of Chemistry and Physics National Environmental Research Center E.P.A. The author also wished to thank Prof.J. Bricard of the University of Paris for many helpful discussions. Particulate Pollutant System Study MRI Contract No. CPA 2269104 (EPA 1971). N. Esmen and M. Corn Atmospheric Environment 1971 5 571. G. Hidy and J. R. Brock The Dynamics of Aerocolloidal Systems (Pergamon Oxford 1970). N. Fuchs The Mechanics of Aerosols (Pergamon Oxford 1964). R. Bird W. Stewart and E. Lightfoot Transport Phenomena (Wiley NY 1960). R. Lamb Atmospheric Environment 1972 6 257. F. Pasquill Quart. J. Roy. Met. Soc. 1971 97 369. J. R. Brock Atmospheric Environment 1971 5 833. J. R. Brock J. Coll. Interface Sci. 1972 39 32. lo G. Hidy and J. R. Brock,Proceedings 2nd IUAPPA Clean Air Congress (Academic Press New York 1971). l1 R. Drake in Topics in Current Aerosol Research (Pergamon Oxford 1972).l2 R. Lee et al Atmospheric Environment 1971 5 225. l3 J. Pich S. Friedlander and F. Lai Aerosol Sci. 1970 1 115. l4 M. Girault Calcul des Probabilities (Dunod Paris 1972). l5 J. R. Brock to appear. l6 A. Kolmogorov Akad. Nauk SSSR 1941 31,99.