International Journal of Computer Mathematics

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1 International Journal of Computer Mathematics "Mathematical study of the selectie remoal of different classes atmospheric aerosols by coagulation, condensation and graitational settling and health impact (SI-CMMSE 00)" Journal: International Journal of Computer Mathematics Manuscript ID: GCOM-00-0.R Manuscript Type: Original Article Date Submitted by the Author: -No-00 Complete List of Authors: Garcia Nieto, Paulino; Uniersity of Oiedo, Mathematics Coz Diaz, Juan Jose del; Uniersity of Oiedo, Construction Martin Rodriguez, Angel; Uniersity of Oiedo, Construction Matías, José; Uniersity of Vigo, Statistics Keywords: Computational fluid dynamics, Aerosol, Precipitation Scaenging, Numerical methods, Respirable dust, Health impact, N, B, S, K

2 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Reiew Article Mathematical study of the selectie remoal of different classes of atmospheric aerosols by coagulation, condensation and graitational settling and health impact (SI CMMSE 00) P. J. GARCÍA NIETO*, J. J. DEL COZ DIAZ, A. MARTÍN RODRÍGUEZ and J. M. MATÍAS FERNÁNDEZ& Department of Mathematics, Uniersity of Oiedo, Faculty of Sciences, 00 Oiedo (Asturias) - Spain. Department of Construction, Uniersity of Oiedo, EPSIG, 0 Gijón (Asturias) - Spain. &Department of Statistics, Uniersity of Vigo, Vigo (Ponteedra) - Spain. Correspondence * Corresponding autor: Paulino José García Nieto. Tel: + Fax: +. pauli@constru.unioi.es Abstract The aim of this paper is to study the scaenging efficiencies of aerosol particles after some gien dynamical mechanisms of remoal known as coagulation, condensation and graitational settling as a function of time. It also analyses the health impact of aerosol before and after the aboe dynamical mechanisms by comparing the respirable dust fractions. The well-known equations of scaenging are applied to eight different classes of atmospheric aerosols: marine background (MB), clean continental background (CCB), aerage background (AB), background and aged urban plume (BAUP), background and local sources (BLS), urban aerage (UA), urban and freeway (UF) and central power plant (CCP). From this work it is inferred that respirable dust is scaenged with relatie difficulty by coagulation, condensation and graitational settling. The deposition of particles in the lungs relies on the same basic mechanisms that cause collection in a filter, but the relatie importance of each mechanism is quite different. While filtration occurs in a fixed system at a steady flow rate, respiratory deposition occurs in a system of changing geometry, with a flow that changes with time and cycles in direction. This added complexity means that predicting deposition from the basic theory is much more difficult, and we must rely to a greater extent on experimental data and empirically deried equations. Therefore, an understanding of how and where particles deposit in our lungs is necessary to ealuate properly the health hazard of aerosols. As compared with the initial olume of respirable aerosol, roughly 0% remains after hours of coagulation, condensation and graitational settling. Besides, graitational settling is the main remoal mechanism of respirable aerosol compared to condensation and coagulation. Keywords: Computational fluid dynamics; Aerosol; Precipitation scaenging; Numerical methods; Respirable dust; Health impact AMS Subject Classification: N; B; S; K Introduction Particle matter (aerosol) processes are emerging as a new frontier in enironmental studies. Aerosols [] negatiely affect human health, reduce isibility, and modify warming through scattering and absorption of solar radiation.

3 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling To study accurately the effects of aerosols it is necessary to resole aerosol number and mass distributions as a function of size. Treatment of aerosol processes leads to an order of magnitude increase in the oerall computational time of an air quality model. This is mainly due to repeatedly soling the aerosol chemistry (or chemical equilibria) for different particle sizes. Therefore there is a clear need for rigorous, reliable and efficient computational techniques for aerosol simulations. In particular, there is a need for methods that accurately sole aerosol dynamics using a small number of size bins (discretization points), such that the time for aerosol chemistry calculations is manageable. Simulation of aerosols plays a significant role in air pollution modelling. For a correct representation of particles in the atmosphere one needs to accurately sole for the size distribution of particle populations. In this paper we propose a discretization method for the aerosol dynamic equation based on piecewise polynomial discretizations of the particle size distribution. This paper presents a discretization technique known as time splitting in order to sole the general dynamic equation (GDE) of aerosol. The growth term is soled using the flux-based modified method of characteristics, and coagulation term by the J-space transform and a cubic spline method. Numerical tests reeal that accurate solutions can be obtained with a ery small number of size bins. We hae implemented an efficient software package to sole the proposed algorithms. Atmospheric aerosols consist of particles ranging in size from a few tens of angstroms to seeral hundreds micrometers [,,, ]. Particles less than. µm in diameter are generally referred to as fine and those greater than. µm diameter as coarse. The fine and course particle modes, in general, originate separately, are transformed separately, are remoed from the atmosphere by different mechanisms, require different techniques for their remoal from sources, hae different chemical composition, hae different optical properties, and differ significantly in their deposition patterns in the respiratory tract []. Therefore, the distinction between fine and coarse particles is a fundamental one in any discussion of the physics, chemistry, measurement, or health effects of aerosols. Fine particles can often be diided roughly in two modes: the nuclei mode and the accumulation mode. The nuclei mode, extending from about 0.00 to 0. µm diameter, accounts for the preponderance of particles by number. Because of their small size, these particles rarely account for more than a few percent of the total mass of airborne particles. Particles in the nuclei mode are formed from condensation of hot apours during combustion processes and from the nucleation of atmospheric species to form fresh particles. They are lost principally by coagulation with larger particles. The accumulation mode, extending from 0. to about. µm, usually accounts for most of the aerosol surface area and a substantial part of the aerosol mass []. The source of particles in the accumulation mode is the coagulation of particles in the nuclei mode and from condensation of apours onto existing particles, causing them to grow into this size range. The accumulation mode is so named because particle remoal mechanisms are least efficient in this regime, causing particles to accumulate there. The coarse mode, from >. µm diameter, is formed by mechanical processes and usually consists of man-made and natural dust particles. Coarse particles hae sufficiently large sedimentation elocities that they settle out of the atmosphere in a reasonably short time. Because remoal mechanisms that are efficient at the small and large particle extremes of the size spectrum are inefficient in the accumulation range, particles in the accumulation mode tend to hae considerably longer atmospheric residence times than those in either the nuclei or coarse mode. Dry deposition is, broadly speaking, the transport of gaseous and particulate species from the atmosphere onto surfaces in the absence of precipitation []. The motion of the particle, howeer, arises in our case because of the action of an external force on the particle called graity. The drag force arises as soon as there is a difference between the elocity of the particle and that of the fluid. The basis of the description of the behaiour of a particle in a fluid is its equation of motion. The hazard caused by inhaled particles depends on their chemical composition and on the site at which they deposit within the respiratory system. Thus, an understanding of how and where particles deposit in our lungs is necessary to ealuate properly the health hazards of aerosols. Such an understanding is also central to the effectie administration of pharmaceutical aerosols by inhalation []. Humans hae eoled effectie defence mechanisms against aerosol hazards, and we consider here the first line of defence: mechanisms that restrict access of particles the sensitie regions of the lungs. In this work we focus on the scaenging efficiencies of aerosol particles after some gien mechanisms of remoal known as coagulation, condensation and graitational settling as a function of time and we also analyse the health impact of aerosol before and after the aboe dynamical mechanisms by comparing the respirable dust fractions [, ] for eight different classes of typical atmospheric aerosols.

4 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Physical-mathematical model The equation describing the behaiour of aerosols is a non-linear, integro-differential equation of considerable complexity [,, 0]. If all the deposition, condensation, and coagulation mechanisms are included, it is clear that the only feasible method of solution is a numerical one. Indeed, in practical calculations this is the procedure employed and many computer codes exist which sole the dynamic equation using finite difference or finite element methods []. A major drawback of numerical methods is the ast amount of computer time necessary to obtain a useful surey of all the releant parameters. This causes difficulty in gaining physical insight into the problem. The Continuous General Dynamic Equation (Continuous GDE) can be written for the three mechanisms (coagulation, condensation and graitational settling) according to the following expression [, -]: N t = 0 0 K ( ~, ~ ) N ( ~, N ( ~, d~ N (, K(, ~ ) N ( ~, ( I N ) g N h where ( smallest stable particle and K ( ~, ~ ) 0 d~ N, (µm cm - ) is the continuous particle size distribution density function, 0 () is the olume of the is the continuous coagulation coefficient between particles of olume ~ and ~. The first two integrals on the right-hand side represent the coagulation, the following term represents the condensation and the last term is due to graitational settling. In order to close this initial boundary-alue problem (IBVP), we hae naturally to impose to the preious equation (GDE) the following initial and boundary conditions []: IC N (,0) = N 0 ( ) () BC N (, 0 < = () where IC means initial condition and BC means boundary condition. Of this form, the final GDE is written as [, -]: N t =. Coagulation 0 K ( ~, ~ ) N ( ~, N ( ~, d~ N (, K(, ~ ) N ( ~, ( I N ) g N h A population of aerosol particles eoles in time as the consequence of colliding with each other in order to adhere. This process is termed coagulation. It is known that two kinds of coagulation exist, depending on the collision mechanism []: (a) one due to the Brownian motion of aerosol particles or Brownian coagulation and (b) another due to turbulent flow or coagulation in turbulence. Coagulation modifies the particle size distribution (PSD) since it reduces the oerall number of particles and it increases their mean size (diameter), which facilitates the remoal of particles by other mechanisms as graitational settling. Either kind of coagulation reduces the particle number but the total particle olume remains constant. 0 0 d~ ()

5 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling. Nucleation. Deposition The formation and growth of an aerosol by condensation require a surface on which the apour can condense initially. This surface can be, for example, a small cluster of apour molecules, an ion or a solid salt particle termed condensation nucleus. Homogeneous nucleation is the nucleation of apour on embryos consisting of apour molecules in the absence of foreign substances, whereas heterogeneous nucleation [] is the nucleation on a foreign substance or surface, such as an ion or a solid particle. Deposition is the remoal of aerosol from the atmosphere to the Earth s surface so that the aerosol PSD suffers modifications in the particle number and in the oerall mass of these. Particle deposition due to dynamic processes is studied, dispensing with deposition caused by chemical processes. In graitational settling, the large particles ( D > 0µm) settle on the Earth s surface due to the force of graity acting on these. If a box p model is applied to study aerosol eolution by graity, calling h the height of the box (this will be the mixing layer height in the atmosphere), the dynamic equation of the graitational settling is []: Numerical method (, dn g N d t h, ( = () The exact solutions sere as benchmarks, and together with the approximate solutions, are useful in parametric studies as well as for proiding insight into the nature of aerosol eolution. Their utility for practical use is limited for seeral reasons: Exact solutions are feasible only for special, and not necessarily realistic, representations of the rate processes and for certain initial and boundary conditions. Aerosol eolution, especially if there are abrupt changes such as source reinforcement, may not follow a prescribed form of distribution. Often bimodal, trimodal, or more complicated distributions may eole, and these can not be dealt with in the context of log-normal, gamma, or other forms. In many applications, those in nuclear industry in particular, particle eolution is strongly coupled to the enironment. Thus, the dynamical equation must be soled simultaneously with equations that describe the thermal hydraulic and chemical conditions of the gaseous/aporous enironment of the particles. For practical applications, we must resort to numerical methods. From a numerical iewpoint, at least in principle, the dynamical equation is too difficult to sole. Since methods for soling partial differential equations (PDEs) hae been studied extensiely and many algorithms and computer programs are aailable, the preferred approach here has been to reduce the general dynamical equation (GDE) to a system of ordinary differential equations by employing some approximation to the integral of the coagulation terms []. The integral can be ealuated (approximated) in arious ways. The approximations are likely to be most successful if they presere the basic properties of coagulation. One such property is the conseration of mass or olume in coagulation. The arious quadratures must either be accurate enough to presere this property or one must presere it forcibly by ealuating the integrals in a special manner or by applying correctie steps to the calculations. The term of coagulation is difficult to integrate since the habitual methods (finite difference, finite elements, etc.) add numerical dispersion (spurious oscillations appear in the solution) and numerical diffusion (the

6 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling spectra are smoothed). This term represents a non-linear integro-differential equation. Since the integrals that appear ary slowly when the particle size distribution is modified, it is possible to apply a classical method of integration for systems of ordinary differential equations []. In the case of stiff systems of equations it is necessary to sole the integrals numerically, so we propose to make it through a Gaussian quadrature ia interpolation with cubic splines [0]. The other term of the GDE is integrated as the coagulation term. In this work we hae used a method of Runge-Kutta-Felhberg (RKF) []. The conjoint application of the RKF method and the denominated flux-based modified method of characteristics [] to sole the condensation equation has been carried out by means of time splitting. This method is based on a supposition of independence of the eolutions of the different processes during a t time (being this is small enough), applying consecutiely the suitable methods to each physical process during the time t.. The J-space transform and cubic spline method and This method has been used by seeral authors [-]. In particular, to describe particle size ariations oer seeral decades in size, we introduce the transformations: b( J ) ( J ) = 0 a () Y ( J, = N(, ( J ) b ln( a) () is the smallest aerosol size corresponding to J = where 0, and a and b are some arbitrary constants. J is an integer and aries from to a fairly large number that would allow consideration of all sizes of interest. With this transformation we hae: Y J, = J d ( J ) dj Y ( J ( J ), K( J, Y( J, c c t c dj Y ( J, K( J, J ) Y ( J, ( () where: and I( ( J )) ~ Y ( J, + S ( ( J ), J ( J ) bln( a) J c ( J c ) ( J ) ( J ) = () J + ln b ln( a) ( J J ) b ( a ) = (0) J d ln() J b ln( a) = () Equation () is conerted to a system of ordinary differential equations by use of quadrature formulas for the integrals and cubic splines for the deriatie on J [0] and we write: ( Y(,, Y (,, Y (,,..., Y( m,, S ~ ( J, )) Y ( J, = f J t t together with the associated initial conditions. Here, J =,,,..., m, where m is chosen sufficiently large so that all particles of interest are coered. The method gies rise to fairly accurate results, but the computational costs can be quite high because of the repeated cubic spline fits. Note that we hae taken a= and b = /. ()

7 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling. Time splitting The integration of the aerosol eolution equation (GDE) can not be carried out by applying a unique method. The basic idea of operator splitting, which is also called time splitting or the method of fractional steps [], is the following one. Suppose we hae an initial alue equation of the form: = Lu t where L is some operator. While L is not necessarily linear, suppose that it can at least be written as a linear sum of m pieces, which act additiely on u, Lu () = L u+ L u+ + Lm u () Finally, suppose that for each of the pieces, we already know a differencing scheme for updating the ariable u from the time step n to time step n +, alid if that piece of the operator were the only one on the right-hand side. We will write these updatings symbolically as: u u u = U = U = U m n ( u, n ( u, n ( u, Now, one form of operator splitting would be to get from n to n + by the following sequence of updatings: u u (/ m) ( / m) u = U = U m = U n ( u, (/ m) ( u, ( m ) / m ( u, For example, a combined adectie-diffusion equation, such as: u = + D t x x use an explicit scheme for the adectie term combined with Crank-Nicholson or other implicit scheme for the diffusion term []. Other form of operator splitting with a slightly different twist would be to reinterpret the preious equations to hae a different meaning. Let U now denote an updating method that includes algebraically all the pieces of the total operator L, but which is desirably stable only for the L piece. LikewiseU,, U m. Then a method of getting from n u to u is: u u (/ m) ( / m) = U = U n ( u, t / m) (/ m) ( u, t / m) () () () ()

8 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling u = U m ( m ) / m ( u, t / m) The time step for each fractional step in equations () is now only/ m of full time step, because each partial operation acts with all the terms of the original operator [].. Numerical method to sole the condensation equation In order to sole the condensation equation we will be based on the denominated flux-based modified method of characteristics [, ]. Now we introduce the notion of first-order partial differential equations and an important technique for soling initial-alue problems: method of characteristics. The problem we will sole is the initial-alue problem: PDE a x, + b( x, + c( x, u= 0 < x<+, 0< t <+ x t ( () IC u( x,0) = φ ( x) < x<+ (0) where PDE means partial differential equation and IC means initial condition. Note that we are soled problems with ariable coefficients. It turns out that if we change coordinates from ( x, to appropriate new coordinates ( s, τ ) (characteristic coordinates), then our differential equation becomes an ordinary differential equation. Hence, we sole the ordinary differential equation (ODE) to find u ( s, τ ), and then the last step is to plug in the alues of s and τ in terms of x and t to get u ( x,. If we recall when the diffusion equations is soled: t = u x x < x<+, 0< t <+ α () the constant α stood for the amount of diffusion, while stood for the elocity of the medium. Hence, if α = 0 (no diffusion), it is clear (since we only hae conection) that the solution trael along the x-axis with elocity. In other words, if the initial solution is u( x,0) = φ( x), then the solution to: t = x < x<+ 0< t <+ would be u( x, =φ ( x. So we can think of the first-order equation: ( x, + b( x, = 0 x t, () a () as the concentration along a stream where the elocity is gien by: Of course, if ( x, a and b( x, a( x, and b( x, a( x, / b( x, = () are constants, then we hae traelling wae solutions. On the other hand, if change in x and t, then the stream elocity aries as we go along the stream and with the time. If we get back to our basic problem (), the solution to this equation is based on a physical fact, namely, that an initial disturbance at some point x propagates along a line in the tx-plane called a characteristic. This phenomenon contrasts with many other equations (such as the heat equation) where an initial disturbance at a point affects the solution eerywhere else later in time. With this in mind, the idea is to introduce two new

9 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling coordinates s and τ (to replace x and that hae the properties: (a) s will change along the characteristic cures, and (b) τ will change along the initial cure (most likely the line t = 0 ). First, we consider the new coordinate s. By choosing s haing the aboe property, the partial differential equation (PDE): ( x, + b( x, + c( x, u = 0 x t a () is transformed into the ordinary differential equation (ODE): du + c( x, u = 0 ds Of course, the question is how to find these characteristics. The answer is simple. We pick the characteristic x( s), t( s) : 0< s<+ so that: cures {[ ] } By doing this, it is clear that: () dx = a( x, ds () dt = b( x, ds () du dx dt = + = a x, + b( x, ds x ds t ds x t In other words, along the cures {[ x s), t( s) ]: 0< s<+ } ( () (, the partial differential equation (PDE) becomes an ordinary differential equation (ODE). A detailed analysis of this method as well as its range of application can be found in Reference []. Analysis of results The selectie scaenging efficiencies hae been determined for coagulation, condensation and graitational settling of atmospheric aerosols with seeral typical size distributions labelled as []: () marine background (MB), () clean continental background (CCB), () aerage background (AB), () background and aged urban plume (BAUP), () background and local sources (BLS), () urban aerage (UA), () urban and freeway (UF) and () central power plant (CPP). These names refer to the atmospheric enironments which characterize them. The size range is diided into three well-defined subinterals referred to the literature as: (a) Aitken nuclei, (b) accumulation mode, and (c) coarse mode. The mean geometrical diameter, the geometrical Standard deiation and the oerall olume for these PSD s can be consulted in Table []. The scaenging efficiency is defined by [, ]: ( D, p n = ( Dp,0) n( Dp, n( D,0) p n = n ( Dp, ( D,0) η () where n( D p,0) is the initial PSD of aerosol particle concentration (particle µm - cm - ) and ( D p n p, is the PSD of aerosol particle concentration at the end of time t. A temperature of K and an atmospheric pressure of atm are assumed in all calculations. The results for other close temperatures and pressures do not differ much from those obtained with the alues taken. In this work, the eolution of olume particle size distribution has been calculated at different time interals (, and hours), for the eight aboe-mentioned enironments, due to graitational settling as the main dynamical mechanism of aerosol remoal in atmosphere.

10 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Table. Size distributions (log-normal) of different classes of atmospheric aerosols. Health impact The effectie PSD was determined considering retention by size in the human pulmonary region (aleolar region and tracheo-bronchial tree) for mouth breathing []. This can be approximately established as a function of inhaled particle diameter. A simplification of retention by size, suitable for our computations, is shown in Fig. [, ]. With mass respirable sampling, an attempt is made to separate the aerosol into two fractions representing the mass that would be deposited in the aleolar region and the mass that would not be deposited in this region. For this purpose, it is necessary to define the size distribution of particles deposited in the aleolar region. This material is defined as respirable dust. There are seeral definitions of respirable dust aailable. In, the British Medical Research Council (BMRC) defined the respirable fraction (RF) in terms of the particle aerodynamic diameter, D (µm), and it is calculated from the expression []: pi RF i BMRC Dpi = () In, the American Conference of Goernment Industrial Hygienists (ACGIH) defined the respirable dust fraction []: RF i D pi BMRC = 0 () In, the United States Enironment Protection Agency (USEPA) [] defined the respirable dust fraction as a function of the particle aerodynamic diameter by means of a semi-empirical cure which is also shown in Fig.. Fig.. Filters: Settled fraction of monodisperse aerosol in the pulmonary region as a function of particle diameter (from data in References [] and U.S. EPA []). The form of the first two filters (BMRC and ACGIH) indicates that for the gien aerosol the BMRC definition will indicate slightly more respirable mass than the ACGIH definition []. Moreoer, because retention under a µm particle diameter is not well defined in the first two filters, the respirable fraction measured according to the BMRC and ACGIH criteria would be smaller than results obtained with the USEPA filter. The effect of retention filters differs greatly. The BMRC respirable dust and the USEPA respirable dust hae been represented in Figs. and, at the initial time and after, and hours of graitational settling in an urban enironment (UA), respectiely. Fig.. Eolution of respirable dust in an urban enironment (UA) as a function of elapsed time by coagulation, condensation and graitational settling for BMRC filter. Fig.. Eolution of respirable dust in an urban enironment (UA) as a function of elapsed time by coagulation, condensation and graitational settling for USEPA filter. With BMRC filter, the accumulation mode of the urban enironment remains roughly unaltered with regard to its mean geometrical diameter and its mass, and the coarse mode decreases in height because of the main effect to the graity force (see Fig. ). Howeer, with USEPA filter, we can obsere that graitational settling affects to the coarse mode mainly while the accumulation mode remains unchanged. In any case, its effect is of an essential importance (see Fig. ). The numerical results obtained regarding oerall efficiencies of remoal can be seen in Table. The first interesting result is: only.% of initial urban aerosol olume (UA) is settled in the pulmonary region

11 International Journal of Computer Mathematics Page 0 of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Conclusions without mechanisms of remoal (scaenging mechanisms), that is to say, without coagulation, condensation and graitational settling (with USEPA filter). After hours of graitational settling, this percentage of respirable dust decreases to.%. This is an important result from a health perspectie. Therefore, graitational settling is the main mechanism yielding lower retention efficiencies in the lungs oer time. Table. Percentages of respirable dust ersus oerall initial olume and ersus respirable initial olume, during, and hours of coagulation, condensation and graitational settling. A simple linear fit of data appearing in Table for the USEPA filter indicates that graitational settling keep on the straight line of equation p (%) =.t+., where p(%) is the percentage of respirable dust and t is the time in hours (, and hours in this case). The correlation coefficient (measuring the goodness of fi is r = 0. so that the fit can be considered as ery good. Hence, the main mechanism to remoe respirable dust is graitational settling since in all eight atmospheric enironments studied, the presence of graitational settling decreases greatly the initial respirable dust olume with respect to that after hours of graitational settling (see Table ). A computational procedure has been deeloped based on the general-purpose code deeloped for us for modelling and simulating the deposition by coagulation, condensation and graitational settling of aerosol particles for eight different enironments and in order to calculate the percentage of respirable dust ersus respirable initial olume as a function of elapsed time. The findings of this study suggest that it may be possible to deise a practical procedure for establishing a remoal s model by using a combined experimental/computational approach. Once deposited, particles are retained in the lung for arying times, depending on their physicochemical properties, their location within the lung, and the type of clearance mechanism inoled. It is known that insoluble particles deposited in aleolar region are cleared ery slowly, oer a period of months or years. Dissoled particles pass through the thin aleolar membrane into the bloodstream. Solid particles may dissole slowly or be engulfed by aleolar macrophages (phagocytic cells) and dissoled or transported to lymph nodes or mucociliary escalator. Fibrogenic dusts, such as silica, interfere with this clearance mechanism and cause gradual scarring or fibrosis of the aleolar region. This study has tried to determine the respirable dust comparing to the initial respirable dust with the respirable dust after,, and hours of coagulation, condensation and graitational settling for eight characteristic atmospheric enironments, gien place to the results exposed aboe. Aerosol remoal by coagulation, condensation and graitational settling is more efficient in urban enironments (UA) as in other enironments characterized as clean areas (CCB), due to the large relatie olume of coarse mode in the urban case with respect to the other modes (accumulation mode and Aitken mode). In an urban enironment (UA) for longer times, many aerosol particles of coarse mode are scaenged, but the Aitken nuclei and accumulation mode are scaenged with a great difficulty by coagulation, condensation and graitational settling. The equation describing the behaiour of aerosols is a non-linear, integro-differential equation of considerable complexity [, ]. If all of deposition, condensation, rainout, coagulation mechanisms, etc., are included, it is clear that the only feasible method of solution is a numerical one. Indeed, in practical calculations this is the procedure and many computer codes exist which sole the dynamic equation using finite difference or finite element methods. Our program can proide useful results when one species is dominant, the particles are nearly spherical, the aerosol is homogeneous, temporal ariations are not large, and the coupling with the surroundings is weak. Use of the aailable programs for analysis or design of experiments or natural/industrial processes must be undertaken with considerable care. It is important to select a method and rate process models appropriate to the

12 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling References problem as the computed results are dependent on these. A major drawback of numerical methods is the ast amount of computer time necessary to obtain a useful of all the releant parameters. This causes difficulty in gaining physical insight into the problem. For this reason a ariety of approximate techniques hae been deeloped which enable analytic solutions to be able. Such techniques use the complementary methods of kernel approximation and equation approximation or a combination of both. The numerical simulation of fluid mechanics problems has become a routine part of engineering practice as well as a focus for fundamental and applied research. Though there are still arious topical areas where our physical understanding and ineffectie numerical algorithms limit the inestigation, a large number of complex phenomena can now be confidently studied ia numerical simulation. Therefore, the model deeloped in this work is certainly incomplete, and it can be used as one part of a more extensie one that takes into account other phenomena (i.e. rainout, washout, phoretic phenomena, etc.) in order to forecast the real eolution of aerosols PSD s [, ]. A big number of computer programs hae been written that attempt to sole arious ersions of the dynamical equations by using some of the methods described aboe [, ]. Improements in computer hardware and system software (e.g., powerful workstations and window enironments) hae contributed significantly to streamlining the numerical process. Thus, these programs proide results with a good numerical accuracy for prescribed forms of the rate processes and for single component aerosols and so to integrate the oerall mathematical model of the atmosphere. Acknowledgements The authors express deep gratitude to Department of Mathematics and Department of Construction at Oiedo Uniersity for their computational support and useful assistance. Helpful comments and discussion are gratefully acknowledged. This work has been partially supported by the basic research grant CN-0-0. [] Baron, P.A. and Willeke, K., 00, Aerosol Measurement: Principles, Techniques, and Applications (New York: Wiley-Interscience). [] Cheng, Y.S., Zhou, Y. and Chen, B.T.,, Particle deposition in a cast of human oral airways. Aerosol Science and Technology, 0, --. [] Dennis, R., 000, Handbook on Aerosols (New York: Uniersity Press). [] Farlow, S.J.,, Partial Differential Equations for Scientists and Engineers (New York: Doer). [] Fernández-Díaz, J.M., García Nieto, P.J., Rodríguez Braña, M.A. and Arganza García, B.,, A fluxbased characteristics method to sole particle condensational growth. Atmospheric Enironment, (0), --. [] Friedlander, S.K., 000, Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics (New York: Oxford Uniersity Press). [] García Nieto, P.J., Fernández-Díaz, J.M., Arganza García, B. and Rodríguez Braña, M.A.,, Parametric study of selectie remoal of atmospheric aerosol by below-cloud scaenging. Atmospheric Enironment, (), --. [] Hinds, W.C.,, Aerosol Technology (New York: John Wiley & Sons). [] Jacobson, M.Z.,, Fundamentals of Atmospheric Modelling (New York: Cambridge Uniersity Press).

13 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling [0] Kincaid, D. and Cheney, W.,, Numerical Analysis: Mathematics of Scientific Computing (California: Brooks/Cole Publishing Company). [] Lutgens, F.K. and Tarbuck, E.J., 00, The Atmosphere: An Introduction to Meteorology (Upper Saddle Rier, New Jersey: Prentice-Hall). [] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P.,, Numerical Recipes in Fortran (New York: Cambridge Uniersity Press). [] Roache, P.J.,, A flux-based modified method of characteristics. Int. J. of Numerical Methods in Fluids, (), --. [] Reist, P.C.,, Introduction to Aerosol Science (New York: Macmillan Press). [] Seinfeld, J.H.,, Atmospheric Chemistry and Physics of Air Pollution (New York: John Wiley & Sons). [] Seinfeld, J.H. and Pandis, S.N.,, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (New York: John Wiley & Sons). [] Stoer, J. and Bulirsch, R.,, Introduction to Numerical Analysis (New York: Springer-Verlag). [] Suck, S.H. and Brock, J.R.,, Eolution of Atmospheric Aerosol Particle Size Distributions ia Brownian Coagulation: Numerical Simulation. Journal of Aerosol Science, 0(), --. [] Williams, M.M.R. and Loyalka, S.K.,, Aerosol Science: Theory and Practice (Oxford: Pergamon Press). [0] Zachmanaglou, E.C. and Thoe, D.W.,, Introduction to Partial Differential Equations (New York: Doer). Figure Caption

14 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling BMRC ACGIH USEPA D p (µm) Fig.. Filters: Settled fraction of monodisperse aerosol in the pulmonary region as a function of particle diameter (from data in References [] and U.S. EPA []).

15 International Journal of Computer Mathematics Page of Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Fig.. Eolution of respirable dust in an urban enironment (UA) as a function of elapsed time by coagulation, condensation and graitational settling for BMRC filter.

16 Page of International Journal of Computer Mathematics Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Fig.. Eolution of respirable dust in an urban enironment (UA) as a function of elapsed time by coagulation, condensation and graitational settling for USEPA filter.

17 International Journal of Computer Mathematics Page of Enironment 0Marine background Clean continental background Aerage background Background and aged urban plume 0Background and local sources Urban aerage Urban and freeway Central Power Plant 0 Mathematical study of the selectie remoal of atmospheric aerosols by coagulation, condensation and graitational settling Table Caption Table. Size distributions (Log-Normal) of different classes of atmospheric aerosol. Nuclei Mode Accumulation Mode Coarse Mode D pg σ V g D (µm (µm) cm - pg σ V g D ) (µm (µm) cm - pg σ g ) (µm) V (µm cm - ) Table. Percentages of respirable dust ersus oerall initial olume and ersus respirable initial olume, during, and hours of coagulation, condensation and graitational settling. Enironment/Filter Initial hours hours hours MB/BMRC./00./../.0./0. MB/ACGIH.0/00./../0../. MB/USEPA./00.0/0../../. CCB/BMRC./00.0/0../../. CCB/ACGIH.0/00./0../. 0./. CCB/USEPA./00./../../. AB/BMRC./00./../../. AB/ACGIH./00./.0 0./.0./. AB/USEPA./00./../../. BAUP/BMRC./00./../../0. BAUP/ACGIH./00./../../0. BAUP/USEPA./00./0../../. BLS/BMRC./00./.0./../. BLS/ACGIH./00./../. 0.0/.0 BLS/USEPA./00./../0.0./. UA/BMRC./00.0/../../. UA/ACGIH.0/00./../../. UA/USEPA./00 0./../../0. UF/BMRC.0/00./..0/../. UF/ACGIH./00./../. 0./. UF/USEPA./00./../../. CPP/BMRC./00./.0./0../. CPP/ACGIH./00./../. 0./. CPP/USEPA.0/00./.0./../.

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21 International Journal of Computer Mathematics Page 0 of Enironment Marine background Clean continental background Aerage background Background and aged urban plume Background and local sources Urban aerage Urban and freeway Central Power Plant Table. Size distributions (Log-normal) of different classes of atmospheric aerosol. Nuclei Mode Accumulation Mode Coarse Mode D pg σ V g D (µm (µm) cm - pg σ V g D ) (µm (µm) cm - pg σ V g ) (µm (µm) cm - )

22 Page of International Journal of Computer Mathematics Table. Percentages of respirable dust ersus oerall initial olume and ersus respirable initial olume, during, and hours of coagulation, condensation and graitational settling. Enironment/Filter Initial hours hours hours MB/BMRC./00./../.0./0. MB/ACGIH.0/00./../0../. MB/USEPA./00.0/0../../. CCB/BMRC./00.0/0../../. CCB/ACGIH.0/00./0../. 0./. CCB/USEPA./00./../../. AB/BMRC./00./../../. AB/ACGIH./00./.0 0./.0./. AB/USEPA./00./../../. BAUP/BMRC./00./../../0. BAUP/ACGIH./00./../../0. BAUP/USEPA./00./0../../. BLS/BMRC./00./.0./../. BLS/ACGIH./00./../. 0.0/.0 BLS/USEPA./00./../0.0./. UA/BMRC./00.0/../../. UA/ACGIH.0/00./../../. UA/USEPA./00 0./../../0. UF/BMRC.0/00./..0/../. UF/ACGIH./00./../. 0./. UF/USEPA./00./../../. CPP/BMRC./00./.0./0../. CPP/ACGIH./00./../. 0./. CPP/USEPA.0/00./.0./../.