United States Environmental Protection Agency Environmental Sciences Research Laboratory Research Triangle Park NC 27711 Research and Development EPA-600/S3-84-088 Sept. 1984 &EPA Project Summary Urban Aerosol Modeling J. R. Brock and T. H. Tsang Accurate numerical schemes for simulation of the coagulation and condensation/evaporation (with vapor conservation) processes of single component aerosols were developed. The schemes were incorporated into modules that permit simulation of these processes in models of atmospheric dispersion and transport. This Project Summary was developed by EPA's Environmental Sciences Research Laboratory. Research Triangle Park, NC, to announce key findings of the research project that is fully docu- mented in a separate report of the same title (see Project Report ordering information at back). Introduction Urban aerosols are associated with various adverse effects such as reduced visibility, inadvertent weather modifica- tion, and increased incidence of respira- tory disease. An understanding of these associations requires, among other factors, knowledge of ambient aerosol size and composition distributions. The dynamic processes that produce these distributions include, in addition to atmospheric dispersion and transport, nucle-ation, coagulation, condensation/ evaporation, and deposition. Models that include these processes are necessary in order to develop aerosol air quality regulations and control strategies. There are currently no validated urban aerosol models that deal with the complexity of these processes. The research project described in detail in the Project Report (see ordering information at back) involved develop- ment of urban aerosol models that include the dynamic processes shaping urban aerosol size and composition distri- butions First, data bases and a K-Theory model for the super- and sub-micrometer aerosol mass concentrations were vali- dated. This Project Summary describes our more recent work on the development of computer modules that accurately describe the coagulation and condensa- tion/evaporation processes of single com- ponent aerosols. These modules can be employed in the K-Theory model or in other models Theoretical Framework Modules that would incorporate aerosol growth processes in a general atmospher- ic dispersion and transport model were developed. In order to understand how growth modules operate, it is useful to illustrate their incorporation in a K- Theory model (as used in this project). For a single component aerosol, it is necessary to study the evolution of the particle number density function, n(m,x,y,z,t), where n(m,x,y,z,t)dm is the number of particles having masses in the range m,dm at downwind position x,y at height z at time t. The evolution equation is: 3n (m,x,y,z,t) + y /z> 3n (m,x,y,z,t) 3t 3x 3m 3z (m,s) n (m,x,y,z,t)] (7) 3n(m,x,y.z.t)+ 3 dz 3y . 3n (m,x,y,z,t) + G^m| 3n (m,x,y,z,t) 3y 3z m b(m-m', m')n(m-m')n(m')dm' (1) - n (m) b(m,m')n(m')dm' ) o This equation is coupled to the following conservation equation for the vapor, ------- given in terms of the saturation ratio, s- 3s(x,y,z.t) + y (z) 9s(x,y,z,t) at 9x the growth rate terms in Eqs. (1) and (2) become, respectively: u dJ' b,(j, J')g(J,t)g(J',t) = JL K(z) 9z K(y) 9z 9y 9s(x,y,z,t) 9y - — f°°*(m,s)n(m,x,y,z,t)dm (2) ( JO In these equations, U(z) is the x compo- nent of the mean fluid velocity, K(z) and K(y) are the vertical and transverse eddy diffusivities, respectively; Gz(m) is the gravitational settling speed for a particle of mass m, ^(m,s) is the condensation/ evaporation growth law for a particle, and b(m,m') is the coagulation coefficient. These quantities have been established based on the literature on growth and Brownian coagulation in noncontmuum regimes. Their forms are not detailed here Eqs. (1) and (2) are subject to appropri- ate boundary and initial conditions. Explicitly the deposition velocity, Vd(m), of a particle with mass m at ground level is: n(m) Vd(m) = K 9n(m)/9z, z = 0 (3) The deposition term plays an important role in aerosol simulation. It is not difficult to incorporate this condition in general atmospheric models; however, there is considerable uncertainty as to the correct parameterization of the deposition velo- city. The other boundary and initial conditions are not detailed here In the future, photochemical models will be included to provide a source term in Eq. (2); this will involve addition of a nucleation term to Eq. (1). These modifi- cations are in progress. In aerosol dynamics simulation, parti- cles with radii covering approximately four orders of magnitude (10 to 10~A cm) must be considered. A logarithmic trans- formation or particle mass was used to avoid problems associated with this large range: m(J) = m(Jo)exp(q(J-J0)) (4) where J is a positive number greater than or equal to J0, rn (J0) is the mass of a particle starting at J0, andq isa numerical parameter that can be selected to give equally spaced integer J values. From the following definition of the density func- tion, n(m(J» = g(J)/qm(J), 9t - g(J,t) dJ'b(J,J')g(J',t) - [¥(J)g(J)/qm(J)] (5) 9J and Ju = J - In2/q, J > 2; J = J + (1/q)ln[1-exp(q(J'-J))]; b,(J,J') = (m(J)/m(J» b(m-m', m'). (6) The adjustable parame ar, q, is a great advantage in "fine tuning" for mass or diameter spacings that increase numeri- cal accuracy. Numerical Solution Strategies Coagulation The coagulation process can be numer- ically simulated with high accuracy. We chose methods that optimize both accu- racy and efficiency. Accuracy was studied in two ways1 comparison of numerical simulations with analytical solutions for restricted forms of b(m,m') and compari- son with Brownian coagulation through tests for conservation of mass. Cubic spline was used for numerical quadrature and interpolation of the coagulation terms. Gear's method was used for time integration. These comparisons showed that simulation by these methods is accurate and reliable and that errors can be reduced to any desired level. We found the method of fractional steps to be essential for incorporating the coagulation rate process into any atmos- pheric dispersion and transport model. A problem associated with this method is the question of where in the time spl itting the source and sink terms (i e., those associated with the aerosol dynamics processes) should be included. In our work, coagulation was included in the dispersion and transport model (Eq. ( 1 )) by the following procedures: (a) Solve the advection equation by a method such as Fromm's at each vertical level (at each interior collocation point) Jmax times, where Jrrmx is the number of size classes. (b) Solve the diffusion equation by a method such as orthogonal collo- cation on finite elements at each grid point in x direction Jmax times. (c) Solve the coagulation equation at each grid point The numerical coagulation results that we obtained prove that the method of fractional steps can accurately simulate aerosol coagulation and transport The desirability of this method can be seen by the fact that a fully implicit scheme for solving a three-dimensional aerosol model requires simultaneous solution of IM-JM-KM-Jmax equations. For IM = 20 in x direction, JM = 20 in y direction, KM = 10 in z direction, and Jmax = 30 size classes a simultaneous solution for the 120,000 unknowns would not be feasible; the method of fractional steps removes the necessity of carrying out simultaneous solutions for these unknowns. We have shown that the method of fractional steps is a reliable treatment for multidimen- sional nonlinear problems, including aerosol coagulation. Condensation /Evaporation The condensation/evaporation term m Eq. (1) is deceptively simple. In the J- space formulation, the condensation coefficient, V, varies by ten order of mag- nitude. The Kelvin effect causes this coefficient to change in sign at a certain particle size for a given supersaturation; it has a negative value for evaporation and a positive value for condensation. These factors make the numerical solution of the condensation/evaporation term difficult. Difficulties in the numerical solution of first-order hyperbolic equations such as the condensation/evaporation terms are evidenced by the numerous reports of attempts at numerical solutions of similar hyperbolic equations (e.g., advection). Most numerical schemes for hyperbolic equations give rise to numerical disper- sion and diffusion. Numerical dispersion, caused by the combination of large phase errors and insufficient short wave damping, manifests itself by the unphysi- cal wakes behind and ahead of the simulated regions of high concentration. Numerical diffusion lowers the peak values of the concentration distribution but increases the values around the peak. A robust numerical scheme for conden- sation/evaporation should be free of numerical dispersion and should mini- mize numerical diffusion. Eulerian nu- merical schemes create numerical dis- persion, Lagrangian schemes do not. We combined Eulerian and Lagrangian methods to produce a numerical scheme in which numerical dispersion and diffu- sion for condensation/evaporation can be reduced to any desired level. Incorporation of the condensation/ evaporation term in an atmospheric dis- ------- persion and transport model is analogous to incorporation of coagulation; however; coupled equations such as Eqs. (1 )and (2) are involved The simultaneous solution of 104to 105 unknowns in a dispersion and transport model by an implicit numerical scheme using a matrix technique (as would occur if a fully implicit solution of Eqs. (1) and (2) were attempted) is difficult. We used the method of fractional steps to incorporate advection, diffusion, and condensation/ evaporation by the following procedures: (1) Solve the advection and diffusion equation Jmax times, where Jmax is the number of size classes. (2) Solve the condensation/evapora- tion equation by our algorithm at each grid point. (3) Solve the advection and diffusion equation for s (4) Calculate the integral term (a source term for vapor due to evaporation) in Eq. (2) and update the saturation ratio at that grid point. This method of fractional steps decou- ples Eqs. (1)and(2). If the saturation ratio of the vapor does not change with position—that is, if it is constant—proce- dures 3 and 4 are unnecessary. The explicit nature of procedure 4 does not pose a problem because the source term is counterbalanced by the dilution effect of advection and diffusion It is particularly important that field and laboratory aerosol data be obtained with sufficient resolution and accuracy to ' permit validation of atmospheric aerosol dynamics models. These data should include time and space resolved size distributions and chemical speciations. Recommendations The general approach and numerical solution schemes outlined here represent one of the first comprehensive studies of aerosol growth in the context of a model of atmospheric dispersion and transport. The modules for coagulation and conden- sation/evaporation that we developed yield highly accurate simulations of aerosol growth processes in the context of an atmospheric model. These growth modules should be compared with proposed approximate methods for simu- lation of aerosol dynamics We feel that currently available approximate methods are not adequate for atmospheric simula- tions on the urban scale. We recommend that new approximate methods be devel- oped. The numerical schemes for coagulation and condensation/evaporation developed in this project need to be extended to multi-component aerosol dynamics. This will be essential to the study of chemical speciation. Similarly, approximate meth- ods for multi-component aerosols should be developed on the basis of accurate simulation methods. J. R. Brock and T. H. Tsang are with the University of Texas. Austin, TX 78712. H. M. Barnes is the EPA Project Officer (see below j. The complete report, entitled "Urban Aerosol Mode/ing," (Order No. PB 84-233 469; Cost: $8.50, subject to change) will be available only from: National Technical Information Service 5285 Port Royal Road Springfield. VA 22161 Telephone: 703-487-4650 The EPA Project Officer can be contacted at: Environmental Sciences Research Laboratory U.S. Environmental Protection Agency Research Triangle Park, NC 27711 ------- United States Environmental Protection Agency Center for Environmental Research Information Cincinnati OH 45268 Official Business Penalty for Private Use $300 U S GOVERNMENT PRINTING OFFICE, 1984 —759-015/71 ------- |