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,

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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-

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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

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