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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