United States
Environmental Protection
Agency
Atmospheric Research and Exposure
Assessment Laboratory
Research Triangle Park NC 27711
Research and Development
EPA/600/S3-91/020 April 1991
vxEPA Project Summary
Modal Aerosol Dynamics
Modeling
Evan R. Whitby, Peter H. McMurry, Uma Shankar and Francis S. Binkowski
The report described by this Project
Summary presents the governing
equations for representing aerosol dy-
namics, based on several different rep-
resentations of the aerosol size distri-
bution. Analytical and numerical solu-
tion techniques for these governing
equations are also reviewed. Described
in detail is a computationally efficient
numerical technique for simulating
aerosol behavior in systems undergo-
ing simultaneous heat transfer, fluid
flow, and mass transfer in and between
the gas and condensed phases. The
technique belongs to a general class
of models known as modal aerosol dy-
namics (MAD) models. These models
solve for the temporal and spatial evo-
lution of the particle size distribution
function. Computational efficiency is
achieved by representing the complete
aerosol population as a sum of additive
overlapping populations (modes), and
solving for the time rate of change of
integral moments of each mode. Appli-
cations of MAD models for simulating
aerosol dynamics in continuous stirred
tank aerosol reactors and flow aerosol
reactors are provided. For the applica-
tion to flow aerosol reactors, the dis-
cussion is developed in terms of con-
siderations for merging a MAD model
v/ith the SIMPLER routine described by
Patankar (1980). Considerations for in-
corporating a MAD model into the U.S.
Einvironmental Protection Agency's
Regional Particulate Model are also
described.
Numerical and analytical techniques
for evaluating the size-space integrals
of the modal dynamics equations
(MDEs) are described. For multimodal
lognormal distributions, an analytical
expression for the coagulation integrals
of the MDEs, applicable for all size re-
gimes, is derived, and is within 20% of
accurate numerical evaluation of the
same moment coagulation integrals. A
computationally efficient integration
technique, based on Gauss-Hermite
numerical integration, is also derived.
This Project Summary was developed
by EPA's Atmospheric Research and
Exposure Assessment Laboratory, Re-
search Triangle Park, NC, to announce
key findings of the research project
that is fully documented in a separate
report of the same title (see Project
Report ordering information at back).
Introduction
For processes in the atmosphere, com-
puter models are indispensable tools for
assessing the environmental impact of
anthropogenic and biogenic sources of
gases, vapors, and particles, because real-
world emission rates are not easily
changed and comprehensive data sets are
difficult to obtain. Atmospheric models that
are calibrated with existing data can be
used to assess the impact of various pro-
posed strategies for altering atmospheric
emissions of air pollutants.
Many atmospheric aerosol models have
been developed using finite-difference
techniques. Finite-difference techniques
are commonly used because they allow
very general models to be constructed,
and the accuracy and computational effort
can be altered by adjusting the grid struc-
Printed on Recycled Paper
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ture. The concept of a "grid-independent"
solution implies a model whose accuracy
has converged within "reasonable" limits
to the true solution. Increased computa-
tional speed with such models, however,
is usually obtained by a deliberate sacrifice
of accuracy, such as by approximating a
truly three-dimensional situation with a two-
dimensional simulation, and/or by imple-
menting a coarse grid system to maintain
the resulting simulation within the capa-
bilities of the computer used.
High computational speed and accu-
racy can also be obtained by sacrificing
the flexible structure of finite-difference
techniques, and building a modeling
structure based on a physical representa-
tion of the dynamic system modeled. For
example, it has often been observed that
aerosol size distributions in the atmosphere
and elsewhere can accurately be approxi-
mated by multimodal bgnormal functions.
This provides a motivation for developing
aerosol dynamics models that assume
distributions to be lognormal, which facili-
tates the derivation of moment equations
that are relatively straightforward to inte-
grate.
Because the computational advantage
of the moment modeling technique is so
great, moment models for a variety of
aerosol systems have been developed and
used by many researchers. This report
surveys the literature on aerosol dynam-
ics models, and presents the general form
of the governing equations used in these
models. It also provides a detailed deriva-
tion of the governing equations used in
modal aerosol dynamics models.
Simulating the Dynamics of
Aerosol Systems
Modeling aerosol dynamics has at least
three challenging aspects: a distribution
of particle sizes must be considered; co-
agulation introduces summations and in-
tegrals over the size distribution; and par-
ticle sizes of interest in many systems
span the free-molecule, transition, and
continuum size regimes, requiring gener-
alized expressions that apply to all size
regimes to be used. Objectives of aerosol
dynamics modeling include determining the
particle size distributions and mixing
characteristics of all chemical compounds
as functions of time and position, and
determining the magnitude and location of
particle deposition.
The dynamics equations for the aerosol
phase are formulated by expressing
mathematically the effect of each process
that represents interactions with the aero-
sol phase. The governing differential
equation is commonly referred to as the
general dynamics equation (GDE)
(Friedlander, 1977). In this report, various
forms of the GDE for representing the
time dependence of the aerosol size dis-
tribution are presented. These are broadly
classified as discrete, sectional, continu-
ous-spline, and modal. Aerosol dynamics
models based on the different forms of
the GDE are reviewed and compared. The
derivation of a computationally efficient
aerosol dynamics modeling technique—
modal aerosol dynamics (MAD) model-
ing—is given in detail, and an application
of this modeling technique for simulating
atmospheric aerosol processes is pre-
sented.
Solving for the temporal and spatial
evolution of the size distribution, n, is tan-
tamount to solving for the coefficients of
the approximating functions. The math-
ematical representation of the size distri-
bution function determines the form of the
GDE solved, and hence the solution algo-
rithms required to construct a complete
model. Because the solution techniques
used to solve for the function coefficients
depend on the mathematical representa-
tion of n, aerosol dynamics models are
commonly named according to the math-
ematical representation of n. Commonly
used aerosol models that approximate the
aerosol size distribution with a set of
mathematical functions are the discrete,
continuous-spline, sectional, and MAD
models. Discrete, continuous-spline, and
sectional models have been described in
the literature, and are reviewed in this
report. To the authors' knowledge, how-
ever, a basic derivation of the governing
dynamics equations for MAD models does
not exist in the literature.
The discrete form is the most general
representation of the size distribution,
where separate equations for the time-
dependent concentration of each cluster
size are formulated. The cluster size in-
crements are determined by the smallest
molecule causing cluster growth. Discrete
models are useful for simulating nucle-
ation processes, where the dynamics of
clusters with relatively few /-mers are im-
portant.
The continuous-spline representation is
so called because particle size-space is
subdivided into a series of contiguous
sections, and the section functions are
splined smoothly together at the section
boundaries. Cubic functions are commonly
used, implying that four differential equa-
tions (one for each coefficient of the cubic
equation) must be solved for each sec-
tion.
The sectional representation is so called
because the particle size space is subdi-
vided into a series of sections and the
distribution function of a prescribed mo-
ment of the distribution (usually the mass
distribution) is assumed constant within
each section. This assumption results in a
size distribution with the appearance of a
histogram. For sectional techniques, one
differential equation is required for each
section.
The modal representation is so called
because the particle size range is divided
into a small number of overlapping inter-
vals (usually three or less), called modes.
Overlapping intervals are used because
the modal approximation originates from
the assumption that the complete distribu-
tion is composed of multiple distinct aero-
sol populations. Within each mode, n is
represented by a continuous distribution
function. Because the modes overlap, the
distribution functions for all modes are
added to yield an approximation for the
entire distribution function. A differential
equation is required for each independent
modal function parameter.
The above modeling techniques are
presented in order of decreasing compu-
tational requirements. It should be noted,
however, that the less computer-intensive
methods are not necessarily less accu-
rate. For example, if an actual system can
be accurately simulated by a highly pa-
rameterized distribution function, then
aerosol dynamics models that employ such
a distribution function should accurately
simulate the system behavior. The associ-
ated savings in computational resources
can make the difference between a prac-
tical model and a model that is too complex
to be solved even with the most sophisti-
cated available supercomputer. This is
especially true for complicated real-world
problems such as the behavior of atmo-
spheric aerosols, which is one reason why
highly parameterized models are still in
use today.
Modal Aerosol Dynamics
Models
Because aerosol distributions often re-
sult from the additive effect of multiple
sources, distributions composed of mul-
tiple distinct aerosol populations (modes)
occur in many systems of interest, and
MAD models are often appropriate for
simulating the behavior of such systems.
The central physical assumption of MAD
models is stated as follows:
An aerosol may be viewed as an
assemblage of distinct populations
of particles, distinguished by size or
chemical composition. The size dis-
tribution of each population is ap-
proximated by an analytical distribu-
tion function.
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In general, aerosol models solve for the
temporal and spatial evolution of the coef-
ficients of the size distribution. For MAD
models, however, differential equations for
the modal function coefficients are re-
placed by differential equations for mo-
ments of each mode (e.g., total number or
mass concentration). Solving for the time
rate of change of a sufficient number of
moments of the distribution allows the size
distribution at every point in the time and
space of interest to be determined. This
report presents a derivation for converting
the differential equations for multiple
overlapping modes into an equivalent set
of differential equations for the moments
of these modes.
To account for the effects of processes
on the size distribution in MAD models,
the net effect of each process on each
mode must be determined. This requires
evaluating integral expressions for the dy-
namics of each process, and including
these effects in the governing dynamics
equations for each mode. One of the prin-
cipal tasks in developing MAD models,
therefore, is to develop solution algorithms
for evaluating these integral terms. Co-
agulation processes are especially diffi-
cult to handle, because they represent
particle-particle interactions, and this
translates into double integrals over the
interacting populations. To maintain com-
putational efficiency with MAD models,
care must be taken when formulating and
solving these integral expressions. This
report discusses in detail the derivation of
these integral terms, and algorithms for
analytically and/or numerically evaluating
the resulting mathematical integrals. Many
analytical expressions for common dy-
namical processes are provided.
A new technique for evaluating multi-
modal coagulation integrals has been de-
rived that is within 20% of accurate nu-
merical evaluations of the same coagu-
lation integrals. Techniques for numeri-
cally evaluating the modal integrals based
on Gauss-Hermite numerical integration
are also described. The report's presenta-
tion on numerical integration techniques
is important because failure to implement
efficient integration techniques can result
in the computational effort of the resulting
model approaching that of one of the more
detailed aerosol dynamics models. The
accuracy and domain of applicability of
the numerical techniques presented in this
work are therefore clearly presented, to
guide the modeler in selecting the tech-
nique most suited to the particular model-
ing application.
References
Friedlander, S.K. (1977) Smoke, Dust and
Haze. John Wiley and Sons, New York.
Patankar, S.V. (1980) Numerical Heat
Transfer and Fluid Flow. Hemisphere,
New York.
. S. GOVERNMENT PRINTING OFFICE: 1991/548-028/20200
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Evan R. Whitby was with Computer Sciences Corporation, Research Triangle Park,
NC 27709, and is now with Hitachi, Ltd., Tsuchiura, IbarakiSOO, Japan; Peter H.
McMurry is with the University of Minnesota, Minneapolis, MN 55455; Uma
Shankar is with Computer Sciences Corporation, Research Triangle Park, NC
27709; Francis S. Binkowski is on assignment to the Atmosperic Research and
Exposure Assessment Laboratory, Research Triangle Park, NC 27711, from the
National Oceanic and Atmospheric Administration.
O. Russell Bullock is the EPA Project Officer (see below).
The complete report, entitled "Modal Aerosol Dynamics Modeling," (Order No.
PB91-161 729/AS; Cost: $31.00, 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:
Atmospheric Research and Exposure Assessment Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
United States
Environmental Protection
Agency
Center for Environmental
Research Information
Cincinnati, OH 45268
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