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

-------
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.

-------
  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

-------
   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
      BULK RATE
POSTAGE & FEES PAID
         EPA
   PERMIT No. G-35
Official Business
Penalty for Private Use $300
EPA/600/S3-91/020

-------