February 1991
MODAL AEROSOL DYNAMICS MODELING
by
Evan R. Whitby
Computer Sciences Corporation
Research Triangle Park, NC 27709
Peter H. McMurry
Mechanical Engineering Department
University of Minnesota
Minneapolis, MN 55455
Uma Shankar
Computer Sciences Corporation
Research Triangle Park, NC 27709
Francis S. Binkowski*
Atmospheric Sciences Modeling Division
Air Resources Laboratory
National Oceanic and Atmospheric Administration
Research Triangle Park, NC 27711
Contract No. 68-01-7365
Project Officer
0. Russell Bullock, Jr.*
Atmospheric Sciences Modeling Division
Air Resources Laboratory
National Oceanic and Atmospheric Administration
Research Triangle Park, NC 27711
*On assignment to the Atmospheric Research and Exposure Assessment Laboratory,
U.S. Environmental Protection Agency
ATMOSPHERIC RESEARCH AND EXPOSURE ASSESSMENT LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 27711
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February 1991
MODAL AEROSOL DYNAMICS MODELING
by
Evan R. Whitby
Computer Sciences Corporation
Research Triangle Park, NC 27709
Peter H. McMurry
Mechanical Engineering Department
University of Minnesota
Minneapolis, MN 55455
Uma Shankar
Computer Sciences Corporation
Research Triangle Park, NC 27709
Francis S. Binkowski*
Atmospheric Sciences Modeling Division
Air Resources Laboratory
National Oceanic and' Atmospheric Administration
Research Triangle Park, NC 27711
Contract No. 68-01-7365
Project Officer
0. Russell Bullock, Jr.*
Atmospheric Sciences Modeling Division
Air Resources Laboratory
National Oceanic and Atmospheric Administration
Research Triangle Park, NC 27711
*On assignment to the Atmospheric Research and Exposure Assessment Laboratory,
U.S. Environmental Protection Agency
ATMOSPHERIC RESEARCH AND EXPOSURE ASSESSMENT LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 27711
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NOTICE
The information in this document has been funded in part by the United
States Environmental Protection Agency under Contract Number 68-01-7365 to
Computer Sciences Corporation. It has been subjected to the Agency's peer and
administrative review, and it has been approved for publication as an EPA
document. Mention of trade names or commercial products does not constitute
endorsement or recommendation for use.
This document was submitted in partial fulfillment of the requirements for
the degree of Doctor of Philosophy by Evan R. Whitby. It was accepted by the
University of Minnesota in August 1990. The draft and final versions of the thesis
received extensive peer review.
11
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PREFACE
In this work we describe a computationally efficient modeling technique for
representing the dynamical behavior of aerosols particles suspended in a gas
that are affected by sources, sinks (e.g., deposition), coagulation, and particle
growth caused by deposition of vapors and/or conversion of gases to nonvolatile
species within aerosol droplets. This work is an extension of a model developed by
the late K.T. Whitby, the lumped mode aerosol growth model (LMAGM). Upon the
death of K.T. Whitby in 1983, the U.S. EPA contracted with Peter McMurry and
Evan Whitby to assist in a comparison between the LMAGM and other aerosol
models (Seigneur et aL, 1986). Upon completion of this work, Peter McMurry and
Evan Whitby decided to continue independent work on the model. In the ensuing
years, they reformulated the model and evaluated the accuracy of the resulting
algorithms. In August 1989, the U.S. EPA again contacted them about
incorporating the new model into the Agency's regional participate models. From
November 1989 to June 1990, the focus of this work was to complete the derivation
of the algorithms for this model, systematically document the formulation of the
governing equations, and begin implementation into EPA's atmospheric chemistry
and particulate models. At this point, Francis Binkowski and Uma Shankar became
involved in the project. As EPA's scientific project coordinator, Francis Binkowski
researched and formulated the physical mechanisms that were included in the
atmospheric models, and reviewed the aerosol model at each development stage to
ensure the computer algorithms for each physical mechanism were accurately
implemented. Uma Shankar was responsible for formulating the computer
algorithms and writing the FORTRAN code for the aerosol dynamics module, as
well as implementing the resulting routines into the EPA's full atmospheric models.
111
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The original goal of this work was to rederive the governing equations of
the LMAGM, and clearly state the physical assumptions and mathematical
approximations required in the derivation, thereby removing some of the limiting
features of the LMAGM, such as time-invariant distribution widths and
time-invariant thermodynamic properties. As part of the reformulation, the name of
the model was changed to the modal aerosol dynamics (MAD) model1 to more
closely reflect the nature of the rederived algorithms. Two papers on this subject
have been published to date (Whitby and Whitby, 1985a; Whitby, 1986).
This report, therefore, is a formalized presentation of MAD modeling
techniques. This work includes the following contributions:
The derivation of a generalized algorithm for analytically
evaluating coagulation integrals, which is an extension of work
performed by Lee et al. (1984) and Pratsinis (1988).
The development of a technique for linking MAD models with
an existing computational flow model, that permits aerosol
dynamics in spatially stratified fields to be simulated. Much of
this work was performed in collaboration with Frank Stratmann
and Professor Heinz Fissan of the University of Duisburg, West
Germany.
The formulation of analytical solution techniques for integrating
the system of differential equations for MAD models. Integration
techniques are usually developed for treating general systems of
equations. By exploiting known characteristics of the governing
equations for MAD models, extremely stable and fast techniques
can be developed.
Perhaps the most important contribution, however, is the formal derivation
of the governing equations for an established modeling technique. To our knowledge,
the derivation provided here for the governing dynamics equations of MAD models
has not appeared in its entirety in any other publication, although components of
the full derivation have appeared elsewhere.
!The name modal aerosol dynamics model was originally used to refer to the particular
model developed by Peter McMurry and Evan Whitby. In the rest of this report,
however, modal aerosol dynamics is used to refer to a class of aerosol models rather than
a specific model.
IV
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ABSTRACT
The governing equations for representing aerosol dynamics, based on several
different representations of the aerosol size distribution, are presented. Analytical
and numerical solution techniques of these governing equations are also reviewed.
Described in detail is a computationally efficient numerical technique for simulating
aerosol behavior in systems undergoing 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 dynamics (MAD) models. These
models solve for the temporal and spatial evolution 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. Applications of MAD
models for simulating aerosol dynamics in continuous stirred tank aerosol reactors
(CSTARs) and flow aerosol reactors (FARs) using the SIMPLER algorithm
(Patankar, 1980) are provided. Considerations for incorporating a MAD model into
the U.S. Environmental Protection Agency's Regional Participate Model are also
described.
Numerical and analytical techniques for evaluating the size-space integrals
of the MDEs for MAD models are described. For multimodal lognormal distribu-
tions, an analytical expression for the coagulation integrals of the MDE, applicable
for all size regimes, 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.
Techniques for analytically integrating (in time) a linearized form of the
MDEs are presented. These techniques are computationally more efficient than
fourth-order Runge-Kutta techniques for situations when global integration errors
on the order of 1% are acceptable.
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CONTENTS
Preface m
Abstract v
Figures xiii
Tables xv
Nomenclature xvii
Acknowledgments xxviii
1 Introduction 1
2 Simulating the Dynamics of Aerosol Systems: A Review of
Current Mathematical Modeling Techniques 3
2.1 Abstract 3
2.2 Introduction 3
2.3 The Task 4
2.4 Aerosol Dynamics Equations 7
2.4.1 Discrete General Dynamics Equation (GDE) 10
2.4.2 Continuous GDE 11
2.4.3 Moment GDE 11
2.4.4 Modal Dynamics Equation (MDE) 13
2.5 Solutions to the Aerosol Dynamics Equations 15
2.5.1 Analytical Solutions to the Aerosol Dynamics Equations 15
2.5.1.1 Discrete GDE 16
2.5.1.2 Continuous GDE 16
2.5.1.3 Moment GDE 17
2.5.1.4 Modal Dynamics Equation 17
2.5.2 Numerical Solutions to the Aerosol Dynamics Equations 17
VI
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CONTENTS (continued)
2.5.2.1 Discrete Representation of the Size Distribution 19
2.5.2.2 Spline, Sectional, and Modal Representations of n 20
Spline representation of n 20
Sectional representation of n 21
Modal representation of n 24
2.5.2.3 Monodisperse Fractions Representation of n 28
2.5.3 Discussion: Numerical Solutions Based on the Spline,
Sectional, and Modal Representations of n 29
3 Modal Aerosol Dynamics (MAD) Modeling 32
3.1 Abstract 32
3.2 Introduction 32
3.3 The Modal Assumption 33
3.4 Modal Representation of the Aerosol Size Distribution 34
3.5 The Task 36
3.5.1 Dynamics Equations for Distribution Function Parameters 37
3.5.2 Dynamics Equations for Multiple Processes 38
3.6 Classification of Aerosol Processes 40
3.7 MDEs for Continuous Stirred Tank Aerosol
Reactors (CSTARs) 41
3.7.1 Internal Processes for CSTARs 42
3.7.1.1 Coagulation 42
3.7.1.2 Particle Growth 46
3.7.1.3 Internal Sources 49
3.7.2 External Processes for CSTARs 50
3.7.2.1 Inflow/Outflow 50
3.7.2.2 Surface Deposition 51
3.8 MDEs for Flow Aerosol Reactors (FARs) 54
3.8.1 The SIMPLER Algorithm 54
3.8.2 Moment Transport Coefficients for the SIMPLER Algorithm:
External Processes 56
3.8.2.1 Fluid Convection 58
vii
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CONTENTS (continued)
3.8.2.2 Transport due to External Forces 58
3.8.2.3 Diffusion 59
3.8.3 Moment Source Terms for the SIMPLER Algorithm:
Internal Processes 60
4 Techniques for Evaluating the Integrals of the Modal Dynamics
Equations 62
4.1 Abstract 62
4.2 Introduction 62
4.3 The Task 63
4.4 Some General Considerations 65
4.4.1 Particle Size Regimes 65
4.4.2 Moments Solved for in MAD Models 67
4.5 Integration Techniques 67
4.5.1 Numerical Integration 67
4.5.1.1 Numerical Integration of Arbitrary Integrands:
Gauss-Legendre Form 68
General Technique 68
Determining the Limits of Integration 69
4.5.1.2 Numerical Integration of Integrands Containing
Lognormal Distribution Functions:
Gauss-Hermite Form 71
General Technique 71
Integrand Centering Technique 72
4.5.1.3 Evaluating a System of Integrals with Similar
Integrands 75
4.5.2 Table Interpolation/Lookup 79
4.5.3 Analytical Integration 79
4.6 Internal Processes 82
4.6.1 Brownian Coagulation 82
4.6.1.1 Brownian Coagulation in the Free-Molecule
Regime (Kn > 10) 85
Vlll
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CONTENTS (continued)
4.6.1.2 Brownian Coagulation in the Continuum/Near-
Continuum Regime (Kn < 1) 86
4.6.1.3 Generalized Brownian Coagulation Coefficient
(all Kn) 87
4.6.2 Particle Growth 88
4.6.2.1 Condensation in the Free-Molecule Regime
(Kn > 10) 88
4.6.2.2 Condensation in the Continuum Regime (Kn < 0.1) .... 91
4.6.2.3 Generalized Condensation Expression (all Kn) 92
4.6.2.4 Surface and Volume Reactions 94
4.6.3 Internal Sources 95
4.7 External Processes 96
4.7.1 Convection 96
4.7.2 Diffusion 96
4.7.3 External Forces 97
4.7.3.1 External Fields 99
Gravitational and Inertia! 99
Electrical 100
4.7.3.2 Phoresis Effects 101
4.8 Discussion 102
5 Time Integration of the Moment Dynamics Equations 105
5.1 The Task 105
5.2 Linearized Moment Equations 106
5.3 Analytical Integration Techniques 107
, 5.3.1 Power-Law Analytical Integration Technique 109
5.3.1.1 General Derivation 109
5.3.1.2 Some Cleanup 114
Processes of Known Order 114
Representing Processes of Known Order,
Other than Zeroth- and First-Order 114
5.3.2 First-Order Analytical Integration Technique 115
IX
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CONTENTS (continued)
5.3.3 Second-Order Analytical Integration Technique
5.4 Interphase/Intermodal Moment Transfer 118
5.4.1 Particle Deposition to Surfaces 120
5.4.2 Intermodal Coagulation 121
5.4.3 Vapor Condensation on Particles 122
5.4.4 Vapor Nucleation to Form New Particles 125
5.5 Integration Time Intervals and Error Control 125
5.5.1 Error Control for the First- and Second-Order
Analytical Integration Techniques 125
5.5.1.1 The Order of the Integration Error 127
5.5.1.2 Estimating the Error of the First- and Second-Order
Analytical Integration Techniques 130
5.5.2 Error Control for the Power-Law Analytical
Integration Technique 132
5.6 Performance of the Analytical Integration Techniques 133
5.6.1 The Simulations Performed 134
5.6.2 Accuracy and Computational Effort 136
5.6.3 The Method Used to Compare the Integration Techniques 137
5.6.4 Assessing the Error of the Integration Techniques 137
5.6.5 The Procedure for Comparing the Integration
Techniques 138
5.6.6 The Behavior of the Integration Techniques 140
5.6.7 Recommendations for Selecting an Integration Technique 141
5.6.7.1 Integration Techniques Used with CSTARs 142
5.6.7.2 Integration Techniques Used with FARs 142
5.6.7.3 Multiphase Models 143
6 Modeling Atmospheric Aerosol Processes and Topics for
Continuing Work 155
6.1 Comparisons of Modal, Sectional, and Spline Models 155
6.2 MAD Modeling of Atmospheric Aerosol Dynamics 157
6.2.1 The Task 157
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CONTENTS (continued)
6.2.2 Background Information on the Regional Participate
Model (RPM) 159
6.2.3 Chemistry 161
6.2.3.1 Aerosol Growth by H2S04 161
Nucleation 162
Condensation 165
6.2.3.2 Aerosol Growth by HN03 165
6.2.4 Cloud Processes 166
6.2.5 Aerosol Transport 168
6.2.6 Meteorology 168
6.3 Multicomponent MAD Models 169
6.3.1 Multicomponent Representation 170
6.3.2 Multicomponent MDEs 172
6.4 Miscellaneous Topics for Future Work 173
6.4.1 Processes Resulting in Lognormal Distributions 173
6.4.2 Effects of Clouds on Stratification 175
6.4.3 Model Comparisons Accounting for Stratification 175
6.4.4 Mixing of Modes 176
6.4.5 Fractal Dimensions 176
6.4.6 Program Modules and Documentation 177
References 178
Appendix A: Method of Moments for Particle Growth A-l
Appendix B: Analytical Solution of the Modal Dynamics Equation
for Particle Growth B-l
Appendix C: Moment Relationships for the Lognormal Distribution C-l
Appendix D: Moment Change of Two Particles During Binary
Coagulation D-l
XI
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CONTENTS (concluded)
Appendix E: Forms of the Modal Dynamics Equation E-l
Appendix F: Mass-Based Aerosol Dynamics Equations F-l
Appendix G: Numerical Evaluation of Double Integrals G-l
Appendix H: Analytical Coagulation Integrals of the Modal Dynamics
Equation using Lognormal Distribution Functions H-l
Appendix I: Correction Factors for the Free-Molecule Intermodal
Coagulation Integrals of the Modal Dynamics Equation .. 1-1
Appendix J: Why a Const ant-Rate Source Term Cannot Be
Represented by a Power Function J-l
Appendix K: Numerical Integration Recipes K-l
Appendix L: Chemical Species List for the RADM L-l
Xll
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FIGURES
2-1 Challenging aspects of aerosol dynamics modeling 5
2-2 Aerosol dynamics processes for a control volume 9
2-3 Approximations of the particle size distribution 18
3-1 Trimodal aerosol size distribution measured at the
General Motors Milford proving grounds 34
3-2 Particle size distribution for a rich propane/air flame 36
3-3 1-D grid structure used in the SIMPLER algorithm 56
5-1 Nomenclature for the integration time intervals 108
5-2 Starting and ending distributions for a 12-hour unimodal
coagulation simulation with a growth factor of 0 145
5-3 Accuracy of various integration techniques for a 12-hour
unimodal coagulation simulation with a growth factor of 0 146
5-4 Starting and ending distributions for a 12-hour unimodal
growth simulation with a growth factor of 500 147
5-5 Accuracy of various integration techniques for a 12-hour
unimodal growth simulation with a growth factor of 500 148
xm
-------
FIGURES (concluded)
5-6 Starting and ending distributions for a 12-hour unimodal
coagulation and growth simulation with a growth factor
of 500 149
5-7 Accuracy of various integration techniques for a 12-hour
unimodal coagulation and growth simulation with a growth
factor of 500 150
5-8 Starting and ending distributions for a 12-hour bimodal
coagulation simulation with a growth factor of 0 151
5-9 Accuracy of various integration techniques for a 12-hour
bimodal coagulation simulation with a growth factor of 0 152
5-10 Starting and ending distributions for a 12-hour bimodal
coagulation and growth simulation with a growth factor
of 500 153
5-11 Accuracy of various integration techniques for a 12-hour
bimodal coagulation and growth simulation with a growth
factor of 500 154
xiv
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TABLES
4-1 Accuracy of evaluating Eq. (4-17b) for k = 3 using Gauss-
Hennite integration and the integrand centering technique 75
4-2 Accuracy of evaluating Eq. (4-21) using six-point Gauss-
Hermite integration and the integrand centering technique 77
4-3 Accuracy of evaluating Eq. (4-22) using six-point Gauss-
Hennite integration and the integrand centering technique
for jfc = 3 and k = 6 78
5-1 Average global error (relative to a 4tk-order Runge-Kutta technique)
of various integration techniques for a 12-hour unimodal
coagulation simulation with a growth factor of 0 146
5-2 Average global error (relative to a 4th-order Runge-Kutta technique)
of various integration techniques for a 12-hour unimodal growth
simulation with a growth factor of 500 148
5-3 Average global error (relative to a 4th-order Runge-Kutta technique)
of various integration techniques for a 12-hour unimodal
coagulation and growth simulation with a growth factor of 500 150
5-4 Average global error (relative to a 4th-order Runge-Kutta technique)
of various integration techniques for a 12-hour bimodal
coagulation simulation with a growth factor of 0 152
5-5 Average global error (relative to a 4th-order Runge-Kutta technique)
of various integration techniques for a 12-hour bimodal
coagulation and growth simulation with a growth factor of 500 154
6-1 Standard RADM Layer Definitions 160
xv
-------
TABLES (concluded)
J-l Intramodal coagulation correction factors H-8
K-l Intennodal coagulation correction factors 1-2
xvi
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NOMENCLATURE
Symbol Units Description
0^,0^0^, [1/s] Transport rate coefficients for the SIMPLER algorithm
(Eq. [3-33])i
Afm [] Factor for free-molecule approximation of GC (Eq. [4-3d])
AuC [] Factor for near-continuum approximation of C~ (Eq. [4-3e])
/
Auc [] Factor for near-continuum approximation of CL, used for
(_>
coagulation integrals (Eq. [H-9c])
A
A, [m ] Area of reactor surface feu (Eq. [3-26])
SU
&£ ' [] Correction factor of approximate integrals for ist-order
processes (Eq. [4-26])
b\ ' [] Correction factor of approximate integrals for 2nd-order
processes (Eq. [4-27])
B [N-m/s] Particle mobility (Eq. [4-68])
B [2] Power-law factor (Eq. [5-8])
B' [2] Power-law factor (Eq. [5-6])
c [m/s] Particle drift velocity due to external forces
c. [m/s] Monomer drift velocity due to external forces
c. [m/s] i-mer cluster drift velocity due to external forces
CA [m/s] &h-moment drift velocity for mode t due to external forces
(Eq. [3-38])
equation number, section number, or appendix letter listed with each definition shows the
usage of each variable.
2The units vary depending on the variable solved for.
XV11
-------
-------
dp [m] Upper bound of Gauss-Legendre integral (Eq. [4-9c])
dp [m] Critical diameter for particle growth (Eq. [4-42])
dp [m] Particle diameter of an agglomerated particle (Eq. [3-9])
D [kg/s] Diffusive mass flux for the SIMPLER algorithm (Eq. [3-34])
n
D [m /s] Monomer diffusion coefficient
n
D. [m /s] i-mer diffusion coefficient
Dp [m2/s] Particle diffusion coefficient (Eq. [4-63])
Dgn [m] Geometric mean size of the number weighted lognormal
distribution (Eq. [C-3])
Dg^ [m] Geometric mean size of the K-weighted lognormal distribution
(Eq. [4-8c])
^gJc [ml Geometric mean size of the ^-weighted lognormal distribution
(Eq.[4-8c])
Z)g- [m] Geometric mean size of the ^-weighted lognormal distribution
(Eq. [4-8c])
A
D [m /s] Binary diffusion coefficient of vapor m in background gas g
(Eq. [4-48b]) .
n
Dk [m /s] ^-moment diffusion coefficient for mode i (Eq. [3-40])
t
e [stC] Charge on an electron = 4.8 x 10" stC
E [stV/m] Electric field strength
fk [2] General moment factor (Eqs. [4-2])
/ (n ,dp) [] Fraction of particles of size dp carrying n units of charge
(Eq. [4-76b])
F [N] General external force vector (Eq. [4-66])
F(dp) [2] Size-dependent component of F (Eq. [4-67])
FT [2] Non-size-dependent component of F (Eq. [4-67])
xix
-------
F [kg/8] Convective mass flux for the SIMPLER algorithm
(Eq. [3-34])
g [N m2/kg2] Gravitational constant
H [-} Thermophoretic transport coefficient (Eq. [4-81])
/ [#/s] Rate of production of stable clusters due to nucleation
(Eq. [4-59])
J [#/s] Molecular flux to a particle (Eq. [4- 41a])
jfca [V/(m2 K)] Thermal conductivity of air
kp [V/(m2 ' K)l Thermal conductivity of particles
£_ [N-m/K] Boltzmann constant
n r
Kfn [m /s] Non-size-dependent terms for free-molecule coagulation
coefficient (Appendix H)
o
Kuc [m /s] Non-size-dependent terms for continuum/near-continuum
coagulation coefficient (Appendix H)
Kn [ ] Knudsen number for air = 2\/dp
Knps [-] Knudsen number for CFS, Knps = 2Aps/dp (Eq. [4-53])
KnD [-] Knudsen number for CD, KnD = 2AD/dp (Eq. [4-54])
n
In (jg [ ] A factor associated with lognormal distribution functions
m [kg] Mass of background gas molecule
o
lc 1
mk [m /m ] fctk-moment frequency distribution function of mode i
t
mk [m*/kg] mk/p
i i 6
mm [kg] Mass of condensing species
TOP [kg] Particle mass
M [kg] Mass
M [#/m3] Total Oth-moment concentration (same as N ) = \ M
t fc ^i=i °t
o
] Oth-moment concentration of mode i (same as N)
t Q ^H **ain
f, [m /m ] Total fcth-moment concentration = > M,
*t ^i=l *
xx
-------
I n
M, [m /m ] fctk-moment concentration of mode i (Eq. [2-3])
i
M [m*/kg] M
g
t
[-] Number of elementary charges (Eq. [4-75])
[] Number of aerosol modes (Eq. [2-4])
n h [ ] Number of nodes for Gauss-Hermite integration (Eq. [4 -lie])
n j [ ] Number of nodes for Gauss-Legendre integration (Eq. [4-4b])
[ ] Number of distinct reactor inflows (Eq. [3-25])
[ ] Number of distinct reactor outflows (Eq. [3-25])
rig [ ] Number of distinct internal particle sources (Eq. [3-23])
[ ] Number of distinct reactor surfaces (Eq. [3-26])
[ ] Total number of time steps for a simulation (Section 5.3)
[#/m ] Number concentration frequency distribution function
(Eq. [C-3])
[#/m ] Number concentration frequency distribution functions of
mode i
[#/m3] n.((ip)/Ar.
M
[#/(m s)l Internal source rate of particles of size dp (Eq. [3-23])
3 T"1 nam
[#/m ] Total number concentration = ^ N.
~ i=l
[#/m ] Number concentration of mode t, also used as the number of
»-mers in the discrete GDE
3
AT'. [#/m ] Value of N. from previous time step
P [atm] System pressure
Pe [-] Peclet number = F/D
P [atm] Vapor pressure of condensable material at r=co
P [atm] Vapor pressure at particle surface
P [atm] Saturation vapor pressure (Eq. [4-42])
s
xxi
-------
Q [m3/s] Reactor inflow /in (Eq. [3-25])
fin
Q [m3/s] Reactor outflow Ut (Eq. [3-25])
*out
R [-] Order of the functional dependence of c on A* (Eq. [5-36])
Rep [-] Reynolds number of the particle (Eq. [4-69a])
5 [2] Simplified nomenclature for the Otb-order coefficient for the
linearized MDEs (same as Sk ) (Eq. [5-3b])
t
2/ 3i
5 [m /m ] Aerosol surface area concentration
Sk [m*/m3] Otk-order coefficient for the linearized MDEs (Eq. [5-2])
5 [-] Vapor saturation ratio (Eq. [4-42])
5_ [2] Constant component of source term for SIMPLER algorithm,
(Eq. [3-33])
Sp [2] Component of source term for SIMPLER algorithm that
linearly depends on 0p (Eq. [3-34c])
SQD [m0-5] (Z)gn)°-5
t [s] Time
T [K] System temperature
T^ [K] Temperature far from the particle surface
Tp [K] Particle temperature
v [m/s] Fluid velocity vector
v, [m/s] #h-moment velocity vector of mode :, v, = v + c,
* *. *.
* t i
4
v, [m ] Section-boundary volume for sectional technique
4
7^ [m ] Molecular volume
Vp [m3] Particle volume (Eq. [2-2])
Vp [m/s] Particle velocity (Eq. [4-68])
Vp [m ] Volume of a coagulating particle (Eq. [2-2])
vp [m/s] Particle velocity vector, vp = v + c
xxii
-------
v [m/s] Fluid velocity far from the particle
CD
n Q
V [m /m ] Aerosol volume concentration
7R [m3] Reactor volume (Eq. [3-25])
W, [] Gauss-Hennite integration weights (Eq. [4-lie])
W^ [] Gauss-Legendre integration weights (Eq. [4-4b])
X h [] Gauss-Hermite dimensionless node values (Eq. [4-lid])
X j [] Gauss-Legendre dimensionless node values (Eq. [4-5])
av>a£>zp H Dimensionless particle size of the K-, «-, or K- weighted
rv /v At
distribution, respectively (Eq. [4-8b])
y [2] Simplified nomenclature for M, (Eq. [5-3a])
y . [] Volume fraction of species s in mode i.
/'' [-] mm/mg(Eq. [4-53])
Greek Symbols
a [] Power-function exponent used to represent the functional
dependence of processes of unknown order (Eq. [5-8])
a' [] Power-function exponent used to represent the functional
dependence of all processes (Eq. [5-6])
A
0(dp ,dp ) [m /s] Coagulation coefficient
$ Im3/s] Approximate form of (3 (Eq. [4-35])
7 [2] General rate coefficient (Eq. [4-2])
7(dp) [2] Size-dependent component of 7 (Eq. [4-2])
[2] Power-law or polynomial approximation of 7 (Eq. [4-25])
[kg/(m-s)] Generalized diffusion coefficient for SIMPLER algorithm
(Eq. [3-29])
xxm
-------
r [2] Terms of the general rate coefficient, 7P, that do not depend
ondp(Eq. [4-2])
tf_ . [] Kronecker delta (equal to 1 for j = 2, otherwise equal to 0)
)J l \
Ai, [s] Allowed time step for the fcth moment of mode i (Eq. [5-37a])
t
A*, [s] Allowed time step for the ith mode (Eq. [5-37b])
A^ [s] Allowed time step for time step 77 (Eq. [5- 37c])
e [-] Relative local integration error (Eq. [5-34a])
( [] Correction factor for continuum regime form of V*p
(Eq. [4-52])
CD [-] Dahneke's form for £ (Eq. [4-54])
CFS [-] Fuchs-Sutugin form for C (Eq. [4-53])
£D [1 Correction factor for Stokes regime form of Cn (Eq. [4-69b])
iVCp L)
TI [] Dimensionless distribution function
rj [] Time step
K The order (with respect to dp) of an integrand (Eq. [4-6])
«j [] Lowermost order (with respect to dp) of an integrand
(Section 4.5.1.1)
«u [] Uppermost order (with respect to dp) of an integrand
(Section 4.5.1.1)
K [] The approximate order (with respect to dp) of an integrand
(Eq. [4-12])
K [] The average order of a system of integrals with similar
integrands (Eq. [4-20])
A [m] Mean free path of background gas molecules
Am [m] Mean free path of vapor molecules
AD [m] Mean free path used with Kn.^ (Eq. [4-54])
Aps [m] Mean free path used with Kn^ (Eq. [4-53])
\i [kg/(m-s)] Gas viscosity
o
p [kg/m ] Gas density
o
o
Pp [kg/m ] Particle density
xxiv
-------
o
Pv [kg/m ] Vapor bulk density
£. [m/s] Particle deposition velocity to reactor surface feu (Eq. [3-26])
<7 [N/m] Surface tension of a liquid (Eq. [4-42])
<7g. [] Geometric standard deviation for mode i (Eq. [C-3])
a. [-} Generalized standard deviation for mode i (Eq. [3 lc])
(j> [2] General Patankar variable (e.g., momentum, enthalpy, mass
fraction) (Eq. [3-29])
o
% [m /s] Particle growth law function (Eq. [3-13])
Tp(dp) [2] Particle-size component of ifa (Eq. [3-13])
VT [2] Terms of the particle growth law that do not depend on dp
(Eq. [3-13])
*T [2] Terms of the particle growth law for surface reactions that do
not depend on dp (Eqs. [4-57])
#T [2] Terms of the particle growth law for internal (volume based)
reactions that do not depend on dp (Eqs. [4-58])
fi [] Function for partitioning condensable material between
modes (Eq. [3-20])
Subscripts
l Monomer
en Continuum regime
e East control-volume face
E East grid point
77 Time step index
770 Beginning of a time step
r/f End of a time step
fm Free-molecule
xxv
-------
g Gas-phase
gb Global
gh Gauss-Hermite integration summation counter
gl Gauss-Legendre integration summation counter
h Process h, which may represent the following processes:
en Intramodal coagulation
cjj Intermodal coagulation
cnv Fluid convection
con Condensation
dif Diffusion
exp System volume expansion
ext External force
gro Particle growth
nuc Nucleation
snk Sink
src Source
i Mode indicator for the MDEs; also used to represent clusters
composed of i monomers in the discrete GDE (Eq. [2- lb]).
;' Mode indicator for the MDEs; also used to represent clusters
composed of jmonomers in the discrete GDE (Eq. [2-lb]).
ki fcith moment (e.g., ki = 3)
£2 *2th moment (e.g., fc2 = 6)
k$ Aath moment
l Section index for sectional representation of the particle size
distribution
L Lognormal
nc Near-continuum/continuum regimes
P Main grid point
5 Chemical species
w West control-volume face
W West grid point
xxvi
-------
Acronyms
AREAL
CSTAR
CPU
EAM
EM
EPA
FAR
GDE
Ihs
LMAGM
MAD
MARS
MDE
RADM
rhs
RPM
wrt
Atmospheric Research and Exposure Assessment Laboratory
Continuous Stirred Tank Aerosol Reactor
Central Processing Unit
Engineering Aerosol Model
Engineering Model
Environmental Protection Agency
Flow Aerosol Reactor
General Dynamics Equation
left-hand side
Lumped Mode Aerosol Growth Model
Modal Aerosol Dynamics .
Model for Aerosol Reacting System
Modal Dynamics Equation
Regional Acid Depostion Model
right-hand side
Regional Parti culate Model
with respect to
xxvii
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ACKNOWLEDGMENTS
This work is as much a product of the efforts of others as it is a record of
our contributions to this field of study. It is in this light that we gratefully
acknowledge the following persons and institutions.
The late Kenneth T. Whit by formulated the fundamental concepts of the
modal representation of aerosol size distributions, the basis on which MAD models
are built. This work represents a significant extension of his early model. Much of
the work described here was performed at the University of Minnesota, first by
Kenneth T. Whitby, and later by Evan R. Whitby under the direction of David B.
Kittelson.
Frank Stratmann of the University of Duisburg, West Germany,
participated in many conversations about simulating aerosol dynamics in fluid
systems. With his keen understanding of thermodynamics, the Navier-Stokes
equation, and numerical flow modeling, he directly contributed to the formulation of
the mathematical expressions in Section 3.8, and indirectly contributed to many of
the other formulations throughout this work.
Thanks are also due to Jeanne Eichinger of Computer Sciences Corporation,
who carefully reviewed this document, and whose job classification of "Technical
Editor" is a misrepresentation of her scientific prowess. Her editorial and technical
contributions to this work have greatly clarified the presentation of this modeling
concept, and are the best hope we have that this document will accurately convey
the concepts presented.
XXVlll
-------
Finally, we thank Dr. William Wilson and the U.S. EPA for their efforts in
bringing this work to fruition. Dr. William Wilson supported this project from the
earliest stages to its completion, and he is largely responsible for this model being
implemented in the EPA's atmospheric participate models.
xxix
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CHAPTER 1
INTRODUCTION
It is often useful to have mathematical models that simulate the behavior of
complex systems. Such models permit the calculation of a system's response to
perturbations of the input variables, with applications to systems design and on-line
process control.
For one-time calculations or repetitive calculations that require
insignificant computational effort, comparatively inefficient algorithms and/or
models are often sufficient. If existing techniques perform sufficiently well, it is a
mistake to expend more effort refining a method than is returned in computational
savings. For highly repetitive simulations or for computationally expensive
simulations, however, the effort of developing compact and computationally efficient
models can often be justified.
To properly construct computationally efficient and accurate models
requires a clear understanding of the desired output to prevent computation of
extraneous quantities. For example, aerosol models that produce detailed
information on size distributions when only integral properties are required may
often be replaced by models that solve directly for the integral quantities. Being
able to clearly specify appropriate computational algorithms and output is
important, because as a rule increased generality results in higher computing costs.
Increased generality does not, however, guarantee increased accuracy.
For example: Systems are often modeled where the distribution of a
variable must be determined. In flow systems, the integral fluid momentum is
distributed across a channel, and the local momentum distribution function is
-------
represented by the fluid velocity. If the momentum distribution can be
approximated by a known analytical function, such as a parabolic profile for laminar
flow, expressions for the integral momentum flux replace calculations for the
distribution of momentum. In coupled heat transfer/fluid flow calculations, the
velocity profile influences the heat transfer rates. However, solving for the velocity
field in laminar flow situations will not improve the heat transfer calculations in
comparison to assuming a laminar velocity profile, and may decrease the model
accuracy if the computational technique for determining the velocity field suffers
from numerical problems.
In analogy to this fluid flow example, we expect that an aerosol dynamics
model based on an assumed time-invariant form of the aerosol size distribution will
yield accurate results for cases where the system modeled produces size distributions
close to those used in the model. In fluid flow the parabolic profile has been derived
analytically as the correct representation of the momentum distribution in laminar
flow. In aerosol science, however, multimodal lognormal functions have been
empirically observed to closely approximate aerosol size distributions in many
systems. Whitby (1978a, 1978b) summarizes measurements of many atmospheric
and combustion aerosols that indicate multimodal lognormal distribution functions
represent measured distributions for many systems, with the same level of accuracy
as the measurement techniques. Roth et al. (1989) cite four laboratory aerosol
generation techniques that produce distributions close to lognormal: the exploding
wire (Phalen, 1972), atomization of an aqueous solution (Raabe, 1976), burning of
metal-organic compounds in a flame (Kasper et al, 1980), and the heterogeneous or
homogeneous condensation of a supersaturated vapor (Kanapilly et al., 1982;
Schiebel and Porstendoerfer, 1983).
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 Cohen and Vaughan (1971), Clark (1976), Whitby (1979), Reed et al. (1980),
Eltgroth (1982), Giorgi (1986), Pratsinis (1988), Brock et al. (1988), Megaridis and
Dobbins (1989, 1990), and others. In this paper we present a general form of the
governing equations used in these models.
-------
CHAPTER 2
SIMULATING THE DYNAMICS OF AEROSOL SYSTEMS:
A REVIEW OF CURRENT MATHEMATICAL MODELING TECHNIQUES
2.1 ABSTRACT
We present the governing dynamics equations for representing aerosol
dynamics in systems, based on several different representations of the aerosol size
distribution. Analytical and numerical solution techniques for these governing
equations are also reviewed.
2.2 INTRODUCTION
Particles are important actors in many atmospheric and industrial
processes; they range from undesirable byproducts to desired products. From man's
perspective, particle concentrations in the atmosphere above biogenic source levels
are generally considered undesirable because of visibility degradation, soiling of
buildings, acid deposition, and transport of toxic compounds. Excessive atmospheric
aerosol concentrations can also reduce the effectiveness of solar collection systems.
Control of atmospheric particle concentrations is therefore important, and
understanding the dynamics of particle sources, transport, and removal is required
to formulate comprehensive atmospheric chemistry models for assessing the impact
of control strategies on resulting pollutant levels.
Particles are also desirable products, such as toner for photocopying
machines, magnetic particles for manufacturing computer storage disks, and carbon
black used in automotive tire manufacturing. Understanding the dynamic behavior
of particles in production processes is essential to optimizing particle physical
-------
characteristics and chemical properties.
In situ measurements are commonly used to assess the physical
characteristics and chemical properties of particles. However, the rapid increase in
the performance of digital computers has allowed mathematical modeling to emerge
as an inexpensive alternative for predicting the dynamical behavior of particles in
complex systems. For processes in the atmosphere, computer models are
indispensable tools for assessing the environmental impact of anthropogenic and
biogenic sources of gases, vapors, and particles, because 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
proposed strategies for altering emissions of toxic compounds into the atmosphere.
Modeling aerosol dynamics has at least three challenging aspects (Figure
2-1): a distribution of particle sizes must be considered; coagulation introduces
summations and integrals over the size distribution; and particle sizes of interest in
many systems span the free-molecule, transition, and continuum size regimes,
requiring generalized expressions that apply to all size regimes to be used. In this
chapter we review the techniques for representing the aerosol size distribution and
models based on these representations.
2.3 THE TASK
An aerosol is defined as a suspension of particles, either solid or liquid, in a
carrier gas. Aerosols are influenced by carrier gas motion, exchange of material with
gases, vapors, and bulk-phase material, and particle-particle interactions. The
objective of aerosol dynamics modeling is to track the migration of material through
the gas (g) phase, vapor (v) phase,1 condensed aerosol (p) phase (hereafter referred
to as the aerosol phase), and condensed bulk (b) phase (hereafter referred to as the
!A gas is a substance at a temperature above its critical temperature and therefore not liquefiable
.by pressure alone (increasing the pressure at constant temperature). A vapor is a substance at a
temperature below its critical temperature and therefore liquefiable by pressure alone (increasing
the pressure at constant temperature) (Webster's Third New International Dictionary,
unabridged, 1967).
-------
mean free path
Mass Distribution
0.01 0.1
1.0 10.0 100.C
Particle sizes range from 0.01 100 |im (Kn = 2mfp/dp).
free-molecule (Kn > 10)
transition (1 < Kn < 10)
near-continuum (0.1 < Kn < 1)
continuum (0.1 < Kn)
Mathematical expressions for the aerosol dynamics in each size regime
must be available.
Aerosol models must follow the evolution of the size distribution.
7/VM
Figure 2-1. Challenging aspects of aerosol dynamics modeling.
bulk phase), and to determine the size distribution of all chemical compounds in the
particle phase for each point in the time and space of interest. The processes
occurring within and between each phase are listed below, and the processes
affecting the aerosol phase are represented pictorially in Figure 2-2.
-------
Gas-Phase Dynamics
(§'§) Chemical conversion
(g»v) Chemical conversion from gaseous to vapor species
(g*P) Adsorption/absorption of gases by particles, with the
potential for conversion to condensed-phase material
(g*b) Adsorption/absorption of gases by bulk-phase material, with
the potential for conversion to condensed-phase material
transport Convection and/or diffusion
VaporPhase Dynamics
(vtp) Transfer of vapor to the aerosol phase by
Adsorption/Absorption
Condensation
Nucleation
(v»b) Adsorption/absorption/condensation of vapor on bulk-phase
surfaces
transport Convection and/or diffusion
AerosolPhase Dynamics
(P~*g) Desorption of gases
(p*v) Evaporation
(P~~*P) Coagulation
(p»b) Deposition to surfaces (e.g., sedimentation)
transport Convection, diffusion, and/or migration due to external forces
Bulk-Phase Dynamics
(b>g) Desorption of gases
(b»v) Evaporation
(b»p) Bulk transfer of material by
Reentrainment of particles from surfaces
Fracture of material to form new particles
-------
A comprehensive model must account for the behavior of material in each
relevant phase. Interphase transfer processes are represented as sources and sinks in
the respective interacting phases. For the aerosol dynamics component of
comprehensive models, rate expressions are formulated with respect to the processes
affecting the aerosol phase. The dynamics equations for the aerosol phase are
formulated by expressing mathematically the effect of each process that represents
interactions with the aerosol phase. In the following sections various forms of the
dynamics equation for representing the time dependence of the aerosol size
distribution are presented. Aerosol dynamics models based on these dynamics
equations are also reviewed and compared. In Chapters 3 through 5 we present a
detailed derivation of a computationally efficient aerosol dynamics modeling
technique (modal aerosol dynamics models). In Chapter 6 we present an application
of this modeling technique for simulating atmospheric aerosol processes.
2.4 AEROSOL DYNAMICS EQUATIONS
The processes affecting aerosol behavior are represented pictorially in
Figure 2-2. The processes are depicted as acting within or at/across a
control-volume boundary. Processes that act within the system boundaries are often
called internal processes, and processes that act at/across system boundaries are
often called external processes. The control volume can represent an entire system
whose contents are assumed to be homogeneously distributed throughout the control
volume, or one control volume from a larger calculation domain.
The processes affecting aerosol behavior are classified as follows:
Internal Processes
Coagulation
Particle growth
Condensation/evaporation of vapors
Adsorption/absorption/desorption of gases, including reactions on or
within the particles
Internal sources
Nucleation
Fracture of bulk-phase material
Reentrainment of particles from surfaces
-------
External Processes
Diffusion
Convection
Particle migration due to external forces
Deposition to surfaces in a closed system (e.g., wall losses such as
sedimentation)
Convection-like fluxes in an open system (e.g., thermophoretic
transport)
Diffusion across a boundary layer near the wall of a closed vessel is treated as a wall
loss term, where the deposition velocity is
where Dp is the particle diffusion coefficient, and £51 is the boundary-layer
thickness.
For clarity in the following discussions, we define the following terms:
Monomer: The smallest building block, either an atom, a molecule, or an
associated group of molecules, that causes growth of an i-mer.
i-mer: A cluster of monomers.
-------
Diffusion
>VN/
Adsorption/Absorption/Desorption
. r~^ Nucleatic
| o O O o
Condensation o°*°°°
| ^| Oo°oo0
1 ^
1 ^ V*
Coagulation §
I Inflow ^ A -
m $
H
o
in ?
n
)
.1
r
.1
s
i
1
^
1
Outflow ^
1 ^
Diffusion ,
/rV^x^~>
1
1
1
Figure 2 -2a
Adsorption/Absorption/Desorption
J
Condensation
Nucleation
o
g
O 00
000
Inflow
Coagulation
Outflow
Diffusion
Figure 2-2b
Figure 2-2. Aerosol dynamics processes for a control volume;
2-2a: open system without walls; 2-2b: open system with walls.
-------
2.4.1 Discrete General Dynamics Equation (GDE)
The equation(s) that describe the time-dependent size distribution of an
aerosol is referred to as the general dynamics equation (GDE) for aerosols
(Friedlander, 1977). For i-mers with fewer than some critical number of monomers
(e.g., 100), a discrete form of the GDE is often used:
GDE for Monomers
|ff,- -V-Tff, - V-C^ +
convection external forces diffusion
Jmax Jmax
coagulation evaporation source
GDE for Clusters
Q
i-N. = -V-vJV. - V-C../V. + V-D3N.
ot i t tt it
convection external forces diffusion
«-l t-i
9. .N.+ E.. ,N., ,
i,j j »+l i+l
^^^^^^^^.^H^^
coagulation
- E.N. + 5. (2-lb)
evaporation source
where N. is the number concentration of i-mers.
10
-------
2.4.2 Continuous GDE
For i-mers that contain many monomers (e.g., : > 100) it is usually possible
to approximate the aerosol size distribution as a continuous function of particle size.
The resulting GDE for the continuous size distribution function is given by Eq.
(2-2) (Friedlander, 1977)2. Equation (2-2) expresses the time rate of change of the
continuous distribution function, n. An important feature of Eq. (2-2) is that nis
completely arbitrary, and may assume any shape in particle size-space.
Q
= -V-vn(t;p) -
convection external forces diffusion
VP
BV
coagulation
particle growth internal sources
2.4.3 Moment GDE
For situations where detailed knowledge of the distribution function is
unimportant, Eq. (2*2) can be converted to an expression for the time rate of
change of integral moments, M,, of the distribution. Expressing Eq. (2-2) in terms
of dp and integrating, the continuous GDE can be written in terms of moments of
the distribution (Friedlander, 1977) to yield the moment GDE:
the continuous GDE has an explicit growth term, the growth term in the discrete GDE
is implicitly included in monomer coagulation (condensation) and evaporation terms.
11
-------
- V-J "
/"
/o
where M, =
For special forms of the rate coefficients, the integrals of Eq. (2-3) can be
expressed in terms of other moments of the distribution, without explicitly
evaluating the integral terms (see Appendix A). This results in expressions for the
time rate of change of moment M, in terms of other moments of the distribution.
For some processes, a closed set of moment equations can be written and solved.
This is referred to as the method of moments, and has been used by Friedlander
(1983), Kodas et al (1986), Pratsinis et al. (1986), and others.
Although it is often possible to express the integrals of Eq. (2 3) directly in
terms of other moments, it is not always possible to obtain closure of the resulting
moment equations (see Appendix A). A method that guarantees closure of the
moment equations is to specify a mathematical form for the aerosol size distribution
function, n.
Strictly speaking, if a functional form for n is specified, Eq. (2 3) is no
longer a "general" equation and the name moment GDE (general dynamics
equation) is misleading. We therefore refer to moment equations (and not moment
GDEs) for situations where Eq. (2-3) is solved by specifying a mathematical form
for n. For cases where the entire distribution is represented by one or more
distribution functions, Eq. (2-3) is converted to the modal dynamics equation
12
-------
(MDE), where a mode is a distinct subset of the total population.
To our knowledge, there is no convention in the literature for naming the
governing dynamics equations for the moments of the distribution. The naming of
the moment GDE and modal dynamics equation (see the next section) was done for
clarity in this thesis, and does not represent standard nomenclature in the
literature.
2.4.4 Modal DynaTnirK Kqnation (MDE)
When one or more distribution functions are used to obtain closure of
Eq. (2-3), the moment GDE is converted to the modal dynamics equation (MDE).
To represent unimodal distributions, a single distribution function is used to
represent n. For multimodal (e.g., bimodal or trimodal) distributions, a
superposition of multiple distribution functions is used to represent n. The modal
dynamics equation is therefore used whenever the entire aerosol distribution can be
viewed as the composition of one or more superimposed aerosol populations.
For a multimodal distribution, the general modal distribution function is
(2-4)
where 7ipm is the number of aerosol modes. By substituting Eq. (2-4) into Eq. (2-3),
the modal dynamics equation for a generalized multimodal distribution can be
written. For all terms except the coagulation term the expansion is straightforward.
The algebraic expansion of the coagulation term, however, leads to many terms.
(The derivation of the multimodal coagulation terms is presented in Chapter 3.) For
systems where coagulation results only from binary collisions, it suffices to consider
the interactions between pairs of distributions. We therefore show the MDE for a
bimodal distribution, and note that the intermodal coagulation terms for a
distribution of three or more modes is a simple extension of the MDE for a bimodal
distribution by considering the interactions between all pairs of modes.
13
-------
The MDEs for a bimodal distribution are shown below.
MDE for Mode i
= -v-
I »
Intramodal Coagulation
Intermodal Coagulation
ddj
MDE for Mode j
Intramodal Coagulation
Intermodal Coagulation
14
-------
The indices of the two modes are i and j; two indices are required because binary
coagulation interactions occur between pairs of modes. Substituting n = n.+ n. in
Eq. (2-3) and separating terms (see Chapter 3 for details) yields the MDE for a
bimodal aerosol. The MDE for mode i is represented by (2-5a) and the MDE for
mode j is represented by (2-5b). The first two coagulation integrals of Eq. (2- 5a)
represent changes due to intramodal (i-i) coagulation, and the third coagulation
integral represents a loss from mode i to mode j due to intermodal (i-j) coagulation.
The first two coagulation integrals of Eq. (2-5b) represent changes due to
intramodal (j-j) coagulation, the third coagulation integral represents gain to mode
j due to intermodal (i-j) coagulation, and the last coagulation integral represents
loss from mode j due to intermodal (i-j) coagulation.
2.5 SOLUTIONS TO THE AEROSOL DYNAMICS EQUATIONS
The task in aerosol dynamics modeling is to simulate the behavior of an
aerosol influenced by various processes, as discussed above. In particular, the
temporal and spatial dependence of either the particle size distribution or the
moments of the distribution are determined. The representation of the size
distribution adopted dictates whether Eq. (2-1), (2-2), (2-3), or (2-5) is used to
simulate aerosol dynamics processes. Analytical solutions of some simple processes
have been derived for each of the dynamics equations, and numerical models for
simulating arbitrarily complex processes have also been developed. Some of the
analytical and numerical solutions are summarized in the following sections.
2.5.1 Analytical Solutions to the Aerosol
Analytical solutions to each of the dynamics equations have been obtained
for special processes and forms of the rate coefficients. Some of the analytical
solutions are summarized in the following sections. Because coagulation is often
important in aerosol dynamics simulations, and because analytical evaluation of the
double integrals associated with coagulation is usually difficult or impossible,
coagulation has received considerable treatment in the literature. A review of
coagulation and some solution procedures is presented by Drake (1972).
15
-------
2.5.1.1 Discrete GDE
For pure coagulation with constant coagulation coefficient, analytical
solutions are presented by Smoluchowski (1917), Fuchs (1964), and Mockros et aL
(1967). McMurry (1980) provides an analytical solution for coagulation with a
constant source rate of monomers.
2.5.1.2 Continuous GDE
Analytical solutions for coagulation have been obtained by several authors,
but these are for special forms of the binary coagulation coefficient. Many authors
have contributed to achieving analytical solutions for coagulation, and Drake (1972)
carefully reviews the literature and methods used. Drake and Wright (1972) extend
the work of Scott (1968) to obtain solutions (using Laplace transforms) for
coagulation kernels expressed as
/3(dpr dp2) = A + B(dPl + dp2)+C dp^
where A, B, and Care nonnegative constants. Lushnikov (1973) also presents
analytical solutions to the coagulation equation.
Analytical solutions for particle growth are presented by Brock (1972), who
derived the expressions as a means for explaining the observed distributions of
airborne aerosols above 0.1 /an. Gelbard and Seinfeld (1979a), in part, extended the
work of Brock (1972) to include source and removal mechanisms. Friedlander (1977)
also presents analytical forms for particle growth.
By extending the work of Scott (1968) and Brock (1972), Ramabhadran et
al (1976) obtained analytical solutions to the continuous GDE for special cases of
simultaneous coagulation and condensation. Peterson et aL (1978) extended this
work to obtain analytical solutions for special cases of simultaneous coagulation,
condensation, nucleation, and removal.
16
-------
2.5.1.3 Moment GDE
Because the moment of a distribution is a weighted integral of n
(Eq. [2-3]), any of the analytical solutions based on the continuous GDE can also be
expressed in terms of M^ by integrating the resulting analytical expression for n.
2.5.1.4 Modal Dynamics Equation
For realistic coagulation and for n represented by a unimodal lognormal
function, Lee (1983) and Lee et al (1990) have derived analytical expressions. It is
also possible to solve the modal GDE for particle growth; a general analytical
solution is provided in Appendix B. To the author's knowledge this solution has not
appeared in the literature, although its derivation is straightforward.
2.5.2 Numerical Solutions to the Aerosol Dynamics Equations
In the previous section we reviewed some of the available analytical
solutions of the dynamics equations for aerosols. As summarized in Figure 2-1, the
main difficulties of obtaining analytical solutions to the dynamics equations are that
coagulation introduces summations or integrals over the size distribution, and that
particle sizes of interest in many practical systems span the free-molecule,
transition, and continuum size regimes, requiring generalized dynamical expressions
that apply to all size regimes to be used. These considerations often preclude
analytical solutions for problems of importance in practical systems. Therefore,
numerical solutions are often required.
Although numerical techniques permit wide latitude in simulating complex
processes, physical assumptions and/or mathematical approximations are usually
required for representing the precise dynamics in order to secure computationally
tractable algorithms. When the computational effort can be afforded, solving Eq.
(2-1) provides the most detailed information on the particle size distribution. For
most cases of interest, however, the discrete representation leads to an intractably
large number of differential equations, so the size distribution must be
approximated by a mathematical function with a smaller number of coefficients
17
-------
than the discrete representation. Mathematical representations are developed by
representing separate portions of the complete distribution by analytical functions,
each of which may contain one or more coefficients. The complete set of functions
provides a representation for the entire size distribution.
The mathematical functions that have been used in aerosol dynamics
models to represent n are shown (from the most general to least general
representation) in Figure 2-3.
Discrete
Spline
Sectional
Modal
Monodisperse
K
Figure 2-3. Approximations of the particle size distribution.
18
-------
Solving for the temporal and spatial evolution of the size distribution is
tantamount to solving for the coefficients of the approximating functions. The
mathematical representation of the size distribution function determines which of
the dynamics equations are solved, and hence the solution algorithms required to
construct a complete model. Because the solution techniques used to solve for the
function coefficients depend on the mathematical representation of n, aerosol
dynamics models are commonly named according to the mathematical
representation of n.
2.5.2.1 Discrete Representation of the Size Distribution
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 increments are determined by the smallest molecule
causing cluster growth. For the discrete representation the discrete GDE is solved,
resulting in the most precise specification of the aerosol distribution function.
Discrete models are useful for simulating nucleation processes, where the dynamics
of clusters with relatively few t'-mers are important.
The discrete equations are difficult to solve numerically because of the large
number of cluster sizes that must be considered in pratical aerosol systems. For
clusters composed of more than about 100 monomers (an arbitrary criterion),3 the
discrete representation rapidly leads to an intractable number of differential
equations. To develop models that can extend into the continuous size regime, a
discrete representation is often used to represent the cluster distribution up to a
limiting size, and then a continuous representation is used for larger particles. A
hybrid discrete/continuous model was developed by Gelbard and Seinfeld (1979b),
and hybrid discrete/sectional models were developed by Wu and Flagan (1986) and
independently by Rao and McMurry (1989).
3For comparison, a 0.01 /m diameter cluster composed of 8A diameter monomers contains about
2000 monomers.
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2.5.2.2 Spline, Sectional, and Modal Representations of n
To accurately represent the size distribution for i-mers with more than
about 100 monomers, nis represented by one or more mathematical functions. A
single analytical function could be used to represent the entire aerosol population (a
unimodal representation of the aerosol size distribution), but most distributions are
more accurately represented by using multiple analytical functions, where an
individual function approximates a portion of the complete distribution. A common
feature of the spline, sectional, and modal representations, therefore, is that the
range of particle sizes of interest is subdivided into intervals, and a mathematical
function is selected to represent n within each interval. The resulting algorithms
solve for the temporal and/or spatial variation of the interval function coefficients.
In subdividing size-space into intervals and specifying interval functions, the
following concepts must be considered:
Interval boundaries may be contiguous or overlap
Interval boundaries may be time variant or time invariant
A continuous function to represent n must be selected for each interval
A set of continuous GDEs, moment GDEs, or MDEs is solved for each
interval, where the number of equations for each interval is equal to the
number of time-variant coefficients of the function used to represent n in
each interval.
The spline, sectional, and modal representations are presented in terms of these
aspects of model design.
Spline representation of nThe 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 (Gelbard, 1978).
Cubic functions are commonly used, implying that four differential equations (one
for each coefficient of the cubic equation) must be solved for each section.
Differential equations for the function coefficients are formulated from Eq. (2-2).
Spline models have been developed by Middleton and Brock (1976), Gelbard and
Seinfeld (1978), Gelbard (1980), Suck and Brock (1979), Tsang and Brock (1982,
1983, 1986), Suck et al (1986), Tsang and Hippe (1988), and others.
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Because relatively detailed section functions are used, models based on the
spline representation provide accurate numerical solutions of the GDE, and these
models are often used as standards of comparison for models based on less precise
representations of n (Seigneur et al, 1986).
The principle disadvantage of spline models is that many differential
equations must be solved,4 resulting in a relatively expensive computational
technique. Also, spline functions may exhibit unrealistic behavior (such as negative
distributions), so solution techniques must be carefully formulated to prevent
unrealistic representations of n. Models based on this representation can therefore be
tedious to construct, and may be unstable.
Another difficulty with some spline techniques is that if time-invariant
sections are used for particle growth simulations with an initially broad distribution,
particle growth causes the distribution to become narrow, resulting in steep
gradients that will not be well represented because of the relatively wide interval
spacing used for the initially broad distribution. For pure particle growth, Tsang
and Brock (1983) developed a spline algorithm to overcome this problem, using the
method of Varoglu and Finn (1980) that allows the sections to "move" with the
distribution. The section boundaries move in response to particle growth dynamics,
so if one part of the distribution peaks sharply, the section boundaries in that part
of the distribution are automatically adjusted to concentrate more spline functions
in that part of the distribution.
Sectional representation of nThe sectional representation is so called
because particle size-space is subdivided into a series of sections, and the
distribution function of a prescribed moment of the distribution (usually the mass
distribution, dM/d[log(dp)]) is assumed constant within each section. This
assumption results in a size distribution with the appearance of a histogram.
Sectional techniques are usually developed with contiguous section boundaries.
However, Gelbard (1990) developed a moving boundary sectional technique for
representing the dynamics of growth/evaporation of a multicomponent aerosol,
4For cubic functions, for example, the number of GDEs is equal to four times the number of
sections.
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where the section boundaries are allowed to overlap. The boundaries are allowed to
overlap because aerosol of different composition grows at different rates, so that
even for initially contiguous boundaries, the boundaries will not, in general, remain
contiguous for growth simulations of a multicomponent aerosol.
For constant frequency distribution functions within each section, the
integral section moments are
M,= I m,dlogdp=m, Adp, (2-6)
*l JAdpj *l */
so that a set of differential equations for the moments of the sections is solved.
Because the section distribution functions contain only one parameter each
(corresponding to a constant frequency function), one differential equation for the
prescribed integral moment of each section is required. The sectional moment solved
for is often mass, because mass is a conserved property.
Sectional models have been widely used, and much of the impetus for their
development originated with the nuclear power industry (Reed et al, 1980; Bunz et
aL, 1983; Bergeron et al, 1985; Fermidjian et al, 1986; Jonas and Bunz, 1987).
Sectional models have also been used for modeling atmospheric aerosol dynamics,
because the sectional technique can easily represent multicomponent aerosol
distributions. Other sectional models have been developed by Dolan (1977), Turco
(1979), Gelbard et al (1980), Gelbard and Seinfeld (1980), Bassett et al. (1981),
Seigneur (1982), Pilinis et al. (1987), Pilinis and Seinfeld (1987, 1988), and others.
Dolan (1977) developed a sectional model for simulating the growth of
aerosols during Diesel engine combustion. Dolan accounted for spatial
inhomogeneity in the engine cylinder by distributing the particulate matter between
a small number of packets (e.g., 10-15), where the total cylinder particulate matter
was apportioned between the packets by a spatial lognormal distribution function.
By accounting for inhomogeneity in this way, Dolan's model was able to reproduce
measured exhaust particulate levels, to within the accuracy of experimental
uncertainty.
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Because relatively simple section distribution functions are used with
sectional techniques, fewer differential equations need to be solved for each section
than with spline techniques using cubic functions to represent n. For sectional
techniques the number of differential equations is equal to the number of sections.
Solving for the integral moments of each section also results in algorithms that are
less likely than cubic functions to admit negative distributions.
Most sectional techniques use section boundaries that are time-invariant,
because the coagulation integrals resulting from interactions within and between
sections must be evaluated whenever the section boundaries are moved. Because
many of the coagulation integrals must be evaluated numerically, the double
integrals for coagulation can represent a substantial computational hurdle. By
maintaining time-invariant section boundaries, the coagulation integrals must be
evaluated only once for a particular choice of section boundaries.
Time-invariant section boundaries may result in an aerosol distribution
that grows beyond the range of sections. Preventing truncation errors requires prior
knowledge about the expected range of particle sizes for a complete simulation.
Time-invariant sections may also cause problems if an initially broad distribution
becomes steep due to particle growth. For sharply peaking distributions, large
differences in M, between sections arise, which may result in numerical diffusion
.
when estimating intersection moment transfer due to particle growth.
A further limitation of some sectional techniques results from a technique
for reducing the computational effort of evaluating the coagulation integrals. By
selecting section boundaries according to the constraint
where v, is the volume of the lower boundary for a section, some of the coagulation
integrals can be eliminated (Gelbard et al, 1980). This reduces the resulting
coagulation calculations, but also restricts the sections to a minimum width. For
example, the minimum section widths resulting from Eq. (2-7) allow 50% of a
lognormal distribution with ag = 1.2, or 80% of a distribution with ag = 1.1, to be
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placed within a single section. This may be a severe limitation if sharply peaking
distributions are modeled, such as for particle growth. Because this criterion is used
to reduce the effort of evaluating the coagulation integrals, it is not needed for
sectional models developed to handle processes not influenced by coagulation. For
pure condensational growth, therefore, Gelbard (1990) developed a sectional
technique that allows the section boundaries to move in response to particle growth.
This technique eliminates numerical diffusion, and exactly recovers an aerosol that
sequentially undergoes condensation of a known quantity of material, followed by
evaporation of the same quantity of material. The task of developing a
computationally efficient moving-boundary sectional technique to handle
comprehensive aerosol dynamics, however, still remains a challenge.
Modal representation of nThe modal representation is so called because
the particle size range is divided into a small number of overlapping intervals
(usually three or less), called modes. Overlapping intervals are used because the
modal approximation originates from the assumption that the complete distribution
is composed of multiple distinct aerosol 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. Models based on the modal representation have been
developed by Cohen and Vaughan (1971), Clark (1976), Eltgroth and Hobbs (1979),
Whitby (1979, 1981), Reed et al. (1980), Brockman et al. (1982), Eltgroth (1982),
Whitby and Whitby (1985a, 1985b), Giorgi (1986), Brock and Gates (1987), Brock
et al. (1988), Pratsinis (1988), Pratsinis and Kim (1989), Megaridis and Dobbins
(1989, 1990), and others.
Whereas relatively simple functions are used with the spline and sectional
techniques, complex distribution functions are used with the modal representation,
consistent with the physical assumption that each interval represents a distinct
population. The function used to represent the aerosol distribution in each mode
may be arbitrarily selected, but there is considerable computational advantage in
selecting lognormal distribution functions. In addition, lognormal distribution
functions have been observed to reasonably represent the distribution of variables in
a wide variety of systems. Shimizu and Crow (1988) state:
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The lognormal distribution (with two parameters) may be defined as the
distribution of a random variable whose logarithm is normally distributed. Such a
variable is necessarily positive. Since many variables in real life, from the sizes of
organisms and the numbers of species in biology to rainfalls in meteorology and
sizes of incomes in economics, are inherently positive, the lognormal distribution
has been widely applied in an empirical way for fitting data. In addition, it has
been derived theoretically from qualitative assumptions; Gibrat (1930, 1931) did
this in 1930, calling it the law of proportionate effect, but Kapteyn (1903) had
described a machine that was the mechanical equivalent. Kolmogoroff (1941)
derived the distribution as the asymptotic result of an iterative process of
successive breakage of a particle into two randomly sized particles.
Lognormal distributions have been used to characterize wind speeds and
pollutant concentrations in the earth's atmosphere (Larsen, 1971; Bencala and
Seinfeld, 1986), to approximate the spatial distribution of soot in Diesel engine
combustion (Dolan, 1977), and to characterize atmospheric aerosol size distributions
(WiUeke et al, 1974; WiUeke and Whitby, 1975; Wilson et al, 1977; Whitby, 1978a;
Whitby and Sverdrup, 1980; Savorie et al, 1987) and combustion aerosol size
distributions (Barsic, 1977; Whitby, 1978b; Dolan and Kittelson, 1979).
Other distributions, such as the gamma distribution function, have also
been used to represent the distributions of variables in ecological and atmospheric
systems. For further reading see Brier (1974), Eriksson (1980), Swift and Schreuder
(1981), Volynets (1982), and Dennis and Patil (1984). Because the gamma
distribution often provides a more precise representation of data, models based on
gamma distributions should allow more accurate simulation of the dynamics in
question than models based on lognormal or other distribution functions. Use of the
gamma distribution in developing numerical models, however, is more difficult, so
that virtually all computational models based on the modal representation
incorporate lognormal functions. Clark (1976) attempted some solutions using
gamma functions, but the number of cases for which satisfactory simulations could
be achieved with the gamma function were few.
The Rosin-Rammler distribution (Rosin and Rammler, 1933) was developed
to characterize the distribution of crushed rock, but particles so generated are
usually larger than 10 microns; therefore, at best this distribution would be
appropriate only for characterizing the distribution of large particles. To the
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author's knowledge, no aerosol models have been developed based on the
Rosin-Rammler distribution function.
For unimodal coagulation, the self-preserving size distribution (Friedlander,
1977) was derived numerically as the asymptotic limit of a coagulating aerosol.
Although the self-preserving distribution is accurate for representing the
distribution of a coagulation aerosol, it is impractical for developing generalized
models. It is fortuitous, however, that the lognormal distribution (which has been
empirically found to reasonably represent aerosol size distributions in many
practical systems) closely approximates the form of the self-preserving size
distribution. As a result, little if any precision is lost by using the lognormal
distribution to represent the evolving aerosol size distribution under the conditions
for which the self-preserving size distribution uniquely applies.
The modal approximation is an attempt to represent the size distribution
with a combination of physically realistic distribution functions. If each distinct
population of the simulated aerosol distribution is represented by a distinct mode,
and the modal distribution functions well represent the actual distribution, modal
models can produce results of comparable accuracy to those from spline techniques.
Because most distribution functions can be characterized by three or four
parameters, relatively few function coefficients are required to represent an entire
aerosol size distribution. This can lead to a reduced set of differential equations and
consequently faster solution algorithms. If a large number of overlapping modes
were required to represent the aerosol size distribution, however, the number of
differential equations for some cases could approach or exceed the number of
differential equations required for sectional techniques of comparable accuracy.
Because the lognormal distribution is well behaved, MAD models based on
this representation of the size distribution function are extremely stable. In
addition, the modes automatically shift in size-space in response to the aerosol
dynamics, so that both broad and steep distributions can be represented equally
well. Stability is ensured because the analytical distribution functions are well
behaved over all sizes. Also, because the modes are free to shift in response to the
26
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aerosol dynamics, truncation errors and numerical diffusion due to boundary fluxes
at the section edges do not occur.
A disadvantage of modal techniques is that relatively few parameters are
used to represent the size distribution. The number of modes and modal functions
must therefore be chosen carefully. Theoretically it should be possible to develop
algorithms that can detect the formation and/or complete disappearance of a mode,
but all modal models developed to date have a predetermined number of modes that
persist throughout the simulation. Prior knowledge of the number of modes required
in a particular simulation is therefore required. For calculations in a model with
multiple control volumes, aerosol from neighboring control volumes is mixed due to
convective transport across the control-volume boundaries. Because the modes are
free to shift in size-space, the aerosol modes in neighboring control volumes may
differ from each other in particle size-space. Aerosol from a particular control
volume that is convected to a neighboring control volume is added to the mode
closest to the aerosol's mean size. There may be some loss of accuracy by combining
two modes of different mean size in this way, but for most situations the difference
in size between two control volumes should not be any more severe than the
gradients of other properties solved for. If the difference in the mean sizes of aerosol
modes between two control volume is uhacceptably large, either a new mode can be
created to receive the incoming material (this might be the case if a plume from a
power plant is convected into a control volume of otherwise background aerosol), or
the grid structure can be made finer to lessen the gradients between control
volumes.
Modal techniques have usually been applied to aerosols for which the
composition is independent of size within each mode, although Brock et al (1988)
developed a multicomponent modal model for simulating aerosol dynamics in a
binary laminar coaxial jet, representing the multicomponent aerosol as a product of
two lognormal distribution functions. They used the modal representation because
dispersion in the data did not warrant more accurate models for simulating the
process for comparison, and the computational times of more precise models (i.e.,
models incorporating spline or sectional representations) was prohibitive, even on a
supercomputer (J.R. Brock, University of Texas, personal communication, 1990.).
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Extension of this unimodal multicomponent model to a multimodal multicomponent
model would be of great utility for multicomponent atmospheric modeling, and is an
area for continued work.
2.5.2.3 Monodisperse Fractions Representation of n
Similar to the discrete representation, the monodisperse fractions form
represents the number of particles, N., at a finite number of discrete sizes. Unlike
the discrete representation, however, the particle sizes for monodisperse fractions
may vary with time and are not intrinsically related to monomer size. The
monodisperse fractions representation may also be interpreted as a modal
representation with monodisperse mode distribution functions.
The solution to the resulting GDEs provides the magnitude and location (in
particle size-space) of each monodisperse fraction. This representation has been used
by Warren and Seinfeld (1984) to model coupled nucleation and growth. This
technique was also used by Adkins et al. (1989) to model particle growth in a
continuous stirred tank aerosol reactor (CSTAR), where the input poly disperse
aerosol was represented as a sum of monodisperse fractions. Because each
monodisperse fraction was intended to represent a portion of the distribution and
not an entire population, their use of monodisperse fractions more closely resembled
a discrete representation, except that the particle sizes were free to assume any
continuous value. For the case of pure growth there is no loss of accuracy with this
technique, because coagulation interactions are assumed negligible for the time
scales of the experiment, meaning each particle grows independently from the
others.
Kyriakides et al. (1986) modeled aerosol dynamics in Diesel engine
combustion, and represented the size distribution by monodisperse fractions.
Because coagulation is very important in Diesel combustion, this form for n
represents a compromise of accuracy for computational speed. However, simulating
complex processes such as aerosol dynamics in engine combustion where many of the
input parameters are only approximately known, the uncertainties that are
introduced by approximating the aerosol as a mixture of several monodisperse sizes
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may be negligible in comparison to other modeling uncertainties.
2.5.3 Discussion- Nnrnprical Solutions Based on the Spline. Sectional, and Modal
Representations of n
Spline techniques are best suited for simulations for which very precise and
detailed size distributions are required. These techniques are also useful for
evaluating the performance of models based on less precise representations of the
size distribution.
Sectional techniques have been developed as very general models, and are
able to represent arbitrary distributions with modest computational effort with
respect to spline techniques. Sectional techniques are also well suited for modeling
complex multicomponent systems. The resolution of the distribution is limited only
by the number of sections used, and, in the limit of section widths decreasing to
zero, these techniques approach an exact solution (ignoring computer roundoff
errors). In practice, however, substantial compromise is often needed between
distribution resolution and computational effort, but as the speed of computers
increases, finer resolution in the distributions should be obtainable.
Modal techniques are best suited for situations where the size distribution
can be reasonably represented by multimodal functions, which for the current state
of development usually means multimodal lognormal functions. Modal techniques
therefore provide the global behavior of the aerosol size distribution by tracking the
parameters of the modal distribution functions in time and space.
Seigneur et al. (1986) and Tsang and Rao (1988) provide comparisons for
the accuracy and computational effort of these three modeling techniques.
Simulations of pure coagulation and condensation simulations were performed for a
homogeneous aerosol, providing a test of each technique's ability to simulate a
particular scenario. It is questionable, however, whether modeling homogeneous
aerosol dynamics of isolated processes represents a meaningful test of algorithms
that are used to model systems that exhibit strong inhomogeneities, such as the
earth's atmosphere. The effect of strong inhomogeneities combined with mixing
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probably smooths otherwise distinct distributions that would result from strictly
homogeneous processes. Since no account of inhomogeneity was made in these
simulations, the comparison can only be used to compare the models, and should
not be used to evaluate their merit in representing aerosol dynamics in complex
systems exhibiting strong inhomogeneities.
The computational times required by the techniques are purposely omitted
here, because the effort required by a particular model depends on whether it is
designed to handle arbitrary aerosol dynamics or is tuned for a specific application.
Computational times depend on many factors, including programmer expertise and
shrewdness, so that citing exact times is meaningless. This is clearly demonstrated
by the fact that Seigneur et al. (1986) reported a difference of three orders of
magnitude in computational effort between the spline and modal techniques,
whereas Tsang and Rao (1988) indicate negligible difference between these two
techniques. Suffice it to say that more general solution techniques, such as the spline
and sectional techniques, should require higher computational effort than more
specialized models, such as modal models.
Each representation of the evolving size distribution, and its associated
class of mathematical model, has advantages and disadvantages. Different situations
require different tools, and aerosol modelers should be cognizant of the
characteristics of each technique. Unfortunately, aerosol models of the sort described
here are rarely developed with sufficient documentation to be implemented easily, so
considerable effort is usually required to learn and implement a particular
technique. This may prevent modelers from choosing between the full array of
available techniques.
Of the three types of techniques, the sectional models appear to have gained
the widest acceptance in the aerosol modeling community, perhaps because sectional
methods are similar to finite-difference techniques with which many modelers have
gained familiarity in other fields of study (such as fluid flow modeling). Sectional
models may also enjoy wide acceptance because the models produced by Gelbard are
well tested, documented, and available. Because of this wide user base, sectional
methods are probably the best developed modeling tools for performing
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comprehensive aerosol dynamics simulations. In many cases the numerical
deficiencies of the sectional techniques may not be any worse than the assumptions
inherent in the modal techniques: that aerosol distributions can be represented by a
small number of distribution functions.
The modal technique presented in this work is therefore targeted towards
highly specialized applications, where the computational cost of implementing
sectional techniques is prohibitive. For complex models requiring on the order of
hours of CPU time on supercomputers, savings of even a factor of two with respect
to sectional techniques may translate into savings of tens of thousands of dollars for
models used to simulate many episode scenarios to assess the effects of regulatory
strategies on resulting atmospheric pollutant levels.
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CHAPTER 3
MODAL AEROSOL DYNAMICS (MAD) MODELING
3.1 ABSTRACT
A computationally efficient numerical technique for simulating aerosol
behavior in systems undergoing simultaneous heat transfer, fluid flow, and mass
transfer in and between the gas and condensed phases has been developed. The
technique belongs to a general class of models known as modal aerosol dynamics
(MAD) models. These models solve for the temporal and spatial evolution 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 instead of the detailed size distribution frequency function. Provided in
this chapter are applications of MAD models for simulating aerosol dynamics in
continuous stirred tank aerosol reactors (CSTARs) and flow aerosol reactors (FARs)
using the SIMPLER algorithm (Patankar, 1980). A systematic development of the
modal dynamics equation (MDE) for aerosols is also presented.
3.2 INTRODUCTION
The distribution of a property may often be approximated by a distribution
function of known form. For such a situation, compact and computationally efficient
numerical techniques can be developed that solve for some integral property, instead
of for a detailed distribution of the property. For modeling the dynamics of aerosol
systems, models of this class have been widely used. Because aerosol distributions
often result from the additive effect of multiple sources, distributions composed of
multiple distinct aerosol populations, called modes, occur in many systems of
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interest (Suck et al, 1977; Whitby, 1978a, 1978b).
In this chapter we describe the formulation of a class of aerosol dynamics
models, called modal aerosol dynamics (MAD) models, that are based on the
physical assumption that aerosol size distributions can be represented by a sum of
distinct aerosol populations. The mathematical terms of the governing dynamics
equation (the modal dynamics equation [MDE]) are derived. The system of
equations used in MAD models for modeling continuous stirred tank aerosol reactors
(CSTARs) and flow aerosol reactors (FARs) according to MAD models are
presented.
3.3 THE MODAL ASSUMPTION
Models are often named according to a central physical assumption and/or
mathematical approximation on which all of the ensuing formulations rest (e.g.,
finite-difference, finite-element, control-volume, sectional). This applies to modal
aerosol dynamics (MAD) models as well. Based on observations of atmospheric
aerosols (Whitby, 1978a), the central physical assumption1 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 distribution of each population is approximated
by an analytical distribution Junction.
Each distinct population is called a mode, giving rise to the name modal aerosol
dynamics model. This assumption is hereafter referred to as the modal assumption.
An example of a trimodal aerosol distribution (Wilson et al., 1977) is shown in
Figure 3-1.
The aerosol size distribution in Figure 3-1 was obtained from measurements
near a roadway at the General Motors Milford Proving Grounds. The distribution
kDther physical assumptions and mathematical approximations are usually imposed when
modeling a specific system, but these reflect choices made by the modeler rather than
fundamental limitations of the model.
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ELECTRICAL AEROSOL ANALYZER
0.001
100.0
Figure 3-1. Trimodal aerosol size distribution measured at the General Motors
Milford proving grounds (Wilson et al, 1977).
clearly exhibits the presence of at least three distinct aerosol populations. The
smallest mode, often called the nuclei mode, represents particles produced in engine
combustion. The second mode is due to aerosol in the air blowing over the roadway,
and represents aerosol from an upwind source. This mode is commonly referred to as
the accumulation mode, because atmospheric processes preferentially remove
particles that are smaller and larger than this size, thus accumulating particles in
this size range. The largest mode is due to coarse particles generated by physical
fracture mechanisms and to resuspension of road dust. Whitby (1978a) provides a
more complete explanation of the formation mechanisms and dynamics of these
modes.
3.4 MODAL REPRESENTATION OF THE AEROSOL SIZE DISTRIBUTION
For the modal representation, one must choose the number of modes
required to represent the initial aerosol distribution and also an analytical
distribution function for each mode. If an experimental data set is used to determine
the form of the initial distribution, a procedure for fitting multiple analytical
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functions to the data is required, such as the personal-computer-based curve-fitting
program DISTFIT (Whitby and Whitby, 1989). See also Raabe (1971, 1978).
In a general sense, a mode is required for each population of particles of a
distinct size and/or chemical composition. In the limit, this criterion leads to an
intractable number of modes if every microscopically distinct source is represented.
Engineering judgment is required, therefore, to reduce the ultimate number of
independent modes to a reasonable limit. For atmospheric aerosols, three modes are
commonly used (Whitby, 1978a), although others have proposed using as many as
seven modes to represent the same data (Davies, 1974). The data of John et aL
(1990) strongly supports the use of two accumulation modes of different composition
to represent the aerosol that was previously represented by a single accumulation
mode. The number of modes chosen, therefore, depends on the specific system and
on the modeler's interpretation of the data and/or processes affecting the system.
The increased computational effort of using multiple modes may also influence a
modeler's decision about how to represent an aerosol size distribution.
For the distribution in Figure 3-1, three modes were used to represent the
distribution: one for the combustion nuclei, one for the background aerosol, and one
for the coarse particles. The particle distribution from each car could be represented
by a separate mode, but because most automotive engines operate under similar
conditions and produce aerosol of a similar size and chemical composition, using a
single mode to represent the combustion aerosol is justified. This is not always the
case, however. Figure 3*2 shows the size distribution for particles produced in a
propane/air flame (Barsic, 1977; Whitby, 1978b), where two combustion nuclei
modes occur; the smaller particles are primarily sulfur containing, while the
intermediate-size particles are of carbon. For this case two modes are used to
represent the combustion nuclei because the generation mechanisms produce two
clearly distinguishable aerosol populations.
In addition to specifying the starting distribution for a model simulation,
during the simulation one must decide when two modes cease to be distinguishable
and should be combined into one. Additionally, when an aerosol source is present,
decisions must be made about whether to introduce the source particles as a new
35
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mode or as a source term in an existing mode. Also, a multicomponent aerosol may
be represented as internally or externally mixed. Externally mixed aerosols are
CNJ
o
^
CO
Second
Nuclei Mode
Accumulation
Mode
First
Nuclei Mode
ran
dp [jim]
Figure 3-2. Particle size distribution for a rich propane/air flame
(Barsic, 1977; Whitby, 1978b).
represented as separate modes of distinct chemical composition, whereas internally
mixed aerosols can be represented as a product of distribution functions (Brock et
aL, 1988). As an aerosol ages, interactions between these modes cause the
distribution to progress towards internally mixed multicomponent modes and a loss
of the modes of distinct composition.
In this work the aerosol is represented as a homogeneous single species.
Generalization of the model to treat internal mixtures of multiple components is a
topic for continued work, and some considerations for developing multicomponent
MAD models are discussed in Chapter 6.
3.5 THE TASK
All numerical solutions to the general dynamics equation (GDE) require
expressions for the time rate of change of the function parameters used to represent
36
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the distribution. For spline representations, differential equations for the spline
coefficients are derived (Gelbard and Seinfeld, 1978). For sectional techniques,
differential equations for a moment of the distribution in each section (usually
number or mass concentration) are formulated (Gelbard and Seinfeld, 1980). For
MAD models, differential equations for the time rate of change of the analytical
function parameters for each mode are derived (Cohen and Vaughan, 1971; Drake,
1972; Lee, 1983; Whitby and Whitby, 1985a). The number of simultaneous
equations to be solved, therefore, is equal to the number of function coefficients
required to represent the entire size distribution.
3.5.1 Dynarnics Equations for Distribution Function Parameters
For MAD models, differential equations for the parameters of each modal
distribution function must be developed. Although distribution functions with any
number of parameters can be used, the MDEs developed in this work are for
three-parameter distribution functions (e.g., gamma, lognormal, Rosin-Rammler,
Wtibull), because they represent the most commonly used functions in aerosol
science. Extension of the MDEs to higher-order distribution functions is
straightforward.
Consider a single three-parameter distribution function. The parameters
can be classified as total number concentration, N. , an average size, H. , and the
variance, a ^ where t is a mode index (these three parameters are chosen as examples
for the following presentation, but in general any parameters of the distribution can
be used). We require expressions for the time rate of change of these parameters for
each mode:
!(.)=... (3-lc)
where t = mode indices a, b, c, ...
37
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By solving these equations, the time rate of change of the distribution function is
specified.
3.5.2 Dynamics Equations for Multiple Processes
For processes that involve a single dynamic mechanism, it is possible to
directly obtain solutions to the differential equations represented by Eqs. (3-1) (see,
for example, Lee [1983, 1990]). A problem occurs, however, if the model is
generalized to account for multiple processes. Consider the effects of combined
coagulation and diffusion. The differential equation for N. (an integral moment) is
For the modal parameters ~d. and a. , however, expressions like Eq. (3-2) cannot be
written easily, because it is not possible to add differential equations for the mean
and variance.2 This problem is avoided if the expressions for the time rate of change
of the mean and variance are replaced with two expressions for the time rate of
change of integral moments. Additional integral moments are calculated as
(3-3)
Two common integral moments are for k = 0 and k = 3:3
= N (3-4a)
M3 = Jn ^n(dp)ddp = (6A)7 (3-4b)
Equations (3 Ib and 3 Ic) can be solved if an independent expression exists for either or both of
these parameters. This approach was adopted by Whitby (1979) when he specified o> for each
mode to be time invariant, in which case the expression was
3The power k can be any real value.
38
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Using the definition of moments from Eq. (3-3), Eqs. (3-lb) and (3-lc) can be
replaced by expressions for the time rate of change of two additional moments of the
distribution. The set of differential equations for each mode is therefore written as
If *y T, P, ..-) (3-5a)
, , Z, a, T,P,...) (3-5b)
M
Z, a, T, P, ...) (3-5c)
Note that in Eqs. (3-5) the functional dependence on the rhs is still expressed in
terms of d. and a. . This choice is made because the size-space integrals are easier to
conceptualize and evaluate for expressions in terms of the mean and variance than
they are for integral moments.
In order to relate the mean and variance to the integral moments,
conversion equations, dependent on the form of the modal distribution functions
(n.), must be formulated:
(3'6b)
where Mk = MJN
For lognormal distribution functions, the relationships between integral moments
and Dgn and crg are shown by Eq. (3-7). The subscripts &i and fc2 refer to specific
moments, whereas the subscript Prefers to any arbitrary moment.
39
-------
M =
*!
exp
(3-7a)
(3-7b)
M. =
(3-7c)
(3-7d)
(3-7e)
where r =
The derivations of Eqs. (3-7) are shown in Appendix C.
3.6 CLASSIFICATION OF AEROSOL PROCESSES
The next step is to develop expressions for the time rate of change of the
moments for each process represented in the GDE. The terms of the GDE exhibit
dependence on one or more of the coordinates: time, space, and particle size. The
particle-size dependence is accounted for by adopting the modal assumption. The
classes of physical problems of interest in aerosol dynamics modeling can therefore
be uniquely classified by their dependence on temporal and spatial variations.
In the absence of spatial variations, a physical system can be classified as a
continuous stirred tank aerosol reactor (CSTAR) (Pratsinis et al, 1986). For
40
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steady-state situations where the size distribution may vary with position but not
with time, the system can be classified as a flow aerosol reactor (FAR). When
temporal and spatial variations both exist, the physical system can be viewed as an
unsteady aerosol reactor. In this chapter we prescribe solution techniques of the
modal dynamics equation (MDE) for the CSTAR and FAR situations.
For FARs, the GDE represents the following internal and external
processes:
Internal Processes
Coagulation
Particle growth
Internal sources
External Processes
Convection
Particle migration due to external force fields
Diffusion
For CSTARs, the same nomenclature is used for internal processes, but a
different nomenclature is used for external processes. Consider a mixing vessel (e.g.,
Figure 2-2b). The bulk fluid is assumed well mixed, and therefore homogeneous
with respect to all properties. Near the wall is a boundary layer across which
gradients in the properties develop. The external processes that act across this
boundary layer are as follows:
General name Name used with CSTARs
Convection > Inflow/outflow
External forces > Surface deposition
Diffusion > Surface deposition across a boundary layer
3.7 MDEs FOR CONTINUOUS STIRRED TANK AEROSOL REACTORS
(CSTARs)
For each process, expressions for the time rate of change of the integral
moments of each mode are required. It is convenient to adopt a framework
consistent with the physical mechanism represented by the process, and to
41
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formulate the mathematical expressions according to that framework. For
coagulation, for example, we consider the time rate of change of number
concentration due to collisions between particles and ask, "For each successful
collision between two particles, how much of the total moment is lost by the
disappearance of two particles, and how much is the total moment increased by the
appearance of the new particle?" For particle growth, which represents volume
transfer from the gas phase to the particle phase, we ask, "What is the change of
moment per unit change of volume of the particle due to particle growth?"
Deposition is viewed as a number loss mechanism and we ask, "How much moment
is lost by each particle that deposits on a surface?"
3.7.1 Internal Processes for CSTARs
3.7.1.1 Coagulation
Because it is convenient to view coagulation as a number transfer process,
the equations for the changes of moments are related to the moment change per
particle collision. The model of coagulation adopted4 is that two spherical particles
collide and are subsequently lost from the system, forming, by coalescence, one new
spherical particle whose volume is the sum of the volumes of the two colliding
particles (sticking coefficients are assumed to be unity, and enhancement factors are
not addressed). Denoting the moment lost during each collision as (dp + dp ) and
the moment gained due to the appearance of the new particle as dp , the net
J.^£
change of the integral moment is
where the factor of \ corrects for overcounting interactions with the double integral.
^Reflecting a choice by this author, and not a limitation of MAD models.
42
-------
Note that Eq. (3 8) is evaluated over all sizes, without any assumptions about n
or /5. For coagulation between spherical particles that results in the formation of a
new spherical particle, the moment of the new particle is
The derivation of Eq. (3-9) is provided in Appendix D.
The effect of applying the modal assumption to Eq. (3-8) is that n = n +
n^ + n. + ... . Because Eq. (3-8) is an integral over two populations (whether or
not they are two distinct populations) it suffices for the following derivation to
substitute n= n.+ n, where the following rules apply:
i = a, b, c, ...
j=b, c, d, ...
Substituting this form for nin Eq. (3-8) and substituting the moment source term
fromEq. (3-9) yields
coag
(3-10)
If three-body collisions were important and an equation similar to Eq. (3-7) with
triple integrals over three distributions were evaluated, then three independent
modes would be required to formulate a general expression similar to Eq. (3-10).
43
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It is convenient to separate the terms of Eq. (3-10) according to the aerosol
modes i and j. In order to expand and group terms, the following conventions are
adopted:
When particles from the same mode collide (intramodal coagulation), the
agglomerated particle remains in that mode.
When particles from two different modes collide (intermodal
coagulation), the agglomerated particle is assigned to the mode with the
larger mean size. This reflects that the particles are growing. In the
derivations here, index jis used to represent the mode with the larger
mean size.
The implication of the second convention is that for intermodal coagulation all
particles from the smaller mode (the mode with the smaller mean size) are
transferred to the larger mode. This convention is arbitrary, but consistent with the
view that the particles are growing (i.e., migrating from the smaller to larger mode).
Furthermore, this convention was used by Eltgroth (1978) and Whitby and Whitby
(1985a and 1985b), and comparisons of the model by Whitby and Whitby (1985b)
with other numerical models (Seigneur et al, 1986) indicate that these conventions
properly account for the migration of particles in particle size-space. Megaridis and
Dobbins (1990), in developing a bimodal aerosol dynamics model, assumed that the
agglomerates resulting from both intramodal and intermodal coagulation
interactions were transferred to the large mode. It appears they adopted this
convention because their small mode was engulfed by one tail of the large mode. For
the aerosol dynamics of interest in their studies, comparisons with a sectional model
(MAEROS [Gelbard, 1984]) indicated that the MAD model so developed provided a
reasonable representation of the aerosol dynamics. It is questionable, however,
whether their convention for distributing agglomerated particles can be applied to
generalized MAD models of more than two modes, or of two modes with
distribution tails that are not strongly overlapping.
Expanding Eq. (3 9) and grouping terms results in the following expression
for mode i:
44
-------
t coag
/* fa,
2Jo Jo
Intramodal Coagulation
(3- lla)
Intermodal Coagulation
The first two integrals on the rhs of Eq (3- lla) represent changes due to intramodal
(z'-i) coagulation, and the last integral represents the loss from mode i to mode j of
particles of size dp 5 due to intermodal (i-j) coagulation. For mode j the terms are
/CD foo
ilo Jo (^1
3 coag
A/3
Intramodal Coagulation
Intermodal Coagulation
5Because the tails of the distributions overlap, dp is not always smaller than dp .
45
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The first two integrals on the rhs of Eq. (3 -lib) represent changes due to
intramodal (j-j ) coagulation, the third integral represents gain to mode j due to
intermodal (i-j ) coagulation, and the last integral represents loss from mode j of
particles of size dp due to intermodal (i-j ) coagulation. In the absence of
intermodal coagulation, the intramodal terms correctly predict that F. and V. do
not change.
Equations (3-11) are completely general, and the only physical assumptions
(other than the modal assumption) are unit sticking coefficients, spherical particles,
and that volume is conserved after two particles collide.
3.7.1.2 Particle Growth
Particle growth is the process of transferring volume from the gas or vapor
phases to particles. Growth processes occur by vapor condensation or by reactions
on or within existing particles.
The total volume change of the particle population during growth is
(3-12)
where Vp represents the rate at which a particle of size dp changes volume, and is
commonly referred to as the growth law. Because particle growth occurs by addition
of gas-phase material, the growth law is often sensitive to thennodynamic
properties. For this reason it is convenient to represent the growth law as
^ = *TV(
-------
The general moment equation is obtained by multiplying the integrand of Eq.
(3 14) by the specific moment change per unit volume change:
at
For spherical particles the change of moment per unit change of volume is
dd*
tlttp nl. in
JA^-3 (3>16)
d[(T/6)4l
Substituting Eq. (3-16) into Eq. (3-15) yields
IT
(3-17)
Note that Eq (3-17) correctly shows that the number moment (i.e., k = 0) does not
change for particle growth.
Equation (3-17) is used to represent particle growth processes in the MDEs.
If particle growth processes are fast compared to the generation of condensable
material, a steady-state develops, where the rate of production of condensable
material is equal to the rate of volume growth of the particles due to the
condensable material. Because volume is proportional to M , a mass balance for the
vapor and particles can be written as
(3-18a)
where
r Q n ro T
(3-18b)
47
-------
r Q 1
and ^(M ) is the production rate of condensable vapor. Equation (3-18) can also
be written as
dt
a* b
(3-19)
The ratios in the brackets represent the fraction of the material that is injected into
each mode, which can also be written as
(3-20a)
where ft. = mode i partition function =
dt
(3-20b)
The partitioning functions, ft., are calculated from Eq. (3-20b), where Eq. (3-17) is
Q
used to evaluate the terms -§^M^ ). Because the partitioning functions depend only
t
on particle size, $T appears as a constant in the numerator and denominator of Eq.
(3-20b) and therefore cancels. This simplifies calculation of the rate coefficients for
rapid condensation, because the deposition rate depends only on the relatively
slowly changing size distribution and the total production rate of condensable
material, and not on rapidly changing thermodynamic properties (such as saturation
ratio).
The change of an arbitrary moment, M,, is expressed as
48
-------
Substituting Eq. (3-20a) for Af) in Eq. (3-21) and simplifying yields
f.
<
^(dp)n.
Jo
(3-22)
3.7.1.3 Internal Sources
Internal sources may occur due to nucleation or a sudden release of particles
within the system boundaries. For example, in a nuclear containment vessel, an
internal explosion may create a burst of particles due to fragmentation of solid
material. Nucleation of gas-phase material would provide an additional internal
source of particles. Internal sources are treated as an injection rate of particles of
size dp into the system. The injection rate of particles from source 5 into mode t is
(3-23)
" u L " J ;
5=1
Changes of the other moments are calculated as
S=l
For multimodal distributions, algorithms must be incorporated that either
direct the source aerosol into one of the existing modes or determine that a new
49
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mode should be created. To determine which of the existing modes should receive
the source particles, the characteristic size of the source particles is compared with
the characteristic size (e.g., D^ for lognormal distribution functions) of each
existing aerosol mode. If the characteristic size of the source aerosol is sufficiently
different (a user-prescribed criterion) than the characteristic sizes of the existing
aerosol modes, a new mode must be created. Often, however, the modes to receive
source aerosol are predetermined (Whitby, 1979; Eltgroth, 1982; Giorgi, 1986;
Pratsinis, 1988). Thus, resuspended dust (i.e., dp> 1 /an) would be included in a
mode with a relatively large mean size (i.e., Dgn « 10 /zm), whereas particles
resulting from vapor nucleation (i.e., dp « 0.01 fan) would be introduced into a mode
with a relatively small mean size (i.e., Dgn » 0.01 /an).
3.7.2 External Processes for CSTARs
A CSTAR is assumed to contain sufficient turbulent energy to maintain the
system in a well-mixed state; this may be caused by the action of a physical stirring
blade or by natural convection processes. The dynamics governing the core fluid,
therefore, are the internal processes of the GDE. Near the wall, a boundary layer
exists that separates the well-mixed core from conditions occurring at or near the
wall. The definition of the boundary layer thickness depends on the specific process,
and also on particle size (Crump et al, 1983). The processes acting across this
boundary layer are considered either as inflow/outflow across the system
boundaries, or as deposition to any of the internal surfaces (see Figure 2-2b).
3.7.2.1 Inflow/Outflow
It is assumed6 that particles are small enough so that inflow/outflow
processes do not discriminate based on particle size. This does not automatically
imply that all of the particles in an outflow stream successfully exit the vessel, only
that the outflow stream contains a representative sample of the entire particle size
spectrum. Some of the particles may impact on surfaces near the outflow boundary,
but they will still be removed from the reactor volume, so this process appears as
6Reflecting a choice by this author, and not a limitation of MAD models.
50
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outflow from the reactor core. The outflow sink term causes a loss of moment M, at
volume flow rate Q- An inflow source introduces moment M, into the vessel at
"out *in
volume flow rate QR. . The expression for the net effect on the total integral
moment of mode i, due to inflows, Q , and outflows, (JL , is
HI. «n
iin lout
Mn
Qr. (3-25)
J\.j
*out
3.7.2.2 Surface Deposition
Surface losses are calculated from size-sensitive deposition velocities due to
processes like gravitational settling, impaction, thermophoresis, and turbulent
diffusion. In simple systems surface deposition is often referred to as wall losses, but
in complex systems there may be additional surfaces (such as stirring blades) that
also contribute to particle losses. The following derivation represents the combined
effects of all processes reflected in the deposition velocity, £ allowing for the
deposition velocity to vary for each reactor surface. The net effect on the total
integral moment of mode i is
"su
lsu=l
/*o>
Jo * J*>
-------
The last term on the rhs of Eq. (3 27) accounts for expansion/contraction of the
vessel. This term is required if the above equations are applied to a system with
expanding/contracting walls.
The complete MDEs for CSTARs and a bimodal aerosol is
R nout R nsu -
y j iisiLJin _ M y -jout.-L- y A
L^ v *. L* v V £* U
/in=l R (out=l R isu=l
Intramodal Coagulation
Intennodar Coagulation
' ns M
Jfc
'*d
52
-------
R n°ut r
""'" -
Intramodal Coagulation
Intermodal Coagulation
M
2fc,T,
-v (3-28b)
'u ", P H t
5=1 J R df
If a vapor deposition rate is specified for particle growth, the particle growth term is
written as
fV3rt
v>] owft = [!((M3.)]vn,-57;; ! (3-28c)
The complete MDEs are also given in Appendix E, and a mass-based form of the
MDEs is given in Appendix F.
Solution to the set of MDEs for all moments provides the time dependence
of the aerosol distribution. Techniques for integrating Eqs. (3-28) are presented in
Chapter 5.
53
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3.8 MDEs FOR FLOW AEROSOL REACTORS (FARs)
For the case of a CSTAR, the external terms of the GDE appear as pseudo-
source/sink terms to/from a homogeneous aerosol. If we model the behavior of an
aerosol in a truly inhomogeneous system, spatial gradients of the aerosol must be
accounted for throughout the calculation domain. This usually requires a numerical
technique, such as finite-element, finite-difference, or control-volume, for handling
spatial gradients. For this work, the application to inhomogeneous systems is
presented for a control-volume-based algorithm developed by Patankar (1981) called
the SIMPLER algorithm. ?
3.8.1 The SIMPLER Algorithm
Patankar (1980) developed an algorithm for solving generalized fluid
flow/heat transfer problems where the governing differential equation is of the form
(3-29)
unsteady convection diffusion source
To use the SIMPLER algorithm, the GDE must be written in the form of
Eq. (3 .29). All terms that do not correspond to the unsteady, convection, or
diffusion terms are considered to be source terms. The GDE for aerosols is similar to
Eq. (3-29), and can be represented as
) + V-vn+ V-cn=V-Z7Vn+ 5^ (3-30)
1 f CP IV I
^ = -JQ /3(vp, Vp-
where
" 'o
+ i«'n-TJi|growth
7This discussion also pertains to the SIMPLE algorithm (Patankar and Spalding, 1972), but the
SIMPLER algorithm is generally favored because of its higher computational efficiency.
54
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Note that the GDE contains two particle convection terms, one due to fluid motion
(V- vn) and the other due to the influence of external forces (V-cn). It is convenient
to combine these terms into a single convection term for the particle velocity:
Siiit (3-31)
where vp = v + c.
Combining the fluid and external force velocities this way is
straightforward, with one additional consideration. Patankar applies a simplifying
assumption in his code, that all velocity fields implicitly satisfy mass continuity.
However, if the fluid and external force convection terms are combined as in Eq.
(3-31), the resulting particle velocity field will not, in general, satisfy mass
continuity. This deviation from a mass-continuity-satisfying velocity field must be
accounted for in the Patankar algorithm. This modification is documented by
Stratmann and Whitby (1989), and is also included in Section 3.8.2 .
Because fluid flow/heat transfer situations often involve density gradients,
Patankar expresses the general 0 variable on a unit mass basis. To use Patankar's
code for modeling FARs, therefore, the MDE is written as
4 ) + v- v peMk = v-r4 lr + s (3-32)
i t i i
where
v, = v + c, = moment convection velocity
t i
c, = moment transport velocity due to external forces
Mk = Mk /ps
i i
T, = D, PZ = moment diffusion coefficient
e e o
n> «v «
t t
source
55
-------
and the terms represented by Sint are calculated with Eqs. (3-11), (3-17) or (3-22),
and (3-24).
3.8.2 Moment Transport Coefficients for the SIMPLER AleorithTn- External
Processes
In order to derive the moment average transport coefficients required for
Patankar's code, some information about the discretization technique used in the
SIMPLER algorithm must be considered. The following discussion applies to a
one-dimensional (1-D) calculation domain; extension to 2-D and 3-D situations is
straightforward and is given by Patankar (1980). Figure 3-3 shows a 1-D grid
structure used in Patankar's solution algorithm. The property information (T, P,
MV etc.) is stored at the control-volume grid points (P, E, W), and the velocities
are stored at the control-volume boundaries (e, w).
H-H
O 1
IT «
W w
W
k?
ID
i
e
e
ki
'
I-l
1+1
Figure 3-3. 1-D grid structure used in the SIMPLER algorithm.
Patankar converts Eq. (3-29) to an algebraic equation of the form
(3-33)
where the coefficients (a^, Og, a.^) depend on the relative strength of convection
and diffusion at the control-volume faces. The interpretation of Eq. (3-33) is that
the property value at a particular grid point is derived as the sum of fractional
56
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contributions from property values at the neighboring grid points. The coefficients
are calculated such that convection/diffusion problems that exhibit flow in one
primary direction are accurately represented by the formulation of Eq. (3-33).
In general, the coefficients of Eq. (3-33) are a complex function of the
relative convection/ diffusion strengths in all directions. For 2-D and 3-D situations,
however, Patankar calculates the coefficients for each respective coordinate
direction from a 1-D Peclet number based on the relative convection and diffusion
strengths in that direction. While this is somewhat approximate, it is accurate when
flow occurs primarily along one of the system coordinates, and it also provides a
faster and more stable algorithm than calculating the coefficients based on a 2-D
Peclet number. The 1-D formulation of the rate coefficients of Eq. (3-33) is
= /(Pee) (3-34a)
J (
where F = psuA = convective mass flux
D = T/AxA = diffusive mass flux
Pe = F/D = Peclet number
5p = source term component that depends linearly on p
and the subscripts e and w refer to the control-volume face to the east and west of
the control-volume grid point, P, respectively. If the velocity field satisfies mass
continuity, then F F = 0, and Eq. (3-34c) reduces to
Equation (3-35) is the form Patankar uses to calculate Op. For velocity fields that
do not satisfy mass continuity, Eq. (3-34c) must be substituted for Eq. (3-35) in
Patankar's code (Stratmann and Whitby, 1989).
57
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3.8.2.1 Fluid Convection
For pure convection, the rate of change of Mk is
(3-36)
Because fluid convection is not a size-sensitive process, Eq. (3-36) can be written as
(3-37)
The form of the fluid convection term for the fcth-moment MDE,
represented by Eq. (3-37), is the same as the form of the fluid convection term in
Eq. (3-29). This implies that by following the normal procedures for using the
SIMPLER algorithm, convection of M, is accounted for properly.
3.8.2.2 Transport Due To External Forces
The formulation for transport of M, due to external forces is not
straightforward, and requires some additional considerations. For pure convection,
the convective velocity carries with it the value of the properties in the upstream
control volume. The difficulty in calculating particle moment transport is that, in
general, external forces are sensitive to particle size, so the moment transport
coefficient due to external forces, c,, must be calculated such that multiplying by
the grid point value of M, yields the correct boundary flux. In the 1-D grid in
Figure 3-3, if 7^ is to the right, then M, is transported from control volume /to
7+1, and c, must be based on the aerosol size distribution in control volume /. If Ue
is to the left, then c, must be based on the aerosol size distribution in control
volume 7+1. In addition, because the velocity is required at the control -volume face,
and external forces may also depend on other properties (e.g., T and P) stored at
the grid points, an interpolation of the pertinent properties to the respective
58
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control-volume face may be required. Patankar provides interpolation factors in his
code, but the user may employ other schemes as well.
Assuming that the interface property values are available, the moment
average convective velocity due to external forces is
fOD
c(T, P , ...,
Jo e e "' ' ]tr ' for flux from/-. 7+1 (3-38a)
c
_ /oVe, p,...,
r*..|
E
[N,
for flux from 1+1*1 (3 38b)
where T , P , ... are the interpolated properties
c is the transport coefficient due to external forces
The complete convection velocity, therefore, is v, = v + c, .
K . n .
t t
3.8.2.3 Diffusion
Patankar treats diffusion as a flux at the control-volume boundary, and
computes a diffusion coefficient for the boundary flux that is an average of the
diffusion coefficients in the neighboring control volumes. By matching the
diffusional resistance in each neighboring control volume, the harmonic mean, F^, is
shown to be the correct averaging technique for calculating the diffusion flux
(Patankar, 1980):
2 P., TD
r = p F' £ (3-39)
where T and T are the diffusion coefficients based on the grid-point property
E P
values. This implies that moment diffusion coefficients calculated using the
59
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grid-point property values will correctly provide Patankar's algorithm with the
diffusion coefficients required to calculate the interface diffusion flux term, F . The
moment diffusion coefficient is
D
k.
j.yp. Pf, . ..
and
i P
Equations (3-28) represent the MDEs. For unsteady situations, the MDEs
are integrated to determine the temporal history of the moments. For FARs the
system is at steady state, so modal GDEs are calculated and used, without
integration, to represent time-invariant source terms in each control volume.
The generalized source term for the SIMPLER solution algorithm is
separated into a constant component, 5_, and a component that linearly depends on
0, 5p (Eqs. [3-33] and [3-34c]). The internal aerosol processes can be accounted for
by introducing the entire MDE as a constant source term. For processes that depend
on the local value of 0, however, faster convergence of the SIMPLER algorithm can
be achieved by splitting the MDE into terms that are best represented as either
constant source terms or terms that depend on .
For most situations it is not trivial to determine the precise order of a given
process. In general, the order of a given process will depend on the other processes
occurring simultaneously. We therefore recommend the following simplified
approach for separating the terms of the MDE between S and Sp.
After computing the moment rate expressions for each process, calculate
I C \
60
-------
t J positive
where the summation is over all active processes, and a7(-Wi) represents
"- i -* positive
processes whose rates are positive. For all negative terms, calculate (Sp)ft as
t
. (3.41b)
M
t
r f\ "i
where Q&M,) represents processes whose rates are negative.
" »' * negative
61
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CHAPTER 4
TECHNIQUES FOR EVALUATING THE INTEGRALS
OF THE MODAL DYNAMICS EQUATIONS
4.1 ABSTRACT
Numerical and analytical techniques for evaluating the integrals of the
moment equations for MAD models are described. Moment rate coefficients for
coagulation, condensation, surface and volume reactions, diffusion, and migration in
external fields (such as sedimentation and thermophoresis) are also presented. For
multimodal lognormal distributions, an analytical expression for evaluating the
coagulation integrals (hereafter referred to as the analytical coagulation integrals) of
the modal dynamics equation (MDE), applicable for all size regimes, is presented.
For coagulation integrals applied to MDEs for M , Af , and M , the analytical
U o u
coagulation integrals are within 20% of accurate numerical evaluation of the same
moment coagulation integrals. If ag of each interacting mode is restricted to a value
greater than 1.5, the analytical coagulation integrals are within 10% of numerical
evaluations of the same integrals.
A computationally efficient numerical integration technique, based on
Gauss-Hermite numerical integration, is presented for cases when moment integrals
must be evaluated numerically. A technique for integrating a system of integrals
with similar integrands is also presented. This method is applied to the set of
moment integrals that results from the multiple MDEs required to represent the
dynamics of each mode's distribution function.
4.2 INTRODUCTION
The task in MAD modeling is to solve a set of modal dynamics equations
62
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(MDEs) for the moments of the modes of an aerosol size distribution. The MDEs
contain integrals of size-dependent rate coefficients for each of the dynamic
processes over the size distribution. The available techniques for solving these
integrals are summarized in this chapter. A new analytical algorithm for solving
multimodal coagulation integrals is presented, and efficient numerical techniques for
evaluating the integrals of the MDEs are discussed in detail.
Also presented are specific recommendations for forms of the rate
coefficients to use in the size and flow regimes commonly encountered in aerosol
systems. Because the integration techniques are discussed in detail, interested
readers should be able to modify any of the expressions to account for other effects
and/or substitute other forms of the rate coefficients.
Coagulation is given special emphasis because the double integrals
belonging to this mechanism represent a significant numerical hurdle in formulating
aerosol dynamics models. For MAD techniques to provide computational advantage
with respect to other aerosol dynamics models, these integrals must be evaluated
analytically and/or with the aid of table interpolation/lookup techniques.
Understanding the techniques for evaluating the coagulation integrals serves as a
basis for evaluating integrals associated with other processes.
In this work, analytical techniques for evaluating the moment integrals are
presented for the lognormal distribution function. Choosing a distribution function
other than lognormal typically requires the use of numerical integration and/or
table interpolation/lookup techniques.
4.3 THE TASK
In MAD models, differential equations for the integral moments of a
distribution must be formulated and solved. The integral moments of a distribution
are defined as
(4-1)
63
-------
and the mathematical expressions for the time rate of change of Mk that are
appropriate for MAD models are called the modal dynamics equations (MDEs).
Solution of the MDEs requires evaluation of single and double integrals of the form
(4-2a)
f QD / OD
*i TJfl Jo * 1? 2 1' 2 i 1 j 2 12
where f,(dp) = moment function (e.g., dp )
TT = non-size-dependent component of the rate coefficient
^dp)1 = size-dependent component of the rate coefficient
n.(dp) = number concentration size distribution function of mode i
In this work, techniques for evaluating these integrals are presented for
each process represented by the MDEs:
Internal Processes
Coagulation
Particle growth
Internal sources
External Processes
Convection
Diffusion
External forces
The techniques for evaluating the integrals represented by Eqs. (4-2) are
classified as numerical, table interpolation/lookup, and analytical. Numerical
integration techniques can handle arbitrarily complex integrands, while analytical
integration is more restrictive and applies only for special forms of the integrands.
compactness, the size-dependent component is written as 7(
-------
4.4 SOME GENERAL CONSIDERATIONS
4.4.1 Particle Size
We are interested in modeling systems with particle sizes ranging from
0.001 100 /an. At standard temperature and pressure, the mean free path of air is
about 0.065 jmi. Aerosol behavior can be related to the ratio of the mean free path
of the suspending gas to particle size. Size regimes based on this ratio are defined in
terms of the Knudsen number (Kn = 2X/dp):
Value of Kn _ Size Regime
Kn > 10 Free-molecule
1 < Kn < 10 Transition
0.1 < Kn < 1 Near-continuum
Kn < 0.1 Continuum
Mathematical expressions for the rate coefficients of each process must be available
for all size regimes. A set of mathematical expressions valid only for a specific
regime may be selected, or generalized expressions valid in more than one regime
may be used. Expressions valid in a single regime tend to be compact and often
permit the resulting integrals of the MDEs to be evaluated analytically, while
generalized expressions tend to be more complex and often do not permit analytical
evaluation of the resulting integrals.
Many rate coefficients for aerosol processes contain the Cunningham slip
correction, CL. An expression that is valid in all size regimes (Friedlander, 1977) is
c
<7_ = 1 + Kn[a + b exp(-c/Kn)] (4 3a)
c
where a = 1.246
b = 0.42
c = 0.87
65
-------
Coefficients for Eq. (4-3a) have been proposed by many authors (Davies, 1945;
Fuchs, 1964; Philips, 1975; Allen and Raabe, 1983). Because we compare some of
the coagulation integrals presented in this paper to values given by Fuchs (1964),
the Fuchs coefficients (shown with Eq. [4-3a]) are used. The mathematical forms of
the expressions developed in this work, however, are not sensitive to the particular
values of a, 6, and c.
The form of CL represented by Eq. (4-3a) does not permit analytical
o
evaluation of the resulting integrals containing it. For this reason, we define the
following power-function approximations to C1-,:
Cc = 1 + 1.43 KnL0488 for Kn > 1 (4-3b)
Cc = 1 + 1.43 Kn1'0814 for Kn < 1 (4-3c)
Equation (4-3b) is within 0.5% of Eq. (4-3a) and Eq. (4-3c) is within 2% of
Eq. (4-3a) in the respective regimes to which each applies. These forms permit
many integrals to be evaluated analytically, but the resulting expressions may
contain multiple terms with real exponents (e.g., Kn1'0814), which are
computationally more expensive to evaluate in a computer code than terms with
f\
integer exponents (e.g., Kn ). Another approximation is
CQ a 1 + AfmK.n for Kn > 1 (4-3d)
Cc » 1 + 4ncKn for Kn < 1 (4 3e)
where Afm = 1.43Kng'0488
A , .01^0.0814
Anc 1.4oKng
Kng = 2A/Z?gI1
When Eqs. (4-3d) and (4-3e) are used to represent CL in the integrals of the MDEs,
the A factors only need to be calculated once. The other terms in the integrals that
depend on C_ will only result in terms with integer exponents of Kn. Because the A
o
66
-------
factors are calculated from Dgn, the accuracy of the resulting integrals that
incorporate these approximations to (7_ depends on the width of the distribution
o
(e.g., on <7g for lognormal distributions).
4.4.2 Moments Solved for in MAD Models
When developing MAD models, the moments to solve for can be selected
arbitrarily. If analytical coagulation integrals are employed, however, the moments
solved for must be selected as multiples of three (i.e., M , M , M , ...), because of
Out)
the form of the coagulation integrals (refer to Section 4.6.1). If the coagulation
integrals are evaluated numerically, other moments may be chosen. Whitby and
Whit by (1985a, 1985b) used numerical integration combined with a table
interpolation procedure to evaluate the coagulation integrals, solving for M , M ,
v 4
and M^. Giorgi (1986) evaluated the integrals numerically and solved for Af , M ,
and Af3-
The number of MDEs required per mode is equal to the number of
time-variant parameters of n. Lognormal distribution functions contain three
time-variant parameters (N, Dgn, and 0g), so three MDEs for each lognormal
distribution must be solved.
4.5 INTEGRATION TECHNIQUES
4.5.1 Numerical Integration
Two numerical integration techniques are presented: a method for
evaluating integrals with integrands of arbitrary form, and a method for cases where
n is represented by a lognormal distribution function. Both methods belong to the
general technique of Gauss numerical integration, and are described in many texts
on numerical techniques (see for example, Atkinson [1985]). We present techniques
for evaluating single integrals. The extension of these techniques for evaluating the
double integrals of the MDEs is presented in Appendix G.
67
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4.5.1.1 Numerical Integration of Arbitrary Integrands: Gauss-Legendre Form
General technique Gauss-Legendre integration is a technique for
numerically evaluating integrals with arbitrary integrands. For logarithmic abscissa
(commonly used in particle statistics), the Gauss-Legendre technique approximates
the integral
(4-4a)
with a summation of the integrand evaluated at prescribed nodes and multiplied by
weight factors, as
) (4-4b)
gl=l
The nodes at which the integrand of Eq. (4 4b) is evaluated are tabulated in
standard references (e.g., Davis and Polonsky [1964]) as dimensionless values, X .
Dimensioned node values, dp ^ , used for calculating /,, 7, and n, are calculated
according to the following relationship:
(4-5)
Gauss-Legendre integration is exact for integrands that can be expressed as
polynomials of degree < (2n . - 1) (Atkinson, 1985). As the degree of an
approximating polynomial increases above (2n , - 1), the error in evaluating the
integral increases. If it is possible to determine the degree of a polynomial that
suitably approximates the integral, the number of nodes may be chosen a priori.
Otherwise, Gauss-Legendre integration should be performed for an increasing
number of nodes until convergence to an acceptably stable and accurate value is
achieved.
68
-------
Determining the limits of integrationThe limits of integration, dp, and
dp , should be selected such that below dp. and above dp the integrand is
sufficiently small that it does not contribute significantly to the integral value. For
this work, sufficiently small implies a user-defined fractional magnitude of the
maximum integrand value. The values of dp where the integrand is sufficiently small
define the limits of the integral. For example, consider the following integral over a
lognormal distribution:
<$ n^lndp) d(lndp) (4-6)
where TL is the lognormal distribution function, defined as
2 T
Ti^lndp) = dJV/d(lndp) = ^ exp[-0.5ln W^gnM (4 7a)
\/2 IT In 0> In ov -"
o o
Defining a dimensionless distribution function, 77, as
77 = TL (Indp) ^TT lnag/N (4- 7b)
Eq. (4-7b) can be written as
"i (4-8a)
where XK = (4-8b)
De = Dgn exp(/c In Og) (4-8c)
At x 0 (i.e., dp = D& ), Eq. (4-8a) has a maximum value of 77 = 1. For example,
for x = ±4//£, 77 = 0.0003. Accepting this as sufficiently small, the limits of
/c
integration are calculated from Eq. (4-8b) as
69
-------
crg + lnDg/c (4 9a)
4prV«4 (4'9b)
dpu = DSKae* (4-9c)
In Eq. (4-6) we expressed the order of the integral explicitly as n. For
complex integrands, the order of the integrand must be approximated. If the
approximate order (represented by K) can be estimated, then Eqs. (4-8) and (4-9)
may be used to calculate dp, and dp . A more accurate technique, however, is to
determine the lowermost order, &, and the uppermost order, K , of the integrand
and calculate dp, based on K. and dp based on K :
dp}=Dg Jo* (4-lOa)
dpu=D&K a* (4-lOb)
u
where Dg and Dg are defined according to Eq. (4-8c). For example, the volume
i u
growth rate for vapor condensation is proportional to dp in the free-molecule regime,
and is proportional to dp in the continuum regime. Particle growth integrals
spanning the free-molecule and continuum regimes therefore contain integrands that
exhibit a different functional order with respect to dp in different regimes, due
partly to the change of growth-law order with size. In general, the order of the
integrand depends on the mathematical form of the rate coefficients for each
process, the size regime and/or flow regime under consideration, and the moment
solved for. Each integrand must be individually analyzed, therefore, to determine
suitable values for K. and « .
I
U
The procedure for performing Gauss-Legendre integration is summarized as
follows:
70
-------
1. Determine dp. and dp from Eqs. (4-8c) and (4-10).
2. Determine the number of nodes to use (flgi), either analytically or by successive
evaluations with an increasing number of nodes.
3. Calculate the dimensioned node values, dp according to Eq. (4-5).
4. Calculate fk at each node/pair of nodes (e.g., Eq. [4-31c])
5. Calculate the rate coefficients at each node/pair of nodes for single/double
integrals.
6. Calculate n at each node.
7. Evaluate the integral according to Eq. (4-4b).
4.5.1.2 Numerical Integration of Integrands Containing Lognormal Distribution
Functions: Gauss-Hermite Form
General tef-hTiigTie Gauss-Legendre integration is a technique for
evaluating integrals with arbitrary integrands. If n is restricted to be a lognormal
distribution function, however, the Gauss-Hermite numerical integration technique
provides a more compact and efficient method of numerically evaluating integrals.
The Gauss-Hermite technique applies to integrals evaluated from -o> to +OD:
(4- lla)
which can also be expressed in terms of a dimensionless size distribution as
(4- lib)
where XQ is defined by Eq. (4-8b). The numerical approximation of Eq. (4- lib) is
71
-------
written as a summation of the integrand evaluated at prescribed nodes and
multiplied by weight factors:
gh=l
Dimensioned node values, dp , , used for calculating fk and 7 are calculated
according to the following relationship:
(4- lid)
where Dg is given by Eq. (4-8c). Note that n(m^p) is not evaluated at the nodes,
but is implicitly accounted for by W,.
For the Gauss-Legendre integration technique, accuracy was controlled by
selecting dp. and dp such that the integrand was evaluated over a sufficiently wide
range. Because the Gauss-Hermite integration technique approximates an integral
with limits that extend from -m to +
-------
For example, the maximum of the number distribution function, n^, occurs at
The integral
J-I
is therefore centered about Dgn. The integral
/_"
however, represents an integral over the volume distribution. The maximum of the
volume distribution occurs at Z)§3 (see Eq. [4-8c]), implying the integrand is
symmetric about Dg .
To center the nodes about Dg~, the integrand of Eq. (4- lla) is transformed
as follows:
fa/A)
(Mjfe) = rT * ^
J-a>
jK
dp
fV
The product d$ n.(lndp) can be written as
(4-13)
exp
-0.5
lnVe
(4-14)
Substituting Eqs. (4-8b) and (4-14) into Eq. (4-13) yields
/x/,(dp)
-^
/-a) ^
(4-15)
The corresponding Gauss -Hermite numerical form of this integral is therefore
expressed as
73
-------
h g gh 16b)
The effect of centering the integrand can be seen from the following
example. Consider the integral
-*o) dzfl (4-17a)
which can also be written as
J-I
Table 4-1 shows the accuracy of evaluating Eq. (4-17b) with Gauss-Hennite
integration for k = 3 and 0 < K < 6. The exact solution of this integral is M^ so the
comparison parameter recorded in Table 4 1 is
J(D (V
(dp I dp) exp(a>*) djp« (4-18)
m
where the integral in the numerator of Eq. (4-18) is evaluated by Gauss-Hermite
integration.
74
-------
TABLE 4-1. ACCURACY OF EVALUATING EQ. (4-17b) FOR
* = 3 USING GAUSS-HERMITE INTEGRATION AND
THE INTEGRAND CENTERING TECHNIQUE
M
K
0
1
2
3
4
5
6
2
0.1794
0.5978
0.9529
1.0000
0.9529
0.5978
0.1794
Number of nodes
4
0.7233
0.9687
0.9998
1.0000
0.9998
0.9687
0.7233
6
0.9610
0.9991
1.0000
1.0000
1.0000
0.9991
0.9610
fv *)
For K = 3 the integrand reduces to exp(x,), which is accurately integrated
O
for any number of nodes. Without centering (i.e., for K = 0), substantial error is
introduced by using two or four nodes. Note also that the error is a function of the
difference between the order of the integrand, k, and the centering power, «, so that
the same error is incurred for K = 6 as for K = 0.
4.5.1.3 Evaluating a System of Integrals with Similar Integrands
In the preceding sections we described numerical techniques for evaluating
a single integrals. When MDEs for a particular process and mode are evaluated, a
system of integrals for multiple moments must be evaluated. The integrals for each
moment contain the same rate coefficient, and in the case of Gauss-Legendre
integration, the same distribution function. Evaluating the integrals for each
moment independent of the others involves repeating the entire integration
procedure for each integral. This results in excessive computations, because
evaluating each moment integral separate from the others does not exploit the
similarity between the integrands.
A more efficient technique for evaluating a system of integrals with similar
integrands is possible. Consider the following example: We are interested in
75
-------
modeling a process for which fk(dp) = dp. The time rate of change of the moments of
mode : are therefore
(4-19a)
a r^u
|(M3)«rT
Jd
(4'19b)
(4-19c)
For this set of integrals, «, (the lowermost order of the integrand) is equal to the
lowermost order of 7, because fk(dp) is equal to one in Eq. (4- 19a), and K^ (the
uppermost order) is equal to six plus the uppermost order of 7, because fk(dp) is
equal to dp in Eq. (4- 19c). The average order of this set of integrals is therefore
(4-20)
where K. = 0 + lowermost order of 7(rfp), and K = 6 + uppermost order of 7(). If
1 11
Gauss-Legendre integration is applied to this system of integrals, then «. and K are
used to determine appropriate limits of the integrals. If Gauss-Hermite integration
is used, then «is used to center the integrand.
For specified integration accuracy, the number of nodes required for
evaluating a system of integrals with similar integrands is larger than the number of
nodes required for evaluating one of the integrals from that same system. The
number of nodes required for evaluating a system of integrals increases as the
difference between «, and K increases.
I U
76
-------
Consider the following system of integrals:
/-I
*/^ /-I
(4'21a)
(4-2lb)
Table 4-2 records the accuracy of evaluating Eqs. (4-21) using Gauss-Hermite
integration and a common set of nodes for both integrals. The integrand for each
integral is centered using values of K ranging from 0 < K < 6. The tabulated accuracy
was calculated from Eq. (4- 18).
TABLE 4-2. ACCURACY OF EVALUATING EQS. (4-21)
USING SIX-POINT GAUSS-HERMITE INTEGRATION
AND THE INTEGRAND CENTERING TECHNIQUE
K
0
1
2
3
4
5
6
Eq. (5-21a)
0.9610
0.9991
1.0000
1.0000
1.0000
0.9991
0.9610
Eq. (5-21b)
0.0613
0.3053
0.7191
0.9610
0.9991
1.0000
1.0000
For the integrals of Eqs. (4-21a) and (4-21b) the proper centering power is
K 4.5. Because the computational algorithms for numerically evaluating integrals
are usually faster if terms with integer powers (e.g., dp) are evaluated instead of
terms with real powers (e.g., d£'31), «in the previous example should be set to
4 or 5.
An example of numerically integrating a system of equations for a realistic
process involves applying the system of integrals for particle growth
77
-------
(4-22a)
to A; = 3 and k = 6 (MQ does not change for particle growth). Neglecting the Kelvin
effect, a generalized growth law valid in all size regimes (see also Eqs. [4-52]
through [4-56]) can be written as
= dp C (4'22b)
where £ is an interpolation function that is equal to 1 in the continuum regime and
changes form in the transition regime to £ « dp in the free-molecule regime.
Evaluating Eqs. (4-22) for k = 3 and k = 6 yields ^ = 1 and «u = 5. The proper
centering power for these two integrals is therefore K = 3. Equations (4-22) were
evaluated using a six-point Gauss-Hermite integration technique, and the accuracy
of the integration evaluated by comparing the six-point integration to a 20-point
Gauss-Hermite evaluation of the same integrals.2 The distribution parameters Z)gn
and 0g were chosen such that the distribution was centered about the transition
regime, where the factor £ exhibits a change of function dependence from zeroth to
first order with respect to dp. Table 4-3 shows the accuracy for various centering
powers.
TABLE 4-3. ACCURACY OF EVALUATING EQ. (4-22)
USING SIX-POINT GAUSS-HERMITE INTEGRATION AND
THE INTEGRAND CENTERING TECHNIQUE FOR Jk = 3 AND k = 6
K
0
1
2
3
4
jfc = 3
1.0000
1.0001
1.0002
1.0000
0.9999
jfc = 6
0.7181
0.9602
0.9990
1.0000
0.9999
2 A 20point GaussHermite integration technique was used to obtain a standard of comparison
for the sixpoint GaussHermite integration, because it is not possible to evaluate Eqs. (5-22)
analytically.
78
-------
4.5.2 Table Interpolation/Lookup
When high accuracy and speed are both required, an alternative technique
is to evaluate the integrals numerically for a discrete set of values for each integrand
parameter (e.g., Dgn and ae for lognonnal functions), and store the results in a table
that is indexed by the specific parameter values used in generating it. Integral
values may be recovered from the table by interpolating between the table's nodes.
The dimension of the table is equal to the number of independent parameters for
which the integrals are evaluated. To generate the table, a range of values for each
parameter is selected, the number of nodes within each respective range at which to
evaluate the integral specified, and the integral evaluated at these nodes.
In general, the accuracy of recovering values with a table interpolation
algorithm depends on the complexity of the multidimensional surface represented by
the integral, the node density of the table, and the sophistication of the
interpolation scheme. Table interpolation is intended for recovering tabular values
when values vary significantly between nodes. For situations where changes in
tabular values between nodes are small, table lookup provides a faster technique for
recovering values than table interpolation. The cost of implementing a table
interpolation/lookup procedure is the cost of storing the table of coefficients in
computer memory and also of recovering tabular values. When sufficient storage is
available to contain tables with a node density fine enough so that values vary only
slightly between nodes, table lookup is substantially faster than and often as
accurate as table interpolation techniques.
4.5.3 Analytical Integration
Because the integrands of Eqs. (4-2) consist of three independent terms
(/,[dp], 7[dp], and n[^p])> securing analytical expressions for the integrals of the
MDEs usually requires simplifying assumptions and/or mathematical
approximations. A common mathematical approximation is that n can be
represented by a lognormal distribution function. With this approximation,
analytical evaluation of Eq. (4-2a) is possible if the product of the remaining terms,
/,(dp)j(dp), can be expressed as a power function or a polynomial of dp. Integrating
79
-------
a power function or a single term of a polynomial over a lognormal function yields
terms like
*d$ nL(dp)ddp = ND%* exp|4n2<7g (4-23)
which can also be represented in terms of a moment of the distribution, as
MK (4-24)
If the rate coefficient, 7, must be approximated by a power function or a
polynomial of dp (represented as 7) to secure an analytical integral, Eq. (4-2a) is
expressed as
faMj « rT/o-/4to)7(iW njdp) ddp (4-25)
where 7 represents an approximation of 7. Equation (4-25) can be converted to an
equality by defining a factor M \ such that
= &[% f "V^TK) ^(^ddp (4-26a)
(4-26b)
The superscript (l) indicates this is a correction factor for first-order
(wrt Mj processes. If a sufficiently accurate approximation function, 7, is obtained
(such as a series expansion of 7[rfp]), & is often set to one (see, for example, the
discussion of series expansion of the integrands given by Eltgroth [1982]). If 6Jp
differs from one enough to cause unacceptable errors, it must be approximated by a
function or obtained by a table lookup technique. For example, the numerator of
Eq. (4-26b) can be evaluated numerically for selected values of the integrand
80
-------
parameters and the resulting values of fr, ' stored in a table (see Appendices H
and I).
The approximation/correction scheme of Eqs. (4-26) can also be used for
the coagulation integrals of the MDE. The coagulation integrals are double integrals
in the MDEs. The algorithms for numerically evaluating double integrals are
provided in Appendix G, and some analytical double integrals for coagulation are
derived in Appendix H. For intramodal coagulation (a second-order process, wrt
M, ) the correction factors are
) = rT6i2) f " f "/^
» i JO JO
TOD fat
/i^r
6(2) = JSLJ* - ! - i - (4 . 27b)
*. /*OD fa> ^ '
4(^1.
/O JO
For intermodal coagulation (a first-order process, wrt M, ) the correction factors are
i
UM^ = rT6ii} r r "/4(^r
t i JO JO
fa> fa>
i^i'
.A.
= ..
' I " f "/rf^r
JO JO
(4 - 27d)
Algorithms for calculating the 6 correction factors for coagulation are given in
Appendices H and I.
81
-------
4.6 INTERNAL PROCESSES
4.6.1 Brownian Coagulation
The coagulation integrals of the MDEs for binary collisions were derived in
Chapter 3 (see Eqs. [3-10] and [3-11]) and are expressed in terms of the interactions
between two arbitrary modes, : and j. For mode i the coagulation integrals are
coag
Intramodal Coagulation
Intermodal Coagulation
The first two integrals on the rhs of Eq. (4 28) represent changes due to intramodal
(i-t) coagulation, and the last integral represents the loss from mode : to mode jot
particles of size dp 3 due to intermodal (i-j) coagulation. The mode j coagulation
integrals are
3Because the tails of the distributions overlap. dD is not always smaller than <£r
M
82
-------
coag
CD ao , */3
o (^1+^)
Intramodal Coagulation
Intermodal Coagulation
The first two integrals on the rhs of Eq. (4-29) represent changes due to intramodal
(j-j) coagulation, the third integral represents gain to mode j due to intermodal (i-j)
coagulation, and the last integral represents loss from mode j of particles of size dp2
due to intermodal (i-j) coagulation.
To evaluate the integrals in Eqs. (4-28) and (4-29), the term
(dp.+dp-)^ ' ' must be expanded. This is possible only if fcis chosen as an integral
power of three. The moments MQ, My Mg, ... are therefore solved for when
analytical integration techniques are used to evaluate the coagulation integrals. The
moments M ., M,, and MR all have physical significance for aerosol systems (see, for
03 o
example, Friedlander [1977]), so it is usually relevant to have information on these
moments in any case. Extension of the following derivations to other moments that
are multiples of three is straightforward.
Equations (4-28) and (4-29) are expanded, and the intramodal and
intermodal terms are separated because the form of the intramodal terms is the
same for modes i and j. For moments MQ, My and Mfi Eqs. (4-28) and (4-29)
become
83
-------
Intramodal Coagulation: Modes * and j
(4-30a)
(4-30b)
(4-30c)
£,' » ' ' ±" 1 ~ ~ A "A ~ *
Intermodal Coagulation: Mode t
Intermodal Coagulation: Mode j
(4-32a)
) (4-32b)
(4-32c)
In Sections 5.6.1.1 through 5.6.1.3, analytical expressions for the integrals
in Eqs. (4-30) through (4-32) are presented, for /3 represented by functions that
apply separately to the free-molecule and continuum/near-continuum regimes. A
general expression valid in all regimes is obtained by averaging the respective
expressions for the free-molecule and continuum/near-continuum regimes.
84
-------
4.6.1.1 Biownian Coagulation in the Free-Molecule Regime (Kn > 10)
For particles much smaller than the mean free path of the surrounding gas,
Friedlander (1977) gives the following expression for the coagulation coefficient:
(4-33)
Equations (4-30) through (4-32) cannot be integrated using /? given by Eq. (4-33)
because it is not possible to separate the resulting size-dependent terms. If the
substitution
i/45)
(4-34)
is made (Lee et al, 1984), Eq. (4-33) can be written as
i 0.5
.5
1
1
.5
1
i
.O jl.O
.5
2
.5
(4-35)
Substituting Eq. (4-35) into Eqs. (4-30) through (4-32) yields approximate
expressions for the time rate of change of M, due to coagulation. The expressions
can be made exact by introducing correction factors of the form given by Eqs.
(4.27). The b factors are calculated for a specific moment and mode. They are
therefore subscripted with k and i or j. For example, Eq. (4-31c) is made exact by
defining
_JO JO
i fat /*
-------
The analytical forms of Eqs. (4-30) through (4-32) using $from Eq. (4-35)
are given in Appendix H, along with recommendations for calculating the b
correction factors.
4.6.1.2 Brownian Coagulation in the Continuum/Near Continuum Regime
-------
Whereas the approximation to /?, required b correction factors in order to
yield accurate results, the approximation to the continuum/near -continuum
coagulation coefficient using the simplified form of Cc does not warrant correction.
The approximation is sufficiently accurate that little if any improvement in
accuracy can be realized by introducing correction factors.
4.6.1.3 Generalized Brownian Coagulation Coefficient (all Kn)
Fuchs (1964) and Dahneke (1983) provide expressions for the coagulation
coefficient that are valid in all size regimes. However, these forms are too complex
to permit analytical evaluation of the resulting coagulation integrals. An alternative
approach used by Pratsinis (1988) for calculating intramodal coagulation coefficients
involves expressing the generalized coagulation integral as an average of the
free-molecule and continuum/near-continuum regime expressions. The averaging
technique Pratsinis used for intramodal coagulation integrals can also be applied to
the intermodal coagulation integrals to yield
, ,
fm I0tv Jb'lnc
(4'39)
where M-^jL.. = free-molecule expressions for Eqs. (4-30) through (4-32)
T5r(Af,) = continuum/near-continuum expressions for Eqs. (4-30)
through (4- 32)
Equation (4-39) reduces to the correct limiting behavior in all size regimes from
free-molecule through continuum.
This technique predicts coagulation rates that are globally within 20% of
those determined by substituting into Eqs. (4-30) through (4-32) the generalized
expression for coagulation coefficients given by Fuchs (1964) and numerically
evaluating the resulting coagulation integrals. If the additional constraints
(0g. > 1.5, C7g. > 1.5) are imposed, the analytical coefficients are within 10% of the
87
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coagulation rates numerically determined with Fuchs' coagulation coefficients.
Because many systems of interest (e.g., the atmosphere) seldom display
distributions narrower than ug = 1.5, coagulation rates evaluated from Eq. (4-39)
will usually be within 10% of values evaluated numerically.
4.6.2 Particle Growth
Particle growth occurs by addition of material to existing particles. This
may occur by condensation of supersaturated vapor-phase material, or chemical
conversion of volatile material that is adsorbed/absorbed from the gas or vapor
phases.
For particle growth, the time rate of change of the fcth moment of a
distribution is (Eq. [3-17])
(4-40)
where *T is the non-size-dependent component of ^ (the particle growth law) and
iftdp) is the size-dependent component of ^p. In the following sections, techniques
for evaluating Eq. (4-40) for specific forms of ^ are described.
4.6.2.1 Condensation in the Free-Molecule Regime (Kn > 10)
In the free-molecule regime, particle growth due to condensation occurs by
bombardment of condensable molecules on existing particles. The expression for the
molecular flux is derived from kinetic theory (Friedlander, 1977):
P
(4-41a)
condensation evaporation
where Pp is the vapor pressure of the condensable material at the particle surface,
-------
Tp is the gas temperature at the particle surface, P is the vapor pressure of the
condensable material far from the particle surface, and T is the gas temperature far
from the particle surface. The difference in temperature between Tp and T may
occur because of energy changes due to the latent heat of vaporization of a
condensing/evaporating material, or due to radiative effects. Because Tp results
from heat transfer effects that are sensitive to particle size, the evaporation term is
size sensitive, and therefore introduces additional complexities. For this work we
consider cases for which the particle is close to thermal equilibrium with the
surrounding environment, so that Tp a T . This will be true when heat transfer
rates within the particles and between the particles and surrounding fluid are fast
compared to radiative heat transfer rates and heat transfer rates resulting from the
latent heat of the material deposited on/removed from the particles. Because heat
transfer with the surrounding fluid is proportional to the surface area of the
particles and the heat storage in a particle is proportional to its volume, the
particles will tend to be closer to thermal equilibrium with the surrounding fluid as
their surface area to volume ratio increases (i.e., as particle size decreases).
For particles in thermal equilibrium with the surrounding fluid Eq. (4 41a)
becomes
(4-41b)
m
Including the Kelvin effect (Friedlander, 1977), Pp is calculated as
Pp = Ps SVPP (4-42)
where < = 4avm/(*B Ta>
P = saturation vapor pressure; pressure over a flat surface having the
same composition as the particle
Sv = P IP = the vapor saturation ratio
OD' s
*
= the Kelvin term, which represents increased vapor pressure due
to curvature of the particle surface
89
-------
Substituting Eq. (4-42) in Eq. (4-41b) and multiplying by the volume of the
condensing molecules, the particle growth law, ^p, is
(4'43)
m B
If the Kelvin effect is important and 5V is much larger than one, Eq. (4-40)
must be integrated numerically when Eq. (4-43) is used to represent ipp. If 5V is
close to one, the Kelvin term in Eq. (4-42) can be expanded in a Taylor series and
truncated after the first-order term (Friedlander, 1977). The resulting expression is
Pp = Ps [1 + (Sv - l)/<*p] (for Sv * 1) (4-44)
Substituting Eq. (4-44) for Eq. (4-42) causes Eq. (4-43) to become
(27T771
v
(4-45)
^
m B OD'
Substituting Eq. (4-45) into Eq. (4-40) and evaluating the integral analytically
yields
(4 ' 46)
1CV
where * =
m B
Neglecting the Kelvin effect (i.e., dp = 0), Eq. (4-46) simplifies to
90
-------
4.6.2.2 Condensation in the Continuum Regime (Kn < 0.1)
In the continuum regime particle growth due to condensation is a diffusion-
limited process, and the particle growth law is
B
p, p>
T T,
CD
dp (4-48a)
where D is the diffusivity of the condensing species in the carrier gas, and the
temperature difference between the particle and surrounding gas is due to latent
heat effects and/or radiative heat transfer. For this work we consider cases for
which the particle is close to thermal equilibrium with the surrounding
environment. Equation (4-48a) therefore reduces to
B OD
(4-48b)
If the Kelvin effect is important, Eq. (4-42) is substituted into Eq. (4-48b)
to yield the general form of the continuum regime growth law:
(4.49)
B CD
If Sv is much larger than one, Eq. (4-40) must be integrated numerically when
Eq. (4-49) is used to represent ifo. For 5V close to one, Eq. (4-44) is substituted
into Eq. (4-48b) and the resulting expression is substituted into Eq. (4-40).
Evaluating the resulting integral analytically yields
(4-50)
where *T
Ten
Neglecting the Kelvin effect (i.e., dp = 0), Eq. (4-50) becomes
91
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4.6.2.3 Generalized Condensation Expression (all Kn)
For the case of thermal equilibrium between the aerosol and surrounding
fluid ( T = Tp), a generalized growth law can be written as
(4-52)
B CD
where (is an interpolation function for the continuum regime growth law. Eq.
(4-52) approximates the behavior of ifa in the transition and near-continuum
regimes, and converges to the correct limiting forms in the free-molecule and
continuum regimes. Two common forms for £ have been proposed by Fuchs and
Sutugin (1971) and Dahneke (1983). The Fuchs-Sutugin form was developed for
condensing molecules that are much lighter than the background gas. For unity
sticking probability this form is
1 + Kn _
CFS = ~ (for z « 1) (4-53)
FS 1 + 1.71Knps+ 4/3 -*
2AFS
where Knps = -3-
AFS = m
z m /TO
m' g
Dahneke developed a form for condensing molecules whose mass is of the
same order of magnitude as the carrier gas. For unit sticking probability, his form is
92
-------
1 +Knn
D = ^ (forzsl) (4-54)
D 1 + 2Kn 1 + Kn
2AD
where KnD =
D mg' m
In this work we show the following derivations using Dahneke's form for £,
because in many systems of interest (e.g., the atmosphere), zis approximately equal
to one. The mathematical forms of the resulting expressions, however, are not
sensitive to this choice. Substituting Eqs. (4-52) and (4-54) into Eq. (4-40) yields
(4-55)
B B
If the Kelvin effect is important and 5V is close to one, Eq. (4-44) is substituted into
Eq. (4-48b) and the resulting expression is substituted into Eq. (4-40) to yield
(for 5V « 1) (4-56a)
Neglecting the Kelvin effect (dp 0), Eq. (4-56a) simplifies to
Because of the complex form of £, Eqs. (4-55) through (4-56) cannot be
evaluated analytically. An approximation for £ could be used, but evaluating
Eq. (4-56a) numerically with six-point Gauss-Hermite integration (see, for example,
Table 4-3) is computationally efficient and accurate enough so that further
approximations to secure analytical integrals are usually not warranted.
93
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4.6.2.4 Surface and Volume Reactions
Conversion of volatile material to stable substances may occur by reactions
on surfaces of solid or liquid particles and/or within droplets. In both cases, the
particle growth rate is controlled by the chemical conversion rate of the volatile
material on and/or inside particles.
Surface reactions occur on the outer surface of the particle, so conversion
rates are proportional to dp. The growth law is therefore written as
g (4-57a)
where * is proportional to the chemical reaction rate for a particular surface
Ls
reaction. Substituting Eq. (4-57a) into Eq. (4-40) and evaluating the resulting
integral analytically yields
Volume reactions are treated the same as surface reactions, except the
growth law is proportional to the particle volume, as
(4-58a)
Substituting Eq. (4-58a) into Eq. (4-40) and evaluating the resulting integral
analytically yields
= (*/3) * M (4-58b)
94
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4.6.3 Internal Sources
Internal sources add new particles to the distribution, so the moments of
the source distribution are added to the moments of the receiving distribution.
Equations of the form
>>],
O'My
-arc
must therefore be developed for each active source. To determine which of the
existing modes should receive the source particles, the characteristic size of the
source particles should be compared with the characteristic size of each of the
existing aerosol modes (e.g., Dgn). If the characteristic size of the source aerosol is
sufficiently different (a user-prescribed criterion) than the characteristic size of the
existing aerosol modes, a new mode must be created. Usually, however, the modes
to receive source aerosol are predetermined (Whitby, 1979; Eltgroth, 1982; Giorgi,
1986; Pratsinis, 1988; Megaridis and Dobbins, 1990). Thus, resuspended dust (i.e.,
dp > 1 /on) would be included in a relatively large mode (e.g., the coarse mode),
whereas particles resulting from vapor nucleation (i.e., dp s 0.01 /mi) would be
introduced into a relatively small mode' (often referred to as the nuclei mode
[Whitby, 1978a]).
A common aerosol source results from vapor nucleation. Many theories for
nucleation rates exist (see, for example, Friedlander [1977,1983], McMurry and
Friedlander [1979], and McMurry [1983]), and these are not discussed here. The
salient feature of nucleation theories, however, is that clusters must attain a
minimum critical size before they are assumed stable enough to survive. Nucleation
rates are often expressed, therefore, as the production rate of clusters at this critical
size. Once clusters attain a stable size, they are introduced into the particle size
distribution as particles. Because nucleation produces very small particles, in MAD
models particles produced by nucleation are usually introduced as a source to the
smallest mode (e.g., a nuclei mode).
95
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Pratsinis (1988) used this approach to formulate nucleation source rates.
His generalized moment source rate is
='(4) (4)* (4-59)
nuc
where /represents the formation rate of stable clusters, and dp is the size of the
smallest stable cluster (see Eq. [4-42]).
4.7 EXTERNAL PROCESSES
4.7.1 Convection
When particles move under the influence of convection, we assume the
particles follow the stream lines perfectly. Processes causing motion normal to
stream lines are represented by external forces. Convective moment transport is
therefore represented by the convective terms of the MDEs:
t J cnv
4.7.2 Diffusion
The particle diffusion flux can be written as
Jn=Dvlfc (4'6°)
An analogous expression can be written for the flux of Af,:
dM,
T r\ K / i
where D, is a moment average particle diffusion coefficient, defined by
96
-------
(4-62)
A form of the particle diffusion coefficient commonly used for particle transport is
(4.63)
Substituting Eq. (4-63) into Eq. (4-62), the moment average particle diffusion
coefficient is
1 k T /* C
Substituting Eqs. (4 -3d) and (4-3e) for C and evaluating Eq. (4-64) yields
1 k T
Dk = WW(Mk-l + 2XAt"Mk.2) (for
Dk = -(M- + ^^ncM) (for Kn < 1) (4-65b)
4.7.3 External Forces
For particles moving under the influence of an external force, the drift
velocity can be represented as
c=FB (4-66)
where B is the particle mobility and F is a force acting on the particle. The moment
average drift velocity is
97
-------
= ~k
where FT is the non-size-dependent component of the force and F(dp) is the
size-dependent component. The mobility is defined as
80
B =
(4-68)
where CL is the drag coefficient on a sphere and (vp v ) is the particle velocity
relative to the gas velocity. For particles in the Stokes regime (Rep < 0.1) the drag
force is purely viscous, and the drag coefficient is therefore
= 24/Rep
(4-69a)
where Rep = pg dp \ Vp v
For higher values of Rep, Eq. (4-69a) is generalized as
= (24/Rep)C
Re
(4-69b)
The moment average drift velocity is therefore calculated as
(4'70)
Seinfeld (1986) reports the following expressions for
Cp =
sRe
1 + (3/16)Rep + (9/160)ln(2Rep)
1 + 0.15ReS'687
Rep/54.55
Rep < 0.1
0.1 < Rep < 2
2 < Rep < 500
500 < Rep<2xl05 (4-71a)
98
-------
We have developed an alternative form for £_. for ReD < 2 x 105:
*
CRe = [1 + exp(-0.0017Rep) VKep"/5-277 + Rep/54.55] (4-71b)
The complex form of £p represented by either Eq. (4- 71a) or (4- 71b)
ivCp
requires that subsequent integrals be evaluated numerically when 0.1 < Rep < 500.
For numerical integrals, Eq. (4-71b) is preferred because of the relative simplicity of
evaluating integrals with a single continuous function compared to multiple discrete
functions. In addition, the computational cost of evaluating expressions using Eq.
(4- 71b) to represent £Re is only slightly higher* than for Eq. (4- 71a).
4.7.3.1 External Fields
Gravitational and inertia! The gravitational force on a particle is
(4-72)
Substituting Eq. (4-72) into Eq. (4-70) yields
For particles in the Stokes flow regime, Eq. (4-73) can be evaluated analytically as
(for Kn > 1) (4- 74a)
c* = W Sp
-------
In ultra-Stokesian (0.1 < Rep) flow regimes, Eq. (4-73) must be evaluated
numerically.
Electrical The electrical force on a particle carrying n elementary
charges is
F£ = neeE (4-75)
In general, particles of the same size carry different numbers of charges. The
fraction of particles of a particular size carrying n units of charge is represented as
/n. Discrete and continuous forms of/n exist, but we confine our attention to
continuous forms of /n because discrete forms, by nature, are not appropriate for
MAD models, which require that distributions be represented with continuous
functional forms5. The moment average drift velocity for electrical forces is
c =
dn (4-76a)
e
For particles in Boltzmann charge equilibrium, the continuous form of /n is (Hinds,
1982)
(4-76b)
where -a> < n < CD. The possibilities for analytical evaluation of Eqs. (4-76) exist
only for very restrictive conditions (e.g., when /n and £R are constant), so
numerical integration is usually required. For a uniformly charged aerosol in the
Stokes flow regime, Eqs. (4-76) can be evaluated analytically as
(4-77.)
n, K-l + 2A^n=Mt-2)
-------
4.7.3.2 Phoresis Effects
When particles are moving in gradients of other gaseous species and/or
gradients of temperature, molecular forces due to these gradients may cause particle
migration. Thermophoresis, for instance, may exist in the presence of temperature
gradients. An example for thermophoresis is provided, but other phoresis effects,
such as diffusiophoresis and photophoresis, are not treated here. Interested readers
are referred to Brock (1962), Fuchs (1964), Preining (1966), Waldmann and Schmitt
(1966), ffidy and Brock (1970), and Derjyagin and Yalamov (1972).
For particles in the transition and free-molecule regimes, the thermo-
phoretic force is given by Waldmann and Schmitt (1966) as
Substituting Eq. (4-78) into Eq. (4-70) yields
Particles in the free-molecule and transition regimes are usually governed by Stokes
flow, so Eq. (4-79) can be evaluated analytically, as
-------
3Kn
1 f kjkn + 2.2Kn ] (4 . 81b)
J [l + 2Jka/fcp + 4 . 4KnJ ^
and ka. and Jfcp are the thermal conductivities of the air and the particle, respectively.
Substituting Eqs. (4-81) into Eq. (4-70) yields
In the continuum regime the Knudsen number goes to zero, and Eq. (4-82) becomes
1) (4-83)
In the Stokes flow regime Eq. (4-83) can be evaluated analytically as
ck = -jff-*T^TMk (for Kn « 1 and Rep < 0.1) (4-84)
In the near-continuum and/or ultra-Stokesian flow regimes, Eq. (4-82) must be
evaluated numerically, or further approximations to the integrand made to yield
analytical integrals.
4.8 DISCUSSION
We have presented techniques for evaluating the integrals of the MDE
moment equations, for specific forms of the rate coefficients for each process. The
choice of integration technique depends on the accuracy required. When selecting a
technique, the modeler should remember that moment rate expressions are
comprised of the non-size-dependent (F ) component, and the size-dependent
integral component. Demanding a 1% error level from the integration technique
(which represents the size-dependent component) while sustaining 20% uncertainty
in F is clearly not warranted.
102
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The same argument applies when selecting how many nodes to use with
numerical integration techniques. Enough nodes should be selected to ensure
convergence of the integral to a stable and accurate value. But "stable" and
"accurate" are loosely defined, and the modeler should not demand convergence at a
confidence level not justified by the accuracy of the model's other parameters.
For coagulation, the analytical evaluation of Eqs. (4-30) through (4-32) is
globally within 20% of accurate numerical evaluations of the same integrals, and
within 10% for crg. and ag. greater than 1.5. It is unlikely, however, that coagulation
rates between discrete particles in real systems are represented this accurately by
the forms of the coagulation rate coefficients assumed in Eqs. (4-33) and (4-37).
Coagulation rates may be substantially affected by other effects, such as image
forces (see, for example, Marlow [1980]) or the shape of the particle (e.g., long chain
agglomerates may coagulate faster than dense spheres). These effects may result in
coagulation rates differing from the rates reported here by more than 20%.
The table interpolation technique presented allows rapid evaluation of
complex integrals, but its use is rarely justified because of the difficulty associated
with setting up and interpolating within such a table. The table lookup techniques
are easy to implement, and are accurate when the tabulated values vary slowly
between the nodes of the table. The table lookup technique was used to correct the
analytical coagulation integral (see Appendix H), because the range of b correction
factors is small (i.e., 0.7071 < b < 1) so that b can be determined by using a table
lookup technique in a relatively sparse table (e.g., 1000 nodes).
One application of table interpolation was the model developed by Whitby
and Whitby (1985a, 1985b), which included multimodal coagulation integrals
evaluated with a table interpolation technique. The resulting algorithms were fast
and allowed development of a program that was two orders of magnitude faster than
a sectional technique of comparable accuracy (Seigneur et al., 1986). The table was
developed for aerosol distributions typical of the atmosphere, but was limited to
constant temperature and pressure. The table was four-dimensional (Z?gn-, crg., Dgn,
0v ) and contained 1280 nodes. Subsequent attempts to generalize the table to apply
3
for all size regimes and variable temperature and pressure led to a five-dimensional
103
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table with more than 10,000 nodes. Even with this number of nodes the accuracy of
recovering values was less than the analytical coagulation integrals presented in this
work. To properly implement table interpolation and/or table lookup techniques,
therefore, the range of parameters for which the table is required should be well
defined, or large quantities of storage should be available if very general tables are
to be produced.
104
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CHAPTER 5
TIME INTEGRATION OF THE MOMENT DYNAMICS EQUATIONS
5.1 THE TASK
In Chapter 3 we developed expressions for the time rate of change of
integral moments of an aerosol size distribution, M, , for processes that affect
i
aerosol behavior. By combining these expressions, differential equations for the time
rate of change of M, for each moment and mode can be written as
(5-1)
h
where h represents specific processes,- such as coagulation and particle growth.
The moment equations represented by Eq. (5-1) express the time rate of
change of the Jfeth moment of the t*h mode, and account for the combined effects of
all processes acting on the aerosol size distribution. To simulate an evolving aerosol
size distribution with MAD models, a system of equations (represented by Eqs.
[2-5]) of the form of Eq. (5-1) must be integrated simultaneously. The number of
equations is equal to the number of independent parameters used to characterize the
aerosol size distribution function.
For CSTARs, Eq. (5-1) represents all processes acting on the aerosol. By
integrating Eq. (5-1), the temporal evolution of the aerosol size distribution can be
simulated.
For FARs, Eq. (5-1) represents the internal processes, such as coagulation
105
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and particle growth. The external processes can be accounted for by using a
computational flow model to solve for the transport of Mk . For FARs, the modeled
«
system is at steady state, so Eq. (5-1) is used as is to represent a time-invariant
source term that is coupled with the transport equations of the computational flow
model. For unsteady FARs (such as the atmospheric aerosol model described in
Chapter 6), the modeled system is time variant, and Eq. (5-1) must be integrated
to estimate an incremental change of the moments for each discrete time interval
simulated by the model. For CSTARs and unsteady FARs, therefore, equations of
the form of Eq. (5-1) must be integrated.
In this chapter we prescribe methods for integrating the set of moment
equations used in MAD algorithms, based on analytical integration of a linearized
form of Eq. (5-1). Comparisons with standard integration techniques are provided
to assess the performance of the analytical integration techniques described here.
5.2 LINEARIZED MOMENT EQUATIONS
To simulate aerosol dynamics in CSTARs and unsteady FARs, Eq. (5-1)
must be integrated. Standard methods, such as the Crank-Nicholson and
Runge-Kutta techniques, can be used to integrate the MDEs. In addition, however,
a family of analytical integration techniques can be developed by converting Eq.
(5-1) to a linear equation of the form
dMJdt = Sk + CkMk (5-2)
t « * t
We demonstrate here that the accuracy of analytical techniques per integration time
step always exceeds or matches the accuracy of the Crank-Nicholson integration
technique, and surpasses fourth-order Runge-Kutta techniques for applications
where only modest accuracy is required (e.g., global errors on the order of 1%). If
global errors less than 1% are required, it is difficult to surpass the accuracy and
simplicity of the fourth-order Runge-Kutta techniques.
We therefore develop the analytical integration technique as a
computationally efficient integration technique when global errors on the order of
106
-------
1% are acceptable.
The key to developing an accurate, computationally efficient, and
mass-conserving solution algorithm for the MAD system of differential equations
(the MDEs) is to properly formulate the coefficients of Eq. (5-2) for each mode.
Three techniques of formulating these coefficients are presented in this chapter; each
technique corresponds to a different analytical integration technique. The most
general integration technique is called the power-law analytical integration
technique. Two subsets of this technique, the first- and second-order analytical
integration techniques, are also described.
In this chapter the nomenclature rapidly becomes encumbered with
subscripts and superscripts that are required to identify the integration time
intervals. The nomenclature for the aerosol model, however, adds nothing to our
understanding of the integration techniques. Unless otherwise required, therefore,
the following simplifying substitutions are used in the derivations:
y E Mk (5-3a)
.5 = Sk (5-3b)
C = Ck (5-3c)
The mode subscripts, : and j, are also dropped unless explicitly required.
5.3 ANALYTICAL INTEGRATION TECHNIQUES
With the simplified nomenclature, the general form of the linearized
moment equation becomes
dy/dt = S + Cy (5-4a)
The integral form is
For constant coefficients, Eq. (5-4b) is readily integrated to yield
107
-------
y = i/o + S At for C = 0 (5-5a)
V = (2/o + 5/C)[exp(C? At)] - 5/C for C # 0 (5-5b)
where 5 and C are constant for the time interval At. Because 5 and C usually vary
with time, the entire simulation time interval, At, is subdivided into an integral
number of time intervals, (At),, which are referred to as time steps. In terms of
(At)_, the total simulation time is
At =
1=1
where nt is the total number of time steps required for a simulation. For each time
step, the coefficients of Eqs. (5-4) are calculated, and Aj/is estimated with
Eqs. (5-5).
Additional nomenclature for the integration time steps is required to
formulate the integration and error control algorithms, and is depicted in
Figure 5-1.
(At), (At)T1+1
-l)o o (l+l)o
i-time step (i-i)j time step i time step ("n+l)
Figure 5-1. Nomenclature for the integration time intervals.
The value of y at the beginning of each time interval is y , and y is an
intermediate value, used in higher-order integration techniques, that approximates y
at the end of time step 77 (e.g., see Eqs. [5-23]). The final value of y from time step 77
is y.^ v , and is an updated value of the intermediate value, y^ . We use the same
system for indexing t, 5, and C.
108
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In terms of this nomenclature, Eqs. (5-5a) and (5-5b) can be written as
for
where 5^ and C^ are constant coefficients for time step 77, and are defined such that
they approximate the values for the time-dependent coefficients 5 and C during the
time interval
The technique of integrating Eq. (5 4b) analytically with constant
coefficients and using Eqs. (5-5) to estimate (Aj/)^ is referred to as the analytical
integration technique. The three analytical integration techniques presented in this
chapter differ in their representation of the coefficients S^ and C^. A formal
derivation of the coefficients for each integration technique is now developed. First,
the power -law analytical integration technique is developed as a general method for
integrating Eq. (5-4b). From this technique two additional integration techniques
are developed: the first- and second-order analytical integration techniques.
5.3.1 Power-Law Analytical Integration Technique
5.3.1.1 General Derivatioi
To develop an analytical integration technique, dy/dt must be represented
by a time-dependent function that can be integrated analytically. Rather than
expressing dy/dt explicitly as a function of t, however, we correlate dy/dt with y by
a power function:
dy/dt = By*' (5-6)
By determining B and a' on the fly, the algorithms developed for the power-law
analytical integration technique automatically adjusts Eq. (5-6) to the approximate
functional dependence of the moment equation for each specific set of processes
acting on the aerosol.
109
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If dy/dt and y are known at two arbitrary points in time, t\ and h, then B
and a' can be calculated as follows:
yi) (5'7a)
(5-7b)
In a forward-time-stepping algorithm, we have available at time ^ all values for
time step (T?-I) and point 770- We can estimate a' for time step 77 in terms of the
values from time step (77 1) as
so that B can be calculated from the values at 770 as
(5-7d)
For processes without sources from other phases and/or modes, the
preceding formulation is very accurate. For situations where mass is transferred
from another aerosol mode and/or phase, the mass transfer is often represented as a
constant-rate source term whose rate is specified for each integration time step. The
precise incremental moment transferred from another phase and/or mode will be
properly accounted for by the receiving mode only if the constant -rate source term
is represented as a constant by the governing moment equation. Equation (5-6),
however, represents the combined effects of all processes as a power function.
Lumping const ant -rate source terms with the power-law approximation results in a
moment equation that does not conserve mass. Similarly, for processes with any
known dependence (such as particle growth due to internal reactions, which is a
first-order process), the calculation procedure used to determine the power-function
coefficient for Eq. (5 6) causes this known dependence to be averaged with the
unknown dependence of other processes. This weakness in the preceding derivation
is illustrated with an example in Appendix J.
To modify the simple power -law representation to account correctly for
processes of known order, we write the moment equation as
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dy/dt = c(0) + c(1)j/ + Bya (5-8)
where r°^ represents zeroth-order processes whose rate is independent of the aerosol
properties (e.g., constant-rate source terms), and c ' represents first-order processes
whose rate coefficient is independent of the aerosol properties (e.g., fluid velocity
and particle growth resulting from chemical reactions within droplets). The last
term on the rhs of Eq. (5 8) represents all processes of unknown order. Although the
formulation of Eq. (5 8) correctly represents the functional form of zeroth- and
first-order processes, it can be integrated analytically only for limited values of a,
and is therefore not useful for developing a generalized solution algorithm. One way
to overcome this problem is to linearize the power-law term. Expanding the
power-law term in a Taylor series and truncating after the second term yields
« (1 - aB + V (5-9)
where B^ and a^ are values of B and a that apply for time step 77. From Eq. (5-8),
B can be approximated for time step 77 as
(5-10)
0)
where (dy/dt)^ - (dy/dt\ - (c, + c), so that Eq. (5-9) becomes
V" * 40) +
x
where
Equation (5-8) is now written as
111
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To complete the derivation of the power -law analytical integration
technique, an estimate for QL is required. The procedure for estimating o^ is
straightforward, but not obvious. If the rhs of Eq. (5-6) is expanded in a Taylor
series and truncated after the second term, a linearized moment equation with
zeroth- and first-order terms results. These linearized coefficients represent the
combined effects of all processes on the aerosol. By subtracting the terms of known
order from these Taylor coefficients, we can derive an expression for o^. The
derivation is as follows:
Expanding the rhs of Eq. (5-6) in a Taylor series yields
« 0)/ + ^'y (5-13)
where c = (1 - a
Equating Eqs. (5-12) and (5-13) yields
4"' = 4°>' - c<0) (5-14»)
4^> = 4^' - 4" (5.i4b)
Equations (5-14) define coefficients for the Taylor series expansion of the term Bya
that are equivalent to the Taylor series coefficients for the expansion of the term
B ya minus the coefficients that represent processes of known order. We note from
Eq. (5-13) that o^ can be interpreted as a function that partitions (dy/dt)^ between
zeroth- and first-order Taylor terms1, and is defined as
!For example, for a functional dependence of the order y ' , 75% of (9y/3t) is grouped
with the first-order term and 25% is grouped with the zeroth-order terms.
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L = (first-order rate from Taylor series)/(total rate)
By analogy, a definition of o^ consistent with the coefficients calculated from
Eqs. (5-14) is
We can now write the moment equations as
dy/dt = S + Cy (5-16)
where5 = q°> + °)
At this point it may appear that we have walked in a circle, because
5 = 4°) + 4°) = 4°)'
and 5 and C are precisely the coefficients obtained by expanding the rhs of
Eq. (5 -6) in a Taylor series. The reason for the previous derivation, however, was to
formulate an expression for o^ consistent with Eq. (5-8). The value of a^ so
obtained is needed to calculate moment transfer between multiple aerosol modes,
such as for intermodal coagulation. (Accounting for intermodal moment transfer is
treated in Section 5.4.2.) We have tried other procedures for estimating o^, but the
above formulation produces coefficients that result in the most accurate integration
techniques.
Appendix K contains the complete algorithm for using the power -law
analytical integration technique.
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5.3.1.2 Some Cleanup
Processes of known order If the preceding derivation for a^ is applied to a
moment equation that contains only processes of known order (e.g., constant-rate
source terms), the calculation procedure breaks down. If no processes are
represented by the term Bya, then applying Eq. (5 15b) to the coefficients
calculated from Eqs. (5-14) will either cause division by zero or, due to machine
roundoff error, result in a very large, unrealistic value for o^. Algorithms must
therefore be coded in a way that ensures B is set to zero and the calculation for o-
circumvented when such a situation occurs.
Representing processes of known order other than zeroth- and
first -order We have developed the moment equations (Eqs. [5-4]) as a linear
equation with zeroth- and first-order terms. In developing the system of linearized
moment equations (Eq. [5-2]) we ensured that zeroth- and first-order processes were
precisely accounted for by the form of Eq. (5-8). It is also possible to represent
processes of known order other than zeroth- or first-order, as follows. Suppose a
process is to be modeled, and we either know or arbitrarily decide that the process
can be represented by
(dy/dt)k = Bhyak (5-17)
where a^ is the order (with respect to y) of process h, and must be specified.
Expanding the rhs in a Taylor series (similar to Eqs. [5-9] to [5-12]) yields the
following zeroth- and first-order coefficients:
,,
General expressions for c^ ' and c^ ' can therefore be written as
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5.3.2 First-Order Analytical Integration Technique
Before continuing, we pause to reflect on the development so far. The
power-law representation of the moment equations contains terms of known order
and a power-law term that represents all processes of unknown order. A
straightforward method of applying the analytical integration technique is to
represent all processes as processes of known order. The term Bya in Eq. (5 *8) is
therefore equal to zero, and the additional expense of calculating a is avoided. The
rate for each process is partitioned into zeroth- and first -order terms (Eqs. [5-18])
according to the value of a, specified for each process, and (Ay)^ is estimated from
Eqs. (5-5c) and (5-5d) using cfj ' and c* ' to represent S^ and C^ respectively:
(Ay),, = (AiX, for C = 0 (5-20a)
(Ay), = (j + cW/CWlcCAO - 1} for C * 0 (5-20b)
Error control formulas (derived from Eqs. [5-5] and described in Section
4.5) are used to calculate (A^ such that the integration error for each time step is
maintained within prescribed limits. This integration technique is referred to as the
first-order analytical integration technique, because the coefficients are evaluated
once for each time step, and the time dependence of the coefficients within each
time interval is not accounted for.
Because this technique does not account for the time dependence of the
coefficients within each time interval, its accuracy is the poorest for the analytical
integration techniques described here, but it is guaranteed to exceed or match the
accuracy of Crank-Nicholson techniques (the reason for this is explained in
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Section 5.6.6). Of the analytical integration techniques described here, the
first-order analytical integration technique is also computationally the least
expensive per time step. It is therefore useful if an aerosol dynamics model is used as
part of a larger model in which the integration time intervals are predetermined by
another component of the full model, and the accuracy of the first-order technique
for the prescribed time intervals is sufficient for the modeling purposes.
For situations where the time intervals are large enough to warrant a
higher-order technique, the accuracy of the first-order analytical integration
technique can be improved by approximating the time dependence of the coefficients
of known order within each interval.
5.3.3 Second-Order Analytical Integration Technique
If we continue to represent all processes as processes of known order, the
accuracy of the first-order analytical integration technique can be improved by
approximating the time dependence of cjj ' and c^ within each time interval by
some functional form. Consider the case in which c°^ is zero, and the
time-dependent behavior of Cis represented by the linear function
(5-21a)
where rr^ = (c^ - cJ^/fA*),, (5-21b)
Substituting Eq. (5-21a) for Cin Eq. (5-4b) and integrating yields
Equation (5-22) can be simplified by substituting Eq. (5-21b) for m^ to yield
3U.Li\ = y-n GXP[CTI (AOJ (5-23a)
( i^p lin I ft *-LI|» ') *
where c^ = (cj1^ + ^)/2 (5-23b)
0 1
116
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Equation (5-23a) indicates that by substituting c^ for Cm Eq. (5-4b), a
linear time dependence of the coefficients S and C within each time interval is
simulated.
To calculate c^ \ an estimate for Cfj ' is required. This estimate is obtained
by integrating the moment equations for time interval (A using the first-order
analytical integration technique, and recomputing the coefficients at the end of this
time step. The coefficient so determined, cjj \ is averaged with c*1 , and the
analytical integration performed a second time using zd .
Equation (5 22) is easily derived for cj, ' equal to zero. However, when both
C* ' and cfj ' are nonzero and time-dependent, a general analytical solution is not
possible. We note, however, that representing ci '(t) with a linear time dependence
is equivalent to using c^ '. In an analogous manner, therefore, cjj \ defined as
cW = (c<°> + c«)/2 (5-24)
is substituted for 5 in Eq. (5-4b) to yield
(Ay), = -c!\M for = 0 (5-25a)
(Ay) = (ft + /{oplCAO - 1} for C, # 0 (5-25b)
This procedure for estimating (Aj/)^ from Eqs. (5-25) is referred to as the
second-order analytical integration technique, because two calculations for the
coefficients are performed in each time step. Although we have not provided a
detailed derivation to demonstrate the validity of representing both 5^ and C^ by
the average values c^°^ and cf. \ our simulations with the second-order integration
technique indicate that, for given computational effort, its accuracy is improved
with respect to the first-order analytical technique. It is also unconditionally stable
when c is positive.
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Higher-order analytical integration techniques may also be possible. We
leave this as an item for future study.
5.4 INTERPHASE/INTERMODAL MOMENT TRANSFER
For simulations where material is transferred between multiple aerosol
modes and/or between a gas, vapor, or bulk phase and an aerosol mode, special
treatment of the moment equations is required to properly account for moment
transfer and guarantee conservation of mass.
To develop a general solution algorithm, the governing equations for all
phases and aerosol modes must be solved as coupled sets of equations. Also,
time-stepping algorithms for a general solution algorithm must consider the
dynamics of all phases and aerosol modes when setting the time steps for each phase
and aerosol mode. Such a generalized solution algorithm is complex, and will not be
discussed in this work. Instead, we note that many models for simulating processes
in complex systems are developed by taking advantage of detailed knowledge of the
modeled processes, so that algorithms designed for specific applications are
developed. We therefore show by example the types of calculations that are required
to properly account for transfer between multiple modes and/or between a gas,
vapor, or bulk phase and an aerosol mode. Four examples of transfer processes
commonly occurring in aerosol systems are
particle deposition to surfaces,
intermodal coagulation,
vapor condensation on particles, and
vapor nucleation to form new particles.
A common feature of the first two processes is that material is transferred
out of an aerosol mode to another phase or to another aerosol mode. Because the
moment equations represent the combined effects of all active processes on an
aerosol mode, the estimate of AM, from solving the set of MDEs must account for
these transfer processes. If one of the processes causes material to be transferred out
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of the aerosol mode, and this material must be introduced into the governing
equations for another phase or aerosol mode, an additional calculation must be
performed to determine the precise amount of material transferred out of the aerosol
mode due to the individual process in question.
Because calculations of distribution moments are specific to aerosol
dynamics and have no meaning for the dynamics of other phases, only the mass
transferred must be estimated when calculating transfer to another phase. For
transfer to another aerosol mode, however, the transfer of each moment must be
determined by solving the moment equations.
The moment transfer for an individual process can be expressed as
(5>26a)
For the linearized moment equation (Eq. [5-4a]), Eq. (5-26a) can be written as
(5-26b)
To integrate Eq. (5-26b), an analytical expression for the time dependence of y(t) is
required. Substituting the integral form of the linearized moment equation
(Eqs. [5-5c] and [5-5d]) for y(t) yields
[(Ay)J =
for C = 0 (5-27a)
+ (SA,(A*L for (L t 0 (5-27b)
Equations (5 27) are general expressions and apply for all three of the
analytical integration techniques described in this chapter. To use Eqs. (5-27) in
conjunction with each integration technique, definitions for (S^, (C^, C^, and
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that are appropriate for the integration technique used must be substituted in
Eqs. (5-27). For processes represented as processes of unknown order with the
power-law analytical integration technique, the coefficients are
For processes of known order a, , the coefficients for all of the analytical integration
techniques described in this work are
(5-28c)
and 5 and C are calculated from Eq. (5-16).
5.4.1 Particle Deposition to Surfaces
Because moment calculations are specific to aerosol dynamics and have no
meaning for surfaces, we consider only mass transport to surfaces. Equations (5-27)
are therefore applied to My and the mass deposited to the surface calculated as
(AMp)dep =
The mass transferred out of the mode is introduced as a source term into the
governing equations for the surface to which the particles are deposited. This might
apply to processes where particles are deposited on surfaces to create coatings or
thin films, and the surface reactions are sensitive to the amount and type of
material deposited.
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5.4.2 Intermodal Coagulation
For intermodal coagulation, we represent the source mode (the mode from
which material is transferred) as mode t, and the receiving mode as mode j. In
Chapter 4 (Eqs. [4-31] and [4-32]) we developed rate expressions for intermodal
coagulation of the form
fyj] = - (S-29a)
For example, the moment equations for M. and M. for intermodal moment
o . 6 .
t j
transfer (see Eqs. [4-31c] and [4-32c]) are
..
ci
cij
Equations (5*29) indicate that one component of the rate of moment change for
mode j is equal in magnitude to the rate of moment change of mode t. We note,
however, that Eqs. (5-29) only ensure that the rate of moment transfer from one
mode to another is equal at the instant when these time derivatives are calculated.
When the rates represented by Eqs. (5«29) are combined with all other processes,
and the resulting equations integrated over a finite time interval, the estimated
incremental moment transfer from mode i due to intermodal coagulation will not be
equal to the estimated incremental moment received by mode j. The discrepancy
between the estimated incremental loss from mode i and the corresponding
incremental increase in mode j occurs because the rate of intermodal moment
transfer changes as a function of time during the integration time interval due to the
effects of other processes.
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Any numerical integration technique used to solve the system of MDEs,
therefore, must be formulated such that the incremental moment transferred from
mode i due to intermodal coagulation is exactly equal in magnitude to the
incremental moment received by mode j. Mathematically, this condition is
(5-30a)
Because the moment is transferred from mode i to mode j, we first integrate the
moment equations for mode i, and from this integration determine the moment
transfer due to intermodal coagulation, (Ay.) .. , using Eqs. (5-27). This
L * ciJJri
incremental moment is then introduced into mode j.
To introduce (Ay.) into the mode j moment equation, we represent it
L * ciJJT)
as a zeroth-order constant rate source term:
[c(°) 1 = - [(Ay.) . 1 /A* (5-30b)
L 7>C1JJT1 L * C1JJT)
With this representation, the incremental moment transferred to mode j will exactly
equal the incremental moment lost from mode :.
5.4.3 Vapor Condensation on Particles
Vapor condensation on particles can be computed from two different
perspectives: by assuming the vapor phase is relatively stable compared to the
aerosol modes, or by assuming the aerosol is relatively stable compared to the vapor
phase. The first perspective implies that the system of equations for the aerosol is
solved first, and the incremental mass transferred from the vapor phase to the
aerosol is determined and included in the vapor-phase equations as a loss term. The
latter perspective implies solving the vapor-phase equations first, and including the
mass transferred from the vapor phase to the aerosol as a source term in the aerosol
equations. If the aerosol equations are solved first, the calculation technique is
similar to the technique for accounting for deposition to surfaces. If the vapor-phase
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equations are solved first, however, the technique for introducing the transferred
mass into the aerosol modes requires some additional considerations.
By performing the vapor -phase calculations and estimating the mass
transferred to the aerosol modes, (AMV) , in time step 77, a source term for M is
written as
(AM ) = [(AMv)con]y[(7r/6)pv] (5-31a)
where M is the total third moment for all aerosol modes.
3t
If the mass is deposited over time interval (AZ)^, then a constant source
rate for the entire aerosol can be defined as
For multimodal aerosol simulations, AM, must be partitioned between the
L 3
-------
[Q *
equal and opposite, -mM
\ (Jv O .
I '.
zeroth-order rate. Because
appropriate coefficients are
is introduced into the moment equations as a
o -I
TsiM is introduced as a constant source rate, the
With this formulation, mass is conserved for this process.
For mass deposition represented as a zeroth-order rate, the orders for the
rates of the other moments are nonzero. For analytical integration techniques, the
order of the moment equations must be represented. If the power-law analytical
integration technique is used, the condensation moment equations for moments
other than M should be grouped with the moment equations of unknown order. For
O
the first- and second-order analytical techniques, an order for the moment transfer
processes must be specified. The precise order will vary for different simulations,
and will also depend on the specific moment solved for (in this work, we solve for N,
M , and M_). The moment AT does not change for condensation, and for M we have
O D ( O
found a value of 0.5 to reasonably represent the order for condensation (i.e., a for
the MDEs for M.).
0'
The preceding discussion of the order of the processes pertains to the
analytical integration techniques. For simulations where global errors on the order
of 0.1% to 1% is acceptable, we recommend using the power-law analytical
integration technique, and grouping the condensation rate for other moments with
the processes of unknown order. This allows the integration technique to
automatically adjust to the dynamics of the specific simulation. When global errors
must be maintained at less than 1% (or when the computational expense can be
afforded to implement a more accurate technique), we recommend a higher-order
integration technique, such as the fourth-order Runge-Kutta technique, which
operates on the total rate of change of the moments and therefore does not require
the order of the processes to be specified.
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5.4.4 Vapor Nndeation to Form New Particles
For nucleation, Pratsinis (1988) gives expressions for the production rate of
aerosol in terms of integral moments of the distribution. Because nucleation is
usually fast compared to all other processes, it is often modeled assuming all other
processes are inactive. The new particles formed are introduced into an aerosol mode
as a constant-rate source term for the time interval for which nucleation is active.
The moment transferred into the interacting mode is equal to the integral moment
of the nucleated particles, and the transfer rate for all moments is zeroth-order.
Assuming nucleation is the dominant process in time step T/, the change of the
moments is
5.5 INTEGRATION TIME INTERVALS AND ERROR CONTROL
The following discussion applies primarily to the first- and second-order
analytical integration techniques, where the orders of all processes are specified and
the power-law term in Eq. (5 8) is zero. For this situation, the order of the error of
integration can be precisely formulated analytically, and appropriate time-stepping
algorithms developed. For the power -law integration technique, the error is a
nontrivial function of the specific processes included in a simulation. A discussion of
error control for the power-law integration technique is presented in Section 5.5.2.
5.5.1 Error Control for First- and Second-Order Analtical Interation Techniues
Equations (5-5c) and (5-5d) are used to estimate (Ay)^ in each time
interval, (Af) . We need a means for determining (Ai)^, subject to constraints on
the allowed error incurred by using Eqs. (5-5c) and (5-5d). The expressions
developed in this section can be used in one of two ways: the error can be
constrained and appropriate time steps calculated, or the time intervals can be
specified and the resulting error evaluated. Whether the allowed error or the time
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step is specified depends on constraints from other parts of a complete model, but a
similar set of expressions must be formulated for each error control strategy.
Time-stepping algorithms are designed to control the error associated with
each time step (the local error), because the error for the entire simulation (the
global error) is a nontrivial function of the local error for the entire simulation
period, and cannot be precisely controlled without repeating a simulation in part or
in its entirety. The objective of developing error control algorithms, therefore, is to
determine time steps such that the estimated relative local error is maintained
below some prescribed value, e, expressed mathematically as
«[(Ay)J/,L = 6
(dy/tyM
/y < f (5-34a)
where S[(^y\} is the estimated local error of integration for interval 77. For
compactness, we represent the estimated relative local error in each time interval as
(5-34b)
The strategy in error control is to estimate an appropriate time interval,
, from (1) the estimated local error for the previous time step, e ; (2) the
time interval over which that error occurred, (Af)T]_1; and (3) the target value for
the local error for the current time step, e_. A simple approach is to estimate (At)
as
(5-35)
For example, if a time step (A<) caused an error e^_ that was one half of the
allowable error e^ for the current time interval, then (A*)^ should be twice as large
as
A problem with this procedure occurs, however, because Eq. (5-35)
represents a linear dependence between c and (A<) . Such a linear dependence is
generally incorrect and may lead to unstable behavior of the system of equations.
Equation (5-35) must therefore be rewritten to account for the functional
dependence of e^ on (Ai)^, as
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R
(5-36)
where R is the functional order of the local integration error, e^, on (Ai)^, which
must be estimated for each functional representation of c^ \t) and c* '(t).
Equation (5 36) can be used to estimate time steps when a single equation
is integrated. Because MAD models require integration of a system of equations, a
time step appropriate to maintain the error of each equation below the target error
level must be determined. Equation (5-36) is therefore rewritten to include the
mode and moment indices as:
(5-37a)
where
(Ai)^ and CA apply to the fcth-moment equation of the tth mode.
By evaluating (Af)^ for each moment, an appropriate time step for the ith
mode is
[(A,),. = MIN
*2t- JT)
(5-37b)
Repeating this process for all modes, the appropriate time step for all moment
equations, Af, is estimated as
(At), = MIN
[(A,),. ] [(A<). ] ...
(5-37c)
5.5.1.1 The Order of the Integration Error
We can estimate R, the functional order of the local integration error, e^,
on (Af)_, by determining the order of the truncation error of a series expansion of
Eqs. (5-5c) and (5-5d) that results from a particular representation of the
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functional form for cjp(£) and S (0- Tne truncation error can be estimated by
comparing the Taylor series expansion of Eqs. (5-5c) and (5-5d) for a low-order
functional approximation of c^ \t] and c^ \t) with the Taylor series expansion for a
higher-order functional approximation of c^ \t) and cfj \t). For the following
formulations we rewrite Eq. (5-8) in terms of arbitrary time-dependent coefficients:
= 4°\t) + 4l\t)y^ (5-38)
and substitute for (~*\t) and c^\t) the mathematical forms used to represent these
coefficients in the analytical integration techniques. For the functional
representation of c°'(<) and c1'(0 in the first- and second-order analytical
integration techniques, Eq. (5-38) is integrated and the truncation error resulting
from each functional representation is determined by Taylor series expansion.
It is not possible, however, to obtain analytical solutions to Eq. (5-38)
when cjj '(t) or cjj '(£) varies within a time step and the other coefficient is nonzero.
We restrict the following analysis, therefore, to cases where either cjj \t) or cjp(i)
is equal to zero.
For c)j \t) equal to zero, the general solution to Eq. (5-38) is
(AyX, = l^JerpCx) - 1] for c^ = 0 (5-39)
where x = f c^\t)dt
Expanding the exponential term in a Taylor series yields
(AyX, = Vnjix + x2/2 + ...) (5-40)
To estimate the order of the first-order analytical integration technique, we
apply Eq. (5-40) in the following way: for c^\t) = c!^l\ Eq. (5-40) becomes
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(Ay)'1' = ycA*) + [c(A*)J2/2 + ...) (5-41)
where (Aj/)^ ' is the estimate for (Ay)^ for the first -order analytical integration
technique. For c- \t) = c ' + rrt^t, Eq. (5-40) becomes
+ (Af2/2 + ...]} (5-42)
where (Ay)^ ' is the estimate for (Ay)^ for the second -order analytical integration
technique.
The difference between (Aj/)^ ' and (Ay)^ ' approximates the truncation
error associated with the first-order analytical integration technique, and provides
an estimate of W ' the order of dependence of e on A£ (the superscript on R
indicates this is the order of the error for the first -order analytical integration
technique). Subtracting Eq. (5-41) from Eq. (5-42) yields,
2/2 + ...] (5-43)
n
The largest term in Eq. (5«43) is of order (A<) , indicating the truncation error of
A
the first-order analytical technique is of order (At) , and a value of two is
appropriate for R^ ' in Eq. (5*37a). Similar analysis indicates that the error for the
second-order analytical integration technique is of order Ar, and a value of three is
appropriate for R^' in Eq. (5-37a).
For ci^(£) equal to zero and c^ '(t) varying within a time step, we found
the order of dependence of e on A t to be the same as for c^ \t) equal to zero and
<~l\£) varying within a time step. And although it is not possible to perform this
analysis for c±Q\t) and c^(i) both expressed as time-dependent functions, the
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example simulations presented in Section 5.6 indicate that the value of R" derived
for the cases of c^°\t) or c^ \t) equal to zero also applies to the case when both
cj:0^) and c^l\t) are nonzero and time-dependent within a time step.
5.5.1.2 Estimating the Error of the First- and Second-Order Analytical Integration
Technique
To use Eqs. (5-37), an expression for c is required. Equations of the
form of Eq. (5-43) could be used, but these are available only for the special cases of
either c^\t) or cjj \t) equal to zero. Another alternative is to differentiate Eqs.
(5-5c) and (5-5d) and estimate the change of (Ay)^ (the local error) for a change in
the coefficients (representing deviation from the assumption of constant coefficients
over the integration time interval).
Equation (5-5c) is for cj, ' equal to zero and Eq. (5-5d) is for c^ ' nonzero.
For c^ ' equal to zero, a differential change in ci ' causes a differential change in
(Ay),
(5-44)
For a finite change of Acf,_{ in time step (A*)^, the error, ^(Aj^J/iu ^ , can
therefore be approximated as
for
For cjj ' and c^ ' both nonzero, similar error analysis applied to Eq. (5-5d) yields
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For first-order analytical integration, the coefficients used in Eqs. (5-45)
are cU,. and cL\. . The changes in the coefficients for interval n-i are estimated
1*1-1 )0 v"-l)o
using the coefficients for interval 77:
O) _ JO) _ (0)
- )o
(5.46b)
For second-order analytical integration, the coefficients used in Eqs. (5-45)
are average coefficients, c and cj, calculated as
where cU,v and cU,x are known values at the beginning of interval OT 1), and
(n-i)0 (^ijo o-o \ /i
c)^.v and c/i-jx are estimates of the coefficient values at the end of the same
interval. After the main integration time step is taken, new coefficients are
calculated. The new set of coefficients are ci ' and c ' for interval Ti, and also
^o ~o
represent updated estimates of cL^ and cU^ (see Figure 5-1). Updated
estimates of the average coefficients for the previous time interval are therefore
131
-------
and the changes in the coefficients appropriate for Eq. (5-45b) are therefore
5.5.2 Error Control for the Power-Law Analytical Integration Technique
Assessing the error of the power-law analytical integration technique is not
straightforward. Equation (5-8) is composed of two parts, and can be represented as
total rate = (rate due to processes of known order)
+ (rate due to processes of unknown order)
The error associated with the rate due to processes of known order behaves the same
as the error for the first-order analytical integration technique, because the
coefficients for this part of Eq. (5-8) are evaluated once for each integration step.
The error associated with the rate due to processes of unknown order behaves like
the error for a higher-order technique, because the time dependence of this rate is
represented as a power function. In terms of the changes in the coefficients within
each time interval, the order of the error is at least as good as the first-order
analytical integration technique. Therefore, the precise error of the power-law
analytical integration technique depends on the contribution (relative to the total
error) of the error from the coefficients that represent processes of known order and
the error from the coefficients that represent processes of unknown order. Assessing
these component contributions is not straightforward, and we do not provide the
analysis here.
The behavior of the power-law analytical integration technique is
demonstrated by Figure 5-9, which represents the accuracy versus number of loops
through the solution algorithm for the three analytical integration techniques for
bimodal coagulation (discussed in Section 5.6.3). For this case intermodal transfer is
represented as a source term in mode j, and the total mass transferred from mode i
132
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to mode j during the entire simulation is comparable to the starting mass of mode j.
The power-law analytical integration technique will represent this moment transfer
as accurately as the first-order analytical integration technique because the moment
transfer is treated as a constant-rate source term. For intramodal coagulation the
power-law analytical integration technique will represent the moment change within
each respective mode as accurately as a higher-order technique because the order of
these processes is implicitly accounted for by the power function in the power-law
analytical integration technique. The curve drawn through the data points in Figure
5-9 is a sum of two functions, one that approximates first-order behavior (the
constant-rate source terms) and one that approximates second-order behavior (the
terms represented by the power function).
By contrast, for intramodal coagulation (Figures 5-2 and 5-3 and Table
5-1), there are no source terms for any of the moment equations, and the power-law
analytical integration technique behaves as a higher-order integration technique.
Because the error analysis is not straightforward, and because the error is
bounded by the first-order analytical integration technique, we use a time
stepping-algorithm based on the error estimate for the first-order analytical
integration technique as the time-stepping algorithm for the power-law analytical
integration technique. This is not the most efficient time-stepping algorithm for the
power-law analytical integration technique, and continuing work should include
development of a more efficient time-stepping algorithm.
5.6 PERFORMANCE OF THE ANALYTICAL INTEGRATION TECHNIQUES
In this section we investigate the performance of the three analytical
integration techniques discussed previously, by comparing them to the Euler,
Crank-Nicholson, and second- and fourth-order Runge-Kutta standard numerical
integration techniques. Appendix K contains detailed algorithms for applying all
seven of the integration techniques compared.
133
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5.6.1 The Simulations Performed
We performed simulations for various combinations of coagulation and
particle growth. Initial and final conditions for 12-hour simulations are shown in
Figures 5-2, 5-4, 5-6, 5-8, and 5-10. The simulations were designed to test the
performance of MAD models for combinations of processes typically encountered
when simulating aerosol behavior. The set of simulations performed was not
exhaustive in terms of the processes that might be important under different
conditions. We examined three cases for unimodal dynamics, and two for bimodal
dynamics. For the bimodal simulations, the modes chosen were relatively close
together, so that both modes would experience comparable volume increases for the
simulations including particle growth, and intermodal coagulation would result in
volume transfer to the large mode of the same order of magnitude as the starting
volume of that mode. These simulations therefore represent severe tests of the
algorithms.
For simulations including particle growth, the amount of particle growth
for the entire simulation was determined by multiplying the initial total aerosol
volume (the volume of all modes) by a growth factor. The resulting incremental
volume was divided by the total simulation time to yield a constant volume growth
rate for the entire aerosol and for the entire simulation time period.
Even though the growth rate for the entire aerosol was constant, the growth
rate for each mode in a bimodal simulation changed for each time step, because the
growth rate for each mode is a function of the competition between modes for the
available vapor. The requirement for treating intermodal/interphase transport,
however, requires that the volume received by a mode from another phase be
represented as a constant-rate source term. The volume added to each mode in each
time step, therefore, was constant, although the rate from time step to time step
varied.
If growth without the Kelvin effect is simulated, the growth process causes
the distribution of each mode to become nearly monodisperse. This is often observed
in systems where particle growth dominates, but results in simulations that are
134
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rather uninteresting and that do not provide a very severe test of the solution
algorithms. It is possible to simulate more severe cases of growth dynamics by
including the Kelvin effect. The essence of the Kelvin effect is that the aerosol below
a critical size, dp (defined by Eq. [4-42]), evaporates, while the aerosol above this
critical size grows. If dp is located near Dgn of a mode, the effect of part of the
aerosol in that mode evaporating and part of the aerosol in that mode growing is
that ae tends towards an asymptotic value of the order of 1.6 (the precise value
depends on the functional form of the particle growth law). Including the Kelvin
effect therefore allows growth scenarios to be simulated that provide more severe
tests of the solution algorithm, because the distribution does not collapse to a
monodisperse distribution.
For unimodal simulations with particle growth that include the Kelvin
effect, we therefore applied the following criterion:
dp = Dgn (5-41)
*
where dp is the critical diameter for nucleation (see Eq. [4-42]). This criterion might
represent a situation where the saturation level in a control volume increased until a
sufficient number of particles were activated to cause growth; Growth would begin
when Eq. (5*41) is satisfied. As the particles grow, the condensing vapor is depleted,
*
thereby decreasing the vapor pressure and causing dp to increase. For very special
conditions, dp would increase at the same rate that Dgn increased due to growth.
For bimodal simulations with particle growth, we also used a scenario that
provided a severe test of the solution algorithms. As for the unimodal case, modeling
particle growth without the Kelvin effect would allow the distributions to become
nearly monodisperse. We therefore imposed the following criterion on the
simulation: For bimodal growth, dp was set equal to Dgn. for the i-mode growth
integrals, and dp was set equal to Dgn. for the j-mode growth integrals. We use this
growth scenario here simply to perform a simulation that provides a relatively
severe test of the solution algorithms, rather than associating any physical
135
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significance with the criterion; it is difficult to imagine a growth scenario where this
criterion would have any physical basis. It is interesting to note, however, that the
widths of the resulting distributions for combined coagulation and condensation are
very close to the widths of distributions often observed in the earth's atmosphere.
Whether or not the criterion is physically meaningful or not the widths of the
distributions resulting from the simulations are physically realistic.
The initial and final modal parameters for the simulations performed are
shown in Figures 5-2, 5-4, 5-6, 5-8, and 5-10.
5.6.2 Accuracy and Computational Effort
We are interested in comparing the accuracy and computational effort
associated with the various integration techniques. Because the global error of
numerical integration techniques can be reduced to any desired level by continually
decreasing the integration time interval, it is not meaningful to report accuracy and
computational effort separately. A meaningful performance parameter is the global
error that results from a specified computational effort for a given technique.
By "computational effort" we usually mean the total CPU time required for
a particular simulation. In order to accurately assess the computational time
required for each integration technique, algorithms optimized for each technique
would have to be developed and run for every test case. An alternative (and much
simpler) approach is to determine the time intervals for a first-order analytical
integration technique with optimized time-step control and then use these time
intervals for all of the other integration techniques. Although this may not always
represent the best performance possible for each technique, the resulting
performance should be close to that obtainable by completely optimized integration
techniques.
To allow us to estimate computational effort of the various integration
techniques, we used the same basic algorithm loop for every technique; the loop
calculated the coefficients, performed an integration step, and recomputed modal
parameters needed to calculate the next set of coefficients. The computational effort
136
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for each technique was proportional to the total number of times through the loop
for a particular simulation. It was difficult to present a single CPU time for each
technique because the actual computational effort per loop depended on the relative
efficiency of a given computer in performing the technique's mathematical
operations. We therefore did not use CPU time as a measure of computational
effort, but instead reported the total number of loops2 through the integration
algorithm for each technique.
5.6.3 The Method Used to Compare The Integration Techniques
We evaluated the three analytical integration techniques by comparing
them with the Euler, Crank-Nicholson, and second- and fourth-order Runge-Kutta
numerical integration techniques. The idea behind the comparison was to determine
directly how much computational effort was required by each integration technique
to achieve a specified level of accuracy in the results of aerosol dynamics
simulations. It was easier, however, to compare the techniques by performing the
simulations using a specified number of time steps, evaluating the resulting
accuracy, plotting accuracy versus computational effort, and determining
graphically the computational effort required to produce a given level of accuracy.
Tables 5-1 through 5-5 and Figures 5-3, 5-5, 5-7, 5-9, and 5-11 show the relative
error of each technique for the simulations performed; the horizontal line in each
figure marks the level for 1% global error. If a simulation with a particular
integration technique resulted in a solution that was two orders of magnitude or
more different from a standard solution (refer to Section 5.6.4), that result was
rejected and a "1" listed in the table of integration accuracy.
5.6.4 Assessing the Error of the Integration TechTriqnes
To evaluate the integration techniques, a standard of comparison is
required. We used a fourth-order Runge-Kutta technique to estimate the standard
results for all cases reported here, with enough time steps to produce results more
2Higher-order integration techniques, such as fourth-order Runge-Kutta, require multiple
integration loops for each main time step.
137
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accurate than the level of accuracy sought for the comparisons. This technique
yielded standard solutions whose accuracy was substantially higher than the other
integration techniques compared except for the case depicted by Figures 5-2 and
5-3. For this case the accuracy of the standard solution appears to have reached a
lower limit. The reason for this is not clear. Because the error of the other
integration techniques are above this apparent lower limit, however, it does not
appear that the indicated dependence of accuracy versus computational effort for
the other integration techniques is affected.
The error of each integration technique was estimated by comparing the
predicted moment values with the predicted moment values from the standard
solution. We used the following comparison parameter for each moment:
Ek = 1 - (MjAfj) (5-40a)
where M, represents the moment value from the standard solution. A composite
error parameter was calculated by averaging the absolute value of E, for all of the
moments, as,
E= (\E \ + \E \ + ... + \Ehi\ + \Ek2\ + ...)/«tot (5-40b)
where ntot is the total number of moment equations with nonzero C. 3 Values of E
are reported in Tables 5-1 through 5-5.
5.6.5 The Procedure for Comparing the Integration Techniques
Five 12-hour simulations were used to evaluate the integration techniques.
They were for various combinations of coagulation and particle growth and for
unimodal and bimodal situations. The simulations performed were as follows:
30nly the moments governed by MDEs with nonzero C were included in the calculation of
E because all integration techniques will correctly integrate an MDE consisting only of an
5-type coefficient. The solutions resulting from the latter case do not provide a
meaningful comparison of the integration techniques' performance.
138
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Unimodal coagulation
Unimodal growth with a growth factor of 500.
Unimodal coagulation and growth with a growth factor of 500.
Bimodal coagulation
Bimodal coagulation and growth with a growth factor of 500.
The modal parameters for these simulations are listed in Figures 5-2, 5« 4, 5-6, 5-8,
and 5 -10.
The procedure for comparing the numerical integration techniques was as
follows:
A target value for the total number of time steps for a particular
simulation was specified (e.g., 8).
For the first-order analytical integration technique, values of e were tried
until the target number of time steps was achieved. Equations (5 37)
were then used to determine (A for each time step.
These values of (Af)^ were stored in a data file.
For each of the other integration techniques, (Af)^ for each time step
was read from the data file created for the first-order integration
technique and used as the other techniques's integration time interval.
In Tables 5*1 through 5-5, the "Loops" column contains the total number
of loops through the solution algorithm for each integration technique. Because
second-order analytical and second-order Runge-Kutta techniques require an
additional substep for each main time step, the number of loops per time step for a
second-order technique is twice the number of loops per time step for a first -order
technique. Likewise, for the fourth-order Runge-Kutta technique, four loops per
time step were required. Therefore, the accuracy is tabulated against the number of
loops instead of the number of time steps because the number of loops through the
solution algorithm is a better measure of the computational effort.
139
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5.6.6 The Behavior of the Integration Techniques
To understand the performance of numerical integration techniques, we
note that there are two time-variant components in Eqs. (5-4): the variable y and
the coefficients 5 and C. The lowest-order numerical integration technique, the
Euler technique, holds the variable and the coefficients effectively constant for each
integration time step. The Crank-Nicholson technique is a modified Euler technique
that maintains constant coefficients but accounts for the time dependence of y in a
simple manner. Algorithms for these techniques are presented in Appendix K.
The Crank-Nicholson technique implicitly evaluates dy/dt at two points in
the time interval. Therefore, as the number of time steps increases, its accuracy
should asymptotically approach the accuracy of the first-order analytical integration
technique, which effectively evaluates dy/dt at an infinite number of points in the
time interval. Tables 5 1 through 5 5 show, as expected, that the accuracy of the
Crank-Nicholson technique approaches the accuracy of the first-order analytical
integration technique as (A£) decreases.
The second-order analytical integration technique simultaneously accounts
for the time variations of y and the coefficients. With this technique, multiple
substeps are taken within the integration time step; at the end of each substep, a
new set of coefficients is calculated. After all of the substeps are completed, a
weighted average of the coefficients determined after each substep is calculated, and
these average coefficients used to perform the integration step. The algorithm for
the second-order analytical integration technique is presented in Appendix K.
Runge-Kutta techniques use weighted averages of the moment equations
that are evaluated at different points in the integration time interval to account for
the time dependence of the moment equations. Multiple substeps are therefore taken
within the integration time step; at the end of each substep, the moment equations
are recalculated. After all of the substeps are completed, a weighted average of the
moment equations determined after each substep is calculated, and these average
moment equations used to perform the integration step. Algorithms for the
Runge-Kutta techniques are presented in Appendix K.
140
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The second-order analytical and second-order Runge-Kutta techniques
should be of comparable accuracy, since each utilizes two substeps within a time
step, and each simultaneously accounts for the time variation of the variable and
the coefficients. This is indeed the observed behavior, as indicated in Tables 5 1
through 5-5. The principal advantage of the analytical technique is that the
analytical integrals guarantee always-positive variables (provided 5is positive). The
Runge-Kutta techniques may admit negative values if some bound on the variable
values is not included in the integration algorithm.
The second-order analytical and second- and fourth-order Runge-Kutta
techniques are more expensive per main integration time step than lower-order
techniques, because of the multiple substeps. However, if the reduction in the local
error more than offsets the increased computational effort per time step, the
resulting technique leads to an overall reduction in computational effort for specified
global error. The computational savings achieved by using a higher-order technique,
with respect to a lower-order technique, therefore depends on the computational
effort required by each technique to maintain the global error required for a
particular simulation.
For the simulations performed in this work, and for global errors less than
1%, the fourth-order Runge-Kutta technique requires less computational effort than
all the other integration techniques. For global errors larger than 1%, however, the
fourth-order Runge-Kutta technique requires more computational effort than the
second-order and power-law analytical integration techniques.
5.6.7 Recommendations for Selecting a*i Integration Technique
In this chapter we have described several analytical integration techniques
and compared them to standard numerical integration techniques. Determining
which integration technique to use for a particular application is crucial to the
overall performance of an aerosol dynamics model the model, because using a
technique of a higher order than required and/or demanding more accuracy from the
integration technique than can be justified by the rest of the aerosol model results in
an unnecessary increase in computational effort. In this section we consider some
141
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factors to use in selecting the proper integration techniques for CSTARs and FARs.
For this discussion it is important to distinguish between two types of
situations: (1) those where time step control can be used to adjust the integration
time step lengths, and (2) those where the time step lengths are set externally to the
aerosol dynamics model, such as for multiphase models, where the integration time
step length suitable for another phase may limit the time step length for the aerosol
phase.
5.6.7.1 Integration Techniques Used with CSTARs
Because a CSTAR is represented as a single homogeneous control volume,
the governing equations are integrated only in time, and spatial gradients are
treated by analytical correlations. The computational effort required to simulate
dynamic processes is therefore substantially less than the effort necessary to
represent the same processes in a finite-difference, finite-element, or control-volume
type calculation. For simulating single-component aerosol dynamics in a CSTAR,
therefore, the fourth-order Runge-Kutta technique with time step control is the
most appropriate choice; the additional programming effort required by the
analytical integration techniques is usually not justified, because in most cases
CSTAR aerosol dynamics models need only modest computational effort.
5.6.7.2 Integration Techniques Used with FARs
For a multiple-control-volume flow model, the benefit of using an analytical
integration techniques depends on the balance between the savings in computational
effort and the effort required to implement the technique. Spending weeks or months
to implement and check a new integration technique is clearly not warranted if only
a few simulations are to be performed. However, for situations where many model
runs will be made and global error on the order of 1% is acceptable, or for situations
where the aerosol-phase time steps are limited by the steps required for another part
of the model, one of the analytical integration techniques may provide an
alternative to traditional integration techniques, with substantial savings in
computational effort.
142
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The main advantage of the analytical integration techniques is manifested
when aerosol dynamics in a calculation domain of multiple control volumes and/or
multiple phases are simulated. In such cases, a substantial portion of the control
volumes may experience relatively mild changes in each time step, so using a
higher-order integration technique is unwarranted. By selecting a lower-order
integration technique that is optimized for the control volumes with weak aerosol
dynamics, a small computational penalty may be incurred when the same
integration technique is used in control volumes experiencing more severe dynamics.
The net result, however, may be a savings over using an integration technique
optimized for the control volumes experiencing severe dynamics.
5.6.7.3 Multiphase Models
The higher-order techniques are computationally more efficient for
situations where automatic time step control can be implemented for the aerosol
phase. For models simulating the dynamics of two or more coupled phases, the
maximum time step for the aerosol dynamics may be controlled by the dynamics in
another phase (e.g., gas-phase chemistry). In such a situation, higher-order
integration techniques may not be needed if global error on the order of 1% can be
achieved with a lower-order technique and is acceptable for the simulations
performed.
When modeling processes in the earth's atmosphere, large uncertainties
exist in representing the meteorology and source strengths of primary pollutants and
gases that participate in secondary aerosol formation. Because atmospheric models
are rather large and complex and require large amounts of CPU time on the world's
fastest computers, we are interested in implementing an integration technique at the
accuracy level with which the other variables affecting aerosol behavior are known
or are predicted by other components of the full model. Demanding higher accuracy
from the aerosol-phase solutions than from the rest of the model results in
unjustified and excessive computational effort.
143
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The Runge-Kutta technique is a workhorse for integrating differential
equations, but it is subject to admitting unrealistic estimates for the integration
substeps because each predictor step is an Euler step. The first- and second-order
analytical integration techniques guarantee always-positive solutions, but (as seen in
Tables 5-1 through 5-5) they also may admit unrealistic solutions; for simulations
with large time steps (i.e., few loops through the solution algorithm), unrealistic
solutions may result because the accuracy of first- and second-order analytical
integration techniques decreases if the actual order of the processes is significantly
different from the order specified by the model user. By contrast, the power-law
analytical integration technique implicitly determines the order of the processes for
each time step, and so maintains reasonable accuracy even for large time steps.
144
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1.00E6 3.39E4
0.0500 0.178
1.500 1.306
p
o.
o
v,
Df
(J
*l
Figure 5-2a
hr 0 hr 12
137 137
0.0819 0220
1.500 1.306
hr 12
0.01
0.1
10
Figure 5-2b
Figure 5-2. Starting and ending distributions for a 12-hour
unimodal coagulation simulation with a growth factor of 0;
5-2a: number distribution; 5-2b: volume distribution.
145
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TABLE 5-1. AVERAGE GLOBAL ERROR (RELATIVE TO BORDER RUNGE-KUTTA
TECHNIQUE) OF VARIOUS INTEGRATION TECHNIQUES FOR A 12-HOUR
UNIMODAL COAGULATION SIMULATION VITH A GROWTH FACTOR OF 0
Loops
8
16
32
64
128
256
512
Euler
0.199
9.62E-2
4.74E-2
2.33E-2
1.16E-2
5.77E-3
2.88E-3
C-N
0.105
4.98E-2
2.42E-2
1.19E-2
5.91E-3
2.93E-3
1.46E-3
A-l
9.82E-2
4.81E-2
2.38E-2
1.18E-2
5.89E-3
2.93E-3
1.46E-3
PL
5.73E-2
l.OOE-2
2.10E-3
4.54E-4
1.12E-4
2.07E-5
3.78E-6
A-2
0.106
1.20E-2
3.32E-3
7.99E-4
2.16E-4
5.18E-5
1.07E-5
RK-2
0.103
5.58E-2
1.46E-2
3.18E-3
7.39E-^
1.84E-4
4.42E-5
RK-4
0.113
3.25E-3
9.49E-5
6.33E-6
4.88E-6
3.87E-6
C-N Crank-Nicholson
A-l lst-order analytical
A-2 2°d.<)rder analytical
PL Power-law analytical
RK-2/RK-4 2°d-/4tii-order Runge-Kutta
"1" Indicates unrealistic solution
O CM.
0>
en
to
<5
.0
1% global error
1"-order analytical
21 -order analytical
Power-law analytical
2ld-order Rnnge-Kutta
4th-order Runge-Kutta
1 ' ' ' i ' 'ii' ' ' > i *
10.0 100.0
Loops Through Integration Algorithm for
1000.0
Figure 5-3. Accuracy of various integration techniques for a 12-hour
unimodal coagulation simulation with a growth factor of 0.
146
-------
1.00E6 1.00E6
0.0500 0.364
1.500 1.603
dp him]
Figure 5-4a
m
uu
o~
cvj
o
Sin
;£-
O)
.o
3^
>
O
0_
I
137 6.85E4
0.0819 0.710
1.500 1.603
hr 12
hrO
0.01
o.i
dp
1.0
10.0
Figure 5-4b
Figure 5-4. Starting and ending distributions for a 12-hour
unimodal growth simulation with a growth factor of 500;
5-4a: number distribution; 5-4b: volume distribution.
147
-------
TABLE 5-2. AVERAGE GLOBAL ERROR (RELATIVE TO BORDER RUNGE-KUTTA
TECHNIQUE) OF VARIOUS INTEGRATION TECHNIQUES FOR A 12-HOUR
UNIMODAL GROWTH SIMULATION VITH A GROWTH FACTOR OF 500
Loops
8
16
32
64
128
256
512
Euler
1.10
0.413
0.185
8.77E-2
4.28E-2
2.09E-2
1.04E-2
C-N
5.93
0.496
0.210
9.57E-2
4.59E-2
2.21E-2
1.10E-2
A-l
0.854
0.387
0.185
8.97E-2
4.45E-2
2.18E-2
1.09E-2
PL
0.166
4.57E-2
1.08E-2
2.72E-3
6.86E-4
1.73E-4
4.40E-5
A-2
1
6.12E-2
3.18E-2
8.79E-3
2.24E-3
5.69E-4
1.40E-4
RK-2
0.652
0.126
3.85E-2
1.12E-2
3.07E-3
8.15E-4
2.07E-4
RK-4
0.189
5.15E-3
5.41E-4
4.12E-5
2.86E-6
1.90E-7
C-N Crank-Nicholson
A-l lst-order analytical
A-2 2nd-order analytical
PL Power-law analytical
RK-2/RK-4 2nd-/4th-order Runge-Kutta
"1" Indicates unrealistic solution
O CNJ.
0>
en
ea
1.0
1% global error
1"-order analytical
2M
-------
1.00E6 4.88E4
0.0500 1.020
1.500 1.572
Figure 5-6a
UJ
o'
CM*
U
n
'IO
;£-
O)
_o
TS"
>
o
o_
137 6.85E4
0.0819 1.88
1.500 1.572
hr!2
hiO
0.01
0.1
1.0
10
.0
Figure 5-6b
Figure 5-6. Starting and ending distributions for a 12-hour
unimodal coagulation and growth simulation with a growth factor of 500;
5-6a: number distribution; 5-6b: volume distribution.
149
-------
TABLE 5-3. AVERAGE GLOBAL ERROR (RELATIVE TO BORDER RUNGE-KUTTA
TECHNIQUE) OF VARIOUS INTEGRATION TECHNIQUES FOR A 12-HOUR UNIMODAL
COAGULATION AND GROWTH SIMULATION VITH A GROWTH FACTOR OF 500
Loops
8
16
32
64
128
256
512
Euler
0.589
0.268
0.136
6.51E-2
3.25E-2
1.61E-2
8.02E-3
C-N
1.19
0.486
0.193
8.29E-2
3.90E-2
1.88E-2
9.25E-3
A-l
0.923
0.368
0.167
7.72E-2
3.75E-2
1.85E-2
9.16E-3
PL
0.168
2.87E-2
5.80E-3
1.52E-3
3.85IM
1.04E-4
2.83E-5
A-2
6.93E-2
7.21E-3
1.12E-3
2.26E-4
5.18E-5
1.24E-5
RK-2
1
3.44E-2
1.45E-2
4.27E-3
1.15E-3
2.95E-i
RK^
1
2.63E-3
7.53E-5
2.74E-6
1.54E-7
C-N Crank-Nicholson
A-l lst-order analytical
A-2 2nd-order analytical
PL Power-law analytical
RK-2/RK-4 2nd-/4tn-order Runge-Kutta
"1" Indicates unrealistic solution
o c\j.
1.0
1% global error
1"-order analytical
2ld-order analytical
* Power-law analytical
A 2ld-order Runge-Kntta
T 4lb-order Runge-Kutta
i i i i i i i i i i i i i |
10.0 100.0
Loops Through Integration Algorithm for
1000.0
Figure 5-7. Accuracy of various integration techniques for a 12-hour
unimodal coagulation and growth simulation with a growth factor of 500.
150
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IN
Figure 5-8a
Dp [|im]
Figure 5-8b
Figure 5-8. Starting and ending distributions for a 12-hour
bimodal coagulation simulation with a growth factor of 0;
5-8a: number distribution; 5-8b: volume distribution.
151
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TABLE 5-4. AVERAGE GLOBAL ERROR (RELATIVE TO 4tM)RDER RUNGE-KUTTA
TECHNIQUE) OF VARIOUS INTEGRATION TECHNIQUES FOR A 12-HOUR
BIMODAL COAGULATION SIMULATION WITH A GROVTH FACTOR OF 0
Loops
8
16
32
64
128
256
512
Euler
0.442
0.194
8.72E-2
4.11E-2
2.01E-2
9.97E-3
4.96E-3
C-N
0.143
6.24E-2
2.90E-2
1.39E-2
6.82E-3
3.39E-3
1.68E-3
A-l
0.125
5.88E-2
2.80E-2
1.37E-2
6.77E-3
3.37E-3
1.68E-3
PL
7.99E-2
1.25E-2
3.44E-3
1.34E-3
5.00E-4
2.83E-^
1.52E-4
A-2
0.174
1.80E-2
4.43E-3
8.82E-4
1.52E-4
3.89E-5
2.12E-5
RK-2
1
2.59E-2
5.19E-3
1.08E-3
2.42E-4
7-74E-5
RK-4
1
6.31E-3
3.20E-4
9.88E-5
2.69E-5
9.26E-6
C-N Crank-Nicholson
A-l lst-order analytical
A-2 2nd-order analytical
PL Power-law analytical
RK-2/RK-4 2nd-/4th-ortier Runge-Kutta
"1" Indicates unrealistic solution
O CM.
1.0
1% global error
1"-order analytical
2ld-order analytical
* Power-law analytical
A 2*d-order Runge-Kutta
T 4th-order Runge-Kutta
' '' id.o ' 106.0
Loops Through Integration Algorithm for
1000.0
Figure 5-9. Accuracy of various integration techniques for a 12-hour
bimodal coagulation simulation with a growth factor of 0.
152
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<0
LU_
CO
O
O)
o
hrO hr!2
N,
Dfl
o..
N
j
>..j
1.00E6
0.0500
1500
1.00E5
0.100
1.800
L58E4
1.20
1.595
3.61E4
1.49
1.567
hr 12
D.01
1.0
dp [tun]
10.0
106.0
Figure 5-lOa
in
LU
p~
in
m
U
n
LU
=Hf>-
?
O)
JO
T3*
>
O
o.
hr 0 hr 12
v,
°«J
137
.0819
1.500
248
0282
1.800
3.81E4
2.31
1.595
1.54E5
2.73
1J67
hr 12
hrO
0.01
0.1
Dp [|im]
1.0
1C
.0
Figure 5-10b
Figure 5-10. Starting and ending distributions for a 12-hour
bimodal coagulation and growth simulation with a growth factor of 500;
5-10a: number distribution; 5-lOb: volume distribution.
153
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TABLE 5-5. AVERAGE GLOBAL ERROR (RELATIVE TO BORDER RUNGE-KUTTA
TECHNIQUE) OF VARIOUS INTEGRATION TECHNIQUES FOR A 12-HOUR BIMODAL
COAGULATION AND GROVTH SIMULATION VITH A GROVTH FACTOR OF 500
Loops
8
16
32
64
128
256
512
Euler
1
0.288
0.130
6.76E-2
2.99E-2
1.49E-2
7.37E-3
C-N
1
0.244
0.102
4.79E-2
2.24E-2
1.10E-2
5.40E-3
A-l
2.41
0.194
9.09E-2
4.50E-2
2.16E-2
1.08E-2
5.36E-3
PL
1.75
3.90E-2
1.14E-2
4.11E-3
1.59E-3
7.09E-^1
3.36E-4
A-2
1.69
2.85E-2
5.43E-3
1.32E-3
2.99E-^
7.41E-5
RK-2
1
6.49E-2
2.01E-2
6.12E-3
1.55E-3
4.08E-^
RK^
1
7.37E-3
1.36E-4
1.14E-5
9.90E-7
C-N Crank-Nicholson
A-l ls*-order analytical
A-2 2n
en
to
2 -order analytical
10.0 '106.0
Loops Through Integration Algorithm for
1000.0
Figure 5-11. Accuracy of various integration techniques for a 12-hour
bimodal coagulation and growth simulation with a growth factor of 500.
154
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CHAPTER 6
MODELING ATMOSPHERIC AEROSOL PROCESSES AND
TOPICS FOR CONTINUING WORK
In this thesis we presented the MAD modeling technique as a
computationally efficient way to model aerosol dynamics based on the modal
representation of aerosol size distributions. The MDEs were derived as the
governing equations used to simulate aerosol dynamics with MAD models. We
described techniques for evaluating the integrals of the MDEs, and gave examples of
how to evaluate these integrals for specific processes commonly occurring in aerosol
systems. A computationally efficient and stable technique for integrating the system
of MDEs was also derived.
In this chapter we make some closing remarks about MAD models, discuss
aspects of a model being developed by the U.S. Environmental Protection Agency
(EPA) for simulating atmospheric aerosol dynamics using MAD modeling
techniques, and indicate topics related to MAD modeling that should be addressed
in the future.
6.1 COMPARISONS OF MODAL, SECTIONAL, AND SPLINE MODELS
To evaluate the computational efficiency of modal models, they should be
compared with more precise models for simulations typical of those for which a
particular model will be used. Comparisons of this sort have been performed,
although the results have often been inconclusive because models have been
compared by simulating aerosol dynamics for single processes in a single
homogeneous control volume (see Seigneur et a/., 1986, and Tsang and Rao, 1988).
We question, however, whether such tests provide meaningful information about the
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suitability of a particular model for representing aerosol dynamics in a complex
system, such as the earth's atmosphere. To properly determine the suitability of a
model for representing the processes in a complex system, either a simulation of the
comprehensive processes occurring in that system should be performed or, if a
simplified set of processes is used to represent the full dynamics, experimental data
for that system should be used to determine the applicability of assumptions used in
simplified models (such as MAD models). For example: Seigneur et al. (1986)
compared spline, sectional, and modal models that were candidates for use in
atmospheric aerosol dynamics models. The comparisons were for simulations of
condensation or coagulation in a homogeneous system. The solution from the spline
model was accepted as the standard solution, and the sectional and MAD models
were evaluated by comparing each one's computational effort and resulting size
distribution to the computational effort and size distribution resulting from the
spline model.
It is important to clarify that the major differences between the models
compared by Seigneur et al. were in the mathematical representations of the aerosol
size distribution. Because this representation influences the computational effort
and accuracy of the resulting model, it is important to account for the major
processes that influence the size distribution. In the atmosphere, growth rates
probably vary significantly within the scale of most numerical grids (on the order of
tens of kilometers), so that mixing of aerosols growing at different rates in these
fluid packets probably leads to broader distributions than those predicted from
simulations of a homogeneous system. To compare numerical models used to
simulate atmospheric aerosol dynamics, therefore, the effects of inhomogeneity and
transport processes on the size distribution should be represented. One technique for
comparing models to be used in atmospheric aerosol dynamics modeling would be to
distribute the rates of processes over a number of air parcels, and to simulate the
mixing of these air parcels in the comparison. Such comparisons should more
accurately assess the performance of models to represent aerosol dynamics in
complex systems such as the atmosphere.
The MAD modeling technique, with the aerosol size distribution
represented by multimodal lognormal distribution functions, is being used by the
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U.S. Environmental Protection Agency (EPA) because of its computational
efficiency, and because multimodal lognormal distribution functions reasonably
represent atmospheric aerosol size distributions. Others, however, have chosen
sectional techniques for developing models to simulate aerosol dynamics in the
atmosphere to avoid the inherent limitation of MAD models: that the size
distribution must be represented with time-invariant functional forms of the
distribution function. There is not a consensus on the best modeling approach for
representing atmospheric aerosol dynamics, and decisions about the type of model to
use may depend as much on personal preference as on objective reasoning.
6.2 MAD MODELING OF ATMOSPHERIC AEROSOL DYNAMICS
The EPA's Atmospheric Research and Exposure Assessment Laboratory
(AREAL) is currently developing the Regional Particulate Model (RPM), a model
for simulating atmospheric chemistry and aerosol dynamics related to the sulfur
dioxide/nitrogen oxide/volatile organic compounds (S02/NOX/VOC) system, to
assess the impact of proposed emission standards on air quality, including visibility
and acid deposition levels. The RPM is being built on the existing Regional Acid
Deposition Model (RADM) framework for simulating gas-phase chemistry and
pollutant advection. Other submodels being included in the RPM are the Model for
Aerosol Reacting System (MARS) to simulate condensed-phase chemical reactions,
and a MAD model to simulate aerosol dynamics due to particle growth/evaporation,
sources/sinks, and coagulation.
6.2.1 The Task
Modeling a complex system such as the earth's atmosphere involves
mathematically representing intraphase and interphase dynamics. Dynamics of the
gas, vapor, and condensed-aerosol phases as well as transport to surfaces (in this
study, the earth) are all relevant. In addition, the influence of meteorology on
thermodynamic properties, cloud formation, and wind fields must be included.
Models of this complexity are often formulated by developing submodels for
representing associated groups of processes.
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The RPM requires separate submodels for the following components:
Gas-phase chemical reactions
Kinetically controlled reactions
Equilibrium reactions
Aerosol-phase reactions of absorbed gases
Kinetically controlled reactions
Equilibrium reactions
Cloud processes
Influence on photolysis rates
Addition of water to particles
Wet removal
Large-scale vertical transport by venting1 clouds
Transport of gases, vapors, and aerosols
Horizontal advection
Vertical advection and diffusion
Meteorology linkage to modeled processes
Thermodynamic properties
Cloud formation
Wind fields
Aerosol dynamics
Coagulation
Vapor nucleation
Vapor condensation
Growth by reactions of adsorbed/absorbed gases
Wet and dry sedimentation
Each of these six groups is typically handled by a single program module.
The subcategories in each group represent processes within a program module that
require mathematical formulation distinct from the other processes in the same
group. The next sections describe the models developed to represent each group of
processes. The discussions focus on the linkage of each subprocess to the aerosol
dynamics module.
Renting clouds transport air from the troposphere to the lower stratosphere.
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6.2.2 Background Information on the Regional Particulate Model (RPMl
The RPM is being built on the RADM platform, which is a
state-of-the-stience model for representing gas- and aqueous-phase processes. The
major program modules of the RADM are
Gas-phase chemistry
Aqueous-phase chemistry
Cloud processes
Advection2 and diffusion
The Penn State/NCAR Mesoscale Model Version 4 (MM4) (Anthes et ai,
1987) is used to supply the RADM with dynamically consistent meteorological
variables that provide the program modules listed above with the necessary
thermodynamic data. The MM4 also provides wind fields for the advection routine.
The RPM is being developed by incorporating aerosol dynamics into the
RADM. This will allow simulation of aerosol dynamics in the presence of the full
S02/NOX/VOC chemistry. A separate but related task of the AREAL is to assess
the impact of controlling emitted SO2 and particles on the resulting atmospheric
levels and deposition of acidic aerosols. After the RADM predicts species fields for
specific S02/NOX/VOC emission levels, modeled S02 emissions can be reduced from
the initial levels to assess how SO2 control strategies affect atmospheric
concentrations and deposition levels of sulfuric acid. A simplified model for
performing these simulations, called the Engineering Model (EM), is being
developed. The simplifications incorporated in the EM result in substantial
reductions of computational effort, allowing the EPA to perform simulations on a
lower class of computers3. The RPM is being developed by adding an aerosol
2 Advection is the transport of material due to the bulk motion of the atmosphere, and is similar
to convection.
3The RADM runs on the CRAY X-MP and typically requires two hours of CPU time, whereas
the simplified model (i.e., the EM) runs on a VAX and typically requires one hour of CPU time.
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dynamics model to the RADM, and the Engineering Aerosol Model (EAM) is being
developed by adding the same aerosol model to the EM.
The RPM and EAM are being developed as Eulerian models based on a
three-dimensional grid structure. The grid covers the entire eastern United States,
and the portions of Canada in the same longitudinal field that have significant
emissions of the species under investigation. The horizontal grid is 35x38, yielding
control volumes with time-invariant boundaries of about 80 km on each side. The
vertical grid contains 6 to 15 levels with time-variant boundaries. The vertical cell
dimension is specified in dimensionless a coordinates as
a = (P-P )/(P, -P ) (6-1)
v top-"v sfc top' v '
where P defines the top of the calculation domain and is always 100 millibars.
top r
P, is the pressure at sea level. The values of a are user specified and are
time-invariant for an entire simulation period. Typical values for a and the
corresponding dimensional control-volume heights are listed in Table 6-1.
TABLE 6-1. STANDARD RADM LAYER DEFINITIONS
Layer
1
2
3
4
5
6
a range
1.0 -0.98
0.98 - 0.93
0.93 - 0.84
0.84 - 0.6
0.6 -0.3
0.3 -0.0
Height [km]
0.0 - 0.15
0.15- 0.55
0.55- 1.3
1.3 - 3.7
3.7 - 7.7
7.7 -16.0
Note: the layer heights calculated from the ff values are based on the NAG A
standard atmosphere, a sealevel surface elevation, and a P. of 100 mb.
top
As the meteorological conditions change during a simulation, both P and the
altitude corresponding to P = 100 millibars vary, resulting in time-variant
top
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control-volume heights4.
Two of the RPM's functions are to estimate budgets for H2S04 and HNOs
in the vapor and aerosol phases, and to estimate acid deposition levels from each
control volume. To track the acidic species, the chemistry modules calculate H2S04
and HNOs formation from emitted 862 and NOX, respectively, and 35 other species
(listed in Appendix L). Chemical reactions occur in the gas phase, on the surface of
wetted particles, and inside droplets. Within aerosol particles the initial oxidizing
reactions of SO 2 are relatively slow, so that chemical kinetics calculations are
required.
H2S04 and HNOs are formed by gas-phase chemical reactions. SO2 and
NOX oxidation rates in the RADM are based on the mechanism of Stockwell (1986),
and a description of the formulation of the RADM gas-phase chemistry module
based on this mechanism is presented by Chang et al. (1987).
6.2.3.1 Aerosol Growth by HaSO4
The conversion rates of S02 to H2S04 in the atmosphere often lead to
highly supersaturated H2S04 concentrations (due to the relatively low saturation
vapor pressure of H2S04), so that once it is formed it is assumed either to condense
on the existing aerosol or to nucleate and form new particles. Because H2S04 is
usually highly supersaturated, the evaporation term in the particle condensational
growth laws (e.g., Eqs. [4-41a] and [4*48a]) can be neglected. Computationally this
implies that the coupling between the aerosol phase and the gas and vapor phases
for H2S04 is as follows:
4The model is formulated in terms of the a coordinates, so the effect of the timevariant grid cell
heights is implicitly accounted for by the calculation procedure. Calculations of the total mass in
each grid cell, however, require the time variation of the grid cell heights to be computed.
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SO2 is converted to H2S04 by gas-phase chemical reactions.
H2S04 is transferred from the vapor phase to the aerosol phase by
nucleation and condensation.
H2S04 does not evaporate from the aerosol.
NncleationMcMurry and Friedlander (1979) developed a theory for
determining new particle5 formation rates as a function of B^SCU monomer
formation rates and the aerosol surface area; because surface area drives
condensation and condensation competes for the available B^SCU monomers. The
key assumption in their work is that nucleation is collision controlled. This
assumption is valid for systems where the critical size of stable clusters, dp, is much
smaller that the nucleating monomer. Because dp decreases as 5 increases (see Eq.
[4-42]), this theory should apply for high supersaturation of the nucleating species.
We have therefore adopted this theory for representing HjSC^ nucleation in the
atmosphere, because we assume that 5 for H2S04 is sufficiently high that nucleation
is collision controlled.
McMurry and Friedlander's theory predicts the generation rate of particles
of arbitrary size, and so can be used to predict the generation rate of a spectrum of
particle sizes. This model was later improved by McMurry (1983) by removing some
of the original limiting assumptions. For our purposes, we select a particle size at
which the particles produced by nucleation are "transferred" to an aerosol mode. In
this way the entire nucleation process is treated by McMurry's model, and the
influence of other processes on the newly formed particles occurs only after the
nucleated particles are transferred to one of the aerosol modes.
Because nucleation and condensation occur simultaneously, the H 28(1)4
monomer production rate must be partitioned between the two competing processes.
In the RPM, therefore, McMurry's theory is used to estimate nucleation rates, and
the vapor that does not nucleate is assumed to condense on the existing aerosol. The
mass generation rate of H2864 monomer is therefore partitioned between nucleation
5McMurry refers to cluster formation rates, but because we are typically interested in clusters with
more than 100 monomers each we refer to these large clusters as particles.
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and condensation such that
[M,] = [M 1 + [M] (6-2)
L 3JH2S04 L Jnuc L ^cnd
A problem with representing nucleation in a general model is that
nucleation often produces large number concentrations of very small particles (i.e.,
dp < 0.01 fan). Introducing high number concentrations into the nuclei mode causes
the numerical time integration algorithm for the MDEs to execute extremely small
time steps to simulate coagulation for this mode. Because McMurry's theory
predicts the number flux for arbitrary particle sizes, however, the flux of a relatively
large particle size (in the range of the existing nuclei mode) is simulated. The
resulting moment source rate is therefore for a nucleated aerosol that has undergone
substantial coagulation after nucleation, resulting in a number concentration of
larger particles that is significantly lower than the number concentration of the
same aerosol at smaller particle sizes. Because nuclei modes are often in the range of
0.01 fan, we apply McMurry's nucleation theory to predict the source rate of 0.01
fan. particles. For a flux of particles at this size, the moments of the nucleating
aerosol are approximated as follows:
The particle size for nucleation is selected as 0.01 fan, and the generation
rate of particles of this size is calculated from the nucleation theory of
McMurry (1983).
The injection rate of M corresponding to this flux is
=I(dp )dp3 (6-3)
v^' ^ v '
where dp is the diameter of the nucleated aerosol, and I(dp ) is the
flux of particles of this size. McMurry's theory is used to estimate the
injection rate of M,, and the other moment source rates are estimated by
O
assuming that the size distribution of the nucleated aerosol can be
represented by a lognormal distribution with (Z>gn) = 0.01 fan and
(<7g) =1.5. The lognormal assumption is reasonable because growth
to this size is dominated by coagulation, which produces distributions
that are approximately lognormal. Selecting (ag) = 1.5 is reasonable
because asymptotic values for ag of a coagulating aerosol (in the absence
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of other effects) are about 1.35 (Lee, 1983). Observed nuclei modes in the
atmosphere are often characterized by o-g a 1.6 (Whitby, 1978a), so
selecting (ag) = 1.5 is a compromise between Lee's theoretical results
and observed values. It is doubtful that the model will be sensitive to the
precise value chosen for ag.
For the moment relationships for a lognormal distribution (Appendix C),
and for a nucleating aerosol characterized by (-Ogn)nuc = 0.01 ^m and
= 1.5, the other moment rates of the nucleating aerosol are
N
fi\ . 1__^ (6-4a)
J mir t \ 3 r . + Z.T / \ n
/nuc
[Ml =\N] (Dgn)^uc exp{181n2[(ag)nuc]} (6-4b)
L J mir L J niir
1 nuc u J nuc
These moment rates are added to the moment rate expressions for the
other processes.
The preceding nucleation theory and resulting modal nucleation rates apply
to situations where nucleation rates are limited by the collisional rate of the
nucleating monomers, and are therefore only applicable for conditions of high
supersaturation. McMurry (University of Minnesota, Minneapolis, Minnesota, 1990)
cautions, however, that this theory was validated against smog chamber data, and
its ability to accurately represent nucleation processes in the atmosphere has not
been demonstrated.
At issue also is the basic question of whether nucleation is important
enough to warrant inclusion in atmospheric aerosol models. A key assumption in the
RPM is that all of the I^SCU formed in the gas phase is transferred to the aerosol
phase. The only benefit of estimating the fraction of the H2S04 that enters the
aerosol phase through nucleation is to introduce a portion of the IhSCU as new
particles. It is likely, however, that direct injection of particles into the atmosphere
from industrial sources is more important than the source of new particles due to
nucleation. Thus, it may be true that nucleation processes are negligible in
comparison to other participate sources, and that additional effort should be
164
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directed at better characterizing the size distribution and magnitude of new particle
injection rates from industrial sources.
Condensation Because the vapor pressure of B^SCU is low, all of the
vapor that does not form new particles by nucleation is assumed to condense on the
existing aerosol. For each time step of the model, therefore, the total H2S04
monomer source rate is partitioned between nucleation and condensation by first
calculating the rate of change of M due to nucleation and then determining the
o
resulting condensation rate for M,, as
o
[M] = |M] -[M] (e-s)
L 3-lcnd L 3-lH2S04 L 3Jnuc
Equation (6-5) is the total amount condensing on all modes. This rate is then
partitioned between the modes. The procedure for partitioning the total
condensation rates between modes is described in Section 3.7.1.2, Eqs. (3-18)
through (3-21). For condensation, N= 0 and the expression for M is derived from
Eq. (3-22).
6.2.3.2 Aerosol Growth by HNOj
Nitric acid vapor is produced by photochemical reactions in the gas phase.
Nitric acid and ammonia (NH3) form ammonium nitrate (NIUNOs) by the
equilibrium reaction
and the NB^NOs so produced exists as a precipitate in the liquid aerosols (in cloud
droplets or in the liquid envelope surrounding wetted particles). The chemistry of
this formation is discussed by Basset and Seinfeld (1983) and Saxeena et al. (1986).
For atmospheric conditions HNOs saturation vapor pressures are about 20
mm Eg, so that HNOs is always subsaturated and nucleation of HNOs does not
occur. The Henry's law coefficient for HN03 is on the order of 105 (units!!!), so it
will be almost entirely absorbed into cloud droplets or into the liquid envelope
165
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surrounding wetted particles, and will only desorb when droplets evaporate. In
general, therefore, HNOs cycles in and out of the aerosol and aerosol growth due to
nitric acid is calculated according to the expressions developed in Sections 4.6.2.1
through 4.6.2.3.
6.2.4 Clond Processes
Clouds are key players in atmospheric chemistry and aerosol dynamics, and
affect the aerosol system as follows:
Addition of water to the particles to form droplets
Photolysis rate changes caused by blocking solar radiation
Wet removal caused by rainout of large droplets or ice crystals
Large-scale vertical transport caused by venting clouds
Clouds are highly inhomogeneous, as evidenced by the nonuniform structure
observable with the naked eye. Clouds may exist as continuous layers or discrete
masses, and are typically on the order of a few hundred to a few thousand meters in
thickness. From empirical correlations of temperature and specific humidity data,
the clouded fraction of each control volume can be computed. The remainder of the
control volume is assumed to be cloud-free. The algorithm for representing cloud
processes is called the cloud processor. The cloud processor operates on the clouded
portion of the control volume, and the aerosol remaining after being processed in the
cloud is mixed with the aerosol that was unaffected by the cloud, by the following
mixing relationship
cloud clouc .Uud + (l ~ 4oudXM* .Wfree t6'
t t t
where / , , = clouded fraction of the control volume
'cloud
(M, ) = ftb moment in the clouded portion of the control volume
t cou after the clould processor is completed
, ) , , , = fcth moment in the cloud-free portion of the
k /cloud-free ,11
control volume
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To determine (AT ) i d> aerosol dynamics are simulated in the cloud
t
processor. As aerosol approaches and enters a cloud, water condenses on the
particles. The volume of water condensing is a function of the water vapor
supersaturation, and the distribution of the condensing water on the aerosol is
controlled by the growth dynamics of condensation (see Section 3.7.1.2).
As a first attempt to represent aerosol dynamics in clouds, empirical
relationships will be used to determine the aerosol size distribution in the clouds.
Because the particles experience pure condensation going into a cloud, the number
concentration does not change. The total mass of water deposited on the particles is
determined from the supersaturation of the moist air. Some technique is required for
apportioning the condensing water vapor between the available modes. To do this,
water vapor condensation on the aerosol is simulated in the absence of other
processes. If the Kelvin effect is neglected,6 then all of the aerosol is activated. Also
in the absence of the Kelvin effect, condensational growth will cause each aerosol
mode to become monodisperse. Because it is not realistic to represent the aerosol in
clouds of an 80 km by 80 km control volume as monodisperse, a lower bound is
placed on ag. From the data of Meischner and Bogel (1988), the distribution widths
of cloud droplets can be reasonably represented by
-------
developed for the RPM.
6.2.5 Aerosol Transport
Transport processes included in the RPM are
horizontal advection,
vertical advection and diffusion, and
venting clouds.
The transport schemes used in the RPM are the same as those used in the RADM,
and are discussed by Change et d. (1987).
Particles in the atmosphere that interact in the physico-chemical processes
leading to acid and visibility-degrading aerosols are sufficiently small that we
assume they are transported the same as any of the other gas-phase species.
Therefore, the RPM does not need to be modified to account for aerosol transport.
6.2.6 Meteorology
Meteorology controls the following processes:
Thermodynamic properties: temperature, pressure, and specific humidity
Cloud formation
Wind fields
For a particular simulation, a meteorology file with the temperature,
pressure, specific humidity, and wind fields is prepared by the MM4 before
beginning a simulation with the RPM. During a particular run, hourly values of
these variables are read and used for calculating the respective properties. The
presence of clouds is deduced from the thermodynamic data by applying empirical
correlations of temperature and specific humidity data.
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Typically, the RPM will be used to simulate episodic events for days when
measurements of all pertinent meteorological and pollutant data are available.
Resulting model predictions can therefore be compared to actual data to evaluate
the model's performance. The evaluated model can then be run for reductions in the
source terms.
For the calculation domain used in the RPM, meteorology data must be
available in every control volume. Experimental measurements are usually available
for only a portion of the control volumes, so the conditions in the other control
volumes must be estimated with a predictive meteorology model, such as the MM4.
The quality of the surface pressure and precipitation fields predicted by the
MM4 is checked by comparing them to observational data in the control volumes,
when such data are available.
6.3 MULTICOMPONENT MAD MODELS
An important aspect of MAD modeling that we have not addressed is the
technique for representing multicomponent aerosols. There are three representations
of multicomponent aerosols:
Externally mixed
Internally mixed, size-independent composition
Internally mixed, size-dependent composition
The formulation of MAD models presented in this thesis represents multicomponent
aerosol as an external mixture, where each distinct chemical species is represented
with a separate mode. However, as aerosols from modes of distinct chemical
composition coagulate, and/or as vapors of different chemical composition condense
on each mode, the modes become an internal mixture of multiple species. The
formulation of the MDEs must therefore be generalized to represent internal
multicomponent mixtures.
169
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In the RPM the aerosol will be treated as an external mixture of modes,
each of which can contain a size-dependent internal mixture of chemical species.
Eventually the model will be extended to allow for size-independent internal
mixtures, similar to the technique used by Brock et cd. (1988), who used
multiplicative lognormal distribution functions to represent the distribution of
chemical species for a single mode. With such a representation, the distribution of
material within each mode can be more accurately represented than by assuming a
size-independent distribution, and the attractive features of working with lognormal
distribution functions are retained.
6.3.1 Multicomponent Representation
For systems with more than one chemical component, moment equations
must be written for each chemical species as well as for the total moments of each
mode. In this section we present one technique for deriving multicomponent
moment equations, assuming size-independent chemical composition within each
mode.
We represent the integral moments of mode i by N., M , and M . The
1 O . 0 .
t t
chemical composition can be represented by the volume fraction of the species in
each mode. Because M is proportional to aerosol volume, we write
"8
M =\ M (6-7a)
O . ^^J O . '
* 5=1 **
\i=\l\ (6'7b)
where 5 represents a particular chemical species and ris is the total number of
chemical species.
It is also useful to separate the volatile and nonvolatile components of the
distribution because during evaporation the volatile components will evaporate until
the aerosol is composed only of nonvolatile species. We therefore index the species
as follows:
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5=1: nonvolatile species 1
5 = 2: nonvolatile species 2
5 = MS : nonvolatile species
5 = ris + 1: volatile species 1
5 = rig +2: volatile species 2
nv
5 = rig = ns + ris : volatile species HS
The fractions of the mode that are volatile and nonvolatile material are therefore
calculated as
The mixture bulk density is
5=1
"S
kl = X
P;~
\, "*
3. 5=1
t
(6-8a)
(6-9)
V ;
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6.3.2 Multicomponent MDEs
Because the chemical composition within a mode is completely specified by
y . , species moment equations are required only for M . In the same way derived
*,« >,
linearized equations for the total moments of each mode (see Chapter 5), moment
equations for each component species can be derived as
M3 =53 +C3 M3 (6-10)
l,t 1,5 t,« t
Equation (6-10) expresses the time rate of change of a single component in terms of
the total moment, because the rates of most processes (e.g., particle growth and
coagulation) are sensitive to the total moment. Equation (6-10), however, cannot be
integrated using the analytical integration techniques derived in Chapter 5, because
a a
-jftM is a function of M (-ijM, would have to be expressed as a function of
M i 1,1
M to permit analytical integration).
The general moment equation for M is now written as
«5 .
t
where h represents specific processes, such as coagulation and particle growth.
Equation (6-11) is linearized according to the techniques described in Chapter 5 to
yield
S3 + C3M (6-12)
This assumes the rates are independent of chemical composition within the time
step. By solving Eq. (6-12) with one of the analytical integration techniques
described in Chapter 5, M (t) is expressed analytically as
O .
172
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M, = M. + 5, A* (for C = 0) (6-13a)
' I M0 '
Mk = \\Mk] + Sk /Ck] [expk Aill - Sk/Ck (for Ck * 0) (6-13b)
' U *J0 * «Jl L « JJ » * »
Because Eqs. (6-13) express M, explicitly as a function of time, the species moment
>
equations represented by Eq. (6-10) can be integrated using a technique like the
Runge-Kutta numerical integration techniques.
6.4 MISCELLANEOUS TOPICS FOR FUTURE WORK
6.4.1 Processes Resulting in Lognormal Distributions
In formulating the foundational assumption of MAD models we noted that
velocity distributions in laminar flow situations could be well represented by an
analytical function (i.e., parabolic functions), and that in an analogous manner we
should be able to represent aerosol size distributions by one or more analytical
distribution functions. From experimental observations we noted that multiple,
additive lognormal distribution functions often reasonably represent atmospheric
and combustion generated aerosol size distributions.
The weakness in this analogy, however, is that parabolic functions can be
derived analytically as the appropriate function with which to represent
laminar-flow velocity distributions, but we have relied on empirical data for
justification of use of multimodal lognormal distribution functions. It would be
desirable to establish a definitive theoretical basis for the conditions under which
lognormal distributions are appropriate. Indeed, some theories have been developed
to explain the origins of lognonnal distributions (see Gibrat 1930, 1931; Kapteyn,
1903; Kolmogoroff, 1941; Shimizu and Crow, 1988). These theories do not enjoy
general acceptance in the aerosol community, however. Further work is required to
clarify these theories and demonstrate their applicability to aerosol systems such as
the atmosphere.
173
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To motivate future work in this area, we note that the central feature of the
theories for the existence of normal distributions (of which lognormal distribution
are one example) is that the governing processes are randomly distributed in some
sense. The functional form of the randomly distributed processes determines
whether the resulting distributions are better represented by a normal or lognormal
distribution. The key to establishing a link between the processes acting on a system
and the resulting form of the size distribution may lie in determining the degree to
which process rates can be viewed as randomly distributed.
Many gaseous species in the atmosphere, as well as wind speeds, are
lognormally distributed (Larsen, 1971; Bencala and Seinfeld, 1986). The often
discrete and dispersed nature of clouds indicate the presence of distributions of
water vapor and thermodynamic conditions in the atmosphere. These distributions
of gases, wind speeds, and water vapor result in a high degree of stratification in the
atmosphere of aerosol growth rates. This stratification may be viewed as a
randomization of aerosol process rates. Turbulence may also play an important role
in mixing packets of fluid that have grown at different rates to produce a
smeared-out distribution.
Clearly demonstrating the link between the degree of stratification,
turbulence levels, and the degree to which the resulting distribution can be
represented by one or more additive lognormal distribution functions, would greatly
aid in objectively determining the conditions under which multimodal lognormal
distributions could be used in MAD models.
The ultimate goal of such a theory would be to formulate one or more
dimensionless groups that would be used to determine the conditions under which
multimodal lognormal distribution functions could be properly applied. This is in
the same sense that the Reynolds number is used to determine the conditions under
which the parabolic function well represents the velocity distribution of a flowing
fluid.
174
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6.4.2 Effect of Clouds on Stratification
In the previous section we focused on the need to identify processes leading
to the existence of multimodal lognormal distributions. We suggested that
stratification of the process rates, combined with turbulence, may be a key factor.
One source of high degrees of stratification may be clouds. Because clouds add
significant amounts of water to the aerosol, and this significantly affects chemical
conversion of gas-phase species to condensed-phase species, clouds may be sources of
high degrees of stratification of growth rates. Combined with the large-scale air
movement and turbulence surrounding clouds, they may strongly influence the
shape of the aerosol size distribution.
6.4.3 Model Comparisons Accounting for Stratification
In the previous sections we raised the issue of the importance of
stratification on the resulting size distribution. Another item for future study,
therefore, is the need to account for stratification of process rates in code
comparisons used to screen aerosol models for use in atmospheric models.
Traditionally, comparisons have been performed for a single process acting in a
homogeneous control volume (see, for example, Seigneur et al [1986]), and from this
comparison, judgments are made about the relative accuracy of each model to
represent atmospheric processes. Because the principle difference between all aerosol
models is the mathematical representation of the aerosol size distribution (the rate
coefficients for representing the dynamics of each process are the same), these
comparisons really serve to demonstrate the suitability of approximating the actual
size distribution by a particular mathematical function. But if the comparisons
really serve to validate a particular mathematical representation of the aerosol size
distribution, the comparison should simulate the the major processes influencing the
size distribution. If the stratification within a system significantly influences the
aerosol size distribution, such comparisons must represent this stratification in some
manner. To our knowledge, however, this has not been done, but must be included if
such comparisons are to provide meaningful guidance in model selection.
175
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6.4.4 Mixing of Modes
For simulations of inhomogeneous systems where fluid packets from
neighboring control volumes mix, determinations must be made about into which
mode material transferred into a control volume should be introduced, or if a new
mode should be initiated. Such decisions have usually been avoided by maintaining
sufficient numbers of modes, so that no new modes were created, and modes were
never allowed to completely disappear. For developing models to simulate specific
systems this approach is acceptable, because the user can develop the program
according to his knowledge of the modeled system. For developing general models,
however, the program should be able to automatically predict the appearance and
disappearance of modes. Algorithms for making these decisions must therefore be
developed.
6.4.5 Fractal Dimensions
For aerosol dynamics models to be of general utility, limiting assumptions
must be removed as the models are applied to systems of increasing complexity. A
potentially important generalization of MAD models that are applied to combustion
systems is to account for the fractal geometry of aggregates. Such a generalization is
not trivial, but may be important for accurately representing diffusion and
coagulation rates between chain-like agglomerates, such as soot produced by Diesel
engines. It is not clear to us how this generalization can best be made, but such a
generalization may be required when MAD models are used to simulate the
dynamics of chain-like agglomerates.
6.4.6 Program Modules a*>d Documentation
A practical item for further work is to develop a library of routines for
MAD modeling, and a corresponding user's manual for constructing such a model.
The goal is to produce computer codes and documentation similar to the codes and
documentation that exist for sectional models. The difficulty in this endeavor,
however, is that development of general-purpose computer codes and the
corresponding documentation is expensive, and usually must be included as part of a
176
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major research program. It is therefore possible that MAD models will remain of
limited use unless they enjoy sufficient acceptance to warrant such a costly
development effort.
177
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187
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APPENDIX A
METHOD OF MOMENTS FOR PARTICLE GROWTH
A.1 GENERAL MOMENT GROWTH INTEGRAL
The moment GDE for particle growth is
CD
-------
The term ddp/dvp is
Substituting Eq. (A-4) into Eq. (A -3) and integrating yields the general expression
for dMjdfc
(A-5)
Substituting k = 0 into Eq. (A-5), the expression for dMQfdt is
j = 0 (A- 6)
By writing the moment equation for M a closed set of equations is obtained
(A-Ta)
(A-7b)
A.3 EXAMPLE OF A SET OF MOMENT EQUATIONS THAT CANNOT BE
CLOSED
For condensational particle growth in the continuum regime, the particle
growth law is
Lcn
Substituting Eq. (A-4) and Eq. (A-8) into Eq. (A-l) and integrating yields
f,) = (2Ar/T)* M (A-9)
* lcn **
The resulting system of equations based on Eq. (A-9) is
A-2
-------
= (e/r^Jlfj (A-lOa)
(A-lOb)
r_2) = (-2/r)*T M_3 (A-lOc)
Lcn
and this system of equations cannot be closed.
A-3
-------
APPENDIX B
ANALYTICAL SOLUTION OF THE MODAL DYNAMICS EQUATION
FOR PARTICLE GROWTH
B.I GENERAL DERIVATION
An analytical solution has been derived for the time variation of the size
distribution of an aerosol undergoing growth, where the size distribution is assumed
to remain lognonnal at all times.
For particle growth in the absence of other effects, the growth law for many
processes can be represented by a power function of the form
where *T represents thermodynamic and gas transport properties, and n is the
growth law order. Situations where Eq. (B-1) apply include condensational growth
in the free-molecule regime (K = 2), condensation in the continuum regime (K = 1),
growth by surface reactions (K 2), and growth by reactions inside droplets
(K = 3). The time rate of change of the total volume can be determined by
integrating the particle growth-law over a size distribution as
Substituting a lognormal distribution function for n in Eq. (B-2) and integrating
yields
= *T */2 N Dn exp[«/2 In(<7g)] (B - 3)
B-1
-------
Another moment equation can be derived for the surface area by noting
that
(B-4)
Substituting Eq. (B-4) and (B-l) into Eq. (B-2) and integrating yields a second
moment equation
exp[(«-l)2/2 In2(ag)] (B-5)
where 5 is the surface area concentration.
An additional moment equation is for the total number concentration,
which does not change for particle growth in the absence of all other dynamical
processes, and is therefore
= 0 (B-6)
Equations (B-3), (B-4), and (B-6) are the three moment equations used to solve for
the time evolution of a lognormal distribution function, and are three equations in
five unknowns. Two additional equations in terms of the five unknowns are needed
to specify the time dependence of Dgn and ag. Two additional equations are
obtained by writing V and 5 in terms of N, Dgn, and ag as
V = T/6 N Dgn exp[4.5 In2(ag)] (B 7)
Differentiating Eqs. (B-7) and (B-8) yields
& V) = ,/2 tf 4- exp[4.5 mVs)] + i.ja (B-9)
B-2
-------
Equations (B-3), (B-5), (B-6), (B-9), and (B-10) are 5 equations and 5 unknowns.
In order to simplify these equations it is useful to make the following substitutions:
i=exp[(3-K)ln (ag)]
(B-lla)
(B-llb)
Equating Eq. (B-3) to (B-9) and Eq. (B-5) to (B-10) and making the
change of variables represented by Eqs. (B-ll) yields
Equating Eq. (B-12) to (B-13) and simplifying, a relationship between
can be written as
11.5
(FT)
For initial conditions
at Z =
Eq. (B 14) can be solved to yield
0.5
Backsubstituting for x yields a relationship between Dgn and
-------
Sgn = 4/(3-"WRnVg)Kexp[(3-K)ln2(crg)] - l}l°-5/(«-3)J (B.lfi)
where BQ = D$f> exp[(3-/c)ln2(ag0)] yexp[(3-«)ln2(ago)] - 1
To determine the relationship between ag and time, Eqs. (B 15) and
(B-14) are substituted into Eq. (B-12) to yield
dT - X T7TT (B-17)
where T = J adt
Equation (B-17) is solved for particular values of K to give relationships
between x (indirectly ag) and time. Substituting the resulting expressions for <7g into
Eq. (B 16) yields relationships between Dgn and time.
B.2 EXAMPLE SOLUTIONS
Consider the solution of Eq. (B-17) for the following cases.
K= 0
« = 1: diffusion limited growth in the continuum regime.
K = 2: kinetic controlled growth in the free-molecule regime.
K = 3: volume growth due to reactions inside a droplet
B.2.1 K = 0 Solution
For K = 0 Eq. (B-17) becomes,
B-4
-------
Substituting
6(1 - I)
x = y
dx = 2y dj/
1'5
(B-19)
(B-20a)
(B-20b)
Equation (B-17) becomes
dr
3(j/ -
Eq (B-21) can be integrated to give
C0nst'
Backsubstituting in terms of x, and letting i = XQ at r = TO
(B-21)
(B-22)
/XQ~ + 3 r
Bo
+ 51
(B-23)
Backsubstitution for x gives an equation for ffe(t). Substituting the
resulting expression for 0g(f) in Eq. (B-16) yields a relationship for D^(f).
B.2.2 K = 1 Solution
Substituting K, = 1 into Eq. (B-17) yields
dr
Bo
4(z - 1)
1.5
(B-24)
B-5
-------
Eq. (B 24) can be integrated directly to give
1 + const. (B-25)
Substituting x = XQ for t = 0 and solving for x
x =
-2
+ 1 (B-26)
Backsubstitution for x gives an equation for ae(t). Substituting the
resulting expression for ag(t) in Eq. (B-16) yields a relationship for Dgn(t).
B.2.3 K = 2 Solution
Substituting « = 2 into Eq. (B-17) yields
dx
2 (x - 1)L5
Making the substitution of Eq. (B-20) allows Eq. (B-27) to be written as
Equation (B-28) can be integrated, and is given by integral #182 from the 58th
edition of the CRC Handbook of Chemistry and Physics as
-£-= y ln(y + \y2 - 1) + const. (B-29)
0 r~2
\y2- i
Backsubstituting for y and setting x = XQ at t = 0 yields
B-6
-------
,
I
(B-30)
It does not appear that an explicit solution of Eq. (B 30) in terms of x is possible, so
an iteration scheme is required to solve for <7g. The time dependence of Dgn can be
determined by substituting the values for ag (determined from the iterative
solution) into Eq. (B-16).
B.2.4 K = 3 Solution
For « = 3, Eq. (B-17) goes to m. To overcome this problem consider Eq.
(B-14). For K = 3 Eq. (B-17) reduces to
, e dx dl /TD o-iN
L5^=3I (B'31)
which has a solution only if
Jf=° (B-32)
Therefore,
ag=
-------
APPENDIX C
MOMENT RELATIONSHIPS FOR THE LOGNORMAL DISTRIBUTION
The integral moments of a distribution are defined as,
which can also be expressed in terms of ln(dp) as
The lognormal number distribution function is
exp
-O.f
In2a
It is convenient to make a change of variables as
(CM)
(C-2)
(C-3)
dp = Dg
di = d(lndp)/lnog
Using this change of variables and substituting Eq. (C-3) into Eq. (C-2) yields
C-l
-------
M, = f "expf-O.S i + fcrlngn exp(A;2/2 In2ag) (C-5b)
By defining average moments M, and M, as
M = D^ exp(Jk2!/2 In2ag) (C 6a)
*l
M = Dg^ exp(Jb2/2 In2(7g) (C-6b)
*2
and solving for Dgn and <7g, the conversion equations between D and <7g; and M,
*
and M, are
*2
(c . 7a)
(C-rb)
where r = ki
Substituting Eqs. (C-7a) and (C-7b) into Eq. (C-5b) yields an expression
for other moments of the distribution in terms of the moments M, and M,
C-2
-------
where
C-3
-------
APPENDIX D
MOMENT CHANGE OF TWO PARTICLES
DURING BINARY COAGULATION
We make the following assumptions about binary coagulation:
The two colliding particles form one new particle
The volume of the new particle equals the sum of the volume of the two
individual particles
The new particle is spherical
Representing the volume of the colliding particles as Vp and vp_, respectively, the
volume of the new particle is
which can be written in terms of particle diameter as
The diameter of the new particle is therefore
-------
APPENDIX E
FORMS OF THE MODAL DYNAMICS EQUATION
The MDEs for a bimodal aerosol are
) = - V-YM - ?. r-d* c(dp)n.(dp)ddp + V- f*4 D(dp^n.(dp)ddp
0 °
Intramodal Coagulation
Intermodal Coagulation
«8
5=1 °
E-l
-------
+
Intramodal Coagulation
Intermodal Coagulation
5=1 ° j
If a vapor deposition rate is specified for particle growth, the particle growth term is
written as
o ^
.v
J
For a CSTAR the MDEs become
E-2
-------
"in < nout
=l R
Intramodal Coagulation
E-3
Intermodal Coagulation
+ ~ ^rr I dp i(\dp)n(dp)
-------
/in=l
-
nout nsu -
Intramodal Coagulation
Intermodal Coagulation
- (E.3b)
5=1 J R d<
E-4
-------
APPENDIX F
MASS-BASED AEROSOL DYNAMICS EQUATIONS
For variable density situations, another form of the continuous GDE is
n
obtained by converting from number concentration (e.g., [#/cm ]) to number
density (e.g., [#/gm]). Noting that
n = pe(n/pg) = psn (F-l)
and substituting Eq. (E-l) into the continuous GDE (Eq. [2-2]) yields
M ^ * . . ~ ,«J . ,1V
O(Vn.
0
(F-2)
growth
The unsteady term can be expanded as
The fluid convection term can be expanded as
V-v/?gn(Vp) = pgv-Vn(flp) + n(Vp)V-vpg (F-4)
F-1
-------
Combining the unsteady and convection terms yields
For incompressible flow, mass continuity can be expressed by the following
relationship
0 (F-6)
so that Eq. (F-5) reduces to
(F-7)
Expanding the term corresponding to motion due to external forces yields
V-c(Vp)pg7i(^) = c(Vp)/7g-Vn(Vp) + n(Vp)V-c(Vp)pg (F-8)
Substituting Eqs. (F-7) and (F-8) into Eq. (F-2) and simplifying yields
= _ v.
+ b&n(«p) + n^Vp) (F-9)
L Jg
The MDEs can also be converted from a concentration based to a mass
based expression by substituting
in Eq. (2 5) and simplifying. Noting that
F-2
-------
c, = ^i/M, )\ dp c(dp)n {dp)ddp (F-ll)
t i
and
the mass based MDEs foi modes t and j are
Intramodal Coagulation
Intennodal Coagulation
F-3
m
i i i i i i i *
-------
= - v.VAf4 - Cjfc -VAf4- (Af4 /pg)V-c4 pg + (M
Intramodal Coagulation
Intermodal Coagulation
F-4
-------
APPENDIX G
NUMERICAL EVALUATION OF DOUBLE INTEGRALS
G.I INTRODUCTION
In the MDEs double integrals of the form
(G-l)
must be evaluated for coagulation. In this appendix we present the mathematical
forms for evaluating this double integral numerically using Gauss-Legendre and
Gauss-Hermite techniques for cases when the independent variable is expressed by a
logarithmic transformation.
G.2 GAUSS-LEGENDRE INTEGRATION
The numerical equivalent of Eq. (G-l) for the Gauss-Legendre integration
technique is
» rT{[ln(dPl /dPl )]/2}{[ln(dp2 /dp )]/2}
u 1 u 'I
(G-2)
where the nodes at which the integrands are evaluated are calculated according to
G-1
-------
p . p (G-3a)
gl u Xl gll u n
= [In(dp2 /dp2 )]/2 *gl + [ln(<*P2 ^ )]/2 (G 3b)
As for single integrals, the limits of integration for each integral are determined
according to relationships of the form of Eqs. (4-8) (4-10), except that for double
integrals the limits of each integral must be set according to the functional
dependence of the independent variable of each integral. For n represented by
lognormal distribution functions, « and K representing the lower and upper order
.1* J.
1 u
of the integrand with respect to dp., respectively, and « and K representing the
1 u
lower and upper order of the integrand with respect to dp , respectively, the limits of
2i
the integral are
dp =DgK 4 (G-4b)
u 1
u
^2=^/4 (G'4C)
! 21
^2 =Z?e« 4 (G-4d)
u 2
u
where D^ = £>gniexp(«1 In2agl) (Q.4e)
- . l ^Si) (G-4f)
1 u
u
lng2) (G.4g)
21 !
Inag2 (G-4h)
u
For example, consider intermodal coagulation in the free-molecule regime.
The system of coagulation integrals for mode i is
G-2
-------
d
(G-5a)
(G-5b)
(G-5c)
For coagulation in the free-molecule regime,
(G-6a)
xO.5
(G-6b)
An approximate form of /? that is instructive for estimating the order of the
integrand with respect to dp and dp. is
J. ^
(G-7a)
5
. O
.5
1
5
O
(G-7b)
The lowermost and uppermost order of Eqs. (G-7) with respect to dp. are 1.5 and
2, respectively. For Eqs. (G-5) this implies
«2j= -1-5 ; *2U = 2
for Eq. (G-5a) «ij = -1.5 ; K^ = 2
for Eq. (G-5b) /tij = 1.5 ; /c^ = 5
for Eq. (G-5c) K = 4.5 ; K^ = 8
(G-8a)
(G-8b)
(G-8c)
(G-8d)
G-3
-------
G.3 GAUSS-HERMITE INTEGRATION
For Gauss-Hennite integration, double integrals of the form
(G-9a)
are evaluated. Equation (G-9a) can also be expressed as
-CD ./-CD
D~ )exp(z~ )da>
/Cf v /C2 &l
(G-9b)
p2
where K\ and /c2 are the average order of the integrand with respect to dPl and dp2,
respectively. For Gauss-Hermite integration Eq. (G-9b) is replaced with a sum over
the integrand as
aW
*V
(G-10)
For the intermodal coagulation integrals (Eqs. [G-5]) evaluated in the free-molecule
regime, the average orders of the integrals using the approximation of /3fm from
Eqs. (G-7) are:
= 0.25
(G-lla)
forEq. (
forEq. (
forEq. (G-5c) M = 6.
(G-llb)
(G-llc)
(G-lld)
If the system of integrals represented by Eqs. (G-5) is evaluated with a common set
G-4
-------
of nodes for all three integrals, the proper centering powers are:
K2 = 0.25 (G-12a)
«i = 3.25 (G-12b)
where K\ and KZ are substituted for KI and £2, respectively, in Eq. (G-10).
G-5
-------
APPENDIX H
ANALYTICAL COAGULATION INTEGRALS
OF THE MODAL DYNAMICS EQUATION
USING LOGNORMAL DISTRIBUTION FUNCTIONS
H.1 THE TASK
To analytically evaluate the integrals of the MDEs when n. and n. are
represented by lognormal distribution functions, the size dependent terms of the
integrand must be expressed as power functions of dp. This results in single integrals
of the form
=-NI%D. exp(«2/2 In2
-------
Intramodal Coagulation: Mode t and j
d
(H-3b)
(H-3c)
***< 4. mt
Intermodal Coagulation: Mode t
(H-4a)
Intermodal Coagulation: Mode j
(H-5a)
) (H.5b)
(H-5c)
The coagulation integrals are evaluated for mathematical forms of /?
applicable in the free-molecule and continuum/near-continuum regimes. The
transition regime is handled by forming the harmonic mean of the integrals resulting
from using free-molecule and continuum/near -continuum regime expressions for /3 in
Eqs. (H-3) through (H-5).
H-2
-------
The analytical expressions for the coagulation integrals of the MDEs
presented here are written in a form that is conducive for implementation into a
computer program. We therefore define the following factors that are used in the
expressions for the coagulation integrals:
A = 1.392Kng'0783
SQGDgn. = SQDgn .
SQTDgn . = SQ6Dgn . SQDgn .
R8=RB
Rn = 1/R
RI2 = (1/R)2
RIS = (1/R)3
ESG = exp[(l/8)ln2ag]
ESG^ = ESG4
ESG16 = ESG16
H-3
-------
H.2.1 Intennodal Coaulation
H.2.1.1 Coagulation in the Free-Molecule Regime (Kn > 10)
For the free-molecule regime, an approximate form of /3 that permits
analytical evaluation of Eqs. (H-3) through (H-5) is given by Eq. (4-35) as
0.5 r , 3. ,2 ,
. Op2 Op2 Op1 Op1
(H-6a)
which was obtained by substituting the approximation
!5 + 1/425)
(H-6b)
into Eq. (5-33). Substituting Eq. (H-6) into Eqs. (H-4) and (H-5), yields
(ESG. + R ESG . + 2Rs ESG . ESG^. + R4 ESGP. ESGlff .
v » 3 » 3 « 3
+ RlS ESGiff. ESGP. -I- IRli ESG-/. ESG . )
« 3 » 31
(H-7a)
= Ni
(ESG^P. + R ESG5ff . ESG . + 2R2 ESG25.
v» i j t
t. ESGP.
t. ESG .
. + R4 ESGP. ESG 16.
^ i j
(H- 7b)
(ESGltfP.-l- J
+ RlS ESG250. ESGP. + 2RI1 ESG196. ESG . )
(ESG^P. ESG5ff. + R
v » j
= AT. AT.
2R2 ESG2J.
*
i ;
(H- 7c)
.) (H-7d)
7 v '
H-4
-------
The b factors are used to correct the integrals obtained with the approximate form
of ft represented by (H-8) (see the discussion in Section 4.5.3). The b factors are
calculated as
'OD r
-------
H.2.1.2 Coagulation in the Continnnm/Near-Continuum Regime (Kn < 1)
For the continuum/near-continuum regime the coagulation rate coefficient
is
where A> = [*Brcrc/(3«/*4p)] (H-9b)
(H-9c)
A = 1.392Kng'0783
Kng =
Substituting Eq. (H-9) into Eqs. (H-4) and (H-5) yields the following expressions:
= N. N. K
[2
+ A. Kng. (ESG^. + R2 ESG16. ESG^. )
.. .
+ (R2 + RlS)(E,SG4i ESG^J ] (H- lOa)
= N. N.
[2 ESGS6.
+ A. Kng|.
^.+ RI2ESG64.ESG16.)
3 i :'
+ R2ESG16.ESGJ.+ AKESGff^ESG^.] (H-lOb)
[2
+ A. Kngj. (ESGlOO. + R2 ESG6J. ESG-/. )
i 3
+ R2 ESGlOO. ESGV-+ ^J^ESGiPtf.ESG^.] (H-10c)
H-6
-------
[2 ESG55. ESGS6.
+ A. KnJ. (ESG1S. ESGS6. + R2 ESGj.
+ A. Kng* (ESG36. ESG16. + Rl2 ESGtf
+ R2 ESG16. ESG 10)
Substituting Eq. (H-6) into Eq. (H-3) yields
(ESG . + ESG25. + 2 ESG5. ) (H lla)
(ESGW. + 2 ESGW. + ESGlo^. ) (H- lib)
Lee et d. (1984) reported correction factors for intramodal coagulation
(6^ ') and showed that b\ ' can be expressed as a unique function of ae. To generate
a table of 6 factors, therefore, Eq. (H-8) is evaluated for a matrix of values of ag. A
recommended set is
ag = 1.0001, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25
For this set of parameters, the index (expressed in standard FORTRAN
nomenclature) to use with the table lookup procedure is calculated as
Iff = MAX(l,MIN(10,NINT((ag-0.75)/0.25)))
6
H-7
-------
We rederived the correction factors for intramodal coagulation using the
Gauss-Legendre numerical integration technique to evaluate the double integral in
the numerator of Eq. (H-8). The correction factors determined using this technique
for the values of ag shown above are compared in Table H 1 to the values reported
by Lee et al. (1984).
TABLE H-l. INTRAMODAL COAGULATION CORRECTION FACTORS
*g
1.0
1.25
1.5
1.75
2.0
2.25
2.5
2.75
3.0
4.0
5.0
6.0
7.0
This
&0
0.7071
0.7261
0.7664
0.8141
0.8617
0.9036
0.9366
0.9601
0.9756
0.9969
0.9996
0.9999
1.0000
Work
&6
0.7071
0.7261
0.7664
0.8141
0.8617
0.9036
0.9366
0.9601
0.9756
0.9969
0.9996
0.9999
1.0000
Lee
bQ
0.7071
0.7663
0.8360
0.8636
0.8755
66
0.7071
0.7637
0.7689
0.7596
0.7550
Because Eq. (H-6b) becomes an equality as one particle size dominates the
other, the intramodal and intennodal coagulation correction factors approach unity
when the integrand is dominated by terms where one particle size is much larger
than the other. This occurs as <7g increases. The values of K ^ reported in this work
and shown in Table H-l confirm this behavior. The values Lee et al. (1984) report
for b^\ however, indicate that b^ ' first increases with increasing ag, and then
decreases. This is contrary to the expected behavior, and the discrepancy in the
values obtained by Lee et al. and ourselves may result from differences in the
numerical integration techniques used to evaluate Eq. (H-8).
H-8
-------
H.2.2.2 Coagulation in the Continuum/Near-Continuum Regime (Kn < 0.1)
Substituting Eq. (H-9) into Eq. (H-3) and integrating yields
[1 + ESG*. + A. Kng fESGto. + ESG-f . )] (H 12a)
ji^'^WMWW** = 2N*. KJ&.
[ESG7«. + ESG80. + A. Kng. (ESG52. + ESG68. )] (H- 12b)
H.2.3 The Generalized Coagulation Coefficients (all Kn)
Pratsinis (1988) found that by averaging the intramodal coagulation
integrals for the free molecule (Eq. [H-ll]) and continuum/near-continuum
(Eq. [H-12]) expressions, generalized coagulation rates of the form
can be calculated that are valid for all size regimes. Applying this averaging method
to the intermodal coagulation integrals of Eqs. (H- 7) and (H-10), and to the
intramodal coagulation integrals of Eqs. (H-ll) and (H-12) results in integral
values that are within 20% of numerical evaluations of Eqs. (H-3) through (H-5)
and using the Fuchs' (Fuchs, 1964) interpolation expression for /?.
H-9
-------
APPENDIX I
CORRECTION FACTORS FOR THE
FREE-MOLECULE INTERMODAL COAGULATION INTEGRALS
OF THE MODAL DYNAMICS EQUATIONS
In Appendix J analytical coagulation integrals for the MDEs were specified
for the free-molecule and continuum/near-continuum regimes. For the free-molecule
regime an approximation to /? was required to secure an analytical integral.
Correction factors for correcting the resulting values calculated from the analytical
integrals are defined as
IT
J^
/"CD fa>
J0 J0 4
where the integral in the denominator of Eq. (! 1) is evaluated from Eq. (H- 7) and
the integral in the numerator is evaluated numerically using Gauss-Legendre or
Gauss-Hermite integration. In this Appendix we report correction factors for the
following integrals:
Intennodal Coagulation: Mode t
(L2b)
1-1
-------
Intermodal Coagulation: Mode j
The integrals are evaluated at the following sets of parameters.
R2 = 1.0, >#, 2, 2v2, 4, 4v2, 8, 8v#, 16,
<7g. = 1.0001, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25
ffg*. = 1.0001, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.25
The intermodal correction factors for
are:
Jo"Jo" ^imUi Uj ddpl
» /*OD /"
-------
R2=2
0.78358,0.79304,0.81445,0.84105,0.86873,0.89491,0.91805,0.93744,0.95300,0.96510
0.78874,0.79822,0.81954,0.84587,0.87312,0.89875,0.92127,0.94005,0.95506,0.96668
0.80106,0.81043,0.83133,0.85686,0.88299,0.90727,0.92835,0.94574,0.95950,0.97006
0.81741,0.82647,0.84653,0.87076,0.89524,0.91766,0.93684,0.95245,0.96466,0.97393
0.83565,0.84423,0.86311,0.88566,0.90810,0.92834,0.94540,0.95908,0.96965,0.97761
0.85434,0.86233,0.87976,0.90034,0.92051,0.93841,0.95328,0.96505,0.97404,0.98075
0.87254,0.87983,0.89564,0.91405,0.93184,0.94740,0.96014,0.97010,0.97764,0.98320
0.88957,0.89611,0.91017,0.92636,0.94177,0.95506,0.96581,0.97413,0.98036,0.98492
0.90503,0.91078,0.92306,0.93703,0.95016,0.96134,0.97028,0.97712,0.98221,0.98591
0.91866,0.92363,0.93416,0.94599,0.95700,0.96626,0.97358,0.97914,0.98324,0.98619
R2=
0.84432,0.85223,0.86990,0.89131,0.91280,0.93223,0.94861,0.96172,0.97185,0.97945
0.84817,0.85608,0.87367,0.89483,0.91591,0.93485,0.95073,0.96339,0.97312,0.98040
0.85728,0.86513,0.88235,0.90278,0.92284,0.94062,0.95535,0.96697,0.97582,0.98241
0.86937,0.87697,0.89346,0.91271,0.93130,0.94753,0.96078,0.97111,0.97890,0.98465
0.88282,0.88999,0.90541,0.92314,0.93999,0.95446,0.96612,0.97509,0.98180,0.98671
0.89644,0.90306,0.91717,0.93317,0.94814,0.96081,0.97088,0.97855,0.98423,0.98836
0.90940,0.91540,0.92807,0.94225,0.95533,0.96626,0.97484,0.98131,0.98606,0.98950
0.92120,0.92654,0.93773,0.95010,0.96137,0.97067,0.97790,0.98330,0.98724,0.99007
0.93153,0.93621,0.94594,0.95659,0.96618,0.97401,0.98004,0.98451,0.98775,0.99006
0.94027,0.94431,0.95266,0.96171,0.96978,0.97631,0.98129,0.98496,0.98760,0.98947
0.89577,0.90190,0.91522,0.93076,0.94575,0.95876,0.96932,0.97751,0.98367,0.98820
0.89877,0.90483,0.91797,0.93323,0.94784,0.96046,0.97065,0.97852,0.98442,0.98875
0.90562,0.91152,0.92418,0.93871,0.95243,0.96414,0.97350,0.98067,0.98600,0.98989
0.91438,0.91999,0.93191,0.94538,0.95792,0.96846,0.97678,0.98310,0.98775,0.99113
0.92382,0.92904,0.94001,0.95221,0.96339,0.97267,0.97991,0.98535,0.98933,0.99220
0.93312,0.93786,0.94773,0.95857,0.96836,0.97639,0.98258,0.98719,0.99054,0.99295
0.94168,0.94591,0.95464,0.96411,0.97256,0.97941,0.98464,0.98851,0.99130,0.99329
0.94918,0.95289,0.96049,0.96865,0.97586,0.98164,0.98603,0.98924,0.99155,0.99319
0.95543,0.95865,0.96519,0.97214,0.97823,0.98307,0.98671,0.98937,0.99127,0.99261
0.96038,0.96313,0.96870,0.97457,0.97967,0.98369,0.98670,0.98888,0.99043,0.99153
R2=
0.93335,0.93777,0.94711,0.95764,0.96741,0.97562,0.98210,0.98701,0.99064,0.99327
0.93560,0.93992,0.94905,0.95929,0.96876,0.97668,0.98290,0.98761,0.99107,0.99358
0.94060,0.94471,0.95332,0.96290,0.97167,0.97893,0.98460,0.98885,0.99196,0.99421
0.94672,0.95053,0.95846,0.96718,0.97506,0.98151,0.98650,0.99021,0.99292,0.99486
0.95304,0.95650,0.96365,0.97141,0.97833,0.98394,0.98823,0.99141,0.99370,0.99535
0.95900,0.96208,0.96840,0.97518,0.98116,0.98596,0.98960,0.99228,0.99420,0.99557
0.96425,0.96696,0.97245,0.97829,0.98339,0.98745,0.99050,0.99273,0.99433,0.99546
0.96860,0.97094,0.97566,0.98064,0.98494,0.98834,0.99088,0.99272,0.99404,0.99497
0.97196,0.97396,0.97798,0.98217,0.98578,0.98860,0.99070,0.99222,0.99330,0.99406
0.97430,0.97600,0.97938,0.98290,0.98590,0.98823,0.98996,0.99121,0.99209,0.99271
1-3
-------
#2=8
0.95858,0.96158,0.96780,0.97460,0.98073,0.98575,0.98963,0.99252,0.99463,0.99615
0.96017,0.96308,0.96911,0.97567,0.98158,0.98639,0.99011,0.99287,0.99488,0.99633
0.96363,0.96634,0.97192,0.97797,0.98337,0.98774,0.99109,0.99357,0.99537,0.99666
0.96772,0.97018,0.97520,0.98061,0.98539,0.98923,0.99215,0.99430,0.99585,0.99696
0.97175,0.97394,0.97838,0.98311,0.98725,0.99055,0.99304,0.99487,0.99618,0.99712
0.97537,0.97729,0.98114,0.98521,0.98875,0.99154,0.99364,0.99517,0.99627,0.99704
0.97837,0.98002,0.98332,0.98679,0.98977,0.99211,0.99386,0.99513,0.99603,0.99667
0.98064,0.98204,0.98485,0.98776,0.99026,0.99221,0.99365,0.99470,0.99544,0.99596
0.98212,0.98331,0.98567,0.98811,0.99018,0.99179,0.99298,0.99384,0.99445,0.99487
0.98281,0.98381,0.98578,0.98781,0.98952,0.99085,0.99183,0.99253,0.99302,0.99337
R2=
0.97470,0.97666,0.98064,0.98491,0.98867,0.99169,0.99399,0.99569,0.99691,0.99779
0.97576,0.97765,0.98149,0.98558,0.98919,0.99207,0.99427,0.99588,0.99705,0.99788
0.97804,0.97977,0.98328,0.98700,0.99026,0.99285,0.99482,0.99626,0.99730,0.99804
0.98065,0.98219,0.98529,0.98857,0.99142,0.99368,0.99538,0.99662,0.99751,0.99815
0.98312,0.98446,0.98715,0.98998,0.99242,0.99434,0.99578,0.99683,0.99758,0.99811
0.98519,0.98635,0.98865,0.99105,0.99311,0.99473,0.99593,0.99680,0.99743,0.99787
0.98675,0.98773,0.98967,0.99169,0.99342,0.99476,0.99576,0.99648,0.99699,0.99735
0.98771,0.98854,0.99017,0.99185,0.99328,0.99439,0.99521,0.99581,0.99622,0.99652
0.98805,0.98874,0.99010,0.99150,0.99268,0.99359,0.99426,0.99474,0.99509,0.99533
0.98775,0.98833,0.98945,0.99060,0.99157,0.99232,0.99287,0.99326,0.99354,0.99373
R2= 16
0.98470,0.98594,0.98844,0.99106,0.99334,0.99514,0.99650,0.99749,0.99821,0.99872
0.98539,0.98658,0.98896,0.99147,0.99365,0.99537,0.99666,0.99760,0.99828,0.99876
0.98682,0.98791,0.99006,0.99232,0.99427,0.99580,0.99696,0.99780,0.99840,0.99882
0.98842,0.98937,0.99125,0.99321,0.99490,0.99623,0.99722,0.99794,0.99845,0.99882
0.98985,0.99067,0.99228,0.99395,0.99538,0.99650,0.99734,0.99794,0.99837,0.99868
0.99094,0.99163,0.99299,0.99440,0.99560,0.99653,0.99722,0.99772,0.99808,0.99833
0.99160,0.99218,0.99332,0.99449,0.99548,0.99625,0.99682,0.99724,0.99753,0.99773
0.99177,0.99226,0.99320,0.99417,0.99499,0.99562,0.99609,0.99643,0.99666,0.99683
0.99142,0.99183,0.99261,0.99341,0.99408,0.99460,0.99498,0.99525,0.99544,0.99558
0.99054,0.99088,0.99152,0.99217,0.99272,0.99314,0.99345,0.99367,0.99383,0.99394
R2=
0.99080,0.99158,0.99311,0.99471,0.99607,0.99715,0.99795,0.99853,0.99895,0.99925
0.99123,0.99197,0.99343,0.99495,0.99625,0.99727,0.99804,0.99859,0.99899,0.99927
0.99210,0.99277,0.99408,0.99544,0.99660,0.99751,0.99818,0.99867,0.99902,0.99927
0.99303,0.99361,0.99474,0.99592,0.99691,0.99769,0.99827,0.99869,0.99899,0.99920
0.99380,0.99429,0.99525,0.99624,0.99708,0.99773,0.99822,0.99857,0.99882,0.99899
0.99427,0.99469,0.99549,0.99632,0.99701,0.99756,0.99796,0.99824,0.99845,0.99859
0.99439,0.99474,0.99540,0.99608,0.99665,0.99710,0.99743,0.99766,0.99783,0.99795
0.99409,0.99438,0.99492,0.99548,0.99595,0.99632,0.99658,0.99677,0.99691,0.99700
0.99334,0.99358,0.99403,0.99448,0.99487,0.99516,0.99538,0.99553,0.99564,0.99572
0.99212,0.99232,0.99268,0.99305,0.99337,0.99360,0.99378,0.99390,0.99399,0.99405
1-4
-------
The intermodal correction factors for
b^ =
IT ^ A°"i "
3' f" f" 41
Jo Jo 1 fm ' j
are:
R2=l
a,. 1.00 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25
5,
1.00 0.70708,0.71681,0.73821,0.76477,0.79350,0.82265,0.85090,0.87717,0.90069,0.92097
1.25 0.72172,0.73022,0.74927,0.77324,0.79936,0.82601,0.85199,0.87637,0.89843,0.91774
1.5 0.78291,0.78896,0.80286,0.82070,0.84022,0.85997,0.87901,0.89669,0.91258,0.92647
1.75 0.87760,0.88147,0.89025,0.90127,0.91291,0.92420,0.93452,0.94355,0.95113,0.95726
2.0 0.94988,0.95184,0.95612,0.96122,0.96628,0.97085,0.97467,0.97763,0.97971,0.98089
2.25 0.98318,0.98393,0.98551,0.98728,0.98889,0.99014,0.99095,0.99124,0.99100,0.99020
2.5 0.99480,0.99504,0.99551,0.99598,0.99629,0.99635,0.99611,0.99550,0.99450,0.99306
2.75 0.99842,0.99848,0.99858,0.99861,0.99850,0.99819,0.99762,0.99674,0.99550,0.99388
3.0 0.99951,0.99951,0.99949,0.99939,0.99915,0.99872,0.99805,0.99709,0.99579,0.99411
3.25 0.99984,0.99982,0.99976,0.99962,0.99934,0.99888,0.99818,0.99719,0.99587,0.99417
R2= fl
0.72957,0.73993,0.76303,0.79178,0.82245,0.85270,0.88085,0.90578,0.92691,0.94415
0.72319,0.73320,0.75547,0.78323,0.81307,0.84287,0.87107,0.89651,0.91852,0.93683
0.74413,0.75205,0.76998,0.79269,0.81746,0.84258,0.86685,0.88938,0.90953,0.92695
0.82588,0.83113,0.84309,0.85825,0.87456,0.89072,0.90594,0.91972,0.93178,0.94203
0.91886,0.92179,0.92831,0.93624,0.94434,0.95192,0.95856,0.96409,0.96845,0.97164
0.97129,0.97252,0.97515,0.97818,0.98108,0.98354,0.98542,0.98665,0.98721,0.98709
0.99104,0.99145,0.99230,0.99320,0.99394,0.99439,0.99448,0.99416,0.99340,0.99217
0.99730,0.99741,0.99763,0.99779,0.99782,0.99762,0.99715,0.99636,0.99519,0.99363
0.99917,0.99919,0.99921,0.99915,0.99895,0.99856,0.99792,0.99698,0.99570,0.99404
0.99973,0.99973,0.99968,0.99955,0.99928,0.99883,0.99814,0.99716,0.99584,0.99415
1-5
-------
0.78358,0.79304,0.81445,0.84105,0.86873,0.89491,0.91805,0.93743,0.95300,0.96510
0.76412,0.77404,0.79635,0.82404,0.85312,0.88101,0.90610,0.92751,0.94500,0.95879
0.74239,0.75182,0.77301,0.79956,0.82809,0.85639,0.88291,0.90658,0.92683,0.94350
0.78072,0.78758,0.80317,0.82293,0.84437,0.86589,0.88643,0.90526,0.92194,0.93625
0.87627,0.88044,0.88981,0.90142,0.91357,0.92524,0.93585,0.94510,0.95285,0.95911
0.95176,0.95371,0.95796,0.96297,0.96792,0.97233,0.97599,0.97880,0.98072,0.98178
0.98453,0.98523,0.98670,0.98833,0.98980,0.99092,0.99160,0.99179,0.99145,0.99058
0.99534,0.99555,0.99597,0.99637,0.99662,0.99663,0.99633,0.99569,0.99465,0.99318
0.99859,0.99864,0.99872,0.99873,0.99860,0.99827,0.99768,0.99679,0.99555,0.99391
0.99956,0.99956,0.99953,0.99942,0.99918,0.99875,0.99807,0.99711,0.99580,0.99412
R2=
0.84432,0.85223,0.86990,0.89131,0.91280,0.93223,0.94861,0.96172,0.97185,0.97945
0.82299,0.83164,0.85101,0.87463,0.89857,0.92050,0.93923,0.95443,0.96629,0.97529
0.77870,0.78840,0.81011,0.83690,0.86477,0.89124,0.91476,0.93460,0.95063,0.96316
0.76386,0.77233,0.79147,0.81557,0.84149,0.86719,0.89126,0.91275,0.93116,0.94637
0.82927,0.83488,0.84756,0.86346,0.88040,0.89704,0.91257,0.92649,0.93857,0.94874
0.92184,0.92481,0.93136,0.93925,0.94724,0:95462,0.96104,0.96634,0.97048,0.97348
0.97341,0.97457,0.97706,0.97991,0.98260,0.98485,0.98654,0.98760,0.98801,0.98777
0.99192,0.99229,0.99305,0.99385,0.99449,0.99486,0.99487,0.99449,0.99367,0.99239
0.99758,0.99768,0.99787,0.99800,0.99799,0.99777,0.99727,0.99645,0.99527,0.99369
0.99926,0.99928,0.99928,0.99921,0.99900,0.99860,0.99795,0.99701,0.99572,0.99405
0.89577,0.90190,0.91522,0.93076,0.94575,0.95876,0.96932,0.97751,0.98367,0.98820
0.87860,0.88547,0.90052,0.91828,0.93557,0.95075,0.96319,0.97292,0.98028,0.98572
0.83381,0.84240,0.86141,0.88425,0.90707,0.92770,0.94510,0.95906,0.96986,0.97798
0.78530,0.79463,0.81550,0.84127,0.86813,0.89367,0.91642,0.93566,0.95125,0.96347
0.79614,0.80332,0.81957,0.84001,0.86190,0.88351,0.90368,0.92169,0.93718,0.95006
0.88192,0.88617,0.89565,0.90728,0.91931,0.93076,0.94107,0.94997,0.95739,0.96333
0.95509,0.95698,0.96105,0.96583,0.97048,0.97460,0.97796,0.98050,0.98218,0.98304
0.98596,0.98660,0.98794,0.98943,0.99074,0.99172,0.99227,0.99235,0.99192,0.99096
0.99581,0.99600,0.99637,0.99672,0.99691,0.99687,0.99653,0.99585,0.99478,0.99329
0.99873,0.99878,0.99884,0.99883,0.99869,0.99834,0.99774,0.99684,0.99558,0.99394
R2=
0.93335,0.93777,0.94711,0.95764,0.96741,0.97562,0.98210,0.98701,0.99064,0.99327
0.92142,0.92646,0.93723,0.94947,0.96096,0.97069,0.97842,0.98431,0.98868,0.99186
0.88678,0.89351,0.90810,0.92508,0.94138,0.95549,0.96693,0.97578,0.98243,0.98731
0.83249,0.84124,0.86051,0.88357,0.90655,0.92728,0.94477,0.95880,0.96964,0.97779
0.79593,0.80444,0.82355,0.84725,0.87211,0.89593,0.91735,0.93566,0.95066,0.96255
0.84124,0.84695,0.85980,0.87575,0.89256,0.90885,0.92383,0.93704,0.94830,0.95761
0.92721,0.93011,0.93647,0.94406,0.95166,0.95862,0.96460,0.96949,0.97326,0.97595
0.97573,0.97681,0.97913,0.98175,0.98421,0.98624,0.98772,0.98860,0.98885,0.98847
0.99271,0.99304,0.99373,0.99444,0.99499,0.99528,0.99522,0.99477,0.99390,0.99258
0.99782,0.99791,0.99807,0.99817,0.99813,0.99788,0.99737,0.99653,0.99533,0.99374
1-6
-------
#2=8
0.95858,0.96158,0.96780,0.97460,0.98073,0.98575,0.98963,0.99252,0.99463,0.99615
0.95091,0.95438,0.96163,0.96962,0.97688,0.98286,0.98751,0.99099,0.99353,0.99536
0.92751,0.93233,0.94255,0.95406,0.96473,0.97366,0.98070,0.98602,0.98994,0.99278
0.88371,0.89075,0.90595,0.92351,0.94028,0.95474,0.96642,0.97544,0.98220,0.98715
0.82880,0.83750,0.85671,0.87980,0.90297,0.92404,0.94195,0.95644,0.96772,0.97625
0.81933,0.82655,0.84279,0.86295,0.88412,0.90449,0.92295,0.93890,0.95215,0.96281
0.89099,0.89519,0.90448,0.91577,0.92732,0.93820,0.94789,0.95616,0.96297,0.96838
0.95886,0.96064,0.96448,0.96894,0.97324,0.97701,0.98004,0.98228,0.98371,0.98435
0.98727,0.98786,0.98908,0.99043,0.99160,0.99245,0.99288,0.99285,0.99234,0.99131
0.99621,0.99638,0.99671,0.99700,0.99715,0.99707,0.99670,0.99599,0.99489,0.99338
R2=
0.97470,0.97666,0.98064,0.98491,0.98867,0.99169,0.99399,0.99569,0.99691,0.99779
0.96996,0.97225,0.97693,0.98196,0.98643,0.99003,0.99279,0.99482,0.99630,0.99735
0.95523,0.95848,0.96522,0.97260,0.97925,0.98468,0.98888,0.99200,0.99427,0.99590
0.92524,0.93030,0.94098,0.95294,0.96397,0.97317,0.98038,0.98582,0.98981,0.99270
0.87576,0.88323,0.89935,0.91799,0.93583,0.95126,0.96377,0.97345,0.98072,0.98606
0.83078,0.83894,0.85705,0.87899,0.90126,0.92179,0.93950,0.95404,0.96551,0.97430
0.85727,0.86294,0.87558,0.89111,0.90723,0.92260,0.93645,0.94841,0.95838,0.96643
0.93337,0.93615,0.94220,0.94937,0.95647,0.96292,0.96840,0.97283,0.97619,0.97854
0.97790,0.97891,0.98105,0.98346,0.98569,0.98751,0.98879,0.98950,0.98961,0.98912
0.99337,0.99367,0.99430,0.99493,0.99541,0.99562,0.99551,0.99501,0.99410,0.99274
R2= 16
0.98470,0.98594,0.98844,0.99106,0.99334,0.99514,0.99650,0.99749,0.99821,0.99872
0.98184,0.98330,0.98624,0.98934,0.99205,0.99420,0.99582,0.99701,0.99787,0.99848
0.97288,0.97498,0.97927,0.98385,0.98789,0.99113,0.99360,0.99541,0.99673,0.99766
0.95403,0.95741,0.96440,0.97202,0.97887,0.98444,0.98872,0.99190,0.99421,0.99586
0.91845,0.92399,0.93567,0.94873,0.96076,0.97079,0.97865,0.98457,0.98892,0.99206
0.86762,0.87533,0.89202,0.91148,0.93027,0.94669,0.96013,0.97062,0.97855,0.98441
0.84550,0.85253,0.86816,0.88721,0.90671,0.92490,0.94083,0.95413,0.96481,0.97314
0.90138,0.90544,0.91437,0.92513,0.93602,0.94615,0.95506,0.96258,0.96868,0.97347
0.96248,0.96415,0.96773,0.97187,0.97583,0.97925,0.98198,0.98394,0.98514,0.98559
0.98837,0.98892,0.99005,0.99127,0.99232,0.99306,0.99339,0.99328,0.99269,0.99161
R2=
0.99080,0.99158,0.99311,0.99471,0.99607,0.99715,0.99795,0.99853,0.99895,0.99925
0.98910,0.99001,0.99182,0.99371,0.99533,0.99661,0.99757,0.99826,0.99876,0.99912
0.98374,0.98506,0.98772,0.99051,0.99294,0.99486,0.99630,0.99736,0.99812,0.99866
0.97238,0.97453,0.97892,0.98361,0.98773,0.99104,0.99354,0.99538,0.99671,0.99765
0.94961,0.95333,0.96103,0.96941,0.97693,0.98303,0.98772,0.99119,0.99371,0.99551
0.90943,0.91550,0.92834,0.94275,0.95608,0.96723,0.97600,0.98263,0.98751,0.99103
0.86454,0.87200,0.88829,0.90749,0.92630,0.94300,0.95687,0.96785,0.97626,0.98254
0.87498,0.88048,0.89264,0.90737,0.92240,0.93642,0.94877,0.95917,0.96762,0.97429
0.93946,0.94209,0.94781,0.95452,0.96111,0.96704,0.97203,0.97602,0.97900,0.98106
0.97977,0.98071,0.98270,0.98492,0.98695,0.98858,0.98970,0.99027,0.99026,0.98968
1-7
-------
The intennodal correction factors for
are:
R2=l
ff,. 1.00 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25
6,
1.00 0.70708,0.71681,0.73821,0.76477,0.79350,0.82265,0.85090,0.87717,0.90069,0.92097
1.25 0.73442,0.74195,0.75907,0.78083,0.80467,0.82906,0.85292,0.87543,0.89599,0.91418
1.5 0.86168,0.86563,0.87474,0.88643,0.89904,0.91150,0.92310,0.93344,0.94230,0.94962
1.75 0.96518,0.96656,0.96955,0.97303,0.97642,0.97939,0.98175,0.98342,0.98437,0.98459
2.0 0.99264,0.99289,0.99340,0.99391,0.99426,0.99436,0.99414,0.99357,0.99259,0.99117
2.25 0.99544,0.99546,0.99547,0.99541,0.99520,0.99481,0.99417,0.99323,0.99195,0.99029
2.5 0.99294,0.99292,0.99286,0.99271,0.99242,0.99196,0.99126,0.99027,0.98896,0.98727
2.75 0.98834,0.98832,0.98824,0.98808,0.98779,0.98732,0.98661,0.98563,0.98431,0.98262
3.0 0.98189,0.98187,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
3.25 0.97347,0.97345,0.97337,0.97321,0.97293,0.97246,0.97176,0.97079,0.96949,0.96783
R2 = V^
0.72957,0.73993,0.76303,0.79178,0.82245,0.85270,0.88085,0.90578,0.92691,0.94415
0.71701,0.72655,0.74774,0.77422,0.80287,0.83182,0.85967,0.88533,0.90803,0.92739
0.80565,0.81099,0.82332,0.83919,0.85653,0.87399,0.89069,0.90605,0.91973,0.93157
0.94296,0.94504,0.94965,0.95519,0.96078,0.96591,0.97028,0.97377,0.97631,0.97791
0.98874,0.98917,0.99006,0.99101,0.99180,0.99230,0.99244,0.99216,0.99143,0.99023
0.99491,0.99496,0.99502,0.99502,0.99489,0.99455,0.99395,0.99306,0.99181,0.99018
0.99287,0.99286,0.99280,0.99266,0.99239,0.99193,0.99124,0.99025,0.98894,0.98726
0.98833,0.98831,0.98824,0.98808,0.98779,0.98732,0.98661,0.98562,0.98431,0.98262
0.98189,0.98187,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97337,0.97321,0.97293,0.97246,0.97176,0.97079,0.96949,0.96783
1-8
-------
R2=2
0.78358,0.79304,0.81445,0.84105,0.86873,0.89491,0.91805,0.93743,0.95300,0.96510
0.74277,0.75292,0.77564,0.80392,0.83397,0.86334,0.89038,0.91402,0.93380,0.94975
0.75686,0.76398,0.78019,0.80083,0.82340,0.84632,0.86853,0.88924,0.90790,0.92422
0.90904,0.91208,0.91890,0.92734,0.93611,0.94443,0.95186,0.95816,0.96323,0.96707
0.98209,0.98280,0.98431,0.98599,0.98751,0.98868,0.98942,0.98966,0.98937,0.98854
0.99399,0.99409,0.99425,0.99436,0.99433,0.99409,0.99358,0.99275,0.99156,0.98998
0.99276,0.99276,0.99271,0.99258,0.99232,0.99188,0.99119,0.99022,0.98891,0.98723
0.98832,0.98830,0.98823,0.98807,0.98778,0.98731,0.98661,0.98562,0.98431,0.98262
0.98189,0.98187,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97337,0.97321,0.97293,0.97246,0.97177,0.97079,0.96949,0.96783
R2=
0.84432,0.85223,0.86990,0.89131,0.91280,0.93223,0.94861,0.96172,0.97185,0.97945
0.79700,0.80630,0.82721,0.85296,0.87949,0.90427,0.92590,0.94381,0.95804,0.96899
0.73790,0.74681,0.76683,0.79206,0.81938,0.84683,0.87300,0.89685,0.91770,0.93528
0.86268,0.86692,0.87655,0.88868,0.90159,0.91419,0.92581,0.93610,0.94487,0.95208
0.97085,0.97200,0.97448,0.97733,0.98005,0.98236,0.98412,0.98525,0.98573,0.98554
0.99241,0.99258,0.99291,0.99321,0.99337,0.99329,0.99292,0.99221,0.99112,0.98962
0.99257,0.99257,0.99255,0.99245,0.99221,0.99178,0.99112,0.99016,0.98887,0.98720
0.98830,0.98828,0.98821,0.98806,0.98777,0.98730,0.98660,0.98562,0.98430,0.98262
0.98189,0.98187,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97337,0.97321,0.97293,0.97246,0.97177,0.97079,0.96949,0.96783
R2=4
0.89577,0.90190,0.91522,0.93076,0.94575,0.95876,0.96932,0.97751,0.98367,0.98820
0.85591,0.86358,0.88056,0.90090,0.92105,0.93905,0.95406,0.96596,0.97507,0.98186
0.75962,0.76940,0.79131,0.81856,0.84735,0.87524,0.90062,0.92256,0.94070,0.95519
0.81089,0.81657,0.82953,0.84596,0.86370,0.88139,0.89814,0.91341,0.92688,0.93843
0.95230,0.95412,0.95807,0.96276,0.96740,0.97155,0.97500,0.97762,0.97940,0.98033
0.98968,0.98998,0.99058,0.99120,0.99167,0.99187,0.99175,0.99125,0.99034,0.98898
0.99223,0.99225,0.99227,0.99221,0.99201,0.99162,0.99099,0.99005,0.98878,0.98713
0.98826,0.98825,0.98818,0.98803,0.98775,0.98729,0.98659,0.98561,0.98430,0.98261
0.98188,0.98186,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97337,0.97321,0.97293,0.97246,0.97177,0.97079,0.96950,0.96783
0.93335,0.93777,0.94711,0.95764,0.96741,0.97562,0.98210,0.98701,0.99064,0.99327
0.90470,0.91051,0.92305,0.93755,0.95135,0.96322,0.97277,0.98011,0.98560,0.98962
0.80971,0.81891,0.83939,0.86430,0.88963,0.91300,0.93313,0.94960,0.96255,0.97243
0.77189,0.77923,0.79589,0.81701,0.83989,0.86287,0.88477,0.90479,0.92243,0.93747
0.92333,0.92607,0.93215,0.93952,0.94703,0.95400,0.96009,0.96510,0.96900,0.97179
0.98495,0.98546,0.98653,0.98768,0.98868,0.98937,0.98967,0.98954,0.98894,0.98784
0.99164,0.99170,0.99177,0.99179,0.99166,0.99133,0.99075,0.98986,0.98863,0.98701
0.98820,0.98819,0.98813,0.98799,0.98771,0.98726,0.98656,0.98559,0.98428,0.98260
0.98188,0.98186,0.98179,0.98163,0.98134,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97337,0.97322,0.97293,0.97246,0.97177,0.97079,0.96950,0.96784
1-9
-------
#2=8
0.95858,0.96158,0.96780,0.97460,0.98073,0.98575,0.98963,0.99252,0.99463,0.99615
0.93968,0.94378,0.95243,0.96208,0.97097,0.97837,0.98418,0.98855,0.99177,0.99409
0.86554,0.87310,0.88965,0.90916,0.92821,0.94498,0.95879,0.96962,0.97784,0.98393
0.76616,0.77500,0.79492,0.81991,0.84660,0.87281,0.89706,0.91841,0.93642,0.95107
0.88259,0.88652,0.89537,0.90629,0.91769,0.92860,0.93846,0.94701,0.95412,0.95981
0.97683,0.97768,0.97949,0.98154,0.98343,0.98495,0.98599,0.98649,0.98643,0.98579
0.99063,0.99073,0.99092,0.99105,0.99104,0.99082,0.99033,0.98952,0.98835,0.98678
0.98809,0.98808,0.98803,0.98791,0.98765,0.98720,0.98652,0.98555,0.98425,0.98258
0.98187,0.98185,0.98178,0.98162,0.98133,0.98087,0.98017,0.97919,0.97788,0.97621
0.97347,0.97345,0.97338,0.97322,0.97293,0.97246,0.97177,0.97079,0.96950,0.96784
R2=
0.97470,0.97666,0.98064,0.98491,0.98867,0.99169,0.99399,0.99569,0.99691,0.99779
0.96281,0.96556,0.97123,0.97739,0.98291,0.98741,0.99086,0.99343,0.99529,0.99663
0.91196,0.91756,0.92955,0.94323,0.95610,0.96701,0.97570,0.98233,0.98725,0.99083
0.79722,0.80653,0.82729,0.85272,0.87890,0.90342,0.92490,0.94277,0.95704,0.96807
0.83558,0.84093,0.85303,0.86817,0.88427,0.90005,0.91473,0.92787,0.93923,0.94879
0.96310,0.96449,0.96750,0.97099,0.97436,0.97727,0.97956,0.98116,0.98203,0.98219
0.98885,0.98904,0.98941,0.98976,0.98995,0.98992,0.98959,0.98892,0.98786,0.98638
0.98788,0.98789,0.98787,0.98776,0.98753,0.98711,0.98644,0.98549,0.98420,0.98254
0.98185,0.98184,0.98177,0.98161,0.98132,0.98086,0.98016,0.97918,0.97788,0.97620
0.97347,0.97345,0.97338,0.97322,0.97293,0.97246,0.97177,0.97079,0.96950,0.96784
R2= 16
0.98470,0.98594,0.98844,0.99106,0.99334,0.99514,0.99650,0.99749,0.99821,0.99872
0.97742,0.97919,0.98279,0.98662,0.98999,0.99268,0.99471,0.99621,0.99730,0.99807
0.94493,0.94879,0.95688,0.96583,0.97398,0.98071,0.98595,0.98986,0.99273,0.99479
0.84770,0.85603,0.87429,0.89592,0.9'1719,0.93612,0.95185,0.96432,0.97386,0.98097
0.79901,0.80593,0.82159,0.84131,0.86244,0.88333,0.90290,0.92044,0.93561,0.94830
0.94089,0.94309,0.94790,0.95362,0.95931,0.96445,0.96878,0.97218,0.97464,0.97618
0.98573,0.98607,0.98675,0.98747,0.98802,0.98831,0.98826,0.98782,0.98697,0.98566
0.98753,0.98755,0.98757,0.98751,0.98732,0.98694,0.98630,0.98538,0.98411,0.98247
0.98182,0.98181,0.98174,0.98159,0.98131,0.98084,0.98015,0.97917,0.97787,0.97620
0.97347,0.97345,0.97338,0.97322,0.97293,0.97246,0.97177,0.97080,0.96950,0.96784
R2 = 16v/S
0.99080,0.99158,0.99311,0.99471,0.99607,0.99715,0.99795,0.99853,0.99895,0.99925
0.98641,0.98753,0.98976,0.99210,0.99413,0.99573,0.99693,0.99781,0.99844,0.99888
0.96642,0.96896,0.97416,0.97978,0.98478,0.98882,0.99191,0.99420,0.99585,0.99704
0.89700,0.90346,0.91732,0.93319,0.94817,0.96094,0.97117,0.97900,0.98482,0.98907
0.79314,0.80148,0.82026,0.84365,0.86830,0.89210,0.91368,0.93229,0.94768,0.95999
0.90795,0.91126,0.91857,0.92744,0.93647,0.94491,0.95234,0.95857,0.96356,0.96735
0.98030,0.98088,0.98209,0.98342,0.98459,0.98544,0.98588,0.98587,0.98536,0.98435
0.98690,0.98696,0.98704,0.98706,0.98695,0.98663,0.98605,0.98518,0.98395,0.98234
0.98176,0.98175,0.98169,0.98154,0.98127,0.98082,0.98013,0.97916,0.97786,0.97619
0.97347,0.97345,0.97338,0.97322,0.97293,0.97246,0.97177,0.97080,0.96950,0.96784
MO
-------
The intennodal correction factors for
f"f" *1
Jo Jo
fe(l) _ JO JO
6. . .
J f(D /*CD ,
^1
Jo Jo
are:
R2= I
ffg.
a*. 1.00 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25
6*
1.00 0.70708,0.72172,0.78291,0.87760,0.94988,0.98318,0.99480,0.99842,0.99951,0.99984
1.25 0.72172,0.72615,0.76957,0.86021,0.94036,0.97976,0.99377,0.99813,0.99943,0.99982
1.5 0.78291,0.76957,0.76643,0.82411,0.91325,0.96897,0.99041,0.99715,0.99915,0.99974
1.75 0.87760,0.86021,0.82411,0.81411,0.87404,0.94729,0.98307,0.99500,0.99853,0.99956
2.0 0.94988,0.94036,0.91325,0.87404,0.86168,0.91495,0.96845,0.99047,0.99723,0.99918
2.25 0.98318,0.97976,0.96897,0.94729,0.91495,0.90360,0.94533,0.98127,0.99450,0.99841
2.5 0.99480,0.99377,0.99041,0.98307,0.96845,0.94533,0.93658,0.96610,0.98891,0.99677
2.75 0.99842,0.99813,0.99715,0.99500,0.99047,0.98127,0.96610,0.96010,0.97945,0.99342
3.0 0.99951,0.99943,0.99915,0.99853,0.99723,0.99450,0.98891,0.97945,0.97565,0.98769
3.25 0.99984,0.99982,0.99974,0.99956,0.99918,0.99841,0.99677,0.99342,0.98769,0.98540
R2= V2~
0.72957,0.76058,0.83853,0.92071,0.97000,0.99019,0.99697,0.99907,0.99970,0.99990
0.72319,0.74716,0.81913,0.90688,0.96404,0.98822,0.99639,0.99891,0.99967,0.99990
0.74413,0.74687,0.78550,0.87072,0.94580,0.98189,0.99448,0.99835,0.99950,0.99985
0.82588,0.81027,0.79383,0.83119,0.91250,0.96839,0.99029,0.99714,0.99915,0.99974
0.91886,0.90521,0.87165,0.84416,0.87621,0.94363,0.98157,0.99458,0.99842,0.99953
0.97129,0.96552,0.94804,0.91727,0.88980,0.91502,0.96503,0.98923,0.99689,0.99909
0.99104,0.98922,0.98334,0.97086,0.94847,0.92652,0.94483,0.97885,0.99368,0.99818
0.99730,0.99677,0.99504,0.99123,0.98333,0.96862,0.95323,0.96565,0.98738,0.99627
0.99917,0.99903,0.99853,0.99744,0.99513,0.99030,0.98112,0.97119,0.97919,0.99251
0.99973,0.99970,0.99956,0.99924,0.99858,0.99719,0.99428,0.98869,0.98260,0.98758
Ml
-------
R2=2
0.78358,0.81902,0.89070,0.95104,0.98229,0.99427,0.99822,0.99945,0.99982,0.99993
0.76412,0.79824,0.87329,0.94172,0.97875,0.99315,0.99789,0.99936,0.99980,0.99994
0.74239,0.76394,0.83140,0.91444,0.96755,0.98951,0.99681,0.99904,0.99971,0.99991
0.78072,0.77650,0.79774,0.87093,0.94473,0.98158,0.99443,0.99835,0.99951,0.99985
0.87627,0.86051,0.83274,0.84190,0.90849,0.96566,0.98943,0.99691,0.99909,0.99973
0.95176,0.94262,0.91719,0.88397,0.88544,0.93922,0.97924,0.99389,0.99823,0.99948
0.98453,0.98135,0.97131,0.95141,0.92379,0.92230,0.96162,0.98765,0.99643,0.99897
0.99534,0.99440,0.99134,0.98466,0.97147,0.95189,0.95005,0.97660,0.99271,0.99790
0.99859,0.99833,0.99745,0.99551,0.99140,0.98316,0.97040,0.96911,0.98602,0.99570
0.99956,0.99949,0.99924,0.99868,0.99750,0.99502,0.98998,0.98205,0.98137,0.99174
R2=
0.84432,0.87527,0.93028,0.97051,0.98959,0.99664,0.99895,0.99967,0.99989,0.99995
0.82299,0.85611,0.91770,0.96474,0.98753,0.99600,0.99877,0.99962,0.99988,0.99996
0.77870,0.81092,0.88243,0.94688,0.98094,0.99392,0.99814,0.99944,0.99983,0.99995
0.76386,0.77767,0.83330,0.91290,0.96686,0.98938,0.99680,0.99904,0.99971,0.99991
0.82927,0.81826,0.81694,0.86896,0.94043,0.97991,0.99397,0.99823,0.99947,0.99984
0.92184,0.90911,0.87961,0.86380,0.90564,0.96169,0.98806,0.99653,0.99899,0.99970
0.97341,0.96803,0.95186,0.92443,0.90578,0.93674,0.97641,0.99297,0.99798,0.99941
0.99192,0.99025,0.98488,0.97353,0.95375,0.93838,0.95987,0.98587,0.99587,0.99881
0.99758,0.99711,0.99554,0.99209,0.98495,0.97202,0.96130,0.97551,0.99166,0.99757
0.99926,0.99913,0.99868,0.99768,0.99558,0.99121,0.98312,0.97633,0.98540,0.99510
#2=4
0.89577,0.91902,0.95703,0.98245,0.99388,0.99802,0.99937,0.99980,0.99992,0.99996
0.87860,0.90486,0.94888,0.97901,0.99269,0.99766,0.99927,0.99978,0.99993,0.99998
0.83381,0.86560,0.92426,0.96813,0.98889,0.99647,0.99892,0.99967,0.99990,0.99997
0.78530,0.81305,0.88060,0.94560,0.98061,0.99388,0.99815,0.99944,0.99983,0.99995
0.79614,0.79925,0.83503,0.90738,0.96387,0.98844,0.99655,0.99898,0.99969,0.99991
0.88192,0.86838,0.84918,0.87183,0.93485,0.97732,0.99320,0.99803,0.99942,0.99983
0.95509,0.94663,0.92359,0.89697,0.90892,0.95741,0.98627,0.99602,0.99885,0.99966
0.98596,0.98304,0.97387,0.95595,0.93335,0.93960,0.97357,0.99187,0.99767,0.99933
0.99581,0.99496,0.99218,0.98613,0.97430,0.95829,0.96195,0.98415,0.99524,0.99863
0.99873,0.99850,0.99770,0.99593,0.99219,0.98478,0.97439,0.97685,0.99068,0.99722
R2=
0.93335,0.94932,0.97401,0.98960,0.99639,0.99883,0.99962,0.99987,0.99995,0.99997
0.92142,0.93995,0.96901,0.98759,0.99571,0.99862,0.99957,0.99987,0.99996,0.99999
0.88678,0.91191,0.95342,0.98117,0.99352,0.99794,0.99936,0.99981,0.99994,0.99998
0.83249,0.86326,0.92227,0.96741,0.98875,0.99646,0.99892,0.99967,0.99990,0.99997
0.79593,0.81569,0.87478,0.94090,0.97884,0.99338,0.99802,0.99941,0.99982,0.99994
0.84124,0.83464,0.84513,0.90160,0.95946,0.98693,0.99613,0.99887,0.99966,0.99990
0.92721,0.91572,0.89091,0.88561,0.93033,0.97410,0.99218,0.99775,0.99934,0.99980
0.97573,0.97080,0.95611,0.93235,0.92205,0.95431,0.98422,0.99540,0.99868,0.99962
0.99271,0.99119,0.98631,0.97604,0.95886,0.94975,0.97164,0.99066,0.99731,0.99923
0.99782,0.99739,0.99596,0.99281,0.98633,0.97508,0.96879,0.98302,0.99456,0.99843
1-12
-------
R2= 8
0.95858,0.96896,0.98442,0.99384,0.99786,0.99930,0.99977,0.99992,0.99996,0.99998
0.95091,0.96311,0.98144,0.99267,0.99748,0.99919,0.99975,0.99992,0.99997,0.99999
0.92751,0.94500,0.97202,0.98893,0.99621,0.99879,0.99963,0.99988,0.99996,0.99999
0.88371,0.90935,0.95218,0.98085,0.99348,0.99794,0.99937,0.99981,0.99994,0.99998
0.82880,0.85693,0.91598,0.96438,0.98777,0.99619,0.99885,0.99965,0.99989,0.99997
0.81933,0.82785,0.87054,0.93448,0.97605,0.99254,0.99779,0.99935,0.99980,0.99994
0.89099,0.87992,0.86895,0.89995,0.95442,0.98496,0.99557,0.99872,0.99962,0.99989
0.95886,0.95117,0.93078,0.91097,0.93016,0.97071,0.99095,0.99740,0.99925,0.99978
0.98727,0.98461,0.97628,0.96031,0.94282,0.95463,0.98215,0.99468,0.99849,0.99956
0.99621,0.99543,0.99289,0.98737,0.97679,0.96435,0.97197,0.98947,0.99691,0.99912
0.97470,0.98122,0.99071,0.99635,0.99873,0.99958,0.99986,0.99994,0.99997,0.99998
0.96996,0.97768,0.98895,0.99567,0.99851,0.99952,0.99985,0.99995,0.99998,0.99999
0.95523,0.96657,0.98338,0.99350,0.99778,0.99929,0.99978,0.99993,0.99998,0.99999
0.92524,0.94342,0.97144,0.98882,0.99621,0.99880,0.99963,0.99989,0.99996,0.99999
0.87576,0.90203,0.94772,0.97911,0.99296,0.99780,0.99933,0.99980,0.99994,0.99998
0.83078,0.85289,0.90810,0.95980,0.98618,0:99574,0.99873,0.99962,0.99989,0.99996
0.85727,0.85477,0.87390,0.92814,0.97255,0.99141,0.99748,0.99926,0.99978,0.99993
0.93337,0.92328,0.90345,0.90723,0.95002,0.98265,0.99487,0.99853,0.99957,0.99987
0.97790,0.97342,0.96019,0.94022,0.93753,0.96792,0.98957,0.99700,0.99914,0.99975
0.99337,0.99198,0.98751,0.97821,0.96359,0.96023,0.98050,0.99391,0.99826,0.99950
R2= 16
0.98470,0.98872,0.99447,0.99783,0.99924,0.99974,0.99991,0.99996,0.99998,0.99998
0.98184,0.98661,0.99344,0.99744,0.99912,0.99972,0.99991,0.99997,0.99999,1.00000
0.97288,0.97995,0.99016,0.99617,0.99869,0.99958,0.99987,0.99996,0.99999,1.00000
0.95403,0.96581,0.98315,0.99348,0.9*9779,0.99930,0.99978,0.99993,0.99998,0.99999
0.91845,0.93808,0.96876,0.98788,0.99593,0.99872,0.99961,0.99988,0.99996,0.99999
0.86762,0.89318,0.94127,0.97633,0.99207,0.99755,0.99926,0.99978,0.99993,0.99998
0.84550,0.85771,0.90170,0.95433,0.98407,0.99512,0.99856,0.99957,0.99987,0.99996
0.90138,0.89290,0.88965,0.92459,0.96868,0.99005,0.99710,0.99916,0.99975,0.99993
0.96248,0.95556,0.93791,0.92495,0.94827,0.98022,0.99407,0.99831,0.99951,0.99986
0.98837,0.98594,0.97835,0.96425,0.95190,0.96708,0.98816,0.99655,0.99902,0.99972
R2=
0.99080,0.99325,0.99670,0.99871,0.99954,0.99984,0.99994,0.99997,0.99998,0.99998
0.98910,0.99200,0.99610,0.99848,0.99948,0.99983,0.99995,0.99998,0.99999,1.00000
0.98374,0.98806,0.99419,0.99774,0.99923,0.99975,0.99992,0.99998,0.99999,1.00000
0.97238,0.97966,0.99010,0.99618,0.99870,0.99959,0.99987,0.99996,0.99999,1.00000
0.94961,0.96257,0.98167,0.99297,0.99764,0.99925,0.99977,0.99993,0.99998,0.99999
0.90943,0.93050,0.96460,0.98631,0.99545,0.99859,0.99957,0.99987,0.99996,0.99999
0.86454,0.88643,0.93402,0.97276,0.99089,0.99722,0.99917,0.99975,0.99993,0.99998
0.87498,0.87622,0.90095,0.94889,0.98157,0.99436,0.99835,0.99952,0.99986,0.99996
0.93946,0.93082,0.91620,0.92734,0.96511,0.98851,0.99665,0.99904,0.99972,0.99992
0.97977,0.97569,0.96384,0.94775,0.95159,0.97808,0.99319,0.99806,0.99945,0.99984
1-13
-------
APPENDIX J
WHY A CONSTANT-RATE SOURCE TERM CANNOT BE REPRESENTED
BY A POWER FUNCTION
In this appendix we demonstrate that representing a constant -rate source
term with a power function leads to incorrect estimates for the resulting change of
Consider a simulation where an aerosol is growing by vapor condensation
and no other aerosol processes are active. We limit our attention to the moment
GDE for My The volume of material lost from the vapor phase in each time
interval is represented as A 7 . This converts to a constant rate source term for
I "Jn
Now suppose that the volume source terms for two consecutive time intervals of
interest are
and that M for time step (?7-l)o is
3.
M.
[A 7J =2(^/6)
= 1
(J.2b)
(J-2c)
J-1
-------
For the conditions of Eq. (J-2a) the change of M, in time step (77-!) is
M3] =[M31 + [AVJ /(7r/6) = 2 (J-3)
With these conditions, we apply Eqs. (4-7) to calculate the moment GDE for time
step 77 according to the power-law representation as
c^ = log(2/l)/log(2/l) = 1
B; = (2V21 = 1
\dMJdtY =M. (J-4)
L 3. j^ 3.
Integrating Eq. (J-4) and evaluating for time step 77 yields
A MJ = 1.7183
M.
3'io
= 3.4366
For the conditions given by Eq. (J-2b), however, the actual change should be
The reason for the discrepancy between the two calculation procedures is
that the power function represents dy/dt due to the source term as a continuous
first-order function (i.e., a^ = 1). The source term represented by Eqs. (J-2a) and
(J-2b), however, is time-variant, and is precisely zeroth order (i.e., constant) within
each time interval.
This example demonstrates that to use the analytical integration
techniques, any formulation of the moment GDEs must preserve the known
dependence1 of source terms if mass conservation is to be assured.
difficulty demonstrated with zeroth-order source terms occurs with any process of known
order that is represented by the power-function term of the power-law analytical integration
technique.
J-2
-------
APPENDIX K
NUMERICAL INTEGRATION RECIPES
K.I INTRODUCTION
Procedures for integrating the moment GDEs are presented for the Euler,
Crank-Nicholson, second- and fourth-order RungeKutta, first- and second-order
analytical, and the power-law analytical integration techniques. The Euler and the
Runge-Kutta techniques integrate equations of the form of Eq. (5-1), where the
total rate of change of the moments is computed at each integration point.
For the Crank-Nicholson and the analytical integration techniques,
equations of the form of Eq. (5 2) are integrated, which means the expressions for
the time rate of change of the moments must be divided into zeroth-order (i.e.,
constant) and first-order terms, according to the procedures outlined in Chapter 5.
For the Crank-Nicholson, first-order analytical, and the second-order analytical
integration techniques, the order of each process, a,, must be specified. For the
power-law analytical integration technique, each term of the moment equations
must either be represented as a term of known order or grouped with the terms of
unknown order.
For the Crank-Nicholson and the first- and second-order analytical
integration techniques the coefficients of Eq. (5-2) are calculated as follows.
K-l
-------
Specify a, for each term of known order
Calculate c[* and cr' for each term.
A A
[C[0)l = (l-aA) by/ft) J (K-la)
L JT1() L JT|0
Calculate c'°' and c1' as
>o
A
The coefficients of Eq. (5-2) are
The additional calculations required for the power-law analytical
integration routine are covered in section K.2.5.
K.2 INTEGRATION RECIPES
K.2.1 The Enter Integration Technique
Calculate the time rate of change of y as the sum of the time rate of change
for each process
(K-4a)
K-2
k
-------
which is integrated as
The moment change due to an individual process is
(K-4b)
(K-4c)
K.2.2 The CraTiV-Nicholson Integration Technique
Calculate the coefficients 5^ and C^ according to Eqs. (K-l) through
(K-3). Express (dy/dt)^ implicitly as the difference of y evaluated at 770 and (77+1)0,
and express y implicitly as an average of the values at r/o and rji
Solving Eq. (K-5a) for y/,,+1)
TH-l)o
(K-5b)
The moment change due to an individual process is
+ [40)]
(K-5c)
(K-5d)
K-3
-------
K.2.3 The First-Orde
Calculate the coefficients C^ and 5^ according to Eqs. (K-1) through
(K-3), and form the differential equation
(K-6a)
Separate variables and integrate
:0=0 (K-8b)
0 (K-6c)
The moment change due to an individual process is
(K-6d)
Substituting Eq. (K-6b) and (K-6c) for y and integrating yields
(K-6e)
(K-6f)
Calculate the coefficients C^ and 5^ according to Eqs. (K-l) through
(K-3), and form the differential equation
K-4
-------
Separate variables and integrate to determine estimates of y at the end of the time
step,
: ci. * ° (K'7c)
With these estimates for y at the end of the time step, calculate the coefficients CL
and Sp and form the differential equation
where 5^(5^ + 5^/2 (R.7e)
Z^,= (^o+cy/2 (K.7f)
Separate variables and integrate
The moment change due to an individual process is
Substituting Eq. (K-7g) or (K-7h) for j/in Eq. (K-7i) and integrating yields
K-5
-------
(K-7J)
K.2.5 Power-Law Analytical Interation Techniue
'
Calculate the coefficients for processes of known order c ' and e'
according to Eqs. (K^l) through (K-2).
From (5j//5f)T_, (dy/dt\o, !/T, and j calculate a as
(K-8a)
Calculate the Taylor coefficients for (dy/dt)^ as
^
(K'8c)
Calculate the rate due to the processes of unknown order as
+ »») (K-8d)
Calculate the Taylor coefficients that correspond to the power-law representation of
as
K-6
-------
(K'8e)
(K-8g)
(K-8h)
'HO
Calculate the power -law coefficient that corresponds to these Taylor coefficients as
NOTE: If there are no processes represented by the power -law term (i.e.,
)^ = 0), then the solution algorithm should set
= 0 (K-8J)
The moment change due to an individual process is calculated as
(K'8m)
If the individual process solved for is one of unknown order, then 5. and C, are
A A
calculated as
K-7
-------
(K'8n)
(K<80)
If the individual process solved for is one of known order, then 5, and C, are
calculated as
, ,
A A
hi =
-------
With the new estimates for the integrated variables, calculate the time rate of
change of y as the sum of the time rate of change for each process
(K-9c)
Form the average derivative from Eqs. (K-9a) and (K-9b)
(K-9d)
and estimate the new value of y
The moment change due to an individual process is calculated as
T1f.
/2 (K-9f)
so that (Aj/), is calculated as
(K-9g)
K.2.7 The Fourth-Order Rnne-Kntta Interation Techniue
Calculate the time rate of change of y as the sum of the time rate of change
for each process
o
(K-10a)
Integrate Eq. (K lOa) to estimate
K-9
-------
(K-lOb)
With the new estimates for the integrated variables, calculate the time rate of
change of y as the sum of the time rate of change for each process
=£ [(dy/ dt) J (K-lOc)
Integrate Eq. (K - lOc) to estimate y^
(K-lod)
With the new estimates for the integrated variables, calculate the time rate of
change of y as the sum of the time rate of change for each process
(K-lOe)
Integrate Eq. (K lOe) to estimate y^
(K-lof)
With the new estimates for the integrated variables, calculate the time rate of
change of y as the sum of the time rate of change for each process
(K-10g)
A ^
Form the average derivative from Eqs. (K-lOa), (K-lOc), (K-lOe), and (K-10g)
3 (K- lOh)
and estimate the new value of y
K-10
-------
The moment change due to an individual process is calculated as
^3
/2
(K-lOj)
so that (Ay) is calculated as
(K-10k)
K-ll
-------
APPENDIX L
CHEMICAL SPECIES LIST FOR THE RADM
Species
number Species name
1 Sulfur dioxide (S02)
2 Sulfuric acid (H2S04)
3 Nitric oxide (NO)
4 Nitrogen dioxide (NO 2)
5 Nitrate radical (N03)
6 Dinitrogen pentoxide (^Os)
7 Nitric acid (HN03)
8 Peroxyacetyl nitrate (PAN)
9 Ammonia (NH3)
10 Ozone (03)
11 Hydrogen peroxide (H202)
12 Carbon monoxide (CO)
13 Formaldehyde (CH20)
14 Aldehydes
15 Organic acid
16 Organic peroxide
17 Peroxyacetic acid
18 Other alkanes
19 Ethane
20 Ethene
21 Propene
22 Butene and other reactive olefins
23 Toluene
24 Xylene
25 Hydroxyl radical (HO)
26 Hydroperoxyl radical (H02)
27 Oxygen singlet D
28 Acetyl peroxy radical
29 Methyl peroxy radical
30 HO-ethylene peroxy radical
32 HO-higher-olefin peroxy radical
33 Alkyl peroxy radical
34 Criegee intermediate (CO
35 Criegee intermediate (Cj)
36 Methane (CH4)
37 Water vapor (H20)
L-l
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