Environmental Monitoring Series
CONTINUED RESEARCH IN MESOSCALE AIR
POLLUTION SIMULATION MODELING:
Volume II - Refinements in the Treatment
of Chemistry, Meteorology, and Numerical
Integration Procedures
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations. It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service. Springfield, Virginia 22161.
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EPA 600/4-76-016
May 1976
CONTINUED RESEARCH IN MESOSCALE AIR
POLLUTION SIMULATION MODELING:
VOLUME II - REFINEMENTS IN THE TREATMENT
OF CHEMISTRY, METEOROLOGY, AND
NUMERICAL INTEGRATION PROCEDURES
S. D. Reynolds
J. Ames
T. A. Hecht
J. P. Meyer
D. C. Whitney
M. A. Yocke
Systems Applications, Incorporated
950 Northgate Drive
San Rafael, California 94903
68-02-1237
Project Officer
Kenneth L. Demerjian
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Office of Research and
Development, U.S. Environmental Protection Agency, and approved
for publication. Mention of trade names or commercial products
does not constitute endorsement or recommendation for use.
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m
CONTENTS
DISCLAIMER ii
LIST OF AUTHORS vi
LIST OF ILLUSTRATIONS vii
LIST OF TABLES . . . * x
LIST OF EXHIBITS xii
I INTRODUCTION . 1
II CHEMISTRY-RELATED DEVELOPMENT STUDIES 4
A. Development of an Automatic Computer Program for the
Evaluation of Kinetic Mechanisms 5
1. Treatment of Chamber Effects 6
2. Computational Considerations . . 8
3. Ease of Changing Reactions . 9
B. Development of an Improved Kinetic Mechanism for
Incorporation in Photochemical Dispersion Models 11
1. General Considerations in .the Design of a
Suitable Mechanism 11
2. Elimination of Unimportant Reactions in the
General Kinetic Mechanism 14
3. Further Modifications To Reduce Computing
Requirements ., 29
4. The Present Status of the Mechanism 32
C. Development of a Kinetic Mechanism Describing S02
Reactions and Sulfuric Acid Formation 35
1. The State of the Art of Gas Phase S02 Kinetics 36
2. The State of the Art Regarding the Oxidation
of S02 in Solution ' 41
3. Efforts To Test the Gas Phase Reaction Mechanism
for S02 51
4. Future Examinations of S09 Chemistry 52
c.
D. Special Considerations Regarding the Treatment of
Temperature, Water, and Hydrogen Peroxide in
the Airshed Model 53
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IV
II CHEMISTRY-RELATED DEVELOPMENT STUDIES (Continued)
1. The Predicted Effects of Changes in Temperature
and Water Concentration on Smog Kinetics 56
2. Specification of the Initial Concentration of FLCL. . . 62
3. Spatial and Temporal Variations in Temperature
and Water Concentration in the South Coast
Air Basin 63
E. Treatment of Organics in the Airshed Model 73
F. Introduction of the Improved Kinetic Mechanism
into the Airshed Model 77
III METEOROLOGY-RELATED DEVELOPMENT ACTIVITIES 87
A. Model Sensitivity to the Inclusion of Wind Shear 87
1. Wind Velocity Profile 89
2. Implementation of the Wind Velocity Profile 90
3. Computer Coding 91
4. Description of the Experiment 91
5. Discussion of the Results 92
B. Treatment of Wind Shear in the Airshed Model. 101
C. Examination of an Algorithm for Deriving Mass-
Consistent Wind Fields 101
1. The Governing Equations 103
2. Tests of the Model 106
3. Discussion of the Results 110
D. Adoption of an Improved Algorithm for Estimating
Turbulent Diffusivities . 121
E. Modified Treatment of the Inversion Layer in
the Airshed Model 123
IV EVALUATION OF ALTERNATIVE TECHNIQUES FOR INTEGRATING
THE SPECIES CONTINUITY EQUATIONS 126
A. Introduction 126
B. Available Methods 130
1. The Price Scheme 131
2. The Crowley Second- and Fourth-Order Methods 132
3. The SHASTA Method 133
4. The Galerkin Method 135
5. Particle-in-Cel 1 Techniques 140
6. The Method of Egan and Mahoney 141
C. A Test Problem 142
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IV EVALUATION OF ALTERNATIVE TECHNIQUES FOR INTEGRATING
THE SPECIES CONTINUITY EQUATIONS (Continued)
D. Results 146
1. The Price Scheme 148
2. The Crowley Second- and Fourth-Order Methods 148
3. The SHASTA Method 149
4. The Galerkin Method 149
5. Particle-in-Cell Methods 149
6. The Method of Egan and Mahoney 150
7. Computational Time 189
E. Conclusions 190
V AIRSHED MODEL MODIFICATION FOR MULTIDAY SIMULATION 192
A. Introduction 192
B. Model Refinements 194
1. Treatment of Photochemistry at Night 194
2. Definition of the Modeling Region 196
3. Use of a Grid with Variable Resolution 197
4. Modification of the Finite Difference Equations . . . 198
5. Modification of the Computer Codes 200
C. Multiday Simulation of the Los Angeles Basin 201
I. Preparation of ^missions and Meteorological
Results 201
2. Discussion of the Multiday Simulation Results .... 203
D. Recommendations for Future Work 216
APPENDIX: A USER'S GUIDE TO MODKIN 218
REFERENCES 283
FORM 2220-1 288
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AUTHORS
CHAPTER I - Steven D. Reynolds
CHAPTER II - Thomas A. Hecht, David C. Whitney, Jody Ames,
Steven D. Reynolds
CHAPTER III - Steven D. Reynolds, Mark A. Yocke, Jody Ames
CHAPTER IV - James P. Meyer
CHAPTER V - Steven D. Reynolds, Jody Ames
APPENDIX - David C. Whitney
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ILLUSTRATIONS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Concentration-Time Profiles for NO, N0?, Oo, and Propylene
at 15°C and 35°C L
Predicted Concentration-Time Profiles for NO, N02, 03, and
Propylene at 0, 50, and 100 Percent Relative Humidity
Locations of Temperature and Relative Humidity
Monitoring Sites
Distribution of the Temperature Aloft Above Rialto on
26-27 July 1973
Temporal Variations in Water Concentration at Five Locations
in the Los Angeles Basin
Distribution of the Water Concentration Aloft Above Rialto
on 26-27 July 1973
Predicted and Measured Concentrations for La Habra Using the
15- and 31-Step Kinetic Mechanisms
Predicted and Measured Concentrations for Anaheim Using the
15- and 31-Step Kinetic Mechanisms
Predicted and Measured Concentrations for Pomona Using the
15- and 31-Step Kinetic Mechanisms .
Predicted and Measured Concentrations for Pasadena Using the 15-
and 31-Step Kinetic Mechanisms
Predicted and Measured Concentrations for Downtown Los Angeles
Using the 15- and 31-Step Mechanisms
Predicted and Measured Concentrations for West Los Angeles
Using the 15- and 31-Step Mechanisms
The Effect— Expressed as Average Deviation— of Variations in
Vertical Wind Shear on NO and N02 . .'
The Effect— Expressed as Percentage Deviation— of Variations
in Vertical Wind Shear on NO and N02 •
The Effect— Expressed as Maximum Deviation— of Variations
in Vertical Wind Shear on NO and NO?
60
. 61
67
. 69
71
. 72
81
82
83
84
85
86
, 93
. 94
, 95
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vm
16 The Effect—Expressed as Percentage Maximum Deviation—of
Variations in Vertical Wind Shear on NO and NO? 96
17 The Effect—Expressed as Average Deviation--of Variations
in Vertical Wind Shear on CO and 63 97
18 The Effect—Expressed as Percentage Deviation—of Variations in
Vertical Wind Shear on CO and 03 98
19 The Effect-Expressed as Maximum Deviation—of Variations in
Vertical Wind Shear on CO and 03 . 99
20 The Effect-Expressed as Percentage Maximum Deviation—of
Variations in Vertical Wind Shear on CO and 03 »,...- 100
21 Distribution of 03 Aloft Between Brackett and Rialto
During the Morning of 11 July 1973 124
22 Concentration as a Function of Downw.ind Distance for the
Explicit Price Scheme 151
23 Concentration as a Function of Downwind Distance for the
Implicit Price Scheme ..... 156
24 Concentration as a Function of Downwind Distance for the
Crowley Second-Order Scheme 161
25 Concentration as a Function of Downwind Distance for the
Crowley Fourth-Order Scheme 166
26 Concentration as a Function of Downwind Distance for the
SHASTA Method ...... 171
27 Concentration as a Function of Downwind Distance for the
Galerkin Method 176
28 Concentration as a Function of Downwind Distance for the
Particle-in-Cell (Smoothed) Methods ... 181
29 Concentration as a Function of Downwind Distance for the Egan
and Mahoney Method 186
30 Comparison of Predicted and Measured Hourly Averaged
CO Concentrations at Downwind Los Angeles 205
31 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at Long Beach 206
32 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at West Los Angeles 207
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33 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at Burbank 208
34 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at Reseda ........ 209
35 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at Whittier ................... 210
36 Comparison of Predicted and Measured Hourly Averaged CO
Concentrations at Azusa ...... 211
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X
TABLES
1 The Reactions Ranked by Amount of Uncertainty ... 18
2 Individual Area and Sensitivity Indices 23
3 The Reactions Ranked by Sensitivity 24
4 Characteristics of the Smog Chamber Runs 27
5 Values of T, H, and M Before and After Removal of the
Six Reactions 29
6 A Lumped Kinetic Mechanism for Photochemical Smog 33
7 Type of Mathematical Representation Required to Predict
Concentrations of Species in the General Mechanism 35
8 The Effect of Different Catalysts on S02 Oxidation 47
9 Activation Energies of Reactions in the General Mechanism .... 57
10 Ground-Level Air Temperatures in the Los Angeles Basin on 28-30
June 1974 65
11 Ground-Level Relative Humidities in the Los Angeles Basin on
28-30 June 1974 66
12 Rate Constants for 0, OH, and 03 Attack on Various Hydrocarbons . 74
13 Hourly Averaged Wind Speed and Direction in the Los Angeles
Basin on 29 September 1969 at 6:00 a.m. PST 107
14 Hourly Averaged Wind Speed and Direction in the Los Angeles Basin
on 29 September 1969 at 3:00 p.m. PST 108
15 Mixing Depths in the Los Angeles Basin on 29 September 1969
at 6:00 a.m. and 3:00 p.m. PST 109
16 Predicted Changes in Wind Speed and Direction for Case 1 Ill
17 Predicted Changes in Wind Speed and Direction for Case 2 112
18 Predicted Changes in Wind Speed and Direction for Case 3 113
19 Predicted Changes in Wind Speed and Direction for Case 4 114
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XI
20 Predicted Changes in Wind Speed and Direction for Case 5 115
21 Predicted Changes in Wind Speed and Direction for Case 6 116
22 Predicted Changes in Wind Speed and Direction for Case 7 117
23 Predicted Changes in Wind Speed and Direction for Case 8 118
24 Conditions Represented in Tables 16 through 23 119
25 Values of Diffusivity and Peclet Number for Three Case Studies . . 147
26 Kinetic Constants for Each Case 147
27 Computing Time Required for Alternative Solution Methods 189
28 Organization of Multiday Input 202
29 Predicted and Measured Hourly Averaged CO Concentrations at the
End of the 29 to 30 September Nighttime Period 213
30 Multiday Ground-Level CO Concentration Map at 5 a.m. PST on
30 September 1969 214
31 Single-Day Ground-Level CO Concentration Map at 5 a.m. PST on 30
September 1969 215
32 Predicted and Measured Hourly Averaged CO Concentrations for
the Last Hour of the Multiday Simulation 216
A-l Input Card Format for MODKIN . . . 221
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XI
EXHIBITS
A-l Listing of Main Program MODKIN 236
A-2 Listing of Subroutine LMPCAL 250
A-3 Listing of Subroutine DIFSUB 253
A-4 Listing of Subroutine DIFFUN 262
A-5 Listing of Subroutine MATINV 266
A-6 Listing of Subroutine PEDERV 268
A-7 Listing of Subroutine PLOT 269
A-8 Sample MODKIN Input 273
A-9 Sample MODKIN Output—Selected Pages 275
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I INTRODUCTION
The SAI urban airshed model was originally developed for the Environmental
Protection Agency (EPA) under Contracts CPA 70-148 and 68-02-0339. Two series
of reports, entitled "Development of a Simulation Model for Estimating Ground-
Level Concentrations of Photochemical Pollutants" and "Further Development
and Evaluation of a Model for Estimating Ground-Level Concentrations of Photo-
chemical Pollutants," describe our models development and evaluation studies.
In concept, the model formulation was general, based on mass conservation rela-
tionships for a reactive species in a turbulent fluid. To implement the model,
however, we assumed that we could do the following:
> Use the gradient transport hypothesis to represent pollutant trans-
port by turbulence.
> Neglect turbulence influences on the net rate of chemical reactions.
> Neglect subgrid-scale concentration variations and their effect on
reaction rates.
Volume III discusses this threefold assumption. In addition, we made
several assumptions with regard to the treatment of various parameters in the
model. For example, we assumed that a 15-step kinetic mechanism could be used
to represent the chemical reaction processes. The nature of these assumptions
reflects not only the time and funding constraints on our work then, but also
the current understanding of the physical and chemical processes that occur in
the urban atmosphere. In this Volume, we discuss efforts carried out under the
present contract to refine further various aspects of the model and its usage.
Basically, our model refinement activities have focused on four areas:
> Chemistry-related model development activities.
> Meteorology-related model development activities.
> An evaluation of alternative techniques for integrating the species
continuity equations.
> Airshed model modification for multiday simulation.
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Each of these areas is the subject of a chapter in this volume.
Chapter II discusses our efforts to improve the treatment of chemical
parameters in the model. Specifically, we began with a 39-step general-
ized kinetic mechanism arid, by eliminating unimportant reactions, by
invoking the steady-state assumption, and by combining reaction steps we
derived a 31-step mechanism suitable for incorporation in the airshed model.
In addition, we examined S(L chemistry and developed an interim 10-step
reaction mechanism for describing both homogeneous and heterogeneous reac-
tions. Although this mechanism has yet to be validated using smog chamber
data, it does provide a starting point for treating S02 chemistry in the
airshed model. We also determined the sensitivity of the kinetic model
predictions to variations in temperature, water concentration, and hLO^
concentration. These results provide guidance with regard to the appro-
priate treatment of the spatial and temporal variations of these parameters
in the airshed model. Finally, the chapter describes our experience to
date in using the new 31-step mechanism in an actual simulation of a smoggy
day in the Los Angeles basin.
Chapter III describes our efforts to improve the treatment of meteo-
rological parameters in the model. We examined the impact on the. model
predictions of wind shear—an effect previously neglected in the model.
Upon finding that wind shear has a significant influence, we extended the
capabilities of the model to treat this parameter. In addition, we devel-
oped a methodology to derive improved diffusivity relationships (discussed
more fully in Volume III) and examined an algorithm for rendering three-
dimensional wind fields mass consistent. We gave special consideration to
the treatment of elevated temperature inversions, especially with respect
to possible importance of pollutant exchange between the stable inversion
layer and the turbulent mixed layer as the inversion is eroded by surface
heating.
Chapter IV presents our evaluation of alternative techniques for inte-
grating the species continuity equations. Because the governing equations
of the photochemical dispersion model are nonlinear, numerical techniques
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must be employed to obtain approximate solutions. Since we must attempt to
solve large systems of coupled, nonlinear partial differential equations,
we have to be careful to choose an appropriate numerical procedure. The
two most important concerns influencing this choice are the following:
> Accurate representation of the horizontal advective transport
of pollutants.
> Efficient solution of large systems of nonlinear equations.
For this contract effort, we restricted our attention to the first of these
areas. We carried out a comparative study of various alternative techniques
that have appeared in the literature and that, if implemented in the airshed
model, would represent a means for minimizing truncation error propagation
effects. The methods examined include finite difference, particle-in-cell ,
and finite element schemes. We applied each method to the same test prob-
lem, and we compared the numerical results with analytical solutions.
Chapter V summarizes our efforts to modify the airshed model for multi-
day simulations. In previous photochemical modeling studies, multiday simu-
lations have been ignored. Model applications have usually been limited to
the simulation of daytime conditions. For example, a model run might start
at some point in the morning preceding the rush hour and extend into the
afternoon to model the buildup of CL. Accurate nighttime simulations are
hindered by the typically small size of wind speeds then and the lack of
available measurements aloft. However, multiple-day simulations may prove
to be extremely useful. For example, in the evaluation of an emission con-
trol strategy that is to be carried out in some future year, the model user
must carefully choose the initial pollutant concentration distribution to be
employed in the simulation. If a multiple-day run is made, the influence of
the initial concentrations on the predictions for the second and subsequent
days will not be as pronounced as it is on the first day. Furthermore, mul-
tiday simulations may uncover errors in the treatment of emission, meteoro-
logical, or chemical parameters that would otherwise remain unnoticed in a
relatively short term simulation. Chapter'V concludes with a presentation
of the results of a 34-hour simulation of the Los Angeles basin for CO using
the SAI model.
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II CHEMISTRY-RELATED DEVELOPMENT STUDIES
Thomas A. Hecht
, David C. Whitney
Jody Ames
Steven D. Reynolds
One of the distinguishing characteristics of models capable of estima-
ting photochemical pollutant concentrations is that chemical reaction pro-
cesses must be represented accurately. Two pollutants treated in such
models for which air quality standards have been established, namely NCL
and Oo, are not emitted from sources in appreciable quantities. These pol-
lutants are formed in the atmosphere as the products of numerous reactions
involving NO, hydrocarbons, and a variety of free radical species. Because
of the inherent complexity of the overall reaction processes, care must be
exercised to incorporate in a photochemical dispersion model a tractable
kinetic mechanism which embodies as much chemical reaction as possible.
In this chapter we discuss efforts to improve the treatment of the
atmospheric chemical reaction processes in the SAI airshed model. Previ-
ously, these processes were represented by a 15-step mechanism developed by
Hecht and Seinfeld (1971). Since this mechanism was developed, however,
additional efforts have been undertaken to design improved mechanisms. One
of the most promising mechanisms to appear in the literature is that reported
by Hecht et al. (1973). This mechanism consists of 39 reaction steps and
treats four classes of hydrocarbons (paraffins, olefins, aromatics, and alde-
hydes). In general, this new mechanism seems to represent an advance of such
importance as to warrant its incorporation in the airshed model.
In the course of reviewing modeling needs with respect to atmospheric
chemistry, a number of issues were raised. These topics, which we address in
this chapter, include the following:
> Development of a computer program to facilitate the evaluation.
of kinetic mechanisms.
> Compaction of the 39-step hydrocarbon/NO /0~ mechanism to reduce
A O
its impact on computing time in the airshed model.
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> Development of an interim mechanism for S0? reactions and
sulfuric acid formation suitable for incorporation in the
airshed model .
> Examination of spatial and temporal variations in temper-
ature, water, and H?CL concentration in an urban airshed
such as the South Coast air basin to provide guidance with
respect to the treatment of these parameters in regional
models.
> Evaluation of alternative means for treating organics in
the airshed model .
> Experience gained in the use of the new kinetic mechanism
to perform an actual airshed simulation of the Los Angeles
Basin.
Each of these issues is discussed in the 'succeeding sections of this chapter.
A. DEVELOPMENT OF AN AUTOMATIC COMPUTER PROGRAM
FOR THE EVALUATION OF KINETIC MECHANISMS
Once a set of reactions for the formation of photochemical smog has been
proposed, it is necessary to demonstrate that the mechanism is correct; i.e.,
that it is able to account for, within experimental error, the actual concen-
tration of each species present in the reaction mixture at any point during
the time span of the reaction. In its simplist form, this evaluation process
involves the formulation and solution of the set of coupled differential equa-
tions that describe the variation in the formation and consumption of each
species in the reaction mixture as a function of time. This set of equations
can be expressed as follows:
R - ^ R , (1)
-JI -s f
j=l i,j k=l i,k
where
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y. = the concentration of species i,
t = time,
R^ = the rate of formation of species i in reaction j of the
i,j set of J reactions in which species i is formed,
R = the rate of consumption of species i in reaction k of
i,k the set of K reactions in which species i is consumed.
The concentrations thus calculated can be compared with those measured experi-
mentally in the reaction mixture.
Unfortunately, the real world presents experimental, computational, and
operational obstacles to the pursuit of this simple validation scheme. First;
for the integrity of the reaction mixture to be preserved, the mixture must be
contained in some sort of reaction chamber, which in turn gives rise to two
side effects: leaks (intentional, as in sampling, or unintentional) and wall
reactions. Second, when the most efficient computer codes are used, the time
needed to solve the coupled differential equations increases as the square of
the number of species increases. Moreover, certain sets of rates lead all too
often to systems of "stiff" equations, for which the solution times can become
quite large. Finally, the urge always exists to "improve" a reaction mechanism,
no matter how closely it approximates the experimental data; the computer code
must allow these changes to be performed with a minimum amount of effort. In
dealing with these realities, the researcher is called upon to display his met-
tle and to tax his ingenuity. The approaches we used in this study are described
in the following subsection.
1. Treatment of Chamber Effects
With few exceptions, reaction chambers are not completely airtight. Under
normal operating conditions, this does not create a serious problem, since al-
most all chambers are maintained at atmospheric pressure, and since the small
amount of interchange by diffusion can usually be ignored. However, a problem
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does arise when samples are removed from the chamber for analysis. Since
the species that comprise smog exist in the atmosphere in minute (1 to 1000
ppb) concentrations, sample sizes on the order of a few liters are commonly
needed to obtain enough material for an accurate analysis. Moreover, sam-
ples must be withdrawn fairly frequently during the reaction to monitor
species concentrations that are changing rapidly. As a result, it is not
unusual for 10 to 20 percent of the chamber volume to be withdrawn through
sampling procedures. The reaction simulation technique must take this
"dilution" of the reaction medium into account.
In the ideal case, the gas used to replace the samples being withdrawn
is inert with respect to the reaction (e.g., pure nitrogen in a smog system),
or it contains only reactive species whose concentrations are so large--
relative to the amounts consumed or produced by the reactions — that they can
be assumed to be constant throughout the reaction (e.g., oxygen or water
vapor in "clean" air). In this case, it is sufficient to apply a "dilution
factor" to the concentrations of all the species (inert diluent) or to those
that do not remain constant (clean air diluent). If samples of an approxi-
mately constant size are removed at reasonably uniform time intervals, the
dilution factor can be considered to be a constant, Q, and the equation for
the rate of change of the concentration of species i becomes
dy.
Rc -yjQ . (2)
j = l i,j k=l ci,k 1
In some chamber experiments, however, the incoming medium is just the
natural atmosphere in the laboratory, which may contain concentrations of
pollutant species as high as or higher than those being followed in the
reaction chamber. Moreover, it may be desirable in some cases to inject
pollutants or pollutant precursors deliberately into the chamber to simulate
the effect of fresh emissions on the reacting mixture. As long as the con-
centration of species i , y . , in the incoming medium is. known, the effect of
such inflowing species on the rate equation can be easily expressed:
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dy-
Rc - (y,. - yin) Q . (3)
j=l i,j k=l ci,k -1 in
Unfortunately, wall effects cannot be handled as neatly. Wall absorption
is best determined by placing the species in question, A, in a "nonreactive"
environment within the chamber and following its decay with time. One can
then include within the reaction mechanism an equation such as
ka
A -> Wall k (4)
with an appropriate rate constant.
Heterogeneous catalysis by the reactor walls is even more troublesome.
However, by determining reaction rates at several different surface-to-volume
ratios for the reactor (e.g., through the use of "artificial" walls to parti-
tion the chamber), the rate constants for the homogeneous and heterogeneous
reactions can be obtained, and both reactions can be included in the mechanism.
kc
Homogeneous: A + B •+ C , (5)
Heterogeneous: A + B -> C (6)
2. Computational Considerations
As mentioned earlier, the computer time required to solve a set of differ-
ential equations increases at least as does the square of the number of equations
to be solved (or, in the present case, as does the square of a number of distinct
chemical species that appear in the reaction mechanism). Thus, any techniques
that can be used to decrease the number of species concentrations that actually
require coupled differential equations for their solution should be applied.
Such techniques include the assumption of constant concentration; the uncoupling
of product-only species; the invocation of the steady-state approximation; and
the aggregation, or "lumping," of. species that. yield similar products. The last
two techniques are the subjects of further discussion in Sections B and E and
are thus not treated here.
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Certain species that appear in the reaction mechanism either are present
at truly constant concentrations (e.g.,. the reactor walls) or have concentra-
tions so high — relative to the amounts of that species formed or consumed
during the reaction—that they remain essentially constant with respect to
time (e.g., oxygen). Since the change in concentration for these species is
only negligibly different from zero, they can be excluded from the differen-
tial equation process.
A second category of species that need not be included in the set of
coupled differential equations is those that appear in the reaction mechanism
only as products (e.g., C0? or HNO.J- The rate of formation of these product
species is, to be sure, represented by the following equation:
A
dy-
However, the presence of this species has no effect on the rate of formation
or depletion of any other species; thus, the differential equation describing
its formation can be uncoupled from the set of all differential equations and
solved independently, at a significant savings in overall computer time.
3. Ease of Changing Reactions
Most computer codes used in the simulation of reaction kinetics incor-
porate, in one form or another, the features described above. The major
advantage offered by the present program is the ease of preparation and,
particularly, the ease of alteration of the mechanism and its associated
species concentrations. The user need know nothing about computer program-
ming or the solution of differential equations, and very little about chem-
ical reaction mechanisms, to obtain meaningful results from the program.
On the first line of input, the user specifies the run identification,
the number of reactions in the mechanism to be studied, how many of these
are lumped reactions, the number of each of the various types of species
described in the previous subsection, and an indication of whether the reac-
tion rates should be printed. The second line continues this specification
of parameters with an indication of the frequency of printout, the time step
sizes, and the dilution factor.
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10
The user then submits his reaction mechanism, one reaction per line,
restricted only by one requirement on ordering: The lumped reactions must
appear last. Each reaction appears as an ordinary chemical equation, with
a list of reactants, a list of products, and a rate constant. The products
can have coefficients (either fractions or integers), but the reactants
cannot—each reactant molecule must be entered separately. The user can
choose any four-letter mnemonic he wishes for the species names.
If there are any lumped reactions, the sets of individual reactions com-
prising each "lump" are then entered. Their formulation is exactly the same
as that of the lumped reaction, except that the name of each species that
contributes to the composition of the lump appears in place of the lumped
species as the first reactant.
The user than provides the list of species and their initial concentra-
tions—one per line. The order of their types must be the same as that given
on the initial parameter line, but no particular order is required within
each species type.
Should any of the species be present in the gas flowing into the reac-
tion chamber, their concentrations in the inflowing stream and, if needed,
the time and new values of any change in this concentration are entered next.
Finally, the user can request concentration-time plots of any species. If
desired, these plots can contain experimental points with which those points
calculated by the proposed mechanism can be compared.
To change a rate constant or chemical reaction, the,user need merely
alter the corresponding input line. New reactions can be added by insertion;
old ones can be removed by deletion. A similarly easy process can be used to.
change an initial species concentration or to add or remove species names from
the list. Species can be transferred among, species types (e.g., differential
to steady-state) by a single interchange of lines.
A complete user's guide to the computer program is included in Appendix
A. This appendix provides detailed information on each of the features de-
scribed above, descriptions and listings of all of the computer routines, and
sample inputs and outputs.
-------
11
B. DEVELOPMENT OF AN IMPROVED KINETIC MECHANISM FOR
INCORPORATION IN PHOTOCHEMICAL DISPERSION MODELS
1. General Considerations in the Design
of a Suitable Mechanism
The selection of a chemical mechanism for inclusion in an atmospheric
diffusion model depends substantially on two factors:
> The accuracy of prediction of the chemistry module.
> The computing time required to evaluate the mathematical
equations representing the mechanism.
From the standpoint of developing a chemical transformation model, the second
factor is subordinate to the first. After a reliable mechanism has been devel-
oped, it can be condensed in several ways to reduce the computing time necessary
to obtain predictions — for example, by eliminating unimportant reactions, by
combining species that react in the same ways and at similar rates into a gen-
eral grouping, and by invoking the steady-state approximation where applicable.
Depending on the degree to which these compaction measures are applied,
the resultant mechanism can be assigned one of three broad categories: detailed
mechanisms, lumped general mechanism^, or parametric models. A detailed mechan-
ism consists entirely of elementary reactions. Because there are hundreds of
different organics in the atmosphere, such a mechanism requires an extremely
large number of mathematical equations to represent the chemical transforma-
tions. Although a detailed mechanism is ultimately the most accurate (in the
limit if all rate constants and reactions are known), it is unsuitable for atmos-
pheric modeling because of the second criterion listed above. A lumped general
mechanism results when reactants and reactions of the same type are combined
into general classes and reactions and those that clearly do not contribute to
the .predictions are eliminated from a detailed mechanism. A lumped general
mechanism strikes a balance between detail of representation and compactness
of form. The elimination of reactions that significantly affect predicted con-
centrations would oversimplify the mechanism to the point where it could not
-------
12
provide accurate predictions unless corrective measures were taken. In par-
ticular, adjustable coefficients would have to be incorporated, forming a
parametric model. For such a model, the values of the adjustable parameters
are selected to minimize the discrepancies between experimental data and cal-
culated values.
When we first began to develop a photochemical airshed model two types
of chemical mechanisms were available for use. The first, a detailed model
for propylene, was unsuitable because it was too narrow in scope. Its pre-
dictions for the atmosphere would have almost certainly been unreliable.
The second was a parametric model, the Hecht-Seinfeld 15-reaction mechanism,
for which values of the adjustable parameters had been determined for several
hydrocarbon-NO systems using smog chamber data. These hydrocarbons included
A
propylene, iso-butylene, toluene, and n-butane; binary mixtures of propylene
and n-butane, propylene and ethane, and toluene and n-butane; and auto ex-
haust. Thus, to the extent that the atmosphere could be represented as a
surrogate consisting of these species, the parametric mechanism could be ex-
pected to provide reasonably accurate predictions over the range of initial
conditions explicitly used in selecting values of the parameters. Moreover,
the mechanism was mathematically compact. The predicted time-varying behav-
ior of the pollutants could be obtained at every time step through the solu-
tion of only four differential equations and six algebraic equations. Given
a choice of these two mechanisms, we selected the parametric model for incor-
poration in the airshed model because it came closest to meeting our two
criteria.
The compact mechanism is far from ideal, however. Recent experimental
studies have demonstrated the key roles of OH and H02 reactions in smog for-
mation, reactions whose importance is understated in the mechanism. Other
studies have shown that 0 and CO are less important that we thought at the
time we formulated the model. And one limitation that is particularly dis-
comforting is the narrow range over which values of the adjustable parameters
are valid. This last shortcoming would limit the accuracy of the results
obtained from the atmospheric dispersion model in such applications as the
evaluation of alternative emission control strategies.
-------
The mechanism most suitable for use in atmospheric models is a lumped
general mechanism. Under EPA Contract 68-02-0580, we recently undertook
the development of such a general kinetic mechanism. In this mechanism,
we incorporated state-of-the-art knowledge of the reaction processes, and
we provided for the rapid and straight forward modeling of organic species
not explicitly evaluated using smog chamber data.
In the new kinetic mechanism, the inorganic reactions common to all
organic-NO systems are treated in great detail. We introduced generality
A
into the model by lumping similar types of organics and free radicals into
several new classes. In particular, olefins, aromatics, paraffins, and
aldehydes constitute four separate classes of organics. We segregated
organic free radicals into alkoxy, peroxyalkyl, and peroxyacyl subgroupings.
Using propylene-NO , n-butane-NO , and propylene-n-butane-NO smog chamber
XX X
data over a wide range of HC/NO ratios, we evaluated the model and showed
X
that its predictions of the dependence of peak ozone on the initial concen-
trations of hydrocarbon and oxides of nitrogen qualitatively agree with
experimental observations. Seinfeld et al. (1973) discussed the rationale
and formulation of this lumped kinetic mechanism, and Hecht et al. (1973)
and Hecht et al. (1974) presented initial and secondary evaluation results
using the mechanism.
This new mechanism appears to be more accurate than the Hecht-Seinfeld
mechanism that we previously employed in the atmospheric simulation model.
In addition, we can easily extend the new mechanism to new organics that
have not been explicitly evaluated (the values of the adjustable parameters
do not need to be determined). Unfortunately, the computing time that is
initially required to carry out a simulation with the new mechanism is much
higher than that needed for the Hecht-Seinfeld model. At the outset of this
project, representation of the chemistry of a system consisting of a paraffin,
an olefin, and NO in air (no aromatics or CO) required 36 reactions and the
X
solution of 16 differential equations and 4 algebraic equations to obtain
predictions. Such mathematical complexity would certainly be excessive if
the mechanism were imbedded in the airshed model, where the kinetics must be
evaluated at every grid point for every time step. We therefore set out to
reduce the computing time necessary to obtain predictions from the mechanism.
-------
14
We approached the problem of long computational time requirements from
two directions. Initially, we sought to condense the mechanism to the small-
est number of reactions required for accurate predictions. We identified
critical reactions by means of a sensitivity analysis, and we subsequently
eliminated insensitive reactions from the model. We also found that taking
a flexible posture toward solving the representative rate expressions resulted
in time savings. Our experience in working with the kinetic mechanism showed
that computation time increased approximately linearly with the number of
reactions, but quadratically with the number of coupled differential equations.
Thus, we identified and verified species for which the pseudo-steady-state
approximation is valid; this step permitted the replacement of three coupled
differential equations by three coupled algebraic equations. Next, we took
advantage of the fact that differential equations describing the concentra-
tions of species that are formed as a result of chemical reactions but do not
themselves enter into reactions can be solved independently of reacting species.
We separated these so-called uncoupled species from the coupled species, elim-
inating three coupled differential equations, but adding three uncoupled dif-
*
ferential equations. Finally, we eliminated one species from the mechanism
by algebraic manipulation, thus reducing the number of coupled algebraic equa-
tions by one.
Since the computing time was the single greatest hindrance to our incor-
porating the improved kinetic mechanism in the airshed model, we focused a
great deal of attention on this problem. As a result, we condensed the mech-
anism in both its physical and mathematical structure to a form that is amenable
to diffusion modeling. We discuss the details, methodology, and results of
this program below.
2. Elimination of Unimportant Reactions
in the General Kinetic Mechanism
During the period in which we first formulated and subsequently modified
the lumped kinetic mechanism to achieve satisfactory predictions, we added and
deleted several reactions. However, we made no attempt to eliminate unimportant
Appendix A discusses the concept of coupled and uncoupled species,
-------
15
reactions. Under EPA Contract 68-02-0580, we recently completed a sensitivity
analysis of the kinetic mechanism; we used the results of this study to help
us select possible reactions for elimination. Our goal in the sensitivity
study was primarily to identify the "critical parameters" in the model, that
is, those whose uncertainties most greatly influence the reliability of pre-
dictions. In essence, we calculated the rate of change in predictions with
changes in the value of each rate constant, holding all other rate constants
fixed at their standard values as a measure of sensitivity. Rate constants
for which the measure has a high value correspond to sensitive reactions.
Low values indicate insensitive reactions that may not have to be included
in the model to make accurate predictions. Before proceeding with a discus-
sion of our results, we describe the procedures and methods that we used in
the sensitivity analysis.
The sensitivity study focused on a binary hydrocarbon-NO system (EPA
X
'Run 352) in which the initial concentrations were as follows:
Concentration
Species (ppm)
N02 0.07
NO 0.27
Propylene 0.265
n-Butane 3.29
We chose this particular experiment for several reasons:
> Both high and low reactivity hydrocarbons were present initially.
> The initial concentrations of total hydrocarbons and oxides of
nitrogen were typical of those found in a polluted atmosphere.
> The accumulation of ozone reached an asymptotic level during the
experiment.
> We had the run-modeled with reasonable success in our evaluation
study.
-------
16
We performed the sensitivity analysis in the following manner. Using
the nominal (or base) values for all parameters reported by Hecht et al .
(1973) (see Tables 14 and 16 in that reference), we obtained base concen-
tration-time profiles for propylene, n-butane, NO, N0?, and CL by integra-
ting the governing rate equations with each parameter at its nominal value.
We then increased (and subsequently decreased) one of the parameters by a
fixed percentage, holding all other parameters at their nominal values. We
integrated the equations twice, once for each of the two new settings (+50
percent and -50 percent) of the selected input parameter. Repeating this
process for each rate constant, we carried out, for n parameters, integra-
tions for a base case and 2n parameter variations. Finally, for each of
the 2n + 1 integrations, we determined the values of the sensitivity measures
or "decision variables." We compared the magnitudes of the decision varia-
bles for each variation in a parameter with those computed for the base case,
and ranked the sensitivities of the parameters by tabulating the magnitudes
of the differences.
a. Measurement of Sensitivity
Central to a sensitivity analysis of a mathematical model is the mean-
ingful quantification of changes in model predictions that result from per-
turbing the input parameters one at a time. As the measure of sensitivity
for each parameter, we chose the absolute area between the concentration-
time profile for the given parameter, with all parameters held at their base
values, and the profile generated when the i-th parameter was perturbed by a
fixed percentage. We denote these parameters as A^, A|\JQ, Ao3> A0lef (Pro~
pylene), and A^^ (n-butane). Since Hecht et al . (1973) discussed these
pard
criteria in some detail, we review their appropriateness only briefly here.
The five indices (A[^OO» ANQ, AQ~, ^olef-' anc' Apara) constitute continuous
measurements of sensitivity determined experimentally over the entire period
of simulation for each species. Mathematically, we represent this relation-
ship as follows:
400 miri
/
lC.(p.t) - C.(p + %p,t)| dt
-------
17
where
= the absolute area between the concentration-time
profile predicted for the i-th species, with all
parameters at their base values, and the profile
obtained when one parameter is perturbed by a
fixed percentage.
C = the concentration of the i-th chemical species.
i = the species index--NCL, NO, CL, olefin, paraffin.
p '= the parameter that is being perturbed.
% = the percentage perturbation in p divided by 100.
If the perturbation of a given parameter greatly alters the time history of
the i-th chemical species, indicating high sensitivity to that parameter, A.|
will have a large value. But if the concentration-time trace remains essen-
tially unchanged, the predictions of the model will be insensitive to varia-
tions in the parameter under evaluation, and |A.| will be small.
To facilitate the comparison and ranking of their sensitivities, we varied
all parameters in turn by the same fixed percentage. Some of the input param-
eters are very poorly characterized, having associated uncertainties of up to
an order of magnitude, whereas the values of other parameters are known within
an uncertainty of 10 percent. The comparative table of rate constants [Table
16 in Hecht et al. (1973)J suggests that a representative "degree of precision"
among the several alternative experimental determinations or theoretical esti-
mates available for any particular parameter is on the order of 50 percent.
Therefore, we used that percentage as the magnitude of perturbation for the
sensitivity calculations. However, because the precision bounds of the rate
constant values for individual reaction rate constants vary greatly, our choice
of the "range of perturbation" was arbitrary. The significance of the 50 per-
cent figure rests only on its approximate division of the very uncertain from
the less uncertain parameters. Table 1 ranks the reactions by the amount of
uncertainty. (We define the uncertainty bound for a rate constant as the range
within which the "true value" of the constant can be presumed to fall with con-
fidence.)
-------
Table-!
THE REACTIONS RANKED BY AMOUNT
OF UNCERTAINTY
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
.'Reaction
N205 + H20
NO + HN03
HN02 + HN03
NO + N02 + H20
HN02 + HN02
HN02 + hv
OH + N02
OH + NO + M
R02 + NO
RC03 + NO
RCOo + N09
o c.
H02 + H02
H02 + R02
R02 + R02
RO + N02
ALD + hv
RO + NO
OLEF + OH
0 + N02 + M
°3 + N02
NO, + NO
Percent
'Uncertainty
± 100 %
100
100
100 ,
100
100
100
100
100
100
100
100
100
100
80
70
65
45
40
40
40
-------
Table 1 (Concluded)
Percent
Reaction Uncertainty
22. PARA +.OH ±40-%
23. ALD +_ OH 40
24. RO + 02 40
25. PARA +0 35
26. H02 + NO 30 -
27. OLEF + 03 30
28. 0 + N02 25
29. N02.+ hv 20
30. 03 + NO . 20
31. NO. + N09 20
0 C.
32. H202 + hv ' 20
33. OLEF +0 20
«
34. 0 + NO + M 15
35. 0 + 62 + M . 10
36. NoOr 5
-------
20
The small amount of experimental data available upon which to base our
estimates limited the procedure we followed to estimate the uncertainty
bound for each rate constant. In essence, we calculated the percentage de-
viation of the highest and lowest expected values of each rate constant,
having surveyed the literature to find independent determinations of these
rate constants. Thus, the so-called estimate of uncertainty bounds is, in
fact, simply an indication of the degree of agreement (or more precisely,
the disagreement) among a number of independent determinations of the same
rate parameter. In three situations, this "definition" does not apply:
> If only a single determination was made for a given rate
constant, the uncertainty bound is an indication of the
precision of the experiment.
> Since photolysis rate constants (e.g., k,) must be deter-
mined in situ for a smog chamber experiment, the bounds
are an indication of the reliability of the experimental
method.
> The uncertainty estimate for k3y (HCL + HCL) was taken from
Lloyd (1974), who reviewed the reactions of the HCL radical.
Because of the imprecision of these estimates, we assumed that the uncertainty
bounds were symmetric about the nominal value. Therefore, although we esti-
mated the uncertainties associated with several parameters to be +100 percent,
the true upper bound may be considerably higher.
b. Results of the Sensitivity Analysis
For the purpose of ranking the parameters by sensitivity, we averaged the
values of the area indices calculated for plus and minus percentage variations
in the parameters to obtain a single characteristic value. Although this pro-
cedure facilitated ranking, some information was lost in the process. Because
each of the measures of sensitivity is based on the difference between nominal
-------
21
and perturbed concentration-time profiles, the magnitude of each difference,
in general, depends upon the degree of perturbation (e.g., 10, 25, 50 per-
cent). Because the equations governing the kinetics are nonlinear, the values
of the sensitivity measures typically are not identical for plus and minus
perturbations in a given parameter. But, in examining the values of the "area"
sensitivity measures for plus and minus perturbations, we found them to agree
sufficiently well to justify using their average values to rank the parameters.
Since we were interested chiefly in quantifying the overall sensitivity
(or insensitivity) associated with each reaction in the model, the use of an
indicator based on the changes in the predictions of several species was ap-
propriate. Thus, we combined the five area sensitivity measures into a single
scalar, which we term the "sensitivity." We defined-the synthesized sensitiv-
ity measure for each rate constant as follows:
Sensitivity.
J
where
i = 1, 2, 3, 4, and 5 refer to the average area sensitivity
measures for NOg, NO, 03, olefin, and paraffin,
respectively.
j = the number of the reaction rate constant in the kinetic
mechanism.
w. = the weighting of the individual measure in the combined
1 sensitivity scalar. (We weighted each of the five mea-
sures equally because, in developing and evaluating the
general kinetic mechanism, we were interested in predict-
ing the concentration-time behavior of each of these
species with equal accuracy. However, we might have
chosen different weights if our goals were-different. For
example, to predict oxidant and NOX for evaluating alter-
native emission control strategies, we might have weighted
03,.NO, and N02 more heavily.)
-------
22
A. . = the average area measurement determined for the i-th
1J criterion and the j-th reaction.
A = the maximum value of the i-th criterion observed for
i any of the reactions. (Dividing by the maximum value
scales each of the arrays of the five individual area
measurements between zero and one.)
Table 2 presents the values of this "sensitivity" index (seventh column), along
with the values of the five averaged individual area indices. Table 3 ranks
the reactions according to the combined sensitivity. As Table 3 shows, the fol-
lowing rate constants, presented in the order of decreasing sensitivity, display
the greateast overall sensitivity:
Rate
Constant Identification
k-, N02 photolysis
kpo Oxidation of n-butane by OH
k20 Oxidation of NO by H02
k.-. Reaction of 0~ with NO
*5 O
k16 Photolysis of HN02
k?~ Oxidation of propylene by 0^
kpq Photolysis of aldehydes
kp* Reaction of OH with propylene
Just as there are critical parameters in the model whose values must be
determined with certainty, some parameters are almost insensitive. Large vari-
ations in the magnitude of these parameters result in small changes in model
predictions. By identifying those reactions that contribute minimally to the
total predicted response, the sensitivity analysis provides the basis for elimi-
nating reactions from the mechanism. Removal, of course, is subject to further
limited individual testing of each reaction over a range of initial conditions
and bounds of uncertainty.
-------
Table 2
INDIVIDUAL AREA AND SENSITIVITY INDICES
NQ.
- 1
"2
3
4
"5
6
7
~ 9
" 9
'10 "
~11
12 '
13
""14"
'15"
16
17~~
18
19
"20"'
'21 '
22
"23"
24
25
~26""
Z7 "
28 -
~29"'
30
31
~32~~
33
34
-35""
36
37
~38~
'39
A (N02)
4.0999994E 00
2.9399997E-01
3.5400000E 00
9.4299972E-02
J.9699997E-01"
9.42.99972E-02
3.4399996E 00
'3.5099993E""Ob~
3.1999998E 00 ""
3.479999SE OQ" '
2.-6999998E""00"-
e.Z599993E 00 :
9.9599999E-01 '•
2-4799995E""00
1.6799994E 00'
5.0799999E 00' "
3.7799997E"'00~
a-559999SE On
0. 0
8.4699993E'"00""
1 .3699999E 00
3.2299995E-0]""
'6.9899998E~'00^~
2.2399998E 00
0. 0
'0,0 "" ~
8.2799971E-02""
5.6199999E 00 "
"6.6099997E"00
1.5799999E 00
1.0400000E 00
'9. 1899997E--01
1 .0499992E 00
6. 1299998E-01 ' "
'3.8299996E-01~
1 .7699999E-0.1
4-1499996E. 00
"2;0400000E-0'1 —
3.1699997E~01"""
A (NO)
3.0100000F-01
l'.5199995E-6r
2.3799998E-OI
5.8300000E-02
'4.7399998E-02'
5.S399998E-02
J.5199995E-01
9.3699992E-02""
8.8999987E-02
9.3499959E-02
9. 5 4 9 '9 9 9 2E- 02."
7.1199954F-0.2
5.7499997E-02
'l.'2599993E""00
7.2599995E-01
1.9299994E 00
r.'0499992E~00"
1.1999998E 00
0.0 ''
6.4899999E-OT
8.7499976E-02
I.2799996E-01
"4'.'5400000E'-OT
8.2699996E-01
'o.o
"0". 0 ":
'4.9599998E-02
'1.5999994E 00
'8.7399995E-01
7.5099945E-02
7.9399943E-02
'r.-'7499995E-01
1.1099994E-01
3.4599996E-OI
T.2399995E-OI
1.0799998E-01
2.7099997E-01
"S;"8300000E-0'2
'6.2099997E-02
A(03)
'• 4.B099991E 01
'5.8699995E-01'
4.2899994E 01
1 .5199995E-01
6.3000000E-01"
- 1 .4699996E-01
'"•li2000000E 0]
5.2399998E"00'
"'4.5699997E 00
5.179999^6! 00
. 3".'5"699997E'"00'
'4. 1699991E. no
8.6199999E-01
~~4T'6699997E"00'
~"3.0799999E "00
"7.5000000F 00
~"6".8199987E~"00'
3.9299994E 00
o.o
" "2.1699997E""OI
'""1.4699993F' 00
- 5.9899990E-01
•~"4":'1"199999E~00'
. 7.7199993E .00
••o.o.
0 . 0
•"8.6S99996E-02
1.2500000E 01
1V7000000E"01
8.6499996E 00
3-2099991E 00
7.7599993E""00
"'8.6499996E no
'"3-5AOOOOOE no
2'.2699995E" no
4.5599997E-01
1 .0200000E 01
4T3599999E-01
"""9".4499999£-01
A(OLEF)
5.9399996E 00
3.7799996E-01"
4.4599991E 00
3.4999996E-02.
~2.5999997Ei-02"
3.4299999E-02
7.7999997E-01
"2'.'8599995E-01"
" 3.4199998E-01"
2.8399998E-01 '
l'.3060000E-01~
3.02.99997E-01
2.6499998E-02
"l'.4899998E'"00"
1.0999994E 00
" 2.799999?E 00
"T2.'3799992E "00"
': 1 .4499998E. 00
0.0
~-"3."2799997E"""00"
'"2.3099995E-01 '
••"' 3.0100000E-01 '
7.1'099997E~00~
: 5.4199991E 00
"0.0
•"0:0 — "
'" 1.6500000E-02'
' 1 .8799996E-01
"~2.4799995E 00'
1.3499994E 00
, 1.7299998E-01
"'"7. 1499997E-01"
'" 7.3699999E-01
' '6.9699997E-01
— 4'.0099996E-or
1.2899995E-01
1.4799QOSE 00
177199997E-02-
"' '2.5099996E-02'
ft(PARA)
• 1.1099999E 01
'7.1099997E-01
9.1799994E 00
6.1699998'E-02
"3.0299997E-or~
6.0699999E-02
_1.9899998E 00
5.9499997EXU '
'4.9599999E-01
S.8999937E-01
"3.6199999E-01'
1.2500000E 00
4-8899996E-01
"2;2900000E '00'
2.0199995E 00
5.15S9998E 00
"5.4899998E' 00 "
2.0299997E 00
0.0
"6.7199993E 00"
'2.0499992E 00
1.6999996E-01
"1 . 1199999E"00'
S.0999994E 00
0.0
"0.0
' 7.9099953E-02
2.0699997E 01
""4 .3899994E 00'
7.7999992E 00
1.1399996E-OJ
~"3.2099998E:-01
3.3099997E-01
' 1.0299997E 00
"'6.1999995E-01'
1.6499996£-01
3.309999-5E 00
-4.149999SE-02
'•7.1799994E-02
SENSITIVITY.
6.0233879E-01
"4.2636652E-02"
5.0078332E-01
1.0480S03E-02
'1 .5842Q65E-0?
1 .0441024E-02
1.8804300E.-01
"1 .2817216E-01"
1 .1538500E-01
1 .2708902E-01'
"9.5649123E-0?"
9.8632284E-0?
3.853H06E-OP
~2.?009088E-Ol~
1.7816830E-01
4.7975492E-01
"3.4641325E-01 '
2.6154286E-01
0.0
' 5.14674aSE-Ol "
7.3833704E-02
'3.3491254E-02
'4.400S167E-01'
3. 7242830E-01
0.0
0.0 ' ' • " -'
9.6847395E-03
5.5576986E-0]
}.9439411E-01
5.2100249E-n2
"9.5314741r-02
9.4606638E-02
'4.860229-4E-n2'
2.2490099E-OZ
"1.3556119E-02
K9249547E-02
CO
-------
24
Table 3
THE REACTIONS RANKED BY SENSITIVITY
Reaction Sensitivity
1. N02 + hv 0.60
2. PARA i OH 0.56
3. H02 + NO '. 0.51
4. 03 + NO 0.50
5. HN02 + hv 0.48
6. OLEF + 03 0.44
7. ALD + hv 0.43
8.. OLEF + OH 0.37
9. OH + N02 0.35
10. NO + N02 + H20 0.27
11. OH + NO ' 0.26
12. H02 + H02 0.24
13. ALD + OH 0.19
14. 03 + N02 0..19
15. HN02 + HN02 0.18
16. N03 + NO 0.13
17. N205 0.13
18. N03 + N02 . 0.12
19. NO + HN03 0.10
20. RC03 + N02 0.10
21. N90, + H90 0.10
c. 0 • c.
22. RCO, + NO 0.10
-------
25
Table 3 (Concluded)
Reaction Sensitivity
23. RO + 02 0.09
24. H2~02 + hv . 0.07
25. R02 + NO ' 0.05
26. RO + N02 ' • 0.05
27. 0 + 02 + M ' ' 0.04
28.. HN02 + HN03 ' 0.04
29. OLEF + 0 0.03
30. RO + NO 0.02
31. R02 + R02 0.02
32. 0 + N0 0.02
33. HO, + R09 0.01
^ t-
34. 0 + NO + M ' 0.01
35. -0 + N02 + M 0.01
36. PARA + 0 0.01
-------
26
Reactions that could potentially be removed from the mechanism, based
on the results of the sensitivity analysis, appear in the lower portion of
Table 3. These reactions included the oxygen atom oxidation of the species
tabulated below:
Species
Paraffins
NO
N09 in the presence of
L a third body (M)
N0? second order reaction
Olefins
Reaction
: k27
>k.
Other candidates included the following:
Species
HN02-HN03
RO-NO
R02-R02
Reaction
36
^38
'39
Finally, we included among the candidates for potential elimination the reac-
tion between NO and HN03 (k,2). After carrying out the sensitivity calcula-
tions, we learned that the experimental value of this rate constant was
several orders of magnitude less than our earlier estimate; this change
sharply decreased the sensitivity of the reaction.
c. Elimination of Insensitive Reactions
We based the tentative conclusions reached thus far largely on the averaged
sensitivity criteria characterizing a single set of initial reactant concentra-
tions. If we were to repeat the calculations using only half the initial hydro-
carbon and twice the initial NO used in the present study, we would expect to
A
-------
27
find the order of parameter ranking to be somewhat different than that given
in Table 3. Thus, we had to scrutinize each reaction carefully prior to its
elimination; our criterion for elimination was that the reaction be "insensi-
tive" (a term defined quantitatively shortly) over the range of initial con-
centrations of interest, as well as over the uncertainty bounds of the
reaction rate constant (Table 1).
We chose three EPA smog chamber runs as a representative set of initial
concentrations and ratios over which to evaluate the reactions for possible
removal. As shown in Table 45 two of these runs are binary hydrocarbon sys-
tems; in the other experiment, propylene was the only hydrocarbon present.
These three runs span a total hydrocarbon-to-NO initial ratio of 0.7 to 10.5
* x
and a reactive hydrocarbon-to-NO initial ratio of 0.2 to 0.8. Air quality
A
data obtained in Los Angeles indicate that the ratios for polluted air there
are often within these ranges. (In the atmosphere, the ratio can vary with
both location and time of day.)
Table 4
CHARACTERISTICS OF THE SMOG CHAMBER RUNS
Concentrations
FPA
Pun [n-butane]0
329
333 3.40
352 3.29
(ppm)
[propylene]Q [NO]Q
0.24
0.23
0.26
0.29
1.25
0.27
[N02JQ
0.06
0.08
0.07
Total
[HC]/[NOxJ
0.7
2.7
10.5
High
Reactivity
[HC]/[NOx]
0.7
0.2
0.8
We considered reactions to be "insensitive" if, upon their removal indi-
vidually and as a group, the remaining set of reactions was able to predict
the following within 10 percent of the values predicted by the complete
mechanism:
Defined as propylene.
-------
28
> The time to the NCL peak (T)
> The height of the N02 peak (H)
> The magnitude of the ozone peak (M).
These three scalars, all of which can be easily quantified, are of interest
because the onset of formation of many secondary products formed in the atmos-
phere accompanies the peak concentration of NCL and because the intensity of
smog is often associated with the ozone and N0? concentrations. Thus, T, H,
and M constitute three major indicators of smog formation and severity. Al-
though the choice of the 10 percent range was arbitrary, this value is lower
than the uncertainty bounds associated with the experimental chamber data
used to evaluate and "tune" the model. Thus, we felt that the choice was
reasonable.
Consideration of the sensitivity values associated with each rate con-
stant led us to select 10 reactions for possible removal from the mechanism,
Of these, we found that only six could actually be eliminated based on the
criteria cited above:
Species Reaction
0 + NO k4
0 + N00 + M kc
<- 0
NO + HN03 k]2
HN02 + HN03 k]3
As shown in Table 5, the values of T, H, and M after removal of these six
reactions were within 10 percent of the values before the reactions were
eliminated. Several other reactions could have been removed for one or two
of the EPA runs, but not for all three. However, since their elimination
would have limited the applicability of the kinetic mechanism to a narrower
range of initial concentrations and ratios, we did not drop them.
-------
29
Table 5
VALUES OF T, H, AND M BEFORE AND AFTER
REMOVAL OF THE SIX REACTIONS '
Concentration
Height of the
NOp Peak (H)
Before After
0.25 0.25
0.75 0.70
0.25 0.25
Magnitude of the
Ozone Peak (M)
Before After
0.39 0.40
0.40 0.41
0.50 0.52
Time to the
NO? Peak (T)
EP/\ (minutes)
Run Before After
329 87 86
333 285 281
352 65 70
The conclusions we reached during this study were based on the lumped
general mechanism. If this mechanism proves to be fundamentally inadequate,
the sensitivity calculations should be repeated withithe corrected mechanism,
and reactions that we eliminated should be examined again to judge their sen-
sitivity in the environment of the corrected mechanism.
3. Further Modifications To Reduce Computing Requirements
Although the elimination of unnecessary reactions saves computing time,
the condensation discussed thus far focused primarily on giving prominence to
the important reactions in the mechanism. Significantly greater reductions in
computing time can be obtained by varying the mathematical representation of
the chemical mechanism. From a purist's point of view, a series of differen-
tial rate equations most accurately represents changes in the concentrations
of reactants with time. (Ideally, one would solve these equations analyti-
cally. We used numerical methods to solve the equations on the computer, but
these techniques were evaluated using test systems of equations for which
analytical solutions were available.) Over the years, scientists have used
the following approximations and simplifications to facilitate the solution of
complex kinetic systems:
-------
30
> Recognizing the fundamental mathematical difference between the
differential equations of species that are produced but do not
enter into reactions and those of species that do react. The
differential equations for reactants are often mathematically
coupled and must therefore be solved simultaneously. If these
coupled species have vastly different characteristic times of
reaction., the equations become "stiff" numerically and must be
solved using very small time steps to preserve accuracy. In con-
trast, the differential equations for species that do not react
are not coupled and can be solved accurately one at a time using
a method as simple as Simpson's rule.
> Applying the steady-state approximation. If the concentration
of a species equilibrates rapidly (relative to many other species
in the system), one can assume that the summation of the rate terms
for formation and consumption of the species is identically zero.
This assumption reduces the differential equation to an algebraic
equation.
> Combining second-order reactions into higher order reactions. In
some special cases, two or more reactions can be combined into a
single reaction, with the elimination of an intermediate as well.
The following subsections summarize the results of applying each of these tech-
niques to the lumped kinetic mechanism.
a. Treatment of Uncoupled Species
In a system containing propylene, n-butane, NO , and air, four species
X
form that do not react subsequently: nitric acid, peroxyacylnitrates, organic
nitrites, and organic nitrates. Because these products do not enter into
reactions with other species present in the system, we can uncouple and solve
the differential equations for each of the four species independently.
-------
31
b. Invocation of the Steady-State Approximation
In earlier work, we demonstrated the validity of the steady-state approxi-
mation for 0, OH, RO, and N03 (Hecht et al . , 1973). Recently, we justified the
application of the approximation to obtain predictions of the concentrations of
H0?, N20r, R02, and RCCL. To demonstrate the validity of the approximation for
any given species, we compared the concentration-time profile for the species
predicted by an algebraic description with that predicted by a differential
expression. In so doing, we found that the profiles generated using the two
mathematical representations agreed to within 0.1 percent for these species.
Thus, we eliminated four additional coupled differential equations, which were
replaced by four coupled algebraic equations.
c. Combination of Reactions into a Single Higher Order Reaction
The species N^Or exists in equilibrium with NO,, and NO,,:
N03
The only important reaction of N205 other than Reaction II is hydrolysis to form
nitric acid, a stable product in the mechanism:
N2°5 + H2° +11 2HN03
If we assume that N205 is in a steady-state (we have established the validity
of this assumption) we can combine these reactions into the single third-order
reaction:
TV
N03 + N02 + H20 iv 2HN03
-------
32
having the rate constant
kiv
The combination of these three reactions eliminates NpO^ as a species, thereby
saving one algebraic equation and removing a net total of two reactions from
the mechanism.
4. The Present Status of the Mechanism
As a result of the procedures described thus far, we added nine reactions
to the mechanism. Thus, a total of 31 reactions are necessary to represent
the chemistry of a system of paraffins, olefins, NO , CO and air. In addition,
A
to facilitate usage of the mechanism in the airshed model, we included two
additional reactions involving 0 and OH reactions with aromatics. It is to be
understood, however, that this is an interim treatment of the chemistry involv-
ing aromatics and is subject to revision at such time as a more suitable mech-
anism is developed. Table 6 presents the revised mechanism. Of the 25 species
included in the mechanism, 10 are represented by coupled differential equations,
7 by algebraic equations, and 4 are constant, as shown in Table 7. Although the
computing time associated with individual sets of initial conditions varies be-
cause of changes in the stiffness of the system of equations, we found that
incorporating the changes presented here reduced the- required computing time by
approximately 50 percent over that required previously (Hecht et al. 1973).
This saving is significant enough to justify the replacement of the simplified
15-step mechanism by the more accurate lumped kinetic mechanism as the kinetics
module in the airshed dispersion model.
-------
Table 6
A LUMPED KINETIC MECHANISM FOR PHOTOCHEMICAL SMOG
33
N0
+ 0
+ M
N02~NO-03 Cycle
0 + N0
-N0
N03 + NO~^2N02
Wo + HoO-J-^2HNO,
Chemistry of NO,
NO + N02 + 2H20-
2HN00-
HN02 +
+ NO
Chemistry of HNO,
OH +
OH +
12
OH + CO
H0
Important Reactions of
OH with Inorganic Species
H02 + NO-11*-OH + N02
Oxidation of NO by H02
H0
Hydroperoxy Radical
Termination
16
Photolysis of
-------
Table 6 (Concluded)
HC1 + 0-*-*
ur 4- n
nu-i T u^j
1 O
HC1 + OH-12-*
HC2 + O-22-*
HC2
II 2
0
RO + 02-^
RO + N02-^-
RO + NO— 1J-
-ROO + aRCOO + (l-a)H02 I
0
^-RCOO + RO + HC,
1 1
0
-ROO + HC3 '
-ROO + OH
\
^ROO + H20 /
^gROO + (2-s)H02 I
^3RCOO + (1-3)'H02 + H20
0
^-ROO + OH
*-ROO + H20
5-RO + N02
^•ROO + N02 + C02
T RCOON02
0
-H02 + HC3
*-RON02
Organic Oxidation
Reactions
HC, = Olefins
HC2 = Paraffins
HC3 = Aldehydes
^ HC, = Aromatics
Reactions of Organic
Free Radicals with NO,
N02 , and 02
-------
35
Table 7
TYPE OF MATHEMATICAL REPRESENTATION REQUIRED TO PREDICT
CONCENTRATIONS OF SPECIES IN THE GENERAL MECHANISM
Coupled
Differential
Equations
Uncoupled
Differential
Equations
HN00
Steady-State
Algebraic
Equations
0
Constant
M
NO
HN0
PAN
RNO,
RNO,
N0
OH
H20
RO
CO
Olefins
RC0
Paraffins
Aldehydes
Aromatics
C. DEVELOPMENT OF A KINETIC MECHANISM DESCRIBING
S02 REACTIONS AND SULFURIC ACID FORMATION
During the past decade, air pollution investigators have focused a substan-
tial amount of scientific attention on SO,,, the precursor of sulfuric acid and
s ill fate, because of its effects on visibility and health. They observed that
the oxidation of gaseous S02 occurs both through reactions with gas phase oxi-
dants and through reactions with liquid aerosol droplets. They demonstrated
that the addition of S0? to a reactor in which atmospheric concentrations of
organics and NO in air are being irradiated (i.e., a smog simulation experiment)
-X
-------
36
results in a substantial decrease in visibility due to the formation of a
sulfuric acid aerosol. And they established that SCL is oxidized in fog.
In this section, we review current knowledge and speculation concerning
the oxidation of SCL through reactions that occur in the gas phase and in
solution. Since Bufalini (1971) has extensively reviewed the oxidation of
SCL in polluted air, our discussion focuses primarily on more recent re-
sults. We conclude this section with a discussion of our efforts to model
a set of dynamic organic-NO -SCL smog chamber experiments and a summariza-
A L.
tion of our future plans to simulate the chemistry of SCL.
1. The State of the Art of Gas Phase SCL Kinetics
Until recently, air pollution SCL research focused primarily on the
qualitative and semi-quantitative characterization of the interaction of
SCL with components of smog. Scientists have been particularly interested
'in evaluating the effect of SCL on oxidant levels and visibility in simu-
lated smog (irradiated mixtures of organics, NO , SCL, and air); they have
X L.
used environmental chambers extensively for this purpose. In these experi-
ments, they observed that the concentration of SCL slowly diminishes with
time. However, most of the early (prior to 1970) experiments were not con-
trolled carefully enough to allow 'an accurate-.estimate to be made of the
rate of S0? oxidation due to gas phase chemical reactions. Variations in
relative humidity, the reactivity of chamber surfaces, and the accuracy of
the analytical instrumentation all served to introduce imprecision into the
data. And, by their very nature, smog chamber experiments provide minimal
insight into the actual individual reactions by which S0? is oxidized in
smog. Observations are limited to macroscopic changes in the concentrations
of the major reactants and products with time. The results of recent chamber
experiments and detailed kinetic studies of elementary reactions have pro-
vided sufficient insight so that we can now postulate a provisional mechanism
for the oxidation of S02 by homogeneous gas phase reactions. We discuss this
10-step mechanism briefly below.
-------
37
Experimental studies have indicated that peroxy radicals, diradicals,
and hydroxyl radicals are the most potent gas phase oxidizing agents with
respect to S(L in photochemical smog. Davis et al. (1973) obtained a pre-
liminary measurement of the rate constant for the reaction
Sl
H02 + S02 -*1 OH + S03
of 0.45 ppm min . The observed rate is sufficiently high to suggest that
the HO?~SO? reaction is important at about the time that N0? reaches its
maximum values and 0., begins to accumulate. Studies of S0? in smog Simula-
•3 L—
tion experiments have shown that this is the time at which the oxidation
rate of S0? is greatest. H0? is, of course, generally regarded as the prin-
cipal oxidant of NO:
H02 + NO ->- OH + N02
Because of the functional similarity of peroxyalkyl and peroxyacyl
radicals to H02, it does not seem unreasonable to presume that these three
species would undergo the same chemical reactions with a given reductant.
Both R0? and RCOo apparently oxidize NO through a reaction similar to the
HOp-NO reaction:
R02 + NO -> RO + N02
RCOO + NO -> R£0 + N02
6 0
Although the rate constants for these reactions are not known yet, the reac-
tions are thought to proceed more rapdily than the H02-N0 reaction. We feel
that, because of the analogies between the structure and behavior of H02,
R09, and RCO-, the last two species oxidize S09 at a rate somewhat faster
-1-1
than that of H09. We therefore estimate that k<. = kc '= 1 ppm min :
-------
38
S2
+ RO
S3
RCOO + S09 _/ RCO + SO,
II ^- II O
0 0
Cox and Penkett (1972) observed that SCL forms with reasonable rapidity
•j
when a system containing 0.,, olefin, and SCL react, and they postulated that
diradicals, products of the 0^-olefin reaction, are the species that oxidize
S0£:
S/L
R2COO + S02 + R2CO + S03
Since diradicals apparently form in smog only as a result of CL-olefin reac-
tions, this reaction, depending on its rate, may be less important that
Reactions S, through S~ in polluted air, where normally less than 20 percent
of the organics are olefinic. O'Neal and Blumstein (1973) recently reconsid-
ered the mechanism of the 0,-,-olefin reaction, and they feel that the interme-
diate complex of the reaction may decompose to form free radicals, including
H. A hydrogen atom formed in this manner could combine with 09 to form H09,
which is known to oxidize SO,, (Reaction S,). Thus, in the Cox and Penkett
experiments, H02, rather than a di radical, may have been the specie generated
by the Oo-olefin reaction that oxidized S09. Consequently, Reaction S. is
very speculative.
Recent measurements of the OH-S02 rate constant have suggested that the
reaction
S, 0
b ,
OH + S09 ^ HO S
L. |
0
may be an important loss mechanism for S09 in photochemical smog. Cox (1974)
-1 -1
obtained a value of 850 ppm min under atmospheric conditions, and Castleman
et al. (1974) found the value to be 600 ppm" min" .
-------
One can only speculate as to subsequent reactions of HOSCL in smog [see.
for example, Smith and Urone (1974) and references therein]. We offer one
possible reaction scheme here, which is largely an analogy to reactions of
organic free radicals.
We assume that CL adds to the HOSCL radical
S
HOS02 + 02 +6 HO S02
0
and that this peroxy radical can oxidize nitric oxide
Q S ?
+ NO J HOSO + NO,
C I t-
The HOSOo might abstract a hydrogen atom from an organic molecule or from an
HO, radical, forming H^O, directly:
0 $ °
HOSO + RH +8 HOSOH + R-
0 0
0 S °
HOSO + H09 +8 HOSOH + 09
i <- i <~
0 0
Or the HOSO,, might undergo a unimolecular decomposition reaction to form OH
and S03:
0 s
HOSO + HO + SO,
i <5
0
-------
40
Sulfur trioxide is, of course, the anhydride of sulfuric acid:
S03
Although we can set forth other reactions for the HSO radicals describing
X
their behavior in the presence of NCL and-other reactive species, we cannot sub-
stantiate such reactions (including S7 through S-.Q) with the results of experi-
ments that have been carried out to date.
Although we did not include several reactions in the core mechanism (S,
through S-.Q) for the oxidation of SCL, some comments about them are in order.
The 0-S02 reaction, for example,
0 + S02 + M -> S03 + M
has a reasonably high rate constant but is, nevertheless, slow because of the
extremely low concentration of oxygen atoms in smog. The direct photolysis of
S02 in otherwise clean air results in the slow disappearance of S02, but the
rate of S02 loss is not comparable to the rates observed in polluted air.
Wilson and Levy (1969) showed that N02 reacts very slowly with SO,,. Calvert
(1975) determined upper limits for the rates of reaction of NO^ and N?0r with
r-in fill O £ D
S02 of 10 ppnf inin" and 6 x 10~ ppnf mirf , respectively. Consequently,
both of the reactions are of negligible importance in photochemical smog. In
addition, Calvert found, in agreement with others, that the 0~-S09 reaction
-8 -1 -1
is very slow, having a rate constant of about 10 ppm min . In summary,
each of this last group of reactions results in the slow oxidation of S02 to
S0~. Although we could have included in the core S02 mechanism, the results
of kinetic studies of these reactions suggest that their combined contribution
to the total predicted loss rate of S02 is minor.
Because kineticists have studied in detail only two of the ten elementary
reactions included in the mechanism for the gas phase oxidation of S02 the
mechanism has an extremely high level of uncertainty. EPA is presently fund-
ing investigations of some of these reactions; therefore, more accurate values
-------
41
of the corresponding rate constants may be forthcoming in the near future.
Despite the uncertainty, we attempted to test this mechanism using smog
chamber data. (Section C-3 describes these efforts.) However, we found
that the chamber data were inadequately characterized in many important
respects and, consequently, were unusable.
2. The State of the Art Regarding the Oxidation
of SQ0 in Solution
2_
A large percentage of the volume of aerosol particles consists of water.
Gas phase borne SCL can dissolve in these particles, especially in the envi-
ronment of a stack plume, where the S0? concentration is often high. Once
S02 is dissolved, both direct and catalyzed reactions apparently lead to the
oxidation of S09 to sulfate. However, it is not now possible to assess the
relative importance of these competitive pathways under conditions of photo-
chemical smog formation. Certainly, the contribution of these two types of
reactions to the total S0? oxidation rate in solution depends on such factors
as aerosol size, oxidant concentration, catalyst concentration, species of
oxidant present, catalyst species present, and other chemical species in the
droplet that might enter into reactions with the oxidants or the catalysts.
In this section, we identify possible important direct and catalyzed reactions
in solution and attempt to explain the mechanisms of these reactions.
a. Reactions of S00 in Solution with
Oxidants Produced in the Gas Phase
Investigators have studied the reactions of S02 in solution with three
products of photochemical smog: N02, 03> and HpSO^.
N09 + S09 »-NO + SO
L *• H0 [t]
S09 ——-^ 09 + S00
L 3
H9SO. + SO., *- H9SO., + S00
24 3 2 3 3
-------
42
The first of these reactions (Wilson et al. , 1972) and the last (Gerhard and
Johnstone, 1955) proceed very slowly. Ozone and SCL, however, react rapidly
in the presence of liquid water, and the reaction probably occurs in solution
(Wilson and Levy, 1969). The rate of this aqueous reaction contrasts sharply
with that of the gas phase 0.,-SCL reaction, which is extremely slow. Thus,
the reaction between SCL and CL could be significant in aerosol particles,
and measurement of the rate constant of the reaction in a simulated atmos-
pheric environment is important.
b. Direct and Catalyzed Reactions of SOg
with Metal Ions in Solution
As reported in the literature, SCL is slowly oxidized when dissolved in
water (probably through a direct reaction with dissolved oxygen); however,
+2 +2 +3 +2 +2
the presence of metal ions, such as Mn , Fe , Fe 5 Ni , and Cu , in the
.solution accelerates the oxidation rate of SCL substantially (Urone and
Schroeder, 1969; Bufalini, 1971). The metal ions can interact chemically with
SCL in either or both of two ways: through direct reaction with SCL or through
catalysis of the (dissolved) air oxidation of SCL. We now turn to a discussion
of each of these classes of reactions.
Direct Oxidation-Reduction Reactions Between Metal Ions and S00. An exam-
j/
ination of half-cell potentials provides a straightforward means of evaluating
whether a given reaction is expected to occur on the basis of purely thermody-
namic considerations. In the context of this discussion, we are particularly
interested in learning whether oxidation-reduction couples (i.e., reactions)
between S00 and metal ions result in the oxidation of S09 to SCL (or.SCL) and
e. <-* 3 lr
the reduction of metal ions to some lower oxidation state .
We first observe that according to predictions, CL, CL, and FLCL should
all oxidize SCL. Noting that the SCL-SCL half-reaction is:
*
We used reduction potentials for these calculations; thus, for a reaction
couple to be favored, the combined potential must be positive.
-------
43
S02° x H20 -v SO"2 + 4H+ + (x - 2) HgO + 2e~ , E° = -0.17 V
we see that S09 is oxidized as a result of any of the following half-reactions
2H+ + 2e~ -> H0 , E° = 0.682 V
2H+ + 2e~ -»-.0 + H0 , E° = 2.07 V
2H+ + 2e~ -> 2\\ , E° = 1.776 V
The coupled potentials.are, therefore, positive by 0.51 V, 1.90 V, and 1.61 V,
respectively.
Of the five metal cations known to "oxidize" S02, only two would be pre-
dicted to enter into direct reaction with ?09 on the basis of thermodynamic
+3 +2
considerations alone: Fe and Cu . Their respective half-cell potentials
are
Cu+2 + 2e~ •* Cu° , E° - 0.34 V
Fe+3 + e" -»• Fe+2 , E° = 0.77 V
+2 +2 +2
Direct reactions between S09 and Mn , Fe , and Ni are extremely unfavored.
Their respective half-cell potentials are:
Mn+2 + 2e~ -> Mn , E° = -2.375 V
Fe+2 + 2e" -> Fe E° = -0.41 V
Ni+2 + 2e" + Ni , E° = -0.23 V
+? +?
These data indicate that for the direct oxidation of S09 by Mn , Fe , and
+?
Ni to occur, one would have to apply 2.54 V, 0.58 V, and 0.40 V, respectively,
of energy to the reacting system.
-------
44
Theoretical results such as these should, of course, be subjected to
experimental scrutiny. In fact, experimenters have observed the direct
reactions between SCL and 0?, CL, and H?0?, in a water solution that are
predicted to take place on the basis of thermodynamic principles. The two
+3 +2
cations Fe and Cu are known to accelerate the rate of oxidation of S0?.
However, it has not yet been shown (to our knowledge) that the mechanism
i Q J_ O _L O
of oxidation of S0? by Fe and Cu is direct. The isolation of Fe and
+3 +2
Cu as products of reactions in an aqueous solution of SCL, Fe , and Cu
would, for example, constitute acceptable evidence for the direct oxidation
mechanism. (It is important to remember the limitations of these electro-
chemical cell calculations. Although half-cell potentials provide a means
of predicting the direction of a chemical reaction, they do not in any way
+2 +2
indicate the rate at which the reaction will proceed.) Mn , Fe , and
+2
Ni do not enter into a direct reaction with SCL unless energy is supplied
to the system; thus, their roles in the oxidation process must be catalytic
or indirect.
Catalytic Oxidation of SO,,. Catalytic oxidation may well be the prin-
cipal process for SCL conversion under conditions of high humidity and high
particulate concentration, such as those that exist in plumes from power
plants. Gartrell et al. (1963) reported, for example, that the rate of SCL
oxidation in a smoke plume was quite low for relative humidities less than
70 percent, but it increased markedly for higher humidities. In one case,
they measured a rate of SCL conversion of 55 percent in 108 minutes. Al-
though such a rate is too high to be accounted for by a photochemical mech-
anism [a conclusion based on early studies of the photochemical oxidation
of S02 by Gerhard and Johnstone (1955)], it is similar to that expected of
oxidation in solution in the presence of a catalyst. Since the metal sul-
fates (and chlorides) emitted in a plume from a coal-burning process are
potential catalysts for the liquid phase oxidation of SCL, a reasonable ex-
planation for this process is that these particles act as condensation nuclei,
producing droplets of metal salt solution, which then act as loci for the SO,,
conversion.
-------
45
The atmospheric catalytic oxidation of SCL involves both water and
••* <-
dissolved CL, .and it requires the presence of a catalyst:
catalyst
Catalysts for this reaction include several metal salts, such as sulfates
and chlorides of manganese and iron, which usually exist in air as suspended
particulate matter. At high humidities, these particles act as condensation
nuclei or undergo hydration to become solution droplets. The oxidation pro-
cess then proceeds by absorption of both SCL and CL by the liquid aerosol,
with a subsequent chemical reaction in the liquid phase.
Early experiments conducted by Johnstone and Coughanowr (1958) and
Johnstone and Moll (I960), in which they measured SCL oxidation in droplets
of MnSCL, confirmed the basic catalytic mechanism. In addition, studies per-
formed by Junge and Ryan (1958) of the oxidation of -SCL in' bulk- catalyst solu-
tions yielded valuable information on the effects of solution acidity on the
rate of SCL oxidation.
Recently, Cheng et al . (1971) reported laboratory results involving the
catalytic oxidation of SCL in aerosol drops containing metal salts. They
developed an aerosol-stabilizing technique in which aerosol particles were
deposited on inert supporting Teflon beads in a fluidized bed. This deposi-
tion process altered neither the physical shape nor the chemical properties
of the aerosol. After packing the Teflon beads with the deposited aerosol
particles into a flow reactor, in which the catalytic oxidation of SCL oc-
curred, the experimenters passed a mixture of SCL and humid air through the
reactor. The SCL concentrations at the reactor entrance ranged from 3 to 18
ppm. To monitor the progress of the oxidation, Cheng et al . continuously
measured the SCL concentration at the reactor exit. They identified reaction
products by analyzing the reactor contents at the completion of an experiment.
_
The rate of the direct reaction of SCL with CL,
2S02 + 02 + 2S03
is too slow at room temperature to be of importance in the atmospheric
oxidation of SO,,.
-------
46
The SCL conversion progressed in three stages. During the initial per-
iod, all of the influent SCL was converted; none appeared at the reactor
exit. A transitional period followed, in which the SCL conversion rate de-
creased from the initial maximum value to a steady value. From then on, a
steady-state conversion of SCL took place. The three-stage process can be
related to the change in solubility of SCL in a water solution as the solu-
tion becomes more acidic. The initially rapid conversion of SCL apparently
results from the high rate of dissolution of gaseous SCL into liquid catalyst
drops. The increase in sulfuric acid in the drops soon affects the initial
stage of rapid conversion. Because HLSCL in a dilute concentration undergoes
+ +
complete dissociation to HSCh and H , the added H concentration diminishes
the solubility of SCL. Finally, as the solution acid concentration exceeds a
*- -j-
certain level, the high H concentration prevents further dissociation of
hLSCL, and the solubility of SCL becomes constant. In this final stage, the
rate of conversion of SCL to sulfate e'quals the rate at which SCL is absorbed
in the drops.
Although NaCl , CuSCL, MnCl?s and MnSCL all exhibited the same general
behavior, each salt differed in effectiveness as a catalyst for the oxida-
tion of SCL. Table 8 shows the steady-state conversions found by Cheng et
al. (1971). in the case of CuClg, Cheng et al . found that, rather than act-
ing as a catalyst, CuCl? reacted directly with SCL according to the following
t- C-
reaction:
SCL + 2CuCl2 + 2H20 i±2CuCl + HgSO, + 2HC1
Although the conversion of SCL proceeded even at very low relative humid-
ities (less than 40 percent), it did so slowly. Above about 70 percent relative
humidity, which is the level at which the transition from solid crystals sur-
rounded by a layer of water to actual solution drops takes place, the rate of
conversion increased dramatically.
The individual steps in the liquid-phase catalytic oxidation of S02 are
as follows:
-------
47
> The gas-phase diffusion of S0? to the drops,
> The diffusion of SCL from a drop's surface to the interior,
> The catalytic reaction in the interior.
Under steady-state conditions, the slowest of these three steps limits the
overall rate of SCL conversion. If the gas phase diffusion of SCL to the
drops is the controlling step, then the rate of SCL conversion should depend
on the gas velocity in the system. If the Tiquid-phase diffusion of SCL con-
trols the conversion rate, then the rate can be expected to be independent of
the type of catalyst. In varying the gas flow rate through their reactor,,
Cheng et al. found that the overall rate of SCL conversion was unaffected.
Since, as the results in Table 8 show, these rates clearly depend on the type
of catalyst, the rate-controlling step is the chemical reaction itself. Foster
(1969) reached similar conclusions.
Table 8
THE EFFECT OF DIFFERENT CATALYSTS ON S02 OXIDATION
Catalyst
NaCl
CuSO*
MnCl2
MnS04
Mean Resi-
Weight dence Time
(nig) (min)
0.36
0.15
0.255
0.51
1.7
1.7
0.52
0.52
Influent S02
Concentrations Fraction
(ppm) Conversion
14.4
14.4
3.3
3.3
0.069
0.068
0.052
0.365
Effective- *
ness Factor
1.0
2.4
3.5
12.2
*
The catalytic effectiveness of the various materials was compared with
that of NaCl. Thus, the effectiveness factor is the product of the
ratio of the weight of the catalyst in the reactor, the ratio of the
reactor mean residence time, and the ratio of the reaction conversion
of SOp in the reactor. The effectiveness factor for MnSO^, for example,
is:
-------
For steady-state conversion in the atmosphere, Cheng et al. derived the
following first-order rate expression from their data for MnSCL:
RSQ = 0.67 x 10"2[S02J
where R$Q9 is the micrograms of SCL converted per minute per milligram of
MnSCL, [SCL] is the gas phase concentration of SCL in micrograms per cubic
meter, and the constant factor is for drops containing 500 ppm of MnSO,.
The factor can be altered for other catalysts using Table 8. We can compute
the rate of conversion of S0? for conditions typical of natural fog in an
urban atmosphere:
> (S02) = 0.1 ppm.
> The average diameter of the fog droplets is 15 \i.
> Half the fog droplets contain a catalyst capable of oxidizing
S0? to HpSO.. The catalyst concentration within these droplets
is equivalent to 500 ppm MnSO,.
> The fog concentration is 0.2 gram of H^O per cubic centimeter of
air.
Under these assumptions, the equivalent catalyst concentration is 50 micro-
grams of MnSO, per cubic meter of air, and the rate of conversion of SOp is
2 percent per hour. Typical concentrations of catalyst metals are tabulated
below:
Concentration
Catalyst (yg m~3)
Mn '10
Cu 10
Zn 58
Fe 74
Pb 17
Thus, the conditions of the sample calculation are reasonable for actual air.
-------
49
The detailed mechanism of the catalized oxidation of SCL is not yet
known; however, the first step in the process may involve the association
of a reactant with the catalyst. If the catalyst is a transition metal
cation, the reactant apparently enters into a coordination complex with
the cation; thus, the reactant occupies a position in the ligand field of
the metal. Matteson et al. (1969) observed that catalyst potency toward
the oxidation of SCL tends to decrease as the number of possible sites at
which SCL can complex on the metal ion decreases. Thus, the configuration
of the ligand field (e.g., square planar, octahedral) of a given metal ion
strongly influences the catalytic behavior of the ion.
If the first step in catalysis is, indeed, the coordination of SCL
with the cation, the rate of displacement of other ligands in the ligand
field by SCL must be examined. Some species form much stronger coordina-
tion bonds with transition metal ions than others do. For example, carbon
monoxide poisoning of the blood results because the binding energy of CO
to the iron in hemoglobin is much greater uhan that of CL. Consequently,
CL cannot displace the CO from the iron, and the body rapidly depletes the
blood of 0?. S0? can, in principle, coordinate with transition metals,
since it contains unshared electrons — a general characteristic of ligands:
S ,S
\ *-* / \\
(Other ligands include, for example, FLO, NO, and CO.) But, if, as a result
of this mechanism involving transition metal cations, S0? is to be catalyti-
cally oxidized in aerosols, it must be able to displace other ligands from
the catalyst. Because of the high concentration of water and the presumably
low relative concentrations of S0? and catalysts in aerosols, the tendency
for S0? to displace water from the ligand field must be especially great.
Thus, an experimental investigation of the rate of hLO displacement by SOp in
the principal catalysts for S02 oxidation is clearly needed.
-------
50
One explanation of the catalytic oxidation of S0? in solution is the
series of four equilibria proposed by Matteson et al. (1969):
S0 + Mn+2 £ Mn • SO2
2Mn • S092 t [(Mn • S092) • 09]
^ ^ <-
[(Mn * S092) • 09J £ 2Mn • SO*2
L 6
7
-»-
8
Mn • SO*2 + HO + Mn+2 + HSO" + H+
Matteson et al. made three crucial assumptions in this mechanism:
+2
> S0? coordinates rapidly with Mn (Step 1)
The association of Mn • S09 complexes is likely (Step 3)
> Oxygen transfer to the [(Mn • S0? )9 • 09] complex occurs (Step 5).
Although Matteson et al. did not address these issues in formulating their
mechanism, the series of reactions provides a construct for further experi-
mental and theoretical inquiries.
It is not possible now to ascertain the extent to which the oxidation of
S09 in solution competes with the gas phase reactions. Very little data per-
taining to the kinetics of the reactions between SO,, and dissolved salts exist
that can be incorporated in a predictive model. Understanding the role of S09
-------
51
in the atmosphere and, indeed, the formulation of effective S0? control stra-
tegies will critically depend on the fundamental investigation of the types
of reactions discussed in this section. Without quantitative data upon which
to build a model, predictions are of little significance.
3. Efforts To Test the Gas Phase Reaction Mechanism for S00
L.
Shortly after the inception of the project, we received the results of a
series of smog chamber experiments from EPA to use to test the 10-step mechan-
ism described in Section C-l as a possible explanation for the oxidation of
SO^ in the gas phase. The experiments were carried out in a dynamic flow
reactor, and propylene, NO , S09, and air were used as reactants. To simulate
X c.
the system, we added the SCL reactions (Reactions S, through S,Q) to a general
mechanism for smog (Hecht et al . , 1974). We had previously performed exten-
sive tests of the organic-NO -air reactions using propylene-NO -air data
A A
obtained in the same smog chamber operated in a static mode.
Unfortunately, we found that the dynamic SCL experiments were unsuitable
for modeling for two reasons. First, the concentration of SCL in the inlet
tube fluctuated substantially during an irradiation, but the inlet concentra-
tions were not measured often enough to permit an accurate inflow profile of
S0? to be generated. Second, the oxidation reactions of S0? are quite slow
relative to the majority of other chemical transformations of interest in this
particular chemical system (e.g., the oxidation of NO and organics, and the
formation of CL). The net effect of these two characteristics of the system
was that the fluctuations in the inlet tube S0? concentrations masked any loss
of S0 due to chemical reactions.
The mechanism evaluation procedure, therefore, became more a test of the
adequacy with which the mixing and flow characteristics of the chamber were
modeled than of the accuracy of the mechanism. In view of the substantial un-
certainties in the inflow data, even very good agreement between the predic-
tions and the data would not be sufficient to demonstrate the validity of the
mechanism. Consequently, we suspended our efforts to test the;, SCL mechanism
-------
52
until more carefully controlled smog chamber data become available. A new
experimental study involving organics, NO , and S09 is now in progress; we
A C-
summarize that program in the following section.
4. Future Examinations of S0? Chemistry
Under the direction of Dr. Arthur Levy, investigators at Battelle
Memorial Laboratory are presently conducting a series of organic-NO -SCL-air
X L-
experiments using propylene (nine runs) and toluene (six runs) as the reac-
tive organic. Under another EPA contract, we expect to employ these data to
test the SCL mechanism proposed in Section C-l. The use of these data offers
several advantages:
> The experiments are being conducted under static conditions.
Consequently, we will not have to' contend with fluctuations
in inflow reactant concentrations as an additional variable
in evaluating the model.
> The chamber is still in operation (the chamber used for the
experiments mentioned in Section C-l has been disassembled).
Thus, any chamber effects that were not yet measured can still
be determined, if needed, for the model testing exercises.
> Dr. Levy's group at Battelle has considerable experience and
expertise in studying SCL in smog chambers. Therefore, the
new SCL data will almost certainly be the best that are
currently obtainable.
The evaluation of a mechanism describing the oxidation of SCL in solution
or in aerosols is more difficult. To our knowledge, no systematic experimental
study of this process suitable for model testing has yet been carried out. Un-
til the oxidation rate of SCL in systems containing aerosol particles has been
determined as a function of particle size (volume, surface area), composition
and concentration of reactants in the particle, pH of the particle, and concen-
tration of SCL in the gas phase, it will be difficult to propose with any con-
fidence a physical model for the oxidation of SCL in solution. As a temporary
-------
53
measure, it may be possible to develop a parametric model in which the oxida-
tion of SCL in particles is described by the first-order reaction
so2
The parameter k can then be estimated from the following:
> Observations of the S0? oxidation rate in droplets under well-
controlled conditions, such as those used in the experiments
of Cheng et al. (1971).
> Knowledge of the composition of atmospheric aerosols.
Although a parametric mechanism is necessarily simplistic, combined with the
gas phase mechanism, it may be sufficient to predict the atmospheric conver-
sion of SOp to sulfate within the uncertainty bounds of aerometric measure-
ments. We expect to analyze the methods for selecting values of k during
Contract 68-02-0580.
D. SPECIAL CONSIDERATIONS REGARDING THE TREATMENT OF TEMPERATURE,
WATER, AND HYDROGEN PEROXIDE IN THE AIRSHED MODEL
In the process of reviewing previous airshed modeling exercises, as well
as considering some of the possible difficulties that might arise in the use
of the latest version of the SAI model, we identified the following three ques-
tions that seemed to need further clarification:
> To what extent should temperature effects on reaction rate
constants be considered in the model?
?• How important are the spatial and temporal variations in
water concentration?
> Will the model predictions be sensitive to the initial con-
centration distribution of hydrogen peroxide?
-------
54
In an attempt to answer these questions, we carried out various sensitivity
studies using the kinetic mechanism, and we reviewed available measurements
for some of these parameters in one of the most severe and persistent photo-
chemical air pollution regions — the South Coast air basin of California.
It is well known that reaction rate constants are a function of temper-
ature. This effect is commonly expressed using the Arrhenius relationship:
k(T) = A
where
k = the rate constant,
A = a constant (sometimes referred to as the frequency factor),
E - the so-called activation energy for the reaction,
a
R = the gas constant,
T = the absolute temperature.
Given k at some temperature T~ and the activation energy, the value of k at
any other temperature can be estimated from
k(T) = k(Tn) exp|-^-(f-f-)| . (8)
'o
Thus, we do not need to determine b. In the computer programs, we input
and the values of E and k(Tn) for each chemical reaction. Then k can be
a u
calculated at any other temperature T using Eq. (8).
Although the algorithm outlined above is not difficult to incorporate in
the model, there is some question of the extent to which spatial and temporal
variations in temperature must be considered. For example, complete specifi-
cation of the temperature as a function of x, y, z, and time would require
significant amounts of additional computer storage, not to mention the extra
-------
55
effort required of the user to assemble sufficient data to estimate the com-
plete temperature field. Thus, we undertook a study to examine the sensitivity
of the kinetic mechanism to variations in temperature that might be found in an
urban airshed. These results can be used as a guide for determining under what
conditions spatial and temporal features of the temperature field must be con-
sidered in the model.
Similar questions arise concerning the distribution of water in the gas
phase over an urban area, especially a region like the South Coast air basin,
in which there are coastal areas as well as inland valleys. We note that
though spatial variations of relative humidity are significant in this airshed,
it is important to examine the variations in water concentration because this
is the parameter entering the kinetic rate expressions. Thus, to determine
the extent to which provisions for treating spatial and temporal variations in
water concentration should be included in the model, we examined the sensitiv-
ity of the mechanism to variations in water concentration.
Finally, incorporation of the 31-step mechanism (excluding SCL chemistry)
in the model will require the user to specify initial and boundary concentra-
tions of HNCL and FLCL, two pollutants that are rarely measured routinely in
most urban areas. To obtain a rough estimate of the concentrations of these
pollutants, we can assume that each is in chemical equilibrium; thus, from the
kinetic mechanism, we can write
-k,n + {k2 + 8kq(2kJNO][NO?][H O]2 + k,?[OH][NO]H 1/2
"] „ 1 \J \ I W «/ \ O L— L~ 1C- I }
k(-[HOj2
[HOJ =
16
If a simulation is to start somewhat before dawn, use of the above relationships
would be tantamount to assuming that chemical equilibrium had been approached
during the preceding nighttime period. Although this assumption may be reason-
able for HNCL, we note that the I^CL photolysis rate constant, k,g, would be
-------
56
essentially zero at night. In fact, from the mechanism we see that there is
no "sink" for FLO? other than the photolysis reaction. Thus, the use of the
equilibrium assumption for HLO,,, especially at night, does not seem desirable.
To examine this issue further, we carried out simulation runs using the mech-
anism to ascertain its sensitivity to the initial hL02 conditions. In the
following sections, we discuss the results obtained from these sensitivity
studies involving temperature, water, and HLO .
1. The Predicted Effects of Changes in Temperature
and Water Concentration on Smog Kinetics
To determine what effect changes in temperature or water concentration
have on the concentration predictions, we carried out simulations of a smog
chamber experiment using the new kinetic scheme incorporated in the airshed
model. The base values used were those of EPA Run 333:
> [N0]c = 1.25 ppm,
> [N02JQ = 0.08 ppm,
> [C3H6]0 = 0.23 ppm,
> [n-C4H10]0 = 3.41 ppm,
> [H20]0 = 16,000 ppm,
> T - 25°C.
For each simulation run, we changed only one parameter from the base values.
We performed the simulations for two different temperatures, 15°C and
35°C, with all other factors kept the same. We ca-lculated the rate constants
at the new temperatures from the base values of the rate constants (25°C) and
from measured or estimated reaction activation energies, as shown in Table 9
(Garvin and Hampson, 1974; Demerjian et al., 1974; Johnston et al., 1970).
Because the majority of the reactions in the mechanism are thermal and because
they have small positive activation energies, raising the temperature acceler-
ated the conversion of NO to N0? and decreased the time to the onset of 0~
-------
57
Table 9
ACTIVATION ENERGIES OF REACTIONS IN THE GENERAL MECHANISM
EA
Reaction ___ kcal mole"
N02 + hv-i-^NO + 0 0
0 + 02 + M-^03 + M -1
03 + NO-WN02 + 02 2.4
0 + NO_-iL*-NO + 02 0.6
03 + N02— 5-^N03 + 02 4.9
N03 + NO-^~2N02 1.4
N03 + N02 + H20-z-^-2HN03 -1.9*
NO + N02 + 2H20-^-*-2HN02 + H20 0
2HN00-2-»-NO + N00 + H00 9
2 2 L
HN02 + hv-^2*-OH + NO 0
OH + N02-^-HN03 -2.2
OH + NO-^^-HNOp -2.2
OH +-CO + (Oj-^-VcO., + N09 0.15
c. c. c.
H02 + NO-ii^-OH + N02 2
HO + HO -lp-^H 0+0 0
Reference
Estimate
Garvin and Hampson (1974)
Garvin and Hampson (1974)
Garvin and Hampson (1974)
Garvin and Hampson (1974)
Johnston et al. (1974)
Davis (1974)
Demerjian et al . (1974)
Demerjian et al . (1974)
Estimate
Garvin and Hampson (1974)
Garvin and Hampson (1974)
Davis (1974)
Estimate
Estimate
Estimate
* The value of E. listed for this composite reaction we determined from the
values of E^ for the three equivalent reactions:
Reaction EA Reference
NO, + N09 ^N,0,- -2 Demerjian et al. (1974)
0 c. (_ D
N2Q5 *-N02 + N03 19.4 Garvin and Hampson (19.74)
N205 + H20 ^2HN03 0 Estimate
-------
Table 9 (concluded)
58
HC1 + 0-ii.
HC, + 0- — -^H
1 3
19
HC1 + OH — i
HC2 + 0 ~^-H
HC2 + OH -iin
HC3 + hv -22-i
HC3 + OH -ilr
24
HC. + 0 _iL
4
HC4 + OH -iL:
ROD + N0_£l^
RCOO + NO + (02)-£2L
0
RCOO + N02-J£-
0
29
RO + 02
RO + N02-2!L
RO + NO -1L
Reaction
-ROO + aRCOO + (1-
II
ii
Q
^RCOO + RO + HC3
0
^-ROO + HC3
,-ROO + OH
^ROO + H20
^gROO + (2- )H02
i^BRCOO + (l-e)H00
8
t-ROO + OH
.ROD + H20
>-RO + N02
>-ROO + N02 + C02
^RCOON02
Q
^H09 + HC.
L. O
^RON02
^RONO
kcal mole
-1
5 Estimated E. for propylene
t Estimated E. for n-butane
0.1.
3.8
§ I
5t
It
0
0
6
0
Reference
Garvin and Hampson (1974)
Garvin and Hampson (1974)
Estimate
Estimate
Estimate
Estimate
Estimate
Estimate
Estimate
Estimate
Garvin and Hampson (1974)
Estimate
Estimate.
-------
59
accumulation, as expected. Conversely, lowering the temperature noticeably
slowed the smog formation process. Figure 1 presents concentration-time
profiles for NO, NCL, CL, and propylene for each of these two runs.
We carried out similar runs at two extreme conditions of relative
humidity--0 and 100 percent—at the base temperature (25°C). These percent
ages correspond to 0 and 32,000 ppm of hLO, respectively. Figure 2 shows a
comparison of the predicted concentration-time profiles for these two cases
with the profile for the base case. Increasing the water concentration
accelerated the conversion of NO to N0?, whereas a complete elimination of
water dramatically slowed down the overall smog kinetics. Both of these
effects are attributable to changes in the production rate and equilibrium
level of nitrous acid, governed by the reactions
NO
Because it is virtually impossible—even with pumping and baking—to
obtain a water concentration of 0 ppm in existing smog chambers, we carried
out one final run at 3.2 ppm of I-LO. The concentration-time profile obtained
under these conditions differed from those of the completely dry run by less
than 2 percent after six hours of simulation time.
In urban areas, ambient temperatures and water concentrations change
considerably during the day and from one day to the next. Thus, the results
of these simulation runs suggest that it may be necessary to account for vari-
ations in temperature and water concentration when modeling urban photochemical
smog. Toward this end, smog chamber experiments conducted at various constant
levels of temperature and water concentration would be most useful in ascer-
taining the effects of variations of these two parameters on smog kinetics.
-------
EPA RUN 333 CONCENTRATIONS (ppm)
E
Q.
CX
o
o
o
0
N0 = 1.25
N02 = 0.08
= 0.23
35°C
!5°C
200
Time (minutes)
300
FIGURE 1. CONCENTRATION-TIME PROFILES FOR NO, N02, 03,
AND PROPYLENE AT 15°C AND 35°C
en
o
-------
E
QL
0.
C
o
c.
o
o
c.
o
o
': EPA RUN 333 CONCENTRATIONS (ppm)
100
200
Time (minutes)
300
FIGURE 2,
PREDICTED CONCENTRATION-TIME PROFILES FOR NO, N02, 03, AND PROPYLENE
AT 0, 50, AND 100 PERCENT RELATIVE HUMIDITY
-------
62
2. Specification of the Initial Concentration of H000
— — . ^—^
With the implementation of the new kinetic scheme in the airshed model,
we must now specify the emission rate and the initial and boundary concentra-
tions of a new reactant., FLCL. To ascertain the accuracy with which these pa-
rameters must be determined, we carried out kinetic simulations of EPA Chamber
Run 333 (under the initial conditions listed in Section D-l) at three differ-
ent initial hL02 concentrations: 0, 0.01, and 0.1 ppm.
The concentration-time profiles obtained for the case in which [HLO^L =
0.01 ppm did not differ appreciably from those for the base case, in which
[O2J = 0 ppm. The small initial HLO^ concentration resulted in a five-
minute reduction in the time to the NO- peak (305 versus 310 minutes) and a
small increase in 0, at 360 minutes (0.32 versus 0.30 ppm).
In constrast, the effect on the predictions of the presence of 0.1 ppm
of H?0 initially was far more visible. The conversion of NO to NO,-, was
accelerated considerably., and the NO- peaked at 264 minutes. As a result of
the substantial reduction in the time to the N0? peak, 0~ accumulated to 0.46
ppm before the simulation was terminated at 360 minutes.
For similar simulations of another smog chamber experiment (EPA Run 349),
the initial conditions were as follows:
> [NO]Q =0.31 ppm,
> [N02]Q =0.03 ppm,
> [propyleneL = 0.44 ppm,
> [n-butane]n = 3.25 ppm,
> [H20]Q = 16,000 ppm,
> T = 25°C.
In these simulations, a maximum in the 0, concentration did occur, and the
results indicate that the asymptotic ozone level is not affected appreciably
(less than 2 percent) by the initial presence of as much as 0.1 ppm of hLCL.
-------
63
However, the H^CL did serve to reduce the time that elapsed before the maximum
was reached. For example, the predicted 0- maximum occurred at 194 minutes
for EPA Run 349 when the initial charge contained 0.1 ppm of H?0?, compared
with 225 minutes when HLCL was absent initially.
On the basis of these simulations, we feel that an effort should be made
to construct an emissions inventory for H?0? only if the sources of such emis-
sions would lead to an ambient hydrogen peroxide concentration of more than
0.01 ppm. Should I-LO,-, sources contribute less than this amount, the error in-
curred by neglecting these sources would be very small, especially prior to
the formation of the N0? peak and at the 0^ asymptote.
With regard to the specification of initial and boundary concentrations
in the airshed model, the sensitivity runs indicate that care should be exer-
cised in specifying HpO? concentrations when they are on the order of 0.1 ppm
or larger. Data presented by Bufalini et al. (1972) suggest that FLOp in the
South Coast air basin may reach levels as high as 0.18 ppm during a very smoggy
day. However, early morning and late afternoon levels were reported to be
about 0.01 to 0.02 ppm, thus indicating that overnight carry-over effects may
not be too significant. We hasten to add that these observations are based on
a very limited number of ambient air measurements. Additional measurements of
the diurnal behavior of H^O,, in an urban airshed would be useful.
3. Spatial and Temporal Variations in Temperature and Water Concentration
in the South Coast Air Basin
Having shown in Section D-l that the kinetic mechanism is somewhat sensi-
tive to changes in temperature and water concentration, we carried out a
limited effort to examine the extent of these variations in an -actual airshed.
We chose the South Coast air basin for this study for two reasons. First,
photochemical smog is particularly severe in this region. Second, we expected
that the spatial and temporal variations in temperature and water concentration
found here would be as large as those found in most other airsheds where the
model might be applied.
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64
During the summer, an onshore flow of moist marine air generally keeps
coastal areas relatively cool [temperatures in the 70s to 80s (°F)]. By the
time the air has traveled to the inland valleys, however, significant heating
has taken place, and the temperature often exceeds 100°F. In addition, rela-
tive humidities near the coast are usually higher than those measured inland.
Of course, since water concentration is the parameter of interest, the effect
of temperature on relative humidity must be considered.
Tables 10 and 11 present hourly ground-level temperature and relative
humidity data for three smoggy days in June 1974. The station location asso-
ciated with each code number is as follows:
Number Station Name
13W Lennox
21W Long Beach
41W Burbank
61W Ontario
75W Downtown Los Angeles
Figure 3 shows the location of each station. Lennox and Long Beach are repre-
sentative of coastal Iocations5 whereas Downtown Los Angeles, Burbank, and
Ontario are representative of inland communities.
To examine variations in temperature and relative humidity with height
above the terrain, we reviewed some of the measurements recently reported by
Blunienthal et al. (1974). They measured the three-dimensional distribution of
pollutants and meteorological parameters throughout the South Coast air basin
using a fully instrumented fixed-wing aircraft. We chose to examine a two-day
period--26-27 July 1973--for which numerous aircraft spirals were made, both
during the day and at night.
-------
Table 10
GROUND-LEVEL AIR TEMPERATURES IN THE LOS ANGELES BASIN ON 28-30 JUNE 1974
• JOS NUMBER =GAMTA8'.S AIR POMUTION CONTROL
PRQCRAM =GAKTABLS
PATF: 07/19/74
• STA
0 1 2 3 4 5 ' A
13W 64
2 in 65
4 1 vi . 60
A 1 W 67
75W _ 67
STA
•0123456
13W 63
21V! .64
4 1 w 65
61W 65
7SW ' 67
.STA
0123456
. 1 3 W • -62
21W 63
41W • 62
61W 57
75W 66
TFMPFR.ATMHF
7
70
70
74
75
70
7
65
65
CO
69
69
8
75
75
79
03
74
H
69
68
74
77
72
9
77
91
85
88
79
9
71
•70
79-
81
74
DISTRICT - COUNTY OF
/ AT HHIIIX / IN
10
76
84
9?
"3
06
10
69
72
04
HA
76
HM,
11
77
97
9fl
93
„„,
12
78
90
99
1
-------
Table 11
GROUND-LEVEL RELATIVE HUMIDITIES IN THE LOS ANGELES BASIN ON 28-30 JUNE 1974
JOS
AIR POLlWri'OH CDKTROL DISTRICT - CQOWr Qf LOS
PROGRAM =GAMTA8LS
DATE: 07/19/74
STA
13V
21W
41 W
61W
75W
STA
13W
21W
41w
61W
75W
STA
13W
?.1W
A 1 w
61W
75W
0123456
87
65
37
38
65
0123456
90
75
43
47
H?
7
76
59
40
30
61
7
90
73
46
47
HO
RFI ATIVE
fl
60
50
31
27
56
H
78
65
43
36
73
9
58
36
28
23
51
9
73
63
' 3H
35
69
HUMIDITY /AT HOUR
10
5H
43
17
20
39
10
76
59
36
2.5
65
11
5?
12
13
35
i—
in-; P<;T
12 13
AH 44
32 31
12 19
io._ 10 _
32 48
Hliim PST
11 1 ? 13
6H
57
40
17
54
66 66
55 52
4? - 40
16 21
60 57'
14
44
31
19
10
46
14
68
50
30
54
f TN
15
44
28
13
..!)..
43
\ 5
6(1
52
31
25
56
PERCENT
16
A?
26
14
_n
40
16
73
52
42
28
60
17
48
79
14
13
39
17
81
57
38
34
65
18
50
31
19
38
1s
84
55
46
36
68
19 20 21
63
40
17
.38
19 ?0 21
87
65 .
56
50
80
22 23 AVE
55.3
38.5
20.9
1 n . o
45.1
22 23 AVF
76.3
59.3
40. fl
3?.l
6S.9
HOUR PST
0123456
90
73
ai
100
83
7
90
73
78
100
82
0
87
70
70
93 •
-31
9'
78
70
63
76
76
10
70
61
60
54
70
11
70
57
58
44
62
12 13
64 66
57 59
52 51
36 32
62 63
14
66
57
54
32
62
15
66
57
51
34
6B
16
6H
63
51
4)
67
17
76
65
53
46
70
18
81
65
61
. 54
77
19 20 21
84
68
68
66
80
22 23 AVfi
75.4
63.9
6O. fl
57.7
71,6
VA 76,
A/28/74
N H I M
14 47
13 26
14 12
12 . 10
14 32
6/29/74*
M HIM
14 66
1 4 50
14 30
13 16
14 54
6/30/74*
N M I H
14 ft 4
14 57
14 51
14 32
14 62
CTl
-------
SAN GABRIEL MOUNTAINS
41W ©
BURBANK,
75W ®
LOS ANGELES
DOWNTOWN
SANTA MONICA MOUNTAINS
61W ©
ONTARIO
21W®
LONG BEACH
FIGURE 3. LOCATIONS OF TEMPERATURE AND RELATIVE HUMIDITY
MONITORING SITES
-------
From the data presented in Table 10, we note that the maximum differ-
ence in temperature at any hour during the day and the variation in average
air temperatures across the basin for each of the three days are as follows:
68
Day
June 28
June 29
June 30
Maximum Spatial
Temperature Difference
°c
15
14
10
°F
27
25
18
Average Spatial
Temperature Difference
°c
10
9
5
°F
18.5
16.5
8.5
Thus, spatial variations in temperature of as much as 15°C may exist in the
Los Angeles area during the middle of the day. However, on the average, the
variations in temperature are somewhat smaller.
To show temperature variations aloft, we plotted in Figure 4 temperature
profiles above Rialto, California, at five times on 26-27 July 1973. The 13:07
sounding on July 26 exhibits a temperature difference of about 9°C. If adia-
batic conditions had persisted, we would have expected the temperature gradient
i
to be -0,01°C tn. Thus, over a 1000m interval, the temperature difference
would be 10°C, which is approximately the amount observed at Rialto at 13:07.
As illustrated below, when an elevated inversion layer is present, the tempera-
ture differences in this situation may be smaller than those that would exist
under adiabatic conditions:
O)
re
Temperature
Inversion layer
-------
69
5000 -
4000 -
3000
2000
1000
9:39 j—
13:07 j—
17:20 (-_.
23:17
^Temperature-- C
FIGURE 4. DISTRIBUTION OF THE TEMPERATURE ALOFT
ABOVE RIALTO ON 26-27 JULY 1973
-------
70
For a modeling region extending to, say, 1000m in height above the terrain,
vertical temperature differences may be as large as horizontal variations.
In considering the distribution of water in the basin, we must first
convert relative humidity measurements to water concentration in ppm. Using
the definition of relative humidity, we can calculate the concentration of
water, [HLO], in ppm from the following formula:
x 10
where
RH = relative humidity (in percent),
P = vapor pressure of water (in mm Hg) at temperature T.
Figure 5 illustrates the temporal variation of water concentration at the
five ground stations on 28 June 1974. The two coastal locations tend to exhibit
similar behavior, as do the two inland locations. Concentrations at the Down-
town Los Angeles site seem to be more characteristic of those found near the
coast than those observed farther inland. In general, the spatial variation
in water concentration is about 7000 to 11,000 ppm.
Examining the temperature and humidity profiles observed at Rial to on 26-27
July 1973, we calculated vertical profiles of water concentration for five times
during this two-day period. These profiles are illustrated in Figure 6. The
maximum variation in concentration measured on these days was about 8000 ppm,
as shown in the 17:20 profile for July 26.
In the analyses described above, we found that spatial variations in tem-
perature and water concentration in the Los Angeles basin can be as large as
15°C and 11,000 ppm, respectively. Of course, since only a very limited number
of days were examined, it is highly probable that even greater variations fre-
quently occur. Considering the sensitivity results presented in Section D-l
and the variations in temperature and water concentration cited above, it is
-------
0600
1000
1200
1400
1600
1800
Time—hour
(28 June 1974)
800
1000
1200
1400
1600
1800
Time—hour
(30 June 1974)
FIGURE 5. TEMPORAL VARIATIONS IN WATER CONCENTRATION
AT FIVE LOCATIONS IN THE LOS ANGELES BASIN
-------
72
5000 r
5,000
10,000 15,000 20,000
Water Concentration--ppm
25,000
FIGURE 6. DISTRIBUTION OF THE WATER CONCENTRATION ALOFT
ABOVE RIALTO ON 26-27 JULY 1973
-------
73
difficult to conclude that these variations can be completely ignored. There-
fore, we recommend that future studies be carried out using the airshed model
itself to test various alternative strategies for treating temperature and
water. Such strategies might include treating temperature or water concentra-
tions as functions of
> Time only
> z and time
> x, y, and time
> x, y, z, and time.
Toward this end we have included provisions in the computer codes to allow the
user to input temperature and relative humidity fields that vary in both space
and time.
E. TREATMENT OF ORGANICS IN THE AIRSHED MODEL
V
Use of the kinetic mechanism discussed in Section B-l requires that the
organic species be grouped into four classes: paraffins, olefins, aromatics,
and aldehydes. To treat a mixture of numerous organics, such as those found
in the atmosphere, "average" rate constants must be estimated for 0, OH, and
0~ attack, as appropriate, for each of the four organic groups. In general,
specification of a single set of average rate constants that are invariant in
space and time is possible only if the individual members of each particular
group are of similar reactivity (neglecting spatial and temporal temperature
effects). Table 12 presents rate constants for 0, OH, and 0, attack on vari-
ous hydrocarbons. Because of the abundance of methane in the atmosphere and
the .wide disparity in reactivities of various paraffins, we conducted a study
to ascertain the best treatment of this hydrocarbon group in the airshed model.
We considered four strategies for grouping paraffins:
(1) One reactive group including all paraffins.
(2) Two reactive groups—C, through C~ low reactive; C.,
C,-, ... high reactive.
(3) Two groups--C, through C,, nonreact'ive; C., Cg, ... reactive.
(4) Two groups—methane nonreactive; C?, C^, ... reactive.
Strategy 3 has been employed in previous applications of the airshed model.
-------
Table 12
RATE CONSTANTS FOR 0, OH. AND 03 ATTACK ON VARIOUS HYDROCARBONS
OH
Hyrdrocarbon
•Paraffin's
Methane
Ethane-
. Propane
Butane
Isobutane
n-pentane
Isopentane
2,2-dimethylbutane
Cyclopentane
2,3-dirnethylbutane
•2-methylpentane
3-methylpentane •' .
n-hexane
Methylcyclopentane
2,4-dimethylpentane
2-methylhexane
3-methylhexane
• 2,2,4-trimethylpentane
' 'n-heptane
Methylcyclohexane
2,4-dimethylhexane.
Rate Constant
1.8 x 10"2
1.37
1.23 x 10
3.2 x 10
8.8
8.5 x 10
1.9 x 102
3.0 x 102
2.9 x 102
1.5 x 102
2.2 x 102
2.2 x 102
1.36 x 102
1.3 x 102
3.3 x 102
2.5 x 102
2.5 x 102
2.5 x 102
1.91 x 102
1.6 x 102
3.7 x 102
Reference
Herron and Hufe (1969
Herron and Huie (1969)
Heicklen (1967)
Herron and Huie (1969)
Wright (1965)
Herron and Huie (1969)
Herron and Huie (1969).
Herron and Huie (1969)
Herron and Huie (1969)
Heicklen (1967)
Estimate
Estimate
Herron and Huie (1969)
Estimate
Estimate
Estimate
Estimate
Herron and Huie (.1969)
Herron and Huie (1969)
Estimate
Estimate
Rate Constant
1.6 x 10
4.5.x 102
1.8 x 103
5.72 x 103 ••
5.12 x 103. ••
5.81 x 103
6.76 x-103 -;.
2.80 x 103 /
1.11 x 104
8.2 x 103
8.41 x 103
8.41 x 103
7.16 x 103
6.87 x 103
1.13 x 104
1.06 x 104
1.06 x 104
7.34 x 103
8.81 x 103
8.5 x 103
1.30 x 1 04
• Reference
.Greiner (1967)
Greiner (1967)
.Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner.'(1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiher (1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Greiner '(1967)
Greiner (1967)
Greiner (1967)
Greiner (1967)
Rate Constant
Reference
-------
Table 12 (Concluded)
OH
Hydrocarbon
2 , 5-d jmethyl hexane
2 ,3 ,4-trimethyl pentane
n-octane
n-nonane
n-decane
Rate Constant • -Reference
3
1
2
2
2
.7
.8
.5
.0
.6
X
X
X
X
X
102
102
102
102 .
102
Estimate
Herron and Huie
Herron and Huie
Estimate.
Estimate
(1969)
(1969)
Rate Constant
1.30
1.58
1.28
1.21
1.38
X
X
X
X
X
104
104
104
104
104
' Reference Rate Constant • Reference
Greiner
Greiner
Greiner
Greiner
Greiner
(1967) -.. " - •
(1967)
(1967)
(1967)
(1967)
Olefins
Ethylene
Propylene
Butenes
1-pentene
Trans-2-pentene
Cis-2-pentene
2-methyl-2"butene
Cyclopentene
1-hexene
Cis-2-hexene
1-heptene
Aldehydes
Formaldehyde
Acetaldehyde
Propionaldehyde
7.72 x 10
4.41 x 103
4.41 x 103
1.69 x 10
6.03 x 104
2.35 x 104
5.00 x 103
4.41 x 10
4.41 x 102
Cvetanovic (1963)
Cvetanovic (1963)
Cvetano'v.ic (1963)
Cvetanovic (1963)
Cvetanovic (1963)
Cvetanovic (1963)
Cvetanovic (1963)
Estimate
Estimate.
2.13 x 10°
2.13 x 104
5.12 x 104
5.33 x 104
1.13 x 106
1.13 x 106
1.49 x 105
1.92 x 104
1.92 x 104
3.84 x TO4
Morris and
Morris and
Morris and
Morris and
Morris and
Morris- and
Morris and
Mikl (1971)
Niki (1971)
Niki (1971)
Niki; (1971)
Niki (1971),
Niki (1971)
Niki (1971)
Morris and Niki (1971)
Morris and Niki (1971)
Morris and Niki (1971)
3.8 x 10
1.6 x 10'
1.3 x 10'
1.3 x 10
S'.O x 10
4.1 x 10
-3
-2
-2
-2
-2
-2
1.5 x 10
4.1 x 10
1.21 x 10
-2
-2
-2
Wei (1963)
Wei (1963)
Wei (1963)
Wei (1963)
Wei (1963)
Wei (1963)
Wei (1963)
Wei (1963)
Cadle (1952)
Aroma tics
Toluene ' 1.1 X 102
M-xylene
p-xylene
0
4.4 x 10^
Estimate
Estimate
Estimate
-------
76
To test these schemes, we performed smog chamber simulations for a mixture
of paraffins, NO , and CO having proportions typical of those found in the Los
X
Angeles atmosphere in 1969. For comparison, we carried out a baseline simula-
tion in which each paraffin was treated as an individual reactive species in
the mechanism. Thus, we compared the predictions for Strategies 1 through 4
with those for the baseline case to determine the errors introduced by each
lumping scheme.
Initial conditions for the simulation runs were derived from air quality
measurements taken at Commerce, California, on 30 September 1969 by Scott
Research Laboratories. In particular, we used the following concentrations,
which were measured at 8 a.m. on that day:
Species
CO
NO
crcs
V
Concentration
(ppm)
10.0
0,4
0.1
1.6 x 10
4.213
4
0.476
3.944
0.269
The predicted values of NO, NCL, and 0~ after 12 hours of irradiation were as
follows:
Predicted Concentration
Strategy
Baseline
1
2
3
4
NO
0.05
0.04
0.05
0.09
0.05
N02
0.30
0.30
0.30
0.28
0.30
°3
0.05
0.07
0.05
0.03
0.05
-------
77
These results indicate that Strategies 2 and 4 led to the best agreement
with the baseline case. Since Strategy 2 uses two reactive species, whereas
Strategy 4 involves only one, we plan to treat the paraffin class according
to Strategy 4 to minimize computing costs.
Thus, five organic classes are considered in the airshed model: non-
reactive hydrocarbons (methane and acetylene), nonmethane paraffins, olefins,
aromatics, and aldehydes. We recommend that future studies be carried out to
ascertain whether the olefins should be treated as a single lumped species or
as several lumped species. In addition, it may be possible to combine the
aromatics with the nonmethane paraffins, since both groups have similar reac-
tivities and may produce similar products (according to the mechanism given
in Section B-l).
F. INTRODUCTION OF THE IMPROVED KINETIC MECHANISM
INTO THE AIRSHED MODEL
In Sections B and C, we delineate efforts aimed at developing improved
mechanisms for describing the chemical interactions of hydrocarbons, NO , 0-,,
X 3
and S0?. With regard to the HC-NO -0., system, the generalized mechanism dis-
C. X O
cussed in Section B represents a significant improvement over the 15-step
mechanism previously employed in the airshed model. Thus, we have incorpo-
rated the expanded mechanism into the model. In addition, we have implemented
in the model the S0? mechanism described in Section C, even though the mechan-
ism has yet to be validated using smog chamber data. In the present section,
we discuss our efforts to use the improved kinetic mechanism in an actual air-
shed simulation.
Installation of the new mechanism in the airshed model required that
numerous changes be made in the computer codes. Particular difficulties arose
because the number of species that must be followed in the airshed model in-
creased from 6 to 12 (NO, N0?, CL, H?02, HN02> nonmethane paraffins, olefins,
aromatics, aldehydes, S0?, CO and unreactive hydrocarbons). Moreover, the
programs were to be exercised on the CDC 7600 computer, which has only a lim-
ited amount of small core memory, at Lawrence Berkeley Laboratory. Thus, we
-------
78
restructured the programs somewhat to make efficient use of available small
core memory, as well as the more abundant amounts of extended core memory.
After the coding changes were made, we checked the programs by running sev-
eral test cases.
To gain some experience in using the new mechanism in airshed simula-
tions, we decided to exercise the model using meteorological and emissions
inputs derived in previous model evaluation efforts. We felt that using the
same meteorological and emissions inputs, to the extent possible would pro-
vide a means for ascertaining how sensitive the model predictions were to the
change in the kinetic mechanism itself. Because of our previous experience
in simulating the Los Angeles basin on 29 September 1969, we chose that day
for our initial model application effort.
Before the simulations could be carried out, we first had to compute new
splits for hydrocarbon emissions and initial and boundary concentrations.
Previously, available hydrocarbon emissions and air quality data were divided
into two groups — reactive and unreactive hydrocarbons. To use the new mech-
anism, we revised the categories to reflect the new definition of the five
organic classes--nonmethane paraffins, olefins, aromatics, aldehydes, and
nonreactive hydrocarbons (methane and acetylene).
Organics are emitted from a variety of sources in the Los Angeles basin,
including motor vehicles, refineries, and numerous other stationary sources.
Although the organic composition of automobile emissions has been documented
by several investigators, very little information is available for use in es-
tablishing guidelines for estimating the compose!tion of the stationary source
emissions. For the purposes of this study, we assumed that the composition of
stationary source emissions is the same as that for automobiles. Although we
recognize that this is not necessarily a good assumption, our main objective
was simply to make "reasonable" estimates of the emission splits to exercise
the model. A more refined inventory can be derived using the results of a
recent study of organic emission control strategies carried out by Trijonis
and Arledge (1975). Unfortunately, their results were not available in time
for inclusion in this study.
-------
79
Using organic composition data derived from tests of 10 automobiles
reported by the Bureau of Mines (1973), we estimated the following mass
emission splits:
Mass Split
^ Group (percent)
Nonmethane paraffins 29
Olefins 30
Aromatics • 23
*
Aldehydes 5
Nonreactive hydrocarbons 18
Thus, we added previous estimates of reactive and nonreactive hydrocarbon-
emissions to estimate the spatial and temporal distribution of total hydro-
carbon emissions. Then, we multiplied the emission splits cited above by
the total hydrocarbon emissions in each grid cell to estimate the distri-
bution of emissions for each of the five classes.
We calculated initial and boundary concentrations using our previous
estimates of reactive and nonreactive hydrocarbon concentrations in conjunc-
tion with gas chromatographic analyses of ambient air in the basin for
29 September 1969 reported by Scott Research Laboratories (1970). We derived
the following relationships:
[Olefins] - 0.211[CR],
[Paraffins] = 0.414[CR] + 0.057[CNRJ,
[Aromatics] = 0.376[CR] + 0.003[CNR],
[Aldehydes] = 0.04 ppm,
[Nonreactive hydrocarbons] = 0.94[CNRJ,
where [CnJ and [C.m] are the original estimates of reactive and nonreactive
K INK
hydrocarbon concentrations, respectively.
Aldehyde emissions are estimated to be about 5 percent of the total hydrocarbon
emissions. Since aldehydes were not included in the original SAI inventory for
Los Angeles, the total percentage adds up to 105 percent. Thus., we increased
the total organic emissions by 5 percent to reflect the additional aldehyde
emissions.
-------
80
Figures 7 through 12 illustrate some of the predictions obtained from
the SAI model using the 31-step kinetic mechanism and the emissions and air
quality inputs described above. These figures also show the predictions
from the analogous simulation in which the 15-step mechanism was employed.
In general, the most obvious characteristic of these results is that the 0~
production seems to have been accelerated, leading to higher predicted 0,
levels. However, in many instances the NCL predictions are in better agree-
ment with the measurements, especially during the late morning and early
afternoon.
It is difficult to make any assessment now of the enhanced reliability
of the model resulting from the incorporation of the new 31-step mechanism.
However, considering the nature of the available model inputs used in this
study, the results are encouraging. We recommend that a greater effort be
expended in future work to assemble an appropriate organic emissions inven-
tory. Furthermore, the enhanced production of CL observed in the results
presented here may be caused in part by inaccuracies in the treatment of
aldehyde photolysis or NO removal in the mechanism. We assumed that alde-
hyde photolysis is proportional to that for NCL. However, shifts in the UV
spectrum throughout the day may invalidate this assumption. Finally, NO may
be removed too rapidly in the mechanism, thus, allowing OQ levels to build
up prematurely. These issues can be resolved only by subjecting the model
to a comprehensive evaluation. We recommend that such an undertaking be
considered in the near future.
-------
81
D_
Q_
•w
CJ
O
o
VJO
30
20
10
LA HABRA
9/29/69
(U MEASURED NO
PREDICTED NO (15)
A MEASURED N02
PREDICTED N02 (15)
© PREDICTED NO (31)
A PREDICTED N0'2 (31)
9 10 11
TIME CPST)
12
13
1U
60
50
•>-
cu
Q-
o
•z.
o
o
30
20
10 -
-6-
J2_
. <: LA HABRA
9/29/69
_ D MEASURED 0,
PREDICTED 03 (15)
© PREDICTED 0, (31)
9 10 11
TIME CPST)
12
13
FIGURE 7. PREDICTED AND MEASURED CONCENTRATIONS FOR LA HABRA
USING THE 15- AND 31-STEP KINETIC MECHANISMS
-------
82
60
50
CL-
OU
CJ
30
20
ANAHEIM
' 9/29/69
[D MEASURED
NO
- PREDICTED NO (15)
A MEASURED N02
• — PREDICTED N02 (15)
© PREDICTED NO (31)
A PREDICTED N0'2 (31)
5 6
60
50
9 10 II
TIME CPST)
12
13
ANAHEIM
9/29/69
IH MEASURED 0-,
PREDICTED 03 (15)
© PREDICTED 03 (31)
1C
O_
Q-
O
110
30
20
10
I I t
8 9 10. 11 12 13
TIME CPST)
1U
FIGURE 8. PREDICTED AND MEASURED CONCENTRATIONS FOR ANAHEIM
USING THE 15- AND 31-STEP KINETIC MECHANISMS
-------
83
POMONA
' 9/29/69
D MEASURED
NO
PREDICTED NO (15)
A MEASURED M02
— PREDICTED N02 (15)
© PREDICTED NO (31)
A PREDICTED N0'2 (31)
31
0-
0_
o
z
o
o
10 -
9 10 11
TIME (PST)
13
POMONA
9/29/69
Q MEASURED 0,
PREDICTED 0~
PREDICTED 0.,
(15)
(31)
9 10 11
TIME (PST)
FIGURE 9. PREDICTED AND MEASURED CONCENTRATIONS FOR-POMONA
USING THE 15- AND 31-STEP KINETIC MECHANISMS
-------
84
50
UO
'£ 30
o
20
- io
PASADENA
9/29/69
MEASURED
NO
PREDICTED NO (15)
A MEASURED XQ
PREDICTED N02 (15)
© PREDICTED NO (31)
A PREDICTED N02 (31)
9 10 11
TIME CPST)
12
13
60
50
o_
Q_
CJ
~z.
ED
CJ
30
20
10 -
PASADENA
9/29/69
tD MEASURED 0,
PREDICTED 03 (15)
© PREDICTED 0, (31)
9 10 11
TIME CPST)
12
13
FIGURE 10. PREDICTED AND MEASURED CONCENTRATIONS FOR PASADENA
USING THE 15- AND 31-STEP KINETIC MECHANISMS
-------
85
DOWNTOWN LA
9/29/69
13 MEASURED NO
PREDICTED NO (15)
A MEASURED N02
PREDICTED N02 (15)
® PREDICTED NO (31)
A PREDICTED N0'2 (31)
15
9 10 11
TIME (PST) .
12
13
DOWNTOWN LA
9/29/69
B MEASURED 0,
— PREDICTED 0,
PREDICTED 0.
(15)
(31)
ED
n:
o_
10
8 9 10 11
TIME (PST)
13
1U
FIGURE IT. PREDICTED AND MEASURED CONCENTRATIONS FOR
DOWNTOWN LOS ANGELES USING THE'15- AND 31-STEP MECHANISMS
-------
86
50
•10
IE 30
o
20
iO
WEST LA
9/29/69
CI MEASURED
PREDICTED
A MEASURED
NO
NO
NO,
(15)
N02 (15)
PREDICTED
© PREDICTED NO' (31)
A PREDICTED
N0'2 (31)
5. 6 7
20
2:
x:
o_
a.
u
2:
o
o
10
9 30 II
TIME (PST)
©
13
WEST LA
9/29/69
13 MEASURED 0,
— PREDICTED 00
(15)
PREDICTED 0 (31)
9 10 11
TIME CPST)
12
1U
FIGURE 12. PREDICTED AND MEASURED CONCENTRATIONS FOR
WEST LOS ANGELES USING THE 15- AND 31-STEP MECHANISMS
-------
87
III METEOROLOGY-RELATED DEVELOPMENT ACTIVITIES
Steven D. Reynolds
Mark A. Yocke
Jody Ames
In our previous model development and application efforts, we made
several assumptions about the treatment of meteorological parameters. Among
these, the most notable are the following:
> Wind shear effects can be neglected.
> A diffusivity algorithm that is solely a function of wind
speed and height can be used.
> The base of an elevated inversion layer is a suitable choice
for the top of the-modeling region.
However, these assumptions clearly introduce inaccuracies. First, the wind
flow field is fully three-dimensional and should be treated accordingly.
Second, the magnitude of the turbulent diffusivity depends on atmospheric
stability and surface roughness, as well as on wind speed and height. Third,
significant quantities of pollutants trapped in an elevated inversion layer
may be injected into the mixed layer as the stable layer is eroded by surface
heating effects. Moreover, ground-based inversions frequently occur at night.
Thus, further consideration needs to be given to the definition of the model-
ing region and the treatment of inversion layers in the model. In the follow-
ing sections, we discuss our efforts to improve the treatment of wind fields,
diffusivities, and inversions in the airshed model.
A. MODEL SENSITIVITY TO THE INCLUSION OF WIND SHEAR
The results of model sensitivity studies reported in Volume I indicate
the importance of accurately specifying the wind speed and direction through-
out the region of interest. In this section, we discuss additional sensitiv-
ity studies that were carried out to assess the importance of characterizing
wind shear effects. The results of this effort will be useful for establish-
ing (1) the need to extend our existing meteorological algorithms to treat
-------
88
wind shear and (2) the extent to which vertical wind soundings should be
taken over urban areas.
Accurate specification of winds aloft is usually-hampered in a grid
model by a dearth of appropriate measurements. Since the full three-
dimensional structure of the wind field must be input to the model, par-
ticular attention must be given to this aspect of model usage. The fol-
lowing are four possible means for establishing the complete wind field:
(1) Assumption of a "flat" velocity profile, where the estimated
ground-level wind speeds and directions are assumed to be
invariant with height (i.e., wind speeds and directions are
a function only of x, y, and t).
(2) Calculation of the winds aloft by scaling the ground-level
winds according to the findings of previous wind shear
studies (i.e., assumption of a form for the wind shear,
such as a power law profile).
(3) Interpolation for the wind speeds and directions using
actual wind soundings aloft.
(4) Prediction of the wind flow field using a numerical simu-
lation model.
The first alternative, which is the simplest, is useful for establishing the
basic characteristics of the flow field. Previous SAI simulations have used
this approach. For more refined estimates of the wind field when no measure-
ments aloft are available, the second technique can be used. The last two
alternatives afford the best means of specifying winds aloft, provided that--
for Alternative 3--the measurement network is sufficiently dense and that—for
Alternative 4--the model has been validated. At present, Alternatives 2 and 3
appear to represent the best means for accurately specifying winds aloft.
An important step in procuring a data base for describing the upper level
winds is being made in the RAPS program for St. Louis. One aspect of this com-
prehensive data gathering study will be the regular monitoring of winds aloft
-------
89
at two to four sites in this metropolitan area. Using the surface wind data
in conjunction with the upper wind measurements, one should be able to esti-
mate with reasonable accuracy the structure of the wind field over this urban
area.
To gain some insight into the importance of wind shear effects on the
predictions obtained from the photochemical airshed model, we carried out a
series of comparative simulations for the Los Angeles basin, using both "flat"
and power law wind velocity profiles. Since vertical wind soundings were not
available for Los Angeles, we used only the surface-based measurements to gen-
erate both wind fields. In the following subsections, we further describe the
treatment of wind shear and discuss the results of the simulations.
1. Hind Velocity Profile
Variations in horizontal wind with height have been the subject of inten-
sive study in meteorology for years. Assuming neutral stability conditions,
von Karman derived a logarithmic relationship for the mean wind velocity in
the sublayer (surface layer) of the atmospheric boundary layer from theoreti-
cal considerations (Plate, 1971):
*' " \ O/
where
U = wind speed at height I,
u* = the friction velocity,
Zfi = the roughness parameter,
K = the von Karman constant.
Subsequently, this relationship was verified through experiment. For diabatic
conditions, Laikhtman (1944) and Deacon (1949) proposed that the expression
-------
90
- g)
be used, where 3 is a function of atmospheric stability.
(10)
Within the remainder of the atmospheric boundary layer (i.e., above the
sublayer), wind profiles are usually characterized by an empirical power law.
Blasius was the first to describe the mean velocity distribution by the fol-
lowing general' relationship:
where LL is the wind velocity vector at a reference height ZR. The exponent
M is a function of ground surface roughness and atmospheric stability.
DeMarrais (1959), Davenport (1965), Shellard (1965), and Jones et al. (1971)
performed experiments to derive quantitative relationships for M. On the
basis of their findings, they estimated that M is likely to be within the fol-
lowing range:
0.4 > M > 0.2
Because of its applicability over the entire boundary layer, we selected
the mean velocity power law relationship [Eq. (11)] as the most suitable avail-
able description of the wind speed shear. We chose 0.2 as a representative
value of M for an urban area, such as Los Angeles.
2. Implementation of the Wind Velocity Profile
The numerical integration scheme used in the grid model requires that the
average wind velocity be specified at each grid cell interface. The integra-
tion of Eq. (11) along the vertical axis from the lower cell boundary to the
upper, followed by division by the cell depth, yields the expression for the
mean horizontal wind velocity over a horizontal cell interface:
-------
91
u = ^R • /ZM+1 _ ZM+1\ ,12)
" (V VZR(M + 1) \t ~ b /
where Z, is the elevation at the top of the cell and Z, is the elevation at
the bottom of the cell. Equation (12) can be used to obtain both the x and
y components of the mean velocity for each horizontal grid cell interface
within the modeling region. Assuming that turbulent atmosphere flow is in-
compressible, the vertical advective velocity, ws can be computed from the
continuity relationship:
3x 3y 3z
3. C o mp u t erjCodjjnj_
To incorporate the wind shear algorithms given by Eqs. (12) and (13),
we made appropriate coding changes in the computer programs embodying the
airshed mode"!. The result of these alterations was a slight increase in
both machine storage requirements and CPU time.
4. Description of the Experiment
After we altered the computer codes, we designed an experiment to ex-
amine the sensitivity of the airshed model to wind shear effects. To insure
that the deviations in predicted concentrations are caused only by dissimi-
larities in the prescribed wind fields, we made test runs using both the
original code (in-which a flat profile was assumed) and the newly revised
code with M set equal to zero. From Eq. (12), if M = 0, we obtain the same
flat wind profile as was used in the original formulation of the model.
Meteorological and emissions data for Los Angeles on 29 September 1969 served
as input data for predicting concentrations of RHC, URHC, NO, N02, 03, and CO,
using both the modified and unmodified programs for the hours 0500 through
1500 PST. The two programs produced identical predictions, thus indicating
that all coding alterations had been implemented properly.
-------
92
Finally, we ran the modified program using a value for M of 0.2, and
we compared the output of this program with the previous unmodified compu-
tations (assuming a flat velocity profile). Figures 13 through 20 show
plots of the average and maximum deviations in ground-level concentrations
for each species as a function of time. Here, the concentration deviations
are defined as the predicted concentrations with wind shear minus the
corresponding concentrations predicted when wind shear is neglected.
5. Discussion of the Results
An examination of the simulation results reveals the significance of
incorporating the power law wind profile in the grid model. Because the
velocities are systematically altered through the application of the wind
profile algorithm, the computed wind velocities at the inversion base and
ground-level heights were increased, relative to the straight profile •
values (M =0), by as much as 70 and 20 percent, respectively. The wind
velocities averaged over the entire mixing depth were consistently much
larger than the uniform profile values. As one would expect, therefore,
the results of the sensitivity experiment, which was performed with a 25
percent increase in all wind velocities (see Chapter IV of Volume I), are
strikingly similar to those shown here.
A characteristic of both the wind speed and wind shear sensitivity
studies is that, when the concentration maps are compared with those gener-
ated for the base case, a perceptible translation of concentration isopleths
toward the northeast, the prevailing wind direction, is observed. In addi-
tion, the majority of maximum concentrations are located in Grid Columns 20
through 25; this result was expected because the translation of concentration
isopleths is greatest when the path of travel is longest. The fact that
average overall deviations for all species are negative also supports the
hypothesis that the net effect of the inclusion of wind shear is similar to
that resulting from a simple increase in wind speeds.
-------
93
1.2
1.0
0.8
0.6
O NO
D .NO,
0.4
cx
cx
I
I
O)
O
CD
O)
to
$_
Ol
0.2
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
_L
_L
500 700 900 1100 1300 1500
Pacific Standard Time--hour
FIGURE 13. THE EFFECT-EXPRESSED AS AVERAGE DEVIATION—OF
VARIATIONS IN VERTICAL WIND SHEAR ON NO AND N00
-------
94
30
O NO
D NO
20
c
O
O)
O
O)
en
c:
QJ
o
S-
II)
D.
Q)
CD
Itf
S_
QJ
>
cf.
10
I
500 700 900 1100 1300 1500
Pacific Standard Time—hour
FIGURE 14. THE EFFECT—EXPRESSED AS PERCENTAGE DEVIATION—OF
VARIATIONS IN VERTICAL WIND SHEAR ON NO AND M09
-------
95
15.0
O NO
D NO,
10.0
Q.
D-
I
I
QJ
O
E
Z!
X
re
s:
5.0
500 700 900 1100 1300 1500
Pacific Standard Time—hour
FIGURE 15. THE EFFECT--EXPRESSED AS MAXIMUM DEVIATION—OF
VARIATIONS IN VERTICAL WIND SHEAR ON NO AND N00
-------
96
-p
ro
OJ
O
-------
CO
o
-G
D-
Q-
I
I
3.0
2.0
1.0
O CO
D °3
i.
Q-
I
C
O
-------
98
30
O CO
D °3
c
O
+J
fO
d)
01
rO
QJ
O
5-
0)
D-
O)
CD
n3
S-
QJ
20
10
_L
500 700 900 "MOO 1300 1500
Pacific Standard Time—hour
FIGURE 18. THE EFFECT—EXPRESSED AS PERCENTAGE DEVIATION—OF
' VARIATIONS IN VERTICAL WIND SHEAR ON CO AND 00
-------
99
CO
o
.c
o.
ex
I
I
i.
Q_
I
I
03
>
QJ
£
X
to
s;
500 700 900 1100 1300 1500
Pacific Standard Time—hour
FIGURE 19. THE EFFECT—EXPRESSED AS MAXIMUM DEVIATION—OF
VARIATIONS IN VERTICAL WIND SMEAR ON CO AND 00
-------
100
+J
ID
O)
Ol
td
4->
C
OJ
O
l-
(L)
D_
E
'a
X
10
200
180
160
140
120
100
80
60
40
20
O CO
D °3
JL
500 700 900 1100 .1300 1500
Pacific Standard Tinre--hour
FIGURE 20. THE EFFECT-EXPRESSED AS PERCENTAGE MAXIMUM DEVIATION-OP
VARIATIONS IN VERTICAL WIND SHEAR ON CO AND 0
-------
101
The results of this study clearly indicate that wind shear phenomena
should be included in the airshed model. Of course, it would be most help-
ful in the construction of velocity profiles to have wind data taken aloft
in cities where the model is to be applied.
B. TREATMENT OF WIND SHEAR IN THE AIRSHED MODEL
On the basis of the sensitivity results presented in the previous
section, we included provisions in the computer programs for treating a
fully three-dimensional wind field. To facilitate usage of either theo-
retical wind shear relationships or actual wind data aloft, we structured
the air quality simulation program to accept the three-dimensional wind
field inputs directly from the meteorological data file. The user assembles
the wind field inputs by employing the Automated Meteorological Data
Preparation Program. Thus, all wind shear algorithms and interpolation
routines are embedded in the meteorological data program. By structuring
the airshed simulation package in this way, we.have enabled changes in the
treatment of wind shear to be accomplished without modifying the photochem-
ical dispersion model code.
In many urban areas, sufficient soundings of the winds aloft are sel-
dom available for use in constructing the complete flow field. Therefore,
we initiated efforts to derive a set of theoretical wind shear relationships
using results obtained from Deardorff's planetary boundary layer model.
These relationships are presented and discussed in Chapter II of Volume III.
For use in those urban areas where numerous pibal or other suitable data are
available, we recommend that an algorithm be developed and installed in the
Automated Meteorological Data Preparation Program for the construction of
wind fields aloft using the available wind soundings.
C. EXAMINATION OF AN ALGORITHM FOR DERIVING MASS-CONSISTENT WIND FIELDS
One of the assumptions commonly invoked in airshed modeling is that the
air flow in the planetary boundary layer is incompressible. Under these con-
ditions, the velocity components satisfy the following continuity relationships
-------
102
3Ji+ 3V + 3W = Q /]
ax ay 3z u ' U
where u, v, and w are the x, y, and z components, respectively, of the wind
velocity vector. In the SAI model, the z coordinate is normalized by the
depth of the modeling region, in which case Eq. (14) becomes
8UAH 3vAH_ SW _ A
~ = U
where
P = [z - h(x,y)J/[Ht(x,y,t) - h(x,y)J,
H. = the elevation of the top of the modeling region
h = the terrain elevation,
AH = Ht(x,y,t) - h(x,y),
W = w -•u[(3h/8x) + p(8AH/8x)] - v [(ah/3y)
In typical airshed model applications, estimates of u(x,y,z,t) and
v(x,y,z,t) are obtained from the available data on wind speed and direction,
both at ground level and aloft. Once the horizontal components are specified,
Eq. (15) can be solved for W. Writing this equation in finite difference
form, we obtain
w. • lxl = w. . , l - -^ (UAH)..! . , - (UAH)
i i Ix -U-L--- n "I [/ _ £>-• A Y I T "ri1*' T K
1->,,J,k]
where the integer triple (i,j,k) designates the center of a grid cell. Equa-
tion (15) is solved subject to the constraint of
W = 0 (17)
-------
103
at p = 0, which simply states that either the wind speed is zero at the
ground or the wind is flowing parallel to the terrain. By first estimating
the horizontal wind components and subsequently solving Eq. (15) for W, we
insure that the net flux of air into each grid cell is zero.
One difficulty associated with the windifield methodology described
above is that nonzero vertical velocities may be calculated at the top of
the modeling region. Thus, pollutants may be advected out of the modeling
region, even though a stable capping inversion layer is present. This situ-
ation is somewhat contrary to the usual belief that an elevated inversion
layer suppresses vertical transport, although buoyant air parcels may pene-
trate the stable layer to some extent. It is important to note that the
calculated vertical motions are, in part, the result of inaccuracies in the
predicted horizontal wind components, especially aloft, where few measure-
ments are generally available.
In previous efforts, we examined means for removing convergence and
divergence areas in the flow field aloft (see Roth et al., 1971). However,
these attempts to force the vertical velocities to obey some specified con-
straint, such as a zero velocity at the inversion base, failed to produce
acceptable wind fields. In many instances, the algorithms generated hori-
zontal wind speeds aloft in excess of 40 mph. Under the present contract,
we revisited this issue of constructing mass-consistent wind fields in light
of the findings of recent studies in this area reported in the literature.
1. The Governing Equations
The problem that we address, here is as follows: Given a set of initial
estimates of u and v over the modeling region, how should these wind speeds
be adjusted to yield vertical wind velocities that not only satisfy Eq. (14),
but also obey some imposed constraint. The methodology described below is
similar to that given by Fankhauser (1974).
-------
104
Let UQ and VQ designate the initial estimates of the horizontal wind
components, which have been obtained through, say, the application of inter-
polation procedures. The values of u and v to be employed in the grid model
are obtained by defining a function $ in the following manner:
UAH = (UAH)Q + |i , (18)
VAH = (VAH)Q + . (19)
Note that we attempt here to adjust only un and vn, not AH. Substituting Eqs.
(18) and (19) into Eq. (15), we obtain
We define the terms on the right-hand side of Eq. (20) as follows:
D = - —
8(uAH)0 8(vAH)Q
0 3x 8y.
Thus, Eq. (20) becomes
A A
d
-------
105
Operationally, Eq. (23) is written in finite difference form ancMs
solved on successive layers of grid cells in the x-y plane. Thus, in
finite difference notation, Eq. (23) becomes
[V2*] = D. . k - (D ) , (25)
1 j |< i >3 >*• U 1 ,J ,K
2
where [v
-------
106
more satisfactory relationship for D - D~. Note that the perturbations to
the flow field near the surface are significantly smaller if one uses
Eq. (27) than they are if Eq. (26) is used, as demonstrated in the next
section.
In specifying boundary conditions, one has two possible choices: the
Dirichlet (= 0) or the Neumann (8/9n = 0) boundary condition. Physically,
the former treatment leaves the u component of the velocity unaltered along
boundaries parallel to the x-axis and the v component unaltered on boundaries
parallel to the y-axis. In the latter case, just the opposite is true.
Fankhauser (1974) employed the Dirichlet condition in his study, and Liu et
al. (1974) report that in simulations using a similar type of model, the
results were not significantly influenced by the choice of one formulation
over the other.
2. Tests of the Model
To test the model described in the previous section, we carried out a
study to determine the magnitude of the alterations that would be predicted
for the typical wind fields previously used as input to the SAI airshed model.
Thus, we rendered the wind fields used in the 29 September 1969 model evalua-
tion study for Los Angeles (see Reynolds et al., 1973) mass consistent; more-
over, we constrained the vertical velocity W to be zero at the base of the
inversion layer. We used a 25 x 25 x 5 grid layout, where Ax = Ay = 2 miles
and Ap= 0.2. Since wind shear was neglected in the Los Angeles study, we
considered UQ and VQ to be functions only of x, y, and time. Tables 13 through
15 illustrate the nominal wind speeds and directions and mixing depths for
6 a.m. and 3 p.m. on 29 September 1969. These maps served as the inputs to
the mass-consistent wind algorithm.
In performing the calculations with the model, we wished to assess the
sensitivity of the predictions to (1) the manner in which D - DQ is approxi-
mated [i.e., the use of Eq. (26) or (27) and (2) the choice of either
Dirichlet or Neumann boundary conditions. Furthermore, we examined the nature
-------
107
Table 13
HOURLY -AVERAGED WIND SPEED AND DIRECTION IN THE LOS ANGELES BASIN
ON 29 SEPTEMBER 1969 AT 6:00 a.m. PST
(a) Hind Speed
I* 15 16 IT 18 19 2« Zl 22 23 24 23
81
24
20
Z2
Zl
20
14
la
ir
16
15
14
13
12
11
10
«
e
T
6
I!
4
8
9
1
23
24
23
22
21
20
19
in
ir
16
19
14
13
12
II
10
9
o '
T
«
8
4
a
1.0
1 ,u
I.e
1 .0
1 .6
2.0
8.0
0.6
2.5
2.6
2.0
2.6
2.6
2.6
2.0
2.0
90
01
60
73
76
77
77
70
79
79
00
01
02
00
04
84
03
C7
08
98
9»
90
90
»
i.e
i.e
1.0
1 .V
I .0
z.e
3.0
3.0
2.5
2.0
2.0
2.0
2.0
2.6
3.0
2.0
Z
103
93
05
T>
60
01
01
02
03
03
04
04
04
03
03
03
06
00
69
91
91
91
91
41
1.0
I.D
2.0
1 ,u
I .D
2.0
3.0
2.6
2.6
2.0
2.0
2.0
2.0
2.8
2.0
0
loa
97
04
03
(16
06
O7
07
no
00
or
07
00
06
07
00
90
92
92
92
93
a.e
2.6
3.0
1 .0
1 . D
2.0
3.0
2.0
2.0
2.0
2.0
2.0
2.6
2.0
4
1 13
106
96
94
93
92
91
91
92
91
90
09
09
07
OB
09
91
93
93
93
93
93
0.0
Z.D
3.0
i.e
1.0
2.0
3.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
B
1 10
113
110
103
102
99
V7
96
94
92
90
00
00
09
90
92
94
94
94
94
»4
3.0
3.0
1. u
1.6
2.0
3.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
6
123
121
110
112
109
163
102
100
97
94
91
09
09
90
92
94
96
96
96
96
96
0. V
3.0
1 .u
1 . o
Z. W
3.6
2.0
2.0
2.6
2.0
2.0
2.0
2.0
T
120
117
1 14
110
106
104
106
90
90
90
92
94
16
90
90
90
90
90
S.u
2.0
2.0
2.6
3.0
4.6
2.6
2.0
2.6
2.0
2.0
2.0
2.0
a
136
124
120
116
11 1
107
163
99
91
90
93
96
90
106
100
100
too
109
ice
2.3
2.0
2.0
2,3
a.u
4.3
3.0
2.5
2.0
2.0
2.0
2.0
2.6
2.5
2.0
a.O
a.o
4.0
4.e
2.5
2.0
2.0
2.0
2.6
2.0
2.6
2.0
3.0
3.G
4.U
4.0
5.0
a.o
2.3
2.0
2,0
2.0
2.0
(b) Wind
9 19 II
141 146 HI
191
126
122
1 17
no
106
102
96
96
90
99
100
102
102
102
102
102
102
130
133
129
12D
113
100
107
193
103
103
103
194
104
104
104
10-4
104
104
146
100
131
I2G
llfl
113
1 I I
ion
ion
107
107
IOC
1 06
106
106
IOG
106
166
2.0
2.0
3,6
4.0
*.W
8.0
o.t»
3.0
2.5
2.5
2.0
2.0
a.e
2.6
2.6
4. «
3.0
0.6
0,0
S.fc
3.0
'2.C
2.5
2. fl
2.0
2.0
2.0
a.e
9.O
O. W
P.O
b.U
3.0
3.0
2.6
2.0
2.0
K.O
^. V
2.0
2.6
4.6
0.0
6.0
o. u
3.0
3.0
2.6
2.0
2.0
2.0
2.0
Direction
.12 is 14 in
149 ICO IGO 130
142
137
139
1 17
116
IIS
113
III
110
109
ion
100
100
loa
100
100
147
141
139
129
123
117
1 17
117
115
113
111
119
110
110
110
no
110
146
13C
133
130
120
1 10
1 18
lie
no
110
117
tie
1 1 1
118
no
no
tic
146
100
133
130
120
126
1 10
1 17
110
123
124
124
•125
1 12
110
no
109
1M
2.0
2.5
6.0
0.0
6.0
5.0
3.0
3.0
2.6
2.9
2.0
2.0
16
150
146
135
130
127
123
120
117
1 16
116
126
120
120
I2S
116
(12
lid
109
169
2.8
3.0
6.0
5.0
4.0
6.0
3.0
2.6
2.0
2.0 '
2.0
2.6
17
101
143
139
123
120
119
110
113
II 1
114
122
121
430
130
127
119
111
110
111
111
2.3
3.6
4.5
4.5
4.0
6.6
3.0
2.6
2.0
2.0
2.0
2.0
2.0
10
161
140
123
123
120
1 10
110
100
107
103
103
113
130
130
124
120
III
100
111
III
2.6
3.3
4. D
4.0
3.0
4.0
2.5
2.0
2,0
2.0
2.0
2.0
19
161
136
120
117
1 10
106
109
105
107
106
102
111
126
120
113
I 16
109
111
111
fi.w 3.6 3.5 J.D 4.0 4.0
3.5 4.6 4.0 4.0 4.0 4.U
*.u •i.u *.U 3.0 y.u 3.0
4.0 y.O 3.0 2.5 2.5 2.S
3.0 *.5 2.5 2.5 2.5 2.5
3.0 2.6 *.« 2.6 2.6 2.0
2.5 2.0 a.u 2.6 2.0 ^.»
2.0 2.6 2.O 2.0 2.0 Z.U
2.0 2.6 H.W 2.0 2.0 K.»
2.0 2.0 2.0 2.0 2.0 2. -
2.0 2.0 2.0 -.0 2.6 2. ft
2.6 2-ft 2.0 2.0 2.0 2.0
20 21 22 23 24 23
160 160 130 156 156 150
136 131 120 126 120 120
115 116 I07 105 163 10-)
1 10 95 93 92 91 90
100 95 93 92 91 9**
100 95 93 92 91 90
90 95 93 92 91 90
100 93 93 92 91 91
101 96 92 90 90 90
101 97 93 90 96 99
109 95 93 93 93 93
100 101 99 9G 92 92
100 96 96 92 90 9(»
115 110 100 93 91 96
104 90 93 90 90 96
169 103 96 92 98 90
104 104 ICO 92 96 90
165 IOO 92 90 90 90
135 109 92 99 9* 90
-------
108
Table 14
HOURLY AVERAGED WIND SPEED AND DIRECTION IN THE LOS ANGELES BASIN
OH 29 SEPTEMBER 1969 AT 3:00 p.m. PST
(a) Wind Speed
22
20
17
15
11
19
9
B
7
D
2
24
£3
21
14
'•
6
1
1
1
6.0
B.O
0.0
6.0
7.0
0,5
B.9
8.0
0.9
0.0
0.0
B.O
131
126
110
7»
71
74
79
6.0
6.0
9.0
7.0
8.5
0.0
0.0
0.0
0.0
B.O
n.e
123
126
110
69
71
74
74
V4
6.0
6.6
6.0
7.0
8,e
0.5
0.0
o.e
o.e
o.o
o.o
o.o
lie
112
no
66
72
74
74
74
6.0
6.5
9.e
7.0
o.e
o.e
B.O
o.e
o.o
s.o
0.0
7.3
IB7
110
no
69
72
79
73
73
6.0
5.5
9.0
7.0
9.0
o.e
B.O
o.e
e.e
o.o
o.o
7.6
90
99
103
65
70
75
73
73
6.0
3.5
C.O
7.0
9.0
0.0
0.0
8.0
0.0
0.0
0.0
7.0
94
93
93
66
60
73
73
73
7
6.0
O.i5
7.0
9.0
0.0
0.0
0.0
0.0
0.0
B.O
7.0
90
«e
90
66
67
73
73
73
e
6.0
8,0
6.e
o.e
8.6
9.0
o.e
8.9
0.0
o.e
o.o
6.0
83
83
79
66
66
V3
73
78
»
9.6 6
9.0 v
7.0 7
B.O 0
9.0 9
10.0 11
9.O 10
9.0 JO
0.9 9
0.9 D
o.e 8
6.6 0
(b]
70
03
73
66
64
va
73
73
10 11 12 13 14
.0 B.O 0.6 G.O 6.0 6
. w o.o (j.w 7,0 D.O 6
.0 0.0 ?.t> 9.0 10.0 12
.0 y.u 10.0 10.0 12.0 11
.0 10,0 11.0 12,0 I (.0 10
.0 12.0 14.0 13.0 11. w 10
.0 12.0 13. 0 13. V 11, e 10
,0 11.0 12.0 13.0 ll.o 10
.0 9.0 10. 0 11 .0 10.0 9
.5 0,6 9.0 10.0 *.6 0
.0 0.0 0.0 D.O 7.6 f
.0 C.O 6,0 e.e S.O 0
Wind Direction
70 66 60 37 S3
65 60 47 00 36
63 DO 40 41 40
.
65 64 62 62 61
•
64 62 64 70 66
'
73 73 73 73 73
73 73 V3 73 73
73 73 73 T3 73
It)
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
. W
36
36
43
61
60
73
73
73
1C
7.0
9.0
12.0
10.0
10. 0
9.0
9.0
0.0
7.0
6.B
0.0
36
36
44
60
72
73
73
73
17 13 19
0.0 B.O 9.0
10.0 10.0 11. w
11.0 10,0 -y.U
10.0 9.0 0.0
9.0 0.0 7.0
9.0 9.0 0.0
9.0 n.O 7.0
0.0 7.0 6,0
7.0 6.0 B.6
6.0 6,6 C.O
6.0 C.O C.O
0,1) D.O 5.0
36 36 36
07 37 37
40 39 37
CO 63 63
72 72 72
73 73 74
73 73 73
73 ra 70
20
v.O
I I.u
0.0
7.W
7.0
7.0
6.0
6.0
5.5
0.0
6.0
6.0
36
37
37
65
72
74
77
77
21 22 23
9.0 9.0 9.0
10. 0 9.0 u.o
O.U 7.0 6.0
7.0 6.0 «.0
6.0 5.U 6.0
6.0 6.0 t.u
5.0 4.0 *.U
4.0 4.0 i.u
4.3 4.6 4.0
6.0 4.E 4.3
B.O 5.0 6.0
5.0 5.0 G.»
39 34 31
06 33 30
37 36 35
65 70 72
73 74 75
77 77 79
79 02 83
79 fl2 03
24 21
9.0 B.O
7.0 7.0
b .U it. «
&.B 6.0
» . 0 5. y
4.0 t.v
4.0 4.0
4.0 4.0
4.tJ 4.P
4.B 4.0
4.5 4.5
o.o S.O
30 30
30 30
35 43
73 73
75 75
79 HO
03 BJ
fi3 O't
-------
109
Table 15
MIXING DEPTHS IN THE LOS ANGELES BASIN
ON 29 SEPTEMBER 1969 AT 6:00 a.m. AND 3:00 p.m. PST
(a) 6:00 a.m.
S3
24
33
23
21
19
IB
IT
ie
in
14
13
12
11
ie
»
D
r
0
«
8
a
1
29
24
23
22
21
20
19
II)
17
16
19
14
19
12
II
1.0
<
8
y
t
I
4
8
2
1
1
100
100
ioe
led
2oe
809
400
DOO
DOO
Doe
BOO
BOO
DOO
DOO
DOO
COO
DOO
DOO
000
COO
000
DOO
1
2599
2509
2509
2409
2000
1300
1200
909
600
650
050
650
530
630
BGO
050
E50
BOO
CCO
D70
060
030
CflO
BOft
ioe 100
100 joe
100 100
160 109
209 £08
000 300
400 400
060 COO
000 600
000 600
009 DOO
660 COO
COO 000
COO 000
COO 600
600 600
600 600
009 BOO
660 COO
60& GOO
500 000
069 600
2 a
2309 259C
2590 2500
2590 2300
2400 24 CIO
2000 2000
1COO 1600
1200 1200
909 909
600 600
059 600
630 550
650 650
000 050
009- COO
630 BOO
650 650
050 050
630 550
030 R00
630 060
650 BOO
630 659
BB0 KliO
660 630
140
100
100
100
200
300
400
400
600
600
669
009
CCO
G09
GOO
coe
006
GOO
000
009
009
060
4
2500
2ceo
2500
2100
2000
1500
1200
900
600
059
630
GOO
500
550
530
030
050
000
650
630
000
039
090
650
104 104
100 160
109 169
J00 166
200 200
300 273
373 323
430 410
496 460
BOO 470
500 400
500 500
BOO 500
G(JO 600
000 600
600 000
GOO 600
500 000
BOO 000
600 500
000 090
000 009
6 6
2306 2300
2500 2300
2500 2300
2400 2200
2000 2000
1500 1500
1200 1200
909 909
600 600
050 676
650 000
550 650
550 050
006 G50
000 BGO
650 530
650 550
559 050
036 530
030 000
OGO 060
030 630
CCO BF»
6170 6BO
106 166
100 100
109 100
109 100
200 200
25 & 230
000 300
350 323
410 330
423 075
450 409
499 450
470 460
490 450
GOO 400
400 356
400 356
000 600
060 GOO
000 GOO
609 BOO
COO 000
7 0
2400 2300
2400 2300
2200 2200
2100 2100
2000 2000
1500 1500
1200 1200
900 909
600 600
600 609
070 600
609 600
OHO 073
650 070
BRO 576
550 670
659 650
G50 500
659 000
000 030
000 069
050 630
000 S50
BOO 600
fr !&
IBO 200
100 ISO
100 [00
100 100
206 200
2S0 225
273 250
300 275
323 300
350 325
370 309
400 378
430 469
400 400
4.39 400
300 400
350 3DO
600 BOO
500 600
300 000
COO 600
600 COO
Cb)
9 10
2200 2000
2200 2000
2209 2000
2200 2000
2200 2000
1000 1700
1500 MOO
1000 1006
O00 900
600 673
630 650
600 630
600 630
600 630
070 610
670 609
050 G73
BOO 073
600 609
050 609
000 000
650 OQO
050 000
500 DOS
a 12
20O 2O«
200 200
160 160
100 100
ISO 140
200 176
223 206
250 223
270 230
300 £70
323 300
3GO 325
375 330
409. 400
423 430
450 450
400 493
DOO 660
600 500
DOO 500
000 GOO
BOO 000
3:00
II 12
1009 IBC0
1900 1700
1900 1000
1900 1900
1000 JOOO
1500 IS00
1300 1300
1 100 1 100
9GO *930
700 723
600 700
660 690
600 700
600 700
6GO 670
625 640
600 600
676 670
039 BSO
GOO 030
OD9 6BO
BOO 839
000 030
CBQ 669
13 14
200 206
206 200
IGO 200
100 100
120 109
100 123-
175 150
200 J73
223 200
250 223
270 236
300 360
350 350
400 400
450 450
4110 473
493 490
COO 600
600 609
500 COO
DOO 600
609 GOft
p..m-
13 14
1200 1209
16 WO/ 16W&
1(100 1(103
1900 2000
1000 2000
1G00 1000
1300 1009
11 09 1209
9HO 930
750 773
729 7+9
709 000
700 700
700 600
709 730
609 700
609 625
G70 075
006 000
050 660
630 600
050 000
600 050
6«0 600
ID
200
200
209
109
100
109
125
150
170
206
230
200
340
400
450
470
4Q5
GOO
600
GPO
609
000
16
1209
iS00
1000
2100
2300
20UO
1700
1409
1000
025
noo
900
000
03d
773
700
623
670
690
069
COD
059
BOO
609
16 ir
20ft 200
200 200
200 200
100 100
100 100
too toe
100 100
125 100
160 123
173 150
223 . 200
270 225
330 250
375 300
400 350
450 400
475 450
660 495
CCO 308
500 305
609 GOO
666 600
I* 17
1Z69 1209
1609 1300
1000 moo
2200 2200
2300 2300
2300 2500
2000 2306
1700 2200
1100 2000
1200 1500
1200 1400
1100 1309
1060 1100
900 1000
ooo nno
700 750
600 650
600 600
GffO 670
000 009
009 5H0
009 IF30
068 060
690 666
18 19
200 20O
200 200
200 209
100 100
166 100
100 106
100 100
100 100
100 100
125 123
IGO 109
200 175
259 200
300 ?,GO
330 300
370 359
400 400
490 4OO
496 495
FOO 500
500 500
COO 009
10 19
1200 1209
1600 1500
1000 1000
2200 2200
2300 2300
2GOO 2300
2500 2500
2300 2GOO
2200 2300
1000 2000
1600 1000
1300 1000
1300 1300
1100 1109
930 1009
900 900
000 (100
700 7GO
600 600
073 670
BOO BOO
BOO BOO
609 050
050 600
20 21
299 200
200 200
200 200
100 100
160 100
109 109
100 100
100 100
100 100
123 123
100 150
173 175
200 200
225 225
273 250
323 000
073 350
450 425
400 460
500 4(10
009 600
ceo 600
20 21
1200 1200
1000 1500
1000 1000
2200 2200
2300 2300
2300 2fiOO
2500 2300
2300 2200
2200 2209
2000 20(10
1000 1000
1500 1500
1300 1300
1190 1100
1000 1000
900 900
000 000
730 750
600 600
073 070
BOD 030
600 600
tfSO GDO
669 606
22 23
200 000
200 200
200 200
173 200
100 100
100 109
100 109
123 149
125 (00
150 200
173 200
200 2dO
223 225
250 250
230 250
300 000
323 323
400 375
423 400
•150 -125
400 400
BOO 400
22 23
1200 1209
1500 1000
1000 1000
2200 2200
2300 2300
2500 2500
2300 2300
2200 2200
2200 2200
:;ooo 2000
1000 1OOO
1600 1600
1400 MOO
1200 1200
1100 MOO
1000 1000
050 (130
750 700
600 609
073 073
BOO 600
023 OOO
623 006
030 020
24 21
200 2
200 20J
200 200
200 209
100 I5d
I 69 1 ')"
120 15U
166 15.)
ISO 15-1
200 200
200 200
200 200
223 25..
230 20 •
273 20*.
300 25 ft
325 32-
373 35l>
400 40H
400 40-1
359 :i5 .
350 309
24 23
1200 1200
1569 1500
10(10 1009
2200 220'1
2000 2:U>'t
2500 2509
2300 230n
2200 2200
2209 2200
2000 20OO
1000 i no t*
1600 1600
1400 140H
1200 1200
1 U>0 I I Oil
1000 1000
03 0 05 0
750 750
600 600
075 B75
630 65»
BOO 009
609 450
coe
-------
no
of the changes in the flow field at two levels on the grid: near the surface
and at the top -of the modeling region (i.e., k = 1 and k = 5) . The results
of these simulations are presented in Tables 16 through 23. Table 24 summar-
izes the nature of the inputs and the treatment of the parameters in the wind
algorithms corresponding to each table. The predicted changes in wind speed
and direction given in these tables are defined as follows:
A6 = 57.2958
where AS and A6 are the reported changes in wind speed and direction, respec-.
tively.
3. Discussion of the Results
Reviewing Tables 16 through 23, we note that wind speed and direction are
altered by no more than about 11 mph and 93°, respectively. To place some persp-eo
tive on these results, we must consider the magnitude of the errors associated
with the input wind fields themselves. Typically, the uncertainties in the
wind fields employed in airshed simulations are on the order of 2 mph in
speed and 60° in direction. For the most part, the predicted perturbations
in wind direction are smaller than 60°. However, significant alterations
in the wind speed are predicted for the 3 p.m. wind inputs. These predic-
tions are the result of generally higher wind speeds and a greater degree
of convergence and divergence in the interpolated wind field.
Since UQ and vn are considered to be independent of z, the perturbations
calculated using Eq. (26) are the same in each layer of grid cells. Thus,
the change in speed and direction is reported only for the bottom layer of
cells (i.e., k = 1).
-------
Table 16
PREDICTED CHANGES IN WIND SPEED AND DIRECTION FOR CASE
(a) Wind Speed
IS 10 17 10 19 2« 21 22 23 24 23
2
S3
24
23
22
21
28
1«
IB
17
16
14
13
12
11
!•
»
e
r
«
e
4
s
a
i
-.0 -.1 -.1 -.1 -.2 -.2
i a e < g <
-2 -2 4 10 19 23
0 1 B 10 23 32
2 2 0 10 22. 31
-1 -1 -2 8 IB 26
-13 -14 -II -8 1 15
-21 -22 -21 -11 6 19.
-IB -20 -21 -14 1 12
-12 -12 -11 -10 -7 -2
-« -6 -1 -1 -1 -B
1 0 0-0-0-0
600011
o e • e i o
-•-000 1 1
000 1 0-0
o o e 0-0-2
-o -o -• -e -i -a
-i -i -i -i -i -e
-2 -2 -a -a -i -o
-a -2 -2 -i -i -o
-1 -1 -1 -1 -0 -0
-i -o -o -e -» -o
_» -e -• -» -o o
,3 .6
-.8 -.3
-.3
-.4 -.4
(b) Wind
21 26
86 30
29 80
24 24
27 23
22 20
3 6
-fl -4
-1 -1
-0 0
-I »
-0 0
-1 -2
-4 -6
-4 -6
-e -3
0 -1
e e
e o
ft 0
6 1
1 1
9
14
24
20
24
01
10
6
0
-2
-0
1
2
1
-3
-3
-2
-3
-a
0
0
1
i
a
19 11
10 «
17 10
26 22
27 2B
29 26
ia 20
B 10
I 0
-3 -2
0 -0
o e
6 -1
1 2
0 9
0 11
4 10
-0 6
-2 1
-1 0
0 2
a 8
2 e
8 8
-.4 -.3
-.3
Direction
12 18
B 7
7 B
16 14
29 32
36 U3
21 22
7 7
-I -3
-2 -2
o e
-0 -1
-3 -4
1 1
9 7
10 7
11 B
9 II
7 14
6 13
6 9
E 7
4 11 '
4 4
14
e
9
13
09
38
27
10
-2
-7
-0
-2
-B.
-8
G
B
4
B
17
IB
12
0
6
f
10
13
13
14
09
30
21
B
-7
-11
-1
-2
-B
-2
1
6
3
7
10
' 16
13
9
6
4
-.2 -.1
16 17
16 20
16 21
IB 17
30 42
34 34
17 15
2 2
-12 -12
-13 -10
-2 -0
-0 -2
-6 -B
-6 -10
-2 -I
3 1
1 -4
B -0
B 1
11 B
10 9
B 7
I 4
0 2
-.1
ID
22
24
10
40
31
9
-e
-i i
-17
-15
-0
-3
-16
-4
6
-2
-1
0
•0
6
7
0
2
-.1
19
22
24
[0
32
23
6
-1
-0
-13
-17
-14
-6
-22
-7
3
-B
-0
2
-1
4
6
2
|
-.2
29
23
26
16
23
16
8
-0
-4
-10
-15
-19
-14
-13
-B
-3
7
-7
2
-2
1
6
2
2
21
23
26
17
19
19
7
2
-1
-4
-II
-10
-16
-11
-fl
-
-------
112
Table 17
PREDICTED CHANGES IN WIND SPEED AMD DIRECTION FOR CASE 2*
(a) Wind Speed
24
17
3
2
1
M
24
23
22
21
2*
19
ID
17
16
19
14
18
!•
»
a
r
6
0
4
»
a
i
,„ ,,j _.p _.o -.0 .e .3 -.3 -.0 -.2
-.0 -.0 -.e -.9 -.1 -,i -.1 -.1 -.1 -.1
-,e -.* -.e -.0 -.1 -.1 -.1 -.1 -.1 -.1
-.B -.e -.» -,o -.1 -.1 -.1 -,i -.1 -.1
(b)
ia04B67BS10
-Z -3 -» 0 B T 0 6 a 2
1 1-08497908
-3 -3-8-1-1 2 7 6 7 18
-7 -7 -0-225 11 11 16 13
-B -4-3-3-1-8-8 I 1 2
-1 -1 -1 -1 -1 -2 -1 1 68
-e -e -» -e » -i -a -e -i -i
1 8-8-8 -1 -2 -3 -2 -8 -2
8 8 8 8-8-8-1-2-8 8
-0-8 8 8 -8 -8 -I -1 -8 1
-8 -8 -8 -8 -8 -0 -8 -6 -8 -8
-1 -1 -1 -1 -8 -» -8 -0 -1 -8
-8 -« -e -8 -8 8 8 8 « 0
-.1
-.3
-.1
.8
-.2
-.1
-.1
Wind
11
B
8
11
11
0
4
8
-0
8
-8
-6
4
«
a
i
8
8
8
1
1
I
-.2
-.1
-.1
a
-.2
-.1
-, I
~. 1
16 17
-.1 -.0
-.1 -.e
-.1 -.1
10 19
-.0 -.e
-.0 -.0
-.0 -.0
26
-. I
». l
21 22
-.1 -.1
-. i -.1
23 24 . 21
-., -., -.,-
.3 .0 -.e
-.ft .« .0
-.e .B ,o
Direction
12 13
ft
1
10
11
id
i
-i
-i
-0
e
0
3
2
3
2
2
I
1
1
1
1
2
2
1 1
12
11
8
-1
-6
2
-8
S
2
2
4
4
4
2
2
1
1
14
8
2
13
14
0
2
-0
-4
I
-e
2
3
0
1
6
4
3
2
1
IS
4
4
11
12
2
«
-2
-4
-0
-a
e
3
e
2
i
6
4
2
2
1
Ifi 17
9 0
2 4
11 12
9 9
1 2
-0 1
-S -1
-3 -4
-1 -4
I «
-1 0
Z 1
0 -3
2 -6
3 -0
4 1
S 4
Z 2
a i
i e
is 19
a 7
4 4
11 0
e 4
0 1
6 -0
-2 -2
-4 -4
-5 -4
-I -3
-e -i
6 2
-2 -5
-e -o
-0 2
-0 -2
9 1
3 3
6 -S
9 6
20
«
3
3
7
3
4
-e
-e
-3
-3
-6
-6
-2
-2
7
-6
3
-2
e
3
-0
e
21 22
7 0
4 B
B 3
4 3
. 8 7
2 S
8 1
-8 1
-1 1
-2 -2
-4 -1
-7 -4
-3 -3
-4 -4
3 -8
-7 -4
3 1
-1 -8
-6 -4
-2 I
2 -8
8 -8
23 24 23
2 e -y
* I -0
0*4
433
4.4 5
a ie s
5 6 5
331
1 2 "•
e -e -h
-i -i -i
-1 -2 -1
-2 -2 -2
-2 -3 -4
-4 -3 -4
-0 -I -J
-3 -1 -1
0 -1 -I
-1 -I -1
-2 -1 -1
-1 -1 -1
-1 -1 -1
-ft -0 -A
* See Table 24 for a description of the experimental conditions.
-------
113
Table .18
PREDICTED CHANGES IN WIND SPEED AND DIRECTION FOR CASE 3*
(a) Wind Speed
u
14
ss
22
91
19
10
16
ID
14
19
ia
1!
10
t
0
I
6
i
4
g
a
i
-.» -.« -.B -l.» -l.T -
-.« -.2 -.9 -1.8 -1.6 -
-,t -.a -i.o -1.6 -i.e -
.w -.0 -1.2 -l.Q -1.3 -
.a -.8 -.6 -l.li -1.0 -
.3 .a -.0 -.3 -.4
.2 .1 -.« -.1 -, I
'.0 -.8 -.1 -.1 -.1
-.0 -.1 -.-I -.2 -.2
-.0 -.1 -.2 -.3 -.3
-.0 -.2 -.2 -.3 -.2
-,« -.1 -.2 -.8 -.2
-.0 -.1 -.2 -.2 -.2
-.0 -.1 -.2 -.2 -.3
-.» -.1 -.2 -.8 -.3
-.« -.1 -.2 -.3 -.4
-.0 -.1 -.2 -.3 -.4
-.0 -.1 -.2 -.3 -.4
-.0 -.1 -.3 -.s -.8
-.0 -.1 -.n -.3 -.3
-.« -.1 -,2 -.2 -.3
-.8 -.1 -.3 -.2 -.a
•1.9 -
-I .9
•l.X
•1.8
•1.0
. 1
-.0
. 1
-.1
-.2
-.2
-.3
-.5
-.*
-.4
•1.1 -.8 -
-.9 -.6 -
-.0 -.4
-.9 -.3 1
-.8 .6 i
.4 -.5 -
-.0 -. j -1
.* -.<, -1
.0 -.6 -
.0 -.1 -
-. 1 .1 -
-. 1 -.1-1
-.3 -.3 -
-.4 -.4 -
-.8 -.5 -
-.t -.0 -
-.4 -.0 -
.4 -.4 -.u -.2 -
.2 -.2 -.S .0 -
.1 1.6 1.4 1.4 1
.. ,., ... ...
.2 -.0 -.3 -.a -
.0 -.7 -I. i -I .2 -I
.3 -1.3 -1.0 -1.4 -2
. 6 - 1 . U -.9-1.4-2
.7 -1.1 -1.6 -2.0 -2
,7 -1.2 -1,8 -2.4 -2
.8 -1.3 -2.2 -1.0 -1
. 1 -1.2 -2.0 -.0 -
.9 -.9 -I .2 -1.0 -1
.6 -,7 -I. 1 -1.0 -I
,8 -.7 -1.0 -J.O -1
.5 -.7 -.9 -.9 -
.6 -.f> -.7 -.7 -
,C -.6 -.6 .-.0 -
.2 -.2 -
.0 -.1 -
. t .2 -
.3 -.9 -1
.B -2.7 -3
. 1 -2. 1 -2
.2 - 1.0 -1
.0 -1.9 -1
, 1 -J.2 -1
.2 -1. 1 -1
.6 -1.3 -1
.6 -1.0 -
,4-1.1 -
.2 -l.u -
.3 - . (t -
.7 -.D -
.6 -.D -
.6 -.B -
.6 -.3 -
.fl -.8 -
.2 -.3 -
. i -.2 -
.0 .3 -
.3 -2.0 -3
,3 -3.0 -4
,2 -2.3 -1
.9 -2.0 -1
.2 -1.7 -1
.6 -1,6 -1
.2 -1 .2 -1
.1 -.0 -
.7 -.4 -
.8 -.6 -
.6 -.3
.2 .«
.3 -.2 -
.4 -.2 -
.a -.4 -
17 10 19 29
.3 -.8 -,» -.4
.* -.3 -.2 -.0
.3 -.9 -.9 -I.I
.1 -.0-1.1 -.9 •
.3 -3.3 -2,6 -1,6 •
. 1 -3.3 -2. 1 -1.6
.7 -.7 -.6 .0
. t -.3 -.2 l.B
.2 .1 .6 .4
.0 .1 ,3 .3
.3 .4 .2 .0
.2 .2 .1 -.0
.6 .0 -.0 -.1
.1 -,l -.1 -.3
:z -.2 -.2 -.4
.a -.2 -.2 -.4
21
-.4
-.4
-.6
-l.o
-1.3
-.0
.3
••>
1.2
1. 1
.9
,7
.2
-.0
-.0
-.2
-.3
-.4
-.4
23
-.7
-.6
-.4
-.9
-.9
-.6
-.0
.t
.3
.4
. 1
-.e
.e
-.2
-.2
-.3
-.4
23
-.4
. 1
-.4
-. 1
-. i
-.1
-•<
.*
. I
.6
-.e
.0
. i
-.0
-•>
24 23
.-» . J
.3 .»
.1 .3
.4 .4
.2 . f
-.0 -.11
-.1 -.«
-.• .»
.0 . 1
-.1 .<-
-.1 ..t
.0 .0
. I
.« ...
.0 .v
.1 . «
(b} Wind Direction
23
24
23
22
21
20
19
18
17
16
14
13
12
11
l»
9
t
T
«
D
4
8
a
i
-2 -1 13 83 48
0 2 11 SG 61
3 4 1 35 09
-8 -3 -D 16 47
r22 -24 -24 -0 18
-20 -32 -35 -36 -1
-19 -19 -19 -17 -13
-6 -6 -7 -8 -9
-0 -0 -1 -2 -a
2 1 0-0-0
0 b 0 I 1
80812
-00112
00111
0000-1
-1 -1 -0 -1 -2
-2 -2 -2 -a -2
-4 -4 -4 -4 -3
-0 -3 -S -8 -2
-i -2 -2 -2 -1
-1 -1 -1 -1 -0
-« -« -0 0 0
49
61
(1
H6
35
19
-4
-18
-0
-0
1
e
i
-0
-4
-S
-0
-e
-e
-0
-e
i
7 e
47 39
60 04
D2 44
44 39
42 39
30 38
6 11
-a -8
-13 -B
-I -2
-0 0
-1 1
-0 0
-2 -'»
-0 -12
-0 -10
-0 -0
1 -1
0 1
0 0
0 1
1 2
a a
9 10 11 12
37 19 16 11
43 80 10 13
30 36 39 36
39 41 39 46
31 31 33 97
-1 6 « »
211-1
3 1 -1 -0
e i 3 i
-fr 1 17 17
-C 7 22 22
-4 0 23 23
-7 -ft 14 22
-4 -fl 2 19
6 -3 I 14
I 1 6 12
a s T 11
8079
4 B T 0
13 14
12 17
14 16
22 22
, 37 07
CO D6
42 61
-6 -7
0 -2
-4 -B
-6 -8
2 -*
13 9
16 14
17 I I
23 17
S* 32
R7 29
19 22
14 16
11 12
9 9
19 16
23 81
23 80
32 34
59 30
KB B2
30 20
-24 -35
-4 -6
-B -3
-9 -II
-4 -10
2 -3
10 +
7 2
13 B
19 13
27 19
23 10
16 13
13 10
e «
17 16 19 20
33 36 37 33
OB 37 30 27
64 64 03 36
43 30 17 13
4 -3 -3 -1
-43 -33 -23 -17
-19 -20 -27 -24
-0 -16 -24 -29
-14 -9 -IB -22
-16 -24 -32 -19
-4 -0 -12 -13
168-4
-5 -3 -B 7
0 -I -I -9
3621
9 1-1-3
IB 10 6 2
12 11 10 8
T 0 4 4
4088
21
33
28
36
34
14
4
-7
-17
-26
-23
-16
-13
-9
1
-12
e
-3
-2
-1
B
9
22
41
20
29
32
21
10
2
-10
-20
-21
-14
-13
-13
-7
-12
-3
-B
-10
-2
-2
-•
23
20
33
2B
28
33
26
16
3
-11
-16
-20
-17
-14
-1C
-9
-I 1
-6
-7
-0
-7
-0
-I
£4 25
4 -J
10 4
22 13
22 10
25 2-T
07 3 1
29 23
IB 15
10 4
-1 -2
-11 -10
-14 -14
-10 -IB
-20 -2.
-19 -m
-13 -n
-O -1
-7 -6
-7 -0
-7 -7
-7 -7
-7 -7
-6 -3
-2
*See Table 24 for a description of the experimental conditions.
-------
114
Table 19
PREDICTED CHANGES IN HIND SPEED AND DIRECTION FOR CASE 4*
(a) Hind Speed
zg
El
£3
22
21
20
19
18
17
16
19
14
13
12
11
IB
9
B
7
6
*
4
3
2
1
23
24
23
22
21
20
19
10
IT
16
13
13
12
II
10
»
0
T
«
B
4
B
-.4 -.5 -.4 -.8 -.4 -
-.4 -.2 -.2 -.1 -.2 -
-.1 .1 .2 .2 .2
.9 .3 .0 .U .9
.7 .6 1.0 1.4 1.6 1
.1 1.2 1.6 2.0 2.0 J
1.0 1.0 2.3 2. B 3.6 4
1.2 S.I 1.1 2.0 3.4 4
.4 1.2 1.3 2.6 2. B 3,
".9 .0 1.0 2.1 3.0 3.
-,S .» 1.3 2.» 2.0 3.
-.3 .U .U 1.9 2.6 y.
-.3 .D l.f) 1,6 2.3 3.
.1 .3 .7 i.i 1,6 I.
.1 .•> ,1 1.0 1.3 1.
.J .3 i« .y ,t I.
-1 I 1 1 «
-6 -7 -2 -0 1
-9 -12 -0 -3 0
-4 -3 -9 -4 1
1 0-2-3 0
23 21 20 10 IB
36 33 33 33 31
39 37 36 34 32
45 43 39 30 30
40 44 37 36 39
3fl 33 32 34 33
26 24 21 23 23
22 23 22 19 10
17 17 17 16 13
13 13 12 11 12
9 10 9 11 9
7 7 7 T 0
44443
23321
« 9 0 » -«
-I -i -i -e -•
-e -o o i i
* T 8 .» .0 .is . •> .3 .0 .» .^
.1 3.0 4.4 4,0 3.G 2.0 1.1 .t 1.6 2.9 2.3 i.z .4 .z .4 .a .1 .4 .* .4
6 l.f Z.2 2.0 2.2 1.0 1.3 .7 1.1 1,1 1.3 1.1 1.0 .8 .6 .4 .3 .2 .1 .4
4 1.2 l.v 1.6 2.1 1.7 1.3 1.0 .B .8 .0 .7 .b ,D .3 .1 .1 .1 .1 .1
(b) Wind Direction
« 70 9 10 It 12 13 14 10 16 17 10 19 20 21 22 23 24 23
0 » 6 2 0 7 9 14 20 9 2 -2 -4 -3 -S -0 1 33 1
2 8 0 B 9 12 17 32 23 10 0 1 1 1 1 3 0 10 6 c
3 1 13 19 20 30 42 47 30 30 24 10 14 II 11 14 17 23 22 23
6 11 21 23 33 43 46 49 41 30 26 16 IB 13 16 19 23 27 20 21
16 17 21 29 34 40 46 40 S3 29 23 17 16 13 10 HI 22 23 23 22
31 32 36 36 39 43 30 34 27 26 22 10 11 12 17 20 21 22 16 17
32 32 35 12 44 37 32 23 22 20 17 11 9 13 13 10 10 10 8 9
3D 43 42 33 31 23 10 10 16 12 9 4 4 3 S 2 1 0 1 3
30 26 23 21 ia 13 0 7 4 1 -2 -0 -11 -0 -6 -3 -1 -3 -1 --•
10 17 16 13 12 B 4 3 1 -3 -6 -10 -14 -10 -It -13 -14 -14 -12 -12
13 13 |3 U 9 6 22-1 -4 -0 -10 -13 -14 -13 -14 -13 -13 -12 -12
, j 0 e | _] -4 -9 -9 -U -12 -13 -10 -16 -16 -16 -13 -15 -13 -in
3 1 -e -3 -7 -10 -12 -13 -12 -12 -13 -14 -1C -14 -12 -12 -13 -14 -13 -16
_» -j -3 -<; -7 -, -n -13 -|2 -11 -11 -11 -12 -12 -11 -11 -11 -12 -13 -14
_„ -i _2 -4 -5 _7 -9 -10 -» -9 -9 -10 -10 -10 -10 -11 -10 -11 -12 -13
-» -1 -2 -3 -4 -0 -6 -7 -6 -5 -6 -7 -0 -9 -19 -U -12 -12 -12 -13
-e -1 -1 -2 -a -3 -4 -4 -2 -2 -3 -4 -» -6 -7 -O -9 -9 -9 -»
i a a o » a a i I i i i • • -» -« -i -i -i -i
*See Table 24 for a description of the experimental conditions-.
-------
115
Table 20
PREDICTED CHANGES IN WIND SPEED AK'D DIRECTION FOR CASE 5*
(a) Wind Spaed
20
24
23
22
21
20
19
10
17
16
10
14
13
12
II
10
9
8
7
«
6
4
9
a
1
25
24
23
22
21
20
19
ia
17
16
10
12
11
6
1
-.2 -.1 -,3 -.1 -.2 -.1 -.1 -.< .2 -.1 .6 .2 .2 .0 -.3 -.4 -.B -.1
-.e .1 .0 .1 .0 .1 .1 -.1 -.6 .*. .9 -.2 -, i -.7 -.6 -.9-1.1 -.7
.1 -.1 ,1 -Z .2 .» .1 .a .a .1 .1 ,| -.0 -,9 -.0-1.0-1.3 -,9 -
.2 .» .8 .4 .5 .8 .4 .6 .4 .2 .5 .1 -.1 -.8 -.3 -.B -1.2 -.7 -
-.8 .1 .2 .3 .4 .5 .6 .B .8 .8 .4 .3 .2 -.1 -,7 -.6 -1.6 -1.4
-.2 ,Z -.1 .0 .4 .V 1.9 1.9 1.2 1.6 1.4 1.4 1.0 1.7 1.0 I.I .2 -.1
-.3 .1 -.1 .4 .7 .9 1.1 1,4 1.9 1.4 l.l .0 1.0 J.0 I.-. 1.1 .3 .0
-.0 -.1 ,2 .-i .r 1.0 J.3 i.O 1.2 1.0 .B .* 1.4 i.« i.a i.l .0 .0
-.2 .a .a .3 ,7 l.e 1.2 1.6 1.3 1.3 .& .a -.2 .7 .b . •* .a .1
.9 .1 .2 .4 .9 .6 .7 .0 .0 .0 .6 .0 .1 .3 .0 .4 .6 .3
,e .1 .2 .3 .4 .6 . B .T . » .7 .« .4 .1 .4 .4 .B .4 .3
.0 .1 ,-t .9 . u .0 .« .3 i.v . v ,o .« .a .3 .4 .a .9 . «
(b) Wind Direction
-4 1 1 ft 0 0 6 6 6 1 3 8 410 2 0-9-1
-4-3-0 6 0 1 I I 8 £ 4 4 14 0 1.2-0 0
-S -6 -1 -1 0 0 2 3 1 8 9 16 1C 9 a 4 1 3
16 9 9 0 6 B 6 6 11 12 20 IB 14 9 B 6 3 4
15 10 14 It 14 1-i IS 1? 14 18 19 14 13 0 0 7 4 2
J7 17 13 • 14 17 13 IS ID 12 9 B 6 O 3 2 1 -0 -2
13 13 11 14 14 13 11 II 0 B 0 3 3 1 « -0 -4 -4
1 | | o e -6 -« -I -3 -3 -3 -4 -3 ' -4 -4 -4 -4 -4'
19 W 51 23 23 34 23
.1 .1 .1 -.1 .<• -.9 -.1
-.3 -.3 -.0 -.1 -./ -.3 -.*
-.B -.6 -.6 -.7 -1.3 -1.2 -l.«
1.2 '-1.0 -1.8 -1.0 -1.0 -.G -I.«
•l.i -l.i -.8 -.6 -.6 -.2 -.ft
-.9 -.w -.6 -.6 -.3 -.7 -. i
-.7 -.0 -.u -.0 .« -.« .1
-.6 -.6 i . z -.v .1 -.0 -.1
-.* -.if -.4 .z -.* .* .-
.2 -.3 .2 -.1 -.1 -.8 .»
.a -.2 .a .a .* ,* .1
.1 .a-.o .a .0 .;i
.4 .Z .1 -.0 ,2 .1 .1
.2 .2 .1 . I -. i .2 . >
,2 .2 .1 ,1 . 1 -.1 .2
.4 .1 .» .t. ... .» .->
.1 .1 ,o -.6 -.0 -.0 .«i
-i -i -o -e i i o
e o i i 3 2 i
2 1 2 1 5 1 4
2234966
9 4 7 6 6 C 6
2*6784
3 4 B 1 3 1
e -o -i -i -i -e »
-a -i a -i -o o -o
-2-2-0 0-1 -0 -1
-4 -5 -4 -4 -3 -4 -»
-4 -4 -4 -4 -4 -4 — *
'
-4 -4 -4 -4 -4 -4
e -» -o -o -o -e -n
*See Table 24 for a description of the experimental conditions.
-------
Table 21
PREDICTED CHANGES III WIND SPEED A!!D DIRECTION FOR CASE 6*
(a) Hind Speed
29
24
fcs
22
21
20
19
la
17
14
IB
14
18
12
11
U
V
B
T
«
a
a
1
29
94
23
22
21
29
19
la
17
16
IS
14
12
II
10
,
B
7
t
B
4
It
»
i
i
-i
-.«
-•>
.0
1.3
1.9
2.6
2.1
1.2
.B
. 1
-.S
.1
.*
1
-2
-12
-IB
r7
4
84
32
03
61
66
96
37
29
22
16
12
7
4
1
-a
-2
-9
-•
« II
-.1 -.9
-.a -.1
.4 .»
1.3 1.9
2.2 2.7
U.I 3.6
I.I 2.3
.9 1.9
.» 1.0
.0 I.O
.6 1.4
.0 1.2
.B 1.0
.3 .«
2 C
e i
-11 -4
-10 -10
-9 -13
2 -I
91 23
92 CO
SO 92
69 91
Bl 47
86 33
20 26
21 19
16 19
12 12
t 7
4 4
1 1
-1 -1
-1 -1
-0 -0
-• 1
-.6 -.T -.T -.» -.9 .1 i
-.2 -.3 -.1 .• -.1 .0 1
.1 .4 .6 .0 .9 I.O 1
1.3 I.O 1.6 1.7 1.7 1.9 *
a. a B.9 B.I 3.4 3.7 3.e 8
9.2 3.7 4.3 0.0 O.I 4.2 9
4.2 9.0 9.9 7.0 7.7 6.2 6,
2.2 3.1 3.0 4.9 6.3 6.3 0.
1.4 I.S 2.6 2.6 3.0 3.0 S.
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47 43 40 35 33 20
2B 26 24 23 22 . 10
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17 17 16 14 12 12
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-------
Table 22
PREDICTED CHANGES IN HIND SPEED AND DIRECTION FOR CASE 7*
(a) Wind Speed
13
10
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*See Table 24 for a description of the experimental conditions-.
-------
Table 23
PREDICTED CHANGES IN WIND SPEED AND DIRECTION FOR CASE 8*
(a) Wind Speed
£9
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(b} Wind Direction
0 9 19 11 12
-5 -6 -B -7 -6
-0 -D -0 -0 7
13 19 3f 42 53
37 40 44 62 44
33 43 47 40 84
34 01 29 22 16
19 15 12 6 0
15 12 0 2-3
12 9 6 0-0
0540-4
4 2 0-3-0
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-14 -19 -26 -30 -33
-19 -23 -27 -31 -36
-2» -23 -27 -31 -33
-23 -23 -Zt -32 -36
-24 -20 '31 -35 -30
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13
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*See Table 24 -for a description of the experimental conditions.
-------
119
Table 24
CONDITIONS REPRESENTED IN TABLES 16 THROUGH 23
Time
D - D
0
r_T_ Grid Level
Case Table 6 a.m. 3 p.m. Eg. (26) Eg. (27) k = 1 k = 5
1
2
3
4
5
6
7
8
16
17
18
19
20
21
22
23
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Boundary
Conditions
= 06
X
X
X
X
X
X
X
X
In addition to the general observations given above, we can also make the
following specific comments:
> Perturbations obtained at ground level from the use of Eq. (26)
to estimate D - D^ are larger than those generated from the use
of Eq. (27) (compare Tables 16 and 17 and Tables 19 and 20).
The opposite is true at the top of the region (compare Tables
16 and 18 and Tables 19 and 21).
> When Eq. (27) is used to estimate D - Dfl,'the perturbations
aloft are much larger than those predicted at ground level
(compare Tables 17 and 18 and Tables 20 and 21).
> The choice of boundary conditions employed does have some
influence on the magnitude of the predicted changes in wind
speed and direction. As expected, this influence is great-
est near the boundary (compare Tables 18 and 22 and Tables
21 and 23).
-------
120
To explain the first two observations cited above, we note that the forcing
function in Eq. (23) follows the same pattern of behavior. This can be illus
trated by considering the ratio of the forcing functions employed in each
case. If we let AD26(K) and AD27(k,K) represent the values of D - DQ calcu-
lated using Eqs . (26) and (27), respectively, then we can write the ratio of
the forcing function corresponding to the comparisons made above as follows:
AD27(1,K)
AD27(K,K) 2K
where K is the number of vertical layers of grid cells. In this study, K has
a value of 5.
The experience gained in this brief study indicates that the use of
algorithms similar to those given in the previous section provides a viable
means of producing mass-consistent wind fields. Although such algorithms are
relatively simple to employ, they are deficient in the treatment of momentum
and energy balance relationships. However, until complete planetary boundary
layer models suitable for predicting flow fields over urban areas can be
developed and validated, photochemical modeling efforts will undoubtedly
continue to rely on wind fields derived from actual field measurements.
Thus, the use of mass-consistent wind algorithms should be considered as
an interim means for removing excessive convergence and divergence ef-
fects in the flow field. The need for such usage may also be enhanced
by the inclusion of wind shear in the airshed model, since the extent of
convergence and divergence in the predicted flow field aloft may be larger
than that previously experienced near the ground.
-------
121
In future work, we recommend that mass-consistent wind algorithms be
employed in conjunction with interpolation procedures for predicting flow
fields over an urban area where a reasonably dense meteorological network
has been established. In this way, tests can be designed to evaluate the
performance of the methodology. The RAPS study in St. Louis may provide
such a data base. In addition, further consideration should be given to
the manner in which the quantity D - DQ is estimated. Examination of the
characteristics of flow fields over urban areas may provide some guidance
in this matter.
D. ADOPTION OF AN IMPROVED ALGORITHM FOR ESTIMATING
TURBULENT DIFFUSIVITIES
Pollutants are dispersed through advection and turbulent diffusion.
In the horizontal directions, the advective mass flux is usually much
larger than the diffusive flux. However, vertical transport is often
dominated by turbulent diffusion. The usual means for treating vertical
diffusion is through the assumption that the turbulent mass flux, F., is
proportional to the gradient of the mean concentration field. That is,
v 9z
where
K = turbulent diffusivity,
= mean concentration.
Turbulent diffusivities are extremely difficult to measure in the field,
and their parameterization has been the subject of numerous studies. Upon
reviewing the algorithm we employed in the 1969 validation study to calcu-
late K , discussed in Roth et al. (1971), we found that there was sufficient
justification to formulate a new algorithm. This new algorithm includes
important atmospheric parameters, heretofore omitted, that are known to have
a significant effect on the value of the diffusivity. Specific criticisms
of the diffusivity algorithm that we used previously are as follows:
-------
122
> The diffusivity is assumed to depend only on the wind speed.
Using measured diffusivity data reported by Hosier (1969),
Eschenroeder et al. (1972) found that the diffusivity does
not correlate well with wind speed alone. This finding is
not surprising,'since we would expect that, for a given wind
speed, the value of Ky for stable atmospheric conditions would
be much less than its value under unstable conditions.
Clearly, an algorithm for K must include the effect of at-
mospheric stability.
> Surface roughness effects are not explicitly included in the
formulation of K . Recent studies by Lissaman (1973) and
Ragland (1973) indicate that ground-level pollutant concen-
trations are significantly influenced by the value of the
surface roughness.
In reviewing previous efforts to parameterize the diffusivity reported
in the literr.ture, we found that guidelines appear to exist that are suffi-
ciently well developed for use in estimating the value of K in the surface
layer (up to about 100 m). However, for the remaining portion of the plane-
tary boundary layer above an urban area, we have not found a definitive
treatment of the diffusivity that is both general and simple enough to include
in an airshed model. Also of concern is the objective for multiday simula-
tions of defining the vertical extent of the modeling region to include the
inversion layer, if present. The "trapping" effect of the elevated tempera-
ture inversion would be treated through the use of the vertical diffusivity
profile. Thus, a relatively sophisticated treatment of K is required aloft,
a region of the.planetary boundary layer where few measurements are generally
available.
Realizing that a completely satisfying treatment of KV may not be attain-
able at the present time, but also recognizing the need to improve the algor-
ithm previously employed in the SAI airshed model, we initiated efforts to
develop an algorithm for K that includes, at a minimum, both atmospheric
-------
123
stability and surface roughness effects. Several schemes for computing K
have been proposed in the literature^ including those described by Blackadar
(1962), Wu (1965), Hino (1968), Pandolfo et al. (1971), Eschenroeder et al.
(1972), Ragland (1973), Bergstrom and Viskanta (1973), and Shir and Shieh
(1973). However, each of these approaches is to some extent heuristic, and
their validity is somewhat uncertain.
To alleviate the difficulties associated with basing a diffusivity algor-
ithm on field measurements, we developed a methodology that uses the predic-
tions of a sophisticated numerical planetary boundary layer model developed
by Deardorff. Although the present K algorithm is applicable for only
neutral and slightly unstable atmospheric stability regimes, the methodology
can be extended to other regimes. For a more detailed discussion of this
algorithm, we refer the reader to Chapter II of Volume III.
E. MODIFIED TREATMENT OF THE INVERSION LAYER
IN THE AIRSHED MODEL
In previous studies, the modeling region has been defined to extend from
the ground level to the base of an elevated temperature inversion. However,
a major difficulty arises when using this approach for multiday simulations:
A significant amount of pollutants can be reintroduced into the mixed layer
from aloft as the inversion is eroded away during each daytime period. Unless
the pollutants that are trapped in the inversion on the previous day are re-
tained in the modeling region, it will be difficult to account properly for
their reintroduction into the mixed layer on a given day. As an example of
the DO levels that have been observed aloft, we present in Figure 21 a cross
section of the pollutant distribution in a portion of the Los Angeles basin on
the morning of 11 July 1973 (Jerskey et al., 1975).
As an alternative definition of the modeling region, we propose to include
the portion of the atmosphere bounded below by the terrain and bounded aloft by
the top of the inversion layer. All governing equations and coordinate trans-
formations used previously still apply, except that the term AH should be
interpreted as
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-------
125
AH = Ht(x,y,t) •- h(x,y)
where
Ht(x,y,t) = elevation of the top of the inversion layer,
h(x,y) = terrain elevation.
The effect of trapping pollutants below the inversion layer can be accounted
for through the height dependence of the vertical diffusivity. Whereas rela-
tively large values of K are used in the mixed layer, the values in the
stable inversion layer are much smaller, reflecting the suppression of turbu-
lent mixing.
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126
IV EVALUATION OF ALTERNATIVE TECHNIQUES
FOR INTEGRATING THE SPECIES CONTINUITY EQUATIONS
James P. Meyer
A. INTRODUCTION
In essence, the SAI atmospheric photochemical simulation program is
based on the solution of the nonlinear, multidimensional species transport
equation
1 + v • vc. = V • Kvc. + R. + S. , (29)
9t
which, for convenience, has been transposed [Reynolds et al . (1973)] into
the form
3(AHc.)
++^^c.} + »~(«c.)
8c 8c 8c
•H""3p / f 5F W + IF (\M;
+ S.AH . (30)
In general, no closed-form analytical solution exists for this highly com-
plex partial differential equation for all possible initial and boundary
conditions. Hence, one is forced to resort to approximation techniques,
most notably finite difference schemes, to find a solution.
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127
The choice of an appropriate numerical technique for inclusion in the
airshed model involves two primary considerations. First, the accuracy of
the solution obtained must be such that the error in the predicted concen-
trations is predominately the result of errors in model inputs rather than
errors introduced by the numerical technique itself. Second, the final
choice between alternative techniques capable of solving the governing
equations to a specified error tolerance should be based on minimizing com-
puting costs. In view of these considerations and the variety of numerical
techniques available for solving the equations of interest, care must be
taken to choose a method that offers an optimal blend of numerical accuracy
and computational efficiency.
Currently, a finite difference approach termed the method of fractional
steps [Yanenko (1969)] is employed in the SAI model. The basic feature of
this method is that the four-dimensional governing equation in (<;,n,p,T) is
split into three two-dimensional equations in (?,T), (n,T), and (P,T). The
details of this analysis are given elsewhere [Reynolds et al. (1973)]. With
this type of approach, errors are introduced into the solution in the follow-
ing ways:
> Through the introduction of truncation errors caused by the
finite differencing of the partial derivatives in the trans-
port equation.
> Through the decomposition of a three-dimensional equation
into a sequence of three two-dimensional equations.
As an example of truncation error effects, Harlow and Amsden (1970)
showed that for the one-dimensional advection equation
(3D
-------
128
a numerical solution involving a first-order finite difference approximation
introduces an error on the order of
(32)
2 ?
into the calculation. Since the term 3 c/9x appears, this error has been
called "numerical diffusion." Its effect is to smooth an initially peaked
distribution over a large portion of the modeling area. This reduces the
resolution of the solution so much that in extreme cases it becomes nonex-
istent. In an attempt to reduce the truncation error effects in the SAI
model arising from the treatment of the horizontal advection terms, we pre-
viously carried out numerical experiments using various second- and fourth-
order difference approximations. Although we observed some reduction in
truncation error using the higher order methods, many of these techniques
also had the undesirable property of producing negative concentrations in
the vicinity of steep concentration gradients. We finally selected an
uncentered second-order method described by Price et al . (1966), which is
somewhat more accurate than the first-order advection approximations, and
which, at the same time, presented no difficulties with regard to the pre-
diction of negative concentrations in the initial application of the model
to the Los Angeles basin.
In the fractional step technique, the decomposition process for the
n,c, and p directions introduces a sequence of higher order partial deriv-
atives that would not normally appear in the transport equation. Although
the effect of these terms is difficult to quantify a priori, their impact
on model predictions can be examined by comparing, predicted pollutant con-
centrations with known analytical solutions of the governing equations.
In the discussion presented in Volume I of the validity of the grid and
trajectory model concepts, we noted that errors introduced into the grid
model predictions by the numerical integration technique can be as large as
50 percent in some situations. These errors are mainly due to the finite
-------
129
difference treatment of the horizontal advection terms. Thus, the objec-
tive of the present study is to test and assess various alternative numer-
ical approximations to the transport terms in the governing equations. In
this analysis, our aim is to provide recommendations regarding the course
of future efforts to improve the numerical integration procedure employed
in the airshed model.
In the work described next, the emphasis was on the development of an
analytical solution to the diffusion equation and on a comparison of the
analytical results with the corresponding results obtained from various ap-
proximate integration schemes. Because of the difficulties involved in
developing solutions to the diffusion equation, only a simplified one-
dimensional, linear, time-dependent result could be obtained. Thus, we
are able to assess the errors associated with various numerical methods
for a one-dimensional flow problem in which the pollutant is allowed to
undergo a first-order chemical reaction. Clearly, this test situation is
not completely representative of a full photochemical airshed simulation.
However, numerical techniques incapable of producing sufficiently accurate
results in a one-dimensional linear problem cannot be expected to perform
better in a multidimensional nonlinear application.
Since it was not possible to carry out the tests for photochemical
pollutants, we are unable to assess the effect of inaccuracies introduced
in the treatment of the transport terms on error propagation, especially
when nonlinear chemical interactions are taking place. In addition, the
test results do not illustrate the errors caused by using the fractional
step methodology to treat a multidimensional problem. In spite of these
limitations, however, we have been able to delineate two numerical methods
that seem to represent a significant improvement over the finite difference
scheme currently employed in the SAI airshed model.
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130
B. AVAILABLE METHODS
A variety of methods have been developed to solve partial differential
equations. In general, they fall into two categories: finite difference
schemes and particle techniques. The former, which are well developed, in-
clude the work of Price et al . (1966), Fromm (1969), Crowley (1968), and,
more recently, Boris and Book (1973). In contrast, particle-in-cell methods
are relatively current; they include the contributions of Sklarew et al.
(1971) and Egan and Mahoney (1972).
For the purpose of analysis, a simplified solution of the diffusion
equation was developed and the results of this calculation were compared to
the results of the suitably programmed approximation schemes. The equation
chosen for this work was the one-dimensional transport equation,
2
8C _ n 8 C 8C ,
— r - U - ^ - U — - KC ,
3t 2 3X
in which u, D, and k were considered constant. The following boundary con-
ditions were imposed:
> Initially, no material is in the modeling region, i.e.
c(05x) = 0 . (34)
> There is zero concentration gradient of infinity, i.e.,
ff (t,-) = 0 • (35)
> There is a uniform concentration at the inlet, i.e.,
c(t,0) - 1 . . (36)
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131
We evaluated the following methods:
> Price scheme
> Crowley second- and fourth-order methods
> SHASTA method
> Galerkin method
> Particle-in-cell techniques
> Egan and Mahoney method.
In'the following subsections we briefly describe each method.
1. The Price Scheme
Currently, the SAI model uses a method proposed by Price, Varga, and
Warren (19661). For the test problem selected, this method has the finite
difference form
cn+1 = cn + c?,, - 2cn + cn . - ^ (3cn. - 4cn . + cn - k5tcn
J J 2 V j+1 j j-} ^6X j j-1 j-
(37)
This approximation has errors that are first order in time (fit) and second
2
order in distance (6x) .
Since the solution is explicit in time, definite limits of stability
exist. These limits can be developed by assuming that the solution of the
transient equation can be written in the complex.Fourier form
c(t.jAx) = *(t) eijAx , (38)
where
i = /T , (39)
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132
After considerable algebraic manipulation and stipulation of the requirement
At)
< 1
(40)
we obtain
k6t < 2
(41)
(42)
26X H
(fix)1
4 2
(43)
as the conditions required for stability to occur.
2. The Growley Second- and Fourth-Order_Methods
To enhance the accuracy of the finite difference approximations used in
formulating the advective terms of the governing equations, Crowley (1965)
developed both second- and fourth-order centered difference algorithms for
these terms. For the test problem, the second-order method has the expansion
•r
a , a
~ 2 2
c" + (1 - 23 - a2 - k6t)c"
J ' \ .. / J
a
2
a" n
where
Ku6t
n
a -
U6t
6X
(44)
(45)
(46)
-------
133
The corresponding fourth-order expansion has the form
n+1 _ /_a
-------
134
where
J- + u!/2 —
» j=l,...,n , (49)
l/2
- UJ
and where u. refers to the velocity at the j-th location at time t + (fit/2)
J
Completion of the antidnffusion step requires the expression
n+1 _ ~n+l l/~n+l ??n+l ~n+l\ . (50)
Cj " Cj " c * ^C
To account for cases in which material may be advected either into or
out of the modeling region, the SHASTA method applies the following rules at
the end points:
> Left-hand side
--If v-j > 0, then CQ and vfi, the upwind boundary conditions,
must be specified.
--If v, < 0, then 3c/5x = 0, and CQ = c, and VQ must be specified.
> Right-hand side
--If vn > 0, then 8c/3x = 0, and c ., - c and v ., must be specified.
--If vn < 0, then cn+^ and vn+-j , the incoming concentration and velocity,
respectively, must be specified.
Since the SHASTA algorithm treats only the advective parts of the continuity
equation, the concurrent diffusion and kinetic steps of the governing equation
must be treated as subsequent operations. .Hence, the system heavily relies on
the method of fractional steps.
-------
135
For the test case, the advective equation has the form
(52)
*n+l _ ~*n+l l/~*n+l -*n+l ~*n+l\
Cj ' Cj " 8
The diffusive and kinetic terms become
+ (1 - 23 - kfit) S + 3 , (54)
K,,,6t
3 - - . (55)
4. The Galerkin Method
Finite element methods represent a significant departure from finite dif-
ference techniques as a tool in solving partial differential equations. Unlike
finite difference equations, which approximate derivatives at specific loca-
tions, finite element techniques approximate functions over an entire domain
[Zienkiewicz (1971), Finder and Gray (1974)J.
To develop finite element solutions, one must follow four steps:
> Subdivide the domain of interest into a finite number of elements
defined by node points.
> Approximate the dependent variables in terms of their unknown node
point values within each element. This insures the continuity of
the dependent variable across the element.
-------
136
> Minimize an appropriate measure of error such that a set of
simultaneous equations results.
> Solve the resulting set of equations for the node point values.
The distinct advantages of finite element techniques are their ability
to model arbitrary geometric areas without a loss of convergence and their
generally greater stability compared with corresponding finite difference
systems.
Two distinct classes of finite element solutions exist. The Rayleigh-
Ritz procedure requires the minimization of a function associated with a de-
fining differential equation. Although this is an extremely useful method,
often one cannot determine the functional form associated with the differential
equation. Thus, the method has limited applicability. A more general, but
somewhat less mathematically elegant, approach is the Galerkin technique
[Keldysh (1964), McMichael and Thomas (1973)]. This method requires merely
that the integral of the approximate solution be orthogonal to each of the
basis functions spanning the solution space. For example, the linear differ-
ential equation
O C , oCr^oC.i r\ f r c \
ni7_^ „ -I- I I , I „_ _ -m i _L I/ f~* — I 1 I ^ r"i I
i LJ "" \J ~f*r ~ |\ L* w \*-^^/
is written in operator form as
and is assumed to have a solution of the form
c = E a.cf,, , (58)
where there are n nodes in the domain of c. Then, the'Galerkin procedure
requires that
-------
137
J U6H.
dv = 0 , . i = 1, 2, ..., n , (59)
so that the coefficients (a.) can be determined.
As an illustration of the technique, consider the following sample
problem:
(60)
Assume a solution of the form
n
c = E a.(t) 6.(x) . (61)
Multiply the differential equation by ., and integrate the result over its
entire domain:
.L A L ^
|f *. dx + u I |£ f. dx - D I ^ *_. dx + | kc*, dx -
fo 3t i JQ 9x i JQ
i=l, ..., n . (62)
By using Green's theorem,
3x
(63)
0
and by substituting the expanded series into the equation, we obtain
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138
f
n aa.
£ -rjr1- 4>. dx + u I I a.
. at Ti I . i
dx
I n
dx + ki i a ..d>. dx
0
J
= D . Ea.
1 J
1=1,2, .... n . (64)
Once the functions $. are selected, a series of matrix equations result.
They are of the form
da
~ dT + ~~ = ~
(65)
where
£i, B = coefficient matrices,
a, S - column vectors.
These equations can be easily solved by using a Crank-Nicholson technique to
approximate a between times t and t + St. The initial conditions an must be
specified.
For the work described in this report, we selected chapeau functions of
the form
"-7- , 0 < x < x,
X] I
elsewhere
(66)
v v ' Xi 1 - X - X1
A • - X • T I ~ I I
(67)
,,,^T ' xi
-------
- x
n-
xn - xn-l
X , < X < X
n-1 - - n
0 , elsewhere
Correspondingly, the matrices A and B had the tridiagonal form
139
(68)
A = ^
1
0
0
4
1
0
1
4
1
0
0
1
0
0
1
1
0
0
... 0
4 1
1 < i < n - 1 (69)
where
B -
-a
0
-a
D , u k6x
H 2 " ~r
0
-Y
-a
0
0
-Y
(70)
- 2D 2K5X
6x 3
_ D u
Y " £T- 2~ ~2
while S was zero everywhere. The boundary condition yielded the terms
(71)
an = an-l
(72)
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140
5. Particle-in-Cell Techniques
Harlow and Welch (1965) first developed particle-in-cell methods for
use in the analysis of free-surface fluid mechanical problems. Since their
initial development, these techniques have been expanded to include such
variations as marker-in-cell (MAC) and HYDRO codes.
An interesting adaptation of the particle-in-cel1 algorithm has been
developed by Sklarew (1971) to mode] mesoscale air pollution problems. In
this variant, pollutant particles representing a fixed weight of material
are generated in quantities proportional to the ambient pollutant concen-
tration. As time passes, the particle positions are tracked in space by
determining the incremental changes in their locations caused by advective
and diffusive forces. Sklarew chose to rewrite the species transport equa-
tion in the form
!§•+ v • (vc - Dvc) - 0 , (73)
0 t ~
where
v = mean velocity,
vc
D— = diffusive velocity.
With these definitions, it is possible to increment the radial position of
each particle during each time step by a corresponding contribution due to
mean fluid flow (v6t) and diffusional motion [(Dvc5t)/cJ.
To account for photochemistry, one must assume that, within each cell,
the particle weights can be summed to form a representative cell concentra-
tion and that the reaction occurs as if the material is homogeneously dis-
tributed throughout the cell. At the end of the reaction sequence, each
particle is reweighted proportionally to the change in the cell concentra-
tion of the individual species:
-------
141
m.(t * At) = m.(t) -c(t) > (74)
and the transport process is subsequently allowed to occur.
Hotchkiss and Hirt (1972) improved the modeling of the diffusional part
of the transport process at Los Alamos. Their contribution was the represen-
tation of the diffusive movement as a random particle motion of the form
6xDIFF = /4D6t $ , (75)
where ^ is a randomly distributed Gaussian variable. Their work indicates
that their method results in substantially better agreement than the method
of Sklarew in areas of strong concentration gradients near point sources.
This modification overcomes the difficulties in computing the finite differ-
ence approximation needed by Sklarew in calculating the gradient of the con-
centration.
Fundamentally, the problem with all particle-in-cell methods is the
essential question of exactly what a particle represents and over what area
should it be considered to have domain—classically an Eulerian-Lagrangian
paradox. A recurrent problem in using this type of analysis is the back-
ground noise that must be accommodated when a particle leaves one cell and
enters another. This quantum jump can be smoothed to some extent by volume-
averaging the particle over the adjacent cells it intercepts. However, such
a procedure may well extend the domain of a pollutant into regions that it
does not actually represent. To circumvent this problem, one can always in-
crease the number of particles associated with a problem, but at the added
expense of dramatically increasing computer storage and computational time.
6. The Meth_od_ of Egan and Mahoney
One of the more interesting developments in the analysis used in air
pollution modeling has been the work of Egan and Mahoney (1970, 1971, 1972).
In essence, their approach is to follow air parcels as they move within a
-------
142
grid network, taking into account the zero, first, and second moments of
the pollutant -distribution. With this type of analysis, it is possible
to maintain extremely high resolution and to eliminate almost entirely
the numerical diffusion caused by errors associated with approximations
for the advection terms.
Unfortunately, the method, which is owned proprietarily by Environmental
Research and Technology in Lexington, Massachusetts, is only paraphrased
in the open literature. Hence, the analysis presented here is cursory and
represents only a superficial evaluation of the utility of this method.
C. A TEST PROBLEM
To provide a common basis of comparison for each of the methods, we
posed the following two-dimensional problem (x - t) .
Consider a semi-infinite strip extending, from zero to infinity over
which the species transport equation is assumed to hold and a first-order
irreversible reaction occurs:
2
, 8C n 8 C , i-,r\
+ u — = D — ~ - kc . (76)
Specify that all parameters (u, D, and k) are constant, and impose the fol-
lowing boundary conditions:
> Initially no material is in the modeling region, i.e.,
c(0,x) =0 , 0 < x < - . (77)
> There is zero flux of infinity, i.e.,
(t,-) = ° • (78)
-------
143
> There is a uniform concentration at the inlet, i.e.,
c(t,0) = 1 . (79)
To develop a solution, take the Laplace transform of the defining
differential equation
2-
sc+ u {j|« D^f- kE . (80)
dx
Then rearrange the equation in the form
2-
D^f - u £- (k + s)c = 0 . (81)
dx^ dx
Next, solve for c:
E = Ae[(Pe/2) - *JA + Be[(Pe/2) + *Jx ^ (82)
where
Pe = ^ , (83)
, (84)
(85)
By imposing the boundary condition
--=0 at x = - , (86)
we find that
B = 0 (87)
-------
144
and, hence,
-c . fle[(Pe/2) - TJ. _ (88)
At the leading edge of the system,
c=j=A at x = 0 , (89)
and the complete solution in transform space becomes
[(Pe/2) -
-------
145
Instead of using x - t space, which involves the derivative of integrals,
it is simpler to compute the derivative in x - s space and invert the ob-
tained transform.
The inversion of the derivative
c (0 s) = -
dx IU'SJ L
s)
2s
is given by
If (t'0) =^F
oX Lu
^m
+Jjferf(nt)|
(95)
(96)
where
Consequently, the total flux
N(0,t) - uc - D
9X
is represented by
(97)
(98)
N(0,t) -
(99)
for a uniform concentration of one at the origin.
During the time interval t to t + at, the amount of material entering
the first cell ,
±+5t
l(0,t')'dt - Q ,
(100)
-------
146
can be approximated by
QO Lx , f U/
= u T +^o
[erf (/n(t + 6t))+ erf(^t)] . (101)
Note that at long times the inflow approaches the quantity
/ . \
Q - (^ + ,/Dn)6t ; (102)
and if k = 0,
Q = u6t , (103)
and pure advection occurs. One effect of having the reactive term is to
enhance the inflow above the purely advective amount.
In Section D, we present figures in which the analytical solution is
always represented by continuous curves.
D. RESULTS
To test each method under conditions similar to those encountered in
atmospheric modeling, we decided to allow the Peclet number (uL/D) and the
kinetic rate constant to vary over a wide range of values. For each run,
the incremental spatial distance was set at 2 miles, and the total length
of the region was assumed to be 50 miles. Each hour was subdivided fnto
12 five-minute segments. In all cases, the free-stream velocity was held
at 4 miles per hour, and the diffusivity was allowed to vary as shown in
Table 25.
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147
Table 25
VALUES OF DIFFUSIVITY AND PECLET NUMBER
FOR THREE CASE STUDIES
Diffusivity
2 -1
Case (m sec ) Pec let Number
1 200 720
2 700 206
3 2000 72
These runs varied from almost a square wave propagation (Pec - 720) to a
smooth diffusion problem (Pec = 721). For each method tested, we executed
a series of 12 runs.
The corresponding kinetic values associated with these transport con-
ditions are given in Table 26.
Table 26
KINETIC CONSTANTS FOR EACH CASE
Kinetic Constant
Case (sec" )
1 0
2 10"4
3 10"3
4 2 x 10~3
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148
These values include the cases of both no reaction (k = 0), and a
tively fast react'
were also considered.
o
relatively fast reaction (k = 2 x 10~ ). Two intermediate reaction rates
In the presentation of the data, we included only those cases in
which no reaction occurs (k = 0; Pec = 720, 206, and 72) and those of
highest Peclet number (Pec = 720, k = 10~4 and 10"3). We chose these
cases because they are somewhat representative of the range of conditions
that can occur in mesoscale modeling systems. The following subsections
present a brief synopsis of the performance of each numerical scheme. In
each figure presenting our results, the analytical solution is given at
3, 6 and 9 hours from the start of the test.
1• The Price Method
As used in the SAI model, the Price scheme is inadequate for accur-
ately modeling mesoscale phenomena. In all cases, the method overpredicts
the actual ground-level concentration and transposes the wave to the left
because of phase shift, as shown in Figures 22 through 23. Although some
improvement occurs in cases having high Peclet numbers, this agreement is
not substantial enough to reduce dramatically the errors involved. Thus,
we rated the method as poor.
2. The Cro_wT_ey Second- and Fourth-Order Methods
The accuracy of prediction can be substantially increased by using
either the Crowley second-order or the Crowley fourth-order approximation,
as shown in Figures 24 through 25. In cases where an extremely strong con-
centration gradient appears (Pec = 720), the second-order scheme exhibits
some rather erratic results near the top of the wave. Aside from such
cases, both methods provide essentially the same results.
In the implementation of these methods in actual simulation programs,
some observers have noticed that higher order methods occasionally predict
negative concentrations in regions having large concentration gradients.
-------
149
Although this result did not appear in our work, one should keep it in mind
as a limitation when using these methods.
3. The SHASTA Method
One of the simplest and yet most efficient methods of solving the spe-
cies transport equation is the SHASTA method. Figure 26 presents its per-
formance results. Not only does the method exhibit a relatively high degree
of accuracy, but also, unlike many of the alternative finite difference
methods, it never predicts negative results. Thus, it is the best choice
available of an explicit solution algorithm.
4. The Galerkin Method
Of all the methods tested, the Galerkin technique provided the most
accurate results over the widest range of conditions selected in this study,
as shown in Figure 27. The predicted results, were always within 1 percent
of the analytical solution, and for many individual points in the analysis,
the results exhibited zero error. Not only could the method be used to model
situations in which extremely strong concentration gradients appeared, but
also it could accurately treat cases involving very fast reaction schemes.
Although the method is implicit and hence iterative in solution, its execu-
tion time appears to be comparable to a corresponding implicit Price scheme
as currently used on the SAI model.
We thus recommend that this technique be used for cases where high
resolution is desirable, even at the expense of increased computing time
and programming effort.
5. Particle-In-Cell Methods
Accuracy in particle-in-cell methods is a strong function of the number
of particles used. In this study, as the particle size was reduced from 80
to 40 to 20 weight units, the average error was reduced from 9.6 to 6.3 to
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150
3.3 percent, respectively. Figure 28 presents the results obtained using
this type of method. These methods proved successful at simulating both
reactive and nonreactive systems, and they were able to treat steep gra-
dients exceptionally well.
A rather interesting aspect of this analysis is that such techniques
show greater accuracy at lower rather than higher Peclet numbers, as would
normally be expected to occur. The reason for this phenomenon is probably
the following: As the diffusivity is increased, the diffusive component
in the Hotchkiss-Hirt analysis displaces the particle by an amount propor-
tional to the square root of the diffusivity. For large values of the dif-
fusion coefficient, this displacement can extend well over several cells.
Hence, the method is best applied to those cases in which the diffusivity
2 -1
is less than 200 m sec .
One drawback of particle-in-cell methods is the amount of computing
time required to solve a particular problem for a given accuracy. Since
a random number must be generated for each particle at each step, comput-
ing costs can be exorbitant as the number of particles increases.
6. The Method of Egan and Mahoney
For the strictly advective case, the Egan and Mahoney method gener-
ates an extremely accurate solution with virtually no error attributable
to numerical diffusion. This accuracy is particularly notable because
advective phenomena have been extremely difficult to simulate using com-
puting methods. Figure 29 presents the results obtained using this method.
Unfortunately, we could not incorporate the diffusion step in this analysis
because of the absence of any clear explanation of the treatment of this
process in the open literature. Without this link, it is difficult to form
an overall critical appraisal of the technique. In light of this limita-
tion, this method should continue to be investigated as more material
becomes available.
-------
" u—cr
O-
o.
I
I
n3
S-
-P
c
O)
o
E
O
o
0.8
0.6
0.4
0.2
o
10 20 30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec
o
-1
40
50
FIGURE 22. CONCENTRATION AS A FUNCTION OF DOWNWIND
DISTANCE FOR THE EXPLICIT PRICE SCHEME
-------
1.0
CL
Q.
I
I
fO
i-
-p
53
u
E
•O
20 30
Downwind Distance—miles
(b) Pec = 206, k = 0 sec
40
-1
FIGURE 22. CONCENTRATION AS A FUNCTION OF DOWNWIND
DISTANCE FOR THE EXPLICIT PRICE SCHEME (Continued)
50
en
rv>
-------
l.O
0.8 -
0.6 -
Q.
CL
i
4J
C
cu
u
c:
o
0.4 -
0.2 -
10 20 30 40
Downwind Distance—miles
(c) Pec = 72, k = 0 sec"1
FIGURE 22. CONCENTRATION AS A FUNCTION OF DOWNWIND
DISTANCE FOR THE EXPLICIT PRICE SCHEME (Continued)
50
tn
OJ
-------
1.0-
Q-
O-
I
0.8
0.6
0.4
0.2
10 20 30
Downwind Distance—miles
(d) Pec = 720, k = 10"4 sec'1
40
50
FIGURE 22. CONCENTRATION AS A FUNCTION OF DOWNWIND
DISTANCE FOR THE EXPLICIT PRICE SCHEME (Continued)
-------
1.0 e
0.8
0.6
CL
CL
0.4
O>
O
O
O
0.2
1234
Downwind Distance—miles
(e) Pec = 720, k - TO'3 sec"1
FIGURE 22. CONCENTRATION AS A FUNCTION OF DOWNWIND
DISTANCE FOR THE EXPLICIT PRICE SCHEME (Concluded)
en
en
-------
7.0, o O
0.8
0.6
CO
S-
0)
0
c
o
0 Q.4
Q.2
o
o
10 20 30
/
Downwind Distance—miles
(a) Pec = 720, k = 0 sec"-1
40
50
FIGURE 23. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE IMPLICIT PRICE SCHEME
en
cr>
-------
1,0
0.8
0.6
ra
S-
-*->
c:
01
o
8 0.4
0.2
10 20 30
Downwind Distance—miles
(b) Pec = 206, k = 0 sec
40
-1
50
FIGURE 23. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE IMPLICIT PRICE SCHEME (Continued)
-------
1.0
0.8
0.6
(O
s_
CD
O
O
O
0.4
0.2
10
20
30
40
Downwind Distance—miles
(c) Pec = 72, k = 0 sec"
FIGURE 23. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE IMPLICIT PRICE SCHEME (Continued)
50
en
OD
-------
1.0
0.8
0.6
4-5
tO
s-
4J
C
O)
o
c:
o
0.4
0.2
10
20
30
40
' 50
Downwind Distance—miles
(d) Pec = 720, k = 10~4 sec"1
FIGURE 23. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE IMPLICIT PRICE SCHEME (Continued)
en
-------
Downwind Distance—miles
(e) Pec = 720, k = 10~3 sec'1
FIGURE 23. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE IMPLICIT PRICE SCHEME (Concluded)
CTl
O
-------
10
20
(a) Pec
30
40
50
= 720, k = 0 sec
.-1
FIGURE 24. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY SECOND-ORDER SCHEME
-------
1,0
0.8 -
O)
o
o
O
0.4
0.2
^ O
10 20 30
Downwind Distance—miles
(b)Pec = 206, k = 0 sec"1
40
50
FIGURE 24. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY SECOND-ORDER SCHEME (Continued)
en
ro
-------
10
20
30
Downwind Distance—miles
(c) Pec = 72, k = 0 sec
-1
40
50
FIGURE 24. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY SECOND-ORDER SCHEME (Continued)
CTi
CO
-------
1.0
20
30
Downwind Distance—miles
(d) Pec = 720, k = TO"4 sec"1
40
50
FIGURE 24. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY SECOND-ORDER SCHEME (Continued)
CTi
-------
1.0
0.8
0.6
O)
o
0.4
0.2
2 3
Downwind Distance—miles
(e) Pec = 720, k = 10"3 sec"1
FIGURE 24. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY SECOND-ORDER SCHEME (Concluded)
-------
1.0
A
0.8
c 0,6
o
fO
o
c.
o
0,4
Q.Z
A
10 20 30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec
40
-1
50
FIGURE 25. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY FOURTH-ORDER SCHEME
-------
1,0
0.8
c:
o
0.6
CD
CJ
O
o
0.4
0.2
20
30
(b) Pec = 206, k = 0 sec
-1
40
50
FIGURE 25. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY FOURTH-ORDER SCHEME (Continued)
CTi
-------
1.0
0.8
o 0.6
(T3
i-
-------
7.0
0.8
0.6
ra
s~
0)
o
O
O
0.4
0.2
20
30
40
Downwind Distance—miles
(d) Pec = 720, k = 10~4 sec"1
50-
FIGURE 25. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY FOURTH-ORDER SCHEME (Continued)
01
-------
1.0
0.8
0.6
fO
s_
OJ
u
0.4
0.2
_L
I
1 2'3 4
Downwind Distance—miles
(e) Pec = 720, k = 10~3 sec"1, 6t = 300 sec, 6x = 2 miles
FIGURE 25. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE CROWLEY FOURTH-ORDER SCHEME (Concluded)
-------
10 20 30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec
40
50
-1
FIGURE 26. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE SHASTA METHOD
-------
10
20
30
(b) Pec =206, k = 0 sec
-1
40
50
FIGURE 26. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE SHASTA METHOD (Continued)
IX)
-------
1.0
30
40
50
Downwind Distance—miles
(c) Pec = 72, k = 0 sec'1
FIGURE 26. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE SHASTA METHOD (Continued)
CO
-------
1.0,
O)
o
O
O
10
20
30
Downwind Distance—miles
(d) Pec = 720, k = 10~4 sec"1
40
50
FIGURE 26. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE SHASTA METHOD (Continued)
-------
1.0
12 3 4
Downwind Distance—miles
(e) Pec = 720, k = 10"3 sec"1, st = 300 sec, 6x = 2 miles
FIGURE 26. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE SHASTA METHOD (Concluded)
-------
7.0
-e a a D D
a a
a on a a ,_, a GL
0.8
0.6
rtf
s_
sr
Of
0.4
0.2
D
' 20 30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec
-1
50
FIGURE 27. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE GALERKIN METHOD
en
-------
fC
S-
O)
O
0,2 -
20 30
Downwind Distance—miles
(b) Pec = 206, k = 0 sec
40
50
-1
FIGURE 27. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE GALERKIN METHOD (Continued)
-------
1.0
0.8 -
c 0.6
o
C
CD
O
0.4 -
0.2 -
20
30
Downwind Distance—miles
(c) Pec = 72, k = 0 sec
-1
40
50
FIGURE 27. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE GALERKIN METHOD (Continued)
CD
-------
1.0
0.8 -
tO
O
O
0.2 -
30
Downwind Distance—Miles
(d) Pec = 720, k - 10~4 sec"1
40
50
FIGURE 27. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE GALERKIN METHOD (Continued)
-------
1.0®
2 3
Downwind Distance—miles
(e) Pec = 720, k = 10~3 sec"1
FIGURE 27. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE GALERKIN METHOD (Continued)
OD
o
-------
1.0
0.8
c.
o
0,6
O)
u
I 0.4
0,2
•o
o o
10
o
o\
20
30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec"1
40
50
FIGURE 28. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE PARTICLE-IN-CELL (SMOOTHED) METHODS
-------
T.Q
0.8
c:
o
c:
O)
o
0.6
0.4
0,2
10 20 30
Downwind Distance—miles
(b) Pec = 206, k = 0 sec
40
50
-1
FIGURE 28. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE PARTICLE-IN-CELL (SMOOTHED) METHODS (Continued)
co
-------
1.0
o
0.8
o
0.6
i-.
c:
O) '
o
c:
o
0.4
0.2
10 20 30
Downwind Distance—miles
(c) Pec = 72, k = 0 sec'1
40
50
FIGURE 28. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE PARTICLE-IN-CELL (SMOOTHED) METHODS (Continued)
-------
1.0
0.8
0.6
(tf
S-
-M
C
cu
u
J 0.4
0.2
10 20 30
Downwind Distance—miles
(d) Pec = 720, k = 10~4 sec"1
40
50
FIGURE 28. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE PARTICLE-IN-CELL (SMOOTHED) METHODS (Continued)
co
-------
7.0
0.8
cr
o
rcf
S-
4->
cr
o;
u
c:
o
o
0.6
0.4
0.2
o
2 3
Downwind Distance—miles
(e) Pec = 720, k = 10"3 sec"
-1
FIGURE 28. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE PARTICLE-IN-CELL (SMOOTHED) METHODS (Concluded)
co
en
-------
10 20 30
Downwind Distance—miles
(a) Pec = 720, k = 0 sec
40
50
-1
FIGURE 29. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE EGAN AND MAHONEY METHOD
CO
en
-------
1.0
10
20
30
Downwind Distance—miles
(b) Pec = 720, k - 10~4 sec"1
40
50
FIGURE 29. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE EGAN AND MAHONEY METHOD (Continued)
co
-------
Downwind Distance—miles
(c) Pec = 720, k = TO"3 sec"1
FIGURE 29. CONCENTRATION AS A FUNCTION OF DOWNWIND DISTANCE
FOR THE EGAN AND MAHONEY METHOD (Concluded)
Co
Co
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189
7. Computational Time
Concurrent with any appraisal of the accuracy associated with alterna-
tive solution techniques must be a comparison of the computing times required
for these methods. Although a particular method may be extremely accurate,
the computational time it requires may be so large that a less accurate, more
efficient algorithm would be a better choice. Table 27 lists the computing
times for the various methods surveyed in this study.
Table 27
COMPUTING TIME REQUIRED FOR ALTERNATIVE SOLUTION METHODS
Computing Time
^Method (sec)
Price—explicit 7.50
Price--implicit 11.10
Crowley--second order 7.40
Crowley--fourth order 7.40
SHASTA 7.95
Galerkin 13.2
Egan and Mahoney 1.10
Particle-in-cell 68.2
Note that all of the explicit finite difference methods use approximately
the same amount of computing time (approximately 7.5 seconds). Of the implicit
schemes, only a slight difference exists between the Galerkin and the Price
methods. Obviously, the accuracy more than compensates for the larger compu-
tational time. Finally, the particle-in-cell methods are extremely costly in
computing time and should be used only as a last resort.
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190
E. CONCLUSIONS
The selection of a solution algorithm for a set of partial differential
equations should be based on considerations of both speed and accuracy. For
the methods surveyed in this analysis, we drew the following conclusions:
> For a rapid explicit scheme, the SHASTA technique should be
chosen. Not only is the method accurate and efficient, but
also it is guaranteed not to predict negative concentrations
dn areas having steep concentration gradients. This latter
quality greatly enhances the appeal of the SHASTA method
over competing schemes.
> In those cases in which extremely high resolution is desir-
able, it is advisable to develop a Galerkin algorithm for
the transport equation. The increase in accuracy, stability,
and ease of modeling irregularly spaced regimes more than off-
sets the increased cost in computational time.
> As more information becomes available in the open literature,
the Egan and Mahoney method should be explored as a possible
supplement or replacement for either the SHASTA or the Galerkin
scheme.
> Finite difference techniques introduce a considerable amount
of numerical diffusion into the calculation, producing an over-
prediction of pollutant concentrations downwind from the source.
The effect is most pronounced using the Price scheme and is
somewhat smaller using the Crowley second- and fourth-order
systems. The Crowley fourth-order scheme tends to be more ac-
curate than the corresponding second-order scheme in regions
having a steep concentration gradient, though the fourth-order
scheme frequently predicts negative concentrations in regions in
which complex flow fields exist. Regardless of the technique
used, finite difference methods are inaccurate for systems in
which extremely fast reactions occur.
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191
> The particle-in-cell technique developed by Sklarew (1971) is
accurate for both reactive and nonreactive systems, provided
that a sufficient number of particles is used in the simula-
tion. However, the time required for the simulation for a
given accuracy can be prohitively excessive and can thus in-
validate the use of the technique.
In conclusion, we wish to caution the reader about interpreting the
results obtained in this study: These results were developed for a simple
one-dimensional, time-dependent problem in which a simple first-order reac-
tion occurs. In a real situation, this idealized model can easily be invali-
dated by a complex flow field, a set of nonlinear reactions, or a complicated
source emissions pattern. In essence, this analysis focused on one aspect of
the complete problem: the identification and assessment of the errors asso-
ciated with the solution of the one-dimensional advection-diffusion equation.
The study did not treat problems that are associated with the method of frac-
tional steps, nor did it consider systems in which nonlinearities occur (as
they frequently do in the real world). Yet, since the numerical diffusion
associated with finite difference techniques is considerable, the results of
this study serve as a benchmark for identifying those schemes that are the
most accurate in a one-dimensional sense. If one can assume that this ac-
curacy is maintained throughout the entire solution, then the application of
the most promising of these techniques to the current SAI model will most
likely produce—but cannot guarantee—an improvement in the results.
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192
V AIRSHED MODEL MODIFICATION FOR MULTI DAY SIMULATION
Steven D. Reynolds
Jody Ames
A. INTRODUCTION
In the pa-st, the application of the SAI airshed model has been limited to
the simulation of one daytime period (usually 5 a.m. to 3 p.m.). During the
present study, we adapted the model for multiday runs. The two most important
benefits to be derived from multiday simulations are the following:
> Treatment of multiday episodes. A primary objective of adapting
models to perform multiday computations is to provide the basis
for evaluating the effectiveness of air pollution control strategies.
For example, difficulties in specifying initial conditions for some
future year can be averted by performing a multiday run, since the
predictions on the second and subsequent days are generally less
sensitive to the choice of the initial concentration distribution
input to the model. Also of interest is the short range prediction
of the ground-level pollutant concentrations for strategies such as
those that might be put into effect when meteorological conditions
conducive to severe pollution episodes occur.
.> Identification of errors. Multiday simulations will be instrumental
in establishing possible sources of error in the airshed model. Errors
incurred in a short term simulation (say, less than 12 hours) would
not accumulate to the extent that they would over a two- or three-day
period. As an example, suppose that the predicted concentrations of
total nitrogen oxides were much higher than measured values after a
simulation of several days. This might suggest that either NOX emissions
are too high or sinks of NO have not been properly accounted for in
A •
the model.
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193
One obvious difficulty that might preclude usage of the model for multiday
runs is the accumulation of errors introduced by the numerical integration
scheme. But as noted in the previous chapter, several promising numerical
techniques are available that—if implemented in the model—should reduce
numerical error propagation significantly.
In modifying the SAI model, we considered the following aspects of
multiday usage:
> Treatment of photochemistry at night.
> Definition of the modeling region.
> Use of a grid with variable resolution.
> Generalization of the finite difference solution technique for
use on a grid with variable vertical resolution.
> Modification of the computer codes.
Furthermore, to obtain some experience in the performance of multiday
runs, we prepared a set of emissions, meteorological, and air quality inputs
applicable to the Los Angeles basin on 29 and 30 September 1969. These are
two days that we studied under a 'previous EPA contract (68-02-0339). Using
these days, we were able to compare results from the multiday simulation with
those obtained from the corresponding 5 a.m. to 3 p.m. runs made previously.
Of particular interest is the comparison of the two sets of predictions at
3 p.m. on 30 September to determine to what extent the two sets of predictions
agree.
In the following sections, we summarize our efforts in each of the pursuits
listed above.
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194
B. MODEL REFINEMENTS
1. Treatment of Photochemistry at Night
The primary objectives of this study were to examine the suitability of
the kinetic mechanism employed in the airshed model for performing nighttime
simulations and to determine whether chemical reaction effects can be ignored
during a portion of the nighttime period to reduce computing costs. In addition,
we wished to obtain some experience in running the model at night, since previous
efforts were limited to the simulation of single daytime periods of 10 hours in
duration. Because of the difficulties we experienced in incorporating the ex-
panded 36-step mechanism in the airshed model, we decided to try to use the
original version of the airshed model, which treats the kinetics using a 15-step
mechanism.
To determine the applicability of the original 15-step kinetic mechanism
employed in the airshed model (see Hecht, 1972) for use at night, we performed
several "numerical" smog chamber simulations with photolysis rate constants set
to only a small fraction of their nominal values. Since sunlight is one of the
most important driving forces in the mechanism, we expected the photochemical
processes to be slowed considerably after sunset. We set k-, (the N02 photolysis
rate constant) equal to 0.01 min [the remaining rate constants and stoichiometric
coefficients were assumed to be equal to those employed in the 29 September 1969
validation study (see Reynolds et al., 1973)] and employed the following initial
conditions:
Initial Concentration
Species (ppm)
RHC 0.4
NO 0.5
N02 0.15
CO 15.0
The model predicted the following concentrations after eight hours:
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195
Concentration
Species (ppm)
RHC 0.30
NO 0.32
N02 0.32
These results clearly indicate that substantial chemical conversion is predicted
in the absence of strong sunlight.
Since we expected the predicted concentrations to change only slightly from
the initial conditions, we hypothesized that the HN02 steady-state assumption
was responsible for the large change in concentrations. The RHC, NO, N02, and
Oo concentrations predicted by the mechanism are independent of the value of kj
(the HN02 photolysis rate constant) used when HN02 is assumed to be in a pseudo-
steady state. We then carried out a second simulation, which was similar in all
respects to the first except that HN02 was not assumed to be in a steady state.
The results of this simulation after eight hours were as follows:
Concentration
Species (ppm)
RHC 0.38
NO 0.45
N02 0.17
These results indicate that considerably less chemical conversion is realized
when it is assumed that d(HN02)/dt ? 0.
As a final check on the old SAI mechanism employed above, we performed an
eight-hour simulation using the new SAI mechanism currently being validated. The
purpose of this test was to use the best available kinetic mechanism to obtain
an estimate of how much chemical reaction takes place in the absence of intense
sunlight. Assuming RHC to be entirely propylene, k^ to be equal to 0.01 min
(other photolysis rate constants were scaled accordingly), and initial condi-
tions to be the same as those cited previously, the new mechanism predicted the
following concentrations after eight hours:
-------
196
Concentration
Species (ppm)
RHC 0.38
NO 0.45
N02 0.18
These results reinforce our initial belief that the HN02 steady-state assump-
tion is responsible for the observed conversion of NO to N0? in the 15-step
mechanism.
It appears from the results of these tests that the 15-step kinetic
mechanism previously employed in the airshed model with HN02 in a steady state
will not be suitable for carrying out photochemical calculations at night. Since
considerable effort would be required to remove the HN02 steady-state constraint
from the old airshed model and, furthermore, since we are replacing the 15-step
mechanism with the new expanded mechanism (in which HN02 is not assumed to be
in a steady state), we decided to defer further study of the treatment of photo-
chemistry at night. This effort should be resumed, however, when further exper-
ience is obtained in using the new mechanism in actual airshed simulations.
After reviewing the results of the smog chamber runs cited above, we found
that we may be able to drop some, or perhaps all, of the reaction terms in the
governing equations during a portion of the nighttime period. If this is possible,
then computing requirements can be reduced significantly. And since we are
concerned with multiday runs, it is especially important to find ways of reducing
the costs of such simulations. To study further the possibility of modifying the
treatment of chemistry at night, we need to perform appropriate nighttime simu-
lations, both with and without chemistry in the model, to determine whether
and when chemical reaction effects can be ignored. If the chemistry cannot
be completely omitted from the model, perhaps the mechanism can be simplified.
2. Definition of the Modeling Region
To minimize errors resulting from the need to specify pollutant concentra-
tions at points of transport into the modeling area, one should choose boundaries
of the region such that either background levels or actual measurements can be
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197
used to estimate the boundary conditions. In previous simulations of pollu-
tants in the Los Angeles basin, we exercised particular care to account properly
for pollutants transported into the region from areas over the ocean and aloft
(i.e., contaminants originally residing in the inversion layer and subsequently
injected into the mixed layer as the inversion was eroded by convective heating).
Although significant amounts of pollutants are often carried out over the ocean
at night, it is usually difficult to estimate the concentration levels in the
returning off-shore air mass because of the absence of appropriate measurements.
To employ background concentrations as the boundary conditions, one must model
both the urban area of interest and a portion of the surrounding environs
(suburban, rural, and ocean areas). In addition, the upper boundary of the
modeling region should be defined at that elevation where background levels
generally exist (1 to 2 kilometers should be sufficient). Thus, we modified
our original treatment of the vertical extent of the model from the region
between the ground and the inversion base to the region between the ground
and a user-specified surface aloft. As an example, one could define the top
of the region to correspond to the top of the inversion layer. The trapping
effect of an elevated inversion layer within the model is treated through the
z-dependence of the vertical diffusivity.
3. Use of a Grid with Variable Resolution
For efficient modeling of an urban area and a portion of its surroundings,
a grid with variable resolution should be used. Choosing the appropriate degree
of resolution in a particular area of the airshed depends primarily on the spatial
characteristics of gradients in the concentration field. In areas where gradients
are large, a relatively fine grid should be used; where gradients are small, a
relatively coarse mesh spacing may be adequate. With respect to horizontal grid
resolution, the mesh spacing in the outlying areas could be, say, two to four times
that used over the urban center. Because many sources are located at "ground
level," the vertical concentration gradient is often greatest near the surface.
Thus, it may be advantageous to use fine vertical spacing near the ground and
coarse spacing aloft. Since the numerical technique currently employed in the
model is not readily adaptable for variable horizontal grid resolution, we devel-
oped only variable vertical grid resolution capabilities during this study.
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198
However, inclusion of a variable horizontal mesh in the model, possibly using
a nested grid approach, should be considered in future efforts to incorporate
improved numerical solution techniques in the model.
4. Modification of the Finite Difference Equations
Because the finite difference equations, previously employed in the model
were derived for an equally spaced grid, it was necessary to modify the differ-
ence expressions involving derivatives of the concentration field in the z (or p)
direction. [See Reynolds (1973) for a discussion of the numerical integration
procedure.] In particular, changes were required in the advective and diffusive
flux terms in Step III of the numerical integration technique. As an illus-
tration of the nature of the changes made, Eqs. (44) through (54) in Reynolds
(1973) become:
cn+l _ ** i , j + AT Fn+l
i>0,k " A isj,k AHn+l 2Ap ARn+l U
F** \ , AT / Rn+1 Hn+l
i,j,k+% " A ri,j,k+% J 2AHn+l UKi,j,k i ,j
R** AHn \ + _^_ / sn+1 , AHn+1 + Sn . , AHn . , (104)
?AH I 1"3' 1'J 1>J»k ls
where
Fn+l _ 'P-i.j.k-ia L rn+l ,
oi T I/ 1 ~" ' 9 ' '- " - - '- n
A- 1 5 J 5 K~-^ c
n+l / cn+l _ cn+l \
, k = 2, 3, .... K . (105)
'
-------
199
** ^i i k_% / ** **
F = —' >J »K 2 I v r + i-\ -\ \ r
-' A L- + u • L
** **
yi,J,k-Js (£Ci,j,k-l " £Ci,j,k ' k = 2, 3, ..., K
(106)
0.5
Kn
i»j.k-is - ^ (108)
Ap.
\-
In view of the discussion given in the previous section, p is now defined by
the following relationship:
-,. z " h(x^!
Hlx,y,t) - h(x,y)
where
h(x,y) = terrain elevation,
H(x,y,t) = elevation of the upper boundary of the modeling region,
Ap, = the dimension less height of the k-th level grid cell.
The boundary conditions at the ground and aloft are the following:
(1) P = 0
Fn+l = Qn+l
d'-i T Is O^n \ »
* 1 >J »-2 * 1 ,J
where k = 1/2 is equivalent to P = 0.
-------
200
(2) p = 1
1f
** ,n
if *i,j,K+Js >
0
where k = K + j is equivalent to p = 1.
Changes in the matrix expressions given in Eqs. (55) through (70) of
Reynolds (1973) follow directly from the difference equations given above.
5. Modification of the Computer Codes
To carry out multiday simulations, we modified several portions of the
computer programs. These changes essentially make the codes more general.
Furthermore, the main code now allows the user to use a grid with variable
vertical resolution, where the spacing interval is treated as a model input.
As an example, in the Los Angeles simulation, which is discussed in the next
section, we used a grid with 10 vertical levels and the following mesh spacing:
Grid Spacing
Level (feet)
10 (top) 1625
9 825
8 425
7 225
6 125
5 75
4 50
3 50
2 50
1 (ground level)" 50
-------
201
Thus, the modeling region is assumed to extend from the ground to an
elevation of 3500 feet above the terrain. In addition, we implemented
appropriate changes, corresponding to the discussion given in the previous
section, in the coding of the finite difference equations.
In addition to the above-mentioned changes, we restructured the input
data deck setup to operate in the following manner. First, all parameters
global to the run--i.e., those parameters that would not be expected to vary
from day to day—are input. Then, the remaining inputs are arranged in
daily packets, one packet for each day to be simulated. When the simulation
reaches midnight, the input packet for the next day is read from the input
file. After the first day, some daily parameters can be omitted from the
input packet, and the values used on the previous day can be used again.
We also included provisions that allow the user to establish multiple
emission files for the input of emissions data to the program. For example,
one might establish two sets of emissions, one applicable to weekdays and the
other suitable for weekends. Once such a set of files is established for an
urban area, multiday runs consisting of any pattern of weekdays and weekends
can be simulated. Table 28 illustrates the deck setup and lists some of the
main parameters included as part of the global and daily inputs.
C. MULTIDAY SIMULATION OF THE LOS ANGELES BASIN
1. Preparation of Emissions and Meteorological Inputs
Since previous applications of the airshed model were limited to the simu-
lation of a 10-hour daylight portion of each of six days in 1969,..little con-
sideration was given to the definition of emissions and meteorological inputs
for use at night. Thus, to gain experience in the performance of multiday runs
with the SAI model, we carried out a necessarily limited effort to estimate
meteorological and emission inputs that would apply during the period from
3 p.m. on 29 September to 5 a.m. on 30 September 1969 (previous simulations
were carried out for the 5 a.m. to 3 p.m. period on both 29 and 30 September
1969). In particular, we performed the following tasks:
-------
202
Table 28
ORGANIZATION OF MULTIDAY INPUT
Global data
Run heading
Simulation options and grid definition
Start and stop times and dates
I/O units
Print options for maps
Region definition
Stations and landmarks
Integration parameters and stoichiometric coefficients
Activation energies
Initial Conditions
Day 1 packet
Date, emissions type, input controls
Rate constants
Light intensity factors
Deposition velocities
Concentrations aloft
Point source emissions
Boundary conditions
Day 2 packet
Date, emissions type, input controls
Rate constants*
Light intensity factors*
Deposition velocities*
Concentrations aloft*
Point source emissions*
Boundary conditions*
Day 3 packet
it
Can be omitted after Day 1.
-------
203
> We used the SAI automated wind field analysis package to generate
hourly wind speed and direction maps spanning the period from 5
a.m. on 29 September to 3 p.m. on 30 September 1969.
> We employed the SAI automated inversion analysis program to esti-
mate hourly mixing depth maps, using actual observed mixing depths
available for the daytime periods to the extent possible. We
examined nighttime temperature profiles measured by Meteorological
Research, Incorporated in the Los Angeles basin during the summer
of 1973 and estimated that pollutants would typically be mixed
throughout a depth of about 60 to 70 meters at night.
> We prepared a set of fixed-source emission maps for hydrocarbons
and NO that are applicable from 6 p.m. to 6 a.m. the following
A
morning; the original SAI fixed-source maps were dervied for the
complementary 12-hour period. Using data presented in Appendix A
of Roth et al. (1971), we estimated that about 25.5 tons of NO
/\ •
and 30 tons of low reactivity hydrocarbons are emitted in the
modeling region between 6 p.m. and 6 a.m.
> We specified boundary conditions at points of horizontal inflow
into the model between 3 p.m. on 29 September and 5 a.m. on 30
September. Boundary conditions at other hours were available from
the model validation studies reported in Reynolds et al. (1973).
At this point, it is appropriate to note that the paucity of meteorological
and emissions data applicable specifically during the nighttime hours makes the
estimation of mixing depths and emission rates highly uncertain. The primary
objective of our present effort was simply to assemble a set of "reasonable"
inputs that can be used in tests of the multiday version of the SAI airshed
model. Further efforts should be made to refine the temporal distribution of
surface street and freeway traffic activity and the spatial and temporal distri-
butions of the HC and NO emissions from stationary sources at night [see Roberts
et al., (1971)].
2. Discussion of the Multiday Simulation Results
To test the various modifications made in both the structure of the model
and the computer codes, we performed a multiday simulation of pollutant concen-
-------
204
trations in the Los Angeles basin. As noted in Section B-l, we decided that
the 15-step chemical kinetic mechanism employed in the model is inappropriate
for use at night. Thus, the simulation reported here was carried out for CO
alone. When an appropriate set of emissions inputs suitable for usage with
the new mechanism can be developed, multiday photochemical simulations should
be undertaken.
Plots of predicted and measured CO concentrations at the downtown Los
Angeles, Long Beach, West Los Angeles, Burbank, Reseda, Whittier, and Azusa
stations are given in Figures 30 through 36, respectively. The simulation
extends from 5 a.m. PST on 29 September to 3 p.m. PST on 30 September 1969.
The results for the 5 a.m. to 3 p.m. period on 29 September are very similar
to those obtained in the previously reported SAI validation effort. (Dis-
crepancies in the two sets of predicted results are due to the manner in
which the meteorological .variables were specified—automatically in the
former case and manually in the latter.) Of greater interest is an examina-
tion of the remaining results, which are best approached by considering the
5 p.m. to 5 a.m. nighttime period and the following 5 a.m. to 3 p.m. daytime
period separately.
In general, the nighttime predictions are reasonably accurate consider-
ing the fact that neither vertical temperature measurements nor refined
temporal distributions for motor vehicle activity were available to estimate
the corresponding meteorological and emissions inputs. Results at the end
of this period (5 a.m.) often fell within a few parts per million of the
measured concentrations, as shown in Table 29. Two notable exceptions, how-
ever, are illustrated in the downtown Los Angeles and Burbank predictions
(Figures 30 and 33, respectively). Upon further examination of the results
for these two stations, we made the following observations:
> Downtown Los Angeles. A rather substantial build-up in the CO
concentration was observed to occur from 9 p.m. to midnight, but
it was predicted to take place two hours earlier, from 7 p.m. to
10 p.m. Since the magnitudes of predicted and'measured concen-
trations during the early morning hours of 30 September agree
-------
40
30
s
Q.
CL.
I
c
o
O)
20
TO
PREDICTED CO CONCENTRATION
MEASURED CO CONCENTRATION
D
D
D
n
12 15 18
29 September 1969
21
0
6 9
30 September 1969
12
Time—hour
15
FIGURE 30. COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT DOWNTOWN LOS ANGELES
no
o
Gi
-------
40
30
Q.
CL
l
I
J 20
to
t-
O)
u
sr
o
O
10
PREDICTED CO CONCENTRATION
MEASURED CO CONCENTRATION
D
12 15 18
29 September 1969
21
0
6 9
30 September 1969-
12
15
Time—hour
FIGURE 31 COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT LONG BEACH
ro
o
CTl
-------
40
30
0.
d.
20
S-
o>
-------
40
30
CL
CL
I
I
20
O)
o
cr
o
10
PREDICTED CO CONCENTRATION
D MEASURED CO CONCENTRATION
D
D
D
9 12 15 18
: 29 September 1969
21 0
3 6 9 12 15
30 September 1969 - •
Time—hour
FIGURE 33. COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT BURBANK
ro
o
CO
-------
40
30
0.
Q.
I
I
* 20
O)
O
10
PREDICTED CO CONCENTRATION
D MEASURED CO CONCENTRATION
D
D
9 12 15 18 21
-29 September 1969
3 6 9 12
30 September 1969
Time—hour
FIGURE 34. COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT RESEDA
15
ro
o
i-D
-------
40
30
ex
CL,
I
I
cr
o
o
tr
o
o
20
10
PREDICTED CO CONCENTRATION
MEASURED CO CONCENTRATION
D
D .
D D D
D
n
9 12 15 18
29 September 1969
21
369
30 September 1969
Time—hour
12
15
FIGURE 35. COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT WHITTIER
ro
o
-------
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CL.
Q.
I
I
* 20
O)
u
cr
o
o
10
PREDICTED CO CONCENTRATION
D MEASURED CO CONCENTRATION
-nan • n
D
D
n n
12 15 18
29 September 1969
21
Time—hour
FIGURE 36. COMPARISON OF PREDICTED AND MEASURED HOURLY AVERAGED
CO CONCENTRATIONS AT AZUSA
n n
3 6 9 12
30 September 1969
15
ro
-------
212
fairly well, the discrepancy in the time of occurrence of the
build-up may well be the result of inaccuracies in the temporal
distribution of motor vehicle emissions (the only source of CO
in the model) during the period from 7 p.m. to midnight. Thus,
the total loading seems correct, but the temporal distribution
appears to be in error by about two hours. Traffic activity at
5 a.m. also seems to be greater .than that predicted by the SAI
emissions model.
> Burbank. Of all the results presented in Figures 30 through 36,
those calculated for the Burbank station are the poorest. Pre-
dicted concentrations after 5 p.m. on 29 September are consistently
low by as much as 10 ppm. Upon examining the results for Reseda
(also located in the San Fernando Valley), we noted that, although
predictions are often low, the discrepancy in the predicted and
measured concentrations is at most only 3 ppm. Meteorological data
for Burbank indicate very light winds (1 mph) from the north. The
high concentrations at this station may thus be the results of local
emissions from the major interstate freeway situated to the north
of the station. A shallow mixing layer coupled with near-calm con-
ditions would certainly limit the extent to which local freeway
emissions would be dispersed.
Basically, the nighttime predictions do not indicate any systematic errant
behavior in the model. In fact, considering the absence of key meteorologi-
cal and emissions data, the 5 p.m. to 5 a.m. results are about as good as
might be expected under the circumstances.
Turning now to an examination of the daytime results for 30 September,
we note that at many stations the multiday predictions are very similar to
those previously reported in the single-day test of the model. Since the
meteorological and emission inputs for these two runs were not significantly
different, discrepancies in the two sets of predictions can be attributed
primarily to differences between the multiday and single-day CO concentra-
tion distributions at 5 a.m. PST on 30 September 1969. In the single-day
run, which began at 5 a.m., the initial CO concentration field, shown in
-------
213
Table 29
PREDICTED AND MEASURED HOURLY AVERAGED CO CONCENTRATIONS
AT THE END OF THE 29 TO 30 SEPTEMBER NIGHTTIME PERIOD*
Predicted Measured
Concentration Concentration
Station (ppm) (ppm)
Downtown Los Angeles 13 13
Azusa 6 8
Burbank 7 16
West Los Angeles 7 6
Long Beach 6 3
Reseda 7 6
Pomona 3 3
Lennox 7 8
Whittier 7 7
*
The figures presented in this .table were averaged from
4 a.m. to 5 a.m. PST on 30 September 1969.
Table 30, was estimated from the appropriate measurements reported by the
Los Angeles and Orange County Air Pollution Control Districts. The concen-
tration field on the grid at 5 a.m. in the multiday run is, of course, the
result of a continuous simulation started at 5 a.m. on the previous morning.
The ground-level concentration map for this case is illustrated in Table 31.
Over much of the modeling region, the two sets of predictions agree within
2 or 3 ppm. However, the discrepancies are much higher in the Pasadena,
downtown Los Angeles, and Burbank areas. The high CO levels in these areas
given in Table 30 are the result of manual interpolation of the measured
values reported at downtown Los Angeles and Burbank (the Pasadena station
did not report CO levels on 30 September). In general, the single-day results
for downtown Los Angeles and Burbank are better than the corresponding multi-
day predictions. At other stations, the model predicted CO levels reasonably
well, especially in view of the significant impact that local roadways may have
on measured concentrations during the peak traffic hours in the morning.
It is also interesting to note the extent to which errors have accumulated
throughout the multiday run. In Table 32, we give both the predicted and mea-
-------
Table 30
MULTIDAY GROUND-LEVEL CO CONCENTRATION MAP
AT 5 a.m. PST ON 30 SEPTEMBER 1969
CO GROUND LEVEL CONCENTRATIONS (PPM 1 AT 500.00 PST
10 U 12 13 1* 15 16 17 18 19 ZQ Zt 22 23 2» 25
25 7.0 7.0 7.0 7.0 7.0 8^0 9iP_lg_..0_ll. P_ 11.. 0_l I . Q_
24 7.3 7.3 7.J 7.3 '8.3 9.0 13.3 12.3 13.0 13.3 12 ^Q_
SAN GABRIEL HTNS
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RESEDA
22 7.0 7.0 7.0 7.0 e.O 10.0 12.0 14.0 16.0 16. 0_16.J)_l._0_\t>.3 16,3J5.3 14.0J2.J 10.iJ_8.p _ 8. 3_8. J_.8.0_ J. 0 __ 6.0_5.3 _*.0
Si.'lTA KCNICA KTNS
19 5.0 6.0 7.0 _ 7.J> _ 7.0 8.0 9.0 II. 0 13.0 15.0 16.0 16.0 16.0_16.p_15. 0 J2.0_IJI.O _8.0 _8.0_ 8.0_8..Q _ 7. p _ 6. 0__5.0 __4.0
~
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-------
Table 31
SINGLE-DAY (GROUND-LEVEL CO CONCENTRATION MAP
AT 5 a,m'. PST ON 30 SEPTEMBER 1969
CO GROUND'L^VEX CONCENTftATrO/^S (PPM ) AT SOO.OO nw 6909JO
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-------
216
sured hourly averaged concentrations over the last hour of the simulation.
Overall, the predicted results tend to be lower than the measurements by 1 to
2 ppm; in only one instance is the discrepancy greater than 2 ppm. Since we
expect the model to predict concentrations somewhat lower than those measured
at stations situated on heavily traveled streets, it is difficult to assess the
cumulative effect of meteorological, emissions, and numerical errors in this
simulation. Thus, it appears that we may have to carry out longer runs (three
or four days) to observe the build-up of modeling errors clearly.
Table 32
PREDICTED AND MEASURED HOURLY AVERAGED CO CONCENTRATIONS
FOR THE LAST HOUR OF THE MULTIDAY SIMULATION
Predicted Measured
Concentration Concentration
Station (ppm) (ppm)
Downtown Los Angeles 5 4
Azusa 3 5
Burbank 4 5
West Los Angeles 4 3
Long Beach 4 6
Reseda 2 3
Pomona 3 6
Lennox 3 5
Whittier 3 4
D. RECOMMENDATIONS FOR FUTURE WORK
During this study, we adapted the SAI airshed model for use in the pre-
diction of inert pollutant concentrations over multiday periods. The predic-
tion of photochemical contaminant concentrations should be undertaken when the
program containing the new kinetic mechanism is fully operational and suitable
hydrocarbon emission inputs are developed. To gain experience in multiday us-
age, we simulated pollutant concentrations in the Los Angeles basin for the
-------
217
34-hour period extending from 5 a.m. on 29 September to 3 p.m. on 30 September
1969. In general, the results obtained from this run agree reasonably well
with available measured pollutant concentration data.
We recommend that the following tasks be undertaken in the future:
> Assembly of an accurate data base for both meteorological and
emissions inputs for a multiple-day period.
> Performance of photochemical simulations as soon as possible.
> Performance of CO (and eventually photochemical) runs over
several consecutive days (say, four or more) to obtain a
better understanding of the cumulative effects of meteoro-
logical., emission, and numerical errors on our ability to
exercise the model on a multiday basis.
Our present experience indicates that, given a suitable input data base, the
model should be capable of producing reasonably good predictions of inert
species concentrations for at least two consecutive days. However, further
testing will be required to establish guidelines regarding the total number
of days that can be simulated before errors accumulate to unacceptable levels.
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218
APPENDIX
A USER'S GUIDE TO
David C. Whitney
-------
219
APPENDIX
A USER'S GUIDE TO MODKIN
David C. Whitney
1. INTRODUCTION
Quantitative description of the rates of chemical reaction of atmospheric
contaminants is a vital ingredient in the formulation of a model capable of
accurately predicting ground-level concentrations of gaseous pollutants. The
formulation of a kinetic mechanism having general validity is, however, an
endeavor beset by several inherent difficulties. First, many stable chemical
species are present in the atmosphere. Most of these exist at very low con-
centrations, thereby creating major problems of detection and analysis. In
fact, a number of atmospheric constituents remain unidentified. Second, the
large variety of highly reactive, short-lived intermediate species and free
radicals further complicates the picture. Finally, the enormous number of in-
dividual chemical reactions that these species undergo creates an even greater
barrier to understanding. Nevertheless, despite our limited knowledge of at-
mospheric reaction processes, it is essential that we attempt to formulate
quantitative descriptions of the processes that are suitable for inclusion in
an overall simulation model.
The formulation and development of a kinetic mechanism that is to be in-
corporated in any airshed model is both delicate and exacting, an undertaking
requiring a blend of science, craftsmanship, and art. On one hand, such a
mechanism must not be overly complex because the computation times for inte-
gration of the continuity equations in which the mechanism is to be imbedded
are likely to be excessive. On the other hand, too simplified a mechanism
may omit important reaction steps and may thus be inadequate for describing
atmospheric reaction processes. Therefore, one major issue is the requirement
that the mechanism predict the chemical behavior of a complex mixture of many
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220
hydrocarbons, and yet do so with a paucity of detail. Thus, in postulating
a mechanism, the formulator must strike a careful balance between compactness
of form and accuracy in prediction.
As an aid in the development of a kinetic mechanism for atmospheric
photochemical reactions, we prepared a computer program that allows the user
to present his proposed mechanism via data, input cards in the same manner as
he would formulate it on paper--i.e,, as a series of chemical equations and
their associated rate constants. Moreover, he can select the method of cal-
culation for determining the concentration of each chemical species in the
'mechanism from among the following choices: the integration of coupled or
uncoupled differential equations, the solution of algebraic equations for
species in a steady-state, or the assumption of a constant concentration.
Either static or dynamic smog chamber observations can be simulated, and plots
of species concentration as a function of time are provided as part of the
printed output. Reactions of similar species can be combined into a single
"lumped" reaction. Changes in the reaction mechanism, rate constants, or
species type designation can be effected by simple input card replacement;
receding or recompilation of the program is not necessary.
Since descriptions of the solution techniques and the development of the
chemical mechanisms have appeared elsewhere (Seinfeld et al., 1971; Hecht,
1972), we do not repeat them here beyond the degree necessary for an under-
standing of the computer program. This appendix is designed to serve primar-
ily as a user's guide to program operation and as a programmer's guide for
such program maintenance and modification as may be needed in the future.
2. USE OF THE PROGRAM
The input to MODKIN consists of two control cards, a set of reaction cards,
a set of species cards, a set of flow cards, and a set of plot cards. The card
formats are described in Table A-l , and additional comments regarding program
input are given below.
-------
Table A-l
INPUT CARD FORMAT FOR MODKIN
Card No.
1
1
1
1
1
1
1
1
1
1
2
2
2
Column No.
1-12
16-20
21-25
26-30
31-35
36-40
41-45
46-50
51-55
56-60
1-10
11-20
21-30
Variable
Name
NTIT(J)
NRXN
NLMP
NDIF
NSTS
NUNC
NREP
.MINT
NFLW
NRAT
TINCR
TEND
HSTART
Item Units of
Format Measure
3A4
15
15
15
15
15
15
15
15
15
F10.0 min
F10.0 min
F10.0 min
Comments
Twelve^character title for run heading
Number of reactions in the mechanism (maximum 99)
Number of reactions that contain species to be
be replaced, i.e., are lumped (maximum 10)
Number of species to be solved in a coupled
differential equation (maximum 40)
Number of species to be solved in a steady-
state approximation
Number of species to be solved via uncoupled
differential equations
Number of species that are replacements for
lumped species
Number of inert (constant concentration) species
(the total count of the above five species types
cannot exceed 50)
Number of species flowing into the reaction chamber
If nonzero, reaction rates will be printed
Time increment for printing and plotting results
Ending time for run :
Initial time step size
ro
IX)
-------
Table A-l (Continued)
Card No.
2
2
2
2
3
3
3
4
4
5-
. Column No.
31-40
41-50
51-60
61-70
1-20
21-50
51-60
1-4
6-10
1-20
Variable
. Name
HMINF
HMAXF
EPSF
Q
NMRC(J.K)
COEFF(J,K)/
NMPD(J,K)
RK(K)
NTEST
NLOC
NMRC(J,K)
Item Units of
Format Measure
F10.0 min
F10.0 min
F10.0
F10.0 min"1
4(A4,1X)
3(F6.0, : —
A4)
F10.0 ppm\~n
min"
A4
15
4(A4,1X)
5
5
21-50
51-60.
COEFF(J.K)/ 3(F6.0,
NMPD(J,K) A4)
RK(K)
F10.0
min
Comments
•Minimum time step size
Maximum time step size
Fractional allowable error for iterative solutions
Dilution or flow rate (sampling and leakage com-
pensation)
Reactant names, up to four per reaction (if
lumped reaction, species to be replaced
must appear first)
Species coefficients and product names,
up to three per reaction
Rate constant; n is the number of reacting
species
Name of species to be replaced in the lumped
reaction
Number of reactions contributing to the
lumped reaction (maximum 10)
Reactant names, up to four per reaction
(the first species name must be the replacement
for the lumped species)
Species coefficient and product names, up
to three per reaction
Rate constant; n is the number of reacting
species
rv>
r\3
ro
-------
Table A-l (Continued)
Card No.
6
6
7
7
8
9
9
9
9
9
9
,.
Column No.
1-4
11-20
1-4
6-10
1-80
1-4
6-7
9
11-14
16-40 '
41-50
Variable
Name
NAME(L)
YAX(L)
NTEST
NTIM
FTIME(J.L)/
FLOW(J.L)
NTEST
NDAT
JSYMB
JFACT
JCONC(J)
CLOW
Item Units of
Format Measure
A4
El 0.0 ppm
A4
15
4(2F10.0) min ppm"1
A4
12
Al
A4
5(A4,1X)
F10.0 ppm
51-60
61-70
CHIGH
TLOW
F10.0
F10.0
ppm
Comments
Species name
Species initial concentration
Flowing species name
Number of flow points (maximum 10}
Time of measurement and concentration of
flowing species
Name of species to be plotted
Number of input data for the plot (maximum 80)
Symbol to be used for the calculated data
Conversion factor for the label
Concentration labels for the y-axix
Minimum concentration value to be considered
for plotting
Maximum concentration value to be considered
for plotting
Minimum time value to be considered for
plotting
ro
IX>
CO
-------
Table A-l (Concluded)
Variable
Card No. Column No. Time
10
11
71-80
1-80
1-4
THIGH
TIME(J)/
DATA(J)
JBLANK
Item
Format
F10.0
Units of
Measure
4(2F10.0) min ppfli
A4
-1
Comments
Maximum time value to be considered for
plotting
Time and concentration input data to be
plotted
Blank in Columns 1-4 stops plotting
ro
-------
225
The first control card contains title and parameter information for the
run. The first field on this card is a 12-character title; the contents of
this field will be printed following "MODULAR KINETICS RUN NO." on the first
page of the printout. The number of reactions is given next; the program
expects this number of reaction cards to follow the control card. The next
entry specifies how many of these reactions represent lumped reactions and
thus need to be recalculated from sets of contributing reactions. The fol-
lowing five entries are the counts of each of the different types of species:
differential, steady state, uncoupled, replacement, and inert. Note that
there are limits on both the number of differential species and the total num-
ber of species. The program expects to find one species card for every species
named on the reaction cards; they must be ordered as shown above (i.e., all
differential species first, then all steady-state species, and so forth). The
next-to-last entry on the control card is the number of species that are flow-
ing into the reaction chamber; there must'be a set of flow cards for each of
these species. The final entry is a request flag governing the printout of the
reaction rates.
The first two entries on the second control card are printout parameters.
The first one determines the time increments (e.g., every five minutes) for
which the current concentration of all species are to be printed and plotted;
the second specifies the time at which the kinetics run is to be terminated.
The next four values are control parameters for the differential equation solu-
tion routine. In order, these parameters are the initial time step (normally
on the order of 10 min), the minimum allowable time step (normally about 10"
min), the maximum time step (about 1 to 10 min), and-the fractional error accep-
table for iterative solutions. The final entry on the second control card is
the rate at which each species concentration would be reduced in the absence of
reaction. This "dilution rate" primarily reflects the loss of material through
sampling; if there is an inflow, it is presumed to occur at this same rate.
The set of reaction cards provides all the reactions and rate constants,
one per card. Each card begins with a list of reactantS's which must appear in
consecutive fields, since a blank stops the scan. For a lumped reaction, the
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226
name of the species to be replaced must appear first; otherwise, the order of
reactant names -is immaterial. Note, incidentally, that the reactants appear
in the printed output in reverse order. If a species reacts with itself, it
must appear twice in the list of reactants. The products, along with their
coefficients, follow. Coefficients can be whole numbers or fractions; the
printout is rounded to two decimal places. Again, products must appear con-
secutively, since a blank stops the scan, but their order is unimportant.
The final entry on the card is the rate constant. The order of the reaction
cards does not matter, except that lumped reactions must follow nonlumped
reactions.
A set of contributing reactions consists of an identification card con-
taining the name of the lumped reactant (the species being replaced) and the
number of contributing reactions, followed by the list of contributing reac-
tions. All of the comments offered above regarding reaction cards apply to
these contributing reactions, except that these reactions cannot themselves
be lumped ones. The contributing reaction must have its reactants and prod-
ucts in the same relative location on the card as they are on the lumped
reaction card; i.e., the replacement species must appear first, and all prod-
ucts must be shown, even those that -have zero coefficients. However, the
order of the reactions within a set does not matter. The order of the sets
of contributing reactions must be the same as the order of the lumped reac-
tions in the set of reaction cards described above, and there must be one
set of contributing reactions for each lumped reaction.
The set of species cards is used to identify the species by type and to
initialize the species concentrations. Each card contains a species name and
concentration. The following is the order of the.species types: differential,
steady state, uncoupled, replacement, and inert. Within a given type, no par-
ticular order is necessary; in fact, some orderings of steady-state species
are clearly preferable to others in terms of elapsed computing time.
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227
A set of flow cards consists of an identification card containing the
species name and the number of input points, followed by cards specifying
the data points themselves. The data are not interpolated; instead, the
inflowing concentration is changed to a new value whenever the progression
of time in the mechanism passes an input time. Note particularly that the
concentration for the inflowing species is zero until its first input time
is passed.
The plot cards control the pictorial representation of concentration as
a function of time for each species. Instead of processing the output using
a plotter, the plot cards map the concentration-time profile onto a page-size
grid of the printout. The first card, which is the plot control card, con-
tains the species name, the number of experimental data points to be read, the
symbol to be used to represent the calculated data (an asterisk is used for
experimental points), the conversion factor, the concentration labels, and the
grid limits. These last three items require comment.
The grid has been divided into four vertical sections and eight horizontal
sections. Aesthetically, therefore, the time (horizontal) limits should be
chosen to give a span divisible evenly by eight (e.g., a limit of 0 to 400 will
result in the printing of a label every 50 minutes). Similarly, the concentra-
tion (vertical) limits should be divisible evenly by four, and they should be
the true rather than the scaled concentrations. The labels for the vertical
axis are not calculated from the concentration span, but rather are read in
from the control card. They can be any multiple of the true concentrations.
The scale factor, which appears in the figure caption along with the run title
and species values, indicates what multiplier was used. For example, if the
data prints were expected to range between 0.08 and 0.16 ppm, a scale factor of
"10+1"; concentration labels of "0.75", "1.00", "1.25", "1.50", and "1.75";
and concentration limits of "0.075" and "0.175" would give a plot containing
all the points. Note that no check is made among the labels, scale factor, and
limits to insure consistency. Also, the limit values themselves will not appear
on the plot; plotted points must fall within the grid boundaries.
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228
Experimental data cards, if any, come after the control card. If desired,
sets of plot cards can be stacked. The end of the plot deck is denoted by a
card with a blank species name; the program will then expect another MODKIN
run control card.
3. PROGRAM DESCRIPTION
The modular kinetics program consists of a main routine labeled MODKIN,
the subroutines LMPCAL, DIFSUB, and PLOT, which are called by MODKIN, and the
subroutines DIFFUN, MATINV, and PEDERV, which are called by DIFSUB. Each rou-
tine is treated in detail below. Listings and samples of program inputs and
outputs appear at the end of this appendix. Symbol, glossaries are included
within each routine that was written especially for this program.
a. MODKIN
The program begins by declaring a number of variables used by DIFSU3 as
being DOUBLE PRECISION. All arrays are identified in DIMENSION statements and
variables needed by LMPCAL and DIFFUN are placed in COMMON. The DATA declara-
tions include the input and output units, a blank word, and the maximum sizes
of the various arrays used for holding user inputs.
The control cards are read (note that this is a return point for stacked
data decks). An initial page is written listing all the control card param-
eters. The number of reactions is checked, and the set of reaction cards is
read. The number of lumped reactions is checked, and the contributing sets
are read. For each set, the number of contributing reactions and the lumped
species name are checked, the reaction counter is incremented and checked,
and the contributing reactions are read. The number of differential species
is checked, the total number of species is calculated and checked, and the set
of species cards is read. The number of flowing species is checked, the flow
variables are cleared to zero, and the flow cards are checked and read.
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229
The numerical identifiers for each species and the uncoupled species
reaction rates are cleared to zero. The initial concentrations of the
differential species are set to their input values, and the input errors
and initial maxima are set. The numerical species indentifiers for the
reactants and products of each reaction are cleared to zero.
The counters for the lumped reactions are set. The reactions are ana-
lyzed, and each species is identified; if it does not match a species list
name, a flag is set. Numerical identifiers are placed in the reaction and
species matrices to allow reference by reaction and location within the
reaction expression. If this is the first of a set of replacement reactions,
a message is printed. The order of the reactants is reversed, and the list
of chemical reactions and rate constants is printed. LMPCAL is called to
adjust the lumped reactions and species concentrations, and the list of
initial species concentrations, broken down by. type, is printed. If the
name flag is set, processing halts. A number of computation parameters are
initialize-ek Initial concentrations are saved, and the incoming concentra-
tions of any flowing species are initialized.
A call is made to the differential equation solution routine DIFSUB
(which in turn calls DIFFUN for solution of the steady-state equations);
note that this is -a return point from the calculation loop. The time values
are updated, and the concentration values are saved and checked for negative
values. If a negative value is found, the time step is reset to one-tenth
of its former value (note that this is done only once and that the time step
must be greater than the user-specified minimum value), the concentrations
from the previous time step are restored, and the call to DIFSUB is repeated.
The replacement species concentrations are calculated using the following
algorithm:
J / K
Y-''^ = Yi(o) exp '"' Z^\mxj " 'k,
K=2
-ATV RK,~|T Y.
(1 - QAT)
-------
230
where
Y.j(n\ = the new concentration of replacement species i,
Y.j/Q\ = the old concentration of replacement species i,
AT = the change in time since the previous calculation,
RK. = the rate constant of the j-th reaction of the set
J' of J contributing reactions that have species i as
the first reactant,
Yk = the concentration of the K-th reactant species in
the set of K reactants in contributing reaction j,
Q = the dilution rate.
LMPCAL is then called to adjust the lumped reactions and species concentrations
to reflect the changes in the replacement species concentrations.
The uncoupled species concentrations are calculated using the algorithm
described below under DIFFUN. Note that the uncoupled species reaction rates
are averaged with those derived during the previous time step. The current
uncoupled species reaction rates are saved.
The time is checked against having passed the user-provided limit, in
which case no plot points are saved. The incremental time since the last
printout is checked against the user-specified value. If appropriate, the
current concentration values are saved for plotting, unless the maximum num-
ber of plot points has been exceeded. The current concentrations are then
printed, regardless of whether the plot points have been saved. If the user
has indicated that the reaction rates are to be printed, they are sorted from
largest to smallest and are listed five per line.
If the time limit has been passed, if a repeated negative concentra-
tion has been encountered, or if an error has occurred in DIFSUB, the error
flag is printed and the PLOT routine is called. Control then passes back to
the beginning of the program, where another set of input cards can be processed
Otherwise, the inflowing species are checked to determine whether any concen-
trations should be updatea; if any updates are made, an appropriate message is
printed. Finally, the current time and concentrations are saved (in case it
is necessary to restart the calculation with a smaller time step), and control
returns to the call of DIFSUB for calculation of the next time step.
-------
231
b. LMPCAL
This subroutine is called by MODKIN to calculate the rate constants and
product species coefficients for the lumped reactions. Two arguments are
provided to this routine by MODKIN: the number of lumped reactions and the
number of contributing reactions for each lumped reaction. COMMON is de-
fined as in MODKIN, array sizes are set with DIMENSION statements, and param-
eters are defined via DATA statements.
The location and number of the current set of contributing reactions are
established, and the corresponding lumped reaction is identified. The rate
constants, product coefficients, and replacement species concentrations for
each contributing reaction are transferred to local arrays, and the replace-
ment species concentration is summed. The concentration of the lumped species
is set to the sum of all the replacement .concentrations, and the mole fraction
of each replacement species is calculated. The rate constant of the lumped
reaction is calculated as the sum of the rate constants for the contributing
reactions multiplied by the replacement species mole fraction. The product
species coefficients are calculated as the sum of the coefficients for the
product species multiplied by the replacement species mole fraction for each
contributing reaction, weighted by the ratio of the rate constant for the
contributing reaction to that of the lumped reaction. Note that as a final
step any product species coefficient below a minimum value is reset to zero
to avoid underflow problems in later computations.
c. DIFSUB
This subroutine, which is called by MODKIN, is a copy of the program for
the integration of coupled first-order ordinary differential equations that
was presented in the Collected Algorithms of the Association for Computing
Machinery (Gear, 1971). Since the algorithm and program are described in de-
tail in the cited reference, they are not discussed further here. The only
change likely to be needed is the alteration of the DIMENSION statement near
the beginning.
-------
232
d. DIFFUN
This subroutine is called by DIFSUB to calculate the rates of change of
the differential species. It also includes, however, the algorithm for the
steady~-state calculations, since the reactions involving species in a steady
state are presumed to be fast relative to the time steps in DIFSUB and must
therefore be updated at every time trial. The three arguments provided to
this subroutine by DIFSUB are the time, the differential species concentra-
tions, and the rates of change of these concentrations; these are all DOUBLE
PRECISION. The arguments are sized in a DIMENSION statement, COMMON is de-
fined as in MODKIN, and several parameters are defined via DATA statements.
The concentrations are transferred to a local array, and the convergence
loop is begun. The reaction rate for each reaction is calculated by using
the following algorithm:
R = RK -fT Y.
1 ] j=l J
where R. is the reaction rate and RK^ is the rate constant of the i-th reaction,
and Y. is the concentration of the'j-th reactant species in the set of J reac-
\j
tants in reaction i.
The steady-state concentrations are calculated by using the following
dynamic mass-balance algorithm:
I
R.C.-
i U
Y = '" ' k J. i
Yj M K K f J
n 4- V vv V v
Q mtl m ^ Yk
where
-------
233
Y.
J
R. and C . .
J
= the concentration of the j-th steady-state species,
= the reaction rate of the i-th reaction and the co-
efficient of species j in the i-th reaction, respec-
tively, of the set of I reactions in which species j
is a product,
= the dilution rate,
= the rate constant for the m-th reaction in the set of
M reactions in which species j is a reactant,
Y, = the concentration of the k-th reactant (except when
k = j) in the set of K reactants in reaction m.
Q
RK
Note that in the case of a species reacting with itself, Y. may be the same as
Yk even though k ^ j ; this case has been explicitly programmed.
If the old and new steady-state values agree within the requisite tolerance,
a convergence counter is incremented; in any event, the new value is saved. If
the value registered on the convergence counter equals the number of steady-
state species, the loop is completed; othei^vise, another pass is made. If the
steady-state concentrations do not converge, a warning message is written and
processing continues.
As a final step, the differentials are calculated according to the follow-
ing algorithm:
I
M
1J
AY = R.C., - R + Q(YF, - Y )
m=l
m
where
AY.
R. and C. . =
1 1J
•R- ••->••
m
the change in concentration of the j-th species
with time,
the reaction rate of the i-th reaction and the
coefficient of species j in the i-th reaction,
respectively, of the set of I reactions in which
species j is a product,
the reaction rate of the m-th reaction in the set
of M reactions in which species j is a reactant,
-------
234
Q = the dilution rate,
YF. = the inflowing concentration of species j,
J
Y. = the concentration of species j.
J
These differential values are returned to the calling program.
e. MATINV
This subroutine is called by DIFSUB to perform matrix inversions. Since
it is a standard matrix inversion routine taken from the utility subroutine
library at the California Institute of Technology, it is not described here.
The only change likely to be needed is the alteration of the DIMENSION state-
ment near the beginning.
f. PEDERV
This subroutine,-which is called by DIFSUB, is used to provide a Jacobian
matrix for the calculation of partial derivatives and is not necessary in this
application. Nevertheless, to preserve the integrity of the Gear routine and
to allow for possible future use of partial derivatives, we retained the rou-
tine in the program as a dummy subroutine. It does contain, however, a DIMENSION
statement; thus, any alteration of array sizes in DIFSUB should also be made in
PEDERV.
g. PLOT
This subroutine is called by MODKIN following completion of the time-
concentration calculations. It maps the results, along with any user input
data, onto a page-sized grid of concentration as a function of time cells for
as many species as the user wishes.
The routine begins by providing DIMENSION statements for some of the argu-
ments and the local arrays. The vertical axis and vertical label are estab-
lished via DATA statements, as are the I/O units, some symbols, and the maximum
array sizes.
-------
235
The grid is cleared to blanks, and the control card is read. If the
name is blank, control returns to MODKIN. If there are input data, the num-
ber is checked and the data are read. The normalization factors are calcu-
lated, and the numerical labels are placed on the axes. The species is
identified; if it is misspelled, the plot is skipped.
The data points, if any, are scaled to the grid; the points are checked;
and, ifithey are acceptable, the appropriate symbol is placed on the grid.
The same procedure is used for the calculated points. A page is skipped, and
the vertical labels and axis and the grid itself are printed. The horizontal
axis and labels and the figure caption are printed, and the routine returns
control to MODKIN.
4. LISTINGS AND SAMPLE REPORT AND OUTPUT
The following pages contain a complete listing of the computer program,
including the main routine MODKIN (Exhibit A-l) and the subroutines LMPCAL,
DIFSUB-r DLFFUN, MATINV, PEDERV, and PLOT (Exhibits A-2 through A-7). These
routines are all written in ASA FORTRAN and should be acceptable without
changes for any computer system that supports FORTRAN.
Following the program listings is an input deck describing a typical
kinetics mechanism (Exhibit A-8) and selected printout from the computer
run using this input deck (Exhibit A-9). We note that, except for the plot
(which uses an entire 11 x 15 inch computer printout page), the output is
contained on a standard 8^ x 11 inch page. With relatively minor changes in
the PLOT routine, the plot can also be reduced to this size.
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN
236
C MAIN PROGRAM *«•«•*** MODKIN
C
C THIS PROGRAM READS AND ANALYZES INPUT FOR MODULAR KINETICS PROGRAM.
C SETS INITIAL VALUES, CONTROLS ITERATION PRINTOUT, AND CALLS THE
C DIFFERENTIAL EQUATION SOLVING AND PLOT ROUTINES.
C
C WRITTEN BY D. C. WHITNEY FOR SYSTEMS APPLICATIONS, INC.
C ORIGINAL DATE 31 AUGUST 1973, LATEST MODIFICATION 25 OCTOBER 1973.
C THIS PROGRAM AND ALL SUBROUTINES (EXCEPT DIFSU8 AND MATINV) ARE THE
C PROPERTY OF AND COPYRIGHT BY SYSTEMS APPLICATIONS. INC.
C 950 NORTHGATE DRIVE; SAN RAFAEL, CALIFORNIA 94903.
C
C SYMBOL DESCRIPTIONS —
C
C COEFF NUMBER OF PARTICLES, ONE PER PRODUCT SPECIES PER REACTION
C DELT DIFFERENCE BETWEEN TWO CONSECUTIVE TIME STEPS
C EPS CONVERGENCE CRITERION, DOUBLE PRECISION, FOR DIFSUB
C EPSF CONVERGENCE CRITERION
C ERROR ESTIMATE OF ERROR IN CONCENTRATIONS, PPM, ONE PER SPECIES,
C DOUBLE PRECISION, FOR DIFSUB
C ESUM SUM OF THE EXPONENTIAL TERMS'FOR THE REPLACEMENT SPECIES
C ETERM TERM IN THE EXPONENTIAL CALCULATION OF THE REPLACEMENT SPEC.
C FLOW SPECIES INFLOWS, PPM/MIN, 10 PER SPECIES
c FTEST TEMPORARY FLOW TIME OP CONCENTRATION -FOR TESTING, MIN OR PPM
C FTIME TIMES AT WHICH INFLOW IS MEASURED, MIN, 10 PER SPECIES
C H NEXT STEP SIZE, MIN, DOUBLE PRECISION, FOR DIFSUB
C HMAX MAXIMUM TIME STEP, MIN, DOUBLE PRECIS-ION, FOR DIFSUB
C HMAXF MAXIMUM TIME STEP, MIN
C HMIN MINIMUM TIME STEP, MIN, DOUBLE PRECISION, FOR DIFSUB
C HMIN MINIMUM TIME STEP, MIN.
C HSTART INITIAL STEP SIZE FOR DIFSUB
C J DO-LOOP INDICES OR LOCAL POINTERS
C JBLANK A HOLLERITH WORD OF FOUR BLANK CHARACTERS
C JFLAG INDICATES NEGATIVE SPECIES FOUND IN DIFFERENTIAL CALCULATION
C JSTART INPUT FLAG, FOR DIFSUB
C K DO-LOOP INDICES OR LOCAL POINTERS
C KCOF COEFFICIENT POINTERS, ONE PER REACTION PRODUCT PER SPECIES.
C KFLAG PERFORMANCE FLAG FOR DIFSUB
C KLMP NUMBER OF CONTRIBUTING REACTIONS TO EACH LUMPED REACTION
C KLOC LOCAL POINTER TO CONTRIBUTING REACTION
C KPRD PRODUCT POINTERS, ONE PER PRODUCT SPECIES PER REACTION
C KRCT RFACTANT POINTERS, ONE PER REACTANT SPECIES PER REACTION
C KRXN REACTION POINTERS, ONE PER REACTION PER SPECIES
C L DO-LOOP INDICES OR LOCAL POINTERS
C LFLAG INDICATES INCORRECT SPECIES NAME IN REACTION -- STOPS JOB
C LRXN POINTER TO FIRST OF A SERIES OF REPLACEMENT REACTIONS
c M DO-LOOP INDICES OR LOCAL POINTERS
c MAXDER MAXIMUM ORDER FOR DERIVATIVES, FOR DIFSUB
c MAXDIF MAXIMUM NUMBER OF DIFFERENTIAL SPECIES
c MAXFLW MAXIMUM NUMBER OF FLOW TIMES
C MAXLMP MAXIMUM NUMBER OF LUMPED REACTIONS
c MAXPNT MAXIMUM NUMBER OF SAVED TIME AND CONCENTRATION POINTS
c MAXPRD MAXIMUM NUMBER OF PRODUCTS
C MAXPRT MAXIMUM NUMBER. OF ENTRIES ON PRINT LINE FOR RATES
ooooooio
00000020
00000030
00000040
00000050
00000060
00000070
OOOOOORO
00000090
00000100
00000110
00000120
00000130
00000140
00000150
00000160
00000170
000001RO
00000190
00000200
00000210
00000220
00000230
00000240
00000250
000002^0
00000270
00000200
00000290
00000300
00000310
00000320
00000330
00000340
00000350
00000360
00000370
000003*0
00000390
00000400
00000410
00000420
00000430
00000440
00000450
00000460
00000470
000004RO
00000490
00000500
00000510
00000520
00000530
00000540
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
237
MAXRCT
MAXREP
MAXPXN
MAXSPC
MF
MRXN
N
NAME
NDIF
NFLW
NIN
NINT
NLMP
NLOC
NMPD
NMRC
NOUT
NP
NPNT
NPRT
NR
NRAT
NREP
C NRESET
NRXN
NS
NSTS
C NTEST
C NTIM
C NTIT
C
C
C
C
C
C
NTOT
Nil
NUNC
PSAVF.
Q
R
C RATE
C
C
C
C
C
RK
RPRT
SAVE
SAVCON
SAVTIM
C T
C
C TCOUNT
C TEND
C TF
C TINCR
C TOL
UNCOLD
TOLD
Y
VAX
MAXIMUM NUMBER OF PEACTANTS
MAXIMUM NUMBER OF REPLACEMENT REACTIONS PER LUMPED REACTION
MAXIMUM NUMBER OF REACTIONS
MAXIMUM NUMBER OF SPECIES
MFTHOD INDICATOR* FOR DIFSUB
TOTAL NUMRER OF REACTIONS, INCLUDING REPLACEMENTS
DO-LOOP INDICES OR LOCAL POINTERS
SPECIES NAMES, ONE PER SPECIES
NUMBER OF DIFFERENTIAL SPECIES
NUMRER OF FLOWING SPECIES
THE FORTRAN INPUT UNIT (NORMALLY 5)
NUMBER OF INERT/CONSTANT SPECIES
NUMBER OF LUMPED REACTIONS
00000550
00000560
00000570
000005*0
00000590
00000600
00000610
00000620
00000630
00000640
00000650
00000660
00000670
C YCALC
C YIN
NUMBER OF CONTIPIJTING REACTIONS PERTAINING TO LUMPED REACTIONO00006P0
PRODUCT N6MES, ONE PER PRODUCT SPECIES PER REACTION 00000690
REACTANT NAMES* ONE PEP REACTANT SPECIES PER REACTION 00000700
THE FORTRAN OUTPUT UNIT NUMBER (NORMALLY 6) 00000710
LOCAL POINTER TO REACTION RATE TO BE PRINTED 00000720
NUMBER OF SAVED TIMES AND CONCENTRATIONS 00000730
HOLDING AREA TO PRINT OUT A LINE OF NAMES OR NUMBERS 00000740
POINTER TO REPLACEMENT SPECIES 00000750
USER INPUT FLAG REQUESTING PRINT OF REACTION RATES 00000760
NUMRER OF REPLACEMENT SPECIES 0.0000770
COUNTER FOR NUMBER OF TIMES STEP,SIZE IS RESET SMALLER 00000780
NUMBER OF REACTIONS 00000790
POINTER TO REACTING SPECIES 00000800
NUMBER OF STEADY-STATE SPECIES OOOOOR10
SPECIES NAME FOR TESTING 00000820
NUMBER OF TIMES AND FLOWS FOR A SPECIES 00000030
USER-INPUT TITLE FOR PRINTOUT, 3 FOUR-CHARACTER WORDS ooooo«4o
TOTAL NUMBER OF SPECIES 00000850
LOCAL POINTER TO UNCOUPLED SPECIES 00000860.
NUMBER OF UNCOUPLED SPECIES 00000870
BLOCK STORAGE, NUMBER OF SPECIES SQUARED, FOR DIFSUB 000008RO
DEGRADATION RATE, /MIN 00000890
REACTION RATES, SEC, ONE PER REACTION 00000900
LOCAL REPRESENTATION OF R, THE REACTION RATE 00000910
REACTION RATE CONSTANTS, PPM-MIN, ONE PER REACTION 00000920
HOLDING AREA TO PRINT OUT A LINE OF REACTION PATES 00000930
BLOCK STORAGE, 12 PER SPECIES, DOUBLE PRECISION, FOR DIFSUB 00000940
SPECIES CONCENTRATIONS, PPM, ONE PER SPECIES AT 80 TIMES 00000950
TIMES THAT CONCENTRATIONS ARE SAVED, MIN, UP TO 80 VALUES 00000960
CURRENT REACTION TIME, SEC, DOUBLE PRECISION, 00000970
FOP AND FROM DIFSUB 000009RQ
NEXT TIME FOR OUTPUT, MIN 00000990
ENDING TIME, MIN 00001000
PREVIOUS TIME OF DIFSUB CALL, MIN 00001010
TIME INCREMENT FOR OUTPUT, MIN 00001020
CONVERGENCE TOLERANCE ON STEADY-STATE ITERATION, PPM 00001030
PREVIOUS VALUES OF RATE OF CHANGE, PPM/MIN, ONE PER SPECIES 00001040
TIME OF PREVIOUS CALL TO DIFSUB 00001050
SPECIES CONCENTRATIONS, 8 PER SPECIES, DOUBLE PRECISION, 00001060
FOR AND FROM DIFSUB 00001070
SPECIES CONCENTRATIONS, PPM, ONE PER SPECIES 00001080
LOCAL REPRESENTATION OF YDOT, THE RATE OF CHANGE 00001090
SPECIES INFLOW RATES, PPM/MIN, ONE PER SPECIES 00001100
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Continued)
238
C YMAX CURRENT MAXIMUM CONCENTRATION VALUES, PPM, ONE-PER SPECIES, 00001110
C DOUBLE PRECISION, FOR AND FROM DIFSUB 00001120
C YOLD CONCENTRATIONS AT PREVIOUS CALL TO DIFSUB, ONE PER SPECIES 00001130
c YUNC RATE OF CHANGE OF UNCOUPLED SPECIES, PPM/MIN 00001140
c ooooiiso
C BEGINNING OF PROGRAM. 00001160
C 00001170
C DECLARE VARIABLES FOR DIFSUB AS DOUBLE PRECISION 000011«0
C 00001190
DOUBLE PRECISION HMIN» HMAX, EPS, YMAX, ERROR, H, SAVE, T, Y 00001200
C . 00001210
C DEFINE VARIABLES AND DIMENSIONS OF COMMON STORAGE WITH DIFFUN 00001220
C 00001230
COMMON RK(99), R(99), YAX(50), YIN(50), COEFF(3999) 00001240
COMMON KPCT(4,Q9)-i KPPD(3,99), KRXN(99,50), KCOF(99,50) 00001250
COMMON Q, TOL, NRXN, NDIF, NSTS 00001260
C 00001270
C DEFINE DIMENSIONS OF LOCAL ARRAYS 000012*0
C . 000012^0
DIMENSION Y(8,40), YMAX(40), SAVE(12,40), ERROR(40), PSAVE(1600) 00001300
DIMENSION NMRC(4j99), NMPD(3,99)» NTIT(3), NPRT(IO), RPRT(IO) 00001310
DIMENSION FTIME (10,50) » FLOW(10»50)» SAVCON(50,80) , SAVTIM(80) 00001320
DIMENSION YOLDC50), NAME(50), UNCOLD(50), KLMP(IO) 00001330
C 0'0001340
C SET MISCELLANEOUS DATA VALUES 00001350
C 00001360
DATA MAXRCT /4/? NIN /5/, NOUT /&/, MAXPRD /3/, JBLANK MH / 00001370
DATA PTAXRXN /99/« MAXDIF /40/. MAXFLW /10/, MAXPNT /80/ 000013/30
DATA MAXSPC /BO/, MAXLMP /10/, MAXREP /10/, MAXPRT /5/ 00001390
C 00001400
C READ CONTROL CARDS — NOTE THIS IS RETURN POINT FROM PLOT CALL 00001410
C 00001420
10 READ (NIN,2,ENO=900) NTIT, NRXN, NLMP, NDIF, NSTS, NUNC, NREP, 00001430
fi, NINT, NFLW, NRAT 00001440
READ (NIN,4) TINCR, TEND, HSTART, HMINF, HMAXF, EPSF, Q 00001450
C ' • 00001460
C PRINT HEADING PAGE AND CONTROL CARD INPUTS 00001470
C 00001480
WRITE (NOUT51002) NTIT, NRXN, NLMP, NDIF, NSTS, NUNC, NREP, NINT, 00001490
& NFLW, NRAT, TINCR, TEND, HSTART, HMINF, HMAXF, EPSF, Q 00001500
C 00001510
C TEST REACTION COUNT 00001520
C • 00001530
IF (NRXN .GT. 0 .AND. NRXN .LE. MAXRXN) GO TO 12 00001540
WRITE (NOUT,1001) MAXRXN 00001550
GO TO 900 00001560
C 00001570
C SET SPECIES NAME FLAG AND OVERALL REACTION COUNT AND READ REACTIONS 000015RO
C 00001590
12 LFLAG = 0 00001600
MRXN = NRXN 00001610
DO 15 K = ItNRXN 00001620
READ (NIN,1) (NMRC(J,K), J = 1,MAXRCT), 00001630
& (COEFF(J,K)» NMPD(J,K), J = 1,MAXPRD), RK(K) 00001640
15 CONTINUE 00001650
C . 00001660
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM M.ODKIN (Continued)
239
TEST NUMBER OF LUMPED REACTIONS
IF (NLMP .LE. 0) GO TO 22
IF (NLMP .LE. MAXLMP) 60 TO 16
WRITE (NOUT*1027) MAXLMP
GO TO 900
READ AND TEST LUMPED SPECIES NAME AND NUMBER OF REPLACEMENT SPECIES
16 DO 21 L = 1tNLMP
READ (NIN,6) NTEST, NLOC
IF (NTEST .EO. NMRCd.NRXN - NLMP + L) ) GO TO 17
WRITE (NOUT;1028) NTEST
LFLAG = 1
17 IF (NLOC .GT. 0 .AND. NLOC .LE. MAXREP) GO TO 18
WRITE (NOUT»1030> MAXREP
GO TO 900
SAVE NUMBER OF REPLACEMENT REACTIONS AND UPDATE AND CHECK TOTAL
18 KLMP(L) = NLOC
MRXN = MRXN + NLOC
IF (MRXN .LE. MAXRXN) GO TO 19
WRITE (NOUT;100l) MAXRXN
GO TO 900
READ REPLACEMENT REACTIONS
19 E>0- 2-9-- K- = 1*NLOC
KLOC = K * MRXN - NLOC
READ (NIN»1) (NMRC(J,KLOC)* J = 1»MAXRCT), (COEFF(J9KLOC)*
Si NMPD(J»KLOC) t J = l.MAXPRDJ* RK(KLOC)
20 CONTINUE
21 CONTINUE
TEST NUMBER OF DIFFERENTIAL SPECIES
22 IF (NDIF .LE. MAXDIF) GO TO 23
WRITE (NOUT»1012) MAXDIF
GO TO 900
SET AND TEST TOTAL NUMBER OF SPECIES
23 NTOT = NDIF + NSTS + NUNC + NREP + NINT
IF (NTOT .LE. MAXSPC) GO TO 25
WRITE (NOUT.1011) MAXSPC
GO TO 900
IF (NFLW .LE. 0) GO TO 50
IF (NFLW .LE. NTOT) GO TO 30
00001670
00001680
00001690
00001700
00001710
00001720
00001730
00001740
00001750
00001760
00001770
00001780
000017QQ
00001800
00001810
00001820
00001830
00001840
00001850
000018^0
00001870
00001880
0-0001890
00001900
00001910
00001920
00001930
00001940
00001950
00001960
00001970
00001980
00001990
00002000
00002010
00002020
00002030
00002040
00002050
00002060
00002070
000020PO
00002090
00002100
00002110
00002120
00002130
00002210
00002220
-------
240
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
WRITE (NOUT, 1023) NTOT 00002230
GO TO 900 00002240
C 00002250
C SET FLOW PATES AND TIMES TO ZERO 000022f>0
IF (NFLW .L.F.. 0) GO TO 50 00002210
IF (NFLW .LE. NTOT) GO TO 30 00002220
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
241
WRITF (NOUT, 1023) NTOT
GO TO 900
C
C SET FLOW RATES AND TIMES TO ZERO
C
30 DO 33 K = 1»NTOT
DO 32 J = 1,MAXFLW
FTIME(J,K) = 0.0
FLOW(J.K) = 0.0
32 CONTINUE
33 CONTINUE
C
C READ FLOW CONTROL CARD
C
DO 45 K = l.NFLW
READ (NIN,6) NTEST, NTIM
C
C IDENTIFY SPECIES NAME — EXIT IF NOT FOUND
C
DO 35 L = 1»NTOT
IF (NTEST .EG. NAME(D) GO TO 40
35 CONTINUE
WRITE (NOUT;1003> NTEST
GO TO 900
C
C CHECK NUMRER OF FLOW INPUTS
C
40 IF (NTIM .LE. MAXFLW) GO TO 42
WRITE (NOUT»1024) MAXFLW
GO TO 900
C
C READ FLOW INPUTS
C
42 READ (NIN,7) (FTIME(J»L)» FLOW(J,L), J = 1»NTIM)
45 CONTINUE
C
C CLEAR REACTION POINTERS, FLOWS, AND UNCOUPLED RATES
C
50 DO 60 K = 1,NTOT
DO 55 J = 1,MRXN
KRXN(J,K) = 0
KCOF(J,K) = 0
55 CONTINUE
YIN(K) = 0.0
UNCOLD(K) = 0.0
60 CONTINUE
C
C MOVE INITIAL DIFFERENTIAL CONCENTRATIONS AND SET ERRORS AND MAXIMA
C
DO 65 J = 1,NDIF
Y(l.J) = YAX(J)
ERROR(J) = 0.0
YMAX(J) = 1.0
65 CONTINUE
C
C CLEAR SPECIES POINTERS
00002230
00002240
00002250
00002260
00002270
00002280
000022«0
00002300
00002310
00002320
00002330
00002340
00002350
00002360
00002370
00002380
00002390
00002400
00002410
00002420
00002430
00002440
00002450
00002460
00002470
000024RO
00002490
00002500
00002510
00002520
00002530
00002540
00002550
00002560
00002570
00002580
00002590
00002600
00002610
00002620
00002630
00002640
00002650
00002660
00002670
00002680
00002690
00002700
00002710
00002720
00002730
00002740
00002750
00002760
00002770
00002780
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
242
70
80
90
C
c
C
c
c
c
DO 90 K = 1
DO 70 J = 1
KRCT(J,K) =
CONTINUE
DO 80 J = 1
KPRD(J,K) =
CONTINUE
CONTINUE
»MRXN
.MAXRCT
0
.MAXPRD
0
SET LUMPED REACTION COUNTERS
LRXN = NRXN
KLOC = 1
DO 190 M =
DO 130 L =
«• 1
1»MRXN
1»MAXRCT
IDENTIFY RFACTANT SPECIES --
FLAG IF MISSING
NTEST = NMRC(L.M)
IF (NTEST .EQ. JBLANK) GO TO 140
DO 110 K = 1»NTOT
IF (NTEST .EQ. NAME(K)) GO TO 115
110 CONTINUE
WRITE (NOUT»1003) NTEST
LFLAG = 1
GO TO 130
C
C FILL IN REACTION AND SPECIES POINTERS AND COUNTERS
C
115
KRCT(L,M) = K
DO 120 J = 1,MAXRXN
IF (KRXN(J,K) .EQ. 0)
120 CONTINUE
1P5 KRXN(J,K) = M
KCOF(J.K) = -1
CONTINUE
GO TO 125
PRODUCT SPECIES — FLAG IF MISSING
130
C
C IDENTIFY
C
140 DO 170 L = 1»MAXPRD
NTEST = NMPDCL.M)
IF (NTEST .EQ. JBLANK) GO TO 180
DO 150 K = 1«NTOT
IF (NTEST .EQ. NAME(K)) GO TO 155
150 CONTINUE
WRITE (NOUT.1003) NTEST
LFLAG = 1
GO TO 170
C
C FILL IN REACTION AND SPECIES POINTERS AND COUNTERS
C
155
160
KPRD(L,M) = K
DO 160 J = 1»MAXRXN
IF (KRXN(J,K) .EQ. 0)
CONTINUE
GO TO 165
00002790
00002800
00002810
00002820
00002830
00002S40
00002350
00002860
00002870
000028«0
00002890
00002900
00002910
00002920
00002930
00002940
00002950
00002960
00002970
00002980
00002990
00003000
00003010
00003020
00003030
00003040
00003050
00003060
00003070
00003080
00003090
00003100
00003110
00003120
00003130
00003140
00003150
00003160
00003170
00003180
00003190
00003200
00003210
00003220
00003230
00003240
00003250
000032^0
00003270
000032HO
00003290
00003300
00003310
00003320
00003330
00003340
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Continued)
243
165 KRXN(J,K) = M 00003350
KCOF(J,K) = L 00003360
170 CONTINUE 00003370
C 000033RO
C SAVE NUMBER OF PRODUCT SPECIES FOR THIS REACTION 00003390
C 00003400
L = MAXPRD + 1 00003410
180 NPRO = L - 1 • 00003420
C 00003430
C IF REPLACEMENT FOR LUMPED REACTION, PRINT. MESSAGE AND UPDATE POINTERS 00003440
C 00003450
IF (M .NE. LRXN) GO TO 183 00003460
N = NRXN - NLMP + KLOC 00003470
WRITE (NOUT<10?9) KLMP(KLOC). N 00003480
LRXN = LRXN + KLMP(KLOC) 00003490
KLOC = KLOC «• 1 00003500
C 00003510
C REVERSE ORDER OF REACTANTS FOR PRINTING 00003520
C 00003530
183 DO 185 J = 1»MAXRCT 00003540
K = MAXRCT - J * 1 00003550
NPRT(J) = NMRC(K>M) 00003560
185; CONTINUE 0.0003570
C 00003580
C PRINT SET OF REACTIONS 00003590
C 00003600
WRITE (NOUT«1004> M» RK(M), (NPRT(J), J=1*MAXRCT), 00003610
&. (COEFF(J»M)» NMPD(JsM)» J = 1,NPRD) 00003620
190 CONTINUE 00003630
C 00003640
C GET INITIAL CONDITIONS FOR LUMPED REACTIONS AND SPECIES 00003650
C 00003660
IF (NLMP ,GT. 0) CALL LMPCALtNLMP. KLMP) 00003670
C 00003680
C PRINT INITIAL SPECIES CONCENTRATIONS — EXIT IF FLAG SET 00003690
C 00003700
. WRITE (NOUT;1013) 00003710
WRITE (NOUT»1005) 00003720
WRITE (NOUT.1006) (NAME(J)« YAX(J)» J=1»NDIF) 00003730
IF (NSTS .GT. 0) 00003740
&WRITE (NOUT»1025) (NAME(J+NDIF)» VAX(J+NDIF)» J=1*NSTS) 00003750
IF (NUNC .GT. 0) 00003760
&.WRITE (NOUT»1007) (NAME (J + NDIF+NSTS) » 00003770
5. YAXCJ + NDIF + NSTS) , J = 1»NUNC) 000037RO
IF (NRF.P .GT. 0) 00003790
&WRITE (NOUT?103l) (NAME(J+NDIF+NSTS+NUNC)9 00003800
81 VAX (J + NDIF + NSTS + NUNC) » J = 1»NREP) 00003810
IF (NINT .GT. 0) 00003820
&WRITE (NOUT;1008) (NAME(J+NDIF+NSTS+NUNC+NREP)t 00003830
81 YAXU + NDIF + NSTS + NUNC + NREP) » J = 1»NINT) 00003840
IF (LFLAG .EQ.-l) GO TO 900 00003850
C 00003860
C SET INITIAL CONDITIONS ' 00003870
C 00003880
HMAX = HMAXF 00003890
HMIN = HMINF 00003900
-------
.EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Continued)
244
TCOUNT = TINCR
EPS = FPSF
TOL = EPSF
H = HSTART
.MF = 2
JSTART = 0
NPNT = 0
TF = 0.0
T = 0.0
TOLD = 0.0
MAXDER = 6
JFLAG = 0
KFLAG = 1
NRESET = 0
SAVE INITIAL CONCENTRATIONS AND FLOWS
DO 195 J = 1»MTOT
YOLD(J) = YAX(J)
IF (NFLW .EQ. 0) GO TO 195
IF (FTIME(I.J) .GT. 0.0) GO TO 195
YIN(J) = FLOW(1»J)
195 CONTINUE
CALL DIFFERENTIAL SPECIES SOLVER — NOTE THIS IS A RETURN POINT
200 CALL DIFSUBtNDIF, J, Y» SAVE* H? HMIN, HMAX, EPS9 MF»
& YMAX, ERROR? KFLAG, JSTART* MAXDER, PSAVE)
UPDATE TIME AND SAVE CONCENTRATIONS, CHECKING FOR NEGATIVITY
TF = T
DELT = TF - TOLD
DO 210 J = 1,NTOT
IF (J .LE. NDIF) YAX(J) = YU,J)
IF (YAX(J) .LT. 0.0) JFLAG = 1
210 CONTINUE
IF (JFLAG .EQ. 0) GO TO 230
JFLAG = 0
NEGATIVE CONCENTRATION — RESET AND TEST STEP SIZE
H = 0.1 * H
IF (H .LT. HMIN) KFLAG = 0
TEST RESET COUNTER FOR RE-ENTRY, THEN SET TO PREVENT SAME—TEST FLAG
IF (NRFSET .GT. 0) KFLAG = 0
NRESET = 1
IF (KFLAG .NE. 1) GO TO 330
RESTORE OLD CONCENTRATIONS AND RECALL DIFSUB WITH SMALLER STEP SIZE
DO 220 J = 1,NTOT
YAX(J) = YOLD(J)
220 CONTINUE
00003910
00003920
00003930
00003940
00003950
00003960
00003970
00003980
00003990
00004000
00004010
00004020
00004030
00004040
00004050
000040*0
00004070
00004080
00004090
00004100
00004110
00004120
00004130
00004140
00004150
00004160
00004170
000041BO
00004190
00004200
00004210
00004220
00004230
00004240
00004250
00004260
00004270
00004280
00004290
00004300
00004310
00004320
00004330
00004340
00004350
00004360
00004370
00004380
00004390
00004400
00004410
00004420
00004430
00004440
00004450
00004460
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Continued)
245
JSTART = -1
GO TO ?00
TEST FOR AMD SET POINTERS TO REPLACEMENT SPECIES
230 IF (NREP .LE. 0) GO TO 265
LRXN = NRXN + ]
DO 260 M = 1,NREP
NR = NDIF + NSTS «• NUNC + M
ESUM = 0.0
FIND REACTIONS CONTAINING REPLACEMENT SPECIES
DO 250 L = LRXN» MRXN
IF (KRCT(1»L) .NE. NR) GO TO 250
MULTIPLY TOGETHER ALL OTHER REACTANT CONCENTRATIONS
ETERM = 1.0
DO 240 K = 2»MAXRCT
MS = KRCT(K»L)
IF (NS .EG). 0) GO TO 245
ETERM = ETERM «• YAX(NS)
240 CONTINUE
MULTIPLY BY RATE CONSTANT AND ADD TO EXPONENTIAL SUM
245 ESUM = ESUM + RK(L) * ETERM
250 CONTINUE
CALCULATE NEW SPECIES CONCENTRATIONS
YAX(NR) = YAX(NR) * EXP(-ESUM * DELT)
260 CONTINUE
(1.0 - DELT
Q)
UPDATE LUMPED REACTIONS AND SPECIES PARAMETERS
CALL LMPCALfNLMP* KLMP)
TEST FOR AND SET POINTERS TO UNCOUPLED SPECIES
265 IF (NUNC .LE,, 0) GO TO 300
DO 290 M = 1»NUNC
, NU = NDIF + NSTS «• M
YUNC = 0.0
DO 270 L = ItNRXN
J = KCOF(L.NU)
K = KRXN(L»NU)
CALCULATE THE RATE OF CHANGE OF THE UNCOUPLED SPECIES
IF (J) 267. 280» 268
267 YUNC = YUNC - P(K)
GO TO 270
268 YUNC = YUNC + R(K) * COEFF(J.K)
270 CONTINUE
00004470
000044RQ
00004490
00004500
00004510
00004520
00004530
00004540
00004550
00004560
00004570
00004580
00004590
00004600
00004610
00004620
00004630
00004640
00004650
00004660
00004670
00004680
Q0004690
00004700
00004710
00004720
00004730
00004740
00004750
00004760
00004770
00004780
00004790
00004800
00004810
00004820
00004830
00004840
00004850
00004860
00004870
00004880
00004890
00004900
00004910
00004920
00004930
00004940
00004950
00004960
00004970
00004980
00004990
00005000
00005010
00005020
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
246
280 YUNC = YUNC + 0 * (YIN(NU) - YAX(NU))
CALCULATE UNCOUPLFD SPECIES CONCENTRATION AND UPDATE OLD RATE
YAX(NU) = YAX(NU) * (YUNC «• UNCOLD(NU)) * DELT » 0.5
UNCOLD(NU) = YUNC
290 CONTINUE
CHECK TIME FOR END AND PRINTING AND SAVING OF PLOT POINTS
300 IF (TF .GT. TEND) GO TO 330
IF (TF .LE. TCOUNT) GO TO 340
INCREMENT TIME AND PLOT POINT COUNTERS
TCOUNT = TCOUNT + TINCR
NPNT = NPNT + 1
CHECK PLOT POINT COUNTER FOR OVERFLOW
IF (NPNT .LE. MAXPNT) GO TO 310
WRITE (NOUT?1019) MAXPNT? TF
NPNT = NPNT - 1
GO TO 330
SAVE PLOT POINTS
310 SAVTIM(NPNT) = TF
DO 320 J = 1»NTOT
SAVCON(J,NPNT) = YAX(J)
3?0 CONTINUE
PRINT INTERMEDIATE RESULTS
330 WRITE (NOUT»1009) TF
WRITE (NOUT-1005)
WRITE (NOUT-1006) (NAME(J)» YAX(J)» J = 1»NDIF)
IF (NSTS .GT. 0)
8.WRITE (NOUT'1025) (NAME (J+NDIF) « YAX(J+NDIF)» J=1»NSTS)
IF (NUNC .GT. 0)
&WRITE (NOUT;1007) (NAME(J+NDIF+NSTS)«
& YAXU + NDIF + NSTS) » J = 1»NUNC>
IF (NREP .GT. 0)
S.WRITE (NOUT»1031) (NAME ( J^NDIF + NSTS^NUNC) ,
& YAX(J+NDIF+NSTS+NUNC), J = 1,NREP)
CHECK RATE PRINT FLAG* PRINT HEADER? AND SET SORT PARAMETERS
IF (NRAT .EQ. 0) GO TO 339
WRITE (NOUT;1032)
N = 0
RATE = -1.0
NP = 0
DO 338 M = 1»NRXN
FIND LARGEST RATE
000050 3D
00005040
'00005050
00005060
00005070
00005080
00005090
00005100
000051 10
000051PO
00005130
0000514-0
00005150
00005160
00005170
00005180
00005190
00005200
00005210
00005220
00005230
00005240
00005250
00005260
00005270
000052^0
00005290
00005300
00005310
00005320
00005330
00005340
00005350
00005360
00005370
00005380
00005390
00005400
00005410
00005420
00005430
00005440
00005450
00005460
00005470
000054BO
00005490
00005500
00005510
00005520
00005530
00005540
00005550
00005560
00005570
00005580
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Continued)
247
DO 336 L = 1»NRXN
IF (P(L> .LT. RATE) GO TO 336
RATE = R(L)
NP = L
336 CONTINUE
SAVE AND FLAG THIS RATE AND RESET SORT PARAMETERS
N = N * 1
NPRT(N) = NP
RPRT(N) = RATE
R(NP) = -1.0
RATE = -1.0
NP = 0
CHECK PRINT COUNT AND PRINT IF FULL LINE OR END OF LOOP
337 IF (N. LT. MAXPRT .AND. M .NE. NRXN) GO TO 338
WRITE (NOUT?1033) (NPRT(K), RPRT(K)? K = 1,N)
N = 0
338 CONTINUE
CHECK FOR ERROR OR FINAL TIME PASSED? IF SO PLOT AND GET NEXT SET
339 I-F C-T-F .LT. TEND .AND. KFLAG .EG. 1) GO TO 340
WRITE (NOUT.1010) KFLAG
CALL PLOTJNTIT, NPNT, NTOT, NAME* SAVTIM, SAVCON)
GO TO 10
CHECK FOR INFLOW UPDATES
340 IF (NFLW .LE. 0) GO TO 380
DO 370 K = 1»NTOT
DO 350 J = 1*MAXFLW
IF (J ,EO. MAXFLW) GO TO 360
FTEST = FTIME(J+1»K)
IF (FTEST .GT. TF) GO TO 360
IF (FTEST .LE. 0.) GO TO 360
350 CONTINUE
UPDATE INFLOWS AND WRITE MESSAGE
360 FTEST = FLOW(J.K)
IF (YIM(K) .EQ. FTEST) GO TO 370
YIN(K) = FTEST
WRITE (NOUT»10?6) NAME(K)* FTEST* TF
370 CONTINUE
UPDATE TIME AND CONCENTRATION AND TAKE NEXT TIME STEP
380 TOLD = TF
NRESET = 0
DO 390 J = l.NTOT
YOLD(J) = YAX(J)
390 CONTINUE
00005590
00005600
00005610
00005620
00005630
00005640
00005650
00005660
00005670
00005680
000056QO
00005700
00005710
00005720
00005730
00005740
00005750
00005760
00005770
00005780
00005790
00005800
00005810
00005820
00005830
00005840
00005850
00005860
00005870
00005880
00005890
00005900
00005910
00005920
00005930
00005940
00005950
00005960
00005970
00005980
000059QO
00006000
000060]0
00006020
00006030
00006040
00006050
00006060
00006070
00006080
00006090
00006100
00006110
00006120
00006130
00006140
-------
EXHIBIT A-1. LISTING OF MAIN PROGRAM MODKIN (Continued)
248
A4), F10.0)
GO TO 200
C
C END OF PROGRAM
C
900 STOP
C
C LIST OF FORMAT STATEMENTS
C
1 FORMAT (4(A4, IX), 3(F6.0,
2 FORMAT (344, 3X, 915)
3 FORMAT (A4, 6X, F10.0)
4 FORMAT (8F10.0)
6 FORMAT (A4, IX, 15)
.7 FORMAT (8F10.0)
1001 FORMAT (33H PROGRAM CANNOT HANDLE MORE
& 26H REACTIONS — JOB ABORTED.)
1002 FORMAT (1H1, 20X, 25HMODULAR KINETICS
29H TOTAL NUMBER OF REACTIONS = ,
NUMBER OF LUMPED REACTIONS = , I
DIFFERENTIAL SPECIES =
STEADY STATE SPECIES =
THAN
14
8,
8,
6,
8,
&
&
&
8.
&
&
RUN
13,
30H
34H
34H
31H
33H
3QH
29H
36H
18H
15H
22H
21H
21H
25H
17H
NUMBER
NUMBER
NUMBER
NUMBER
NUMBER
NUMBER
OF
OF
OF
OF
OF
OF
NO- , 3A4, ////,
//,
•> //<,
. 13, //,
, 13, //,
UNCOUPLED SPECIES = , 13, //,
REPLACEMENT SPECIES = , 13, //•
INERT OP CONSTANT
FLOWING SPECIES =
REACTION RATE PRINT REQUEST
TIME INCREMENT = , 1PE12.3,
ENDING TIME = , 1PE12.3, 9H
STARTING STEP SIZE =
MINIMUM STEP SIZE =
SPECIES =
, 13, //,
13, //
FLAG = . 13, //,
9H MINUTES , //,
MINUTES , //,
1PE12.3, 9H MINUTES
1PE12.3, 9H MINUTES
, //•
//,
//,
MAXIMUM STEP SIZE = , 1PE12.3* 9H MINUTES
CONVERGENCE TOLERANCE = , 1PE12.3, //,
DILUTION RATE = , 1PE12.3, 12H MINUTES(-l), //,
& 1H1, 26X, 18H LIST OF RE ACTION'S, //•
81 14H R. CONST., 8X, 9HREACTANTS, 12X, 8HPRODUCTS,
1003 FORMAT (14H SPECIES NAME , A4, 21H NOT IN SPECIES LIST,
8. 21H JOB WILL BE ABORTED.)
1004 FORMAT (13, 1PF11.3, 4(1X, A4), 2H =,
8, 3(OPF6.2, IX, A4M
1005
1006
1007
1008
1009
1010
1011
81
FORMAT d
FORMAT (/
FORMAT (/
FORMAT (/
FORMAT (/
FORMAT (/
FORMAT (/
24H
', 4(18H SPECIES VALUE ))
, 18H DIFFERENTIAL(PPM) , //. (4(3X, A4, 1PE11.3)))
, 15H UNCOUPLED(PPM) , //, (4(3X, A4, 1PE11.3)))
, 20H INERT/CONSTANT (PPM) , //, (4(3X, A4, 1PE11.3)))
/. 20X? 8H TIME = , 1PE12.3, 8H MINUTES, /)
, 33H THIS RUN TERMINATED WITH KFLAG = , 13)
/, 33H PROGRAM CANNOT HANDLE MORE THAN , 14,
SPECIES -- JOB ABORTED.)
14
1012 FORMAT (//, 33H PROGRAM CANNOT HANDLE MORE THAN
8. 30H DIFFERENTIALS — JOB ABORTED.)
1013 FORMAT (/, 1H1, 20X, 30HINITIAL SPECIES CONCENTRATIONS
1019 FORMAT (/, 31H MAXIMUM NUMBER OF PLOT POINTS , 13,
R- 19H HAS BEEN EXCEEDED., /. 15H POINT AT TIME , F8.2,
& 21H WILL NOT BE PLOTTED.)
1023 FORMAT (//, 33H PROGRAM CANNOT HANDLE MORE THAN , 14,
8, 22H FLOWS — JOB ABORTED.)
1024 FORMAT (//, 33H PROGRAM CANNOT HANDLE MORE THAN , 14,
/)
00006150
00006160
00006170
00006lflO
00006190
00006200
00006210
00006220
00006230
00006240
00006250
00006260
00006270
00006230
00006290
00006300
00006310
00006320
00006330
00006340
00006350
00006360
00006370
00006380
00006390
00006400
00006410
00006420
00006430
00006440
00006450
00006460
00006470
000064RO
00006490
00006500
00006510
00006520
00006530
00006540
00006550
00006560
00006570
00006580
00006590
00006600
00006610
00006620
00006630
00006640
00006650
00006660
00006670
00006680
00006690
00006700
-------
EXHIBIT A-l. LISTING OF MAIN PROGRAM MODKIN (Concluded)
249
J. ?7H FLOW TIMES — JOB ABORTED.) 0000671.0
1025 FORMAT {/, IflH STEADY STATE(PPM), //, (4(3X, A4? 1PE11.3))) 00006720
1026 FORMAT (/, 10H INCOMING i A4» 26H CONCENTRATION CHANGED TO , 00006730
& 1PE11.3, 4H AT « 1PE11.3, 5H MIN.) 00006740
1027 FORMAT (//, 33H PROGRAM CANNOT HANDLE MORE THAN t I4» 00006750
& 33H LUMPED REACTIONS -- JOB ABORTED.) 00006760
1028 FORMAT (16H LUMPED SPECIES , A4- 24H IS NOT FIRST SPECIES IN, 00006770
f. 54H CORRFSPONDING LUMPED REACTION — JOB WILL BE ABORTED.) 00006780
1029 FORMAT (/. 21H THE FOLLOWING SET OF , I3» 00006790
f, 4?H REACTIONS CORRESPONDS TO REACTION NUMBER , 13, /) 00006800
1030 FORMAT (//» 33H PROGRAM CANNOT HANDLE MORE THAN 9 14* 00006810
8, 39H CONTRIBUTING REACTIONS — JOB ABORTED.) 00006«?0
1031 FORMAT (/, 17H REPLACEMENT(PPM)» //. (4(3X» A4» 1PE11.3))) 00006830
1032 FORMAT (/. 15X, 44HREACTION RATES (SORTED INTO DECREASING SIZE)* 00006640
&, //» 5(15H NO. RATE )» /) 00006850
1033 FORMAT (5(15, 1PE10.2)) 00006860
END 00006870
-------
EXHIBIT A-2. LISTING OF SUBROUTINE LMPCAL
250
C SUBROUTINE ****** LMPCAL «****« 00000010
C 00000020
C THIS SUBROUTINE CALCULATES THE CONCENTRATIONS OF LUMPED SPECIES AND 00000030
C THE COEFFICIENTS AND RATE CONSTANTS FOR THE CORRESPONDING REACTIONS 00000040
C 00000050
C SYMBOL DEFINITIONS — • 00000060
C 00000070
C ALPHA LOCAL VALUES OF COEFF, THE PRODUCT COEFFICIENTS 000000*0
C COEFF NUMBER OF PARTICLES* ONE PER PRODUCT SPECIES PER REACTION 00000090
C COLOC LOCAL VALUE OF PRODUCT COEFFCIENT 00000100
c COMIN MINIMUM ALLOWABLE COEFFICIENT SIZE 00000110
C J DO-LOOP INDICES OR LOCAL POINTERS 000001PO
C K DO-LOOP INDICES OR LOCAL POINTERS 00000130
C KCOF COEFFICIENT POINTERS, ONE PER REACTION PRODUCT PER SPECIES 00000140
C KLMP NUMBER OF CONTRIBUTING REACTIONS TO EACH LUMPED REACTION 00000150
C KPRD PRODUCT POINTERS? ONE PER PRODUCT SPECIES PER REACTION 00000160
C KRCT REACTANT POINTERS, ONE PER REACTANT SPECIES PER REACTION 00000170
C KRXN REACTION POINTERS? ONE PER REACTION PER SPECIES 00000130
C L DO-LOOP INDICES OR LOCAL POINTERS 00000190
C LRXN POINTER TO LUMPED REACTION 00000200
C M DO-LOOP INDICES OP LOCAL POINTERS .00000210
C MAXPRD MAXIMUM NUMBER OF PRODUCTS 00000220
C N DO-LOOP INDICES OR LOCAL POINTERS 00000230
C NDIF NUMBER OF DIFFERENTIAL SPECIES 00000240
C NLMP NUMBER OF LUMPED REACTIONS 00000250
C NLOC NUMBER OF REPLACEMENT REACTIONS FOR THIS LUMPED REACTION 000002^0
C NOUT THE FORTRAN OUTPUT UNIT NUMBER (NORMALLY 6) 00000270
C NR POINTER TO REPLACEMENT REACTION 000002RO
C NRXN NUM-B-ER OF REACTIONS % 00000290
c NS POINTER TO REACTANT SPECIES * 00000300
C NSTS NUMBER OF STEADY-STATE SPECIES ' 00000310
C Q DEGRADATION PATE? /MIN 00000320
C R REACTION RATES? SEC, ONE PER REACTION 00000330
c RK REACTION RATE CONSTANTS? PPM-MIN, ONE PER REACTION 00000340
C RKLMP LOCAL VALUE OF LUMPED RATE CONSTANT 00000350
C RKLOC LOCAL VALUES OF RK? THE REACTION RATE CONSTANTS 00000360
C SUM SUM OF CONCENTRATIONS OF ALL THE REPLACEMENT SPECIES 00000370
C TOL CONVERGENCE TOLERANCE ON STEADY-STATE ITERATION? PPM 000003RQ
C VAX SPECIES CONCENTRATIONS* PPM, ONE PER SPECIES 00000390
C YF THE MOLE FRACTIONS OF THE REPLACEMENT SPECIES 00000400
C YIN SPECIES INFLOW RATES. PPM/MIN? ONE PER SPECIES 00000410
'C YLOC' LOCAL VALUES OF YAX, THE SPECIES CONCENTRATIONS 00000420
C 00000430
C SUBROUTINE ENTRY POINT 00000440
C 00000450
SUBROUTINE LMPCAL(NLMP, KLMP) 00000460
C 00000470
C DECLARE COMMON STORAGE " 00000480
C 00000490
COMMON RK(99), R(99), YAX(50), YIN(50), COEFF(3,99) 00000500
COMMON KRCT(4,99)t KPRD(3,99), KRXN(99,50), KCOF(99»50) 00000510
COMMON 0, TOL» NRXN, NDIF, NSTS 00000520
C 00000530
C SET DIMENSIONS . 00000540
-------
EXHIBIT A-2. LISTING OF SUBROUTINE LMPCAL (Continued)
251
DIMENSION YLOC(10)» PKLOC(IO). ALPHA(3*10), YF(10), KLMP
SET DATA STATEMENT PARAMETERS
DATA MAXPRD /3/» COMIN /o,oooi/
SET CONTRIBUTING REACTION POINTER AND NUMBER OF CONTRIBUTING
NR = NRXN + 1
DO 70 N = 1»NLMP
NLOC = KLMP(N)
LRXN = NRXN - NLMP + N
SAVE RATE CONSTANT AND SAVE AND SUM SPECIES CONCENTRATIONS
SUM =0.0
DO 20 K = 1»NLOC
NS = KPCT(l.NR)
RKLOC(K) = RK(NR)
YLOC(K) = YAX(NS)
SUM = SUM + YLOC(K)
SAVE PRODUCT COEFFICIENTS
DO 10 J = l^MAXPRD
ALPHA(J,K) = COEFF(J,NR)
10 CONTINUE
ADVANCE REACTION POINTER AND SAVE OVERALL SUM
NR = NR * 1
20 CONTINUE
NS = KRCT(1«LRXN)
YAX(NS) = SUM
CALCULATE THE MOLE FRACTIONS
DO 30 K = 1?NLOC
YF(K) = YLOC(K) / SUM
30 CONTINUE
CALCULATE LUMPED RATE CONSTANT
RKLMP =0.0
DO 40 K = 1
-------
EXHIBIT A-2. LISTING OF SUBROUTINE LMPCAL (Concluded)
252
50 CONTINUE
C
C NORMALIZE COEFFICIENT AND CHECK FOR UNDERFLOW
C
COLOC = COLOC / RKLMP
IF (COLOC .LT. COMIN) COLOC = 0.0
COEFF(J,LRXN) = COLOC
60 CONTINUE
70 CONTINUE
C
C END OF PROGRAM — RETURN TO CALLER
C
RETURN
END
00001110
00001120
00001130
00001140
00001150
00001IftO
00001170
000011RO
00001190
00001200
00001210
00001220
00001230
00001240
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB
253
C» THE PARAMETERS TO THE SUBROUTINE DIFSUB HAVE 00000020
C* THE FOLLOWING MEANINGS: 00000030
C* 00000040
THE NUMBER OF FIRST ORDER DIFFERENTIAL EQUATIONS. N 00000050
MAY BE DECREASED ON LATER CALLS IF THE NUMBER OF 00000060
ACTIVE EQUATIONS REDUCES, BUT IT MUST NOT BE 00000070
INCREASED WITHOUT CALLING WITH JSTART = o oooooo«o
THE INDEPENDENT VARIABLE. 00000090
AN 8 BY N ARRAY CONTAINING THE DEPENDENT VARIABLES AND 00000100
THEIR SCALED DERIVATIVES. Y(J+I,D CONTAINS 00000110
THE J-TH DERIVATIVE OF Y
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
254
C* ARRAY WHERE M IS THE VALUE OF N USED ON
C* THE FIRST CflLL TO THIS PROGRAM.
C* 2 THE SAME AS CASF 1, EXCEPT THAT THIS
C* SUBROUTINE COMPUTES THE PARTIAL
C* DEVIVATIVES BY NUMFRICAL DIFFERENCING
C* OF THE DEVIVATIVES. HENCE PEDERV IS
C* NOT CALLED.
C» YMAX AN ARRAY OF N LOCATIONS WHICH CONTAINS THE MAXIMUM
C* OF EACH Y SEEN SO FAR. IT SHOULD NORMALLY BE SET TO
C* 1 IN EACH COMPONENT BEFORE THE FIRST ENTRY. (SEE THE
C* DESCRIPTION OF EPS.)
C* ERROR AN ARRAY OF N ELEMENTS WHICH CONTAINS THE ESTIMATED
C* ONE STEP ERROR IN EACH COMPONENT.
C* KFLAG A COMPLETION CODE WITH THE FOLLOWING MEANINGS:
c* +1 THE STEP WAS SUCCESSFUL.
C* -1 THE STEP WAS TAKEN WITH H = HMIN, BUT THE
C* REQUESTED ERROR WAS NOT ACHIEVED.
C* -2 THE MAXIMUM ORDER SPECIFIED WAS FOUND TO
C* BE TOO LARGE.
C* -3 CORRECTOR CONVERGENCE COULD NOT BE
C* ACHIEVED FOR H .GT. HMIN.
C* -4 THE REQUESTED ERROR IS SMALLER THAN CAN
C* BE HANDLED FOR THIS PROBLEM.
C* JSTART AN INPUT INDICATOR WITH THE FOLLOWING MEANINGS:
C« -1 REPEAT THE LAST STEP WITH A NEW H
C* 0 PERFORM THE FIRST STEP. 'THE FIRST STEP
C* MUST BE DONE WITH THIS VALUE OF JSTflRT
C* SO THAT THE SUBROUTINE CAN INITIALIZE
C* ITSELF.
c* +1 TAKE A NEW STEP CONTINUING FROM THE LAST.
C* JSTART IS SET TO NQ, THE CURRENT ORDER OF THE METHOD
C* DERIVATIVE USED, THIS RESTRICTS THE ORDER. IT MUST
C* BE LESS THAN 8 FOR ADAMS AND 7 FOP STIFF METHODS.
C* PSAVF. A BLOCK OF AT LEAST N**2 FLOATING POINT LOCATIONS.
C*
C*
C*
DERIVATIVE USED, THIS RESTRICTS THE ORDER. IT MUST
BE LESS THAN 8 FOR ADAMS AND 7 FOR STIFF METHODS.
PSAVE A BLOCK OF AT LEAST N**2 FLOATING POINT LOCATIONS.
***
DOUBLE PRECISION A , D,E , H,R, T , Y , Rl , R2, BND,EPS, EUP, EDWN, ENO 1
1 . *ENQ2, ENQ3 » HM AX » HMIN «HNEW» HOLD » SAVE? TOLD , YMAX , ERROR * RACUM
2*SDOT1,SDOT2
DIMENSION Y(8»40)» YMAX(40), SAVE(12*40)» ERRORUO)* PSAVE(
DIMENSION A(H), PERTST ( 7 » ? , 3 ) , SDOT1(40)* SDOT2(40)
00000550
00000560
00000570
00000580
00000590
00000600
00000610
00000620
00000630
00000640
00000650
00000660
00000670
00000680
00000690
00000700
00000710
000007.PO
00000730
00000740
00000750
00000760
00000770
•00000780
00000790
00000800
00000810
000008 ? 0
OOOOOH30
00000840
00000850
00000900
00000910
000009PO
innnnnntQ 0000930
00000900
00000910
00000920
innnnnni-00000930
KFLAG, 00000940
00000950
00000960
00000970
00000980
1600) 00000990
00001000
C* THE COEFFICIENTS IN PERTST APE USED IN SELECTING THE STEP AND
C* ORDER, THEREFORE ONLY ABOUT ONE PERCENT ACCURACY IS NEEDED.
DATA PERTST
/2.0«4.5.7.333,10.42,13.7,17.15,1.0,
2.0,12.0,24.0,37.89,53.33., 70.08,87.97,
3.0,6.0,9.167,12.5,15.98,1.0,1.0,
12.0,24.0,37.89,53.33,70.08,87.97,1.0,
!.,!., 0.5, 0.1667,0.04133,0.008267,1.0,
1.0,1.0,2.0,1.0,.3157,.07407,.0139/
00001020
00001030
* 00001040
00001050
00001060
00001070
00001080
00001090
00001100
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
255
A(2)=-l.
IRET = 1
KFLAG = 1
IF(JSTART.LF.O)
GO TO 140
C*
c»
C*
C*
c*
C*
BEGIN BY SAVING INFORMATION FOR POSSIBLE RESTARTS AND CHANGING
H BY THE FACTOR R IF THE CALLER HAS CHANGED H. ALL VARIABLES
DEPENDENT ON H MUST ALSO RE CHANGED.
E IS A COMPARISON FOR ERRORS OF THE CURRENT ORDER NO. EUR IS
00001110
00001120
00001130
00001140
nxnnn>0000 1 150
00001160
00001170
00001130
00001190
TO TEST FOR INCREASING THE ORDER, EDWN. FOR DECREASING THE ORDER. 00001200
HNEW IS THE STEP SIZE THAT WAS USED ON THE LAST CALL. 00001210
100D0110I=1,N
DO 110 J = 1 ,K
110 SAVE(J,I) = Y(J,I)
HOLD = HNEW
IF ( H.EO.HOLD) GO TO 130
120 RACUM = H/HOLD
IRET1 = 1
GO TO 750
130 NQOLD = NO
TOLD = T
RACUM = 1.0
IF (JSTART.GT.O) GO TO 250
GO TO 170
140 IF (JSTART.EQ.-l) GO TO 160
C* ON THE FIRST CALL* THE ORDER
C* DERIVATIVES ARE CALCULATED.
IS SET TO i AND THE INITIAL
NQ =
N3 =
Nl =
N2 =
N4 =
N5 =
N6 =
CALL
1
N
N*10
Nl + 1
N«.»2
Nl + N
N5 + 1
DIFFUN
(T,Y»SDOT1)
150
DO 150 I = 1,N
Y(2»I) = SDOT1(I)*H
HNEW = H
K = ?
GO TO 100
C* REPEAT LAST STEP BY RESTORING SAVED INFORMATION.
160 IF (NO.EQcNQOLD
IF(KFLAG.GE.-l)
NQ = NOOLD
K = NQ + 1
GO TO 120
i JSTART = 1
T = T - HOLD
00001230
000012^0
00001250
00001260
00001270
000012*0
00001290
00001300
00001310
000013PO
00001330
'00001340
00001350
00001360
1370
000013*0
00001390
00()01400
00001410
00001420
00001430
00001440
00001450
00001460
00001470
000014^0
00001490
00001500
00001510
00001520
00001530
•innnnnnnnnnnnnni-00001540
00001550
1560
00001570
000015HO
00001590
00001600
C*
C#
C*
C*
00001610
j#tttt#«tt*fttt«#«#«-tt«*#««tt
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
256
C» STEP IF IT HAS NOT YET PEEN DONE (IRET = 1) OR SKIP TO A FINAL
C« SCALING BEFORE EXIT IF IT HAS BEEN COMPLETED (IRET = 2).
C«HHHHHHHHHHt tf«#^tttt#Ctt1»tt##tt##tta#fc##tt#«H><»««&«tttt«tttt-H-<
170 IF(MF.EQ.O) GO TO 180
IF (NQ.GT.6) GO TO 190
GO TO (221,222*223*224,225,226),NQ
180 IF(NO.GT.7) GO TO 190
GO TO (211*212,213,214*215,216,217)9NQ
190 KFLAG = -2
RETURN
Q»ft»«*»*»«*»«»*»*««#*««*«»««»»«««»««»*»«»tt«*»»«»«*««»««»«-»«'»-»-»»-»a'«'{
C# THE FOLLOWING COEFFICIENTS SHOULD BE DEFINED TO THE MAXIMUM
C» ACCURACY PERMITTED BY THE MACHINE* THEY ARE IN THE ORDER USED:
C»
c« -i
C*
C*
c»
C*
C*
c»
C*
c»
C*
C*
c»
C*
-3/8, -11/1?, -1/3, -1/24
-251/720, -25/24, -35/72, -5/48 9 -I/ 120
-95/288, -137/1 20 .-5/8, -17/96, -1/40, -1/720
-19087/6 048 0, -49/40, -203/270* -49/1 92 » -7/1 44, -7/1 440, -1/5040
-6/119-6/11, -1/11
- 12/25 » -7/10 «-l/5 » -1/50
-120/274, -225/274 , -85/274 ,-15/274? -1/274
-180/441, -58/63 ,-15/36, -25/252,-3/252.-l/1764
211
212
213
214
215
A(l) = -1
GO TO 230
A(l) = -0
A(3) = -0
GO TO 230
Ad) = -0
A(3) = -0
A(4) = -
GO TO 230
A(l) = -0
A<3) = -0
A(4> = -0
A(5) = -0
GO TO 230
216
Ad) =
A(3) =
A(4) =
A(5) =
A(6) =
GO TO
Ad) =
A(3) =
A(4) =
A(5) =
A(6) =
A(7) =
GO TO
: -0
' -1
: -0
: -0
• -o
230
• -0
-1
-0
-0
-0
-0
230
.0
.500000000
.500000000
.4166666666666667
.750000000
0.166666666666667
.375000000
,9166666666666667
.3333333333333333
.0416666666666667
.34861111111111
.04166666666667
.48611111111111111
.104166666666666667
.00833333333333333
.3298611111111111
.14166666666666667
.625000000
.1770B33333333333333
.02500000000
.001388888888888889
00001*70
000016PO
01690
00001700
00001710
000017PO
000017^10
000017^0
00001750
000017*0
1770
00001780
00001790
00001800
00001810
00001820
00001830
00001840
00001850
00001860
00001870
00001HPO
00001890
00001900
00001910
00001920
00001930
00001940
1950
00001960
00001970
000019«0
00001990
00002000
00002010
00002020
00002030
00002040
00002050
000020*0
00002070
000020*0
00002090
00002100
00002110
00002120
00002130
00002140
00002150
00002160
00002170
00002180
00002190
00002200
00002210
00002220
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
257
217
221
222
223
224
225
226
230
240
A(l)
A(3)
A(4)
A(5)
A(6)
A(7)
A(8)
GO TO
-0,
-1,
-0.
-0.
-0,
-0.
-0,
230
3155919312169312
235000000
7518518518518519
2552083333333333
0486111111111111
,0001984126984126984
,000000000
,6666666666666667
,3333333333333333
,5454545454545455
I)
, 09090909090909091
A(l! = -1,
GO TO ?30
Ad) = -0,
A(3) = -0,
GO TO 230
Ad) = -0,
A(3) = AC
A(4) = -0.
GO TO 230
Ad) = -0.480000000
A(3) = -0.7000000000
A(4) = -0.2000000000
A(5) = -0.0200000000
GO TO 230
Ad) = -0.437956204379562
-0.8211678832116788
3102189781021898
05474452554744526
0036496350364963504
A(3)
A(4)
A(5)
A(6)
GO TO
A(3) =
,4081632653061225
,9206349206349206
,4166666666666667
0992063492063492
0119047619047619
000566893424036282
= -0
= -0
= -0
230
=. -0
-0
A(5) = -0
A(6) = -0
A(7) = -0
K = NQ+1
IDOUB = K
MTYP = (4 -MF)/2
ENQ2 = ,5/FL.OAT(NO + 1)
ENQ3 = .5/FLOAKNQ +2)
ENQ1 = ,5/FLOAT(NQ)
PEPSH = EPS
EUP = (PERTST(NQ»MTYP,2)»PEPSH)**2
F = (PFRTST(NO,MTYP» 1)"PEPSH)**2
EDWN = (PERTST(NQ,MTYP,3)«PEPSH)**2
IF (EDWN.EO.O) GO TO 780
BND = EPS«ENQ3/FLOAT(N)
IWEVAL = MF
GO TO (250 .680 )»IRET
C* THIS SECTION COMPUTES THE PREDICTED VALUES BY EFFECTIVELY
C* MULTIPLYING THE SAVED INFORMATION BY THE PASCAL TRIANGLE
C* MATRIX.
250 T = T + H
DO 260 J = 2.K
DO 260 Jl = J.K
00002230
00002240
00002250
00002260
00002270
00002280
00002290
00002300
00002310
00002320
00002330
00002340
000023SO
00002360
00002370
00002380
000023QO
00002400
00002410
00002420
00002430
00002440
00002450
.00002460
00002470
000024RO
000024QO
00002500
00002510
00002520
00002530
00002540
00002550
00002560
00002570
000025^0
000025QO
00002600
00002610
00002620
00002630
00002640
00002650
00002660
00002670
00002680
000026QO
00002700
'10
000027?0
00002730
00002740
00002760
00002770
00002780
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
258
J2 = K-J1 * J - 1
00 260 I = 1,N
260 Y(J2*I) = Y(J2,I)
n
C* UP TO 3 CORRECTOR ITERATIONS ARE TAKEN* CONVERGENCE IS TESTED
C« BY REQUIRING CHANGES TO BE LESS THAN BND WHICH IS DEPENDENT ON
C* THE ERROR TEST CONSTANT.
C* THE SUM OF THf CORRECTIONS .IS ACCUMULATED IN THE ARRAY
C* ERROR(I). IT IS EQUAL TO THE I-TH DERIVATIVE OF Y MULTIPLIED
C* BY H**K/(FACTORIAL(K-1)*A(K))» AND IS THEREFORE PROPORTIONAL
C* TO THE ACTUAL ERRORS TO THE LOWEST POWER 'OF H PRESENT. (H**K)
DO 270 I = 1,N
270 ERROR(T) = 0.0
DO 430 L = 1*3
CALL DIFFUN(T?Y?SDOT1)
+ PSAVE(I*(N3+1)-N3>
C» IF THERE HAS BEEN A CHANGE OF ORDER OR THERE HAS BEEN TROUBLE
C* WITH CONVERGENCE., PW IS PE-EVALUATED PRIOR TO STARTING THE
C* CORRECTOR ITERATION IN THE CASE OF STIFF METHODS. IWEVAL IS
C* THEN SET TO -1 AS AN INDICATOR THAT IT HAS BEEN DONE.
IF (IWEVAL.LT.l) GO TO 350
IF (MF.EQ.2) GO TO 310
CALL PEDERV(T,Y»PSAVE,N3>
R = A(1)*H
DO 280 I = 1,M4
PSAVE(I) = PSAVE(I)*R
DO 300 I = 1»N
PSAVE(I*(N3+1)-N3) = 1.0
IWEVAL = -1
CALL MATINV(PSAVE7N3»N3»J1)
IF(Jl.GT.O) GO TO 350
GO TO MO
DO 320 I = 1,N
SAVE(9,I) = Y(1»I)
DO 340 J = 1,N
R = EPS*DMAX1(EPS-DABS(SAVE(99J)))
Y(1»J) = Y(1»J) + R
D = A(1)*H/R
CALL DIFFUN(T,Y»SDOT2)
00 330 I = 1»N
PSAVE(I+(J-1)*N3) = (SDOT2(I).-SDOT1 (I)
Y(1,J) = SAVE(9»J)
GO TO 290
IF (MF.NE.O) GO TO 370
DO 360 I = 1»N
SAVE(9,I) = Y(2»I)-SDOT1(I)*H
GO TO 410
DO 380 I = liN
= Y(2,I)-SDOT1(I)*H
280
290
300
310
320
330
340
350
360
370
380
390
400
SDOT2(I)
DO 400 I = 1»N
D = 0.0
DO 390 J = l.N
D = D + PSAVE(IMJ-1)*N3>»SDOT2(J)
SAVE(9,I) = D
00002790
00002800
00002810
00002830
00002840
00002B50
00002880
00002870
00002880
00002890
*»00002900
00002910
00002920
00002930
00002940
00002^^0
00002970
00002980
00002990
103000
00003010
•00003020
00003030
00003040
00003050
000030^0
00003070
00003080
00003090
00003100
00003110
00003120
00003130
00003140
00003150
00003160
00003170
00003180
00003190
00003200
00003210
0000322C
0000323C
00003240
0000325C
0000 32 7C
0000328C
0000329C
0000330C
0 0 0 0 3 3 1 C
0000332C
0000333C
0000334C
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
259
410 NT = N
HO 420 I = 1,N
Yd»I> = Y(l, I) «• Ad)*SAVE(9,I)
Y(2.I) = Y(2,I) - SAVE(9,I>
ERRORd) = ERROR d) + SAVE(9,I)
IF (DABS(SAVE(9»I) ) .LE. (BND*YMAX (I) ) )
420 CONTINUE
IF (NT.LE.O) GO TO 490
430 CONTINUE
NT = NT - 1
C» THE CORRECTOR ITERATION FAILED TO CONVERGE IN 3 TPIESo VARIOUS
C* POSSIBILITIES ARE CHECKED FOR. IF H IS ALREADY HMIN AND
C* THIS IS EITHER ADAMS METHOD OR THE STIFF METHOD IN WHICH THE
C* MATRIX PW HAS ALREADY BEEN RE-EVALUATED, A NO CONVERGENCE EXIT
C* IS TAKEN. OTHERWISE THE MATRIX PW IS RE-EVALUATED AND/OR THE
C* STEP IS REDUCED TO TRY AND GET CONVERGENCE.
( (H.LE. (HMIN*1. 00001 ) ) .AND. ( (IWEVAL - MTYP ) .LT . -1 ) }
( (MF.EQ.O) .OR. (IWEVAL. NE.O) ) RACUM = RACUM**2*0 . 5
440 IF
IF
IWEVAL = M
T = T - H
IRET1 = 2
GO TO 750
460 KFLAG = -3
•470 DO 4flO I =
00 480 J =
480 Y(J*I) :• SAVE(J»I)
H = HOLD
NQ = N&OLD
JSTART = NQ
RETURN
GO TO 460
ItN
1,K
C* THE CORRECTOR CONVERGED AND CONTROL IS PASSED TO STATEMENT 520
C* IF THE ERROR TEST IS O.K., AND TO 540 OTHERWISE.
C* IF THE STEP IS O.K. IT IS ACCEPTED. IF IDOUB HAS BEEN REDUCED
C* TO ONE* A TEST IS MADE TO SEE IF THE STEP CAN BE INCREASED
C* AT THE CURRENT ORDER OR BY GOING TO ONE HIGHER OR ONE LOWER.
C* SUCH A CHANGE IS ONLY MADE IF THE STEP CAN BE INCREASED BY AT
C* LEAST 1.1. IF NO CHANGE IS POSSIBLE IDOUB IS SET TO 10 TO
C* PREVENT FUTHER TESTING FOR 10 STEPS
C* IF A CHANGE IS POSSIBLE* IT IS MADE AND IDOUB IS SET TO
C# NO * 1 TO PREVENT FURTHER TESTING FOR THAT NUMBER OF STEPS.
C* IF THE ERROR WAS TOO LARGE* THE OPTIMUM STEP SIZE FOR THIS OR
C* LOWER ORDER IS COMPUTED* AND THE STEP RETRIED. IF IT SHOULD
C* FAIL TWICE MOPE IT IS AN INDICATION THAT THE DERIVATIVES THAT
C* HAVE ACCUMULATED IN THE Y ARRAY HAVE ERRORS OF THE WRONG ORDER
C« SO THE FIRST DERIVATIVES ARE RECOMPUTED AND THE ORDER IS SET
C* TO 1.
490 D = 0.0
DO 500 I = 1 »N
500 D = D f (ERROR (I)/YMAX(I) )
IWEVAL = 0
IF (D.GT.E) GO TO 540
IF (K.LT.3) GO TO 520
DO 510 J = 3,K
00003350
00003360
00003370
00003380
00003390
00003400
00003410
00003420
00003430
00003450
00003460
00003470
00003430
0000349U
00003500
000035PO
00003530
00003540
00003S50
00003560
00003570
00003580
000035QO
00003600
00003610
00003620
00003630
00003640
00003650
00003670
00003680
00003690
00003700
00003710
00003720
00003730
00003740
0000375C
00003760
00003770
0000378C
00003790
0000380C
0000381C
000038PC
00003B4C
0000385C
0000386C
0000337C
0000388T
0000389C
0000390C
-------
EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Continued)
260
DO 510 I = 1,N
510 Y(J,I) = Y(J,I) + A(J)*ERROR(I)
520 KFLAG = +1
HNEW = H
IF (IDOUP.LE.l) GO TO 550
IDOUR = IDOUB - 1
IF (IDOUB.GT.l) GO TO 700
DO 530 I = 1,N
530 SAVE(10»I) = FRROR(I)
GO TO 700
540 KFLAG = KFLAG - 2
T = TOLD
IF (H.LE.(HMIN«1.00001)) GO TO 740
IF (KFLAG.LE.-5) GO TO 720
550 PR2 = (D/E) *-*ENQ2*l c2
PR3 = l.E+20
IF ((NQ.GE.MAXDER).OR.(KFLAG.LE.-1)) GO TO 570
D = 0.0
DO 560 I = 1,N
560 D = D + UF.RROP(I) - SAVE (1 0 » I) ) /YMAX (I) ) **2
PR3 = (D/EUP)»*ENQ3<*1.4
570 PR1 = l.E+20
IF (NQ.LE.l) GO TO 590
D = 0.0
DO 5BO I = 1»M
580 D = D + (Y(K*I)/YMAX(I))»«2
PR1 = (D/EDWN)**ENQ1*1.3
590 CONTINUE
IF (PR2oLE.PR3) GO TO 650
IF (PR3.LT.PR1) GO TO 660
600 R = 1 „ 0/AMAX1 (PR1;1.E-4)
HfrWQ = NO - 1
610 IDOUB = 10
IF ((KFLAG.EQ.l).AND.(R.LT.(1.1))) GO TO 700
IF (NEWQ.LE.NO) GO TO 630
DO 620 I = 1»N
620 Y(NEWQ+1,I) = ERROR(I)*A(K)/FLOAT(K)
630 K = NEWQ + 1
IF ( KFLAG.EG. 1 ) GO TO 670
RACUM = RACUM*R
IRET1 = 3
GO TO 750
640 IDOUB = K
IF (NEWQ.EQ.NQ) GO TO 250
NQ = NFWQ
GO TO 170
650 IF (PR2.GT.PR1) GO TO 600
NEWQ = NO
R = 1.0/AMAX1(PR2.1.E-4)
GO TO 610
660 R = 1.0/AMAX1(PR3.1.E-4)
NEWQ = NO + 1
GO TO 610
670 IRET = 2
R = DMIN1(R»HMAX/DA8S(H))
H=H*R
0000391.0
000039PO
0000^930
00003940
000039SO
00003960
00003970
00003980
00003990
0000^000
00004010
00004020
00004030
00004040
000040?0
00004060
00004070
000040HO
00004090
00004100
00004110
00004120
000041 TO
00004140
00004150
00004160
00004170
0 0 0 0 4 1 B 0
00004190
00004200
00004210
00004220
00004230
00004240
0000425H
00004260
00004270
000042*0
000042^0
0000430C
00004310
0000432C
0000433C
00004341
000043SC
0000436r
0000437T
000043«f
0000439f
0000440C
000044K
0000442C
0 0 0 0 4 4 3 C
0000444C
0000445f
0000446C
-------
•EXHIBIT A-3. LISTING OF SUBROUTINE DIFSUB (Concluded)
261
HNEW = H
IF (NQ.EO,
NEWQ) GO TO 630
6BO
690
700
710
730
740
NO
GO
Rl
DO
Rl
DO
= NFWQ
TO 170
= 1.0
690 J =
= R1»R
690 I =
2»K
1,N
1,N
IDOUB = K
DO 710 I =
YMAXCI) = DMAX1 (YMAXd) *DABS(Yd»I) ) )
JSTART = NO
RETURN
720 IF (NO.EQ.l) GO TO 780
CALL DIFFUN(T,Y?SDOT1)
R = H/HOLD
DO 730 I = 1,N
Yd,I) = SAVE(1,I)
SAVE(2.I) = HOLD*SDOtld>
Y(2»I) = SA\
NQ = 1
KFLAG = 1
GO TO 170
KFLAG = -1
HNEW = H
JSTART = NQ
RETURN
*«•*««••» •& t
C* THIS SECTION SCALES ALL VARIABLES CONNECTED WITH H AND RETURNS
C* TO THE ENTERING SECTION.
H
750 RACUM = DMAX1(0ABS(HMIN/HOLD).RACUM)
RACUM" = DM INI (RACUM,DABS(UMAX/HOLD))
Rl = 1.0
DO 760 J = 2«K
Rl = R1*RACUM
DO 760 I = 1»N
760
770
780
H = HOLD*RACUM
DO 770 I = 1»N
Yd. I) = SAVEd.I)
'GO TO d30«250,640) 9IRET1
KFLAG = -4
GO TO 470
END
00004470
00004480
00004490
00004500
00004510
000045?0
00004530
00004540
00004550
00004560
00004570
000045PO
00004590
00004600
00004610
000046PO
00004630
00004640
00004650
00004660
00004670
00004680
00004690
00004700
00004710
000047?0
00004730
00004740
00004760
00004770
00004790
00004800
00004810
000048?0
00004R30
00004840
00004850
00004360
00004870
00004880
00004890
00004900
00004910
000049PO
-------
262
EXHIBIT A-4. LISTING OF SUBROUTINE DIFFUN
C SUBROUTINE
DIFFUN
C THIS SUBROUTINE CALCULATES THE RATE OF CHANGE OF DIFFERENTIAL AND
C STEADY-STATE SPECIES CONCENTRATIONS -- CALLED BY DIFSUB.
C '
C
C SYMBOL DESCRIPTIONS —
C
C COEFF NUMBER OF PARTICLES? ONE PER PRODUCT SPECIES PER REACTION
C J DO-LOOP INDICES OR LOCAL POINTERS
C JFLAG INDICATES SPECIES HAS BEEN SEPARATED FROM SDEN CALCULATION
C K DO-LOOP INDICES OP LOCAL POINTERS
C KCOF COEFFICIENT POINTERS, ONE PER REACTION PRODUCT PER SPECIES
C KPRD PRODUCT POINTERS, ONE PER PRODUCT SPECIES PER REACTION
C KRCT REACTANT POINTERS* ONE PER PEACTANT SPECIES PER REACTION
C KRXN REACTION POINTERS? ONE PER REACTION PER SPECIES
C L DO-LOOP INDICES OR LOCAL POINTERS
C M DO-LOOP INDICES OR LOCAL POINTERS
C MAXPRD MAXIMUM NUMBER OF PRODUCTS
c MAXPCT MAXIMUM NUMBER OF REACTANTS
C N DO-LOOP INDICES OR LOCAL POINTERS
C MCNV NUMBER OF CONVERGED STEADY-STATES
C NDIF NUMBER OF DIFFERENTIAL SPECIES
C NOUT THE FORTRAN OUTPUT UNIT NUMBER (NORMALLY 6)
C NRXN NUMBER OF REACTIONS
C NS LOCAL POINTER TO STEADY-STATE SPECIES
C NSTS NUMBER OF STEADY-STATE SPECIES
C NTRY NUMBER OF ITERATION ATTEMPTS FOR STEADY-STATE CONVERGENCE
C 0 DEGRADATION RATE, /MIN
C R REACTION RATES. SEC? ONE PER REACTION
C RATE LOCAL REPRESENTATION OF R, THE REACTION RATE
C RK REACTION RATE CONSTANTS* PPM-MIN* ONE PER REACTION
C SDEN DENOMINATOR IN STEADY-STATE CALCULATION? /MIN
C SNUM NUMERATOR IN STEADY-STATE CALCULATION, PPM/MIN
C STEST TEST VALUE FOR STEADY-STATE CONVERGENCE-. PPM
C T CURRENT REACTION TIME; SEC? DOUBLE PRECISION?
C FOR AND FROM DIFSUB
C TOL CONVERGENCE TOLERANCE ON STEADY-STATE ITERATION? PPM
C Y SPECIES CONCENTRATIONS? 8 PER SPECIES? DOUBLE PRECISION?
C FOR AND FROM DIFSUB
C YAX SPECIES CONCENTRATIONS. PPM, ONE PER SPECIES
C YCALC LOCAL REPRESENTATION OF YDOT? THE RATE OF CHANGE
c YDOT RATES OF CHANGE OF SPECIES CONCENTRATION, PPM/MIN. ONE PER
C DIFFERENTIAL SPECIES? DOUBLE PRECISION? FOR DIFSUB
C YIN SPECIES INFLOW RATES? PPM/MIN? ONE PER "SPECIES
C
C BEGINNING OF PROGRAM.
C
C ENTRY POINT
C
SUBROUTINE DIFFUNCT? Y, YDOT)
C
C DECLARE INPUTS FROM DIFSUR TO BE DOUBLE PRECISION WITH DIMENSIONS
C
0000001
000000?
0000003
0000004
oonooo<=.
0000006
0000007
OOOOOOR
0 0 0 0 0 0 Q
0000010
0000011
000001?
0000013
0000014
000001?
0000016
000001T
000001H
000001^)
OOOOOPO
00000?!
00000??
0000023
00000?4
00000?S
00000?6
00000?T
000002A
00000?.^
00000.30
0000031
000003?
0000033
0000034
0000035
0000036
0000037
000003^
0000039
0000040
0000041
000004?
0000043
0000044
0000045
000004A
0000047
000004B
0000049
0000050
0000051
000005?
0000053
0000054
-------
EXHIBIT A-4. LISTING OF SUBROUTINE DIFFUN (Continued)
263
DOUBLE PRECISION T, Y» YDOT
DIMENSION Y(8*40)» YDOT(40)
C
C DEFINE VARIABLES AND DIMENSIONS OF COMMON STORAGE WITH MODKIN
C
COMMON RK(99), R(99), YAX(50), YIN(50)? COEFF(3,9<5)
COMMON KRCT(4,99)« KPRD(3,99), KRXN(99t50)» KCOF(99*50)
COMMON Q, TOL, NRXN« NDIF, 'NSTS
C
C DEFINE MISCELLANEOUS DATA VALUES
C
DATA NTRY /25/, MAXRCT /4/9 MAXPRD /3/, NOUT /6/» NWARN /O/
C
C MOVE DIFFERENTIAL CONCENTRATIONS TO LOCAL ARRAY
C
DO 110 J = 1,NDIF
YAX(J) = Y(1?J)
110 CONTINUE
C
C SET ITERATION LOOP AND CALCULATE REACTION RATES
C
DO 260 N = 1?NTRY
DO 140 L = 1?NRXN
RATE = RK(L>
DO 120 K = 1,MAXPCT
J = KRCT(K,L)
IF (J 0EQo 0) GO TO 130
RATE = RATE * YAX(J)
120 CONTINUE
130 R(L) = RATE
140 CONTINUE
C
C SET CONVERGENCE COUNTER AND BEGIN STEADY-STATE CALCULATION LOOP
C
NCNV = 0
IF (NSTS .LE. 0) GO TO 255
DO 250 M = 1»NSTS
SDEN = 0
SNUM =0.0
NS = NDIF + M
STEST = YAX(NS)
C
C IDENTIFY STEADY STATE SPECIES IN REACTION
C
DO 230 L = 1»NRXN
J = KCOF(L»NS)
K = KRXN(L?NS)
C
C SKIP OVER LUMPED MECHANISM REPLACEMENT SPECIES
C
IF (K .GT. NRXN) GO TO 230
IF (J) 205* 235* 203
C
C CALCULATE NUMERATOR OF STEADY STATE EQUATION
C
203 SNUM = SNUM * R(K) « COEFF(J,K)
0000055
000005^
0000057
000005H
0000059
OOOOOiSO
0000061
000006?
000006?
0000064
0000065
0000066
0000067
000006*
000006^
0000070-
0000071
000007?
0000073
OOOOOT&
0000075
0000076
0000077
0 0 0 0 0 7.«
000007Q
0 0 0 0 0 K 0
0000081
OOOOOR?
OOOOOP3
OOOOOB^-
OOOOOB5
0000086
0000087
OOOOOBH
0 0 0 0 0 fi 9
0000090
0000091
000009?
0000093
0000094
0000095
OOOOOQ6
0000097
0000091*
0000099
0000100
0000101
0000102
0000103
0000104
0000105
0000106
0000107
000010*
0000109
0000110
-------
FXHIBIT A-4. LISTING OF SUBROUTINE DIFFUN (Continued)
264
GO TO 230 0000111
c o o o o 11;
C START REACTION PATE CALCULATION AND SET SPECIES FLAG 000011;
C 0000114
205 RATE = RK(K) 0000111-
JFLAG = 0 OOOOllf-
DO 210 NR = 1,MAXRCT 0000117
J = KRCT(NR,K) 000011-'
IF (J .EG. 0) GO TO 220 000011C
IF (J .NE. WS) GO TO 208 000012C
C 0000121
C CALCULATE RATE* SKIPPING FIRST OCCURRENCE OF SPECIES IN REACTION 000012?
c 000012:
IF (JFLAG .EQ, 1) GO TO 208 0000124
JFLAG = 1 00001?c
GO TO 210 OOOOlSf:
208 RATE = RATE * YAX(J) 0000127
210 CONTINUE 000012*
C 000012S
C CALCULATE DENOMINATOR OF STEADY STATE EQUATION 0000130
C 0000131
220 SDEN = SDEN + RATE 0000132
230 CONTINUE 0000133
•C 0000134
C TEST VALUES FOR ZERO -- SKIP CONVERGENCE TEST IF SO 000013^
C 000013*=
235 IF (SDEN .LE. 0.0) GO TO 240 0000137
IF (SNUM .LE. 0.0) GO TO 240 000013^
C 000013^
C CALCULATE STEADY-STATE CONCENTRATION AND CHECK FOP CONVERGENCE 0000140
C 0000141
STEST = SNUM / SDEN 0000142
IF (ABS((STEST - YAX(NS)) / STEST) .GT. TOL) GO TO 245 0000143
C 0000144
C UPDATE CONVERGENCE COUNTER AND SPECIES CONCENTRATION 0000145
C 0000146
240 NCNV = NCNV «• 1 0000147
245 YAX(WS) = STEST 000014P
250 CONTINUE 000014Q
C 0000150
C TEST FOR CONVERGENCE OF ALL STEADY-STATES — WRITE MESSAGE IF FAILED 0000151
C 0000152
255 IF (NCNV «,EQ* NSTS) GO TO 300 0000153
260 CONTINUE 0000154
WRITE (NOUT»1031> NTRY 0000155
1031 FORMAT (« STEADY STATE FAILED TO CONVERGE IN '» I3» 0000156
S. • ITERATIONS.') 0000157
C 000015H
C INCREMENT WARNING COUNTER AND STOP IF TOO MANY 0000159
C 0000160
NWARN = NWARN + 1 0000161
IF (NWARN .GT. NTRY) STOP 000016?
C 0000163
C CALCULATE RATE OF CHANGE OF CONCENTRATION FOR DIFFERENTIAL SPECIES 0000164
C 0000165
300 DO 330 M = 1»NDIF ' 0000166
-------
EXHIBIT A-4. LISTING OF SUBROUTINE DIFFUN (Concluded)
265
YCALC = 0.0
DO 310 L = 1«NPXN
J = KCOF(L«M)
K = KRXN(L»M)
C
C SKIP OVER LUMPED MECHANISM REPLACEMENT SPECIES
C
IF (K .GT. NRXN) GO TO 310
IF (J) 305» 320* 307
305 YCALC = YCALC - R(K)
GO TO 310
YCALC = YCALC * R(K)
CONTINUE
YDOT(M) = YCALC * Q
CONTINUE
0000167
OOOni6-J
307
310
3?0
330
c
C END OF ROUTINE — RETURN TO CALLER
C
RETURN
END
COEFF(J»K)
(YlN(M) - YAX(M))
0000170
0000171
000017^
000017"
0000174
0000175
000017^
0000177
000017f-
000017s
00001«0
00001^1
OOOOIR;
00001H2
0000184
000018C
OOOOlflf
-------
EXHIBIT A-5. LISTING OF SUBROUTINE MATINV
266
67
10
15
20
30
40
45
50
60
70
80
85
90
95
100
105
110
130
HO
150
160
170
200
260
?70
310
320
330
340
350
380
390
400
420
430
450
550
600
610
620
630
640
650
660
670
SUBROUTINE MATINV(PSAVE,N->MM,J1)
DIMENSION A(40,40), INDEX(40»?)»
DIMENSION PSAVF(1600)
EQUIVALENCE (IROW,JROW), (ICOLUM,JCOLUM)
KK = 0
DO 67 I = 1,N
DO 67 J = 1,N
KK = KK «• 1
A(J,I) = PSAVE(KK)
OETERM=1.0
DO 20 J=1,M
IPIVOT(J)=0
DO 550 1 = 1,N
AMAX=0.0
DO 105 J=1,N
IF (IPIVOT(J)-l)
100 K=1,N
(IPIVOT(K)-l) 80,
PIVOT(40), IPIVOT(40)
(AMAX, T, SWAP)
105? 60
100, 740
(AMAX)-ABS (A(J9KM) 85* 100, 100
60»
DO
IF
IF (ABS
IROW=J
ICOLUM=K
AMAX=A(J,K)
CONTINUE
CONTINUE
IFCAMAX .EQ. 0.) GO TO 760
IPIVOT(ICOLUM)=IPIVOT(ICOLUM)+1
IF (IROW-ICOLUM) 140, 260, 140
DETERM=-DETERM
DO 200 L=1,N
SWAP=A(IROW.L)
A (IROW,L)=A(ICOLUM,L)
A(ICOLUM,L)=SWAP
INDEX(I91)=IROW
INDEX(I,2)=ICOLUM
P I VOT(I)=A(ICOLUM,ICOLUM)
DETERM=DETERM*PIVOT(I)
A (ICOLUM, ICOLUM )=KO
DO 350 L= 1,N
A(ICOLUM,L)=A(ICOLUM,L)/PIVOT(I)
DO 550 L1=1,N
'IF(LI-ICOLUM) 400* 550» 400
T=A(L1,ICOLUM)
A(L1»ICOLUM)=0.0
DO 450 L=1,N
CONTINUE
DO 710 I=1,N
L=N+1-I
IF (INDEX(L,1)-INDEX(L92))
JROW=INOEX(L,l)
JCOLUM=INDFX(L,2)
DO 705 K=1,M
SWAP=A(K.JROW)
A(K9JROW)=A(K,JCOLUM)
630, 710» 630
00000010
000000?0
ooonoo TO
OOOOOOAO
ooooooso
OOOOOOnO
000 (J 0070
000000^0
OOOOOOQO
00000100
000001 in
000001?u
00000130
00000140
000001SO
000001^0
0 0 0 0 0 1 « 0
000001^0
00000200
OOOOOR10
'000002? 0
00000230
OOOOOPAO
000002^0
OOOOOP60
00000270
000002HO
000002^0
00000300
00000310
000003PO
00000330
00000340
00000350
00000360
00000370
00000 3 flO
00000390
00000400
00000410
00000420
00000430
00000440
00000450
000004AO
00000470
000004PO
000004QO
0000050C
00000510
ooooo5?n
0000053C
0000054C
-------
267
EXHIBIT A-5. LISTING OF SUBROUTINE MATINV (Concluded)
700 A(K,JCOLUM)=SWAP
705 CONTINUE 00000550
.11 = 1 00000560
00000570
710 CONTINUE
740 'GO TO 780 000005«0
760 DETERM = 0. 00000590
Jl - -1 00000600
780 KK = 0 00000610
DO 68 I = 1,N 000006PO
00 68 J = 1,N 00000630
KK = KK * 1 000006AQ
68 PSAVE(KK). = A(J,I,
RETURN 00000660
'E 00000670
00000630
-------
268
EXHIBIT A-6. LISTING OF SUBROUTINE PEDERV
SUBROUTINE PEDERV(T»Y«PSAVE»N) 0000001C
DOURLE PRECISION T, Y OOOOOOPC
DIMENSION Y(8,40)« PSAVE(1600) OOOOOO^f
RETURN OOOOOO^C
0000005C
-------
EXHIBIT A-7. LISTING OF SUBROUTINE PLOT
269
C SUBROUTINE *****•» PLOT *»»*«•»
C
C THIS SUHROUTINF READS THE PLOT CARDS AND PLOTS THE RESULTS AS PART
C OF THE PRINTED OUTPUT -- IT DOES NOT DRIVE A PLOTTER.
C
C SYMBOL DESCRIPTIONS —
C
C CGRID THE LENGTH OF THE VERTICAL AXIS, PPM
C CriIGH HIGHEST CONCENTRATION VALUE? PPM
C CLOW LOWEST CONCENTRATION VALUE, PPM
C CSPAN CONCENTRATION NORMALIZATION FACTOR
C DATA CONCENTRATION DATA POINTS? PPM, UP TO 80
C J DO-LOOP INDICES OR LOCAL POINTERS
C JBLANK A HOLLERITH WORD OF FOUR BLANK CHARACTERS
C JCONC CONCENTRATION LABELS
C JFACT CONVERSION FACTOR FOR LABEL
C JGRID THE PLOTTING GRID
C JSTAR THE CHARACTER «*»
C JSYMB SYMBOL TO BE USED FOR PLOTTING SAVED POINTS
C JVERT VERTICAL LEGEND
C K DO-LOOP INDICES OR LOCAL POINTERS
C KCON CONCENTRATION COORDINATE ON GRID
C KTIM TIME COORDINATE ON GRID .. ., ..-.;•,
C L DO-LOOP INDICES OR LOCAL POINTERS " "*
C M DO-LOOP INDICES OR LOCAL POINTERS
C MAXCON LIMIT ON NUMBER OF VERTICAL POINTS
C MAXPNT MAXIMUM NUMBER OF SAVED TIME AND CONCENTRATION POINTS
C MAXTIM LIMIT ON NUMBER OF HORIZONTAL POINTS
C N DO-LOOP INDICES OR LOCAL POINTERS
C NAME SPECIES NAMES, ONE PER SPECIES
C NDAT NUMBER OF CONCENTRATION DATA POINTS
C NIN TH-E FORTRAN INPUT UNIT (NORMALLY 5)
C NOUT THE FORTRAN OUTPUT UNIT NUMBER (NORMALLY 6)
C NPNT NUMBER OF SAVED TIMES AND CONCENTRATIONS
C NTEST SPECIES NAME FOR TESTING
C NTIT USER-INPUT TITLE FOR PRINTOUT? 3 FOUR-CHARACTER WORDS
C NTOT TOTAL NUMBER OF SPECIES
C SAVCON SPECIES CONCENTRATIONS? PPM? ONE PER SPECIES AT 80 TIMFS
C SAVTIM TIMES THAT CONCENTRATIONS ARE SAVED? MIN, UP TO 80 VALUES
C TGRI.D THE LENGTH OF THE HORIZONTAL AXIS? MIN
C THIGH HIGHEST TIME VALUE? MIN
C TIME TIMFS AT WHICH CONCENTRATIONS ARE INPUT? MIN? UP TO BO
C TLOW LOWEST TIME VALUE? MIN
C TPRINT TIMES FOR PRINTOUT ON HORIZONTAL AXIS? MIN
C TSPAN TIME NORMALIZATION FACTOR
C
C BEGINNING OF PROGRAM.
C
C ENTRY POINT
SUBROUTINE PLOTtNTIT, NPNT, NTOT? NAME? SAVTIM? SAVCON)
C
C SET DIMENSIONS OF INCOMING ARRAYS
C
DIMENSION SAVCON(50?80)? SAVTIM(80), NTIT(3)» NAME(50)
oooooon
000000?!
00000031
00000041
00000051
OOOOOOM
00000071
00000081
00000091
00000101
oooooi n
00000121
00000131
0 0 0 0 0 1 * i
00000151
0 0 0 0 0 1 f- 1
00000171
oooooia<
00000191
0000020(
000002K
0000022(
0000023(
0000024C
0000025'
00000261
0000027(
00000?P(
0000029(
0000030(
0000031 '
0000032(
0000034(
0 0 0 0 0 3 5 (
0000036(
0000037(
0 0 0 0 0 3 B (
00000391
0000040(
0000041C
0000042C
00000431
0000044(
0000045f
0000046C
00000471
0000048C
0000050C
0000051C
0000052(
0000053C
0000054C
-------
EXHIBIT A-7- LISTING OF SUBROUTINE PLOT (Continued)
270
SET DIMENSIONS OF LOCAL ARRAYS
DIMENSION JVEPT(52,2>, JCONCC5), TIMEC80),
.DIMENSION JGRID<121,52)» TPRINT(9)
DEFINE THE VERTICAL AXIS VIA DATA STATEMENTS
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DATA
DEFINE THE VERTICAL LABEL
DATA(80:
JGRIDd ;1) /1H-/, JGRID (1,2) /1H I/, JGRID (1,3) /1HI/
JGRIDd »4) /1H| /, JGRIDd ,5)/lHI/,JGRID(1.6)/lHI/
JGRIDd, 7) /I H| /, JGRID (1,8) /I HI /» JGRID (1,9) /1H|/
JGRIDd ?10)/1H
JGRI"D(1,13)/1H
JGRID ( 1, 16) /1H
JGRIDd, 19 )/lH
JGRIDd, 2?)/lH
JGRIDd, 25) /1H
JGRIO (1,28) /1H
JGRIDd »31) /1H
JGRIDd, 34J/1H
JGRID(1;37)/1H
JGRID (l»4n) /1H-
JGRIDd ,43) /1H
JGRID N'OUT /6/, JBLANK
MAXTIM /121/, MAXCON /52/,
DATA JSTAR /lH*/» TGRID /1HO./,
/4H /
MAXPNT /80/
CGRID /52./
SET LOOP FOR ALL SPECIES AND CLEAR GRID
P40
250
C
C READ
C
DO 360 N =
DO 250 K =
DO 240 J =
JGRIDU, K)
CONTINUE
CONTINUE
1,
1,
2,
=
PLOT CONTROL
NTOT
MAXCON
MAXTIM
JBLANK
CARD
READ (NIN54,END=900) NTEST, NDAT,
S, CLOW, CHIGH, TLOW, THIGH
TEST FOR END OF PLOTTING
JSYMB, JFACT, JCONC,
OOOOOBSr
OOOOOS6C
0000057C
0 0 0 0 0 5 P f.
0000059f
ooooo^or
0000061C
000006PC
0000063f
0000064f
0000065C
0000066C
0000067C
0000069f
0000070f
000007H
0 0 0 0 0 7 2 (
0000073f
ooono74(
ooooo7c;f
0 0 0 0 0 7 6 f.
0000077f
0000079C
OOOOOROt
0 0 0 0 0 R 1 f.
OOOOORPC
0 0 0 0 0 R 3 r
OOOOOR5C
oooooe^c
OOOOOB7f
OOOOORRr
OOOOOR9(
OOOOOQOf
000009K
000009?f
0000093f
000009M
000009S(
0 0 0 0 0 9 7 (
000009Ri
000009Q(
OOOOlOOf
000010K
ooooin?(
oooni03(
00001041
OOOOIOM
0000106(
00001071
000010R1
0000109(
00001101
-------
EXHIBIT A-7. LISTING OF SUBROUTINE PLOT (Continued)
271
IF (NTEST .EQ. JBLANK) GO TO 800
C
C TEST NUMBER OF DATA POINTS AND READ DATA
C
IF (NDAT .IE. 0) GO TO 308
IF (NDAT .LE. MAXPNT) GO TO 305
WRITE (NOUT»10?0) MAXPNT
GO TO 900
C
C READ DATA POINTS
C
305 READ (NIN»5) (TIME(J), DATA(J)» J = 1»NDAT)
C
C SET NORMALIZATION FACTORS AND VERTICAL CONCENTRATION LABELS
C
308 CSPAN = CGRID / (CHIGH - CLOW)
TSPAN = TGPID / (THIGH - TLOW)
JVERTtl,?) = JCONC(5)
JVERT(14?2) = JCONC(4)
JVERT(27»2) = JCONC(3)
JVERT(40»2> = JCONC(2)
C
C SET HORIZONTAL TIME LABELS
C
DO 310 J = 1,9
TPRINT(J) = FLOATtJ - 1) / 8. * (THIGH - TLOW) + TLOW
310 CONTINUE
C
C TEST FOR CORRECT SPECIES NAME
C
DO 320 L = 1«NTOT
IF (NTEST .EQ. NAME(D) GO TO 325
320 CONTINUE
WRITE (NOUT?1021) NTEST
GO TO 360
C
C IF THERE ARE DATA POINTS, GET THEIR COORDINATES
C
325 IF (NDAT .LE. 0) GO TO 335
DO 330 J = 1»NDAT
KTIM = IFIX((TIME(J) - TLOW) * TSPAN + 1.5)
"KCON = IFIX((DATA(J) - CLOW) * CSPAN - 1.5)
KCON = MAXCON - KCON
C
C CHECK FOR BEING WITHIN GRID? THEN PLACE ON GRID
C
IF (KTIM .LT. 2) GO TO 330
IF (KCON .LT. 1) GO TO 330
IF (KTIM .GT. MAXTIM) GO TO 330
IF (KCON.GT. MAXCON) GO TO 330
JGRID(KTIM»KCOM) = JSTAR
330 CONTINUE
C
C IF THERE ARE CALCULATED POINTS, GET THEIR COORDINATES
C
335 IF (NPNT .LE. 0) GO TO 345
0000111C
0000112C
ooonii3C
00001141'
OOOOll^f
0000117*:
000011«(
OOOOll^C
0000120f
000012H
0000122C
0000123T
000012?'
0 0 0 0 1 ? 6 f
0000127C
000012PL
0000 IP^f.
0 0 0 0 1 3 0 f.
0000131C
0000132C
0000133'".
000 01 34 C
000013SC
0 0 0 0 1 3 6 C
0 0 0 0 1 3 7 r
0000139C
0000140 c:
ooooi4ir
0000142C
0000143C
0000 144 r,
000 01 45 C
0000146C
0000147C
0 0 0 0 1 4 P C
0000149C
0000150C
0000151C
0000152C
0000 153C
0000154C
0000155C
0000156f
0000157C
000015BC
0000159C
0000160f
0000161C
000016?C
0000163C
000016fC
0000165C
000 01 66 C
-------
EXHIBIT A-7. LISTING OF SUBROUTINE PLOT (Concluded)
272
DO 340 J = 1,NPNT
KTIM = IFIX((SAVTIM(J) - TLOW) *
KCON = IFIX((SAVCON(L,J) - CLOW)
KCON = MAXCON - KCON
TSPAN + 1.5)
* CSPAN - 1.5)
CHECK FOR BEING WITHIN GRID, THEN PLACE ON GRID
IF (KTIM .LT. 2) GO TO 340
IF (KCON .LT. 1) GO TO 340
IF (KTIM .GT. MAXTIM) GO TO
IF (KCON ,GT. MAXCON) GO TO
JSYM8
340
340
340
C
C SKIP
C
345
JGRID(KTIM.KCON) =
CONTINUE
A PAGE? THEN PRINT THE VERTICAL AXIS AND GRID
WRITE (NOUT;1014)
DO 350 K = 1,MAXCON
WRITE (NOUT,1015) JVERT(K,1), JVERT(K,2>»
& (JGRir>( J,K) , J = 1,MAXTIM)
350 CONTINUE
C
C PRINT THF HORIZONTAL AXIS AND LABELS
C
WRITE
WRITE
(NOUT, 1016) JCONC(l)
(NOUT.1017) TPRINT
WRITE (NOUT.1018) NTIT, NAME(L), JFACf
CONTINUE
360
C
C END OF SUBROUTINE -- RETURN TO CALLER
C
800 RETURN
900 STOP
C
C LIST OF FORMAT STATEMENTS
C
4
5
1014
1015
1016
1017
1018
FORMAT (A4, IX, 12, IX. Al» IX, 6(A4?
FORMAT (8F10.0)
FORMAT (1H1)
FORMATdX, 2A4, 121A1)
FORMAT (5X. A4, 1H+. B(15H ——
FORMAT (F12.2-. 8F15.2, /, 62X, 14HTIME
FORMAT (27X* 11HFIGURE . , 3A4, 12H.
8, 5X» PRHCONCENTRATION SCALE FACTOR
1020 FORMAT (33H PROGRAM CANNOT HANDLE MORE
fii 2RH PLOT POINTS — JOB ABORTED.)
1021 FORMAT (14H1SPECIES NAME , A4, 21H NOT
& 23H SKIPPING TO NEXT PLOT.)
END
4F10.0)
— I ) )
(MINUTES).
SPECIES:
: 9 A4)
THAN , 14,
/)
A4,
IN SPECIES LIST.
00001
0 0 0 0 1 ft R (
0000169(
0000171K
0000171C
00001 7P(
0000173f
0000174C
0 0 0 0 1 7 is f
0000177C
n000178(
OOOOlBOf
0 0 0 0 1 8 1 1
00001«?(
0 0 0 0 1 « ? (
00001«5f
00001flf>(
ooooiRrr
OOOOlRHt
00001fi9(
o o o o i q o (
0000191C
0000 19? f
0 0 0 0 1 9 3 (
0000194C
00001 95 (
000019AC
0000197C
0000198C
0000199C
OOOOPOOt
000020K
000020?f
0000203f
0000204(
000020S(
0000?06(
0000207C
OOOOPORf
0000209(
0000210(
000021K
0000212C
00002131
0000214(
000021S!
-------
273
EXHIBIT A-8. SAMPLE MODKIN INPUT
SAMPLE DECK
5.0
N02
0
03
03
!M03
N03
N205
N205
NO
HN02
HNI02
OH
H02
H02
H202
ALD
ALD
R02
OH
RC03
RC03
RO
RO
RO
R02
R02
0
0
0
NO
HN02
H02
R02
N03
N205
OLEF
OLEF
OLEF
OLEF
PROP
ETHY
OLEF
PROP
ETHY
OLEF
PROP
ETHY
N02
NO
03
N205
H02
R02
RC03
HN02
375,0
02 M
NO
N02
NO
NO?
H20
N02 H20
HN02
N02
NO
H02
OH
NO
NO
NO
N02
02
NO
N02
R02
H02
NO
N02
N02
HN03
HN03
502
S02
S02
S02
0
03
OH
2
0
0
2
03
03
2
OH
OH
0.08
0.27
0.022
38 3 13
0.0001 0
1NO
103
1N02
1N03
2N02
1N205
1N02
2HN03
2HN02
1NO
10H
1HN03
10H
1H202
20H
0.63R02
0.63PC03
IRQ
1HN02
1R02
1PAN
1H02
1RN02
1RN03
2RO
IRQ
1N02
1NO
1N03
1HN02
1H20
1S03
1S03
1S03
1S03
1R02
1RC03
1R02
1R02
IRQ?
1RC03
1RC03
1R02
1R02
4 4- 2 4 5 1
.00001 5.0
10
1M
102
102
1N03
1N02 1H20
1NO
1N02
102
1.37H02
0.37H02 1H20
1N02
1N02 1C02
1ALD
10H
102
1N02
2N02
10H
1RO
1N02
2N02
0.5RC03 0«5H02
1RO 1ALD
1ALD
0.5RC03 0.5H02
ORC03 1H02
IRQ 1ALD
1RO 1ALD
1ALD
1ALD
0.001 0.00835
2c66E-l
2.00E-5
2.08E+1
4865E-2
lc50E+4
4.50E+3
2.70E+1
l.OOE-5
2.10E-ft
4.50EOO
1.30E-2
1.50E+4
7«OOE*2
5.30E+3
1.06E-3
2.50E-3
2.30E+4
9.10E*2
1.20E+4
9.10E+2
l.OOE+2
2.40E-2
2.50Ef2
4.90E+2
1.38E+4
4.5000E-1
6.0000E-1
1.5000E+4
4.0000E-1
1 .9776E*3
0.64E-2
0.70E+4
6.80E+3
7.72E+2
1.60E-2
4.00E-3
2.50E+4
2..50E+3
-------
274
EXHIBIT A-8. SAMPLE MODKIN INPUT (Concluded)
H202
ALD
S02
OLEF
HN03
0
N03
OH
RO
PAN
RNO?
RN03
503
ETHY
PROP
M
02
H20
C02
S02
N02
NO
ETHY
PROP
N02
NO
03
S02
PAN
OLEF
0.10
0.334
1.90
l.OE-9
l.OE-10
5.0E-10
0.0
0.19
1.71
1000000.0
210000.0
20000.0
lOOOOcO
5
0
231
7
0
157
7
0
157
7
a
177
7
0
177
9 0 10*0
8
117
297
7 0 10+0
8
117
10 0 10+0
11
.118
298
10 0 10+0
11
108
305
7 0 10+0
30
225
16 0 10+0
16
66
116
237
0.291
0*328
0.035
0.040
0.35
0.36
0.184
0.197
1.656
1.773
0.00 0.15
0.09
0.26
0.090
0.00 0.15
0.26
0.00
0.00 0.15
0.025
0.54
0.45
0.00 0.15
0.353
0.136
0.241
0.00 0.15
0.000
0.211
0.00 0.50
1.R5
1.16
0.71
0.69
49
29
187
29
187
35
271
35
271
0.30 0.45
29
157
0.30 0.45
29
157
0.30 0.45
32
158
298
0.30 0.45
11
173
305
0.30 0.45
101
266
1.00 1.50
31
71
132
261
0.241
0.039
0.043
0.36
0.39
0.193
0.200
1.737
1.800
0.60
0.27
0.12
0.60
0.16
0.010
0.60
0.036
0.46
0.43
0«60
0.384
0.291
0.415
0.60
0.054
0.220
2.00
1.78
1.09
0.69
0.70
108
64
297
64
297
91
330
91
330
0.0
44
187
O.'O
44
187
0.0
47
188
0.0
47
231
0.0
162
311
0.0
42
86
151
301
0.396
0.040
0.034
0.36
0.31
0.201
0.204
1.809
1.836
0.6
0.40
0.14
0.60
0.05
0.00
0.6
0.314
0.41
0.6
0.285
0.173
0.6
0.109
0.213
2.0
1.63
0.88
0.70
0.68
173
117
117
121
121
0.0
64
228
0.0
64
0.0
67
229
0.0
108
231
0.0
185
0.0
51
102
191
323
0.409
0.048
0.43
0.202
1.818
400.0
0.33
0.12
400.0
0.00
400.0
0.496
0.41
400.0
0.186
0.080
400.0
0.197
400.0
1.56
0.72
0.69
0.65
-------
275
EXHIBIT A-9. SAMPLE MODKIN OUTPUT—SELECTED PAGES
MODULAR KINETICS RUN NO. SAMPLE DECK
TOTAL NUMBER OF REACTIONS = 38
NUMBER OF LUMPED REACTIONS = 3
NUMBER OF DIFFERENTIAL SPECIES = 13
NUMBER OF STEADY STATE SPECIES = 4
NUMBER OF UNCOUPLED SPECIES = 4
NUMBER OF REPLACEMENT SPECIES = 2 '
NUMBER OF INERT OR CONSTANT SPECIES = 4
NUMBER OF FLOWING SPECIES = 5
REACTION RATE PRINT REQUEST FLAG = 1
TIME INCREMENT = 5.000E 00 MINUTES
ENDING TIME = 3.750E 02 MINUTFS
STARTING STEP SIZE = l.OOOE-04 MTNUTES
MINIMUM STEP SIZE = l.OOOE-05 MINUTES
MAXIMUM STEP SIZE = 5.000E 00 MINUTES
CONVERGENCE TOLERANCE = l.OOOE-03
DILUTION RATE = 8.350E-03 MINUTES<-1>
-------
276
EXHIBIT A-9. SAMPLE MODKIN OUTPUT—SELECTED PAGES (Continued)
P. CONST.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
2.660E-01
2..000E-05
2.080E 01
4.650E-02
1.500E 04
4.500E 03
2.700E 01
l.OOOE-05
2.100E-06
4.500E 00
1.300E-02
1.500E 04
7.000E 02
5.300E 03
1.060E-03
2.500E-03
2.300E .04
9.100E 02
1.200E 04
9.100E 02
l.OOOE 02
2o400E-02
2.500E 02
4.900E 02
0.0
OoO
0.0
1.380E 04
0.0
0.0
0.0
4.500E-01
6.000E-01
1.500E 04
4.000E-01
1.978E 03
6.400E-03
7.000E 03
LIST OF REACTIONS
REACTANTS PRODUCTS
H20
02
NO
N02
NO
N02
H20
N02
HN02
N02
NO
H02
OH
NO
NO
NO
N02
02
NO
N02
RO?
H02
NO
N02
N02
HN03
HN03
S02
S02
S02
S02
0
03
OH
N02
0
03
03
N03
N03
N205
N205
NO
HN02
HN02
OH
H02
H02
H202
ALO
ALD
R02
OH
RC03
RC03
RO
RO
RO
R02
R02
0
0
0
NO
HN02
H02
R02
N03
N205
OLEF
OLEF
OLEF
=
=
=
=
=
t:
=
=
=
=
=
=
=
=
z
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
—
=
1
1
1
1
2
1
1
2
2
1
1
1
1
1
2
0
0
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
«
e
9
e
«
a
©
e
e
o
o
o
e
a
»
a
0
e
o
e
0
0
e
a
»
o
«
a
o
e
o
o
e
a
e
*
a
c-
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
A3
63
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
NO
03
N02
N03
N02
N205
N02
HN03
HN02
NO
OH
HN03
OH
H202
OH
R02
RC03
RO
HN02
R02
PAN
H02
RN02
RN03
RO
PO
N02
NO
N03
HN02
H20
503
S03
S03
S03
R02
RC03
R02
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
2
1
1
1
2
0
1
1
.00
.00
»00
.00
.00
.00
.00
.00
.00
.37
.37
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.50
.00
.00
0
M
02
02
N03
N02
NO
N02
02
H02
H02
N02
N02
ALD
OH
02
N02
N02
OH
RO
N02
N02
RC03
RO
ALD
1.00 H20
1.00 H20
1.00 C02
0.50 H02
1.00 ALD
THE FOLLOWING SET OF 2 REACTIONS CORRESPONDS TO REACTION NUMBER 36
39 6.800E 03
40 7-720E 02
0 PROP = 1.00 R02 0.50 RC03 0.50 H02
0 ETHY = 1.00 R02 0.0 RC03 1.00 H02
THE FOLLOWING SET OF 2 REACTIONS CORRESPONDS TO REACTION NUMBER 37
41 1.600E-02
42 4.000E-03
03 PROP = 1.00 RC03 1.00 RO 1.00 ALD
03 ETHY = 1.00 RC03 1.00 RO 1.00 ALD
THE FOLLOWING SET OF 2 REACTIONS CORRESPONDS TO REACTION NUMBER 38
43 2.500E 04
44 2.500E 03. '
OH PROP = 1.00 R02 1.00 ALD
OH ETHY = loOO R02 1.00 ALD'
-------
277
EXHIBIT A-9. SAMPLE MODKIN OUTPUT-SELECTED PAGES (Continued)
INITIAL SPECIES CONCENTRATIONS
SPECIES VALUE SPECIES VALUE SPECIES VALUE SPECIES VALUE
DIFFERENTIAL(PPM)
NO? 8.000E-0? NO 2.700E-01 03 2.200E-02 N205 0.0
HO? 0.0
H202 0.0
HM03 0.0
R02 0,0
RC03 0.0
HN02 0.0
ALD l.OOOE-01 S02 3.340E-01 OLEF 1.900E 00
STEADY STATE(PPM)
0 l.OOOE-09 N03 loOOOE-10 OH 5oOOOE-10 RO 0.0
UNCOUPLED(PPM)
PAN 0.0 RN02 0.0 RN03 0.0 S03 0»0
REPLACEMENT(PPM)
ETHY 1.900E-01 PROP 1.710E 00
INERT/CONSTANT(PPM)
M l.OOOE 06 02
2.100E 05 H20 2.000E 04 C02 l.OOOE 04
TIME = 5.021E 00 MINUTES
SPECIES VALUE SPECIES VALUE SPECIES VALUE SPECIES VALUE
DIFFERENTIAL(PPM)
N02 1.856E-01 NO
HO? 1.367E-04 R02
H202 1.937E-04 ALD 1.904E-01
HN03 2.405E-03
1.505E-01 03 1.436E-02 N205 5.113E-07
9.789E-05 RC03 80061E-06 HN02 1.142E-02
S02 3.319E-01 OLEF 1.773E 00
STEADY STATE(PPM)
0 1.172E-08 N03 1.706E-08 OH 2.960E-07 RO 2.670E-06
UNCOUPLED(PPM)
PAN 2.988E-04 RN02 5.147E-04 RN03 7.154E-04 S03 3.069E-04
REPLACEMENT(PPM)
ETHY 1.816E-01 PROP 1.5S91E 00
-------
278
EXHIBIT A-9. SAMPLE MODKIN OUTPUT-SELECTED PAGES (Continued)
REACTION RATES (SORTED INTO DECREASING SIZE)
NO.
RATE
NO.
RATE
NO.
RATE
NO.
RATE
NO.
RATE
1
18
1?
24
23
32
8
27
4.94E-02
1.34E-02
8.24E-04
2.43E-04
l.OOE-04
2.04E-05
1.02E-07
0.0
2
38
10
21
14
33
35
26
4.92E-02
1.19E-02
5.87E-04
1.50E-04
9.91E-05
le95E~05
6.79E-08
0.0
3
17
19
11
34
6
31
25
4.49E-02
1.30E-03
5.34E-04
1.48E-04
8.50E-05
1.43E-05
0.0
0.0
13
9
16
36
5
7
30
1
1
4
1
3
1
0
.44E-02
.17E-03
.76E-04
.28E-04
.85E-05
.38E-05
.0
22
20
37
4
28
15
29
1.
1.
3.
1.
3.
2.
0.
35E-02
10E-03
76E-04
24E-04
OOL-05
05E-07
0
TIME =
SPECIES VALUE SPECIES
DIFFERENTIAL(PPM)
1.007E 01 MINUTES
VALUE
SPECIES
VALUE
SPECIES
VALUE
N02
K02
H202
HN03
2.753E-01
4.046E-04
2.104E-03
7.896E-03
NO
R02
ALD
5.053E-02
2.890E-04
3.073E-01
03
RC03
S02
5.492E-02
3.710E-05
3.285E-01
N205
HN02
OLEF
5.508E-06
1.311E-02
1.634E 00
STEADY STATE(PPM)
1.738E-08 N03 1.230E-07
OH
2.982E-07
UNCOUPLED(PPM)
PAN
2..737E-03 RN02 8.319E-04 RN03 2.355E-03
RO
S03
2.827E-06
2.064E-03
REPLACEMENT(PPM)
ETHY 1.733E-01 PROP 1.A61E 00
REACTION RATES (SORTED INTO DECREASING SIZE)
NO.
RATE
NO.
RATE
NO,
RATE
NO.
RATE
NO.
RATE
1
18
12
4
36
28
8
27
7.32E-02
1.33E-02
1.23E-03
7.03E-04
1.75E-04
6.60E-05
1.10E-06
0.0
' 2
38
21
34
11
32
35
2ft
7.30E-02
1.10E-02
1.02E-03
6«06E-04
1.70E-04
5.98E-05
7.24E-07
0.0
3
17
14
9
6
33
31
25
5.77E-02
2.11E
8.68E
5.84E
1.52E
5.70E
0.0
0.0
-03
-04
-04
-04
-05
13
20
10
24
7
23
30
K43E-02
1.71E-03
7.74E-04
3.81E-04
1.49E-04
3.57E-05
OoO
22
37
16
19
5
15
29
1
1
7
1
9
2
0
.43E-02
.3PE-03
.68E-04
.81E-04
.32E-05
.23E-06
.0
-------
279
EXHIBIT A-9. SAMPLE MODKIN OUTPUT-SELECTED PAGES (Continued)
INCOMING S02 CONCENTRATION CHANGED TO 3.960E-01 AT 1.103E 02 WIN.
TIME =
1.160E 02 MINUTES
SPECIES VALUE SPECIES VALUE SPECIES VALUE SPECIES VALUE
DIFFERENTIALtPPM.)
N02
H02
H202
HN03
1.451E-01
5.998E-04
1.795E-01
3.199E-02
NO
R02
ALD
7o451E~03
4.193E-04
6.646E-01
03
RC03
S02
2.239E-01
1.155E-04
lo532E-01
N205
HN02
OLEF
1.497E-05
2.416E-03
3.383E-01
STEADY STATE(PPM)
0 9.179E-09
UNCOUPLED(PPM)
PAN 1.826E-01
REPLACEMENT(PPM)
N03
6.252E-07
OH
1.454E-07
RO
7.696E-07
RN02 6.014E-04 RN03 1.037E-02 S03 1.405E-01
ETHY 6.209E-02 PROP 2.762E-01
REACTION RATES (SORTED INTO DECREASING SIZE)
NO.
RATE
NO.
RATE
NO.
RATE
NO.
RATE
NO,
RATE
1
18
4
6
24
10
23
27
3.86E-02
2.84E-03
1.51E-03
4.08E-04
5.47E-05
2.63E-05
1.43E-06
0.0
2
17
34
7
9
2fi
35
26
3.86E-02
2.22E-03
1.44E-03
4.04E-04
4.54E-05
1.84E-05
9.17E-07
0.0
3
14
37
12
32
36
31
25
3.
1.
1.
3.
4.
1.
Oo
0.
47E-02
91E-03
05E-03
16E-04
14E-05
78E-05
0
0
22
21
38
15
33
19
30
3.88E-03
1.68E-03
1.03E-03
1.90E-04
3.86E-05
1.30E-05
0.0
13
16
20
5
11
8
29
3
1
7
6
3
2
0
.13E-03
.66E-03
.84E-04
.99E-05
.14E-05
.99E-06
.0
INCOMING N02 CONCENTRATION CHANGED TO 40flOOE-02 AT 1.185E 02 WIN.
INCOMING NO CONCENTRATION CHANGED TO 4.300E-oi AT I.ISSE 02 MIN.
TIME =
1.204E 02 MINUTES
SPECIES VALUE SPECIES VALUE SPECIES VALUE SPECIES VALUE
DIFFERENTIAL(PPM)
N02 1.458E-01
H02 5.840E-04
H202 1.802E-01
HN03 3.225E-02
NO 7.718E-03 03 2.208E-01 N205 1.46RE-05
R02 3.974E-04 RC03 1.112E-04 HN02 2.422E-03
ALD 6.492E-01 S02 1.554E-01 OLEF 3.173E-01
-------
280
EXHIBIT A-9. SAMPLE MODKIN OUTPUT—SELECTED PAGES (Continued)
STEADY STATE(PPM)
0 9.225E-09
UNCOUPLED(PPM)
N03
6.102E-07
OH
1.514E-07
PAN 1.831E-01 RN02 5.857E-04 RN03 1.023E-02
REPLACEMENT(PPM)
ETHY 5.951E-02 PROP 2.57BE-01
RO
S03
REACTION RATES (SORTED INTO DECREASING SIZE)
NO.
RATE
NO,
RATE
NO,
RATE
NO.
RATE
7.428E-07
1.420E-01
NO.
RATE
1 3.88E-02
18 2.79E-03
4 1.50E-03
6 4.00E-04
24 5.31E-05
10 2»64E-05
23 1.43E-06
27 0.0
2 3.87E-02
17 2.26E-03
34 1.42E-03
7 3.96E-04
9 4.73E-05
28 1.86E-05
35 9.12E-07
26 0»0
3 3*54E-02
14 lo81E-03
38 1,OOE-03
12 3.31E-04
32 4.08E-05
36 1.67E-05
31 0»0
25 0»0
22
16
37
IS
33
19
30
3.74E-03
1.62E-03
9.70E-04
1.91E-04
3.71E-05
1.40E-05
0.0
13
21
20
5
11
8
29
3.16E-03
1.62E-03
7o81E-04
7.06E-05
3.15E-05
2.93E-06
0.0
INCOMING ETHY CONCENTRATION CHANGED TO
INCOMING PROP CONCENTRATION CHANGED TO
2.020E-01 AT
1.818E 00 AT
1.226E 02 MIN.
1.226E 02 MIN.
TIME =
1.298E 02 MINUTES
SPECIES
VALUE
SPECIES
DIFFERENTIAL(PPM)
VALUE SPECIES
VALUE
SPECIES
VALUE
N02
H02-
H202
HN03
1.514E-01
5.474E-04
1.801E-01
3.307E-02
NO
R02
ALD
8.386E-03
3.493E-04
6»146E-01
03 2.122E-01
RC03 1.003E-04
S02 1.600E-01
N205 1.474E-05
HN02 2.508E-03
OLEF 2.763E-01
STEADY STATE(PPM)
0 9.581E-09
N03
5.899E-07
OH
1.647E-07
RO
6.859E-07
-------
281
EXHIBIT A-9. SAMPLE MODKIN OUTPUT-SELECTED PAGES (Continued)
TIME =
3.760E 02 MINUTES
SPECIES VALUE SPECIES VALUE' SPECIES VALUE SPECIES VALUE
DIFFERENTIAL(PPM)
N02
HO?
H202
HN03
1.518E-01
6.287E-06
3.019E-02
6.088E-02
NO
R02
ALD
1.044E-01
2.154E-06
5«619E-02
03
RC03
S02
K856E-02
1.025E-06
2.635E-01
N205
HN02
OLEF
5.930E-07
1.166E-02
1.002E-02
STEADY STATE(PPM)
0 9.607E-09
UNCOUPLED(PPM)
PAN 5«204E-02
REPLACEMENT(PPM)
N03
2e370E-08
OH
1.367E-07
RN02 1.897E-04 RN03 2.607E-03
RO
S03
4.048E-08
7.403E-02
ETHY 5.277E-03 PROP 4.743E-03
NO.
REACTION RATES (SORTED INTO DECREASING SIZE)
RATE NOo RATE NO. RATE NO. RATE
MO,
RATE
1
13
19
34
6
23
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288
TECHNICAL REPORT DATA
(Please read fatiniciions on tlic reverse before completing)
1. REPORT NO. ~~~
EPA-600/4-76-016b
2.
4. TITLE CONTINUED RESEARCH IN MESOSCALE AIR POLLUTION
SIMULATION -MODELING. VOLUME II. Refinements
in the Treatment of Chemistry, Meteorology,
and Numerical Integration Procedures
6. PERFORMING ORGANIZATION CODE
3. RECIPIENT'S ACCESSION-NO.
5. REPORT DATE
May 1976
'. AUTHOR(S)
S. D. REYNOLDS, J. P. MEYER, T. A. HECHT,
D. C. WHITNEY, J. AMES, AND M. A. YOCKE
8. PERFORMING ORGANIZATION REPORT NO.
EF75-69
9. PERFORMING ORG ^.NIZATION NAME AND ADDRESS
SYSTEMS APPLICATION
950 NORTHGATE DRIVE
SAN RAFAEL, CALIFORNIA 94903
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
68-02-1237
12. SPONSORING AGENCY NAME AND ADDRESS
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U. S. ENVIRONMENTAL PROTECTION AGENCY
.RESEARCH ..TRIANGLE PARK, N.C. 27711
13. TYPE OF RE PORT AND PERIOD COVERED
FINAL REPORT 6/74-6/75
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This report describes the refinement of a .mesoscale photochemical air quality
simulation model through studies of selected chemical and meteorological phenomena
that contribute to air pollution. The chemistry activities focused on the design of
an automatic computer program for evaluating kinetic mechanisms, the improvement of
a photochemical mechanism for incorporation in mesoscale models, and the development
of a chemical mechanism for describing S02 oxidation. The meteorology studies ex-
amined the sensitivity of the model to the inclusion of wind shear, algorithms for
deriving mass-consistent wind fields, -and the treatment of turbulent diffusivities
and elevated inversion layers. Alternative numerical techniques for solving the
advection/diffusion equation in grid models are evaluated, including various finite
difference, particle-in-cell, and finite element methods, in an attempt to find a
suitable methodology for accurately calculating the horizontal transport of pollu-
tants. Finally, the report considers the problem of multiday model usage and pre-
sents results from a two-day CO simulation for the Los Angeles basin.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. cos AT I Field/Group
*Air Pollution
*Photochemical Reactions
*Reaction Kinetics
*Numerical Analysis
^Mathematical Models
^Meteorological Data
13B
07E
07D
12A
14B
04B
8. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (Tills Report)
UNCLASSIFIED
1. NO. OF PAGES
287
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (3-73)
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