November 1983
' i r ^
' / s
PLUME CONCENTRATION ALGORITHMS WITH
DEPOSITION, SEDIMENTATION, AND CHEMICAL TRANSFORMATION
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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PLUME CONCENTRATION ALGORITHMS WITH
DEPOSITION, SEDIMENTATION, AND CHEMICAL TRANSFORMATION
by
K. Shankar Rao
Atmospheric Turbulence and Diffusion Division
National Oceanic and Atmospheric Administration
Oak Ridge, Tennessee 37830
IAG-AD-13-F-1-707-0
Project Officer
Jack H. Shreffler
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for pub-
lication. Approval does not signify that the contents necessarily re-
flect the views and policies of the U.S. Environmental Protection Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
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ABSTRACT
A gradient-transfer model for the atmospheric transport, diffusion,
deposition, and first-order chemical transformation of gaseous and particu-
late pollutants emitted from an elevated continuous point source is formulated
and analytically solved using Green's functions. This analytical plume model
treats gravitational settling and dry deposition in a physically realistic
and straightforward manner. For practical application of the model, the eddy
diffusivity coefficients in the analytical solutions are expressed in terms of
the widely-used Gaussian plume dispersion parameters. The latter can be speci-
fied as functions of the downwind distance and the atmospheric stability class
within the framework of the standard turbulence-typing schemes.
The analytical plume algorithms for the primary (reactant) and the
secondary (product) pollutants are presented for various stability and mixing
conditions. In the limit when deposition and settling velocities and the
chemical transformation rate are zero, these equations reduce to the well-
known Gaussian plume diffusion algorithms presently used in EPA dispersion
models for assessment of air quality. Thus the analytical model for estimating
deposition and chemical transformation described here retains the ease of appli-
cation associated with Gaussian plume models, and is subject to the same basic
assumptions and limitations as the latter.
A new mathematical approach, based on mass budgets of the species, is
outlined to derive simple expressions for ground-level concentrations of the
primary and secondary pollutants resulting from distributed area-source emissions.
These expressions, which involve only the point-source algorithms for the well-
mixed region, permit one to use the same program subroutines for both point
and area sources. Thus the area-source concentration equations developed in
this report are simple, efficient, and accurate.
The new point-source algorithms are applied to study the atmospheric transport
and transformation of SCL to SO,, and deposition of these species. Calculated
variations of the ground-level concentrations are presented and discussed. The
results of a sensitivity analysis of the concentration algorithm for the secon-
dary pollutant are given. The specification of gravitational settling and depo-
sition velocities in the model is discussed.
The work described in this report was undertaken to develop concentration
algorithms for the Pollution Episodic Model (PEM). This report was submitted
by NOAA's Atmospheric Turbulence and Diffusion Division in partial fulfillment
of Interagency Agreement No. AD-13-F-1-707-0 with the U. S. Environmental
Protection Agency. This work, covering the period September 1981 to March 1983,
was completed as of March 31, 1983.
111
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CONTENTS
Abstract iii
Figures vi
Symbols and Abbreviations viii
Acknowledgements xi
I. Introduction 1
2. Literature Survey 4
3. The Gradient-Transfer Deposition Model 7
Mathematical formulations 8
Analytical solutions 11
Parameterization of concentrations 28
Well-mixed region 35
Plume trapping 40
Summary of point source concentration equations 42
Surface deposition fluxes 43
Area sources 43
4. Results and Discussion 60
Sensitivity analyses 61
Ground-level concentrations 67
Chemical transformation rate 71
Effects of atmospheric stability 74
5. Summary and Conclusions 79
References 82
Appendix
Settling and Deposition Velocities 84
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FIGURES
Number
1 Schematic diagram for area-source algorithms showing (a) a grid
square with emissions and a grid square with receptor, and the
distances; (b) a cross-section of the calculation grid square,
and the incoming and outgoing normalized fluxes of pollutant .... 45
2 Schematic diagram showing a single grid square with emissions
and four downwind calculation grid squares with receptors, and
the distances used in area-source algorithms 48
3 Schematic diagram showing a single grid square with receptor
and four upwind grid squares with emissions, and the dis-
tances used in area-source algorithms 50
4 Variation of the three major terms in the point-source
algorithm for the GLC of the secondary pollutant. A
non-zero concentration results from the small positive
imbalance between the terms 62
5 Variation of the weighting function, in the point-source
algorithm for the GLC of the secondary pollutant, as a
function of the effective source height 64
6 Variation of the weighting function, in the point-source
algorithm for the concentration of the secondary pollutant,
as a function of the receptor height 65
7 Variation of the weighting function, in the point-source
algorithm for the GLC of the secondary pollutant, as a
function of the deposition velocity of the primary
pollutant 66
8 Variation of the weighting function, in the point-source
algorithm for the GLC of the secondary pollutant, as a
function of its deposition velocity 68
9 Variation of the calculated GLC of the primary pollutant
for different values of the parameter V.../U, and W = 0 69
10 Variation of the calculated GLC of the secondary pollutant
for different values of the parameter Vj2/U, and VL = VL = 0 .... 70
11 Comparison of the calculated GLC of the secondary pollutant
when it is made of~(a) particles with V,2 = W_ = 1 cm/s
or VJ2/U = 5 x 10 , and (b) gas with V^ = 1 cm/s and
W2 = 0 or V12/U = 10"2 72
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FIGURES (Continued)
Number Page
12 Variation of the calculated GLC of the secondary pollutant
for an arbitrary variation of the chemical transformation
rate by two orders of magnitude 73
13 Variation of the calculated GLC of the primary pollutant
for an arbitrary variation of the chemical transformation
rate by two orders of magnitude 75
14 Variation of the calculated GLC of the primary pollutant as
a function of the P-G atmospheric stability class 76
15 Variation of the calculated GLC of the secondary pollutant
as a function of the P-G atmospheric stability class 78
VI1
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SYMBOLS AND ABBEEVIATIONS
SYMBOLS
C-, C2 mean concentrations of primary and secondary pollutants
for point source
C.j , C.2 mean concentrations of primary and secondary pollutants
for area source
D. , D? surface deposition fluxes of primary and secondary pollutants
F- , F2 Weighting functions in secondary pollutant concentration
algorithms
G, , G2, G_ Green's functions
H effective height of source
k chemical transformation rate
K , K eddy diffusivities in y and z directions
L height of the inversion lid
L , L length scales of concentration distribution
in y and z directions
p probability density of concentration distribution
in y direction
Q1 , Q2 source strengths or emission rates of primary
and secondary pollutants
q- , q2 probability densities of concentration distributions
of primary and secondary pollutants in z direction
U mean wind speed
deposition velocities of primary and secondary
pollutants
V V - W /2
11 dl 1/Z
V12 Vd2 - V2
Vi3 vn - (wi
Vlll
-------
V V - W
V21 Vdl Wl
V V - W
22 d2 2
W1, VL gravitational settling velocities of primary and
secondary pollutant particles
x, y, z horizontal downwind, horizontal crosswind,
and vertical coordinates
x downwind distance at which cr = 0.47L
m z
a , a Gausssian dispersion parameters in y and
z directions
t characteristic time scale of chemical transformation
NONDIMENSIONAL QUANTITIES
g.. crosswind diffusion function
82 x < x
vertical diffusion function for point source when
< x
= m
gi-, , g~2 89 m°dified (for deposition and chemical transformation)
for primary and secondary pollutants
g_ vertical diffusion function for point source in plume-
trapping region (x < x < 2 x )
gi^ , g' g.- modified (for deposition and chemical transforma-
tion) for primary and secondary pollutants
g, vertical diffusion function for point source in
well-mixed region (x > 2 x )
g! 1 , gj . g, modified (for deposition and chemical transforma-
tion) for primary and secondary pollutants
H H/V2~a
Z
L/V2 a
Vdl/U ' Vd2/U
vn , v12 vn/u , v12/u
IX
-------
v13 v13/u
v21 , v22 v21/u , v22/u
, w2 Wj/u , w2/u
x, z x/V2 CT , z/«/2 a
Z ^
y y/V2 a
c
ratio of molecular weight of secondary pollutant
to molecular weight of primary pollutant
ABBREVIATIONS
ATDL Atmospheric Turbulence and Diffusion Laboratory
EPA Environmental Protection Agency
GLC Ground-level concentration
KST Atmospheric stability class index
P-G Pasquill-Gifford
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ACKNOWLEDGEMENTS
This report was prepared for the Office of Research and Development,
Environmental Sciences Research Laboratory of the U.S. Environmental
Protection Agency to support the needs of the EPA's Office of Air Quality
Planning and Standards in urban particulate modeling. This work was accomplished
under interagency agreements among the U.S. Department of Energy, the National
Oceanic and Atmospheric Administration, and the EPA. The author is grateful to
Dr. Jack Shreffler of ESRL for the opportunity to do this work, and for his
interest and patience. The author expresses his appreciation to the following
members of the Atmospheric Turbulence and Diffusion Laboratory: Director Bruce
Hicks for his understanding and support, Dr. Ray Hosker for many useful sugges-
tions during the course of this work; Ms. Louise Taylor for making the model test
runs and plotting the results on computer; and Ms. Mary Rogers for her expert
technical typing and patient revisions. Special thanks are due to Dr. Frank
Gifford for his interest and useful discussions.
XI
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SECTION 1
INTRODUCTION
Pollutant gases and suspended particles released into the atmosphere are
transported by the wind, diffused and diluted by turbulence, and removed by
several natural processes. An important removal mechanism is dry deposition of
pollutants on the earth's surface by gravitational settling, eddy impaction,
chemical absorption, and other effects. Another significant removal mechanism
is chemical transformation in the atmosphere. Depletion of airborne pollutant
material by these natural processes affects pollutant concentrations and
residence times in the atmosphere. Moreover, the product of a chemical
reaction may be the pollutant of primary concern, rather than the reactant
itself. Surface deposition of acidic and toxic pollutants may adversely impact
on local ecology, human health, biological life, structures, and ancient
monuments. Furthermore, large concentrations of particulate products resulting
from chemical reactions may lead to significant deterioration of atmospheric
visibility. It is important, therefore, to obtain reliable estimates of the
effects of dry deposition and chemical transformation.
This report presents an analytical plume model for diffusion, dry
deposition, and first-order chemical transformation of gaseous or particulate
pollutants released from an elevated continuous point source, based on gradient-
transfer or K-theory. The method solves the atmospheric advection-diffusion
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equation subject to a deposition boundary condition. The model includes
similar and complementary sets of equations for the primary (reactant)
and the secondary (product) pollutants. These equations are analytically
solved using Green's functions.
In order to facilitate practical application of the model to air pollution
problems, the K-coefficients are expressed in this report in terms of the
widely-used Gaussian plume dispersion parameters which can be easily obtained
from standard turbulence-typing schemes. The parameterized diffusion-deposi-
tion-transformation algorithms for various atmospheric stability and mixing
conditions are simplified, and presented as analytical extensions of the well-
known Gaussian plume-diffusion algorithms presently used in EPA models for air
quality assessment.
For practical application of the model to urban air pollution problems,
a new mathematical approach, based on mass balance considerations, is developed
to derive simple expressions for ground-level concentrations of the reactant
and the product pollutants resulting from distributed area-source emissions.
These novel expressions for area sources involve only the point-source algorithms
of the well-mixed region, thus allowing one to use the same program subroutines
for both point and area sources. In the limit when deposition rates approach
zero, the concentrations calculated by these expressions agree with the corres-
ponding values given by the area-source algorithms without deposition, currently
used in urban air pollution models.
This report gives a brief review of the literature on gradient-transfer
models with chemical transformation. Details of the mathematical formulations
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and analytical solutions of the present model with deposition, sedimentation,
and chemical transformation are given, and the parameterized concentration
algorithms for the primary and the secondary pollutants are listed. Calculated
variations of the ground-level concentrations and results of a sensitivity
analysis are presented and discussed. Some guidance is provided for the
specification of the settling and deposition velocities in the model.
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SECTION 2
LITERATURE SURVEY
Applied air pollution models used in industry and regulation are generally
based on the Gaussian plume formulation. These models have been extensively
modified over the years to include pollutant removal mechanisms such as dry
and wet deposition, and chemical decay. Rao (1981) gave a brief review of the
existing methodologies of Gaussian diffusion-deposition models, including a
comprehensive literature survey of gradient-transfer (K-theory) models. For
the latter case, Rao (1981) also gave the mathematical formulations (for a non-
reactive pollutant), analytical solutions, parameterized concentration algorithms
for various atmospheric stability and mixing conditions, and expressions for net
deposition and suspension rates of the pollutant.
In this section, we briefly review the literature on K-theory or Gaussian
models for chemically reactive pollutants. Only a first-order chemical trans-
formation is considered, and expressions for both the primary (reactant) and
secondary (product) pollutants are given in the references cited.
Heines and Peters (1973) studied the diffusion and transformation of
pollutants from a continuous point or infinite line source. The effect of
a temperature inversion aloft was also included through multiple eddy reflec-
tions. The eddy diffusion coefficients were assumed to be power functions of
the downwind distance. The expression for concentration of the secondary
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pollutant was derived from a simple component mass balance. The concentrations
of the reactant and product were presented in terms of dimensionless plots.
Deposition and sedimentation were not considered for either species.
Rao (1975) adapted the analytical solution of Monin (1959) to study the
dispersion, deposition, and chemical transformation of the SO- plume from a
power plant stack represented by an elevated continuous point source. The
eddy diffusivities were expressed in terms of the Gaussian dispersion para-
meters. A constant first-order transformation rate of SO- to SOT was assumed.
Concentrations of both species were calculated, and compared with observations
at several downwind receptors. Izrael, Mikhailova, and Pressman (1979) used
Monin1s (1959) instantaneous source solution to estimate the long-range transport
of sulfur dioxide and sulfates, assuming constant eddy diffusivities and non-
equal deposition velocities for the two species.
Ermak (1978) described a multiple point-source dispersion model which
considered a chain of up to three first-order chemical transformations. The
source-depletion approach (e.g., Van der Hoven, 1968) was used to include ground
deposition. The gradient-transfer plume model of Ermak (1977) was used to
incorporate the effects of gravitational settling of particles. For the latter
case, however, the chemical-transformtion option could not be used, and the
effects of an inversion layer were assumed to be negligible.
Lee (1980) gave analytical solutions of a gradient-transfer model similar
to that described in this report, in terms of constant eddy diffusivity coeffi-
cients. These solutions apply only to gases or very small particles emitted
from an elevated continuous point source, since the gravitational settling
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effects were not considered. A first-order chemical transformation was con-
sidered, and direct emission of the secondary pollutant was assumed to be zero.
Lee used this model, which included wet deposition processes, to study the
= 2
atmospheric transport and transformation of SO- to SO,, assuming K = K = 5 m /s.
Details of the analytical solutions were not available.
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SECTION 3
THE GRADIENT-TRANSFER DEPOSITION MODEL
This section gives the mathematical formulations and analytical solutions
of the gradient-transfer (K-theory) model for the most general case that includes
transport, diffusion, deposition, sedimentation, and first-order chemical
transformation of pollutants. We consider two chemically-coupled gaseous or
particulate pollutant species; the primary (species-1 or reactant) pollutant
is assumed to transform into the secondary (species-2 or reaction product)
pollutant at a known constant rate. For generality, the two species are
assumed to have known non-equal deposition and settling velocities.
First, we derive analytical solutions for concentrations of the two
pollutant species emitted from an elevated continuous point source. Then
we express the K-coefficients in these solutions in terms of the widely used
Gaussian plume dispersion parameters. The resulting expressions are parame-
terized, simplified, and presented as extensions of the Gaussian plume algorithms
currently used in EPA air quality models for various atmospheric stability and
mixing conditions. Further simplifications of the new algorithms are indicated
for gaseous or fine suspended particulate pollutants with negligible settling,
and for ground-level sources and/or receptors. Limiting expressions of the
algorithms are derived for large particles, when gravitational settling is
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the dominant deposition mechanism. Finally, we utilize these new point source
concentration algorithms to derive expressions for the concentrations of the
two species due to emissions from area sources, using an innovative approach
based on mass balance considerations.
No assumptions are made here regarding the nature of the pollutant
species. The formulations and the solutions are, therefore, general enough
to be applicable to any two gaseous or particulate pollutants that are
coupled through a first-order chemical transformation. Either of the two
species may be a gas, or particulate matter with a known average size.
Molecular weights of the two species are assumed to be known. Direct emission
of the secondary pollutant is permitted from both point and area sources. A
direct emission of secondary pollutant, if present, may contribute to its
concentration significantly more than the chemical transformation.
In the absence of a chemical coupling, expressions for concentrations of
two chemically-independent pollutants, each subject to deposition and/or
sedimentation, can be obtained as degenerate cases of the concentration
algorithms for the general case. The notation used in this section is similar
to that of Rao (1981), which is consistent with the notation presently used
in the user's guides for EPA's atmospheric dispersion models.
MATHEMATICAL FORMULATIONS
We consider the steady state form of the three-dimensional atmospheric
advection-diffusion equation for the primary pollutant (denoted by subscript 1)
with deposition, sedimentation, and first-order chemical transformation:
-------
D 3C /3x = K 32C /3y2 + K 32C_/3z2 + W. 3C../3Z - C./T (la)
x y x z i xx x c
Here, x, y, z are the horizontal downwind, horizontal crosswind, and vertical
coordinates, respectively; U is the constant average wind speed, and W is the
gravitational settling velocity (taken as positive in the downward or negative
z-direction) of the primary pollutant particles, C- is the primary pollutant
concentration at (x, y, z), K and K are constant eddy diffusivities in the
crosswind and vertical directions, respectively, and t = 1/k is the time scale
C t-
associated with the chemical transformation which proceeds at a given rate k .
The last term of the equation, - CL/T , represents the chemical sink, or loss
of the primary pollutant due to transformation.
For a continuous point source, with an emission rate or strength Q1 of
pollutant species-1, located atx=0, y=0, z=H, the initial and boundary
conditions are given by
C1(0, y, z) = QX/U 6(y) 6(z-H) (Ib)
C.(x, ±00, z) = 0 (Ic)
Cj(x, y, oo) = 0 (le)
In the initial condition (Ib), which is the limiting form of the mass continuity
equation at the source, 6 is the Dirac delta function. Boundary condition (Id)
-------
states that, at ground-level, the sum of the turbulent transfer of pollutant
down its concentration gradient and the downward settling flux due to the
particles' weight is equal to the net flux of pollutant to the surface resulting
from an exchange between the atmosphere and the surface; V,.. is the deposition
velocity which characterizes this exchange of the primary pollutant. When
deposition occurs, from Eq. (Id) the turbulent flux at the surface (z=0) is
given by
-w'cj = KZ - acyaz = (vdl - wp c1 > o,
which is in agreement with the standard micrometeorological notation. This
obviously requires that V, > W > 0. If 0 < V, < W , then the direction of
the turbulent flux at the surface is reversed, implying re-entrainment of the
particles from the surface into the atmosphere. Thus, the deposition boundary
condition (Id), originally suggested by Monin (1959) and discussed by Calder
(1961), adequately describes the exchange between the atmosphere and the sur-
face. Eq. (Id) is analogous to the so-called 'radiation' boundary condition
used in the theory of heat conduction (see, e.g., Carslaw and Jaeger, 1959,
p. 18) to describe the temperature distribution in a body from the boundary
of which heat radiates freely into the surrounding medium, when the latter is
at zero degrees temperature.
The corresponding formulations for the secondary pollutant (designated
by subscript 2) can be written as follows:
U 3C./3x = K 32C0/3y2 + K 82C./3z'i + W0 9C0/3z + Y C./T (2a)
2 y2 z2 i 2. 1C
10
-------
C2(0, y, z) = Q2/U - 6(y) - 6(z-H) (2b)
C,(x, ±», z) = 0 (2c)
(2d)
C0(x, y, «) = 0 ' (2e)
In Eq. (2a), VL is the gravitational settling velocity of the secondary
pollutant, and y is the ratio of its molecular weight to that of the primary
pollutant; the term yd-/! represents the chemical source for the secondary
pollutant. Eq. (2b) describes the direct emission of species-2 from the point
source, located at x=0, y=0, z=H, with an emission rate or strength 0~. Eq. (2d)
is the deposition boundary condition, and V,^ is the deposition velocity for the
secondary pollutant. For generality, we assume here V,_ ^ V,- and W~ ^ W...
ANALYTICAL SOLUTIONS
The solution of Eqs. (1) and (2) can be expressed as
C1(x, y, z) = Qj/U p(x,y) q-^x.z) (3)
C2(x, y, z) = Qj/U - p(x,y) q2(x,z) (4)
where p, q1, and q_ are probability densities of the concentration distributions.
It should be noted that the concentration of the secondary pollutant, C_, is
11
-------
expressed in terms of the emission rate of the primary pollutant, Q , in Eq. (4).
This is due to the likelihood that, for many practical applications of the
model, the direct emission rate Q_ of the secondary pollutant may be zero;
C~ would then be zero if Q? were used instead of Q in Eq. (4), thus incorrectly
ignoring the non-zero contribution to C_ by the chemical source. The proba-
bility density p(x,y) of the concentration distribution in the horizontal
crosswind direction is unaffected by deposition, sedimentation, and chemical
transformation. Therefore, it is identical for both species.
Substituting Eq. (3) in Eq. (1) and using the separation of variables
technique, two independent systems of equations and boundary conditions in
p and q, can be obtained as follows:
U 3p/3x = K 32p/3y2 , o
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The analytical solution of (5) can be written as
p(x,y) =
g1(x,y)
r i I -z-2 u
g (x,y) = exp - ^
I y
(7)
L = 2
x/U
where L is a length scale characteristic of the horizontal crosswind diffusion,
and g.(x,y) is a nondimensional function. This is one of the fundamental
solutions of the diffusion or heat conduction equation (e.g., see Carslaw and
Jaeger, 1963, p. 107); p(x,y) represents the probability that a particle released
from a source of unit strength located at x=0, y=0 will be at a crosswind loca-
tion y after travelling a distance x downwind with a speed of U.
Equation (6) cannot be solved in its present form because of the sedimenta-
tion term, W- dq^/dz, and the chemical sink term, -q,/! , in the differential
equation. In order to remove these terms, we apply the following transformation:
exp( -p1 - ~-
c
(8a)
where
WjCz-H)
2K
z
-L * ,
4K U
z
1/2
(8b)
is a nondimensional parameter representing the effect of sedimentation of the
primary pollutant particles on the primary pollutant concentration, and q,(x,z)
13
-------
is the transformed new variable. Substituting Eq. (8) in Eq. (6) and simplifying,
we obtain the following:
U 8q /3x = K 92q-/3z2 , 0 < x , z < » (9a)
JL Z J.
[ W (z-H) 1
q^O.z) = exp - | 6(z-H) = $ (2) (9b)
I z J
(9c)
qCx,*) = 0 (9d)
where
hl = Vll/Kz > Vll = Vdl - V2 (9e)
The homogeneous boundary condition (9c) expresses the relationship between
the variable q. and its normal gradient at the surface. This type of equation
is usually referred to as a boundary condition of the third kind; (A more
general form of this boundary condition is 3q../3z - h1q1 = f_. The Dirichlet
boundary condition of type q. = f1 and the Neumann boundary condition of type
3q1/3z = f_, are boundary conditions of the first and second kind, respectively.
Here, f f , and f., are functions of x given at the boundary, z = 0, of the region
1- * -J
in which the solution is sought). Equations (9a, c, d) constitute a homogeneous
boundary-value problem of the third kind. The solution of this problem, also
known as its Green's function G(x,z,4), can be obtained by Laplace transform
methods (Carslaw and Jaeger, 1963, p. 115), and written as follows:
z
ex? { "^V } + exp { "(£?i }
Z ' Z
(10a)
14
-------
where
L = 2
z z
(10b)
is a length scale characteristic of the vertical diffusion. Equation (10)
describes the stationary diffusion in a semi-infinite region from a unit source
located at x=0, z=|. The subscript 1 of the Green's function G corresponds to
the subscript of h appearing in Eqs. (9c) and (lOa).
The solution of Eq. (9) can be now written in terms of its Green's function
(see, e.g., Tychonov and Samarskii, 1964), as follows:
00
J
0^,2,4) d4,
where, from Eq. (9b),
W, (4-H)
(11)
(12)
Substituting (10) and (12) in (11), and noting that
/
-*0
f(4)
= f(a),
where f(|) is any arbitrary function of f, term-by-term integration of (11)
gives
qi(x,z) = r
exp
f ? \
- L h exp \ h (z+H) + h^ K x/U f erf c
21 l X i z j
x/U
x/U
(13)
15
-------
In order to simplify the equations, and considerably reduce the difficulty
in typing them, we define the following nondimensional parameters:
_ (z-H)2U _ (z+H)2!!
r ~ 4KZX ' S ~ 4KZX
______
+ h. ,/SIi/U (14)
Noting that
exp (h^z+H) + h2K2x/U) = exp(^-s) = A
Eq. (13) can be now written as
= - [e"r + e"S (1 - a] (15)
From Eqs. (8) and (15), the solution of Eq. (6) can be written as
qi(x,2) = g'2l I L2 , (16a)
where
-p^ - x/Ulc) [ e"r < e"S (1 - c^) J (16b)
is a nondimensional function. The prime indicates modification of function
g_(x,z) of the primary pollutant (denoted by the second subscript 1) for the
effects of deposition, sedimentation, and chemical transformation. In the
absence of these effects, i.e., when or, = p. = 0 and k = 0 or T = °°, Eq. (16b)
reduces to
16
-------
g21(x,z) = e"r + e~s = g2(x,z), (17)
as in the familiar Gaussian plume model. The notation used here is an extension
of Rao's (1981) notation for one pollutant, and is consistent with the notation
presently used in the user's guides for EPA air quality models.
We now turn our attention to the solution of Eq. (2). Substituting
Eqs. (3) and (4) in Eq. (2) and using the separation of variables technique,
two independent systems of equations and boundary conditions in p and q«
can be obtained. The p-equations and their solution are the same as those
given in Eqs. (5) and (7). The q--equations can be written as follows:
U 3q2/3x = Kz 32q2/3z2 + W£ 3q2/3z + Y q^T,, > 0 < x , z <
q2(0,z) = (yQj - 6(z-H) = t|>2(z) (18)
q2(x,») = 0
In order to eliminate the sedimentation term, W_ 3q_/3z, from the differen-
tial equation, we use the following transformation:
q2(x,z) = q2(x,z) exp(-p2) (19a)
17
-------
where
P2U,z) =
W2(z-H)
^z
W^x"
4K U
z
1/2
(19b)
is a nondimensional parameter representing the effect of sedimentation of the
secondary pollutant particles on the secondary pollutant concentration, and
q_(x,z) is the transformed new variable. Substituting Eqs. (8) and (19) in
Eq. (18), we obtain the following transformed equations:
U ZqJBx = K 32i,/az2 + y -i exp
£ Z £ L
c
q2(0,z) =
J- + x/Utc)
) ,
0 < x , z < »
W,(z-H) |
exp l -55 [ - 6(z-H)
(20a)
(20b)
(20c)
where
q2(x,o») = 0
=V/K V =V
V127 z ' 12 Vd2
(20d)
(20e)
Equation (20a) is a nonhomogeneous partial differential equation for the
secondary pollutant, coupled to Eq. (9) for the primary pollutant through
the chemical source term,
f(x,z) = (Y/Tc)-q1(x,z)-exp J -(p2- {J2 + x/Utc
(21)
18
-------
Unlike Q~, this chemical source is not a constant; in addition to x and z,
it depends on H, U, K , V , W1, W0, y, and t,. As shown by Budak et al. (1964),
Z u 1 1 ^ C
the analytical solution of Eq. (20) may be directly obtained as follows:
00
G2(x,z,|)
q2(x,z) =
X
+ J dx' J fCx',4) G2(x-x',z,|) d|
(22)
Here ijL and f are the functions defined in Eqs. (20b) and (21), respectively.
The first term in Eq. (22) gives the contribution of the initial condition,
and the second term gives the contribution of the inhomogeneity in Eq. (20a).
G2(x,z,|) is the Green's function of the homogeneous differential equation,
subject to the boundary condition (20c) of the third kind; the Green's function
is given by
G2(x,z,|) = £-
exp
f -(2-£)2u 1 + I -(z+)2u I
1 4K x f + exp 1 4K x J
c
exp
Jn
4K x
(23)
G?(x-x',z,4), needed to solve Eq. (22), is the associated source function,
obtained by replacing x in the Green's function G_(x,z,£) of Eq. (23) with x-x'
19
-------
Though straightforward in principle, the solution given by Eq. (22) is difficult
to derive in practice. This is because the contribution of the inhomogeneity,
given by the second part of Eq. (22), involves evaluation of several integrals
whose integrands are complicated functions of £. Some of these integrations are
mathematically intractable, thus rendering this direct application of the
Green's function solution to the inhomogenous differential equation fruitless.
In order to remove the inhomogeneity from the differential equation (20a),
we introduce the following transformation:
q2(x,z) = q3(x,z) - y q1(x,z) (24)
where
.z) = qjCx.z) exp j -(^- ^ + x/UTc) j
(25)
and q»(x,z) is a new (unknown) variable. Substituting (24) in Eq. (20a)
and separating the variables, we obtain two independent homogeneous differential
equations in q- and q_. The initial and boundary conditions to solve the
differential equation for q.. are obtained by substituting Eq. (25) in Eqs.
(9b,c,d). The initial and boundary conditions for the q^-equation can be
obtained by substituting Eq. (24) in Eqs. (20b,c,d), and using the initial
and boundary conditions for q- to simplify the resulting expressions. The
final expressions for the two sets of equations thus obtained can be written
as follows:
20
-------
U
0 < X , Z < «
(26a)
= exp
W2(z-H)
2K
6(z-H) =
(26b)
(26c)
(26d)
where
and
h3 = V13/Kz ' V13
U 3q,./3x = K 32q /3z2 , 0 < x , z <
j Z j
(26e)
(27a)
q3(0,z) = (Q2/QJ + Y) * exp
W2(z-H)
2K
6(z-H) =
(27b)
(27c)
= 0
(27d)
where
*3(X) = - t I hl * "2 2K
(28)
is the inhomogeneity in the boundary condition (27c). Thus, the transfor-
mation (24), which is used to remove the inhomogeneity in the differential
equation (20a), transforms the homogeneous boundary condition, Eq. (20c), into
21
-------
a nonhomogeneous boundary condition of the third kind. Nevertheless, it will
be shown later that, unlike the nonhomogeneous differential equation set (20),
we can solve Eq. (27) with relative ease, if q (x,0) in Eq. (28) is given from
the solution of Eq. (26).
The solution of Eq. (26) can be now expressed as
°»
qi(x,z) = exp(-x/UTc) - J ^(4) G3(x,z,|)d4,
(29)
where $,(|) is obtained from the initial condition (26b), and the Green's
function G~ is given by
G3(x,z,4) = ±-
- 2v/
exp
.
*
| -(z+|)2U \
Her- J
(30)
After term-by-term integration, the final form of the solution, Eq. (29),
can be written as
where
:,z) = exp(-x/UT ) f- [e"r+ e'S-
»- Ju
z
(31)
"3 = Lz h3
erfc
2VK x/U
z
and r and s are defined in (14)
(32)
22
-------
Now we can proceed to solve Eq. (27). First, we set z=0 in Eq. (31)
and substitute the resulting equation for q.(x,0) in Eq. (28). After
simplification, the final expression for the inhomogeneity, <{»~(x), in the
boundary condition (27c) is given by
A fir}
gives the contribution
JO
of the inhomogeneity in the boundary condition (27c), to the solution q_(x,z)
23
-------
The evaluation of q.,, is straightforward. The final expression can be
written as
q3A(x,z) = O^/Qj + Y ) £- [e"r + e"s (1 - c^)] (35a)
where
*2
"2 = Lz h2 e * erfc ^2 (35b)
To solve for > first we integrate with respect to r) in Eq. (34c) , substitute
for ~(x') from Eq. (33), and considerably rearrange and simplify the resulting
equation. The final expression for qOT> can be written as
Ji>
(V - V )
i(x, z) = - v ^ 2l 22 ' (x^) (36a)
where
V9i = VHI " Wi = Vn " V2
21 dl 1 11 1 (36b)
V =V -W =V - W /?
V22 Vd2 W2 V12 W2^
and F (x,z) is a function given by
1 rx l j -z2U H^ x'
= . , . exp 1 4K (x_xf) - -4^^, - UT
z z c
I fX 1
- I
" Jo Vx'(x-x')
e erfc
45 1
3 <*~e~ i
\l - VrtK (x-x')/U h_ e'° erfc |. i dx' (36c)
L z 2 5 J
24
-------
Here, H + 2h0 K x'/U
- * z
(36d)
2 VK x'/U
Z
z + 2h0 K (x-x')/U
, = 2 2
3 2 ,/K (x-x')/U
Z
are aondimensional functions. The integration with respect to x1 in Eq. (36c)
cannot be done analytically. Therefore, F..(x,z) needs to be numerically
evaluated.
The expression for q (x,z), from Eqs. (34a), (35a), and (36a), can be now
written as
i3(x,z) = (Q2/Q1 + Y) ' £- [ e"r + e"s (1 - a^ J
(v21 - v )
- Y -^ ^ F7(x,z) (37)
This is the analytical solution of Eq. (27). From Eqs. (24), (31), and (37),
we obtain
q2(x,z) = (Q^Qj + Y) £- [ e"r + e"S (1 - a2
exp(-x/U-Cc) ~ [ e"r + e"S
(V21 - v )
Y * K F1(x,z) (38)
z
25
-------
This is the analytical solution of Eq. (20). Finally, from Eqs. (19) and (38),
the analytical solution of Eq. (18) can be written as
q2(x,2) = g22 / Lz (39a)
where
g^2(x,z) = e
(Q/Q + Y) e~r + e~s
1
exp(-x/UTc) - { e"r + e"S
(V21 - V22)(Lz/Kz). FI(X,Z)
(39b)
is a nondimensional function. The prime indicates the modification of function
g-(x,z) of the secondary pollutant (denoted by the second subscript 2) for
the effects of deposition, sedimentation, and chemical transformation.
The physical meaning of the various terms in Eq. (39b) is as follows: The
first term has contributions from Q?/Q-, and y; the Qo/Qn part of this term accounts
for the direct emission of the secondary pollutant, and the Y part of this
term assumes that all the primary pollutant is transformed into the secondary
pollutant. The second term corrects this spurious assumption by subtracting
the equivalent of the primary pollutant still available for transformation at
any given location (x,z). The third (last) term accounts for the effect of
differences in the removal rates of the two pollutant species by deposition
and sedimentation; F-(x,z) in this term essentially represents a weighting
function such that 0 < FI < 1, as will be shown later. It can be easily seen
the contribution of this term is zero when V? = V2_ > 0, i.e., when the
deposition and sedimentation velocities of the two pollutants are equal.
26
-------
In the absence of a chemical coupling between the two species, all the
terms with y will be identically zero, and Eq. (39b) reduces to
g22(x,z) =
(39c)
which is the concentration algorithm for a chemically-independent, non-
reactive pollutant produced only by direct emission from a source of strength
Q,,. For this case alone, we can rewrite Eqs. (4) and (39c) as
C2(x,y,z) = Q2/U p(x,y) q2(x,z)
g22(x,z)
= e
*
"S
(40a)
where p and q_ are given by Eqs. (7) and (39a), respectively. This expression
for gI2 is similar in form to that of g* given in Eq. (I6b) for the primary
pollutant.
For W = W = 0 and Q = 0, Eq. (39b) reduces to
g22(x,z) =
(e"r
"S
- exp(-x/TJTc)}
-s
FI(X,Z)
(40b)
This equation agrees with the corresponding expression given by Lee (1980),
except that the latter paper incorrectly shows 2H/K instead of L /K in
Z 22
the last term.
27
-------
For Vdl =
and
= W, Eq. (39b) reduces to
>22(x,z) =
e"r + e"s (1 - a)
Q2/Q1 + * I1 * exp(-s/Utc) }
(40c)
This agrees with the expression given by Rao (1975), who assumed equal non-zero
deposition and sedimentation velocities for the two species for simplicity.
For V = 0 and W = 0, Eq. (40c) reduces to
g22(x,z) = (e~r + e"S)
VQ1 + Y {l - exp(-x/UTc)|
(40d)
which agrees with the expression given by Heines and Peters (1973), who
considered only chemical transformation and derived this result based on a
simple component mass balance. Thus, we can obtain all the known solutions
for simpler problems from the analytical solution (39b) of the general problem
considered here.
PARAMETERIZATION OF CONCENTRATIONS
In order to facilitate the practical application of the analytical
solutions, the eddy diffusivities K and K are expressed here in terms of
a and a , the standard deviations of the crosswind and vertical Gaussian
y z'
concentration distributions, respectively, as follows:
K -
y 2 dx
TT
_ U z
J\ 7T ' j '
z 2 dx
(41)
Thus, for Fickian diffusion, K and K can be expressed by the relations
y ^
28
-------
K = a2 U/2x , K = a2 U/2x, (42)
y y ' z z ' * v '
in order to utilize the vast amount of empirical data on the Gaussian plume
parameters cr and or available in the literature for a variety of meteorological
and terrain conditions. An excellent review and summary of these data can be
found in Gifford (1976). Equation (42), in combination with the Gaussian
assumption (see, e.g., Gifford, 1968), forms the basis for the practical
plume diffusion formulas that are found in the applications literature.
In order to parameterize the expressions for concentrations under
various stability conditions and to considerably reduce the difficulty in
typing the equations, we adopt the following nondimensionalization scheme:
All velocities are nondimensionalized by U, the constant mean wind speed.
The horizontal downwind distance x and all vertical height quantities are
nondimensionalized by V2 CT ; the transformation time scale T is nondimen-
Z C
sionalized by -/2 a /U. The horizontal crosswind distance y is nondimen-
sionalized by ''jn'jO/^1
dl dl d2 d2
yv A
W= W /IT W = W /TT
- f* - / \J y O **O/ *"^
V =V - V II V =V -W/2
*ni JT " t / ** > "10 "JO " o/ *
11 dl 1 12 d2 2
V13 = Vll - (W1 - V/2
V =V -W V =V -W
V21 Vdl *1 ' V22 Vd2 W2
x = x/V2 a , z = z/V2 a
Z A
H = H/V2 a , L = L/V2~ a
Z Z
= TcU/V2 az , y = y/V2" ay
29
(43)
-------
where the capped quantities denote the nondimensionalized variables, and
L is the mixing height.
From Eqs. (3), (7), (8), (14), (16), (42), and (43), we can now rewrite
the expression for the primary pollutant concentration as follows:
Q g g'
(^(£,7,2) =iT'r'r^ (44a)
y z
gl(x,y) = exp(-y2) (44b)
,z) = exp(-p2 - x/Tc) [ e"r + e"s (1 - ce (44c)
L = v2n cr , L = V27t a (44d)
where
r = (z - H)2 , s = (z + H)2
2 = 2Wx(z - H) x + W^ x2 (44e)
x el erfc
4X = z + H + 2 Vn x
Equation (44) clearly shows that concentration C depends on the ratios
VL/U and V /U, not on W and V per se. Thus, the effect on Cj of large
values of W.. and V,- at high wind speeds is the same as that of small values
of W and V,1 at low wind speeds, provided W_/U and V,./U remain constant.
The effect of deposition can be seen as a multiplication of the contribution
30
-------
of the image-source or reflection term, e , by a factor -1 < (1 - or ) < 1.
In the absence of chemical decay (t = °°) , Eq. (44) agrees with the expression
for concentration given by Rao (1981) . In the trivial case of negligible
* /*,
deposition and chemical decay (W = 0, V, = 0, t = °°) , this equation reduces
to the well-known Gaussian plume model with g' = g_, where g_(x,z) is defined
in Eq. (17).
For a ground-level receptor (z=0), the concentration algorithm for the
primary pollutant reduces to
= exp(-p2 - H2 - x/tc) (2 - aj (45a)
where p2 = - 2 V x H + W^ x2
or1 = 4Vn V1X x e erfc ^ (45b)
I, = H + 2
Further simplifications of this equation are possible by setting H = 0 for
«s ^\ A
ground-level sources, and W = 0 and V = V, for gases and small particles
with negligible settling.
From Eqs. (4), (7), (19b), (32), (35b), (39), (42), and (43), the
parameterized expression for the secondary pollutant concentration can be
written as follows:
31
-------
- ~ gl 81
C2(x,y,z) = yi ^ -
8
22
L
y z
(46a)
>22(x,z) = e
3
V) e"r + e"s
(1 - «2) j
-Y exp(-x/tc) J e"r
"s
-Y 4/n
21 22'
F,(x,z)
(46b)
where g1(x,y), L , L , r, and s are defined in Eq. (44), and the remaining
I y 2
quantities are as follows:
P = 2 W (z - H) x + W x2
V12 x e erfc
42 = z + H + 2 V12 x
(46c)
x e erf c
63 = z + H + 2 V13 x
From Eqs. (36c), (36d), (42), and (43), the nondimensional function FI(X,Z)
in Eq. (46b) can be parameterized as
32
-------
1 - 2 V13 x vnt e erf c
I2
r * i ^s l
1 - 2 V,0 x vW-t) e erfc |_ dt (46d)
L 12 5 J
where |,
* (46e)
and t = x'/x is the normalized integration variable.
Equation (46b) calculates gl» as the sum of contributions from the three
major terms on the right-hand side of this equation. The physical signifi-
cance of these terms has been already explained in the discussion following
Eq. (39b). In the absence of a direct emission of the secondary pollutant
(i.e., Q- = 0), a delicate balance exists between these three terms, as shown
in the next section. The third term, which arises due to the differences in
the deposition rates of the two pollutant species, becomes important at large
x and, therefore, cannot be ignored. The weighting function in this term
F~(x,z), given by Eq. (46d) , should be evaluated by numerical integration
to a sufficiently high degree of accuracy.
For a ground-level receptor (z=0), the concentration algorithm for the
secondary pollutant reduces to:
33
-------
(x,0) = e
H2
(Q2/Q! + Y) e"11^ (2 - 02)
-Y exp(-H2 - x/t) (2 -
-Y
x
(47a)
where
~9 ~
= -2VLxH + Wrx
12
x e erfc
a3 = Wt V13 x e erf c |3
(47b)
= H + 2 Vu x , 43 = H + 2 V13 x
and
*-0) = ~n / 7^=
71 'o ./rrn
o Vt(l-t)
exp
1 - VJF (4-
s
) e 4 erf c
e J erfc E, dt
(47c)
where
e, = H/VF-»- 2 V., x v'
13
^5 = 2 V12 S
(47d)
34
-------
Further simplifications of Eq. (47) are possible by setting H = 0 for ground-
level sources, and W = W,, = 0, V 3 = V,., and V - = V,2 when both species
are gases or small particles with negligible settling.
WELL-MIXED REGION
Under unstable or neutral atmospheric conditions, when the plume travels
sufficiently far away from the source, the pollutant is generally well-mixed
by atmospheric turbulence, resulting in a uniform vertical concentration
profile between the ground and the stable layer aloft at a height L. This
concentration, which is independent of source height as well as the receptor
height, can be calculated as the average concentration in a mixed layer of
depth L. Following Turner (1970), we assume that the plume is well-mixed
for x > 2 x , where x is the downwind distance at which o (x )=0.47 L.
= or m z m
The primary pollutant concentration in the well-mixed region can be
calculated as follows:
. . Ql 81 8
C (x v z1 =
SU,y,z; u L L
y
g1(x,y) = exp (-y )
r
Jri
41
°°
dz = i-
H=0
(48a)
(48b)
dz (48c)
H=0
35
-------
Substituting for gi..(x,z) from Eq. (44) and carrying out the integration in
Eq. (48c) , we obtain the following:
For Vdl * W1 or V21 # 0,
I2
= exp(-62 - x/tc) [(Vn/V21) e X erfc ^
erfc p (48d)
where £_ = 2 V-_ x and B = W.. x.
This algorithm applies to gases or small particles. This equation is indeter-
A t oS
minate when V,- = W_ or V_. = 0. For this case, g!..(x) can be determined by
setting W = V, in the expression for g' in Eq. (44) and then integrating
as indicated in Eq. (48c). Alternately, one can take the limit of Eq. (48d)
as W > V.s. Thus, we obtain the limiting expression for gj-(x) for
large particles as follows:
*. A A
FnT V =W nr-V =0
sue v,1 w_ or v_- u,
- x/tc) (1 * 2 |) e erfc ^ - 2 ^/^T (48e)
where 4, =27,, x = V,, x = W, x.
31 11 dl 1
Equations (48d) and (48e)
-------
The secondary pollutant concentration in the well-mixed region can be
calculated as follows:
QI 8, gA2
c2(x,y,z) = tr * r * r^ <49a)
">00 /»03
/»00 /»O3
f r 8221 i I r
T /--A. v I I ^i^t I , ! I I I / ^ '
2 t X J -~ I I - I Q.Z ' I I gr (X '
42 J0 L Lz JH=O Jn. J0 L 22
(49b)
H=°
Substituting for gi-(x,z) from Eq. (46) and carrying out the integration in
Eq. (49b), we obtain the following:
For V2l £ 0 and V22 £ 0,
R2 T , ?2 S2
P7 ( * - £2 P2
o ' TY'I o I (n in + v~\ ! ( V IM \ r* erf r t - ( W /9V } (* erfc S
g42(xj - e ^ W2/Q1 -1- YJ j C V12/V22 ^ e ertc ^2 C VZV22; e erIC P2
/v ,v\ f.2 02 \
-x/T / * * ^., P2 1
-vo CJfV /V 1 *« *»rfr ? - fW /? V 'i <> prfc 6 I
y e < IV13'V21J e er±C ^3 L 2/Z 21J P2 j
- Y W« (V21 - V22) x - F2(x) (50a)
where = 2 V x ,
= 2 V13 ^ ' V13 = Vll - (W1 - W2)/2 (5°b)
f00 r
F (x) = |F (x,z) "L
J0 L JH=O
37
-------
Substituting for F (x,z) from Eq. (46d) and integrating, we obtain
F/-*\ X I -1- f ** M- /~ \ I T / * ^ f
2(x) = 27T I ~7^- exp(-xt/T ) j 1 - ^n £, e erfc
JQ ^ C
(V12/V22} e erfc ^5 " (W2/2V22} 6 6rfc ^ dt'
where 4 = | Vt , 4 = 4 Vl^t , l = P Vl1^ (50d)
Equation (50) is applicable when the two pollutant species are either gases
or small particles. In the large-particle limit of W > V, for one or
both of the species, three other forms of this algorithm can be derived as
follows:
For V21 # 0 and V22 = 0,
v) (i + 2 c « erfc C - 2
e"X/Tc I (V13/V21) e 3 erfc ^ - (W2/2V21) e 2 erfc
x F2(x) (51a)
where £2 = 2 V12 x = Vd2 x = W2 x = 02
2 V13 x
38
-------
e erfc
[
(1 + 2 £) e erfc ^ - 2
n ] dt,
(51b)
with
4, = 4, Vt~ , 4r = i
Equation (51) is applicable when species- 1 is a gaseous or small-particle
pollutant, and species-2 consists of large particles.
For V01 = 0 and V0_ ? 0,
21
22
,2
'3
42
erfC
^3
e erfc
2 43) e erfc |3 - 2
/ I TT ^ T^ f \
V 4^ V22 X - F2(x)
(52)
where
= 2 V12 x , 43 = 2 V13 x = W2 x
and F?(x) is given by Eq. (50c). This algorithm applies when species-1
consists of large particles, and species-2 is a gaseous or small-particle
pollutant.
39
-------
For V21 = 0 and V22 = 0,
[ Vi+
,2
-4
e 2 (1 + 2 £) e erfc _ 2 /^ (53)
where £2 = W2 x = P2-
This algorithm applies when both species consist only of large particles.
The physical situations represented by Eqs. (52) and (53) are unlikely to
occur in reality, since any primary pollutant consisting only of large particles
would not generally reside in the atmosphere long enough to produce significant
concentrations of the secondary pollutant by chemical transformation (i.e.,
x /U « T ) . These algorithms are included here primarily for completeness
max c
of the solutions. Since the well-mixed region concentration algorithms are
A
independent of z, they apply to all heights 0 < z <[ L.
PLUME TRAPPING
For x < x < 2 x , where x is the downwind distance at which the plume
m m m
upper boundary (corresponding to an isopleth representing one-tenth of the plume
centerline concentration) extends to the height of the inversion lid, the mixing
depth L should be included in the concentration algorithms . This is usually
done through calculation of multiple eddy reflections (Turner, 1970) from both
the ground and the stable layer aloft, when the plume is trapped between these
two surfaces. For the general case which includes deposition, sedimentation,
and chemical transformation, the concentration of the species at any height z
in the plume trapping region can be expressed as
-------
Ql gl S31
C.(x,y,z) s-i.-i.-4i (54a)
1 U Jj Jj
y z
C2(x,y,z) = - . -
One can write the equations for gl-(x,z) and g' (x,z), following Rao
(1981). These expressions look similar to those given for g' (x,z) and
/\
g22(x>z), respectively, except that the effective plume height H will be
A yv /\,
replaced here by H- - H + 2NL and the equations are summed over N from - » to
In general, a maximum of N = ± 10 eddy reflections are adequate to obtain
convergence of the sum within a small tolerance.
A simpler alternate procedure suggested by Turner (1970) may be adopted
if one is interested only in ground-level concentrations: We may calculate
the ground-level centerline concentrations of each species at x and 2x
r m m
using the appropriate algorithms given earlier in this section, and then
linearly interpolate between these values on a log-log plot of concentration
versus downwind distance to obtain the ground-level centerline concentration
at any x in the plume- trapping region.
«.
41
-------
TABLE 1
SUMMARY OF POINT SOURCE CONCENTRATION EQUATIONS:
Applicable Algorithms and Equation Numbers
Primary
Pollutant
Secondary
Pollutant
1. Near-source region
(0 < x < x )
m
U L L
y 2
- -
2~ U L L
y 2
(a) Elevated receptor
(2 > 0)
(b) Ground-level receptor
(z = 0)
Eq. (44)
Eq. (45)
Eq. (46)
Eq. (47)
2. Plume trapping region
(x < x < 2x )
m m
(a) Elevated receptor
(2 > 0)
1 " U L L
y
Eq. (54a)
II !i _
U " L ' L
y 2
Eq. (54b)
32
(b) Ground-level receptor
(2 = 0)
Interpolation
Interpolation
3. Well-mixed region
c - ..
1 ~ U L L
y
2 U L L
y
any 2 > 0
Eq. (48)
Eqs. (50 to 53)
42
-------
SURFACE DEPOSITION FLUXES
The surface deposition fluxes of the primary and the secondary pollutants
at ground-level receptors are calculated directly from Eqs. (Id) and (2d) as
(55a)
D2(x,y) = Vd2 - C2(x,y,0) (55b)
D gives the amount of pollutant deposited per unit time per unit surface area,
2
and is usually calculated as kg/km -hr, while seasonal estimates are expressed
2
as kg/km -month. The estimation of the monthly or yearly surface deposition
fluxes at a given downwind distance x from the source in a given wind-directional
sector requires the knowledge of the fraction of the time that a mean wind of a
given magnitude blows in that direction in a month or a year, respectively. To
2 3
obtain D in kg/km -hr when V, is given in cm/s and C in g/m , the right-hand
2
side of Eq. (55) should be multiplied by 36000. To obtain D in Mg/m ~nr when
3
V. is given in cm/s and C in (Jg/m , the corresponding multiplication factor is
36. For D calculations, the ground-level receptor is generally defined as any
receptor which is not higher than 1 meter above the local ground-level
elevation.
AREA SOURCES
Urban air pollution results from (a) elevated large point sources such
as tall stacks of electric power plants and industries, (b) isolated line
43
-------
sources such as highways, and (c) distributed area sources such as industrial
parks, clustered highways, and busy interchanges, parking lots, and airports.
Numerous small low-level point sources distributed over a broad area, such as
smoke from chimneys of dwellings in an urban residential area, can be treated
as an area source. Therefore, an urban diffusion model should be able to
account for point, line, and area sources.
The line and area source problems are generally treated by integrating
the point-source diffusion algorithms over a crosswind line or over an area.
Differences between urban air pollution models occur only in the details of
how the area source summation is carried out and in how various meteorolo-
gical paremeters are included. The simple ATDL Area-Source Model described
by Gifford (1970), Gifford and Hanna (1970), and Hanna (1971) is widely used
in many of the current practical urban air pollution models, such as the
Texas Episodic Model (TAGS, 1979). We briefly describe below the derivation
of the Gifford-Hanna algorithm for estimating the ground-level concentrations
due to urban area sources.
Consider two equal grid squares with side Ax, one of them containing
the ground-level area source emissions (Q), assumed to be located at the
center of the square; the second square, also known as the "calculation
grid square," contains a ground-level receptor (R) at its center. The wind
(U) blows along the line from Q to R, as shown in Figure l(a). The distances
from the receptor R to the downwind and upwind edges of the emission grid
square are denoted by x., and x~ , respectively. Since the two grid squares
are equal in size, x_ and x_ are identical to the distances measured from Q
to the upwind and downwind edges, respectively, of the calculation grid square,
44
------- An error occurred while trying to OCR this image.
-------
as shown in Figure l(a).
Neglecting deposition, sedimentation, and chemical transformation, the
surface (z=0) concentration due to a ground-level (H=0) point source of
strength Q (units : M T ) is given, from Eq. (44), by
where
O Or
C(x,y,0) = § |7 * |7 (56a)
y z
g2(x,0) = 2 (56b)
The surface concentration due to a crosswind infinite line source can be
obtained by integrating Eq. (56a) with respect to y:
00
C(x,0) = / C(x,y,0) dy = § - ^ = Jl £r (57)
-OB z
where Q now has units of M L T and represents the emission rate of the line
source.
The surface concentration at the receptor R due to an area source Q
can be obtained by integrating Eq. (57) with respect to x:
2
C,
= C(x,0) dx = ,/F g f C5B)
46
-------
Assuming a is given by a power law of the form,
Z
a (x) = a xb (59)
Z
where a and b are constants depending only on the atmospheric stability, we
obtain
CA = VF ^TT^T i <" ~ <~" \ (60)
1-b 1-b
Ua(l-b) \ 2 " xl
-2 -1
Here Q has units of M L T , representing the emission rate of the area
source, and 0 < x. < x». This equation gives the concentration of a single
area source Q located upwind of a single receptor R. If the latter were located
at the center of the emission grid square itself, then x. =0, x~ = Ax/2, and
Eq. (60) becomes
c _ _
CA - \H Ua(l-b)l
If N receptors are located downwind of a single area source Q , then
the concentration at the receptor R. in the i th grid square (for i = 0, 1,
- - -, N) is given by
Ai
where x. . and x« . are the distances measured from Q to the upwind and downwind
edges, respectively, of the calculation grid square with the i th receptor.
Figure 2 illustrates this for N = 4.
If N area sources are located upwind of a single receptor R , then the
contribution to the concentration at the receptor by the source Q. in the i th
47
-------
CO
^
to
00
2E
^
(0 3 co
3 §
a1 w J3
ca 4) *J
S-l -H O
O aj M O»
H 3 o
W C71 00 4)
_ 00 CO r-4 U
CVl -^ 4) T3 3
00 M CJ W
d oo >-i
H 3 41
co (4 O X5
O 00 4->
<0 -H 1
4J a B
00 (0 4) O
fl r-l M V-l
H 3 «J "4-1
3 U
0 t-4 fl T3
^J « -i-l S
^, co o -i-i
Xx RJ O co 3
t-l -H 3 O
00 3 'O
nj fl w
H 3 41 -O
O O O 41
T3 0 S-l
U eo 3
H S-l 4-> M
4-> 3 CO (0
«J O -H 4)
S «w -0 S
4)
O J3 T3 4) 4)
iri -^- u ca ^a >-i
1X1 * en co 4J (0
CN
2 00
J% rt
48
-------
grid square (for i = 0, 1, , N) is given by
?i / 1-b . 1-b \ .
n Ua(l-b) X2i xli j (63)
where x.. and x^. are the distances measured from R to the downwind and upwind
edges, respectively, of the i th emission grid square. This is illustrated in
Figure 3 for N = 4. As noted earlier, the distances x.. . and x-. in Eq. (63)
are identical to those used in Eq. (62), since all grid squares are equal in
size. The total surface concentration at the receptor R can be obtained by
summing up the individual contributions of all N area sources:
i-0 /
This algorithm for urban area sources was given by Gifford and Hanna (1970) in
a slightly different form by using
xu = (2i - 1) Ax/2 , x2i = (2i + 1) Ax/2 ,
(65)
° = xli < X2i
A more general form of this algorithm was first derived by Gifford (1970),
assuming that the mean wind and vertical eddy diffusivity are functions of
height given by power laws.
It should be noted that the Gifford - Hanna algorithm, Eq. (64), ignores
horizontal diffusion. Gifford (1959) postulated that air pollutant concentration
at a receptor due to the distributed area sources depends only on sources located
in a rather narrow upwind sector. The angular width of this sector, derived from
49
-------
to
^
ro
00
2
o
S
< 0
o
o
x
O 01
4-> H
O4 1 ^9
01 CO
O -H CO
^4 CJ
oi a
K) ^- J3 j3 eo
i\i _ jj AJ jj
X X -i-l W
* S T3 -H
U CO
CO « 09
3 to 01
W O fcH
rl
*O 03
H CO
W -H CO
oo s a o
OJ. ro ^ « .a «
X ** 00 4-» M O
a -i-» o *>
r4 5 00 O»
CO rH 01
03 CO U
CO 0) 0)
M 0} M
00 CO U
a 3 n 01
H O» 3 J3
3 tn O -U
0 «J
J3 T3 1 5
_ , Cfl -i-l CO O
T: cu M o) )-i
X x CO CO
oo a a a
CO -H -H -i-l
H 3 S
3 Ol 3
U CO
H 14 3 'O
4-> 3 01
CO O 03 >-l
S m 01 3
oi u co
Ojs *o a co
«- U ft « O>
OJ v CO CO 4-) S
x " «
2 00
x 2J
50
-------
known values of the horizontal diffusion length scale L , is less than the
usual 22.5° resolution of observed wind directions. Consequently, horizontal
diffusion can be ignored. This assumption is referred to as the "narrow plume
hypothesis." The crosswind variations in the source-strength patterns can be
similarly ignored, since the urban source-inventory box areas are quite large,
usually 5 x 5 km or more, and do not vary much in strength from box to box;
therefore, the contribution of the source box containing the receptor is generally
the dominant one, and the contributions of more remote upwind area sources to
this receptor concentration are comparatively small. For this reason, it is
generally adequate to consider only four area sources immediately upwind of
each receptor grid square, i.e., N = 4 in Eq. (64). For the same reason,
a and b are assumed to be independent of the upwind distance x.
Gifford and Hanna (1973) noted that, for the usual case of receptors
within a city, the area source component of the urban air pollution is
strongly dominated by the source-strength pattern and by transport by the
mean wind. Atmospheric diffusion conditions in cities tend to be of the
near-neutral type, without the strong diurnal variations found elsewhere.
For these reasons, they suggested a simple approximation of the area source
formula, Eq. (61), as
CA = k Q/U (66)
where k is given by
1-b
X (67)
a(l-bT
and x is the distance from a receptor to the upwind edge of the area
source. The parameter k is a weak function of city size and should be
51
-------
approximately constant. Using a large quantity of air pollution data, average
annual emissions, and concentrations of particles for 44 U.S. cities and S0_
data for 20 cities, Gifford and Hanna (1973) found k = 225 for particles, and
50 for SCL, with standard deviations of roughly half their magnitude. They
noted that Eq. (66) works better for longer averaging periods.
None of the equations given above for area sources consider deposition,
sedimentation, and chemical transformation/decay of gaseous or particulate
pollutants. However, removal and transformation processes can be important for
obtaining reasonable estimates of the pollutant deposition fluxes in urban
residential areas. The area source concentration algorithms for the
general case that includes deposition, sedimentation, and chemical trans-
formation can be obtained by integrating the point-source concentration
algorithms for pollutant species-1 and 2 emitted at ground level, following
a procedure similar to that shown in Eqs. (56) to (60), as follows:
C = 2 C(x'y'0) dy
Al J J "1
x -oo
Q, f2
= ~ J qj(x,0) dx (68a)
Xl
X2
CA2 = C2(x,y,0) dy dx
xl -°°
52
-------
In the above, we utilized Eqs. (3) and (4), and integrated with respect to y.
Though straightforward in principle, the x-integrations required in Eq. (68)
are very difficult to carry out in practice, especially for pollutant species-2.
This is due to the obvious fact that the integrand functions represented by
the probability densities q.(x,0) and q?(x,0) are complicated functions of x,
unlike the case in the derivation of the Gifford-Hanna algorithm. Even if
one is successful in evaluating these difficult integrals analytically, one
finds it nearly impossible to physically interpret the terms of the resulting
complicated expressions. Following this experience, we started exploring for
an alternative to this direct approach for deriving the area source concentra-
tion algorithms in the general case that includes deposition, sedimentation,
and chemical transformation. These efforts were successful, culminating in the
derivation of an elegant alternate approach, which can be physically explained
in terms of mass balance considerations, as outlined below.
We rewrite the differential equations and the deposition boundary conditions
of Eqs. (6) and (18) as follows:
D 3qa/3x = 8(Kz Sq^Bz + V^ q^/Sz - q^T^ (69a)
+ Wx qj = V q(x,0) (69b)
U 3q2/3x =
(70b)
53
-------
Integrating Eqs. (69a) and (70a) with respect to z from 0 to », and substituting
Eqs. (69b) and (70b), we obtain
/*oo s~ 00
U fj I q dz = - V q (x,0) -\ q dz (71a)
-*<1 *~ o Jn
U IE J % dz = - Vd2 12(X>0) + I J
'o c 'o
dz (71b)
For ground-level sources (H = 0),
J ^ dZ = J
Jf\ Jn
? dz = g' (x) (72a)
\ q2 dz = I 22£X?Z) dz = g; (x)
-'n Jn 7.
(72b)
'o "
Substituting Eq. (72) in Eq. (71), and integrating the latter with respect to x
from x.. to x^ and rearranging, we obtain
Vdl
fX2 , , fX2
q (x,0) dx = U g' (x ) - gl^x,,) - gin^) dx (73a)
» A I ^ A 1 ^A A> I v I "A
Jxl L J Jxl
rx2 rx2
Vd q2(x,0) dx = U [ g^ (xa) - g;2(x2) 1 + f g;2U) dx (73b)
-'x, L J r -'x.
54
-------
where V.. ^ 0 and V,0 ?t 0.
ul d2
Noting from Eq. (68) that
ft
J qi(x,0) dx = U CM/Q1 , i = 1 or 2 , (74)
Xl
we can now write the area source concentration algorithms for pollutant
species-1 and 2 as follows:
QI i r2
C = -= f g* (x ) - g1 (x ) / g* (x) dx | (75a)
dl ^ c xi
X2
CA2 = V7 [ «42<*1> - *42(X2> + lit- J «il« dx J <75a>
d2 C Xl
Here g' (x) and g!?(x) are the nondimensional algorithms derived for the point-
source concentrations (see Table 1) in the well-mixed region. Thus the
alternate approach, outlined above, is unique in that it allows one to use
the same algorithms (and the same program subroutines) to compute the concentrations
due to both point and area sources.
Rao (1981) showed that, for H = 0,
g£00 = £00 (76a)
where
u f°° r°°
£00 -Til I C(x,y,z) dy dz (76b)
55
-------
is the suspension ratio, representing the proportion (fraction) of the pollutant
released at a rate Q by a source located at (0,0,H) that still remains air-
borne at downwind distance x. Therefore, Q * g!(x) in Eq. (75) represents
the flux of pollutant passing through an imaginary vertical plane at downwind
distance x. Referring to Figure l(b), we can now physically interpret the
area source concentration algorithms, Eq. (75), as follows: For each calcu-
lation grid square box formed by the ground surface and two imaginary vertical
planes at x = x. and x = x», the pollutant mass balance is given by
Incoming flux - outgoing flux ± flux gain/loss due to chemical
transformation = surface deposition flux
where
Incoming flux = Q.. g].(x1), i = 1 or 2
Outgoing flux - Qj g^Cxj),
Flux loss due to chemical transformation (species-1)
/"',,<.
Flux gain due to chemical transformation (species-2)
YQ
U
c
1
dx'
Surface deposition flux = V,. C..
The only unknowns in the above are the surface concentrations, CA- , which can be
56
-------
calculated as shown in Eq. (75). Thus the area source concentration algorithms
in the alternate approach, derived mathematically from the governing differential
equations and the deposition boundary conditions, can be explained in terms of
physically realistic mass budgets of the pollutant species.
Note that the functions g! . are parameterized and given in terms of
x = x/V2cr instead of x. Further, we did not specify any particular form of
z
variation for O (x) in the derivation of the area source algorithms using the
z
alternate approach. Therefore these algorithms should be valid for any specified
type of variation (e.g., power law, exponential, polynomial, etc.) of a (x).
Z
For a power law of form a = a x , used by Gifford and Hanna (1970), the
Z
algorithms given by Eq. (75) are valid when the value of b is not signifi-
cantly different from 0.5. This latter value follows from the relations
given in Eq. (42), which allowed us to express the exact analytical solutions
in terms of the empirical Gaussian dispersion parameters. For urban area
sources, however, Gifford and Hanna (1970) used values of b ranging from
0.91 to 0.71, depending on the atmospheric stability class. These values
are based on extensive observational data summarized by Smith (1968) and
Slade (1968). Thus when b is significantly different from 0.5, Eq. (75)
should be modified for use with a power law variation of a as follows:
z
-X2
C
Al 2(l-b)Vdl |_ S41^1' 641V~2' UT(
(77a)
JA2 2(l-b)Vd2
f" g^Cx,) -842(x2)+uH
(77b)
57
-------
For V..-
di
0 and t > «, the concentrations calculated by these algorithms
agree with corresponding values given by the Gifford-Hanna algorithm, Eq. (60)
The latter can be easily extended for two chemically-coupled pollutants as
follows:
"Al
-b -X/UT ,
x e c dx
(78a)
rt Ua(l-b)
1-b 1-b1
X X
,2 xl ;
"Al
(78b)
For V -
di
0 and any finite value of T , the concentrations evaluated by
Eqs. (77) and (78) show remarkable agreement, even though these two sets of
equations axe derived using different approaches. This good agreement may
be considered as verification of the area source algorithms, Eq. (77), based
on the mass balance approach.
Equations (77) can be easily extended, as shown before, to the case of
N receptors downwind of a single area source (see Fig. 2); the concentrations
at the receptor R. in the i th grid square can be obtained by using the
appropriate downwind distances x. . and x_. in Eq. (77). If N area sources
are located upwind of a single receptor R , as shown in Fig. 3, then the total
surface concentrations at the receptor can be obtained by summing up the
individual contributions of all N area sources, as follows:
58
-------
p .
CA1
r. - .
1
1
v
dl I
1=0
N
V
i=0
tO-H
]
In Eq. (79), it should be noted that for i = 0,
= Q2/Q1 (80a)
specify the initial conditions. Further, we note that for i > 0,
where j = i - 1.
In Eq. (79), therefore, one needs to compute the functions g! - (x) and g!2(x)
only at x = x? . , i = 0, 1, --- , N, since their values at x = x- . are known.
The subroutines that compute g! 1 and g!_ are common to both point and area
sources. Thus the area source algorithms, Eq. (79), are simple, accurate, and
computationally efficient.
59
-------
SECTION 4
RESULTS AND DISCUSSION
In this section, we consider the well-known problem of the atmospheric
transport and transformation of SO^ (species-1 or primary pollutant) to
SO, (species-2 or secondary pollutant). The diffusion-deposition algorithms
developed in the previous section for various stability and mixing conditions
for an elevated rural continuous point source were tested using the following
nominal values for the model parameters:
U = 5 m/s , H = 30 m , KST = 5 (P-G stability class E)
Vdl = 1 cm/s , Wx = 0 , Qj = 1 g/s
Vd2 = 0.1 cm/s , W2 = 0 , Q2 = 0
k. = 1% per hour (T = 36xl06 s) , y = 1.5
t c
Some of the important results, calculated up to a downwind distance of 20 km
from the source, are presented and discussed in this section. Any variations
of these nominal values of the model parameters are clearly shown in the
figures and noted in the text. The parameters a and a used in the calcu-
lations are the P-G values, which appear as graphs in Turner (1970) and in
Gifford (1976, Figure 2). These values, which are widely used for continuous
point sources in rural areas, are most applicable to a surface roughness of
0.03 m (Pasquill, 1976).
The diffusion over cities is enhanced, compared to that over rural areas,
due to increased mechanical and thermal turbulence resulting from the larger
60
-------
surface roughness and heat capacity of the cities. This is reflected in the
urban dispersion parameter curves based on interpolation formulas given by
Briggs (see Gifford, 1976, Figure 7). Some urban air pollution models, such
as the Texas Episodic Model (TACB, 1979), simulate the increased surface
layer turbulence over urban areas by decreasing the P-G atmospheric stability
class index by one, for all classes except Class A. In any case, the algo-
rithms given in the previous section can be applied to sources in urban as
well as rural areas by using the appropriate dispersion parameters.
SENSITIVITY ANALYSES
The algorithm for the ground-level concentration of the secondary
pollutant given in Eq. (47a) can be written as
g' (x,0) = Term 1 + Term 2 + Term 3
The physical interpretation of these terms was discussed in the previous
section. Figure 4 shows the delicate balance that exists between the
three terms; Term 2 and Term 3 together nearly balance Term 1. A non-zero
concentration for the secondary pollutant results from the small positive
imbalance of the three terms. Figure 4 shows that Term 3, which accounts
for the differences in the deposition rates of the two pollutant species,
becomes increasingly important as x increases.
Because of this tenuous balance between the three terms in g' (x,0),
the weighting function F-(x,0) in Term 3 must be evaluated by numerical
integration to a very high degree of accuracy. An examination of F,(x,0)
in Eq. (47c) shows that the integrand function has singularities at the
61
-------
en *
I I
a a
in
cn
£
O
cn CD
n it
X
cu
n
D
\
X
cu
n
13
\
A
54 < -T-I
=! 4J
O -H
co ca
i *-> o
+j e &,
d ns
H 4J r-l
O 3 rH
a
O 00
O<
qj
d
P eu y
(-1 U 4J
4-1 o d « o ,JQ
u
da cu
o J2 o u
H 4-> t> d
*j -H eu «J d -H
00
rl
OJ
I
62
-------
end-points t = 0 and t = 1 of the integration domain in t. A computer-
library subroutine named D01AJF, developed by the Numerical Algorithms
Group (NAG), was utilized in this study for the numerical integration.
This routine, which is capable of handling the singularities, has been
selected because of its accuracy and applicability. It estimates the
value of a definite integral of an externally defined function over a
finite range to a specified absolute or relative accuracy, using Gauss-
Kronrod rules in an adaptive strategy with extrapolation.
The behavior of F.(x,0) for four different effective plume heights is
explored next. Noting that
dt = 1,
we can easily see from Eq. (46d) that the value of the weighting function
F (x,z) is always between 0 and 1. Figure 5 shows that the value of F-(x,0)
is close to 1 for ground-level non-buoyant sources (H=0), but decreases
rapidly as H increases, especially for small values of x. For all non-zero
H values, F (x,0) increases with x as shown in Fig. 5. This clearly illus-
trates that Term 3 becomes important as x increases.
The behavior of F (x,z) for four different receptor heights is shown
in Fig. 6. The weighting function decreases rapidly as z increases. At
large x, the calculated values of F are of the same order of magnitude for
all z values. Figure 7 shows the variation of F..(x,0) for three different
values of the deposition parameter V - of the primary pollutant. Though F..
63
-------
1,1,1.1 1., , 1,0,
u
u
14 (8
O
W
W (Q
I
*J -
d -u
H d
O (0
w m o
X! w
00 -U
d «
H
U O -H
43 4J
oou u
H H? 0»
« O M-l
3 <44
O
4-> -i-l -r-l
(0 V4 4-1
f-l O O
n oo c
t8 r-4 d
> «J M-i
60
H
'o
o
o
64
-------
a i i
to o a
» X X
31 CM n
* I B
in
H
o o o o
co
w a
3 r-»
o ^
CO O
d >»
H U
O CO
d
co CQ
00
H
o
LaT
65
-------
0)
u
u eg
d
o as
CO CO
I K*.
4-> - 1-1
U 4-> «J
H c e
O <0 -M
O4-U k
3 04
4) r-l
fl r-4 U
M O 45
E
_£
u o
« CO
a -o >,
o d -u
rt O -H
+J U U
U (U O
§ca r »
4)
d o
H m -r^
+J O +J
J3 -H
QOU 09
H 1-3 O
o 3 a a
H J-> C «
.(J -rl -H 4J
(0 M 4-> 3
H O O r-t
!4 00 a r-4
fl iH 3 O
00
H
o
O
66
-------
increases with x initially for all V_1 values, as shown in previous plots,
F- reaches a maximum at a certain value of x, and then starts decreasing as
x increases. This is illustrated by an order of magnitude increase in the
/s.
deposition parameter V.. 1 in Fig. 7. A similar behavior is shown in Fig. 8
for the secondary pollutant. This plot suggests that the effects of the
deposition parameter in the evaluation of F..(x,0) may be ignored if
V < 2 x 10"4, i = 1 or 2.
GROUND-LEVEL CONCENTRATIONS (GLC)
The variation of ground-level (z=0) plume-centerline (y=0) relative
concentrations, UC1(x,0,0)/Q , of the primary pollutant are shown in Fig. 9
for Vdl = 0, 2 x 10"3, and 2 x 10"2; Wj = 0 in all three cases. The
V,- = 0 value corresponds to the zero-deposition case in which the new
concentration algorithms reduce to the well-known Gaussian plume algorithms
but with a first-order chemical decay of pollutant. This case is included here
for comparison. As V,- increases, the peak GLC's and the downwind distances
-2
where they occur decrease. For V, = 2 x 10 , representing moderately
strong deposition, the concentration at x = 20 km is about an order of magni-
* -3
tude smaller than the corresponding value for V,- = 2 x 10
Figure 10 shows the GLC variation for the secondary pollutant. Since
the direct emission of species-2 is zero, the concentrations shown are
entirely due to the chemical transformation. For k = 1% per hour, the
peak concentration of species-2 is about three orders of magnitude smaller
than the corresponding value of species-1, and the downwind-distances of
67
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these peak values are larger than those of species-1. It should be noted
that C« depends on the deposition parameters of both species as well as on
the chemical transformation rate.
Figure 11 shows a comparison of calculated ground-level centerline concen-
trations of the secondary pollutant, when the latter is composed of (a) particu-
-3
late matter with V,_ = W_ = 1 cm/s (V..- = 5 x 10 ), or (b) gaseous species with
* _o
Vd- = 1 cm/s, W = 0 (V - =10 ). The wind speed U is identical in both
cases. The peak GLC for case (a) exceeds the corresponding value for the
non-deposition case (V,? = VL = 0) and occurs closer to the source. This is
a result of gravitational settling of particles which tends to move the plume
toward the ground as it travels downwind. The effect of deposition in this
case can be seen as an increase of GLC near the source, and a compensating
decrease farther downwind. The GLC variation in case (b) for the gaseous
pollutant is similar to that shown in the previous figure. These results
are consistent with those given by Rao (1981).
CHEMICAL TRANSFORMATION RATE
The effect of varying the chemical transformation rate by two orders of
magnitude on the secondary pollutant concentration is shown in Fig. 12.
Since Q_ = 0 here, chemical transformation accounts for the entire concen-
tration of species-2. For all k values shown, the peak GLC's occur at the
same downwind location, since the meteorology and the deposition parameters
are kept constant. An order of magnitude increase in k increases the
71
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concentration by about the same magnitude. The GLC of the secondary pollu-
tant increases sharply with x until its peak value is attained, and then it
gradually decreases primarily due to a decrease in the reactant concentration.
At small x, the rate of increase in the GLC is directly proportional to k .
The corresponding plot for the primary pollutant concentrations calcu-
lated over the same range of k values is shown in Figure 13. The results
shows that C. is not significantly altered by even a value of k = 10% per
hour, which corresponds to I = 36 x 10 s. The pollutant-transport time
3
scale, given by x/U, is 4 x 10 s at x = 20 km; this is much smaller than
t , and so exp(-x/UT ) ~ 1. Even for a value of k = 100% per hour, GI
decreases by only about 1.1% at x = 20 km.
EFFECTS OF ATMOSPHERIC STABILITY
Figure 14 shows the primary pollutant concentrations calculated for
P-G stability classes A to F (KST = 1 to 6). The peak GLC's decrease slightly
with increasing stability, and their downwind locations move farther away from
the source. The effects of deposition on the GLC generally increase markedly
as the stability increases. However, under convective conditions, atmos-
pheric turbulence enhances the mixing of the plume and, in the well-mixed
region, distributes the pollutant uniformly through the entire mixing depth.
Therefore, as the stability decreases, the GLC's decrease as shown. For KST = 1,
the interpolation for concentration in the plume-trapping region (1 < x < 2 km)
can be clearly seen in the figure.
74
-------
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The corresponding variations of the secondary pollutant concentrations
are shown in Fig. 15. It can be seen that the peak GLC's of species-2
increase with increasing stability. Their values are generally about three
orders of magnitude smaller than the corresponding values for species-1.
The downwind distances of the peak GLC's of the secondary pollutant are
generally larger than those for the primary pollutant.
The concentration curves shown in Figs. 14 and 15 do not appear smooth
for x < 0.5 km because of the finite steps in x used in the calculations. Smoother
curves can be obtained by decreasing the step size Ax and increasing the number
of steps in this region.
77
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78
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SECTION 5
SUMMARY AND CONCLUSIONS
A gradient-transfer (K-theory) model that accounts for the atmospheric
diffusion and deposition of two gaseous or particulate pollutant species,
which are coupled through a first-order chemical transformation, has been
formulated. This model, which includes a deposition boundary condition for
each species, treats the pollutant removal mechanisms in a physically realistic
and straightforward manner. The exact analytical solutions for concentrations
of the two pollutants released from an elevated continuous point source have
been derived. For practical application of the model to a variety of atmospheric
stability conditions, the eddy-diffusivity coefficients in the analytical
solutions have been expressed in terms of the widely-used Gaussian plume disper-
sion parameters, which are functions of downwind distance and stability class.
This approximation allows one to utilize the vast amount of empirical data on
these parameters, for a variety of diffusion conditions, within the framework
of the standard turbulence-typing schemes.
In order to facilitate comparison, the new diffusion-deposition-transforma-
tion algorithms for various stability and mixing conditions have been parame-
terized and presented as analytical extensions of the well-known Gaussian plume
dispersion algorithms presently used in EPA air quality models. In the limit
when the deposition and settling velocities and the chemical transformation rate
are zero, the new algorithms reduce to the standard Gaussian plume equations.
79
-------
Thus the model outlined here retains the ease of application--and is subject
to the same basic assumptions and limitationsassociated with Gaussian plume-
type models.
The formulations and the solutions are general enough to permit either
(or both) of the two pollutant species to be a gas or particulate matter (of
a known size). Direct emission of the secondary (reaction product) pollutant
from the source is permitted. Simplifications of the new algorithms for
ground-level sources and/or receptors, and very slow chemical reactions,
are indicated. Limiting expressions of the algorithms for large particles
are derived.
An innovative mathematical approach has been outlined to derive elegant
expressions for ground-level concentrations of the two species due to emissions
from area sources. These simple expressions, derived from the governing
equations and the deposition boundary conditions, involve only the point source
algorithms for the well-mixed region. This permits one to use the same program
subroutines to compute the concentrations due to both point and area sources.
The new area source concentration algorithms are physically explained in
terms of the mass budgets for the pollutant species in each calculation grid
square. For practical application to urban air pollution models, these algo-
rithms are extended to multiple area sources and receptors. For power-law
variation of a (x), simple modifications of the algorithms are suggested to
z
establish agreement with the Gifford-Hanna area-source algorithms in the limit
when the deposition loss of species is negligible. Thus the area-source
80
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algorithms, derived here through a new mathematical approach are physically
realistic, simple, efficient, and accurate.
A sensitivity analysis of the concentration algorithm for the secondary
pollutant is given to illustrate the delicate balance of its terms and the
behavior of the numerically-evaluated weighting function in one of the
terms. For an elevated point source in a rural area, the variations of
the ground-level concentrations of the two species are calculated by varying
the assigned nominal values of each of the model parameters. The results
are presented and discussed with reference to the Gaussian plume concentra-
tions .
The point and area source concentration algorithms developed in this
report may have wide applicability in practical rural and urban air pollu-
tion and particulate models, such as the Pollution Episodic Model (Rao
and Stevens, 1982), for use in research and regulation.
81
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REFERENCES
Budak, B. M., A. A. Samarskii and A. N. Tikhoiiov, 1964: A Collection of
Problems on Mathematical Physics, Pergamon Press, The MacMillan Co.,
New York, 770 pp.
Calder, K. L, 1961: Atmospheric diffusion of particulate material consi-
sidered as a boundary value problem. J. Meteorol. 18, 413-416.
Carslaw, H. S., and J. C. Jaeger, 1959: Conduction of Heat in Solids,
2nd ed. Oxford University Press, London, 510 pp.
, , 1963: Operational Methods in Applied Mathematics,
Dover Publications, New York, 360 pp.
Ermak, D. L., 1977: An analytical model for air pollutant transport and
deposition from a point source. Atmos. Environ. 11, 231-237.
, 1978: Multiple Source Dispersion Model. Report No. UCRL-52592,
NTIS, Springfield, VA, 68 pp.
Gifford, 1959: Computation of pollution from several sources. Int. J.
Air Pollut. 2, 109-110.
, 1968: An outline of theories of diffusion in the lower layers
of the atmosphere. Chapter 3, Meteorology and Atomic Energy 1968, D. H.
Slade, ed.; available as TID-24190. NTIS, Springfield, VA, 65-116.
, 1970: Atmospheric diffusion in an urban area. 2nd IRPA Con-
ference, Brighton, UK. ATDL Contribution File No. 33, 7 pp.
, and S. R. Hanna, 1970: Urban air pollution modelling. In Proc.
of 2nd Int. Clean Air Congress, Washington, D.C. ATDL Contribution
File No. 37, 6 pp.
, , 1973: Modelling urban air pollution. Atmos. Environ.
7, 131-136.
, 1976: Turbulent diffusion-typing schemes: A review. Nuclear
Safety 17, 68-86.
Hanna, S. R., 1971: A simple method of calculating dispersion from urban
area sources. APCA Journal 21, 774-777.
Heines, T. S., and L. K. Peters, 1973: An analytical investigation of the
effect of a first-order chemical reaction on the dispersion of pollutants
in the atmosphere. Atmos. Environ. _7, 153-162.
82
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Izrael, Y. A., J. E. Mikhailova and A. J. Pressman, 1979: A model for
operative evaluation of transboundary flows of pollutants. WMO
Symposium on the Long-Range Transport of Pollutants and its Relation
to General Circulation Including Stratospheric/Tropospheric Exchange
Processes. WMO No. 538, Geneva, Switzerland, 271-279.
Lee, H. N., 1980: A study of analytical mesoscale nodel for atmospheric
transport, diffusion and removal of pollutants. Proc. Symp. on Interme-
diate Range Atmospheric Transport Processes and Technology Assessment,
Gatlinburg, TN, 289-297. CONF-801064, NTIS, Springfield, VA.
Monin, A. S., 1959: On the boundary condition on the earth surface for
diffusing pollution. Adv. Geophys. 6_, 435-436.
Pasquill, F. 1976: Atmospheric dispersion parameters in Gaussian plume
modeling. Part II. Possible requirements for change in the Turner
Workbook values. EPA-6QO/4-76-030b, U.S. Environmental Protection
Agency, Research Triangle Park, NC, 44 pp.
Rao, K. S., 1975: Models for sulfur oxide dispersion from the Northport
power station. The LILCO/Town of Huntington Sulfates Program, Project
Report P-1336, Environmental Research & Technology, Inc., Concord, MA.
, 1981: Analytical solutions of a gradient-transfer model for
plume deposition and sedimentation. EPA-600/3-82-079, U.S. Environmental
Protection Agency, Research Triangle Park, NC; NOAA Tech. Memo.
ERL ARL-109. 75 pp. ATDL Contribution File No. 81/14.
, and M. M. Stevens, 1982: Pollution Episodic Model User's Guide.
EPA- , U.S. Environmental Protection Agency, Research Triangle
Park, NC; NOAA Tech. Memo. ERL ARL- ATDL Contribution File No.
82/28, 186 pp.
Slade, D. H., 1968: Ed., Meteorology and Atomic Energy, USAEC; available
as TID-24190, NTIS, Springfield, VA, 455 pp.
Smith, M. E., 1968: Ed., Recommended Guide for the Prediction of the Dispersion
of Airborne Effluents. ASME, New York, 1st ed., 94 pp.
Texas Air Control Board, 1979: User's Guide: Texas Episodic Model. Permits
Section, Austin, TX, 215 pp.
Turner, D. B., 1970: Workbook of Atmospheric Dispersion Estimates. Public
Health Service Publication No. 999-AP-26, U.S. Environmental Protection
Agency, Research Triangle Park, NC, 84 pp.
Tychonov, A. N., and A. A. Samarskii, 1964: Partial Differential Equations
of Mathematical Physics. Vol. I. Holden-Day, Inc., San Francisco, 380 pp.
Van der Hoven, I., 1968: Deposition of particles and gases. Section 5-3, in
Meteorology and Atomic Energy 1968, D. H. Slade, ed.; available as
TID-24190. NTIS, Springfield, VA, 202-208.
83
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APPENDIX
SETTLING AND DEPOSITION VELOCITIES
For a monodisperse particulate cloud, the individual particles have a con
stant gravitational settling velocity. This terminal velocity is given by
Stokes' equation (Fuchs, 1964):
18
where d is the diameter of the particle, g is acceleration due to gravity, p is
the particle density, and p is the dynamic viscosity of air. However, for
d > 100 |Jm, the terminal fall velocity is sufficiently great that turbulence
in the wake of the particle cannot be neglected, and the drag force on the
particle is greater than the viscous drag force given by the Stokes' law,
F, = 3nd|jW. For a particle with d = 400 pm, the actual value of W is about
one-third the value given by Eq. (A-l) . Furthermore, Stokes' expression for
the drag force describes the effects of collisions between air molecules
and a particle, assuming air to be a continuum. This assumption is not valid
for very small particles, since the mean free path between molecular collisions
is comparable to the particle size; under these conditions "slippage" occurs,
and the particles undergo Brownian motion and diffusion, which give a terminal
velocity greater than that predicted by Eq. (A-l). A discussion of the slip
correction factor for the Stokes' equation can be found in Fuchs (1964) and
Cadle (1975).
84
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The values for the terminal gravitational settling velocities for different
particulate materials are given in a tabular form by Lapple (1961) based on
particle diameter and Reynolds number. These values, which account for
the deviations from Stokes1 equation discussed above, are given for spherical
particles with a specific gravity of 2.0 in air at 25°C and 1 atm. pressure.
This table has been reprinted in Sheehy et al (1969) and Stern (1976).
The dry deposition pollutant-removal mechanisms at the earth's surface
include gravitational settling, turbulent and Brownian diffusion, chemical
absorption, inertial impaction, thermal, and electrical effects. Some of the
deposited particles may be re-emitted into the atmosphere by mechanical resus-
pension. Following the concept introduced by Chamberlain (1953), particle
removal rates from a polluted atmosphere to the surface are usually described
by dry deposition velocities which vary with particle size, surface properties
(including surface roughness z and moisture), and meteorological conditions.
The latter include wind speed and direction, friction velocity lu, and thermal
stratification of the atmosphere. Deposition velocities for a wide variety
of substances and surface and atmospheric conditions may be obtained directly
from the literature (e.g., McMahon and Denison, 1979; Sehmel, 1980). Sehmel and
Hodgson (1974) give plots relating deposition velocity V, to d, z , u^., and the
Monin-Obukhov stability length, on the basis of wind tunnel studies.
Considerable care needs to be exercised in choosing a representative
deposition velocity, since it is a function of many factors and can vary by
two orders of magnitude for particles. Generally, V, should be defined
relative to the height above the surface at which the concentration measurement
is made. The particle deposition velocity is approximately a linear function
85
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of wind speed and friction velocity, and its minimum value occurs in the particle
diameter range 0.1-1 fjm.
In the trivial case of W = V, = 0, settling and deposition effects are negligible
For very samll particles (d < 0.1 }Jm), gravitational settling can be neglected,
and dry deposition occurs primarily due to the nongravitational effects mentioned
above. In this case, W = 0 and V, > 0. For small to medium-sized particles
(d = 0.1~50 [Jim), 0 < W < V.; deposition is enhanced here beyond that due to
gravitational settling, primarily because of increased turbulent transfer
resulting from surface roughness. For larger particles (d > 50 |jm), it is
generally assumed that V, = W > 0, re-entrainment of the deposited particles
from the surface into the atmosphere is implied as, for example, in a dust
storm. The first four sets of model parameters given above are widely used
in atmospheric dispersion and deposition of particulate material. The deposition
of gases is a special case of the particulte problem with W = 0.
86
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REFERENCES
Cadle, R. D., 1975: The Measurement of Airborne Particles. John Wiley & Sons,
New York, 342 pp.
Chamberlain, A. C., 1953: Aspects of travel and deposition of aerosol and vapor
clouds. A.E.R.E. Report H.P.-1261, Atomic Energy Research Estab., Harwell,
Berks., U.K., 32 pp.
Fuchs, N. A., 1964: The Mechanics of Aerosols. The Macmillan Co., New York,
408 pp.
Lapple, C. E., 1961: J. Stanford Res. last. 5, p. 95.
McMahon, T. A., and P. J. Denison, 1979: Empirical atmospheric deposition
parameters - a survey. Atmos. Environ. 13, 571-585.
Sehmel, G. A., and W. H. Hodgson, 1974: Predicted dry deposition velocities.
Atmosphere-Surface Exchange of Particulate and Gaseous Pollutants.
Available as CONF-740921 from NTIS, Springfield, VA, 399-423.
Sehmel, G. A., 1980: Particle and gas dry deposition: a review. Atmos.
Environ. 14, 983-1011.
Sheehy, J. P., W. C. Achinger, and R. A. Simon, 1969: Handbook of Air
Pollution. Public Health Service Publication No. 999-AP-44, U.S.
Environmental Protection Agency, Research Triangle Park, NC.
Stern, A. C., 1976: Air Pollution. Academic Press, New York, Vol. I, 3rd ed.,
715 pp.
87
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TECHNICAL REPORT DATA
- - - (Please read Instructions on the reverse before completing)
1. REPORT NO. 2.
4. TITLE ANO SUBTITLE
PLUME CONCENTRATION ALGORITHMS WITH DEPOSITION,
SEDIMENTATION, AND CHEMICAL TRANSFORMATION
7. AUTHOR(S)
K. Shankar Rao
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Atmospheric Turbulence and Diffusion Division
National Oceanic and Atmospheric Administration
Oak Ridge, Tennessee 37830
12. SPONSORING AGENCY NAME ANO ADDRESS
Environmental Sciences Research Laboratory - RTP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
3. RECIPIENT'S ACCESSION-NO.
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
CDTA1D/02-1606 FY-84
11. CONTRACT/GRANT NO.
IAG-AD-13-F-1-707-0
13. TYPE OF REPORT AND PERIOD COVERED
Final 9/81-3/83
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT
A gradient-transfer model for the atmospheric transport, diffusion, deposition,
and first-order chemical transformation of gaseous and particulate pollutants emitted
from an elevated continuous point source is formulated and analytically solved using
Green's functions. This analytical plume model treats gravitational settling and dry
deposition in a physcally realistic and straightforward manner. For practical appli-
cation of the model, the eddy diffusivity coefficients in the analytical solutions
are expressed in terms of the widely-used Gaussian plume dispersion parameters. The
latter can be specified as functions of the downwind distance and the atmospheric
stability class within the framework of the standard turbulence-typing schemes.
The analytical plume algorithms for the primary (reactant) and the secondary
(product) pollutants are presented for various stability and mixing conditions. In
the limit when deposition and settling velocities and the chemical transformation
rate are zero, these equations reduce to the well-known Gaussian plume diffusion
algorithms presently used in EPA dispersion models for assessment of air quality.
Thus the analytical model for estimating deposition and chemical transformation
described here retains the ease of application associated with Gaussian plume models,
and is subject to the same basic assumptions and limitations as the latter.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
13. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport)
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UNCLASSIFIED
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EPA Form 2220-1 (9-73)
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