oEPA
United States
Environmental Protection
Agency
Environmental Sciences Research EPA 600 4 79 037
Laboratory June 1 979
Research Triangle Park NC 27711
Research and Development
Analytical
Diffusion Model for
Long Distance
Transport of Air
Pollutants
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
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The nine series are:
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2 Environmental Protection Technology
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This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
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and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-79-037
June 1979
ANALYTICAL DIFFUSION MODEL FOR
LONG DISTANCE TRANSPORT OF AIR POLLUTANTS
James A. Fay
Jacob J. Rosenzweig
Fluid Mechanics Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
Grant No. R804891-01-2
Project Officer
Kenneth L. Demerjian
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, N.C. 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, N.C. 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 publication.
Approval does not signify that the contents necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does mention of trade
names of commercial products constitute endorsement or recommendation for use.
AFFILIATION
Dr. Demerjian, the Project Officer, is on assignment to the Meteorology
and Assessment Division, Environmental Sciences Research Laboratory, from the
National Oceanic and Atmospheric Administration, U.S. Department of Commerce.
ii
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ABSTRACT
A steady-state two-dimensional diffusion model suitable for predicting
ambient air pollutant concentrations averaged over a long time period (e.g.,
month, season or year) and resulting from the transport of pollutants for
distances greater than about 100 km from the source is described. Analytical
solutions are derived for the primary pollutant emitted from a point source
and for secondary pollutant formed from it. Depletion effects, whether due
to wet or dry deposition or chemical conversion to another species, are
accounted for in these models as first order processes. Thus, solutions for
multiple point sources may be superimposed.
The analytical theory for the dispersion of a primary pollutant is compared
with the numerical predictions of a plume trajectory model for the case of
steady emission from a point source. Good overall agreement between the two
models is achieved whether or not depletion by wet and dry deposition is
included.
The theory for the dispersion of a secondary pollutant is compared with
measurements of the annual average sulfate concentration in the U.S. Calcula-
tions are carried out using S0£ emissions from electric power plants in the
United States as a source inventory. Using optimum values of the dispersion
parameters, the correlation coefficient of observed and calculated ambient
concentrations for the United States is 0.46. However, when the observed data
is smoothed to eliminate small scale gradients the best correlation coefficient
achieved is 0.87 for the eastern United States and 0.69 for the western region.
The optimum dispersion parameters used are comparable to values quoted in the
literature.
The horizontal length scale characterizing the sulphate concentration
distribution from a single source is about 500 km, being noticeably larger
than that for primary (sulfur dioxide) distribution. Using optimum dispersion
parameters, a point source of 33 kg/s of sulfur dioxide would give rise to a
maximum annual average sulfate concentration of 1
A calculation of annual average S02 concentrations in the United States
is carried out using previously derived optimal values of the parameters from
the sulfate calculation. The resulting isopleths are similar to measured
values in the eastern U.S.
iii
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SECTION 1
INTRODUCTION
The transport of air pollutants to great distances from a source becomes
a matter of concern when the aggregate effects of many sources may produce
harmful ambient pollutant levels or depositions far downwind (10^ to 10-* km)
of the source region. Most attention has focussed on sulfur compounds and
acidic rainfall (see Symposium on sulfur in the atmosphere, 1978), but the
long distance transport of ozone (Wolff et al., 1977), photochemical oxidants
and their precursors (Cleveland et al., 1976), pesticides (Glotfelty, 1978)
and nitrates (National Research Council, 1978) also may have significant envi-
ronmental effects far downstream of their sources. A principal difficulty in
understanding quantitatively such phenomena is the limited ability of models
to describe accurately the transport, dispersion, transformation and deposition
at greater distances and far longer travel times than has customarily been
treated in the past when examining pollutant transport close to the source.
This paper describes a relatively simple analytical model which may help to
provide estimates of long term average pollutant levels at large distances
from a large collection of sources. It also permits a rapid assessment of the
contribution to the pollutant level at a given receptor of a single source or
class of sources.
In one way or another, all long distance transport models provide for the
principal physical effects considered to be significant in the process: advec-
tive motion of the atmosphere, dispersion caused by atmospheric turbulence,
transformation of primary to secondary pollutants, and wet and dry deposition
of pollutants. In the case of sulfur compounds, all of these processes are
thought to be significant, at least in some region of the flow field, and
the corresponding models exhibit considerable complexity. But the understand-
ing of the contributory processes is limited and simplifications (e.g., model-
ling of the transformation as a first order process) are often used.
Current long distance transport models can be classified conveniently
into two categories. In the first are models for which the diffusion equation
is integrated over a limited domain of time and space, such as several days
arid distances of the order of 10^ km. These models are probably most useful
in studying short duration episodes of high pollutant levels (Hidy et al.,
1978). The second category uses a calculation of air parcel trajectories to
determine advective motion, with various methods for treating the dispersion
about the mean motion of the parcel. These latter models are most often used
to determine ambient levels or deposition rates averaged over a period of a
month to a year, and over travel distances up to 10^ km (Eliassen, 1978,
Johnson et al., 1978, Ottar, 1978).
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A model of the first variety, described by Eidy et al. (1976), considers
concentration variation in three dimensions and time. The governing three
dimensional unsteady diffusion equation is averaged over the period of 24 hours.
Depletion of pollutants is viewed as first order in pollutant concentration.
Since the governing equations are solved numerically the horizontal wind speci-
fied at each grid point can approximate realistically the actual short term
averaged wind field. In another model, Gillani and Husar (1976) assume a
steady state diffusion equation for transport scales from 1 - 100 km. The
mean wind is assumed uniform while the horizontal cross wind eddy dlffusivity
varies with downwind distance. Again the rate of primary pollutant loss is
assumed to be first order.
Trajectory diffusion models can be further categorized as puff or plume
models. Eliassen and Saltbones (1975) developed a receptor-oriented puff
model. Daily trajectories terminating at a receptor point are determined back-
ward in time for 48 hours. A parcel of air moving along the trajectory is
assumed to be subjected to an instantaneous Infusion of pollutant from adjacent
area sources while the rate of loss of pollutant occurs as a continuous first
order process. The concentration of pollutant within the air parcel is assumed
uniform. Four puffs arriving at the sampling point are appropriately averaged
to determine mean daily pollutant concentration. Start and Wendell (1974)
developed a source-oriented trajectory puff model, where source emissions are
approximated by a series of puff releases. The center of mass of each puff Is
determined by trajectory ends while the concentration distribution, about the
center of each puff, is assumed to be Gaussian in the vertical and horizontal
directions. Rates of diffusion about the puff centers may vary along the tra-
jectory length. Concentrations at a given point are determined by superposing
concentrations due to all puffs. In a similar model, Shelh (1977) assumed the
concentration distribution of each puff as Gaussian in the horizontal direction.
However, the vertical concentration distribution is determined by numerical
integration of nonsteady diffusion equations for primary and secondary pollu-
tants. In this case emissions from elevated sources and depletion of pollutant
to the ground can be treated more appropriately.
A trajectory plume model developed by Heffter et al. (1975) was applied
using length scales between one hundred and several thousand kilometers and
averaging times of one month or greater. Trajectories initiated from the
source four times daily are assumed to be the instantaneous centerlines for a
quasi-steady Gaussian plume. Contributions from each plume are superposed
and averaged over an appropriate time interval. Depletion due to various
mechanisms is incorporated in the model by adjusting the source strength term
appropriately. Draxler (1976) added a finite difference scheme for vertical
diffusion to this trajectory plume model, thus giving a more realistic concen-
tration profile in the vertical direction and treating wet or dry deposition
as a boundary condition.
Validation of these models is limited by the sparsity of suitable field
observations. The most extensive comparison of a model with field data has
been that for sulfur compounds in western Europe (Ottar, 1978). Since the
models take into account the major mechanisms involved in the long distance
transport of pollutants, namely, the large scale variability of the wind
velocity and depletion effects, they require detailed knowledge of the wind
field and precipitation over large regions for the duration of the time-averaging
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interval. Thus their major disadvantage is the need for substantial computer
capacity in order to account for the details of meteorology.
The diffusion model discussed below differs from those mentioned above in
that it does not require detailed knowledge of the history of the wind field.
By averaging over a time period of one month or greater, it is assumed that the
governing diffusion equations are time independent and that the wind field can
be usefully approximated as being uniform over a large spatial scale. To
account for the advective motions developed in detail in other (unsteady)
models we employ a horizontal diffusion coefficient suitable for the large
time and length scales involved.
Based upon the very limited comparison with a trajectory model, which
we describe below, our model successfully describes the advective effects of
the variable wind field for an averaging time of a month. When applied to a
determination of annual average ambient sulfate levels in the U.S., this model
(incorporating first order transformation and deposition effects) correlates
the observations about as successfully as do the more elaborate trajectory
models for sulfur compounds in the western European region (Ottar 1978).
Because of its simplicity, this model may be a useful adjunct to the repertoire
of models needed to understand better the problem of long distance transport.
In what follows, we present in Section 2 the theory and analytical solu-
tions of our diffusion model. In Section 3 we carry out calculations for a
point source and compare the results with trajectory-plume model predictions.
Section 4 describes a calculation of sulfate dispersion in the United States
due to emissions from electric power plants of SC^- This calculation is com-
pared with data and parameters are adjusted for the best possible agreement
with data. Using the best values of the parameters an S02 calculation for the
United States is also carried out.
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SECTION 2
THEORY AND ANALYTICAL SOLUTIONS
As an alternative to the models described above, we propose a steady
state two dimensional diffusion model to describe the long distance transport
of primary and secondary air pollutants. It is assumed that primary emissions
occur from an ensemble of continuous point sources. The source of secondary
pollutant is distributed continuously throughout the entire region due to the
volumetric conversion of primary pollutant. Both types of pollutant are
assumed to mix uniformly in the vertical direction within a well-mixed region
of the atmospheric boundary layer. Thus ambient concen Orations vary as a
function of horizontal coordinates only. Removal of pollutant occurs due to
wet and dry deposition and chemical transformation to other species.
The large scale horizontal dispersal of atmospheric pollutants may be
thought of as analogous to a two dimensional random walk process. A contin-
uous source of pollutant may be viewed as a serial emitter of infinitesimal
puffs, each puff being transported in a random manner by the turbulent wind
fluctuations. Given that the scale of our problem is on the order of several
thousand kilometers it becomes apparent that the large turbulent eddies (e.g.
on the order of several hundred km.) are dominant in dispersing the puffs about
the point of emission while smaller turbulent eddies are responsible for dis-
persing each puff about its center of mass. We neglect the latter small scale
distortion of the puff and instead consider each puff as if it were an infini-
tesimal point mass. This simplification is appropriate in determining the
spatial concentration distribution averaged over long time periods since it
superposes the contributions from each puff. Thus we are interested in deter-
mining the horizontal puff dispersion primarily due to the large scale turbu-
lent wind motions. In the limiting case of infinitely many puffs averaged over
a long time our problem reduces to a solution of the steady state diffusion
equation with an appropriately determined horizontal diffusion coefficient.
Taylor (1921) showed formally how a diffusion coefficient may be deter-
mined by extending the statistical analysis of the random walk process to the
continuous motions of turbulent flow. We choose to avoid the formalism of
Taylor's analysis but take note of the conclusion reached that for large travel
times an ensemble of diffusing particles has an rms displacement about their
common center of mass which is proportional to the square root of the travel
time. Therefore, for large travel times the diffusion coefficient approaches
a constant value, as it should for a random walk process. As a consequence,
we assume a constant horizontal diffusivity Djj that is determined to an order
of magnitude by the relation
D « V2 T (1)
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where V is the Eulerian standard deviation of the wind velocity and rp is the
Eulerian mean persistence time of the wind with respect to direction, which is
analogous to the mean free time between collisions of a gas molecule. An order
of magnitude equivalence is assumed between the diffusivity as used in an
Eulerian sense with an ensemble average of Lagrangian measurements. Although
a more precise relationship for the horizontal diffusivity might be derived
with great difficulty we do not now consider such an improvement. Instead,
we intend to adjust Tp or D^ in order to achieve the best agreement with other
calculations or with measured data.
Limited data exists on typical values of T'. Shirvaikar (1972) measured
at ground level the persistence time for a given direction within a sector of
22.5°. He found at several sites that the distribution of persistence time is
lognormal with a standard deviation of 2 to 3 hours. Pasquill (1974) noted
that the Lagrangian time scale in the crosswind direction is on the order of
5 to 15 hours. Thus we expect Tp to be on the order of several hours. Assum-
ing V is 5 m/s, Dh would then be about 5 x 10^ m^/s.
Estimates of D^ have been made by several investigators using geostrophic
trajectory calculations. Murgatroyd (1969) used two independent methods for
calculating the horizontal diffusivity. In the first method, the standard
deviation a of the end points of the trajectories launched from a fixed point
were determined as a function of travel time and the corresponding diffusion
coefficient, also a function of time, was determined from
Dh = o2/2t (2)
The asymptotic value of D^ was extrapolated as t -»• °°. The second method assumed
an autocorrelation coefficient of the damped harmonic form and values of the
parameters where chosen by regression with trajectory data. The correlation
coefficient was then substituted into the equations derived by Taylor (1921)
to determine the asymptotic value for the diffusion coefficient. The resulting
calculations showed D^ to be about 2 x 10^ m2/s at the lowest elevation (700mb).
Durst et al. (1959) also used trajectories to calculate the diffusion coeffi-
cient by three independent methods and found values of D^ in the range of 10^
to 10 7 m^/s at 700 mb. Thus, the order of magnitude of D^ ~ 10& m2/s seems to
be well established.
Because our model describes the dispersion process averaged over a long
period of time (e.g., month, season or year), time-averaged values are used
for the process parameters such as precipitation rate, dry deposition rate,
transformation rate, wind speed/direction, etc. A more substantial question
is whether these time-averaged parameters vary also with position.
As a first approximation we have assumed that these parameters are con-
stant over the geographic region of interest, permitting an analytical form
for the solution of the horizontal diffusion equation. An alternative, which
we have not attempted, is to consider these parameters to be position dependent
and obtain a numerical integration of the diffusion equation. Not only is the
simplicity of the analytical method lost in this alternative, but the values of
5
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most of the parameters are not known with enough accuracy to justify the con-
siderable additional complication. A more intuitively appealing variation is
to select values of the principal parameters suitable to the conditions at
each source, so as to more accurately determine concentrations near the source
where they will be highest and hence most significant. (This we have tried in
our sulphate calculation described below). Finally, it can be shown that var-
iations of mixing height can be well approximated by selecting as the parame-
tric value that at the receptor point.
Depletion effects are simulated by assuming a mass loss rate per unit
volume of the form C/T, where C is the mass concentration of the pollutant per
unit volume and T"-*- is the first order rate constant for the loss of pollutant
due to all causes. Three modes of depletion are taken into account: dry depo-
sition, wet deposition caused by precipitation scavenging, and the loss of
pollutant by chemical conversion of the primary species to a secondary form,
(e.g., the conversion of S0£ to sulfates). Thus the rate constant which takes
into account all forms of depletion of a primary pollutant can be written as
I' = CT^ + T^ + Tj) (3)
where T^ , TW and TC are the dry deposition, wet deposition and chemical
conversion rate constants respectively. Of course, only the chemical conver-
sion term is truly a volume loss term, whereas the other depletion terms simu-
late the loss of pollutant to the ground, and must be suitably averaged through
the mixing layer. Thus TC may be interpreted as the average time interval dur-
ing which a molecule of primary pollutant drifts in the air before chemically
converting to a secondary species.
Bearing these assumptions in mind, the horizontal distribution of concen-
tration for a primary pollutant is determined by
where u and v are the components of the mean wind velocity in the x and y
directions and D^ is the horizontal diffusivity.
Equation (A) can be solved for a steady point source having an emission
rate Q and assuming the boundary condition of zero concentration at infinity.
The horizontal distribution of concentration becomes:
exp
K
V
(5)
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where the -x' axis is aligned in the direction of the mean wind velocity w, K
is the modified Bessel function of the zeroth order, and r is the radial dis-
tance from the source to the receptor. It should be noted that (5) is a solu-
tion of a linear equation satisfying homogeneous boundary conditions. Thus,
the principle of superpositioning the multiple sources applies, which we have
utilized below.
Since (5) is a solution for a point source, we must disregard the solu-
tions very near the source where predicted concentrations approach infinity.
Because the function KQ is only weakly singular about r = 0, for (5) to be
valid it is sufficient that r be large enough that the logarithm of the argu-
ment of K not be a large negative number. Also, the assumption of uniformly
mixed pollutants within the mixing layer breaks down close to the source. In
order to avoid the aforementioned difficulties we calculate concentrations at
points no closer than about 10 kilometers from the source, a distance past
which we assume that the pollutants have dispersed uniformly in the vertical
direction.
The formation and dispersion of secondary pollutants can be treated by a
diffusion model, similar to (4), which we now derive. Unlike primary pollu-
tants which are emitted from point or area sources secondary products are
formed continuously, within the atmosphere itself. The loss term in (4) due
to chemical conversion of primary pollutant is proportionate to the source
term in the secondary pollutant differential equation, namely BC/TC, where C
is the mass concentration of the primary pollutant, g is the mass ratio of
secondary pollutant formed per unit mass of primary pollutant, and TC is the
rate constant for chemical conversion to a secondary species. Thus, the govern-
ing equation for the secondary pollutant becomes
9C
S _. i 8 . e i B . ^w /f\
w = D. \ ^ + o / - — + — (6)
-1 -1 -1
where Cs is the concentration of the secondary pollutant and TS = (TQ +Tr )
is the net rate constant for loss of the secondary pollutants due to wet and
dry deposition, whose respective rate constants are Tq-l and Tr~l.
Equation (6), although nonhomogeneous, may be solved in closed form for
a distributed source of secondary pollutants formed from a point source of
primary pollutant at r = 0. Thus the horizontal concentration profile becomes
w p - Ko(ar)
8 2 2 2 2
. T h a - y
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where
(8)
4Dh
Again, solutions due to multiple sources of primary pollutant can be super-
posed because the boundary condition at infinity specifies an ambient concen-
tration of zero.
Considering the forms of (5) and (7), the spatial distribution of primary
pollutant concentration C is characterized by two length scales, l± = 2 D^/w
and £,£ = (Dh1)1^2* while that of secondary pollutant Cs involves as well a
third length scale £3 « (D^s)1'2- The estimates of Djj given above lead to
the conclusion that l± is of the order of 10^ m. (It will be seen below that
SL2 and &3 have the same magnitude for sulfur oxide dispersion). Thus the
characteristic scale of the concentration pattern from a point source will be
of the order of 10^ m.
8
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SECTION 3
COMPARISONS WITH TRAJECTORY-PLUME MODEL CALCULATIONS
As a first step in assessing the accuracy of the diffusion model for
primary pollutants we compare our results with those of the trajectory-plume
model developed by Heffter et al., (1975). For the purpose of this comparison,
Heffter provided us with a sample computer calculation of the two-dimensional
model which assumes a spatially uniform mixing height with pollutants well
mixed in the vertical direction. For a steady point source in Illinois, mean
concentration levels were calculated for trajectories averaged over the month
of March 1975. Trajectories were initiated four times daily at six hour inter-
vals and tracked for a maximum period of ten days. Contour maps depicted
ambient concentration levels for two cases, the first without depletion and
the second with depletion due to dry and wet deposition. Also calculated
were deposition contours and individual trajectories.
The input parameters and data used in the comparative diffusion calcula-
tion were the same as these used in the trajectory calculation. The deposi-
tion velocity parameter used in the trajectory calculations was converted to
its equivalent dry deposition rate constant by the relation
(10)
where v, is the dry deposition velocity. Likewise an equivalent wet deposi-
tion rate constant was derived from the relation
T - (C/Oh/EP (11)
w
where C is the ambient concentration at gjround level, which in this case is
assumed uniform within the mixing layer, C is the mean concentration of the
pollution in the column of air from the ground to the top of the rainbearing
layer, P is the precipitation rate and E is an empirically derived scavenging
ratio. Trajectory data at the source was used to calculate the values of the
mean wind speed and direction. In these calculations various values of D^
were assumed and the diffusion model concentration contours were compared with
these of the trajectory calculation. It was found that Djj - 6.4 x 105 m2/s
gave the best overall agreement for all comparisons made. This value of the
diffusion coefficient is near that given by Murgatroyd (1969) and Durst et al.
(1959).
9
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The comparative calculations for two-dimensional diffusion without and
with depletion effects are shown in Figs. 1 and 2, respectively. For these
calculations the mixing layer height was 1500 m, the mean wind velocity was
3.2 m/s at the point of emission with a wind direction from 265° true. For
the calculation displayed in Fig. 2, depletion was assumed to have occurred
by two mechanisms, dry deposition with a deposition velocity v^ = 0.01 m/sec
and depletion due to precipitation scavenging with a rate constant TW = 3 x 105
sec. The net depletion time T was consequently determined to be 10-> sec. An
obvious consequence of including depletion effects is that the contour profiles
tend to decrease in area and also close at lesser distances downwind. Compared
with Fig. 1, there is better agreement between the two models when depletion
effects are included, as in Fig. 2.
Judging from the graphical comparison of Figs. 1 and 2, it is evident
that there are some differences between the results of each model. In the
case of no depletion it is specially evident that the trajectory model contours
tend to close at lesser distances downwind from the source than the diffusion
model contours. In the trajectory model, trajectories are computed for a maxi-
mum of ten days travel time so as not to exceed computer capacity limitations.
Thus, regions outside the range of these trajectories experience zero values
for the ambient concentration. The diffusion model, however, desribes a
steady state situation where the ambient concentration approaches zero at
infinity, a condition that would probably be approximated by the trajectory
model if the individual trajectories were followed for a infinite time inter-
val and to an infinite distance.
It is also evident that the trajectory model exhibits puckering of the
contour lines while the diffusion model exhibits smooth contours that are
symmetric about the axis of the direction of the mean wind velocity. The
puckering probably is a consequence of averaging concentrations along traject-
ories for the relatively short period of one month, for which 120 trajectories
were not fully random in character and some wind directions were more likely
than others, causing some of the trajectory paths to be grouped and contour
profiles to become puckered. It is believed that averaging concentrations
for a longer period of time (e.g. annually) would result in "smoother" contour
profiles because the trajectories would not be lined up in any preferred
direction.
10
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-14
0 500
Kilometers
1000
Figure 1. Comparison of model calculations of isopleths of primary pollutant
concentration for a point source (+) without depletion effects.
Dashed lines are determined from a plume-trajectory model and solid
lines from the analytical diffusion model. Source strength is
1 kg/s. Isopleth value is the logarithm of the concentration
-11
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0 500 1000
Kilometers
Figure 2. Same comparison as in Figure 1, but including depletion by wet and
dry deposition.
i '
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SECTION 4
COMPARISON WITH U.S. ATMOSPHERIC SULFATE MEASUREMENTS
Because most current interest has centered on the long distance transport
of sulfur compounds, we have compared the results of our model calculations
with measurements of atmospheric sulfate concentrations within the continental
U.S. In making this comparison, we have assumed that the major contributors
to ambient sulfate levels are the elevated sources of sulfur dioxide, following
the conventional argument that sulfur dioxide emitted from low or gound level
sources is significantly depleted close to the source by deposition to the
ground before it can be converted to sulfate in the atmosphere. As a conven-
ient elevated source inventory we use the emissions from fossil fuel electric
power plants, which comprise the major proportion of such emissions. However,
in using our model for the purpose of this comparison, we assume that there
exists a geographically uniform background level of sulfates whose level is
to be determined empirically, added to which is the contribution from the
power plant emissions. Low level anthropogenic and biogenic S02 emissions can
thus be considered as the principal source of this sulfate background.
As there are about 700 power plants to be included in this inventory, we
adopted the following simplification in order to reduce the length of the
computation. The power plant emissions, as estimated by the Federal Power
Commission (1973) for the year 1970, were aggregated within each air quality
control region, as defined by the Environmental Protection Agency (1972).
These aggregated emissions were then considered to issue from a single source
located in the center of each air quality control region. Our aggregated
inventory thus consisted of 106 point sources comprising 96% of total U.S.
power plant emissions, the remainder being judged too small to justify their
inclusion.
An advantage of our model is the relative ease with which multiple sources
may be superposed. Trajectory-plume model calculations for 700 power plants,
or even 100 air quality control region sources, would be very expensive,
especially if averaged over a season or year. Some aggregation of the sources,
as we have done, is not essential to the use of the model, but is a practical
convenience having negligible effect on the precision of the results.
Using trial values for the parameters in equation (7), a calculation was
made of the annual ambient sulfate concentrations for comparison with the
measurements described by Hidy et al. (1976), which comprise a network of 222
stations located as shown in Fig. 3. A disappointingly low correlation coeffi-
cient of 0.46 was obtained between the observations and the model predictions.
Our model could not account for the significant variations in concentrations
within short distances which were observed.
13
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Others have encountered similarly poor correlations. Wendell et al. (1977)
compared trajectory-puff model calculations with these same measurements aver-
aged for the month of April 1974, finding a correlation coefficient of 0.40.
Similarly, the spatial variability of observed data could not be duplicated by
their model. They suggested that uncertainties in the observed data might
be responsible for the poor correlation. For example, they noted that most air
quality monitoring stations were located in urban areas where the depletion
rates might be higher than in rural areas and that S02 interference on high
volume samplers may introduce errors of unknown amount in the sulfate measure-
ment. In addition to these suggested effects, we believe that the significant
spatial variability in the observed data may be also a result of locally induced
phenomena such as the unique terrain features upon which the various sensing
instruments are located and sulfur emissions from small local sources which
have not been accounted for in the model calculations.
In discussing the use of a trajectory-puff model to describe sulphur trans-
port and deposition in western Europe, Eliassen (1978) noted that the correla-
tion coefficient for model and measured average daily sulfate levels varied
between 0.4 and 0.8, and showed that the annual average sulfate contours pre-
dicted by the model gave less spatial variability than the measured values.
Those effects which give rise to local variability in the measurements
may be diminished by smoothing out the measured data. Accordingly, we fitted,
by least squares, a polynomial surface to the data points, using latitude and
longitude as independent variables. A best correlation coefficient of 0.75
was obtained for a fifth order polynomial when comparing the measured data with
the polynomial representation. In all of the subsequent comparisons with our
model, the smoothed representation of the measured data was used.
In our calculations the mixing height h used in (7) was evaluated at each
receptor point as the harmonic mean of the values given by Holzworth (1972) for
mean annual morning and afternoon mixing heights. At most, this variation did
not exceed 25% of a national average of about 650 m.
In the first comparisons between the model calculation and observation with
respect to the spatial distributions of concentrations across the U.S., it was
noticed that the west coast region might be better described by a set of para-
meters which differ from those in the east. Therefore, we divided the United
States into two separate regions at the 100th meridian and attempted to corre-
late the model with the data separately for each region. The east contained
173 data points while the west had 49.
As previously mentioned, the solution (7) for the spatial distribution of
secondary pollutant concentration depends parametrically upon three dimensional
length scales. In addition, the wind direction and the scaling factor D^ TC
are also parameters, the mixing height being determined as described above.
Concentrating on the eastern U.S., we varied these five parameters so as to
maximize the correlation coefficient for the model calculation as compared
with the smoothed representation of the data, obtaining a value of 0.87 when
using the optimum choice of dimensional parameters listed in Table 1. A back-
ground level of 5.95 yg/m^ was obtained from this linear regression.
14
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For the western half of the U.S. a smaller correlation coefficient was
obtained (0.69) and the optimum dimensional length scales were smaller
(Table 1) as might be expected from the higher gradients of concentration
observed on the southwest coast where most measuring stations are located. In
an alternate trial, using a different west coast source inventory compiled by
the U.S. Environmental Protection Agency (1974) for the year 1972, a still
lower correlation coefficient of 0.53 was found. This particular inventory
included estimates of emissions from all sources, not just those from power
plants.
Figure 3 depicts the diffusion model concentration contours (solid lines)
resulting from use of the optimal values of the parameters for the eastern and
western United States as given in Table 1. Also shown are the contours of the
smoothed sulfate data (dashed lines) and the location of sulfate monitoring
stations. It can be seen that good agreement, both quantitative and qualita-
tive, is achieved in the eastern portion of the United States. In the western
portion, however, agreement is less favorable. In southern California espe-
cially peak values of the ambient concentration are underpredicted while the
contrary is true for the northwest mountain region. These less favorable re-
sults in the west may be due in part to our arbitrary division of the United
States at the hundredth meridian into eastern and western regions and in part
to real effects of regional topography. In addition, since there are fewer
measuring stations in the western region, the correlation there is less signi-
ficant than in the east.
3
The sulfate background level of about 6 vg/m (Table 1) determined from
the linear regression is a little higher than values reported for rural regions.
Ottar (1978) notes that annual sulfate concentrations in western Europe are
about 1 iig/m3 in the more remote (western) regions. Meszaros (1978) summarizes
values of 3 to 5 ug/m^ for rural western and eastern U.S. sites. Recognizing
the spatial variability of the measurements, the agreement is satisfactory.
The four optimum dispesion parameters listed in Table 1 incorporate five
physical parameters: w, D^, T, T and TS. While these latter values are thus
not determinable from the four dispersion parameters, we suggest a set of
values for them, shown in Table 2, which is consistent with Table 1 and various
estimates given in the literature. Also given in Table 2 are specific esti-
mates for or bounds upon these values as given by other investigators. Of
these, the values of Johnson et al. (1978) and Eliassen (1978) were suggested
by examination of the comparison of trajectory model calculations with obser-
vations. There is general agreement among the suggested values for the signi-
ficant time scales. The advection parameters, w and D^» are used in our model
to replace trajectory calculations.
Additional model calculations were carried out for the eastern U.S. illus-
trating several interesting points. A calculation using optimal parameters as
determined previously but assuming a uniform mixing height resulted in a
correlation coefficient of 0.87, equal to that for the case incorporating
variable mixing height. Another calculation using the same optimal parameters
as above and a 1972 inventory of all sulfur sources (U.S. Environmental Protec-
tion Agency 1974) gave an equal correlation coefficient of 0.87. We thus
conclude that the model results are not sensitive to the spatial distributions
of mixing height (and probably other dispersion parameters as well) and source
inventory.
15
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TABLE 1 OPTIMUM SULFATE DISPERSION PARAMETERS
TD (m )
T D, (m )
s h
? 2 7
Dh /w On )
Wind Direction
Dh2 TC (m4/s)
Correlation
Coefficient
3
Background (yg/m )
Eastern U.S.
1.3 x 1011
2.5 x 1011
4.0 x 1012
270° true
9.78 x 1017
.87
5.95
Western U.S.
1.0 x 109
1.3 x 109
6.4 x 109
270° true
1.02 x 1013
.69
5.65
16
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Figure 3. A comparison of calculated (solid line) and measured (dashed line)
annual average sulfate concentrations (yg/m^) in the U.S. Small
circles give location of monitoring stations. Measured isopleths
are determined from a smoothed correlation of the raw data.
17
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TABLE 2 SULFATE DISPERSION CONSTANTS
w D, T T T
h s c
(m/s) (m2/s) (s) (s) (s)
This study, eastern 2.0 2.0 x 106 8 x 104 1.6 x 105 2.4 x 105
U.S.
Johnson et al. (1978) <5 x 105 <5 x 105 3.6 x 105
Garland (1978) <105 >104 4 x 105
Eliassen (1978) 2.5 x 104 2.5 x 105 2.8 x 105
Rodhe (1978) 9 x 104 3 x 105 3 x 105
Prahm et al. (1976) 5 x 104 2.5 x 105 3 x 105
Murgatroyd (1969) 2 x 10
Durst et al. (1959) 5 x 106
18
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As can be seen in Fig. 3, the contours of annual average sulfate concen-
tration show only gradual changes over large distances. Eliassen (1978) also
notes this consequence of his transport model. In both instances, since the
secondary pollutant sulfate is produced over a broad area, even when the
primary pollutant sulfur dioxide issues from a point source, it is to be
expected that no small regions of high local concentration will exist.
A more precise statement can be made by examining the nature of the
solution (7) for the secondary pollutant concentration for a single point
source of primary pollutant. For radial distances r which are small compared
with the length scales 2D,/w, a~^and y~^ tne maximum value (C ) of C is
found to be: s m s
' 27?
m 2-nh D, T (a - j
n c
- [3 x 10"11 s/m3] Q (12)
where the numerical value follows from the use of the parameter values given
in Table 2 for sulfate dispersion. What is also evident from an examination
of the form of (7) is that Cs does not decrease by a factor of two below this
maximum until r exceeds about 500 km. This is in marked contrast to the
substantial reduction in primary pollutant concentration within a comparable
distance from the source (see Fig. 2, for example). It is also interesting
to note from (12) that a single source (or a group of closely spaced sources)
of 33 kg/s of sulfur dioxide produces an average increment of 1 yg/m^ of sul-
fate within this large region.
Finally, we determined the annual average S02 concentration from equation
(5) using the optimum dispersion parameters of Table 1 and the power plant
source inventory. The resulting diffusion model concentration contours are
illustrated in Fig. 4 (smooth lines) along with approximate contours derived
from measurements of annual average S02 levels in urban areas for 1970-71
(dashed lines) as depicted by the National Academy of Sciences (1975). The
diffusion model contours are similar in shape to the measured values, but tend
to overpredict ambient concentrations in the northeastern U.S. A more careful
analysis and comparison of this case seems warranted. It is interesting to
note that Eliassen (1978) also found that his model calculations give a higher
ratio of sulfur dioxide to sulfate than was observed, as is evident from com-
paring our Figs. 3 and 4.
19
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Figure 4. A comparison of annual average SC>2 concentrations (ug/m )
determined from the analytical diffusion model (solid lines)
and measurements (dashed lines) for the same model parameters
as were used in obtained the sulfate concentrations of Fig. 3.
-------
SECTION 5
CONCLUSIONS
Our diffusion theory for the dispersion of primary and secondary pollutants
shows promise as an alternate and simple analytical method for evaluating the
time-averaged effects of long distance pollutant transport on regional air
quality. By assuming that large scale wind fluctuations act in a random manner
over a sufficiently long period of time and that depletion and transformation
processes are of first order, we find analytical solutions for the concentrations
of both primary and secondary pollutants. For multiple point sources, these
solutions can be superposed. Thus the computational effort involved in multiple
source calculations is only a fraction of that necessary for the analogous tra-
jectory calculation.
Limited calculations using the diffusion model for a single point source
have been shown to be in good overall agreement with a more detailed trajectory
calculation when the horizontal diffusivity is appropriately selected and identi-
cal values of other parameters are used. This resulting diffusivity was found
to be comparable with values derived from other trajectory studies.
Comparisons of our model calculation of the annual average concentrations
of sulfate formed from U.S. fossil fuel power plant emissions with ambient
measurements yield a good correlation when model parameters are optimally chosen
and the measured data are smoothed to reduce variations on a small geographic
scale. These comparisons appear to be as good as those obtained from trajectory
models describing sulfur transport in western Europe. The optimum values of the
diffusion model parameters we have determined are consonent with the rate con-
stants for sulfur oxide atmospheric behavior and diffusivities as reported in
the literature.
21
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REFERENCES
Cleveland, W.S., Kleiner, B., McRae, J.E. and Warner, J.L. (1976) Photochemical
air pollution: transport from the New York City area into Connecticut and
Massachusetts. Science, 101, 197.
Draxler, R.R., (1976) A diffusion deposition scheme for use within the ARL
trajectory model. NOAA TM ERL ARL 63, U.S. Department of Commerce, National
Oceanic and Atmospheric Administration, Washington, B.C.
Durst, C.S., Crossley, A.F. and Davis, N.E., (1959) Horizontal diffusion in
the atmosphere as determined by geostrophic trajectories. J. Fluid Mech., 6^,
401.
Eliassen, A., (1978) The OECD study of long range transport of air pollutants:
long range transport modelling. Atmospheric Environment, 12, 479.
Eliassen, A., and Saltbones, J., (1975) Decay and transformation rates of S02
as estimated from emission data, trajectories and measured air concentrations.
Atmospheric Environment, 9^, 425.
Federal Power Commission, (1973) Steam electric plant air and water quality
control data for the year ended December 31, 1970, Washington, D.C.
Garland, J.A., (1978) Dry and wet removal of sulfur from the atmosphere.
Atmospheric Environment, 12, 349.
Gillani, N.V. and Husar, R.B., (1976) Mesoscale model for pollutant transport,
transformation and ground removal. Presented at the third symposium on
atmospheric turbulence, diffusion and air quality, October 19-22, 1976, Raleigh,
North Carolina.
Glotfelty, D.E. (1978) The atmosphere as a sink for applied pesticides. J. Air
Pollution Control Assoc., 23>, 917.
Heffter, J.L., Taylor, A.D. and Ferber, G.J., (1975) A regional continental
scale transport diffusion and deposition model. NOAA TM ERL ARL 50, U.S.
Department of Commerce, National Oceanic and Atmospheric Administration,
Washington, D.C.
Hidy, G.M., Mueller, P.K. and Tong, E.Y. (1978) Spatial and temporal distribu-
tions of airborne sulfate in parts of the United States. Atmospheric Environ-
ment, 12, 735.
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Hidy, G.M., Tong, E.Y. and Mueller, P.K., (1976) Design of the sulfate regional
experiment (SURE). EPRI EC-125, Electric Power Research Institute, Palo Alto,
California.
Holzworth, C.G., (1972) Mixing heights, wind speeds and potential for urban
air pollution throughout the contiguous United States. AP-101, U.S. Environ-
mental Protection Agency, Research Triangle, N.C.
Johnson, W.B., Wolf, D.E. and Mancuso, R.L. (1978) Long term regional patterns
and transfrentier exchanges of airborne sulfur pollution in Europe. Atmospheric
Environment, 12, 511.
Meszaros, E., (1978) Concentration of sulfur compounds in remote continental
and oceanic areas. Atmospheric Environment, 12, 699.
Murgatroyd, R.J. (1969) Estimations from geostrophic trajectories of horizontal
diffusivity in the mid-latitude troposphere and lower stratosphere. Q. Jl. R.
Met. Soc., 95, 40.
National Academy of Sciences, (1975) Air quality and stationary source emission
control. Serial No. 94-4, Committee on Public Works, U.S. Senate, U.S. Govern-
ment Printing Office, Washington, B.C.
National Research Council, (1978) Nitrates: an environmental assessment. Nation
National Academy of Sciences, Washington, D.C.
Ottar B., (1978) An assessment of the OECD study on long range transport of air
pollutants (LRTAP). Atmospheric Environment, 12, 455.
Pasquill, P., (1974) Limitations and prospects in the estimation of dispersion
of pollution on a regional scale. Advances in Geophysics, 18B, 1.
Prahm, L.P., Torp, U. and Stern, R.M., (1976) Deposition and transformation
rates of sulfur oxides during atmospheric transport over the Atlantic. Tellus,
28, 4.
Rodhe, H., (1978) Budgets and turn-over times of atmospheric sulfur compounds.
Atmospheric Environment, 12. 671.
Sheih, C.M., (1977) Application of a statistical trajectory model to the simula-
tion of sulfur pollution over northeastern United States. Atmospheric Environ-
ment. 11, 1973.
Shirvaikar, V.V., (1972) Persistence of wind direction. Atmospheric Environment.
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Start, G.E. and Wendell, L.L., (1974) Regional effluent dispersion calculations
considering spatial and temporal meteorological variations, NOAA TM ERL ARL 44,
U.S. Department of Commerce, National Oceanic and Atmospheric Administration,
Washington, D.C.
Symposium on sulfur in the atmosphere (1978). Atmospheric Environment, 1-3, 1.
Taylor, G.I., (1921) Diffusion by continuous movements. Proc. London Math. Soc..
20, 196. ~
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U.S. Environmental Protection Agency, (1972) Federal air quality control regions,
AP-102. Rockville, Md.
U.S. Environmental Protection Agency, (1974) 1972 National emissions report.
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Wendell, I.L., Powell, D.C. and McNaughton, D.J., (1977) A multi-source compari-
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Wolff, G.T., Lioy, P.J., Wight, G.D., Meyers, R.E. and Cederwall, R.T., (1977)
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24
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-79-037
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
ANALYTICAL DIFFUSION MODEL FOR LONG
DISTANCE TRANSPORT OF AIR POLLUTANTS
5. REPORT DATE
June 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
James A. Fay and Jacob J. Rosenzweig
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Fluid Mechanics Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
10. PROGRAM ELEMENT NO.
1AA603 AB-015 (FY-79)
11. CONTRACT/GRANT NO.
Grant No. 804891
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory—RTP.NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
is. ABSTRACT
steady-state, two-dimensional diffusion model suitable for predicting ambi
ent air pollutant concentrations averaged over a long time period (e.g., month, season
or year) and resulting from the transport of pollutants for distances greater than
about 100 km from the source is described. Analytical solutions are derived for the
primary pollutant emitted from a point source and for the secondary pollutant formed
from it. Depletion effects, whether due to wet or dry deposition or chemical conver-
sion to another species, are accounted for in these models as first order processes.
Thus, solutions for multiple point sources may be superimposed.
The analytical theory for the dispersion of a primary pollutant is compared with the
numerical predictions of a plume trajectory model for the case of steady emission from
a point source. Good overall agreement between the two models is achieved whether or
not depletion by wet and dry deposition is included.
The theory for the dispersion of a secondary pollutant is compared with measurements
of the annual average sulfate concentration in the U.S. Calculations are carried out
using S02 emissions from electric power plants in the U.S. as a source inventory.
Using optimum values of the dispersion parameters, the correlation coefficient of
observed and calculated ambient concentrations for the U.S. is 0.46. However, when
the observed data is smoothed to eliminate small scale gradients, the best correlation
coefficient achieved is 0.87 for the eastern U.S. and 0.69 for the western region.
optimum dlsperison parameters used are rnmparahlP to valnpg gnntpH in fhP
17.
KEY WORDS AND DOCUMENT ANALYSIS
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b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
* Air pollution
* Mathematical models
* Atmospheric diffusion
Deposition
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