Annex 5.1 Urban dispersion models
This annex documents simple models for first estimation or screening.
An urban area contains thousands, or even millions, of individual sources. The application of a diffusion model to each source is impractical. Consequently most of the small sources are combined into larger area sources of strength Q_{a }(mass per unit time per unit area), and it is assumed that the emissions from the ground surface are uniform over that particular area.
Diffusion from the largest point sources can
be calculated individually and the resulting concentrations at a receptor point can be
added to the contribution from the area sources.
A. AREA SOURCE MODEL
In order to estimate by hand calculations the 1hr average air concentration at an arbitrary receptor point (o) due to area source emissions, a modified expression of the ATDL urban diffusion model (Hanna, 1972; Gifford and Hanna, 1973) may be used:
(A.1)
where
C, the concentration (micrograms m^{3})
o, denotes the location of the receptor point
n, is the number of grid blocks (of size), necessary to reach the upwind edge of the urban area, starting from the receptor point.
Q, for i=o,1,2,...,n, are source strengths (microgram sec^{1} m^{2}), constant over a distance .
u, is the wind speed, assumed constant within the mixing layer.
c,d are the Brookhaven National Laboratory parameter values, (Smith, 1968), as listed in Table 1.
Table A.1. Brookhaven National Laboratory parameter values a, b, c and d in equation (A.1) and in the formulas for the dispersion parameters , and
atmospheric conditions 
insolation 
wind speed 
a 
b 
c 
d 

very unstable 
strongmoderate 
2 
0.40 
0.91 
0.40 
0.91 
unstable 
strongmoderate 
23 
0.36 
0.86 
0.33 
0.86 
neutral 
moderateslight 
34 
0.32 
0.80 
0.22 
0.80 
estimated Pasquill D 
moderateslight or night 
4 
0.32 
0.75 
0.15 
0.75 
stable 
night 
24 
0.31 
0.71 
0.06 
0.71 
Note that these values represent one choice; alternative datasets exist. Note also the dependence of the dispersion parameters on averaging time (see end of this annex).
Basic assumptions
 The pollutants are assumed to be uniformly mixed in a layer, whose height is proportional to the vertical dispersion parameter , where x is the total distance from the urban area.
Simpler approach
When the distribution of emissions is quite smooth, as it is often the case in residential urban areas, the calculated concentration (C) at any receptor point is usually proportional to the emissions Q_{ao }in the grid square in which the receptor is located. In this case it is sufficient to use the following simpler relation:
(A.2)
the expression denotes the distance to the edge of city.
The following values for the dimensionless factor A are suggested:
atmospheric conditions 
A 

neutral or average  200 
stable  600 
unstable  50 
Note, however, that A is slightly dependent on .
Additional contribution from other sources
The urban area source model can give the average concentration over a broad area. In a street canyon or adjacent to a highway in an urban area, there is an additional contribution to the concentration from local sources. In this case the total concentration C_{t} is the sum of the spatial average C (calculated from equation (A.1)) and the local C_{l} component. Finally, the concentrations resulting at a receptor point from large point sources, C_{p}, can also be added to the spatial average concentration C.
B. ELEVATED POINT SOURCES
In order to estimate the contribution from an elevated point source C_{p}, (Fig. 1) of strength Q, the following Gaussian relation can be used:
(A.3)
where
C is the concentration (micrograms m^{3});
Q is the source strength (micrograms sec^{1});
u is the wind speed at the plume height;
y refers to the horizontal direction at right angles to the plume axis with y equal to zero on the axis;
z is the height above the ground;
are standard deviations of the concentration distribution C, in the y and z direction and are calculated from table A.1;
h is the effective plume
height (stack height plus plume rise). The plume rise can be calculated by using the
appropriate formulas, summarised in table A.2
.
Figure A.1 Diagram of plume, illustrating concepts important in the Gaussian plume formula.
Table A.2. Plume rise formulas according to the plume characteristics and atmospheric conditions
plume type  1 
atmospheric conditions 
formulas 
bentover 
buoyant jet 
stable
strong wind neutral,
unstable strong wind neutral


vertical  jet
buoyant 
low wind stable 

where
is the buoyancy flux , ;
M is the momentum flux, M=wV;
s is the atmospheric stability, ;
V is the plume volume flux (V=wR^{2} for vertical plume and V=uR^{2} for bent over plume);
w is the plume vertical speed;
x is distance from the stack;
D is the stack diameter;
T is the temperature.
Subscripts p and e denote plume and environment.
Note that alternative formulations for the plume rise exist.
Limitations
Although the Gaussian plume formula in general
is appropriate to calculate the dispersion of elevated continuous major point sources, it
has been demonstrated that it can lead to misleading results in special cases, such as in
inhomogeneous terrain. For other simple models which could be used, see for instance
Kretzschmar et al.,1994; Kretzschmar and Cosemans, 1996.
Longer averaging times
The diffusion parameters are directly related to the standard deviations of the turbulent velocity fluctuations. Thus, as averaging time increases, the turbulent velocity fluctuations increase and hence increase. Gifford suggests accounting for the effects of sampling time trough the empirical formula:
(A.4)
where,
d and e represent two different averaging times, and q is in the range 0.25 to 0.3 for 1hr_{sd}<100hr and equals approximately 0.2 for 3min_{sd}<1hr.
The standard dispersion parameters given in table A.1, represent a sampling time T_{se} of about 10 min.
Extension to longer averaging times is made by
solving the above equations for a variety of wind directions and then weighting each
result by the frequency with which the wind blows from that direction.
C. STREET CANYON SUBMODEL
Consider the street canyon in figure A.2, where the important variables are defined. Depending on the wind direction, at roof level, the following relations can be used.
Figure A.2 Schematic of crossstreet air circulation in a street canyon. [From Johnson et al, 1977)
Wind direction normal to the street axis
If the wind direction is nearly normal to the street, the equations for the concentration C_{l} in the street canyon are:
 lee side, (A.5)
 windward side, (A.6)
where,
C_{l}  is the concentration (µg/m^{3}); 
N  is the traffic flow (vehicles/hr); 
q  is the emission factor (g/km); 
u  is the wind speed at roof level (m/sec); 
W  is the street width(m); 
x and z  are horizontal distance and height (both in m) of the receptor point relative to the traffic lane; 
K  is a dimensionless "best fit" constant (K7 is suggested). 
Wind direction parallel to the street axis
If the wind direction is nearly parallel to the street axis, the equations for the concentration C_{l} in the street canyon are:
(A.7)
Limitation: The model as such is not suitable for calculation of NO_{2} concentrations, which are mainly determined by chemical reaction of NO with ozone.
D. HIGHWAY SUBMODEL
The excess concentration C_{l }contributed by a major highway in an urban area is important for a distance less than 300m downwind of the highway. Consider the highway in figure A.3, the concentration at some distance x from the highway can be estimated from the relation:
(A.8)
where
C is the concentration (µg/m^{3});
Q is the line source strength (µg/s/m);
h is the effective height of emissions due to initial dispersion (23m);
is the angle between the wind direction and the highway;
is the vertical dispersion parameter.
F() Function of ; for around 90 degrees, F() is close to 1.
Figure A.3 Infinite line source pattern.
Limitation: This formula cannot be used to calculate concentrations on the highway, or in case the wind blows in the direction of the highway .
Meteorological data and emissions
All the models in this annex are best suitable for estimation of long term average concentrations. They should not be used for short term high percentile values, which are highly dependent on critical meteorological conditions.
For long term average concentrations, calculations can be made with average meteorological conditions. For wind speed, the annual average can be used. For wind direction, the most frequent direction can be taken. For point sources, neutral atmospheric conditions should be selected.
Appropriate choices must also be made for
emission estimates, to make sure that they reflect typical conditions.
References
Gifford F.A. and Hanna S.R., (1973) Modelling urban air pollution. Atmospheric Environment, 7, 131136.
Hanna S.R., (1972) Description of ATDL computer model for dispersion from multiple sources. Proc. of second annual industrial air pollution control conference. Knoxville, Tn. ATDL Report 56, NOAA, Oak Ridge.
W. B. Johnson, R. C. Sklarew, and D. B. Turner, Urban Air Quality Simulation Modeling in Air Pollution, vol. 1, 3rd ed. Chapter 10, p. 530, A. G. Stern (Ed.), Academic Press, New York, 1977.
Kretzschmar, J.G., Cosemans G.(1996) 4th workshop on harmonization within atmospheric dispersion modelling for regulatory purposes, vol. 1 and 2. E&M.RA9603, VITO, Mol, Belgium.
Kretzschmar, J.G., Maes, G. Cosemans G.(1994) Operational short range atmospheric dispersion models for environmental impact assessment in Europe, vol 1 and 2. E&M.RA9416, VITO, Mol, Belgium.
Smith, M. E. (ed.) (1968) Recommended Guide for the Prediction of the Dispersion of Airborne Effluents, Am. Soc. of Mech. Engineers, New York
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