Air dispersion modeling has been evolving since before the 1930s. Over the last 15-25 years, strict environmental regulations and the availability of personal computers have fueled an immense growth in the use of mathematical models to predict the dispersion of air pollution plumes. Beychok's recently published book, "Fundamentals Of Stack Gas Dispersion", details the evolution of the widely used Gaussian air dispersion models and their inherent assumptions and constraints.
Unfortunately, many users of such models are completely unaware of those assumptions and constraints and mistakenly believe that the precision achievable with computers equates to accuracy. This article discusses how the propagation of seemingly small errors in the Gaussian model parameters can cause very large variations in the model's predictions.
In most dispersion models, determining the pollutant concentrations at ground-level receptors beneath an elevated, buoyant plume of dispersing pollutant-containing gas involves two major steps:
First, the height to which the plume rises at a given downwind distance from the plume source is calculated. The calculated plume rise is added to the height of the plume's source point to obtain the so-called "effective stack height", also known as the plume centerline height or simply the emission height.
Then, the ground-level pollutant concentration beneath the plume at the given downwind distance is predicted using the Gaussian dispersion equation. (1)
Assumptions and Constraints
A host of assumptions and constraints are required to derive the Gaussian dispersion equation for modeling a continuous, buoyant plume from a single point-source in flat terrain ... which is still a long way from the more sophisticated models now in use for multiple sources in complex terrain. The most important assumptions and constraints are related to:
- The accuracy of predicting the plume rise since that affects the emission height
used in the Gaussian dispersion equation.
- The accuracy of the dispersion coefficients (i.e., the vertical and horizontal
standard deviations of the emission distribution) used in the Gaussian dispersion
equation.
- The assumption of the averaging time period represented by the calculated
ground-level pollutant concentrations as determined by the dispersion coefficients used in
the Gaussian equation. In other words, do the calculated ground-level concentrations
represent a 5-minute, 10-minute, 15-minute, 30-minute or 1-hour average concentration?
Besides the assumptions and constraints in deriving the Gaussian equation, the methods for obtaining certain parameters used in the Gaussian models are also subject to many assumptions and constraints. Those methods include: obtaining the atmospheric stability classifications (which characterize the degree of turbulence available to enhance dispersion), determining the profiles of windspeed versus emission height, and converting ground-level short-term concentrations from one averaging time to another. This discussion of shortcomings in the Gaussian dispersion models is not unique. The literature abounds with such discussions. (2,3,4,5,6) Unfortunately, despite those discussions, there is a widespread belief that dispersion models can predict dispersed plume concentrations within a factor of two or three of the actual concentrations in the real world. Indeed, there are some who believe the models are even more accurate than that.
Deriving the Gaussian dispersion equation requires the assumption of constant conditions for the entire plume travel distance from the emission source point to the downwind ground-level receptor. (1) Yet we cannot say with any reasonable certainty that the windspeed at the plume centerline height and the atmospheric stability class are known exactly or that they are constant for the entire plume travel distance. Whether such homogeneity actually occurs is a matter of pure chance, particularly for large distances. Also, determining the exact windspeed and atmospheric stability class at the plume centerline height requires (a) the prediction of the exact plume rise and (b) the exact relation between windspeed and altitude ... neither of which are achievable.
Plume Rise Uncertainty
Most Gaussian dispersion models use the Briggs plume rise equations (1, 7, 8, 9) to predict buoyant plume rise. There are few knowledgeable dispersion modelers who would dispute that the Briggs equations could over- or under-predict actual plume rises by 20 percent.
Dispersion Coefficients
Most Gaussian models use modifications of the dispersion coefficients derived experimentally by Pasquill (10) in a rural area of fairly level, open terrain and for relatively moderate plume travel distances. There are few knowledgeable dispersion modelers who would dispute that Pasquill's coefficients could be in error by plus or minus 25 percent, especially when used for non-level, complex terrain and for large distances ranging up to 50 kilometers or more. Pasquill himself has proposed a re-examination of his coefficients (11) and has suggested they be revised.
Concentration Averaging Periods
As mentioned above, there is the question of what averaging time period the calculated ground-level concentration (i.e., C) represents when using Pasquill's dispersion coefficients. Turner (12) states that C is a 3- to 15-minute average. An American Petroleum Institute publication (13) believes C is a 10- to 30-minute average. An American Institute of Chemical Engineers publication, written by Hanna and Drivas (14), states that C represents a 10-minute average. The Tennessee Valley Authority (15) attributes a 5-minute average to their C values.
Despite that body of opinion, many of the dispersion models ... whose use is mandated by most of our federal and state regulatory agencies ... assume the Gaussian dispersion equation yields 1-hour average concentrations. It can be shown (1) that assuming the C values represent a 1-hour average, rather than a 10-minute average, constitutes a "built-in" over-prediction factor of as much as 2.5.
Other Assumptions and Constraints
Deriving the Gaussian dispersion equation also assumes the following:
- Windspeed and wind direction are constant from the source point to the receptor
(for a windspeed of 2 m/s and a distance of 10 km, 80 minutes of constant conditions would
be needed).
- Atmospheric turbulence is also constant throughout the plume travel distance.
- All of the the plume is conserved, meaning: no deposition or washout of plume
components; components reaching the ground are reflected back into the plume; no
components are absorbed by bodies of water or by vegetation; and components are not
chemically transformed. [Some of the more complex dispersion models do adjust for
deposition and chemical transformation. However, such adjustments are separate from
the basic Gaussian dispersion equation.]
- Only vertical and crosswind dispersion occurs (i.e., no downwind dispersion).
- The dispersion pattern is probabilistic and can be described exactly by Gaussian
distribution.
- The plume expands in a conical fashion as it travels downward, whereas the ideal
"coning plume" is only one of many observed plume behaviors.
- Terrain conditions can be accommodated by using one set of dispersion coefficients for rural terrain and another set for urban terrain. The basic Gaussian dispersion equation is not intended to handle terrain regimes such as valleys, mountains or shorelines.
In short, the Gaussian models assume an ideal steady-state of constant meteorological conditions over long distances, idealized plume geometry, uniform flat terrain, complete conservation of mass, and exact Gaussian distribution. Such ideal conditions rarely occur.
Sensitivity Study
A sensitivity study was performed by assuming reasonable degrees of error in some of the key variables used in the Gaussian models and determining the propagated end-result effect of those errors on the calculated, ground-level pollutant concentrations. Several comparative models were defined as follows:
- Base Model A The base model uses Briggs' plume rise equations,
power-law conversion of surface windspeeds to obtain windspeeds at the source height (for
use in the plume rise equations) and at the plume centerline height (for use in the
Gaussian dispersion equations), and the calculated ground-level concentrations are taken
to be 1-hour averages as per the U.S. EPA.
- Adjusted Model B Same as model A except that the calculated
plume rises were increased by 20 percent and the Pasquill vertical dispersion coefficients
were decreased by 25 percent.
- Adjusted Model C Same as model B except that the calculated
ground-level concentrations reflect an assumed wind direction shift of 10 degrees.
- Adjusted Model D Same as model C except that an over-prediction factor (C10/C60) of 2.5 was included to account for the EPA's assumption that the calculated ground-level concentrations represent 1-hour averages rather than 10-minute averages.
Table 1 presents the results of the sensitivity study. Comparing the ground-level concentrations calculated by the base model A to the concentrations calculated by the adjusted model D, it is seen that the base model A over-predicts model D by a factor ranging from 6 (at downwind distances of 6 to 10 km) to a factor of 80 (at a downwind distance of 2 km). Thus, seemingly minor changes in some of the key variables can result in a propagated over-prediction factor ranging from 6 to 80.
This study was not intended to downgrade the value of Gaussian dispersion models. They are very useful tools. However, we should be aware that they are merely tools and do not provide the ultimate truth. This study has shown that it is unrealistic to expect the Gaussian models to predict real-world dispersing plume concentrations consistently by a factor of two or three. It is probably more realistic to expect consistent predictions of real-world dispersing plume concentrations within a factor that may be as high as ten.
Table 1: Calculated ground-level concentrations (micrograms/cubic
meter)
at downwind distances ranging from 2 km to 10 km
| Receptor Downwind Distance (km) | (A)
Base Model |
(B) Adjusted Model | (C) Adjusted Model plus Wind Shifts |
(D)
Adjusted Model Plus Wind Shift and C10/C60 Corrections |
Over-Prediction
Ratio (A)/(D) |
|---|---|---|---|---|---|
| 2 | 16.1 | 1.1 | 0.5 | 0.2 | 80 |
| 3 | 24.3 | 9.2 | 4.4 | 1.7 | 14 |
| 4 | 20.5 | 13.8 | 6.0 | 2.4 | 9 |
| 5 | 15.8 | 13.7 | 5.8 | 2.3 | 7 |
| 6 | 12.0 | 12.0 | 4.7 | 1.9 | 6 |
| 7 | 9.4 | 10.1 | 3.9 | 1.5 | 6 |
| 8 | 7.4 | 8.4 | 3.1 | 1.2 | 6 |
| 9 | 6.0 | 7.0 | 2.5 | 1.0 | 6 |
| 10 | 5.0 | 5.9 | 2.0 | 0.8 | 6 |
References
(1) Beychok, M.R., "Fundamentals of Stack Gas
Dispersion", published by author, Irvine, Calif., 1994
(2) "Atmospheric dispersion modeling, a critical
review", JAPCA, Sept. 1979
(3) Ellis, H.M. et al, "Comparison of predicted and measured
concentrations for 58 alternative models of plume transport in complex terrain",
JAPCA, June 1980
(4) American Petroleum Institute, "An evaluation of
short-term air quality models using tracer study data", API Report No. 4333, Oct.
1980
(5) Bowne, N.J. et al, "Overview, results and conclusions for
the EPRI plume model validation and development project: Plains Site", Electric Power
Research Institute, Final Report 1616-1 for Project EA-3704, 1983
(6) Benarie, M.M., "Editorial: The limits of air pollution
modeling", Atmospheric Environment, 21:1-5, 1987
(7) Briggs, G.A., "Plume rise", USAEC Critical Review
Series, 1969
(8) Briggs, G.A., "Some recent analyses of plume rise
observation", Proc. Second Internat'l. Clean Air Congress, Academic Press, New York,
1971
(9) Briggs, G.A., "Discussion: Chimney plumes in neutral and
stable surroundings", Atmospheric Environment, 6:507-510, 1972
(10) Pasquill, F., "The estimation of the dispersion of
windborne material", Meteorology Magazine, Feb. 1961
(11) Pasquill, F., "Atmospheric dispersion parameters in
Gaussian plume modeling. Part II: Possible requirements for change in Turner
Workbook values", U.S. EPA Publication 600/4-76-030b, June 1976
(12) Turner, D.B., "Workbook of atmospheric dispersion
estimates", U.S. EPA Publication AP-26, revised 1970
(13) American Petroleum Institute, "Gaussian dispersion
models applicable to refinery emissions", API Publication 52, Oct. 1977
(14) Hanna, S.R. and Drivas, P.J., "Guidelines For The Use
Of Vapor Cloud Dispersion Models", Center For Process Safety, American
Institute of Chemical Engineers, 1987.
(15) Montgomery, T.C. and Coleman, J.H., "Empirical
relationship between time-averaged SO2 concentrations", Environmental Science &
Technology, Oct. 1975
Contact Information
The author may be contacted regarding this article at:
1126 Colony Plaza, Newport Beach, CA 92660
Phone & Fax: 949-718-1360
E-mail (1): mbeychok@air-dispersion.com
E-mail (2): mbeychok@home.com
Web site: http://www.air-dispersion.com
