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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.1
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
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:
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.
Most Gaussian dispersion models use the Briggs plume rise equations1, 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.
Most Gaussian models use modifications of the dispersion coefficients derived experimentally by Pasquill10 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 coefficients11 and has suggested they be revised.
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. Turner12 states that C is a 3- to 15-minute average. An American Petroleum Institute publication13 believes C is a 10- to 30-minute average. An American Institute of Chemical Engineers publication, written by Hanna and Drivas14, states that C represents a 10-minute average. The Tennessee Valley Authority15 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 shown1 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.
Deriving the Gaussian dispersion equation also assumes the following:
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 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.
Figure 1 depicts the profiles, as predicted by Models A, B, C and D, of the pollutant ground-level concentrations versus downwind distance.
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
much more realistic to expect consistent predictions of real-world dispersing plume concentrations within a
factor that may be as high as ten. In fact, a recently released report by the Amarillo National Research Center16 states that field research by Texas A&M University has shown that the U.S. EPA's widely used ISCST model overpredicts downwind concentrations by a factor of 10 or more.
Although this article was first written in about 1988, the comments excerpted from the 1991 to 2002 references17, 18, 19, 20, 21, 22 presented in the Appendix to this article make it abundantly clear that very little has yet been done to quantify the uncertainty involved in air dispersion modeling.
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