Some Computational Problemls In the Simulation of Stack Gas Dispersion
Authors: Wallis J. , Thusoo A. , Donnelly J. , Aziz K. 
Published: March 2010

Abstract
This paper examines some aspects of the numerical simulation of plume dispersion. Several difficulties associated with numerical solutions are considered, and practical methods for the handling of these problems are provided. Some analytical solutions which are extremely useful for the testing of numerical methods are also presented. It is shown that, with adequate care, it is possible to simulate numerically the dispersion of pollutants from isolated stacks.

Introduction
THERE HAS BEEN a growing concern amongst the public over the quality of air it breathes. This has resulted in a number of air pollution studies to determine where an effluent goes. In particular, there have been a number of numerical studies of pollutant dispersion from stacks: Ragland (1973), Carr (1973), Egan and Mahoney (1972), Lantz at or. (1972), Randerson 11970), and Shir (1970).
Previous investigators have concerned with problems varying complexity, but with the exception of Carr (1973) they do not discuss computational problems. Consequently, one is led to believe that there are no serious problems associated with the solution air pollution dispersion problems; all one needs to do is to set up the governing equations, find appropriate coefficients, define boundary conditions and integrate the problem to find the solution. Although no single part of the problem is actually easy, it is the integration that causes the greatest difficulty. Some of the studies referenced above report the maximum allowed integration interval for explicit numerical techniques (e.g. Randerson, 1970), but fail to discuss the discretization error caused by their differencing schemes and the allied problem of grid spacing and its effect on the error. Also they do not discuss the method used for the introduction of a point source into the numerical calculations.
In this paper, a practical look is taken at these problems. In addition, the use of characteristic curves for the solution of transient pollutant dispersion problems is considered. A grid spacing guide related to stack height has been developed as an aid in the numerical solution of diffusion problems. Some of the results should also be useful in allied engineering applications.
(equations in full paper)

Numerical Techniques
Analytical solutions are difficult or impossible to obtain when irregular terrain and diffusivity variation with position must be accounted for. Numerical techniques are ideally suited to the handling of these problems. Before using these methods, it is important to have some understanding of their limitations. Much insight can be gained by comparing the results predicted by numerical procedures with available analytical solutions. In the following discussion, some of the problems associated with numerical computation are considered.
All finite differences used, excepting the integration procedure, are second-order central-difference approximations to the governing equation.
Wallis and Aziz (1975) outline some advantages of explicit methods over implicit methods for the numerical solution of steady-state plume dispersion problems. For this reason, only explicit methods are considered in this paper, Because Equation 5 contains first-order derivatives with respect to time and space, the solution may be obtained by marching either in time or in the downwind direction.