12.2. Theory
As stated above, the goal of all plume models is an accurate simulation of the distribution of a gas which has been released from a source under a given set of environmental conditions. Due to effects such as wind, atmospheric turbulence, and the general nonlinear nature of viscous fluid flow, this is a complicated problem. As such, several approximate methods have been developed to predict the distribution and temperature of the gas as a function of both space and time. Some of these methods will be briefly described here. The method chosen for implementation into DIRSIG will be discussed in more detail at the end of this section.
Several methods exist to predict the density and temperature spatial distribution resulting from the release of a gas. Some methods address only point sources while others consider extended sources. The most accurate, but computationally "expensive", are Computational Fluid Dynamics (CFD) models. These typically treat the full hydrodynamic and thermodynamic properties of the fluid through the conservation of mass, momentum, and energy. While these methods provide accurate simulations of the plume shape and temperature, other methods which are less computationally intensive can approximate real plumes that are sufficiently accurate for many applications. Among these are various analytical approximations to the gas flow and cooling problem considering the separate flow regions. Approximations exist describing the jet region of the plume, where the plume momentum due to its release velocity is dominant, as well as the buoyancy region, where the buoyancy forces on the plume determine the rise/fall of the plume centerline. In addition, empirical approximations for the dispersion and cooling of the plume can be made and matched to time-averaged observations of real gas plumes.
Several of these methods have been integrated into DIRSIG successfully [Kuo 1997] [Bishop 2001] [Kuo 2000]. DIRSIG requires a plume model to produce a three-dimensional representation of the gas density and temperature, through which it will cast rays and compute the radiative transfer along the rays. Currently, the plume model supported in DIRSIG is an approximate model based on a Gaussian approximation for the cross-sectional shape of the gas density and temperature. This model is described in further detail below.
12.2.1. Gaussian Model
DIRSIG currently supports an analytic, Gaussian plume first developed by the EPA and enhanced by JPL. It will be referred to here as the JPL model. In general, near the stack exit, where the flow is momentum-dominated, the particle density distribution is similar in form to the particle velocity distribution, which is Gaussian. Far down-wind, small-scale turbulence in the plume causes uncorrelated particle trajectories leading to a Gaussian distribution of particle density as well. The transition regions between these two extremes are less well understood and, for simplicity, treated in the JPL model as Gaussian.
In general, Gaussian models assume some form for a plume centerline height and downwind dilution. This is combined with a Gaussian density distribution in the vertical and horizontal directions perpendicular to the downwind direction. The dispersion coefficients for the cross-wind directions are related to the atmospheric stability and are typically based on empirical studies. The temperature distribution is closely related to the density distribution. The JPL formulation used in DIRSIG includes effects such as a temperature-dependent thermal diffusivity to more closely model the effects of ambient air entrainment by the plume [Kuo 1997].
The primary limitation of using a Gaussian model is the smoothness of the spatial distribution of points. The analytical forms for the density and temperature distributions result in conservative estimates for those quantities in the sense that there are no high sigma outliers in the distributions. In other words, there will be no voids, dense clumps, or hot spots in the plume, all of which result from turbulence in the plume and are obviously observed in real gas releases. However, the density and temperatures computed by the JPL model are realistic. In this sense, the JPL model produces a time-averaged picture of the plume over a period of several minutes or tens of minutes. Caution should be used when treating sensors with integration times long enough to be influenced by motion within the gas. Phenomenology seen in real observations under these conditions are not produced in the (static) JPL model.
12.2.1.1. Brigg's model rise
The Brigg's model is an empirical estimate to the height of the centerline of a plume [Kuo 1997] [Bishop 2001] [Kuo 2000]. The model approximates the competing effects of plume momentum, dominant in the near-stack jet region, and buoyancy forces of the gas in the downwind region [Halitsky 1989]. In addition, effects of ambient air entrainment on the rate of buoyant rise are included. The air entrainment rate is assumed to be proportional to the plume velocity and cross-sectional area.
In the Brigg's model, the height of the centerline, h is related to the stack height, h_0 and the downwind distance x through the following equation [Kuo 1997]:
Here, r_0 is the stack radius and m is the ratio of the gas release velocity to the wind velocity.
12.2.1.2. Density and Temperature Distributions
In the Gaussian model, the gas temperature and density distributions are related to each other and follow similar spatial distributions. Initially, for a given downwind position the distribution of gas away from the centerline is computed as:
where Q is the source term (in units of [g/s]),
[y,z] are the horizontal and vertical directions
orthogonal to the downwind direction,
are the dispersion coefficients in those directions, and
is the mean wind speed. Note that this distribution includes
a second vertical term that is an "image" of the plume
at negative heights. This allows the model to correctly
account for particles that reflect off the ground surface.
In this model, all particles experience perfect reflection
off the surface.
Once the concentration distribution is computed, a dilution factor can be computed as well. This dilution factor can be used to then compute the temperature distribution of the gas. The dilution factor as a function of position in the scene is simply computed as
12.2.1.3. Halitsky Formulation
The Halitsky modification to the original Brigg's model as applied here, was to incorporate density variation as a function of temperature in the downwind distribution [Halitsky 1989]. Consequently, the temperature distribution is computed from the dilution factor in the following way:
where T_s is the gas stack exit temperature, T_a is the ambient air temperature, and T is the temperature at a point in the plume. At a given point in the plume, the temperature of the gas is computed as
12.2.2. Radiative transfer through the plume
The exact solution to the radiative transfer problem through a cloud of gas requires the full treatment of absorption, transmission, and scattering of electromagnetic radiation by the gas particles. This is complicated more by the effects of multiple scattering within the cloud. DIRSIG currently does not perform scattering calculations within the plume. Radiation from sources external to the plume can be absorbed, and the gas itself can emit, but the scattering properties of the plume are not included in the calculation. Consequently, scattering off water-droplets in cooling-tower plumes can not be addressed within the current DIRSIG model. Single scattering will be included in future versions of the simulation and numerical procedures for computing the multiple scattering effects are under investigation.
DIRSIG is essentially a ray-tracing radiative transfer simulation. When a ray, cast from the sensor, encounters a region containing gas plume particles, the simulation determines how many integration steps to process through the plume. The radiative transfer computation is then performed as a stepwise integration along the ray path, through the plume over the stepsize just determined. When the integration point does not fall on a grid point that was pre-computed by the (external) plume model, a bi-linear interpolation is performed to compute the gas density and temperature at the integration point.