Conservation equations plumes

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

The third level of complexity in airshed modeling involves the solution of the partial differential equations of conservation of mass. While the computational requirements for this class of models are much greater than for the box model or the plume and puff models, this approach permits the inclusion of chemical reactions, time-varying meteorological conditions, and complex source emissions patterns. However, since this model consists only of the conservation equations, variables associated with the momentum and energy equations—e.g., wind fields and the vertical temperature structure—must be treated as inputs to the model. The solution of this class of models will be examined here. 

Chracterization of plume rise in terms of the exhaust gas properties and the ambient atmospheric state is a complex problem. The most detailed approach involves solving the coupled mass, momentum, and energy conservation equations. This approach is generally not used in routine calculations because of its complexity. An alternate approach, introduced by Morton et al. (1956), is to consider the integrated form of the conservation equations across a section normal to the plume trajectory [e.g., see Fisher et al. (1979) and Schatzmann (1979)]. 

In CFAST, a set of equations that predict state variables (pressure, temperamre, etc.) are solved based on the enthalpy and mass flux over small increments of time. These equations are derived from the conservation equations for mass, momentum, energy and the ideal gas law together with plume models, vent flow equations, radiation and combustion models. Forney and Moss reviewed that there are 11 variables to be solved the mass, internal energy, density, temperature and volume for the upper and lower layers (Mu, Eu, qu, Tu, Vu and ML, EL, qL, TL, VL), and the pressure R Because there are seven constraints, any four of those variables have to be chosen as solution variables. The four variables solved are the pressure... 

The Gaussian Plume Model is the most well-known and simplest scheme to estimate atmospheric dispersion. This is a mathematical model which has been formulated on the assumption that horizontal advection is balanced by vertical and transverse turbulent diffusion and terms arising from creation of depletion of species i by various internal sources or sinks. In the wind-oriented coordinate system, the conservation of species mass equation takes the following form ... 

The height of the rising plume can be determined from a conservation of mass for a control volume, enclosing the plume as it rises. From Equation (3.15), it follows that... 

The box model is closely related to the more complex airshed models described below in that it is based on the conservation of mass equation and includes chemical submodels that represent the chemistry more accurately than many plume models, for example. However, it is less complex and hence requires less computation time. It has the additional advantage that it does not require the detailed emissions, meteorological, and air quality data needed for input and validation of the airshed models. However, the resulting predictions are... 

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