Clouds mathematical description

Because of the inadequacies of the aforementioned models, a number of papers in the 1950s and 1960s developed alternative mathematical descriptions of fluidized beds that explicitly divided the reactor contents into two phases, a bubble phase and an emulsion or dense phase. The bubble or lean phase is presumed to be essentially free of solids so that little, if any, reaction occurs in this portion of the bed. Reaction takes place within the dense phase, where virtually all of the solid catalyst particles are found. This phase may also be referred to as a particulate phase, an interstitial phase, or an emulsion phase by various authors. Figure 12.19 is a schematic representation of two phase models of fluidized beds. Some models also define a cloud phase as the region of space surrounding the bubble that acts as a source and a sink for gas exchange with the bubble. 

In-cloud chemical processes transform soluble trace gases into various ionic products. In the case of acid precursors, such as SO2 and NO2, definitions of the significant chemical reactions in aqueous cloud droplets are necessary for the mathematical description of acid deposition. These significant reactions can be inferred from measurements in the real atmosphere (1,2), and they can be identified in controlled laboratory experiments (3,4). Since measurements in the real atmosphere may be characterized by large uncertainties (1), laboratory simulation of aqueous phase chemical systems supplement... 

Humans have always dealt with and been fascinated by the properties of our atmosphere. In ancient times, the motivation to observe the atmosphere was clearly the driving force which increased the understanding of nature. Atmospheric (weather) observations were closely associated with astronomy, and everything above the earth s surface was named heaven or ether . The weather phenomena - fog, mist and clouds, precipitation (rain, snow, and hail) and dew - have been described since Antiquity. A phenomenological understanding of the physical (but not the chemical) processes associated with hydrometeors was complete only by the end of the nineteenth century. Today the physics and chemistry in the aerosol-cloud-precipitation chain are relatively well understood - also with relation to climate. However, it seems that because of the huge complexity a mathematical description of the processes (i. e., the parameterization of the chemistry and also for climate modeling) is still under construction. 

In the latter way of looking at the build-up of molecules, the successive addition of electrons to a positively charged system is reminiscent of the manner in which the atoms of the Periodic Table were considered in Chapter 1. Here again there are certain configurations permitted the electron clouds, and these cloud shapes (or probability density functions) can be described using quantum numbers. Such probability density descriptions are called molecular orbitals in analogy to the much simpler atomic orbitals. Although the initial setup and subsequent mathematical treatment for molecules are much more complicated than for atoms, there arise certain similarities between the two types of orbitals. 

In an electrolyte solution the ions are interspersed by water molecules, moreover they are subject to thermal motion. Debye and Huckel simplified the description of this problem to a mathematically manageable one by considering one isolated ion in a hypothetical, uniformly smeared-out sea of charge, the ionic cloud, with the total charge just opposing that of the ion considered. For this case the Poisson equation in terms of spherical coordinates is given by... 

The description of atoms and molecules which has been adopted—that of one or more positive nuclei surrounded by a cloud of electrons which, for many purposes, is equivalent to a smeared-out negative charge—presents in pictorial form the results of thirty years of quantum mechanics. All available evidence suggests that this general picture is unlikely to suffer substantial modification. Theoretical chemistry accepts the Schrodinger equation and is largely concerned with finding the most direct mathematical path to a unified and transparent explanation of the physico-chemical prop-... 

FIGURE 17.12 Schematic description of the cloud formation mathematical framework. 

Understanding absorption is the key to all these operations. This understanding is usually clouded by presenting the ideas largely in mathematical terms. All chemistry and all simple limits are implied rather than explained. As a result, novices often understand every step of the analysis but have a poor perspective of the overall problem. To avoid this dislocation, we begin in Section 10.1 with a description of the gases to be absorbed and the liquid solvents that absorb them. A few of these liquids depend only on the solubility of the gas many more liquids react chemically with the components of the gas. 

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