Mathematical Treatment of the Process

The following assumptions are made in order to describe the process precisely  

Following assumptions (iii) and (iv), the transport of contaminant through the two layers is governed hy the unidirectional equation of diffusion with a constant of diffusivity  

Following assumptions (ii) and (v), the initial and boundary conditions are expressed by  

The analytical solution of the problem is obtained by using the method of separation of variables (shown in Chapter 1.3). 

The concentration of pollutant at position x is thus expressed as a function of time  

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Bi-layer Package with Various Relative Thicknesses 2.3.1 Mathematical Treatment of the Process [1, 8]... 

In an effort to optimize the solvent-containing passive sampler design, Zabik (1988) and Huckins (1988) evaluated the organic contaminant permeability and solvent compatibility of several candidate nonporous polymeric membranes (Huckins et al., 2002a). The membranes included LDPE, polypropylene (PP), polyvinyl chloride, polyacetate, and silicone, specifically medical grade silicone (silastic). Solvents used were hexane, ethyl acetate, dichloromethane, isooctane, etc. With the exception of silastic, membranes were <120- um thick. Because silicone has the greatest free volume of all the nonporous polymers, thicker membranes were used. Although there are a number of definitions of polymer free volume based on various mathematical treatments of the diffusion process, free volume can be viewed as the free space within the polymer matrix available for solute diffusion. 

Whether the prediction scheme is a simple chart, a formula, or a complex numerical procedure, there are three basic elements that must be considered meteorology, source emissions, and atmospheric chemical interactions. Despite the diversity of methodologies available for relating emissions to ambient air quality, there are two basic types of models. Those based on a fundamental description of the physics and chemistry occurring in the atmosphere are classified as a priori approaches. Such methods normally incorporate a mathematical treatment of the meteorological and chemical processes and, in addition, utilize information about the distribution of source emissions. Another class of methods involves the use of a posteriori models in which empirical relationships are deduced from laboratory or atmospheric measurements. These models are usually quite simple and typically bear a close relationship to the actual data upon which they are based. The latter feature is a basic weakness. Because the models do not explicitly quantify the causal phenomena, they cannot be reliably extrapolated beyond the bounds of the data from which they were derived. As a result, a posteriori models are not ideally suited to the task of predicting the impacts of substantial changes in emissions. 

More detailed mathematical treatments of the diffusion process can be found in Crank (1975). 

Solute equilibrium between the mobile and stationary phases is never achieved in the chromatographic column except possibly (as Giddings points out) at the maximum of a peak (1). As stated before, to circumvent this non equilibrium condition and allow a simple mathematical treatment of the chromatographic process, Martin and Synge (2) borrowed the plate concept from distillation theory and considered the column consisted of a series of theoretical plates in which equilibrium could be assumed to occur. In fact each plate represented a dwell time for the solute to achieve equilibrium at that point in the column and the process of distribution could be considered as incremental. It has been shown that employing this concept an equation for the elution curve can be easily obtained and, from that basic equation, others can be developed that describe the various properties of a chromatogram. Such equations will permit the calculation of efficiency, the calculation of the number of theoretical plates required to achieve a specific separation and among many applications, elucidate the function of the heat of absorption detector. 

One of the relevant examples in chemistry which can be treated in terms of the application of the rules of mathematics is the Law of Mass Action. This simple and beautiful mathematical treatment of equilibrium processes is of great scientific and economic importance (e.g. for the calculation of yields ). The synthesis of esters is a typical reversible process and occurs as shown in eqn. I ... 

It is necessary to note that (44) is an approximation, because the value of y is lower than unity. This approximation is widely used in qualitative discussions, because it permits the simple mathematical treatment of electrochemical processes with relatively small errors and with clear physical meaning. If y 1 is included in the derivation of the general polarization curve equation, simple analytical solutions are not available and numerical solutions are required. 

Benedict, Pigford, and Levi have carried out mathematical analysis of the GS process. An exhaustive treatment of the process, including calculations for flow rates, dependence of composition on number of stages, effect of solubility and humidity on process analysis, temperature profile in cold towers, simultaneous heat and mass transfer in heat transfer section, concentration reversal in heat transfer section, corrosion, materials of construction, feed purification, and safety, etc. have been reviewed by Dave, Sadhukhan and Novaro. ... 

The spectrophotometer monitors infrared absorbance in the sample cell by carrying out repetitive scans for the predetermined wavelength or frequency range at a scan rate of up to 20 scans s . Consecutive scans can be added prior to the Fourier transformation. The fast Fourier transform process is a mathematical treatment of the recorded data that converts time domain data into frequency domain spectra [19,20]. That is, the intensity values... 

Thin-film evaporators have proved particularly useful on a semi-technical scale [135]. The performance of various types of evaporators with rotors and the causes and degrees of resistance to mass transfer were studied bj Dieter [136], Billet has reported methods for the mathematical treatment of the distillation process in thin-film evaporators with rotating elements [137]. 

Mathematical treatment of the total catalytic process is complicated by strong coupling of the physical and chemical reaction steps, and by the heat of reaction of the chemical reactions. This leads to temperature and pressure gradients that are difficult to solve mathematically. 

A mathematical treatment of this process has been given by Taylor The continuous phase with a viscosity of and applied shear rate of dv /dy deforms a sphere of radius R and viscosity r)- with the force... 

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