Solution of Demchik's Model of Water Filtration Using the Method of Separation of Variables (original) (raw)
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AIChE Journal, 1983
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Review of the Role of Analytical Modelling Methods in Riverbank Filtration System
Jurnal Teknologi, 2014
Riverbank filtration (RBF) technology is applied in several countries around the world as one of the main sources of drinking water supply both from quantitative and qualitative point of view. Consequently, several analytical modelling methods, mostly based on the transformation techniques, are developed in literature to describe different processes which occur in RBF system. An extensive overview of these analytical methods, their uses and limitations are discussed. The review disclosed that most analytical models usually are concerned in evaluating stream depletion rate rather than contaminants transport especially the transportation of pesticides and pathogens. Laplace and Fourier methods are more popular methods used by researchers to solve the system of partial differential equation that developed to simulate the RBF problem.
Analytical Solutions for Filtration Process Based on the Constriction Size Concept
Geo-Congress 2014 Technical Papers, 2014
An analytical model is proposed to describe the filtration process applicable to a base soil-filter system. The Navier-Stokes equations for porous media are used to capture the hydrodynamic behavior, whereas, numerically, a new algorithm is proposed to solve the Navier-Stokes equation in a nonlinear form. The various mixtures of base soil particles eroded and water flow within the system are computed using the workenergy principle incorporating the constriction size of the filter.The model can assess the filtration process through the flowrate and the accumulation and redistribution of fine particles within the filter. By discretizing the base soil and filter domains into discrete elements, the model can predict the time-dependent particle gradation of the filter for each element. Laboratory tests reported in other studies and those conducted by the authors validate the model in relation to other available models.
A.S FOKAS SEPARATION OF VARIABLES METHOD: OVERVIEW
IAEME PUBLICATION, 2023
In 1750, D'Alembert demonstrated how to solve linear partial differential equations through separation of variables, a method of decomposing PDEs into a set of ODEs. This method has served as the basis for the development of many branches of modern analysis, from function spaces to spectral analysis of operators and the theory of special functions. In the present paper, the detail of separation of variables is discussed.
The modifications of customary filtrational equation
The usable limits of the customary and relaxational filtrational theories are considered. The questions of applicable the locality and local thermodynamical equilibrium principles to depict the nonstationary flows are discussed. The experimental procedures are proposed to determine the filtrational flows relaxation times.
On the Applicability of the Method of Separation of Variables for Partial Difference Equations
Journal of Difference Equations and Applications, 2002
In this paper it will be shown that the method of separation of variables for partial difference equations (as for instance described by R.E. Mickens [1] and R.P. Agarwal [2]) can be applied to a much larger class of problems as is generally assumed. To show how the method should be applied two problems are treated and solved. In [1] and [2] these two problems were considered and claimed to be not solvable by applying the method of separation of variables.
Filtration solutions for variable inputs
Solutions of the deep bed filtration equations are sought for situations involving spatially varying initial porosites and temporally varying superlicial velocities and inlet concentrations. Results of a general nature are obtained for both a quasistatic model and a diffusionless model. In the latter case it is shown that one numerical solution can sometimes be used to infer information about a class of problems.
Mathematical modeling of diafiltration
2009
The main objective of this study is to provide a general mathematical model in a compact form for batch diafiltration techniques. The presented mathematical framework gives a rich representation of diafiltration processes due to the employment of concentration-dependent solute rejections. It unifies the existing models for constant volume dilution mode, variable volume dilution mode, and concentration mode operations. The use of such a mathematical framework allows the optimization of the overall diafiltration process. The provided methodology is particularly applicable for decision makers to choose an appropriate diafiltration technique for the given separation design problem.