Solution of Demchik's Model of Water Filtration Using the Method of Separation of Variables (original) (raw)
AIChE Journal, 1983
systems is the large dimensionality of the process model. This paper is concerned with simple (reduced-order) steady-state and dynamic models for processes such as distillation, absorption and extraction. The model reduction procedure is based on approximating the composition and flow profiles in the column using polynomials rather than as discrete functions of the stages. The number of equations required to describe the system is thus drastically reduced. The method is developed using a simple absorber system. In the second part of this paper, the application of the method to nonlinear multicomponent separation systems is demonstrated.
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.
A SIMPLIFIED EMPIRICAL MODEL FOR THE ONE-STAGE DIRECT FILTRATION
A new configuration of the one stage direct filtration process has been investigated by Prof. Fadel (correspondent author) and his team in Mansoura University, Egypt. In this configuration, deep media with fine gravel particles (size 2-4 mm) has been used under low filtration rate (5-8 m 3 /m 2 /hr). For this system a simplified empirical model is proposed to describe the one stage of direct filtration process under steady-state conditions. The model is obtained based on curve fitting of true field data, which is obtained from several plants employing direct filtration process. The model considers all the essential concepts that describe the two processes of flocculation and filtration in one tank and the competition between them for limiting depth. Further, the model has considered all the major factors, which affect on the filtration process such as filter depth, filtration rate, run length, run time, size of media, and alum dose. The present model has an explicit solution, which may be useful for many applications of such filters. The application of the model has been explained for a given set of data and verified by comparison with another filed scale data. Also using the present model, the optimum operation conditions for running this type of filter have been investigated. Compared with other solutions for such system, the model is simple, easy to use, and provides a quick tool for describing such system.
Journal of Applied Sciences and Environmental Management, 2021
Riverbank filtration (RBF) is a natural technology that is used for river water treatment. This research seeks to investigate the effect of pumping rate on the transport of colloids in RBF. However, this work considered Dissolved Organic Matter (DOM) as a nutrient for bacteria. The mathematical model consists of groundwater flow equation and colloids concentration equations. The equations were solved analytically using parameter expanding method and Eigen function expansion techniques. The results obtained are presented graphically and discussed. It was observed that increase in pumping rate value enhance both the hydraulic head and concentration of colloids which slightly reduces the quality of pumped water from RBF. Keywords: Riverbank filtration, analytical model, colloids, hydraulic head and pumping rat
Mathematical modelling of contaminant transport in riverbank filtration systems
2017
Analytical study of contaminant transport in riverbank filtration (RBF) systems is significant in providing a guide for managing and operating drinking water supplies from pumping wells. The pumping process and the distance of the pumping well from the river are two important factors for producing permissible drinking water from the system. Simulation of the impact of pumping rate and pumping time on contaminant transport based on analytical studies are not yet extensive. Thus, there is a lack of mathematical models for RBF systems to determine the shortest distance of the pumping well to the river, that produces quality water. This research aimed to provide a mathematical model based on advection dispersion equation and Green’s function approach to determine the potential effects of pumping rate and pumping time, on one and two-dimensional contaminant transport models in RBF systems. The model would be able to show how the pumping time and pumping rate affect the contaminant concen...
Mathematical Model of Self-purification of River Benue Water Quality Control
In this study, nonlinear differential equation of order one model of River Benue water quality control was formulated. The fourth-order Runge-Kutta method was used to study the self-purification in the River Benue. The results showed that waste going down the River Benue becomes non-active to both human being and animals. We conclude that selfpurification of waste in a river is possible as the waste go down the river. It is advisable to dump waste about 300km away from human being and animal water usage.
Modelling of Water Quality: An Application to a Water Treatment Process
Applied Computational Intelligence and Soft Computing, 2012
The modelling of water treatment processes is challenging because of its complexity, nonlinearity, and numerous contributory variables, but it is of particular importance since water of low quality causes health-related and economic problems which have a considerable impact on people’s daily lives. Linear and nonlinear modelling methods are used here to model residual aluminium and turbidity in treated water, using both laboratory and process data as input variables. The approach includes variable selection to find the most important factors affecting the quality parameters. Correlations of∼0.7–0.9 between the modelled and real values for the target parameters were ultimately achieved. This data analysis procedure seems to provide an efficient means of modelling the water treatment process and defining its most essential variables.