Coupled Differential Equations (coupled + differential_equation)

Distribution by Scientific Domains


Selected Abstracts


Synthesis of charged ultrafiltration poly(styrene- co -divinyl benzene) composite membrane

JOURNAL OF APPLIED POLYMER SCIENCE, Issue 1 2008
Sonny Sachdeva
Abstract A ceramic supported crosslinked polystyrene composite membrane has been prepared from its monomers using a dual initiator system. The nonionic hydrophobic membrane so prepared has been chemically modified by a low temperature (50°C), single step reaction with chloroacetic acid. The carboxylated membrane has acid functional groups on its surface making it negatively charged and highly hydrophilic in nature. The membranes (unmodified and carboxylated) have been used for the separation of hazardous chromium (VI) salt solution where observed and intrinsic rejection has been studied as a function of pressure and concentration of the feed solution. The intrinsic rejection has been determined by calculating the concentration at the membrane surface (Cm) using Speigler-Kedam model and osmotic pressure model. The observed rejection for the chemically modified membrane decreases with increasing pressure but the intrinsic rejection is found to be more than 80% for all concentrations in the range of study. The experimental results have been fitted using Space-Charge model to obtain the membrane wall potential and the membrane surface concentration which are difficult to measure directly. The transport through the membrane capillaries has been described by the two dimensional model using Nernst-Planck equation for ion transport, Navier-Stokes equation and Poisson-Boltzmann equation for the radial distribution of potential. We have then presented a semianalytical series solution to the highly nonlinear Poisson-Boltzmann equation to reduce the computational time required to solve the set of coupled differential equations. The effective wall potential of the carboxylated membrane was found to be ,28.07 mV. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008 [source]


Kinetics of separating multicomponent mixtures by nondispersive solvent extraction: Ni and Cd

AICHE JOURNAL, Issue 4 2001
Inmaculada Ortiz
A model for nonsteady metals separation using nondispersive solvent extraction presented explicitly accounts for selective separation of multicomponent solutions and concentration of separated components in a back-extraction phase. The separation process comprises extraction and back-extraction steps carried out in two different hollow-fiber modules, connected through the organic phase, together with three homogenization tanks. The model is based on a set of coupled differential equations describing mass balances of metallic solutes in the fluid phases. To be applied in its simplest form, it requires the knowledge of three characteristic parameters, two parameters related to the chemical reactions, and one mass-transport parameter. The latter describes the mass-transport rate through pores of the hollow-fiber membrane filled with the organic phase. The mathematical model was checked against the kinetic results of the separation of Cd/Ni mixtures working with high-concentration solutions and obtaining the characteristic parameters of this system. [source]


Material Modelling of Porous Media for Wave Propagation Problems

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
M. Schanz PD Dr.-Ing.
Under the assumption of a linear geometry description and linear constitutive equations, the governing equations are derived for two poroelastic theories, Biot's theory and Theory of Porous Media (TPM), using solid displacements and pore pressure as unknowns. In both theories, this is only possible in the Laplace domain. Comparing the sets of differential equations of Biot's theory and of TPM, they show different constant coefficients but the same structure of coupled differential equations. Identifying these coefficients with the material data and correlating them leads to the known problem with Biot's ,apparent mass density'. Further, in trying to find a correlation between Biot's stress coefficient to parameters used in TPM yet unsolved inconsistencies are found. For studying the numerical effect of these differences, wave propagation results of a one-dimensional poroelastic column are analysed. Differences between both theories are resolved only for compressible constituents. [source]


Close coupling approach in optically allowed atomic transitions

ANNALEN DER PHYSIK, Issue 12 2007
S. Bougouffa
Abstract In general the calculations of the cross sections in atomic collisions theory need a treatment of a system of coupled integro-differential equations. We perform a numerical technique for calculations of electrons scattering with sodium atom. The cross sections are evaluated in the close coupling approach, where the problem is formulated in three coupling channel approximation. It is found that the three-channel problem results are typically in good agreement with experiment and two-channel calculations for intermediate energy range. The difference in the other range of energy can be assigned to the number of the used set of coupled differential equations in the 32S , 32P transition of sodium. [source]


Electron excitation cross section of atomic transitions in the two-state approximation

ANNALEN DER PHYSIK, Issue 6 2004
A. Kamli
Abstract The cross section of the 3s , 3p transition of sodium produced by electron impact has been calculated by performing a numerical integration of the coupled differential equations. The potential functions have been calculated exactly using the hydrogen-like wave functions for the valence electron of the sodium with an effective charge adjusted to fit the experimental 3s , 3p line strength. The results compare very well with experimental data and with those obtained using more elaborate and sophisticated models. [source]


Gekoppelter Wärme- und Stofftransport einschließlich der Korrosionsprozesse in porösen Baustoffen mit dem Simulationsprogramm AStra

BAUPHYSIK, Issue 3 2007
Rosa Maria Espinosa Dr.-Ing.
Zur Beschreibung von Wärme- und Feuchtetransportvorgängen gekoppelt mit Stofftransportprozessen in porösen Baustoffen ist ein Differentialgleichungssystem bestehend aus der Energieerhaltungsgleichung und den Massenerhaltungsgleichungen aller beteiligten Stoffe einschließlich des Wassers aufzulösen. Hierzu ist die Modellierung der stattfindenden Phasenumwandlungen der vorliegenden Stoffe und der chemischen Reaktionen der Porenlösung mit der Baustoffmatrix erforderlich. Zu unterscheiden sind dazu inerte, nicht reaktive Baustoffe und reaktionsfähige, zementgebundene Baustoffe. Für die numerische Simulation dieser Vorgänge bzw. die praktische Handhabbarkeit der Problemlösung wurde eine benutzerfreundliche Programmoberfläche AStra geschaffen, die neben dem eigentlichen Berechnungsmodul die benötigten Pre- und Postprocessing Möglichkeiten beinhaltet. Für die Berechnung des zeitlichen Verlaufs eines lösenden oder treibenden Angriffs auf zementgebundene Baustoffe wird durch eine in Abhängigkeit der beteiligten Spezies geeignete Vorauswahl von ablaufenden Reaktionen der Rechenaufwand für die Vorhersage von Nichtgleichgewichtszuständen optimiert und damit die Möglichkeit geschaffen zeitliche Abhängigkeiten mit vertretbarem Berechnungsaufwand zu beschreiben. Zusätzlich können mechanische Beanspruchungen als Folge von Kristallisationsvorgängen zumindest qualitativ vorhergesagt werden. Im Folgenden werden eine Übersicht der implementierten Berechnungsmodelle sowie drei Anwendungsbeispiele von AStra vorgestellt. Coupled heat and mass transfer simulation including corrosion in porous building materials with the program AStra. For the description of corrosion processes of porous building materials, it is necessary to solve a system of coupled (non-linear) differential equations, which consists of a conservation equation for the energy and one for the mass of each substance (including water and air), whose content may change within the simulated time period. Indeed, it must be distinguished between degradation of reactive and of inert materials. The computation program AStra simulates a coupled transport of heat, moisture, air and chemical substances in porous materials. AStra consists of a user interface for pre- and post-processing and a computation module (solver), which contains the necessary algorithms to solve the system of coupled differential equations. Mathematical models for phase changes of salts and for chemical reactions between substances, including the components of the material matrix in case of cementitious materials, were developed. Furthermore, the computational cost for the prediction of the corrosion of cementitious materials was optimized by means of an adequate pre-selection of chemical reactions. Thus, it is possible to simulate corrosion processes within justifiable simulation time. On the other hand, a simulation of the mechanical stress resulting from crystallization processes is possible. This paper presents some applications of the programs and a brief introduction into their theoretical basis. [source]


Modellierung und Simulation von Chemiereaktoren , Aspekte einer zeitgemäßen Ingenieurausbildung

CHEMIE-INGENIEUR-TECHNIK (CIT), Issue 1-2 2005
J. Hagen Prof.
Abstract Die Durchführung chemischer Prozesse in der Technik unter möglichst optimalen Bedingungen setzt ein Grundverständnis für die Arbeitsweise von Chemiereaktoren voraus. Grundlage der Modellierung von Chemiereaktoren sind die Stoff- und Wärmebilanz und die Kinetik der Reaktion. Nur in einfachen Fällen lassen sich analytische Lösungen für Problemstellungen aus der Praxis durch Integration der Differentialgleichungen finden. Das in der Chemieingenieurausbildung eingesetzte Softwarepaket POLYMATH ist sehr anwenderfreundlich und einfach zu erlernen. Es wird u.,a. eingesetzt, um gekoppelte Differentialgleichungen simultan zu lösen und Datenanalyse durch Regression durchzuführen. Mit einem einmal aufgestellten Modell lässt sich der Einfluss verschiedener Reaktionsparameter auf den Gesamtprozess leicht nachvollziehbar simulieren. Modeling and Simulation of Chemical Reactors , Aspects of a Modern Education of Engineers Understanding how chemical reactors work lies at the heart of almost every chemical processing operation. Basis information for modeling and simulation of chemical reactors is needed from mass transfer, heat transfer and chemical kinetics. Only simple problems from practice can be solved analytically by means of integration of the differential equations. POLYMATH is a extremely user-friendly software package which makes modeling easy for the education of chemical engineers and chemists. POLYMATH is used to numerically solve coupled differential equations simultaneously or to find kinetic parameters in rate expressions by regression. Using an identified model the influence of various reaction parameters on the overall process can be simulated easily. [source]