scholarly journals Modeling and numerical analysis of the effect of dissociation/recombination of water molecules on the transport of salt ions in diffusion layer

2019 ◽  
Vol 19 (3) ◽  
pp. 268-280
Author(s):  
N. O. Chubyr ◽  
A. V. Kovalenko ◽  
M. Kh. Urtenov ◽  
A. I. Sukhinov ◽  
V. A. Gudza
Keyword(s):  
Mathematical Model ◽  
Numerical Analysis ◽  
Boundary Value Problem ◽  
Joule Heating ◽  
Diffusion Layer ◽  
Temperature Effects ◽  
Boundary Value ◽  
Water Molecules ◽  
Recombination Reaction ◽  
Salt Ions

Introduction. The paper presents a theoretical study on binary salt ion transport considering the water dissociation/recombination reaction. The work objectives are as follows: to build a mathematical model; to develop an algorithm for the numerical solution to the boundary value problem corresponding to the mathematical model; to work out the similarity theory including the transition to a dimensionless form using characteristic quantities; to determine a physical meaning of trivial similarity criteria; to find nontrivial similarity criteria; to build and analyze the volt-ampere characteristic (VAC).Materials and Methods. The theoretical study and numerical analysis of the transport of binary salt ions consider the dissociation/recombination reaction of water. In this case, the heat transfer equation and the mathematical model of electrodiffusion of four types of ions simultaneously (two salt ions, as well as ????+ and ????????−ions) in the diffusion layer of electromembrane systems with a perfectly selective membrane are used. For the first-order differential equations, a singularly perturbed boundary-value problem is set. In the equation for the electric field, the right side is independent of the intensity. In the numerical solution to the digitized system of equations by the Newton-Kantorovich method, this causes the stability of the method. In this regard, the boundary-value problem is reduced for numerical solution: a transition to a system of the second-order equations is provided, and the missing boundary conditions for the electric field strength are calculated.Research Results. A new mathematical model, a numerical algorithm to solve a boundary value problem, and software are developed. A numerical analysis is carried out, and fundamental laws of the transport of salt ions are determined considering the dissociation/recombination reaction of water molecules, temperature effects, and Joule heating. The VAC is built and analyzed.Discussion and Conclusions. The transport of binary salt ions through a diffusion layer near a cation exchange membrane is considered. A mathematical model of this process is proposed. It takes into account the temperature effects due to dissociation/recombination reactions of water molecules and Joule heating in a solution. The basic laws of the transport of salt ions are established considering the dissociation/recombination reaction of water molecules and temperature effects. The temperature effects of the dissociation/recombination reaction and the Joule heating in the electroneutrality region (ENR) are almost imperceptible (with the exception of the recombination region, RR). The Joule heating in the space-charge region (SCR) is by two orders of magnitude larger than the cooling effect of the water dissociation reaction. Upon recombination, approximately the same heat is released in the RR as during Joule heating in the expanded SCR. However, due to the small size of the RR, the effect of this heat is imperceptible. Therefore, we can assume that there is only one heat source at the interface in the SCR, which, due to its noticeable size, causes a significant increase in temperature in the entire diffusion layer. It follows that the emergence and development of gravitational convection is possible. General conclusions, following from the results obtained, open up the possibility of intensifying the process of transport of salt ions in the electrodialysis machines.

scholarly journals Mathematical modelling of space charge breakdown in membrane systems taking into account the non-catalytic dissociation/ recombination reaction of water molecules

2020 ◽  
Vol 224 ◽  
pp. 02009
Author(s):  
V Gudza ◽  
M Urtenov ◽  
N Chubyr ◽  
I Shkorkina
Keyword(s):  
Ion Exchange ◽  
Boundary Value Problem ◽  
Space Charge ◽  
Theoretical Study ◽  
Boundary Value ◽  
Water Molecules ◽  
Recombination Reaction ◽  
Ion Transfer ◽  
Ion Exchange Membranes ◽  
Poisson Equations

In electromembrane systems, a theoretical study of salt ion transfer usually uses mathematical models of salt ion transfer in the depleted diffusion layer of ion-exchange membranes. In this paper, a new mathematical model of ion transport in the cross-section of the desalination channel formed by two ion-exchange membranes – anion-exchange (AEM) and cation-exchange (CEM), taking into account the non-catalytic dissociation/recombination reaction of water molecules. The model is a boundary value problem for a non-stationary system of Nernst-Planck and Poisson equations. A numerical analysis of the boundary value problem is performed and the main regularities of the 1:1 salt ion transfer process are established, in particular, the occurrence and development of space charge breakdown is shown. The interaction of the space charge and the noncatalytic dissociation/recombination reaction of water molecules are theoretically investigated.

Mathematical model implementation in conditions of uncertainty and possible risks

2021 ◽  
pp. 137-145
Author(s):  
A. Kravtsov ◽  
◽  
D. Levkin ◽  
O. Makarov ◽  
◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Layer Structure ◽  
Boundary Value ◽  
Goal Function ◽  
Cauchy Problems ◽  
System Of Differential Equations

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

scholarly journals Theoretical Analysis of the Stationary Transport of 1:1 Salt Ions in a Cross-Section of a Desalination Channel, Taking into Account the Non-Catalytic Dissociation/Recombination Reaction of Water Molecules

Membranes ◽  
2020 ◽  
Vol 10 (11) ◽  
pp. 342
Author(s):  
Makhamet Urtenov ◽  
Vitaly Gudza ◽  
Natalia Chubyr ◽  
Inna Shkorkina
Keyword(s):  
Ion Transport ◽  
Space Charge ◽  
Cross Section ◽  
Mutual Influence ◽  
Double Electric Layer ◽  
Limiting Current ◽  
Water Molecules ◽  
Recombination Reaction ◽  
Salt Ions ◽  
The One

In electromembrane systems, the theoretical study of salt ion transport usually uses mathematical models of salt ion transport in the depleted diffusion layer of ion-exchange membranes. This study uses a one-dimensional mathematical model of salt ion transport in a cross-section of a desalination channel formed by anion-exchange and cation-exchange membranes, taking into account an effect of a dissociation/recombination reaction of water molecules. The reaction on the one hand leads to an overlimiting mass transfer due to the effect of exaltation of the limiting current. On the other hand, an appearance of new electric charge carriers (hydrogen and hydroxyl ions) can reduce the space charge that occurs in membranes and suppress an electroconvective mechanism of overlimiting transport. Thus, there is a problem of studying these phenomena together, taking into account their mutual influence, and this article is devoted to the solution of this problem. Theoretically, using a method of mathematical modeling and numerical research, main regularities are established; in particular, it is shown that the dissociation/recombination reaction of water molecules does not lead to the destruction of the double electric layer at the membranes, but also creates a new double electric layer in the middle of the desalination channel. Thus, the space charge and the dissociation/recombination reaction significantly affect each other and simultaneously the transport of salt ions.

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

Mathematical model of magnetic field for a sectorial ferromagnetic cylinder

2021 ◽  
Vol 2021 (49) ◽  
pp. 19-25
Author(s):  
R. M. Dzhala ◽  
◽  
V. R. Dzhala ◽  
B. I. Horon ◽  
B. Ya. Verbenets ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Mathematical Model ◽  
Spatial Distribution ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Secondary Field

The solution of the boundary-value problem of magnetostatics for a circular ferromagnetic cylinder with a longitudinal sectorial cutout is described. The external primary magnetic field is orthogonal to the cylinder and directed at arbitrary azimuth relative to the cutout. A system of algebraic equations for finding the amplitudes of azimuthal expansions of the spatial distribution of the secondary field of the outer and sectorial partial regions of the cylinder is obtained by the method of rearrangement of functions.

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