scholarly journals ON MATHEMATICAL MODELLING OF THE SOLID-LIQUID MIXTURES TRANSPORT IN POROUS AXIAL-SYMMETRICAL CONTAINER WITH HENRY AND LANGMUIR SORPTION KINETICS

2018 ◽  
Vol 23 (4) ◽  
pp. 554-567 ◽  
Author(s):  
Ilmars Kangroa ◽  
Harijs Kalis
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Axial Direction ◽  
Initial Boundary Value Problem ◽  
Liquid Mixtures ◽  
Sorption Kinetics ◽  
Rate Of Change ◽  
Filtration Problem ◽  
Initial Boundary Value

In this paper we study diffusion and convection filtration problem of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances. As an example we consider round cylinder with filtration process in the axial direction. The cylinder is filled with sorbent i.e. absorbent material that passed through dirty water or liquid solutions. We can derive the system of two partial differential equations (PDEs), one expressing the rate of change of concentration of water in the pores of the sorbent and the other - the rate of change of concentration in the sorbent or kinetical equation for absorption. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM). This procedure allows reducing the 2-D axisymmetrical mass transfer problem decribed by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order. We consider also model 1-D problem for investigation the depending the concentration of water and sorbent on the time.

scholarly journals ON MATHEMATICAL MODELLING OF THE 2-D FILTRATION PROBLEM IN POROUS AXIAL SYMMETRICAL CYLINDER

2017 ◽  
Vol 3 ◽  
pp. 129
Author(s):  
Ilmārs Kangro ◽  
Harijs Kalis ◽  
Ērika Teirumnieka ◽  
Edmunds Teirumnieks
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Axial Direction ◽  
Initial Boundary Value Problem ◽  
Rate Of Change ◽  
Filtration Problem ◽  
Dirty Water ◽  
Round Cylinder ◽  
Initial Boundary Value

In this paper we study diffusion and convection filtration problem of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances. As an example we consider round cylinder with filtration process in the axial direction. The cylinder is filled with sorbent i.e. absorbent material that passed through dirty water or liquid solutions. We can derive the system of two partial differential equations (PDEs). One equation is expressing the rate of change of concentration of water in the pores of the sorbent and the other - the rate of change of concentration in the sorbent or kinetically equation for absorption. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM). This procedure allows reducing the 2-D axis-symmetrical mass transfer problem described by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order.

scholarly journals SPECIAL SPLINES OF HYPERBOLIC TYPE FOR THE SOLUTIONS OF HEAT AND MASS TRANSFER 3-D PROBLEMS IN POROUS MULTI-LAYERED AXIAL SYMMETRY DOMAIN

2017 ◽  
Vol 22 (4) ◽  
pp. 425-440
Author(s):  
Harijs Kalis ◽  
Andris Buikis ◽  
Aivars Aboltins ◽  
Ilmars Kangro
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Circular Cross Section ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rate Of Change ◽  
Wood Block ◽  
Initial Value ◽  
Special Integral ◽  
Initial Boundary Value

In this paper we study the problem of the diffusion of one substance through the pores of a porous multi layered material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. As an example we consider circular cross section wood-block with two layers in the radial direction. We consider the transfer of heat process. We derive the system of two partial differential equations (PDEs) - one expressing the rate of change of concentration of water vapour in the air spaces and the other - the rate of change of temperature in every layer. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM) with special integral splines. This procedure allows reduce the 3-D axis-symmetrical transfer problem in multi-layered domain described by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order.

CALCULATION OF HEAT AND MOISTURE DISTRIBUTION IN THE POROUS MEDIA LAYER

2007 ◽  
Vol 12 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Harijs Kalis ◽  
Ilmars Kangro
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rate Of Change ◽  
Moisture Distribution ◽  
Initial Boundary Value ◽  
Volume Method ◽  
Heat And Moisture ◽  
Media Layer

In this paper we study the problem of the diffusion of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. The transfer of moisture and the heat are described by the model. The system of two partial differential equations (PDEs) is derived, one equation expresses the rate of change of concentration of water vapour in the air spaces and the other the rate of change of temperature. The obtained initial‐boundary value problem is approximated by using the finite volume method. This procedure allows us to reduce the 2D transfer problem described by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order.

scholarly journals SIMPLE METHODS OF ENGINEERING CALCULATIONS FOR SOLVING TRANSFER PROBLEMS OF MULTI – SUBSTANCES IN HORIZONTAL LAYER

2005 ◽  
Vol 1 ◽  
pp. 40
Author(s):  
Harijs Kalis ◽  
Ilmārs Kangro
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Horizontal Layer ◽  
Initial Value ◽  
Difference Vector ◽  
Initial Boundary Value ◽  
Volume Method ◽  
Transfer Problems

We consider the simple algorithms in the modelling of the transfer problem of different substances (concentration, heat, moisture, and e. c.) in plate. The approximations of corresponding initial – boundary value problem of the system of the partial differential equations (PDE) is based on the finite volume method. This procedure allows one to reduce the 2-D transfer problem described by a PDE to initial value problem for a system of ordinary differential equations (ODE) of the first or second order. In the stationary case the exact finite – difference vector scheme is obtained.

DYNAMICS OF MULTISECTION SEMICONDUCTOR LASERS

2005 ◽  
Vol 9 (1) ◽  
pp. 51-66 ◽  
Author(s):  
J. Sieber ◽  
M. Radžiūnas ◽  
K. R. Schneider
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Semiconductor Lasers ◽  
Invariant Manifolds ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Low Dimension ◽  
Global Existence And Uniqueness ◽  
Numerical Bifurcation ◽  
Initial Boundary Value

We investigate the longitudinal dynamics of multisection semiconductor lasers based on a model, where a hyperbolic system of partial differential equations is nonlinearly coupled with a system of ordinary differential equations. We present analytic results for that system: global existence and uniqueness of the initial‐boundary value problem, and existence of attracting invariant manifolds of low dimension. The flow on these manifolds is approximately described by the so‐called mode approximations which are systems of ordinary differential equations. Finally, we present a detailed numerical bifurcation analysis of the two-mode approximation system and compare it with the simulated dynamics of the full PDE model.

scholarly journals The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

2015 ◽  
Vol 3 ◽  
pp. 186
Author(s):  
Ērika Teirumnieka ◽  
Ilmārs Kangro ◽  
Edmunds Teirumnieks ◽  
Harijs Kalis
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Elliptic Type ◽  
Exponential Spline ◽  
Initial Boundary Value

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

scholarly journals Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations

2020 ◽  
Vol 100 (4) ◽  
pp. 5-16
Author(s):  
A.T. Assanova ◽  
◽  
Zh.S. Tokmurzin ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Periodic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Hyperbolic Type ◽  
Order System ◽  
Partial Differential ◽  
Initial Boundary Value

A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

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