scholarly journals SPECIAL SPLINE APPROXIMATION FOR THE SOLUTION OF THE NON-STATIONARY 3-D MASS TRANSFER PROBLEM

2021 ◽  
Vol 2 ◽  
pp. 69-73
Author(s):  
Ilmārs Kangro ◽  
Harijs Kalis ◽  
Ērika Teirumnieka ◽  
Edmunds Teirumnieks
Keyword(s):  
Mass Transfer ◽  
Transfer Problem ◽  
Spline Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fourier Series Method ◽  
Mass Transfer Problem ◽  
Mass And Heat Transfer ◽  
Initial Boundary Value ◽  
Transfer Problems

In this paper we consider the conservative averaging method (CAM) with special spline approximation for solving the non-stationary 3-D mass transfer problem. The special hyperbolic type spline, which interpolates the middle integral values of piece-wise smooth function is used. With the help of these splines the initial-boundary value problem (IBVP) of mathematical physics in 3-D domain with respect to one coordinate is reduced to problems for system of equations in 2-D domain. This procedure allows reduce also the 2-D problem to a 1-D problem and thus the solution of the approximated problem can be obtained analytically. The accuracy of the approximated solution for the special 1-D IBVP is compared with the exact solution of the studied problem obtained with the Fourier series method. The numerical solution is compared with the spline solution. The above-mentioned method has extensive physical applications, related to mass and heat transfer problems in 3-D domains. 

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.

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.

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.

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 Parabolic-hyperbolic transmission problem in disjoint domains

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6911-6920
Author(s):  
Zorica Milovanovic-Jeknic
Keyword(s):  
Mass Transfer ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Model Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Similar Type ◽  
Transmission Problems ◽  
Initial Boundary Value ◽  
Disjoint Domains

In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper we investigate a mixed parabolic-hyperbolic initial-boundary value problem in two nonadjacent rectangles with nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problem the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.

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.

Numerical Modelling of Mass Transfer Problems in Combined Domains with Non-Linear Imperfect Transmission Conditions

2009 ◽  
Vol 283-286 ◽  
pp. 527-532
Author(s):  
Michal Wrobel ◽  
Gennady Mishuris
Keyword(s):  
Mass Transfer ◽  
Boundary Condition ◽  
Transfer Problem ◽  
Contrast Material ◽  
Local Boundary ◽  
Local Boundary Condition ◽  
Mass Transfer Problem ◽  
Type Boundary ◽  
Transfer Problems ◽  
Type Boundary Condition

This work deals with a nonlinear mass transfer problem in an infinite combined domain, consisting of half-space matched to a bounded part via a thin intermediate layer. The latter exhibits high contrast material properties, whereas its thickness is assumed to be negligible in comparison with the dimensions of the bounded subdomain. The corresponding problem is reduced to an auxiliary one, defined only in the bounded region with a non-local boundary condition on the transmission surface, which is solved numerically by means of FEM. To introduce the boundary condition, a special iterative subroutine based on the classic Robin type boundary condition is constructed. The accuracy of the procedure and the range of its applicability are investigated for an analytical benchmark. Numerical results for an axisymmetrical stationary mass transfer problem are presented.

scholarly journals SPECIAL SPLINES OF EXPONENTIAL TYPE FOR THE SOLUTIONS OF MASS TRANSFER PROBLEMS IN MULTILAYER DOMAINS

2016 ◽  
Vol 21 (4) ◽  
pp. 450-465 ◽  
Author(s):  
Andris Buikis ◽  
Harijs Kalis ◽  
Ilmars Kangro
Keyword(s):  
Boundary Value Problem ◽  
Exponential Type ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonstationary Problem ◽  
Alternating Direction ◽  
Adi Methods ◽  
Initial Boundary Value ◽  
Transfer Problems

We consider averaging methods for solving the 3-D boundary-value problem of second order in multilayer domain. The special hyperbolic and exponential type splines, with middle integral values of piece-wise smooth function interpolation are considered. With the help of these splines the problems of mathematical physics in 3-D with piece-wise coefficients are reduced with respect to one coordinate to 2-D problems. This procedure also allows to reduce the 2-D problems to 1-D problems and the solution of the approximated problemsa can be obtained analytically. In the case of constant piece-wise coefficients we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem. The solution of corresponding averaged 3-D initial-boundary value problem is also obtained numerically, using the discretization in space with the central diferences. The approximation of the 3-D nonstationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.

scholarly journals Nonstationary Radiative–Conductive Heat Transfer Problem in a Semitransparent Body with Absolutely Black Inclusions

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1471
Author(s):  
Andrey Amosov
Keyword(s):  
Heat Transfer ◽  
Existence And Uniqueness ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conductive Heat ◽  
Stability Of Solutions ◽  
Complex Heat Exchange ◽  
The Stability ◽  
Initial Boundary Value

The paper is devoted to a nonstationary initial–boundary value problem governing complex heat exchange in a convex semitransparent body containing several absolutely black inclusions. The existence and uniqueness of a weak solution to this problem are proven herein. In addition, the stability of solutions with respect to data, a comparison theorem and the results of improving the properties of solutions with an increase in the summability of the data were established. All results are global in terms of time and data.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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