Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains

Buikis, Andris; Kalis, Harijs; Kangro, Ilmars

doi:10.3846/13926292.2016.1182594

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…Such parameters are used i0 (z) = 0 with the independent variable z. The functions m iz = m iz (t), e iz = e iz (t) are found, as in previous publications [2], from boundary conditions of (1.1) at z 0 = 0, z = z 1 , z = L Z , but in addition a condition q i = r i + 1z /r iz that determines the "jump-off" effect of the solution is used. The solution of the system (1.3) is in the form of vector-matrices: / a F f = .…”

Section: Averaging In Z-direction Applying Integral Spline Of Exponenmentioning

confidence: 99%

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CAM with special splines for solving of diffusion-convection problems with discontinuous coefficients for layered materials exposed to fire

Aboltins

Kalis

Kangro

2019

Engineering for Rural Development

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In this paper we consider the conservative averaging method (CAM) for solving the diffusionconvection initial-boundary value problem (IBVP) with piece-wise coefficients in two layered domains. The created special exponential and hyperbolic type approximation functions-splines interpolate the middle integral values of the piece-wise smooth function, and thus the studied IBVP is reduced to the problem of the system of ordinary differential equations. The solution of the corresponding 1-D IBVP with diffusion was obtained also numerically, using the Matlab routine "pdepe" and the numerical solution was compared with the analytical solution. The 1-D diffusion-convection and 1-D diffusion boundary problems were solved and the solutions obtained were analysed with CAM (in case of exponential and hyperbolic splines) and also with Matlab "pdedpe" programme. Higher accuracy of the calculations in solving the 1-D diffusion-convection IBVP (case of discontinuous coefficients with a "jump-off" effect of the solution of IBVP) was identified by CAM. A practical task was solved: the temperature was calculated through the two layered material of gypsum products exposed to fire and the calculations were compared with the results obtained at the Faculty of Environment and Civil Engineering of the Latvia University of Life Sciences and Technologies. The data processing and analysis tools MS Excel and IBM SPSS Statistics20 tested the coincidence of the numerical calculations performed with the above mentioned experimental results.

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Section: Averaging In Z-direction Applying Integral Spline Of Exponenmentioning

confidence: 99%

“…It should be noted that, in a limited case, when the parameters of the spline function tend to zero, we have the integral parabolic spline, obtained from A.Buikis [1]. The solving of 3-D IBVP in Nlayer domain is considered in [2].…”

Section: Introductionmentioning

confidence: 99%

CAM with special splines for solving of diffusion-convection problems with discontinuous coefficients for layered materials exposed to fire

Aboltins

Kalis

Kangro

2019

Engineering for Rural Development

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“…CAM can be applied both to linear processes (Kalis, 2016) and non-linear processes (the dependency of mathematical model equation coefficients on the process characteristics, such as temperature in the combustion process) (Aboltins, 2017), (Weber, 2012).…”

Section: Introductionmentioning

confidence: 99%

A Modified Method to Solve the One-Dimensional Heat Conduction Problem

Kangro¹

2018

BJMC

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Abstract. The article focuses on the tasks of the mathematical physics -one-dimensional diffusion-convection boundary-value problem (BVP) for solving the heat conduction equation with piece-wise smooth coefficients in the multi-layer media. For this purpose the conservative averaging method (CAM) is using with special created integral splines of exponential type that interpolate the middle integral values of piece-wise smooth function through averaging in zdirection. Thus BVP is reduced to the system of ordinary differential equations (ODE) dependent on time -this enables to find out the averaged solutions of BVP -non-stationary and stationary.

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“…The method of conservative averaging by using the special integral splines with hyperbolic trigonometrical functions is openly discussed in papers [8]. We consider the following model problem…”

Section: Cam With Integral Hyperbolic Splines In Two Layersmentioning

confidence: 99%

On mathematical modelling of heat and moisture distribution in the drying process for porous two layered gypsum board products

Aboltins¹,

Kalis²,

Kangro³

2017

Engineering for Rural Development

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Abstract. In this paper we study the problem of the heat and diffusion of one substance through the pores of two layered material of gypsum board products, which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. This paper proposes a thermal conductivity model for gypsum plate and gypsum carton at high temperatures, treating gypsum as a porous material consisting of solid and pores. We shall further assume linear dependence on both the temperature T and the moisture content in every layer M=const+aC-bT, where C is the concentration of water vapour in the air spaces, M is the amount of moisture absorbed by unit mass of fibre, a and b are positive constants. For two processes, the transfer of heat and moisture, we derive the system of 4 non-stationary partial differential equations (PDEs), 2 expressing the rate of the change of concentration of water vapour C in the air spaces and the other 2 the rate of the change of temperature T in every layer. The approximation of the corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM) by using special splines with hyperbolic functions. This procedure allows reducing the 2-D heat and mass transfer initial-boundary problem described by a system of 4 PDEs to initial value problem for a system of 4 ordinary differential equations (ODEs) of the first order. The results of calculations are obtained by MATLAB.Keywords: gypsum board products, heat and diffusion distribution, conservative averaging method, special splines, analytic and numerical solution. The mathematical modelThe study of heat and mass transfer through a porous media becomes much more interesting due to its vast applications. Many mathematical models are developed for the analysis of such processes, for example, mathematical models of moisture movement in wood, when the wood is considered as porous media.The study of hydrodynamic flow and heat transfer through a porous media becomes much more interesting due to its vast applications, such as drying in porous solids and soils, drying of wood and paper, soil mechanics, porous heat pipes, paper machines, liquid composite moulding. Many mathematical models are developed for the analysis of such processes, for example mathematical models of moisture movement in wood, when the wood is considered as porous media [1; 2]. This paper proposes a thermal conductivity model for gypsum at high temperatures, treating gypsum as a porous material consisting of solid and pores. We study the heat and moisture transfer processes in the two porous layers of gypsum material. We consider the gypsum board material with two layered plates: gypsum plate (0.0525 m) with density 300 kg·m -3 and gypsum carton plate (0.0125 m) with density 1000 kg·m -3 . The gypsum plate on one border is heated with temperature 20 + 345lg(8t +1) ºC, where t is the time in minutes. In one layer the heat and mass transfer process are analysed and described in [1; 2; 5; 6].The article [4] has viewed ...

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Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains

Cited by 9 publications

References 5 publications

CAM with special splines for solving of diffusion-convection problems with discontinuous coefficients for layered materials exposed to fire

CAM with special splines for solving of diffusion-convection problems with discontinuous coefficients for layered materials exposed to fire

A Modified Method to Solve the One-Dimensional Heat Conduction Problem

On mathematical modelling of heat and moisture distribution in the drying process for porous two layered gypsum board products

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