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.
Abstract. In this paper we study the problem of the heat transfer through the two layered material of gypsum (with different densities) board products exposed to fire. This paper proposes a thermal conductivity model in two gypsum product layers for foam gypsum plate and gypsum board with different density and at high temperatures. For transfer of heat the system of 2 non-stationary partial differential equations (PDEs) is derived expressing the rate of the change of temperature T in every layer. The approximation of the corresponding initial boundary value problem of this system is based on the conservative averaging method (CAM) by using special splines with hyperbolic functions. This procedure allows reducing the 2-D heat transfer initial-boundary problem described by a system of 2 PDEs to the initial value problem for a system of 2 ordinary differential equations (ODEs) of the first order. The results of calculations are obtained by MATLAB.Keywords: gypsum, heat, averaging method, splines, numerical solution. IntroductionThe standards in building construction increase and it leads to the development of new materials. Fire protection becomes a prime requirement of building regulations in many countries in the construction industry. New composite materials with good properties as thermal insulators at elevated temperatures are developed. Commercial gypsum boards are widely used in the building industry as facing materials for walls and ceilings due to their very good mechanical and thermal properties. Thermal response of gypsum materials has been experimentally and numerically studied during the past years [1][2][3][4][5]. Gypsum based materials are known for excellent properties in fire protection and are used as general materials to protect building structures against fire. Most of the gypsum based materials are made as low or high density boards, but it is not common to make composite materials combining low and high density layers. Boards made with different layer density can achieve better results in fire safety, thermal insulation and acoustics [6; 7]. The endothermic dehydration process that takes place at high temperatures is capable of slowing down the fire spread through gypsum board based systems. A very important role here is played by the heat transfer processes in the same material. Experimentally it is very difficult to determine, therefore, it uses the heat transfer process of mathematical modeling. There are several computation models made to predict the thermal behaviour of gypsum boards under fire conditions. They revealed the significance of using appropriate physical properties for simulating the temperature evolution inside a gypsum board when exposed to fire conditions [8][9][10][11][12]. As noted "no significant influence of vapour transport on temperature is observed when the phase change is omitted" [12].The objective of the present study is to develop a mathematical model of heat transfer thought the two layered material of gypsum (foam gypsum and gypsum board) with different densiti...
<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>
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.
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.
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