ENGINEERING FOR RURAL DEVELOPMENT Jelgava, 24.-26.05.2017. 1369 MATHEMATICAL MODELLING OF HEAT TRANSFER PROBLEM FOR TWO LAYERED GYPSUM BOARD PRODUCTS EXPOSED TO FIRE Aivars Aboltins 1 , Harijs Kalis 2 , Kristaps Pulkis 1 , Juris Skujans 1 , Ilmars Kangro 3 1 Latvia University of Agriculture; 2 Institute of Mathematics and Computer Science, University of Latvia; 3 Rezekne Academy of Technologies aivars.aboltins@llu.lu,Kalis@lanet.lv,k.pulkis@gmail.com,ilmars.kangro@ru.lv 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. Introduction The 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 [1-5]. 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 [8-12]. 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 densities at high temperature on one of the walls. 1. The mathematical model The study of heat and mass transfer through a different media becomes much more interesting due to its vast applications. We study the heat transfer processes in the two layers of gypsum material at high temperatures. We consider a gypsum board material with two layered plates in x-direction: foam 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 T = 20 + 345lg(8t + 1) ºC (1) where t – the time in minutes. In one gypsum layer the heat and mass transfer process is analysed and described in [12; 13]. In this paper the hybrid experimental numerical method is proposed and the specific heat c p and thermal conductivity K coefficient dependence on temperature T is obtained: K decreases from the value DOI: 10.22616/ERDev2017.16.N312