Modelling and analysis of heat transfer through 1D

Authors: E Szymanek T Blaszczyk M R Hall P Keikhaei Dehdezi J S Leszczynski

Publish Date: 2014/07/05

Volume: 16, Issue: 5, Pages: 687-694

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Abstract

This article demonstrates the solution to the problem of the passage of air through the external wall barrier and the influence of the materials type and its layer arrangement on heat conductivity in respect of building heat losses It shows how the temperature changes inside the wall barriers and in a room while the external temperature changes Also this article presents the mathematical model based on fractional differential equation describing the analysed phenomenonThe process of heat transfer being present in many technological disciplines is very common as it occurs wherever temperature differences appear Heat transfer is achieved in three physicallydifferent ways that is through heat conduction convection and radiation Without any doubt the heat conduction is the most important form of the heat transfer process Next to the alreadymentioned three types of heat transfer there are also combinations of those types in the forms of heat transmission penetration and heat permeability This work will investigate the heat flow through a complex system understood as a system made of the granulated material air and water At the core of the heat transfer phenomenon in a complex system lies temperature difference Therefore one should look for solutions which allow properly defining the temperature distribution in its interior A more precise explanation of the phenomenon of heat transfer between the constituents of a given process may allow the better control of temperature changes The empty spaces between the granular material may be filled with for example gas or water Additionally many connected phenomena take place between the granular material In the conditions of fluctuating pressure and temperature chemical reactions between bodies and the material filling the empty space might occur The number of phenomena and the multiplicity of parameters cause problems with their description and because the phenomenon of the heat flow is connected both with everyday life and with many technological disciplines interest in the said subject is still current In literature we can find models describing heat flow through complex systems The threephase porous material consisting of the granule material framework and water and air filling the pores of the material 3 15 18 as well as the system consisting of two materials solid bodies and fluid or air 10 21 are taken into consideration In order to describe the heat flow computational analysis for example the finite difference method is applied Some problems with the method application are connected with the boundary conditions and with the irregular shape of the boundaries The other method applied to describe the phenomenon of heat flow is the finiteelement method This method is used to examine heat flow and to investigate other issues described with a differential equation of the first and second order 9 The common application of the said methods in solving technical problems is a result of the possibility of obtaining an equation result which cannot be solved in an analytical way or its solution is too complex and timeconsuming 36 The idea of the method of boundary elements also used to describe the analysed phenomenon is to reduce the given boundary element to a simultaneous integral equation which is defined on the boundary of the given closed and connected set and is equivalent to the reduction of the problem size The next method applied to the examination of heat flow is a model prepared by Rajagopal and Massoudi 30 where the material density is of great significance This model was used to examine various problems like the heat flow in a vertical pipe 13 or the heat flow occurring due to natural convection 27 A very common method used in granular media for prediction of global heat flux isotherms and isofluxes is the Lattice Boltzmann Method LBM is a relatively new simulation technique for complex fluid systems Due to its particulate nature and local dynamics LBM has several advantages over other conventional methods especially in dealing with complex boundaries incorporating of microscopic interactions and parallelization of the algorithm 11 14 The necessity to create a model of the heat flow through complex systems has been already indicated in the former publications The granular materials were very often treated as systems of barriers separated by air layers 19 To define the heat flow through granular materials Prasolov considered the granular structure as a system of a solid body and a gas 29 while Smolukovsky 32 examined the granulated materials in loweredgaspressure conditions assuming that the grains of the granular material are round Kelly and Schwarz 16 in their work analysed models of heat transfer In these models the physical geometry which is reproduced in all porous materials is idealised and shown in a simplified formOn the basis of the literature it can be stated that taking into account the classical approach to heat flow we apply the Fourier–Kirchoff equation But when we manage a complex system the classical approach is not the best solution One of the basic problems which appears while modelling the heat flow in a complex system by classical equations is the multiplicity of interrelations factors and coefficients necessary to describe the structure of the examined system Defining general interrelations which will allow the defining of heat conductivity in complex system is complicated because of the heat qualities of the particular elements and porosity as well as the moisture content Therefore it is necessary to create new modelsIn this paper has been proposed mathematical model based on fractional calculus which is used in many fields such as engineering chemistry electrical and electromechanical systems etc The basic mathematical ideas of fractional calculus were developed by the mathematicians Leibnitz Liouville Riemann and others and today is fastgrowing part of mathematics Podlubny Magin Mainardi West Fractional calculus is particulary useful in describing the dynamics of complex systems During the last decades of the nineteenth century 1892 Heaviside introduced the idea of fractional derivatives in his study of electric transmission lines Sebaa 31 used fractional calculus describe the viscous interactions between fluid and solid structure in human cancellous bone Kulish 20 used fractional calculus to fluid mechanics Assaleh 2 proposed a novel approach for speech signal modelling using fractional calculus in presented Fellah and Depollier 12 used application to the sound waves propagation in rigid porous materials Soczkiewicz 33 fractional calculus used in the theory of viscoelasticity Podlubny’s work 28 contains information about the applications of fractional calculus to various problems of mechanics physics and engineering West in 35 describes how to use the fractional operators in the modelling of complex phenomena After the analysis of this work we have observed that the spatial fractional derivatives have longrange interactions and may have deep physical implications when modelling a complex phenomenon The application of fractional calculus in bioengineering was presented by Magin in 24 25 Such fractional order models provide an improved description of the observed bioelectrode behaviour Mainardi in work 26 shows how the fractional calculus provides a suitable method of describing dynamical properties of linear viscoelastic media with the problems of wave propagation and diffusion The book by Uchaikin 34 presents detailed motivation for fractional differential equations in various branches of physicsThe growing number of such applications indicates that there is a significant demand for better mathematical models of real objects and that the fractional calculus provides one of many possible approaches regarding the way to more adequate mathematical modelling of real objects and processes especially when we are dealing with complex systems Some process can describe models approach in classical equation but suggest that additional mathematical tools may be needed to better describe this complex system Fractional derivatives have many properties in common with the classical ones but not all the properties are the same These differences can be used to describe complex phenomena that arise due to nonlocal interactions and system memory Classical approach requires large number of coefficient and constitutive relations in order to minimize uncertainty in mathematical modelling However large number of coefficient generate uncertainty in determination of their values To reduce this uncertainty we decided to apply fractional calculus which requires low number of coefficients describes the analysed process We limited our consideration only to steady state where the analysed phenomenon depends only on location This paper proposes a description which does not investigate the structure but assumes some degree of its heterogeneity When abandoning the classical equation and substituting it with an ordinary differential equation including the left and right fractional derivatives 4 5 7 8 22 we arrive at a model which has such a quality that it does not investigate the structure and does not include such a number of factors Such equations are the result of the modification of the principle of least action and the application of the fractional rule of integration by parts 5 The considered problems faced in the process of heat flow during the works will be analysed from the practical and theoretical points of view Computer simulations were additionally supported by experimental tests in the research station using a climate chamber in order to simulate the temperature conditions

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