Keywords
heat storage
multilayer wall
numerical modelling
Finite Volume Method
How to Cite
Abstract
The paper presents details of discretisation of a thermal diffusion equation in one-dimensional space in terms of the Finite Volume Method. In the following sections, the method of space discretisation is discussed along with the approximation of a spatial derivative, matrix notation of a system of equations, special cases, approximation of three types of boundary conditions and derivative approximation over time. Much attention is also given to the issue of averaging material properties which can generally be different in adjacent cells.The study aims to analyse various multilayer structures for their suitability as heat storage. The launch of studies described in the paper has been driven by the lack of methods for effective heat storage, which is currently one of the key problems faced by the renewable energy industry.
References
Ashgriz N., Mostaghimi J.: An Introduction to Computational Fluid Dynamics. Chapter 20 in Fluid Flow Handbook. McGraw-Hill Publisher, J. Saleh Editor, United States 2002.
Bradji A., Herbin R.: Discretization Of The Coupled Heat And Electrical Diffusion Problems By The Finite Element And The Finite Volume Methods. IMAJournal of Numerical Analysis, Vol. 28(3), 2008, 469-495.
Burger M.: Numerical Methods for Incompressible Flow. Lecture notes [on-line], URL: ftp://ftp.math.ucla.edu/pub/camreport/cam04-12.pdf (available at 2010-01-12), Darmstadt University of Technology, Germany, March 2004.
Burch D.M., Thomas W.C., Mathena L.R., Licitra B.A., Ward D.B.: Transient Heat and Moisture Transfer in Multi-Layer, Non-Isothermal Walls - Comparison of Predicted and Measured Results. Proceedings of Thermal Performance of the Exterior Envelopes of Buildings IV, Clearwater Beach, FL, 4-7 Dec. 1989, 513-531.
Carrillo J.A., Chertock A., Huang Y.: A Finite Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure. Communications in Computational Physics, Vol. 17(1), 2015, 233-258.
Cebo-Rudnicka A.: The influence of selected parameters of spray cooling andthermal conductivity on heat transfer coefficient (in Polish). PhD Thesis. AGH University of Science and Technology, Kraków (Poland) 2011.
Chandrashekar P.: Finite volume discretization of heat equation and compressible Navier-Stokes equations with weak Dirichlet boundary condition on tri-angular grids. International Journal of Advances in Engineering Sciences and Applied Mathematics, Vol. 8(3), 2016, 174-193.
Das A., Alagirusamy R., Kumar P.: Study of heat transfer through multilayerclothing assemblies: a theoretical prediction. AUTEX Research Journal, Vol.11(2), 2011, 54-60.
Díaz J.J.C., Rabanal F.P.A., Nieto P.J.G., López M.A.S.: Sound transmissionloss analysis through a multilayer lightweight concrete hollow brick wall by FEM and experimental validation. Building and Environment, Vol. 45, 2010, 2373-2386.
Després B.: Non-linear schemes for the heat equation in 1D. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48(1), 2014, 107-134.
Douglas V.N.: Finite Volume algorithms for heat conduction. Technical Report for the period December 2009 - May 2010 AFRL-RW-EG-TR-2010-049.
Feistauer M.: Finite Volume and Finite Element Methods in CFD (Numerical Simulation of Compressible Flow) [on-line], Faculty of Mathematics and Physics, Charles University, Prague 2007.
Ferone C., Colangelo F., Frattini D., Roviello G., Cioffi R. di Maggio R.: Finite Element Method Modeling of Sensible Heat Thermal Energy Storage with Innovative Concretes and Comparative Analysis with Literature Benchmarks. Energies, Vol. 7, 2014, 5291-5316.
Gantenbein P., Rindt C.: Collection of experimental data on the behavior of TCM/PCM-materials to benchmark numerical codes. Report A3.2 of the Working Group on Numerical Modelling, December 2012.
Germi S.S., Rahimi B., Sadat S.A.K.: The moisture influence on a brick wall. International Journal of Advances in Engineering and Technology, Vol. 8(3), 2015 256-260.
Gómez H., Colominas I., Navarrina F., Casteleiro M.: A hyperbolic model forconvection-diffusion transport problems in CFD: Numerical analysis and applications. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemticas (RACSAM), Vol. 102(2), 2008, 319-334.
Gong J., Xuan L., Ming P., Zhang W.: An Unstructured Finite-VolumeMethod for Transient Heat Conduction Analysis of Multilayer Functionally Graded Materials with Mixed Grids. Numerical Heat Transfer, Part B: Fundamentals, Vol. 63, 2013, 222-247.
Johnson N., J.: Review of Numerical Methods. 5th Annual Conference of the CFD of Canada, CFD97, May 1997, Victoria, British Columbia.
Jouhaud J.C., Sagaut P., Labeyrie B.: A Kriging Approach for CFD/WindTunnel Data Comparison. Journal of Fluids Engineering, Vol. 128(4), 2006, 847-855.
Kadioglu S.Y., Nourgaliev R.R., Mousseau V.A.: A Comparative Study ofthe Harmonic and Arithmetic Averaging of Diffusion Coefficients for Non-Linear Heat Conduction Problems. Idaho National Laboratory, Idaho Falls (US), March 2008.
LeVeque R.J.: Finite Difference Methods for Differential Equations. Lecture Notes. University of Washington. Version of September 2005.
Majchrzak E., Turchan Ł.: The finite difference method for transientconvection-diffusion problems. The Finite Difference Method for transientconvection-diffusion problems. Scientific Research of the Institute of Mathematics and Computer Science, Vol. 11(1), 2012, 63-72.
Martin K., Escudero C., Erkoreka A., Flores I., Sala J.M.: Equivalent wallmethod for dynamic characterisation of thermal bridges. Energy and Buildings, Vol. 55, 2012, 704-714.
Matusiak M., Kowalczyk S.: Thermal-insulation properties of multilayer tex-tile packages. AUTEX Research Journal, Vol. 14(4), December 2014.
Mavromatidis L., Bykalyuk A., Mankibi M., Michel P., Santamouris M.: Numerical investigation of a wall’s optimum multilayer thermal insulation position. Proceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association, Sydney, 14-16 November.
Mierzwiczak, M.: Steadystate heat conduction in a plate with temperature d-pendent thermal conductivity. Scientific Papers University of Technology. Engineering and Production Management, Vol. 9, 2008, 67-79.
Miyakita T., Hatakenaka R., Sugita H.: Evaluation of Thermal Insulation Performance of a New Multi-Layer Insulation with Non-Interlayer-Contact Spacer. 45th International Conference on Environmental Systems, 12-16 July 2015, Bellevue, Washington.
Monteiro E., Almeida R., Rouboa A.: Finite Volume Method Analysis of Heat Transfer in Multi-Block Grid During Solidification. Chapter 5 in book ”Heat Transfer - Mathematical Modelling, Numerical Methods and InformationTechnology”, edited by Aziz Belmiloudi, 2011. ISBN 978-953-307-550-1.
Navarro L., de Gracia A., Colclough S., Browne M., McCormack S.J., Griffiths P., Cabeza L.F.: Thermal energy storage in building integrated thermalsystems: A review. Part 1. Active storage systems. Renewable Energy, Vol. 88, 2016, 526-547.
Olsen-Kettle L.: Numerical solution of partial differential equations. The University of Queensland, School of Earth Sciences, Centre for Geoscience Computing. ISBN: 978-1-74272-149-1.
Pakanen J.: Conduction of heat through slabs and walls: A differential-difference approach for design, energy analysis and building automation applications. VTT Publications, Finland 2009.
Patankar S.V.: Numerical heat transfer and fluid flow. Hemisphere, NewYork, 1980.
Patil P.V., Prasad J.S.V.R.K.: The unsteady state finite volume numerical gridtechnique for multidimensional problems. International Journal of Advances Inapplied Mathematics and Mechanics, Vol. 2(2), 2014, 78-87.
Pietkiewicz P., Miąskowski W., Nalepa K., Neugebauer M., Miazio Ł.: Ideas and construction of the model heat repository. Mechanics, Vol. 85(7), 2012, 687-695.
Pietkiewicz P., Nalepa K., Miąskowski W.: Test study of the model thermalenergy storage. Mechanics, Vol. 86(7), 2013, 625-631.
Puszkarz A.K., Krucińska I.: Study of Multilayer Clothing Thermal Insulation Using Thermography and the Finite Volume Method. FIBRES and TEXTILES in Eastern Europe, Vol. 24(6), 2016, 129-137.
Ramin H., Hanafizadeh P., Behabadi M.A.A.: Comparative study betweendynamic transient and degree-hours methods to estimate heating and coolingloads of building’s wall. JCAMECH, Vol. 46(2), 2015, 153-165.
Recktenwald G., W.: Finite-Difference Approximations to the Heat Equation[on-line] (available 2011-05-11), Mechanical Engineering Department PortlandState University, Portland, Oregon, United States 2011.
Sambou V., Lartigue B., Monchoux F., Adj M.: Thermal optimization of multilayered walls using genetic algorithms. Energy and Buildings, Vo. 41, No. 10(2009), 1031-1036.
Singhal U.M, Dixit R., Ary R.K.: Drying of Multilayer Polymeric Coatings, Part I: An Experimental Study. Drying Technology, Vol. 32, 2014, 1727-1740.
Sobieski W.: Possibility to use the diffusion equation to heat flow modellingin a composting process. Journal of Applied Computer Science, Vol. 19(2), 2011, 111-123.
Sobieski W.: Thermodynamics in experiments (in Polish). Olsztyn (Poland), 2015.
Sobieski W.: The basic equations of fluid mechanics in a form characteristicof the finite volume method. Technical Sciences, Vol. 14(2), 2011, 299-313.
Sobieski W.: The basic closures of fluid mechanics in a form characteristicf or the Finite Volume Method. Technical Sciences, Vol. 16(2), 2013, 93-107.
Sobieski W.: The application concept of the Finite Volume Method for modelling heat flow through multilayer structures with variable material parametersand different thickness (submitted).
Straube J.F., Ueno K., Schumacher C.J.: Measure Guidelines: Internal Insulation of Masonry Walls. Building Science Corporation, Somerville, Massachusetts, United States, July 2012.
De Sterek H., Ullrich P.: Introduction to computational PDES. Department of Applied Mathematics, University of Waterloo, 2009.
Tamene Y., Abboudi S., Bougriou C.: Numerical and Economical Study of Thermal Insulation in Multilayer Wall Exposed to Real Climatic Conditions. Athens Journal of Technology Engineering, Vol. 1(2), 2014, 137-148.
Tesch K., Collins M.W., Karayiannis T.G., Atherton M.A., Edwardsc P.: Heatand Mass Transfer in Air-Fed Pressurised Suits. Applied Thermal Engineering, Vol. 29(7), May 2009, 1375-1382.
Tsilingiris P.T.: Parametric space distribution effects of wall heat capacity andthermal resistance on the dynamic thermal of walls and structures. Energy and Buildings. Vol. 38 (2006), 1200-1211.
Tveito A., Langtangen H.P., Nielsen B.F., Cai X.: Elements of Scientific Computing. Computational Science and Engineering, Vol. 7. Springer-Verlag Berlin Heidelberg 2010.
Wongpanyo W., Charoensawan P., Rakwichian W., Seetapan P.: Improving Heat Transfer Performance of Concrete Thermal Energy Storage with Use of Local Material. International Journal of Renewable Energy, Vol. 3(2), 2008, 15-26.
Vírseda P., Pinazo J.M.: Thermal behaviour in multilayer products by response factors. Journal of Food Engineering, Vol. 33(3-4), 1997, 347-358.
Verma S.; Kulkarni V. S., Deshmukh K. C.: Finite element solution to transient asymmetric heat conduction in multilayer annulus. International Journal of Advances in Applied Mathematics and Mechanics, Vol. 2(3), 2015, 119-125.
Verma S., Kulkarni V. S., Waghmare G. L.: Heat conduction in multilayercomposite circular rod. Pure and Applied Mathematics Letters, Volume 2015, 8-12.
Yuan J., Ren R., Sundén B.: Analysis of chemical-reaction-coupled mass and heat transport phenomena in a methane reformer duct for PEMFCs. International Journal of Heat and Mass Transfer, Vol. 50(3-4), 2007, 687-701.