International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 12, December 2017, pp. 890 898, Article ID: IJMET_08_12_096 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=12 ISSN Print: 0976-640 and ISSN Online: 0976-659 IAEME Publication Scopus Indexed HEAT BALANCE INTEGRAL METHOD FOR CYLINDRICAL AND SPHERICAL ENCAPSULATED PHASE CHANGE THERMAL ENERGY STORAGE SYSTEM Mayank Srivastava Department of Mechanical Engineering, NIT Jamshedpur, India Dr. M.K.Sinha Department of Mechanical Engineering, NIT Jamshedpur, India ABSTRACT Heat balance integral method (HBIM) is used to analyse interface position and temperature variation of phase change thermal energy storage system with constant heat flux for melting process. The phase change process is mainly governed by conduction. Phase change material is kept inside a cylindrical and spherical cavity under fixed temperature boundary condition, applied only on one wall. All other walls are thermally insulated. The numerical results are obtained for melting of solid initially at its fusion temperature by using Matlab. Keywords: Conduction, HBIM, Interface position, Melting, Phase change materials, Stefan s number. Cite this Article: Mayank Srivastava and Dr. M.K.Sinha, Heat Balance Integral Method for Cylindrical and Spherical Encapsulated Phase Change Thermal Energy Storage System, International Journal of Mechanical Engineering and Technology 8(12), 2017, pp. 890 898. http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=12 1. INTRODUCTION Thermal energy storage (TES) using phase change materials (PCM) have been main area of research in the last two decades and more. Continuous depletion of non-renewable energy resources led to raise serious issues on their future availability and create a huge difference between supply and demand of energy. This problem can be solved by either using the renewable energy like solar, wind, tidal etc. or by reusing the waste energy. Renewable energy is the most promising, clean and safe source of energy. The main problem associated with the use of renewable energy is its oscillating nature e.g. solar energy is only available at daytime and to utilize waste heat, we need a device which can store waste energy at a faster rate and release heat on demand hours. During peak hours the consumption of energy is http://www.iaeme.com/ijmet/index.asp 890 editor@iaeme.com
Heat Balance Integral Method for Cylindrical and Spherical Encapsulated Phase Change Thermal Energy Storage System higher than its generation and at night energy generation is higher than its consumption, this excess heat at night can be stored in a suitable energy storage system and may use in many industrial & domestic purpose. Both the problem can be solved by using thermal energy storage system. Likewise any other conventional energy storage system (electrical energy storage - battery, mechanical energy storage - flywheel), thermal energy storage system is used to store energy as sensible heat, latent heat, and thermo-chemical heat or combination of these, as a change in internal energy of a phase change material. In Sensible TES, thermal energy is stored by changing the temperature of storage medium while Latent TES is stored energy available during phase change. Phase change materials (PCMs) for TES are materials which provide a suitable way of charging and discharging of thermal energy due to their large heat storage capacity at constant temperature. PCM are broadly classified into main categories: organic, inorganic and eutectic [1]. Selection of PCM is basically based on their melting temperature & applications. Materials, which have melting temperature below 15 C are used for cooling purpose while materials which melts above 90 C are used for absorption refrigeration. Nomenclature C = Heat capacity T m = melting temperature, K k = constant, k=1,2 for cylinder & sphere respectively T s = surface temperature, K K = thermal conductivity, W/m K α = thermal diffusivity, m 2 /s L = latent heat, J/Kg Q τ = non-dimensional total heat non-dimensional radial distance of ƞ = absorbed phase front, δ/r i R = radius, m ƞ time rate of non-dimensional radial = distance of phase front, d ƞ/d τ r i = inner radius, m ξ = non-dimensional radial distance within phase change, r/r i S t = Stefan number, = density, Kg/m non-dimensional temperature, t = time, s θ = T = temperature, K τ = non-dimensional time, All other materials that melt in between these two temperatures are used in heating purpose [2]. Encapsulation of PCM is the technique used to keep the PCM in a container to avoid any direct contact between the PCM and HTF. To ensure long-term thermal performance of any PCM system, the size and shape of the PCM container must correspond to the melting time of the PCM. Two geometries commonly employed as PCM containers are the rectangular and cylindrical containers []. A number of experimental and analytical investigations have been carried out by the researchers on phase change thermal energy storage system of different geometry under different boundary conditions. A phase change problem has to be solved separately due to the non-linear nature of the problem. There is wide range of different kinds of methods available for solving phase change problems. The most common methods are the heat balance integral method [7, 1, 16], enthalpy method [1, 16] and finite difference methods [12]. G.E.Bell [4] has numerically solved melting and freezing problem by HBIM and improved the accuracy of http://www.iaeme.com/ijmet/index.asp 891 editor@iaeme.com
Mayank Srivastava and Dr. M.K.Sinha HBIM by subdividing dependent variable temperature into equal intervals. A system of first order, non-linear differential equations is produced to calculate the position of each isotherm. G.Poots [5] investigated the location and time history of the one dimensional solid-liquid interface during the inward solidification of semi-infinite region, circular cylinder and sphere by approximate integral method which is similar to solution of boundary layer equations. Semi-infinite solid of constant cross section area under 1-Dimensional convective heating was numerically analysed by Anant Prasad [6] using Biot variational method to calculate temperature build-up, thermal penetration depth and melt fraction. Figure 1 Thermal Energy Storage System James Caldwell & C.K.Chiu [7] found that in spherical case the solidification front moves slightly slower and a smooth solution for solidification front can be obtained by using the front-tracking method. Spherical outward melting of PCM in radial direction was investigated by A.Kumar et al. [8] by employing Variational, Integral and Quasi-steady method. He-Sheng Ren [9] obtained the solutions of the one-dimensional inward solidification in Cartesian and spherical encapsulation problem by heat balance integral method. A.Prasad et al. [10] numerically investigated melting phenomena of PCM encapsulated in a rectangular container under radiative heat injection by Biot variational method, HBIM & Quasi-static method. The result obtained with all these methods was almost same. R.K.Sharma et al. [11] provided the solution of a two-dimensional solidification problem in isosceles trapezoidal cavity by using CFD software. The heat transfer mainly occurs due to the conduction. K.Morgan [12] used explicit finite difference method to solve freezing and melting phenomenon in a cylindrical thermal cavity. The method takes into account the combined effect of conduction and convection heat transfer. The method used was validated by comparing the result with the experimental result. The accuracy of heat balance integral method has been considerably improved by successive sub division of the dependent variable [4, 7] and/or by choosing suitable temperature profile like cubic [6, 9], exponential, quadratic [1] profile. A.S.Wood [1] gave six alternative ways to develop the original quadratic HBI solutions to solve melting of a semi-infinite slab, initially at its melting temperature by HBIM. A new method has been developed by T.G.Myers [14] to find out the best possible value of power of highest order term in the approximating function used in HBIM by minimizing the square of the difference of the terms in the heat equations. http://www.iaeme.com/ijmet/index.asp 892 editor@iaeme.com
Heat Balance Integral Method for Cylindrical and Spherical Encapsulated Phase Change Thermal Energy Storage System R.S.Gupta and Dhirendra Kumar [15] used modified variable time step method to obtain numerical results for solidification of a liquid initially at its fusion temperature. The result obtains for the movement of the interface & temperature distribution was compared with the result found by HBIM. PCM Solidification by enthalpy method & HBIM over a wide range of the Stefan number except for very small value of Stefan number were numerically analysed by James Caldwell and Ching-chuen chan [16]. Anica Trp [17] numerically and experimentally analysed the shell and tube type latent heat thermal energy storage system during melting and solidification process to evaluate the heat transfer during process. The experimental energy and exergy analysis during the charging process has been reported by Dinu G Thomas, Sajith babu et al. [18] by using sodium thiosulfate pentahydrate as phase change material. Piia Lamberg, Reijo Lehtiniemi et al. [19] numerically & experimentally investigated the PCM storage devices with and without heat transfer enhancement structure by enthalpy method and heat capacity method. The result obtained with numerical analysis is compared with the experimental results. Stetislav Savovic et al. [20] applied finite difference approach to analyse the one-dimensional Stefan problem with periodic fixed boundary condition. In the present paper, the behaviour of PCM melting, in a spherical and cylindrical encapsulation under dirchilet boundary condition has been numerically analysed by using MATLAB. 2. MATHEMATICAL MODELLING A schematic drawing of a spherical and cylindrical vessel containing solid PCM used in the present study is shown in fig.2. For simplicity, we consider a single phase 1-dimensional problem of melting of PCM inside a spherical and cylindrical capsule, whereas practical melting problem is rarely one dimensional, initial and boundary conditions are always complex. Initially, PCM is at its melting temperature T m. The temperature at the vessel boundary T s is higher than the PCM melting temperature T m. Since the term moving boundary problems is associated with time-dependent boundary problems, also referred as Stefan problems, where the position of the moving boundary must be determined as a function of time and space. As the time passes solid PCM will melt due to the boundary temperature applied at vessel and the governing equations for this process may be described by: = Liquid Solid (1) Solid-Liquid Interface Figure 2 Schematic drawing of encapsulated PCM undergoing outward melting http://www.iaeme.com/ijmet/index.asp 89 editor@iaeme.com
Mayank Srivastava and Dr. M.K.Sinha Where k = 1 for cylindrical vessel k=2 for spherical vessel Boundary Conditions: r = r i, t > 0, T = T s (1) r = δ, t > 0, T = T m (2) r r i, t = 0, T = T m () Energy balance at the solid-liquid interface: = To reduce dependent variables we introduce the non-dimensional variables: = = δ $ = %& r # ' ( = ) ) * ) + ) *, = - ) + ) * Now governing equations (1) becomes, /... 0 (5) Boundary Conditions becomes, = 1, $ > 0, ( = 1 (6) =, $ > 0, ( = 0 (7) 1, $ = 0, ( = 0 (8) Now Energy balance at the solid-liquid interface, /. = /.. 0 Heat Balance Integral Method (HBIM): 1. /. / 2 = 1 2 0 41 (25 ( 0.6 = /..6 /..6 2.1. Interface location Now assume a suitable linear temperature profile with negligible temperature drop within the solid layer, which satisfies the boundary conditions: θ = 1 9 : ; : = (12) < Substituting eq. (1) into eq. (12) leads to For spherical geometry (k=2), > '? @ A, + ' = $ = C D ' @ + E ' log I > J K @ A (14) For cylindrical geometry (k=1), > '?J K ' A = (4) (9) (10) (11) (1) (15) http://www.iaeme.com/ijmet/index.asp 894 editor@iaeme.com
Heat Balance Integral Method for Cylindrical and Spherical Encapsulated Phase Change Thermal Energy Storage System 2.2. Heat Transfer Analysis X Q = 1 Latent heat + Sensible heatdξ For sphere X Q Z = XC + 1 ξ ' θ dξ (16) [ \ Substituting eq. (1) into eq. (16) leads to ^0 = C + D 1 > J K A>C A (17) D ' For cylinder ^0 = + 2 1 ( 2 (18) J K Substituting eq. (1) into eq. (18) leads to ^0 = C + D 1 > J K A>C D A (19) ' With the rise of high-speed digital computers, mathematical modelling and computer simulation often become the most economical and fastest approaches to provide a broad understanding of the practical processes involving the moving boundary problems.. RESULTS AND DISCUSSIONS In this part we present the numerical results obtained by using heat balance integral method in cylindrical and spherical melting process of phase change materials used in thermal energy storage. Figure Behaviour of the non-dimensional interface location with non-dimensional time, Stefan s number is taken as a parameter. In Fig., the plot shows the variation of interface location/depth with time for different values of Stefan s number. For each values of Stefan number during initial time, interface depth rapidly increases with time and as the time passes, interface depth becomes almost linear with time. This happens because during starting time heat transfer rate increases more sharply. The solution obtained for present problem by using HBIM depends upon Stefan s number. It is observed that at any time instant as St decreases interface depth increases. http://www.iaeme.com/ijmet/index.asp 895 editor@iaeme.com
Mayank Srivastava and Dr. M.K.Sinha Figure 4 Non-dimensional heat transfer with non-dimensional time, Stefan s number is taken as a parameter Fig.4 shows the heat transfer variation with time in spherical and cylindrical melting on different values of Stefan s number. During initial time of melting because of large temperature difference available between two phases, the heat transfer increases more rapidly. But as the time passes, heat transfer becomes linear with time. This happens because temperature difference between two phases also decreases. This means that, except the initial time interval, the heat transfer rate is very slow. The time taken to absorb a fixed amount of heat in spherical melting process is lower than that of cylindrical melting process. Figure 5 Behaviour of the non-dimensional rate interface position with non-dimensional time, Stefan s number is taken as a parameter The rate of interface position is a function of Stefan s number (Eq.14. 16). The plot between rate of interface position and time is almost a single curve for all values of Stena s number (Fig.5). Rate of interface position decreases with time for all values of Stefan s number. http://www.iaeme.com/ijmet/index.asp 896 editor@iaeme.com
Heat Balance Integral Method for Cylindrical and Spherical Encapsulated Phase Change Thermal Energy Storage System 4. CONCLUSIONS The melting phenomenon in Cylindrical and Spherical encapsulated thermal energy system with phase change materials at fixed temperature heat transfer is investigated. In present work, only HBIM is used for single phase change models in cylindrical and spherical melting. This method can be further applied on more complicated and realistic melting/solidification problems. Heat transfer between different phases occurs only by conduction. The effect of convection and radiation is neglected. REFERENCE [1] Belen Zalba, Jose Ma Marın, Luisa F. Cabeza, Harald Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, Applied Thermal Engineering, 2, 200, 251 28. [2] Mohammed M. Farid, Amar M. Khudhair, Siddique Ali K. Razack, Said Al-Hallaj, A review on phase change energy storage: materials and applications, Energy Conversion and Management, 45, 2004, 1597 1615. [] Francis Agyenim, Neil Hewitt, Philip Eames, Mervyn Smyth, A review of materials, heat transfer and phase change problem formulation for latent heat thermal energy storage systems (LHTESS), Renewable and Sustainable Energy Reviews, 14, 2010, 615 628. [4] G.E.Bell, A refinement of the heat balance integral method applied to a melting problem, Int. J. Heat Mass Transfer, 21, 1978, 157-162. [5] G.Poots, On the application of integral methods to the solution of the problem involving the solidification of liquids initially at fusion temperature, Int. J. Heat Mass Transfer, 15, 1962, 525-51. [6] Anant Prasad, Melting of solid bodies due to convective heating with the removal of melt, American Institute of Aeronautics and Astronautics, Inc., 16, 1979, 445-448. [7] J.Caldwell and C.K.Chiu, Numerical solution of one phase Stefan problems bt the heat balance integral method, Past-I cylindrical and spherical geometries, Communications in Numerical Methods in Engineering, 16, 2000, 569-58. [8] A.Kumar, A.Prasad, S.N.Upadhaya, Spherical Phase change energy storage with constant temperature heat injection, Journal of Energy Resources Technology, 109, 1987, 101-104. [9] He-Sheng Ren, Application of the heat balance integral to an inverse Stefan problem, Int. J. of Thermal Sciences, 46, 2007, 118-127. [10] A.Prasad, S.P.Singh, Conduction controlled phase change energy storage with radiative heat addition, Transactions of the ASME, 116, 1994, 218-22. [11] R.K.Sharma, P.Ganesan, J.N.Sahu, H.S.C.Metselaar, T.M.I Mahila, Numerical study for enhancement of solidification of phase change materials using trapezoidal cavity, Int. J. of Powder Technology, 268, 2014, 8-47. [12] K.Morgan, A numerical analysis of freezing and melting with convection, Computer Methods in Applied Mechanics and Engineering, 28, 1981, 275-284. [1] A.S.Wood, A new look at the heat balance integral method, Applied Mathematical Modelling, 25, 2001, 818-824. [14] T.G.Myers, S.L.Mitchell, G.Muchatibaya, M.Y.Myers, Acubic heat balance integral method for one dimensional melting of a finite thickness layer, Int. J. Heat Mass Transfer, 50, 2007, 505-517. [15] R.S.Gupta, Dhirendra Kumar, Variable time step methods for one-dimensional Stefan problem with mixed boundary condition, Int. J. Heat Mass Transfer, 24, 1981, 251-259. [16] James Caldwell and Ching-Chuen Chan, Spherical solidification by the enthalpy method and the heat balance integral method, Applied Mathematical Modelling, 24, 2000, 45-5. http://www.iaeme.com/ijmet/index.asp 897 editor@iaeme.com
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