CC BY
5
SRSH73.31.09
https://doi.org/10.48081/JFBH2786
*V. S. Yessaulkov
Toraighyrov University, Republic ofKazakhstan, Pavlodar *e-mail: yessaulkov.val@gmail.com
ON CHOOSING APPROPRIATE EQUATIONS FOR MATHENATICAL MODELING OF PROCESSES IN PCM-BASED ENERGY STORAGE SYSTEMS FOR VEHICLES
With regards to storing thermal energy, latent heat storage are the matter of growing attention over the past years due to their straightforward design, affordable production and maintenance costs, as well as universal applicability. The phase-change materials (PCM) are well-known for their widespread use in aforementioned systems primarily because of their high thermal storage density. Many studies concerning analysing and optimising the design of latent heat storage systems (LHHS) have been carried out in the last years. In relatively recent researches, different research teams have investigated different types of LHHS, using mathematical models of different levels of complexity.
The wide variety of different approaches to the mathematical modeling (including different methods, algorithms and applications from different field of pure and applied research) sometimes make it challenging to choose and implement a proper system of equations and criteria -without excessive complicacies.
This very paper describes and elaborates on the system of equations that can and -will be applied to the mathematical modeling of charging/recharging process in PCM-based LHHS for commercial vehicles. After a brief review of previous models, author presents the model and asses its accuracy and possibilities for further development and exploration.
Keywords: phase-change materials, mathematical modeling, latent heat, energy storage systems, thermal efficiency, non-stationery heat conduction, computational stability.
Introduction
Latent heat storage is appealing as it provides a high energy storage density and latent heat storage capacity at a constant temperature matching the phase transition temperature of PCM. Some of the most important criteria for choosing PCM are melting point, latent heat and volumetric latent heat storage capacity, since they restrict the parameters and size of the system.
The objective of this study is to develop a digital heat transfer model for predicting thermal performance of a PCM-based LHTES (latent heat thermal energy storage system) and to better understand the thermal behavior of the PCM.
A large number of publications on PCM research describing modeling and/or experiments are available in the current literature.
Analysis of the models is quite important in the development of a proper understanding of thermodynamics behavior of PCM-based energy storage systems as it takes into account lost heat and temperature in storage applications [1]. A properly performed analysis results in optimal functioning of the thermal system. The study of the problems of heat transfer in fusion and solidification processes is particularly complex due to the fact that the solid-liquid boundary shifts depending on how fast latent heat is absorbed or lost at the aforementioned boundary [2].
Materials and methods
A model of polyproplyene PCM (mixture of sodium sulphate, sodium chloride and water) tube in annular air flow was developed by Kurklu et al [3]. The model was based on the energy balance or the concept of energy conservation. It was found that the thermal properties of the PCM tube in this study had a significant effect on heat transfer, since they had a higher thermal resistance than the PCM, due to its lower thermal conductivity. Their model suggested the possibility of removing some complex assumptions such as the change of thermal properties of the PCM with temperature and the existence of convection in the liquid phase maintaining an acceptable level of accuracy and compatibility with the experimental results. The base energy equation was used for a check volume around an inner node after dividing the tube into five volumes and twenty-five units of equal length.
Model by Hamdan et Elwerr [4] considered thermal energy storage using a PCM in the rectangular enclosure the sides of which are were insulated, with the exception of the left vertical side where heat was supplied. The enclosure contained a pure PCM (n-octadecane) which was assumed to be initially in solid phase and at an initial temperature. Later in the melting process, the experimental data diverged due to the assumption that the inclination angle 9 was independent of the altitude y, and due to the assumption that the walls of the enclosure were all adiabatic, except the heated one.
In the general model for heat charging and discharging processes of various LHHS having encapsulated phase change materials developed by Zhang [5] for given conditions, the system thermal behavior is shown to depend upon dimensionless solid
PCM volume (Vp) of the capsule which in turn depends on Fourier number (FQ) and dimensionless capsule surface (Qs). The Fourier number was defined by the equation (1)
Fo=~ir d)
Lc
where ap is thermal diffusivity, m2 • s1; t is time, s;
lc is the characteristic length of a PCM capsule, m [6].
The thermal efficiency of the PCM component thus determined the thermal efficiency of the whole system. Thus, LHTES thermal performance analysis could be
simplified to problem of determining of Vp(Fol 0S) for a capsule for the given operating conditions. That model was not limited to a specific system or to a specific PCM.
Hed and Bellander [7] in their model of a PCM air heat exchanger introduced a Active heat transfer coefficient. That coefficient included aspects of the geometry and the airflow in the heat exchanger as well as the material properties of the PCM. The PCM temperature and the external temperature were calculated both for rough and smooth surfaces.
Within the model designed to depict the physical system with ongoing processes the following assumptions were used:
- heat transfer is modeled irrespective of the heat source and final heat receiver;
- heat transfer is considered to occur in the ideal displacement mode;
- temperature in any grid is equal;
- inlet velocity and temperature of water are constant and its flow is laminar;
- initial temperature of the PCM is uniform, the PCM itself is in the solid phase;
- thermophysical properties of water, the tube wall and the PCM are not subject to change during the process.
The PCM has a transition temperature where it during an increase in temperature and during a decrease in temperature will solidify. During melting process, the previously solid PCM in an enclosure sinks downwards or floats upwards, depending on the gravitational force, so that buoyancy is a result of differences in solid and liquid densities.
There are two key properties that should be known to the PCM equipment designer:
- heat conductivity and its variation with the temperature Xp(T);
- heat capacity and its variation with the temperature cp(T). (G. Hed, 2006)
The follow thermal properties of paraffin wax (acting PCM) where used to ensure computational stability:
- melting point 324.15K (51 °C);
-latentheat 168 kJ kg1;
- thermal conductivity 0.22 W (m- K)"1 [8];
- specific heat 21kJ (kg- K)"1 [9];
-density 900 kg -m3 [10].
Results and discussion
Energy balance equation:
dE dt
— = Qg- Qt> (2)
where E is the energy of the system, J;
Qg is the heat generated in the system, W;
Qt is the heat transferred outside the system, determined by the boundary conditions, W.
Heat release equation:
Qn = mCn—, P dr.'
(3)
where m is the mass of the system, kg;
C is the specific heat capacity of the system, J - (kg - K)"1;
dT is the change of temperature, K. Boundary condition equations should be as follows:
(4)
Qt=~dT,
kA
where h con is the coefficient of convective heat transfer, W- (ml- K)"1; S is the surface area, m2; T is the ambient temperature, K; T is the transition temperature of the system, K; k is the coefficient of thermal conductivity, W (m • K)"1; A is the length of the cross section, m; e is the thickness, m.
The general equation for heat generation and removal in thermal elements is based on the first equation. Thus, the volumetric heat release in the cell non-stationary sums the reversible and irreversible heat. It is possible to try to describe the heat transfer of systems with phase change materials by adding a term to the right side of the equation, taking into account heat generation and latent heat in phase change materials:
One of the most widely used methods for solving phase transition problems is the enthalpy method. The purpose of this method is to solve the energy (temperature field) equation for the solid and liquid regions in one equation. With the introduction of the enthalpy method, the constitutive equation, expressed in terms of enthalpy, allows for an interface condition that is automatically satisfied at the solid-liquid interface, as well as the creation of a softened zone between the two phases. This implies a much simpler phase transition problem. As a consequence, the crux of this fixed-mesh method is to attempt to create a mesh to solve this softened zone.
The energy conservation during a phase transition (melting or solidification) is determined by the equation:
(6)
where T is the temperature of the system, K ;
H is the total volumetric enthalpy, J - kg1.
Enthalpy is the sum of latent and apparent heats and is written by the equation:
H(T) = h(T) + pif(T)A, (7)
Гт
h
pkckdT, (8)
jtl.
where A. is the latent heat of fusion, J - kg1;
p is the phase density of the material with the phase transition, kg - m3.
H depends on the temperature of the phase change material:
H = f pscsdT, T < Tm (solid)
H — Pif(T)À, T — Tm (melting) (9)
i T
[н = L Pi^idT + p[À, T > Tm(liquid) After solving the equations (9), we obtain:
H
Трем = Tm +---, when 0 < H
PpcmlPCM
Трем = Tm> when о <H <pA (10)
H — Ррсм^
Трем = Тт +-г-- when 0 > Я
Ppcmlpcm
It follows from the above equations that if a phase change material liquefies, the total volumetric enthalpy is equal to the latent and sensible heat. If the phase change material is solid, the enthalpy is zero. Then we can write an alternative equation for two-dimensional heat transfer in a material with a phase transition: [12]
Sh S ( Sh\ S ( Sh\ Sf
Conclusions
The equation of non-stationary heat conduction must be solved in both phases to estimate the temperature distribution and the position of the interface during the phase transition. Analytical solutions are limited by simple boundary conditions, while the numerical solutions available for a two-dimensional model provide detailed information not only on the temperature distribution in the radial and axial positions in a cylindrical capsule, but also on the convection flow in the melt layer. It is possible to simplify heat transfer with a phase change in a vertical cylinder using a one-dimensional model and using effective thermal conductivity, which takes into account the influence of natural convection in the melt layer.
To study the thermal behavior of the engine and thermal management systems, there are several methods, including experimental and numerical simulation. The most popular approaches for studying phase transition materials in thermal systems are computational fluid dynamics and the finite element method. These are advanced methods of mathematical and numerical modeling. In these models of the heat transfer mechanism in phase change materials, the numerical solution equations are taken directly from reference books.
Such a model could be used for similar applications as long as the required changes are made to the model.
REFERENCES
1 Verma P. et al. Review of mathematical modeling on latent heat thermal energy storage systems using phase-change material //Renewable and sustainable energy reviews. - 2008. - T. 12. - №. 4. - P. 999-1031.
2 Goto T., Suzuki M. Analysis of transient heat conduction with phase changes by a boundary integral equation method //Nuclear engineering and design. - 1996. -T. 162. - №. 2-3. - P. 317-324.
3 Kurklu A., Wheldon A., Hadley P. Mathematical modelling of the thermal performance of a phase-change material (PCM) store: cooling cycle //Applied Thermal Engineering. - 1996.-T. 16.-№. 7.-P. 613-623.
4 Hamdan M. A., Elwerr F. A. Thermal energy storage using a phase change material //Solar Energy. - 1996. - T. 56. - №. 2. - P. 183-189.
5 Zhang Y. et al. A general model for analyzing the thermal performance of the heat charging and discharging processes of latent heat thermal energy storage systems //J. Sol. Energy Eng. - 2001. - T. 123. -№.3.-P. 232-236.
6 Dhanaraj G. et al. (ed.). Springer handbook of crystal growth. - Berlin: Springer, 2010.-T.2.
7 Hed G., Bellander R. Mathematical modelling of PCM air heat exchanger II Energy and Buildings. - 2006. - T. 38. - №. 2. - P. 82-89.
8 Sari A., Karaipekli A. Thermal conductivity and latent heat thermal energy storage characteristics of paraffin/expanded graphite composite as phase change material //Applied thermal engineering. - 2007. - T. 27. - №. 8-9. - P. 1271-1277.
9 Trigui A. et al. Development and characterization of composite phase change material: thermal conductivity and latent heat thermal energy storage //Composites Part B: Engineering. -2013.-T. 49.-P. 22-35.
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Material received on 01.06.23.
*В. С. Есаулкое Торайгыров университет, Казахстан Республикасы, Павлодар к. Материал 01.06.23 баспага туст1.
АВТОКЭЛ1К УШШ ФАЗАЛЬЩ АУЫСУЫ БАР МАТЕРИАЛДАР НЕГ131НДЕ ЭНЕРГИЯ АККУМУЛЯТОРЛАРЫНДАЕЫ ПРОЦЕСТЕРД1 МАТЕМАТИКАЛЬЩ МОДЕЛЬДЕУ УШШ ТЕНДЕУЛЕРД1 ЖАСАУ МЭСЕЛЕС1 БОЙЫНША
Жылу энергиясын сацтауга келетт болсац, жасырын жылуды сацтау олардыц царапайым дизайны, цол жетгмдг энЫр1с жэне техникалъщ цызмет корсету шыгындары, сондай-ац эмбебап цолдану мумктдггте байланысты соцгы жылдары ост келе жатцанмэселе болып табылады. Фазалъщ ауыспалы материалдар жогарыда аталган жуйелерде кецтен цолданылуымен, ец алдымен, жогары жылу сацтау тыгыздыгына байланысты белгш. Соцгы жылдары жасырын жылу сацтау жуйелертщ цурылымын талдау жэне оцтайландыру бойынша коптеген зерттеулер журггзшдг. Салыстырмалы турде жацында жург1з1лген зерттеулерде эртурлг зерттеу топтары эртурл1 курделшк децгейлертщ математикалыц улгшерт пайдалана отырып, жасырын жылу сацтау жуйелертщ эртурлг турлерт зерттедг.
Математикалыц моделъдеудщ эртурлг тэсшдертщ алуан турлшгг (соныц ¡штде таза жэне цолданбалы зерттеулердщ эртурлг салаларындагы эртурл1 эдгстер, алгоритмдер жэне цолданулар) кейде шамадан тыс циындыцтарсыз тецдеулер мен критерийлердщ дурыс жуйест тацдауды жэне енггзудг циындатады.
Бул мацала коммерциялыц колжтерге арналган аазалыц ауыспалы материалдар неггзтдегг жасырын жылу сацтау жуйелертщ жуйестде цуаттау жэне асыра цуаттау процест математикалыц модельдеуге цолданылатын жэне цолданылатын тецдеулер жуйест сипаттайды жэне эзгрлейдг. Алдыцгы улгыерге цысцаша шолу жасаганнан кейт автор модельдг усынып, оныц дэлдт мен одан эрг дамыту жэне барлау мумктджтерт багалайды.
Кшттг создер: фазалыц ауыспалы материалдар, математикалыц моделъдеу, жасырын жылу, энергия сацтау жуйелерг, жылу тшмдшш, стационарлыц емес жылу откгзггштж, есептеу турацтылыгы.
*В. С. Есаулкое Торайгыров университет, Республика Казахстан, г. Павлодар. Материал поступил в редакцию 01.06.23.
К ВОПРОСУ СОСТАВЛЕНИЯ УРАВНЕНИЙ ДЛЯ МАТЕМАТИЧЕСКОГО МОДЕЛИРОВАНИЯ ПРОЦЕССОВ В АККУМУЛЯТОРАХ ЭНЕРГИИ НА ОСНОВЕ МАТЕРИАЛОВ С ФАЗОВЫМ ПЕРЕХОДОМ ДЛЯ АВТОМОБИЛЕЙ
Системы аккумулирования скрытого тепла в последние годы привлекает все большее внимание из-за простоты конструкции, относительно невысоких затрат на производство и техническое обслуживание, а также универсальности применения. Материалы с фазовым переходом широко используются в вышеупомянутых системах, в первую очередь из-за их высокой плотности накопления тепла. В последние годы было проведено множество исследований, касающихся анализа и оптимизации конструкции систем хранения и аккумулирования скрытого тепла. В относительно недавних исследованиях различные группы рассматривали некоторые типы подобных систем, используя математические модели разного уровня сложности.
Большое разнообразие подходов к математическому моделированию (включая различные методы, алгоритмы и приложения из разных областей научных и прикладных исследований) иногда затрудняет выбор и реализацию правильной системы уравнений и критериев.
В статье описывается и разрабатывается система уравнений, которая может и будет применяться для математического моделирования процесса зарядки и разрядки в системе накопления скрытой теплоты на основе материала, подвергающего фазовому переходу, для коммерческих автомобилей. После краткого обзора предыдущих моделей автор представляет модель и оценивает ее точность и возможности для дальнейшего развития и исследования.
Ключевые слова: материалы с фазовым переходом, математическое моделирование, скрытая теплота, системы накопления энергии, тепловой КПД, нестационарная теплопроводность, расчетная устойчивость.
