Thermal Storage Performance of a Shell and Tube Phase Change Heat Storage Unit with Different Thermophysical Parameters of the Phase Change Material

Abstract

The thermal storage performance of shell and tube phase change heat storage units is greatly influenced by the thermophysical parameters of the phase change material (PCM). Therefore, we use numerical simulations to examine how the thermal storage capability of shell and tube phase change heat storage units is affected by thermophysical parameters such as specific heat capacity, thermal conductivity, and latent heat of phase change. The findings indicate that while the rate of temperature increase and the rate of the PCM melting both slow down as specific heat capacity increases, the overall heat storage increases. Within the specified range of parameters, the average rate of heat storage increases by approximately 4% for every 50% increase in specific heat capacity. The PCM’s rate of temperature rise slows down and its overall heat storage capacity rises throughout the middle stage of the phase change heat storage process as the latent heat of phase change grows. The average heat storage rate increases by approximately 6% and 22% for every 50% increase in latent heat and thermal conductivity, respectively. Notably, when the thermal conductivity is enhanced by a factor of 1.5, the average heat storage rate experiences an almost 50% increase. The thermal conductivity of the PCM has a negligible impact on total heat storage. The choice and use of the PCM in shell and tube phase change heat storage heat exchangers has a theoretical and empirical foundation thanks to this work.

1. Introduction

The exponential progress of human civilization has given rise to an escalating need for energy, whereas conventional fossil energy resources such as oil, natural gas, and coal, which are presently consumed in substantial volumes, are finite and unsustainable. The overuse of conventional fossil fuels will undoubtedly result in contamination of the environment and global warming, which will threaten ecological security and energy supplies worldwide. Vigorous developments, uses of new energy resources, and improved energy utilization have become important initiatives to solve this series of problems. However, in the field of new energy exploitation and development, there is a fundamental paradox in the temporal and spatial aspects of energy production and demand. In addition, various sectors, including industry and daily life, face the challenge of low efficiency in energy conversion and use. There is an urgent need to address these pressing issues [1]. The discrepancy between energy production and consumption can be effectively resolved and the energy structure can be optimized with the use of energy storage technology. Thermal energy storage (TES) stands out as a prominent large-scale energy storage technology with extensive future prospects amidst the array of energy storage technologies available [2].

At present, there are three primary categories of thermal energy storage techniques: sensible heat thermal energy storage [3,4], latent heat thermal energy storage (LHTES) [5,6,7], and thermochemical energy storage [8,9]. Among these methods, LHTES stands out due to its notable attributes, such as its capacity to store substantial amounts of energy while maintaining a consistently steady temperature during the process of phase change [10]. An extensively utilized industrial application of LHTES involves the implementation of a shell-and-tube heat exchanger (STHE) in combination with phase change materials (PCMs) [11].

The current focus of research in the field of STHEs revolves around optimizing the energy storage capabilities through the implementation of PCMs in the shell while allowing the heat transfer fluid (HTF) to circulate through the tubes. Enhancing the thermal conductivity of the PCM and incorporating fins into the design of the shell-and-tube heat exchanger are the primary areas of investigation aimed at boosting the properties of latent heat storage [12].

Paraffin, one of the more commonly used PCMs [13], is hampered in its thermal performance in the STHE and energy storage during melting due to its low thermal conductivity [14,15,16]. Therefore, scholars at home and abroad have further investigated paraffin waxes and found that adding nanoparticles [17,18], porous matrices [19], metal foams [20,21], and fins [22,23] to PCMs can both improve the overall thermal energy storage efficiency of the STHE to a certain extent and be an effective way to improve the heat transfer performance of paraffin waxes.

In the STHE, it has been found that the addition of fins on one side of the PCM is a very effective way to improve heat transfer. This method is unique because it is both efficient in heat transfer and reasonably priced [24]. The STHE for LHTES systems has been the subject of previous studies aimed at optimizing the design by employing different types of fins, including spiral fins [25], annular fins [26], corrugated fins [27], triple fins [28], pinned fins [29], and dendritic fins [30], in order to further improve the heat transfer performance of the STHE. A comparative analysis of the previous research results revealed that annular fins are the most effective fin forms for improving the heat transfer performance of the STHE [26,31]. In order to maximize the advantages of heat transmission and energy storage capabilities, Wu et al. suggested a unique shell-and-tube heat exchanger design that includes uneven fins. To assess the impacts of the layout, quantity, and overall length of fins with different lengths, they carried out numerical investigations. It has been demonstrated that the performance of heat transmission and energy storage can be further improved using fins of increasing length, with short fins at the fluid’s entrance and long fins at its outlet [32].

PCMs undergo a transition in phase at a consistent temperature due to the absorption or release of heat energy in the form of latent heat. It is crucial to remember, though, that the thermophysical characteristics of PCMs are a contributing factor to a number of restrictions on the thermal storage capabilities of STHEs. Because PCMs have limited thermal conductivity, thermal energy must be released and stored over an extended period of time, which reduces the energy storage device’s overall performance. Furthermore, because of the small amount of specific heat and latent heat of fusion, a larger volume is needed for the thermal energy storage device. Most of the previous studies have been on the qualitative analysis of the physical parameters of PCMs, including their thermal storage properties and techniques to enhance their specific heat, latent heat, and thermal conductivity [33,34,35]. Nevertheless, there is a limited number of quantitative studies available that examine the impact of the physical characteristics of PCMs on their thermal storage capabilities.

This paper conducts a numerical simulation examination of the resultant impact of thermophysical parameters, such as thermal conductivity, specific heat capacity, and latent heat of phase change, on the thermal storage performance of shell and tube phase change heat storage units. This study is based on the previously mentioned STHE model with unequal fins [32]. This study offers a foundation in theory and empirical evidence to assist in making informed decisions concerning the selection and implementation of PCMs within shell and tube phase change heat storage heat exchangers.

2. Methodology

2.1. Geometric Description

A physical model of the STHE with uneven fins, comprising two concentric cylinders measuring 400 mm in height, is illustrated in Figure 1. The exterior cylinder is constructed of Plexiglas and is 44 mm in inner diameter and 4 mm in thickness. The inner cylinder measures 15 mm in inner diameter and 2.5 mm in thickness, and it is made of copper. The copper tube is equipped with copper fins, which are evenly distributed along its outer surface at regular intervals of 12 mm. The thickness of the fins remains constant at 2 mm, while their length varies between 2 mm, 4 mm, and 6 mm. The annular space is entirely occupied by the PCM.

Table 1. Thermophysical parameters of the materials.

Material	Paraffin	Copper	Water	Perspex
Density (ρ)/(kg/m3)	785	8978	998	800
Specific heat (cp)/(J/(kg·K))	2850	381	4182	1900
Thermal conductivity (λ)(W/(m·K))	0.2(s)/0.1(l)	387.6	0.6	0.2
Dynamic viscosity (μ)(Pa·s)	3.65 × 10−3	-	1.003 × 10−3	-
Latent heat (r)/(J/kg)	102,100	-	-	-
Solidus/Liquidus temperature (K)	323/328	-	-	-

displays the thermophysical parameters for paraffin, water, copper, and perspex.

2.2. Governing Equations

Because the shell-and-tube heat exchanger has an axisymmetric construction, the calculation is made simpler by turning the three-dimensional problem into a two-dimensional issue, which is then simulated using a two-dimensional axisymmetric model. In the meantime, the following basic assumptions have been established to simplify the computation and the discussion since the STHE operates using heat conduction, heat convection (natural convection, forced convection), and thermal radiation with phase change.

• The liquid PCM is a non-compressible fluid, and its flow is laminar flow;
• The thermal characteristics of the PCM are consistent, other than thermal conductivity, which varies with phase state and temperature;
• The Boussinesq approximation is employed to account for density fluctuations in the natural convection simulation of the PCM;
• Neglect the changes in volume of the PCM during phase change and the impact of viscous dissipation during flow;
• Neglect the thermal radiation during heat transfer.

Therefore, following the physical model assumptions discussed earlier, we can express the two-dimensional control equations of the latent thermal energy storage unit (LTESU) used for mathematical modeling in the following manner:

For the HTF:

Continuity equation:

$$\nabla \cdot \mathit{u}=0$$

(1)

Momentum equation:

$${\rho }_{\mathrm{HTF}}\frac{\partial \overrightarrow{u}}{\partial t}+{\rho }_{\mathrm{HTF}}\overrightarrow{u}\cdot \left(\nabla \cdot \overrightarrow{u}\right)=-\nabla p+{\mu }_{\mathrm{HTF}}{\nabla }^2\overrightarrow{u}$$

(2)

Energy equation:

$${\rho }_{\mathrm{HTF}}{c}_{\mathrm{HTF}}\frac{\partial T}{\partial t}+{\rho }_{\mathrm{HTF}}{c}_{\mathrm{HTF}}\overrightarrow{u}\cdot \nabla T=\nabla \cdot \left({\lambda }_{\mathrm{HTF}}\nabla T\right)$$

(3)

For the PCM:

Continuity equation:

$$\nabla \cdot \mathit{u}=0$$

(4)

Momentum equation:

$$\begin{array}{l}{\rho }_{pcm}\frac{\partial \mathit{u}}{\partial t}+{\rho }_{pcm}\mathit{u}\cdot \left(\nabla \cdot \mathit{u}\right)=-\nabla p+{\mu }_{pcm}{\nabla }^2\mathit{u}+{\rho }_{pcm}{\alpha }_v(T-T_m)\mathit{g}\\ +A_{mush}\frac{{\left(1-\beta \right)}^2}{{\beta }^3+\epsilon }\mathit{u}\end{array}$$

(5)

The CFD model incorporates the concept of a mushy zone as a porous medium using the enthalpy–porosity approach. It assumes that the liquid fraction of the PCM is consistent across all elements, where A_mush is 10⁵ kg/m³·s and $\epsilon$ is a tiny number (0.001), to avoid being divided by zero, and β is the melting fraction of the PCM. The linear function of this value with respect to the PCM temperature in the viscous zone is as follows:

$$\beta =\left\{\begin{array}{l}0,T<T_s\\ \frac{T-T_s}{T_l-T_s},T_s\le T\le T_l\\ 1,T>T_l\end{array}\right.$$

(6)

Energy equation:

$${\rho }_{pcm}\frac{\partial H}{\partial t}+{\rho }_{pcm}\nabla \cdot \left(\mathit{u}H\right)=\nabla \cdot \left({\lambda }_{pcm}\nabla T\right)$$

(7)

The solid zone:

$${\rho }_S{c}_S\frac{\partial T}{\partial t}=\nabla \cdot \left({\lambda }_S\nabla T\right)$$

(8)

where t represents time. The density is denoted by ρ, and λ stands for thermal conductivity. The coefficient of thermal expansion is denoted by α_v, and β represents the melting rate of the PCM. Meanwhile, the specific enthalpy of the PCM is denoted by H, and T stands for the temperature of the PCM. P stands for the pressure, while the dynamic viscosity is represented by the symbol μ. Velocity is denoted by u, and g stands for gravitational acceleration. Thermally conductive fluids are denoted by the subscript “HTF”, and PCMs are represented by the subscript “pcm”. In addition, the subscript “S” stands for solid materials.

The sum of the sensible and latent enthalpies results in the specific enthalpy, denoted as H:

$$H=h_{\mathrm{ref}}+{\displaystyle {\int }_{T_{\mathrm{ref}}}^T{c}_p\Delta T}+\beta L$$

(9)

2.3. Initial and Boundary Conditions

In the initial state, the entire STHE unit was subjected to a uniform temperature of 298 K. The left wall of the computational domain representing the HTF was presumed to have an axis of symmetry in order to set the conditions for the boundary. The velocity intake, as well as the pressure outlet, of the HTF domain were assigned to the top and bottom walls, respectively. The HTF, which had a temperature of 343 K, was fed into the inlet at a speed of 0.01 m/s. Because of its low Reynolds number (2300), laminar flow occurred inside the HTF. Both the top and bottom walls of both the HTF tube and PCM domain were assumed to be adiabatic. Thermally coupled conditions and the implementation of no-slip/penetration assumptions have been applied at the interface of the PCM and the fin, as well as the HTF and the inner tube wall. This ensures continuous thermal energy transport and facilitates the flow of liquid PCM and HTF, respectively. Furthermore, numerical analysis has been used to account for the convection of heat that exists between the STHE wall and the ambient air surrounding it. The environmental temperature was kept at 298 K, and the coefficient for convection heat transfer was fixed at 3 W/m².

2.4. Computational Strategies

The governing equations, which describe the unsteady heat transfer and flow involving phase change, are solved numerically using ANSYS FLUENT 21R1, a commercial computational fluid dynamics (CFD) software package. The finite volume method (FVM) is employed to calculate these equations.

A completely implicit schema is utilized for the formulation of the unsteady term. To address the phase change phenomenon of PCMs during the melting process, the enthalpy–porosity approach is employed. The Pressure Linked Equations algorithms (known as SIMPLE) are used to solve the pressure–velocity coupling phenomenon. The PRESTO! interpolation scheme is employed to obtain the pressure field. A second-order upwind scheme is utilized for all the governing equations.

To enhance the convergence of the continuity, momentum, and energy equations (with convergence thresholds of 10⁻⁶, 10⁻⁶, and 10⁻⁸, respectively), specific under-relaxation factors are assigned. These factors, specifically 0.7, 0.3, 0.8, and 0.9, are assigned to the momentum, pressure correction, energy, and liquid fraction variables, respectively.

Three distinct sets of mesh systems with roughly 13 k, 97 k, and 220 k numerical cells for the computational domain were used to analyze the independence of grid resolution. The curves displayed in Figure 2a exhibit a satisfactory level of agreement, leading to the selection of the 97 k grid for the current study due to its cost-effectiveness. Furthermore, based on the calculated courant number, the sensitivity of the time step size is examined with values of 0.1 s, 0.2 s, and 0.25 s. The findings are shown in Figure 2b. There is a small difference between the curves regarding the liquid phase fraction. Thus, the current study takes into account situations with 97 k grids and a time step size of 0.2 s.

2.5. Model Validation

To validate the accuracy of the numerical model, Figure 3 exhibits a comparative analysis of the simulated and experimental outcomes [36]. It was found that the maximum difference between them was 8.7% by comparing and analyzing the simulated data with the measured data. In addition, the average difference between the simulated and measured data was 1.3%. These discrepancies can be ascribed to two factors. First, the convective heat transfer coefficient (3 W/(m²·K)) was assumed to be constant when performing the numerical simulations, while 298 K was considered to be the temperature at which the ambient temperature was maintained at all times. Some simplification of the boundary conditions was calculated. Second, the treatment of the specific heat of the PCM as a constant in the simulations does not align with the experimental data. Thus, it can be inferred that the current numerical outcomes align with the experimental outcomes in the literature [36]. The numerical model is suitable for further computation in this research.

3. Results and Discussion

3.1. Effects of the Specific Heat Capacity of the PCM

The change in thermophysical parameters will have some effect on the heat transfer of shell-and-tube latent heat thermal energy storage devices, and only the specific heat capacity of the PCM is changed, and the other physical parameters are unchanged, through which the effect of six groups of different heat capacities (0.5c_p, c_p, 1.5c_p, 2c_p, 2.5c_p, 3c_p) on the heat transfer performance of shell-and-tube latent heat thermal energy storage devices is quantitatively analyzed. The average temperature and melting fraction of the PCM over time throughout the heat storage process are illustrated in Figure 4. Regarding the overall trend of the PCM temperature change during thermal storage, it was observed that the temperature rise trend of the six different heat capacities of the PCMs was consistent. The PCM temperature gradually increased during the process, while the rate of temperature rise gradually decreased. Eventually, the temperature stabilized at approximately 340 K. This behavior can be attributed to the initial stage of the process, where there is a notable disparity in temperature between the HTF and the PCM. This results in a fast convective heat transfer rate. In the meantime, the average PCM temperature does not reach its phase change temperature. As the progression of the process unfolds, the speed of convective heat transfer gradually decelerates owing to the diminishing temperature disparity existing between the HTF and the PCM. The average PCM temperature reaches its phase change temperature, causing the PCM temperature growth rate to slow down. During the concluding phases of the heat storage procedure, the temperature of the PCM gradually attains a stable state of approximately 340 K. This temperature proximity to the HTF signifies the completion of all phase transitions.

Through a comparison of the temperature and liquid phase rate curves of six PCMs with varying specific heat capacities (c_p), it was observed that an increase in c_p resulted in a decrease in both the rate of temperature increase and the rate of PCM melting. The complete melting of the PCM, which accounts for 99.9% of the liquid phase, needed approximately 75, 88, 97, 106, 117, and 123 min, respectively.

Figure 5 illustrates the temporal variation of sensible heat storage and latent heat storage of the PCM for different c_p values during the heat storage process. It is evident that both sensible heat storage and latent heat storage of the PCM gradually increase over time, eventually reaching a stable condition. Notably, the sensible heat storage capacity exhibits a linear relationship with c_p. As the c_p of the PCM increases, its sensible heat storage capacity also increases, with the change ratio of c_p matching the final sensible heat storage capacity. Conversely, the latent heat storage capacity is solely influenced by the liquid fraction of the PCM and the latent heat of phase change, rendering the c_p change inconsequential to the final latent heat storage energy. In the thermal energy storage procedure, the specific heat capacity (c_p) influences both the temperature alteration of the PCM and its melting rate, ultimately resulting in variations in the latent heat storage.

3.2. Effects of the Phase Change Latent Heat of the PCM

In order to quantitatively investigate the effect of latent heat on the heat transfer properties of PCMs, Figure 6a,b showcase the temporal progression of the mean temperature and melting proportion of the PCM throughout the heat storage procedure. By comparing the temperature and liquid phase fraction variations over time for six PCMs with different latent heat values (0.5r, r, 1.5r, 2r, 2.5r, 3r), several observations can be made. Initially, during the early stage of the thermal storage process, the curves of the average temperature for all six PCMs exhibit similar trends, with the temperature rising from 298 K to approximately 324 K. This is attributed to the lower average temperature and liquid fraction during this stage, where the change in latent heat has minimal influence on the thermal storage process. In the middle stage of the thermal storage process, the impact of latent heat on the PCM’s warming rate becomes apparent, as the warming rate decreases with higher latent heat values. As the heat storage process nears its conclusion, the temperature of the PCM gradually converges with the HTF, and the liquid fraction approaches 1. This convergence results in a decrease in the impact of latent heat on the PCM’s heat storage process. Consequently, the curves depicting the average temperature and liquid fraction changes for PCMs with varying latent heat values almost coincide with the passage of time.

Figure 7 exhibits the temporal evolution of sensible heat accumulation for PCMs possessing distinct latent heat characteristics during the process of thermal storage. Moreover, this figure also illustrates the temporal behavior of latent heat storage for PCMs with varying latent heat properties. Throughout the course of the process, it becomes apparent that the sensible heat storage and latent heat storage of the PCM exhibit a gradual increase over time before reaching a state of stability. Among them, sensible heat storage is a linear function of c_p and temperature, so the shapes of the graphs and plots are similar in the case of constant c_p. For the latent heat storage heat, since it is related to the PCM liquid phase rate and the latent heat of phase change, the latent heat storage heat increases with the increase in the latent heat of the phase change in the PCM.

3.3. Effects of Thermal Conductivity of the PCM

The figure presented in Figure 8 illustrates the curves depicting the average temperature variation over time during the process of thermal storage for PCMs with varying thermal conductivities, specifically 0.5, 1.5, 2, 2.5, and 3. Furthermore, the figure also displays the changes in the melting rate of the PCMs over time for the aforementioned thermal conductivities. The findings of the analysis reveal a noteworthy correlation between the rate of melting of the PCM and its thermal conductivity, indicating a substantial acceleration in the melting process as the thermal conductivity rises. Additionally, the maximum temperature that the PCM can attain progressively rises, approaching the temperature of the HTF. This is corroborated by Figure 9, which illustrates that the quantity of sensible heat storage obtained rises in tandem with the elevation of the PCM temperature, while the final quantity of latent heat storage remains constant.

3.4. Comprehensive Comparative Analysis of Influencing Factors

To conduct a comprehensive comparison and analysis of the effect of the thermal storage properties of PCMs with different specific heat, latent heat, and thermal conductivity, this study examined the effects of variations in PCM’s thermal properties on its average thermal storage rate and total heat storage. The calculation of the overall heat storage was conducted by accounting for the heat storage quantity during instances when the heat storage rate is below 0.5 J/s. This specific duration is then identified as the designated heat storage time. The graphical representation in Figure 10 illustrates the correlation between the average rate at which heat is stored and the specific heat. Additionally, the graph shows the impact of latent heat on the average thermal storage rate, as well as the alterations in the average thermal storage rate when the thermal conductivity is modified. It is worth mentioning that there is a noticeable upward trend in the average heat storage rate as these properties increase. Notably, the average thermal storage rate demonstrates a relatively slower growth when influenced by the increase in specific heat and latent heat. Specifically, a 50% increment in specific heat leads to an approximate 4% increase in the average heat storage rate. Conversely, a 50% rise in latent heat results in an approximately 6% increase in the average heat storage rate, slightly surpassing the impact of specific heat. The impact of thermal conductivity on the average thermal storage rate is more obvious. With each 50% increase in latent heat, the average thermal storage rate increases by approximately 22%, especially when the thermal conductivity increases to 1.5 times the most significant, and the average thermal storage rate increases by nearly 50%. At this time, the thermal conductivity is 0.4(s)/0.2(l), and the average thermal storage rate is 52.7 J/s.

The variation in the total heat storage with the specific heat, latent heat, and thermal conductivity of the PCM is shown in Figure 10. Based on the depicted figure, it is evident that the fluctuation in thermal conductivity exerts a negligible impact on the overall heat storage capacity of PCMs, and its total heat storage increases linearly with increasing specific heat and latent heat. The growth rate with latent heat is greater than the growth rate with specific heat.

4. Conclusions

In this research, an extensive numerical analysis was conducted to examine the thermal storage efficiency of shell and tube phase change heat storage units with varying thermophysical parameters of PCMs. Specifically, the impact of the specific heat capacity, latent heat, and thermal conductivity of the PCM was investigated. Based on the findings obtained, several noteworthy conclusions can be drawn:

• Selecting a logical value for the specific heat capacity of the PCM proves to be a viable approach in enhancing the thermal storage capabilities of shell and tube phase change heat storage systems. As the specific heat capacity (c_p) of the PCM increased, the total heat storage exhibited a linear increase. However, the rate of temperature increase and the PCM melting rate both experienced deceleration. Within the specified range of parameters, the average rate of heat storage rises alongside the increase in the c_p of the PCM. Specifically, for every 50% rise in specific heat, the average heat storage rate expands by approximately 4%;
• The latent heat of the PCM is of great importance in improving the thermal storage efficiency of shell and tube phase change heat storage units. During the intermediate phase of the thermal storage process, the impact of changes in latent heat on the rate of temperature rise in the PCM is investigated. It is evident that an increase in latent heat leads to a decrease in the rate of temperature increase. However, the overall heat storage capacity exhibits a linear growth with the increase in latent heat. For each 50% increase in latent heat, the average heat storage rate increases by approximately 6%;
• The impact of thermal conductivity on the average thermal storage rate is more obvious. With each 50% increase in thermal conductivity, the average thermal storage rate increases by approximately 22%, especially when the thermal conductivity increases to 1.5 times the most significant, and the average thermal storage rate increases by nearly 50%. The total heat storage is minimally impacted by the thermal conductivity of the PCM.

The findings of this study demonstrate that the heat storage capacity is primarily influenced by the specific heat capacity and latent heat of the phase change material. Furthermore, the heat storage rate is significantly impacted by the thermal conductivity of the phase change material. Hence, for the purpose of enhancing both heat transfer and storage capability, PCMs that possess elevated thermal conductivity are chosen while ensuring they meet the prerequisites for storage density.