Numerical Simulation Study of Heat Transfer Fluid Boiling Effects on Phase Change Material in Latent Heat Thermal Energy Storage Units

Abstract

The innovation in thermal storage systems for solar thermal power generation is crucial for achieving efficient utilization of new energy sources. Molten salt has been extensively studied as a phase change material (PCM) for latent heat thermal energy storage systems. In this study, a two-dimensional model of a vertical shell-and-tube heat exchanger is developed, utilizing water-steam as the heat transfer fluid (HTF) and phase change material for heat transfer analysis. Through numerical simulations, we explore the interplay between PCM solidification and HTF boiling. The transient results show that tube length affects water boiling duration and PCM solidification thickness. Higher heat transfer fluid flow rates lower solidified PCM temperatures, while lower heat transfer fluid inlet temperatures delay boiling and shorten durations, forming thicker PCM solidification layers. Adding fins to the tube wall boosts heat transfer efficiency by increasing contact area with the phase change material. This extension of boiling time facilitates greater PCM solidification, although it may not always optimize the alignment of bundles within the thermal energy storage system.

1. Introduction

The pursuit of low-carbon, clean, safe, and efficient renewable energy stands as an imperative for achieving sustainable development in human society. Leveraging solar energy and waste recovery systems offers a means to diminish reliance on fossil fuels among manufacturers. Nonetheless, these technologies exhibit drawbacks such as inconsistent power output and intermittent operation periods [1]. Thermal energy storage (TES) technology presents a solution by capturing surplus energy from transitional processes in the form of cold, heat, or a combination thereof, holding significant promise for both renewable energy consumption and industrial waste heat utilization [2,3]. Particularly in manufacturing sectors like civil building, food processing, automotive, and chemical industries, which demand heat at low temperatures (up to 100 °C) and medium temperatures (100–400 °C) [4,5,6], these processes exhibit a high tolerance for both heat transfer rates and temperatures, with thermal energy storage meeting the demands of supply. Notably, recent research has directed attention towards employing water/steam as a heat transfer fluid (HTF) in lieu of synthetic oil or molten salt, a concept known as TES for direct steam solar thermal power plants [7]. By utilizing water/steam as the working medium, this approach eliminates the need for synthetic oil heat transfer processes in heat exchangers, consequently enhancing system efficiency and reducing costs [8]. Among the three modes of TES, latent heat thermal energy storage using phase change materials (PCMs) is promising in the field of medium and low temperatures due to its advantages of high energy storage density and providing stored energy in a constant or narrow temperature range [9,10]. Additionally, it enhances the reliability, capacity factor, and schedulability of heat sources, thereby improving energy management [11].

However, the low thermal conductivity inherent in PCMs significantly impacts the charge and discharge performance of LHS systems, limiting their application in scenarios requiring rapid heat storage and release [12]. Enhancing the heat transfer characteristics of PCMs and improving heat transfer efficiency are currently prominent research areas. In the study of shell-and-tube latent heat thermal energy storage, various approaches are explored. These include modifying the PCMs themselves, such as using mixtures of materials with high thermal conductivity, like graphite matrix composite PCMs [13], or incorporating nanoparticles [14]. Additionally, combining PCMs with different melting points to create a cascade thermal energy storage system is being investigated [9]. Starting from the storage vessel’s structure, various approaches have been explored. These include incorporating metal foam into the PCM [15], adding external fin structures to enhance heat transfer [16,17,18], and employing other structural methods to address heat transfer challenges. For instance, helical heat transfer tubes have been introduced to augment heat transfer by increasing the contact surface [19], while heat pipes have been utilized to enhance heat transfer [20,21,22]. In a typical shell-and-tube system, the PCM is contained either in the shell or the tube, with the HTF flowing through the other component. This arrangement facilitates heat transfer to the PCM, and the tubes are designed accordingly to investigate the system’s heat transfer mechanism [23].

In shell-and-tube heat exchangers with a multi-pipe structure, the utilization of multiple pipes serves to enhance the heat transfer area. Consequently, research on heat exchangers with a multi-tube configuration primarily focuses on various aspects, including different tube arrangements such as tube spacing and the number of tubes [24]. Given that the melting process of PCM is influenced by natural convection, the heat transfer mechanisms differ between vertically and horizontally arranged tubes [25]. Additionally, Entezari et al. [26] introduced an innovative triplex-tube helical-coil storage unit, which integrates various geometrical parameters such as coil pitch, diameter, and inclination angle.

Further studies have explored the heat transfer properties of HTF. Abbasi et al. [27] investigated the impact of continuous porous media with varying porosities and localized porous media in three distinct regions of the phase change system on related parameters of the phase changing process. Sriram et al. [28] examined a triple-cylinder TES system, identifying an optimal inner cylinder diameter. They found that the mass flow rate and inlet temperature of HTF significantly affect the melting rate, while the effect of tube length is negligible. The outer boundary of most shell-and-tube thermal energy storage models is typically simplified as insulated. Wang et al. [29] introduced an innovative heat transfer calculation method for single-tube units within tube-and-shell bundle latent heat TES devices, enhancing the simulation speed of shell-and-tube bundle units. Fornarelli et al. [30] addressed the conjugate heat transfer process between HTF and PCM, incorporating conduction within the tube via unsteady Navier–Stokes equations, enabling effective assessment of melted PCM’s natural convection compared to pure heat transfer calculations. Seddegh et al. [25] investigated heat transfer behavior in shell-and-tube heat storage units using a combined convection–conduction model. Their results highlighted the significant impact of HTF inlet temperature on heat transfer, with minimal influence from inlet flow rate. Additionally, Mekrisuh et al. [31] explored the impact of system parameter settings on total PCM melting time in a vertically oriented concentric-tube TES system, including HTF inlet temperature and aspect ratio.

Despite extensive research on TES systems, a specific articulation of the knowledge gap remains. Current studies primarily focus on the total melting time of PCM as a performance indicator, with limited research on the widely utilized HTF parameters and their impact on system performance. This gap is particularly evident in the context of vapor-water two-phase flow and PCM interactions in latent heat thermal energy storage systems. The parameter configuration of the HTF significantly impacts the shell-and-tube TES system, yet detailed investigations into the effects of HTF inlet temperature, flow rate, and tube length on PCM solidification and heat transfer efficiency are lacking. Therefore, this paper investigates the heat transfer characteristics between vapor–water two-phase flow and a PCM in a latent heat thermal energy storage system using water/steam as the HTF. The selected phase change material is a binary nitrate molten salt (60%NaNO₃–40%KNO₃) with a melting temperature of approximately 221 °C [32,33,34]. Utilizing a two-dimensional shell-and-tube model, CFD effectively evaluates heat transfer factors between water/steam and the PCM. This study aims to fill the knowledge gap by providing a comprehensive analysis of the coupled heat transfer process involving PCM solidification and HTF flow boiling, thereby enhancing the understanding and design of efficient TES systems.

2. Physical Models

The physical problem examined in this study involves a vertically oriented shell-and-tube TES system. In this system, the PCM fills an annular space with a diameter of 108 mm (2D_o), while the HTF, specifically water/steam, flows upward through a tube with a diameter of 10 mm (2D_i). The pressure in the tube is maintained at 0.2 MPa, as depicted in Figure 1. The geometric parameters of the annular finned tube remain consistent with those of a smooth tube, with both the tube wall and fins constructed from copper, each with a thickness of 1 mm (δ and t_f). The fins have a height of 20 mm (h_f), a spacing of 19 mm (d_f), and a total of 25 fins in the shell-and-tube model spanning a length of 500 mm (L). To conserve computational resources, a simplified two-dimensional axisymmetric model was employed, taking advantage of the rotational symmetry inherent in shell-and-tube TES systems. The thermophysical properties of the materials utilized are outlined in

Table 1. Thermophysical properties of the different materials employed in this work.

Material	Thermophysical Property	Value
60%NaNO3–40%KNO3	Viscosity (Pa·s)	0.00693
	Thermal conductivity (W/m·°C)	0.55
	Density (kg/m3)	1925
	Specific heat (J/kg °C)	1600
	Enthalpy of fusion (kJ/kg)	102
Water	Viscosity (Pa·s)	0.001003
	Thermal conductivity (W/m °C)	0.6
	Density (kg/m3)	998.2
	Specific heat (J/kg °C)	4182
Vapor	Viscosity (Pa·s)	1.28 × 10−5
	Thermal conductivity (W/m·°C)	0.0265
	Density (kg/m3)	1.122
	Specific heat (J/kg·°C)	2119.8
	Latent heat of vaporization (kJ/kg)	2202.3
Copper	Thermal conductivity (W/m·°C)	387.6
	Density (kg/m3)	8978
	Specific heat (J/kg·°C)	381

.

To streamline the mathematical model, the following simplifications and assumptions are employed in this study:

(1)

All materials utilized are assumed to be homogeneous and isotropic across all phases [35].

(2)

Fixed-wall, no-slip boundary conditions are applied to all boundaries.

(3)

Volume changes and heat dissipation due to viscosity in the PCM are disregarded, as well as heat transfer via radiation.

(4)

The PCM region is assumed to exhibit laminar flow, and only natural convection under gravity is considered. Volume changes resulting from PCM solidification are neglected.

(5)

Inlet velocity and temperature of the HTF are held constant, and both liquid and gas phases of the HTF in the tube are treated as incompressible. This approach ignores pressure drops during tube flow and does not account for changes in density.

3. Mathematical Models

3.1. Governing Equations

For the PCM domain:

The solidification process of the PCM was simulated using the enthalpy–porosity technique proposed by Voller et al. [36]. This method treats the fluid domain of the PCM as a porous medium, with the porosity representing the liquid volume fraction of the cell. The phase change paste region is considered as a pseudo porous region, where the porosity is represented by the value of the liquid fraction ${\beta }_l$ ranging between 0 and 1.

The liquid fraction ${\beta }_l$ is defined as

$${\beta }_l=\left\{\begin{array}{c}0 \mathrm{i}\mathrm{f} T<T_s\\ {\displaystyle \frac{T-T_s}{T_l-T_s}} \mathrm{i}\mathrm{f} T_s<T\le T_l\\ 1 \mathrm{i}\mathrm{f} T\ge T_l\end{array}\right.$$

(1)

The source term S in the momentum equation acts as a flow resistance when the PCM is solid. S is defined as follows:

$$S={\displaystyle \frac{{\left(1-{\beta }_l\right)}^2}{\left({{\beta }_l}^3+\epsilon \right)}}{A}_{mush}\overrightarrow{u}$$

(2)

where ε is a small number specified as 0.001 to avoid division by zero, when ${\beta }_l$ is equal to zero. The mushy region constant $A_{mush}$ used in the current study is 10⁵ [37], which is in good agreement with experimental results.

The total enthalpy H is defined as follows:

$$H=h_{ref}+{\int }_{T_{ref}}^T{c}_p\mathrm{d}T+∆H$$

(3)

where the latent heat is represented by the latent heat $L_h$ of the PCM.

$$∆H=\beta L_h$$

(4)

For the liquid phase part, considering the convection caused by buoyancy, the Boussinesq hypothesis is adopted, which is calculated by adding a source term to the momentum equation, where the density varies with temperature.

$$\rho ={\rho }_0\left[1-\beta \left(T-T_0\right)\right]$$

(5)

Therefore, the governing equations considered in this study are defined as follows:

Continuity equation:

$$\nabla ·\overrightarrow{u}=0$$

(6)

Momentum equation:

$${\rho }_0\left({\displaystyle \frac{𝜕\overrightarrow{u}}{𝜕t}}+\overrightarrow{u}·\nabla \overrightarrow{u}\right)=-\nabla P+\nabla ·\left[\mu \left(\nabla \overrightarrow{u}+\nabla {\overrightarrow{u}}^T\right)\right]+\rho g+S$$

(7)

Energy equation:

$${\displaystyle \frac{𝜕}{𝜕t}}\left({\rho }_{0}H\right)+\nabla ·\left(\rho \overrightarrow{u}H\right)=\nabla ·\left(k\nabla \mathrm{T}\right)$$

(8)

For the HTF domain:

For water flow boiling in a tube, the transient VOF method was used in conjunction with the Lee model [38]. The VOF method is suitable for analyzing two and more immiscible liquids and tracking the volume fraction of each fluid. In this study, liquid water serves as the continuous first phase and water vapor as the second phase; the sum of volume fractions of phases must equal unity.

$${\mathsf{\alpha }}_l+{\mathsf{\alpha }}_v=1$$

(9)

To ensure volume fraction continuity, accounting for mass transfer between the two phases, and tracking interfacial behavior during flow boiling, the continuity equation of the gas–liquid phase is expressed as follows:

$${\displaystyle \frac{𝜕{\mathsf{\alpha }}_l}{𝜕t}}+\nabla ·\left({\mathsf{\alpha }}_l{\overrightarrow{u}}_l\right)={\displaystyle \frac{{\dot{m}}_{lv}}{{\rho }_l}}$$

(10)

$${\displaystyle \frac{𝜕{\alpha }_v}{𝜕t}}+\nabla ·\left({\alpha }_v{\overrightarrow{u}}_v\right)={\displaystyle \frac{{\dot{m}}_{vl}}{{\rho }_v}}$$

(11)

where ${\dot{m}}_{lv}$ and ${\dot{m}}_{vl}$ represent the mass transfer rate of liquid and gas, respectively, and ${\dot{m}}_{lv}=-{\dot{m}}_{vl}$.

Momentum equation:

$${\displaystyle \frac{𝜕}{𝜕t}}\left(\rho \overrightarrow{u}\right)+\nabla ·\left(\rho \overrightarrow{u}\overrightarrow{u}\right)=-\nabla P+\nabla ·\left[\mu \left(\nabla \overrightarrow{u}+\nabla {\overrightarrow{u}}^T\right)\right]+\rho \overrightarrow{g}+{\overrightarrow{F}}_{vol}\mathrm{ }$$

(12)

Energy equation:

$${\displaystyle \frac{𝜕}{𝜕t}}\left(\rho E\right)+\nabla ·\left(\overrightarrow{u}\left(\rho E+P\right)\right)=\nabla ·\left(k_{eff}\nabla T\right)+S_h$$

(13)

where the Continuum Surface Force model (CSF) [39] is employed to represent surface tension force as part of the source term in the momentum equation.

The energy conversion during boiling and condensation is calculated as a source term added to the energy equation.

$$S_h={\dot{m}}_{vl}{h}_{vl}$$

(14)

The mass transfer rate of gas–liquid phase change at the interface is determined using the Lee model, a widely utilized approach known for its good convergence and accuracy. The mass source term is derived from the following equations:

$${\dot{m}}_{vl}=r_{i,e}{\alpha }_l{\rho }_l{\displaystyle \frac{\left(T_l-T_{sat}\right)}{T_{sat}}} \mathrm{for} \mathrm{evaporation} (T\ge T_{sat})$$

(15)

$${\dot{m}}_{lv}=r_{i,c}{\alpha }_v{\rho }_v{\displaystyle \frac{\left(T_{sat}-T_v\right)}{T_{sat}}} \mathrm{for} \mathrm{condensation} (T<T_{sat})$$

(16)

The time relaxation factor parameter $r_i$ is a simplified empirical value, which is set to 0.1 according to the literature [40].

For piping wall domain:

The energy equation is simplified to the transient heat conduction equation for a solid:

$${\rho }_w{c}_{p,w}{\displaystyle \frac{𝜕T}{𝜕t}}=k_w{\nabla }^{2}T$$

(17)

3.2. Boundary and Initial Conditions

As shown in Figure 1, the PCM is initially assumed to be in a liquid state with temperature $T_0$ set at 495 K, above its solidus temperature. Initially, the tube contains water at a temperature $T_{in}$ as 353 K. The tube’s inlet is specified as a velocity inlet and the outlet as an outflow condition, with the pressure value equal to the saturation pressure of the steam.

The initial conditions for the PCM and pipe wall computing region are

$$t=0, T=T_0, u=0$$

The initial conditions for the HTF computing region are

$$t=0, T=T_{in}, u=0$$

The inlet boundary condition of the HTF is

$$u=u_{in}, T=T_{in}$$

The symmetric axis boundary condition is

$$u=0, {\displaystyle \frac{𝜕u}{𝜕n}}=0,{\displaystyle \frac{𝜕T}{𝜕n}}=0$$

Both sides of the tube wall are coupled heat transfer conditions:

$$u=0, T_{\mathrm{H}\mathrm{T}\mathrm{F}}=T_w, {\lambda }_{\mathrm{H}\mathrm{T}\mathrm{F}}{\displaystyle \frac{𝜕T_{\mathrm{H}\mathrm{T}\mathrm{F}}}{𝜕n}}={\lambda }_w{\displaystyle \frac{𝜕T_w}{𝜕n}}$$

$$u=0, T_w=T_{\mathrm{P}\mathrm{C}\mathrm{M}}, {\lambda }_w{\displaystyle \frac{𝜕T_w}{𝜕n}}={\lambda }_{\mathrm{P}\mathrm{C}\mathrm{M}}{\displaystyle \frac{𝜕T_{\mathrm{P}\mathrm{C}\mathrm{M}}}{𝜕n}}$$

3.3. Computational Methods

In this study, ANSYS Fluent 2020 R2 was utilized to simulate the transient heat transfer process of the model. Three phases were established: liquid water, water vapor, and PCM. The solid and liquid states of the PCM were regarded as fluid states and differentiated only by porosity. The implicit scheme was employed to solve the volume fraction equation, allowing stable calculations with larger time steps and lower costs, independent of the Courant number. To balance convergence and computational cost, the mass and velocity component residuals were set to 10⁻³, and the energy component residual was set to 10⁻⁶. Other solver settings are presented in

Table 2. Solver settings in this work.

Solver Variables	Settings
Pressure—velocity coupling	Viscosity (Pa·s)
Gradient	Least Squares Cell Based
Pressure	PRESTO!
Momentum	Second Order Upwind
Volume fraction	Compressive
Turbulent kinetic energy	First Order Upwind
Specific dissipation rate	First Order Upwind
Energy	Second Order Upwind
Transient formulation	First Order Implicit

.

3.4. Grid Independence and Model Verification

To conduct the grid independence analysis, four distinct grid sizes (0.008 m, 0.006 m, 0.004 m, and 0.001 m) were established within the PCM region, corresponding to grid counts of 4325, 7900, 10,600, and 30,325, respectively. Simulations were executed with a time step of 0.01 s. The outcomes revealed that the comparison of the liquid fraction between the grid sizes of 0.004 m and 0.001 m yielded an error margin of less than 5%. Consequently, a grid size of 0.004 m was utilized for the PCM region in this study.

The grid size near the tube wall is evaluated using the dimensionless distance from the wall, defined as

$$y^{+}={\displaystyle \frac{y{u}_{\tau }}{\nu }}$$

(18)

where $y$, $u_{\tau }$, and $\nu$ are the absolute distance between the fluid and the wall, the friction velocity, and the kinematic viscosity, respectively. For the viscous sublayer, when $y^{+}\ge 5$, the heat transfer of the fluid in the viscous sublayer can be accurately described, capturing the vapor nucleation near the wall, which is crucial for accurately predicting the formation and growth of bubbles. Through calculations, a grid size of 0.00004 m is adopted in this study to ensure that the value of $y^{+}$ is less than 5 throughout the entire heating length.

To validate the mathematical model proposed in this study, it was compared with experimental data reported by Yuan et al. [35]. The simulation was configured with a water tube inlet temperature of 80 °C, allowing for a comparison of the paraffin melting process’s liquid fraction variation over time. Due to the initial thermal resistance between the PCM and the tube wall in this experiment, there was a preheating period required, causing a delay in the onset of phase change. Consequently, the simulation curve was shifted to the right by 150 s to align with the experimental data. As shown in Figure 2a, there is an error between the 2D numerical simulation results and the experimental data. This error primarily stems from the 2D model’s assumption that the flow and heat transfer are symmetric along the axial direction, thereby neglecting three-dimensional effects such as lateral flow and non-uniform heat distribution. The error is 4.08%. Therefore, it is concluded that the present numerical calculation model is capable of analyzing the phase change heat transfer process in a vertical shell-and-tube heat exchanger. To further verify the accuracy of the boiling model in the HTF fluid domain, we compared the numerical simulation results of the wall temperature during the boiling process with the experimental data from the literature. As shown in Figure 2b, due to the continuous generation of vapor bubbles on the heated wall and the subsequent detachment of the growing vapor bubbles from the wall, the wall temperature experiences slight fluctuations. The error between the numerical results and the experimental data is less than 1%. Therefore, this model can be used for numerical simulation of the boiling process inside the tube.

4. Results and Discussion

4.1. Effect of Tube Length

In the TES heat exchanger, varying tube lengths result in different heat transfer rates, directly impacting the total boiling time, as depicted in Figure 3. It is evident that, at any given moment, the length of single-phase flow within the tube remains constant. Consequently, it is anticipated that the boiling time for the 1.5 m long case will significantly exceed that of the 0.5 m case. Based on the calculation of the single-phase flow length inside the tube, the boiling duration inside a 1.5 m tube is approximately eight times that of a 0.5 m tube.

Meanwhile, heat transfer remains consistent at identical cross-sectional positions across different pipe lengths during the heat transfer period, as illustrated in Figure 4. The variation in average liquid fraction of the PCM at identical positions is nearly uniform, while the solidification layer thickness of the PCM gradually diminishes towards the upper portion of the tube shell. At 300 s, the liquid fractions of PCM at the 0 m and 0.5 m sections are about 0.775 and 0.825, respectively. So, in this model, when the HTF flows through a tube length of 0.5 m during 300 s, the heat transfer can increase the solidification of PCM by about 5%. Additionally, the temperature change in the PCM reveals that as water flows through the pipe and boils due to heat, the heat transfer effect on the PCM at corresponding positions remains consistent over the same duration.

At 300 s, the average temperature of PCM at the 0 m section dropped from the initial 494.3 K to 482.1 K, and at the 0.5 m section, it fluctuated downward to 490.9 K. The average temperature change of the PCM at 0 m remains constant under the single-phase flow of the HTF throughout the entire process. In contrast, at 0.5 m, the HTF transitions from initial single-phase flow heat absorption to two-phase flow with vapor. Following the heat absorption and vaporization phase change of the HTF in the tube, the heat transfer coefficient decreases in the vapor–water two-phase flow. Concurrently, the solidification process of the PCM outside the tube, corresponding to the presence of vapor within the tube, decelerates. The temperature curve of the PCM begins to shift after 20 s of boiling, with the PCM temperature drop gradually leveling off. This behavior is attributed to the heat absorption and boiling of the HTF, marking a significant departure from the behavior observed in single-phase fluids.

4.2. Effect of HTF Inlet Temperature

The influence of the HTF inlet temperature on the PCM is multifaceted. As depicted in Figure 5a, the average PCM liquid fraction decreases more rapidly at lower HTF inlet temperatures compared to higher temperatures. The impact is more pronounced at the outlet cross-section (1.5 m). At 200 s, for the HTF with an inlet temperature of 353 K, the average liquid fraction difference in the PCM between the 0 m and 1.5 m sections is 4.6%, while for the HTF with an inlet temperature of 313 K, this value is 3.2%. It indicates that the solidified PCM thickness difference between the inlet and outlet cross sections is smaller at lower inlet temperatures. The occurrence of this phenomenon is primarily attributed to the boiling of the HTF. The relatively high inlet temperature causes the HTF within the tube to commence boiling and generate steam at an earlier stage. Compared with liquid water, steam exhibits a significantly lower thermal conductivity. Hence, the formation of steam in the HTF leads to a reduction in the heat transfer rate, thereby decreasing the solidification rate of the PCM.

In Figure 5b, the influence of HTF inlet temperature on average PCM temperature is evident, with the PCM in the low inlet temperature case maintaining a lower temperature. The temperature curve at 1.5 m indicates an inflection point around the onset of boiling. Following this, the PCM undergoes a period of latent heat release with minimal temperature change, succeeded by progressively lower temperatures due to the influence of HTF temperature on the solidified PCM portion. Moreover, the temperature difference between 0 m and 1.5 m remains consistent across all three cases over time, maintaining approximately 9 K at the boiling endpoint.

In Figure 6, during the HTF boiling process, the HTF temperature is relatively low while the average wall heat flux is relatively high, necessitating enhanced heat transfer from the PCM to ensure HTF boiling. The boiling initiation times varied among cases with inlet temperatures arranged from high to low, occurring at 30 s, 50 s, and 90 s, respectively, and the inflection point of the wall heat flux curve coincides with the initiation of HTF boiling. Prior to HTF boiling, the average wall heat flux decreases sharply when the HTF in the tube remains in a single-phase flow state. This decline is attributed to the PCM solidifying outward from the wall, resulting in heat transfer occurring solely between the outer wall of the tube and the PCM, without convective heat transfer. The average wall heat flux during boiling remains approximately 10,000 W/m² across different HTF inlet temperatures, with temperature changes having no discernible impact on the thermal conductivity of the PCM.

4.3. Effect of HTF Inlet Flow Rate

As shown in Figure 7a, the HTF inlet flow rate scarcely impacts the PCM liquid fraction. In the same cross-sectional area, the PCM liquid fraction variation curves versus time align across different inlet flow rates. Yet, the PCM temperature changes differ significantly with varying HTF inlet flow rates, especially at the outlet cross-section. As shown in Figure 7b, the higher the HTF inlet flow rate, the faster the PCM temperature drops over time. Though boiling starts around 30 s for all flow rates (marked by the outlet PCM temperature curve inflection), the time for complete outlet boiling is much shorter (about 130 s) at high inlet flow rates. This is because the increased flow rate strengthens convective heat transfer, more evidently for steam containing the HTF. At 130 s, when the HTF inlet velocity is 0.008 m/s, the average PCM temperature at the 0.5 m section is 489.1 K. When the inlet velocity is 0.006 m/s and 0.004 m/s, the corresponding average PCM temperatures are 491.6 K and 492.1 K. At the outlet, where the HTF undergoes phase change, an increase in the flow rate of the gaseous HTF significantly enhances the heat transfer rate. This causes the PCM temperature to decrease more rapidly.

Nevertheless, this rapid temperature drop in the PCM occurs primarily in the PCM solidification layer adjacent to the HTF tube, lowering the temperature of the solidification layer while having a smaller effect on the PCM fluid layer. As a consequence, the variation in PCM liquid fraction remains small. This phenomenon is more clearly illustrated by PCM liquid fraction and temperature field contour plots under different HTF flow rates. As shown in Figure 8, although the thickness of the PCM solidification layer is almost the same under different HTF flow rates, the temperature gradient at the outlet increases significantly when the HTF flow rate is high. Therefore, adjusting the inlet HTF flow rate can effectively regulate the quality of the outlet HTF.

4.4. Effect of Finned Tubes

The fins serve to augment heat transfer between the PCM and HTF. Figure 9a illustrates that the decline in PCM liquid fraction in the smooth tube gradually decelerates over time. Conversely, the finned tube exhibits varying decreasing trends in PCM liquid fraction at different locations. At 0 m, the PCM liquid fraction initially decreases rapidly, resembling the trend observed in the smooth tube case during the 10 s preceding vapor generation. However, after this initial period, the rate of decline slows down, with a notable acceleration resuming only after 600 s. Similarly, at 0.5 m, the PCM liquid fraction experiences a slow decline from 0 s to 160 s, followed by a rapid decrease between 160 s and 250 s, and then another slow decline thereafter.

As depicted in Figure 9b, the temperature difference between the PCM at the 0 m and 0.5 m sections widens during the heat transfer process. A greater portion of the PCM at 0 m solidified, resulting in a lower temperature during the HTF boiling period. Conversely, the temperature of the PCM at 0.5 m remained higher compared to the smooth tube case, gradually decreasing until its rate of decrease accelerated after 600 s.

The enhanced heat transfer efficiency of the finned tube is closely associated with the substantial contact area between the fins and the PCM. As illustrated in Figure 10, during the PCM solidification process, the average heat flux on the outer wall surface of the fins and the tube is lower compared to that of the smooth tube. However, the total heat flux on the outer wall is elevated due to the extensive contact area of the finned tube. Moreover, the heat flux on the inner wall is significantly higher than that of the smooth tube. The heat flux on the inner wall is a key indicator of the thermal energy storage unit’s thermal efficiency. In this model, the finned tube’s inner wall average heat flux, at around 12,800 W/m², is 1.3 times that of the smooth tube’s 9900 W/m². Therefore, finned tubes lead to more effective heat transfer.

As depicted in Figure 11a, the fins become enveloped by PCM solidification rapidly at 10 s, initiating boiling. The temperature of the PCM section, which interacts with the steam, gradually declines. Once the solidified PCM surrounding the fins converges between them, the HTF at the corresponding cross section transitions to a single-phase flow, ceasing boiling. By 600 s, the solidified PCM in the lower part of the tube extends beyond the fin height. A comparison with Figure 11b reveals a notable temperature difference trend in the solidified PCM of the finned tube case, attributed to the influence of the HTF’s single-phase flow temperature. The solidification range extends beyond the fins, leading to a severe temperature drop in the solidified PCM section, insufficient to meet the heat absorption required for HTF boiling. Solidification thickness diminishes outside the fins, with boiling duration determined by fin height. Consequently, the available PCM volume for the total boiling time is less than in the smooth tube case.

5. Conclusions

This paper examines the impact of a HTF on the solidification of a PCM under conditions where boiling can occur. Specifically, it investigates the scenario of two-phase flow involving vapor and water passing through the tube. This study analyzes the heat transfer mechanisms under varying system parameters, leading to the following conclusions:

(1)

Tube length predominantly influences the boiling duration, subsequently affecting PCM solidification, with longer durations resulting in thicker solidification layers. In this model, the boiling duration inside a 1.5 m tube is approximately eight times that of a 0.5 m tube.

(2)

The inlet temperature of the HTF determines the amount of heat absorbed by the HTF before reaching the boiling point. The solidified PCM thickness difference between the inlet and outlet cross sections is smaller at lower inlet temperatures. At 200 s, the difference in the solidification thickness of the PCM at the inlet and outlet under an HTF inlet temperature of 313 K is 1.4% lower than that under an HTF inlet temperature of 353 K.

(3)

The HTF inlet flow rate affects the temperature of the solidified PCM layer. While solidification thickness remains largely unchanged in cases with high HTF velocity, the temperature gradient within the solidified layer increases. At 130 s, the average temperature of the PCM under an HTF inlet velocity of 0.008 m/s is 3 K lower than that under an HTF inlet velocity of 0.004 m/s.

(4)

The installation of fins on the outer tube wall increases the contact area between the tube wall and the PCM, thereby augmenting the PCM’s heat transfer capacity. Compared to the smooth tube configuration, the presence of fins leads to a thicker solidified PCM layer, accompanied by lower temperatures in the solidified region. In this model, the finned tube’s inner wall average heat flux, at around 12,800 W/m², is 1.3 times that of the smooth tube’s 9900 W/m².

Factors such as the solidification thickness of the phase change material, boiling time, and heat transfer are comprehensively analyzed. It is concluded that tube length and temperature are key design parameters. Increasing the heat transfer efficiency of the PCM requires consideration of PCM solidification thickness during boiling, corresponding to the spacing of tube bundles in the TES system. This study provides a solid foundation for the subsequent design of molten salt latent heat TES for direct steam generation.