Performance enhancement of a shell-and-coil ice storage enclosure for air conditioning using spiral longitudinal fins: A numerical approach

Abstract

This study carries out a series of complex unsteady three-dimensional (3D) computational analyses that were validated by experimental tests to examine the performance enhancement of a cylindrical shell-and-coil ice storage enclosure by employing various spiral longitudinal fin geometries covering the heat transfer fluid (HTF) tubes throughout the transient process of solid-liquid phase change. The phase change material (PCM) in this work was selected to be water and the tubes and fins were chosen to be made of copper alloy. To study how this type of fin accelerates the charging process, several dimensionless parameters related to the spiral longitudinal fin geometries were defined, which included dimensionless fin length (Ω), count (Ψ), and thickness (Φ) parameters. Additionally, the impact of fin orientation was evaluated. The findings suggested that raising Ω from 0 to 0.719 reduces the solidification time by 42.29 %. Similarly, changing the Ψ from 0 to 6 expedites the process by 50 %. Regarding the changes of Φ, it was observed that even though the freezing dynamic was not sensitive to this parameter in the first half of the process, the effect was more pronounced throughout the second half. Furthermore, with identical heat transfer areas, horizontal fins outperformed the vertical ones by a marginal difference of 3.58 %, which was found to be because of the better distribution of fins within the container.

1. Introduction

Continuous population growth around the globe has led to escalations in the demand for reliable energy sources; however, several challenges considerably pose threats to the development of such sources. The most important impediments in this regard are the unpredictable nature of renewable energy sources [1] and the imbalance between energy production and demand [2]. Because of these limitations, the energy sources cannot provide reliable energy access to people in some countries, which leads to problems like power outages and related problems. This emphasizes the critical need for energy efficiency as a key driver for sustainability, aligning with global initiatives such as the European Green Deal, which targets climate neutrality through enhanced energy efficiency and increased renewable energy integration across all sectors [3]. Thermal energy storage enclosures with phase change materials (PCMs) are thermal batteries based on latent heat, which can be considerably helpful in this regard [4]. By saving renewable energy from their sources, they can make them more reliable, and also, they can be significantly effective in making the production and demand of energy more balanced [5]. Fig. 1 illustrates the placement of the cold variant of these devices in charging and discharging cycles and shows how they contribute to the air conditioning of buildings. These devices can be charged during off-peak hours by storing cold energy, and during peak hours they can help reduce the pressure on air conditioning units by releasing their cold energy.

Fig. 1.

The application of a cold thermal energy storage enclosure and the cycles involving this device.

The fact that the enclosures based on PCMs can store a significant amount of energy without a considerable temperature difference with the surrounding area, makes them markedly more useful than the systems working using sensible heat [6]. Regardless of all the advantages these systems offer, they often suffer from a sluggish rate of phase change, which necessitates the use of heat transfer enhancers in them. For this reason, both the industry and researchers are constantly trying to improve such systems by testing different approaches.

Geometrical enhancement of the heat transfer fluid (HTF) tubes and their arrangement is known as one of the effective techniques to improve such systems. Bahrami and Allahdadi [7] investigated various arrangements of HTF tubes to expedite the melting process in a tubular enclosure containing N-eicosane PCM using two-dimensional (2D) computational assessments. They recognized the number of HTF tubes as the most influential parameter, capable of improving the charging dynamic by up to 80 %. Ou et al. [8] studied tube configuration and flow temperature and speeds in a shell-and-tube PCM storage unit. They reported that spiral tubes result in the best performance in terms of energy consumption and heat removal. Afsharpanah et al. [9] used various dimensionless numbers to optimize the spiral coil design in an ice storage unit during the freezing process. According to their report, geometrical enhancement can lead to up to around 19 % improvement in the enclosure performance. The optimal design in their study had a tube thickness of 1.25 mm, an outer diameter of 19.05, and a spiral pitch of 54 mm. In the study undertaken by Liu et al. [10], various temperature and flow speeds were assessed to evaluate the phase change rate of palmitic acid in a double spiral tube unit. According to the research findings, despite the fact that the flow rate vastly improves the freezing speed, it does not alter the melting performance considerably. They highlighted that the PCM side suffers from considerable thermal resistance, which necessitates the use of heat transfer augmentation techniques. Various methods are usually used to augment the rapidness of the phase change in PCM enclosures, the most practical of which are the use of multiple PCMs, nano additives (NAs) [11,12], metallic foams [13,14], and expanded surfaces [15,16]. Different approaches that are used by researchers in this field are comprehensively discussed in the review article presented by Rahman et al. [17].

Zhang and Zhu [18] worked on the utilization of multiple PCMs to expedite the melting dynamic in a shell-and-tube PCM enclosure. They revealed that by using the cascade form of multiple PCMs, not only can the melting dynamic be expedited by around 36 %, but also the efficiency of the unit can be elevated by 20 %. Zhang et al. [19] assessed the performance of a PCM enclosure with an inner spiral coil tube. By using two different PCMs in the enclosure, they managed to decrease the charging rate by 56 % compared to when only one PCM was employed. Al-Amayreh [20] et al. worked on a double-cylindrical shell enclosure containing two PCMs and alumina NAs. The examined PCMs were wax and petroleum jelly located in segregated regions within the enclosure. According to the results, the process was virtually 5 % improved by using two PCMs and dispersing NAs. In an attempt to improve the phase change pace, Alhashash and Saleh [21] used hybrid NAs and a rotating wavy enclosure containing PCMs based on N-nonadecane and polyurethane in a 2D computational assessment. They stated that rotating the enclosure can considerably affect the phase change pace and pattern in both charging and discharging directions. Additionally, the addition of NAs could result in up to 38 % better heat exchange rate. Mithu et al. [22] computationally worked on the improvement of a triplex-tube enclosure containing RT82 PCM enhanced with different concentrations of alumina NAs in a 2D study. The unit was designed for air-conditioning systems. Based on the study, up to 33 % faster energy release rate could be acquired by using these NAs. To solve the problem regarding the intermittence of solar sources, Ben Khedher et al. [23] investigated the melting behavior of PCMs enhanced with graphite NAs in a trapezoidal-shaped container in a 2D computational assessment. They explored different enclosure configurations, having various trapezoidal angles in the existence of various graphite NA concentrations. Based on their report, the trapezoidal angle of 40° resulted in the fastest process. In another research, Ben Khedher et al. [24] employed both computational and experimental tools to investigate a shell-and-tube PCM enclosure containing A58H PCM modified with several types of NAs. They revealed that dispersion of one of the NAs by only 1 % of concentration can expedite the melting dynamic by around 90 %. They also considered using longitudinal fins to expedite the process even further.

Basem et al. [25] made some effort to numerically analyze the melting dynamic of paraffin wax in a 2D cavity. They employed V-shaped fins connected to one of the walls to accelerate the phase shift process. According to their results, the addition of these fins can expedite the melting dynamic by around 200 %. Kirincic et al. [26] investigated the phase change dynamic in a shell-and-tube PCM enclosure. To enhance the unit, they evaluated various enclosure aspect ratios as well as different numbers and thicknesses of longitudinal fins. According to their analysis, using twelve fins leads to up to 15 % improved phase change dynamic. In the 2D computational research of Jaberi et al. [27], porous fins were utilized to expedite the melting rate in a shell-and-tube PCM enclosure. They explained that the Darcy number is considerably influential on the melting dynamic and the use of porous fins can result in up to around 47 % of expedition in the melting dynamic compared to the benchmark case. Waqas et al. [28] used 2D numerical tools to assess the utilization of leaf-vein-shaped fins to accelerate the melting dynamic in a triplex tube enclosure containing molten salt PCM. They evaluated various fin configurations to study their impact on temperature, velocity, and melting front changes over time. According to their findings, the use of these fins can expedite the melting rate by up to 91 % compared to the benchmark case. Additionally, they used metallic NAs to further enhance the process, which underlined their potential to be an effective method in combination with fins for expediting the phase change more significantly. In the 2D investigation carried out by S et al. [29], an ice storage unit was thoroughly studied. They examined the influence of using various tube arrangements and diameters, as well as adding fins to enhance the melting dynamic. According to their study, raising the fin count from 4 to 15 expedites the melting rate by around 35 %. Moreover, raising the HTF temperature can improve the process by more than two times. Rashid et al. [30] employed fins with various shapes and numbers to expedite the PCM melting dynamic in a shell-and-tube enclosure in a 2D computational investigation. They reported that using eight straight fins provides better results in terms of the melting front uniformity and quicker melting dynamic. Fahd et al. [31] utilized angled and curved branched fins to expedite the melting dynamic in a 2D horizontal PCM-containing enclosure. The enclosure contained N-eicosane PCM. They reported that while curved branched fins could expedite the melting only by 40 %, the angled branched fins were capable of reducing the melting time by virtually 85 %. Piran et al. [32] investigated different parameters related to spiral V-shaped fins on the HTF tube in a double-pipe PCM enclosure. They studied different heights, pitch, and numbers of these fins to enhance the melting dynamic. As explained in their research, utilizing four fins can lead to around 66 % quicker melting within the enclosure. NematpourKeshteli et al. [33] employed NAs, porous materials, and Y-shaped fins to improve the melting dynamic of RT54HC in a triplex-tube unit with a lobed design. They revealed that the simultaneous use of NAs and foams can expedite the phase change dynamic by around 74 %. Huang et al. [34] proposed an innovative cooling system integrated with PCM to improve energy efficiency in data centers. They designed a biomimetic LHTES device with palmate leaf-shaped fins and developed a three-dimensional (3D) phase-change heat transfer model to analyze its performance. Their study revealed that the biomimetic design significantly enhanced charge and discharge rates by 26.4 % and 61 %, respectively, compared to traditional designs. Zhang et al. [35] introduced a fractal tree-shaped fin design for shell-and-tube PCM units to enhance energy discharge performance. Using a 2D unsteady model, they explored the transient temperature distribution, solid-liquid interface evolution, and dynamic changes in sensible and latent heat. Their results demonstrated that tree-shaped fins significantly improve energy discharge rates and temperature uniformity, reducing solidification time by 66.2 % compared to radial fins. Abdeldjalil et al. [36] worked on various types of fins including normal, tilted, and long-arm swastika fins to accelerate the melting dynamic in a PCM enclosure in a 2D computational investigation. Based on their study, long-arm swastika fins provided the best performance, increasing the melting dynamic by more than two times compared to conventional rectangular fins. Liu et al. [37] studied several longitudinal fin designs in an attempt to expedite the melting dynamic in a 2D energy storage unit filled with RT42 PCM. Triangular, trapezoidal, and rectangular fins were tested. According to their plots, triangular-shaped fins managed to raise the melting pace by around 16 %. In the investigation carried out by Peng et al. [38], a 2D rectangular enclosure containing organic PCMs was computationally modeled to assess the influence of fin installation height and intersectional angle. They stated that optimization of these parameters can expedite the melting dynamic by around 10 %. In the 2D computational analysis carried out by Sharma et al. [39], the water PCM solidification process was modeled in a tube-in-tank enclosure. They tested different tube orientations and diameters to enhance the enclosure and also considered the addition of various numbers of longitudinal fins. Based on their report, using fifteen fins expedites the heat exchange rate by around 20 %. In the study of Basem et al. [40], the enthalpy-porosity method was employed to model the phase change process in a 2D half-cylindrical container. They analyzed the impact of varying the number of copper rods on the melting process and found that the addition of copper rods significantly reduces the melting time by up to 52 % compared to a container without rods. Kadhim et al. [41] used a 2D numerical analysis to study the phase change heat transfer in a square PCM container. They evaluated four configurations with copper fins of varying lengths and different orientations. Their findings showed that longer fins significantly accelerate the melting process, reducing melting time by up to 55.56 %, compared to the configuration without fins.

As illustrated in this literature review, it is clear that researchers are constantly trying their best to explore suitable fin designs in different configurations and forms of PCM enclosures. However, it is important to note that the majority of the research studies are focused on simple shell-and-tube geometries or rectangular cavities which can be easily simplified as 2D models. Aside from several exceptions in this regard, even most of the limited fraction of available 3D studies are usually focused on simple tube designs like double-pipe and shell-and-tube geometries of PCM containers, and the evaluation of fin utilization in complex PCM enclosures like shell-and-coil units is quite rare to find. Especially, research studies on ice storage systems are even more sparse, and the studies usually consider the melting behavior of other PCMs like paraffin and fatty acids. In this study, for the first time, the effects of various spiral longitudinal fin geometries are studied in a shell-and-coil ice storage system for applications in air-conditioning of buildings. With this goal in mind, a set of complex 3D transient computational models was created to evaluate several dimensionless parameters related to the length, orientation, thickness, and number of spiral longitudinal fins on the solidification front and liquid fraction reduction dynamic in shell-and-coil ice storage enclosures. Several dimensionless parameters related to the spiral longitudinal fin geometries were defined, which included dimensionless fin length (Ω), count (Ψ), and thickness (Φ) parameters. Additionally, the impact of fin orientation was evaluated. The influence of these parameters on the phase change rate will be later discussed throughout the manuscript.

2. Model definition

As discussed previously, this study tries to investigate how various dimensionless parameters related to spiral longitudinal fin characteristics can affect the phase change dynamic in a cold PCM enclosure in a transient computational model. Fig. 2 depicts one of the examined PCM enclosures and provides the general sizing information needed for the recreation of the system. It is necessary to note that the shell diameter was 260 mm. The enclosure is replenished with water as the PCM, and the tubes as well as its associated spiral longitudinal fins are made of C12200 alloy. Additionally, the ethylene glycol solution is selected as the cold fluid flowing through the tubes.

Fig. 2.

One of the assessed PCM enclosures and its sizing information.

To be able to carry out the assessment, several geometries with various spiral longitudinal fin characteristics were defined, which are tabulated in Table 1. It is noteworthy that the dimensions of the spiral fins were selected through a combination of geometric constraints, material availability, and thermal performance considerations, like preventing the fins from becoming too close to each other to avoid serious challenges while meshing.

Table 1.

Dimensionless variables created to study the spiral fin characteristics.

Parameter/Purpose	Dimensionless fin length				Dimensionless fin count				Dimensionless fin thickness				Fin orientation
	Ω = Fin length/Tube diameter				Ψ = Number of fins				Φ = Fin thickness/Tube thickness				–
Geometry	A	B	C	D	A	E	C	F	A	C	G	H	A	E	I
Value/State	0	0.359	0.719	1.078	0	2	4	6	0	1	2	3	–	Vertical	Horizontal
Fin length [mm]	0	3.4225	6.845	10.2675	6.845				6.845				6.845
Fin count	4				0	2	4	6	4				2
Fin thickness [mm]	1.25				1.25				–	1.25	2.5	3.75
Fin orientation	Vertical				Vertical				Vertical				–	Vertical	Horizontal
Additional heat exchange surface [cm2]	0	352.4	702.4	1052.5	0	349.8	702.4	1053.4	0	702.4	709.1	719.5	349.8
Additional heat exchange surface [%]	0	43.3	86.4	129.5	0	43	86.4	129.6	0	86.4	87.3	88.5	43

Moreover, the data related to each of the substances involved in the simulation (Table 2) was fed into the model.

Table 2.

Thermophysical characteristics of substances employed [42,43].

Material/Properties	Thermal Capacity [J/(kg·K)]		Viscosity [Pa·s] × 102	Thermal Conductivity [W/(m·K)]		Thermal Expansion Coefficient [1/K] × 107	Density [kg/m3]	Freezing point [K]	Latent heat of fusion (kJ/kg)
	Solid	Liquid		Solid	Liquid
C12200 alloy	386	–	N/A	330	–	N/A	8940	N/A	N/A
Ethylene glycol solution	–	3546	0.735	–	0.408	N/A	1055.51	N/A	N/A
Water	2217	4180	0.162	1.918	0.578	−6.733	958.4	273.15	334

The simulations were conducted by ANSYS Fluent 2023 R2, where the governing equations were solved utilizing a numerical approach, after considering the following assumptions:

• •The flow is 3D, transient, incompressible, and laminar.
• •Small vibrations of the tube throughout the freezing prevent supercooling of the PCM.
• •Except for density, phases of PCM possess separate characteristics.
• •Fins and the tube are considered completely integrated, eliminating the necessity to consider the contact resistance between them.
• •Phase change dynamic is modeled through the enthalpy-porosity approach [44].
• •Phase change phenomenon is only considered for the PCM because the other materials like the HTF are far from their phase change temperature.
• •Enclosure outer boundaries are well-insulated; therefore, no thickness or material was selected for the shell [9].
• •The Boussinesq approximation is accountable for considering the impact of natural convection.

3. Equations and solver adjustments

The governing equations for the PCM include:

The mass conservation [45]:

$$\frac{\partial }{\partial t}\rho +\nabla \cdot \left(\rho \overrightarrow{v}\right)=0$$ (1)

In which, t, ρ, and $\overrightarrow{v}$ stand for time, density, and velocity vector.

Navier–Stokes [24]:

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\rho \overrightarrow{v}\cdot \left(\nabla \cdot \overrightarrow{v}\right)=\mu {\nabla }^2\overrightarrow{v}-\nabla P+\rho \overrightarrow{g}\beta \left(T-T_0\right)+{\overrightarrow{S}}_v$$ (2)

Where, μ, $\overrightarrow{g}$, β, T, and ${\overrightarrow{S}}_v$ denote viscosity, gravitational acceleration, thermal expansion coefficient, temperature, and Darcy law source term:

$${\overrightarrow{S}}_v=R_{mush}\frac{{\left(1-\xi \right)}^2}{{\xi }^3+\epsilon }\overrightarrow{v}$$ (3)

$R_{mush}$, ε and ξ indicate the mushy zone constant (which is considered 10⁵ based on previous studies [9,46]), an insignificant value for preventing the denominator from becoming zero, and the liquid fraction parameter. Some studies like [47] have examined the influence of ε on the phase change; however, based on the majority of previous studies focusing on the solidification of water [9,46], it is suggested that the value of 10⁵ offers a balance between numerical stability and physical accuracy. The simulation of the phase change process is performed using the enthalpy-porosity method which was first introduced by Voller and Prakash [44]. The liquid fraction can be written as:

$$\xi =\{\begin{array}{cc}0& T<T_s\\ \frac{T-T_s}{T_l-T_s}& T_s\le T\le T_l\\ 1& T>T_l\end{array}$$ (4)

And the energy conservation equation:

$$\frac{\partial }{\partial t}\left(\rho h\right)+\nabla \cdot \left(\rho \overrightarrow{v}h\right)=\nabla \left(k\nabla T\right)$$ (5)

Where, k and h are the thermal conductivity and total enthalpy, which is defined using:

$$h=h_{ref}+{\int }_{T_{ref}}^T{C}_{p}dT+\xi L$$ (6)

In which, $C_p$, and L is the specific thermal capacity and latent heat of fusion. In the previous equations, the density was defined using the Boussinesq approximation:

$$\rho ={\rho }_0\left(1-\beta \left(T-T_0\right)\right)$$ (7)

Where, the ${\rho }_0$ and $T_0$ are employed to express the initial density and temperature of the PCM. It is clear that in all of the above, and the below equations, PCM and HTF data from Table 2 should be applied, respectively.

For the HTF, the mass conservation, Navier–Stokes, and energy conservation equations are simpler:

$$\nabla \cdot \overrightarrow{v}=0$$ (8)

$$\rho \frac{\partial \overrightarrow{v}}{\partial t}+\rho \left(\overrightarrow{v}\cdot \nabla \right)\overrightarrow{v}=-\nabla P+\mu {\nabla }^2\overrightarrow{v}$$ (9)

$$\frac{\partial \mathrm{T}}{\partial t}+\nabla \cdot \left(\overrightarrow{v}T\right)=\nabla \cdot \left(\frac{k}{\rho C_p}\nabla T\right)$$ (10)

Regarding the model conditions, it is necessary to note that the HTF flow is selected to be flowing at the Reynolds number of 2000 at 253.15 K at the inlet, while for the outlet 0 Pa pressure is imposed. The solution is commenced in a condition where the entire regions are considered to be stationary at 273.25 K. Additionally, the precision of the solver in each time step was defined to be 10⁻⁶ for energy and 10⁻³ for continuity and momentum equations.

4. Validation and sensitivity examinations

To verify that the simulations progressed as meticulously as expected, several crucial procedures were followed. Initially, the geometry from the experimental study of Ezan et al. [48] was computationally modeled to take a closer look at the freezing dynamic and compare the predictions originating from the current model against the real-world findings from their experiments. The system utilized water (in the Plexiglass shell) and ethylene glycol at 273.15 K (in the polythene tube) as the operating fluids in the shell-and-tube container which was employed for the experimental test. Essential dimensions of this container included a 400 mm length, a 190 mm shell diameter, and a 15 mm tube inner diameter. The stored cold energy parameter was considered as the criteria for comparing the findings, which is observable in Fig. 3a. The similarity in the progression pattern and the maximum difference of 5.64 % reveal the validity of the model in precisely anticipating the results.

Fig. 3.

(a) Model validation with the experimental work of Ezan et al. [48], (b) grid, and (c) time step size sensitivity assessment, and (d) the final grid for geometry D.

Following this step, the computational model sensitivity toward grid (Fig. 3b) and time-step (Fig. 3c) sizes were determined. As illustrated, 3.10 million elements and a time-step size of 0.5 s provide trends that are close enough to their finer counterparts, verifying that the generated grid (Fig. 3d) and time-step size are logically selected, and are precise enough to carry out the remaining computations.

5. Results and discussion

This part discusses the impact of different dimensionless numbers defining the spiral longitudinal fin geometries as well as their orientation on the quantitative and qualitative aspects of the charging process of the PCM enclosure. Freezing time, as well as liquid fraction (ξ) and average PCM temperature (T) plots and contours, are considered decisive factors in analyzing the fin performance, and they will be properly discussed in this section.

5.1. Dimensionless fin length (Ω)

To assess how the freezing dynamic is altered by the length of the longitudinal spiral fins, the Ω dimensionless parameter is studied. A higher number for this variable shows a longer fin as compared to the tube diameter where the fins are installed. Fig. 4a and b provide the alternations of the volume-averaged liquid fraction (ξ) and average PCM temperature (T) parameters throughout the freezing, respectively. According to the plot, the completion of the process by reaching the ξ = 0 for the geometries with the Ω of 0, 0.359, 0.719, and 1.078 is observed to happen after 266, 186.3, 153.5, and 146.5 min, showing that the fins with Ω of 0.359, 0.719, and 1.078 offer an improvement of 29.96, 42.29, and 44.92 %, respectively. The reason for this enhancement is the rise of the heat transfer surface which enables the heat from the enclosure to be carried to the HTF tube more conveniently. The presence of fins, which are made of thermally conductive material presents a faster way for the coldness to infiltrate the parts of the enclosure which are initially beyond the reach of the HTF tube. Based on the aforementioned values, it is apparent that while the presence of spiral longitudinal fins and their growth in length is a considerable factor in the acceleration of the freezing dynamic, after Ω = 0.719, the growth in length is just not that significant anymore, suggesting that keeping the length at moderate levels seems logical. Regarding the changes in the T value, it can be said that the average temperature somehow obeys the freezing dynamic within the enclosure, because without freezing the temperature cannot go beyond the freezing point, but as the freezing is carried out, the average temperature can immediately drop.

Fig. 4.

Time-dependent alternations of the average (a) ξ, and (b) T parameters throughout the freezing with different Ω values.

To qualitatively monitor the freezing dynamic within the enclosure, the ξ contours can be informative. For this purpose, they are provided in Fig. 5 for the models with various Ω values of 0, 0.359, 0.719, and 1.078.

Fig. 5.

The alternations of the ξ contours around the HTF tube with different Ω values.

According to the contours, the presence of spiral longitudinal fins not only alters the freezing dynamic in the enclosure but also changes the freezing front pattern around the HTF tube. As the dimensionless length parameter grows, the circular form of ice around the tubes changes its shape to a rotated square, as the sides covered by fins face faster freezing compared to finless portions of the tube, creating a less uniform solidification around the tube, which obeys the location of the fins as well. Comparing the geometries with various fin length ratios, it can be inferred that longer fins infiltrate the enclosure better, and the difference regarding the shape of the freezing front becomes more obvious as the Ω value is raised. Moreover, it is observed that the geometries with Ω = 0.719 and Ω = 1.078 are very similar in terms of the solidification pattern and its advancement in the enclosure. It is also evident that when the frozen areas reach each other and make contact, the region in the middle of the enclosure solidifies immediately, and reduces the liquid fraction value fast, which is observed in the liquid fraction plot. Even though the effects of natural convection are considered throughout the process, it can be inferred from the liquid fraction contours that no significant difference in the freezing fronts in the top and bottom of the enclosure can be found. More information on the influence of natural convection in shell-and-coil ice storage units is given in the work of Afsharpanah et al. [9].

5.2. Dimensionless fin count (Ψ)

To measure the impact of dimensionless fin count on the freezing dynamic within the enclosure, the fin counts of Ψ = 0, 2, 4, and 6 were tested. Fig. 6a and b illustrate the alternations in the average ξ and T values throughout the freezing, respectively. According to the plot, the complete freezing (ξ = 0) with the fin counts of Ψ = 0, 2, 4, and 6 needs 266, 203.5, 186.3, and 133 min. This indicates that compared to the bare tube, the presence of 2, 4, and 6 fins can enhance the dynamic by 23.49, 29.96, and 50 %, respectively. Based on this trend, it can be inferred that, unlike the previous parameter (Ω), raising the fin count within the studied range can constantly improve the freezing dynamic and does not lose its impact after the addition of several fins. It is also understood that this parameter can be even slightly more influential than the Ω. When the fin count is increased, the highly-conductive fins can better distribute the HTF coldness throughout the enclosure, facilitating the freezing dynamic considerably. The average T alternations follow a similar pattern to the liquid fraction plots. It starts to gradually decrease as the ice is formed around the finned tubes, and after some time, a considerable drop in the T value is observed, which as will be shown in the next figure, belongs to when the middle region between the spiral tubes is completely frozen.

Fig. 6.

Time-dependent alternations of the average (a) ξ, and (b) T parameters throughout the freezing with different Ψ values.

To better examine the freezing dynamic, it is essential to observe the ξ contours. Fig. 7 depicts the alternations in the freezing behavior with various fin counts of Ψ = 0, 2, 4, and 6 installed on the HTF tube.

Fig. 7.

The alternations of the ξ contours around the HTF tube with different Ψ values.

As can be seen, the amount of formed ice as well as its shape are significantly altered by changing the number of spiral longitudinal fins installed on the tube. While the finless geometry exhibits circular patterns of ice in the initial stages, the model with two fins shows a vertical elliptical pattern. On the other hand, the geometry with four fins provides a square shape pattern. Nonetheless, it is evident that when the number of spiral longitudinal fins is continuously increased, the frozen shape approaches a circular pattern again, because the addition of a high number of fins allows the coldness to distribute in all directions from the HTF tube more evenly, making the tube to act like a tube with a larger diameter with a circular freezing pattern around it. By comparing the ξ contours and plots, it can be clearly inferred that the abrupt change in the slope of the ξ plot corresponds to the moment when the freezing fronts formed on the spiral tube finally encounter each other in the middle region of the enclosure. In this situation, a significant amount of ice is formed in the enclosure, which is reflected as a rapid advancement in the liquid fraction decrease and an abrupt reduction in the average T which were observed in the previous plots.

5.3. Dimensionless fin thickness (Φ)

To assess how the freezing dynamic can be influenced by the spiral longitudinal fin thickness, simulations were carried out with various dimensionless fin thicknesses. The results of the time-dependent alternations of the average ξ and T parameters with various Φ values are provided in Fig. 8a and b, respectively. Based on the figure, it is understood that the complete freezing of the enclosure (ξ) with the dimensionless fin thickness values of Φ = 0, 1, 2, and 3 takes 266, 153.5, 137.7, and 114.5 min respectively. This indicates that the use of fin thicknesses of equal, double, and triple the tube thickness can improve the freezing dynamic by 42.29, 48.23, and 56.95 %, respectively, showing that even though making the fins thicker can improve the freezing dynamic by creating more significant thermal conduction in the enclosure, the thin fins offer considerable improvement as well. Additionally, as can be observed, unlike the previous cases, the results are close within the first 90 min. This can be explained through the ξ contours showing the process of phase shift around the finned tubes (Fig. 9).

Fig. 8.

Time-dependent alternations of the average (a) ξ, and (b) T parameters throughout the freezing with different Φ values.

Fig. 9.

The alternations of the ξ contours around the HTF tube with different Φ values.

As can be observed, while the difference in the freezing front with Φ values of 1, 2, and 3 are considerably similar at first, after around 90 min, the ice completely engulfs the area around the tubes and fins, and subsequently, because of the difference between the thermal conductivity of water and ice, the improvement of thermal conduction using fins becomes more important, making the conduction more reliant on the fins. Based on this, the geometries with a thicker fin could transfer heat in this frozen block more efficiently, which led to faster freezing. In conclusion, the influence of fin thickness is mostly concentrated in the final stages of the freezing process. This is of importance especially because if the enclosure is being designed for an external melting scenario (where a complete solidification is prevented and during the melting the HTF is directly injected into the shell [9]) using fins with Φ values of 1, 2, and 3 offer an almost equal improvement. On the other hand, if the enclosure is designed for an internal melting scenario, where complete freezing is possible, thicker fins can offer slightly better results.

5.4. Fin orientation

Considering that in cases with fewer fins, such as Ψ = 2, the fin orientation might be affecting the freezing dynamic as well, in this section, an attempt was made to assess its impact on the process. To carry out this examination, the geometries with horizontal and vertical orientations of the spiral longitudinal fins with Ω = 0.719, Ψ = 2, and Φ = 1 dimensionless specifications were compared. Fig. 10a and b show the alternations of the average ξ and T parameters throughout the freezing, respectively. Based on the plot, the ξ = 0 which corresponds to a complete freezing within the enclosure can be acquired after 266, 203.5, and 196.2 min for the finless geometry, and the geometry with vertical and horizontal spiral longitudinal fins, respectively. This indicates that the vertical and horizontal spiral longitudinal fins offer a freezing acceleration of 23.49 and 26.24 compared to the finless geometry, respectively. In other words, the horizontal spiral longitudinal fins offer 3.58 % better performance compared to that of the vertical spiral longitudinal fins. This is especially worthy of attention because, in this comparison, the added heat transfer surfaces were identical. The reason for this difference will be later explained using the ξ contours. The alternation of the average T value also follows the changes in the ξ, because, without phase shift in the enclosure, the temperature of each cell cannot be decreased beyond the freezing point, making this parameter heavily reliant on ξ.

Fig. 10.

Time-dependent alternations of the average (a) ξ, and (b) T parameters throughout the freezing with different fin orientations.

To understand the difference and also take a closer look at the freezing dynamic within the enclosure, the ξ contours are presented in Fig. 11 for the finless model and the models with vertical and horizontal fins.

Fig. 11.

The alternations of the ξ contours around the HTF tube with different fin orientations.

As observed in the contours, it can be inferred that while a finless geometry exhibits a circular pattern of freezing around the HTF tube, the geometries with vertical and horizontal spiral longitudinal fins show vertical and horizontal elliptical freezing fronts, respectively. Based on the geometry specifications provided previously, it is clear that the vertical distance between each turn of the HTF tube centerline is 54 mm, and this distance is traversed by two solidification fronts from the top and bottom tubes, while the horizontal distance between the HTF tube centerline and the enclosure center is 65 mm. This underlines the fact that the infiltration of the HTF coldness in the horizontal direction needs more help and explains why the horizontal fins offer slightly better results in terms of the complete freezing time within the enclosure.

5.5. Temperature uniformity analysis

In this section, temperature contours are used to analyze the temperature distribution of the containers in the base and the best configurations examined in the current study. Based on the previous stages, it was found that while the case without any fins (Case A) provides the slowest rate of phase change, Case F with Ω = 0.719, Ψ = 6, and Φ = 1 provides the fastest solidification rate among the cases with thin fins. Fig. 12 compares the temperature and liquid fraction distribution of PCM in these two cases over time, so that the liquid fraction and temperature contours are compared and also the effect of the fins on the temperature distribution becomes clearer.

Fig. 12.

The comparison of the T and ξ contours changes around the HTF tube in the base and best cases.

By taking a close look at the temperature and liquid fraction of each case, it is apparent that the fluid region of the container is completely uniform, having a constant temperature of 273.15 K, equal to the solidification temperature. Of course, in the first several minutes of the process, there is a temperature distribution in the fluid section as well, but the temperature becomes uniform shortly after the start of the process. This is because, in the current research, the initialization temperature of the container is 273.65 K, meaning that the PCM temperature at the start of the process is 0.5 K higher than the freezing temperature of the container. After the process starts and the coolant flows into the spiral coils, the PCM temperature immediately drops to the solidification temperature, where it is faced with the latent heat of fusion as a barrier that does not allow the temperature to drop any further. In this condition, the PCM stays uniform, and ice starts to form layer by layer on the HTF tubes and fins, while the fluid region of the PCM possesses an absolute temperature uniformity. Thus, the temperature front and the solidification front look somehow similar, because the temperature distribution only happens within the region where solidification has already happened. Now, comparing the two cases, it is evident that the presence of fins accelerates the ice formation, and therefore, helps the fin more easily inject their coldness deep into the container. This can be easily inferred by comparing the temperature contours of Cases A and F. While in Case A the temperature variation is limited to a small region around the HTF tubes even 160 min after the start of the process, in Case F, the freezing front has occupied the entire computational domain and has created a temperature distribution within the frozen region.

6. Conclusions

The current investigation was concentrated on using transient and 3D simulations for examining the fin-assisted enhancement of PCM freezing in a cylindrical enclosure in the presence of spiral longitudinal fins. Different dimensionless variables were defined for the spiral longitudinal fins to be able to assess the influence of various geometrical parameters in these fins using the numerical model. The dimensionless parameters included the fin length/tube diameter (Ω), fin count (Ψ), fin thickness/tube thickness (Φ), and fin orientation. The effect of these variables was closely monitored on the liquid fraction (ξ) and temperature (T) alternations to elucidate their influence on the freezing dynamic of the ice storage unit. The highlights of the research are summarized below:

• •Raising the dimensionless fin length by up to Ω = 0.719 considerably expedited the freezing dynamic by 44.92 % compared to the finless condition. Even though the rise of this value to Ω = 1.078 reduced the complete freezing time by an additional 4.56 % compared to that of Ω = 0.719, the benefit was found to be marginal compared to the big leaps in the enhancement observed with previous stages of Ω value growth.
• •The addition of fins with the dimensionless fin count of Ψ = 6 raised the fin freezing dynamic by 50 % compared to the geometry lacking fins. The difference between the freezing time with Ψ = 4 and Ψ = 6 was 13.35 %, which shows that increasing the fin count within the studied range consistently enhances the freezing dynamic.
• •The use of fins changed the freezing front pattern around the HTF tubes from a circular shape with Ψ = 0 to an oval, a rotated square, and a hexagon shape with Ψ = 2 and Ψ = 4, and Ψ = 6, respectively. A higher number of fins approached the freezing front towards a circular shape again.
• •Raising the dimensionless fin thickness beyond Φ = 1 did not create a meaningful difference in the solidification of around 60 % of the enclosure; however, the freezing dynamic of the remaining 40 % was considerably altered by thickening the fins.
• •Even though the heat transfer surface in the horizontal and vertical orientations of the geometries with Ψ = 2 was the same, the horizontal spiral longitudinal fins outperformed their vertical counterparts by 3.58 %, because of the more suitable distribution of fins within the enclosure.
• •The natural convection did not change the freezing front shape at the top and bottom of the enclosure in a meaningful way.

For future works in this field, it is recommended that the researchers focus on the second law of thermodynamics analysis of such systems during the charging and discharging stages. Also, considering a cycle analysis can lead to fruitful results which were out of the scope of the current research. The usage of different innovative fins like anchor-type or tree-shaped ones in such systems and comparing their performance would probably yield interesting findings. In addition, providing a long-term economic analysis on the cost-efficiency of different fin types in thermal energy storage systems is recommended to readers specialized in this field.

CRediT authorship contribution statement

Masoud Izadi: Visualization, Software, Resources, Methodology, Investigation, Data curation, Conceptualization. Farhad Afsharpanah: Writing – original draft, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. Ali Mohadjer: Writing – review & editing, Visualization, Software, Resources, Data curation. Mostafa Omran Shobi: Resources, Formal analysis, Data curation. Seyed Soheil Mousavi Ajarostaghi: Writing – review & editing, Validation, Supervision, Investigation, Formal analysis. Federico Minelli: Writing – review & editing, Supervision, Resources, Investigation, Funding acquisition, Formal analysis.

Data availability

Data will be made available on request.

Nomenclature

$C_p$	Specific heat capacity [J·(kg·K)−1]
$\overrightarrow{g}$	Gravitational acceleration [m·s−2]
$h$	Enthalpy [J·kg−1]
$h_{ref}$	Reference enthalpy [J·kg−1]
$k$	Thermal conductivity [W·(m·K) −1]
$L$	Latent heat of fusion [J·kg−1]
$P$	Pressure [Pa]
$R_{mush}$	Mushy zone constant [kg·(m−3 s)−1]
$t$	Time [s]
$T$	Temperature [K]
$T_0$	Initial temperature [K]
$T_l$	Liquid temperature [K]
$T_{ref}$	Reference temperature [K]
$T_s$	Solid temperature [K]
$\overrightarrow{v}$	Velocity [m·s−1]
Greek Symbols
$\alpha$	Angle of attack [radians]
$\beta$	Thermal expansion coefficient [K−1]
$ϵ$	Small numerical constant (to avoid division by zero)
$\mu$	Viscosity [Pa·s]
$\xi$	Liquid fraction
$\rho$	Density [kg·m−3]
${\rho }_0$	Initial density [kg·m−3]
$\mathrm{\Phi }$	Dimensionless fin thickness
$\mathrm{\Psi }$	Dimensionless fin count
$\mathrm{\Omega }$	Dimensionless fin length
Subscripts
0	Reference conditions
s	Solid phase
l	Liquid phase
in	Inlet boundary
out	Outlet boundary
Abbreviations
2D	Two-dimensional
3D	Three-dimensional
HTF	Heat transfer fluid
NA	Nano additive
PCM	Phase change material

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.