Development of freezing process of phase change materials in cylindrical thermal energy storage tanks with various fin configurations

Abstract

One of the most effective methods for thermal energy storage relies on the latent heat property of phase change materials (PCMs). Fins are widely employed as an efficient technique to enhance heat transfer. However, precise modeling of the freezing process remains challenging due to the dynamic nature of the phase change boundary. This study introduces a novel semi-analytical approach based on Green’s function to investigate the impact of radial and longitudinal fins on the freezing process within cylindrical thermal energy storage tanks under time-dependent first and third type boundary conditions. Unlike previous studies that primarily focused on steady-state conditions or simplified geometries, this research provides a more realistic and comprehensive analysis of transient thermal behavior. The findings reveal that while fins initially accelerate freezing, their effectiveness diminishes over time due to thermal resistance effects. Radial fins consistently outperform longitudinal fins, reducing total freezing time by up to 75.31% and demonstrating a 19.68% efficiency improvement. These results offer new insights for optimizing the design of thermal energy storage systems, particularly in applications requiring efficient cold energy storage under varying boundary conditions.

Introduction

In modern district heating networks, an efficient thermal energy storage (TES) system is a key component. As thermal management becomes increasingly important in urban and industrial areas, the need for well-designed and planned TES systems is more pronounced. TES systems do not typically produce or consume thermal energy on their own; instead, their role in heat distribution networks is similar to that of batteries or capacitors in electrical distribution networks. They bridge the temporal gap between supply and demand, allowing excess production at one time to cover deficits and meet energy demand at another time¹.

Olfat and Talati² investigated how the number of fins influences the freezing of PCM within a cylindrical container with radial fins under fixed and time-dependent boundary conditions. They utilised the finite volume method based on the enthalpy formulation, they demonstrated that fins significantly enhance heat transfer under the first type of boundary condition. However, under the second and third types, the effect is less due to the heat returning from the fin to the PCM.

Soltani et al.³ conducted a numerical simulation using the enthalpy-porosity technique to improve heat transfer equipped with longitudinal fins and a rotating mechanism. Their findings indicated that the combined use of fins and rotation significantly reduced the melting and freezing duration of the PCM by 74% and 84%, respectively. Moreover, an increase in rotational speed enhanced the heat transfer rates during the melting and freezing.

Qaiser et al.⁴ performed an experiment to improve the melting performance of PCM-based thermal energy storage systems. They considered three shell types cylindrical, elliptical, and triangular around Y-shaped finned copper tubes. In a base case with a Y-shaped finned copper tube under fixed temperature boundary conditions inside an adiabatic cylindrical steel shell, they modelled and experimentally validated the heat transfer from the fluid to the surrounding PCM using ANSYS Fluent 19.0. Doubling the tubes with vertical configuration and tripling with V-shaped configuration improved the average heat transfer rate by 34% and 24%, respectively, compared to the base case, and reduced complete melting time by 28% and 22%. Both configurations with elliptical and triangular shells increased the average heat transfer rate by up to 85% and halved the complete melting time of the PCM.

Tiari et al.⁵ compared various annular fin configurations’ impact on enhancing thermal energy storage numerically. They simulated six different configurations using ANSYS Fluent 17.0 under fixed temperature boundary conditions. The 20-fin configuration with variable lengths (longest at the bottom of the vertical tube) reduced charging time by 84.4%, while the 20-fin configuration with uniform lengths optimised discharging time, reducing it by 79.2%. The 20-fin uniform length configuration was the most efficient, reducing overall time by 76.3%.

Khan and Khan⁶ investigated how different longitudinal fin configurations impact the efficiency of a horizontal cylindrical latent heat energy storage unit. Their study focused on the two-dimensional melting behaviour of stearic acid, used as the PCM, situated between an isothermal finned tube and a shell from steel. They evaluated various angular positions, ranging from ⅄ to Y configurations. The results revealed that the melting rate significantly increased with the Y configuration and decreased with the ⅄ configuration. Additionally, they found that a higher fin length-to-thickness ratio, greater thermal conductivity of the shell, and elevated heat transfer fluid temperature all significantly enhanced the heat transfer rate and reduced the melting time, thereby improving the overall system performance. Zare et al.⁷ introduced an innovative thermal system for lithium-ion batteries based PCM. They integrated both internal and external longitudinal fins to enhance the conductivity of PCM surrounding the battery. By employing an integrated thermal capacity model to simulate battery heat generation and the enthalpy-porosity method to analyse PCM melting.

Hasnain et al.⁸ performed a numerical simulation to explore the influence of branched longitudinal fins and nanoparticles on the melting process of PCM within a horizontal cylindrical latent heat system. They simulated the problem in two dimensions under convective boundary conditions and validated their findings with experimental data. Their results indicated that dual-branched longitudinal fins reduced the melting time by 45.9% compared to the Y-shaped base fin configuration. The addition of nanoparticles accelerated the melting and solidification rates of the PCM while maintaining stable heat transfer in the base scenario. Similarly, Liu et al.⁹ conducted a study on the impact of T-shaped fins on PCM’s energy storage. Fins with a vertical-to-horizontal length ratio (T-shaped) greater than 1 outperformed the longitudinal planar fins, reducing the melting time by 34.5%.

Rozenfeld et al.¹⁰ conducted an experimental study with numerical and analytical modelling on thermal energy storage tanks with helical fins. They tested melting under ambient air exposure (ordinary melting) and with slight shell heating to achieve the close-contact melting (CCM) phenomenon. With CCM, the melting time tripled. They numerically and analytically analysed CCM on the helical surface, finding good agreement with experimental data by determining the fin temperature variation and extending the analytical model to various helical fin pitches and non-isothermal fin conditions. Rabinejad Darzi et al.¹¹ investigated the effects of geometry, longitudinal fins, and nanoparticles on PCM melting and solidification in a cylindrical tank. They considered four tank models with circular shell and elliptical horizontal, elliptical vertical, circular, and finned circular internal channels. Results showed higher melting rates at the top due to natural convection, with vertical elliptical channels reducing complete melting time compared to non-finned channels but being less effective during solidification. Adding nanoparticles increased melting and solidification rates while maintaining stable heat transfer. The most efficient geometry was the finned circular channel, which enhanced solidification efficiency more than melting by reducing natural convection.

As for analytical research, Alsulami et al.¹² simplified freezing processes under third-type boundary conditions (convective surface cooling) with volumetric heating in both phases using a quasi-steady assumption (St ≪ 1) and solved the Stefan problem for planar, cylindrical, spherical, and semi-infinite geometries analytically and approximately. They solved the nonlinear ODEs for the time-dependent dimensionless interface location and steady-state interface position using Runge–Kutta and Newton–Raphson methods, matching previous works. Krishnan et al.¹³ studied heat transfer in PCMs under unsteady heat flux boundary conditions in flat, cylindrical, and spherical geometries. Assuming negligible free convection, constant phase change temperature, and small Stefan numbers, they solved the governing equations using the special function expansion method, showing excellent agreement with previous works and numerical simulations. Xu et al.¹⁴ analytically and approximately studied PCM freezing in a hollow cylinder under Dirichlet boundary conditions, solving the two-phase Stefan problem using the asymptotic method with four spatial and three temporal scales (four-layer and three-regime). They validated results with numerical solutions from the enthalpy method.

Mostafavi et al.¹⁵ investigated the effect of straight fins in a rectangular tank under time-dependent first-type boundary conditions using the perturbation method. They split heat transfer into two separate one-dimensional problems for fins and walls, finding an optimal fin size as excessive fin size reduced direct PCM-wall contact. They neglected temperature-dependent thermal properties and natural convection in the liquid phase. Parhizi and Jain¹⁶ presented a solution for a one-dimensional PCM problem in Cartesian coordinates under unsteady conditions. Their solution showed increased accuracy without divergence at higher times, unlike previous works.

Parhizi and Jain¹⁷ theoretically solved a heat transfer problem with an initial pre-melted or pre-frozen region using a perturbation-based method, coupling time-dependent first-type boundary conditions with phase change, and found good agreement with numerical simulations. Hazrati and Jain¹⁸ semi-analytically studied heat transfer with phase change in a thick multi-layered cylindrical body encapsulating PCM using special function expansion under third-type boundary conditions, predicting PCM phase change rates. Assuming negligible natural convection and constant properties, they showed good agreement with experimental data and finite element simulations, closer to real conditions.

Parhizi and Jain¹⁹ used perturbation-based techniques to study thermal properties in Cartesian and cylindrical coordinates under fixed and time-dependent first-type boundary conditions, finding that increased thermal conductivity improved overall energy storage but decreased energy density in cylindrical systems due to faster melting, whereas Cartesian systems showed no change. Thus, high thermal conductivity PCM may not be ideal for compact energy storage. Mostafavi and Jain²⁰ analytically examined heat transfer between a PCM bed and a laminar convective flow. Using the perturbation method, they solved the phase change heat transfer and governing convective heat transfer equations for fluid over the bed, validating results with numerical simulations and showing invariance with initial guess.

Despite extensive research on TES, most studies have been limited to simple geometries and steady-state boundary conditions, neglecting the influence of realistic, time-dependent conditions on phase change behavior. Additionally, existing numerical methods often lack analytical validation, making precise modeling challenging.

The main contributions of this study are:

• 1. Developing a semi-analytical approach using Green’s function to model the freezing process in cylindrical TES tanks under realistic, time-dependent boundary conditions.

• 2. Investigating and comparing the effects of radial and longitudinal fins on PCM freezing, providing a quantitative assessment of their efficiency.

• 3. Demonstrating that radial fins outperform longitudinal fins, reducing total freezing time by up to 75.31% and improving efficiency by 19.68%.

• 4. Providing practical design insights for optimizing TES systems in applications such as HVAC, industrial cooling, and renewable energy storage.

The primary aim of this work is to develop an accurate and computationally efficient semi-analytical model that can predict the freezing front position in finned TES tanks. The key objective is to determine the most effective fin arrangement for maximizing PCM freezing efficiency under different thermal boundary conditions. This research provides new insights into PCM-based energy storage optimization, particularly in applications requiring efficient cold energy storage.

Model description

In this research, two concentric cylindrical containers are considered, where the inner cylinder has radial and longitudinal fins (Fig. 1). The effect of these fins on the solidification under time-dependent first and third type boundary conditions is examined. Due to the non-linearity of the governing equations, an exact solution for phase change problems in finned energy storage tanks is not feasible in two dimensions. Therefore, the problem is simplified to a semi-analytical one, which will indicate the position of the solidification front at any given time. The semi-analytical approach involves dividing the two-dimensional problem into two one-dimensional problems, a method that has yielded acceptable results in previous studies²¹.

Fig. 1

Cylindrical energy storage tank: (a): without fin, (b): with radial fins, and (c): with longitudinal fins. The image was created by the authors using COMSOL Multiphysics® software, version 5.5 (https://www.comsol.com).

The following assumptions are made to develop an accurate mathematical model and simplify the problem due to the non-linearity of the solidification equations^21,22,23:

• Initially, all the PCM is in a liquid state at the melting condition of ${T}_{m}$. The fins’ temperature is also equal to ${T}_{m}$.

• Convective heat transfer during solidification is negligible, and heat transfer is assumed to be uniform and only through conduction.

• The process of phase change occurs at a specific temperature (isothermal), and the PCM is homogeneous.

• All thermophysical properties are assumed to be constant and independent of temperature.

• Due to high thermal conductivity, uniform cross-section, and thin fins, heat transfer in the fins considers it as a one-dimensional problem confined to the radial direction $r$.

Based on these assumptions, control volumes from these radial and longitudinal finned tanks containing PCM are considered, as shown in Figs. 2 and 3. These control volumes are subject to thermal boundary conditions applied to the inner wall. By dividing each control volume into two separate regions, each one is solved as one-dimensional problems. In each control volume of radial or longitudinal fins, the thermal boundary condition applied to the inner wall is the only source of heat transfer, and the solidification front grows only in the radial direction. In the second region, heat transfer occurs only through the fins, and since energy travels the shortest path, the solidification front moves perpendicular to the fins. Finally, by superimposing the one-dimensional solidification fronts from the two regions, the two-dimensional solidification of the PCM in the respective control volumes is obtained.

Fig. 2

(a) Dimensions of the control volume of a cylindrical storage tank with radial fins based on the reference²⁴, and (b) Division of the two-dimensional control volume into two separate regions based on the reference²⁴.

Fig. 3

(a): Control volume of a cylindrical storage tank with longitudinal fins based on the reference²⁴, and (b): Division of the two-dimensional control volume into two separate regions based on the reference²⁴.

Radial fin tank (semi-analytical definition and solution of the freezing problem in region 1)

In this region, heat conduction occurs only in the radial direction $r$. The governing equations for the one-dimensional Stefan problem with different first and third type boundary conditions are as follows^2,25:

$$\frac{1}{r}\frac{\partial }{\partial r}\left[ {r\frac{{\partial T_{s} }}{\partial r}} \right] = \frac{1}{{\alpha_{s} }}\frac{{\partial T_{s} }}{\partial t},\;\;R_{in} < r < S\left( t \right),\;\;t > 0$$

(1)

$$\rho {\mathcal{L}}\frac{dS\left( t \right)}{{dt}} = k_{s} \frac{{\partial T_{s} \left( {S\left( t \right),t} \right)}}{\partial r}$$

(2)

$$S\left( 0 \right) = R_{in}$$

(3)

$$T_{s} \left( {r,0} \right) = T_{m}$$

(4)

$$T_{s} \left( {S,t} \right) = T_{m}$$

(5)

Dirichlet boundary condition (first type):

$$T_{s} \left( {R_{in} ,t} \right) = T_{w} \left( t \right),\;\;T_{w} \left( t \right) < T_{m}$$

(6)

Third type boundary condition:

$$- k_{s} \frac{{\partial T_{s} \left( {{ }R_{in} ,t} \right)}}{\partial r} + h\left[ {T_{s} \left( {{ }R_{in} ,t} \right) - T_{\infty } \left( t \right)} \right] = 0,\;\;T_{\infty } \left( t \right) < T_{m}$$

(7)

Radial fin tank (semi-analytical definition and solution of the freezing problem in region 2)

In Region 2, heat transfer occurs solely through the fins, and the movement of the freezing front is only along the direction perpendicular to the fins (in the $z$ direction). According to Fig. 4, a small annular element with a rectangular cross-section and thickness is selected ($dr$) from the radial fin and write the energy balance for this element:

$$\dot{E}_{st} = q_{in}^{\prime \prime } + q_{r + dr}^{\prime \prime } - q_{r }^{\prime \prime }$$

(8)

Fig. 4

One-dimensional element of a radial fin with thickness $dr$.

The terms in the above energy balance equation are as follows:

• Rate of change of internal energy per unit cross-sectional area of the fin:

$$\dot{E}_{st} = mc\frac{\partial T}{{\partial t}} = \rho_{f} (2\pi r\delta dr)c_{f} \frac{{\partial T_{f} }}{\partial t}$$

(9)

• Rate of heat transfer from the freezing front to the fin:

$$q_{in}^{\prime \prime } = - k_{s} \left( {2\pi rdr} \right)\frac{{T_{f} - T_{m} }}{S\left( t \right)}$$

(10)

• Net rate of heat transfer through conduction in the fin:

$$q_{r + dr}^{\prime \prime } - q_{r}^{\prime \prime } = \left[ {k_{f} \left( {2\pi r\delta } \right)\frac{{\partial T_{f} }}{\partial r} + \frac{\partial }{\partial r}\left( {k_{f} \left( {2\pi r\delta } \right)\frac{{\partial T_{f} }}{\partial r}} \right)dr} \right] - \left[ {k_{f} \left( {2\pi r\delta } \right)\frac{{\partial T_{f} }}{\partial r}} \right]$$

(11)

Now, by substituting the above expressions into the above equation and simplifying it, the governing equation for the temperature distribution of the radial fin is derived, and its initial and boundary conditions are considered as follows²⁶:

$$T_{f} \left( {r,0} \right) = T_{m}$$

(12)

$$\frac{{\partial T_{f} }}{\partial r}\left( {R_{out} ,t} \right) = 0$$

(13)

First type boundary condition:

$$T_{f} \left( {R_{in} ,t} \right) = T_{w} \left( t \right)$$

(14)

Third type boundary condition:

$$- k_{f} \frac{{\partial T_{f} \left( {R_{in} ,t} \right)}}{\partial r} + h\left[ {T_{f} \left( {R_{in} ,t} \right) - T_{\infty } \left( t \right)} \right] = 0$$

(15)

Longitudinal fin reservoir (definition and semi-analytical solution of the freezing problem in region 1)

The freezing problem in Region 1 for a reservoir with longitudinal fins is identical to that of a reservoir with radial fins, as described in Section “Radial fin tank (semi-analytical definition and solution of the freezing problem in region 1)”.

Longitudinal fin reservoir (definition and semi-analytical solution of the freezing problem in region 2)

In the longitudinal fin arrangement, the movement of the freezing front in Region 2 remains perpendicular to the fins. As illustrated in Fig. 5, a small cubic element is selected with a rectangular cross-section and thickness $dr$ from the longitudinal fin and write the energy balance for this element:

$$\dot{E}_{st} = q_{in}^{\prime \prime } + q_{r + dr}^{\prime \prime } - q_{r }^{\prime \prime }$$

(16)

Fig. 5

One-dimensional element of a longitudinal fin with thickness $(dr)$.

The terms in the above energy balance equation are as follows:

• Rate of change of internal energy per unit cross-sectional area of the fin:

$$\dot{E}_{st} = mc\frac{\partial T}{{\partial t}} = \rho_{f} (\delta dr)c_{f} \frac{{\partial T_{f} }}{\partial t}$$

(17)

• Rate of heat transfer from the freezing front to the fin:

$$q_{in}^{\prime \prime } = - k_{s} \left( {dr} \right)\frac{{T_{f} - T_{m} }}{S\left( t \right)}$$

(18)

• Net rate of heat transfer through conduction in the fin:

$$q_{r + dr}^{\prime \prime } - q_{r}^{\prime \prime } = \left[ {k_{f} \left( \delta \right)\frac{{\partial T_{f} }}{\partial r} + \frac{\partial }{\partial r}\left( {k_{f} \left( \delta \right)\frac{{\partial T_{f} }}{\partial r}} \right)dr} \right] - \left[ {k_{f} \left( \delta \right)\frac{{\partial T_{f} }}{\partial r}} \right]$$

(19)

By substituting the above expressions into Eq. (12) and simplifying it, the governing equation for the temperature distribution of the longitudinal fin is derived, and its initial and boundary conditions are considered exactly similar to those of the radial fins, as given in Eqs. (12) to (15).

Model description

In this section, the semi-analytical solutions is coded using Matlab to predict the position of the freezing front, which is coupled with time and temperature distribution in each region. By determining the position of the freezing front, the impact of using fins and their arrangement on reducing the freezing time of PCM will be identified.

To compute the solutions of the equations solved in Section “Model description” and find the position of the freezing front, an algorithm has been written in Matlab for each region. As mentioned, the temperature distribution equations and the solid–liquid interface position in both regions are coupled and must be solved simultaneously.

• Region 1:

• 1. Initially, considering related conditions, the position of the freezing front at zero time is one. Using this value, the temperature distribution equation is derived.

• 2. The temperature gradient is calculated from the obtained temperature distribution and substitute it into the Stefan condition to find the new position of the freezing front.

• 3. The new freezing front position is compared with the previous one. If the convergence criterion is not met, the temperature distribution is recalculated using the new interface position.

• 4. This process is repeated until the convergence criterion for the freezing front position is satisfied.

• 5. Finally, with the stabilized position of the freezing front in the nth time step, it is moved to the (n + 1)th time step.

• 6. The convergence criterion of $10^{ - 4}$ and a time step of 1s are considered.

• Region 2:

• 1. To calculate the solid–liquid interface position at a specific moment, it is assumed that the base fin temperature, where the fin is attached to the inner cylinder, is either equal to the fluid temperature passing through the inner cylinder at that moment or the temperature reached by the first type boundary condition at that time. Using this temperature in the Megerlin relation and finding ${\varvec{S}}({\varvec{t}})$, the fin’s base temperature is calculated.

• 2. The new base temperature is compared with the previous value. If the temperature convergence criterion is not met, we recalculate the freezing front position and the new temperature.

• 3. This process is repeated until the temperature convergence criterion is satisfied.

• 4. Using the stabilized temperature at the base or any other point, the final freezing is derived front position from the Megerlin relation.

• 5. The next spatial step is proceeded and this process is repeated using the fin temperature from the previous step.

• 6. This process is repeated until reaching the end of the fin.

The spatial step size is 0.06 for the third type boundary condition and 0.02 for the first type boundary condition, according to the problem conditions.

Geometric and material specifications of the tank and fins

As mentioned previously, in this study, two cylindrical energy storage tanks are examined with fins arranged differently in the tank, as shown in Fig. 1. The aim is to determine the reduction in freezing time due to the addition of fins to the system and to select the optimal fin arrangement under the most realistic boundary conditions. To compare these two types of fins, the system dimensions are chosen as shown in Table 1, ensuring equal PCM volume and heat transfer surface area from the fins. Thus, using four longitudinal fins with the specified dimensions in Table 1 is equivalent to using one radial fin in the system.

Table 1 Geometric specifications of the tank, fins, and each PCM cell.

In latent heat energy storage systems, selecting the appropriate PCM is one of the most crucial aspects of design. These materials must be chemically compatible with the tank material (i.e., non-corrosive) and should have a melting and freezing temperature range suitable for the intended application of the tanks. Therefore, for this study, with the tank designed for building comfort applications, the PCM is chosen to have a melting and freezing temperature suitable for air conditioning applications. Additionally, the tank and fin materials are selected to ensure high thermal conductivity, affordability, and availability. Thus, the PCM is Salt Hydrate Climsel C23, and the fin material is aluminum. Table 2 shows material specifications.

Table 2 Thermophysical properties of C23 and aluminum fin.

Validation

As mentioned previously, Mosaffa et al.²¹ conducted a study on a finned cylindrical tank under boundary conditions held constant over time, similar to the present research. They divided a control volume into two distinct regions and solved the semi-analytical problem, achieving good results compared to numerical solutions. Since experimental verification of the present study is not feasible and this particular problem has not been investigated in a time-dependent manner with fins before, validation of the modelling accuracy can be assessed against studies conducted on non-finned cylindrical tanks. A comparison of Caldwell and Kwan’s results²⁷ with the current method (Green’s function approach) under the time-dependent temperature boundary condition (${\theta }_{0}\left(\tau \right)=1+{\tau }^{2}$) is shown in Fig. 6.

Fig. 6

Results of the present work and Caldwell and Kwan’s results²⁷ under the boundary condition ${\theta }_{0}\left(\tau \right)=1+{\tau }^{2}$ in a cylindrical tank.

By comparing the results of the present study with previous works, it is found that the maximum difference with the Green’s function method used in the current research compared to the perturbation method employed by Caldwell and Kwan’s results²⁷ is 7.84%, and with the eigenfunction expansion method used by Krishnan et al.¹³, it is 5.5%, both of which are acceptable deviations.

The validation of the results of this study is also presented for a specific case with a constant wall temperature. The temperature profile of the fins over 500 s (corresponding to τ = 0.418) is compared with the approximate analytical and numerical study conducted by²⁸, as illustrated in Fig. 7. The maximum discrepancy of 9.6% demonstrates the high accuracy of the findings in this study.

Fig. 7

The results of the present study and those of Mostafavi et al.²⁸ using a PCM with a melting point of 28 °C, a density of 780 kg/m³, a specific heat capacity of 2300 J/kg K, a thermal conductivity of 0.15 W/m K, and a latent heat of 244,000 J/kg, with the inner wall temperature maintained at a constant value 20 K below the melting point.

For a more comprehensive comparison and validation of the results, the findings of the present study have been examined against the experimental and numerical study by Kozak et al.²⁹, as presented in Fig. 8. Although the present study employs a time-dependent boundary condition, which introduces a slight difference compared to the previous work, the comparison indicates that the overall trend remains similar.

Fig. 8

Comparison of the present study’s results with the experimental and numerical findings of Kozak et al.²⁹.

Examination of the freezing front advancement in region one

The Fig. 9 represents the position of the freezing front at each moment in time, occurring in Region One under Dirichlet boundary conditions. Note that both the position of the freezing front and the time are dimensionless.

Fig. 9

Position of the freezing front relative to time under Dirichlet boundary condition.

Examination of the freezing front advancement in region two and superposition with region one

In this section, the movement of the freezing front from radial fins under the first type of boundary condition is examined. The freezing front positions from the fins at various times are determined and superimposed with the corresponding freezing front positions from Region One (the inner wall of the tank) to determine the freezing of a PCM cell. Afterward, the impact of the fins on PCM freezing at different times can be assessed. Figures 10, 11, and 12 show the freezing front position in each PCM cell of a single-finned tank with a radial arrangement over 350 s, 700–900 s, and 1024 s, respectively, and Figs. 13 and 14 show the freezing front position in a two-finned tank over 200 s and 350–423 s, respectively.

Fig. 10

Prediction of the freezing front position in each PCM cell of a single-finned tank with radial arrangement over 350 s under the first type of boundary condition with temperature.

Fig. 11

Prediction of the freezing front position in each PCM cell of a single-finned tank with radial arrangement over 700 and 900 s under the first type of boundary condition with temperature.

Fig. 12

Prediction of the freezing front position in each PCM cell of a single-finned tank with radial arrangement over 1024 s under the first type of boundary condition with temperature.

Fig. 13

Prediction of the freezing front position in each PCM cell of a two-finned tank with radial arrangement over 200 s under the first type of boundary condition with temperature.

Fig. 14

Prediction of the freezing front position in each PCM cell of a two-finned tank with radial arrangement over 350 and 423 s under the first type of boundary condition with temperature.

It should be noted that the oscillatory nature of the freezing front positions obtained from the fins is due to the singularity of the eigenvalue equations at boundary points. These oscillations essentially reflect the behaviour of Bessel functions. Based on the results obtained in the Table 3, it is shown that the effect of the fins on increasing the freezing rate of PCM is more significant in the initial stages. Over time, this effect diminishes due to the increase in the temperature of the wall and fins, as well as the thermal resistance provided by the newly formed phase. In the best scenario (with two radial fins), there is a 132.58% increase in the fraction of frozen PCM within 423 s, leading to complete freezing in the tank and reducing the total freezing time by 75.31%.

Table 3 Segmentation of frozen PCM values and the percentage impact of fins on its solidification in a radial-finned tank with a first-type boundary condition.

Freezing in cylindrical energy storage tank with longitudinal fin arrangement

For longitudinal fins, considering the dimensions and properties presented in Tables 1 and 2, the time-dependent temperature $T_{w} \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$ is applied to the inner wall of the tank, while the outer wall is assumed to be insulated.

Examination of freezing front advancement from the first region

Given that there is no change in the structure of the first region, and in this region, the freezing front progression originates from the inner cylinder for both fin arrangements, the freezing front position at any moment for tanks with 4 and 8 longitudinal fins corresponds to Figs. 14 and 15, respectively.

Fig. 15

Prediction of freezing front position in each PCM cell of a tank with 4 longitudinal fins over 350 and 700 s under the first type boundary condition with temperature $T_{w} \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$.

Examination of freezing front advancement from the second region and superposition with the first region

The movement of the freezing front from the longitudinal fins under the first type boundary condition is examined in Figs. 15 and 16. The freezing front position from the fins is determined at different times and superimposed with the corresponding freezing front position from the first region (the inner wall of the tank), resulting in the freezing of a PCM cell. Using the obtained results, the impact of longitudinal fins on PCM freezing at various times is determined.

Fig. 16

Prediction of freezing front position in each PCM cell of a tank with 8 longitudinal fins over 200 and 350 s under the first type boundary condition with temperature $T_{w} \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$.

Based on the results obtained from the Figs. 15 and 16 and presented in Table 4, it can be understood how the fins impact the progression of the freezing front under the first type boundary condition at different times. The effect of fins on increasing the freezing rate of PCM is more significant initially and diminishes over time due to the increase in temperature of the wall and fins and the thermal resistance provided by the newly formed phase. In the best case scenario (with 8 longitude fins), there is a 69.99% increase in the fraction of frozen PCM within 760 s, leading to complete freezing in the tank and reducing the total freezing time by 55.63%.

Table 4 Segmentation of frozen PCM values and the percentage impact of fins on its solidification in a longitudinal-finned tank with a first-type boundary condition.

Third type boundary condition

To analyze the freezing process of the tank under this boundary condition, following the same approach as in the previous section, with the key difference being the application of freezing relations specific to the third-type boundary condition in the solution algorithm. The temperature of the fluid passing through the inner cylinder is considered as a time-dependent function $T_{\infty } \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$. Additionally, the convective heat transfer coefficient of the fluid is taken as $h = 80\;{\text{W}}/{\text{m}}^{2} \;{\text{K}}$.

Examination of freezing front advancement from the second region and superposition with the first region

Now, by predicting the position of the advancing freezing front from the second region at various times under the convective boundary condition and superimposing it with the corresponding predicted freezing front position in the first region, the freezing of a PCM cell with longitudinal fins is examined in Figs. 17 and 18 over 700–900 s and 1200–1600 s, respectively.

Fig. 17

Prediction of freezing front position in each PCM cell of a tank with 4 longitudinal fins over 700 and 900 s under the third type boundary condition with temperature $T_{\infty } \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$.

Fig. 18

Prediction of freezing front position in each PCM cell of a tank with 4 longitudinal fins over 1200 and 1600 s under the third type boundary condition with temperature $T_{\infty } \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2}$.

Examination of freezing front advancement from the second region considering temperature gradient along the Z-axis of the fin

To ensure the validity of the two-dimensional solutions performed for the longitudinal fins, the equations solved in this section are implemented at both the beginning and end of the tank according to the described algorithm. The fluid temperature is considered time-dependent as $T_{\infty } \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2} + 50z$ to ensure a significant temperature difference at the beginning and end of the tank, despite its short length. Based on Fig. 19, despite the temperature difference of approximately 1.25 degrees at the beginning and end of the tank, the freezing front advancement from the fins remains nearly identical. Therefore, it is concluded that the previous two-dimensional solutions have provided entirely reasonable results.

Fig. 19

Prediction of the freezing front position in region 2 at the beginning and end of each PCM cell in a 4-fin tank with longitudinal arrangement over 1600 s under the third type boundary condition with temperature $T_{\infty } \left( t \right) = 1 + \left( {{\raise0.7ex\hbox{${\alpha_{s} t}$} \!\mathord{\left/ {\vphantom {{\alpha_{s} t} {R_{in}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{in}^{2} }$}}} \right)^{2} + 50z$.

Conclusion

This study analyzed the freezing behavior of PCMs in cylindrical TES tanks with radial and longitudinal fins under time-dependent boundary conditions. A semi-analytical Green’s function approach was employed to model the freezing front progression, providing insights into the effectiveness of different fin configurations.

The key findings are summarized as follows:

• 1. Radial fins significantly outperform longitudinal fins in all examined cases, achieving up to a 75.31% reduction in total freezing time and demonstrating 19.68% higher efficiency compared to longitudinal fins.

• 2. Fins are most effective in the early stages of freezing, where they accelerate phase change from liquid to solid. However, over time, their effectiveness declines due to increasing thermal resistance from the newly formed solid phase.

• 3. Boundary conditions strongly influence fin performance:

  • Under first-type (Dirichlet) boundary conditions, fins substantially enhance heat transfer.

  • Under third-type (convective) boundary conditions, fins become practically ineffective over time due to the low convective heat transfer coefficient (h), making their use less cost-effective in such applications.

• 4. Validation through three-dimensional modeling confirmed the reliability of the two-dimensional approach. Despite a 1.25 °C temperature variation along the z-axis, freezing front advancement remained uniform, supporting the accuracy of the 2D model for predicting PCM solidification.

• 5. Practical Implications: These findings provide critical design insights for optimizing TES systems in HVAC, industrial cooling, and renewable energy storage applications, where maximizing freezing efficiency is essential.

In conclusion, radial fins are the optimal configuration for enhancing PCM freezing performance in TES systems, particularly under first-type boundary conditions. Future research should explore variable fin geometries, porous fins, and rotational effects to further improve TES efficiency under real-world operating conditions.

Recommendations for future works

• Investigate the freezing of PCM under various boundary conditions in cylindrical energy storage tanks with longitudinal fins of variable cross-section.

• Examine the freezing of PCM in cylindrical TES tanks with porous radial fins.

• Study the phase change process of PCM in spherical and rectangular finned tanks under time-dependent boundary conditions.

• Investigate the freezing of PCM in cylindrical TES tanks with time-dependent boundary conditions in a two-phase manner, considering heat transfer in the liquid phase.

• Examine the effect of the rotation of the cylindrical TES tank on the melting and freezing of PCM under time-dependent boundary conditions.

• Investigate the freezing problem in energy storage tanks considering variable thermophysical properties.