1 Introduction

The use of latent heat thermal storage (LHTS) systems has been increasing rapidly due to their wide range of applications in solar thermal systems (Tian and Zhao 2013), shaving energy demand peaks times (Chen et al. 2009), wasted heat recovery (Fernández et al. 2014), desalination (Mazraeh et al. 2018), energy-efficient buildings (Zhou et al. 2012), etc. There are three main thermal energy storing approaches: sensible storing (Salomoni et al. 2014), latent storing (Agyenim et al. 2010) and thermochemical storing (Aydin et al. 2015). Sensible heat thermal storage systems are usually associated with temperature rise or fall in storage media, and thermochemical ones use a large quantity of enthalpy change during endothermic/exothermic chemical reactions in order to release or absorb energy. However, a large amount of energy is transferred during the solid–liquid phase change process in latent heat thermal storage systems. Apart from constant working temperature during solidification and melting procedures, phase change materials (PCMs) as LHTS systems have been widely employed in a diversity of applications because of their many other benefits such as high density of energy, low vapor pressure at operating temperature and non-corrosiveness (Sefidan et al. 2017). Therefore, several researchers have observed the effectiveness of phase change materials for different storage systems. However, PCMs are suffering from their low thermal conductivities which limits their usage. Thus, various techniques have been introduced to overcome this problem including fins (Taghilou and Talati 2016; Talati and Taghilou 2015; Erek et al. 2005), adding nanoparticles (Mahdi and Nsofor 2016; Darzi et al. 2016), saturating PCMs in porous matrix (Gopalan and Eswaran 2016; Du and Ding 2016) and employing multi-layer (cascaded) PCMs (Pasupathy and Velraj 2008; Brousseau and Lacroix 1996; Shaikh and Lafdi 2006).

Since fins are simple and inexpensive to use, it is a common way to enhance the heat transfer rate within the PCMs. Wang et al. (2016) performed a numerical investigation on the effects of the geometry of fins in a cylindrical container on the melting procedure of PCMs. Mosaffa et al. (2012) analytically studied the rate of PCM solidification in a vertical shell and tube heat exchanger by using radial fins. Rathod and Banerjee (2015) experimentally studied the solid–liquid phase change processes of paraffin with the presence of longitudinal fins in a vertical shell and tube heat exchanger. Their results indicated decrease in the freezing and melting cycles up to 43.6%, and 24.5%, respectively. Taghilou and Talati (2016), Talati and Taghilou (2015) carried out a numerical study based on the lattice Boltzmann method (LBM) to investigate the phase transition of a PCM through a rectangular finned container. Their reports showed considerable enhancement in the phase change rate. They also took the effects of free convection into account and discussed the melting process for different solid phases describing the Rayleigh number.

The multi-layer technique can be used in order to provide a comfortable temperature in buildings throughout the whole year round. Pasupathy and Velraj (2008) employed two kinds of PCMs in building components in order to design a double-layer PCM that stores and releases the heat in its latent form during summer and winter weather conditions. Hamza et al. (2016) numerically investigated the impacts of multi-layer PCMs on maximizing energy savings inside an air-conditioned space. They found that combinations of two PCMs can decrease the energy consumption in the room without considering annular weather change effects. According to Farid and Kanzawa (1989), the heat transfer rate saw a significant rise when different PCMs with various freezing temperatures are used instead of a single PCM within a tubular container. Sefidan et al. (2017), Taghilou et al. (2017) numerically prepared a study into the effects of multi-layer PCMs on the performance of tubular and rectangular-shaped heat exchangers and reported desirable effects on heat transfer efficiency of heat exchangers.

Since it has a wide range of applications, recently a large number of research works on the cascaded thermal storage have been done. In a theoretical work, Wang et al. (1999) studied the cascaded composite PCMs assuming linear temperature distribution of PCMs. They found a 25–40% reduction in phase transition cycle on their system. Yang et al. (2014) compared the performance of a cascaded latent heat thermal storage system with single PCM and that with multi-PCM. According to their results, employing multi-layer PCMs improved system efficiency. By applying a numerical approach based on a one-dimensional finite difference method, Seeniraj and Narasimhan (2008) simulated a cascaded solar dynamic latent heat storage unit. They reported a considerable increase in the melting rate using multiple PCMs. Li et al. (2013) developed a mathematical model for shell and tube LHTES based on a two-dimensional finite volume method using three cascaded PCMs. They provided a design optimization of this system for solar applications. Michael and Pitz-Paal (2007) carried out an experimental study into a phase change feature of three-stage cascaded PCMs in a shell and tube heat exchanger under realistic operational conditions. They reported that the low thermal conductivity of PCMs makes it impossible to use cascaded PCMs on a large scale. Peiró et al. (2015) experimentally studied the advantages of using cascaded PCMs instead of single PCM in thermal energy storage systems. They indicated a 19.36% of heat transfer elevation by using cascaded PCMs compared to single PCM. Chiu and Martin (2013) provided a numerical analysis to study the freezing and melting rate of single PCM in comparison with a cascaded design of multi-layer PCMs within a finned tubular heat exchanger. They found an improvement in the thermal performance of the system by using cascaded PCMs. Gong and Mujumdar (1997) presented a thermodynamic optimization of the phase change process in LHTES based on the lumped model. Their analytical results showed a considerable improvement in terms of the exergy efficiency of the system by using cascaded PCMs.

Shamsi et al. (2017) employed a method of characteristics and genetic algorithm to model and optimize cascaded latent heat storage system and declared that attributed performance of the final optimized system improved 5% concerning single PCM storage. Wu et al. (2016) numerically found out that thermal storage systems in non-cascade form have the disadvantage of lower charging ratio and longer charging time, while cascaded systems periodic process can be repeated with higher energy storage efficiency.

A case study of CSP plant analyzing cyclic performance of cascaded and multi-layer PCM shell and tube heat exchanger was carried out by Tehrani et al. (2018). Noting that the most important design consideration is melting point selection of cascaded thermal energy storage and annual cyclic investigations in various seasons can reveal applicability of the elevated thermal energy storage. Tessier et al. (2016) developed a new insulated compressed air energy storage unit incorporating cascaded PCMs and illustrated an increment of almost 15% efficiency of the energy storage and relevant heat recovery from the system.

Zhao et al. (2016) presented the thermal behavior of thermocline storage incorporated with cascaded solid PCMs. The influence of different packed bed configuration was identified, and results demonstrated cost-effective when applying multi-layer solid PCMs. Exergy analysis and optimization were performed by Xu and Zhao (2016) for cascaded latent heat/cold storage. They concluded that heat transfer enhancement (an increase of NTU) is essential for cascaded thermal energy storage and remarked that the quality of attributed exergy in these thermal systems is higher for cold units rather than that of heat. COP (coefficient of performance) of cold storages enhanced during increased temperature of ambient, while it decreased for heat storages at the same condition.

Analysis and optimization of cascaded thermal energy storage from the viewpoint of exergy, entropy and entransy for steady and unsteady cases were performed by Xu and Zhao (2015, 2017a, b, 2019). They realized that temperature with entropy showed geometric distribution, while that with entransy is linear. In case of steady analyses, entransy and entropy are reasonable for heat transfer and heat work conversion, respectively. Thermal efficiency and exergy efficiency of transient cases are less than those in steady cases. A three-stage cascaded cold storage unit was designed by Cheng and Zhai (2018), and the solidification process of the unit was numerically and experimentally tested to obtain the thermal performance of the storage. They stated that although exergy stored in the cascaded unit is less than single stage, the cold charging rate increased at about 11–35% during the solidification process showing great advantage of the cascaded storage unit.

The cascaded configuration has been applied for high-temperature storages such as molten salt latent heat thermal energy storages (Yuan et al. 2018; Prasad et al. 2018; Bayon et al. 2019). Inlet temperature of heat transfer fluid (HTF) and its flow rate considerably affect on charging or discharging rate, and the number of cascades should be properly evaluated to design a cost-optimized thermal storage unit. Teameh et al. (2018) used cascaded PCM storages of different melting points to increase the performance of solar domestic hot water systems. They used hybrid tanks containing hot water and phase changing materials. PCMs were inserted as vertical cylindrical modules, and water was flowing along the length of the tank as HTF. They numerically modeled the hybrid system and concluded that employing a multi-tank hybrid system reduces over 50% of storage volume. Zhao et al. (2018) developed a three-stage latent heat thermal storage system for the heat charging process. They remarked that on behalf of weak heat transfer performance, the solid–liquid phase change in the three stages does not take place simultaneously and a higher melting point difference. Moreover, with the increase of stages number energy storage efficiency, exergy storage efficiency and entransy storage efficiency reach higher values. Higher entering fluid temperature and its flow rate lead to enhanced rates of energy, exergy and entransy, but the storage efficiency of energy, exergy and entransy could only be obviously improved by higher HTF inlet temperatures. To model the phase change problems several techniques including: (1) fixed grid method, (2) variable grid methods, (3) front fixing method, (4) adaptive grid generation method and (5) enthalpy method based on the theoretical and experimental approaches are introduced in previous studies (Jourabian et al. 2013, 2016 Darzi et al. 2013; Zivkovic and Fujii 2001; Esen et al. 1998; Horbaniuc et al. 1999). For instance, Jourabian et al. (2013, 2016) numerically simulated the natural convection melting of PCM in a cavity using of enthalpy-based lattice Boltzmann method. Based on the (Jourabian et al. 2013), enthalpy-based method as a regular scheme for solving solid–liquid phase change problems, makes the problem become simpler. The behavior of the phase change material is similar to the behavior of an incompressible fluid in a porous media, since during solid–liquid phase change simulation the Darcian force is modeled in the phase change area. The Darcian force proportionates to the relative velocity of the moving phase and the permeability of the mushy region. Enthalpy–porosity scheme associated with finite volume method is a powerful tool to simulate complex flow and heat transfer in PCMs applications (El Ganaoui et al. 1999).

Most of the reviewed researches about cascaded thermal energy storage were analyzed from the perspective of thermodynamics and related topics. This paper carries out a two-dimensional unsteady numerical approach to study the solidification process of cascaded LHTES systems within a channel under various cold fluid flow strategies to evaluate the system performance under each methodology. The system includes 80 identical PCMs with length (l) and height (h) which are located in a row within a channel in five different ways separating with aluminum fins (with the thickness of t). Cold air flows through the unit with constant bulk temperature of T0 and bulk velocity of U0 with four different Reynolds numbers. System’s responses to these flows are assessed in terms of liquid fraction history and will be discussed for a complete freezing cycle.

2 Physical Model

2.1 Problem Definition and Boundary Condition

Figure 1a–e demonstrates schematics of the proposed cascaded thermal storage system within a channel under different airflow strategies. In type (a), air flows in a channel with the height of H = 2 cm, while 80 cascaded PCMs are embedded above the channel and are separated by thin aluminum fins of t = 0.5 cm thickness (thermophysical properties of utilized materials are given in Table 1). For AR= 1, height and width of the PCM cavity are h = l=19 mm for L = 2 m length of the channel, but for other AR besides 1 (including AR= 0.5 and 1.5), due to variation of cavity width, length of channel differs to keep 80 cascaded PCM cavities with the same overall area of cavities. Aluminum thicknesses for upper and lower parts of cavities are the same and equal to 0.5 mm. RT30 was used as PCM and related thermophysical properties are available in Table 1. Based on the airflow strategy, cooling air with T0 = 288 K and uniform velocity of U0 enters the channel and exited into the atmosphere. The bulk temperature of the airflow at the inlet is less than the PCMs melting point to guarantee the solidification process. The value of U0 can be obtained from air properties (from Table 1), channel height and Re for various cases. The outlet condition is set as atmospheric pressure (P = P0). All of the channel external wall is thermally insulated, and hydrodynamically no-slip condition is applied on them. All of the system including PCM and fins are kept at Tinitial = 301 K at the beginning of the simulation (t = 0 s). Regard the fins and PCMs interface, the temperature and heat flux continuity are assumed at the interface.

Fig. 1
figure 1

Physical models for air flow strategies a one-side channel, b parallel flow, c counter flow, d U type, e middle inlets

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Table 1 Thermophysical properties of PCM and aluminum
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2.2 Assumptions

In this paper, to investigate the unsteady freezing process of cascaded LHTS systems inside a channel under different forced airflow strategies, the following assumptions have been taken into account (Sefidan et al. 2017; Taghilou et al. 2017):

  1. 1.

    The steady airflow inside the channel and unsteady solidification process are considered laminar and 2D including natural convection in the liquid and gas phases.

  2. 2.

    All PCM properties are independent of temperature, except liquid phase density of PCMs which obeys the Boussinesq approximation, as below:

    $$ \rho = \rho_{\text{ref}} \left[ {1 - \alpha \left( {T - T_{\text{ref}} } \right)} \right] $$
    (1)

    where α is the coefficient of thermal expansion in PCMs, and ρref and Tref are the reference density and temperature, respectively.

  3. 3.

    Supercooling and viscous dissipation effects are imperceptible.

  4. 4.

    Volume changes of PCMs resulted from the phase change process are negligible.

  5. 5.

    Aluminum walls and fins have a certain thickness, and then their thermal resistance is taken into account.

  6. 6.

    To be initially liquid status in PCMs, the initial temperature of the aluminum and PCMs is assumed to be higher than PCMs’ melting point.

  7. 7.

    No-slip boundary conditions are applied in walls.

2.3 Governing Equations

Based on the aforementioned assumptions, and considering the continuity (Eq. 2), momentum (Eq. 3) and energy (Eq. 4) equations, the velocity and pressure fields and heat transferred via conduction and convection mechanisms can be calculated. The enthalpy-based scheme was used for solid–liquid phase change simulation. The applied scheme provides advantages such as omitting the requirement of satisfying boundary condition at the solid–liquid interface, thermophysical properties and temperature determining at each node of the interface, permission of mushy zone during phase change occurrence, obtaining the solid–liquid interface by temperature field implementation.

$$ \frac{\partial \rho }{\partial t} + \nabla \cdot \left( {\rho u} \right) = 0 $$
(2)
$$ \rho \frac{\partial u}{\partial t} + \rho u \cdot \nabla u = - \nabla p + \nabla \cdot \left( {\mu \left( {\nabla u + \left( {\nabla u} \right)^{T} } \right)} \right) $$
(3)
$$ \rho \frac{{Dh_{\text{total}} }}{Dt} = k\nabla^{2} T $$
(4)

where ρ, k and T are density, thermal conductivity and temperature, respectively. In Eq. (4) \( h_{\text{total}} \) consists of sensible enthalpy and latent heat:

$$ h_{\text{total}} = h + \Delta H $$
(5)

The liquid fraction (Ψ) and ΔH are defined as:

$$ \varPsi = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {T < T_{\text{solid}} } \hfill \\ \begin{aligned} \frac{{T - T_{\text{solid}} }}{{T_{\text{liquid}} - T_{\text{solid}} }}, \hfill \\ 1, \hfill \\ \end{aligned} \hfill & \begin{aligned} T_{\text{solid}} < T < T_{\text{liquid}} \hfill \\ T_{\text{liquid}} < T \hfill \\ \end{aligned} \hfill \\ \end{array} } \right. $$
(6)
$$ \Delta H = \varPsi \cdot L $$
(7)

L in Eq. (7) indicates phase change process heat of fusion. The heat storage capacity of the system can be calculated by the following equation:

$$ Q = \sum\limits_{t = 0}^{{t = t_{s} }} {\sum\limits_{i = 1}^{i = N} {m_{i} \cdot C_{i} \cdot \Delta t_{i} } } $$
(8)

here mi and Ci represent mass and heat capacity for the ith stage thermal storage unit, in the given order. Also, \( \Delta t_{ti } \) expresses the charging time from beginning of solidification process (t = 0) to end of it (t = ts) (Shamsi et al. 2017).

The transient conduction heat equation is governed on the mechanism of heat transfer inside the fins:

$$ \rho_{\text{fin}} C_{\text{pfin}} \frac{{\partial T_{\text{fin}} }}{\partial t} = \nabla \cdot \left( {k_{\text{fin}} \nabla T_{\text{fin}} } \right) $$
(9)

Temperature and heat flux continuity are assumed at the interface of the PCM and moving airflow:

$$ T_{f} = T_{\text{fin}} ,\quad k_{f} \frac{{\partial T_{f} }}{\partial n} = k_{\text{fin}} \frac{{\partial T_{\text{fin}} }}{\partial n} $$
(10)

The transient pressure-based technique is employed for the numerical modeling of the problem utilizing the finite volume method (FVM) (Sefidan et al. 2017; Taghilou et al. 2017). Modeling mushy zone, this FVM-based code assumes ‘‘pseudo”-porous medium for PCM parts with porosity equal to the liquid fraction. The porosity descends from 1 (liquid) to 0 (solid) as the material solidifies and the velocities also fall to zero. Triangular-shaped (unstructured) cells are selected for meshing the geometry which is finer close to the walls due to higher gradients of quantities. Second-order accuracy is selected for the discretization of momentum and energy equations (Sefidan et al. 2017; Taghilou et al. 2017). The PRESTO technique is employed for the discretization pressure term. The value of the convergence criterion for all equations is set to be 106 (Sefidan et al. 2017; Taghilou et al. 2017). The under-relaxation factors for the continuity, momentum and thermal energy are 0.5, 0.3 and 0.8, respectively (Sefidan et al. 2017; Taghilou et al. 2017).

2.4 Grid and Time Step Independency Tests and Verification

In order to analyze the effect of grid size and time step value on the calculated results, time series of the liquid fraction obtained by three grid systems as well as three different time steps are illustrated in Fig. 2. It can be inferred from Fig. 2 that the calculated results neither vary with the grid system nor the time steps. Hence, considering the computing time, the biggest grid system and time step (i.e., 113,688 cells and 0.2 s, respectively) are selected for the present study.

Fig. 2
figure 2

Liquid fraction history of counter flow with three different mesh and time step sizes

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For verification of the current numerical simulation, an experimental study of Al-Abidi et al. (2013) is considered. All parameters, such as the heat exchanger dimensions as well as the boundary conditions, are selected the same as the parameters used by Sefidan et al. (2017), Taghilou et al. (2017), Al-Abidi et al. (2013). Figure 3a, b compares the time series of average temperature and liquid fraction obtained by the current methodology and Sefidan et al. (2017), Taghilou et al. (2017), Al-Abidi et al. (2013). As can be seen from Fig. 3, the calculated numerical results are in fair agreement with those of experimental study.

Fig. 3
figure 3

Comparison of calculated a liquid fraction, and b temperature with results given by Al-Abidi et al. (2013)

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3 Results and Discussion

Figure 4 demonstrates liquid fraction contours for different Re numbers when air flows from one side of the channel (type-1 in Fig. 1). Due to forced convection heat transfer from cascaded LHTES systems to the cooling air, PCMs solidify as time passes (red color displays fully liquid phase, and blue color displays fully solid state). Conduction mechanism happens for heat transfer through interconnected PCM units and aluminum fins, since PCMs nearby the entrance of the channel solidify sooner against other parts. By increasing the Re number, the total heat transferred from cascaded LHTES systems to the cooling air rises. Consequently, the total solidification time which is about 2000s for Re = 300 reduces to almost 1300 s for Re = 1000.

Fig. 4
figure 4

Liquid fraction contours for different Re of one-side channel (red color displays fully liquid phase, and blue color displays fully solid state)

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Liquid fraction contours for different Re numbers of parallel flow channel strategy are depicted in Fig. 5. The general trend of the solidification process is similar to the one-side channel strategy. For a constant Re number, a major part of the cascaded PCMs solidifies due to convection and conduction heat transfer mechanisms over time, as expected. Moreover, the increment in Re numbers decreases the total solidification time from about 3000 s (for Re = 300) to about 1000 s (for Re = 1000). In comparison with one-side channel strategy, parallel flow channel case has lower solidification time for all Re numbers except Re = 300. Since the cooling air flows through the channel from both sides in parallel flow strategy, heat is transferred from both sides of PCMs which leads to sooner solidification.

Fig. 5
figure 5

Liquid fraction contours for different Re of parallel flow channel. (Red color displays fully liquid phase, and blue color displays fully solid state)

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For counter flow channel strategy, liquid fraction contours are presented in Fig. 6 for different Reynolds numbers. In this case, PCM units start to solidify from both sides of the channel, and therefore, middle parts freeze at the end. Like one-side and parallel flow channel strategies, Re number increment leads to reduction in total solidification time. However, total freezing time for counter flow channel strategy (which is 700 s for Re = 1000) is lower than that for one-side and parallel flow channel strategies (that are 1300 s and 1000 s at Re = 1000, respectively). In this strategy which the geometry is the same as parallel type, counter airflow from both sides of the channel improves the heat transfer from the PCMs.

Fig. 6
figure 6

Liquid fraction contours for different Re of counter flow channel. (red color displays fully liquid phase, and blue color displays fully solid state)

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Figure 7 illustrates liquid fraction contours for different Re numbers when the U-type channel is utilized for cooling cascaded LHTES systems. For all Re numbers, the amount of solidified PCM grows by time. As can be seen, the freezing process of U-type and counter flow channel strategies has an almost similar trend. In this type, again the cascaded LHTES units start to freeze from channel sides. Furthermore, by increasing the Re number, the discharging time decreases, due to increment in total heat transfer from PCM sets to the cooling air.

Fig. 7
figure 7

Liquid fraction contours for different Re of U-type channel. (Red color displays fully liquid phase, and blue color displays fully solid state)

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Liquid fraction contours for different Re numbers of middle inlets channel strategy are shown in Fig. 8. The whole trend of the phase changing process is in contrast with counter flow channel strategy, in which the middle of the channel begins to solidify, and then freezing grows to the sides. As time goes by, more PCM cavities become solid for constant Re number. Moreover, by increasing the Re number, total discharging time decreases (which is 3000 s and 900 s for Re = 300 and 1000, respectively).

Fig. 8
figure 8

Liquid fraction contours for different Re of middle inlets channel. (Red color displays fully liquid phase, and blue color displays fully solid state)

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Figures 9, 10, 11 and 12 depict timewise variations of the liquid fraction for different airflow strategies at Re = 300, 500, 700 and 1000, respectively. All diagrams have a similar trend as the cascaded PCM sets are completely liquid at t = 0 s and then become solid due to heat transfer from the PCM units to the cooling air.

Fig. 9
figure 9

Comparison solidification time for different air flow strategies at Re = 300

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Fig. 10
figure 10

Comparison solidification time for different air flow strategies at Re = 500

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Fig. 11
figure 11

Comparison solidification time for different air flow strategies at Re = 700

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Fig. 12
figure 12

Comparison solidification time for different air flow strategies at Re = 1000

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In the counter flow strategy, the temperature difference between the air and PCMs is larger than the parallel one. Also, in the U-type, cold air has more time to influence the PCMs and get thermal energy from them. Therefore, U-type and counter flow strategies lead to quick solidification among others in all Re numbers studied.

For all Re numbers, counter flow type is the best way to quickly solidify the whole system except Re = 300 in which U-type works better. Since the convective heat coefficient is small for low Re numbers, cooling airflow has enough time in the U-type strategy to freeze the LHTES. Then for the lowest Re number (Re = 300), the total solidification time for the U-type channel is the lowest.

It can be seen that the parallel flow and middle inlets channel types have almost the same liquid fraction histories during the freezing process almost for all Re numbers. This resemblance stems from the similar airflow upside and downside of cascaded PCM set at middle inlets strategy, which is in the same direction as parallel flow channel.

As it is clear from the figures, by increasing the Re number, the middle inlets and parallel flow strategies lead to a shorter solidification process in comparison with one-side channel. For low Re numbers, one-side channel type is better in terms of freezing time, but at high Re numbers as the heat transfer surface area doubles in the parallel flow and middle inlets strategies, one-side channel results in the longest solidification time.

To demonstrate the effect of the cavity aspect ratio (AR), we performed some other simulations for constant Re = 1000 of counter airflow strategy. Because this strategy presented that liquid fraction can descend faster for most of the cases, the aspect ratio is defined as the ratio of the height of the cavity to its width. Figure 13 shows the time variation of the average liquid fraction of counter flow for AR= 0.5, 1 and 1.5 when Re = 1000. It can be seen that for lower AR, faster solidification happens. This fact is related to the increased length of airflow and cavity connection. But note that for lower AR, the length of the channel is longer (as explained in part 2) and a higher value of pressure drop for airflow will be experienced. So it is important for thermal system designer that to reach a desirable condition of outlet temperature, number of cavities, their AR, length of the channel and employed PCM should be well selected.

Fig. 13
figure 13

Comparison solidification time for different AR of counter flow strategy at Re = 600

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The outlet temperature of airflow is a very important parameter that should be considered during thermal system designing. In this research, air should remove much heat from PCM cavities to reach a higher temperature for winter conditions. The higher outlet temperature (closer to melting temperature of PCM), the higher performance for a designed system. Table 2 illustrates the outlet average temperature of five described strategies for three AR for Re = 1000. It can be concluded that lower AR of counter flow has better performance of heat removal due to higher exit average temperature. This fact is attributed to the longer thermal connection of flowing air and the PCM cavity for lower AR. Table 3 demonstrates required time to complete solidification process for all AR and configurations, and it can be obviously concluded that it takes longer for higher AR, while it is shorter for higher Re. Moreover, counter flow configuration with AR= 0.5 and Re = 1000 depicted very short time (t = 649 s) among all other cases.

Table 2 Outlet temperature of flowing air for various AR and air flow strategies at time 1000 s and Re = 1000
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Table 3 Time in seconds needed for all solidification processes
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4 Conclusions

In the present study, the unsteady freezing process of cascaded latent heat thermal energy storage (LHTS) systems inside a channel is developed. Five various types of the channel have been studied in order to figure out the effects of different forced air flow strategies on solidification process. Based on the results obtained, main conclusions could be drawn as below;

  1. 1.

    The Re number as one of the contributing parameters effects the performance of the system. By increasing the cooling air Re number, the solidification process completes sooner.

  2. 2.

    Among all Re numbers, Re = 300 has the longest total solidification time, at which the U-type channel and parallel air flow strategies lead to a minimum and maximum freezing time, respectively.

  3. 3.

    U-type and counter flow strategies are effective in terms of total solidification time.

  4. 4.

    The parallel flow and middle inlets channel types have almost the same liquid fraction histories.

  5. 5.

    For all Re numbers, counter flow type is the best way to quickly solidify the whole system except Re = 300 in which U-type works better.

  6. 6.

    Lower aspect ratio of counter flow strategy could solidify the PCM faster and could remove higher value of heat from cavities (higher outlet temperature).

  7. 7.

    As it is clear from the figures, by increasing the Re number, the middle inlets and parallel flow strategies lead to shorter solidification process in comparison with one-side channel.