Numerical analysis of measures to minimize the thermal instability in high temperature packed-beds for thermal energy storage systems

A transverse temperature variation may occur in packed-bed systems during the charging/discharging/standby processes due to heat losses of preferential areas, causing a lower pressure drop in these areas. In addition, the lower pressure drop in these areas results in an increased mass flow rate and further cooling, leading to an increased transverse temperature variation, in a positive feedback loop – a phenomenon called thermal instability. The thermal instability results in a difference between the maximum rock temperature of the packed-bed and the mean outlet temperature of the heat transfer fluid, which significantly can deteriorate the performance of the packed-bed. The numerical investigation presented in this paper addresses high temperature applications of large-scale, incorporating a thermal energy storage system utilizing air-rock packed-bed. As minimizing thermal instability is crucial, the study aim is to identify methods that can limit this in these storage systems. Transient 2-dimensional axisymmetric computational fluid dynamics models developed and validated in a previous work were used. The effects of different parameters, including inlet mass flow rate of the air, type of rocks, heat transfer coefficient of the packed-bed tank walls, inlet temperature of the air, porosity of the packed-bed, diameter of the rocks, and geometry of the packed-bed on the thermal instability during the discharging process initiated after one hour of standby process, were analyzed. Furthermore, the effect of various standby durations (1 h, 3 h, 5 h and 10 h) succeeded by a discharging process for two different cases – case 1 and case 2 – were carried out. Case 1 represents a benchmark case with initial geometric and operating parameters, while the parameters for case 2 were selected to minimize thermal instability in the packed-bed. The results suggest that the thermal instability, that is, the temperature difference between the maximum of the packed-bed rock and the mean of the outlet air, decreases by 41 %, 45 %, and 56 % with an increase in the inlet mass flow rate of the air from 0.25 kg/s to 0.45 kg/s, decreases in the heat transfer coefficient from 0.77 W/m 2 ⋅ K to 0.37 W/m 2 ⋅ K, and increases in the inlet temperature of air from 300 K to 400 K, respectively. Moreover, the results indicate that the thermal instability is 200 K and 29 K for case 1 and case 2, respectively, for a discharging process following 10 h of standby duration.


Introduction
The climate crisis is a major threat for the society, and many dedicated efforts are required for its mitigation.Energy conversion is both the major cause and a key solution to the climate crisis, as it accounts for two-thirds of greenhouse gas emissions, which result from the burning of fossil fuels [1].To combat the climate crisis and reverse its effect, transitioning from fossil fuels to renewable energy worldwide for energy needs is essential [2].The intermittent availability of sources of renewable energy, such as wind and solar, makes it very difficult to maintain a secure supply in the electric grid and mitigate the imbalance between production and demand of electricity [3,4].The use of thermal energy storage enables the widespread use of sources of renewable energy as well as the feasibility to maintain electric grid reliability.
The widely used mature technology for energy storage is pumped hydro-electricity storage.However, its use to cater the growing energy needs of the world is limited due to its geographical dependence, and low energy and power densities [5].Promising alternative large-scale technologies include pumped thermal energy storage [6], liquid air energy storage [7,8], and compressed air energy storage [9,10].In all these technologies, there is a need for a thermal energy storage system, in many cases including a packed-bed.
Several studies are available in open literature that use a packed-bed thermal energy storage (PBTES) loosely compacted with heat storage materials, forming a porous structure [11][12][13].Due to their availability, low cost and usability for a wide temperature range, rocks are commonly used as heat storage materials in PBTES [14].
During the charging process of the PBTES system, the high temperature heat transfer fluid, being in direct contact with the heat storage material, is pumped from the PBTES system top, transferring its heat to the heat storage materials and leaving from the PBTES system bottom at a reduced temperature.During the discharging process, the low temperature heat transfer fluid is pumped from the PBTES system bottom, reducing the temperature of the heat storage materials as it passes through the PBTES system and leaves from the PBTES system top at a higher temperature.A standby period between the charging and discharging, implying no air flow from the bottom and the top of the PBTES system and no use of the stored thermal energy, can also occur.The circulation of the heat transfer fluid inside the PBTES system during the charging/discharging processes creates two isothermal zones in the tank, the hot zone and cold zone, separated by a transition region with a significant thermal gradient called the "thermocline zone".A transverse temperature variation may occur in packed-bed systems during the charging/discharging/standby processes due to heat losses of preferential areas, causing a lower pressure drop in these areas.In addition, the lower pressure drop in these areas results in an increased mass flow rate and further cooling, leading to an increased transverse temperature variation, in a positive feedback loopa phenomenon called thermal instability [15,16].The thermal instability results in a difference between the maximum rock temperature of the packed-bed and the mean outlet temperature of the heat transfer fluid, which significantly can deteriorate the performance of the packed-bed.
In the available literature, the heat transfer fluids reported for PBTES systems are thermal oil [17], water/steam [18], molten salt [19], and air [20].Compared to other heat transfer fluids, air has advantages such as free availability, chemical stability, environmental friendliness, no degradation over time, and its ability to be used for a wide temperature range [21,22].Air is used as a heat transfer fluid in packed-bed systems for various industrial applications such as drying processes [23], catalytic reactors [24], gas-cooled nuclear reactors [25], air preheating in steel and glass industries [26,27], and air purification systems in industries [26,27].The available studies on PBTES systems with air as the heat transfer fluid mainly include small-scale/laboratory-scale setups [28][29][30], while the studies on air-based PBTES for large-scale energy storage systems (> 1 MWh th ) were reported in open literature to a limited extent [31][32][33][34][35][36][37].Previous works suggest that in order to improve the performance and increase the economic viability of PBTES systems [15,16], there is a need to understand the causes and to have preventive strategies at hand for instabilities occurring in packed-bed systems.
The thermal instability phenomenon in single-tank thermocline systems is rarely reported in the open literature.The majority of the studies available considered a liquid as the heat transfer fluid [38][39][40][41][42], and only three of them [15,16,43] considered air as the heat transfer fluid.Zavattoni et al. [43] and Davenne et al. [16] reported that the transverse temperature variations occurring in packed-bed systems causing thermal instability were due to non-uniform flow of the heat transfer fluid at the inlet and/or non-uniformities in void fraction.Zavattoni et al. [43] carried out a numerical investigation and developed a 2-dimensional computational fluid dynamics (CFD) model by considering variation of porosity in the transverse direction for charging/discharging processes.The authors reported that the radial porosity variation is insignificant for tank-to-particle diameter ratios higher than 25-30.Also, a 1-dimensional and a CFD model were developed by Davenne et al. [16] to investigate the growth of a small perturbation possibly causing a major disruption in the thermocline zone.
The viscosity of the air increases with the temperature; hence, the pressure drop that is required to be overcome in order to push the air through the PBTES system increases with the temperature of the air [16].Results of previous works indicate that packed-beds are generally more stable while undergoing the charging process in comparison to the discharging process [15,16].Both during charging and discharging, the pressure drop decreases in the preferentially colder areas, which leads to a higher mass flux in these areas compared to the heated areas.During the discharging process, the higher mass flux in the preferentially colder areas further reduces the temperature in a positive feedback loop, increasing the transverse temperature variation, and thereby making the packed-bed instable.On the other hand, during the charging process, the increase in mass flux in the preferentially colder areas increases the temperature of these areas more than the central hotter areas, leading to a lower transverse temperature variation compared to that occurring during the discharging process, and thereby making the packed-bed less prone to thermal instability.The standby process is another important process that can deteriorate the thermal performance of the PBTES systems and increase their instabilities.Depending on the application of the packed-beds, the timespan of the standby can change spanning from minutes to several days.In our previous pioneering work on large-scale air-rock PBTES systems, we predicted the thermal instability using numerical models and observed it in experiments for the discharging and standby processes [15].In addition, the effect of different storage sizes and standby durations on the thermal instability of packed-beds was reported for the first time.It was concluded that both the standby duration and the storage size significantly affect the thermal instability in PBTES systems.There are no previous works reported in the open literature that analyze methods to minimize the thermal instability in high temperature PBTES systems.This paper presents a numerical investigation of a PBTES system using air as the heat transfer fluid and magnetite rocks as the heat storage materials for high temperature and large-scale applications.The objective of the study is to identify methods to minimize the thermal instability, that is, to minimize the temperature difference between the maximum of the packed-bed rock and the mean of the outlet air.Transient 2-dimensional axisymmetric CFD models developed and validated in a previous work were used.Typically, the discharging and standby processes are less stable than the charging process, thus the focus of the current study is on the discharging and standby processes.The analysis looks at various parameters and their effects, including the inlet mass flow rate of the heat transfer fluid, type of rocks, heat transfer coefficient of the PBTES tank walls, inlet temperature of the heat transfer fluid, porosity of the PBTES system, diameter of the rocks, and geometry of the PBTES on the thermal instability during the discharging process initiated after one hour of standby process.Furthermore, the effect of various standby durations (1 h, 3 h, 5 h and 10 h) succeeded by a discharging process for two different casescase 1 and case 2 -were carried out.Case 1 represents a benchmark case with initial geometric and operating parameters, while the parameters for case 2 were selected to minimize thermal instability in the packed-bed.The parameters considered in the analysis were selected because they are the parameters that can be affected in practice when designing, building, and operating thermal energy storage systems including packed-beds.No previous work in the open literature reports the effects of these parameters, while aiming at reducing the instability of high temperature air-based packedbed rock thermal energy storage systems.
A reduction in the thermal instability in packed-bed systems can significantly increase their exergetic efficiency, increasing the roundtrip efficiency and thereby the economic viability of storage systems employing packed-beds.The findings of the paper provide a firm basis for system level analyses and the design of air-rock PBTES systems for high temperature applications, benefiting both industry and academia.
This paper is divided into four sections.A description of the numerical modelling is presented in Section 2. Section 3 shows the results and discussion, and Section 4 contains a summary of the study conclusions.

Numerical modelling
Davenne et al. [16] carried out a 3-dimensional CFD analysis of thermal instability in a PBTES system and found that the thermal instability occurring due to cold tunnel or non-uniform porosity spread symmetrically over time along the centerline.Based on these findings and to save computational time, transient 2-dimensional 2-phase axisymmetric CFD models using the Ansys Fluent 2022 R1 [44] software were developed and validated in our previous study, both for the discharging and the standby processes [15].The models consider the packed-bed as a continuous porous medium and assume uniform inlet flow conditions and uniform porosity.In the present work, the same models and modelling approach as those used in Ref. [15] were used.
The simulation domain for the benchmark analysis of the packedbeds considered in this study is shown in Fig. 1; it comprises a 2-dimensional axisymmetric cylindrical tank with axis boundary condition at the centerline of the simulation domain.The simulation domain consists of an inner diameter of 1.00 m and height of 5.32 m.The study looks at the mass flow rate inlet and the pressure outlet boundary conditions during the discharging process.The heat loss from the bottom of the PBTES system was omitted, as well as from the top during the discharging process, and at the walls a convective heat loss boundary condition was introduced.Since the air, during the standby process, does not flow in/ out of the PBTES system, there was the application of a convective heat loss boundary condition at the walls, top, and bottom of the PBTES system.
The following assumptions were made: • Axisymmetric simulation domain • The packed-bed consists of a homogeneous, continuous, and isotropic porous medium • Spherical rocks with equal average diameter (10 mm for the benchmark case) • Intraparticle temperature gradients neglected (Bi << 1) • Inlet plug flow for the heat transfer fluid Based on the assumptions outlined earlier, the following are the governing equations.
Continuity equation: where ρ f is the density of the air, v → is the superficial velocity, and ε (the fraction of volume of voids of the storage tank to its total volume) is the porosity of the PBTES system.Momentum equation: where the source term S i is given as To study the inertial and viscous momentum dissipations in the PBTES system, the source term is part of the momentum equation.The source term (Eq.( 3)) contains the viscous loss term or Darcy term, found on the right side of Eq. ( 3) as the first term and the inertial loss term, also found on the right side of Eq. ( 3), as the second term.Here, μ f and p represent the dynamic viscosity of the heat transfer fluid and the pressure, respectively.
C 1 and C 2 are inertial loss and permeability coefficients.The permeability coefficient (C 2 ) and the viscous loss coefficient are inverses of each other.As stated by Ergun [45], C 1 and C 2 are given as where d s is the rock diameter.Energy equation: Solid phase: where k eff,f is the effective thermal conductivity of air, k eff ,s is the effective thermal conductivity of the rocks, h is the heat transfer coefficient between the air and rocks, h w is the heat transfer coefficient on the PBTES tank wall surface, and a s is the superficial area of rocks per unit storage volume.The subscripts w, f and s are used for wall, fluid, and solid, respectively.The equations and correlations representing k eff,s k eff,f , h, h w , and a s were given in our previous study [15], and the same set of equations was used in this study.The temperature dependent thermophysical properties of the air and rock were estimated following Ref.[15].The finite volume method was used to solve the energy, momentum, and continuity equations.The effects of turbulence were considered by employing the k-ε model with a standard wall function.The pressurevelocity coupling was handled using the SIMPLE method [46], while second-order schemes were utilized for discretizing the energy and momentum equations, as well as for pressure corrections.The absolute residuals of 10 − 6 and 10 − 4 were obtained for the convergence criteria for the energy equations and for the continuity and momentum, respectively.There is significant variation across the rocks and the air in the local rate of change in temperature, because rocks and air have differing heat capacities and thermal conductivities.Therefore, in order to determine the thermal performance, it is necessary to consider the local temperature of the air and rocks.In this scenario, the local thermal non-equilibrium (LTNE) approach, employing distinct energy equations for the air and the rocks, was implemented to simulate the heat transfer within the PBTES system.Other functions as defined by users were added to incorporate the pressure losses in the PBTES system, the effective thermal conductivity of the air and rocks, and the heat transfer coefficient.Additionally, custom field functions were utilized to patch the initial temperature profiles to the PBTES system for the discharging and standby processes.

Results and discussion
To address the aim of identifying means to minimize the thermal instability (the temperature difference between the maximum of the packed-bed rock and the mean of the outlet air) in the PBTES system, this section presents the effects of inlet mass flow rate of air, type of rocks, heat transfer coefficient on the PBTES tank walls, inlet temperature of air, porosity of the PBTES, diameter of rocks, and geometry of the PBTES on the outlet temperature of the PBTES system during the discharging initiated after 1 h of standby.
As a benchmark case, the same parameters as selected in our previous study [15] were used.The porosity and dimensions in terms of diameter and height of the packed-bed are 0.36, 1 m and 5.32 m, respectively.Magnetite rocks of spherical shape and 10 mm diameter were considered, and the heat transfer coefficient on packed-bed tank walls was 0.77 W/m 2 ⋅K.The inlet temperature of air and mass flow rate during the discharging process were set to 300 K and 0.35 kg/s.In all cases, the discharging process was initiated after 1 h of standby.
Experimental results [15] indicate that after the completion of the charging process, the difference in temperature between the center and the PBTES tank walls for a given axial distance is 70 K, due to the heat losses from the walls of the PBTES tank during the charging process.The standby process is initiated after the end of the charging process.The temperature readings obtained from the thermocouples at the start of the standby process were curve fitted; see Eqs. ( 8) and ( 9).Here, T denotes the temperature in Kelvin (K), while Z signifies the distance along the axis measured from the bottom of the PBTES.Due to potential curve fitting errors, the temperature difference between the center and wall of the PBTES system may deviate from 70 K at some locations of the packed-bed (for e.g. at Z = 0).Therefore, the temperature difference between the center and the wall of the PBTES system for a given axial distance was considered as given by Eqs. ( 8) and ( 9) in this study during the start of the standby process.In the experimental setup [15], the thermocouples (measuring the temperature) were located at the center and the wall of the PBTES system.No thermocouples were located at different radial locations inside the packed-bed.Therefore, the temperature of the PBTES system was assumed to be the same in the radial direction, except for the wall temperature, for a given axial position.The total storage volume was kept the same for all the cases.

Model validation
In our previous paper [15], the transient axisymmetric 2-dimensional 2-phase CFD model used in the present paper was validated using experimental data obtained by the authors for the discharging and standby processes.The CFD model was validated by comparing the numerical and experimental results in terms of variation of the temperature inside the PBTES system with location and time.It was found that the mean absolute percentage deviation between the numerical and experimental results is < 5 % and < 1 % for the discharging and the standby processes, respectively, indicating that our model provides reasonable results.

Influence of inlet mass flow rate of air
Fig. 2 shows the thermal instability in the PBTES system at different time steps during the discharging process for three different inlet mass flow rates, 0.25 kg/s, 0.35 kg/s, and 0.45 kg/s.The mass flow rate of 0.35 kg/s was considered as a reference case corresponding to 7.5 h discharging time for a packed-bed of 5.32 m height and 1.0 m diameter.In order to analyze the influence of mass flow rate, the lower and higher values (0.25 kg/s, and 0.45 kg/s) were considered.All other parameters were kept the same as those of the benchmark case.The findings suggest that the temperature difference follows the same pattern for all three mass flow rates.The temperature is lower at the PBTES system top at the beginning of the discharging process, than the maximum temperature of the rocks inside the PBTES system, because of the PBTES tank losses occurring in the standby period.As the discharging process advances, the air drawn from the PBTES bottom absorbs the heat from the rocks at high temperature and rejects some of the heat to the rocks at the PBTES system top, resulting in an increase in the temperature at the PBTES system top and a decrease in the outlet air temperature.Therefore, the thermal instability is higher at the beginning of the discharging process, and it decreases with time until 120 min of discharging time for 0.45 kg/ s and 0.35 kg/s, and until 150 min of discharging time for 0.25 kg/s.
When the temperature of rocks at the PBTES system top reaches the maximum temperature inside the PBTES, the thermal instability is minimum.Afterwards, the outlet temperature starts decreasing because of the thermal instability in the PBTES arising due to heat losses from the walls of the PBTES.Therefore, the thermal instability increases until it reaches its maximum value.After this point, the thermal instability again decreases with time.This is because the PBTES system approaches full discharged conditions, and since there are small maximum and average temperatures in the whole PBTES tank, the result is that there is less temperature difference between the maximum of the packed-bed rock and the mean of the outlet air.Furthermore, the results presented in Fig. 2 indicate that the maximum thermal instability decreases with the increase in the inlet mass flow rate of air, being 108 K, 80 K and 64 K for 0.25 kg/s, 0.35 kg/s and 0.45 kg/s, respectively.There are two reasons for this.First, the ratio of heat transfer due to convection in the axial direction to the heat transfer because of conduction in the radial direction increases with the inlet mass flow rate, resulting in a lower variation of temperature in the radial direction at a given axial location in the packed-bed and a lower temperature difference between the center and wall of the PBTES system at higher mass flow rates.Second, a decrease in the discharging time with the increase in mass flow rate results in lower heat losses from the walls of the PBTES system.Fig. 3 depicts how the change in mass flow rate affects the pressure drop in the PBTES system.The findings indicate that the maximum pressure drop increases with the inlet mass flow rate.The higher the inlet mass flow rate, the higher the velocity of air through the packedbed, increasing the pressure loss.Hence, though the thermal instability in the packed-bed decreases with increasing inlet mass flow rate of air, also the pressure drop through the packed-bed needs to be considered to avoid excessive pressure loss resulting in a high power demand to push the air through the packed-bed.

Effect of rock types
A comparison of different rock types was carried out to investigate the effect of different thermophysical properties of heat storage materials on the thermal instability of the PBTES system.Granite and basalt rocks were considered in addition to the benchmark case of magnetite rocks; their thermophysical properties are reported in Table 1.Apart from the type and thermophysical properties of the rocks, all other operating and geometric parameters were considered the same as those of the benchmark case.
Fig. 4 shows the thermal instability in the PBTES system versus time during the discharging process.The volume of the PBTES tank is kept the same in all cases.The results indicate that the thermal instability increases with the use of granite rocks and basalt rocks in comparison to the benchmark case (80.4 K) with magnetite rocks, by 67 % and 60 %, respectively.The main reason for the lower thermal instability for magnetite rocks is their higher density and higher specific heat capacity compared with the other rock types, resulting in a lower temperature drop throughout the PBTES.Because of higher density and higher Fig. 2. Thermal instability for different inlet mass flow rates of air.

Table 1
Thermophysical properties of the rocks.

Thermophysical properties
Granite rocks [47] Basalt rocks [48,49] Magnetite rocks [15,50] Density (kg/m  Kothari et al. specific of magnetite rocks, the temperature drop of the rocks close to the walls for magnetite rocks is lower than granite and basalt rocks for the same heat transfer coefficient on the outside of the walls.As a result, the temperature difference between the center and the regions close to the walls becomes smaller for magnetite rocks compared to granite and basalt rocks, thereby causing a smaller thermal instability and a more uniform outlet temperature in packed-bed with magnetite rocks.In addition, a longer discharging time is obtained for magnetite rocks than those of granite and basalt rocks.Overall, the results suggest that heat storage materials with high energy density can be useful in reducing the thermal instability, size, and investment cost of packed-bed units for large-scale energy storage technologies.

Effect of heat transfer coefficient
Both the insulation material and its thickness affect the loss of heat through the walls of the PBTES tank and thereby affect the thermal instability in the PBTES system.In order to analyze the effect on the thermal instability of the heat losses from the walls of the PBTES system, the thermal instability versus time during the discharging process initiated after 1 h of standby is estimated for three different heat transfer coefficients, 0.37 W/m 2 ⋅K, 0.57 W/m 2 ⋅K, and 0.77 W/m 2 ⋅K; the results are presented in Fig. 5.An HTC of 0.77 W/m 2 ⋅K on the walls of the PBTES tank corresponds to 0.13 m thick rock wool insulation and 283 K of ambient temperature.The lower values of the heat transfer coefficient (0.37 W/m 2 ⋅K and 0.57 W/m 2 ⋅K) can be achieved in practice by increasing the thickness of the insulation (lower than the critical insulation thickness) and by using lower thermal conductivity material compared to that of rock wool for a fixed ambient temperature.The results suggest that the maximum thermal instability is 44 K, 58 K, and 80 K for the heat transfer coefficient values 0.37 W/m 2 ⋅K, 0.57 W/m 2 ⋅K, and 0.77 W/m 2 ⋅K, respectively.It may be concluded that the insulation of the packed-bed plays a significant role for the thermal instability.

Effect of geometry of the packed-bed
The effect of geometry of the PBTES system on the thermal instability was investigated by considering three different height (H)/diameter (D) ratios of the PBTES system, 0.5, 1, 2 and 5.The height (H) and diameter (D) for each case are reported in Table 2.All other parameters except for the heat transfer coefficient (see below) were kept the same as those of the benchmark case.
The initial temperature profile used in the benchmark case, given by Eq. ( 8) and Eq. ( 9), was non-dimensionalized and used for the different cases.The non-dimensional initial temperature profile for the packedbed and for the wall are given by Eq. ( 10) and Eq. ( 11), respectively.A temperature difference between the center and wall of the PBTES system of approximately 70 K was imposed at the start of the standby process for all the cases. where Here, T Z=0 = 501.4K for the packed-bed and 421.4K for the wall.T is the temperature in K, H is the total height of the packed-bed in meters, and Z is the axial location inside the packed-bed in meters.Since the non-dimensional temperature profile is the same for all the cases, the heat transfer phenomena should remain the same for all the cases.However, the heat transfer coefficient on the walls of the PBTES system varies with the length of the PBTES system due to variation in heat transfer fluid flow velocity and temperature distribution near the walls along the length [51].Therefore, the Biot number (h L c /k packed-bed tank ) is kept the same, rather than the heat transfer coefficient, for all the cases.By assuming an heat transfer coefficient (h) from the PBTES tank wall of 0.77 W/m 2 ⋅K, a thermal conductivity of the material of the PBTES tank (k packed-bed tank ) of 30.12 W/m⋅K, and a characteristic length (L c ) of 5 m, a Biot number of 0.13 was obtained.
Figs. 6 and 7 depict the temperature contours and velocity contours of the PBTES system, respectively at different discharging times initiated after 1 h standby duration for different H/D ratios.The contour plots are   shown only for half of the PBTES system due to the symmetry of the PBTES tank.The results indicate that the temperature and velocity at the center of the PBTES system is higher than at the outer radial locations of the PBTES system for all the cases.Moreover, the difference in temperature increases with time for all the cases.Fig. 8 shows the thermal instability versus time during the discharging process initiated after 1 h standby duration for different H/D ratios.Initially the temperature difference is highest for the H/D ratio of 0.5 and 1 followed by 2 and 5, because of a higher heat transfer  coefficient on the walls of the PBTES tank for an H/D ratio of 0.5 and 1, which is a consequence of keeping the Biot number the same for all cases.This trend continues up to 390 min of discharging time where the thermal instability increases for the H/D ratio of 5 and reaches the maximum temperature difference.The maximum thermal instability is 48 K, 50 K, 57 K, and 90 K for the H/D ratios 0.5, 1, 2, and 5, respectively.The higher the H/D ratio, the greater the heat loss, and thereby the higher temperature difference between the center and the peripheral areas and the greater the thermal instability.However, the smaller the H/D ratio, the greater the volume occupied by the thermocline (due to larger diameter and lower air velocity, thus higher cross-sectional area of the PBTES tank and lower heat transfer between the rocks and air), and thereby the greater the degradation of the performance after numerous repeated charge/discharge cycles.It needs to be stressed that the losses from the bottom and top of the PBTES system during the discharging process were not considered in this investigation, which may slightly increase the thermal instability for the large diameter PBTES system (H/D = 0.5 and 1).
Fig. 9 presents the influence of the H/D ratio on the pressure loss in the packed-bed.The findings indicate that the pressure drop decreases with decreasing H/D ratio due to the decreasing height of the PBTES system.The maximum pressure drop decreases from 3844 Pa to 90 Pa when the H/D ratio is decreased from 5 to 0.5.Overall, since lower H/D ratio results in both lower thermal instability and lower pressure drop, we recommend to use a low H/D ratio for air-rock PBTES systems, while still using an H/D ratio high enough to ensure keeping a proper thermocline profile during numerous repeated charging/discharging cycles.

Effect of inlet temperature of air
In the context of large-scale energy storage technologies, like pumped thermal energy storage, the inlet temperature should be kept close to the bottom temperature of the PBTES (411 K in this analysis for all cases) during discharging and close to the top temperature of the PBTES during charging.This is because higher differences in these temperatures may cause design complexities for other components, for example, the turbomachinery.Operating the PBTES based on fixed top and bottom temperatures (design operating temperatures), as done when evaluating the storage capacity by design engineers, would lead to rapid degradation of the thermocline, reducing the effective thermal energy storage capacity during cyclic operation [52,53].Off-design temperature operation, deviating from fixed top and bottom temperatures, reduces the thermocline degradation [52,53].For example, in concentrated solar power plants, the minimum discharge temperature (top temperature) is restricted by the minimum permissible temperature at the turbine inlet, and the maximum charging temperature (bottom temperature) is restricted by the minimum permissible temperature at the inlet of the solar collector field [54,55].These minimum and maximum allowable temperatures are about 20 % of the design operating temperature range [54,55].
The present analysis looks at the effect of the inlet air temperature on the thermal instability of the PBTES system.Three different inlet temperatures were considered, 300 K, 350 K, and 400 K, while keeping all other geometric and operational parameters the same as those of the benchmark case.The results are reported in Fig. 10.The maximum thermal instability is 80 K, 49 K, and 37 K for inlet temperatures 300 K, 350 K, and 400 K, respectively, that is, the lower the inlet air temperature, the higher the thermal instability in the PBTES.When the inlet temperature is low (for e.g. the 300 K case), the air absorbs heat from already cooled rocks near the walls of the PBTES tank, leading to further cooling of the rocks near the walls.This increases the mass flux near the wall region and gives a positive feedback effect, resulting in a lower air outlet temperature and higher thermal instability in the PBTES system.On the other hand, when the air inlet temperature is high (for e.g. the 400 K case), the air transfers heat to the cold rocks near the walls, leading to a negative feedback effect, in turn maintaining the outlet air temperature close to the maximum rock temperature.
Furthermore, the findings depicted in Fig. 10 indicate that the influence of the inlet temperature is relevant only close to the completion of the discharging process, that is, when the thermocline approaches the top of the PBTES system.Before that there is practically no difference among the different cases.This is because the radial temperature variation is significantly higher when the PBTES system is close to discharged, leading to higher thermal instability in the PBTES system.Overall, a higher inlet air temperature reduces the thermal instability in the PBTES system.However, maintaining the inlet temperature of the air and the bottom temperature of the tank very close to the design conditions, decreases the discharged energy from the PBTES system, and therefore, the selection of the inlet temperature of the air is a tradeoff between minimizing the thermal instability and maximizing the recovered energy from the PBTES system.

Effect of porosity of the packed-bed and diameter of the rocks
Fig. 11 shows the effect of rock diameter (5 mm and 10 mm are applied) and porosity (0.36 and 0.4 are applied) on the thermal instability of the PBTES system.The results suggest that there is no significant influence of the rock diameter nor of the porosity on the temperature difference.This is because these parameters only have a limited effect on the heat loss from the packed-bed.However, a little decrease in the thermal instability is observed (the temperature difference is decreased from 92 K to 76 K) when the porosity is increased from 0.36 to 0.4 and the diameter is increased from 5 mm to 10 mm.This is due to increased permeability of the PBTES system resulting from the combined effect of increased diameter and higher porosity, enabling enhanced mixing of the air in the PBTES system.With the increase in porosity, the height of the PBTES system increases, leading to an increase in cost of the PBTES system.Therefore, there is a tradeoff between the reduction in thermal instability and increase in porosity.
Fig. 12 presents the influence of the porosity of packed-bed and diameter of rocks on the pressure loss in the packed-bed.The findings indicate that the smaller the diameter particles and the lower the porosity, the higher the pressure drop.The maximum pressure drop decreases from 13,952 Pa to 2485 Pa when the porosity is increased from 0.36 to 0.4 and the diameter is increased from 5 mm to 10 mm.Overall, since high packed-bed porosity and rock diameter results in both low thermal instability and low pressure drop, we recommend to use a high porosity and high diameter for air-rock PBTES systems, while still ensuring a proper H/D ratio and considering possible manufacturing constraints of the tank and the cost for the packed-bed governed by its size.

Evaluation of a tailored case
Two different cases, case 1 (benchmark case) and case 2, are evaluated for various standby durations succeeded by the discharging process.The parameters for the two cases are reported in Table 3.The mass flow rate of air considered in both cases is 0.35 kg/s, while the inlet temperature of air for case 1 and case 2 is 300 K and 350 K, respectively.Based on the results presented in Section 3.4-3.6,the inlet temperature of air, H/D ratio, and heat transfer coefficient of the walls of the PBTES tank, respectively, were selected for case 2 to minimize the thermal instability in the PBTES system.The findings presented in Section 3.7 indicate that the rock diameter and the porosity of PBTES systems do not significantly affect the thermal instability; hence, these are kept the same for case 2 as the benchmark case.Moreover, since the use of magnetite rocks as storage material results in a lower thermal instability than those of the other rock types (granite and basalt) (see Section 3.3), magnetite rocks are used as storage material in case 2. Fig. 13 depicts temperature contours of the PBTES system at different discharging times for both the cases for the discharging process following 10 h of standby duration.As explained earlier, contour plots are shown only for half of the PBTES system owing to the symmetry of the PBTES system.The findings suggest that the temperature gradient in the transverse direction is much higher for case 1 than for case 2.Moreover, the temperature gradient increases more with respect to time for case 1 than for case 2.
Fig. 14 shows the thermal instability for the discharging process following different standby durations (1 h, 3 h, 5 h, 10 h) for both the cases.The results suggest that all the curves follow a similar trend (as discussed in Section 3.2).The figure shows that for both cases, the thermal instability increases with the standby duration, as can be explained by the loss of heat through the walls of the PBTES system, as well as from the top, and bottom, and such losses increase with the 11.Thermal instability for different porosity of packed-bed and diameter of rocks.standby duration, raising the transverse temperature variation in the PBTES at the beginning of the discharging.In addition, the results suggest that there is a significant reduction in the thermal instability of the PBTES system in case 2 compared to case 1 for all standby durations.The maximum thermal instability is 200 K and 29 K for case 1 and case 2, respectively, for the discharging process following 10 h of standby duration.In conclusion, there is a significant effect of the H/D ratio of the PBTES system, the inlet temperature of the air, and the insulation (heat transfer coefficient) of the PBTES system on the thermal instability of a high temperature air-rock PBTES system.However, the thermal instability is present in the PBTES system throughout the discharging process initiated after 10 h of standby duration also for the conditions for case 2.

Design guidelines
The inlet mass flow rate, height to diameter (H/D) ratio, and inlet air temperature are the main design and operating parameters that affect the thermal instability of high temperature thermal energy storage systems utilizing an air-rock packed-bed.The inlet mass flow rate of air reported in the literature depends on the application and the charging/ discharging time of the packed-bed thermal energy storage system.The present study suggests that the deviation from the design condition (lower mass flow rate compared to design condition) leads to higher thermal instability.Therefore, while deciding the minimum flow rate for air-rock packed-bed the thermal instability should be considered.The typical value of H/D considered in the literature is 0.5 to 2 [56], while the results of our analysis suggest that the H/D ratio should be between 1 and 2 to minimize the thermal instability.The literature reports that the inlet temperature of heat transfer fluid in packed-bed can differ up to 20 % from the design operating temperature range [54,55] to avoid rapid degradation of the thermocline.From thermal instability perspective, our study suggests that maintaining the inlet temperature of the air and the bottom temperature of the tank very close to the design conditions reduces the thermal instability of the packed-bed.
In addition, the standby duration of the packed-bed plays a significant role in the thermal instability of the packed-bed.However, the standby duration is not a design parameter, and the duration of the standby period can vary depending on the application of the packedbed.

Conclusions
The numerical investigation presented in this paper addresses high temperature applications of large-scale and incorporates a thermal energy storage system utilizing air-rock packed-bed.Given such storage systems, the study aimed to identify methods to limit the thermal instability.The approach involved transient 2-dimensional axisymmetric computational fluid dynamics models that we developed and validated in a previous work.The key findings of the present study are outlined as follows: 1.The thermal instability, that is, the temperature difference between the maximum of the packed-bed rock and the mean of the outlet air, decreases by 41 %, 45 %, and 56 % with the increase in inlet mass

Fig. 3 .
Fig. 3. Pressure drop for different inlet mass flow rates of air.

Fig. 5 .
Fig. 5. Thermal instability for different heat transfer coefficient at the outside of the walls.

Fig. 12 .
Fig. 12. Pressure drop for different porosity of the packed-bed and diameter of the rocks.

Fig. 13 .
Fig. 13.Temperature contours for 10 h of standby process succeeded by discharging process at various time intervals for (a) Case 1 and (b) Case 2.

Fig. 14 .
Fig. 14.Thermal instability for various standby durations succeeded by a discharging process for case 1 and case 2.

Table 2
Different height to diameter (H/D) ratios and their corresponding height (H) and diameter (D).

Table 3
Parameters considered for comparing case 1 and case 2.