# Experimental study of thermocline behavior under high propagation velocities in a water based thermal energy storage

Alexis Ferre, Sylvain Serra, Jérôme Pouvreau, Rémi Manceau, Arnaud Bruch, Olivier Soriano, Mathilde Bois

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Alexis Ferre, Sylvain Serra, Jérôme Pouvreau, Rémi Manceau, Arnaud Bruch, et al.. Experimental study of thermocline behavior under high propagation velocities in a water based thermal energy storage. Journal of Energy Storage, 2025, 125 (116921), 10.1016/j.est.2025.116921. hal-05063852

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Research papers

# Experimental study of thermocline behavior under high propagation velocities in a water based thermal energy storage

Alexis Ferre<sup>a</sup>, Sylvain Serra<sup>b</sup>, Jérôme Pouvreau<sup>a</sup>, Rémi Manceau<sup>c</sup>, Arnaud Bruch<sup>a,*</sup>, Olivier Soriano<sup>a</sup>, Mathilde Bois<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Thermal energy storage
Thermocline
Experimental setup
Stratification
Richardson number
Hydraulic distributors

## ABSTRACT

This experimental study investigates the development of the thermocline under varied operating conditions and particularly at high propagation velocities ranging from 0.5 mm.s⁻¹ up to 8 mm.s⁻¹ and temperature differences (ΔT) of 10 °C, 30 °C and 65 °C. For this purpose, a new experimental setup, designed to investigate thermocline formation, propagation, and withdrawal is implemented. The setup meets essential requirements for acquiring controlled validation data for CFD models and enables the collection of novel high-velocity flow data. The storage system features extensive instrumentation, including temperature measurements at curved tank bottoms and radial positions. Dynamic experiments validated literature findings, demonstrating stronger stratification with slower axial propagation velocities and larger temperature differences. High propagation velocities remain feasible with adequate ΔT, i.e. with a Richardson number based on the penetration height of the inlet fluid, $Ri_{0,h_i} \ge 0.25$ ensuring a thin thermocline for this test section. Variation of the extraction distribution system revealed the influence of fluid withdrawal on thermocline thickness, underscoring the need to consider outlet dynamics in distributor design. These findings establish the relevance of $Ri_{0,h_i}$ for predictive performance correlations and provide a foundation for optimizing thermal energy storage systems through refined distributor designs. Finally, an original definition of thermocline thickness was introduced, based on the slope of the thermocline, offering an alternative to the classical boundary-temperature approach and limiting the dependence on temperature limits.

## 1. Introduction

Thermocline thermal storage is a key technology for enhancing energy efficiency across various applications, such as district heating networks, solar thermal power plants, and industrial processes. Single medium thermocline storage (SMTS) is characterized by three distinct zones: a hot zone, a cold zone, and a thermal gradient zone, commonly referred to as the thermocline. For optimal performance, a narrow thermocline must be maintained during operating cycles. Thermocline development relies on three phases: formation, propagation and withdrawal.

In the ideal case of a tank initially at uniform temperature, the thermocline begins forming during the injection of fluid. The thickness of the thermocline is largely determined by turbulent mixing during this stage [1-4]. To achieve a fine and stable thermocline, it is recommended

to inject the fluid horizontally. The inlet flow is thus characterized by a density difference between the incoming fluid and the surrounding medium, also theorized as a density current [1]. The initial penetration height of these currents has been identified as the main responsible for the thermocline thickness [5]. Applied to thermal storages, recent studies using Large Eddy Simulation (LES), have further elucidated the governing dynamics [6-9] of density currents. The geometry of the distributor plays a critical role in this phase, as it significantly impacts the uniformity and stability of the velocity profile. Hence, numerous studies have been testing and developing new designs (see review of the designs in [10]). Among the studies in the literature, Zurigat et al. (1991) [11] tested a lateral pipe, a perforated plates, and an elbowed pipes but did not explicitly compare their effectiveness, focusing instead on deriving 1D correlations. Davidson et al. (1994) [12] found that porous media distributors outperformed axial pipes in maintaining stratification, based on the MIX number. Shah and Furbo (2003) [13]

<footnote><sup>a</sup> Univ. Grenoble Alpes, CEA, Liten, DTCH, 38000 Grenoble, France</footnote>

<footnote><sup>b</sup> Université de Pau et des Pays de l'Adour, E2S UPPA, LaTEP, Pau, France</footnote>

<footnote><sup>c</sup> Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, Inria, équipe CAGIRE, LMAP, Pau, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: arnaud.bruch@cea.fr (A. Bruch).</footnote>
Nomenclature

Latin symbols

D Storage diameter [m]

E Stored energy [J]

H Storage height [m]

g gravity acceleration [m.s⁻²]

h Specific enthalpy [J.kg⁻¹]

Re Reynolds number [−]

Ri Richardson number [−]

t Time [s]

T Temperature [K]

U<sub>ax</sub> Mean axial velocity of the fluid inside the storage [m.s⁻¹]

Greek symbols

α Diffusivity [m².s⁻¹]

β Thermal expansion coefficient [K⁻¹]

δ Thermocline thickness [m]

ν Kinematic viscosity [m².s⁻¹]

ρ Density [kg.m⁻³]

θ Dimensionless temperature [−]

τ Restitution rate [−]

ξ<sub>t</sub> Slope obtained from a dimensionless temporal temperature evolution [m⁻¹]

ξ<sub>z</sub> Slope obtained from a dimensionless temperature profile [s⁻¹]

Indices and exponents

0 globally

c cold

diff. diffusive

h hot

h<sub>i</sub> Inlet height

in Inlet

nom Nominal

max Maximum

ss Stainless steel

w water

(t) Obtained thanks to a temporal temperature evolution

(z) Obtained thanks to an axial temperature profile

θ Based on temperature limits

∇ Based on thermocline slope

Abbreviations

SMTS Single Medium Thermal Storage

Δ Difference

Temporal average

◊ Spatial average

showed that a single-plate deflector better preserved exergy compared to axial pipes, which were the most destructive. Chung et al. (2008) [14] demonstrated that double-plate and modified double-plate distributors produced thinner thermoclines than H-split pipes. Panthalookaran et al. (2008) [15] concluded that conical axial pipes with low diffusion angles offered higher global efficiency using a thermodynamic entropy-based indicator. García-Marí et al. (2013) [16] confirmed the superiority of porous pipes over elbowed pipes for various flow rates. Li et al. (2014) [17] and Chandra and Matuska (2020) [2] both found that impacting jet pipes enhanced discharge efficiency better than a showerhead type distributor or lateral configurations. Hosseinnia et al. (2021) [18] reported that double-plate deflectors achieved thermoclines approximately 2.5 times thinner than impacting jet pipes.

Once the thermocline is established, the thermal front propagates vertically within the tank. Local turbulent motions generated by the injection, along with diffusion, contribute to thermocline thickening [18].

Finally, the thermocline is withdrawn out of the storage. However, the specific impact of the withdrawal on thermocline systems using water remains underexplored.

While the described phases affects the thermocline, various dimensionless numbers can be used to characterize its quality in the storage (up to 11 according to [19]). Among these, the Richardson number (Ri) is widely used [20–27] to quantify the balance between buoyancy forces and turbulent stresses, which directly influences stratification quality. This parameter can be expressed locally (Ri<sub>g</sub>), based on density gradients and shear forces:

$$Ri_g = \frac{\frac{g}{\rho} \left( \frac{\partial \rho}{\partial z} \right)}{\left( \frac{\partial u}{\partial z} \right)^2} \quad (1)$$

or globally (Ri<sub>0</sub>), using macroscopic storage parameters and operating conditions:

$$Ri_0 = \frac{g\beta\Delta TL_{ref}}{U_{ref}^2} \quad (2)$$

High Ri<sub>0</sub> values favor stratification by enhancing the buoyancy effect

relative to turbulence. Experimental studies have attempted to establish threshold Ri<sub>0</sub> values for stratification performance. Based on the average velocity of the flow in the tank and the storage height, Zurigat et al. (1991) [11] reported that when the Richardson number exceeds 3.6, the influence of distributor geometries on thermal stratification becomes negligible. Since it affects the thermocline formation, the global Richardson number derived from tank height and distributor outlet velocity (Ri<sub>0,H</sub>) is commonly employed to assess stratification. Guo and Wu (1985) [28] observed that when Ri<sub>0,H</sub> > 1, thermal stratification is promoted due to a reduction in forced convection and an enhancement of the temperature gradient. Zurigat et al. (1988) [29] concluded that for a Ri<sub>0,H</sub> > 5, the geometry of the distributor has minimal effect on stratification. Ghajar et al. (1991) [30] reported that for Ri<sub>0,H</sub> > 10, mixing effects at the inlet can be neglected. Hence, no consensus has been reached in literature. Indeed, Ri<sub>0</sub> does not explicitly characterize the physical phenomena within the storage, unlike Ri<sub>g</sub>, which assesses the competition between shear, which generates turbulence, and the local buoyancy that dampens it vertically.

Additionally, a Richardson number based on the penetration height of the inlet fluid (h<sub>i</sub>) has been used to optimize distributor designs [5,31–34]:

$$Ri_{0,h_i} = \frac{g\beta\Delta T h_i}{U_{in}^2} \quad (3)$$

The penetration height depends on the geometry of the distributor. For a single-plate distributor, this height corresponds to the distance between the plate and the bottom of the storage tank. For other distributor geometries, it is defined as the vertical distance occupied by the incoming flow or by the density current as it exits the distributor [33]. Therefore, for the double-plate distributor used in this study, h<sub>i</sub> corresponds to the spacing between the two plates.

In practice, the inlet Froude number (Fr<sub>i</sub>) is used; however, the two numbers are related by the following equation Ri<sub>0,h<sub>i</sub></sub> = Fr<sub>i</sub><sup>2</sup>. For efficient stratification, the Froude number should remain below 2, corresponding to Ri<sub>0,h<sub>i</sub></sub> > 0, 25 [1,35].

At a fixed geometry, temperature difference (ΔT) and axial propagation velocity of the fluid (U<sub>ax</sub>) are key operating parameters affecting thermal stratification. Hence, numerical sensitivity studies on flow rate
have been carried out [2,4,36]. Researchers agree that increasing the flow rate disrupts stratification due to mixing at the inlet and outlet ports. Conversely, if the flow rate is too low, fluid transit time becomes excessive, and conduction-induced diffusion may become significant. Regarding fluid temperature, ΔT generates density difference that stabilizes the flow which aim to reduces thermocline thickness [2,4]. However, a larger ΔT also enhances conduction within the fluid, leading to a broader thermal gradient.

In addition to simulations, experimental studies are conducted to validate the models used. These experimental tests are either performed by the authors themselves [3,24,37–40] or sourced from third-party publications [4,25,41–44]. For most of the validation figures provided, the thermal gradient is generally well captured far from the injection point, while it can be poorly predicted in the region near the distributor. This issue is usually explained by noting that the flow has not yet fully developed in this area [2,4]. In other words, the mixing phenomena associated with the turbulent flow at the storage inlet are not well modeled, highlighting the importance of local validation. Indeed, if the simulation struggles to accurately reproduce thermocline formation, it will compromise the overall validity of the model. Hence, characterization techniques such as Particle Image Velocimetry (PIV) [45] or the use of dyes [13,46,47] have been employed to visualize the flow. In terms of temperature, a high density of thermocouples near the distributor is required to capture locally the thermal repartition.

For cylindrical flat-bottomed storage, Xu et al. (2022) [48] recently extensively instrumented the area close to the distributor, with thermocouples (TC) spaced vertically by 5 cm. For tanks with domed bottoms, no fine instrumentation in the lower and upper parts of the storage is currently known by the author in the literature.

Yet experimental results are available in the literature and have been mapped in both dimensional and dimensionless quantities (Fig. 1). The mean fluid velocity in purely liquid thermocline thermal energy storage systems has been investigated in relatively few studies, resulting in a limited range of reported values, typically below 1 mm.s⁻¹. To the authors' knowledge, some studies explore velocities up to 1.5 mm.s⁻¹, but does not exceed 2 mm.s⁻¹ (Fig. 1.a). It is important to note that this commonly cited threshold of 1 mm.s⁻¹, often observed in industrial applications, is not derived from a comprehensive or systematic study but rather appears to be based on practical experience, suggesting that thermocline formation remains acceptable below this value. However, most studies do not justify this limitation nor identify any specific behavior or physical phenomenon to support it. Increasing this velocity allows for enhanced discharge power, meeting the needs of industrial processes. Moreover, most experimental studies employ ΔT values

ranging from 10 °C to 30 °C, which are relatively small for industrial applications. For example, urban heating networks typically operate with ΔT values exceeding 50 °C.

As for the dimensionless quantities, the ones considered are the Reynolds number within the tank ($Re_{tank} = U_{ax}D/ν$) and $Ri_{0,H}$ (Fig. 1.b). These parameters both characterize the flow regime: the influence of turbulence and of buoyancy, respectively. Their ranges, as reported in the literature, will be discussed later in the paper.

Hence, this paper presents a new experimental setup that enables the acquisition of controlled experimental data for CFD validation purposes, while also covering a broad range of temperatures and velocities. A parametric study varying these parameters will be conducted to investigate their influence on the storage performance. Finally, the impact of all phases of the thermocline development will be examined: formation, propagation, and extraction.

## 2. Experimental set-up

### 2.1. Storage and distributors

Before detailing the specific features of the storage, its environment is described here. The test section is integrated into an experimental loop

<figure><img src="image_3.png" /><figcaption>Fig. 2. Simplified PID around the test section. The heating and cooling circuits are not shown due to space limitations.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) Dimensional and (b) dimensionless mapping of available experimental studies for SMTS using water as heat transfer fluid in dynamic phase (charging or discharging). Articles referenced in caption: [11,13,17,20,38,47,49–56].</figcaption></figure>
(Fig. 2) which consists of three independent circuits:

* **Test section circuit** (Circuit 200): Contains the storage and uses soft water as the heat transfer fluid. Soft water prevents equipment fouling while supporting electromagnetic flow meter operation;

* **Heating Circuit** (Circuit 100): Supplies heat to Circuit 200 using demineralized water via a 250 kW electric boiler and a heat exchanger.

* **Cooling Circuit** (Circuit 400): Cools Circuit 200 via heat exchanger, using industrial water.

The test section system consists of a 3.2 m straight section (ferrule), two domed bottoms, and two hydraulic distributors, for a total height of 3.5 m. The ferrule has a diameter of 711 mm which results in an aspect ratio (H/D) of 5. This aspect ratio allows to maximize the storage volume while ensuring compatibility with the existing built environment. This choice also meets literature recommendations, which advocate for aspect ratio of 3 or higher to promote thermal stratification [4,18,36,57,58].

Made of stainless steel, the storage has a net volume of 1.35 m³ and a capacity of 100 kWh for a temperature difference of 65 °C. The domed bottoms can be dismantled to modify the instrumentation or change the hydraulic distribution system. Flat gaskets are installed between the flanges in the experimental setup to prevent water leakage. The storage is positioned on a metal support (Fig. 3).

A layer of insulation is applied around the storage and its support. The insulating material is glass wool ($\lambda \approx 0.04 \text{ W.m}^{-1}.\text{K}^{-1}$ at 20 °C). Its thickness is 100 mm on the storage, while a thickness of 50 mm surrounds the metal support. In addition, to limit conduction losses between the storage and its support, 4 cm insulating wedges (Materials:

<figure><img src="image_5.png" /><figcaption>Fig. 3. STEROID storage in its environment.</figcaption></figure>

MONOLUX®, $\lambda \approx 0,20 \text{ W.m}^{-1}.\text{K}^{-1}$ at 20 °C) are placed at the junction between the two.

Among all the geometries of distributors found in the literature, the double-plate dispenser is selected because it seems to reconcile performance and simplicity of design. The dimensions are as follows: plate diameter ($D_p$) is 200 mm, inter-plate spacing ($e_p$) is 20 mm, height relative to the tank bottom ($h_p$) is 75 mm, and the external pipe diameter ($D_e$) is 44.3 mm (Fig. 4).

Furthermore, to study the different aspects of distribution, global design choices concerning distributors were made:

* In order to perform parametric studies on the double-plate distributor (not presented in this paper), the geometric parameters $D_p$, $e_p$, $h_p$ can be varied;

* To study the impact of another distributor, the actual distributor can be uninstalled, and other geometries can be tested;

* To study the impact of the distributor on the withdrawal, there are two possibilities: the fluid exits through the distributor (Fig. 5.a) or through the pipe at the end of the domed bottom, also known as the “high/low point” (Fig. 5.b). Fluid injection is always carried out by the distributor.

## 2.2. Operating range

At low velocities, where the flow regime is generally favorable for the thermocline, experimental data is well-documented in the literature; however, no experimental data are available for higher velocities exceeding $2 \text{ mm.s}^{-1}$ (Fig. 6.a). These higher velocity regimes are particularly relevant both from an applicative standpoint to reach higher power, and from a scientific perspective, as they challenge the limits of thermocline destabilization and model accuracy. Thus, two operating ranges can be mentioned:

* the desired operating range: $U_{ax} \in [0.5; 8] \text{mm.s}^{-1}$ and $\Delta T \in [10; 65]^{\circ}\text{C}$;

* the possible operating range: dictated by the thermal and hydraulic stresses of the experimental loop.

Therefore, the operational range achievable by the test section is the intersection of the targeted and feasible ranges.

In dimensional terms, the proposed experimental setup operates within a range consistent with the literature and allows to explore novel high-velocity scenarios up to $8 \text{ mm.s}^{-1}$ for a maximum $\Delta T$ of 20 °C (Fig. 6.a). However, not all velocities can be tested for every $\Delta T$ due to the limitation of the heating power. The extended operating range of the experimental setup is shown in dimensionless quantities (Fig. 6.b).

## 2.3. Instrumentation

To measure the temperature, T-type thermocouples (TCs) (copper/constantan, 1 mm diameter) are arranged in the test section along several axis (Fig. 7). TCs are arranged as follows:

* **Domed bottoms:** to examine the formation and deformation of the thermocline, as well as the transverse temperature distribution during fluid injection and withdrawal, 31 TCs are arranged in each domed bottom along six vertical axes (shown in blue in Fig. 8). The instrumentation setup is identical for both the upper and lower domed bottoms (Fig. 9);

* **The ferrule:** To study the propagation of the thermocline, TCs are placed every 10 cm along four axes (A1, B1, C1, and D1). Generators C1 and D1 are used to verify azimuthal symmetry, while A1, B1, and either C1 or D1 serve to confirm radial homogeneity. Additionally, in the center of the main axis, TCs are spaced 5 cm apart over a height of
<figure><img src="image_6.png" /><figcaption>Fig. 4. Double plate distributor of the test section.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. 2D plan of the lower distributor to illustrate the withdrawal (a) by the distributor and (b) from the low point.</figcaption></figure>

1 m to capture the thermocline spread more precisely during the standby phase (indicated in red in Fig. 10);

* Near-internal wall: to study near-wall behavior, especially during the standby phase, five groups of five TCs (each 0.5 mm in diameter) are positioned radially at 2 azimuthal positions and five vertical locations (K3 and L3) (shown in green in Fig. 10).

* Outer wall: To analyze the effect of storage inertia, 11 TCs are positioned on the outer wall of the storage tank (shown in yellow in Fig. 10).

Altogether, the storage instrumentation includes a total of 273 TCs. After calibrating the sensors, the temperature measurements of the thermocouples are accurate to ±0.20 °C, those measured by platinum probes to ±0.03 °C, and the flow rates to ±7.2 × 10⁻² m³·h⁻¹. To monitor the entire instrumentation of the setup, measurements are recorded every 20 s.

## 3. Criteria definition

Various criteria are developed in the literature to characterize

stratification in water thermocline storages (see reviews [59,60]). In industrial systems, the evolution of the outlet temperature is important because it allows evaluating the energy restored. However, to evaluate the influence of injection on the formation of the thermocline, the thickness of the thermocline is chosen because, in addition to being a commonly used criterion, it allows to characterize the area of the thermal gradient in a relevant and easily understandable way. Hence, two criteria are developed in the following section: the thermocline thickness and the restitution rate.

In order to facilitate the understandings, explanatory plots in this section are presented as a function of the height z and the dimensionless temperature θ is used in the sketches:

$$ \theta(z) = \frac{T(z) - T_c}{T_h - T_c} \qquad (4) $$

### 3.1. Thermocline thickness

Here we introduce two different ways of estimating the thermocline thickness, one based on the gradient at the inflection point (δᵣ) and the other on the distance between points where limit temperatures are
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Possible and desired operating range of the test section in (a) dimensional and (b) dimensionless quantities. Dots represent experimental tests from the literature (same references as those used in Fig. 1); solid black lines represent the thermal and hydraulic limits imposed by the experimental loop; the gray shaded area indicates the range of possible tests; green dashed lines show the desired test range; the green shaded area represents the achievable test range. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Mapping of TCs generators (measurements shown in black are radii in mm).</figcaption></figure>

reached ($\delta_\theta$). To avoid calculated thickness being arbitrarily dependent on the definition used, we will impose the constraint that the definitions give the exact thickness of the thermocline in the simplified case of the piecewise linear temperature profile illustrated in Fig. 11 for which the thickness is unambiguously defined.

### 3.1.1. Thickness based on the thermocline slope

The first thickness definition is based on the thermocline slope and therefore focuses on its central region. While the maximum temperature gradient has been used to calculate the volume fraction of the mixed zone [61], it has not been directly used to measure the thermocline thickness.

To estimate the thermocline slope, given the limited number of experimental points and associated noise, it is preferable to perform a regression rather than a gradient estimation. Thus, the temperature distribution is assumed to be linear around the mean temperature. The thickness of the thermocline is defined by the intersection between the extreme temperatures and the linear function (Fig. 11.a). For the experimental case studied, and to reliably capture the thermocline slope, points satisfying $\theta = 0.5 \pm 0.15$ are selected to perform the regression.

This approach corresponds to identifying the simplified profile in 3

<figure><img src="image_11.png" /><figcaption>Fig. 8. Schematic scaled representation of TCs in the lower domed bottom (a) on the OE axis and (b) on the SN axis.</figcaption></figure>

linear zones that approximate the real profile (Fig. 11.c). Based on a dimensionless temperature profile, the thickness is expressed as:

$$ \delta_{\nabla}^{(z)} = \frac{1}{\varepsilon_z} \qquad (5) $$

with $\varepsilon_z$ (m$^{-1}$) being the slope obtained from the dimensionless temperature profile. The exponent “(z)” indicates that this thickness is calculated from an axial profile. With a temporal evolution of temperature at a given point, which is experimentally easier to obtain and use, a constant average axial liquid velocity is assumed, and the exponent “(z)” replaced by an exponent “(t)”:
<figure><img src="image_12.png" /><figcaption>Fig. 9. Photograph of the instrumentation in the lower domed bottom.</figcaption></figure>

$$ \delta_{\nabla}^{(t)} = \frac{U_{ax}}{\xi_t} \qquad (6) $$

with $\xi_t$ ($s^{-1}$) as the slope. The thickness $\delta_{\nabla}$ is based on the central part of the thermocline and does not necessarily capture the concavities near the hot and cold temperatures on a temperature profile. Therefore, a second criterion based on top and bottom edges is used.

### 3.1.2. Thickness based on temperature limits

The second thickness definition is based on the temperature limits i. e. the admissible hot and cold limits of temperature and focuses on thermocline edges (Fig. 11.b). The industrial process in which the storage is included imposes the fraction of the tolerable temperature difference.

In this study, the chosen dimensionless temperature range is $0.15 \le \theta \le 0.85$. With a temperature profile alongside a vertical line,

thickness is calculated thanks to $z_h$ and $z_c$ which are the heights where $\theta_h = 0.85$ and $\theta_c = 0.15$ respectively. This specific definition of thickness, along with various alternative threshold definitions, has been used in the literature (Table 1) highlighting the absence of standardized criteria. This suggests that thermocline thickness is highly dependent on user-defined limits, and no clear consensus currently exists.

Hence, in this paper, since 70% ($\theta_h - \theta_c = 0.70$) of the temperature difference is considered for the thickness calculation, a coefficient $1/\gamma$ is applied (Eq. (7)). This coefficient is introduced to minimize the impact of the choice of limiting temperatures on the calculated thickness. The coefficient is chosen so that, in the case of a simplified piecewise linear profile, definition (8) gives the same thickness regardless of the choice of limiting temperature and also the same thickness as definition (6), as illustrated in Fig. 11.c.

$$ \delta_{\theta}^{(z)} = \frac{1}{\gamma} \Delta z \text{ with } \gamma = \theta_h - \theta_c \qquad (7) $$

With the evolution of temperature at one sensor, the thickness is calculated thanks to $\Delta t = t_h - t_c$ where $t_h$ and $t_c$ are times when $\theta_h = 0.85$ and $\theta_c = 0.15$ respectively. In addition, an ideal plug flow is assumed to reconstruct the thickness:

$$ \delta_{\theta}^{(t)} = \frac{U_{ax}}{\gamma} \Delta t \qquad (8) $$

In the following study, the temperature evolution is used. The two definitions of the thickness are used to quantify the impact of the flow rate and the temperature difference.

### 3.2. Restitution rate

Each test is characterized by its:

* Mean initial temperature ($\langle T_{in} \rangle$): the arithmetic mean of the temperature of all TCs in the storage at the initial time;

* Mean inlet temperature ($T_{in}$): the time average of the temperature entering the storage;

<figure><img src="image_13.png" /><figcaption>Fig. 10. Schematic representation of thermocouples in the storage (a) in the ferrule on SD axis and (b) on the inner and outer wall on the K generator. 1 dot = 1 TC.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Thermocline thickness (a) based on thermocline slope and (b) obtained on temperature limits. Both thicknesses are calculated on a simplified temperature profile in (c).</figcaption></figure>

<table><caption>**Table 1**

Thresholds used in the literature to calculate thermocline thickness.</caption><thead><tr><td>Threshold (θ<sub>c</sub> - θ<sub>h</sub>)</td><td>References</td></tr></thead><tbody><tr><td>0.10-0.90</td><td>[14,62-64]</td></tr><tr><td>0.14-0.86</td><td>[65]</td></tr><tr><td>0.15-0.85</td><td>[56,62]</td></tr></tbody></table>

* Mean outlet temperature (T<sub>out</sub>): the time average of the temperature exiting the storage.

The initial and inlet temperatures define the hot (T<sub>h</sub>) and cold (T<sub>c</sub>) temperatures of each test. On charge: T<sub>c</sub> = <T<sub>t0</sub>> and T<sub>h</sub> = T<sub>in</sub>; on discharge: T<sub>c</sub> = T<sub>in</sub> and T<sub>h</sub> = <T<sub>t0</sub>>. The maximum temperature difference for a test is therefore:

$$ \Delta T = T_h - T_c \qquad (9) $$

Energy is stored by the module during the charging phase; this energy is released during the discharging phase.

The power (P) depends on the inlet flow rate and the outlet temperature of the storage:

$$ P = \dot{m}_w [h_w(T_{in}) - h_w(T_{out})] \qquad (10) $$

with Ḍ<sub>w</sub>, water mass flow rate (kg.s<sup>-1</sup>) and Ḍ<sub>w</sub>, specific enthalpy of water (J.kg<sup>-1</sup>). While water's specific heat capacity remains nearly constant within the operating temperature range, specific enthalpy was chosen to avoid relying on simplifying physical assumptions, ensuring greater accuracy and providing a more versatile approach. Indeed, this method allows for broader applicability to other fluids, whose physical properties may vary more significantly. Additionally, this approach did not entail higher computational costs. The entire thermocline cannot be discharged due to system constraints, which limits the duration of the discharge process to a time t<sub>lim</sub> which is defined thereafter. Hence, the power is maximal as long as the outlet fluid maintains a constant temperature, and it decreases once the thermocline begins to exit (Fig. 12). This power is used to calculate the stored energy (E):

$$ E = \int_{t_{0}}^{t_{\text{lim}}} P(t)dt \qquad (11) $$

With t<sub>0</sub> being the initial moment of the test. In the experimental data processing presented here, the instant t<sub>lim</sub> corresponds to the instant from which the outlet temperature reaches a limit temperature (Eq. (12)). This energy does not represent the instantaneous energy of the module, which is derived from the axial temperature profile within the storage, but rather the energy exchanged between the storage and the process.

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Stored energy.</figcaption></figure>

In practice, it is the system in which the storage is integrated that imposes these limit temperatures according to its constraints. Here, to be consistent with the definition of δ<sub>θ</sub><sup>(t)</sup> it is defined from a portion of ΔT:

$$ T_{\text{lim}} = T_c + \theta_{\text{lim}} \Delta T \qquad (12) $$

with θ<sub>lim</sub> ∈ {θ<sub>min</sub>; θ<sub>max</sub>} , where θ<sub>min</sub> = 0.15 and θ<sub>max</sub> = 0.85. Under a charge, θ<sub>lim</sub> = θ<sub>min</sub>: the fluid is used by the process as long as its temperature is lower than T<sub>lim</sub>. Under a discharge, θ<sub>lim</sub> = θ<sub>max</sub>: the fluid is used as long as its outlet temperature is higher than T<sub>lim</sub>.

The maximum energy is the energy that can be stored in the storage i. e. in the water, the walls and any element internal to the tank, for the maximum temperature difference of a test. In the module, the mass of water is 1301 kg, that of the walls is 244 kg and that of the flanges is 343 kg. Therefore the flanges are considered in the solid structure made of stainless steel (ss). Consequently:

$$ E_{\text{max}} = E_w + E_{\text{ss}} \qquad (13) $$

Where:

$$ E_{\text{mat}} = m_{\text{mat}} [h_{\text{mat}}(T_h) - h_{\text{mat}}(T_c)] \qquad (14) $$

Where mat ∈ {water; stainless steel}. The properties of stainless steel (304 l) (ss) are taken from the REFPROP database and the ones of water from the NIST database [66]. For a temperature difference of 30 °C: E<sub>w</sub> = 45 kWh and E<sub>ss</sub> = 2.4 kWh. In other words, E<sub>ss</sub> represents approximately 5 % of the total energy. The vessel wall acts as an additional thermal storage medium, which slows down the thermal front propagation. Due to its higher thermal conductivity compared to water, the wall can also introduce significant transverse temperature gradients,
potentially degrading stratification. At high flow velocities, thermal cycles are shorter, limiting the wall's ability to absorb heat, but increasing the risk of transverse inhomogeneities. Conversely, at low velocities, thermal exchange between the fluid and the wall may reach an equilibrium, further slowing the thermal front while reducing the risk of transverse mixing.

Finally, the restitution rate, which quantifies the ratio of stored energy compared to the maximum energy storable in the test section, is defined as:

$$ \tau = \frac{E}{E_{max}} \qquad (15) $$

This criterion ranges from 0 to 1:

- when $\tau \to 1$, it indicates strong stratification (thin thermocline) or a low constraint on $T_{lim}$;

- when $\tau \to 0$, it indicates weak stratification (thick thermocline) or a high constraint on $T_{lim}$.

This criterion is highly dependent on the limit temperature chosen to calculate the stored energy: if $T_{lim} \to T_c$ on discharge, then $\tau \to 1$ even with a thick thermocline.

## 4. Thermal characterization of the flow

To characterize the stages of thermocline development inside the storage, the temperature is represented at various stages of the filling process. This process constitutes a loading test with operating conditions within the range documented in the literature: $U_{ax} = 1$ mm.s$^{-1}$, $\Delta T = 25$ C – 55 C. Water is withdrawn via the distributor.

### 4.1. In the upper domed bottom

Initially, hot water is injected into the upper domed bottom of the storage tank:

* Above the distributor (z = 3510 mm and z = 3460 mm): the zone shows a temperature homogeneity (Fig. 13.a). A temperature delay is noticeable at the highest point (TC G2–3510 mm), which is consistent with the time required for the fluid to reach the top;

* Below the distributor: At 3360 mm, thermal homogeneity is also observed, although slightly more pronounced deviations can be detected. In this regard, the axial temperature profile at 100 s and z

= 3360 mm shows a deviation of 5 °C (Fig. 13.b). At 3210 mm, the thermocline is formed.

Overall, the temperature is homogeneous within the upper domed bottom section.

### 4.2. In the ferrule

The flow continues within the cylindrical shell:

* At a fixed position: The temperature evolution highlights the thermocline propagation, confirming its formation (Fig. 14);

* At different heights or at different time intervals (Fig. 14 or Fig. 15): The shape of the curves is consistent, indicating a piston-like flow. This ideal flow is characterized by a uniform fluid displacement with minimal mixing between adjacent regions;

* Along generators at the same radius (C1, D1): The temperature behavior is consistent in temporal evolution and axial profiles, demonstrating azimuthal symmetry (Fig. 14, Fig. 15);

* Along generators at different radii (A1, B1, C1): The temperature behavior is similar in both temporal evolution and axial profiles. Furthermore, the radial profiles at the tank's mid-height during the thermocline passage show a temperature variation of only 1 °C across the section (Fig. 16). Therefore, the temperature is considered homogeneous across a section.

### 4.3. In the lower domed bottom

In the lower part of the storage tank, water is extracted:

* Above the distributor (z = 310 mm and z = 160 mm): Stratification is observed (Fig. 17.a). The flow is homogeneous across the section and exhibits azimuthal symmetry;

* Below the distributor (z = 60 mm and z = 10 mm): The cold temperature increases uniformly. The thermocline is not observed because it is extracted by the distributor. Additionally, the delayed temperature rise at 10 mm indicates stratification, which is confirmed by the axial profiles (Fig. 17.b). A slight transverse inhomogeneity is noticeable at 60 mm and 10 mm (Fig. 17.a) and in the profile at 3400 s (Fig. 17.b). Finally, the persistence of a cold zone below the distributor supports the need for liquid to exit through the lowest point of the domed bottom, which is feasible with the experimental setup;

<figure><img src="image_16.png" /><figcaption>Fig. 13. Temperature in the upper domed bottom: (a) temporal evolution and (b) axial profiles.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 14. Temperature evolution in the ferrule.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. Axial temperature profiles in the ferrule.</figcaption></figure>

*   **At the outlet of the storage tank:** The temperature evolution downstream of the distributor is similar to that upstream. Therefore, qualitatively, the extraction seems to have minimal impact on the thermocline.

### 4.4. Synthesis

The temperature evolution described above was analyzed at multiple positions, revealing:

*   **Radial homogeneity:** While slight temperature variations are observed near the distributor, the temperature is homogeneous across a section within the cylindrical shell. Therefore, at a given vertical position sufficiently distant from the distributor, (3210 mm < *z* < 60 mm), the temperature evolution at a single point is representative of the temperature across the entire section. This observation will be relevant for calculating the thermocline thickness in subsequent analyses.

*   **Azimuthal symmetry:** The temperature evolution at points located on the same radius is identical. Consequently, the flow can be considered axisymmetric with respect to the central axis of the storage tank. This can be useful for performing 2D-axisymmetric simulations.

Finally, conditions unfavorable for stratification (high velocity, low Δ*T*) have been tested and these conclusions remain valid.

## 5. Parametric study

Operating parameters impacting the stratification, $U_{ax}$, $\Delta T$ and withdrawal distribution system, will be varied here. Thermocline will be characterized during three stages: formation, propagation, and extraction. The experiments considered involve charging (injection through the upper distributor) and discharging (injection through the lower distributor). Extraction is typically conducted via the distributor, except in Section 4.3, where it will be done via the low point of the curved bottom.
<figure><img src="image_19.png" /><figcaption>Fig. 16. Radial temperature profiles at the middle of the ferrule (z = 1760 mm) during the thermocline passage.</figcaption></figure>

Initially, the storage is expected to be thermally homogeneous, although slight temperature variations are present due to experimental realities. Analysis of representative test cases consistently shows initial temperature deviations of approximately 0.5 °C across all thermocouples, and about 0.4 °C along the axial profile. These minor variations are mainly observed near the distributor zones, especially at the top and bottom of the tank, where flow geometry induces localized non-uniformities.

Unless otherwise specified, all thicknesses presented in this section are calculated along the A1 generator, at the center of the storage (z = 1750 mm).

### 5.1. Thermocline formation

A sensitivity study on the nominal flow rate and ΔT is presented. As a reminder, not all velocities can be tested for each ΔT due to heating power limitations (Fig. 6). Thus, the highest velocities (6 mm·s⁻¹ and 8 mm·s⁻¹) are tested exclusively at a temperature difference of 10 °C.

Fig. 18.a and Fig. 18.b show similar results, indicating that the thicknesses definitions are robust. This suggests that there is no bias associated with the thickness definitions, and the observed differences are solely due to the expected variations in profile shapes. Therefore, the thickness can be analyzed with confidence.

At a given ΔT, the general trend indicates that the thermocline
<figure><img src="image_20.png" /><figcaption>Fig. 17. Temperature in the lower domed bottom: (a) axial profiles and (b) temporal evolution</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. Impact of mean axial velocity on thickness based (a) on thermocline slope and (b) on temperature limits. Criterion on the limit temperatures: $\theta_{min} = 0.15$ and $\theta_{max} = 0.85$. Thicknesses are calculated at TC A1-1750mm.</figcaption></figure>

thickens with increasing velocity. Indeed, higher flow rates enhance mixing near the curved surfaces, which in turn affects thermocline formation. Specifically:

* For a ΔT of 10 °C, the thermocline reaches up to 2.5 m in a 3.5 m storage tank (Fig. 18). This unusually high value at 8 mm.s⁻¹ can be attributed to the fact that for these low ΔT values, the stabilizing effect of buoyancy at the thermocline is insufficient to counteract the destabilizing effects of turbulence;

* For ΔT values of 30 °C and 60 °C, the thermocline thickness does not increase with flow velocities between 0.5 mm.s⁻¹ and 2 mm.s⁻¹. In these cases, the density differences are sufficient to stabilize the flow. A limiting thermocline thickness of 25 cm is observed for this specific system.

The competition between the stabilizing effects induced by buoyancy

and the destabilizing effects caused by turbulence can be quantified by examining $Ri_{0,h_i}$. For this type of distributor, the injection height $h_i$ is defined as the gap between the plates $e_p$ (Fig. 4). For all tests, a notable common trend for thickness is observed (Fig. 19). In particular for the charging tests at ΔT = 30 °C:

* At 2 mm.s⁻¹ et 3 mm.s⁻¹, i.e. $Ri_{0,h_i}$ of 0.57 and 0.26 respectively: thermocline thickness $\delta_{\theta}^{(t)}$ measures 35 cm and 45 cm (Fig. 19.b). This represents 10 % and 13 % of the total storage height respectively;

* At 4 mm.s⁻¹, i.e. $Ri_{0,h_i}$ = 0.14: the thickness $\delta_{\theta}^{(t)}$ increases and represents 26 % of the total storage height.

Therefore, for this experimental device, it is possible to operate at high speed and obtain reasonable thermocline thicknesses, here <13 %
<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Impact of $Ri_{0,h_i}$ on thermocline thickness based on (a) the temperature slope and (b) temperature limits. Criterion on the limit temperatures: $\theta_{min} = 0.15$ and $\theta_{max} = 0.85$. Thicknesses are calculated at TC A1-1750mm. Various $U_{ax}$ have been used for the tests.</figcaption></figure>

of the total height, as long as the $Ri_{0,h_i}$ is sufficiently high ($Ri_{0,h_i} \ge 0.25$). Thus, $Ri_{0,h_i}$ is a relevant parameter that effectively represents the competing physical phenomena in this problem. Defining the Richardson number based on the ferrule height lacks physical meaning, as the resulting value arbitrarily depends on a parameter that has no influence on thermocline formation. Indeed, for the current storage system, $Ri_{0,H} \ge 43$ allows to obtain an appropriate stratification which corresponds to an order of magnitude much higher than the values obtained by the studies carried out in the literature (factor of at least 4).

Moreover, regarding the restitution rate, an increase in thickness leads to its reduction as the flow rate increases (Fig. 20). This is coherent: the thinner the thermocline, the greater the energy released by the storage. Furthermore, as for the thickness, a common trend for all tests is also observed for the restitution rate (Fig. 20.a). In addition, when plotted as a function of $Ri_{0,h_i}$, it is revealed that the threshold of 0.25 is relevant to characterize the performances: below this threshold the restitution rate increases, above the threshold the rate hardly varies any more (Fig. 20.b).

<figure><img src="image_23.png" /><figcaption>**Fig. 20.** Impact of (a) axial velocity and (b) $Ri_{0,h_i}$ on restitution rate. Criterion on the limit temperatures: $\theta_{min} = 0.15$ and $\theta_{max} = 0.85$. Various $U_{ax}$ have been used for the tests.</figcaption></figure>

## 5.2. Thermocline propagation

Since the beginning of this section, thicknesses have been calculated at a single point within the storage unit. However, to visualize thermocline propagation, thickness can be calculated at various positions within the reservoir. To assess thermocline thickening during its transport, thickness is measured at five locations:

* Outside the storage unit: at the lower and upper outlets;

* Inside the storage unit: along the central axis, at the bottom ($z = 560$ mm), middle ($z = 1760$ mm), and top ($z = 3160$ mm).

The behavior during charging and discharging is similar (Fig. 21). The x-axis is oriented so that the thermocline is propagating from left to right for both charging and discharging.

To simplify explanations, observations are focused on the charging phase:

* For $U_{ax} \in \{0.5;1;2\}$ mm.s$^{-1}$, thermocline thickness increases over time (top → bottom). It forms at the top of the storage unit and continuously thickens as it travels downwards;
<figure><img src="image_24.png" /><figcaption>Fig. 21. Evolution of thermocline thickness based on temperature limits ($\theta_{min} = 0.15$; $\theta_{max} = 0.85$) during a (a) charge and a (b) discharge, with $\Delta T = 30$ °C. Criterion on the limit temperatures: $\theta_{min} = 0.15$ and $\theta_{max} = 0.85$. The arrow indicates the flow direction. White/gray area: inside/outside the storage.</figcaption></figure>

* For $U_{ax} \in \{3;4\}$ mm.s$^{-1}$, thermocline thickness initially decreases (top → middle) and then increases (middle → bottom) over time. This indicates that thermocline is not fully formed at the chosen TC at the top of the storage unit, as this region lies within the mixing zone, which grows with higher injected flow rates. Once fully formed (middle → bottom), the thermocline thickens until it reaches the outlet.

Finally, significant thermocline thickening is observed as the thermocline exits the storage unit (bottom → lower outlet) (Fig. 21). This thickening is associated with mixing induced by extraction at the distributor. It is thus relevant to pay special attention to the distributor used during water extraction. Accordingly, two fluid extraction methods are explored in the following section.

### 5.3. Thermocline extraction

The extraction via the distributor (Fig. 5.a) is compared with extraction from the low point (Fig. 5.b). For this study:

* Thicknesses are calculated based on PT-100 sensors at the upper (discharge) or lower (charge) outlets;

The test operating conditions are: $U_{ax} = 1$ mm.s$^{-1}$; $\Delta T = 25$ °C – 55 °C.

For both calculated thicknesses, the thermocline is thinner when extracting from the bottom than when using the distributor (Fig. 22). However the restitution rate is slightly higher compared to extraction from the distributor, although the difference is not significant. The slight discrepancy in the restitution rate is minimal because it corresponds to a $Ri_{0,h_t} > 0.25$, at which the criterion shows little variation.

## 6. Conclusion

A new experimental setup has been designed and implemented to meet the requirements for acquiring:

* Controlled validation data, including a known and homogeneous initial temperature state, measured variations in flow rate and inlet temperature during the test, and constant temperature injection;

<figure><img src="image_25.png" /><figcaption>Fig. 22. Impact of extraction on (a) the thermocline thickness based on the temperature limits ($\theta_{min} = 0.15$; $\theta_{max} = 0.85$; thermocline calculated at the lower outlet during charging and the upper outlet during discharging) and (b) the restitution rate ($\theta_{min} = 0.15$; $\theta_{max} = 0.85$) (1 bar = 1 trial).</figcaption></figure>
New data for unexplored parameters, such as high-velocity flow data, $U_{ax} \in [0.5: 8] \text{ mm.s}^{-1}$.

The storage system is distinguished by its extensive instrumentation, which includes temperature measurements at the tank's curved bottoms and various radial positions. Special attention was given to hydraulic distributors with adjustable geometry, enabling the future study of different flow configurations at the inlet and outlet of the storage system.

This experimental setup allowed studies of thermocline development:

* An original definition of thermocline thickness has been proposed, based on the slope of the thermocline. The classical definition, which relies on boundary temperatures, has been adapted to eliminate dependence on arbitrarily chosen limits;

* Variations in flow rate and $\Delta T$ confirmed findings from the literature: stratification becomes stronger as axial propagation speed decreases and $\Delta T$, and thus $\Delta \rho$, increases;

* With sufficient temperature differences, reasonable thermocline thicknesses can be achieved. Thus, high propagation velocities are achievable as long as $\Delta T$ remains sufficient. For this storage, a Richardson number $Ri_{0,h_i} \ge 0.25$ ensures thermocline thicknesses below 15 % of the total storage height. Since the value of $h_i$ can be varied for this distributor, it will be possible to assess whether the 0.25 threshold remains valid in different configurations;

* The Richardson number based on the penetration height of the inlet fluid $Ri_{0,h_i}$ was validated as a relevant parameter for establishing a consistent trend across tests with different characteristics, proving its applicability in predictive performance correlations for this system;

* Adjustments to the distribution system highlighted the impact of fluid withdrawal on thermocline thickening at the outlet. While thickening is less pronounced during withdrawal than during injection, it is advisable to account for fluid outlet dynamics when designing distributors.

Finally, this experimental setup could enable further experimental studies, including the study of standby and cycling conditions, the generation of data for CFD validation, and the testing of optimized distributors.

### CRediT authorship contribution statement

**Alexis Ferre:** Writing – review & editing, Writing – original draft, Methodology, Data curation, Conceptualization. **Sylvain Serra:** Writing – review & editing, Supervision, Methodology, Investigation, Conceptualization. **Jérôme Pouvreau:** Writing – review & editing, Supervision, Methodology, Investigation, Conceptualization. **Rémi Manceau:** Writing – review & editing, Supervision, Methodology, Conceptualization. **Arnaud Bruch:** Writing – review & editing, Supervision, Methodology, Investigation, Conceptualization. **Olivier Soriano:** Supervision, Software, Methodology, Investigation, Conceptualization. **Mathilde Bois:** Investigation, Formal analysis, Data curation.

### Declaration of competing interest

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

### Data availability

Data will be made available on request.

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