2 nd law analysis of thermocline -heat storage system

Baptiste Rebouillat, Quentin Falcoz, Pierre Neveu

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Baptiste Rebouillat, Quentin Falcoz, Pierre Neveu. 2 nd law analysis of thermocline -heat storage system. THE 32ND INTERNATIONAL CONFERENCE ON EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS, Jun 2019, Wroclaw, Poland, Poland. hal-05109693

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# 2nd law analysis of thermocline
– heat storage system

Baptiste Rebouillat<sup>a</sup>, Quentin Falcoz<sup>b</sup> and Pierre Neveu<sup>c</sup>

**Abstract:**

Thanks to its relative low cost and good performances, thermocline tank storage system is a promising way to store sensible heat. The proposed work aims at finding an optimal design for both the filler material and the tank size that minimises the exergy destruction occurring in the thermocline. To this end, the following study uses a 2 phases 1-D model able to evaluate the entropy production associate to a discharging phase. The method is illustrated through two examples: a thermocline designed for an Andasol-type CSP and working with a liquid heat transfer fluid; a thermocline pilot used to recover industrial process heat and working with a gaseous heat transfer fluid.

**Keywords:**

Exergy, Optimisation, Thermal storage, Thermocline, Porous medium

# 1. Introduction

This study deals with thermocline tank system used as sensible heat storage. The study covers two different applications: the concentrated solar plant, and the process heat valorisation. In both cases, the storage system is placed between 2 components. One that gathers thermal energy from a source (industrial heat or sun) and one other that transforms the thermal energy into a utility (mechanical energy, heating ...). The storage handles the mismatch between the energy availability and the energy demand.

Maximizing energy efficiency of such processes can be performed using 2^nd^ Law analysis. That consists in evaluating the entropy generated (or exergy destroyed) by each sub-system (energy collector, energy storage and energy conversion) and detect which one is mainly responsible of the entropy production, and thence, has to be optimized in priority. Such methodology requires quite accurate evaluation on the entropy production for each sub-system involved in the process. Numerous works have focuses on 2^nd^ law analysis for conversion systems and concentrated solar systems, operating in quasi-stationary regime but only a few concern thermal storage systems. For these latter, transient behaviour has to be taking into account, and numerical integrations required to compute the entropy production can lead to big uncertainties. The paper aims to evaluate the entropy generated by a thermocline system, and to performed its geometry optimisation.

A thermocline system is composed of a tank filled with filler materials (e.g rocks). The thermal energy is injected/extracted in the tank thanks to a heat fluid transfer (HTF). This HTF can be a gas, usually air, or a liquid, usually oil or molten salt. When the HTF is a liquid, the structure must be designed

<footnote><sup>a</sup> Laboratoire PROMES-CNRS, Perpignan, France, Baptiste.Rebouillat@promes.cnrs.fr</footnote>

<footnote><sup>b</sup> Laboratoire PROMES-CNRS, Odeillo, - Université de Perpignan Via Domitia - UPVD, France, Quentin.Falcoz@promes.cnrs.fr</footnote>

<footnote><sup>c</sup> Université de Perpignan Via Domitia - UPVD, Perpignan, France, neveu@univ-perp.fr</footnote>
for a vertical fluid flow. If the HTF is a gas, the structure can be horizontal, thus reducing the stresses on the foundations of the tank. While operating, the hot HTF flows in/out from one side and the cold HTF flows out/in from the other side. Consequently 3 different zones appear (Cf. Fig. 1). There are 2 zones with quasi-uniform temperature, one hot and one cold. These zones are separated by a zone with an important temperature gradient called the thermocline zone.

<figure><img src="image_2.png" /><figcaption>Figure 1. Thermocline tank working principle.</figcaption></figure>

Designing a thermocline tank implies to take into account several phenomena. Indeed, the system's performance depends on the flow of matter through the packed-bed, the quality of the exchange between the HTF and the solid or the conductive diffusion in the thermocline zone. The presented thermodynamic study aims at finding a thermocline tank optimal geometry that fulfil design specifications. These design specifications impose characteristics such as the energy capacity and the discharging time (or the average power). The method is based on the minimisation of the entropy generation. It is applied in this study to two industrial cases. Firstly, the replacement of a 2-tank system by a thermocline in an Andasol type concentrated solar plant (CSP) and secondly an industrial pilot named Eco-Stock which recover industrial process heat.

## 2. Model

The model depicted below uses strong hypotheses in order to analyse a wide range of configurations in a reasonable computing time. It aims at defining trends in order to limit the studied field, for further analysis through classical CFD method.

### 2.1. Hypotheses

- Side walls are adiabatic.

- The medium is compound of a solid porous phase (the packed-bed) and fluid phase (the heat transfer fluid).

- The properties are considered uniform along a radius (1-dimension model).

- Thermophysical properties of the solid filler and the fluid are constant.

- Unidirectional and steady flow, Darcy's flow.

The Representative Elementary Volume (REV) is defined as a slice of the tank. It is composed of a fluid subdomain ($REV_f$) and a solid subdomain ($REV_s$). All the properties are considered uniform into the REV.

### 2.2. Equation

Thanks to the 1D flow hypothesis, the normal component of the matter flow is null. The equation system (1) presents the transient balances of energy and entropy, as well as the Gibbs equation applied to the $REV_f$ and the $REV_s$.
$$
\left\{
\begin{aligned}
\frac{dE_{\Delta\varphi}}{dt} &= \iint_{\Sigma_{\Delta\varphi}} -(\vec{q}_{n,\varphi} + \vec{q}_{ta,\varphi} + \vec{m}_{ta,\varphi}\Delta h_{\varphi}) \cdot \overline{d\Sigma_{\varphi}} \\
\frac{dS_{\Delta\varphi}}{dt} &= \iint_{\Sigma_{\Delta\varphi}} \left(\frac{\vec{q}_{n,\varphi}}{T_{\varphi}} + \frac{\vec{q}_{ta,\varphi}}{T_{\varphi}} + \vec{m}_{ta,\varphi}\Delta s_{\varphi}\right) \cdot \overline{d\Sigma_{\Delta\varphi}} + \iiint_{V_{\Delta\varphi}} \dot{s}_{irr} \overline{dV_{\Delta\varphi}} \\
\frac{dU_{\Delta\varphi}}{dt} + T_{\varphi}\frac{dS_{\Delta\varphi}}{dt} &= 0
\end{aligned}
\right.
\quad (1)
$$

$E_{\Delta\varphi}$, $U_{\Delta\varphi}$ and $S_{\Delta\varphi}$ are respectively the energy (J), the interne energy (J) and the entropy (J.K⁻¹) of the REV$_\varphi$. $\vec{q}_{n,\varphi}$ and $\vec{q}_{ta,\varphi}$ names the heat flux densities (W.m⁻²) into the phase $\varphi$ in the normal and tangential direction to the matter flow. $\vec{m}_{ta,\varphi}$ is the mass flux density (kg.s⁻¹.m⁻²) in the tangential direction, into the phase $\varphi$. The latter term is null for the solid phase ($\varphi = s$).

$\Delta h_{\varphi}$ and $\Delta s_{\varphi}$ represent the variations of the specific enthalpy (J.kg⁻¹) and specific entropy (J.K⁻¹.kg⁻¹) of the fluid crossing the REV$_F$. $T_{\varphi}$ names the temperature of the phase $\varphi$. $\Sigma_{\Delta\varphi}$ is the surface of the REV$_\varphi$, $\overline{d\Sigma_{\Delta\varphi}}$ (m²) being an oriented element of this surface.

The Gibbs equation expresses the hypothesis of local balance for each phase. This equation enables to assess local entropy production (W.m⁻³.K⁻¹) of the REV, into the following form [5]:

$$ \dot{\sigma}_{irr} = \dot{q}_n \left(\frac{1}{T_f} - \frac{1}{T_s}\right) s_{sp} + \dot{q}_{ta,s} \frac{\partial}{\partial z} \left(\frac{1}{T_s}\right) (1 - \varepsilon) + \dot{q}_{ta,f} \frac{\partial}{\partial z} \left(\frac{1}{T_f}\right) \varepsilon - \frac{\dot{m}_{ta}}{\rho_f T_f} \frac{\partial p}{\partial z} \varepsilon \quad (2) $$

with $p$: pressure into the fluid phase (Pa), $s_{sp}$: maximal energy capacity (m².m⁻³), $\varepsilon$: porosity of the porous medium, $\rho_f$: fluid's density (kg/m³).

This expression enables to identify the four irreversible processes contributing to the total entropy production.

i) Heat transfer in the tangential direction to the mass flow in each phase, modelled by the Fourier's law.

$$ \dot{q}_{ta,\varphi} = \dot{q}_{cd,\varphi} = -\lambda_{\varphi} \frac{\partial T_{\varphi}}{\partial z} \quad (3) $$

with $\lambda_{\varphi}$: heat conductivity into the considered phase $\varphi$ (W/(m¹.K¹)).

ii) Heat transfer in the radial direction, convective exchange between fluid and solid, modelled by the Newton's law.

$$ \dot{q}_{n,\varphi} = \dot{q}_{cv} = h_{sf} (T_f - T_s) \quad (4) $$

with $h_{sf}$: fluid/solid convective exchange coefficient (W/(m².K)).

iii) The mass transfer in the tangential direction, depicted by the Darcy's law

$$ \frac{\partial p}{\partial z} = -\frac{\mu}{B_0} v_d \quad (5) $$

with $\mu$: dynamic viscosity in Pa.s⁻¹, $v_d$: Darcy's velocity in m.s⁻¹ (volume flow/tank section). $B_0$: Darcy's permeability (m²).

The transfer parameters $h_{sf}$ and $B_0$ can be estimated thanks to correlations available in the literature.
In this study, Carman-Kozeny correlation has been used to estimate the Darcy's permeability of the porous medium:

$$B_0 = \frac{D_r^2}{180} \frac{\epsilon^3}{(1 - \epsilon)^2} \qquad (6)$$

with D<sub>r</sub> (m) the diameter of the rocks composing the packed-bed. The exchange coefficient h<sub>sf</sub> is estimated from the Nusselt's number:

$$h_{sf} = Nu \frac{\lambda_f}{D_r} \qquad (7)$$

with D<sub>r</sub> (m) the equivalent hydraulic diameter of the pores. In the case of a packed-bed with a spherical structure, the hydraulic diameter of the pores can be simply calculated as a function of the rocks diameters D<sub>p</sub> = D<sub>r</sub>/4. At first approximation, a constant Nusselt's number equal to 3.66 is assumed, corresponding to a laminar flow into a capillary with a constant temperature on the side walls. Other correlations could be subsequently used, particularly during the experimental validation phases.

## 2.3. Problem definition

The equation system to be solved (1), the entropy production (2) and the expressions (3) to (7) describe heat and mass transport involve several physical and geometrical quantities (λ<sub>φ</sub>, ρ<sub>φ</sub>, ε, s<sub>sp</sub>, D<sub>r</sub>, D<sub>p</sub>). Other indicators also imply to evaluate integrated quantities within the volume or the section of the tank. An analysis of the mathematical relations between the different quantity shows that the problem can be depicted by 7 independent variables defined below. Moreover, the thermophysical properties of filler material (ρ<sub>s</sub>, c<sub>s</sub>, λ<sub>s</sub>) and heat transfer fluid (ρ<sub>f</sub>, c<sub>f</sub>, λ<sub>f</sub>, μ<sub>f</sub>) have to be added to the 7 variables to completely depict the system. These defining quantities can be separated into two categories. On the one hand, the quantities set by the design specifications, that are treated as designing parameters. On the other hand, the optimisation variables that can be used for the optimisation.

### 2.3.1 Designing parameters

The parameters are set by the specification related to the heat input and the energy demand.

* The energy density d<sub>E</sub> (J/m<sup>3</sup>) or (kWh/m<sup>3</sup>).

* The maximal energy capacity Q<sub>max</sub> (J) or (kWh).

* The discharging time t<sub>ds</sub> (s).

* The two operating temperatures, high T<sub>H</sub> and low T<sub>L</sub> (K).

### 2.3.2 Optimisation variable

The optimisation variables concern both the materials and the geometric configuration of the tank.

* Material variables:

* The packed-bed material (ρ<sub>s</sub>, c<sub>s</sub>, λ<sub>s</sub>).

* The heat transfer fluid material (ρ<sub>f</sub>, c<sub>f</sub>, λ<sub>f</sub>, μ<sub>f</sub>).

* Geometrical variables:

* The external shape factor, F<sub>e</sub> defined by F<sub>e</sub>=D<sub>t</sub>/L<sub>t</sub>.

* The internal shape factor F<sub>i</sub>, defined by F<sub>i</sub>=D<sub>p</sub>/D<sub>t</sub>.

### 2.3.3 Optimisation criterion

Optimum integration of a storage system in energy processes can be performed by minimising the exergy destruction (or the entropy generated) related to the storage device. This minimisation must
be realized in the frame of the design specification (dE, Q, tds, TH, TL), the expression (5) allows estimating the generated entropy during the discharging phase.

$$S_{irr} = \int_{0}^{t_{cutoff}} \dot{S}_{irr} dt \qquad (8)$$

with $\dot{S}_{irr}$ the entropy production of the storage system, expressed in W/K. This latter can be estimated by integrating the local entropy production (2) on the whole volume.

$$\dot{S}_{irr}(t) = \iiint_V \dot{\sigma}_{irr} dV \qquad (9)$$

## 2.4 Quality indicators

In order to quantify the difference between the performance of the designed tank and the target specifications, two quality indicators are evaluated according to the date t cutoff, defined as the time when the output temperature reaches a limited value T cutoff. This limit is set by the other components of the process that can't work below this temperature.

**The discharging ratio** compare the ratio between the energy Q(t cutoff) extracted of the system and the maximum energy storable in the system.

$$\tau_c = \frac{Q(t_{cutoff})}{Q_{max}} \qquad (10)$$

**The utilisation ratio** compares the duration of the discharging phase and duration of a perfect thermocline (or a 2 tank system).

$$\tau_t = \frac{t_{cutoff}}{t_{ds}} \qquad (11)$$

# 3. Results/discussion

## 3.1 Case studies

The following results deal with 2 case studies. The studies consist in using the materials and design specification from industrial situations and comparing the structure of the existing design with the structure of the optimised design.

One reference of case study is a thermocline tank designed to replace a 2 two tank system in an Andasol type CSP plant [1]. Such a replacement is interesting because thermocline system can lead to a cost reduction estimated between 25 and 40% [3], [4]. The HTF is oil and the system work in closed circuit. The other reference of case study is an industrial thermocline pilot named Eco-Stock developed by EcoTech-Ceram. The system is used to value industrial process heat. It works in open circuit with air as HTF.

### 3.1.1 Andasol-type

The parameters are defined by the T. Fasquelle's estimation [1] on a Andasol-like concentrated solar power plant. The material properties, the tank configuration taken as reference as well as the specification data are listed in Table 1.
<table><caption>Table 1. Characteristics of the Andasol type thermocline tank, by [1]</caption><thead><tr><th colspan="2">Variable</th><th colspan="3">Heat transfer fluid (Dowtherm A)</th><th colspan="3">Packed-Bed (alumina)</th><th colspan="4">Parameters</th></tr><tr><th>F<sub>e</sub></th><th>F<sub>i</sub></th><th>ρ</th><th>c</th><th>λ</th><th>ρ</th><th>c</th><th>λ</th><th>Q<sub>max</sub></th><th>d<sub>e</sub></th><th>t<sub>ds</sub></th><th>T<sub>H</sub></th><th>T<sub>l</sub></th></tr></thead><tbody><tr><td>/</td><td>/</td><td>Kg/m<sup>3</sup></td><td>J/kg<sup>1</sup></td><td>W/m<sup>1</sup>.K<sup>1</sup></td><td>Kg/m<sup>3</sup></td><td>J/kg<sup>1</sup></td><td>W/m<sup>1</sup>.K<sup>1</sup></td><td>MWh</td><td>kWh/m<sup>3</sup></td><td>h</td><td>°C</td><td>°C</td></tr><tr><td>2.36</td><td>1.5e<sup>-4</sup></td><td>753</td><td>2507</td><td>0.08</td><td>3670</td><td>1028</td><td>21</td><td>1010</td><td>84.4</td><td>7.5</td><td>293</td><td>393</td></tr></tbody></table>

Discharging phase is investigated. The tank is initially totally charged ($T = T_H$ uniform at $t=0$). In the frame of the solar plant, this strategy is considered as optimal to maximise de load factor of the concentrated solar plant [5]. The discharging phase end when the output temperature reach $T_{cutoff}$. $T_{cutoff}$ is arbitrarily set at 20% of $\Delta T_{max}$.

The limits condition are Heaviside's type: for $t>0$ m=constant and $T(0,t) = T_L$.

### 3.1.2 Eco-Stock

The parameters are defined according to an industrial pilot system. The firm Ecotech-Ceram own this pilot. The system purpose is to valorise the industrial process heat. The material's properties, the tank configuration used as reference as well as the specification data are listed in the table below.

<table><caption>Table 2. Characteristics of the Eco-Stock thermocline tank</caption><thead><tr><th colspan="2">Variable</th><th colspan="3">Heat transfer fluid (air)</th><th colspan="3">Packed-Bed (Bauxite)</th><th colspan="4">Parameters</th></tr><tr><th>F<sub>e</sub></th><th>F<sub>i</sub></th><th>P</th><th>C</th><th>λ</th><th>ρ</th><th>c</th><th>λ</th><th>Q<sub>max</sub></th><th>d<sub>e</sub></th><th>t<sub>ds</sub></th><th>T<sub>H</sub></th><th>T<sub>l</sub></th></tr></thead><tbody><tr><td>/</td><td>/</td><td>Kg/m<sup>3</sup></td><td>kJ/kg<sup>1</sup></td><td>W/m<sup>1</sup>.K<sup>1</sup></td><td>Kg/m<sup>3</sup></td><td>kJ/kg<sup>1</sup></td><td>W/m<sup>1</sup>.K<sup>1</sup></td><td>MWh</td><td>kWh/m<sup>3</sup></td><td>h</td><td>°C</td><td>°C</td></tr><tr><td>0.47</td><td>5.1e<sup>-3</sup></td><td>0.5</td><td>1000</td><td>0.03</td><td>3050</td><td>750</td><td>21</td><td>1.2</td><td>236</td><td>2</td><td>20</td><td>600</td></tr></tbody></table>

Discharging phase is investigated. The tank is initially totally charged ($T = T_H$ uniform at $t=0$).

The limit conditions are Heaviside's type: for $t>0$, m=constant and $T(0,t) = T_L$.

The discharging phase end when the output temperature reaches $T_{cutoff}$. $T_{cutoff}$ is arbitrarily set at 450°C ($T_h-T_{cutoff} = 26\%$ of $\Delta T_{max}$).

## 3.2 Optimal geometry

### 3.2.1 Andasol-type

The Fig. 2 allows to identify the optimal geometric conditions that minimise the irreversibility for given filler material and HTF (Cf. Table. 1). The optimal configuration generates 2,9 MJ.K⁻¹ of entropy, to be compared with 20 MJ.K⁻¹ for the reference tank (approx. 7 time less).

Fig. 3a and Fig. 3b show that this configuration answer correctly to the specifications. $\tau_c \approx 98\%$ and $\tau_t \approx 98,5\%$ for the optimal configuration when $\tau_c \approx 92\%$ and $\tau_t \approx 92,5\%$ was obtained for the reference tank.

Qualitatively, optimising a thermocline system imply to adopt a slender tank shape filled with small rock. This kind of trend has ever been observed with other optimisation methods [5]. Note that technological limits prevent from constructing tanks with a too slender geometry. The main ways to overpass these constraints are to bury the tank or to use several shorter tanks connected in series.
<figure><img src="image_3.png" /><figcaption>Figure 2. Entropy generated according to $F_e$ and $F_i$, Andasol type case</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3. Discharging ratio (a), Utilisation ratio (b) according to $F_e$ and $F_i$, Andasol type case</figcaption></figure>

### 3.2.2 Eco-Stock

The Fig. 4 allows to identify the optimal geometrical condition that minimise the irreversibility for given filler material and HTF (Cf. Table. 2). The optimal configuration generates 0.41 MJ.K⁻¹ of entropy when 0.78 MJ.K⁻¹ entropy for the reference tank. (approx. 2 time less).

Fig. 5a and Fig. 5b show the quality factor are $\tau_c \approx 88.3\%$ and $\tau_t \approx 90.8\%$ for the optimal configuration when $\tau_c \approx 68.1\%$ and $\tau_t \approx 72.5\%$ was obtained for the reference tank. The optimisation leads to a better correspondence between the tank's behaviour and the design specifications. However, the difference between the specifications and the simulated system is still significant, that mean that the system has to be significantly over-sized to reach the targeted performance.
<figure><img src="image_5.png" /><figcaption>Figure 4. Entropy generation according to $F_e$ and $F_i$, Eco-Stock case.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. Discharging ratio (a), Utilisation ratio (b) according to $F_e$ and $F_i$, Eco-Stock case</figcaption></figure>

## 3.3 2nd Law analysis

This section aims at analysing the entropy's repartition and the temperature profile during a discharging phase.

### 3.3.1 Andasol type

#### Reference system

Fig. 6a shows the profiles of temperature inside the reference tank. The profiles are plotted for different moment of the discharging phase between the start time, t=0, to the phase's end, t=tcutoff (i.e when outlet temperature reaches $T_{cutoff}$). The continuous lines represent the fluid temperature and the dotted lines represent the packed-bed temperature. Fig. 6b displays the entropy production ($\dot{S}_{irr}$) evolution according to time. The entropy is mostly produced by the convective heat transfer and, in a smaller part, by the conduction through the solid (respectively 85% and 15%). The others phenomena are negligible.
<figure><img src="image_7.png" /><figcaption>Figure 6. Profile of temperature (a), Entropy production (b), Andasol (reference)</figcaption></figure>

### Optimal system

The optimal system approaches an ideal thermocline behaviour. The Figure 7a shows that the thermocline zone is smaller than in the reference system present in Fig. 6a. The Figure 7a shows that three irreversible processes mainly generate entropy: advective transfer (pressure drop), conductive within the solid and convective heat transfer (respectively 16%, 45% and 39%). Compared to the reference system, the optimal configuration reduces irreversibilities due to conductive and convective heat transfer. To this end, it is necessary to reduce the rocks size and to adopt a slenderer shape but pressure drop increases. Indeed, increasing the convective heat exchange quality by reducing both $F_i$ and $F_e$ implies reducing the advective transfer quality. It has been demonstrated in both Andasol type and Eco-Stock case that the great increase in entropy production observed for low $F_i$ and $F_e$ values is mainly due to the pressure drop increase.

<figure><img src="image_8.png" /><figcaption>Figure 7. Profile of temperature (a), Entropy production (b), Andasol (optimal)</figcaption></figure>

### 3.3.2 Eco-Stock

#### Reference system

The Fig. 8a present the profile of temperature inside the tank. The thermocline zone's length represents approximately a half of the length of the tank. Besides, there is an important difference between the temperature of the fluid and the solid. This results in an important entropy production due to the convective transfer, as shown in the Fig. 8b that appears as the main source of irreversibility.
(92%), a smaller part being related to the conduction through the solid (7.6%). The others phenomena have negligible impact.

<figure><img src="image_9.png" /><figcaption>Figure 8. Profile of temperature (a), Entropy production (b), Eco-Stock (reference)</figcaption></figure>

## Optimal system

The Fig. 9a present the profile of temperature inside the tank. The thermocline zone's length represents approximately a quarter of the length of the tank. The difference between the temperature of the fluid and the solid is reduced compared to the reference case. The Fig. 9b shows that the entropy is generated by the advective transfers (pressure drop), the conductive transfers through the solid and the convective transfers (respectively 5.9%, 7.8% and 87%). As well as in the liquid case, the entropy production minimisation implies to adopt a slenderer shape and to reduce the rock's size. However, even in the optimal case, convective heat transfer remains here the major entropy source. Indeed, to transfer heat, gas has less adapted thermophysical properties than liquid. That explains why an air systems have significantly lower performance in term of quality indicators than a liquid system, the temperature profile being always lesser steep, even after optimisation.

<figure><img src="image_10.png" /><figcaption>Figure 9. Profile of temperature (a), Entropy production (b), Eco-Stock (optimal)</figcaption></figure>

## 5.Conclusion

The study develops a 1-dimension and 2-phases model with constant thermophysical properties. The model estimates the entropy generated during a discharging cycle, according to 2 parameters (Fe and Fi) and identified the contribution of each phenomenon.
The study shows the existence of an optimal configuration with given filler material and HTF. It appears that it is possible to approach the performances of an ideal system with thermocline using liquid as HTF. Besides, optimised thermoclines using air as HTF have lower performances than ideal system. Further analysis is necessary to conclude on those systems.

This study should be taken as a first step of guidance, before developing more accurate design through classical CFD tools, taking into account 2D or 3D effects, thermo-dependent physical properties and/or thermal insulation default.

Moreover, the estimation of the entropy production during a charging/discharging cycle will allow quantifying the system's impact on the exergy performances of a whole process (energy receiver, storage system, valorisation system, connections, and even final user demand). This kind of study can estimate the performance according to different filler material coming from notably industrial waste material, thus allowing reducing the system's cost. This work will lead to an optimal integration method of the storage systems.

## Nomenclature

D diameter, m

d<sub>E</sub> energy density, J/m

E energy, J

F<sub>e</sub> external shape factor

F<sub>i</sub> internal shape factor

h specific enthalpy, J/kg

h<sub>sf</sub> fluid/solid convective exchange coefficient, W/(K.m<sup>2</sup>)

L length, m

ṁ mass flow, kg/s

p pressure, Pa

q̇ heat flux densities, W/m<sup>2</sup>

Q<sub>max</sub> maximal energy capacity, J

S entropy, J/K

s volume entropie, J/(kg.K)

s<sub>sp</sub> maximal energy capacity, m<sup>2</sup>/m<sup>3</sup>

t<sub>ds</sub> discharging time, s

T temperature, K

U internal energy, J

v fluid speed, m/s

## Greek symbols

λ Thermal conductivity, W/(m.K)

ε porosity

μ dynamic viscosity, Pa/s

ρ density, kg/m<sup>3</sup>

σ volume entropie, J/(m<sup>3</sup>.K)

## Subscripts and superscripts

d Darcy

Δ REV

f fluid

H high
irr irreversible

L low

n normal to the matter flow

p pore

r rock

s solid

ta tangent to the matter flow

t tank

## Abbreviations

2D/3D 2 dimensions / 3 dimensions

CFD Computational fluid dynamics

CSP Concentrated solar plant

HFT Heat transfer fluid

REV Representative elementary volume

## Acknowledgment

This work was supported by the National Agency for Research (ANR) in the framework of the project Opticline (ANR-17-CE06-0013), program ANR : Energie, propre, sûre et efficace (DS02) 2017.

## Reference

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