Guillaume Merle
email: merle@lurpa.ens-cachan.fr

Jean-Marc Roussel
email: roussel@lurpa.ens-cachan.fr

Jean-Jacques Lesage
email: lesage@lurpa.ens-cachan.fr

Andrea Bobbio
email: bobbio@mfn.unipmn.it

Algebraic Expression of the Structure Function of a subclass of Dynamic Fault Trees

Keywords: Dynamic Fault Tree, Algebraic approach, Qualitative analysis, Quantitative analysis

This paper focuses on a subclass of Dynamic Fault Trees (DFTs), called Priority Dynamic Fault Trees (PDFTs), containing only static gates and Priority Dynamic Gates (PAND and FDEP) for which a priority relation among the input nodes completely determines the output behavior. We define events as temporal variables and we show that, by adding to the usual Boolean operators new temporal operators denoted BEFORE and SIMULTANEOUS, it is possible to derive the structure function of the Top Event with any cascade of Priority Dynamic Gates and repetition of basic events. A set of theorems are provided to express the structure function in a sum-of-product canonical form. We finally show through an example that the canonical form can be exploited in order to determine directly and algebraically the failure probability of the Top Event of the PDFT without resorting to the corresponding Markov model. The advantage of this approach is that it provides a complete qualitative description of the system and that any failure distribution can be accommodated.

INTRODUCTION

Fault Tree Analysis (FTA) is one of the oldest and most diffused techniques in industrial applications, for the dependability analysis of large safety-critical systems [START_REF] Henley | Reliability Engineering and Risk Assessment[END_REF]; [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF]). FTA is usually carried out at two levels: a qualitative level in which the list of all the possible combinations of events that lead to the Top Event (TE) is determined (the minimal cut sets). A quantitative level, in which the probability of occurrence of the TE, and of the other nodes of the tree, is calculated; the quantitative level requires the additional knowledge of the time-to-failure probability distributions of all the basic events. One of the main restrictive assumptions in FTA is that basic events must be assumed as statistically independent (s-independent) and their interaction is described by means of boolean OR/AND gates, so that only the combination of events is relevant and not their sequence. We refer to this model as Static Fault Tree (SFT). Several attempts have been reported in the literature to remove these constraints and include various kinds of temporal and s-dependencies in the model. A Priority-AND (PAND) gate has been introduced in [START_REF] Fussel | On the quantitative analysis of priority-and failure logic[END_REF]) to model situations in which the failure of the gate occurs if the inputs fail in a preassigned order. However, the model that has received the greatest attention is the Dynamic Fault Tree (DFT), proposed by [START_REF] Dugan | Dynamic fault-tree models for fault-tolerant computer systems[END_REF], 2000)). The DFT is based on the definition of new gates that induce temporal as well as s-dependencies: Priority-AND (PAND), Functional Dependency (FDEP), Warm Spare (WSP) and Sequence enforcing (SEQ). Some compositional techniques have been later envisaged to build DFTs, either in terms of Stochastic Petri Nets [START_REF] Bobbio | Parametric faulttrees with dynamic gates and repair boxes[END_REF]), or in terms of Input/Output Interactive Markov Chains [START_REF] Boudali | Dynamic Fault Tree analysis through input/output interactive Markov chains[END_REF]), in order to include chains of dynamic gates. The quantitative analysis of the DFT consists in exploding minimal modules [START_REF] Dutuit | A linear-time algorithm to find modules of fault tree[END_REF]) of dynamic gates into their state-space representation and computing numerically the related occurrence probability by means of a Continuous Time Markov Chain [START_REF] Dugan | Dynamic fault-tree models for fault-tolerant computer systems[END_REF]), thus assuming exponential time-to-failure distributions. A new approach, able to include any probability distribution, has been presented in [START_REF] Amari | A new approach to solve dynamic fault-trees[END_REF]), where closed form expressions are determined as a function of the generic probability distributions of the basic events, and a numerical integration is proposed to solve them.

In the present paper, we restrict the consideration of classical dynamic gates to priority gates PAND and FDEP, only, for which a temporal relation completely defines the output, and we refer to this restriction as Priority DFT (PDFT). In order to build up an algebraic framework for PDFTs, we define events as temporal binary variables and we introduce, beside Boolean operators OR and AND, temporal operators BEFORE (BF) and SIMULTANEOUS (SM) [START_REF] Roussel | Algebraic modelling of fault trees with Priority AND gates[END_REF]). We include the possibility that basic events are repeated without restriction and we allow any cascade of Priority Dynamic Gates. We show that it is possible to provide a complete qualitative description of the PDFT through an algebraic expression of the structure function that can be reduced to a sum-of-product canonical form. Each product term of the canonical form contains basic events connected by Boolean and temporal operators. Finally, we show how to compute the probability of occurrence of the TE from the canonical form, by assigning to basic events any failure time distribution.

The PDFT model with repeated events is formalized in Section 2, and the new temporal variables and operators are introduced in Section 3. Section 4 shows how to derive the canonical form of the structure function whereas the probabilistic analysis, with a completely developed example, is reported in Section 5.

PRIORITY DYNAMIC FAULT TREES WITH REPEATED EVENTS

According to [START_REF] Dugan | Developing a low-cost high-quality software tool for dynamic fault-tree analysis[END_REF]), DFTs comprise basic events, static gates (OR, AND and K-out-of-N) and dynamic gates (PAND, FDEP, WSP and SEQ). Dynamic gates can be divided into two categories according to their temporal and statistical behavior:

• gates PAND and FDEP have sequential or preemptionbased behaviors and can be modeled by means of discrete mathematics, as presented in Section 3.3. • Warm Spare (WSP) and Sequence enforcing (SEQ)

gates are s-dependent on event duration, and their probability of occurrence is not completely defined by an order relation.

We have retained the term of Priority Dynamic Gates for gates PAND and FDEP since both gates express a semantics of "priority": a priority between input events for gate PAND; a preemption priority for gate FDEP. FTs containing Priority Dynamic Gates are denoted as Priority DFTs (PDFTs) and constitute a subclass of DFTs. The formal definitions of gates PAND and FDEP [START_REF] Coppit | Formal semantics of models for computational engineering: a case study on dynamic fault trees[END_REF]; [START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF]) is reminded in Table 1. Table 1. Definitions of Priority Dynamic Gates from [START_REF] Coppit | Formal semantics of models for computational engineering: a case study on dynamic fault trees[END_REF])

In a FT, simultaneity among events may arise in two ways. Basic events can occur simultaneously if they have a discrete probability distribution with a non-null probability mass exactly at the same time. Since usually, the failure probabilities distributions are considered as continuous functions with infinite support, the simultaneous occurrence has null probability and can be neglected.

A second case of simultaneity may arise at any level of a FT when there are repeated basic events and has not yet been explored in its full generality.

Let us consider the PDFT in Figure 1, in which event A is a repeated basic event. If basic events A, B, C and 2007)) and has been solved resorting to the concept of "non-determinism", concept that is hardly acceptable in the engineering practice. We assert that a choice must be made regarding the semantics of simultaneous events and priority dynamic gates. For instance, in the case of simultaneous events in input to a PAND gate, two choices are possible (Figure 1):

• if the order relation is considered strictly, when intermediate events G and H occur simultaneously, TE1 does not occur. Gate PAND would then be considered as being "non-inclusive". • if the order relation is not considered strictly, when intermediate events G and H occur simultaneously, TE1 occurs at the same time as G or H. Gate PAND would then be considered as being "inclusive".

Both interpretations of the order relation can be taken into account and algebraically modeled.

ALGEBRAIC FORMALIZATION OF PDFTS

Temporal events

In SFTs, basic events are considered as Booleans. However, the Boolean model cannot render the order of occurrence of events as previously defined for Priority Dynamic Gates.

In order to take into account this temporal aspect, we consider the TE, the intermediate events and the basic events as temporal functions, which are piecewise rightcontinuous on R + ∪ {+∞} and whose range is B = {0,1}.

Since we consider non-repairable events, only, a generic timing diagram of an event a is given in Figure 2, where d(a) is the unique date of occurrence of a. We denote by E nr the set of non-repairable events.

The definition of Boolean operators OR and AND can be extended to E nr . The identity elements of these operators in E nr , equivalent to 0 and 1, are denoted by ⊥ and to which the following dates can be assigned:

d(⊥) = +∞ ; d( ) = 0 6 - 0 1 t £ ¢ £ ¢ a d(a) Fig. 2. A non-repairable event (E nr , +, •, ⊥,
) is an Abelian dioid, like ({0, 1} , +, •, 0, 1), so that the properties of Boolean algebra that are commonly used for the simplification of SFTs can still be applied with our model, and their structure functions can be determined as usual. A complete description of the algebraic framework developed for temporal events can be found in [START_REF] Merle | Algebraic Framework for the Modelling of Priority Dynamic Fault Trees[END_REF]). Because of the notation difference between the identity elements of E nr and the identity elements of {0, 1} for operators + and •, the rewriting of four common theorems of Boolean algebra is necessary:

a + ⊥ = a a • = a a + = a • ⊥ = ⊥

Temporal operators

In order to model priority relations among temporal events, we introduce a temporal operator non-inclusive BEFORE (BF, with symbol ¡) and a temporal operator SIMULTANEOUS (SM, with symbol ), whose formal definitions, based on the dates of occurrence of a and b, are as follows:

a ¡ b =    a if d (a) < d (b) ⊥ if d (a) > d (b) ⊥ if d (a) = d (b) a b =    ⊥ if d (a) < d (b) ⊥ if d (a) > d (b) a if d (a) = d (b)
Based on the previous two operators, we can introduce a non-strict or INCLUSIVE BEFORE (IBF, with symbol ¢) operator:

a ¢ b = a ¡ b + a b (1)
whose definition based on the dates of occurrence of a and b is:

a ¢ b =    a if d (a) < d (b) ⊥ if d (a) > d (b) a if d (a) = d (b)
Operator is commutative, while ¡ and ¢ are not. These three operators satisfy the following theorems, which will be used later in the paper (A more complete set of theorems and their proofs can be found in [START_REF] Merle | Algebraic Framework for the Modelling of Priority Dynamic Fault Trees[END_REF])).

(a ¢ b) + b = a + b (2) a • (a ¢ b) = a ¢ b (3) a ¢ (b • c) = (a ¢ b) + (a ¢ c) (4) a ¢ (b ¢ c) = (a ¡ b) + (a • b • (c ¡ b)) + (a b) • (b ¢ c) (5) (a • b) ¢ c = (a ¢ c) • (b ¢ c) (6) (a ¢ b) ¢ c = (a ¢ b) • (a ¢ c) (7) (a ¢ b) • (b ¢ c) • (a ¢ c) = (a ¢ b) • (b ¢ c) (8) a ¡ a = ⊥ (9) a • (a ¡ b) = a ¡ b (10) (a ¢ b) • (b ¡ a) = ⊥ (11)

Algebraic model of Priority Dynamic Gates

In Section 2, we have shown how both a strict and a non-strict order relation can be taken into account and algebraically modeled. However, a non-strict inclusive interpretation of priority dynamic gates seems more coherent with the designers' expectations. For this reason, in the remainder of this paper, we define an algebraic model of gates PAND and FDEP by means of operator IBF (¢), only. Furthermore, we assume that basic events are s-independent and have a continuous failure time distribution so that they cannot occur simultaneously. Hence, for any two basic events A and B with the above characteristics, the following relation holds:

Q = (A • B) • (A ¢ B) A T = (A ¢ T ) + T (2) = A + T (3) = B • (A ¢ B) B T = (B ¢ T ) + T (2) = B + T (a) (b)
A B = ⊥ (12)
In order to arrive to the determination of the structure function of any PDFT, special attention should be payed to the cascades of PAND gates.

Cascading PAND gates

Two elementary combinations of cascading PAND gates are possible, as represented in Figures 4.a

T E2 = C • (F ¢ C) = C • ((B • (A ¢ B)) ¢ C) (6),(7) = C • (B ¢ C) • (A ¢ B) • (A ¢ C) (8) = C • (A ¢ B) • (B ¢ C) (13)
The second possible combination of cascading PAND gates is given in Figure 4.b and its structure function can be determined and developed thanks to the theorems of Section 3.2. Note, in particular, that theorem (5) is somewhat counterintuitive, but simply states that a

¢ (b ¢ c) is true iff (a ¡ b) or if (b ¢ c) = ⊥ is true. T E3 = J • (A ¢ J) = C • (B ¢ C) • (A ¢ (C • (B ¢ C))) (4) = C • (B ¢ C) • ((A ¢ C) + (A ¢ (B ¢ C))) = C • (B ¢ C) • (A ¢ C) +C • (B ¢ C) • (A ¢ (B ¢ C)) (5) = C • (A ¢ C) • (B ¢ C) + C • (B ¢ C) • (A ¡ B) +C • (B ¢ C) • (A • B • (C ¡ B)) +C • (B ¢ C) • (A B) • (B ¢ C) (12) = C • (A ¢ C) • (B ¢ C) + C • (A ¡ B) • (B ¢ C) +A • B • C • (B ¢ C) • (C ¡ B) (11) = C • (A ¢ C) • (B ¢ C) +C • (A ¡ B) • (B ¢ C) (14)

CANONICAL FORM OF THE STRUCTURE FUNCTION

The algebraic models of Priority Dynamic Gates (Figures 3.a and 3.b) allow us to determine the structure function of any PDFT as a function of basic events that can be repeated without restrictions.

Given a PDFT with n basic events {b i , i ∈ (1, ..., n)}, the structure function for the TE becomes an expression containing at most the n basic events and operators +, •, ¡, and ¢. The structure function can then be developed and simplified thanks to the theorems presented in Section 3.2, to arrive to a standardized sum-of-product canonical form where each product term contains operator • and ordered pairs of variables linked by operator ¡, only. The steps to be followed to arrive to the canonical form are:

(1) Starting from the TE, in a top down fashion, replace each FDEP gate by its algebraic expression in Figure 3.b and each PAND gate by its algebraic expression in Figure 3.a. (2) In the case of cascading PAND gates, apply theorems ( 5) and ( 7). ( 3) Eliminate the parenthesis by applying distributivity theorems, such as theorems ( 4) to ( 7). ( 4) The structure function is then expressed in a sum of product terms as in ( 15):

T E = b i • b j ¢ b k • (b l ¡ bm) • bo bp (15)
(5) Since b o and b p are basic events, in virtue of theorem ( 12), function ( 15) can always be simplified to the following form:

T E = b i • (b j ¢ b k ) • (b l ¡ b m ) (16)
(6) Taking into account theorems ( 1) and ( 12), we can write b j ¢ b k = b j ¡ b k . Hence the expression in ( 16) becomes:

T E = b i • (b j ¡ b k ) (7) According to theorem (9), j = k ⇒ b j ¡ b k = ⊥, then
the structure function can be simplified to:

T E = b i • (b j ¡ b k ) , j = k (8) Finally, according to theorem (10), i = j ⇒b i •(b j ¡ b k ) = b j ¡ b k
, so we get the structure function in canonical form:

T E = b i • (b j ¡ b k ) , j / ∈ {i, k} (17)

PROBABILISTIC ANALYSIS OF PDFTS

In the case of DFTs, the determination of the failure probability of the TE from the failure probabilities of the basic events is computed numerically by developing dynamic modules into the corresponding Markov chain. Close form expressions for the dynamic gates with any distribution function are given in [START_REF] Amari | A new approach to solve dynamic fault-trees[END_REF]).

In this section, we show that the TE probability of any PDFT can be evaluated in a purely algebraic way from the canonical form of the structure function, for any possible time-to-failure distribution of basic events.

The quantitative analysis of PDFTs is illustrated by means of an example taken from [START_REF] Fussel | On the quantitative analysis of priority-and failure logic[END_REF]). First, the traditional approach consisting in the generation and solution of the Markov chain is applied. Then the algebraic solution with exponential distributions is proposed starting from the canonical form, showing that the same procedure can be extended to any probability distribution (the Erlang distribution is considered as an example).

Figure 5 shows the PDFT of a non-repairable electrical supply system that has a principal power supply (P), a parallel spare (S), and a switch (C) that commutes on S when P fails [START_REF] Fussel | On the quantitative analysis of priority-and failure logic[END_REF]). We assume that the principal power supply and the parallel spare fail with failure rates λ p and λ s , respectively, and that the switch fails with failure rate λ c . Markov chain are obtained by solving the following system of differential equations:

dP (t) dt = P (t) • Q (18)
where P (t) is the state probability vector and Q the transition rate matrix given by:

- Solving Equation ( 18) with transition rate matrix (19) provides the following close form expression for the probability of state 8:

P r {T E4} (t) = P r {8} (t) = λ p λ c + λ p e -(λc+λp+λs)t -e -λpt - λ p λ c + λ p e -λst + 1 (20)

Calculation of the failure probability with the algebraic approach

When the structure function of the TE is expressed in canonical form (17), the probability can be calculated starting from the expressions of the product terms. Given an event x with distribution function F x (t) and density function f x (t), the following expressions hold under the hypothesis of s-independence by extension of [START_REF] Amari | A new approach to solve dynamic fault-trees[END_REF]; [START_REF] Fussel | On the quantitative analysis of priority-and failure logic[END_REF]):

P r {a • b} (t) = F a (t) × F b (t) P r {a + b} (t) = F a (t) + F b (t) -F a (t) × F b (t) P r {a ¡ b} (t) = t 0 f a (u)(1 -F b (u)) du P r {b • (a ¡ b)} (t) = t 0 f b (u) F a (u) du (21)
The canonical form of the structure function of the PDFT shown in Figure 5 is:

T E4 = (P • S) + (P • (C ¢ P ))
(1),( 12

= λ p λ c + λ p e -(λc+λp)t -e -λpt + λ c λ c + λ p Hence: P r {T E4} (t) = λ p λ c + λ p e -(λc+λp+λs)t -e -λpt - λ p λ c + λ p e -λst + 1 (24)
The result in ( 24) coincides with the one in (20). However, structure function ( 22) is suited to evaluate the TE probability with any distribution.

Case of non-exponential distributions

If the components of the studied systems do not exhibit an exponential behavior, application of the Markov chain procedure is unfeasible, whereas algebraic manipulation remains a viable solution.

In the case of mechanical systems, for instance, exponential distribution is not the most suitable one and other distributions, such as the Erlang distribution, are more commonly used. We show that the failure probability of such systems can be determined algebraically by resorting to the expressions (21). The Erlang distribution has the expression shown in Equation (25).

F (t) = 1 - k-1 n=0 (λt) n n! e -λt f (t) = λ k t k-1 e -λt (k -1)! (25)
Starting from the TE probability expression in ( 23) we obtain:

P r {P } (t) = 1 - kp-1 n=0 (λ p t) n n! e -λpt P r {S} (t) = 1 - ks-1 n=0 (λ s t) n n! e -λst P r {P • (C ¡ P )} (t) = t 0 λ kp p u kp-1 e -λpu (k p -1)! 1 - kc-1 n=0 (λ c u) n n! e -λcu du = 1 - kp-1 n=0 (λ p t) n n! e -λpt - kc-1 n=0 n + k p -1 k p -1 λ n c λ kp p (λ c + λ p ) n+kp - kc-1 n=0 n+kp-1 q=0 n + k p -1 k p -1 λ n c λ kp p t q e -(λc+λp)t q! (λ c + λ p ) n+kp-q
Consequently: P r {T E} (t) = 1 - The calculation of the failure probability of the TE can be performed with any other non-exponential distribution. If the considered failure distribution is not analytically integrable (as for instance the Weibull distribution), the probabilistic relation deducted from the canonical form of the structure function can still be used by resorting to numerical integration.

CONCLUSION

In this paper, we have defined a subclass of DFTs, called Priority Dynamic Fault Trees (PDFTs), comprising Priority Dynamic Gates, PAND and FDEP, only. We have modeled both gates by means of new temporal operators called BF, SM and IBF defined on a set of temporal variables and allowing the simultaneity of intermediate events which can be caused by the use of repeated basic events. The definition of an algebraic model allows the determination of the structure function of any PDFT. Thanks to the theorems that we presented, this structure function can always be simplified to a sum-of-product canonical form, which can then be reduced by removing redundant terms. On the one hand, this canonical form can be used for the qualitative analysis of PDFTs. On the other hand, we presented a quantitative approach allowing the direct algebraic determination of the failure probability of the TE from the canonical form, whatever the failure distributions.

On-going work is now addressed to the determination of an algebraic model for WSP and SEQ gates in order to extend the work presented in this paper to the whole DFT formalism.

  dependency -that the failure of the trigger event causes the immediate and simulta--neous failure of the dependent basic events.

Fig. 1 .

 1 Fig. 1. An example of PDFT with one repeated basic event D occur according to sequences [B, C, A], [C, B, A] or [D, A], intermediate events G and H occur simultaneously at the same time as A occurs. This example shows that intermediate nodes of a FT can occur simultaneously because of the presence of repeated basic events. The simultaneity problem has been briefly addressed in (Boudali et al. (2007)) and has been solved resorting to the concept of "non-determinism", concept that is hardly acceptable in the engineering practice. We assert that a choice must be made regarding the semantics of simultaneous events and priority dynamic gates. For instance, in the case of simultaneous events in input to a PAND gate, two choices are possible (Figure1):

Fig. 3 .

 3 Fig. 3. Algebraic models of gates PAND and FDEP The algebraic expression of gate PAND is in Figure 3.a, whereas the expression for gate FDEP is in Figure 3.b. Regarding gate FDEP, basic events A and B can fail by themselves or are forced to fail by the trigger event T . We choose to denote the global behavior of basic events A and B by the substituted variables A T and B T to explicitly indicate the effect of trigger T . As already noticed in[START_REF] Stamatelatos | Fault tree handbook with aerospace applications[END_REF]), the algebraic formalization proves that gate FDEP can be represented by Boolean OR gates, only.

Fig. 4 .

 4 Fig. 4. Basic PDFTs made of a cascade of PAND gates

Fig. 5 .

 5 Fig. 5. Example of sample logic model from (Fussel et al. (1976))5.1 Calculation of the failure probability with Markov chainsThe state transition diagram of the corresponding Markov chain is shown in Figure6, where state 8 is the only failure state and represents the TE. The state probabilities of the

  ) = (P • S) + (P • (C ¡ P )) (22) We then calculate P r {T E4} as: P r {T E4} = P r {P • S} + P r {P • (C ¡ P )} -P r {(P • S) • (P • (C ¡ P ))} = P r {P • S} + P r {P • (C ¡ P )} -P r {S • (P • (C ¡ P ))} = P r {P } × P r {S} + (1 -P r {S}) × P r {P • (C ¡ P )} (23) In the case of exponential distributions, we obtain from (21): P r {P } (t) = 1 -e -λpt P r {S} (t) = 1 -e -λst P r {P • (C ¡ P )} (t) = t 0 λ p e -λpu (1 -e -λcu ) du

  λ c + λ p ) q+kp t n e -λst + s t n+r e -(λc+λp+λs)t n!r! (λ c + λ p ) q+kp-r


Noe Beaupere
email: noe_beaupere@ens.univ-artois.fr

Ulrich Soupremanien
email: ulrich.soupremanien@cea.fr

Laurent Zalewski
email: laurent.zalewski@univ-artois.fr

Influence of Water Addition on the Latent Heat Degradation of Sodium Acetate Trihydrate

Keywords: Thermal energy storage, Sodium acetate trihydrate, Aging study, Latent heat degradation and Water addition 1. Abbreviations

To promote the development of thermal energy storage (TES), the sodium acetate trihydrate (SAT) presents interesting thermal properties. However, this material may suffer from aging throughout thermal cycles. Various solutions were explored in the literature to limit this aging, mainly based on the use of additives such as water. In this study, two samples were prepared.

The first one consisted of raw SAT material whereas 3 wt.% of supplementary water has been added to the second one. They were then poured into 350 cm 3 bricks, which were placed in an experimental test bench. After 35 cycles between 20 and 70 °C, a drop of about 10 % of the latent heat was observed for the first sample. This behavior was similar to the literature data. For the second sample, the decrease of latent heat was more rapid (about 30%). Contrary to our expectations, the water addition seems not beneficial for the improvement of thermal stability. Interestingly, we noticed that the drop of the latent heat was fully reversible after sample stirring. This degradation might not be related to a thermal destructive mechanism but rather to a global segregation phenomenon. This segregation may be due to the breaking of hydrogen bonding between anhydrous sodium acetate and water, resulting in the separation of the two species.

Introduction

Phase change materials (PCMs) appear as an attractive solution for thermal energy storage and for the development of renewable energies. Indeed, they can be used for both domestic hot water production [1] and passive heating/cooling of buildings [START_REF] Cabeza | Materials Used as PCM in Thermal Energy Storage in Buildings: A Review[END_REF]. For a better ratio of production vs. volume, daily melting and solidification are generally wished. Thus, many thermal cycles may be applied to the material that may suffer from aging, resulting in a decrease of the latent heat.

In particular, salt hydrates have an appropriate range of melting points (between -5 and +120 °C) for building applications and a range of interesting latent heat capacity (100-350 J.g -1 ) [START_REF] Beaupere | Nucleation Triggering Methods in Supercooled Phase Change Materials (PCM), a Review[END_REF]. The solid phase of this class of PCMs presents a density of approximately 1700 kg.m -3 and a specific heat capacity around 1650 J.kg -1 .K -1 [START_REF] Beaupere | Nucleation Triggering Methods in Supercooled Phase Change Materials (PCM), a Review[END_REF], which may ensure relevant heat storage capabilities. One of the most interesting materials issued from this class of PCMs is sodium acetate trihydrate (SAT, CH 3 COONa3H 2 O). However, the SAT may present a large supercooling degree up to 100 K [START_REF] Wada | Studies on Salt Hydrates for Latent Heat Storage. IV. Crystallization in the Binary System CH3CO2Na-H2O. Bull[END_REF]. This high value of supercooling is generally perceived as a drawback but could be turned into a benefit if a reliable solidification triggering technique could be developed [START_REF] Beaupere | Nucleation Triggering Methods in Supercooled Phase Change Materials (PCM), a Review[END_REF]. This would lead to the discharge of the stored heat on demand [START_REF] Beaupere | Experimental Measurements of the Residual Solidification Duration of a Supercooled Sodium Acetate Trihydrate[END_REF]. The heat release amplitude will depend on the triggering temperature (as losses may occur during the cooling of the supercooled SAT). This quantity of stored heat in the supercooled SAT may be degraded by aging throughout successive thermal cycles, a process already explored for other PCMs [START_REF] Mazzeo | Experimental Validation of the Exact Analytical Solution to the Steady Periodic Heat Transfer Problem in a PCM Layer[END_REF][START_REF] Mazzeo | Thermal Field and Heat Storage in a Cyclic Phase Change Process Caused by Several Moving Melting and Solidification Interfaces in the Layer[END_REF]. As the SAT is heated until 58 °C, an incongruent melting is observed according to the phase diagram [START_REF] Wada | Heat Storage Capacity of Sodium Acetate Trihydrate during Thermal Cycling[END_REF] due to the break of hydrogen bonds. The SAT decomposes into two components: sodium acetate (CH 3 COONa) and water (H 2 O) that have different densities. If a temperature of 77 °C is reached, different behavior is observed upon cooling, with a solution (45.7wt% of water instead of 39.7wt%) and the segregation of sodium acetate. Kong et al. showed that the concentration of a non-stirred SAT mixture depends on the height in the container [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF]. The separation of species will increase with cycles, as the mobility is limited both in liquid (no agitation) and in solid state as the remaining SAT in the middle-height will crystallize and hinder the contact between the two species. This would decrease the latent heat released by the material, leading to a reduction of its storage capacity. It was observed by Kong et al. [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF] that a deeper container (more than 5 cm in height) would limit the gathering of the two components to again form a trihydrate material. The aging of the SAT was analyzed by several authors such as Pálffy et al. [START_REF] Pálffy | Thermal Properties of a Heat Storage Device Containing Sodium Acetate Trihydrate[END_REF] who experienced that a raw sample of SAT had lost about a third of its initial value after 300 cycles.

The evolution throughout more cycles (600 cycles) was also performed by Wada et al. [START_REF] Wada | Heat Storage Capacity of Sodium Acetate Trihydrate during Thermal Cycling[END_REF], showing that aging could be reduced thanks to impurities. Several solutions were investigated to limit this aging [START_REF] Wada | Heat Storage Capacity of Sodium Acetate Trihydrate during Thermal Cycling[END_REF][START_REF] Kong | Experimental Investigations on Heat Content of Supercooled Sodium Acetate Trihydrate by a Simple Heat Loss Method[END_REF][START_REF] Kong | Ageing Stability of Sodium Acetate Trihydrate with and Without Additives for Seasonal Heat Storage[END_REF], and one of them (addition of water) was found interesting as it also decreases the solidification temperature and thus promotes the supercooling degree [START_REF] Wada | Studies on Salt Hydrates for Latent Heat Storage. IV. Crystallization in the Binary System CH3CO2Na-H2O. Bull[END_REF]. Indeed, aging would be limited by an excess of water from the extra water principle, a concept developed by Furbo et al. [START_REF] Furbo | Heat Storage with an Incongruently Melting Salt Hydrate as Storage Medium Based on the Extra Water Principle[END_REF]. Then, this phenomenon was observed by Kong et al. [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF][START_REF] Kong | Experimental Investigations on Heat Content of Supercooled Sodium Acetate Trihydrate by a Simple Heat Loss Method[END_REF] as they increased the quantity of water, moving its mass concentration in SAT from 39.7 to 45.7wt% (0 to 6wt% extra water). A constant quantity of latent heat was released by the solidification of material even after 130 days of supercooling. Dannemand et al. [START_REF] Dannemand | Experimental Investigations on Prototype Heat Storage Units Utilizing Stable Supercooling of Sodium Acetate Trihydrate Mixtures[END_REF][START_REF] Dannemand | Laboratory Test of a Prototype Heat Storage Module Based on Stable Supercooling of Sodium Acetate Trihydrate[END_REF] have shown that a sample of SAT with 48.7wt% of water (9wt% of extra water) can suffer from aging during thermal cycling. Indeed, it was observed a decrease of 8% of its latent heat, from 194 to 179 kJ.kg -1 after 14 cycles. The same authors observed in [START_REF] Dannemand | Experimental Investigations on Cylindrical Latent Heat Storage Units with Sodium Acetate Trihydrate Composites Utilizing Supercooling[END_REF] that a sample with 43.5wt% mass concentration of water (3.8wt% extra water) presented a decrease from 177 to 140 J.g -1 (-21%) after 17 cycles, but no comparison was made with raw material. Such a comparison was performed in this study. First, the properties of raw material, and their evolution throughout thermal cycles, were investigated. A sample containing an addition of water was then analyzed. In the one hand, because water addition can increase the supercooling degree, this would limit the appearance of a non-whished solidification and a subsequent heat release what is relevant for an application. The monitoring of aging was based on the measurement of latent heat accumulated and released throughout the successive phase change. In the other hand, as the supercooling was so stable in the sample having extra-water, a way of solidification triggering needed to be developed. Thus, a seeding method by contacting a welding rod to the supercooling SAT was finally selected to induce solidification. As solidification could be triggered, it was possible to compare the degradation of the latent heat in both samples.

Materials and Methods

Preparation of Macro-encapsulated Sodium Acetate Trihydrate Samples

The material characterized in this study was SAT (Sigma Aldrich, BioXtra, 99 % purity). Two different samples have been prepared:  A reference sample (SAT60%) with a raw fraction of 60.3wt% of CH 3 COONa for a comparison with the data from the literature.

 A second sample (SAT58%) where water was added. In addition to a limitation of aging, this concentration was expected to bring stability regarding the supercooling phenomenon [START_REF] Kong | Experimental Investigations on Heat Content of Supercooled Sodium Acetate Trihydrate by a Simple Heat Loss Method[END_REF].

The samples were prepared from about 330 g of the raw material. For the second sample, the material was heated at a temperature of 70 °C for melting. This would allow for a better homogenization of the added deionized water (35 g). The sample was vigorously shaken to homogenize the repartition of water in the volume. The second sample was then solidified for transportation.

As they were solid, the real concentrations of both samples were measured. To do so, a ThermoGravimetric Analysis device (Netzsch, Pegasus 449 F1) was used. About 50 mg of material was taken from preparation and heated until 200 °C that was above the evaporation temperature of the water and allows for the analysis of a fully dry sample. For the SAT60% sample, the experiment was repeated 3 times for reliability, and the residual mass 𝑤 𝑆𝐴 was included between 59.6wt% and 61.5wt% (in comparison with a theoretical concentration of 60.3wt%). In the second sample, the experiments were repeated two times for reliability the residual mass of sodium acetate was found to be between 57.2 and 58.3wt%, for an average value of 57.7wt%. Such differences are assumed to be due to uncertainties.

After transportation, the samples were again melted for pouring by a syringe into 250 cm3 bricks (210×140×18 mm 3 , 4 mm of thickness) that can contain 350 g of SAT. They were made of Polymethyl methacrylate (PMMA) that was found to have good chemical compatibility with SAT [START_REF] Zalewski | Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material[END_REF]. The bricks were sealed in order to limit the evaporation of water. The quantity of material was higher than the value used in the usual devices such as DSC (up to 40 µL) or calorimeter (about 10 mL). Its behavior is assumed to be more representative of that of a real system, having usually more than 1 L of PCM [START_REF] Desgrosseilliers | Design and Evaluation of a Modular, Supercooling Phase Change Heat Storage Device for Indoor Heating[END_REF][START_REF] Dannemand | Compact Seasonal PCM Heat Storage for Solar Heating Systems[END_REF]. Once the material was solid, the brick was put into the experimental bench, and the analysis was launched.

Experimental Set-up Description

The behavior of the material was studied by thermal cycling, which was considered to be representative of the behavior of a real application. This method is also the most currently used to study aging, according to Ferrer et al. [START_REF] Ferrer | Review on the Methodology Used in Thermal Stability Characterization of Phase Change Materials[END_REF]. However, these authors showed that mainly DSC devices or calorimeters were used. Therefore, a test ring that allows for better measurement of latent heat [START_REF] Zalewski | Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material[END_REF] was preferred in this work, (Figure 1 (a)). The sample was surrounded by fluxmeters where both heat flux and temperatures were measured (Figure 1 (b)). Therefore, the heat released by the sample was precisely recorded, which permits an adequate interpretation of the material behavior. Two thermo-regulated water baths were used for temperature regulation of the exchange plates.

From preliminary experiments, it was observed that a heating and cooling rate of 20 °C.h -1 gives repeatable conditions between left and right plates (±0.1 °C.h -1 ) and throughout cycles. This cooling rate was chosen and thermal cycles of 8 hours were applied from 20 to 70 °C. The maximal temperature was chosen to be above the melting temperature, approximately 58 °C for the SAT compositions considered here according to Green [START_REF] Green | Melting-Point" of Hydrated Sodium Acetate: Solubility Curves[END_REF]. This temperature was also chosen below 77 °C [START_REF] Johansen | Thermal Conductivity Enhancement of Sodium Acetate Trihydrate by Adding Graphite Powder and the Effect on Stability of Supercooling[END_REF] to ensure the occurrence of solidification on the next cooling, which allows for the experimental determination of the latent heat. In parallel, it was observed that the SAT generally solidifies around 50 °C in the prior experiments, therefore a minimal temperature close to the ambient conditions (20 °C) was arbitrarily chosen.

Data Analysis

The thermal behavior of the material during a full cycle (heating, cooling, and constant temperature steps), using the previously presented experimental bench, is described in Figure 2.

Whereas all the surfaces are covered by sensors, only the record of the left and right fluxmeters (140 × 210 mm²) is given. Indeed, the measurements of the four other surfaces are not considered significant as they represent about 4% of the total heat flux recorded. Here, the analysis was performed between two points (1 st and 2 nd point) on both sides of a temperature variation (heating or cooling). These two points were chosen at the end of the thermal plateau in order to have a stabilized heat flux. Hence, a baseline below 0.01 W could be obtained. The value of latent heat could then be obtained from an integration of the heat flux between these two points, after subtracting the sensible heat.

This sensible heat was stored in the container, made of PMMA, whose specific heat capacity 𝐶 𝑃,𝑏𝑟𝑖𝑐𝑘 has been determined by blank tests to be approximately 1.45 kJ.kg -1 .°C -1 . The sensible heat was also stored in the solid 𝐶 𝑃,𝑆 and liquid 𝐶 𝑃,𝐿 SAT's specific heats. They can be calculated using linear approximations, given in Eq. 1, based on the data from Araki et al. [START_REF] Araki | Measurements of Thermophysical Properties of Sodium Acetate Hydrate[END_REF]. These equations were considered valid for a mass fraction 𝑤 𝑆𝐴 from 54.3 to 60.3wt% and a temperature between 27 and 58 °C for solid and between 27 and 77 °C for liquid (valid also in supercooled state), with an error of 5% maximum.

𝐶 𝑃 = 𝐴 + 𝐵 𝑇 + (𝐶 + 𝐷 𝑇) 𝑤 𝑆𝐴 (1)
Where 𝑇 is the temperature (°C). The empirical parameters A, B, C, and D are given in Table 1 for both solid and liquid specific heats.

Table 1. Empiric parameters for solid and liquid specific heat determination.

A (J.kg -1 .°C -1 ) B (J.kg -1 .°C -2 ) C (J.kg -1 .°C -1 ) D (J.kg -1 .°C

Where 𝑡 1 and 𝑡 2 are the time for the 1 st and 2 nd point (Figure 2), and 𝑇 𝑚𝑒𝑙𝑡 is the melting temperature (58 °C). Eq. 2 is focused on the melting of the material because it was assumed that the energy released during solidification was influenced by the possible presence of a non-repeatable supercooling degree. This equation could be compared with an empirical relation from Araki et al. [START_REF] Araki | Measurements of Thermophysical Properties of Sodium Acetate Hydrate[END_REF], given in Eq. 3 that is valid for a mass fraction of sodium acetate 𝑤 𝑆𝐴 from 47.5 to 60.3wt% with an error of 5% maximum. This procedure was then applied to the study of the two samples (SAT60% and SAT58%).

Aging Study of Sodium Acetate Trihydrate

Experimental Validation of the Empirical Equations

The validity of the previous equations was verified by the application of heating and cooling cycles on a fully solid (between 20 and 28 °C) and fully liquid (between 63 and 70 °C) SAT60% sample.

During these tests, the supercooling state was not observed in this sample as the necessary conditions were not reached. However, the supercooling degree in the SAT58% was increased by the addition of water (reduction of the temperature of solidification) as noted by Wada et al. [START_REF] Wada | Studies on Salt Hydrates for Latent Heat Storage. IV. Crystallization in the Binary System CH3CO2Na-H2O. Bull[END_REF]. Experimentally, the SAT58% sample did not crystallize even at a temperature as low as 5 °C. As the solidification was not obtained by a temperature reduction, it was decided to trigger it by an external mean (insertion of a copper welding rod), as presented in Figure 4. The crystallization of the supercooled SAT was observed after the introduction of this additional metal rod. In addition, it continues appearing in the following cycles even after the rod had been removed. As SAT may corrode copper [START_REF] Cabeza | Immersion Corrosion Tests on Metal-Salt Hydrate Pairs Used for Latent Heat Storage in the 48 to 58°C Temperature Range[END_REF], some particles were assumed to be spread in the liquid PCM. These particles may act as nucleating agents as a new surface was provided from where the nuclei would grow as lower energy is required [START_REF] Beaupere | Nucleation Triggering Methods in Supercooled Phase Change Materials (PCM), a Review[END_REF]. Therefore, the addition of a small number of particles can strongly promote the solidification of the SAT. This repeatable crystallization allows performing preliminary thermal cycles with a phase change. The values of the latent heat, completed by the specific heats measured for the SAT60% (solid and liquid) and the SAT58% (supercooled), are given in Table 2. Indeed, the characterization of SAT58%, which was performed before the triggering of the solidification by a metal rod, only gave properties for the supercooled state. The uncertainties were calculated to include 99.7% of the values ( ± 3𝜎 ). The number of experiments for each measurement was also added. The measured specific heat was found consistent with the literature data [START_REF] Araki | Measurements of Thermophysical Properties of Sodium Acetate Hydrate[END_REF] as a difference of less than 10% was found regarding Eq. 1. The same observation was done for the latent heat determined from Eq. 2 (maximal difference 2% regarding Eq. 3 from the literature). Nevertheless, only a few experiments could be performed, as they may induce the aging of the sample.

The Initial Drop of Latent Heat During Preliminary Cycles

In order to verify the repeatable apparition of solidification and melting before the aging analyses and compare the measured latent heat with Eq. 3, preliminary cycles were performed on SAT58%. From Eq. 2, the latent heat of melting could be determined and its evolution is given in Figure 5. The uncertainties are given in Table 2. It was observed that a phase change (melting and solidification) occurred at every preliminary cycle. In the 3 first cycles (left blue rectangle), the latent heat determined by Eq. 2 differed by a maximum of 2% from Eq. 3 from Araki et al. [START_REF] Araki | Measurements of Thermophysical Properties of Sodium Acetate Hydrate[END_REF]. However, it was suspected that these preliminary thermal cycles had induced the aging of the material. Indeed, a decrease of the latent heat from 220 to 200 J.g -1 was observed after 12 thermal cycles. In parallel, the bricks containing the different samples were taken out of the experimental bench at the end of the preliminary stage. The segregation of anhydrous salt could be observed at the bottom of the brick. This segregation can either be due to the melting before the filling of the brick (SAT60%, Figure 6 According to the phase diagram of Sodium Acetate -Water [START_REF] Green | Melting-Point" of Hydrated Sodium Acetate: Solubility Curves[END_REF], a peritectic transformation exist during melting of the material for a concentration above 𝑤 𝑆𝐴 = 0.57 . This transformation is associated with the formation of anhydrous salt and water. Indeed, these two molecules are linked in SAT by a hydrogen bond [START_REF] Efremov | Refinement of the Crystal Structure of Sodium Acetate Trihydrate[END_REF] that can easily be broken. The anhydrous sodium acetate, denser, may fall at the bottom of the brick, whereas water is generally found in the upper part [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF]. Thus, the quantities of water and anhydrous salt are increased in the global volume, as presented in Figure 7.

This phenomenon of aging continues with cycles and reduces the capacity of the global volume to release the latent heat. Thus, it can be considered that the height of the sample also influences aging. As experimented by Kong et al. [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF], a sample with a height of 8 cm was more subjected to aging than other samples of 4 or 5 cm. Indeed, the gravity phenomenon and the separation of species would be accentuated for samples of important heights. In existing application prototypes (250 L) the heights of the cells were limited to 5 cm height to reduce the separation phenomenon [START_REF] Dannemand | Compact Seasonal PCM Heat Storage for Solar Heating Systems[END_REF].

As it was observed that the latent heat could be regenerated by agitation, the bricks were then vigorously shaken by hands. This results in a visual elimination of segregation as presented in Figure 6 (c). This can be correlated to the fact that the latent heat measured on the following thermal cycle (after agitation) was similar to the initial value of the sample. The agitation, that was here manual, could be done by mechanical means such as nucleation triggering technique like ultrasounds [START_REF] Beaupere | Nucleation Triggering Methods in Supercooled Phase Change Materials (PCM), a Review[END_REF] or bubbling [START_REF] Duquesne | Nucleation Triggering of Highly Undercooled Xylitol Using an Air Lift Reactor for Seasonal Thermal Energy Storage[END_REF]. This solution may also allow the release of heat on demand by inducing the solidification of a supercooled SAT. After the segregated salt had been dissolved by shaking, the bricks containing the different SAT samples were then ready for aging analysis.

Study of the Raw Material (SAT60%)

To evaluate the potential of the use of water in order to attenuate the aging of the SAT, the behavior of raw material was first evaluated. Therefore, 35 cycles between 20 and 70 °C were applied to the sample. These cycles may be summarized in Figure 8. It may be noted that a discontinuity appears between the 5 th and 6 th cycle that corresponds to the change of heating and cooling rate. The 1 st cycle of the series was not considered here because of a different thermal history. Besides, some cycles were applied with a different heating and cooling rate (5 th cycle) whose results were used in another study [START_REF] Dannemand | Compact Seasonal PCM Heat Storage for Solar Heating Systems[END_REF]. This lead to a different thermal history for the 6 th cycle. Therefore, 1 st , 5 th , and 6 th were not considered in the analysis and plotted as stars in the following graphs to perceive the influence of different experimental parameters.

The latent heat of the remaining cycles could be determined from Eq. 2, leading to the monitoring of aging. However, some crystallizations that occurred at a lower temperature would influence the latent heat measurement, such as the 34 th cycle of this experiment, which is plotted in Figure 9. During this 34 th crystallization, the heat flux was not stabilized and the core of the material may still be cooling as the temperature started to re-increase. As the material remained hot, the energy needed to bring the material back to 70 °C for the 35 th melting may be lowered. In parallel, it was assumed in the calculation of the sensible heat subtracted (Eq. 2) that the whole material was heated from 20 to 70 °C. Therefore, the latent heat, determined from Eq. 2 during the 35 th melting, may be underestimated. Even if this is an extreme case, the next melting may be impacted by a delay in the solidification of singular points. Besides, a faster solidification was observed when the supercooling degree was larger [START_REF] Munakata | Electrical Initiation of Solidification and Preservation of Supercooled State for Sodium Acetate Trihydrate[END_REF]. This may induce a larger number of porosities [START_REF] Dannemand | Porosity and Density Measurements of Sodium Acetate Trihydrate for Thermal Energy Storage[END_REF] and influence its properties and the progression of the melting front on the next heating step. For these reasons, the melting that follows a low-temperature crystallization has been considered as singular data.

Before the determination of latent heat, the first step was to measure the crystallization temperatures, based on the procedure given in Figure 3. The histogram of crystallization temperatures is given in Figure 10 with a step of 1 °C, where the values for the 1 st , 5 th , and 6 th cycles are excluded. It can be seen that the crystallization temperatures are not homogeneous, but they can be approximated by a normal distribution (in red). The mean value 𝑇 𝐶,𝑚𝑒𝑎𝑛 is here equal to 44.64 °C and the standard deviation 𝜎 is about 0.97 °C. The crystallization temperatures out of a range of 𝑇 𝐶,𝑚𝑒𝑎𝑛 ± 3𝜎 (𝑇 𝐶 < 41.7 °C) were considered as singular data. In the present case, about 11 over a total number of 32 solidifications (34%) were concerned, after the exclusion of 1 st , 5 th , and 6 th cycles.

The second step consisted of the latent heat monitoring for these non-singular points. Its evolution throughout cycles is given in Figure 11 (a). The singular data (open symbols) were separated from other values of latent heat, whose decrease should only be related to aging. However, a fluctuation in the remaining crystallization temperatures (between 41 and 47 °C) was still noticed, and the latent heat of the next melting may still be influenced. It was assumed from Figure 7 that the quantity of SAT was decreasing throughout cycles. This led to the assumption that the probability of a SAT's molecule to be separated (into water and anhydrous sodium acetate) was constant. This is characteristic of an exponential decrease fit. To find a trend of aging, the latent heat for non-singular data were fitted by an exponential decrease law (dashed curve), given in Eq. 4.

𝑑𝐿(𝑥) 𝑑𝑡 = -𝑏 𝐿(𝑥) (4)
where 𝐿(𝑥) is the latent heat for any cycle 𝑥. The crystallization temperatures are also plotted at the In addition to the 5 th cycle that was cooled more slowly (2.5 °C.h -1 instead of 20 °C.h -1 ) and crystallized at a higher temperature (50 °C), several singular points (𝑇 𝐶 < 41.7 °C) were observed.

The gap between those temperatures and the mean one seems to increase throughout cycles. Indeed, it was low at the beginning, with a crystallization temperature of 38 °C (a supercooling degree ∆𝑇 ≈28 °C) different from the mean value of 45 °C (∆𝑇 ≈ 13 °C). Then, an increase in the gap was found over the number of cycles as a crystallization temperature of 30 °C (∆𝑇 = 28 °C) is reached after the 15 th cycle and is minimal after the 34 th cycle (crystallization temperature of 20 °C (∆𝑇 ≈ quantity of material presenting a stoichiometric concentration (𝑤 𝐴𝑆 = 0.603) decreases over cycles. It is assumed that this reduction of volume would lead to the reduction of the number of potential sites for nucleation of the SAT and the rise of the supercooling degree. This was supported by the work of Guion et al. [START_REF] Guion | Nucleation of Sodium Acetate Trihydrate in Thermal Heat Storage Cycles[END_REF], who noticed an increase of this supercooling degree throughout the aging cycle of the SAT.

Without considering these singular crystallizations, the latent heat on the second cycle was about 240 J.g -1 , close to literature data [START_REF] Araki | Measurements of Thermophysical Properties of Sodium Acetate Hydrate[END_REF][START_REF] Zalba | Review on Thermal Energy Storage with Phase Change: Materials, Heat Transfer Analysis and Applications[END_REF]. Then, the latent heat decreases and stabilizes around 213 J.g -1 after 35 cycles, which represents a drop of about 10%. A comparison was done with the results from other studies, performed over more cycles. Figure 12 presents the evolution of latent heat throughout aging from Pálffy et al. [START_REF] Pálffy | Thermal Properties of a Heat Storage Device Containing Sodium Acetate Trihydrate[END_REF] and Wada et al. [START_REF] Wada | Study of Sodium Acetate Trihydrate for Latent Heat Storage[END_REF], combined with the results obtained from SAT60%.

The exponential decrease fits from the data of the different authors are represented by dashed curves. The symbols represent the value obtained by authors. The dashed line corresponds to the exponential fit that was performed in this study. An inset is given from 0 to 50 cycles.

The sample tested here presents an initial latent heat around 240 J.g -1 that reaches approximately 213 J.g -1 (-11%) after 35 cycles applied between 20 and 70 °C. In comparison, an analysis performed by Wada et al. [START_REF] Wada | Heat Storage Capacity of Sodium Acetate Trihydrate during Thermal Cycling[END_REF](green curve) showed that the latent heat of the SAT technical-grade sample seemed to stabilize around 200 J.g -1 (-17.4%) after 300 cycles applied between 40 and 70 °C. The results were comparable for Pálffy et al. [START_REF] Pálffy | Thermal Properties of a Heat Storage Device Containing Sodium Acetate Trihydrate[END_REF] (red curve) where a drop of the latent heat from 280 to 200 J.g -1 (-28.5%) of a SAT technical grade sample was found after 300 cycles applied between 23 and 65 °C.

Besides, Wada et al. have tested a laboratory-grade sample (blue curve) where aging was stronger, as latent heat in the range from 150 to 170 J.g -1 was reached after 300 cycles. It was assumed by [START_REF] Wada | Heat Storage Capacity of Sodium Acetate Trihydrate during Thermal Cycling[END_REF] that the impurities, which are more present in the technical-grade sample, may be absorbed onto the surface of anhydrous salt particles and limit their aggregation.

A high drop of latent heat was obtained in the first cycles for the sample tested (99% purity), similar to the laboratory-grade sample (blue curve) of Wada et al. Then, it differs from the blue curve after about 20 cycles, as the exponential fit stabilized around 213 J.g -1 . Moreover, it can be observed that the R² values are higher in this study. This shows the efficiency of the particular experimental bench in the determination of a consistent value of the latent heat stored.

As a conclusion, the aging behavior of SAT60% is roughly similar to the literature data. It appeared that this decrease of the latent heat is important and cannot be neglected in the development of applications. The addition of water as a solution to aging would then be studied.

Resistance to the Aging of a Sample Having Additional Water

To evaluate the influence of water on the aging of SAT, similar cycles were applied to the SAT58% sample. For better clarity, the temperature evolution of this experiment is given in Figure 13. As before, the 3 rd , 4 th and 36 th cycles, whose heating and cooling rate were about 2.5 °C.h -1 , were dedicated to the study of latent heat evolution under different conditions [START_REF] Dannemand | Compact Seasonal PCM Heat Storage for Solar Heating Systems[END_REF]. Due to their different thermal history, the data could not be included in the global analysis and will be later plotted as stars.

This also concerns the 5 th and 37 th cycles, whose thermal history was impacted by the previous heating and cooling steps. These five cycles were not considered for the probability distribution of the crystallization temperatures that are presented in Figure 14 with a 1 °C step. This sample presents an initial latent heat between 220 and 230 J.g -1 while the first cycles are difficult to consider due to a different thermal history. The trend is more obvious for this sample containing an addition of water, with a coefficient of determination 𝑅 2 = 0.99. The decrease of latent heat is also more important, up to 25% of its initial value throughout 35 cycles, and continues for the 5 remaining cycles. Besides, this trend was verified for a slower cooling rate (2.5 °C.h -1 ) for the 3 rd , 4 th and 36 th cycles, plotted as stars. Their crystallization temperatures were closed to each other (respectively 49.7 °C, 51.1 °C and 50.9 °C). This behavior was similar to the observations of Dannemand et al. [START_REF] Dannemand | Experimental Investigations on Prototype Heat Storage Units Utilizing Stable Supercooling of Sodium Acetate Trihydrate Mixtures[END_REF] where a decrease of 8% (from 194 to 179 J.g -1 ) was found after 20 cycles for a concentration 𝑤 𝑆𝐴 =56.5wt%.

The water addition, rather than increasing the aging stability of the SAT as proposed by Kong et al. [START_REF] Kong | Experimental Investigations on Phase Separation for Different Heights of Sodium Acetate Water Mixtures under Different Conditions[END_REF], tends to diminish it. Then, to explain the behavior of the different samples, visual observations of the aged samples were performed.

Discussion on the Visual Aspect of the Samples

After the thermal cycles were applied to the different samples, they were taken out of the experimental bench, without shaking to avoid the contact between anhydrous salt and water. It was first noted that a large quantity of anhydrous salt had segregated at the bottom of the box, as presented in Figure 16. For SAT58%, some cycles were done at a higher temperature (80 °C). To take this into account, a picture of SAT60% that has undergone similar temperature cycles (up to 80 °C) was also added for better comparison. A reduction of the anhydrous sodium acetate volume and an increase of its porosity with a higher temperature can first be qualitatively observed from Figure 16 (b). Indeed, the solubility of sodium acetate increases with temperature, from 149 g/100 g of water at 72 °C to 154.3 g/100 g of water at 83 °C [START_REF] Green | Melting-Point" of Hydrated Sodium Acetate: Solubility Curves[END_REF]. After heating above 80 °C, the SAT60% (Figure 16 (b)) seems to contain a lower quantity of segregated anhydrous salt (about 9 cm) than the SAT58% (11 cm, Figure 16 (c)). In addition, the SAT60% appears more porous that may accentuate the difference between the heights of segregated salts in both samples. Measuring the height of the segregated sample may help to quantify aging. However, it should be used with caution, as a part of sodium acetate trihydrate may be mixed with the porous anhydrous sodium acetate.

Conclusion and Outlook

This study was dedicated to the analysis of the aging of sodium acetate trihydrate. It was found that the reference sample (SAT60%) had lost 10% of its latent heat quantity, from 235 to 215 J.g -1 after about 35 cycles. After such cycles, the evolution seemed asymptotic, reaching a constant value that was supposed to be close to the final latent heat after aging. Besides, some crystallization occurred at a lower temperature and were considered as singular behaviors. The supercooling degree of these solidifications increased throughout cycles that may be considered as a factor of aging, unwanted for applications. Indeed, a reduction of the SAT quantity at stoichiometric concentration would result in a degradation of the properties. This would limit the use of this material as a storage medium.

Therefore, the addition of water as a solution to aging was investigated. In contrary to our expectations, the sample containing water in excess (SAT58%) was less stable after 35 cycles. Indeed, the latent heat had lost 25% of its initial value moving from about 225 to 170 J.g -1 . In addition, the achievement of the stability is slower, as the exponential fit gives a final value of 96 J.g -1 (vs 213 J.g -1 for SAT60%).

Moreover, preliminary experiments have also shown that the agitation of the sample could "reset" the value of latent heat. This agitation could be induced by any mechanical movements in the liquid, such as shaking, bubbling, or ultrasound that may also present interest as nucleation

Figure 1 .

 1 Figure 1. (a) Global description of the experimental bench and (b) focus on the sample, adapted from [21,22].

Figure 2 .

 2 Figure 2. Description of a full thermal cycle.

  𝐿 = -462 + 11.8 𝑤 𝑆𝐴 (3) Moreover, the determination of the crystallization temperatures was necessary for a correct analysis of the material's solidification and its impact on the next melting. For each cycle (as presented in Figure 2), the onset temperature was determined in three steps. (1) The onset time was determined by the intersection of 2 tangents represented by black dash lines in Figure 3. (2) The crystallization temperature was then identified by a blue arrow and (3) its value was read on the left axis.

Figure 3 .

 3 Figure 3. Method for crystallization onset temperature determination.

Figure 4 .

 4 Figure 4. Seeding of SAT with a welding rod (a) Schematic draw of the brick and detailed pictures (after the brick was taken out of the experimental bench) (b) initial state; (c) insertion throughout the grey tape; (d) beginning of crystallization and (e) growth of the crystal.

Figure 5 .

 5 Figure 5. Preliminary evaluation of latent heat of SAT58%. The arrow symbolizes a reset of the properties by agitation (the two rectangles allow for comparison of the latent heat at the beginning of the experiment and after agitation).

  (a)) or the preliminary aging cycles (SAT58%, Figure6 (b)).

Figure 6 .

 6 Figure 6. Formation of anhydrous salt segregation in (a) SAT60% (the colors were modified for contrast accentuation), (b) SAT58%, and (c) SAT58% after agitation, before insertion in the experimental bench.

Figure 7 .

 7 Figure 7. Schematic diagram of SAT aging throughout melting cycles.

Figure 8 .

 8 Figure 8. Temperature evolution along the experiment performed on SAT60%.

Figure 9 .

 9 Figure 9. Singular crystallization at 20 °C for the 34th cycle of the experiment. Cryst.: Crystallization. Melt.: Melting.

Figure 10 .

 10 Figure 10. Probability distribution of crystallization temperatures for SAT60%.

  bottom (Figure 11 (b)). Thus, the relation between singular crystallization temperatures (below the black dashed curve) and lower latent heat values on the next cycle (open symbol) would be highlighted.

Figure 11 .

 11 Figure 11. Evolution of (a) the latent heat and (b) the supercooling degree over 35 cycles of a raw SAT sample (w AS = 0.603). Values for the 5 th and 6 th symbol (from different thermal history) are plotted as stars. The open symbols (following a low-temperature crystallization) will not be considered for exponential fit and aging analysis. 𝑇 𝑀 and 𝑇 𝐶 respectively represent melting and crystallization temperatures.

Figure 12 .

 12 Figure 12. Evolution of latent heat adapted from Pálffy et al. [10] (in red), Wada et al. [8] for a technical grade (in green), and laboratory-grade (in blue) combined with the results obtained here (in black).

Figure 13 .

 13 Figure 13. Evolution of temperature throughout aging cycles.

Figure 14 .

 14 Figure 14. Probability distribution of crystallization temperatures for SAT58%. In the inset, the crystallization temperature has a 0.1 °C temperature step.

Figure 15 .

 15 Figure 15. Evolution of (a) the latent heat and (b) the supercooling degree over 40 cycles of a SAT58% sample (w SA = 0.577). Values for 3 rd , 4 th , 5 th , 36 th , and 37 th are plotted as stars. As the open symbols (following a low-temperature crystallization), they will not be considered for exponential fit and aging analysis. 𝑇 𝑀 and 𝑇 𝐶 respectively represent melting and crystallization temperatures.

Figure 16 .

 16 Figure 16. Visual appearance of (a) SAT60% after 35 cycles at 70 °C, (b) the same sample after 46 more cycles until 80 °C and (c) SAT58% after 41 cycles until 70 °C and 10 cycles until 80 °C.

Table 2 .

 2 Initial properties of the sample before aging.

		SAT58%	Number of experiments	SAT60%	Number of experiments
	Mean L value (J.g -1 ) Difference with Eq. 3 (%)	214 ± 3 2	5	253 ± 2 1	3
	Mean 𝐶 𝑃,𝑆 value (20; 28 °C, J.kg -1 .°C -1 ) Difference with Eq. 1 (%)			2021 ± 19 0.2	4
	Mean 𝐶 𝑃,𝐿 value (63; 70 °C, J.kg -1 .°C -1 ) Difference with Eq. 1 (%)			3260 ± 35 9.3	4
	Mean 𝐶 𝑃,𝑠𝑢𝑝𝑒𝑟𝑐𝑜𝑜𝑙𝑒𝑑 value (5; 70 °C, J.kg -1 .°C -1 ) Difference with Eq. 1 (%)	3199 ± 15	10

triggering techniques. Indeed, the limitation of the supercooling degree is generally targeted in phase change materials to improve their heat release capability.

Finally, for an application, the use of a high and narrow container could induce a more rapid aging behavior. Indeed, after the separation of the two species (sodium acetate and water) during heating, it may be more difficult to put them in contact during cooling. It would then be recommended to use a container of low height (h < 5 cm) to limit the separation of species due to gravity in order to facilitate the rehydration of the SAT.