Thermodynamic Efﬁciency of Water Vapor/Solid Chemical Sorption Heat Storage for Buildings: Theoretical Limits and Integration Considerations

Featured Application: The paper provides theoretical limits of chemisorption heat storage in buildings and discusses solutions for an efﬁcient system integration. Abstract: The theoretical limits of water sorbate-based chemical sorption heat storage are investigated in this study. First, a classiﬁcation of thermochemical heat storage is proposed based on bonding typology. Then, thermodynamics of chemical solid/gas sorption is introduced. The analysis of the reaction enthalpy from the literature indicates that this value is only slightly varying for one mole of water. Using this observation, and with the help of thermodynamic considerations, it is possible to derive conclusions on energy efﬁciency of closed and open heat storage systems. Whatever the salt, the main results are (1) the energy required for evaporation of water is, at least, 65% of the available energy of reaction, and (2) the maximum theoretical energy efﬁciency of the system, deﬁned as the ratio of the heat released to the building over the heat provided to the storage, is about 1.8. Considering the data from literature, it is also possible to show that perfectly working prototypes have an energy efﬁciency about 49 %. Based on those results, it is possible to imagine that for the best available material, a perfect thermochemical heat storage system would correspond to 12 times water with a temperature difference about 50 ◦ C. Such solution is deﬁnitely competitive, provided that some difﬁcult issues are solved—issues that are discussed throughout this paper.


Introduction
It is obvious that everything has a limit.However it is better to know its value in order to avoid wasting effort to find means of going beyond that limit !Chemical solid/gas sorption has a relatively high potential of heat storage density and is a subject of interest to both scientists and engineers [1][2][3][4][5][6][7][8].
Among possible reactants, water sorbate based chemical reactions have retained attention because of non-toxicity and availability of water.Whatever the system, closed or open reactor, one of the main design criteria is the energy efficiency of the thermal storage system.In the present work, only reactions involving temperatures below 150 • C are considered, which corresponds to a building application [1].However, the ideas developed in the present paper can be straightforward adapted to other applications or other sorbates.
MgCl 2 • 6 H 2 O was selected by [9] to be tested in a 17 L open reactor.Under realistic operating condition, the energy storage capacity of the reactor reached about 139 kWh m −3 .The energy efficiency was characterized by the instantaneous electrical COP of the system evaluated via: COP el = P heating P f an (1) where P heating is the heat released and P f an the fan electrical energy.An instantaneous electrical COP of 12 was found, but the authors expected to reach up to 30 with an optimization of heat recovery and pressure drop in the system.Of course this high COP value is a direct consequence of its "home-made" definition.For the sake of clarity, the present study will only deals with thermodynamics efficiency of the system.
A lab-scale closed thermochemical heat storage reactor was developed in [10], capable of holding about 974 g of material.The storage material used was SrBr 2 • 6 H 2 O and 13 dehydration-hydration cycles were conducted.The energy storage capacity of the reactor was about 65 kWh.The reactor thermal energy efficiency is 0.77 meaning a global heat loss of 23%.
It is worth mentioning that other reactors have been experimentally tested but no data are available concerning the energy efficiency: • Na 2 S • 5 H 2 O in a closed reactor - [12], • SrBr 2 • 6 H 2 O in an open reactor - [13,14], • CaCl 2 • 2 H 2 O in an open reactor - [16].
The experimental investigation was focused on the reactor alone and the storage capacity and efficiency were based on a perfect system integration.For example, heat required for water vapor generation is never evaluated nor discussed.
On the other side, the theoretical COP of chemical heat pump, i.e. closed system, was studied in [17].Calculated standard enthalpy of reaction of 34 salts were used to evaluate the influence of thermophysical properties on energy and exergy efficiency of a perfect system with recovery of condensation heat and "energy-free" heat of vaporization.The main results were 1) the maximum COP is about 1.84 for CaSO 4 • 2 H 2 O and 2) the behavior of the energetic efficiency and the exergetic efficiency is opposite.The calculations were based on the evaluation of the perfect thermodynamic cycle.
We propose, in this paper, to evaluate the theoretical limits of the thermodynamic efficiency of water sorbate / salt chemical sorption heat storage system.To our knowledge, it is the first attempt to evaluate such limits.The starting point of our work is simple: regarding data from the literature, the enthalpy of reaction of one mole of water varies little from one reaction to the other.Then, with the help of chemical thermodynamics, it is possible to derive general considerations concerning the efficiency of open and closed heat storage systems.

Classification of reaction heat storage
In the literature, thermochemical heat storage is employed for a family of reactions involving both, physical and chemical processes.A tentative of classification is given in [18] and is presented in Fig. 1.However, in this classification, sorption is used to integrate different physical phenomena and can lead to misunderstandings.Then, we propose, in this section, to derive a classification of thermochemical heat storage based on a physical phenomena typology.We deliberately limit the classification development to heterogeneous 1 reactions as homogeneous reactions are seldom used for thermal energy storage.
Let's first define sorption: according to [19], sortpion is the process by which a substance (sorbate) is sorbed (adsorbed or absorbed) on or in another substance (sorbent).The process can be caused by physical bonding, i.e. physical sorption, or chemical bonding, i.e. chemical sorption.The main difference between physical and chemical sorption lies in the nature of created bonds.Physical sorption is weak, long range bonding mostly Van der Walls interactions and hydrogen bonding.Chemical sorption is strong, short range bonding involving orbital overlap and charge transfer.Another main difference between physical and chemical sorption is that the latter requires activation energy whereas it is not the case for the first process.
Sorption can be absorption or adsorption.Definitions of both processes can be found in [19]: • Absorption is the process of one material (absorbate) being retained by another (absorbent); this may be the physical solution of a gas, liquid, or solid in a liquid, attachment of molecules of a gas, vapor, liquid, or dissolved substance to a solid surface by physical forces, etc. • Adsorption is an increase in the concentration of a dissolved substance at the interface of a condensed and a liquid phase due to the operation of surface forces.Adsorption can also occur at the interface of a condensed and a gaseous phase.
While molecule undergoing absorption are taken up through the bulk of the absorbent (for example by diffusion), adsorption is a surface process.It is sometimes difficult to find the difference between adsorption and absorption.Taking for example the dehydration of lithium sulphate monohydrate [20] (i.e. chemical sorption), nucleation is supposed to occur at the surface of the grain (adsorption) and then the growth proceeds towards the center of the grains by diffusion (absorption).Physical sorption can be split into absorption and adsorption: • In physical absorption, the mass transfer takes place at the interface between the absorbate and the absorbent.This type of absorption depends on the solubility of absorbate, the pressure and the temperature.The rate and amount of absorption also depend on the surface area of the interface and its duration in time.
• Physical adsorption is called physisorption.Physisorption is adsorption in which the forces involved are intermolecular forces (Van der Waals forces) of the same kind as those responsible for the imperfection of real gases and the condensation vapors, and which do not involve a significant change in the electronic orbital patterns of the species involved.The term van der Waals adsorption is synonymous with physical adsorption, but its use is not recommended.[19].
Similarly to physical sorption, chemical sorption can be split into absorption and adsorption: • Chemical absorption or reactive absorption is a chemical reaction between the absorbed and the absorbing substances.Sometimes it combines with physical absorption.This type of absorption depends upon the stoichiometry of the reaction and the concentration of its reactants.
• Chemical adsorption is called chemisorption.Chemisisorption is Adsorption which results from chemical bond formation (strong interaction) between the adsorbent and the adsorbate in a monolayer on the surface.[19].
From the previous definition, we propose in Fig. 2 a classification of heat storage systems based on the physical/chemical phenomena involved.For heat storage, the split between physical and chemical sorption is important as the heat related to these reactions is quite different.For example, heat of adsorption is different for physisorption and chemisorption: • Chemisorption: 80 − 400 kJ mol

Energy change in chemical sorption
Let's consider the water sorbate heterogeneous chemical sorption reaction process expressed under the following general form: The first law of thermodynamics states that the change in the internal energy of a system ∆U is equal to the sum of the heat gained/lost by the system Q and the work done by/on the system W: The amount of work of expansion done by the reaction during any transformation is given by: At constant volume (i.e.W = 0), the heat given off or absorbed by the reaction is equal to the change in the internal energy that occurs during the reaction: Such configuration is close to closed chemical sorption heat storage systems.
At constant pressure, the change in the internal energy occurring the reaction is given by: Let's introduce the enthalpy of the system H related to the internal energy by: Then, the heat given off or absorbed during a chemical reaction at constant pressure is equal to the change in the enthalpy of the system: Such configuration is close to open chemical sorption heat storage systems.
The relationship between the change in internal energy and the change in enthalpy, assuming an ideal gas, is given by: where R = 8.31 J K −1 mol −1 and ν the stoichiometric coefficient defined in Eq. 2. For a temperature of 273.15 K,, R × T = 2.27 kJ mol −1 .This value is very low compared to heat of reaction given per mole of water 2 , less than 4%, and then can be neglected.Consequently, it is possible to assume that the change in internal energy is more or less equal to the change in enthalpy.

Enthalpy of reaction
For the reaction given in equation 2, dn H 2 O , dn A and dn AνH 2 O are respectively the mole variation of water, solid A and solid AνH 2 O.Then, the degree of advancement of the reaction ξ is given by: The variation of enthalpy can be written under the following form: where ∆ r H is the enthalpy of reaction at constant temperature and pressure.Of course, for a transformation at constant pressure and temperature 3 , the variation of enthalpy is given by The standard enthalpy of reaction (denoted ∆ r H 0 ) is the enthalpy change that occurs in a system when one mole of matter is transformed by a chemical reaction under standard conditions, i.e. a temperature of 273.15K and a pressure of 100000 Pa.The standard enthalpy of reaction can be measured or computed using the standard enthalpy of formation of the reactants and products.
The enthalpy of reaction for a a temperature T is related to the standard enthalpy of reaction via: where C is the specific heat of AνH Moreover, the chemical sorption reaction being monovariant, the equilibrium is given by the Clausius-Clapeyron relation: where P e is the equilibrium water vapor pressure [Pa] and T e the equilibrium temperature [K].

Material considerations
In the remaining of the study, we will consider only reactions involving temperatures below 150 • C which corresponds to a building application [1].However, the ideas developed in the present paper can be straightforward adapted to other applications or other sorbates.
The dehydration of cobalt(II) chloride-6-hydrate (CoCl2.6H2O)was investigated in [23].The enthalpy of formation was calculated using thermodynamic values.The results show that one mole of water corresponds to a variation of the reaction enthalpy about 55.2 kJ mol −1 .It is worth mentioning that the latter value is very close to the enthalpy of vaporization of ice at 25 • C i.e. 52 kJ mol −1 .This conclusion is also validated by [24].
Enthalpy of reaction given in the literature are summarized in table 1.The main result is that the enthalpy of reaction of one mole of water only varies between 55.1 kJ mol −1 and 67.8 kJ mol −1 .Of course, this value is close to the observations made in the literature and given above.This specific feature is the basis of the system maximum theoretical efficiency limits.

Concepts
The schematic diagram of a perfect sorption thermal battery for energy storage using solid-water vapor chemical reaction is presented in Fig. 3.The two curves present the solid/water vapor equilibrium of the sorbent and the vapor/liquid equilibrium of the water sorbate.Under the solid/water vapor line, the sorbent is under the A solid form.Above the solid/water vapor line, the sorbent is under the (AνH 2 O) solid form.Basically, the two main concepts of system design are closed and open.
The principle of a closed chemical sorption heat storage system is given in Fig. 4. Initially, sorbent is under the (AνH 2 O) solid form.During the storage phase, Q in heat is transferred to the material at the temperature T in .Then, the water vapor pressure is increasing and the gas moves from the material to the condenser where the pressure is P h .Then, the vapor condensates 4 and heat of condensation is released, Q cond .During the release phase, liquid is evaporated 5 at temperature T e , requiring a quantity of heat, Q evap .As pressure is higher in the evaporator than in the material, a gas flow occurs.A quantity of heat Q out is then released during the sorption process in the material.
The other option lies in the use of an open system, presented in Fig.  phase, the carrier (or the mixing) is passing through the material under the (AνH 2 O) solid form.Q in heat is then transferred to the material at the temperature T in , resulting in the desorption process and a potential heat of condensation in the carrier, Q cond .During the release phase, the mixing is passing through the material under the A solid form and may requires a quantity of heat to evaporate liquid water, Q evap .The water vapor is sorbed and heat is Q out heat is transferred to the carrier (or the mixing).

Thermodynamic Efficiency
Whatever the system open or closed, it is important to answer the issue related to the thermodynamic efficiency limit of the heat storage.Basically, the designer of such system must know the thermodynamics limit to evaluate the enhancement possibilities of its prototype.
Let's first define Q + as the supplied energy per mole of salt to the storage system and Q − as the recovered energy per mole of salt.The efficiency of the system η is defined, for the sake of universality, as: This efficiency can also be found as COP in the literature [3].However, the definition of COP may vary from an author to the other: an example is the definition of COP given by [9] which is completly different from the one given by [17].
The maximum reachable efficiency η max is calculated with the maximum Q − , called Q − max , and, of course, the minimum Q + min .
The assumptions used to evaluate the theoretical limits of the energy efficiency are: • Heat losses in the system are not taken into account.
• Sensible heat of materials and parts of the reactor are neglected.
• The energy taken off or absorbed by the reaction is approximated by the standard enthalpy of reaction.
• Only total hydration / dehydration processes are considered.
Given the previous assumptions, if the heat of condensation is not recovered and if the heat of evaporation is energy-free 6 , the trivial efficiency is η max = 1 (see Fig. 6 - Energy-free means that no additional energy is required for producing water vapor.System integration: heat of condensation is not recovered and heat of evaporation is energy-free: η max = 1.Considering the previous assumptions, if non-free heat of evaporation, the maximum efficiency becomes (see Fig. 7): The quantity of energy required to evaporate liquid water per mole of salt is evaluated with: where L v is the water heat of vaporization in standard conditions taken as 2456 kJ kg −1 and M H 2 O the molar mass of water equal to 18 g mol −1 .Considering data from table 1, and considering that ∆ r H 0 /ν is few varying from one salt to another, the ratio Q evap /∆ r H 0 can be evaluated as: The previous equation also shows that energy required for evaporation is, at least, 65% of the available energy of reaction!The maximum energy efficiency of the integrated system η max is then varying between 55% and 60% only.
Let's now consider the case with a total recover of condensation heat and free-energy heat of evaporation.Then, the energy efficiency becomes (see Fig. 8): System integration: heat of condensation is recovered and heat of evaporation is energy-free: The quantity of energy (per mole of salt) recoverable from condensation is evaluated by: Then, the energy efficiency of the system is evaluated via the simple equation: The maximum theoretical energy efficiency of the system is about 1.8, whatever the salt!This conclusion is clearly in accordance with the results of [17] where the maximum value is 1.84.However, our study extends these results to the integration of an open system.

Conclusions
Regarding data from the literature, the reaction enthalpy of one mole of water only varies between 55.1 kJ mol −1 and 67.4 kJ mol −1 .Considering an open or closed single-stage system, the two main conclusions are: • Energy required for evaporation of water is, at least, 65% of the available energy of reaction.
• For a perfect system, the maximum theoretical energy efficiency of the system is about 1.8.
The previous conclusions don't depend on the adsorbent material considered.Then, a special attention must be paid from a system point of view for: • Developing water evaporation system "energy-free" or low-energy for the discharging phase.
• Developing or using water condensation recovering systems: examples of such recovering systems are the cascaded thermal battery [25,26] or the integration of a heat-pump.
Of course, these theoretical limits remain valid for the operating conditions given in this work and further studies must be carried to extend the conclusions to higher temperature storage.Moreover, numerical modeling is also under investigation to evaluate the potential improvement of the system integrated in the building.

Conflicts of Interest:
The author declares no conflict of interest.

Figure 2 .
Figure 2. Classification of heat storage by physical phenomena.
Figure 6.System integration: heat of condensation is not recovered and heat of evaporation is energy-free: η max = 1.

Figure 7 .
Figure7.System integration: heat of condensation is not recovered and heat of evaporation is not free: η max =1 1+Qevap ∆r H 0 Figure 8.System integration: heat of condensation is recovered and heat of evaporation is energy-free:η max = 1 + Q cond ∆ r H 0 .

Table 1 .
15O, A or H 2 O.It is worth mentioning that, usually, the quantity AνH 2 O − C A − ν × C H 2 O ) × dt is small compared to ∆ r H 0 273.15.Measured enthalpy of reaction extracted from the literature.
2Values of enthalpies of reaction are given in table13 Constant temperature or neglected variation of enthalpy due to temperature change.