Alexis Godefroy
email: alexis.godefroy@univ-perp.fr

Maxime Perier-Muzet
email: maxime.perier-muzet@univ-perp.fr

Nathalie Mazet
email: mazet@univ-perp.fr

Thermodynamic analyses on

Keywords: thermochemical cycles, sorption, hybrid cycles, power and refrigeration cogeneration, thermal storage, thermodynamic analysis

come

Introduction

Renewable energy sources are an attractive solution to the problem of scarce fossil resources and the challenge of cutting greenhouse gas emissions, but industrial waste heat (typically from steel and glass industry or power plants) is another huge but under-exploited energy pool [START_REF] Ammar | Low grade thermal energy sources and uses from the process industry in the UK[END_REF]. The low-grade heat resource available and several ways of recovering this energy were thoroughly investigated by Ling-Chin et al. [START_REF] Ling-Chin | State-of-the-Art Technologies on Low-Grade Heat Recovery and Utilization in Industry[END_REF]. However, the availability of these energy sources is very often variable.

Needs also fluctuate strongly on the energy demand side, not only over time but also in energy form: demands such as electricity, cooling and heating need to be covered at different time-scales with different energy levels. Efficiently matching these sources and needs requires suitable energy storage systems. Stutz et al. [3] reviewed existing thermal solar energy storage systems, and Nadeem et al. [START_REF] Nadeem | Comparative Review of Energy Storage Systems, Their Roles, and Impacts on Future Power Systems[END_REF] recently completed a detailed comparative review of existing energy storage systems.

The net result is that we have a major energy management issue to address in order to make the best use of these renewable energy sources and industrial waste heat to meet wider form and quantity of energy demands. This paper focuses on these under-exploited low-grade heat sources, chiefly solar thermal sources (concentrated or not) and industrial waste heat, targeting temperatures up to Thot,max = 250°C. The aim is to store the thermal energy of these sources and convert it into one or more useful effects, chiefly electric power and cooling according to demand. The technical solutions presented here could be used in embedded applications, especially for the delivery of refrigerated goods in urban areas using electric vehicles.

The conversion of low-grade heat into cold has been widely investigated, leading to tri-thermal cycles involving sorption processes as a relevant way forward, as evidenced in [START_REF] Srikhirin | A review of absorption refrigeration technologies[END_REF] and [START_REF] Elsheniti | Adsorption refrigeration technologies[END_REF]. Two types of sorption cycles can meet cold production requirements:

 Liquid/gas absorption cycles (typically using LiBr/H 2 O or H 2 O/NH 3 solutions)  Solid/gas adsorption cycles (typically using NH 3 /activated carbon or H 2 O/silica gel) or thermochemical cycles (using ammonia salt/NH 3 working pairs for example)

Note that the storage function can only be intrinsically achieved by the second type of sorption cycles (solid/gas sorption cycles). Indeed, the absorption process works continuously (the liquid solution circulates between a gas absorber and a desorber) whereas the solid/gas sorption process intrinsically works discontinuously as the reactive solid is usually implemented in a fixed-bed reactor. These solid sorption cycles involve two operation steps that are shifted in time, thus enabling storage of the input low-grade heat in the first working step (by endothermal desorption: charging step) and cooling production in the second step (discharging step). Moreover, due to the wide diversity of available reactants, solid/gas thermochemical cycles can use lower-temperature heat sources than other sorption cycles. Among them, systems based on ammonia salt/NH 3 pairs afford the added advantage of using a refrigerant with a zero Global Warming Potential (GWP) and zero Ozone Depletion Potential (ODP).

Low-grade heat can also be converted into mechanical work (for power production purposes) using an Organic Rankine Cycle (ORC) or a similar power cycle.

Therefore, one possible way to implement a process combining low-grade heat storage and conversion into several forms is to create a hybrid thermodynamic cycle. Basically, the hybridization of several independent thermodynamic cycles consists in sharing their common physical components (such as evaporator, condenser, chemical reactor, and working fluid) in a single cycle, thus combining the advantages and features of each cycle at potentially less cost than when using two separate systems. Hybrid thermodynamic cycles are attracting increasing attention from research aiming to develop novel multi-purpose systems, but the concept is still only emerging and the Technological Readiness Level (TRL) is relatively low (at 2-3). Here we bring innovation to the current body of knowledge on hybrid cycles by developing a broader approach of hybrid cycles based on solid/gas thermochemical and power cycles. Section 2.2 reviews the state of the art in the field of hybrid sorption cycles providing power and refrigeration cogeneration. Early investigations focused on the hybridization of a liquid/gas absorption refrigeration cycle with a power cycle. As detailed in §2.2., several systems have been theoretically and experimentally studied, some of them using a low-grade heat source (see especially [START_REF] Xu | A combined power / cooling cycle[END_REF] and [START_REF] Tamm | Theoretical and experimental investigation of an ammonia-water power and refrigeration thermodynamic cycle[END_REF]). However, as mentioned above, liquid/gas absorption cycles operate continuously and so cannot provide a storage feature. Their maximal thermal and exergy efficiencies are around 0.25 and 0.65, respectively. Later investigations proposed other hybrid systems based on solid/gas sorption cycles (adsorption or thermochemical cycles) that bring together power and cold cogeneration with an intrinsic storage feature. The state of the art in these systems (see §2.2. for details) reveals a very broad range of energy and exergy performances, as the existing literature considers very different assumptions and operating conditions. Thus, maximal efficiency values are not provided for hybrid thermochemical cycles, since any comparison with other sorption systems would be unfair. The very few studies carried out on hybrid thermochemical cycles are based on a thermodynamic analysis focusing only on two main performance criteria (energy and exergy efficiencies) and a few reactive salts (see especially the theoretical studies [START_REF] Wang | A resorption cycle for the cogeneration of electricity and refrigeration[END_REF][START_REF] Jiang | Analysis on innovative resorption cycle for power and refrigeration cogeneration[END_REF] and the experimental setup investigated in [START_REF] Jiang | Experimental study on a resorption system for power and refrigeration cogeneration[END_REF]). Nevertheless, these cycles prove suitable for low-grade heat use and demonstrate relatively good energy and exergy performances. This paper expands the concept of hybrid thermochemical cycle by defining three ways of hybridizing a solid/gas thermochemical refrigeration cycle with a vapor power cycle in order to engineer innovative cycles combining a refrigeration and power cogeneration cycle with an intrinsic energy storage function. A wider thermodynamic analysis of these three hybrid cycles (referred to as modes) is developed, based on an original approach: a set of five relevant performance criteria is investigated to account for the various features of the cycles (power production, cold production, and storage). Moreover, a large set of solid reactants (103 reactive salts) is screened. This study leads to an exhaustive comparative assessment of the performances for the whole set of criteria, modes and reactants. This thermodynamic analysis serves to identify the most relevant modes and reactive salts for various fields of application (according to low-grade heat source temperature and specific needs). The paper thus addresses a significant gap with respect to the current body of knowledge in terms of configurations, reactants and performance criteria for hybrid thermochemical cycles for power and refrigeration cogeneration.

The paper starts by overviewing the working principle and state of the art in hybrid sorption cycles for power and cold production. Then, we depict the three modes of hybridizing solid/gas thermochemical and power cycles in order to build a cogeneration cycle. Next, the thermodynamic analysis model and methodology are presented and the relevant parameters and performance criteria are defined. The paper goes on to report the results and analyses of this thermodynamic study, leading to different energy purposes: for the three modes (separated power and cold production mode, simultaneous power and cold production mode, combined mode), the operating conditions and performances are summarized and analyzed. These results are then discussed through a sensitivity study of some key variables, and we interpret and compare the operating conditions and performances of all the hybrid cycles investigated. In the conclusion, the most relevant applications for each hybrid cycle are proposed, and the prospects for these investigations are sketched out.

State of the art and configurations of hybrid cycles involving a sorption process

General working principle

The principle can be described by starting from a sorption refrigeration cycle, which is illustrated in Fig. 1 and Fig. 2 for hybridization of a liquid/gas absorption cycle and a solid/gas thermochemical cycle, respectively, with a power cycle. As a rule, these hybridizations lead to configurations whose main components are:

 2 vapor generators, where endothermal processes occur: one operating at high pressure (with a heat supply Qin from a heat source at Thot), and the other operating at low pressure (removing Qcold from the medium to be cooled at Tcold).

 2 vapor absorbers, where exothermal processes occur: one at high pressure and the other at low pressure. Both release heat at Tamb or at an intermediate temperature Tm.

 1 or several expanders for power production.

This general principle leads to two different cycles, depending on the type of sorption process:  A hybrid cycle based on a liquid/gas absorption cycle (Fig. 1) works continuously, as the liquid solution containing the sorbent circulates between vapor absorber and vapor generator. The expander can be located either (i) on the vapor flow between the high-pressure vapor generator and the high-pressure vapor absorber, resulting in power production then cold production in series; or (ii) between the highpressure vapor generator and the low-pressure vapor absorber. In this configuration, the working fluid flow is split between the expander and the vapor absorber and generator on the left side of Fig. 1, leading to parallel power and cold production.  A solid/gas sorption hybrid cycle (Fig. 2) is intrinsically discontinuous, as it uses a solid sorbent. It involves two time-shifted operating steps, which provides its intrinsic storage function: a charging step when a heat source is available, then a discharging step. The expander can operate in one or both steps. Furthermore, the low-temperature endothermal process (in the discharging step) may lead to cold production. This kind of hybrid cycle can therefore operate according to three modes: (i) separated power and cold production mode, when the expander operates in the charging step while cold is produced in the discharging step; (ii) simultaneous cold and power production mode, when the expander operates only in discharging step alongside cold production; (iii) combined mode, when the two previous modes are combined, leading to power productions in both steps and cold production in the discharging step.

Note that the vapor absorber and desorber can be chosen in different ways:  For the absorption hybrid cycle: the right-side components on Fig. 1 are a liquid/gas desorber and absorber, while the left-side components are a condenser and an evaporator;  For the solid/gas sorption hybrid cycle (Fig. 2): the right-side component, operating at high temperatures, must be a solid/gas reactor, while the left-side component can be either another solid/gas reactor operating at lower temperature (this process is named resorption cycle) or a reactive fluid condenser/evaporator (named single sorption cycle). For a general description of the various hybrid cycles (in the following sections of this paper), the material in the component requiring the high temperature source will be called High Temperature Material (HTM, right-side material on Fig. 2, Fig. 3, Fig. 5 and Fig. 7) while the material in the other component operating with the low temperature source will be called Low Temperature Material (LTM, in the left-side component).

State of the art

The first hybridization of a sorption cycle with a power cycle was implemented by Goswami et al. [START_REF] Xu | A combined power / cooling cycle[END_REF]. Their combined cooling and power cycle is based on a liquid/gas (H 2 O/NH 3 ) absorption cycle coupled with an expander for power generation. It is suitable for use with a low-grade solar heat source (heat source temperature used for calculations is 140°C). The main output is mechanical work (about 74% of the total output energy), and an additional cooling effect occurs at the expander outlet by exploiting the low temperature flow of the working fluid (cold production temperature is set at -10°C). The energy efficiency is ηI = 0.24, and the exergy efficiency (computed with data available from [START_REF] Xu | A combined power / cooling cycle[END_REF]) is ηex = 0.59. Moreover, a cogeneration performance optimization study on this cycle is performed in [START_REF] Hasan | First and second law analysis of a new power and refrigeration thermodynamic cycle using a solar heat source[END_REF], using a second-law-based criterion as the objective function. Moreover, an experimental setup [START_REF] Tamm | Theoretical and experimental investigation of an ammonia-water power and refrigeration thermodynamic cycle[END_REF] was built and a good agreement was found with theoretical predictions: energy and exergy efficiencies computed from experimental data were respectively 0.25 and 0.65. Later, several hybrid absorption cycles developed for power and cold cogeneration were compiled by Ayou et al. [START_REF] Ayou | An overview of combined absorption power and cooling cycles[END_REF] for systems with H 2 O/NH 3 as working pair. This paper highlights the broad diversity of performances with available cycles (various heat-source temperatures and power-to-cooling ratios).

More recently, Ventas et al. [START_REF] Ventas | Performance analysis of an absorption double-effect cycle for power and cold generation using ammonia/lithium nitrate[END_REF] investigated the performances of a double-effect hybrid cycle based on liquid/gas (NH 3 /LiNO 3 ) absorption coupled with an expander for power production. Their system makes it possible to adjust vapor flow through the expander, and thus the power-to-cooling ratio. Working under 150°C heat source temperature and providing cold at -5°C, it achieves energy and exergy efficiencies of 0.61 and 0.56, respectively. At these performance levels, the main output is cold (about 90% of the total useful energy).

On the other hand, Ziegler et al. [START_REF] Ziegler | Re-evaluation of the Honigmann-process: thermo-chemical heat store for the supply of electricity and refrigeration[END_REF] borrowed the Honigmann system to develop a broader hybrid cycles approach involving any kind of sorption process (absorption, adsorption or chemical reaction). They proposed a generalized cycle configuration including four main components (condenser, evaporator, desorber, ab-or adsorber) and an expander, allowing flexible inputs (heat or work) and outputs (work or cold) as well as a possible storage function. Then, using liquid/gas absorption (with LiBr/H 2 O and NaOH/H 2 O as working pairs), they performed dynamic simulations of a power cycle with a storage function (storage of the working fluid) [START_REF] Ziegler | First cycle simulations of the Honigmann process with LiBr / H 2 O and NaOH / H 2 O as working fluid pairs as a thermochemical energy storage[END_REF].

Concerning hybrid cycles involving solid/gas chemical reaction, the system that has been addressed most so far is the resorption cycle (case (a) on Fig. 2). This cycle associates two reactive beds involving two different solid salts reacting with the same gas, i.e. a Low Temperature Material (LTM) and a High Temperature Material (HTM), whose reaction equilibrium temperature is higher than that of the LTM at a given pressure. See §2.3. for an in-depth description of hybrid solid/gas thermochemical cycles. This state of the art review focuses on the use of ammonia as reactive vapor.

Several studies have investigated resorption cycles for power production only. Bao et al. [START_REF] Bao | Integrated chemisorption cycles for ultra-low grade heat recovery and thermo-electric energy storage and exploitation[END_REF] designed a cycle using ultra-low-grade heat (source temperature from 30 °C to 100 °C) to generate power, using a compressor to drive the decomposition. Overall energy efficiency ranged from 0.47 to 0.62 and exergy efficiency ranged from 0.60 to 0.90, depending on heat source temperature. Later, the same authors compared a pumpless ORC with resorption power cycles [START_REF] Bao | Chemisorption power generation driven by low grade heat -Theoretical analysis and comparison with pumpless ORC[END_REF], and evaluated the output work per mass unit of working fluid as 100 to 550 kJ/kg for the resorption cycle against 10 to 100 kJ/kg for a pumpless ORC. They highlighted that the difference between these specific values was linked to the small molecular weight of ammonia compared with the organic fluid used in ORC. Using a low-grade heat source between 60 °C and 180 °C, the energy efficiencies ranged from 0.08 to 0.25 for pumpless ORC but were much lower (0.06-0.17) for the resorption cycle, due to the wet property of ammonia which prevents exploiting the whole available pressure ratio (considering only one expansion stage). This can be overcome by the multiple-expansion resorption power generation that Bao et al. also investigated in [START_REF] Bao | A chemisorption power generation cycle with multi-stage expansion driven by low grade heat[END_REF]. Using a heat source between 30 °C and 150 °C and 2 to 4 expansion stages, they achieved a slightly higher work output per unit mass of working fluid (from 100 to 600 kJ/kg) but similar energy efficiencies (0.06-0.15) as previously. One of the main findings of this work was the need to find a trade-off between more total work output (more expansions) and a more compact system with higher average output per expansion device.

Lu et al. [START_REF] Lu | Study of a novel dual-source chemisorption power generation system using scroll expander[END_REF] designed a system involving two resorption cycles operating in phase opposition, recovering low-grade heat from two heat sources, at 180-200 °C and 80-90 °C, and enabling continuous operation. The reactive ammonia salts MnCl 2 and SrCl 2 were chosen as working pair and led to a thermal efficiency of about 0.11 in each mode. Their detailed dynamic study [START_REF] Lu | Investigation and performance study of a dualsource chemisorption power generation cycle using scroll expander[END_REF] stressed the importance of the coupling between expander and chemical reactor and the need to manage coupling pressure in order to control the power output. Jiang et al. [START_REF] Jiang | Exploration of ammonia resorption cycle for power generation by using novel composite sorbent[END_REF] proposed an improvement of resorption cycles using a new composite sorbent that enhances the reactor kinetics, and they obtained energy efficiencies ranging from 0.11 to 0.14 and exergy efficiencies from 0.62 to 0.81, with a heat source temperature between 80 °C and 110 °C.

Closer to our focus here, a few resorption cycles have been investigated for power and cold cogeneration. Wang et al. [START_REF] Wang | A resorption cycle for the cogeneration of electricity and refrigeration[END_REF] evaluated the energy and exergy performances of this kind of resorption cycle for 9 different reactive pairs and for a heat source temperature between 100 °C and 400 °C, with cold production temperature set at 10°C. The best performances were obtained with SrCl-BaCl 2 as working pair, the Coefficient Of Performance (COP: performance indicator for cold production) ranged from 0.55 to 0.78, and exergy efficiency (considering both power and cold production) ranged from 0.81 to 0.90. Jiang et al. [START_REF] Jiang | Experimental study on a resorption system for power and refrigeration cogeneration[END_REF] proposed both theoretical and experimental investigations for a system using MnCl 2 -CaCl 2 as reactive salts, and a heat source temperature between 120 °C and 180 °C. Moreover, they added a Phase Change Material tank to provide an energy storage function [START_REF] Jiang | Performance prediction on a resorption cogeneration cycle for power and refrigeration with energy storage[END_REF], and employed a scroll expander as it suits smaller power generation levels. Nevertheless, the highly dynamic behavior of the chemical reactions makes it hard to control the power output of this system. The experimental COP ranged from 0.27 to 0.37 and total exergy efficiency from 0.12 to 0.16. Lu et al. [START_REF] Lu | Analysis of an optimal resorption cogeneration using mass and heat recovery processes[END_REF] performed a thermodynamic analysis of a resorption cycle using mass and heat recovery processes in order to improve cycle performances. After setting refrigeration temperature at 10°C and heat source temperature at 100-300 °C, one of the best working pairs was MnCl 2 -BaCl 2 (COP 0.74-0.82). Finally, Jiang et al. [START_REF] Jiang | Analysis on innovative resorption cycle for power and refrigeration cogeneration[END_REF] recently demonstrated the huge potential of hybrid thermochemical cycles for low-grade heat utilization: they proposed a novel resorption cycle, integrating internal heat recovery to improve energy performances in comparison with existing resorption systems. Using heat source temperatures in the range 200-360°C and providing a refrigeration effect at 0 °C, this cycle gives a COP of 1.3 and exergy efficiencies between 0.41 and 0.74.

The second hybrid thermochemical cycle configuration, named single sorption cycle (case b. on Fig. 2), has been less investigated. Unlike the resorption cycle, a single sorption cycle involves only one solid/gas chemical reaction coupled with the liquid/vapor phase change of the same gas (ammonia). Consequently, the low-pressure vapor generator is an evaporator and the high-pressure vapor absorber is a condenser. This kind of configuration favors low-temperature cold production. One of the main analyses was completed by Bao et al., who first proposed a theoretical study [START_REF] Bao | Modelling of a chemisorption refrigeration and power cogeneration system[END_REF] then an experimental investigation using a scroll expander [START_REF] Bao | Chemisorption cooling and electric power cogeneration system driven by low grade heat[END_REF]. Their cycle (see Fig. 2) runs in separated mode, and the heat source temperature range is 150-200°C. The results again highlight the difficultly involved in coupling expander to chemical reactor: instead of obtaining the expected power output at 1000 W with a 0.01 kg/s mass flow rate of ammonia, power generation proved unstable and power output only reached a maximal value of 490 W.

Among hybrid thermochemical cycles, very few configurations have been investigated and they are mainly related to resorption cycles. The present work therefore investigates novel configurations that can provide significant improvements over the existing systems, such as easier technical implementation (especially in single sorption cycles) or the ability to store and/or increase power production. In terms of applications, the studied systems show a huge potential for cold production (cooling represents more than 70 % of produced useful effects).

Details on hybridizing a solid/gas sorption cycle with a power cycle

The thermodynamic analyses are performed for both resorption cycles (2 sorption reactors -see Fig. 2, case a.) and single sorption cycles (1 sorption reactor only -see Fig. 2, case b.). Ammonia is chosen as the working fluid (it is the reactive gas of the chemical reactions). The general layout of solid/gas sorption hybrid cycles, as presented on Fig. 2, defines the location of the heat input (Qin) and cold production (Qcold) in the cycle. In addition, expansion devices operation can define three working modes for the hybrid cycle, that are detailed in different sections of this paper:

 Separated power and cold mode: work output occurs in the charging step (the discharging step is isobaric) (see §4.).  Simultaneous power and cold mode: work output occurs in the discharging step (the charging step is isobaric) (see §5.).  Combined mode: work output occurs in both the charging and discharging steps (see §6.).

The following sections 4 to 6 present the thermodynamic analyses of these three modes (developed in a similar manner) and discuss their advantages and difficulties in practice. Both resorption and single sorption cycles are considered for each mode. These various modes and configurations are then compared in §7.

2.3.a. Separated power and cold production mode: description of the cycle and thermodynamic path

For the separated power and cold production mode, the pattern of the cycle (thermodynamic path of the working fluid) is plotted on a Clausius-Clapeyron diagram on Fig. 3. This diagram highlights the two distinct operating steps and their corresponding temperature and pressure levels, as well as input and output energies. The thermodynamic path is also plotted on the schematic T-s diagram of ammonia (Fig. 4) to give a more complete picture of the cycle, as Fig. 3 does not account for entropy variations of the working fluid. As described in §2.1, the hybrid sorption cycle uses two active materials: the LTM and the HTM. They are involved either in a solid/gas chemical reaction or in a liquid/gas phase change according to cycle configuration (either a resorption or a single sorption cycle). The corresponding thermodynamic equilibrium curves of these reactions or phase changes are plotted on both diagrams.

During the charging step (points 13 to 6), a heat source is needed to heat the reactor containing HTM to Thot (heat supply Qin). Opening valve V1 (see Fig. 2) between the vapor generator, expander and absorber brings HTM out of its thermodynamic equilibrium state (point 1), therefore triggering the endothermal decomposition reaction and generating a superheated ammonia vapor at high pressure Pdec,H (point 2). The vapor flow leaving the reactor is then expanded to a lower pressure (point 3), producing the mechanical work W. Expanded ammonia at pressure P4 finally flows towards the vapor absorber containing LTM, where the nonequilibrium temperature and pressure conditions allow the exothermal synthesis reaction (or condensation) to occur (points 4-5-6). Heat Qamb is rejected at ambient temperature level Tamb.

During the discharging step (points 7 to 12), the respective roles of vapor generator and absorber are reversed. Opening valve V2 between these two components imposes a pressure level P7 in both components, which brings LTM and HTM out of their thermodynamic equilibrium states: a refrigeration effect Qcold is provided by the endothermal decomposition reaction (or evaporation) of LTM (points 7-8-9), then the generated ammonia vapor flows towards the vapor absorber containing HTM, where the non-equilibrium temperature and pressure conditions enable the exothermal synthesis reaction to occur (points 10-11-12). Heat Qm is rejected at an intermediate temperature level Tm.

The transient step (switching from charging to discharging step) is achieved as follows: (i) vapor generator and absorber are disconnected, (ii) they are brought respectively to intermediate heat sink temperature Tm and cold source temperature Tcold and the temperature decreases result in pressure decreases, since LTM and HTM thermodynamic equilibria are monovariant, (iii) the components are connected again and the pressure equalization (at P9 = P10) brings them out of their thermodynamic equilibria, which enables carrying out the discharging step. The reverse transient step (switching from discharging to charging step) is achieved in a similar way.

Note that the T-s diagram (Fig. 4) serves to discriminate between several pairs of points with the same pressure and temperature but different specific entropies of the working fluid (points 4-5, 7-8, 11-12 and 13-1). On this diagram, the LTM and HTM equilibrium curves give the temperature and entropy of ammonia in absorbed (or saturated liquid) state (left) and desorbed (or saturated vapor) state (right). They are obtained using thermochemical data for the salt and suitable assumptions. Neveu et al. [START_REF] Neveu | Diagrammes thermodynamiques relatifs aux équilibres solide / gaz[END_REF] detailed a method to build these thermodynamic equilibrium curves of sorption reactions for ammonia salts in several thermodynamic diagrams.

Moreover, Fig. 3 and Fig. 4 feature several temperature pinches:

 Temperature pinches ΔTHX1 and ΔTHX2 for heat exchange with the heat sources or sinks.  Temperature deviations from thermodynamic equilibrium curves, ΔTr-eq and ΔTLV-eq.

These temperature pinches have to be considered in real processes. More details on these parameters are given in §3.2.a.

2.3.b. Simultaneous power and cold production mode: cycle and thermodynamic path

For the simultaneous power and cold production mode, as in §2.2.a., the pattern of the cycle is given on a Clausius-Clapeyron diagram in Fig. 5 and the thermodynamic path of the working fluid is also plotted on a T-s diagram on Fig. 6.

In contrast to the separated mode, the charging step (points 9 to 14) is isobaric and does not provide any power production. Heat Qin is supplied at the HTM reactor to achieve its endothermal decomposition reaction (points 9-10) and generate a superheated ammonia vapor (point 11); this vapor at pressure P11 directly flows towards the LTM vapor absorber where the exothermal synthesis reaction (or condensation) takes place (points 12-13-14). Like in the separated mode, heat Qamb is thus rejected at the ambient temperature Tamb.

During the discharging step (points 15 to 8), both useful effects (refrigeration effect and power generation) are produced. When the valve V1 between LTM vapor generator and HTM vapor absorber is opened, the in-component pressures change, which brings LTM and HTM out of their equilibrium states. This triggers the endothermal decomposition reaction (or evaporation) of LTM (points 15-1-2), providing a refrigeration effect Qcold and releasing ammonia vapor. However, instead of flowing towards the HTM vapor absorber, this vapor at pressure P2 is superheated before entering the expander (point 3) and providing mechanical work W (points 3-4). The expanded ammonia at Psynth,H (point 4) often reaches very low temperatures, which may serve for additional cold production (points 4-5). Finally, the exothermal synthesis reaction of HTM occurs (point 6) and absorbs this ammonia flow. Note that heat Qm released by this exothermal reaction is partially used to superheat the ammonia vapor upstream the expander (from Tamb to point 3) in order to reduce the liquid fraction at the expander outlet (point 4), with the surplus heat getting rejected at intermediate temperature Tm.

Compared with the separated mode, we would expect a lower cycle high pressure value (P11 = P12), because there is no expansion stage in the charging step. Since the expansion device is located in the discharging step, the low pressure of the cycle (P4 = P7 = Psynth,H) should reach low values. This low pressure may cause mass transfer limitations inside the porous reactive bed in this step, which the HTM reactor design has to prevent.

2.3.c. Combined power and cold production mode: cycle and thermodynamic path

Finally, the pattern of the cycle in combined power and cold production mode is given in a Clausius-Clapeyron diagram in Fig. 7, and Fig. 8 adds information on entropy variations within the cycle. This cycle combines the non-isobaric charging step of the separated mode (points 9 to 15) with the non-isobaric discharging step of the simultaneous mode (points 16 to 8), which enables increased total power production (for a given heat input Qin) to W = W1 + W2. On the other hand, the reactor design in this mode will have to address technical barriers regarding both the high pressure (P9 = P11 = Pdec,H) and low pressure (P4 = P7 = Psynth,H) of the cycle.

Thermodynamic study

3.1. Model

3.1.a. Thermodynamic and process assumptions

The assumptions for the thermodynamic study are defined below:  The system has reached a steady state (this assumption is needed for this energy-related study, despite the dynamic behavior of chemical reactions).  Kinetic and potential energy variations are neglected.  Heat losses are neglected in all components throughout the cycle.  Pressure drops are also neglected.

 The reference temperature used in exergy calculations is T 0 = Tamb.  For the expander, an isentropic efficiency, ηis, is introduced, according to the assumptions of Table 1.

 Different temperature pinches are defined in the process:

-For (liquid/liquid) or (liquid/vapor) heat exchange: ΔTHX1.

-For (vapor/vapor) heat exchange: ΔTHX2.

-For the deviation from thermodynamic equilibrium required for a chemical reaction or a liquid/vapor phase change to proceed, respectively ΔTr-eq and ΔTLV-eq. This thermodynamic equilibrium deviation has to be taken into account for chemical reactions, but it is not usually rate-limiting for a liquid/vapor phase change, thus ΔTLV-eq = 0 K.

3.1.b. Model equations

As illustrated in Fig. 3 to Fig. 8 and explained in §2.1., the thermodynamic path and other characteristics of hybrid cycle operation can be calculated once the two thermodynamic equilibrium curves are defined (straight lines in Fig. 3, Fig. 5 and Fig. 7; bell-shaped curves in Fig. 4, Fig. 6 and Fig. 8). Each of these thermodynamic equilibria is monovariant, and can be expressed as P = Π(T):

-For ammonia liquid-vapor equilibrium:

𝑙𝑛 ( 𝑃 𝑃 0 ) = - 𝐿 𝑣𝑎𝑝 𝑅.𝑇 + Δ𝑆 𝑣𝑎𝑝 𝑅 (1)
-For a chemical reaction equilibrium:

𝑙𝑛 ( 𝑃 𝑃 0 ) = - Δ 𝑟 𝐻 0 𝑅.𝑇 + Δ 𝑟 𝑆 0 𝑅 (2)
The thermodynamic states of the fluid at each point of the cycle are calculated according to the transformations described in §2.3. and Fig. 3 to Fig. 8. Once the thermodynamic path is determined, input and output energies are computed in order to evaluate the cycle performances. Three equations are needed: one for work output, another for heat input, and another for cold production. Each is detailed below for each of the three modes:

 Separated power and cold mode (see Fig. 3 and Fig. 4):

W = 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 2 -ℎ 3 ) (3)
 Simultaneous power and cold production (see Fig. 5 and Fig. 6):

W = 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 3 -ℎ 4 ) (6)
 Combined power and cold production (see Fig. 7 and Fig. 8):

W = 𝑚
are respectively the molar heat capacities of the High Temperature Material and Low Temperature Material at the end of the synthesis reaction. Moreover, 𝜂 𝑖𝑠 is linked to the cycle points, for example in separated power and cold mode (Fig. 3 and Fig. 4), by:

𝜂 𝑖𝑠 = (ℎ 3 -ℎ 2 ) (ℎ 3,𝑖𝑠 -ℎ 2 ) ⁄ (14)
For the chemical reactors, three parameters are introduced:

 The range of reaction advancement: ΔX = Xmax -Xmin, such that

𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 = 𝜈 𝑠𝑎𝑙𝑡 . Δ𝑋. 𝑛 𝑠𝑎𝑙𝑡 (15)
 The ratio of wall metal of the reactor to reactive composite volume, τmet (-), such that:

𝑉 𝑚𝑒𝑡 = 𝜏 𝑚𝑒𝑡 . 𝑉 𝑐𝑜𝑚𝑝 (16
) This parameter is related to the thermochemical reactor configuration, and is involved in the sensible heat terms of Eq. (4), (5a.), ( 7), (8a.), ( 10) and (11a.).

 The porosity ε of the composite bed implementing the reactive salt in reactors, which is set to 0.7.

This parameter is such that 𝑉 𝑝𝑜𝑟𝑜𝑢𝑠 = 𝜀. 𝑉 𝑐𝑜𝑚𝑝 and 𝑉 𝑠𝑎𝑙𝑡 = (1 -𝜀). 𝑉 𝑐𝑜𝑚𝑝 .

Finally, regarding cold production, Qcold, two expressions are given for each mode (respectively (5a.)-(5b.), (8a.)-(8b.) and (11a.)-(11b.)) in order to differentiate between resorption (a.) and single sorption (b.) cycles.

3.1.c. Sets of assumptions for 'perfect' and 'real' cases

For each cycle, two cases are simulated: 'perfect' case for favorable assumptions and 'real' case for unfavorable assumptions: the perfect case assumes ideal transformations and gives the maximal performances of the cycle (case usually considered in the previous studies -see §2.2.) while the real case sets realistic values for the parameters, based on experience with ORC and thermochemical machines. The two corresponding sets of values for the parameters are reported in Table 1.

Parameters

'Perfect' case 'Real' case

𝜼 𝒊𝒔 (-) 1 0.8 𝜟𝑿 (-) 1 0.8 𝝉 𝒎𝒆𝒕 (-) 0 0.1 𝜟𝑻 𝑯𝑿𝟏 (K) 0 5 𝜟𝑻 𝑯𝑿𝟐 (K) 0 10 𝜟𝑻 𝒓-𝒆𝒒 (K) 0 20
Table 1 -Values of the main parameters of the model (defined in §3.1.a. and §3.1.b.) for the two simulated cases (𝜀 and Δ𝑇 𝐿𝑉-𝑒𝑞 are assumed constant)

Operating conditions and technological boundary values

To study these thermodynamic processes, it is essential to define the main temperature levels in order to set the framework for the performance calculations:  Focusing on refrigeration, the cold production temperature is set at Tcold = 0°C.

 The ambient temperature (heat sink temperature for the release of condensation or synthesis heats) is set at Tamb = 20 °C.  Heat source temperature Thot is not fixed but instead results from calculations for a given cycle (see §3.3.). However, a boundary value is set at Thot,max = 250°C in order to fulfil the target of low-grade heat utilization.

Furthermore, some technological limit values are set for different variables:

 Ammonia pressure is bounded by Pmin = 0.1 bar and Pmax = 30 bar, for technical safety and cost criteria;  A minimal vapor quality xmin = 0.8 is required at the expander outlet, in order to avoid damaging it;  A maximal volumetric ratio Rv,max = 10 is assumed for expanders, and it complies with the nominal range of scroll expanders, which are particularly targeted and suitable for small facilities. Taking Fig. 3 and Fig. 4 as an example, the volumetric expansion ratio is defined by: 𝑅 𝑣 = 𝑣 3 𝑣 2 ⁄ (17)

Methodology

The calculation process described hereafter was performed using EES software [START_REF] Software | Engineering Software EES: Engineering Equation Solver[END_REF]. This thermodynamic study is based on energy calculations. For the sake of clarity, electrical production at the expander outlet is fixed at W = 1 kWh. Then, all other extensive quantities related to one cycle (Qin, Qcold, masses, volumes, …) are computed, and specific values can then be calculated by ratioing extensive values to the mass of cycled working fluid mNH3,cycled.

The purpose of this study is to explore the potential of the hybrid sorption cycles described in §2.3. for different solid reactive salts, and to emerge the most promising reactants. To this end, we use a database of thermochemical data (reaction enthalpy ΔrH 0 and entropy ΔrS 0 ) for 103 reactive ammonia salts. These salts are mainly metallic chlorides, bromides and iodides, such as CaCl 2 , MnCl 2 , FeBr 2 , or SrI 2 . The database comes from values collected and computed by Touzain [START_REF] Touzain | Thermodynamic values of ammonia-salts reactions for chemical sorption heat pumps[END_REF] and from CNRS-PROMES research. Other available data (molar mass, heat capacity and density of the salts) has been added. Note that at this stage, only a general thermodynamic study is required, so it does not integrate any external issues (such as cost, toxicity, corrosiveness or specific operating characteristics).

First, we set the LTM thermodynamic equilibrium: either a chemical reaction equilibrium (for resorption cycles) or the ammonia liquid-vapor equilibrium (for single sorption cycles). In the next sections, the LTM for resorption is BaCl 2 (8/0)NH 3 , as it enables cold production at target temperature Tcold = 0°C under a decomposition pressure (see Fig. 3, Fig. 5 and Fig. 7) of Pdec,L = 0.09 bar, which is close to the limit value Pmin = 0.1 bar, and there are no other candidate ammonia salts able to operate at a higher pressure for the same cold production. Note that implementing an expander downstream of the LTM decomposition reactor to generate power is unrealistic due to this low Pdec,L value. Hence the most appropriate configurations for separated and combined modes have to mobilize ammonia as LTM and its liquid/vapor thermodynamic equilibrium. This means that only the single sorption cycles are studied in §5. (separated mode) and §6. (combined mode).

Each of the 103 reactive salts is successively selected as HTM, and the thermodynamic path (i.e. pressure, temperature, specific enthalpy and entropy at each key point of the cycle) is computed. Then, the energy amounts Qin (see Eq. ( 4), ( 7) and (10)) and Qcold (see Eq. ( 5), ( 8) and (11)) are calculated, and relevant variables such as Rv (or Rv,1 and Rv,2, when several expansion stages are considered), mNH3,cycled (for W = 1 kWh, see Eq. (3), ( 6) and (9)), nNH3,cycled and Thot are deduced.

Then, in order to pick out the most relevant reactive salts from among the large set databased, five relevant performance criteria are defined and computed for each HTM:

 Energy efficiency 𝜂 𝐼 = 𝑊+𝑄 𝑐𝑜𝑙𝑑 𝑄 𝑖𝑛 (18)
 Exergy efficiency

𝜂 𝑒𝑥 = 𝑊+𝐸𝑥 𝑐𝑜𝑙𝑑 𝐸𝑥 𝑖𝑛 (19)
 Power production ratio

𝜏 𝑤 = 𝑊 𝑊+𝑄 𝑐𝑜𝑙𝑑 (20)
 Specific work output 𝑤 = 𝑊 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 (21)
 Specific exergy output

𝑤 + 𝑒𝑥 𝑐𝑜𝑙𝑑 = 𝑊+𝐸𝑥 𝑐𝑜𝑙𝑑 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 (22)
with

𝐸𝑥 𝑖𝑛 = 𝑄 𝑖𝑛 . (1 - 𝑇 0 𝑇 ℎ𝑜𝑡 ) (23)
and

𝐸𝑥 𝑐𝑜𝑙𝑑 = 𝑄 𝑐𝑜𝑙𝑑 . ( 𝑇 0 𝑇 𝑐𝑜𝑙𝑑 -1) (24)
Energy efficiency ηI (Eq. ( 18)) is the ratio of the total useful energy to input energy, where the useful effects are both work and cold production. This cogeneration efficiency is suitable for comparison with other cogeneration cycles working under the same temperatures (Tcold, Tamb, Thot).

Nevertheless, it is not possible to run a fair comparison between cogeneration cycles using only the firstlaw efficiency ηI. Therefore, τw (Eq. ( 20)) represents the part of mechanical work among the two useful effects of the cycle. In addition, in order to take better account of the part of mechanical work produced and the temperature required for the heat source, Thot, the exergy efficiency ηex (Eq. ( 19)) is computed. It accounts for the differences between work output and heat at several temperature levels, through the Carnot factors which depend on Thot (Eq. ( 23)) and Tcold (Eq. ( 24)).

Finally, the last two criteria, w (Eq. ( 21)) and w+excold (Eq. ( 22)), are introduced to provide a more quantitative assessment of both useful effects: work generation and useful exergy, respectively, are ratioed to mass unit of the working fluid.

Once the computations have been done for all reactive salts, a first selection is made and hybrid cycles that cannot meet limit values Rv,max, xmin, Pmin, Pmax or Thot,max defined in §3.2. are excluded. The remaining salts are ranked according to each of the previous 5 criteria. For each of these criteria, the 10 best salts are selected. At the end of this process, about 15 to 30 usable reactive salts are selected for each mode. This general approach thus covers all the possibilities offered by existing solid salts, once we have a fixed LTM. The following sections 4, 5 and 6 gather all performance results for the selected salts in the three operating modes, and for the 'real' set of parameters (see Table 1).

Separated power and cold production mode

As explained in §2.3.a. and depicted in Fig. 3 (Clausius-Clapeyron diagram) and Fig. 4 (T-s diagram), the separated power and cold production mode consists in generating power during the charging step, whereas the refrigeration effect is produced during the discharging step.

Following the methodology described in §3.3., once the LTM is set (ammonia or reactive salt BaCl 2 (8/0)NH 3 ), the 103 solid reactive salts in the database are individually tested with EES software, and the thermodynamic characteristics of the cycles, as well as their performances, are retrieved.

Fig. 9 presents cycle operating conditions (required heat source temperature and volumetric expansion ratio) for each of the salts selected in this mode, for both configurations (i.e. resorption, Fig. 9a., and single sorption, Fig. 9b.). Note that the selected salts are ranked, on the x-axis, by increasing equilibrium temperature at P = P 0 = 1 bar.

For resorption cycles (Fig. 9a.), the volumetric expansion ratio Rv remains constant at 4.7 for the first selected salts (from CaBr 2 (6/2)NH 3 to SrI 2 (6/2)NH 3 ), because the high pressure P1 = Pdec,H is bounded by the maximum value (Pdec,H = Pmax). On the other hand, for the last 6 salts, Rv decreases from 4.7 to 2.1 because the maximal heat source temperature Thot,max is reached, and the pressure P1 = Pdec,H consequently decreases. For all reactive salts, the low pressure of the charging step is set at P3 = Psynth,L = 4.8 bar, due to the constraint of rejecting synthesis heat at ambient temperature level Tamb.

For single sorption cycles (Fig. 9b.), the required heat source temperatures are lower than for resorption cycles, and the volumetric expansion ratio Rv is also frozen (at 2.5) because Pdec,H = Pmax. For all reactive salts, the low pressure of the charging step is set at P3 = Pcond = 10.0 bar, due to the constraint of rejecting condensation heat at ambient temperature level. Pressure Pmax is often reached due to the fact that Psynth,L and Pcond are already relatively high: the available pressure ratio to carry out an expansion stage between Pmax and Psynth,L (or Pcond) is low (between 3 and 6.25). Fig. 10 gathers the performance results (ηI, ηex) in separated power and cold mode for resorption (left side -Fig. 10a.) and single sorption (right side -Fig. 10b.) cycles. This plot highlights the higher range of performances for resorption compared to single sorption cycles: the ranges are respectively [0.37; 0.61] and [0.26; 0.49] for energy efficiencies, and [0.16; 0.28] and [0.13; 0.29] for exergy efficiencies.

Several reasons explain these differences: first, for a given cycled mass of ammonia, the refrigeration effect Qcold is greater for resorption than for single sorption, because the reaction enthalpy of a solid ammonia salt is far greater than the vaporization enthalpy of ammonia (ΔrH 0 ≈ 2.Lvap). Moreover, the specific work output w is higher in the resorption cycle, as pressure P3 = Psynth,L (minimal pressure required to reject heat of the exothermal LTM synthesis reaction at ambient sink temperature, see Fig. 3 and Fig. 4) is lower than Pcond (minimal pressure required to reject heat of the exothermal ammonia condensation). This is due to the respective positions of the ammonia liquid-vapor equilibrium curve and sorption equilibrium curve of the LTM used in resorption cycles, BaCl 2 (8/0). Exergy efficiencies follow a similar trend (with lower values for single sorption cycles) but the differences are lower because the heat source temperatures are higher for resorption cycles, as Fig. 9 shows: [212; 250°C] for resorption (Fig. 9a.) against [138; 250°C] for single sorption (Fig. 9b.).

In addition to Fig. 10, Table 2a. and Table 2b. give an overview of the differences between 'real' and 'perfect' efficiencies (corresponding to the sets of assumptions listed in Table 1): minimal, maximal and mean values are provided for the absolute deviations (ηI -ηI,perfect) and (ηex -ηex,perfect), as well as for the relative deviations (ηI -ηI,perfect)/ηI,perfect and (ηex -ηex,perfect)/ηex,perfect. These tables show a significant drop in energy efficiency (relative deviations reach -37.5% in resorption and -30.5% in single sorption) and even more in exergy efficiency (maximal relative deviations: -57.6% in resorption, -45.3% in single sorption).

From perfect to real case, various parameters are modified (see Table 1), of which the most influential for cycle performances are τmet and ΔTr-eq: the first one increases sensible heat in Qin (Eq. (4)) and Qcold (Eq. (5)), while the second one strongly modifies cycle temperatures, in particular temperature differences (T2 -T10) and (T6 -T9) which affect Qin and Qcold, respectively However, some other parameters related to each salt can also explain the performance gap when switching from perfect to real case. From the equations Eq. (3) to (5) (providing detailed expressions of the input and output energies), a development of the expressions of ηI (Eq. ( 18)) and ηex (Eq. ( 19)) shows that the factors driving the gap between 'perfect' and 'real' efficiencies are heat capacities 𝐶 ̅ 𝐻𝑇𝑀 and 𝐶 ̅ 𝐿𝑇𝑀 , reaction stoichiometry (νH, νL), reaction enthalpy (Δ 𝑟 𝐻 𝐻𝑇𝑀 0 , Δ 𝑟 𝐻 𝐿𝑇𝑀 0 ), density of reactive salts (ρHTM, ρLTM) and operating parameters ηis, ΔX, ΔTHX1, ΔTr-eq.

Alongside the dimensionless efficiencies of Fig. 10, useful effects (cold and power production) are detailed in Fig. 11 by means of two specific quantities: w and w + excold. Note that these values are ratioed to the total mass of ammonia involved in one complete cycle, mNH3,cycled (see Eq. ( 21) and ( 22)).

The overview of useful effects given by Fig. 11 supports the previous observations on performances: indeed, work output (green bars) and refrigeration output (included in blue bars) are much higher for resorption than for single sorption cycles. Specific work output w ranges from 161 to 296 kJ/kg NH3 for resorption, and from 139 to 190 kJ/kg NH3 for single sorption.

However, the power production ratio τw (red points and right axis) appears to be slightly lower for resorption ([7.2; 12.5 %]) than single sorption cycles ([10.9; 14.3 %]). This is one of the reasons for the low exergy efficiency differences between resorption and single sorption cycles.

Simultaneous power and cold production

This section addresses an innovative cycle configuration in which both cold and mechanical work are produced during the discharging step (step 2 on Fig. 2). In this case, the evaporator is located upstream of the expander, and it also produces the refrigeration effect by operating at a relatively low pressure (due to the LTM equilibrium). Hence the expander works at a lower pressure in this mode than in the previous mode. Only the single sorption cycle is studied in this section, for the reasons stated in §3.3.

The pattern of the hybrid cycle in simultaneous mode is plotted in thermodynamic diagrams in §2.3.b.

(Fig. 5 and Fig. 6). However, when volumetric expansion ratio between P1 = Pevap and P4 = Psynth,H is too high (i.e. Rv > Rv,max), we can implement two expansion stages: more details on this configuration are given in Appendix A.

As in §4.2, five performance criteria are used to pick out the most relevant reactive salts: ηI, ηex, w, τw and w+excold.

First, the operating conditions (heat source temperatures and volumetric expansion ratios) are plotted in Fig. 12. (similar to Fig. 9). The reactive salts flagged '*' require a configuration with two expanders to fulfil the boundary value Rv,max = 10, while the other salts have been implemented in single expander configurations. The range of heat source temperatures is wider than for separated power and cold mode: [107; 245°C]. Indeed, as the charging step is isobaric, there is no need to aim for higher HTM decomposition pressure and temperature. Furthermore, starting from the salt FeCl 2 (6/2)NH 3 , volumetric expansion ratios are frozen at Rv,1 ≈ Rv,2 ≈ 4.3, because the pressure P4 = Psynth,H has reached Pmin. Fig. 13 gathers the performances (ηI, ηex) for the simultaneous power and cold production mode. As in Fig. 10, only real cases are plotted, and Table 3 is added to give an overview of the performance differences between perfect and real cases.

The energy efficiency ηI is of the same order of magnitude as for the separated mode and single sorption (Fig. 10b.), and ranges from 0.25 to 0.48. This is consistent with the fact that both results relate to single sorption cycles: for both cycles, the refrigeration effect Qcold is mainly due to the vaporization enthalpy of ammonia Lvap.

Regarding exergy aspects, the performances are better for the simultaneous power and cold mode than for the separated mode: as Fig. 13 shows, ηex ranges from 0.18 to 0.34. This results from the high values of specific work output w , at a maximum of 399 kJ/kg NH3 , and power production ratio τw , at a maximum of 25% (Fig. 14), both of which are much higher values than for the separated mode (see §4.). This mode allows a higher power output because it uses a higher pressure difference at the expander boundaries, i.e. between P3 = Pevap and P4 = Psynth,H in Fig. 5. Indeed, the evaporation pressure required to produce cold at Tcold = 0°C is approximately 3.5 bar, while the minimal achievable pressure is set at Pmin = 0.1 bar. Fig. 14 -Simultaneous power and cold mode: exergy densities and power production ratio

Combined mode

Combining the non-isobaric steps of the two previous modes (separated power and cold production mode and simultaneous mode) leads to the cycle presented in Fig. 7 and Fig. 8, in which both the charging and discharging steps produce power. Again, for reasons set out in §3.3., only the single sorption cycle is studied here; a configuration using 2 expansion stages is also considered for some reactive salts (see Appendix A for more details on this configuration).

Firstly, Fig. 15 shows that the range of usable heat source temperatures is the same as for separated power and cold mode in single sorption ([138; 250°C]) but narrower than for simultaneous mode ([107; 245°C]). This is due to the use of the same non-isobaric charging step as in separated mode.

Regarding the useful effects of the cycle, Fig. 17 shows that τw ranges from 14.6 to 30.1%, which is higher than all previous cycles. Inserting an expander in both operating steps obviously produces more power, which is highlighted by the high values of specific work output w (from 197 to 547 kJ/kg NH3 ).

Consequently, the exergy efficiencies are higher in combined power and cold mode than all other cycles: ηex ranges from 0.20 to 0.41 (see Fig. 16). These high values reflect a higher exergy content in the outputs (produced useful effects).

The energy efficiency ηI ranges from 0.26 to 0.51: these are the best values for a single sorption cycle.

Only the resorption cycle exceeds these values because of the higher cold production (see Fig. 10a., §4.). We ran a sensitivity analysis on the 'real' case results taken as nominal case. Table 1 lists the nominal values of the main parameters, and the variation ranges were chosen in such a way as to keep the extreme values of the parameters realistic. Moreover, the sensitivity study was carried out on combined power and cold mode as it emerged as the most relevant and general mode. The six parameters listed below seem particularly relevant for this study as potential control parameters:

 The following three parameters affect the expansion stages, and the specific work output w:

-the technological boundary value Pmin is varied from 0.1 to 1 bar, which may limit the expansion in discharging step.

-the limit value Pmax is varied between 30 and 100 bar, which defines an upper limit for the expansion in charging step.

-the isentropic efficiency ηis (see Eq. ( 14)) is varied from 0.7 to 0.9.  The metal ratio τmet (see Eq. ( 16)) is varied from 0 to 0.3. This parameter mainly affects efficiencies ηI and ηex, through the sensible heat terms in Qin and Qcold.

 The following two parameters affect temperatures and pressures of the points of the cycle:

-ΔTr-eq (i.e. the temperature deviation from the chemical reaction equilibrium) is varied between 10 and 40 K, which changes the thermodynamic states of points 3 to 12 on Fig. 7 and Fig. 8, and thus affects specific work output w and exergy. -Tamb is varied from 15 °C to 25°C, which also modifies the thermodynamic path of the cycle.

7.1.b. Results

The results are presented on Fig. 18 for 3 representative salts: CaCl 2 (8/4)NH 3 , SrCl 2 (8/1)NH 3 and SrBr 2 (8/2)NH 3 . For each salt and each of the 6 parameters, we evaluated the sensitivity of two performance criteria: energy efficiency ηI (red lines) and specific exergy output w+excold (green dotted lines). The rationale for choosing these criteria is that they give a good qualitative and quantitative overview of the cycle performances.

Sensitivities are plotted as relative deviations with respect to nominal value of the variable: Δ(ηI)/ηI,nom and Δ(w+excold)/(w+excold)nom. Symbols (respectively red filled and green empty symbols) flag extreme deviations.

From this graphical representation, several observations can be made:  Most of the results fall between -10 and +10% (especially for parameters τmet, ηis and Tamb).  Concerning the technological boundary value Pmax, the specific exergy output w+excold is very sensitive to this parameter with all three salts, due to the ceiling effect it has on the thermodynamic path of the cycle: the potential improvement ranges from 26.2 % to 29.7 % according to the salt. 15.8%] for w+excold. This is due to the fact that it changes the thermodynamic path. There is huge potential for improving this parameter if heat transfer is correctly managed. However, a dynamic study is needed for this point.

Finally, note that ηI (red lines and symbols) is less sensitive to the studied parameters than w+excold, with relative deviations ranging from -13.2 % to 9.3 %. ηI is an energy ratio while w+excold is an exergy-related criterion, and the chosen parameters have a more direct influence on operating temperatures and pressures than energy content.

Fig. 18 -Sensitivity analysis for combined power and cold production mode.

Symbols are the extremal relative deviations of ηI and w+excold with respect to their nominal values.

Summary of the results

The main performance criteria for all selected salts from sections 4, 5 and 6 are collected as a function of the cycle regeneration temperature Thot on Figs. [START_REF] Bao | A chemisorption power generation cycle with multi-stage expansion driven by low grade heat[END_REF] to 21. These plots serve to discriminate between the three studied modes for achieving effective cogeneration of power and cold for a given heat source:

 The resorption cycle in separated power and cold mode (green triangles) cannot be run with a heat source temperature under 212°C. It provides higher energy efficiency than the other cycles ([0.34; 0.61], see Fig. 19) due to its high cold production, but its exergy efficiency is lower than the simultaneous and combined modes ([0.13; 0.30], see Fig. 20) due to its low power production ratio. Finally, in relation to the high cold production but low power production ratio, the specific exergy output w+excold (see Fig. 21) lies between the single sorption cycle in separated mode and the two other modes.

 The single sorption cycle in separated mode (blue triangles) is suitable for using a lower-temperature heat source, as it requires a heat source temperature higher than 138°C. However, despite a comparable power output, it provides a lower cold output than the resorption cycle, thus underperforms on energy efficiency ([0.24; 0.49], see Fig. 19) and exergy efficiency ([0.06; 0.29], see Fig. 20).

 The simultaneous power and cold mode (black squares) is relevant for the low-temperature sources, as the minimal required regeneration temperature Thot is only 107°C. Its energy efficiency is enormously dependent on the reactive salt used, and lies within the range [0.25; 0.54] (see Fig. 19), i.e. lower values than for the resorption cycle in separated mode (due to the higher cold production of resorption cycle). Its higher power production ratio means that exergy efficiency and specific exergy output are higher than for the two previous cycles (exergy efficiency range is [0.18; 0.34], see Fig. 20).

 The charging step of the combined mode (red circles) is the same as for the single sorption cycle in separated mode, so they share the same required heat source temperature. The energy efficiency ([0.26; 0.51], see Fig. 19) is higher than for simultaneous mode and for the single sorption cycle in separated mode, but remains lower than for the resorption cycle in separated mode which offers higher cold production; furthermore, driven by the higher exergy content of the outputs (higher power production ratio), this cycle presents the highest exergy efficiency ([0.20; 0.40], see Fig. 20) and specific exergy output ([281; 640 kJ/kg NH3 ], see Fig. 21).

At 30 °C to 400 °C heat source temperatures, energy efficiencies of existing hybrid thermochemical cycles (detailed in §2.2.) lie in the range [0.06; 0.62], while exergy efficiencies are in the range [0.12; 0.90]. Here, heat source temperatures ranged from 107 °C to 250 °C, energy efficiencies from 0.24 to 0.61 and exergy efficiencies from 0.06 to 0.40. However, we cannot offer a fair comparison, as different studies have used different sets of assumptions and operating conditions (cold production temperatures, heat sink temperatures, etc.).

Conclusions

This paper investigated three ways of hybridizing the solid/gas sorption cycles with the power cycle, leading to three operating modes for power and cold cogeneration with a storage function of either cold or cold and power. The potential of these three modes was explored using an innovative methodology to select the most promising reactive salts for each mode.

The energy efficiencies of the proposed hybrid cycles ranged from 0.24 to 0.61, the highest values being reached for the resorption cycle in separated mode. Exergy efficiencies ranged from 0.06 to 0.40, and the best values were obtained in combined mode, due to its high power production ratios. Note that the innovative simultaneous and combined modes emerge as promising developments for applications where a power storage is required. In comparison with the separated mode, they offer particularly high power production ratios (the maximal value is 30.1%) and exergy densities (the maximal value is 640 kJ/kg NH3 ).

Note too that these hybrid thermochemical cycles can also focus on energy storage and power production. In such case, during the discharging step, the temperature of the LTM vapor generator is no longer set at Tcold and so an auto-thermal system can be performed in this step (see Fig. 2) by recovering the surplus heat of the exothermal vapor absorption (Qm) to achieve the endothermal vapor generation upstream of the expander. This leads to several other configurations that have also been the subject of a thermodynamic study, with results that will be presented in a separate paper.

Building on this first study, the next step is to analyze the dynamic behavior of the cycles, where the coupling between the chemical reactor and the expander appears to pose a number of challenges: indeed, antagonistic purposes have to be achieved in order to simultaneously improve both chemical reaction kinetics and power generation at the expander. For example, on Fig. 5 and Fig. 7 (in the discharging step):

 Improving chemical reaction kinetics for the HTM synthesis reaction entails increasing the temperature gap between ammonia vapor coming from the expander (point 6) and the thermodynamic equilibrium of the reaction (points 7-8). If temperature Tm is fixed, this must be achieved by increasing the low pressure P4 = Psynth,H.

 Improving power generation entails increasing the pressure difference between the inlet (point 3) and outlet of the expander (point 4). Since pressure P3 is fixed, this must be achieved by reducing pressure P4 = Psynth,H.

For the dynamic study, discussion should revolve around choosing one solid reactant, and the criteria considered should go beyond performances to encompass factors such as the toxicity, corrosiveness and/or cost of the salts. Finally, an experimental setup will be required to validate the theoretical predictions and assess the feasibility of such hybrid thermochemical cycles. Monitoring the changes in pressure and mass flow rate conditions during operation of this machine will be of great importance to validate the above-mentioned dynamic study of the coupling between expander and chemical reactor. The design of a prototype is in progress, and a scroll expander is already available and operable with ammonia and under the predicted temperature and pressure conditions.

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Liquid/gas absorption hybrid cycle for power and cold cogeneration: operating principle. Power and cold production in series (valve V1 open, V2 and V3 closed) or in parallel (V1 closed, V2 and V3 open)

Fig. 3 -

 3 Fig. 3 -Separated power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in a Clausius-Clapeyron diagram.

Fig. 4 -

 4 Fig. 4 -Separated power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in the T-s diagram of ammonia (see Fig. 3 for captions).

Fig. 5 -

 5 Fig. 5 -Simultaneous power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in a Clausius-Clapeyron diagram (see Fig. 3 for captions).

Fig. 6 -

 6 Fig. 6 -Simultaneous power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in the T-s diagram of ammonia (see Fig. 3 for captions).

Fig. 7 -

 7 Fig. 7 -Combined power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in a Clausius-Clapeyron diagram (see Fig. 3 for captions).

Fig. 8 -

 8 Fig. 8 -Combined power and cold production mode of a hybrid solid/gas chemical reaction cycle: thermodynamic path in the T-s diagram of ammonia (see Fig. 3 for captions).

Fig. 9 -Fig. 10 -

 910 Fig. 9 -Separated power and cold mode: operating conditions of the hybrid cycle for selected salts

Fig. 11 -

 11 Fig. 11 -Separated power and cold mode: exergy densities and power production ratio

Fig. 12 -Fig. 13 -

 1213 Fig. 12 -Simultaneous power and cold mode: operating conditions of the hybrid cycle for selected salts *: 2-expansions configuration

Fig. 15 -

 15 Fig. 15 -Combined power and cold mode: operating conditions of the hybrid cycle for selected salts

Fig. 17 - 7 . Discussion of the results 7 . 1 .

 17771 Fig. 17 -Combined power and cold mode: exergy densities and power production ratio 7. Discussion of the results 7.1. Sensitivity of the results to key parameters 7.1.a. Methodology



  Pmin can prove influential depending on the reactive salt: in this example, only the SrBr 2 (8/2)NH 3 cycle has lower performances when Pmin is set at higher values than the nominal value 0.1 bar (-21.6 % for w+excold). The reason is that the SrBr 2 (8/2)NH 3 nominal low pressure is P4 = Psynth,H = 0.3 bar, which is close to the limit value, whereas Psynth,H values for CaCl 2 (8/4)NH 3 and SrCl 2 (8/1)NH 3 , are 1.9 and 1.4 bar, respectively, i.e. far from Pmin.  The most influential operating parameter is ΔTr-eq, as it leads to a relative variation range of[-26.1;

Fig. 19 -

 19 Fig. 19 -Energy efficiencies: summary of all selected salts in the 3 cogeneration modes

Figure 22 -

 22 Figure 22 -Overview of the strengths and weaknesses of each mode

Fig. B2 -

 B2 Fig. B2 -Simultaneous power and cold mode: energy and exergy efficiencies of the cycle for selected salts (both 'perfect' and 'real' cases are plotted)

  Q 𝑖𝑛 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 1 , 𝑃 1 ) + [𝑉 𝑚𝑒𝑡,𝐻 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐻𝑇𝑀 . 𝐶 ̅ 𝐻𝑇𝑀 ]. (𝑇 2 -𝑇 10 ) (4) Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 8 , 𝑃 8 ) -[𝑉 𝑚𝑒𝑡,𝐿 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐿𝑇𝑀 . 𝐶 ̅ 𝐿𝑇𝑀 ]. (𝑇 6 -𝑇 9 ) (5a) or Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝐿 𝑣𝑎𝑝 (𝑇 8 ) -𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 5 -ℎ 7 )

  Q 𝑖𝑛 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 10 , 𝑃 10 ) + [𝑉 𝑚𝑒𝑡,𝐻 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐻𝑇𝑀 . 𝐶 ̅ 𝐻𝑇𝑀 ]. (𝑇 11 -𝑇 6 ) (7) Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 1 , 𝑃 1 ) + 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 5 -ℎ 4 ) -[𝑉 𝑚𝑒𝑡,𝐿 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐿𝑇𝑀 . 𝐶 ̅

		𝐿𝑇𝑀 ]. (𝑇 14 -𝑇 2 )	(8a)
	or	Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝐿 𝑣𝑎𝑝 (𝑇 1 ) + 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 5 -ℎ 4 ) -𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 13 -ℎ 15 )

  𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 10 , 𝑃 10 ) + [𝑉 𝑚𝑒𝑡,𝐻 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐻𝑇𝑀 . 𝐶 ̅

			𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 3 -ℎ 4 ) + 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 11 -ℎ 12 )	(9)
		Q 𝑖𝑛 = 𝐻𝑇𝑀 ]. (𝑇 11 -𝑇 6 )	(10)
		Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝛥 𝑟 𝐻(𝑇 1 , 𝑃 1 ) + 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 5 -ℎ 4 )
			-[𝑉 𝑚𝑒𝑡,𝐿 . 𝜌 𝑚𝑒𝑡 . 𝑐 𝑚𝑒𝑡 + 𝑛 𝐿𝑇𝑀 . 𝐶 ̅ 𝐿𝑇𝑀 ]. (𝑇 15 -𝑇 2 )	(11a.)
		or	Q 𝑐𝑜𝑙𝑑 = 𝑛 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . 𝐿 𝑣𝑎𝑝 (𝑇 1 ) + 𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 5 -ℎ 4 ) -𝑚 𝑁𝐻3,𝑐𝑦𝑐𝑙𝑒𝑑 . (ℎ 14 -ℎ 16 )	(11b.)
	where
	and	𝐶 ̅ 𝐻𝑇𝑀 = 𝑋 𝑚𝑎𝑥 . 𝐶 𝐻𝑇𝑀,𝑟 + (1 -𝑋 𝑚𝑎𝑥 ). 𝐶 𝐻𝑇𝑀,𝑝 𝐶 ̅ 𝐿𝑇𝑀 = 𝑋 𝑚𝑎𝑥 . 𝐶 𝐿𝑇𝑀,𝑟 + (1 -𝑋 𝑚𝑎𝑥 ). 𝐶 𝐿𝑇𝑀,𝑝	(12)

Table 2a

 2a Single sorption Mean value Max. value Min. value SrCl 2 (8/1)NH 3

		Absolute deviation	-0.09	-0.17	-0.06	-0.07
	𝜼 𝑰	Relative deviation	-20.2%	-30.5%	-13.1%	-13.1%
		Absolute deviation	-0.12	-0.17	-0.08	-0.15
	𝜼 𝒆𝒙	Relative deviation	-37.9%	-45.3%	-31.4%	-34.7%
		Absolute deviation	-0.19	-0.27	-0.15	-0.16
	𝜼 𝑰	Relative deviation	-28.4 %	-37.5 %	-20.5 %	-20.6 %
		Absolute deviation	-0.19	-0.23	-0.17	-0.18
	𝜼 𝒆𝒙	Relative deviation	-47.6 %	-57.6 %	-39.2 %	-39.2 %

-Separated power and cold mode (resorption): comparison between 'perfect' and 'real' efficiencies.

-Absolute deviations: ηI -ηI,perfect and ηex -ηex,perfect.

-Relative deviations: (ηI -ηI,perfect)/ηI,perfect and (ηex -ηex,perfect)/ηex,perfect.

(b)

Table 2b -

 2b

Separated power and cold mode (single sorption): comparison between 'perfect' and 'real' efficiencies

Table 3 -

 3 [START_REF] Bao | Chemisorption power generation driven by low grade heat -Theoretical analysis and comparison with pumpless ORC[END_REF] Simultaneous power and cold mode: comparison between 'perfect' and 'real' efficiencies

		Relative deviation	-28.4 %	-38.8 %	-20.0 %	-27.4 %
		Absolute deviation	-0.30	-0.55	-0.14	-0.51
	𝜼 𝒆𝒙	Relative deviation	-52.5 %	-72.2 %	-38.3 %	-65.7 %

Table 4 -

 4 Combined power and cold mode: comparison between 'perfect' and 'real' efficiencies

		Absolute deviation	-0.17	-0.27	-0.12	-0.20
	𝜼 𝑰	Relative deviation	-29.7%	-39.4%	-21.3%	-27.7%
		Absolute deviation	-0.28	-0.46	-0.18	-0.44
	𝜼 𝒆𝒙	Relative deviation	-47.5%	-58.9%	-36.5%	-53.4%

Acknowledgements

The authors thank Pierre Neveu for his invaluable support in thermodynamics. Alexis Godefroy was supported by a PhD grant from the French Ministry of National Education (doctoral contract #2017-09-ED.305).

 Combined power and cold production with two expansion stages (see Fig. A3 and Fig. A4):

Appendix B. Additional results

On Fig. B1 to B3, energy and exergy efficiencies using both 'perfect' (white symbols) and 'real' (solid symbols) sets of assumptions (see Table 1) are plotted for each of the three modes.

Finally, similar to §7.2., the main results are summarized in Fig. B4 to B7. Instead of the regeneration temperature Thot, the x-axis contains all selected salts for the three modes. These plots chart both one-expansion (white symbols) and two-expansions (solid symbols) configurations. Note that regeneration temperatures Thot are plotted on Fig. B4.

Note that the configurations using only one expansion stage have slightly higher energy efficiencies but slightly lower exergy efficiencies than the configurations involving two expansion stages. These small differences are explained by the fact that the cold effect provided at the outlet of the expander(s) (transformation 4-5 on Fig. 5 and Fig. 7, transformations 4-5 and 7-8 on Fig. A1 and Fig. A3) is greater with only one expansion than in two-expansions configurations. On the other hand, two-expansion configurations deliver a greater power output than one-expansion configurations, and hence have higher exergy efficiencies.