SECOND LAW ANALYSIS OF A HYBRID
THERMOCHEMICAL CYCLE FOR COLD AND
ELECTRICITY COGENERATION

Hasan Ghazale, Pierre Neveu, Nathalie Mazet, Maxime Perier-Muzet

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Hasan Ghazale, Pierre Neveu, Nathalie Mazet, Maxime Perier-Muzet. SECOND LAW ANALYSIS OF A HYBRID THERMOCHEMICAL CYCLE FOR COLD AND ELECTRICITY COGENERATION. 38th INTERNATIONAL CONFERENCE ON EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS, Jun 2025, Paris, France. hal-05144240

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Distributed under a Creative Commons Attribution 4.0 International License
SECOND LAW ANALYSIS OF A HYBRID THERMOCHEMICAL CYCLE FOR COLD AND ELECTRICITY COGENERATION

Hasan Ghazale<fnref n="1" />, Pierre Neveu<fnref n="1" />,<fnref n="2" />, Nathalie Mazet<fnref n="1" />, Maxime Perier-Muzet<fnref n="1" />,<fnref n="2" />

ABSTRACT

A hybrid thermochemical process aimed at valorizing heat waste from industrial processes has been designed and tested. Following the experimental campaign, this work presents numerical modeling of the cycle to clarify the dynamic behavior of the entire process. Moreover, a comprehensive second Law analysis is undertaken, quantifying the exergy destruction related to each component for further optimization of the overall process. Results highlight the significant role of heat transfer in the reactor, expansion process, and leaks in the scroll expander on the exergy performance of the system.

1 INTRODUCTION

Energy policies include important issues such as the use of low-temperature heat sources (renewable, industrial waste heat) and the management of variabilities in both sources and demands. Hybrid thermochemical processes combining energy multigeneration and storage functions could be an attractive solution to address this issue (Chan et al., 2013). Such thermochemical cycles are based on a solid-gas reversible chemical reaction, involving uncharged solid S₀, charged solid S₁ and the reactive gaz G:

$$S_0 + vG \rightleftharpoons S_1 \quad (1)$$

where v is the stoichiometric coefficient. In the forward direction, known as the synthesis reaction, heat is released from the reactive bed whereas in the forward direction, known as decomposition reaction, heat is supplied to the reactive medium. Thermochemical cycles have been extensively studied for refrigeration, heat pumps or heat transformer applications (Wongsuwan et al., 2001). These thermally driven processes involve a solid-gas reactor and an evapo-condenser, or two different solid-gas reactors (resorption cycle) exchanging a gas flow. Hybridization of such systems consists in integrating an expander between these two components for valorizing the gas flow into mechanical power. Although numerous architectures have been proposed and studied numerically (Thiele et al., 2020; Wang et al., 2013), only a few experiments are reported in literature and have revealed malfunctions or poor performances (Bao et al., 2014), (Jian et al., 2015). Ghazale et al. (2024) have recently tested a prototype (Figure 1) based on the following reactive pair:

$$MnCl_2, 2NH_3 + 4NH_3 \rightleftharpoons MnCl_2, 6NH_3 \quad (2)$$

Driven by a low-grade heat source (150 to 180°C), the process produces cold and electrical energy while providing a storage function. The expander was a commercial oil-free scroll expander with a nominal power of 1 kW. The other components (reactor, etc.) were sized accordingly. Except for the reactor, specifically designed for this prototype, all other components were commercial products (expansion valve, plate heat exchangers, reactor tubes and NH₃ tank, piping.). The reagent (MnCl₂) was

<footnote n="1">¹ CNRS-PROMES. Laboratoire PROcédés, Matériaux et Energie Solaire, Tecnosud, Rambla de la Thermodynamique – 66100 Perpignan</footnote>

<footnote n="2">² UPVD Université de Perpignan Via Domitia, 52 Avenue Paul Alduy – 66100 Perpignan</footnote>

<footnote>*Corresponding Author: maxime.perier-muzet@univ-perp.fr</footnote>
implemented with a conductive graphite binder. The reactor contained 30 kg of solid reagent, cycling
23 kg of NH3.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Schematic description of the prototype.</figcaption></figure>

The experiment consists of two main stages, called charging and discharging stages, separated by a stand-by period (storage stage). The cycle can be described in Clapeyron and entropic diagrams (Figure 2). During the charging phase, heat is provided to the closed reactor, initially at ambient temperature (point 0). Its temperature and pressure rise, following the equilibrium line and reaches the condensing pressure (1). The reactor is then connected to the condenser. The desorption reaction (1→2) generates superheated vapour (2) that is condensed in the condenser (3). The liquid is stored in the ammonia tanks at ambient temperature. This phase ends when approx. 90% of S₁ (i.e. *MnCl*₂, *6NH*₃) is converted in S₀ (*MnCl*₂, *2NH*₃). Heating is then stopped, and reactor is closed. Its temperature then decreases to the ambient temperature.

During the storage phase, NH₃ liquid and regenerated solid S₀ are stored separately at ambient
temperature.

During the discharging (or production) phase, evaporator is connected to the liquid tanks (3) through the throttling valve. NH₃ enters in the evaporator as a liquid/gas mixture (4) that vaporizes (4→5), producing cold. The vapour is then superheated (6) and expands in the expander (7) producing mechanical work. The exhaust vapour is then absorbed in the reactor (8→9).

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Hybrid thermochemical cycle for simultaneous production of cooling and electricity. Thermodynamic paths of NH3 in (a) Clapeyron diagram (-1/T,P), (b) Entropic diagram (s,T).</figcaption></figure>

This study established the experimental proof-of-concept of a hybrid thermochemical cycle for heat
storage and conversion into cooling and electricity, demonstrating continuous mechanical and cooling
production during the synthesis phase. The dynamic behavior and performance of the prototype were
analyzed under nominal operating conditions (Ghazale et al., 2025). However, experimental
performances show a low production of mechanical work compared to cold production: the mechanical energy produced was 68.3 kJ for a cooling capacity around 15.2 MJ. The main limitations of the prototype were identified as being i) internal leakage in the expander, ii) poor heat transfer at reactor level, both playing a major role in the couplings between components and therefore affect the cycle performances. Further investigations are then required to refine the design of future prototypes. To assess the possible improvements, the aim of this work is to assess the impact of every irreversible process occurring throughout the system on exergy efficiency. That implies two steps:

1) Building a reliable dynamic model of the process that can reproduce the experimental data

2) Quantifying the exergy destructions related to every component.

Section 2 is devoted to the dynamic model based on previous works available at PROMES lab. Section 3 presents the methodology developed to evaluate the entropy production and thence, to perform an exergy analysis. Results will be then discussed in section 4.

## 2 DYNAMIC MODELLING

The dynamic model developed to simulate dynamic behavior and assess the performances of the process is based on matter and energy balances, kinetic equations and convective heat transfer equations. The model was developed by Godefroy (Godefroy, 2020), and adapted by Ghazale (Ghazale, 2024) to fit with the experimental data. Evaporator, condenser, NH₃ tanks and reactor are modelled through transient balance, whereas quasi-static regime is assumed for expander, throttling valve, pipes and ducts. To illustrate the method, only reactor and expander models are detailed below. One can refer to the original thesis and the upcoming paper (Godefroy et al., 2025) for more details.

### 2.1 Thermochemical reactor

The reactor model comprises two control volumes (Figure 3):

* The porous reactive mixture V<sub>r</sub>, at uniform but time-dependent temperature T<sub>r</sub>, pressure P<sub>r</sub> and composition X, including solid S<sub>0</sub>, solid S<sub>1</sub>, graphite binder and reactive gaz G,

* The reactor wall V<sub>w</sub> including heat exchanger wall, heat transfer fluid (HTF) and reactor shell at temperature T<sub>w</sub>

The dynamic modelling is based on mass and energy balances and a kinetics model for the solid/gas reaction.

2.1.1 Matter balances: Molar balances are written for the three reactive components,

$$ \text{Solid S}_1 (MnCl_2, 6NH_3): \quad \frac{dn_1}{dt} = n_s \frac{dX}{dt} \quad (3) $$

$$ \text{Solid S}_0 (MnCl_2, 2NH_3): \quad \frac{dn_0}{dt} = -n_s \frac{dX}{dt} \quad (4) $$

$$ \text{Reactive gaz G (NH}_3): \quad \frac{dn_g}{dt} = -v \cdot n_s \frac{dX}{dt} + \dot{n}_g^e \quad (5) $$

With n<sub>0</sub>, n<sub>1</sub> and n<sub>g</sub>: molar quantities of solids S<sub>0</sub>, S<sub>1</sub> and gas G (mol), dX/dt: reaction rate (s<sup>-1</sup>), v: stoichiometric coefficient (v = 4 from eq. (2)) and ḍ<sub>g</sub><sup>e</sup>: gas molar flow exchanged with the surrounding, negative for charging phase ( ḍ<sub>g</sub><sup>e</sup> = - ḍ<sub>g</sub><sup>out</sup> < 0), and positive for discharging phase ( ḍ<sub>g</sub><sup>e</sup> = ḍ<sub>g</sub><sup>in</sup> > 0). The reaction rate is computed by the following kinetics equation:

$$ \frac{dX}{dt} = f(X) \frac{P_r - P_{eq}(T_r)}{P_r} \quad (6) $$

where f(X) = K<sub>cin,dec</sub>·X for the decomposition reaction (charging stage), and f(X) = K<sub>cin,syn</sub>·(1 - X) for the synthesis reaction (discharging stage). Kinetics coefficients K<sub>cin,X</sub> were fit from experiments.
$$ \dot{W} + \dot{n}_{eff} \cdot (\bar{h}^{in} - \bar{h}_{eff}^{out}) \approx 0 \quad (12) $$

The effective molar flow rate ($\dot{n}_{eff}$) is calculated from the rotational speed and the swept volume of the expander $\dot{n}_{eff} = \frac{\omega \cdot V_{swept}}{\bar{v}^{in}}$ and the effective molar enthalpy of the fluid at the outlet ($\bar{h}_{eff}^{out}$) is calculated using the isentropic efficiency $\eta_{is} = \frac{\bar{h}^{in} - \bar{h}_{eff}^{out}}{\bar{h}^{in} - \bar{h}_{is,eff}^{out}}$.

The leak molar flow rate $\dot{n}_{leak}$ is computed assuming a constant pressure loss coefficient $k_{leak}$:

$$ \dot{n}_{leak} = \frac{k_{leak}}{\bar{v}^{in}} \cdot (P^{in} - P^{out})^{0.5} $$

An isenthalpic expansion is considered: $\bar{h}_{leak}^{out} = \bar{h}^{in}$.

$$ 0 = \dot{n}_{eff} + \dot{n}_{leak} - \dot{n}^{out} \quad (13) $$

$$ 0 = \dot{n}_{eff} \cdot \bar{h}_{eff}^{out} + \dot{n}_{leak} \cdot \bar{h}_{leak}^{out} - \dot{n}^{out} \bar{h}^{out} \quad (14) $$

and permit to compute the total molar flow rate and the outlet specific enthalpy

$$ \dot{n}^{out} = \dot{n}_{eff} + \dot{n}_{leak} \quad (15) $$

$$ \bar{h}^{out} = (\dot{n}_{eff} \cdot \bar{h}_{eff}^{out} + \dot{n}_{leak} \cdot \bar{h}_{leak}^{out}) / \dot{n}^{out} \quad (16) $$

The ammonia outlet state (temperature and vapor quality) of the swept flow, leak flow and total flow are calculated from the outlet specific enthalpies and pressure. The volumetric efficiency, ratio between effective ($\dot{V}_{eff}^{out}$) and total ($\dot{V}^{out}$) volumetric flow rates at the expander outlet, can be then computed:

$$ \eta_v = \frac{\dot{V}_{eff}^{out}}{\dot{V}^{out}} = \frac{\dot{n}_{eff} \cdot \bar{v}^{out}}{\dot{n}^{out} \cdot \bar{v}_{eff}^{out}} \quad (17) $$

$\bar{v}_{eff}^{out}$ and $\bar{v}^{out}$ are the molar volume of the gas respectively after the polytropic expansion and after the mixing of effective and leak flows

<figure><img src="image_4.png" /><figcaption>**Figure 4** scheme of the modelled volumetric expander (left). Mass flow rate division approach in the expander (right).</figcaption></figure>

## 3 SECOND LAW ANALYSIS
## 3.1 Entropy generation

2nd Law analysis requires computing the entropy production related to each irreversible process in every unitary component. The method differs depending on the operating regime assumed for the modelling of the device: transient or quasi-static regime.

For the components operating in transient regime (reactor, condenser, evaporator, tanks), Prigogine methodology (Prigogine, 1968) is applied.

As each phase is assumed at uniform pressure and temperature, Gibbs equation is valid:

$$dU - T \, dS + P dV - \sum \mu_k dn_k = 0 \quad (18)$$

The method consists in:

- Expressing component quantities dn_k/dt; internal energy dU/dt and entropy dS/dt time derivatives according to the heat and matter flows (i.e transient balance).

- Substituting these expressions in the Gibbs equation.

- Expressing the entropy production as a sum of products flow x driving force.

For components operating in quasi-steady state (expander, junctions, valves), Gibbs equation is identically null, and the previous method is not applicable. In that case, entropy production is directly expressed from the matter, energy and entropy balances.

### 3.1.1 Components operating in transient regime: Explicit expression of the entropy production is derived here for the reactor. For the other components (evaporator, condenser, tanks), the same method applies and details can be found in (Neveu and Mazet, 2002). As seen in section 2.1, two subsystems are involved in the reactor: reactive medium, and the set (reactor wall + heat transfer fluid), both assumed at uniform temperature and pressure.

#### 3.1.1.1 Reactive medium (subscript r): The volume of the three phases S₀, S₁ and G are assumed constant (dV_R = 0). Gibbs equation then writes:

$$dU_r - T_r dS_r - \bar{\mu}_g dn_g - \bar{\mu}_1 dn_1 - \bar{\mu}_0 dn_0 = 0 \quad (19)$$

Introducing the affinity of reaction (de Donder, 1927)

$$A_{T,p} = - \sum \bar{\mu}_j v_j = -(\bar{\mu}_0 - v_g \bar{\mu}_g + \bar{\mu}_1) = \bar{\mu}_0 + v_g \bar{\mu}_g - \bar{\mu}_1 \quad (20)$$

and using matter balances (3), (4) and (5), we get:

$$\frac{dU_r}{dt} - T_r \frac{dS_r}{dt} + n_s A_{T_r, P_r} \frac{dX}{dt} - \bar{\mu}_g \dot{n}_g^e = 0 \quad (21)$$

The reactive medium exchanges heat flow with the exchanger wall at temperature T_w (cf. 2.1). The entropy balance related to the reactive media then writes:

$$\frac{dS_r}{dt} = \frac{\dot{q}_{w-r}}{T_r} + \dot{n}_g^e \cdot \bar{s}_g^e + \dot{\sigma}_{irr,r} \quad (22)$$

with $\bar{s}_g^e$ molar entropy of the gas flowing in or out (J/kg.K) and $\dot{\sigma}_{irr,r}$, entropy generation in the reactive medium. Introducing $dU_r/dt$ (7) and $dS_r/dt$ (22) in Gibbs equation (21) yields:

$$\dot{\sigma}_{irr,r} = \dot{q}_{w-r} \cdot \left( \frac{1}{T_r} - \frac{1}{T_w} \right) + \frac{A_{T_r, P_r}}{T_r} \frac{dX}{dt} + \dot{n}_g^e \cdot \frac{\frac{(\bar{h}_g^e - T_r \bar{s}_g^e - \bar{\mu}_g)}{T_r}}{\Delta Y} \quad (23)$$

Reminding that the chemical potential $\bar{\mu}_g$ refers to the conditions (P_r, T_r), the driving force $\Delta Y$ related to the matter exchange can be split into two contributions:

$$\Delta Y = \frac{\int_{P_r, T_r}^{P_r, T_e} (d\bar{h}_g - T_r d\bar{s}_g)}{\Delta_p Y} + \frac{\int_{P_r, T_e}^{P_e, T_e} (d\bar{h}_g - T_r d\bar{s}_g)}{\Delta_T Y} \quad (24)$$

The first term ($\Delta_p Y$: variation at constant P) expresses the contribution of the thermal mixing of the gas entering in the reactor at a temperature $T_g^e$ different from $T_r$. For ideal gas, it simplifies in:

$$\Delta_p Y = \bar{c}_{p,g} \left( \frac{T_g^e}{T_r} - 1 - \ln \frac{T_g^e}{T_r} \right) \quad (25)$$
Notice that this term is always positive when the gas enters the reactor ($\dot{n}_g^e > 0$), whatever the temperature difference ($T_g^e - T_r$) is. For existing flows ($\dot{n}_g^e < 0$), $\Delta_p Y$ is null as the gas is desorbed by the reactive medium at $T_g^e = T_r$.

The second term ($\Delta_T Y$ : variation at constant T) expresses the contribution due to pressure losses. For ideal gas, it simplifies in:

$$ \Delta_T Y = R \ln \left( \frac{P_e}{P_r} \right) \qquad (26) $$

3.1.1.2 Reactor wall and heat transfer fluid (subscript w). Wall and heat transfer fluid are supposed to form a unique system at temperature $T_w$ (cf. 2.1). It exchanges heat $\dot{q}_{rw} = -\dot{q}_{wr}$ (Equ. (9)) with the reactive medium at $T_r$. Since there is no change in mass or volume for this system, the Gibbs equation writes:

$$ \frac{dU_w}{dt} - T_w \frac{dS_w}{dt} = 0 \qquad (27) $$

where $dU_w/dt$ is given by Equ. (6) and $dS_w/dt$ can be expressed from the entropy balance:

$$ \frac{dS_w}{dt} = \dot{m}_{HTF} \cdot (s_{HTF}^{in} - s_{HTF}^{out}) - \frac{\dot{q}_{w-r}}{T_r} + \dot{\sigma}_{irr,w} \qquad (28) $$

Introducing the mean entropic temperature $\tilde{T}_{HTF}$:

$$ \tilde{T}_{HTF} = \frac{h_{HTF}^{in} - h_{HTF}^{out}}{s_{HTF}^{in} - s_{HTF}^{out}} \qquad (29) $$

we get, using (10),

$$ \frac{dS_w}{dt} = \frac{\dot{q}_{HTF-w}}{\tilde{T}_{HTF}} - \frac{\dot{q}_{w-r}}{T_r} + \dot{\sigma}_{irr,w} \qquad (30) $$

Introducing (6) and (30) in (27) yields:

$$ \dot{\sigma}_{irr,w} = \dot{q}_{HTF-w} \left( \frac{1}{T_w} - \frac{1}{\tilde{T}_{HTF}} \right) + \dot{q}_{w-r} \left( \frac{1}{T_r} - \frac{1}{T_w} \right) \qquad (31) $$

3.1.1.3 Reactor (subscript R). The entropy balance related to the whole reactor writes:

$$ \frac{dS_R}{dt} = \dot{m}_{HTF} \cdot (s_{HTF}^{in} - s_{HTF}^{out}) + \dot{n}_g^e \cdot \bar{s}_g^e + \dot{\sigma}_{irr,R} \qquad (32) $$

$S_R$ being an extensive variable, it is equal to the sum of $S_r$ and $S_w$. Then, using the mean entropic temperature $\tilde{T}_{HTF}$, we have:

$$ \frac{dS_r}{dt} + \frac{dS_w}{dt} = \frac{\dot{q}_{HTF-w}}{\tilde{T}_{HTF}} + \dot{n}_g^e \cdot \bar{s}_g^e + \dot{\sigma}_{irr,R} \qquad (33) $$

Substituting the expression of $dS_r/dt$ (22) and $dS_w/dt$ (28) in (33) leads to:

$$ \dot{\sigma}_{irr,R} = \dot{\sigma}_{irr,r} + \dot{\sigma}_{irr,w} + \dot{q}_{w-r} \left( \frac{1}{T_w} - \frac{1}{T_r} \right) \qquad (34) $$

and gives the explicit expression of the entropy generation in the whole reactor:

$$ \dot{\sigma}_{irr,R} = \left\{ \begin{array}{l} \overbrace{\dot{\sigma}_{irr,w-r}}^{\dot{\sigma}_{irr,w-r}} \left( \frac{\dot{q}_{w-r}}{T_w} - \frac{1}{T_r} \right) + \overbrace{\dot{q}_{w-HTF}}^{\dot{\sigma}_{irr,w-HTF}} \left( \frac{1}{T_w} - \frac{1}{\tilde{T}_{HTF}} \right) + \overbrace{n_s \cdot \frac{A(T_R, P_R)}{T_R} \cdot \dot{X}}^{\dot{\sigma}_{irr,kinetics}} \\ + \underbrace{\dot{n}_g^e \cdot \Delta_p Y(T_R, T_g^e)}_{\dot{\sigma}_{irr,Th,mix}} + \underbrace{\dot{n}_g^e \cdot \Delta_T Y(P_R, P_g^e)}_{\dot{\sigma}_{irr,\Delta_p}} \end{array} \right. \qquad (35) $$

Equation (35) permits to differentiate the impact of every irreversible process occurring in the reactor on the entropy generation: heat transfer between exchanger wall and reactive mixture ($\dot{\sigma}_{irr,w-r}$) or HTF ($\dot{\sigma}_{irr,w-HTF}$), chemical reaction out of equilibrium ($\dot{\sigma}_{irr,kinetics}$), thermal mixing of the gas entering in the reactor ($\dot{\sigma}_{irr,Th,mix}$) and pressure drop inside the reactor ($\dot{\sigma}_{irr,\Delta_p}$).
3.1.2 Components operating in quasi-steady state. Explicit expression of the entropy production is derived here for the expander. For the other components (superheater, liquid pipes and gas ducts), the same method applies and details can be found in (Ghazale, 2024). As seen in section 2.2, the expander is divided in three subsystems: the effective flow, the leak flow and the mixing zone. Since steady regime is assumed for these three subsystems, second law writes:

$$ \dot{n}_{eff} \cdot (\bar{s}^{in} - \bar{s}_{eff}^{out}) + \dot{\sigma}_{irr,eff} = 0 \quad (36) $$

$$ \dot{n}_{leak} \cdot (\bar{s}^{in} - \bar{s}_{leak}^{out}) + \dot{\sigma}_{irr,leak} = 0 \quad (37) $$

$$ \dot{n}_{eff} \cdot \bar{s}_{eff}^{out} + \dot{n}_{leak} \cdot \bar{s}_{leak}^{out} - \dot{n}^{out} \bar{s}^{out} + \dot{\sigma}_{irr,mix} = 0 \quad (38) $$

where $\bar{s}_j^{out}$ is the molar entropy of the gaseous ammonia at the outlet related to flow j; j being eff, leak or mix. The total entropy production due to the expander is the sum of three entropy generations appearing in equations (36) to (38):

$$ \dot{\sigma}_{irr,exp} = \left\{ \begin{array}{ll} \frac{\dot{\sigma}_{irr,eff}}{\dot{n}_{eff} \cdot (\bar{s}_{eff}^{out} - \bar{s}^{in}) + \dot{\sigma}_{irr,leak} \cdot (\bar{s}_{leak}^{out} - \bar{s}^{in})} & (39) \\ + \frac{\dot{n}_{eff} \cdot (\bar{s}_{eff}^{out} - \bar{s}^{out}) - \dot{n}_{leak} \cdot (\bar{s}_{leak}^{out} - \bar{s}^{out})}{\dot{\sigma}_{irr,mix}} & \end{array} \right. $$

Equation (35) shows that three irreversible processes are involved in the expander: irreversible polytropic expansion for the effective flow ($\dot{\sigma}_{irr,eff}$), isenthalpic expansion for the leak flow ($\dot{\sigma}_{irr,leak}$) and mixing of the two flows at the exhaust ($\dot{\sigma}_{irr,mix}$)

## 3.2 Exergy Analysis

The evaluation of entropy production presented above enables exergy analysis to be conducted, aimed at identifying which components and which irreversible processes are primarily responsible for the degradation in exergy efficiency.

3.2.1 Exergy efficiency: Exergy efficiency should be first defined for the cogeneration prototype. During the charging stage, the reactor receives desorption heat $\dot{q}_{des}$ at $T_{hot}$ and the condenser releases $\dot{q}_{cond}$ to the ambient. During the discharging stage, the evaporator receives $\dot{q}_{evap}$ at $T_{cold}$, the superheater consumes heat from the ambient $\dot{q}_{sh}$, the expander produces mechanical power $\dot{w}_{exp}$, and the reactor releases heat of synthesis $\dot{q}_{syn}$ to the ambient. During the three steps, all the components involve thermal losses $\dot{q}_{loss}$. The exergy balance over the whole cycle writes:

$$ \oint_{cycle} \frac{dEx}{dt} dt = \begin{cases} \int_{charg.} (\dot{q}_{des}\theta_{hot} + \dot{q}_{cond}\theta_{amb} + \dot{q}_{loss}\theta_{amb} - \dot{e}x_{d,ch})dt \\ + \int_{stand-by} (\dot{q}_{loss}\theta_{amb} - \dot{e}x_{d,sb})dt \\ + \int_{discharg.} (\dot{q}_{syn}\theta_{amb} + \dot{q}_{evap}\theta_{cold} + \dot{q}_{sh}\theta_{amb} + \dot{w}_{exp} - \dot{e}x_{d,ch})dt \end{cases} \quad (40) $$

With $\theta_i = 1 - T^0/T_i$: Carnot factor related to thermostat i (-), $\dot{e}x_{d,j}$ exergy destruction related to stage j (kW). As the prototype begins and ends a cycle in the same state (ambient temperature, reactive mixture fully converted in S1, same level of NH3 in the tank), the left-hand term is zero. Thence, choosing the ambient temperature as reference temperature T0, equation (40) simplifies in:

$$ 0 = \int_{\substack{\text{charg.} \\ Ex_{hot}>0}} \dot{q}_{des}\theta_{hot}dt + \int_{\substack{\text{discharg.} \\ Ex_{cold}<0}} \dot{q}_{evap}\theta_{cold}dt + \int_{\substack{\text{discharg.} \\ Ex_{w}<0}} \dot{w}_{exp}dt - \int_{\substack{\text{cycle} \\ Ex_{d}>0}} \dot{e}x_{d}dt \quad (41) $$
The exergy destruction term can be computed from the entropy generation related to every component k, i.e Reactor (R), Evaporator (evap), condenser (cond), throttling valve (thr), expander (exp), superheater (sh) and tanks (tank).

$$Ex_d = \int_{cycle} \dot{e}x_d = \int_{cycle} \left( \sum_k T_0 \dot{\sigma}_{irr,k} \right) dt = \sum_k Ex_{d,k} \quad (42)$$

Dividing equation (41) by the fuel exergy $Ex_{hot}$ yields:

$$0 = 1 - \frac{|Ex_{cold}|}{\underbrace{Ex_{hot}}_{\eta_{ex}}} - \frac{|Ex_{w}|}{\underbrace{Ex_{hot}}_{\psi_k}} - \sum_k \frac{Ex_{d,k}}{\underbrace{Ex_{hot}}_{\psi_k}} \quad (43)$$

showing that the cogeneration exergy efficiency $\eta_{ex}$ can be expressed according to the dissipation factors $\psi_k = Ex_{d,k}/Ex_{hot}$ related to every component k:

$$\eta_{ex} = \frac{|Ex_{cold}|}{Ex_{hot}} + \frac{|Ex_{w}|}{Ex_{hot}} = 1 - \sum_k \psi_k \quad (44)$$

or, introducing the work and cold exergy efficiencies for the cogeneration system:

$$\eta_{ex} = \eta_{ex,cold} + \eta_{ex,w} = 1 - \sum_k \psi_k \quad (45)$$

with $\eta_{ex,w} = Ex_w/Ex_{hot}$, and $\eta_{ex,w} = Ex_{cold}/Ex_{hot}$

In the following, we will focus on the discharging stage. Equation (45) is then written in the following form:

$$\eta_{ex} = \eta_{ex,w} + \eta_{ex,cold} = 1 - \psi_{ch} - \psi_{R,syn} - \psi_{exp} - \psi_{evap} - \psi_{thr} - \psi_{sh} \quad (46)$$

With:

$- \psi_{ch} = \psi_{R,dec} + \psi_{cond} + \psi_{tank}$, the dissipation factor of the charging stage.

# 4 RESULTS AND DISCUSSIONS

The 2nd Law analysis is performed by post-processing the results of the dynamic model. This latter was firstly fit with the experimental results(Ghazale, 2024; Godefroy et al., 2025). Fitting parameters are the kinetics coefficients $K_{cin,dec}$ and $K_{cin,syn}$, the reactive medium/wall heat exchange coefficient $\alpha_{w-r}$, the pressure loss coefficient $k_{leak}$ and the expander isentropic efficiency $\eta_{is}$. After identifying these parameters, errors are less than 10% for the reaction advancement, temperatures, pressures, mechanical and thermal power, flow rates, etc. The post-processing of the dynamic model consists in evaluating, the exergy destroyed by all the irreversible processes occurring inside each component along a complete cycle. Thence, dissipation factors $\psi_k$ and exergy efficiencies $\eta_{ex,w}$ and $\eta_{ex,cold}$ appearing in (46) can be computed.

**Figure 5** shows the distribution between exergy efficiencies and dissipation factors. According to equation (46) the whole pie represents 100% of the fuel exergy ($Ex_{hot}$).supplied to the process. The cogeneration exergy efficiency is 13%, sum of $\eta_{ex,w}$ (1%) and $\eta_{ex,cold}$ (12%). Consequently, irreversible processes destroy 87% of the fuel exergy. Such pie diagram clearly exhibits which components are mainly responsible for this destruction and have to be improved in priority : the reactor and the expander, destroying respectively 36% and 5% of the fuel exergy. Exergy destruction related to these two components can also be more deeply analysed in terms of irreversible processes From equations (35) and (39), we have

$$\psi_{R,syn} = \psi_{w-r} - \psi_{w-HTF} + \psi_{kinetics} + \psi_{Th,mix} + \psi_{\Delta P} \quad (47)$$

$$\psi_{exp} = \psi_{leak} + \psi_{eff} + \psi_{mix} \quad (48)$$

Concerning the reactor, pressure drop ($\psi_{\Delta P} \approx 0$), thermal mixing ($\psi_{Th,mix} = 1\%$) and out of equilibrium kinetics ($\psi_{kinetics}=4\%$) have low impact on the exergy efficiency whereas heat transfer should be improved, first of all on the HTF side ($\psi_{w-HTF} = 20\%$), and to a lesser extend the reactive medium side ($\psi_{w-r} = 11\%$).

Concerning the expander, leakage between exhaust and suction chambers appears to be the main issue to address ($\psi_{leak} = 4\%$), whereas mixing contribution is negligible ($\psi_{mix} \approx 0$). This feature also
explains the low value of the electrical exergy efficiency ($\eta_{ex,w} = 1\%$). To demonstrate this point, let's write the expander exergy balance:

$$0 = \dot{w} + \dot{n} \overbrace{(\bar{e}x^{in} - \bar{e}x^{out})}^{\dot{e}x_{exp}} - \dot{e}x_{d,exp} \quad (49)$$

Integrating (49) on the charging stage duration and dividing the results by the process fuel exergy ($Ex_{hot}$) gives

$$0 = -\frac{\eta_{ex,w}}{1\%} + \frac{Ex_e^{in}}{Ex_{hot}} - \frac{\psi_{leak}}{4\%} - \frac{\psi_{eff}}{1\%} - \frac{\psi_{mix}}{0\%} \quad (50)$$

One can deduce that the input exergy that should be converted into work in the reversible expander was approx. 6% of $Ex_{hot}$. From this amount, 4% is destroyed by the internal leakage, i.e. 67% of the available exergy in the gas flow.

<figure><img src="image_5.png" /><figcaption>Figure 5. Distribution between dissipation factors and exergy efficiencies</figcaption></figure>

## 5 CONCLUSION

The 2nd law analysis clearly demonstrates that improving the cycle performance requires to intensify heat transfer in the reactor and reducing internal leakage in the expander. The prototype overall heat transfer coefficient $\alpha_{r-HTF} = (1/\alpha_{r-w} + 1/\alpha_{w-HTF})^{-1}$ was 23 W/(m²K) and the volumetric efficiency of the expander was only 10%. Both values are low regarding the present know-how. To assess performance in the near future, simulations have been carried out with $\alpha_{r-HTF}$ and $k_{leak}$ values leading to more realistic values for $\alpha_{r-HTF}$ and $\eta_v$: 23 W/(m²K) < $\alpha_{r-HTF}$ < 55 W/(m²K), 0.4 < $\eta_v$ < 0.9. Figure 6 shows the evolution of the exergy efficiency as a function of the evolution of these two parameters. It can be seen that exergy efficiency could be approximately doubled with a better expander ($\eta_v$ = 85%) and better design of the reactor $\alpha_{r-HTF}$ = 53 W/(m²K).

Same analysis will be performed for the charging step and should exhibit other route for optimizing this hybrid thermochemical process.
<figure><img src="image_6.png" /><figcaption>**Figure 6** : Cogeneration exergy efficiency vs expander volumetric efficiency η<sub>v</sub> and reactor overall heat transfer coefficient α<sub>r-HTF</sub></figcaption></figure>

## 6 NOMENCLATURE

<table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>A</td><td>Affinity or Area</td><td>J/mol or m<sup>2</sup></td><td>subscript</td></tr><tr><td>Ex</td><td>Exergy</td><td>J</td><td>0</td><td>solid S<sub>0</sub></td></tr><tr><td>e'x<sub>d</sub></td><td>exergie destruction</td><td>W</td><td>1</td><td>solid S<sub>1</sub></td></tr><tr><td>h</td><td>specific enthalpy</td><td>J/kg</td><td>ch</td><td>charging</td></tr><tr><td>h&#773;</td><td>specific molar enthalpy</td><td>J/mol</td><td>coge</td><td>cogeneration</td></tr><tr><td>K<sub>cin</sub></td><td>kinetics coefficient</td><td>1/s</td><td>cond</td><td>condenser</td></tr><tr><td>k<sub>leak</sub></td><td>pressure loss coef.</td><td>Pa/(m<sup>3</sup>/s)<sup>2</sup></td><td>dec</td><td>decomposition</td></tr><tr><td>m&#775;</td><td>mass flow</td><td>kg/s</td><td>eff</td><td>effective</td></tr><tr><td>n</td><td>number of moles</td><td>Mol</td><td>eq</td><td>equilibrium</td></tr><tr><td>n&#775;</td><td>molar flow</td><td>mol/s</td><td>evap</td><td>evaporator</td></tr><tr><td>P</td><td>pressure</td><td>Pa</td><td>ex</td><td>exergy</td></tr><tr><td>q&#775;</td><td>heat flow</td><td>W</td><td>g</td><td>reactive gas</td></tr><tr><td>S</td><td>entropy</td><td>J/K</td><td>is</td><td>isentropic</td></tr><tr><td>s&#773;</td><td>specific molar entropy</td><td>J/(mol.K)</td><td>leak</td><td>leak</td></tr><tr><td>t</td><td>time</td><td>S</td><td>r</td><td>reactive mixture</td></tr><tr><td>T</td><td>temperature</td><td>K</td><td>R</td><td>reactor</td></tr><tr><td>T&#773;</td><td>mean temperature</td><td>K</td><td>s</td><td>solid</td></tr><tr><td>T&#773;&#773;</td><td>entropic mean temperature</td><td>K</td><td>syn</td><td>synthesis</td></tr><tr><td>U</td><td>internal energy</td><td>J</td><td>v</td><td>volumetric</td></tr><tr><td>V</td><td>volume</td><td>m<sup>3</sup></td><td>w</td><td>wall</td></tr><tr><td>v&#773;</td><td>molar volume</td><td>m<sup>3</sup>/mol</td><td>superscript</td><td></td></tr><tr><td>V&#775;</td><td>volumetric flow</td><td>m<sup>3</sup>/s</td><td>D</td><td>decomposition</td></tr><tr><td>X</td><td>convention ratio</td><td>-</td><td>e</td><td>exchanged</td></tr><tr><td>Y</td><td>mass flow driving force</td><td>J/(mol.K)</td><td>in</td><td>inlet</td></tr><tr><td></td><td></td><td></td><td>out</td><td>outlet</td></tr><tr><td>Greek</td><td></td><td></td><td>S</td><td>synthesis</td></tr><tr><td>α</td><td>convective heat transfer coef.</td><td>W/(m<sup>2</sup>.K)</td><td></td><td></td></tr><tr><td>Δ</td><td>difference</td><td></td><td></td><td></td></tr><tr><td>ν</td><td>stoichiometric coefficient</td><td></td><td></td><td></td></tr><tr><td>μ&#773;</td><td>Molar chemical potential</td><td>J/mol</td><td></td><td></td></tr><tr><td>σ&#775;<sub>irr</sub></td><td>entropy generation</td><td>W/K</td><td></td><td></td></tr></tbody></table>

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8 ACKNOWLEDGEMENT

This work has been financially supported by the French National Research Agency (ANR) for the purpose of the ThermHyVal project ANR-20-CE05-0036 (https://anr.fr/Projet-ANR-20-CE05-0036) and by a prematuration program of the French National Centre for Scientific Research (CNRS).