Thermodynamic Assessment of Heat Pump Configurations for Waste Heat Integrated Carnot Batteries

Abstract

Carnot batteries based on the coupling of high-temperature heat pumps (HTHPs) and Organic Rankine Cycles (ORCs) emerge as promising solutions for large-scale and long-duration energy storage, enabling sector coupling and the valorization of industrial waste heat. In such systems, the charging subsystem plays a dominant role, as variations in heat pump performance influence the round-trip efficiency more strongly than comparable variations in the ORC. This work presents a thermodynamic assessment of Rankine-based HP–ORC Carnot batteries focusing on the influence of heat pump configuration and working fluid selection. System performance is evaluated using the heat pump coefficient of performance, volumetric heat capacity, ORC efficiency, and Carnot battery round-trip efficiency through a grid-search optimization over a wide range of storage outlet and waste heat source temperatures. The results show that single-stage configurations are optimal at low to moderate temperature lifts, while two-stage and cascade systems become advantageous at higher lifts. Among the investigated fluids, R-601 provides the highest round-trip efficiencies at elevated storage temperatures, whereas R-600 enables more compact systems due to its higher volumetric heat capacity. These findings provide design guidance for selecting heat pump configurations and working fluids in industrial waste-heat-assisted Carnot battery applications.

1. Introduction

The increasing penetration of variable renewable energy sources and the urgent need to decarbonize both the electricity and heat sectors have intensified interest in large-scale, long-duration energy storage technologies [1]. Among the emerging solutions, Carnot batteries (CB) have gained considerable attention as a pumped thermal energy storage (PTES)-based power-to-heat-to-power concept, enabling the storage of electrical energy as thermal energy and its subsequent reconversion into electricity through established thermodynamic cycles [2]. By relying on mature heat pump and heat engine technologies combined with scalable thermal energy storage, Carnot batteries offer a flexible and potentially cost-effective pathway for grid-scale energy storage, while enabling sector coupling between electricity and heat [3]. Their relevance is particularly pronounced in industrial contexts, where large thermal capacities and available waste-heat streams can be exploited to enhance overall system efficiency and reduce greenhouse gas emissions [4].

Heat pump–Organic Rankine Cycle (HP–ORC) Carnot batteries have emerged as a promising solution for large-scale and long-duration energy storage. Their performance, however, is strongly dependent on the thermodynamic efficiency of both the charging and discharging subsystems, as well as on the temperature levels at which thermal energy is stored and recovered [5]. Within waste heat integrated Carnot batteries, the high-temperature heat pump (HTHP) plays a central role as the charging unit and represents the primary driver of overall system efficiency [6]. The heat pump determines the achievable storage temperature, the electrical energy required during charging, and the extent to which low- and medium-temperature heat sources can be effectively upgraded. Operating under large temperature lifts and high delivery temperatures, HTHPs face significant thermodynamic and technological challenges, making their configuration and design critical for Carnot battery performance [7]. In particular, the choice of heat pump architecture has a decisive influence on both efficiency and system compactness. As a result, optimizing the heat pump configuration is a key step in the development of high-performance HP–ORC Carnot batteries, especially for applications involving high-temperature thermal storage and waste heat recovery.

Three primary factors influence CB performance: (1) the type of thermal energy storage medium, (2) the selection of working fluids for the heat pump and Organic Rankine Cycle, and (3) the system configuration [8].

Considering the Rankine-CB, multiple studies have examined optimal working fluids based on thermodynamic performance and environmental impact. Frate et al. [9] analyzed 17 working fluids, including both artificial and natural options, and found that R1233zd(E) yielded the highest round-trip efficiency, followed by R1234ze(Z) and R717. Dumont and Lemort [10] mapped the performance of a thermally integrated reversible CB with 16 working fluids, determining that R1233zd(E) and R1234yf were most suitable for lower temperature sources, while R245fa provided greater adaptability across a broader temperature range. Zhou et al. [11] investigated working fluid selection for a thermal-integrated Carnot battery, showing that R1233zd(E)-based zeotropic mixtures improve power-to-power efficiency, reaching 81.26% in the PR-PTES configuration. The study also emphasizes the importance of optimizing fluid selection for HP and ORC subsystems to minimize exergy losses.

Numerous studies have compared PTES-based Carnot battery architectures combining basic and recuperated heat pumps with basic and recuperative ORCs, revealing a strong dependence of power-to-power efficiency on boundary conditions, working fluids, and thermal storage approaches. Niu et al. showed that recuperated PTES systems integrated with solar collectors outperform basic configurations, with efficiency increasing as the solar collector area grows. Frate et al. [12] optimized configurations with and without a regenerator and showed that regeneration increases power-to-power and exergy efficiencies by up to 32% and 19%, respectively. Zhang et al. [13] applied an artificial bee colony optimization to Rankine-based Carnot batteries operating with R245fa and demonstrated that the inclusion of regeneration reduces the levelized cost of storage by up to 10%, with a minimum value of 0.293 USD/kWh at 120 °C. Fan et al. [14] applied a genetic algorithm to optimize Carnot battery configurations using LCOS as the objective function, showing that systems with regeneration in both the heat pump and ORC achieve superior thermo-economic performance, with a minimum LCOS of 0.29 USD/kWh and a maximum exergy efficiency of 19% for regeneration in the heat pump subsystem. Bellos et al. [15] analyzed a solar-assisted PTES system with recuperated heat pump and ORC using cyclopentane-based latent thermal storage (120–220 °C) supplied by flat-plate collectors (30–80 °C), reporting power-to-power efficiencies ranging from 35% to 117% depending on operating conditions. Eppinger et al. [16] investigated recuperated HP–ORC Carnot battery systems using cyclopentane, R365mfc, and R1233zd(E), combined with sensible and latent thermal storage, reporting power-to-power efficiencies between 33% and 82% for a heat source temperature of 80 °C.

The performance of Carnot batteries strongly depends on the temperature level and availability of the heat source used during the charging process. In industrial environments, large quantities of waste heat are typically available over a wide temperature range, from low-temperature streams below 100 °C to medium-grade heat exceeding 200 °C, as shown in Figure 1. Several studies have highlighted the importance of matching Carnot battery operating conditions to these temperature levels to maximize system performance and economic viability [17,18]. Waste heat sources within the range of approximately 40–100 °C are widely available in sectors such as chemical processing, food production, and metal manufacturing, making them attractive candidates for integration with high-temperature heat pumps. When upgraded through heat pump technologies, such thermal streams can supply the temperature levels required by thermal energy storage systems and subsequently drive Organic Rankine Cycle (ORC) units for electricity generation.

Consequently, while ORC architectures are commonly implemented in recuperative configurations, the heat pump subsystem offers substantially greater potential for performance improvement through architectural design. Building on these considerations, it becomes evident that, within HP–ORC Carnot batteries, the charging subsystem plays a dominant role in determining the overall round-trip efficiency. Variations in the heat pump coefficient of performance have a substantially larger impact on the round-trip efficiency than comparable variations in the ORC efficiency, as the electrical energy input during the charging phase directly sets the denominator of the power-to-power conversion. Although existing studies on Carnot batteries have extensively investigated thermodynamic performance, techno-economic aspects, and parameter optimization, the explicit inclusion and systematic evaluation of advanced high-temperature heat pump configurations within Carnot battery architectures remain limited. To the best of the authors’ knowledge, the systematic comparison of multiple high-temperature heat pump architectures within HP–ORC Carnot batteries, combined with an updated set of low-GWP working fluids and the use of off-design compressor and expander models to capture realistic component interactions, has not yet been comprehensively investigated at the Carnot battery system level.

In this context, the objectives of this work are:

• To systematically assess the impact of high-temperature heat pump configurations and low-GWP working fluid selection on Carnot battery charging performance, by comparing single-stage, multi-stage, and cascade architectures with internal heat recovery under representative operating conditions and evaluating their effects on the coefficient of performance and volumetric heat capacity.
• To develop and apply a more rigorous modelling of the ORC discharging subsystem, in which the turbine isentropic efficiency is not assumed to be constant but varies with operating conditions, allowing off-design behavior and over-expansion effects to be realistically captured.
• To evaluate the resulting system-level performance of HP–ORC Carnot batteries in terms of round-trip efficiency, through grid-search optimization across a wide range of storage outlet and waste heat source temperatures, providing operating maps and design guidance for industrial Carnot battery applications.

The paper is organized as follows. Section 2 presents the system description, modelling assumptions, and working fluid selection. Section 3 discusses the parametric analysis and optimization results, including heat pump configurations, refrigerant effects, and Carnot battery round-trip efficiency. Finally, Section 4 summarizes the main conclusions and outlines directions for future work.

2. Carnot Battery System and Model

This chapter presents the materials, system configurations, and modelling framework adopted in this study. First, the characteristics and temperature levels of industrial waste heat sources considered in the analysis are described. The overall Carnot battery concept is then introduced, followed by a detailed description of the heat pump investigated layouts and recuperative Organic Rankine Cycle. Subsequently, the thermodynamic modelling approach is outlined, including the definition of the key performance indicators and the models used for the heat pump configurations and the ORC. Finally, the working fluids considered in this work are presented and justified.

2.1. System Description

Based on the industrial mapping of Figure 1, the present work focuses on low-temperature waste heat sources in the range of 40–100 °C, which are representative of residual heat streams commonly rejected to the environment through cooling systems, exhaust air, warm effluents, or low-grade thermal loops. These temperature levels are especially relevant for industrial facilities where deeper heat integration is constrained by process requirements, intermittency, or economic considerations, making them attractive candidates for recovery through high-temperature heat pumps (HTHPs). In this context, waste heat acts as the cold reservoir of the Carnot battery charging process, supplying thermal energy to the heat pump evaporator.

The upgraded heat is delivered at outlet temperatures between 90 and 150 °C, corresponding to a wide spectrum of industrial heat demands, including hot water, low-pressure saturated steam, thermal oil loops, and intermediate-temperature process heating. This temperature interval is also suitable for driving Organic Rankine Cycle (ORC) systems during the discharge phase of the Carnot battery, enabling the partial reconversion of stored thermal energy into electricity.

A typical Rankine-based CB operation process is illustrated in Figure 2. During periods of grid electricity surplus, the electrical energy ($E_{\mathrm{in}}$) drives the heat pump, upgrading low-temperature heat from an external source to the high-temperature thermal energy storage, which transfers heat from a cold tank to a hot tank. During the discharge process, the heat engine utilizes the stored heat/cold to produce electrical power ($E_{\mathrm{out}}$) through ORC, rejecting heat to a cold sink.

The thermodynamic operation of the simple heat pump (Figure 3a) during the charging phase of the Carnot battery can be described as follows. Heat from the waste heat source is transferred to the working fluid in the heat pump evaporator, where the fluid evaporates (process 4–1). The vapor is then compressed in the compressor (process 1–2) using excess electricity, resulting in an increase in both pressure and temperature. The high-pressure working fluid then flows into the condenser, where it releases heat (process 2–3) to the thermal energy storage, thereby raising the temperature of the storage water from the low-temperature tank to the high-temperature tank. Finally, the condensed working fluid expands through a throttling valve (process 3–4), resulting in a reduction in pressure and temperature before returning to the evaporator, and the cycle repeats.

A series of modifications were implemented to improve the performance of the heat pump. These include the incorporation of a regenerator (Figure 3b), a two-stage heat pump (Figure 3c), a two-stage heat pump with regenerator (Figure 3d), a two-stage heat pump with flash tank and regenerator (Figure 3e), and a cascade configuration (Figure 3f). The heat-pump regenerator (Figure 3b,d,e) preheats the working fluid entering the compressor by recovering heat from the fluid exiting the condenser. This internal heat recovery enables higher compressor outlet temperatures, increasing the specific heat transferred in the condenser and consequently reducing the required working-fluid mass flow rate. The two-stage HP (Figure 3c–e) comprises two compression stages to improve efficiency and extend the operating range of the heat pump under large temperature lifts. Finally, the cascade configuration in heat pumps (Figure 3f) is particularly advantageous for addressing performance, efficiency, and safety limitations that arise when a single refrigerant or a single compression stage is required to operate over a wide temperature range.

Following the charging stage, the stored thermal energy is converted back into electricity during the discharging phase when electricity availability is limited. In Rankine-based Carnot batteries, the heat-pump charging unit is coupled with an organic Rankine cycle for the discharging phase in most reported designs, forming a fully integrated thermodynamic loop. The ORC is commonly implemented in a recuperative configuration, where a regenerator transfers heat from the hot turbine exhaust to the liquid working fluid before evaporation. This internal heat recovery significantly reduces the required external heat input, lowers irreversibilities, and improves cycle efficiency compared with a basic ORC. As a result, the HP–recuperative-ORC pairing, shown in Figure 4, enhances the overall round-trip efficiency and is widely adopted in state-of-the-art Carnot battery designs.

In the discharge phase, the hot water from the high-temperature storage tank acts as the heat source and flows through the ORC evaporator, where it evaporates the working fluid, (process 2–3). Subsequently, the working fluid enters the turbine, where it expands (process 3–4), thereby generating electricity that is fed back to the grid. After expansion, the fluid is slightly cooled in the internal heat exchanger (IHX), which recovers heat from the turbine outlet to preheat the fluid prior to its entry into the evaporator. The working fluid then flows to the condenser, releasing heat to the cooling water (process 5–6). The working fluid, now in the liquid state, is pumped from the condenser to the IHX (process 6–1) pressurized and preheated, enabling completion of the cycle.

2.2. Thermodynamic Model

A steady-state model of the complete system was developed in Matlab 2025a to determine the nominal operating conditions of the different components. The thermophysical properties of the working fluids were obtained using CoolProp [20]. The main modelling assumptions adopted in this work are:

• Steady-state operation of both HTHP and ORC;
• Negligible thermal losses from the storage tanks and heat exchangers;
• No pressure drop in pipes and heat exchangers;
• Kinetic and potential energy variations of the working fluid are neglected;
• No auxiliary electricity consumption from pumps, fans, and control systems,
• Pressurized water is used as the heat-transfer fluid in the heat source, thermal storage, and cold sink.

The performance of the HP subsystem is evaluated based on the coefficient of performance (COP):

$$COP={\displaystyle \frac{{\dot{Q}}_{cond,HP}}{{\sum }_i{{\dot{W}}_{comp}}_i}}$$

(1)

In addition, the volumetric heat capacity (VHC) is considered a key parameter for the heat pump, as it relates the heating capacity to the required compressor size by normalizing the thermal output with the total suction volumetric flow rate. Its formulation is given by:

$$VHC={\displaystyle \frac{{\dot{Q}}_{cond,HP}}{{\sum }_i{{\dot{V}}_{comp,in}}_i}}={\displaystyle \frac{{\dot{Q}}_{cond,HP}}{{\sum }_i\frac{{{\dot{m}}_{comp,in}}_i}{{{\rho }_{comp,in}}_i·{{\eta }_{vol}}_i}}}$$

(2)

where ${{\dot{V}}_{comp,in}}_i$ is the suction volumetric flow of compressor i, ${\dot{m}}_{comp,in}$ is the corresponding mass flow rate, ${\rho }_{comp,in}$ is the refrigerant density at suction conditions, and ${\eta }_{vol}$ is the volumetric efficiency estimated using an empirical correlation [21] as a function of the compressor pressure ratio, $r_p$:

$${{\eta }_{vol}}_i=1.0455-0.0184·r_{p_i}-0.0011·{r_{p_i}}^2$$

(3)

The summation is performed over all active compressors in the configuration, resulting in a single term for single-stage (SS) layouts and two terms for two-stage (TS) and cascade configurations. The VHC thus provides a direct measure of the heating capacity achievable for a given compressor size, linking thermodynamic cycle performance with practical machine constraints.

The performance of the ORC subsystem is assessed using the ORC thermal efficiency (${\eta }_{ORC}$), defined as:

$${\eta }_{ORC}={\displaystyle \frac{{\dot{W}}_{turb}-{\dot{W}}_{pump}}{{\dot{Q}}_{evap,ORC}}}$$

(4)

The overall energy efficiency of the Carnot battery is evaluated through the round-trip efficiency, also referred to in the literature as the power-to-power efficiency:

$${\epsilon }_{rt}=COP\cdot {\eta }_{sto}\cdot {\eta }_{ORC} ={\displaystyle \frac{{\dot{Q}}_{cond,HP}}{{\sum }_i{{\dot{W}}_{comp}}_i}}\cdot {\eta }_{sto}\cdot {\displaystyle \frac{{\dot{W}}_{turb}-{\dot{W}}_{pump}}{{\dot{Q}}_{evap,ORC}}}\cdot {\displaystyle \frac{{\tau }_{dis}}{{\tau }_{ch}}}$$

(5)

where ${\tau }_{ch}$ and ${\tau }_{dis}$ represent the charge and discharge durations, respectively. For the two-tank sensible thermal storage configuration considered in this work, the hot and cold storage tanks operate at fixed temperature levels during the charge and discharge processes. Under these conditions, the mass flow rates of the storage fluid are equal and the heat delivered by the heat pump condenser during charging is equal to the heat absorbed by the ORC evaporator during discharge. As a result, the terms ${\dot{Q}}_{cond,HP}$ and ${\dot{Q}}_{evap,ORC}$ cancel in Equation (5).

Furthermore, assuming a complete charge and discharge cycle of the storage system, the durations ${\tau }_{ch}$ and ${\tau }_{dis}$ are equal, which allows Equation (5) to be simplified as:

$${\epsilon }_{rt}={\displaystyle \frac{{\dot{W}}_{turb}-{\dot{W}}_{pump}}{{\sum }_i{{\dot{W}}_{comp}}_i}}$$

(6)

The storage efficiency ${\eta }_{sto}$ is assumed to be 1 in this study, as the thermal storage tank is considered to be well insulated and heat losses are neglected. It should be noted that the Carnot battery configuration analyzed in this work is assisted by an external waste-heat source, which supplies thermal energy to the heat pump evaporator during the charging phase. In the definition of the round-trip efficiency adopted here, only the electrical power input to the compressor is included in the denominator, while the waste heat is treated as an external thermal co-input. Consequently, values of ${\epsilon }_{rt}$ greater than unity do not violate the first law of thermodynamics but indicate that part of the electrical energy recovered during discharge originates from the upgraded waste heat. The reported metric therefore represents an electric-to-electric efficiency with waste-heat assistance.

The basic Carnot battery configuration couples the heat pump and the ORC subsystems and comprises compressors, condensers, evaporators, expansion valves, pumps, and a turbine. The governing thermodynamic relations used to evaluate the performance of each component are described below.

The compressor power consumption ${\dot{W}}_{comp,HP}$ is obtained by

$${\dot{W}}_{comp,HP}=\sum _i\left[{\displaystyle \frac{{{\dot{m}}_{comp}}_i\left(h_{{comp,out,is}_i}-h_{c{omp,in}_i}\right)}{{{\eta }_{OU}}_i·{{\eta }_{em}}_i}}\right]$$

(7)

where ${\dot{m}}_{comp,i}$ is the refrigerant mass flow rate through compressor $i$, $h_{comp,in}$ and $h_{comp,out,is}$ are the inlet enthalpy and the isentropic outlet enthalpy, respectively, ${\eta }_{OU}$ is the adiabatic over/under-compression efficiency, and ${\eta }_{em}$ represents the electromechanical efficiency accounting for electrical and mechanical losses. The adiabatic over/under-compression efficiency is adapted from the compressor model proposed by Winandy et al. [22], Cuevas et al. [23] and, Lemort [24] and defined as:

$${{\eta }_{OU}}_i=\sum _i\left[{\displaystyle \frac{h_{{comp,out,is}_i}-h_{c{omp,in}_i}}{\left(h_{{ad}_i}-h_{c{omp,in}_i}\right)+{\nu }_{{ad}_i}\left(P_{{comp,out}_i}-P_{{ad}_i}\right)}}\right]$$

(8)

The power consumed by each compressor accounts for the overall electromechanical efficiency (${\eta }_{em}$) which represents the mechanical and electrical losses, as well as the over-under compression efficiency (${\eta }_{OU}$), which converts the isentropic specific compression work into the real compression work. This efficiency compares the ideal (isentropic) enthalpy rise between suction and discharge with a penalized denominator that accounts for mismatches between the cycle discharge pressure and the compressor built-in volume ratio. The term ($h_{{comp,out,is}_i}-h_{c{omp,in}_i}$) represents the isentropic specific compression work of compressor i, $P_{{comp,out}_i}$ is the discharge pressure, while $h_{{ad}_i}$ and $P_{{ad}_i}$ denote the enthalpy and pressure of the adiabatic target state, respectively. The corresponding specific volume at that state is denoted by ${\nu }_{{ad}_i}$.

The heat transferred in the heat pump condenser is determined from the refrigerant enthalpy variation which is equal to the heat transferred to the storage fluid:

$${\dot{Q}}_{cond,HP}={\dot{m}}_{f,HP}\left(h_{cond,in}-h_{cond,out}\right)={\dot{m}}_{sto}{c}_{p_{sto}}\left(T_{HT}-T_{LT}\right)$$

(9)

where ${\dot{m}}_{sto}$ is the mass flow rate of the storage fluid, $c_p$ is its specific heat capacity, and $T_{HT}$ and $T_{LT}$ represent the hot and cold tank temperatures, respectively.

The throttling process in the expansion valve is assumed to be isenthalpic:

$$h_{tv,in}=h_{tv,out}$$

(10)

The heat absorbed from the waste heat source in the evaporator which corresponds to the heat extracted from the waste heat stream is calculated as:

$${\dot{Q}}_{evap,HP}={\dot{m}}_{f,HP}\left(h_{evap,out}-h_{evap,in}\right)={\dot{m}}_{src}{c}_{p_{src}}\left(T_{src,in}-T_{src,out}\right)$$

(11)

where ${\dot{m}}_{src}$ is the mass flow rate of the waste heat source fluid.

Starting now on the ORC mode, the pump power consumption and his isentropic efficiency are given by:

$${\dot{W}}_{pump}={\displaystyle \frac{{\dot{m}}_{f,ORC}\left(h_{pump,out}-h_{pump,in}\right)}{{\eta }_{em,pump}}}$$

(12)

$${\eta }_{is,pump}={\displaystyle \frac{(h_{pump,out,is}-h_{pump,in})}{(h_{pump,out}-h_{pump,in})}}$$

(13)

The heat supplied to the ORC working fluid in the evaporator which corresponds to the heat extracted from the thermal storage system is calculated as:

$${\dot{Q}}_{evap,ORC}={\dot{m}}_{f,ORC}\left(h_{evap,out}-h_{evap,in}\right)={\dot{m}}_{sto}{c}_{p_{sto}}\left(T_{HT}-T_{LT}\right)$$

(14)

The power produced by the turbine is given by:

$${\dot{W}}_{turb}={\dot{m}}_{f,ORC}\left(h_{turb,in}-h_{turb,out}\right)·{\eta }_{em}$$

(15)

Accurate modelling of the turbine is essential for a realistic assessment of ORC performance, as the turbine often operates under off-design conditions in Carnot battery applications. In contrast to the common assumption of a fixed isentropic efficiency, which may mask important performance losses associated with varying pressure ratios and operating regimes, a variable-efficiency approach allows the turbine behaviour to be captured more faithfully [25]. In the ORC subsystem, the isentropic efficiency of the turbine is calculated using the empirical correlation developed by Declaye et al. [26], which was originally inspired by Pacejka’s equation [27] and adapted to represent the off-design behaviour of volumetric expanders. The correlation provides a smooth and continuous representation of the turbine efficiency as a function of the pressure ratio and is expressed as:

$${\eta }_{i,turb}=y_{max}·sin\left\{\xi ·arctan\left[B·\left(r_p-r_{p0}\right)-E·\left(B·\left(r_p-r_{p0}\right)-arctan\left(B·\left(r_p-r_{p0}\right)\right)\right)\right]\right\}$$

(16)

For brevity, the intermediate expressions used to evaluate these coefficients are not reproduced here. The interested reader is referred to the original work of Declaye et al. [26], where the development of the correlation, its underlying assumptions, and its experimental validation are presented in detail. In this formulation, the inlet pressure ($p_{evap}$), the rotational speed ($N_{rot}$), and the turbine pressure ratio ($r_p$) are used as input parameters, as they were identified as the most representative variables governing turbine operating conditions.

Finally, the heat rejected in the condenser which is transferred to the cooling water is expressed as:

$${\dot{Q}}_{cond,ORC}={\dot{m}}_{f,ORC}\left(h_{cond,out}-h_{cond,in}\right)={\dot{m}}_{snk}{c}_{p_{snk}}\left(T_{snk,out}-T_{snk,in}\right)$$

(17)

While the equations above describe the thermodynamic behavior of the basic Carnot battery configuration, additional relations are required to model the specific components introduced in the advanced heat pump layouts.

In the heat-pump subsystem, the regenerator is placed upstream of the expansion valve. The liquid working fluid exiting the condenser flows through the regenerator, where it is subcooled before expansion, while the working fluid leaving the evaporator is simultaneously preheated prior to compression. The outlet temperature of the cold stream leaving the regenerator ($T_{cold,reg,out}$) is modelled by a minimum temperature approach difference (${\Delta T}_{pp,reg,hp}$) according to:

$$T_{cold,reg,out}=T_{hot,reg,in}-{\Delta T}_{pp,reg,HP}$$

(18)

The pinch point is assumed to occur at the cold-side outlet, since the vapor phase exhibits a lower heat-capacity flow rate than the hot-side liquid. Consequently, the vapor temperature increases more rapidly, making the minimum temperature difference more likely at this location. Neglecting thermal losses to the environment, the regenerator thermal load is calculated as:

$${\dot{Q}}_{reg,HP}={\dot{m}}_{f,HP}\left(h_{cold,reg,out}-h_{cold,reg,in}\right)={\dot{m}}_{f,HP}\left(h_{hot,reg,in}-h_{hot,reg,out}\right)$$

(19)

The two-stage heat-pump cycle with flash tank employs an intermediate vessel to separate the two-phase stream produced after the first expansion stage. The intermediate pressure associated with the flash tank configuration was determined so that the two compressors operate with the same pressure ratio. The key parameter governing this process is the vapor quality (x), which is determined from the enthalpy of the stream entering the flash tank ($h_{ft,in}$) and the saturated liquid and vapor enthalpies at the intermediate pressure:

$$x={\displaystyle \frac{h_{ft,in}-h_{liq,sat}}{h_{vap,sat}-h_{liq,sat}}}$$

(20)

The vapor quality defines the split of the total mass flow rate into a vapor stream injected into the intermediate compression stage and a liquid stream directed to the evaporator:

$$\begin{gathered} {\dot{m}}_{ft,vap,out}=x·{\dot{m}}_{total}; \\ {\dot{m}}_{ft,liq,out}=\left(1-x\right)·{\dot{m}}_{total} \end{gathered}$$

(21)

The superheated vapor leaving the ORC turbine enters the regenerator, where it is cooled before flowing to the condenser. Simultaneously, the liquid working fluid exiting the pump is preheated by the superheated vapor at turbine exhaust. Analogously to the heat pump regenerator, the ORC regenerator is modelled using a minimum temperature approach difference (${\Delta T}_{pp,reg,ORC}$) as follows:

$$T_{hot,reg,out}=T_{cold,reg,in}-{\Delta T}_{pp,reg,ORC}$$

(22)

In this case, the pinch point is assumed to occur at the hot side outlet, since the vapor stream exhibits a lower heat capacity flow rate than the cold side liquid. As a result, the vapor temperature decreases more rapidly. The thermal load of the regenerator is given by:

$${\dot{Q}}_{reg,ORC}={\dot{m}}_{f,ORC}\left(h_{hot,reg,in}-h_{hot,reg,out}\right)={\dot{m}}_{f,ORC}\left(h_{cold,reg,out}-h_{cold,reg,in}\right)$$

(23)

The operating conditions and simulation parameters adopted in the thermodynamic model are summarized in

Table 1. Operating conditions and main simulation parameters adopted for the thermodynamic modelling of the Carnot battery system.

System	Parameters	Value	Units
HP	Heat source heat load (${\dot{Q}}_{src}$)	100	kW
	Heat source inlet temperature ($T_{src,in}$)	40–100	°C
	Heat source temperature glide ($∆T_{src}$)	10	°C
	Superheat degree ($∆T_{sh,HP}$)	15	°C
	Subcooling degree ($∆T_{sc,HP}$)	10	°C
	Pinch point heat exchangers ($∆T_{pp,hx}$)	5	°C
	Storage outlet temperature ($T_{HT}$)	90–150	°C
	Storage temperature glide ($∆T_{TES}$)	15	°C
	Built-in volume ratio (${BVR}_{comp}$)	3.5	-
	Compressor/turbine electromechanical efficiency (${\eta }_{em}$)	95	%
ORC	Cold sink outlet temperature ($T_{snk,in}$)	20	°C
	Cold sink temperature glide ($∆T_{snk}$)	10	°C
	Pinch point heat exchangers ($∆T_{pp,hx}$)	5	°C
	Superheat degree ($∆T_{sh,ORC}$)	5	°C
	Subcooling degree $∆T_{sc,ORC}$)	0	°C
	Isentropic efficiency of the pump	60	%

.

2.3. Working Fluids

Due to the progressive restrictions on hydrofluorocarbons (HFCs) imposed by European environmental regulations, alternative low-global warming potential (GWP) working fluids have gained increasing attention for high-temperature heat pump (HTHP) applications. In such systems, the critical temperature is a key selection criterion, as it defines the maximum achievable heat delivery temperature under subcritical operating conditions. Accordingly, the working fluids selected in this study exhibit critical temperatures comparable to or higher than that of the reference fluid HFC-245fa, which is commonly used in high-temperature heat pump systems [28].

Table 2. Thermophysical and environmental properties of low-GWP working fluids [29].

Working Fluid	Formula	Critical Temperature, °C	Critical Pressure, Bar	Boiling Point, °C	ODP	GWP	Class (ASHRAE)
R245fa	C3H3F5	154.1	36.5	15.2	0	858	B1
R600	C4H10	152.0	38.0	−0.5	0	4	A3
R601	C5H12	196.6	33.7	36.1	0	20	A3
R1224yd(Z)	CF3CF=CHCl	155.5	33.4	14.6	0	<1	A1
R1233zd(E)	CF3CH=CHCl(E)	166.5	36.2	18.3	0	1	A1
R1234ze(Z)	CHF=CHCF3(Z)	150.1	35.3	9.7	0	1	A2L
R1336mzz(Z)	CF3CH=CHCF3 (Z)	171.4	29.0	33.5	0	2	A1

summarizes the main thermophysical, environmental, and safety-related properties of the investigated working fluids, including synthetic HFOs and HCFOs, as well as natural hydrocarbons. Among the synthetic alternatives, HCFO-1233zd(E) and HFO-1336mzz(Z) stand out due to their relatively high critical temperatures and low environmental impact, making them attractive candidates for integration with high-temperature heat sources. Hydrocarbons such as R-600 and R-601 also present favorable thermodynamic characteristics. In particular, n-pentane (R-601) offers a high critical temperature, enabling operation at elevated temperature levels, albeit with additional safety constraints related to flammability.

Overall, the working fluids listed in Table 2 comply with current environmental regulations regarding ozone depletion potential (ODP) and global warming potential, while providing a representative basis for assessing the performance of high-temperature heat pump cycles under demanding heat source conditions.

Hydrocarbons such as R-600 and R-601 belong to the ASHRAE A3 safety class and therefore require appropriate safety measures due to their flammability. However, the Carnot battery systems considered in this work are primarily intended for industrial applications operating in controlled environments, where the system is not exposed to combustion sources and appropriate safety standards can be implemented. Under such conditions, hydrocarbons may remain viable working fluids from a thermodynamic perspective, although safety considerations must be addressed in practical implementations.

3. Results

Section 3.1 presents a parametric analysis of the heat pump key performance indicators (KPIs), namely the coefficient of performance and the volumetric heat capacity, as a function of the temperature lift under two operating scenarios: (i) a fixed high-temperature storage level and (ii) a fixed waste-heat source temperature. In addition, the influence of the ORC efficiency (${\eta }_{ORC}$) and the overall round-trip efficiency (${\epsilon }_{rt}$) is investigated. Subsequently, in Section 3.2, a single-objective optimization is performed to evaluate the performance of the different system configurations and working-fluid combinations, using the round-trip efficiency as the objective function, for three representative waste-heat source temperatures typical of different industrial sectors.

3.1. Parametric Analysis

3.1.1. Comparison of HTHP Configurations

This section begins with a comparative performance analysis of the six heat pump configurations operating with R-1233zd(Z), with the high-temperature storage outlet temperature fixed at 120 °C. Figure 5a and Figure 5b show, respectively, the variation of the COP and the VHC as a function of the temperature lift. Together, these indicators allow a simultaneous assessment of energetic performance and system compactness, since the VHC is directly linked to compressor sizing.

As shown in Figure 5a, the COP decreases with increasing temperature lift for all configurations. This decline is mainly driven by the rise in specific compression work. A higher temperature lift leads to a lower evaporation pressure and, consequently, a lower suction pressure. For fixed condensation pressure, this results in a higher pressure ratio, which increases the isentropic compression work. More importantly, when the volume ratio required by the thermodynamic cycle exceeds the built-in volume ratio of the compressor, under-compression losses arise. These losses reduce effective compressor efficiency and lead to a sharp increase in the real compression work. Since the increase in the condensation enthalpy difference does not compensate for this effect, the overall COP decreases.

At low temperature lifts, the SS-IHX configuration achieves the highest COP, reaching approximately 8.0 at a temperature lift of 20 K, followed closely by the simple SS configuration. Two-stage configurations exhibit lower COP values in this range, typically between 5.8 and 6.0. In this range, the pressure ratio remains moderate, allowing the compressor to operate close to their design point with high efficiency. Consequently, the benefits of splitting compression into two stages do not outweigh the additional complexity and associated irreversibilities introduced. The internal heat exchanger is particularly advantageous, as the added subcooling increases the condensation enthalpy difference without significantly penalizing the compression work.

A crossover behavior is observed between temperature lifts of 40 and 60 K, beyond which two-stage architecture becomes the most efficient solution. Their superior performance at high temperature lifts is attributed to the mitigation of compression losses. By splitting the total pressure ratio between two compressors, each stage operates with a lower pressure ratio and closer to its optimal condition, thereby improving the overall compressor efficiency and reducing the total compression work. Configurations incorporating a flash tank and an internal heat exchanger, as well as cascade layouts, further benefit from intercooling effects, which reduce the inlet temperature of the second compression stage and reduce the overall work. At a temperature lift of 80 K, these configurations converge to COP values of approximately 3.0 to 3.1, while the SS-IHX and TS decrease to around 2.4 and 2.6, respectively, and the simple single-stage configuration drops to approximately 2.0.

Figure 5b shows the corresponding variation of the volumetric heat capacity. A clear decreasing trend with increasing temperature lift is observed for all configurations. This behavior is directly related to the decrease in suction pressure required to achieve higher temperature lifts. Lower suction pressure leads to a decrease in suction density, while the simultaneous increase in the pressure ratio deteriorates the volumetric efficiency of the compressor. The combined effect outweighs any increase in the condensation enthalpy difference, resulting in an overall decrease in the VHC.

For low to intermediate temperature lifts, the SS-IHX configuration consistently achieves the highest VHC values, reaching around 5.45 MJ·m⁻³ at a temperature lift of 20 K, closely followed by the simple single-stage configuration. The advantage of single-stage cycles in this range is associated with the still moderate pressure ratio required. The superiority of the SS-IHX over the basic SS cycle arises from the additional subcooling provided by the internal heat exchanger, which increases the condensation enthalpy difference and, consequently, the volumetric heat capacity. In contrast, two-stage architectures exhibit lower VHC values, with cascade configuration being the least favorable, with approximately 3.35 MJ·m⁻³ at a temperature lift of 20 K.

In Figure 5a, the crossover around a temperature lift of approximately 50 K reflects the point where the performance penalty associated with high compression ratios in single-stage configurations becomes significant, making two-stage architectures with internal heat recovery more advantageous. In contrast, the crossover observed in Figure 5b occurs at a higher temperature lift (around 70 K) because volumetric heat capacity is primarily influenced by suction density and volumetric efficiency. As a result, the relative advantage of multi-stage configurations in terms of volumetric performance appears at higher temperature lifts compared to the efficiency crossover.

At higher temperature lifts, another crossover behavior emerges. At a temperature lift of 80 K, the cascade configuration with internal heat exchanger achieves the highest VHC, with values around 1.0 MJ·m⁻³. In contrast, the SS-IHX and the simple SS configuration decrease to approximately 0.8 and 0.65 MJ·m⁻³, respectively. This crossover can be explained by the strong degradation of volumetric efficiency in single-stage compressors operating far from their design point. Conversely, in two-stage architectures, the split of the total pressure ratio allows each compressor to operate with a less penalized volumetric efficiency, enabling superior volumetric performance at high temperature lifts.

3.1.2. Comparison of Low-GWP Refrigerants

This section analyzes the impact of working fluid selection on the performance of the SS-IHX configuration. The comparative assessment is carried out for several low-GWP refrigerants under two operating scenarios: (i) a fixed high-temperature storage outlet temperature of 120 °C with varying temperature lift, and (2) a fixed temperature lift of 50 °C with varying absolute temperature levels of both the waste heat source and the thermal storage. Figure 6a and Figure 6b present the COP and the VHC, respectively, as a function of the temperature lift.

The COP trends presented in Figure 6a, show that for low lifts (20–40 K), R-601 achieves the highest performance, reaching values of approximately 8.4 at 20 K and 5.6 at 40 K, closely followed by R1336mzz(Z). In contrast, R-600, R1224yd(Z), R1233zd(E), R-1234ze(Z) and R-245fa exhibit lower COP in the operating range. As the temperature lift increases, the performance curves progressively converge, and at higher lifts (>60 K) R-600 slight outperforms the other fluids, attaining a COP of 2.55 at a lift of 80 K. The superior performance of R-601 at low lifts is primarily due to its thermophysical properties, which result in reduced specific compression work under moderate pressure ratios.

Figure 6b illustrates the corresponding trends in VHC. In contrast to the COP ranking, the hierarchy observed for VHC is almost reversed. R-600 exhibits the highest VHC, reaching approximately 7.0 MJ·m⁻³ at a lift of 20 K, followed by R-1234ze(Z). Conversely, R-601 shows significantly lower VHC, with 3.3 MJ·m⁻³ at a lift of 20 K. This behavior is mainly explained by differences in suction vapor density. R-600, being more volatile, exhibits a higher vapor density at a given evaporation temperature, leading to greater volumetric capacity. In contrast, the lower volatility of R-601 results in reduced suction density, which penalizes VHC and implies larger compressor displacement requirements. Notably, this ranking remains essentially unchanged as the temperature lift increases, indicating that the influence of refrigerant volatility on volumetric performance is preserved across the entire operating range.

To isolate the effect of absolute temperature levels from that of the temperature lift, a second parametric analysis is performed. In this case, the temperature lift is kept constant at 50 °C, while both waste heat source inlet temperature and heat storage outlet temperature are varied simultaneously. This approach enables the assessment of system performance under identical temperature lifts applied at different absolute temperature levels, which is particularly relevant for industrial applications where waste heat availability and heat demand may shift in parallel. Figure 7a,b present the corresponding variations of volumetric heat capacity and coefficient of performance for SS-IHX configuration.

The efficiency analysis presented in Figure 7a reveals more complex behavior and deepens the conclusions from the previous analysis (Figure 6a). Unlike VHC, the performance ranking for COP is not constant across operating conditions. While the earlier analysis R-601 and R-1233zd(E) stood out at a lift higher than 50 K, this plot clarifies that such an advantage only materializes at higher outlet temperatures. For outlet temperatures below approximately 110 °C, R-600 achieves the highest COP, a behavior that was not evident in the previous parametric study.

The peak-and-drop behavior observed for R-600, R-245fa and R-1234ze(Z) arises from a balance between two competing thermodynamic effects. On the one hand, the decrease in pressure ratio with increasing temperature is beneficial and tends to raise the COP. On the other hand, as the cycle operates at higher temperatures and pressures, approaching the fluid’s critical region, vapor properties change unfavorably. In this regime, despite the lower compression ratio, the specific compression work increases significantly, as reflected by the growing enthalpy rise across the compressor.

For these refrigerants a threshold temperature exists beyond which the increase in compression work outweighs the benefit associated with the reduced pressure ratio. At this point, the COP reaches a maximum and then declines. In contrast, fluids such as R-601 and R-1233zd(E), characterized by higher critical temperatures and, or more favorable thermodynamic behavior in the investigated range, continue to operate in a regime where the reduction in pressure ratio remains the dominant effect. This allows their COP to increase over the entire temperature range considered. These results demonstrate that the selection of the most efficient refrigerant is strongly dependent on the application temperature level, highlighting a clear engineering trade-off. R-600 is preferable for lower delivery temperatures, whereas R-601 and R-1233zd(E) are superior under more demanding higher-temperature operating conditions.

Figure 7b examines the VHC, a fundamental parameter for equipment sizing. The results show that the VHC increases with condenser outlet temperature for all refrigerants, a behavior directly linked to the increase in refrigerant density at the compressor suction. In terms of comparative performance, the ranking of refrigerants in terms of VHC remains consistent over the entire temperature range and mirrors that observed in the previous analysis (Figure 6b), where temperature lift was varied. In both cases, R-600 exhibits the highest volumetric capacity, while R-601 emerges the least favorable option from a volumetric standpoint.

To analyze the behavior of the discharging subsystem, the performance of the recuperative Organic Rankine Cycle is evaluated as a function of the storage outlet temperature for different working fluids. Figure 8a illustrates the variation of the ORC thermodynamic efficiency with increasing storage temperature, while Figure 8b provides additional insight into the underlying turbine behavior by showing the evolution of turbine work output, pressure ratio, and isentropic efficiency. Together, these results allow the observed ORC performance trends to be interpreted considering the turbine off-design characteristics.

Figure 8a shows that the ORC thermodynamic efficiency generally increases with the storage outlet temperature for all investigated working fluids. This trend is primarily driven by the increase in available temperature difference across the ORC evaporator, which enhances the specific turbine work and raises the cycle efficiency. Differences among fluids are observed, with R-601 exhibiting the highest efficiency at elevated storage temperatures, reflecting its favorable thermodynamic properties and compatibility with high-temperature operation. R-600 presents the lowest ORC efficiency at low storage temperatures; however, for storage temperatures above approximately 110 °C, its performance approaches that of R-601, remaining the second-best option across the upper temperature range considered.

However, the efficiency increase is not linear, and a progressive flattening of the curves can be observed at higher storage temperatures. This behavior is clarified by the turbine performance trends shown in Figure 8b. As the storage temperature rises, the pressure ratio across the turbine increases, leading to higher turbine work output. At the same time, the turbine isentropic efficiency exhibits a peak followed by a gradual decline, as predicted by the adopted off-design efficiency correlation. This decline occurs when the turbine operates under over-expansion conditions, when the pressure ratio imposed by the cycle exceeds the turbine’s optimal built-in expansion ratio, leading to increased internal losses and a reduction in isentropic efficiency.

The combined effect of these competing mechanisms explains the observed ORC efficiency trends. At moderate storage temperatures, the simultaneous increase in pressure ratio and turbine efficiency results in a strong improvement in ORC performance. At higher temperatures, although the turbine work continues to increase, the reduction in isentropic efficiency partially offsets this gain, leading to diminishing marginal improvements in cycle efficiency. These results highlight the critical role of turbine off-design behavior in ORC-based Carnot batteries and underline the importance of matching turbine characteristics to the targeted storage temperature range when optimizing system performance.

3.2. Configurations and Refrigerants: A Combined Comparison

The joint analysis of the different heat pump configurations and working fluids is carried out under fixed operating conditions, namely a storage outlet temperature of 120 °C, and a temperature lift of 50 K. These conditions are representative of high-temperature industrial heat recovery applications. The results highlight a fundamental trade-off between maximizing energy efficiency, quantified by the coefficient of performance (COP), and ensuring system compactness, quantified by the volumetric heat capacity (VHC), as illustrated in Figure 9a,b. Consequently, the optimal combination of system architecture and working fluid strongly depends on the relative importance assigned to efficiency versus compactness in the targeted application.

From an efficiency perspective, the advanced two-stage configurations, namely TS-IHX, TS-FT-IHX, and Cascade-IHX, exhibit clear superiority under the selected high temperature lift. By mitigating compression losses through pressure ratio splitting and intercooling effects, these architectures maintain higher compressor efficiencies at demanding operating conditions. Among the investigated working fluids, R-601 consistently achieves the highest COP values across all configurations, exceeding a COP of 4.5 in the best-performing cycles. This behavior is consistent with its favorable thermophysical properties, which lead to reduced specific compression work at elevated temperature levels.

In contrast, the VHC analysis reveals an almost inverse ranking. The SS-IHX configuration consistently delivers the highest VHC values, reflecting its superior compactness and reduced compressor displacement requirements. In terms of working fluids, R-600 clearly stands out as the most advantageous option for maximizing VHC, owing to its high vapor density at compressor suction. The resulting trade-off is therefore evident: R-601, while offering the highest efficiency, leads to larger system sizes due to its low volumetric capacity, whereas R-600 enables the most compact designs at the expense of a lower COP.

Under the operating conditions considered, the system selection therefore requires a clear design choice. If maximum energetic efficiency is prioritized, a two-stage configuration combined with R-601 represents the most suitable solution. Conversely, if compactness and reduced compressor size are the primary objectives, a single-stage configuration with internal heat exchange operating with R-600 is the preferred option.

To provide a synthetic overview of the relative performance improvements, Figure 10 presents radar charts comparing all configurations and working fluids with respect to a reference system, defined as the SS-IHX configuration operating with R-1233zd(E). The results are expressed as percentage deviations from this baseline, allowing the most effective upgrade strategies to be readily identified. The radar charts clearly illustrate that no single solution dominates all performance metrics simultaneously, reinforcing the necessity of a multi-criteria perspective when selecting Carnot battery charging configurations.

The radar chart in Figure 10a makes clear that performance improvements are primarily driven by architectural changes rather than by refrigerant substitution alone. Advanced two-stage configurations and the use of IHX consistently expand outward relative to the reference, indicating systematic COP gains under medium/high temperature-lift conditions. Among refrigerants, R-601 exhibits the largest relative efficiency improvements, especially when combined with two-stage architectures, confirming its suitability for high-temperature operation. In contrast, fluids such as R-600 show more limited relative gains in COP, reinforcing the observation that its advantages lie elsewhere. Overall, Figure 10a confirms that maximizing energetic efficiency requires a combination of favorable thermodynamic properties and architectures capable of mitigating compression losses.

Figure 10b presents the corresponding radar chart for volumetric heat capacity. In contrast to the COP results, the VHC radar chart is dominated by the influence of the working fluid rather than the system architecture. R-600 consistently exhibits the largest positive deviation from reference across all configurations, reflecting its high vapor density at compressor suction and the resulting compact compressor requirements. Conversely, R-601 shows pronounced negative deviations, indicating significantly lower volumetric performance and larger required compressor displacement. Architectural changes play a secondary role in this case: while SS-IHX remains the most favorable configuration in terms of compactness, two-stage and cascade layouts generally incur a VHC penalty due to increased complexity and reduced suction density. These results therefore reinforce the fundamental trade-off between efficiency and compactness, clearly illustrating that refrigerants optimized for high COP are not necessarily suitable when system size and compressor scaling are critical design constraints.

3.3. Carnot Battery Roundtrip Efficiency

While the previous sections focused on the performance of the heat pump and ORC subsystems independently, the overall viability of a Carnot battery is ultimately determined by its round-trip efficiency. This metric captures the combined effect of the charging and discharging processes and therefore provides a system-level perspective that is directly relevant for energy storage applications. In this section, the round-trip efficiency is evaluated as a function of the storage outlet temperature and the waste heat source temperature, allowing the optimal integration of heat pump configuration and working fluid to be identified across a wide operating envelope.

To this end, a single-objective optimization based on a grid-search approach is performed at each operating point. the storage outlet temperature $T_{HT}$ (90–150 °C) and the waste heat source inlet temperature $T_{src,in}$ (40–100 °C) were varied within the ranges defined in

Table 3. Best-performing operating points for the investigated working fluids and KPI of the Carnot battery.

Fluid	${\mathit{T}}_{\mathit{s}\mathit{r}\mathit{c},\mathit{i}\mathit{n}}$ [°C]	${\mathit{T}}_{\mathit{H}\mathit{T}}$ [°C]	VHC	COP [-]	${\mathit{\eta }}_{\mathit{O}\mathit{R}\mathit{C}}$ [%]	${\mathit{\epsilon }}_{\mathit{r}\mathit{t}}$ [-]	HP Config
R245fa	100	95	7.67	14.81	8.12	1.2035	SS-IHX
R600	100	102	7.94	11.52	9.06	1.0435	SS-IHX
R601	100	97	3.70	13.96	8.75	1.2205	SS-IHX
R1224yd(Z)	100	96	6.83	14.13	8.22	1.1608	SS-IHX
R1233zd(E)	100	97	6.24	13.46	8.48	1.1424	SS-IHX
R1234ze(Z)	100	97	7.85	13.42	8.30	1.1138	SS-IHX
R1336mzz(Z)	100	95	4.48	15.49	8.39	1.2989	SS-IHX

with a step of 1 °C. The resulting operating domain was discretized using a uniform grid to evaluate the round-trip efficiency for each heat pump configuration and working fluid. Only thermodynamically feasible operating points were retained, ensuring consistent cycle operation and avoiding unrealistic conditions such as near-critical operation or excessive compression ratios. In addition, compressor operating conditions were constrained to remain within reasonable pressure ratio limits to avoid unrealistic off-design operation.

For every combination of storage outlet temperature and waste heat source temperature, all candidate heat-pump configurations are systematically evaluated, and the round-trip efficiency is maximized by selecting the best-performing architecture. This approach enables the identification of both the optimal heat-pump configuration and the corresponding maximum achievable round-trip efficiency under the prescribed thermal boundary conditions.

The first column of Figure 11 illustrate the optimal heat pump configuration selected through grid-search optimization as a function of the storage outlet temperature and waste heat source temperature. Across all three working fluids, a consistent pattern emerges. R-600 and R-601 were chosen to represent the extreme behaviors observed in the previous results, whereas R-1233zd(E) was selected as the reference working fluid employed throughout the analysis. At low storage outlet temperatures and high waste heat source temperatures, single-stage configurations, particularly SS and SS-IHX, are predominantly selected. Under these conditions, the required temperature lift is moderate, pressure ratios remain relatively low, and the added complexity of multi-stage architecture does not provide a sufficient efficiency benefit to offset their additional irreversibilities.

As the storage outlet temperature increases and/or the waste heat source temperature decreases, the optimal configuration progressively shifts towards more advanced architectures, including TS-IHX, TS-FT-IHX, and cascade configurations. This transition reflects the growing importance of mitigating compression losses at high temperature lifts. By splitting the total pressure ratio across multiple stages and benefiting from intercooling effects, these configurations maintain higher compressor efficiencies and become increasingly advantageous under demanding operating conditions. The resulting configuration boundaries follow diagonal trends in the temperature space, highlighting that it is the combined effect of storage temperature and heat source quality, rather than either parameter alone, that governs the optimal system architecture.

The second column of Figure 11 presents the corresponding maximum round-trip efficiency achieved at each operating point. For all working fluids, the round-trip efficiency increases with increasing waste heat source temperature and generally decreases with increasing storage outlet temperature. Higher source temperatures reduce the required compression work during charging and improve the heat pump COP, while higher storage temperatures impose larger temperature lifts and increase irreversibilities in both the heat pump and ORC subsystems. The contour shapes clearly show regions of diminishing returns at high storage temperatures, where improvements in ORC efficiency are progressively offset by the degradation of heat pump performance.

Comparing the three working fluids, clear differences in both achievable efficiency levels and optimal operating regions are observed. R-600 exhibits its highest round-trip efficiencies at moderate storage temperatures and high waste heat source temperatures, but its performance deteriorates more rapidly at higher storage temperatures due to limitations in heat pump efficiency. R-601 consistently achieves the highest peak round-trip efficiencies, particularly at elevated storage temperatures, confirming its suitability for high-temperature Carnot battery applications where efficiency is prioritized over compactness. R-1233zd(E) displays an intermediate behavior, offering robust round-trip efficiency across a wide operating range and smoother transitions between optimal configurations. This balanced performance highlights R-1233zd(E) as a versatile low-GWP working fluid, providing a compromise between efficiency, operating flexibility, and system design constraints.

Following the operating maps presented in Figure 11, the optimal operating point identified for each working fluid is summarized in Table 3. For each refrigerant, the table reports the combination of waste heat source inlet temperature and storage outlet temperature that maximizes the round-trip efficiency within the investigated operating domain, together with the corresponding heat pump configuration and key performance indicators of the Carnot battery. This representation complements the spatial information provided in Figure 11 by condensing the optimization results into a set of representative best-performing cases.

The results show that, under the selected operating envelope, the optimal solutions for all investigated fluids correspond to high waste heat source temperatures, reflecting the strong influence of heat source quality on overall system performance. In all cases, the SS-IHX configuration is selected as the optimal heat pump architecture at the identified operating points, indicating that the temperature lifts associated with the optimal conditions remain moderate enough for single-stage compression with internal heat recovery to outperform more complex multi-stage layouts. This observation is consistent with the configuration maps presented in the first column of Figure 11, where single-stage solutions dominate regions characterized by high source temperatures.

From a comparative perspective, the table highlights clear differences among working fluids in terms of volumetric heat capacity, COP, ORC efficiency, and resulting round-trip efficiency. Fluids such as R-600 and R-1234ze(Z) achieve high VHC values, reflecting their favorable suction vapor densities and compact compressor requirements, whereas fluids like R-601 and R-1336mzz(Z) exhibit lower VHC but higher COP values, indicating a shift towards efficiency-oriented operation. The resulting round-trip efficiencies span a relatively wide range, illustrating how variations in heat pump and ORC performance combine at system level, even when the same heat pump configuration is selected.

It should be noted that the storage efficiency was assumed equal to unity in the present analysis, neglecting heat losses from the thermal storage tanks. In practical systems, sensible two-tank thermal storage typically exhibits efficiencies between approximately 90% and 98%, depending on insulation quality and storage duration. Accounting for such losses would proportionally reduce the reported round-trip efficiencies; for example, assuming a storage efficiency of 95% would decrease the reported values by approximately 5%. However, since this reduction would affect all configurations in a similar manner, it is not expected to significantly modify the comparative ranking of the investigated heat pump architectures or working fluids, which is the primary objective of the present benchmarking study.

4. Conclusions

This work presented a comprehensive thermodynamic assessment of a Rankine-based Carnot battery integrating multiple configurations of high-temperature heat pump (HTHP) and a recuperative Organic Rankine Cycle (ORC), with explicit consideration of waste heat recovery and low-GWP working fluids. A steady-state modelling framework was developed to investigate how heat pump architecture, refrigerant selection, and operating temperature levels jointly influence component-level performance and overall round-trip efficiency under realistic industrial conditions. The main findings of the study can be summarized as follows:

• Heat pump configuration plays a dominant role in system performance, with single-stage configurations with internal heat exchange providing the best performance at low to moderate temperature lifts, combining simplicity with favorable volumetric characteristics. As the temperature lift increases, two-stage and cascade configurations become more advantageous, as they mitigate compression losses through pressure ratio splitting and intercooling, enabling efficient operation under demanding high-temperature conditions.
• Refrigerant choice has a strong and non-uniform impact on performance. R-601 and R-1233zd(E) exhibit superior energetic performance at elevated temperature levels, while R-600 consistently achieves the highest volumetric heat capacity due to its high suction vapor density. These trends highlight a fundamental trade-off between efficiency-oriented designs and compact system layouts.
• The recuperative ORC benefits from increasing storage outlet temperatures, which enhances turbine work and cycle efficiency. However, turbine off-design behavior plays a critical role at high operating temperatures, where over-expansion leads to a reduction in isentropic efficiency and limits marginal gains in ORC performance.
• Grid-search optimization of the complete Carnot battery revealed that no single configuration or working fluid is universally optimal. The optimal solution depends strongly on the combination of storage temperature and waste heat source temperature, confirming the need for integrated system-level optimization rather than isolated component selection.
• The resulting operating maps show that R-600 is best suited for applications with moderate storage temperatures and high-quality waste heat sources, whereas R-601 achieves the highest round-trip efficiencies at elevated storage temperatures, at the expense of increased compressor size. R-1233zd(E) provides a balanced compromise, offering robust performance across a wide operating range while complying with stringent environmental constraints.

Under the operating conditions investigated, the electric round-trip efficiency of the Carnot battery ranges approximately between 1.04 and 1.30, depending on the working fluid and operating temperatures. It should be noted that this metric represents an electric-to-electric efficiency with waste-heat assistance, where only the electrical input to the heat pump compressor is considered in the denominator while the waste heat supplied to the evaporator acts as an external thermal co-input. Consequently, values exceeding unity do not violate thermodynamic principles but reflect the additional electrical power generated from the upgraded waste heat.

Overall, this study demonstrates that the design of high-performance Carnot batteries requires a holistic approach in which heat pump architecture, working fluid selection, and operating conditions are jointly optimized. The modelling framework and performance maps developed provide practical guidance for tailoring Carnot battery systems to specific industrial waste heat sources and storage temperature requirements.

Future work will extend the present analysis to dynamic and part-load operation, including realistic thermal storage losses, and incorporate techno-economic and environmental assessments to support deployment-oriented design and decision-making.