On the efficiency of caloric materials in direct comparison with exergetic grades of compressors

Efficiency improvements in heat pump can drastically reduce global energy demand. Caloric heat pumps are currently being investigated as a potentially more efficient alternative to vapor compression systems. Caloric heat pumps are driven by solid-state materials that exhibit a significant change in temperature when a field is applied, such as a magnetic or an electric field as well as mechanical stress. For most caloric materials, the phase transition results in a certain amount of power dissipation, which drastically impacts the efficiency of a caloric cooling system. The impact on the efficiency can be expressed by a figure of merit (FOM), which can directly be deduced from material properties. This FOM has been derived for 36 different magneto-, elasto-, electro and barocaloric material classes based on literature data. It is found that the best materials can theoretically attain second law efficiencies of over 90%. The FOM is analogous to the isentropic efficiency of idealized compressors of vapor compression systems. The isentropic efficiency can thus be directly linked to the theoretically achievable efficiency of a compressor-based refrigeration system for a given refrigerant. In this work a theoretical comparison is made between efficiency of caloric heat pumps and vapor compression systems based on the material losses for the caloric heat pump and the efficiency of the compressor for vapor compression systems. The effect of heat regeneration is considered in both cases. In vapor compression systems, the effect of the working fluid on the efficiency is also studied.


Introduction
Cooling and heating of our living space, food and water make up about 29% of global energy consumption [1][2][3]. The amount of energy for cooling is expected to grow rapidly [1]. In the future most of this heating and cooling will be performed by heat pumps [1,3]. Therefore any increase in the efficiency of heat pumps will drastically reduce global energy needs. Vapor compression systems, which are the most common type of heat pumps, achieve exergetic efficiencies of up to 50% [4,5], indicating that there is still room for improvement. Caloric heat pumps are being researched as a more efficient alternative to vapor compression systems. They use specific solid-state materials that exhibit a caloric effect i.e. magnetocaloric-, elastocaloric-, electrocaloric-or barocaloric-effect. In some of these materials this effect is highly reversible, suggesting that it should be possible to build more efficient heat pumps with these materials. However, with the exception of Chaudron et al [6], all of these systems reported an efficiency of less than 50% up to now. This can be explained by the fact that these types of systems are far less mature than classical vapor compression systems.
In this work we compare the efficiency potential of the different caloric technologies with that of vapor compression systems. For caloric systems, this maximum efficiency potential is determined by properties of the caloric materials. Hereby, a figure of merit (FOM) is derived, which directly links these material properties to the maximum attainable systems efficiency. For vapor compression systems, the corresponding property is the isentropic efficiency of the compressor. The aim of this work is to have a direct comparison between the FOM of caloric materials and the isentropic efficiency of a compressor.
It should be noted that several different figures of merits have been defined for caloric materials over the years that focus on different desirable properties. The reversible heat defined by Gottschall et al [7] focuses on the size of the caloric effect. Similar figures of merit were defined by Griffith et al [8] and by Wood and Potter [9]. The relation between the work of the field driving the caloric effect and the thermal work of the caloric effect was also proposed as a FOM by Wood and Potter [9,10]. This relation indirectly affects efficiency as field creation and the recovery of work from the field leads to additional losses [10]. This is especially important for electrocaloric materials where the smallest amount of field work is converted to thermal work [10,11]. Some other notable figure of merits include a benchmark simulation of a caloric material in a single bed regenerator proposed by Niknia et al [12] and a FOM defined by Suchaneck et al [13] that combines dissipative losses, heat transfer properties and effect size.
For the comparison, some simplifications have to be assumed: A number of different losses like irreversible heat transfer losses at the hot-and cold side, parasitic heat fluxes and losses when converting electrical to mechanical power occur in both, vapor compression and caloric cooling systems. Therefore, we will focus on efficiency losses that are specific to each technology and ignore those that are shared between the technologies.
Furthermore, at this point the efficiency of the field generation device (magnetic resp. electric field, force) is not included in the efficiency evaluation. Here it is assumed that these devices can attain a very high efficiency such that they hardly influence the total system efficiency. In fact, for electrocalorics it has already been experimentally shown that the efficiency of the field generation device can be as large as 99.74% [11]. For a caloric system, the efficiency mainly gets diminished due to hysteresis of the phase transition. Masche et al [14] showed that a small increase in hysteresis from 0.5% to 1% could reduce efficiency in active caloric regeneration systems by as much as 50%. Hess et al [15] provided a theoretical maximum performance of a caloric material in a Carnot-like cycle. Zeggai et al [16] later came to the same conclusion when looking at the efficiency of electrocaloric materials. Both of these predictions essentially rely on a more general description of dissipation in thermodynamic cycles made by Leff and Jones [17]. Carnot-like cycles can be found in cascaded systems using thermal diodes or switches [18][19][20]. However, most caloric systems such as [6,[21][22][23][24] use active heat regeneration. The present work expands the theory by Hess et al [15] by including maximum efficiency in regeneration cycles and compares the results to the efficiency of vapor compression cycles with different working fluids and isentropic efficiencies.
Neglecting heat transfer losses, the efficiency of vapor compression systems under idealized conditions is mainly influenced by the isentropic efficiency of the compressor. Thus, losses are directly linked to the cooling cycle such as throtteling and overheating of the gas during compression. These depend mainly on the working fluid of the system. Therefore the theoretical efficiency limit of vapor compression systems using a compressor with an internal heat exchanger (IHX) is provided.
The efficiency of a heat pump is defined by the coefficient of performance (COP). The COP is defined as the ratio between heat that is being removed/added ∆Q c , ∆Q h by the system and the required work W that needs to be put into the system during one closed cycle: The Carnot process has the maximum theoretical possible COP (COP c ) and is given by [25]: It only depends on the temperature of the hot side T h and the temperature of the cold side T c . Real cooling processes always suffer from irreversibilities that reduce the COP. The exergetic efficiency η is given by the ratio between the COP of the system and the Carnot efficiency COP c : For simplicity, we will only discuss cooling cycles. But all these considerations apply to heating cycles as well.

Caloric cooling cycles
During phase change of first-order caloric materials, dissipative heat q diss is produced due to the hysteresis of the material. Hess et al [15] showed that q diss can be approximated by: (4) where ∆T hyst corresponds to the temperature hysteresis and ∆s iso to the isothermal entropy change. Based on this it can be shown that the efficiency of a cascaded caloric cooling cycle can be expressed as follows [15]: with ∆T ad being the adiabatic temperature change of the caloric material. Thus, the ratio ∆T ad /∆T hyst is solely depending on properties of the caloric material. In analogy to Mönch et al [11], this defines a FOM for the caloric material. This can be used to directly quantify the maximum possible systems efficiency from the material properties.
Using equation (4) this FOM can also be written as It should be noted that for caloric materials, most parameters like ∆T hyst , ∆s iso , q diss and ∆T ad are field and temperature dependent. Therefore, it is important that all parameters used for calculating the FOM must correspond to the same field change and temperature.
This FOM can also be expressed in the form of the COP of the caloric material COP mat = qc q diss [26]: From this equation it can be seen, that for a given COP mat , the FOM gets larger, the larger the adiabatic temperature change is. As explained in Hess et al [15], the factor four in equation (5) is resulting from the optimal working point, where in the cascaded system half of ∆T ad is used for generation of the temperature lift and half of ∆s iso is used for heat transfer.
For a system based on a regenerator concept, this factor of four can be reduced to a factor of one, if ideal heat regeneration is assumed: in that case, for a single caloric element the total ∆T ad is used for generation of the temperature lift as well as the total amount of ∆s iso is used for heat transfer.
Factor a sys indicates how efficiently the material gets used in the system where q c and ∆T are the heat removed during one cycle and temperature lift generated by a single caloric element in the system and q c,max = T c ∆s iso is the maximum possible heat than can be removed where the full amount of ∆s iso is used for heat transfer. Furthermore, w rev is the reversible part of the input work w, with w = w rev + q diss .
For an ideal regenerator this leads to α sys = 1 and α sys = 1 4 for an ideal cascaded system. Thus, equation (5) can be rewritten in a more general form to: This difference in factor α sys for a cascaded system and a regenerator is visualized in figure 1. The input work of an ideal regenerator system is four times larger than that of an ideal cascaded system, giving twice as much removed heat (q c = T c ∆s iso ) as well as twice as much of a temperature lift. The effect of hysteretic losses in these cycles is shown in figure 1(b). Figure 2 shows the correlation between FOM and efficiency for ideal regenerators (green) and ideal cascades (blue). As shown in figure 2 the FOM of a material in a cascaded system needs to be four times higher to achieve the same efficiency as a regenerator. At a FOM < 10, this results in a large drop in performance, when cascading is used over regeneration. For a FOM of two, for example, this halves the exergetic efficiency. However, since several different losses occur in real systems this does not mean that a system using regeneration will always be more efficient than a cascaded system.

Efficiency of different caloric materials
Many studies on caloric materials have been made from which the FOM can be estimated. One common method is differential scanning calorimetry DSC measurements at zero field strength, measuring both the cooling and heating curves. Here ∆T ad and ∆T hyst are derived for a full phase transition as described by Hess et al [15]. Field hysteresis measurements can also be used to estimate the FOM. Here q diss can measured for a specified field change. This than needs to be coupled with the entropy and temperature change corresponding to that field change.
Many factors have been observed that have an impact on hysteresis, such as chemical composition, temperature [27,28], microstructure [29][30][31], field strength [32], field directionality [33] as well as dynamic effects [34,35]. For example, electrocaloric materials exhibit a smaller hysteresis when an unidirectional field is applied instead of a bidirectional field [36]. For most materials, only indirect measurements are available. For accurate quantification of the FOM, the caloric material should be characterized in conditions as close as possible to the conditions in the system.  It should be noted that for some of the LaFeSiX compounds the FOM was within the detection limit, so the actual FOM could be higher than shown is this figure. The FOM of gadolinium (Gd) lies outside of the graph. This is marked by the blue arrow.

FOM of magnetocaloric materials
The most common magnetocaloric materials are gadolinium and lanthanum-iron-silicon (LaFeSi) based alloys. gadolinium has a second order phase transition and should therefore have no thermal hysteresis, i.e., ∆T hyst and q diss are zero. For some of the LaFeSi based alloys (table 1), ∆T hyst was within the detection limit of the experimental setup, so the exact upper limit is unknown. The magnetocaloric materials in table 1 typically require fields of more than 10 T for a full phase transition. Current magnetocaloric demonstrators are based on permanent magnets which produce fields of around 1 T. Hence some differences between values reported in this study and actual performance in a system can be expected. The effect of the FOM ranges of the material on the maximum exergetic efficiency can be seen in figure 3. Table 2 is based on DSC measurements of more than 100 different elastocaloric alloys from 14 different publications ( figure 4). In comparison to magnetocaloric materials, elastocaloric materials show much lower FOMs. It should be noted that many of these alloys were originally developed for the use in actuators. These alloys are thus not yet optimized towards elastocaloric cooling and the requirement of a small hysteresis.

FOM of electrocaloric materials
For electrocaloric materials such as polyvinylidene fluoride (PVDF) based polymers and lead magnesium niobate-lead titanate (PMN-PT) the adiabatic temperature change remains constant over a wide temperature range while hysteresis can increase strongly at lower temperatures. Data published by Chen et al [69] show a reduction of hysteresis by a factor of 7 by increasing temperature from 20 • C to 100 • C.
Several field hysteresis loops have been published for electrocaloric materials. The dissipative heat of the electrocaloric material in a cooling cycle corresponds to a unipolar loop. Table 3 shows an overview of the FOM of different electrocaloric materials calculated from data published in literature ( figure 5). In cases where only bipolar hysteresis loops were published,q diss was assumed to be half of the full hysteresis. This is a conservative estimate and might overestimate dissipative heat generated by unipolar cycling. Nair et al [36] [36].  published both bi-and unipolar field hysteresis measurements for lead scandium titanate (PST). In that case q diss was reduced by more than a factor of four.

Vapor compression system
Just like the efficiency of a caloric cycle depends to a large part on the FOM, the efficiency of a vapor compression cycle depends on the isentropic efficiency η c of the compressor. The vapor compression cycle shown in figure 7(a) differs from the Carnot-process. Due to technical limitations compression has to take place in the gas-phase [88]. This results in-depending on the applied refrigerant-some overheating of the discharge gas. Ideal compression for refrigeration cycles is assumed to be adiabatic and loss free and thus isentropic. However, compression is associated with some dissipation described by the isentropic efficiency of the compressor. Typically, a throttle valve is used to expand the fluid from high pressure to low pressure side. This causes additional losses. Depending on the refrigerant, the use of an IHX can further increase the cycle efficiency and will be described later.
The idealized vapor compression cycle (figure 7(a)) can be divided into the following steps: 1 → 2 s /2: Starting from the evaporation curve, vapor gets compressed by the compressor. This causes vapor to reach a temperature above the condenser temperature. In an ideal and adiabatic compressor this step is isentropic and the vapor gets compressed to the point 2 s. Real compressors exhibit losses due to friction, causing the entropy to increase slightly during this step so that point 2 is reached at the end of this step 2 s/2 → 2 ′ : The overheated gas gets cooled down isobarically by the condenser until it reaches the evaporation curve.
2 ′ → 3: After reaching the wet steam area the fluid gets liquified. This process is isothermal and isobaric. 3 → 4: The liquid working fluid gets expanded by throttling. This process step is in good approximation isenthalpically.
4 → 1: Heat intake to the liquid/vapor and evaporates it. Like 2 ′ -3 this process is isobaric an isothermal as well.
The irreversibility of a real compressor can be considered by the isentropic efficiency η c of the compressor [88]. This is in accordance with international standards for performance measurements for refrigeration compressors, such as DIN EN 12 900 [89] or AHRI 540 [90], respectively. Assuming an adiabatic compressor for the considerations done here, the isentropic efficiency can be defined by: where h 1 , h 2 , h 2s and h 4 are the specific enthalpies at the points specified in figure 7. As a rough range of numbers, typical compressors have an isentropic efficiency of 40%-80% depending on type and operational conditions. The COP of the vapor compression system COP VC is defined as the cooling powerQ c divided by the compressor power input P v . Hence, with the help of equation (10) the COP VC as a function of the isentropic efficiency can be derived [25]   In analogy to the caloric regenerator concept, the efficiency in a vapor compression system might be further increased by using an IHX. The IHX transfers thermal energy from the working fluid before expansion to the vapor prior to being compressed. This causes liquid to be further subcooled before expansion and the vapor to be more superheated before compression. The resulting cooling cycle can be seen in figure 7(b). It has to be mentioned that IHX application not necessarily leads to better cycle COP. Whether an IHX is beneficial or not is dependent on the refrigerant and the operational conditions. For the here discussed refrigerant isobutane, an IHX is beneficial, for others not, such as ammonia. Figure 8 shows the theoretical performance limits of different fluids in vapor compression cycles with a cold side of 5 • C and a hot side of 55 • C. For the calculations the fluid properties provided by NIST [91] were used and are provided in table 5. A hot side of 55 • C is typical for vapor compression systems an based on the norm EN 12 900 [89]. The cold side of 5 • C is on the warm end for most cooling applications. For applications requiring colder temperatures, the efficiency will further decrease, as losses due to throttling increase. At an isentropic efficiency of 100%-no losses in the compressors-the efficiency of the vapor compression system stays below 80% due to throttling losses and overheating during compression. As shown  Figure 9. Comparison of the achievable efficiencies of the different caloric technologies with classical compressors. FOM ranges of the best magnetocaloric materials (green), barocaloric materials (blue), electrocaloric (yellow) and elastocaloric materials (orange). The black curve shows the maximum achievable exergetic efficiency of a compressor system using isobutane (T h = 55 • C, Tc = 5 • C) as a function of the isentropic efficiency. The hatched area indicates the typical range of isentropic efficiency for today's systems. Some experimental caloric systems of magnetocalorics (green), electrocalorics (yellow) and elastocalorics (orange) are marked as asterisks.
in figure 8(b) an IHX can further improve the efficiency, but the effect is much smaller compared the effect regeneration has in a caloric system.

Discussion
Based on the calculations made in the previous sections a comparison can be made between the efficiency potentials of vapor compression systems and caloric cooling systems. In this comparison, loss types that are shared between the two technologies are ignored. The remaining efficiency reducers are hysteresis losses described by the FOM for caloric systems and losses during compression described by η c for vapor compression systems. Figure 9 shows a comparison of the maximum achievable exergetic efficiency as a function of the FOM for caloric systems and as a function of the isentropic efficiency η c for vapor compression systems. The diagram assumes that caloric systems would already be able to overcome the temperature spread (5 • C…55 • C) associated with the refrigeration cycle. The FOM ranges of the best materials of their respective group are marked in green (magnetocaloric), yellow (electrocaloric), orange (elastocaloric). The range of today's compressors is marked by the hatched pattern. Additionally, figure 9 shows a selection of experimentally realized caloric systems and the exergetic efficiency they achieved. The highest experimental efficiency of a caloric system is 60% published by Chaudron et al [6]. The elastocaloric system with the highest reported efficiency is that of Kirsch et al and Welsch et al [23,24]. Based on simulations it reached an exergetic efficiency of about 30%. This shows how regeneration can positively impact efficiency, since it would be impossible to reach the same efficiency with the same material in a cascaded system. The highest published value for an electrocaloric system with 16% is shown by Wang et al [92]. With optimizations to the system an efficiency of 56% is predicted. To date no efficiency for a system using barocaloric materials has been published.
As shown in figure 9 baro-and magnetocaloric materials have already reached a point where the systems could reach efficiencies that are not even theoretically possible in a vapor compression system. Even with elasto-and electrocaloric materials, exergetic efficiencies comparable to those of a compressor should be possible when active regeneration is used. However, real caloric systems with heat transfer losses will be more sensitive to hysteresis losses of the material. This is because heat transfer losses reduce cooling power and reversible work while q diss stays constant [93].
It is also important to note that the efficiency of caloric materials is still improving. Highest reported FOMs for both, electro-and barocaloric materials, almost tripled in the last three years [32,80].

Conclusion
A method was developed to compare the maximum efficiency of vapor compression cooling systems with caloric cooling systems, considering the isentropic efficiency of the compressor for compression systems and the FOM for caloric systems.
For caloric cooling systems the maximum efficiency was derived for both, cascaded systems and systems using active regeneration. For FOMs < 10 it was shown that active regeneration can significantly increase the maximum efficiency.
Based on data found in literature, the material efficiency was derived for several different magneto-, elasto-, electro-and barocaloric materials. In a cascade configuration the best caloric materials can reach exergetic efficiency of almost 100% for magnetocaloric materials due to very low hysteresis losses, 53% for elastocaloric materials, 78% for electrocaloric and 83% for barocaloric materials. For an ideal regenerator this increases up to 100% for magnetocaloric materials, 82% for elastocaloric materials, 93% for electrocaloric materials and 96% for barocaloric materials. All the gained values neglect heat transfer losses within regenerator and cascade. These additional losses further reduce the effective efficiency of a system.
Magneto-and barocaloric materials are already at a point where systems are possible, that would even surpass a vapor compression system with a 100% efficient compressor. Nevertheless, up to now no system has been published where the efficiency exceeds 60% of Carnot. Thus, further effort in the development of caloric systems is required to get closer to the potential efficiency limit of caloric materials.

Data availability statement
No new data were created or analysed in this study.