Numerical modelling and experimental validation of a regenerative electrocaloric cooler Modélisation numérique et validation expérimentale d’un refroidisseur électrocalorique régénératif

This paper reports on the research and development of a cooling device with an active electrocaloric regenerator (AER) based on the bulk ceramic material (1-x)Pb(Mg (PMN-100xPT). For the purposes of the study a 2D transient numerical model of the AER was developed. This model makes it possible to investigate the cooling characteristics of a device with an AER while considering the effect of the hysteresis of the electrocaloric material and the effect of the electric-energy recovery related to the polarization/depolarization process. The results of the numerical analyses show that the degree of electric energy recovery has a major impact on the eﬃciency of the device. By considering an idealised system for electric-energy recovery the energy eﬃciency (expressed by the coeﬃcient of performance) of the device could be increased by up to ten times. A validation of the numerical model was performed through the design, construction and experiments on an improved AER cooling device. The results revealed a maximum speciﬁc cooling power of 16 W kg − 1 and a maximum temperature span of 3.1 K for the new device.


Introduction
Electrocaloric (EC) cooling is a technology based on the properties of certain dielectric materials that tend to heat or cool under the influence of an external electric field change. The technology is considered as an alternative to vapour-compression refrigeration, Peltier cooling and some emerging technologies such as magnetic refrigeration ( Ožbolt et al., 2014a;Correia and Zhang, 2014 ). Kobeko and Kurtschatov (1930) were first to report about the electrocaloric effect (ECE). Later, in the 1960 s and 1970 s researchers ( Hegenbarth, 1961( Hegenbarth, , 1965Radebaugh et al., 1979 ) sug-gested the use of electrocaloric materials for cryogenic refrigeration. However, since the EC effect for known materials was less than 1 K, no attempt was made to develop a prototype ( Radebaugh et al., 1979 ).
In the early 1990 s Sinyavsky (1995) argued that EC cooling is one of the most promising alternatives for new cryogenic and refrigeration units. The work of Sinyavsky and Brodyansky (1992) resulted in the development of the first EC proof-of-concept device. The discovery of the so-called giant EC effect by Mischenko et al. (2006) and the characterization of numerous EC materials in the subsequent years ( Valant, 2012;Moya et al., 2015 ) opened up new opportunities for the design of EC cooling devices. Until now, different device designs emerged and some of them were also experimentally tested ( Kitanovski et al., 2015;  span, cooling power) of the first prototype devices do not yet match the real application conditions ( Kitanovski et al., 2015;. One of the concepts of EC cooling devices is based on the active electrocaloric regenerator (AER), which is well known in the field of magnetic refrigeration ( Kitanovski et al., 2015 ) and was first introduced to electrocaloric cooling in the early 1990 s by Sinyavsky and Brodyansky (1992) . So far, three prototypes of EC cooling devices with an AER have been developed Plaznik et al., 2015a;Sinyavsky and Brodyansky, 1992 ). To the best of our knowledge, EC materials with the largest EC effects (up to a few 10 K of temperature change), such as thin-film ceramics and thick-film polymer materials ( Valant, 2012;Moya et al., 2015 ), are not yet suitable for use in devices with an AER. The first devices with an AER were constructed from EC materials with an EC temperature change in the range from 0.5 K to 1.5 K ( Kitanovski et al., 2015;. The temperature span (i.e., the temperature difference between the heat sink and the heat source of the device) of these devices was in the range of a few K, usually by a factor greater than the temperature change of the EC material. We found no reports about the experimentally measured cooling power or the coefficient of performance (COP) for a cooling device with an AER. However, there are theoretical studies on such performance ( Aprea et al., 2016a;Guo et al., 2014;Ožbolt et al., 2014b ). In the theoretical analyses of AER's performance carried out so far the irreversibilities due to the electric-polarization hysteresis and the irreversibilities of the system for electric-energy recovery were neglected ( Aprea et al., 2016a;Plaznik et al., 2015b ). Therefore, their influence on the AER's performance is not known. The literature review also shows that a comparison between the theoretical and experimental analyses of an AER has not been carried out so far.
The aim of this study was to present a 2D numerical simulation model of the AER's performance, which includes the effect of electric-polarization hysteresis and the effect of the irreversibilities of the system for electric-energy recovery. The general description of the two effects can be found in Moya et al. (2015) and in Plaznik et al. (2015c) . Heat gains/losses are included in the model and a comparison between the theoretical and experimental analyses of the AER is presented. For the purposes of the study an improved experimental setup was built and the performance of the AER device was tested for different operating parameters. At the end of the paper the numerical model is validated by a comparison between the theoretical analysis and experimental results.

Modelling of a cooling device with an AER
A simple cooling device with an AER is schematically presented in Fig. 1 (a). The core part of the device is the AER, which is located in a closed housing. The housing is filled with the working fluid, which, in the absence of any electrical insulation, has to be a dielectric fluid (i.e., silicone oil). The electrodes of the EC material are connected to a power source. A pumping mechanism forces the fluid from the heat exchanger adjacent to the ambient (HHX), to the heat exchanger adjacent to the cooled chamber (CHX) and vice versa. The device's operation consists of four main steps: (1) polarization (charging) of the EC material, (2) fluid flow from the CHX to HHX, (3) depolarization (discharging) of the EC material and (4) fluid flow from the HHX to CHX. A more detailed description of the operation of a device with an AER can be found in Kitanovski et al. (2015) .
To the best of our knowledge all the EC materials have so far been prepared in the shape of plates or sheets. Other geometries, such is a packed bed of electrocaloric particles, are at the moment too challenging to make. Such a configuration should namely consist of numerous electrodes providing the electric field to the EC particles with rather complex connections to the power source. Therefore, here we consider the modelling of a device with a parallel-plate AER.
The energy efficiency of a cooling device can be described by its coefficient of performance (COP). For a simple cooling device with an AER, operating under steady-state conditions, the COP can be expressed as: where ˙ Q c is the time-averaged cooling power of the device, ˙ W PM is the time-averaged power needed to drive the fluid-pumping mechanism and ˙ W PS is the time-averaged power needed to drive the power source. The term ˙ W PS can be further expressed as the sum of the time-averaged power need to polarize (charge) the EC material ( ˙ W pol . ) and the time-averaged power which can be regenerated when the EC material is depolarized ( ˙ W reg . ). A more detailed explanation of the energy flows during the proces of polarization and depolarization is given in Section 2.2 .

Model of the AER
The modelling of an AER is analogous to the modelling of an active magnetic regenerator (AMR). The approaches used to model an AMR can also be used to model an AER. A number of papers dealing with modelling an AMR can be found in the literature ( Kitanovski et al., 2015;Nielsen et al., 2011 ). The model presented in this paper is a 2D numerical model of a parallel-plate AER. When modelling a parallel plate AER (because of the symmetry) the AER geometry can be reduced to half of the EC plate and half of the fluid-flow channel as described by Nielsen et al. (2009a) and Petersen et al. (2008) (see also Fig. 1 (b) and (c)). The governing energy equations for the EC plate ( Eq. 2 ) and the fluid ( Eq. 3 ) are: where c E is the specific heat capacity of the EC material at a constant electric field, s gen is the specific entropy, generated due to the hysteresis of the EC material, s T is the specific entropy of the EC material at a constant temperature, c f is the specific heat capacity of the fluid, ρ is the density, T is the temperature, t is the time, u is the velocity, λ is the thermal conductivity, μ is the dynamic Table 1 General boundary conditions.
˙ q x = 0 at x = 0 and at x = L ˙ q y = 0 at y = 0 and at y = d f / 2 + d s / 2 viscosity and ˙ Q ab is the heat transfer from the ambient to the AER (heat gains). The index "s" denotes the solid EC material and the index "f" denotes the fluid.
The general boundary conditions, which are applied to the Eqs. (2 ) and ( 3 ) are listed in the Table 1 . In order to simulate the four steps of the cooling device with an AER, additional boundary conditions (listed in Table 2 ) are applied, depending on the step type. A more detailed explanation is given below for the processes of polarization and depolarization, since they are specific to the field of electrocaloric energy conversion. In these two steps we assume that the fluid is at rest and that the time step is zero. The consequence of these two conditions is a zero specific heat flux, therefore the processes of polarization and depolarization are considered to be adiabatic. We also assume that the electrocaloric material is subjected to an electric field change (positive during the process of polarization and negative during the process of depolarization) which results in the adiabatic temperature change of the EC material. By considering Eq. (2) the adiabatic temperature change is expressed in terms of the change of the specific entropy of the EC material when the material is exposed to the electric field change. A more detailed description of coupling the adiabatic temperature change and specific entropy change is given in Brey et al. (2014) , Ožbolt et al. (2014a) and Pirc et al. (2011) . The governing equations are solved numerically using the control-volume method ( Patankar, 1980 ). For the spatial discretization the centraldifference scheme was used, and for the time discretization the fully implicit scheme was applied. The velocity u f profile of the fluid was calculated using the expression for the fully developed laminar flow between two parallel plates ( Shah and London, 1978 ). The heat gains were modelled by the adoption of the approach presented by Nielsen et al. (20 09a,20 09b ).
For the given input parameters (e.g., AER length, ambient temperature, temperature of the cooled chamber, etc.), the cooling power in the steady state can be calculated as: where ˙ m f is the mass flow rate of the fluid and T f is the bulk temperature of the fluid exiting or entering the CHX. The bulk Table 2 Boundary conditions specific to the four steps of the AER thermodynamic cycle.

Energy equation for fluid ( Eq. 2 ) Energy equation for solid ( Eq. 3 ) Electric field
Polarization temperature of the fluid is calculated as: where d f is fluid channel height (see also Fig. 1 (b)).

Power source with a system for electric-energy recovery
The power source is connected to the electrodes of the individual EC elements and provides the energy needed to polarize (i.e., to charge) the EC material. Once the EC material is depolarized a part of the charging energy can be recovered. In order to utilize this energy, the energy must be stored and used in the next charging cycle of the EC material. A special system for electric-energy recovery should be developed for this purpose. It was shown that such a system is feasible ( Campolo et al., 2003 ); however, to the best of our knowledge, there is no report of such a system for electricenergy recovery designed/constructed for EC cooling devices. Due to the irreversibilities related to the electric-energy recovery not all of the energy released in the process of the EC material's depolarization can be utilized in the next cycle. The charging and discharging of the electrical energy of the EC material depends on the dielectric properties of the EC material and on the type of the thermodynamic cycle that is performed by the EC material. Fig. 2 (a) presents the thermodynamic cycle of a part of an EC material in a temperature -specific entropy ( T -s ) diagram, and Fig. 2 (b) presents the cycle of an electric field displacement -electric field ( D -E ) diagram. The process 1-2 is the charging of the EC material, the process 2-3 is the heat transfer from the AER to the fluid, the process 3-4 is the discharging of the EC material and the process 4-1 is the heat transfer from the fluid to the AER. The electric power needed to drive the power source can be defined as: where V EC is the volume of the EC material in the AER, ν is the frequency (number of thermodynamic cycles per unit time), E is the electric field, D is the electric-field displacement, and η er is the degree of electrical energy recovery expressed in percentages ( Plaznik et al., 2015c ). From the general laws of thermodynamics we also know that the work per unit of time that needs to be done on the EC material can be written as: where s is the specific entropy of the EC material and s gen the specific entropy generated in the EC material due to the hysteresis. By combining Eqs. (6) and (7) we obtain the expression for the electric power needed to drive the power source:

Fluid-pumping mechanism
The power required to drive the system for fluid pumping is calculated as the power needed to overcome the fluid pressure drop over the AER divided by the efficiency of the fluid-pumping mechanism: where p is the pressure drop over the AER, V is the amount of the fluid pumped through the AER in step (2) and step (4) of the device's operation and η PM is the overall efficiency of the fluidpumping mechanism. The pressure drop is calculated using correlations for the fully developed laminar flow between two parallel plates ( Shah, 1978 ).

Implementing the properties of the electrocaloric material
The s T , s gen and c E of the EC material needed for the numerical model (see Eqs. (2) and (3) ) can be calculated from the adiabatic temperature change of the EC material, which is the most commonly reported EC material property ( Kitanovski et al., 2015;Liu et al., 2016;Valant, 2012 ). By considering the general laws of thermodynamics the entropy change of the EC material at constant temperature can be written as (see also Fig. 2 (a)): The generated specific entropy ( s gen ) due to the hysteresis in the EC material can be calculated as presented by Brey et al. (2014) : where P irr is the irreversible polarization. By considering Eq. (11) and rewriting s 1-4 , Eq. (10) can be expressed as: where c E 0 is the zero-field specific heat capacity of the EC material. Finally, the heat capacity of the EC material at a constant electric field can be calculated as: where s is the entropy of the EC material obtained by adding s T to the zero-field specific entropy of the EC material (see also Fig. 2 (a)). The zero-field specific entropy can be calculated from the measured zero-field specific heat capacity of the EC material.

Analysis of the influence of the hysteresis and the degree of electric-energy recovery on the performance of a device with an AER
The numerical model makes it possible to analyse the influence of the electric-polarization hysteresis and the influence of the efficiency of the system for electric-energy recovery on the performance of the AER. The geometrical parameters of the AER, the boundary and the operating conditions considered in the theoretical study are presented in Table 3 . Heat gains from the environment were neglected for the purposes of the analyses and the bulk Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) ceramic material was selected as the EC material. The EC properties of the PMN were calculated based on the experimental data ( Roži č et al., 2011a;Urši č et al., 2016 ) via the procedure presented in Section 2.4 . The calculated EC properties of the PMN material are presented in the Appendix.
First, we analysed the influence of the hysteresis on the performance of the device with an AER. The specific cooling power and COP were calculated for a wide range of the ratios of the displaced fluid ( V * ). The V * is defined as the ratio between the fluid pumped through the AER in step (2) and step (4) of the device's operation and the volume of the fluid in the AER. For selected operating conditions (device frequency, electric field change) and for the selected geometrical parameters (AER length, EC plate thickness) the maximum cooling power ( ˙ q c , mx ) and the maximum COP (COP mx ) as a function of V * were calculated. The results of the analysis are presented in Fig. 3 . The ˙ q c , mx and COP mx as a function of frequency were calculated for two cases. In the first case, the irreversibilities due to the hysteresis were neglected and in the second case, the irreversibilities due to the hysteresis were taken into account.
The difference in ˙ q c , mx between the two cases was found to be less than 2.5% for all frequencies analysed, thus showing that the hysteresis has a marginal impact on ˙ q c , mx . We can observe that a peak value of ˙ q c , mx is achieved at the frequency of 1 Hz. In general, by increasing the frequency, the number of thermodynamic cycles per unit of time, to which the EC material is subjected, increases, and to some extent this results in an increase of the ˙ q c , mx . However, by increasing the frequency, the time of the fluid flow in steps 2 and 4 of the device operation is decreasing, which means that there is less and less time for the heat transfer from the EC material to the fluid and vice versa. The decreased time for the heat transfer can result in a lower cooling power of the device. The combination of these two factors gives the optimum operating frequency in terms of ˙ q c , mx . It is expected that by increasing the effectiveness of the heat transfer between the EC material and the fluid flow, for example, by decreasing the EC plate thickness and by decreasing the fluid channel height, the optimum frequency will be higher. Similar findings were observed by authors investigating the performance of devices with magnetocaloric and elastocaloric active regenerators ( Aprea et al., 2016b;Lionte et al., 2015;Qian et al., 2015;Tušek et al., 2015 ).
On the other hand, we observed that the COP mx decreases between 10% and 15% when the hysteresis is considered. This is an unexpected result, since the PMN ceramic has a rather slim hysteresis loop (see Appendix), especially when compared with other EC materials, such as BaTiO 3 ( Qian et al., 2014 ), Pb(Zr 0.95 Ti 0.05 )O 3 ( Mischenko et al., 2006 ) and polyvinylidene-fluoride-based polymers ( Chen et al., 2013 ). This leads to the conclusion that in order to realistically predict the energy efficiency of a cooling device with an AER the irreversibilities due to the hysteresis must be taken into account, even when the hysteresis looks slim and could be mistakenly considered negligible. We also note that the COP mx is decreasing by increasing the frequency. To provide an explanation for the decrease of COP mx with an increase in frequency let us first note that for the parameters analysed the specific cooling power changes with frequency by a relatively small amount ( Fig. 3 (a)). On the other hand, by increasing the frequency, the work input per unit of time, which is the sum of the work needed to drive the fluid pumping mechanism and the work needed to drive the electric power source, increases. Since the COP is the ratio of the cooling power and the work input per unit of time, the COP mx is decreased by increasing the frequency.
Next, we analysed the impact of the electric-energy recovery efficiency on the COP mx . The results are presented in Fig. 4 . The analysis was performed only for the case in which the impact of the hysteresis was taken into account. It should be pointed out that the efficiency of the electric-energy recovery does not influence the specific cooling power of a device with an AER, but it affects significantly the COP, as reported here. In Fig. 4 COP mx is plotted as a function of the degree of electric-energy recovery for three different frequencies. For all three frequencies analysed the COP mx increases by an order of magnitude when the degree of electricenergy recovery is increased from 0 to 100%. For example, when the degree of electric-energy recovery when the device operates at 2 Hz is increased from η er = 0% to η er = 100% the COP mx increases from 0.53 to 5.45, which is more than 10 times.
In order to better understand why the degree of electric-energy recovery has such a strong influence on the energy efficiency  of the device let us have a look at Fig. 4 (b), which shows the relative shares of the power inputs needed to drive the cooling device with an AER without electric-energy recovery. A part of the total power input is dissipated as heat due to the hysteresis; a part of the power input is needed to pump the fluid; a part of the total power input is needed to drive the thermodynamic cycle of the EC material; and the rest of the power input is the energy dissipated during the EC material's discharging. Fig. 4 (b) shows that the amount of electrical energy dissipated when discharging the EC material is 154 times larger than the fluid pumping energy, 88 times larger than the energy dissipated due to the hysteresis and 11 times larger than the energy needed to drive the thermodynamic cycle of the EC material. In order to increase the energy efficiency of the device as much as possible, the energy released when discharging the EC material should be recovered and reused in the next cycle of the device's operation. This can be achieved with an efficient system for electric-energy recovery. The development of an efficient system for the electric-energy recovery and a realistic estimation of the achievable degrees of electric-energy recovery are very important challenges for future work.

Experimental validation of the numerical model
For the purpose of this study an improved experimental cooling device with an AER was built. The device is an upgraded version of a device previously presented by Plaznik et al. (2015a) . The AER was assembled from five stacks of 0.9Pb(Mg 1/3 Nb 2/3 )O 3 -0.1PbTiO 3 (PMN-10PT) bulk ceramic in the form of thin plates (10 mm × 20 mm × 0.20 mm) with golden electrodes deposited on the faces. Each stack was constructed from 9 individual EC plates positioned on top of one another. Between the EC elements, distance holders with a thickness of 125 μm were inserted to create a void for the fluid flow and to create the electric connections between the electrodes of the EC material and the power source. The length of the individual EC element was 20 mm and the total length of the AER, including the spacings between stacks, was 110 mm. Fig. 5 is a schematic presentation of the device with its auxiliary parts. On one side of the AER a simple copper tube heat exchanger was inserted to act as the heat sink, and on the other side of the AER a thin-film electric heater was inserted into the housing to act as the heat load. The heat exchanger was connected to a thermostatic bath to ensure a constant temperature of the fluid entering the heat exchanger. The housing of the AER was filled with silicone oil and made from transparent material (see also Fig. 6 (b)) so that we were able to detect any irregularities, such as air bubbles in the silicon oil or cracks in the EC elements, which can occur during assembly.
To measure the temperature span of the AER, i.e., the temperature difference between the hot and the cold sides of the AER, K type thermocouples were inserted into the housing, as presented  in Fig. 6 (a). The signals from thermocouples were recorded using NI 9214 data aqusition device. The temperature measuring system was calibrated using a thermostatic bath Julabo FPW 50 with the temperature stability of ± 0.01 °C and using a reference PT100 1/3 DIN B sensor. The ratio of the displaced fluid (for the definition of the ratio of the displaced fluid see Section 3 , second paragraph) was determined based on the number of turns of the stepper motor (Watson-Marlow 114ST stepper motor) which was used to drive the peristaltic pump. The volume of the displaced fluid per stepper motor turn (we named it K factor) was determined by measuring the mass of the displaced fluid per stepper motor turn with the Exacta 3600EB laboratory weight scale. Measurement uncertainty analysis of the K factor was performed and was later used to estimate the measurement uncertainty of the ratio of the displaced fluid. The electric power of the electric heater was measured using EX 520 and Victor VC97 measuring devices. The measured parameters (temperature span, ratio of the displaced fluid and specific cooling power) are determined indirectly and therefore we estimated their relative combined measuring uncertainty. The number of repeated experiments was not large enough to evaluate the statistical uncertainty (type A) and we only evaluated the uncertainty of type B based on the specifications of the accuracy of the measuring equipment. We assumed uniform (rectangular) probability distribution. Measuremet uncertanties for typical values of measured paramters are given in Table 4 .
The temperature span for the different values of the ratio of the displaced fluid ( V * ) was measured when the device was operated at a frequency of 0.65 Hz. The result of a typical temperature measurement performed at a selected V * is presented in Fig. 7 . The temperature of the fluid coming from the thermostatic bath and entering the HHX was set to 296 K. The temperature of the fluid on the hot side of the AER in contact with the HHX was in general 0.5 K higher than the temperature of the fluid entering the HHX from the thermostatic bath. In the steady state a temperature difference T AER between the hot and cold sides of the AER can be determined, as shown in Fig. 7 . The temperature span T AER of

Table 4
Measurement uncertanties for typical values of measured parameters.

Paramter
Typical value Temperature span ( T AER ) 1 K ± 0.25 K 2 K ± 0.25 K 3 K ± 0.25 K Ratio of the displaced fluid ( V * ) 0.8 ± 0.016 0.5 ± 0.025 0.2 ± 0.006 Specific cooling power ( ˙ q c ) 4 W kg − 1 ± 0.075 W kg −1 8 W kg − 1 ± 0.12 W kg −1 16 W kg − 1 ± 0.20 W kg −1 the AER was measured for an electric field change of 28 kV cm −1 and 57 kV cm −1 . The results of the experimental analysis are presented in Fig. 8 (data with solid line). The maximum temperature span of 3.3 K was achieved at V * = 0.2 under electric field change of 57 kV cm −1 .
Next, we fixed the operating conditions ( ν = 0.65 Hz, V * = 0.4) and measured the temperature span T AER of the device with the electric heater turned on, simulating the cooling power of the device. Again, the temperature of the fluid entering the HHX from the thermostatic bath was set to 296 K, and the temperature of the fluid in the AER in contact with the HHX was in general 0.5 K higher. The results of the analysis are presented in Fig. 9 (data with solid lines). The maximum specific cooling power, measured for an electric field change of 57 kV cm −1 and at a temperature span of 1 K accounted for 16 W kg −1 (per mass of electrocaloric material). As shown in Fig. 9 , the specific cooling power is linearly decreasing with the increasing temperature span, which is typical for devices with an active regenerator ( Tušek et al., 2013;Engelbrecht et al., 2011 ). Next, the temperature span and the specific cooling power were calculated by using the numerical model. The input parameters of the numerical model were set to match the properties of the experimental device. Data from literature Vrabelj et al., 2016 ) was used to calculate the EC properties of the PMN-10PT ceramic EC material using the procedure described in Section 2.4 . The results of the numerical analysis ( Fig. 8 , data with dashed lines) show that the heat transfer from the device to the environment has a strong impact on the temperature span of the device, especially when V * are smaller than 0.5. For example, when E = 57 kV cm −1 and V * = 0.1, the calculated temperature span accounted for 20 K when heat gains were neglected and the calculated temperature span was 6.4 K when heat gains were taken into account. When heat gains from the environment are considered in the simulations, the measured and calculated data show a good match, with a similar trend of changing the temperature span as a function of V * for both E analysed. The comparison between the measured and the calculated specific cooling power (heat gains included) of the device is presented in Fig. 9 . It is again possible to observe a deviation between the absolute values of the measured and calculated specific cooling powers. However, the trends describing the relation between the specific cooling power and the temperature span of the device are similar for both cases.
In general, the numerical model correctly predicts the variations of the cooling characteristic of a device with an AER with respect to different parameters. On the other hand, we observed a deviation between the measured and the calculated values of the cooling characteristic. The reason for this deviation could lie in a number of effects that were not included in the numerical model. The 2D numerical model with the simplified AER geometry does not allow for the simulation of the flow maldistribution effect ( Nielsen et al., 2014 ) and the dead volume effect ( Kitanovski et al., 2015;Liu et al., 2014;Zimm et al., 2018 ). The active part of the EC material (the part of the EC material under the external electrodes) accounts for 80% of the total EC material volume. Furthermore, distance holders are glued to the EC material. The inactive part of the EC material and the distance holders act as a sort of par- asitic thermal capacitance. It is expected that part of the heat generated due to the EC effect is quickly transferred from the active part of the EC material to the parasitic thermal masses. Therefore, the temperature change of the EC material with respect to the fluid is lowered, which leads to a loss in the efficiency of the heat transfer between the EC material and the fluid. Since interpolation and extrapolation methods are used to describe the EC material properties, the properties used in the simulations do not completely match the actual properties of the EC material. Lastly, some constructional characteristics, such as the gap between the stacks of the EC material, were not implemented in the model. To better predict the absolute values of the cooling characteristics of the AER, the above-described effects should be implemented in the numerical model.

Discussion and conclusions
A numerical model of a cooling device with an AER, which serves for the simulation of the dynamic operation of the electrocaloric cooler with the AER is presented. The model considers the geometrical and operating parameters of the AER, the influence of the material hysteresis, the influence of the efficiency of the system for electric-energy recovery and the influence of heat transfer from/to the environment.
For the purposes of the validation of the numerical model, an improved experimental cooling device with an AER was built. The temperature span of the device and its specific cooling power were measured.
We demonstrated that the hysteresis and the efficiency of a system for electric-energy recovery have a strong impact on the COP of a device with an AER. Therefore, they should be considered with all types of EC cooling devices, especially when investigating the device's efficiency. The energy recovery was not implemented in the experimental device presented in the paper and for now only the theoretical studies investigating the impact of the energy recovery on the energy efficiency of the device with an AER have been performed. We are planning to implement the energy-recovery system in the actual device when developing the next generation of the AER device.
The experimental results show a maximum temperature span of 3.3 K and the maximum specific cooling power of 16 W kg −1 for a temperature span of 1 K. This is, to the best of our knowledge, the first report of a measured specific cooling power for an EC cooling device with an AER.
The comparison of the results of the numerical model with the experimental results shows that if heat gains to the environment are considered in the numerical model, the model correctly predicts the changing of the cooling characterises of a device with re-spect to different operating parameters. On the other hand, effects such as flow maldistribution and parasitic thermal masses should be implemented into the model to improve its capability to accurately predict the absolute values of the cooling characteristics.
The measured cooling characteristics of the EC cooling device presented in this paper do not match the characteristics needed for practical applications. In order to reveal the true potential of cooling devices with an AER the optimization of their performance should be carried out. The optimization should address geometrical properties of the AER (EC material thickness, fluid flow channel height, AER length, etc.) and operating conditions of the device (mass flow rate, electric field changes, etc.). Furthermore, different EC materials should be investigated, including thin film ceramic materials and thick film polymer materials (their ECE in terms of adiabatic temperature change can reach up to a few tens of degrees Kelvin). The study should take into account the efficiency of electric energy recovery and the influence of EC material hysteresis, which, as presented in this paper, have a significant impact on the device performance. In the future, we plan to use the numerical model presented in this paper to carry out the optimisation and to provide guidelines for AER design and EC material selection.   ten different electric field changes between 8.4 kV cm −1 and 90 kV cm −1 (note that the T EC could also be calculated for a larger number of electric field changes). Next, a second set of polynomic functions was fitted to obtain the relation between the T EC and the temperature of the material for different electric field changes (plotted in Fig. A.1 (a) with solid lines). The zero-field specific heat capacity of PMN was obtained from Urši č et al. (2016) . To obtain the data in the temperature region from 270 K to 300 K, the measured specific heat capacity of PMN was extrapolated (see Fig. A.1 (b)) over the temperature region of interest.
The hysteresis of the PMN ceramic, measured at room temperature, is presented in Fig. A.2 (a). The description of the hysteresis measurement can be found in Plaznik et al. (2015c) . For simplicity, we have assumed that the energy dissipation due the hysteresis is constant in the temperature region of our interest. The s T was calculated according to Eq. 12 and the results are presented in Fig. A.3 (a). Next, the total specific entropy of the PMN was calculated ( Fig. A.3 (b)). Finally, using Eq. (13 ), the specific heat capacity as a function of the temperature for different electric field changes was calculated. The calculation procedure is universal and can be applied to any given electrocaloric material. The only information needed is the T EC data, the zero-field specific heat capacity and the hysteresis loops.