4.1. EC figure of merit

2. Heat rejection: heat rejection to a heat sink under a constant electric field E2.

material experiences EC cooling by (�ΔTEC).

sition. The relative refrigerator efficiency then comes out as

state.

values of COP > 8.

refrigerators is given by [48]

given by

28 Refrigeration

3. Adiabatic depolarization by decreasing the electric field to a value E1, e.g. zero, the

4. Heat absorption from a load under the constant electric field E<sup>1</sup> returning to the initial

The amount of heat transferred from the load to the heat sink per cycle and per unit volume is

where c(E2) is the volumetric-specific heat at E = E2. The cooling power is then 〈q\_〉 ¼ q=τ<sup>c</sup> with τ<sup>c</sup> the cycle time. The difference between the Brayton and Ericsson cycles is that the Brayton cycle uses adiabatic steps instead of using isothermal ones. Compared to the Carnot cycle, the mean temperature during heat rejection to the heat sink will be less than T2, whereas the mean temperature during heat absorption from the load is higher than T4. For an ideal gas and a specific heat independent on temperature, the relative refrigerator efficiency amounts to Φ ≈ 1=4 [45]. Ferroelectrics themselves exhibit a strong temperature dependence of the volumetric-specific heat, particularly in the vicinity of the ferroelectric-paraelectric phase tran-

<sup>Φ</sup> <sup>¼</sup> <sup>c</sup>ðE2ÞðT<sup>3</sup> � <sup>T</sup>2<sup>Þ</sup>

cðE1ÞðT<sup>4</sup> � T1Þ � cðE2ÞðT<sup>3</sup> � T2Þ

This means that it is determined by the ratio cðE1Þ=cðE2Þ > 1. Consequently, it leads to Φ < 0:25. Here, Φ is also largely affected also by the ratio of Tspan to ΔTEC, i.e. ðT<sup>2</sup> þ T3Þ=2 � ΔTEC, which are functions of the EC material, the EC element design and the thermal interfaces. However, this estimation does not account for the losses described in detail below. With regard to losses, Φ is limited to a value of approximately 10–15%, which is comparable to thermoelectric energy converters. Efficiencies of Φ ≥ 0:5 have been reported in literature for a micro-EC cooling module comprising a micro-electromechanical heat switch [46], a chip scale EC oscillatory refrigerator (ECOR) [28] and a EC refrigerator with intrinsic regenerator [47]. It seems that such a Φ value originates from unreasonably large

Considering only heat losses caused by heat switches, the maximum relative efficiency of EC

where K = κon/κoff is the conductivity contrast of the heat switches and, κon and κoff are the thermal conductivities of the heat switch in the on and off states, respectively. Thus, if K > 10, then EC cooling exceeds the efficiency of thermoelectric cooling. For K > 100, it offers an

ffiffiffi K <sup>p</sup> � <sup>1</sup> ffiffiffi K <sup>p</sup> <sup>þ</sup> <sup>1</sup> !<sup>2</sup>

Φswitch ¼

q ¼ cðE2Þ�ðT<sup>2</sup> � T3Þ, ð16Þ

: ð17Þ

, ð18Þ

One way to characterize the performance of devices is the derivation of appropriate figures of merits FOM, that is, of appropriate combinations of physical properties affecting device efficiency. The EC device performance is determined by (i) the performance of the refrigeration cycle, (ii) the refrigeration capacity (RC) and (iii) the heat transfer efficiency. The different design of EC refrigerators makes a general treatment difficult. Therefore, we will consider first a FOM of cooling power based on materials performance instead of system performance. Our FOM accounts only for the thermal resistance at the interfaces of the EC material. It does not take into account the thermal mass of the heat switch or the heat transfer agent. Moreover, we assume that the heat transfer does not limit device efficiency, i.e. in case of a heat switch the thermal contrast becomes infinite: K!∞ (cf. Eq. (18)). We denote this model as an ideal EC element.

In order to characterize the maximum potential of the refrigerant for each cooling technology, an energy conversion efficiency originating from the material, COPmat, was derived in Ref. [7]. It does not include the system details such as limitations in the driving system efficiency (from compressor, motor, etc.), system dynamics, regenerator effectiveness, heat or mass transfer and component geometries. Therefore, it can be regarded as the maximum potential the material has for the cooling technology. In the case of EC cooling, COPmat yields [7]:

$$\text{COP}\_{\text{mat,EC}} = \frac{T\_s \Delta S - A\_{\text{EC}}}{(T\_l - T\_s)\Delta S + 2A\_{\text{EC}}},\tag{20}$$

where AEC is a materials constant appearing for hysteresis and dielectric losses. In Eq. (20), TsΔS represents the heat transferred from the load to the heat sink within one refrigeration cycle and ðTl � TsÞΔS the work supplied by the EC material within this cycle. For AEC ! 0, Eq. (20) turns into Eq. (14) describing a thermodynamically reversible process. The COPmat of EC materials (~0.35) is inferior to the ones of MC (~0.86) and elastocaloric materials (~0.65) which both exhibit a much larger latent heat of the corresponding first-order phase transitions [7]. Actually, EC refrigerators benefit much less from the large entropy change induced by driving the material through the phase transition by means of applying of releasing relatively small fields (cf. Table 1).

Eq. (20) includes a device-related parameter—the temperature span ðTl � TsÞ, a thermodynamic parameter ΔS and a physical parameter AEC. For practical use, it would be extremely helpful to implement a refrigerant materials criterion which characterizes the efficiency of the physical cooling process and which is therefore independent on the performance of different thermodynamic cycles. For this purpose, COPmat may be written as [7]

$$\text{COP}\_{\text{mat}} = \text{COP}\_{\text{C}} \cdot \text{O}\_{\text{mat}}.\tag{21}$$

where Φmat is the dimensionless material efficiency:

$$\mathfrak{O}\_{\text{mat}} = 1 - \frac{\varepsilon \varepsilon\_0 E^2 \cdot \tan \delta}{c \Delta T\_{\text{EC}}},\tag{22}$$

where tanδ is the sum of dielectric and hysteresis losses during E cycling. The temperature difference ΔT� ¼ ΔTEC � ðTl � TsÞ of the heat transfer steps was replaced by its maximum value ΔTEC since we consider the maximum cooling power (cf. Eq. (26)). Then Φmat becomes independent on cycle parameters, and it receives its minimum value. The second term on the right-hand side of Eq. (22) represents the inverse of the EC efficiency, i.e. the ratio between the heat transferred from the load to the heat sink and the dissipated electrical energy, introduced in Ref. [49].

The RC is a measure of how much heat can be transferred between the load and the heat sink in one ideal refrigeration cycle [50]:

$$R\mathcal{K} = \int\_{T\_1}^{T\_2} \Delta S dT \approx \Delta S \cdot \delta T = \frac{c\Delta T\_{\rm EC}}{T} \delta T\_{\prime} \tag{23}$$

where δT is the full width at half maximum of the ΔS versus T curve. Estimations of δT for different EC materials were given in Ref. [51]. Another physical estimate is the distribution width of the local Curie temperatures considered in Refs. [22, 37].

Complete heat transfer from and to the surroundings requires a Fourier number Fo ¼ αt=d<sup>2</sup> > 1, where α is the thermal diffusivity, t is the time and d is the thickness of the EC material [46]. Since power is the subject of interest, we have to consider the Fourier number per cycle time τ<sup>c</sup>

$$\frac{Fo}{\tau\_c} = \frac{a}{d^2} = \frac{\kappa}{cd^2},\tag{24}$$

with κ is the thermal conductivity. The FOM of an ideal EC element proposed in Ref. [37], combines the materials efficiency Φmat, the refrigeration capacity RC and the Fourier number per cycle time:

$$FOM = \frac{1}{1 - \Phi\_{\text{mat}}} \cdot RC \cdot \frac{Fo}{\tau\_c} = \frac{\kappa c \cdot \Delta T\_{\text{EC} \cdot}^2}{\varepsilon \varepsilon\_0 \cdot \tan \delta} \cdot \frac{1}{T \cdot E^2 d^2} \,. \tag{25}$$

This FOM consists of a term describing the properties of the EC material and a term of operational parameters applied to the material. It increases not only with the square of the EC coefficient (Φ∝½ΔTEC=ΔE� 2 ), but also with increasing EC efficiency (Φ∝cΔTEC=εε0E<sup>2</sup> tanδ), increasing COP of the EC effect (Φ∝ΔTEC=T) (the latter two values were proposed separately as EC figures of merit in Refs. [49, 52], respectively), and increasing heat transfer rate (Φ∝κ=d<sup>2</sup> ). The larger the FOM, the better the cooling performance will be. For an ideal refrigerant, FOM becomes infinite: FOM ! ∞.
