3.1. Carnot cycle

amounts to about 10 K. The EC temperature of a first- or second-order phase transition

Table 1. EC temperature change corresponding to the latent heat calculated using Eqs. (9) or (10) from experimental

tions. For a second-order phase transition in PZT thin films, ΔTEC amounts to 4.0–5.3 K at 665 K [36]. A phase-transition independent upper bound of the EC effect was proposed in Ref. [37] based on the fact that only a certain energy density might be stored in a dielectric—equivalent to a limit in electrostatic pressure. Electrical breakdown of metal oxide dielectrics is fixed by the arising local electric field and the chemical bond strength leading to Emax∝ε�1=2, with Emax the

For a relaxor ferroelectrics exhibiting a huge temperature dependence of dielectric permittivity

cpðTÞ c ph <sup>p</sup> ðTÞ

ph is the portion of volumetric-specific heat at constant pressure p due to lattice vibra-

∂εð0, TÞ

K�<sup>1</sup> [39], we estimate an ultimate EC temperature change of ΔTEC,max ≈ 50K.

<sup>∂</sup><sup>T</sup> � <sup>ε</sup>ð0, TÞE<sup>2</sup>

� 1 " #

dT, ð12Þ

max: ð13Þ

ð<sup>T</sup><sup>2</sup> T1

Composition TPT, �C ΔTEC,PT, �C Ref. PbTiO3 493 10 [4] KNbO3 435 6 [4] PbZrO3 230 13 [4] Pb(Zr0.95Ti0.05)O3 230 12 [4] Pb0.99Nb0.02(Zr0.75Sn0.20Ti0.05)O3 163 3.0 [4] [111] 0.705PbMg1/3Nb2/3O3-0.295PbTiO3 127 0.55 [4] BaTiO3 120 1.5 [4]

P(VDF-TrFE)68/32 ~105 ~12 [30] (Suppl.) P(VDF-TrFE)65/45 81 9.5 [8], p.112 P(VDF-TrFE)55/45 ~65 ~41 [30] (Suppl.)

0.87PbMg1/3Nb2/3O3-0.13PbTiO3 18 0.2 [31] 0.9PbMg1/3Nb2/3O3-0.1PbTiO3 5 0.8 [32, 33] PbSc0.5Ta0.5O3 2 1.6 [4] 0.95PbMg1/3Nb2/3O3-0.05PbTiO3 �28 0.13 [33, 34] [111] PbMg1/3Nb2/3O3 �55 0.33 [34] NH4H2PO4 �123 8.2 [4] KH2PO4 �153 0.7 [4]

dielectric strength of the EC material [38]. According to Eq. (7), this results for Pr!0 in:

<sup>2</sup><sup>c</sup> � <sup>1</sup> εð0, TÞ

<sup>Δ</sup>TEC,max <sup>&</sup>lt; � <sup>ε</sup>0<sup>T</sup>

ΔTEC ¼

deduced from the specific heat curves yields:

where cp

1

values.

24 Refrigeration

Calculated using Eq. (12).

1 <sup>ε</sup>ð<sup>0</sup>, <sup>T</sup><sup>Þ</sup> <sup>∂</sup>εð<sup>0</sup>, <sup>T</sup><sup>Þ</sup> <sup>∂</sup><sup>T</sup> <sup>≈</sup> <sup>10</sup>�<sup>2</sup> An EC refrigerator working under the Carnot cycle will reach the highest efficiency possible. The Carnot cycle describes a reversible change of an ideal gas, which allows to convert a given amount of thermal energy into work, or, conversely, to provide cooling using a given amount of work. It consists of four steps of operation: two adiabatic and two isothermal ones. During the adiabatic steps, no heat is transferred while the refrigerant absorbs heat from the load at its minimum temperature and expels heat to the heat sink at its maximum temperature in the isothermal steps.

The EC Carnot cycle is demonstrated in Figure 2. The cycle starts from point 1 where the electric field on the EC material is E1. In steps 1–2, the electric field is increased adiabatically to E3. Here, the entropy of the EC material stays constant, and therefore, the temperature increases. At point 2, the EC material starts to experience an isothermal process. The electric field will be increased until it reaches its maximum value E<sup>4</sup> at point 3. In order to conserve isothermal conditions, heat should be simultaneously rejected to the heat sink. In the adiabatic steps 3–4, the electric field is decreased to E<sup>2</sup> while the temperature of the EC material decreases until reaching point 4. In the second isothermal steps 4–1, the electric field decreases to E<sup>1</sup> while heat should be absorbed from the load. Thus, the Carnot cycle requires a minimum of four different electric fields. Since the heat rejected to the heat sink amounts to Q ¼ T1ΔS, the cooling power depends significantly on the chosen working point (cf. cycles 1-2-3-4 and 5-6-7-8).

Figure 2. Reverse Carnot cycle for EC refrigeration.

The implementation of the Carnot cycle into a practical refrigeration system is challenging, since the isothermal steps and the transition from an adiabatic process to an isothermal one are not easy to realize. In the two isothermal steps, the refrigerant is in thermal contact with the load or the heat sink, respectively. Here, the rate of electrical field change is limited by the relatively large thermal relaxation time of the thermal interfaces (heat switch or heat transfer agent) of the system. This significantly lowers cycle time. Moreover, the maximum temperature span Tspan = Tl � Ts of the whole refrigerator will be less than the ΔTEC of the EC material. On the one hand, a small temperature span provides large cycle efficiency (cf. Eq. (14)). The temperature span Tl � Ts might be increased by means of a cascaded structure of m units where the unit n ejects heat to unit n + 1, while this unit absorbs heat from unit n (1 < n < m) covering the desired Tl � Ts. Such a cascade system does not require large ΔTEC. However, in order to reach high efficiency, the heat ejected from the previous step should be completely absorbed by the following step. In general, since the EC-induced entropy change is not a constant, and the specific heat of the EC material also changes with temperature, this requirement is hard to meet. Consequently, the performance of the cascaded refrigerator is further reduced.

The coefficient of performance COP is defined as the ratio between the useful heating or cooling provided to work required. Considering an ideal Carnot cycle, the corresponding COPC can be written as

$$\text{COP}\_{\text{C}} = \frac{T\_s}{T\_l - T\_s} \tag{14}$$

where Ts and Tl indicate the temperature of heat sink and load, respectively. COPC establishes an upper bound for the COP. Since the EC effect is a thermodynamically reversible process, EC refrigerators could reach the limit of the Carnot efficiency. The relative efficiency of a refrigerator with respect to an ideal Carnot cycle is defined as

$$
\Phi = \frac{\text{COP}}{\text{COP}\_{\text{C}}} \tag{15}
$$

where Φ is determined by the EC material hysteresis, the heat losses of the heat transfer processes through heat switches or a regenerator, the thermal resistance of the heat switches, the regenerator efficiency (the ratio of actual heat exchange in the regenerator to an ideal one), the heat flow from the environment to the load, the deviation of the isothermal steps from the ideal case, Joule heating at the contacts, etc. The total efficiency is then the product of separate efficiency coefficients. The current state-of-the-art commercial vapour compression cycle has a COP of about 3.6 [7].
