3.2. Alternative refrigeration cycles

Similar to MC cooling, also alternative refrigeration cycles might be employed for EC refrigeration. The Stirling and Ericsson cycles were considered in Ref. [40]. The Stirling refrigeration cycle consists of two isothermal and two isopolarization steps, while the Ericsson refrigeration cycle consists of two isothermal and two isofield steps. The heat is released and absorbed in the two isothermal steps. In the Stirling cycle, polarization would change during heating and cooling due to a strong temperature dependence of the dielectric permittivity of polar dielectrics. Therefore, this cycle is not suitable for EC refrigeration from an experimental point of view. The Ericsson cycle is more readily applicable than other thermodynamic cycles such as the Stirling cycle. An isofield condition is easily realized keeping the EC material connected to the voltage source. The Ericsson cycle requires heat regeneration, i.e. the heat rejected from the hot refrigerant is intermittently stored in a thermal transfer medium, before it is transferred to the cool refrigerant. The efficiency of the regenerator is significantly affected by the heat transfer conditions (heat transfer surface, heat transfer coefficient, boundary layers, heat conduction, fluid viscosity, etc.). Generally, the coefficient of performance of a ferroelectric Ericsson refrigeration cycle is smaller than that of a Carnot cycle for the same temperature range [40].

The Olsen cycles was proposed for application in pyroelectric energy harvesting [41]. Therefore, it will not be considered in this work. It replaces adiabatic polarization and depolarization of a ferroelectric by corresponding isothermal steps.

A gas refrigeration cycle where a gas is compressed and expanded, but does not change phase, is called Bell-Coleman cycle. It consists of two adiabatic (compression and expansion) and two isobaric processes (heat addition and heat rejection). This cycle corresponds also to the reverse Brayton cycle widely used for subcooling in the liquid nitrogen industry. For EC refrigerators, even the reverse Brayton cycle is predominantly chosen [42–44]. It includes the following steps (Figure 3):

1. Adiabatic polarization by increasing the electric field to a value E2, the EC material experiences EC heating (+ΔTEC).

Figure 3. Reverse Brayton 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

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 <sup>¼</sup> Ts

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 refriger-

> <sup>Φ</sup> <sup>¼</sup> COP COPC

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

Similar to MC cooling, also alternative refrigeration cycles might be employed for EC refrigeration. The Stirling and Ericsson cycles were considered in Ref. [40]. The Stirling refrigeration cycle consists of two isothermal and two isopolarization steps, while the Ericsson refrigeration cycle consists of two isothermal and two isofield steps. The heat is released and absorbed in the two

Tl � Ts

, ð14Þ

, ð15Þ

of the cascaded refrigerator is further reduced.

ator with respect to an ideal Carnot cycle is defined as

COPC can be written as

26 Refrigeration

COP of about 3.6 [7].

3.2. Alternative refrigeration cycles


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

$$
\eta = \mathfrak{c}(E\_2) \cdot (T\_2 - T\_3),
\tag{16}
$$

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 transition. The relative refrigerator efficiency then comes out as

$$\Phi = \frac{c(E\_2)(T\_3 - T\_2)}{c(E\_1)(T\_4 - T\_1) - c(E\_2)(T\_3 - T\_2)}. \tag{17}$$

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 values of COP > 8.

Considering only heat losses caused by heat switches, the maximum relative efficiency of EC refrigerators is given by [48]

$$\mathfrak{O}\_{\text{switch}} = \left(\frac{\sqrt{K} - 1}{\sqrt{K} + 1}\right)^2,\tag{18}$$

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 efficiency comparable to magnetic cooling (about 70%) with much smaller and cheaper equipment. With regard to the thermal conductivity of the EC element κEC [23], the thermal contrast is

$$K'=K\cdot\frac{\kappa\_{\text{off}}+\kappa\_{\text{EC}}}{\kappa\_{\text{on}}+\kappa\_{\text{EC}}}.\tag{19}$$

For applications, usually κEC ≈ κon ≫ κoff holds.

To cyclically store and release energy to the refrigerant, cyclic operating EC systems require a regenerator. In vapour compression refrigeration, the refrigerant is also the circulating fluid. Similarly, an EC material is used as both the refrigerant and the regenerator. However, an exchange fluid is needed to transport heat to and from the refrigerant since the refrigerant is a solid. Such an active EC regenerator (AER) consists of a porous structure of an EC material and voids or channels through which the heat transfer fluid can flow [42]. Regenerators are also a source of heat loss [40].
