4.4. Cooling power of the refrigeration system

An EC refrigerator, i.e. a heat pump, is able to transport thermal energy against a temperature gradient from Tl to Ts where Tl > Ts. Here, the heat flows from the load to the EC layer and from the EC layer to the heat sink. Both are controlled by thermal connections that have to be opened and closed appropriately as the layer is heated or cooled. Heat is transferred from the load or to the heat sink either


The heat switch or the heat transfer (HT) agent acts as an additional cycle-average thermal mass CHT of the system. Correspondingly, the cycle time τ<sup>c</sup> � RthðCEC þ CHTÞ is increased.


Table 3. Specific cooling power of EC devices in dependence on the dominating thermal resistance.

by the ratio of ΔT\* to the total thermal resistance Rth of the device. Taking into account that there are two heat transfer steps (heat absorption and heat rejection) with equal time constants and equal heat fluxes within both steps, the average cooling power per cycle yields [44]

Table 2. Material characteristics, the materials efficiency Φmat and the figure of merit selected EC refrigerants.

BaTiO3 400 1.5 1 4.2 5000 2.6 5 0.05 0.9996 13.875 BaTiO3 295 0.5 30 2.5 500 2.6 60 0.07 0.554 0.119 BaZr0.2Ti0.8O3 310 4.5 15 3.4 800 2.6 30 0.05 0.994 21.739 0.7PMN-0.3PT 420 2.5 10 2.8 6000 1.5 100 0.08 0.949 1.765 0.9PMN-0.1PT 350 5 90 3 1250 1.3 100 0.1 0.986 0.311 PLZT8/65/35 385 2.5 10 3 5000 2.3 80 0.1 0.991 1.320 PLZT8/65/35 318 40 120 3 1000 2.3 80 0.07 0.963 20.295

constants with m > 2 (two isothermal steps at least with a duration τRC), the EC material's temperature decays to almost the steady-state value. Here, the cooling power increases linearly with frequency f. At smaller values of m, a temperature offset appears decreasing the effective ΔT\*. The maximum specific cooling power is obtained at a value of m ¼ ln2 ≈ 0:7 yielding

> 〈q\_〉max <sup>¼</sup> <sup>0</sup>:36ΔT� X i R<sup>00</sup> th,i

Table 3 compiles estimated specific cooling powers of hypothetical EC devices in dependence

An EC refrigerator, i.e. a heat pump, is able to transport thermal energy against a temperature gradient from Tl to Ts where Tl > Ts. Here, the heat flows from the load to the EC layer and from the EC layer to the heat sink. Both are controlled by thermal connections that have to be opened and closed appropriately as the layer is heated or cooled. Heat is transferred from the

ii. by pumping a gaseous or liquid heat transfer agents through the solid refrigerant [59]. The heat switch or the heat transfer (HT) agent acts as an additional cycle-average thermal mass CHT of the system. Correspondingly, the cycle time τ<sup>c</sup> � RthðCEC þ CHTÞ is increased.

i. via controlled heat switches [58] as well as uncontrolled thermal rectifiers, or

½1 � exp ð�mÞ� 2mRth

305 20 200 2.7 60 0.2 50 0.15 0.941 1.111

, ð26Þ

K ε κ, W/mK δT, K tanδ Φmat FOM,

: ð27Þ

/KW. For thermal time

mW/cm3

〈q\_〉 <sup>≈</sup> <sup>Δ</sup>T�

th <sup>¼</sup> <sup>A</sup> � Rth is the area-specific thermal resistance given in m<sup>2</sup>

on the dominating thermal resistance of possible heat-releasing parts.

4.4. Cooling power of the refrigeration system

Refrigerant T, K ΔTEC, K ΔE, V/µm c, MJ/m<sup>3</sup>

load or to the heat sink either

where R<sup>00</sup>

P(VDF)-based polymers

32 Refrigeration

Usually, CHT > CEC, i.e. the cycle time is primarily determined by CHT. Thus, the response times of the heat switches or the gas/liquid delivery systems limit significantly the cycle time of EC refrigerators. The thermal time constants of releasable solid-solid, liquid-solid and solidliquid (hybrid)-solid contacts are 350, 135 and 75 ms, respectively [56]. In an AER, a secondary heat transfer agent (gas or fluid) is used to transfer heat from the cold to the hot end of the regenerator. The heat transfer agent pumped through the EC material substantially enhances the heat flow and, thus, increases the specific cooling power as well as the device efficiency. Here, the Biot number Bi ¼ d=ðκ � R<sup>00</sup> th, <sup>b</sup>Þ, characterizing the ratio of the thermal resistances of the EC material volume and the boundaries, will be small for thicknesses d below 100 μm. Assuming a uniform heat flux across the interface, a height of a very long rectangular duct of 0.5 mm and a thermal conductivity of the heat transfer agent of 0.15 W/mK (silicon oil), the corresponding heat transfer coefficient h ¼ 1=R<sup>00</sup> th, <sup>b</sup> yields a value of h ≈ 1250 W/m2 K. At Bi < 0.1, the temperature of the EC element during heat transfer remains nearly constant, enabling a lumped system approximation [60]. The time constant amounts then to τ<sup>i</sup> ¼ c � d=h. It is in the order of the response time of piezoelectric valves for gas or liquid supply amounting to a few milliseconds [61]. Table 4 compiles the time constants of hypothetical thermal interfaces and the corresponding operational frequency limits. Note that oxide thermal rectifiers made of two oxides with different thermal conductivities [62] possess a thermal contrast of K = 1.43 which is still too low for EC applications.


Table 4. Time constants of thermal interfaces and the associated frequency limit of EC devices.

Currently, the operational frequency of EC refrigerators is limited to about 10 Hz providing cooling powers of a few W/cm2 .
