## Abstract

To date, most of electrocaloric devices reported can be simplified as a multilayer structure in which thermal source and sink are different materials at two ends. The thermal conduction in the multilayer structure is the key for the performance of the devices. In this paper, the analytical solutions for the thermal conduction in a multilayer structure with four layers are introduced. The middle two layers are electrocaloric materials. The analytical solutions are also simplified for a hot/cold plate with two sides being different media—a typical case for thermal treatment of materials. The analytical solutions include series with infinite terms. It is proved that these series are convergent so the sum of a series can be calculated using the first N terms. The equation for calculating the N is introduced. Based on the case study, it is found that the N is usually a small number, mostly less than 40 and rarely more than 100. The issues related to the application of the analytical solutions for the simulation of real electrocaloric devices are discussed, which includes the usage of multilayer ceramic capacitor, influence of electrodes, and characterization of thin film.

## 1 Introduction

From air conditioners to refrigerators, cooling systems and devices are essential for our daily life. Vapor compression (VC) technology has played a critical role in cooling systems [1]. However, the VC technology faces some critical challenges as it emits greenhouse gases responsible for global warming [2,3]. Therefore, it is critical to find new technologies to replace the VC technology for cooling process. For example, a route map to regulate materials responsible for ozone depletion potential and global warming potential [2] was developed [3,4], which emphasizes the importance to replace the VC technology with new cooling technologies. In pursuit of alternatives for VC technology, about 12 not-in-kind technologies were considered to be the potential candidates for next generation cooling, refrigeration, and/or heat pumping systems [5]. The term not-in-kind refers to any refrigerating/air-conditioning technology that is not based on the VC technology.

The discovery of ferroelectrics with a giant electrocaloric effect (ECE) [6,7] revived research on the ECE as an alternative for the VC technology. The ECE is a primary coupling effect between the electric and thermal energy in polar dielectrics, such as ferroelectrics. A polar dielectric with a useful ECE is also named as electrocaloric material (ECM). For an ECM, a change in the external electric field, applied on it, can induce a change in its temperature under an adiabatic condition or a release/absorption of heat under an isothermal condition. The temperature change induced in an ECM by changing electric field under an adiabatic condition is mostly small to medium [810], but it can be as high as 40–50 °C [11,12]. Therefore, one can use ECMs to build heat pumps. During the past decade, several ECE-based cooling devices have been reported that can transfer heat from source (SO) (cold end) to sink (SI) (hot end) [1323].

The basic principle of an ECE-based heat pump is based on the Brayton cycle [24,25] that works with using two adiabatic processes and two isofield processes. In the adiabatic processes, the temperature of the ECM is changed by changing electric field, while in the isofield processes the heat is absorbed from the SO and is rejected into the SI through the thermal conduction. In all the reported prototype devices to date [13,20,21,26], alternative thermal coupling and decoupling between the ECM and SI/SO is critical. The performance of a heat pump is directly determined by the heat transfer through the interfaces between the ECM and SI/SO due to the thermal conduction during the isofield processes. For each of these thermal conduction steps (i.e., isofield process), the device can be simplified as a one-dimensional (1D) thermal conduction in a multilayer structure. For example, for the step in which the ECM absorbs heat from SO, a high thermal conduction is desirable for the interface between the ECM and SO, while a low thermal conduction is required for the other end of the ECM that is usually done by using air. That is, there are at least three layers: SO, ECM, and air [13,20,21,26]. In other words, each of these devices can be simplified as a three-layer structure with the ECM layer in the center and, usually, the thickness of the ECM layer is much smaller than that of the SO/SI/air. Regarding to the ECM parts of a heat pump, multiple layers of the same ECM have been used for two different purposes: (1) reducing the thickness of each ECM layer results in an enhanced electric breakdown field for the ECM layer so that a higher electric field can be applied on the ECM layer, and (2) for an ECM operated at the same electric field, a smaller thickness means a lower voltage to be used. Although a multilayer ECM is used, these ECM layers are usually operated with the same electric field profile. That is, to date, most of the reported ECE-based heat pumps can be simplified as a three-layer structure, in which the ECM layer is in the center. More recently, an ECE-based heat pump using two independently operated ECM layers was reported [27,28]. This unique structure results in some unique performances, such as an ECE-based heat pump without any moving part [27,28].

For the performance of an ECE device, there are two key output parameters: (1) amount (Q) of heat transferred during one cycle; and (2) the time (Δt) required to perform one cycle. Both of these two factors are directly dependent on the time dependence of the temperature profiles in all the layers of a heat pump. Unfortunately, the performance of the reported ECE-based devices was only experimentally determined for some special cases, which makes it difficult to optimize: (1) an design of ECE-based device, and (2) the operation parameters, such as the time for each step, for an ECE-based device to achieve its best performance. Although the finite element analysis, which is powerful, has been widely used to simulate the temperature profile [15,1923], finite element analysis requires too much details in the materials' properties and dimensions of these layers and is strongly dependent on the skill and experience on selecting mesh used. Additionally, the temperature span and coefficient of performance (COP) are critical to the overall performance of a heat pump. The temperature span (i.e., the temperature difference between the SO and SI) of an ECE-based device is limited by the temperature change induced in the ECM by changing the electric field, which is usually a small to medium value, but it can be as high as 40–50 °C, as mentioned above. The COP is defined as the ratio of the heat transferred to work (energy) used. The real input energy for an ECE-based device is the electric energy that is used to induce the change in the temperature of the ECM in an adiabatic process. In the calculation or simulation of the COP of an ECE-based device, the electric energy is mostly not included to date. In this case, the COP is actually the COP of the thermal cycle used in the ECE-based device. This COP is dependent on the exact design and operation of the thermal cycle, in which the heat transfer among different parts is included. The thermal conduction is the key to calculate the heat transfer.

To deepen the understanding of the heat conduction process in a multilayer structure, it is important to determine the time dependence of heat transfer through an interface in the multilayer structure. To optimize the performance and operation parameters of an ECE-based device, it is interested to know the temperature profile and its time dependence in each part of the device. For both of above, analytical solutions of temperature profile in a multilayer structure and heat flux through interfaces in a multilayer structure would be a very powerful tool. For example, very recently [27,28], the analytical solution was developed for a four-layer structure with a center symmetry to simulate the cascade-based devices [15,17,29]. Using the analytical solution, the detailed time dependence of the heat transfer was determined, and a new concept for the development of ECE-based devices without any moving part was devised [28].

Regarding the heat conduction in a multilayer structure, the classical case is a three-layer structure in which a finite-thick layer with a uniform initial temperature is sandwiched into two same semi-infinite layers that have a uniform initial temperature. The analytical solution for this case can be found in textbook [30]. For a four-layer structure, the analytical solution was recently introduced for a special case [27]: two semi-infinite layers are the same material. To the best of our knowledge, no analytical solution was reported for a four-layer structure in which two semi-infinite layers are different materials.

To simulate the performance of most ECE-based devices reported to date, the analytical solution of heat conduction is introduced for a four-layer structure with the SO and SI using different materials. There are two ECM layers and these two ECM layers are independently operated, which provide the unique opportunities for designing ECE-based devices. The usage of the SO and SI with different materials provides much more flexibility for the application of the analytical solution. The solutions introduced here can be used to determine following key parameters of an ECE-based device: the amount of heat transferred from the SO in one cycle and the time needed for one cycle, both of which can be used to calculate the heat power generated by the device. Additionally, the amount of heat transferred among different parts of ECE-based devices can be calculated using the solutions introduced here. All of these can be used to determine the COP of the device. In all these calculations, the exact design and operation parameters of an ECE-based device are needed. The analytical solution introduced here can not only be used to reveal the detailed process of heat transfer through any interface, such as SO/SI and ECM, of many reported ECE-based devices so that the performance of an ECE-based device can be optimized but also to quantify the characterization of ECMs. For example, for the characterization of the ECE of a material, measuring surface temperature of an ECM has been widely used [1,31,32]. The analytical solution provides the relationship between the surface temperature and body temperature of an ECM so that a better characterization of body temperature of the ECM can be performed. The analytical solutions can also account the influence of environment such as the substrate used for ECM thin film [33]. Additionally, in engineering practices, such as thermal treatment process in steel plants, a heat/cold plate is placed between two different materials, in which the analytical solution of the heat is obtained by simplifying the solutions introduced here.

## 2 Model and Analytical Solution

### 2.1 Physical Model.

As discussed in Sec. 1, to date, most of the reported ECE-based heat pumps can be simplified as a multilayer structure with an ECM in the middle. Usually both sides of the ECM layer(s) are the thermal SO and SI made of different materials, and the SO and SI media have a much bigger dimension/thickness than the ECM layer. To simplify the process, both SO and SI are assumed as homogenous semi-infinite materials [34] as shown in Fig. 1, where two layers of an ECM are used and are independently controlled to represent more general cases. The temperature in these two ECM layers can be changed adiabatically by changing the electric field applied on the ECM layers. That is, by using the ECE, a temperature profile can be established in the structure, then the temperature profile changes with time due to the thermal conduction. This thermal conduction in this multilayer structure is simplified as 1D conduction (i.e., heat flux is only along x-axis in Fig. 1).

Fig. 1
Fig. 1
Close modal

### 2.2 Mathematical Model and Governing Equations.

To determine the heat flux in 1D structure shown in Fig. 1, the following assumptions are used: (1) only heat conduction along x-direction is considered, no temperature gradients along y- and z-direction are considered, (2) no heat loss to the environment is considered, (3) the contacts between different layers (i.e., SI|ECM1, ECM1|ECM2, and ECM2|SO interfaces) are thermally perfect (i.e., zero thermal resistance), and (4) SO and SI are semi-infinite. Each layer is a uniform/homogenous material, in which thermal conduction is described as q = −kT (qx = − kdT/dx, for 1D condition), where q is the heat flux, T is the temperature, and k is the thermal conductivity. For an isotropic material, Fourier's law leads to the heat equation, $∂T/∂t=α((∂2T/∂x2)+(∂2T/∂y2)+(∂2T/∂z2))$ or $(∂T/∂t)=α(∂2T/∂x2)$ for 1D condition, which is also known as heat kernel [35], where α (= k/ρcp, m2/s) is the thermal diffusivity, in which k (W m−1 K−1), ρ (kg m−3), and cp (J K−1 kg−1) are the thermal conductivity, mass density, and gravimetric heat capacity, respectively, of the material. The ρcp (J K−1 m−3) is the volumetric heat capacity. Using the heat equation (i.e., heat kernel) to each layer of the device described in Fig. 1, one can get the following equations:
$∂TSI(x,t)∂t=αSI∂2TSI(x,t)∂x2−∞
(1)
$∂TECM1(x,t)∂t=αEC∂2TECM1(x,t)∂x2 −R
(2)
$∂TECM2(x,t)∂t=αEC∂2TECM2(x,t)∂x2 0
(3)
$∂TSO(x,t)∂t=αSO∂2TSO(x,t)∂x2 R
(4)

where TSI(x,t), TECM1(x,t), TECM2(x,t), and TSO(x,t) are the temperatures at time t and location x for the SI, ECM1, ECM2, and SO, respectively. The materials' properties of two ECM layers are the same, but are different for the SI and SO (i.e., αEC ≠ αSI ≠ αSO).

#### 2.2.1 Initial Conditions.

It is assumed that at the beginning, the temperature in each layer is a constant as shown in Eq. (5), but it can be any value so that there is a flexibility of the mathematical model to be implemented in different scenarios
$TSI(x,0)=TSIiTECM1(x,0)=TECM1iTECM2(x,0)=TECM2iTSO(x,0)=TSOi$
(5)

The initial temperature (i.e., $TECM1i$ and $TECM2i$) in ECM1 and ECM2 can be set adiabatically using the ECE by changing the electric field applied on the ECM layer so that an initial temperature profile in the multilayer structure can be established. The initial temperatures (i.e., $TSIi$ and $TSOi$) in SI and SO are also set as different to make the final solution more flexible.

For the application of the solution introduced here to an ECE-based device, it has to be mentioned that the $TECM1i$ and $TECM2i$ are the temperatures in ECM1 and ECM2, respectively, right after the change in the electric field applied. In other words, the ECE induced temperature change in an ECM is treated as an instantaneous change under an adiabatic condition. This is very close to the reality. For example, the response time of the polarization in dielectric materials upon the application of an electric field is typically less than 10−8 s [3638]. After the change, the electric field applied on both ECM1 and ECM2 is maintained as a constant (i.e., isofield). For example, when the temperature of ECM1 and ECM2 is $TECM1,0i$ and $TECM2,0i$, the electric field on both ECM layers is changed at the same time, which results in an instantaneous change in the temperature of ECM1 and ECM2 as ΔTECM1 and ΔTECM2, respectively. Then, the electric field applied on both of ECM1 and ECM2 is maintained as the constant. In this case, $TECM1i=TECM1,0i+ΔTECM1i$ and $TECM2i=TECM2,0i+ΔTECM2i$.

#### 2.2.2 Boundary Conditions.

The solutions of differential Eqs. (1)(4) are also dependent on the boundary conditions at all three interfaces. First of all, the temperature at two surfaces of each interface should be the same. In other words, the temperature is continuous in the structure. Second, due to first law of thermodynamics, the heat flux at two sides of each interface should be the same. Mathematically, these two boundary conditions result in two equations as Eq. (6) for the interface between the SI and ECM1.

SI|ECM1 interface (at x = −R)
$TSI(−R,t)=TECM1(−R,t)−kSI∂TSI(x,t)∂xx=−R=−kEC∂TECM1(x,t)∂xx=−R$
(6)

Similarly, two equations are obtained for each of the next two interfaces (i.e., interface between ECM1 and ECM2 and interface between ECM2 and SO) as Eqs. (7) and (8).

ECM1|ECM2 interface (at x = 0)
$TECM1(0,t)=TECM2(0,t)−kEC∂TECM1(x,t)∂xx=0=−kEC∂TECM2(x,t)∂xx=0$
(7)
ECM2|SO interface (at x = R)
$TECM2(R,t)=TSO (R,t)−kEC∂TECM2(x,t)∂xx=R=−kSO∂TSO(x,t)∂xx=R$
(8)

#### 2.2.3 Physical Conditions.

For semi-infinite SI and SO, they can be treated as continuous SO and SI with infinite capacity. Their perimeters are perfectly insulated so that no heat transfer to the surrounding is allowed at $x=±∞$. That is,
$∂TSI(x,t)∂xx=−∞=0=∂TSO(x,t)∂xx=∞$
(9)

### 2.3 Analytical Solution.

Laplace transform method [30,39] is used here to solve the system of Eqs. (1)(4). All boundary conditions, Eqs. (6)(8), and the initial conditions, Eq. (5), are used along with the physical conditions, Eq. (9), to determine the constants of the integration. The final analytical solution of the temperature profile in each layer as a function of both location x and time t is determined and given in Eqs. (10)(13), in which both the thermal diffusivity (α) and thermal effusivity (e) that is defined as e = (k⋅ρ⋅cp)0.5 (W m−2 K−1 s−0.5) are used to represent the relative changes in the materials property across the interface, which are $KαSI=αEC/αSI$ and $KαSO=αEC/αSO$ being the relative thermal inertia of ECM1 to SI and ECM2 to SO, respectively [30], and $KSI=kEC/kSI(αSI/αEC)=eEC/eSI$ and $KSO=kEC/kSO(αSO/αEC)=eEC/eSO$ being contacting coefficients of ECM1 to SI and SO to ECM2, respectively. Both KSI and KSO characterize the thermal activity of one layer relative to next layer [30]. hSI = (1 − KSI)/(1 + KSI), hSO = (1 − KSO)/(1 + KSO), and h = hSIhSO so that |hSI| < 1, |hSO| < 1, and |h| < 1.

#### 2.3.1 Temperature Profiles.

The detailed steps for the derivation of temperature profiles are shown in the Supplemental Materials on the ASME Digital Collection.

##### 2.3.1.1 Temperature profile in sink
$TSI(x,t)−TSIi=KSI1+KSI×(TECM1i−TSIi)hSO×∑n=1∞hn−1erfc−x−R+(4n−0)KαSI−1/2R2αSIt−(TECM1i−TECM2i)hSO×∑n=1∞hn−1erfc−x−R+(4n−1)KαSI−1/2R2αSIt−2(TECM2i−TSOi)1+KSO×∑n=1∞hn−1erfc−x−R+(4n−2)KαSI−1/2R2αSIt−(TECM1i−TECM2i)×∑n=1∞hn−1erfc−x−R+(4n−3)KαSI−1/2R2αSIt+(TECM1i−TSIi)×∑n=1∞hn−1erfc−x−R+(4n−4)KαSI−1/2R2αSIt$
(10)
##### 2.3.1.2 Temperature profile in ECM1
$TECM1(x,t)−TECM1i=(TECM1i−TECM2i)h2×∑n=1∞hn−1erfc(4n−0)R+x2αECt+∑n=1∞hn−1[hSO1+KSI(TECM1i−TSIi)×erfc(4n−1)R−x2αECt +hSI1+KSO(TECM2i−TSOi)×erfc(4n−1)R+x2αECt]−12∑n=1∞hn−1[hSO(TECM1i−TECM2i)×erfc(4n−2)R−x2αECt −hSI(TECM1i−TECM2i)×erfc(4n−2)R+x2αECt]−∑n=1∞hn−1[11+KSO(TECM2i−TSOi)×erfc(4n−3)R−x2αECt +11+KSI(TECM1i−TSIi)×erfc(4n−3)R+x2αECt]−(TECM1i−TECM2i)2×∑n=1∞hn−1erfc(4n−4)R−x2αECt$
(11)
##### 2.3.1.3 Temperature profile in ECM2
$TECM2(x,t)−TECM2i=−(TECM1i−TECM2i)h2×∑n=1∞hn−1erfc(4n−0)R−x2αECt+∑n=1∞hn−1[hSO1+KSI(TECM1i−TSIi)×erfc(4n−1)R−x2αECt +hSI1+KSO(TECM2i−TSOi)×erfc(4n−1)R+x2αECt]−12∑n=1∞hn−1[hSO(TECM1i−TECM2i)×erfc(4n−2)R−x2αECt −hSI(TECM1i−TECM2i)×erfc(4n−2)R+x2αECt]−∑n=1∞hn−1[11+KSO(TECM2i−TSOi)×erfc(4n−3)R−x2αECt +11+KSI(TECM1i−TSIi)×erfc(4n−3)R+x2αECt]+(TECM1i−TECM2i)2×∑n=1∞hn−1erfc(4n−4)R+x2αECt$
(12)
##### 2.3.1.4 Temperature profile in source
$TSO(x,t)−TSOi=KSO1+KSO×(TECM2i−TSOi) hSI×∑n=1∞hn−1erfcx−R+(4n−0)KαSO−1/2R2αSOt+(TECM1i−TECM2i) hSI×∑n=1∞hn−1erfcx−R+(4n−1)KαSO−1/2R2αSOt−2(TECM1i−TSIi)1+KSI×∑n=1∞hn−1erfcx−R+(4n−2)KαSO−1/2R2αSOt+(TECM1i−TECM2i)×∑n=1∞hn−1erfcx−R+(4n−3)KαSO−1/2R2αSOt+(TECM2i−TSOi)×∑n=1∞hn−1erfcx−R+(4n−4)KαSO−1/2R2αSOt$
(13)

#### 2.3.2 Temperature at Interface.

For the heat conduction in a multilayer structure, the temperature $(Ts)$ at an interface depends on the thermal properties of the materials across the interface and can be given at any time by Eq. (14) as follows [30,40]:
$TsSI|ECM1(t)=TECM1(x,t)×KSI+TSI(x,t)1+KSIx=−Rand$
$TsECM2|SO(t)=TECM2(x,t)×KSO+TSO(x,t)1+KSOx=R$
(14)

#### 2.3.3 Heat Flux Through the Interfaces.

The heat fluxes through the interfaces SI|ECM1 (x = −R), ECM1|ECM2 (x = 0), and ECM2|SO (x = R) are given in Eqs. (15)(17), respectively,
$qSI|ECM1(t)=−kSI∂TSI(x,t)∂xx=−R$
$=−kEC∂TECM1(x,t)∂xx=−R$
(15)
$qECM1|ECM2(t)=−kEC∂TECM1(x,t)∂xx=0$
$=−kEC∂TECM2(x,t)∂xx=0$
(16)
$qECM2|SO(t)=−kEC∂TECM2(x,t)∂xx=R$
$=−kSO∂TSO(x,t)∂xx=R$
(17)

The heat fluxes through the interfaces described by Eqs. (15)(17) may be determined by differentiating Eqs. (10)(13). The resulting heat fluxes, qSI|ECM1(t), qECM1|ECM2(t), and qECM2|SO(t) at interfaces SI|ECM1 (x = −R), ECM1|ECM2 (x = 0), and ECM2|SO (x = R), respectively, as the function of time are given in Eqs. (18)(20), respectively.

##### 2.3.3.1 Heat flux through SI|ECM1 interface
$qSI|ECM1(t)=−kSI∂TSI(x,t)∂x|x=−R=−kSIπαSItKSI1+KSI×{(TECM1i−TSIi) hSO×∑n=1∞hn−1|exp(−[(4n−0)R2αECt]2)−(TECM1i−TECM2i) hSO×∑n=1∞hn−1exp(−[(4n−1)R2αECt]2)−2(TECM2i−TSOi)1+KSO×∑n=1∞hn−1exp(−[(4n−2)R2αECt]2)−(TECM1i−TECM2i)×∑n=1∞hn−1exp(−[(4n−3)R2αECt]2)+(TECM1i−TSIi)×∑n=1∞hn−1exp(−[(4n−4)R2αECt]2)}$
(18)
##### 2.3.3.2 Heat flux through ECM1|ECM2 interface
$qECM1|ECM2(t)=−kEC∂TECM2(x,t)∂x|x=0=−kECπαECt×{−(TECM1i−TECM2i)h2×∑n=1∞hn−1exp(−[(4n−0)R2αECt]2)+∑n=1∞hn−1[hSO1+KSI(TECM1i−TSIi)×exp(−[(4n−1)R2αECt]2)−hSI1+KSO(TECM2i−TSOi)×exp(−[(4n−1)R2αECt]2)]−12∑n=1∞hn−1[hSO(TECM1i−TECM2i)×exp(−[(4n−2)R2αECt]2)+hSI(TECM1i−TECM2i)×exp(−[(4n−2)R2αECt]2)]−∑n=1∞hn−1[11+KSO(TECM2i−TSOi)×exp(−[(4n−3)R2αECt]2)−11+KSI(TECM1i−TSIi)×exp(−[(4n−3)R2αECt]2)]−(TECM1i−TECM2i)2×∑n=1∞hn−1exp(−[(4n−4)R2αECt]2)}$
(19)
##### 2.3.3.3 Heat flux through ECM2|SO interface
$qECM2|SO(t)=−kSO∂TSO(x,t)∂x|x=R=kSOπαSOtKSO1+KSO×{(TECM2i−TSOi) hSI×∑n=1∞hn−1exp(−[(4n−0)R2αECt]2)+(TECM1i−TECM2i) hSI×∑n=1∞hn−1exp(−[(4n−1)R2αECt]2)−2(TECM1i−TSIi)1+KSI×∑n=1∞hn−1exp(−[(4n−2)R2αECt]2)+(TECM1i−TECM2i)×∑n=1∞hn−1exp(−[(4n−3)R2αECt]2)+(TECM2i−TSOi)×∑n=1∞hn−1exp(−[(4n−4)R2αECt]2)}$
(20)

#### 2.3.4 Heat Energy Through the Interfaces.

The amount, Q(t), of heat transferred through any interface over one unit area of the cross section from the beginning (t = 0) to any time (t) is the integration of the heat flux with the time as
$Q(t)=∫0tq(t)dt$

## 3 Applications of Analytical Solutions

### 3.1 Calculations Using Analytical Solutions.

Each of the solutions, temperature profiles (i.e., Eqs. (10)(13)) and heat fluxes (i.e., Eqs. (18)(20)), obtained above has series with infinite number of terms, which cannot be directly used for any calculation. Therefore, it is important to find the summation of each series.

All the series in the solution for heat fluxes can be represented by the series Sn(x,t) (n = 1, 2, 3…) or Sn(x,t) (n′ = 1, 2, 3…) (i.e., n′= n − 1 or n = n′ + 1) as
$Sn(x,t)=hn−1exp(−[(4n−m)R2αECt]2)$
(21a)
$Sn′(x,t)=hn′exp(−[(4n′+4−m)R2αECt]2)$
(21b)
where m = 0, 1, 2, 3, and 4, respectively. Based on the nature of the problem and the physics, the sum of the series shown in Eq. (21) should be a finite number. It is also found that
$Sn+2Sn+1=hn+1exp(−[(4(n+2)−m)R2αECt]2)hnexp(−[(4(n+1)−m)R2αECt]2)$
$=hexp(−12R2−2mR2αECt−8R2αECtn)$
Therefore, it can be simplified as
$Sn+2Sn+1=h·B·exp(−A·n)$
(22)
where $A=(8R2/αECt)>0$ and $B=exp(−((12−2m)R2/αECt))<1$ are constants that depend on the device (i.e., R and αEC) and the time (t). As mentioned in Sec. 2.3, |hSI| < 1, |hSO| < 1, and |h| < 1, one can get from Eq. (22) that |Sn+2/Sn+1| < 1 and, more importantly, |Sn+2/Sn+1| decreases with increasing n, which means that these series are convergent. That is, there should be an integer N such that the sum of terms from N + 1 to has no significant effect on the overall summation value of the series. That is,
$∑n=1∞Sn(x,t)=∑n=1NSn(x,t)+∑n=N+1∞Sn(x,t)=∑n=1NSn(x,t)+Δ≅∑n=1NSn(x,t)$
(23)

where Δ is a small number or very close to zero so that the sum of first N terms of the series can be used as the summation.

To determine the value of N, we can set |Δ| = δ |S1(x,t)|, where S1(x,t) (>0) is the value of the series' first term and δ (0 < δ ≪ 1) is the control of the approximate calculation using Eq. (23). The smaller is the δ, the better is the approximation using Eq. (23). That is, as long as δ is small enough, Δ can be ignored for the calculation of series' summation. Using Eq. (21b), the Δ can be manipulated as follows:
$Δ=∑n′=N∞Sn′(x,t)=∑n′=N∞hn′exp(−[4R2αECtn′+(4−m)R2αECt]2)=∑n′=N∞hn′exp(−[an′+b]2)=∑n′=N∞hn′exp(−a2n′2)exp(−2abn′)exp(−b2)$
where $a=(4R/(2αECt))$ (>0) and $b=((4−m)R/(2αECt))$ (≥0). Considering that the value of h can be positive or negative as defined in Sec. 2.3, the calculation of Δ is replaced by the calculation of |Δ|. That is, as long as |Δ| is small enough, Eq. (23) can be used to calculate the summation of a series. Considering that $n′2≥3n′$ when n′ ≥ 3 (i.e., N ≥ 4), we can get
$|Δ|=|∑n′=N∞hn′exp(−a2n′2)exp(−2abn′)exp(−b2)|≤∑n′=N∞|hn′exp(−3a2n′)exp(−2abn′)exp(−b2)|=C∑n′=N∞|hexp(−γm)|n′=C∑n′=N∞(|h|exp(−γm))n′$
(24)
where γm = 3a2 + 2ab =(10 − m)2R2/(αECt) > 0, and C = exp (−b2) > 0. For the series, [|h exp(−γm)|]n, it is a geometric progression with a common ratio of |h| exp(−γm) < 1. Therefore,
$∑n′=N∞(|h|exp(−γm))n′=|h|Nexp(−Nγm)1−|h|exp(−γm)$
Then, inequality (24), can be rewritten as
$|Δ|≤C|h|Nexp(−Nγm)1−|h|exp(−γm)≤C|h|Nexp(−Nγ)1−|h|exp(−γ)$

where $γ=γm|m=4=12R2/(αECt)=3a2$.

Now, we can set
$C|h|Nexp(−Nγ)1−|h|exp(−γ)=S1(x,t)·δ$
That is,
$Δ≤C|h|Nexp(−Nγ)1−|h|exp(−γ)=S1(x,t)·δ$
Or
$Δ≤exp(−b2)|h|Nexp(−Nγ)1−|h|exp(−γ)=δ·exp(−[a1]2)$
where $a1=b$ so that
$(|h|)Nexp(−γN)==δ·[1−|h|exp(−γ)]$
(25)
Therefore, the N ($≥4$) can be written as
$N=−lnδ−ln(1−|h|exp(−γ)) γ−ln(|h|)$
(26)

Equation (26) can be used to determine the N. As mentioned after Eq. (5), the temperature change introduced by the ECE using the change in the electric field is included in the initial temperature of ECM1 and ECM2. Equation (26) indicates that the temperature change induced by the ECE using an electric field does not affect the value of N. Equation (26) indicates that the value of N is dependent on the R since $γ=12R2/(αECt)$.

From Eq. (26), one can find that if γ → ∞ (i.e., either R is very big or t is very small), N → 0, and that if |h| → 0 (i.e., the contacting coefficient at SI|ECM1 and ECM2|SO interface is close to one), the N → 0 for all values of $γ$. For both cases, the sum of the first four terms of the series (i.e., N = 4) shown in Eq. (23) can be used as the sum of the series. For a constant |h|, the N decreases as the γ increases (i.e., for a device, the N increases with time increases).

As shown in Eq. (26), to determine the value of N, the value of h (i.e., hSI and hSO) is important. To find the typical value of h and hSI/hSO, the properties of three typical ECMs—BaTiO3 (BT), Pb(Mg1/3Nb2/3)O3–4.5%PbTiO3 (PMNT) ceramics, and poly(vinylidene fluoride) (PVDF) polymer—and two metals—aluminum (Al) and copper (Cu)—as well as air are listed in Table 1, in which the materials are ordered using the thermal diffusivity (k) and effusivity (e). The thermal diffusivity of a material reflects its ability to absorb heat, while the thermal effusivity represents the speed to reach thermal equilibrium. Based on the properties of these materials, the values of hSI and hSO for different combinations of SI/ECMs/SO are calculated and presented in Table 2.

Table 1

Properties of some EC and non-EC materials at room temperature

Materialρ (kg m−3)cp (J K−1 kg−3)k (W m−1 K−1)α (m2 s−1)e (W m−2 K−1 s−0.5)
Air1.1610070.0262.226 × 10−55.51
PVDF180015000.27.407 × 10−87.348 × 102
PMNT81002000.251.543 × 10−76.364 × 102
BT606052761.879 × 10−64.377 × 103
Al2689951237.59.287 × 10−52.464 × 104
Cu89333854001.163 × 10−43.709 × 104
Materialρ (kg m−3)cp (J K−1 kg−3)k (W m−1 K−1)α (m2 s−1)e (W m−2 K−1 s−0.5)
Air1.1610070.0262.226 × 10−55.51
PVDF180015000.27.407 × 10−87.348 × 102
PMNT81002000.251.543 × 10−76.364 × 102
BT606052761.879 × 10−64.377 × 103
Al2689951237.59.287 × 10−52.464 × 104
Cu89333854001.163 × 10−43.709 × 104
Table 2

Determination of N for different combinations of ECMs and sink/source materials, δ = 10−5, t = 20 s

SIECMSOKSIhSIKSOhSO|h|R (mm)N
AirBTAl793.8−0.99750.17760.69830.69660.5/1.025/17
AlBTAir0.17760.6983793.8−0.99750.69660.5/1.025/17
AlBTCu0.17760.69830.11770.78940.55120.5/1.019/14
AirBTCu793.8−0.99750.11770.78940.78740.5/1.035/20
AirPMNTAl115.4−0.98280.02580.94970.93340.5/1.011/4
AirPMNTCu115.4−0.98280.01710.96640.94980.5/1.011/4
AlPMNTCu0.02580.94970.01710.96640.91780.5/1.012/4
AirPVDFAl133.3−0.98510.02980.94210.92810.05/0.1115/70
AirPVDFCu133.3−0.98510.01980.96130.94700.05/0.1146/81
AlPVDFCu0.02980.94210.01980.96130.90560.05/0.1115/74
SIECMSOKSIhSIKSOhSO|h|R (mm)N
AirBTAl793.8−0.99750.17760.69830.69660.5/1.025/17
AlBTAir0.17760.6983793.8−0.99750.69660.5/1.025/17
AlBTCu0.17760.69830.11770.78940.55120.5/1.019/14
AirBTCu793.8−0.99750.11770.78940.78740.5/1.035/20
AirPMNTAl115.4−0.98280.02580.94970.93340.5/1.011/4
AirPMNTCu115.4−0.98280.01710.96640.94980.5/1.011/4
AlPMNTCu0.02580.94970.01710.96640.91780.5/1.012/4
AirPVDFAl133.3−0.98510.02980.94210.92810.05/0.1115/70
AirPVDFCu133.3−0.98510.01980.96130.94700.05/0.1146/81
AlPVDFCu0.02980.94210.01980.96130.90560.05/0.1115/74

From the results shown in Table 2, one can find that the typical value of hSI/hSO for metals as SI/SO materials is 0.70–0.96, which results in the h of 0.55–0.95. Therefore, as long as the γ = 12R2/αECt is more than 3 (i.e., 4R2 > αECt), the N obtained from Eq. (26) is less than 4 for the δ being 10−5 or bigger. Again, in this case, the N = 4 can be used for the calculation of the series. When air is used as SI/SO material, the value of hSI/hSO is close to −1.0. In this case, the N is mainly determined by γ. The value of N for these combinations at special cases (i.e., two different values of R and at time of 20 s) is also presented in Table 2. Clearly, for the most of cases, the value of N is less than 30. For very thin PVDF films, the value of N is less than 150.

It has to be mentioned that the upper limit N given in Eq. (26) is obtained based on the heat flux solutions. That is, one can use the N determined from Eq. (26) to the calculation of heat fluxes given by Eqs. (18)(20). As discussed in the Supplemental Materials on the ASME Digital Collection, the series in Eqs. (10)(13) are also convergent and even convergent faster than the series in Eqs. (18)(20). Therefore, the N determined by Eq. (26) can also be used to calculate the temperature profiles given in Eqs. (10)(13).

Using Eq. (26), the dependence of N on both γ and h is illustrated in Fig. 2.

Fig. 2
Fig. 2
Close modal

From Table 2, one can find that the typical value of |h| is 0.55–0.95. From Eq. (26) and Fig. 2, one can find that for a constant h, the N decreases as increasing γ and that for a constant γ, the N decreases with decreasing h. Dependence of N on h and γ for some typical values is shown in Fig. 3.

Fig. 3
Fig. 3
Close modal

As shown by Eq. (26), the value of N is dependent on the values of δ, hSI/hSO, and γ, while the γ is determined by R, αEC, and t (i.e., size of ECM body, thermal diffusivity of the ECM, and time). As an example, using BT as the ECM, air as SI, and Al as SO to draw the values of N with R and time. Figure 4 shows the dependence of N on both t and R for δ = 10−5.

Fig. 4
Fig. 4
Close modal

Clearly, the longer the time, the bigger the N. But, as time gets longer, the increase in the N becomes slow. Also, the smaller the R, the bigger the N. However, the results shown in Fig. 4 demonstrate that the increase in the N is pretty slow at smaller R. It seems that for this case, for the numerical calculation, the N is no more than 40. The N is also dependent on the tolerance δ. The dependence of N on δ is shown in Fig. 5 on a semilogarithmic scale for the case. As expected, the smaller the tolerance (δ), the bigger the N is.

Fig. 5
Fig. 5
Close modal

### 3.2 Special Cases

#### 3.2.1 Only One Electrocaloric Material Layer.

If an ECE-based device uses only one layer of an ECM, all one needs to do is to set $TECM1i=TECM2i=TECMi$, which results in the temperature profile described by Eqs. (11) and (12) can be combined as given in Eq. (27), while Eqs. (10) and (13) can be simplified as Eqs. (28) and (29), respectively. The heat flux through SI|ECM and ECM|SO interface, Eqs. (18) and (20), can be simplified as shown in Eqs. (30) and (31), respectively, while Eq. (26) is still valid for the calculation of N used to calculate the sum of the series.

##### 3.2.1.1 Temperature profile in electrocaloric material (−R < x <R)
$TECM(x,t)−TECMi=∑n=1Nhn−1[hSO1+KSI(TECMi−TSIi)×erfc(4n−1)R−x2αECt +hSI1+KSO(TECMi−TSOi)×erfc(4n−1)R+x2αECt]−∑n=1Nhn−1[11+KSO(TECMi−TSOi)×erfc(4n−3)R−x2αECt +11+KSI(TECMi−TSIi)×erfc(4n−3)R+x2αECt]$
(27)
##### 3.2.1.2 Temperature profile in sink (−∞ < x < −R)
$TSI(x,t)−TSIi=KSI1+KSI×{(TECMi−TSIi) hSO×∑n=1Nhn−1erfc−x−R+(4n−0)KαSI−1/2R2αSIt−2(TECMi−TSOi)1+KSO×∑n=1Nhn−1erfc−x−R+(4n−2)KαSI−1/2R2αSIt+(TECMi−TSIi)×∑n=1Nhn−1erfc−x−R+(4n−4)KαSI−1/2R2αSIt}$
(28)
##### 3.2.1.3 Temperature profile in source (R < x < ∞)
$TSO(x,t)−TSOi=KSO1+KSO×{(TECMi−TSOi) hSI×∑n=1Nhn−1erfcx−R+(4n−0)KαSO−1/2R2αSOt−2(TECMi−TSIi)1+KSI×∑n=1Nhn−1erfcx−R+(4n−2)KαSO−1/2R2αSOt+(TECMi−TSOi)×∑n=1Nhn−1erfcx−R+(4n−4)KαSO−1/2R2αSOt}$
(29)
##### 3.2.1.4 Heat flux through SI|ECM interface
$qSI|ECM(t)=−kSIπαSItKSI1+KSI×{(TECMi−TSIi) hSO×∑n=1Nhn−1exp(−[(4n−0)R2αECt]2)−2(TECMi−TSOi)1+KSO×∑n=1Nhn−1exp(−[(4n−2)R2αECt]2)+(TECMi−TSOi)×∑n=1Nhn−1exp(−[(4n−4)R2αECt]2)}$
(30)
##### 3.2.1.5 Heat flux through ECM|SO interface
$qECM|SO(t)=kSOπαSOtKSO1+KSO×{(TECMi−TSOi) hSI×∑n=1Nhn−1exp(−[(4n−0)R2αECt]2)−2(TECMi−TSIi)1+KSI×∑n=1Nhn−1exp(−[(4n−2)R2αECt]2)+(TECMi−TSOi)×∑n=1Nhn−1exp(−[(4n−4)R2αECt]2)}$
(31)

Application notices: The solutions, Eqs. (27)(31), represent the heat conduction in a system where a hot/cold plate is placed between two different media. This is very close to the thermal treatment processes used in many engineering practices, such as the steel rolling process. Therefore, the analytical solution introduced here can also be used for these engineering practices.

For an ECE-based device with only one ECM layer, an analytical solution was introduced in Ref. [41], in which the temperature of the ECM layer was simply set as uniform all the time (i.e., the temperature in the ECM layer is only dependent on time, but independent of the location). The analytical solution reported in Ref. [41] has a simple exponential function for the time dependence of the temperature of the ECM layer. Clearly, the boundary conditions and assumptions used here are much more close to the reality than the model reported in Ref. [41]. For example, the temperature profile in the ECM layer is shown in Eq. (27). Based on Eq. (27), the temperature at any point in the ECM is dependent on the time by multiple error functions rather than one exponential function.

#### 3.2.2 Characterization of Thin Film of Electrocaloric Material.

For the direct characterization of the ECE of a material [31,42,43], the key is to accurately measure the temperature change in the ECM induced by changing electric field under an adiabatic condition, which can be a great challenge [33]. For example, when the electric field applied on an ECM is changed, the temperature of the ECM is changed due to the ECE, but the temperature will continuously change immediately due to the thermal conduction between the ECM and its environment, such as the substrate of an ECM thin film. That is, the ECE should be determined by the temperature of the ECM before the thermal conduction starts. Therefore, the sensor used to measure the temperature should have a very fast response time and also a very small thermal mass. The latter is more critical for the characterization of a thin film of an ECM. To overcome the challenges facing normal temperature sensors, such as thermocouples and thermistors, infrared (IR) cameras have been used to measure the temperature change in a thin film induced by changing electric field, which is illustrated in Fig. 6(a) for the characterization of a thin film on a substrate. The setup shown in Fig. 6(a) can be simplified as Fig. 6(b) as a three-layer structure with the middle ECM layer. It has to be mentioned that the IR sensor only measures the temperature of the surface of an ECM film. Therefore, the temperature measured by the IR sensor should be the interface temperature described by Eq. (14) rather than the temperature of the ECM layer described by Eq. (28). In other words, the current IR measurements underevaluate the ECE of a thin film. Based on the properties of thin film and air, the real temperature change in a thin film can be calculated using Eq. (14) so that the real ECE of a thin film can be determined.

Fig. 6
Fig. 6
Close modal

### 3.3 Other Issues.

It has to be mentioned that the multilayer structure of an ECM has been widely used as an ECE unit, such as multilayer ceramic capacitor [41,4448], in the development of ECE-based devices. There are three basic reasons for using the multilayer structure of an ECM: (1) a smaller thickness of ECM results in a higher electric breakdown field so that a higher electric field can be applied on the ECM, which would result in a higher temperature change or a stronger ECE, (2) for an ECM to be operated at the same electric field, a smaller thickness of the ECM means a lower electric voltage to be used, and (3) increasing the number of ECM layers results in a higher thermal mass. However, the electric field applied on each layer in the multilayer structure is the same. Therefore, the multilayer structure of an ECM can be treated as one ECM body so that the solutions introduced in Sec. 3.2.1 can be used.

When the thickness of ECM is small enough, the influence of electrodes cannot be ignored anymore. For example, an multilayer ceramic capacitor structure comprised of BaTiO3 layers of thickness 6.5 μm and the electrodes of thickness 2 μm has been reported [44]; the thin film with a thickness of ∼350 nm was studied [6], in which the thickness of electrode would be close to 100 nm. In these cases, the ECM/electrode structure can be treated as a 2–2 composite [49]. For this composite, the effective materials' properties can be used to replace the materials' properties of the ECM used in the solutions introduced here. For example, for a multilayer structure with M layers of ECM and M + 1 layers of electrodes, the effective density (ρeff) of the multilayer structure would be as
$ρeff=M·ρEC·dEC+(M+1)ρEL·dELM·dEC+(M+1)dEL$
(32)

where ρEC and ρEL are the density of ECM layer and electrode, respectively, and dEC and dEL are the thickness of one ECM layer and one electrode layer, respectively. Equation (32) can also be used to calculate the other materials' properties used in the solutions introduced above.

It has to be mentioned that in reality both the SI and SO in an ECE-based device are not semi-infinite. As demonstrated in Ref. [28], regarding the temperature change in the SI and SO, what is important is actually a thin layer of the SI/SO next to the ECM layer. This thin layer is usually less than five times of the thickness of the ECM layer. In other words, as long as the thickness of the SI/SO in an ECE-based device is more than five times of that of the ECM layer, the solution introduced here can be used as a very good approximation.

Mathematically, the solutions introduced here can be used for ECE-based devices with any thickness of ECM layer (R). However, it should be mentioned that the analytical solutions are obtained for ideal interfaces in the multilayer structure (i.e., the contact resistances are not included). It has to be mentioned that the smaller the thickness of ECM layer, the higher the influence of the contact resistances. Therefore, one should be cautious when use the analytical solutions in cases where a very thin ECM layer is used.

## 4 Concluding Remarks and Future Work

A set of versatile and flexible analytical solutions, the temperature profiles versus the time and the heat flux through the interfaces versus the time, have been introduced for the heat conduction in a four-layer structure to simulate the performance of ECE-based devices. The middle two layers are ECM and are sandwiched into the semi-infinite thermal SO and SI made of different materials. Although each solution includes the series with infinite terms, it is proved that these series are convergent so that the first N terms of the series can be used to calculate the sum of the series. The equation to calculate the N is introduced. By the case study, it is found that the N is usually small, mostly less than 40 and rarely more than 150. Additionally, the issues related to the applications of the solutions for real ECE-based devices are discussed.

The analytical solutions introduced here can be used to determine the time dependence of both the temperature distribution in each layer and the heat flux through each interface in an ECE-based using structure shown in Fig. 1. Similar to what was reported in Ref. [28], when these two ECM layers are independently operated, there would be a critical time for each interface, which can be calculated using the analytical solutions reported here. At this critical time, the heat flux changes its direction. This phenomenon can be used to improve the performance of ECE-based heat pumps.

## Acknowledgment

This research was partially supported by funding from the Southeastern Regional Sun Grant Center at the University of Tennessee through a grant provided by the U.S. Department of Agriculture under Award No. 2020-38502-32916. F.N. appreciates the financial support from the U.S. Department of State, Bureau of Educational and Cultural Affairs (ECA) and the U.S. Agency for International Development (USAID), as well as the administrative support from the U.S. Educational Foundation in Pakistan (USEFP) and the Institute of International Education (IIE), throughout his research as a Ph.D. candidate under the Fulbright Foreign Student Program.

## Funding Data

• U.S. Department of Agriculture (Award No. 2020-38502-32916; Funder ID: 10.13039/100000199).

## Data Availability Statement

The data that support the findings of this study are available within the article.

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