Numerical and thermal analysis of a caloric refrigeration device operating near room temperature

The application of external stimuli such as the magnetic and electric field in magnetocaloric and electrocaloric materials, and stress and pressure in elastocaloric and barocaloric materials give rise to a new generation of a refrigeration technology based on caloric materials which are considered an emerging alternative to classical refrigeration. Active caloric regenerator (ACR) made in parallel plates is studied under a large number of materials with Comsol multiphysics for a 2D numerical model. In this work, we compare various types of caloric materials, in terms of their thermodynamic properties, working mechanisms, and potential applications as solid refrigerant on caloric refrigeration devices. For this purpose, the energy equation, Navier-Stocks equation, and continuity equation are considered to study the heat transfer phenomena in refrigerator. The water was used as a carrier fluid to transport the thermal energy from the solid refrigerants to heat exchanger. This study is performed at velocity 0.06 m/s and the frequency 2 Hz at room temperature. Among them, Gadolinium show the best results in term temperature span, coefficient of performance, and the cooling power, higher than every other caloric materials, conferring to magnetocaloric cooling globally the most promising system. Our analysis provides insights into the selection and optimization of caloric materials for caloric refrigeration, which can contribute to the development of sustainable energy systems.


Introduction
The area of refrigeration and air conditioning faces several major challenges today.We have already mentioned the immediate and growing need for the use of cold in very diverse fields such as food security, health, and transport, as well as for the development of information technologies and microelectronics, to name just a few [1][2][3].In the medium and long term, the need to lower gas emissions with a negative impact on the environment was obvious to the national and international scientific community [2,3].
In view of the various existing refrigeration systems, caloric refrigeration is a serious alternative to the current system and vapor compression refrigeration system.Indeed, they do not require the use of a mechanical compressor which is substituted by the application of a electric and magnetic field or a pressure.They do not use fluids that are sources of greenhouse gases but are likely to use cleaner fluids (water, silicone).These systems arise from an intrinsic property of certain types of materials [4,5], so understanding this phenomenon and precisely identifying the parameters influencing it are necessary for the optimization of these materials.It also has another major advantage which is the high thermodynamic efficiency.Indeed, with the caloric effect, the efficiencies can reach 60% (40% in the best conventional thermodynamic systems).The coefficient of performance is theoretically 10 whereas it is 5 for a conventional thermodynamic cycle [6].Other advantages can also be mentioned such as the possibility of making systems compact, the material being solid and not gaseous, and an easy adjustment of the power or the external field [6,7].
Solid-state refrigeration is a promising technology for the production of cold.The main element of solid-state refrigeration is the active caloric regenerator (ACR) made of magnetocaloric, electrocaloric, elastocaloric, and barocaloric.Magnetocaloric cooling, based on magnetocaloric effect exhibiting an entropy change under a magnetic field (electromagnets, superconducting magnets, permanent magnets), interest in magnetocaloric cooling began in 1918 [6,7]; Barclay [8] introduced the use of active magnetic regenerator (AMR).AMR is proven to achieve higher coefficient of performance and cooling power.Electrocaloric cooling is based on ferroelectric material ceramics and polymer [9,10].The electrocaloric effect is detected in electrocaloric materials especially ferroelectric showing an entropy and temperature change while applying electric field, dually to magnetocaloric cooling, reference thermodynamic cycle is AER (active electrocaloric regenerators), and electrocaloric refrigerator allows to reach 50% coefficient of performance (carnot cycle) [10], by pressing or stretching the materials with mechanical properties, BCE (barocaloric effect) or eCE (elastocaloric effec).Such principles are the fundamental of mechanocaloric cooling, eCE employs shape memory alloys [11,12].The exploitation of caloric effect near room temperature is limited by the existing caloric materials which do not make it possible to reach high temperature span [13], and a sample of Gd near room temperature produces a magnetocaloric effect of about 10 K under B=5 T. Knowing that Gd is considered to be one of best caloric materials for the refrigeration systems [13,14].This problem has been overcome through the exploitation of active caloric regenerator (ACR) [15].Regeneration (ACR) in caloric devices allows the energy thermal rejected by system to be restored and returned to system in the cycle.The capacity used for network load refrigeration can be effectively used to increase the effective entropy and the resulting temperature span [15].
The performance caloric refrigeration depends on the selection and optimization of caloric materials, which can convert mechanical work into thermal energy.In recent years, various types of caloric materials have been developed and investigated for solid-state refrigeration, including shape memory alloys, magnetocaloric materials, electrocaloric materials, and elastocaloric materials [16,17].Each type of caloric material has its unique working mechanisms, thermodynamic properties, and potential applications [18].Therefore, it is essential to compare and evaluate different types of caloric materials for solid-state refrigeration to identify the most promising candidates for practical applications.This works provides a comprehensive comparison between various caloric materials for solid-state refrigeration, highlighting their advantages, limitations, and potential for future development.The review aims to provide insights into the selection and optimization of caloric materials for solid-state refrigeration and to promote the development of sustainable energy systems.
The year 2015 marked an important advancements in the field of caloric refrigeration the second generation [19].Since then, works on the study of the caloric effect and its use for refrigeration has progressed considerably as shown in Fig. 1.This works concerns as much the development of a technique for characterizing the caloric effect as the synthesis of new caloric materials and the development of prototypes of caloric refrigeration.
The study of caloric refrigerator is essential for the optimization of ACR refrigeration systems.Furthermore, the numerical model of ACR is complex; it is characterized by nonlinear and transient operation and by mico-scale and macro-scale models [20].The objective of this paper is to compare the energy efficiency of an ACR based on MCE, ECE, eCE, and BCE using a 2D numerical model based on COMSOL multiphysics in term the temperature span (∆T), the caloric effects, and COP.
This analysis of the functioning of the model and the sensitivity of the parameters helps us to improve understanding, to highlight the influencing parameters, and to study the interactions between the parameters and their impacts on the total performance of the system.It makes it possible to draw parametric behavior maps to help design future ACR refrigeration systems at room temperature with better performance and at lower cost.The main objective of this study is to compare different types of caloric materials in terms of their performance and potential applications as solid refrigerants in caloric refrigeration devices operating near room temperature.By evaluating the performance and applications of various caloric materials, this research aims to provide insights into their suitability for efficient and environmentally friendly cooling at room temperature.

Second generation of an active caloric prototype
Active caloric regenerator (ACR) is based on the same physical phenomenon for all types, such as temperature or entropy change, due to a variable electric and magnetic field, pressure, and stress under adiabatic or isothermal process.The state of art of an active caloric device secondgeneration built near room temperature is presented in Tables 1, 2, and 3.

Active caloric regenerator description
A fluid, usually water, alternately passes through the regenerator bed, based on caloric material, to allow heat exchange between the different thermodynamic stages as shown in Fig. 2. The objective of the model is to better understand the functioning of the ACR cycle to optimize the performance and to determine the efficiency of system.Six main parameters define our system: the fluid velocity, the frequency, the thickness of water and caloric material, the temperature span, and the length of ACR.
The main operational of ACR device are as follows: • ACR made in parallel plates.
• The thickness of a single channel is 45x0.125mm(53 channels), and the geometry of ACR is shown in Fig. 3. • Fluid flow velocity and frequency: V=0.06 m/s, f=2 Hz.

Thermodynamics of caloric effects
Caloric effects can be described by thermodynamic equations [40,41].Although the responsible mechanisms are not yet clearly identified, they show great similarities.These equations were developed for ceramic materials.However, they are applicable to polymeric materials [42].
The difference in Gibbs free energy to understand the caloric effects of caloric materials can be expressed by Eq. ( 1) [42,43].
Where, ΔU: The internal energy; ΔS: isothermal entropy change µHΔM: Applies to magnetocaloric effect (MCE) EΔP: Applies to electrocaloric effect (ECE) σΔu: Applies to elastocaloric effect (eCE) pΔV: Applies to barocaloric effect (BCE) Equation ( 2) for calculating the entropy change, denoted as ΔS, within a system can be represented as follows [44,45]: (1)  The expression for the adiabatic temperature change can be represented as follows (Eq.( 3)): where T represents the temperature, C denotes the heat capacity, X represents the extensive property related to temperature changes, Y represents a variable related to the system's state, and (∂X/∂T) signifies the partial derivative of X with respect to T.
The equations of Navier-Stokes, continuity, and energy are fundamental in studying various phenomena that occur between solid materials and the carrier fluid.These equations provide a mathematical framework for understanding and analyzing the behavior of fluid flow and the interactions between the fluid and solid surfaces [46,47].
The Navier-Stokes (Eq.( 4)) describe the conservation of momentum in fluid flow, taking into account factors such as pressure (p), viscosity (U), and inertial forces.These equations enable us to model and predict the velocity and pressure distributions within the fluid as it interacts with the solid material [46,47]. ( The continuity (Eq.( 5)) ensures that mass is conserved within the fluid.It relates the divergence of the velocity field to the rate of change of density, providing insights into how the fluid flows around and interacts with the solid material.By solving the energy equation, we can study the temperature distribution and heat transfer phenomena between the solid material and the carrier fluid.
The energy (Eq.( 6)) can be used in order to characterise the heat transfer phenomena between the solid material and the carrier fluid [47,48].

Results and discussions
In general, our numerical model respects the successive stages which are the introduction of the parameters and the input functions, the solver for the resolution of the equations, and the output parameters during the numerical simulation.In the following, we will present the input data of the model as well as our approach by using comsol multiphysics software.The two major categories of input parameters are the solid matrix represented by the caloric materials and their variable characteristics over time and as a function of the fluid velocity and the applied field in the solid matrix.In a very important way, the influence of the intensity and the frequency of applied field is to be taken into account as input variable (6) Fig. 3 The geometry of ACR (Table 4).We used the reference materials of each type of solid state refrigeration.

Numerical model validation
The  2017), and the results obtained from the numerical model demonstrate a high level of agreement with only minor differences.This indicates that the numerical model is accurate and reliable.The successful validation of the numerical model implies that it can be confidently utilized to predict the behavior of the refrigeration system under various operating conditions.Its predictions can serve as a dependable basis for making informed decisions regarding the system's design and optimization.Overall, the findings highlight the suitability and efficacy of the numerical model as a valuable tool in the study and development of mechanocaloric refrigeration systems.

The 2D temperature distribution and adiabatic temperature
The simulation model of an ACR with thermophysical parameters is presented in Table 4.In order to see the temperature gradient generated by the ACR, Fig. 5 gives the comparison between the temperature profiles along the length of the material for the four caloric materials for the last cycle.In addition to the temperature profiles, our model allows us to obtain the profiles of the different energies involved in the ACR process.We place ourselves in the case where the plates systematically have a longitudinal thermal gradient of 278 to 293 K.They can also be the seat of a second transverse thermal gradient in the thickness which is superimposed on the first.This thermal gradient reproduces the situation of exchange of caloric material with the fluid.Fig. 4 Comparison between numerical used model and experimental data reported in literature Figure 6 shows the evolution of temperature of the different caloric materials as a function of time with a zoom representing an elementary ACR cycle.Note that after 25 s, the four curves reach steady state.Moreover, the temperature span ∆T obtained is more than the initial caloric effect.From this figure, we can deduce the hot and the cold temperature according to Table 4; the positive curve corresponds to what we called hot energy.Similarly, the negative curve corresponds to cold energy.This shows that with the principle of ACR, based on thermal regeneration, we can obtain systems with temperature differences which are close to those required by industrial applications.The maximum obtained values for different ACR are presented in Table 5.
In order to highlight the caloric effect in of this various materials, the different parameters of the caloric effect have been represented and presented in Fig. 7.This type of representation makes it possible to have a synthetic vision of the caloric effect.The observation of the caloric effect as a function of the transition temperature, according to the natures of materials, makes it possible to highlight the category of materials presenting the most interesting cooling capacity at ambient temperature (≈ 293K) with a view to an application in a cooling system.Indeed, as shown in Fig. 5, magnetocaloric material has a higher temperature span ∆T than other categories of materials near room temperature, which allows magnetocaloric material to be considered as promising materials for room temperature application.

Evolution of temperature in caloric regenerator
The system operates at startup in transient mode, and a temperature gradient is thus formed along the regenerator following the alternating movement of the fluid.Thus, HHEX and CHEX are at different temperatures.Figure 8 shows a typical evolution of temperature in caloric materials as a function of the   regenerator length with the system operating at zero charge.In the initial state, caloric materials are at a temperature of 293 K.During operation, the temperature increases and approaches a curved profile with a temperature span which depends on the flow velocity and regenerator geometry.The numerical simulation was conducted for 100 s with a time step of 0.1 s for the transient regime.We obtained 1000 lines of temperature gradients.To make it easier to observe the evolution of gradients in the solid material, we suggest representing only 10 gradient lines.can observe the orientation of the current lines which is parallel to the horizontal axis, which confirms the nature of flow previously mentioned, laminar flow in all the cases studied.

The 2D velocity field and pressure drop
Figure 10 shows pressure drop as a function of microchannel length under velocities of fluid range from 0.05 to 0.15 m/s.Pressure drop is influenced by the flow velocity and is directly proportional to the length of the microchannel.The smallest value of pressure drops at the outlet of the microchannel 1200 Pa is obtained with the smallest value of flow velocity 0.05 m/s.The greatest value of pressure drops at the outlet of the microchannel 1800 Pa is obtained with the greatest value of flow velocity 0.150 m/s.For the other intermediate values of the flow velocity, the evolution of the pressure drops is linear and rises with the growth of the fluid velocity.

Coefficient of performance and cooling power
Figure 11 shows the evolution of cooling power as a function of temperature span between the HHEX and CHEX in a system having a flow velocity of 0.05 m/s for several caloric materials.The best performances are obtained, in our case, with a magnetocaloric material and with a fluid velocity 0.05m/s which corresponds to a frequency of 2 Hz.For all four situations, cooling power shows maximum in temperature span of 3-6 K, after the cooling power decreases with the increase of the temperature span between HHEX and CHEX.The system performance increases with decreasing fluid velocity in the range of 0.150 to 0.05 m/s.In addition, the coefficient of performance as a function of time COP is directly proportional with time; except one of the cases studied.The coefficient of performance for the fluid velocity 0.150 m/s is the lowest at frequency 0.30 Hz and not correlated with the flow velocity, which leads to the decrease in the gradient thermal, as also seen in Fig. 10.

Conclusion
The objective of this work is the design of a 2D multiscale and multiphysics numerical model with COMSOL mutliphysics intended to simulate the behavior of an active caloric regenerator system near room temperature using caloric materials.The improvements made in the numerical model and the validation of the functioning of the model by the study of comparison between several caloric materials (Gd, BaTiO 3 , PbTiO 3 , (NH 4 ) 2 MoO 2 F 4 ) allowed us to reproduce in a relatively faithful way the behavior of an ACR device.PbTiO 3 was employed as elastocaloric, electrocaloric, or multicaloric materials in an active caloric refrigerator.The nonlinear 2D model requires long calculation times but is fully operational (taking nonlinearities into account, convergence criterion, field profile, etc.), a very useful tool for dimensioning, still needs to be improved, especially in taking into account of viscous friction.The perspectives of this work is the identification of the type of the order-disorder phase transition.For this, various characterization methods can be considered, such as characterization by temperature infrared spectroscopy.Regarding caloric materials, a research proposal can be made on an in-depth study of caloric materials.It is estimated that at this level, the progress of basic research will be the most important, because there is a real boom in this direction.

21 Fig. 1
Fig. 1 The papers published recently in caloric refrigeration since 2000 to 2022

Fig. 2
Fig. 2 Schematic diagram of the ACR numerical model validation was conducted using the work of Engelbrecht et al. (2017) as a reference [39].Figure 4 illustrates the comparison between the experimental data provided by Engelbrecht et al. (

Fig. 5 2D
Fig. 5 2D Temperature distribution for different ACR implemented in COMSOL multiphysics software

Fig. 6
Fig. 6 The Temperature evolution as function of time at frequency 2 Hz for different ACR

Fig. 7
Fig. 7 Adiabatic temperature change for different ACR

Figure 9 Fig. 8 Fig. 9
Figure 9 shows the theoretical velocity in the numerical model.We can observe in the figure.The fluid velocity decreases from the maximum in the center of the micro-channel 0.06m/s to the zero value of the boundary layer.For all the flow velocities studied, we

Fig. 10
Fig. 10 Pressure drop evolution as a function of regenerator length

7 NomenclatureB
Magnetic field induction [T] C Specific heat [J/kg K] E Electric field [V/m] f Frequency [Hz] H Magnetic field [A/m] k Thermal conductivity [W/m K] P Polarization [C/m 2 ] p Pressure [Pa] S Entropy [J/K] T Temperature [K] U Fluid flow velocity [m/s] V Volume [m 3 ]

Table 4
Operational parameters introduced in numerical model of ACR

Table 5
The maximum values obtained of hot temperature, cold temperature, and temperature span for different ACR under comsol multiphysics software