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Solidstate is the new frontier of NotInKind technologies arisen as possible future replacement of vaporcompressionbased systems. Caloric cooling and heatpumping found their operation on caloric effect; a class of thermophysical effects detected in caloric materials following an adiabatic change of the intensity of an applied external field, that results in a variation of the temperature in the material itself. A Braytonbased thermodynamical cycle, called Active Caloric Regenerative cycle, is used to build caloric cooling systems or heatpumps. In ACR cycle the caloric material acts both as refrigerant and as regenerator and an auxiliary HeatTransfer Fluid (HTF) is employed to vehiculate heat fluxes between the cold and hot environments. The most common HTF is water but advanced solutions could be adopted to enhance the heat exchange coefficients, like nanofluids. Nanofluids are suspensions consisting of solid highthermalconductivity nanoparticles dispersed in a base fluid to enhance the global thermal conductivity of the fluid.
In this paper we investigate, numerically, the energy performances of a caloric heat pump employing waterbased Al_{2}O_{3} nanofluids as HTF. The analysis is perpetuated changing both the nanofluid volume concentrations and the caloric materials employing electrocaloric, elastocaloric and barocaloric ones.
caloric, cooling, heatpumping, nanofluid
Solidstate caloric heatpumping is the new frontier of NotInKind (NIK) technologies arisen as possible future replacement of vaporcompressionbased systems [13]. This technology has the strong point to not employ greenhouse gases, that can result toxic or damaging for the environment and that can contribute to increase global warming, together with presenting improvements in energy efficiency and exhibiting the potential of recycling its components [47].
Caloric effect, an intrinsic property of caloric materials, is the physical phenomenon which solidstate heatpumping is based on: a temperature variation in the material is detected if an applied external field changes its intensity, adiabatically. Measures of caloric effect are:
$\Delta s=\int{_{{{Y}_{0}}}^{{{Y}_{1}}}}{{\left( \frac{\partial x}{\partial T} \right)}_{Y}}dY$ (1)
$\Delta {{T}_{ad}}=\int{_{{{Y}_{0}}}^{{{Y}_{1}}}}\frac{T}{C}{{\left( \frac{\partial x}{\partial T} \right)}_{Y}}dY$ (2)
The nature of the driving field Y particularizes the caloric effect. Magnetic fields applied to a magnetocaloric materials give rise to magnetocaloric effect (MCE) where Y=H and X=M, electric fields to electrocaloric effect (ECE) [8] where Y=E and X=P, mechanical stress to elastocaloric effect (eCE) [9], where Y=σ and X=ε, pressure field to barocaloric effect (BCE) [10] where Y= p and X=V.
Caloric effects are applied in a Braytonbased thermodynamical cycle experienced by an Active Caloric Regenerator (ACR), made of the caloric refrigerant which acts both as refrigerant and regenerator to recover heat fluxes. ACR counts four processes, executed sequentially and cyclically, experimented by the regenerator to which the external field is applied and crossed by a thermovector fluid. The latter connect thermally a cold and a hot reservoir (CHEX and HHEX). The four processes are:
(1) Adiabatic field decreasing, with consequent reduction of the caloricmaterial temperature;
(2) Fluid flows from hot to cold side, cooling itself and then reaching the cold heat exchanger, where it absorbs heat from the latter, producing a cooling load.
(3) Adiabatic field increasing, during which the intensity of the external field is increased causing the increasing of the material temperature, due to caloric effect;
(4) Fluid flows from cold to hot side, cooling the regenerator and rejecting heat in the HHEX, thus producing a heating load. This process realizes the desired effect for heatpumping operation mode.
As seen from the above described ACR cycle, the auxiliary heat transfer fluid plays a keyrole since it is responsible of transferring heat fluxes and therefore the more efficient is the solidtofluid heat exchange, greater are the energy performances of the caloric heatpump. The most common HTF employed in Active Caloric Regenerator is water but advanced solutions could be adopted to enhance the heat exchange coefficients [11]. Among them, an innovative HTF could be represented by nanofluids [12]. Nanofluids are suspensions consisting of solid highthermalconductivity nanoparticles (1100 nm) dispersed in a base fluid to enhance the thermal conductivity of the resulting fluid. The conception of nanofluid, formulated by dispersing metallic or nonmetallic nanometersize particles in base liquids such as water and ethylene glycol, was proposed first by Choi [13] in 1995 and, a few years later, Choi et al. [14] showed that the addition of a small amount (less than 1 % by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. Ever since, there have been great researchinterests in exploring the effectiveness and feasibility of using nanofluids as convective heat transfer fluids. As a matter of fact, nanofluids are potential heat transfer fluids with enhanced thermophysical properties and heat transfer performance. They can be applied in many devices for better performances (i.e. energy, heat transfer and other performances), so the number of potential applications of this technology is extremely vast. Specific application of nanofluids in engine cooling, solar water heating, cooling of electronics, cooling of transformer oil, improving diesel generator efficiency, cooling of heat exchanging devices, improving heat transfer efficiency of chillers, domestic refrigeratorfreezers, cooling in machining, in nuclear reactor and defense and space have been studied and investigated by nanofluidsscientific community [15]. Among them, extremely exiguous is the number of investigations on nanofluids as auxiliary heattransfer fluid of caloric cooling and heatpumping devices: literature accounts only two works conducted in Active Magnetocaloric Refrigerators field. Chiba in his work [16] tested the energy performance of an Active Magnetocaloric Regenerative refrigeration (AMR) cycle operating near room temperature using nanofluids as heattransfer fluid, in order to enhance the heat transfer in the regenerator bed during the fluidflow phases. Mugica et al. [17], analyzed the energetic and exergy performances of parallelplate AMR refrigerator though a 1D model. Both of the investigations [16, 17] employed waterbased Al_{2}O_{3} nanofluids as HTF. Therefore, literature did not account of investigations conducted on caloric effect different from magnetocaloric one. This paper aims to fulfill this gap, presenting the results of a numerical investigation on the energy performances of a caloric heat pump employing waterbased Al_{2}O_{3} nanofluids as HTF while the regenerator works with electrocaloric, elastocaloric and barocaloric materials. The tests were performed through a 2Dimensional model solved with Finite Element Method and the analysis is perpetuated changing both the nanofluid volume concentration and the caloric materials.
This section describes all the aspects regarding the physics and the structure modelling: Active Caloric Heat Pump modelling, the physics and the modelling of: the caloriceffect materials as refrigerants; the nanofluids as heat transfer fluids.
2.1 Modelling active caloric regenerative heat pumps
The behavior of a caloric heat pump is described through a 2D model, already introduced [18, 19] and validated [2022] in previous investigations. The geometry of the heat pump sees a parallelplate regenerator made of caloric material, separated by channels in which the Heat Transfer Fluid (HTF) crosses the regenerator. The desired effect is to add heat to the indoor room connected with a hot heat exchanger at T_{H}. The cold heat exchanger is coupled with the outdoor environment, whose temperature is T_{C}. Below there is the mathematical structure that the 2D model is based on:
$\left\{ \begin{matrix} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \\ \frac{\partial u}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=\frac{1}{\rho nf\partial x}\frac{\partial p}{\partial x}+v\left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right) \\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=\frac{1}{\rho nf\partial x}\frac{\partial p}{\partial y}+v\left( \frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{y}^{2}}} \right) \\ \begin{align} & \frac{\partial Tnf}{\partial t}+u\frac{\partial Tnf}{\partial x}+v\frac{\partial Tnf}{\partial y}=\frac{kf}{\rho nfCnf}\left( \frac{{{\partial }^{2}}Tnf}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}Tnf}{\partial {{y}^{2}}} \right) \\ & \frac{\partial Ts}{\partial t}=\frac{ks}{\rho sCs}\left( \frac{{{\partial }^{2}}Ts}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}Ts}{\partial {{y}^{2}}} \right) \\ \end{align} \\\end{matrix} \right.$ (3)
$\left\{ \begin{matrix} \rho nfCnf\frac{\partial Tnf}{\partial t}=Knf\left( \frac{{{\partial }^{2}}Tnf}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}Tnf}{\partial {{y}^{2}}} \right) \\ \rho sCs\frac{\partial Ts}{\partial t}=Ks\left( \frac{{{\partial }^{2}}Ts}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}Ts}{\partial {{y}^{2}}} \right)+Q \\\end{matrix} \right.$ (4)
$Q=Q(field,Ts)=\frac{\rho sCs(field,Ts)\Delta \text{T}ad(field,Ts)}{\tau }$ (5)
Eq. (3) models the fluidflow processes of the ACR cycle; Eq. (4) describes the field rising/falling phases. Caloric effect is modeled through the Qterm reported in Eq. (5). It converts the caloric effect into a power density. Since Q is a function of the field and the temperature, its mathematical expression has been obtained by a mathematical finder software, as a result of elaboration and manipulation of experimental data of C_{s }(field, T_{s}) and ∆T_{ad }(field, T_{s}), coming from scientific literature. Different Qterms correspond to the tested caloric materials.
The model is solved with Finite Element Method and the ACR cycle runs cyclically several times until reaching steadystate conditions.
2.2 The selected caloric materials
To select the caloric refrigerants for the present investigation we based on highcaloric effects and diversity criteria, in order to test the most performing ones, in room temperature range, among the electrocaloric, elastocaloric and barocaloric effect materials.
Pb_{0.97}La_{0.02}(Zr_{0.75}Sn_{0.18}Ti_{0.07}) O_{3} (PLZST) [23], deposited on LaNiO3/Si (100) substrate, is the electrocaloric material under test. It exhibits a maximum Giant ECE at 278 K but ∆T_{ad} remains very high in the range 278÷298 K, making PLZST definitely suitable for heat pump applications. Specifically, we considered it under electric field changes of 90 MVm^{1} and 70 MV m^{1 }whose ∆T_{ad} peaks are 54 K and 43 K, respectively.
We also considered the elastocaloric NiTi polycrystals [24], benchmark of elastocaloric systems, showing a peak at 350 K but anyway exhibiting a remarkable elastocaloric effect in temperature range devoted to heat pump applications. The considered stress field change is 0.9 GPa which results in a peak of 25 K as adiabatic temperature change due to elastocaloric effect.
The barocaloric oxyfluorides (NH_{4})_{2}MoO_{2}F_{4} [25] showing a maximum direct barocaloric effect at 272 K which remains remarkable until 360 K, ensures a good applicability for heat pumps application. The considered drop of applied pressure field is 0.9 GPa that guarantees a maximum of 18 K due to barocaloric effect.
As barocaloric we also focused on Acetoxy Silicon Rubber (ASR) exhibiting a supergiant [26] barocaloric effect investigated for ∆p = [0.273; 0.390] GPa under which associated maximum ∆T_{ad} of 30 and 41 K occur, respectively.
2.3 Modelling nanofluids and their properties
Basing on the only two previous studies about nanofluids for magnetocaloric applications, we decided to employ the same nanofluid, the alumina–water nanofluid. It is composed of water, as base fluid, in which nanometric particles of alumina (Al_{2}O_{3}) were dispersed. Moreover, to conduct the investigation with nanofluids containing variable volume fraction of alumina, we considered different concentrations of nanoparticles dispersed on equal volume of base fluid as:
$\varphi \text{=}\frac{{{V}_{np}}}{{{V}_{nf}}}=\frac{{{m}_{np}}}{{{m}_{nf}}}$
The thermophysical properties of the aluminawater nanofluids are dependent on the volume fraction of nanofluids φ; therefore, to include such dependence in our caloric heatpump model, we adopted the following correlations demonstrated in previous investigations [17, 27]:
${{\rho }_{nf}}=\frac{{{m}_{nf}}}{{{V}_{nf}}}=\frac{{{m}_{bf}}+{{m}_{np}}}{{{V}_{nf}}}={{\rho }_{bf}}\frac{{{V}_{bf}}}{{{V}_{nf}}}+{{\rho }_{np}}\frac{{{V}_{np}}}{{{V}_{nf}}}=(1\varphi ){{\rho }_{bf}}+\varphi {{\rho }_{np}}$
$Cnf=(1\varphi ){{C}_{bf}}+\varphi {{C}_{np}}$ (6)
${{K}_{nf}}={{K}_{bf}}\frac{{{k}_{np}}+2{{k}_{bf}}2\varphi ({{k}_{b}}_{f}{{k}_{np}})}{{{k}_{np}}+2{{k}_{bf}}+\varphi ({{k}_{b}}_{f}{{k}_{np}})}$ (7)
Μ_{nf}=(1+7.74φ) (8)
Table 1 lists the properties of the base fluid and the nanoparticles of aluminawater nanofluids at T=293 K and p=1 atm.
Table 1. Thermophysical properties of aluminawater base fluid and nanoparticles
Substance 
ρ $\frac{kg}{m^3}$ 
C $\frac{J}{kgK}$ 
k $\frac{W}{mK}$ 
μ $\frac{kg}{ms}$ 
Water 
998.2 
4182 
0.597 
9.93*10^{4} 
Al_{2}O_{3} 
3970 
765 
36 
 
The investigation was performed employing the above described set of caloric materials in the 2D model of the ACR working in the range 278÷298 K in heatpump operation mode at fixed ACR frequency (1.25 Hz) and fluid velocity (0.2 m*s^{1)}. The auxiliary heattransfer fluid was aluminawater nanofluids with variable concentration φ = [0; 0.02; 0.04; 0.06; 0.08; 0.1].
The results are reported in terms of power of the heatpump, that measures the power at which the system pumps heat and Coefficient of Performance (COP) defined, respectively, as:
$\dot{Q}H=\frac{1}{\theta }\int{\begin{matrix} 2\tau +n\theta \\ \tau +n\theta \\\end{matrix}}{{\dot{m}}_{nf}}{{C}_{nf}}({{T}_{nf}}(L,y,t){{T}_{H}})dt$ (9)
$\text{COP=}\frac{{{{{\dot{Q}}}}_{\text{H}}}}{{{{{\dot{W}}}}_{\text{TOT}}}}$ (10)
OP is the coefficient of performance of the heat pump and it is conceived as the ratio between the heating power of the pump and the total energy expense made to get it. $\dot{W}_{TOT}$ embraces both the contribution due to the external field variation and the one connected to the mechanical power required for the fluid motion.
In Figure 1 one can appreciate the heating power of the caloric heatpump, for the tested materials occurring under the operative conditions of section 3, when the heattransfer fluid is aluminawater based nanofluid with variable concentration. General considerations arising through examining the data plotted are that heating power of the heatpump increases when incrementing nanofluid concentration for all the tested materials, with a medium increment of 19 %. PLZST under ∆E=90 MV m^{1} as well as NiTi under ∆σ=0.9 GPa give the highest Q_{H }with a maximum of around 580 W in correspondence of φ=0.1. This is due to the Giant ECE shown at 278 K by the former, and to the elastocaloric effect that is quite constant in the operating temperature range. On the contrary the lowest heatingpowers are detected for (NH_{4})_{2}MoO_{2}F_{4} since they do not exceed 240 W. Such results are due both to the ∆T_{ad} smaller than PLZST and to the peak located at 272 K out of the working temperature range.
Figure 1. Heating power vs nanofluid concentration for the tested caloric materials
Figure 2 reports the coefficients of performances of the caloric materials under test, evaluated with respect to the variable concentration of the aluminawater based nanofluid.
The situation is quite different from the data about heatpumping powers, since the highestCOP material is still NiTi but not anymore PLZST that exhibits, for both the 90 and 70 MV m^{1} cases, the lowest coefficients of performances. Such turnarounds are caused by the expenses needed for electricalfield changing, higher than mechanical stretching ones. The NiTi based caloric heat pump presents 14.8 as maximum COP if it works with 10 % alumina90 % water nanofluid. A middle COPincrement of 15 % is detected if the nanofluid concentration increases from 0 % up to 10 %. The barocaloric Acetoxy Silicone Rubber presents also satisfying COPs with a maximum of 9.3 registered for φ=0.1 under ∆p=0.39 GPa and a middle increment of 19 % while increasing nanofluid concentration.
Figure 2. Coefficient of Performance vs nanofluid concentration for the tested caloric materials
In this paper we investigated, numerically, the energy performances of a caloric heat pump employing waterbased Al_{2}O_{3} nanofluids as HTF. The analysis was perpetuated changing both the nanofluid volume concentrations and the materials employing electrocaloric, elastocaloric and barocaloric ones.
The investigation was performed testing a wide set of materials showing remarkable electrocaloric, elastocaloric or barocaloric effects in the temperature range of interest (278÷298 K) for heat pump operation mode. The ACR frequency was fixed (1.25 Hz) as well as the fluid velocity (0.2 m. s^{1}). The concentration of the auxiliary heattransfer nanofluid was varied in the range φ=[0; 0.02; 0.04; 0.06; 0.08; 0.1].
From the results, reported in terms of power of the heatpump and coefficient of performance, we detected that the effect of working with aluminawater based nanofluids and increasing the concentration, results in an improvement of the energy performances of the caloric heat pump.
General considerations arising through examining the data plotted are that heating power of the heatpump increases when incrementing nanofluid concentration for all the tested materials, with a medium increment of 19 %. The highest ones were measured for PLZST under ∆E=90 MV m^{1} and NiTi under ∆σ=0.9 GPa with a maximum of around 580 W in correspondence of φ=0.1. Examining the coefficient of performances, PLZST presents the lowest ones due to the high electrical power needed for electricalfield changing, whereas NiTi presents the highest COPs, too. Anyhow globally, the effect of enhancing nanofluid concentration results in a performance improvement also in terms of COP.
Basing on the results collected, the final considerations that can be drawn are that employing aluminawater nanofluids as auxiliary HTF carries to an upgrading of the energy performances of the caloric heatpump and that, in this investigation, the best combination of caloricrefrigerant + HTF is given by NiTi + 10 % alumina90 % water nanofluid.
C 
specific heat, J. kg^{1}. K^{1} 
E 
electric field, V.m^{1} 
H 
magnetic field, A. m^{1} 
k L 
thermal conductivity, W. m^{1}. K^{1} length of the regenerator, m 
M m P 
magnetization, A. m^{1} mass, kg polarization C. m^{2} 
p Q Q ̇ 
pressure, Pa power density, W. m^{3} power, W 
s T t u V v W X x Y y 
entropy, J. kg^{1}. K^{1} temperature, K time, s longitudinal fluid velocity, m. s^{1} volume, m^{3} orthogonal fluid velocity, m. s^{1} mechanical power, W conjugate field longitudinal spatial coordinate, m applied driving field orthogonal spatial coordinate, m 
Greek symbols 

Δ 
finite difference 
ε θ 
elongation, % period of ACR cycle, s 
μ ν φ ρ σ τ 
dynamic viscosity, kg. m^{1}. s^{1} cinematic viscosity, m^{+2}. s^{1} nanofluid concentration density, kg. m^{3} stress, Pa period of each step of ACR cycle, s 
Subscripts 

0 1 ad bf C 
initial final adiabatic base fluid cold 
H nf 
hot nanofluid 
np p s TOT 
nanoparticles constant pressure solid total 
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