# Numerical Modeling of Shell-and-Tube-like Elastocaloric Regenerator

^{*}

## Abstract

**:**

^{−1}and COP of about 4 (at zero temperature span).

## 1. Introduction

_{2}). However, these are not faultless since all HCs are at least slightly flammable, and CO

_{2}has rather low efficiency in hot climates [2,3]. Moreover, large vapor-compression systems achieve relatively high exergy efficiencies (>50%), but this is not true for small systems, which are characterized by relatively low exergy efficiencies (<30%) [4,5]. Due to the rapid progress of developing countries and global warming trends, the number of small refrigeration and especially air-conditioning systems is expected to increase significantly over the next two decades. Elastocaloric cooling is considered a promising alternative to vapor-compression technology used for cooling/heating, and its development could provide a solution for environmentally friendly cooling and heat-pumping systems [6]. It has two important advantages over vapor-compression technology: it is environmentally benign during operation and can potentially achieve better efficiency (especially compared to small refrigeration units). In 2014 and 2016, the US Department of Energy [7] and the European Commission [8], respectively, recognized elastocaloric cooling as the most promising future non-vapor-compression refrigeration technology. In addition to traditional HVAC applications, elastocaloric cooling can play an important role in micro-cooling applications for thermal management of electronic devices [9,10], such as batteries and supercapacitors [11,12].

^{−1}of eCM) [35]. The concept of active regeneration was originally developed in the field of magnetocaloric cooling [41] and was later applied also in other caloric technologies [31,42]. An active elastocaloric regenerator (AeCR) is essentially a porous structure (matrix) made of eCM through which a heat transfer fluid (HTF), such as water, oscillates between the heat sink and the heat source. The eCM in the AeCR has a dual function, acting as both a refrigerant and a heat regenerator enabling an increase in the temperature span. The active regenerative caloric cycle is a unique thermodynamic cycle in which a temperature profile is established along the length of the regenerator under steady-state conditions. Therefore, each part of the caloric material undergoes its own thermodynamic cycle at different temperatures, so the temperature span between the heat source and the heat sink can be larger than the adiabatic temperature change of the eCM itself.

## 2. Methods

#### 2.1. Experimental Determination of Superelastic and Elastocaloric Properties

_{f}= −7.3 °C from Memry Corporation, with an outer diameter of 2.5 ± 0.01 mm and an inner diameter of 1.5 ± 0.025 mm, was used. The tube was cut and polished to a length of 29 ± 0.01 mm and inserted into a specially designed holder [66] that provides uniaxial compressive loading. The holder contained two ER collets that held the tube, prevented its movement in the holder (outside the gauge length), and assured uniaxial loading. The gauge length of the tube between the ER collets was 8 mm. Characterization of the superelastic behavior was performed on the Zwick/Roell Z050 Universal Testing Machine (UTM), equipped with a 5 kN load cell (maximum measurement uncertainty of ±0.4% of full scale), and a Zwick/Roell thermostatic chamber. Since the UTM measures the displacement of the crosshead, the measurements included a complex response of the entire system between the crossheads (i.e., tube, holder, clamps). Therefore, the use of an extensometer or a strain gauge was required to accurately measure the strain of the tube. Due to the short gauge length of the specimen (8 mm) and poor accessibility for a mechanical extensometer due to the design of the holder and the size limitations of the thermostatic chamber, a video extensometer was used. The video extensometer was developed in-house and comprised carefully positioned lighting and a Panasonic DC-GH5L camera that recorded the movement of the tube during loading and unloading (a photo of the experimental setup can be found in Figure S2 in the Supplementary materials). Details of the strain measurements with the video-extensometer and their uncertainties, which can be up to 0.2% of the measured value, can be found in [66]. The schematic representation of the experimental setup is shown in Figure 1. The gauge length of the tube was divided into three sections (Figure 1). The middle section was painted with a thermographic paint with high emissivity (ϵ = 0.92), which allows for accurate IR temperature measurements. The upper and lower sections of the tube were painted with white base paint, onto which black speckles were randomly sprayed for video extensometer measurements [66]. The force measurements are made using the UTM’s load cell, where the stress is calculated based on the initial cross-section of the tube.

^{−3}s

^{−1}between 6.5 MPa and 1275 MPa for 100 cycles. During the training, a decrease in the transformation stress, hysteresis loop area, and the recoverable strain can be observed, and the residual strain increases (see Figure 2a), until the response stabilizes towards the end of the training, as previously shown in [67,68]. Figure 2a shows the first and last training cycles. Once the tube was fully stabilized, a series of isothermal and adiabatic tests were performed at room temperature (about 24 °C) and 45 °C. Four strain ranges were applied at each temperature.

^{−5}s

^{−1}(corresponding to the isothermal conditions according to [69]) between 6.5 MPa and a selected upper limit, which was 1100 MPa at 24 °C and 1275 MPa at 45 °C. Figure 2b shows the experimentally measured isothermal superelastic response of the tube at both evaluated temperatures, based on which the material properties (Young’s modulus, transformation strain, martensite peak temperature, and Clausius-Clapeyron coefficient) required for the phenomenological modeling were extracted as explained below. The isothermal superelastic responses at all strains evaluated are shown in the Supplementary materials (Figure S3). As explained and discussed later, the phenomenological model is only used to model the loading curves. Therefore, only the material parameters related to the forward transformation (i.e., the loading curve) are extracted from the experiments. The transformation strain was determined by extrapolating the elastic region of the martensite to zero stress, as shown in Figure 2b. The martensite peak temperature (M

_{p}) and Clausius-Clapeyron coefficient (C

_{Mp}) were determined from the phase diagram of forward transformation (Figure 2c). It was constructed based on the critical stresses defined in the middle of the isothermal transformation plateau for each temperature, as shown in Figure 2b. The hysteresis loop area was calculated using Equation (1) and is shown in Figure 2d as a function of strain, where we can see that the hysteresis loop area increases with both temperature and strain (but only until the end of the transformation). An approximation function was fitted to the experimentally determined hysteresis loop areas, based on which the entropy irreversibilities (Equation (4)) were further calculated for a wider range of strains and temperatures.

^{−2}s

^{−1}(corresponding to adiabatic conditions according to [69]) between 6.5 MPa and selected upper limits, which were the same as for the isothermal tests (i.e., 1100 MPa at 24 °C and 1275 MPa at 45 °C). A holding time of 60 s was applied after each adiabatic loading and unloading to allow the temperature of the tube to adjust to the ambient temperature. In this way, we ensured that the temperature conditions for loading and unloading were the same. To assess the repeatability of the measurements, three adiabatic cycles were performed at each applied strain and each temperature and based on this the standard deviations were calculated and included in Figure 3b as error bars. Since the video extensometer and the UTM were not synchronized, the actual strain of the tube could not be controlled (held constant) during the hold time but was instead controlled by the crosshead displacement. As a result, the actual strain of the tube could change to some extent during the holding time (due to the heating or cooling of the tube by heat transfer to the environment). Thus, since the strain of the tube during unloading was not the same as during loading, the measured negative adiabatic temperature changes were not evaluated and considered in the modeling. However, as explained later in the text, the negative adiabatic temperature changes included in the AeCR model were determined by subtracting the temperature irreversibilities due to hysteresis losses from the positive adiabatic temperature changes.

#### 2.2. Phenomenological Modeling

_{1}(well below the transformation temperatures where transformation can be neglected) and was assumed to be 430 J·kg

^{−1}·K

^{−1}[69]. A complete total entropy diagram (Figure 3c) is obtained by adding the values of the isothermal entropy changes (second term in Equation (5)) and subtracting half of the entropy irreversibilities:

_{tot}diagram, the specific heat (c) and adiabatic temperature changes ($\Delta {T}_{\mathrm{ad}}$) can be calculated as follows (both as a function of strain and temperature):

#### 2.3. Numerical Modeling of the AeCR

- the HTF flow is incompressible, with no flow maldistributions,
- the HTF properties are defined according to the mean temperature,
- the stress throughout the AeCR is constant,
- the strain within the segment of the AeCR is constant,
- the mechanical loading and unloading are adiabatic,
- a step on and off function of the fluid flow period is assumed,
- the strain is kept constant during the HTF flow period,
- it is assumed that the energy released during unloading is fully recovered.

_{s}) of NiTi, which is calculated using Equation (6). We assumed that the density (ρ) of NiTi is the same for the martensite and austenite phases, although the thermal conductivity (k) of the two phases is different, as shown in Table A1 in Appendix A. The HTF properties included in the model (c, k, ρ, ν) are calculated according to the average temperature of the HTF before each cycle. The heat transfer area of the eCM (A

_{s}) included in the model is only the one in contact with the water, ignoring the portion of the tubes in the baffles since they do not directly contribute to the heat transfer. On the other hand, the mass (m

_{s}) and volume (V

_{s}) of the eCM taken into account in the model also include half of the mass of the tubes in the baffles because, based on our preliminary analysis, we estimate that only about half of the eCM in the baffles transforms (as it is constrained by the baffles) and thus only half of the tubes contributes to active cooling/heating (by heat conduction through the tubes). The AeCR length (fluid flow path) is defined by the active length, i.e., only the length that the HTF passes through the tube bundle (excluding the turns between segments). Since the housing and the baffles are complex in design (see [35] for details), the effective material properties of the housing (ρ

_{H}, c

_{H}, k

_{H}) and its volume (V

_{H}) were calculated based on the volume fraction of each material in the housing assembly (see Table A1 in Appendix A). Heat losses to the surroundings are accounted for in Equation (10) by defining the thermal resistance of the housing (R

_{a}) through each element, as shown in Table A1 in Appendix A.

_{ef}) and the friction factor (F) were taken from [65] where the shell-and-tube-like regenerators were tested for their thermohydraulic properties at operating conditions relevant for such applications (Re < 2000). It should be noted that the governing equations include the effective heat transfer coefficient (h

_{ef}), which takes into account the finite thermal conductivity of the eCM (see Table A1 in Appendix A).

^{3}on each side of the AeCR is included in the model. The dead volume is defined as the HTF volume that exits the AeCR but returns to the AeCR in the following fluid flow period instead of flowing toward the heat exchangers, as shown in Figure 4a [73]. In addition, the heat losses/gains of the HTF to/from the environment in the fluid connections between the AeCR and the hot/cold heat exchangers are considered as a heat load on each side of the AeCR (based on the measured values). The latter is applied in the numerical model only for comparison with the experimental results (Section 3.1) and is determined based on the temperature difference between the HTF inlet and outlet temperatures on the cold and hot sides of the AeCR. Heat loads of the fluid connections are omitted from the subsequent analysis of the effects of operating conditions, hysteresis, and heat transfer area on the performance of the AeCR (Section 3.2).

## 3. Results and Discussion

^{i}) with $\Delta {s}_{\mathrm{irr}}=0$. A similar effect occurs during unloading, except that the hysteresis irreversibilities reduce the negative adiabatic temperature changes compared to the ideal transformation. Figure 6 shows the temperature distribution during all four operating steps (a–d) along the HTF path of the eCM in the AeCR after a steady state has been reached.

#### 3.1. Model Verification against the Experimental Results

#### 3.2. Impact of the Operating Conditions and Hysteresis Losses

^{−1}) and 12 W for the cooling (corresponding to a specific heating power of 876 W·kg

^{−1}), both at the frequency of 4 Hz (i.e., the maximum frequency analyzed). The optimal displaced volume ratio in the heat-pumping mode was around 0.6 and increased slightly with frequency; in the cooling mode it was around 0.5 and increased with frequency up to 2.5 Hz and then started to decrease when the frequency increased further. On the other hand, the COP in the heat-pumping mode increased with frequency, as did the optimal displaced volume ratio (see Figure 12b), and the COP in the cooling mode increased with frequency up to 3.5 Hz and started to decrease at a frequency above 4 Hz (see Figure 12d). A similar trend was expected also for the heat-pumping mode, only at higher frequencies (due to the larger adiabatic temperature changes during loading compared to unloading). The maximum COP values were 1.65 and 0.43 for heat-pumping and cooling, respectively.

## 4. Conclusions

^{−1}of eCM) and COP of around 3 were obtained at zero temperature span, operating frequency of 2 Hz, displaced fluid volume ratio of 1, and applied strain of 2.7%, and in the heat-pumping mode, the maximum heating power of 54 W (corresponding to the specific cooling power of 3940 W·kg

^{−1}of eCM) and the maximum COP of around 4 were obtained under the same operating conditions. At a temperature span of 15 K, cooling power of 8.5 W (620 W·kg

^{−1}of eCM) and COP of 0.45 were obtained in the cooling mode, and in the heat-pumping mode, the heating power of 27 W (1970 W·kg

^{−1}of eCM) and COP of 1.4 were obtained at a frequency of 4 Hz and applied strain of 2.7%. We demonstrated that the main source of irreversibility that results in rather low COP values is the hysteresis of the eCM. The COP could be increased up to four times if the material with no hysteresis but the same eCE could be applied. Therefore, it is crucial to study an eCM with smaller hysteresis, which can be achieved by using a different eCM (e.g., Cu-based SMAs or some Heusler alloys) and modifying the microstructure of the NiTi alloys (e.g., grain size).

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Description | Units |

Roman | ||

A | area | (m^{2}) |

c | specific heat | (J·kg^{−1}·K^{−1}) |

COP | coefficient of performance | (/) |

d | inner diameter | (m) |

D | outer diameter | (m) |

d_{h} | hydraulic diameter | (m) |

E | Young’s modulus | (GPa) |

F | friction factor | (/) |

f | frequency | (Hz) |

h | convective heat transfer coefficient | (W·m^{−2}·K^{−1}) |

H | height | (m) |

k | thermal conductivity | (W·m^{−1}·K^{−1}) |

L | length | (m) |

m | mass | (kg) |

$\dot{m}$ | mass flow rate | (kg·s^{−1}) |

M_{p} | martensite peak temperature | (K) |

n_{t} | number of tubes | (/) |

n_{s} | number of segments | (/) |

Nu | Nusselt number | (/) |

p | pressure | (Pa) |

Pr | Prandtl number | (/) |

Q | heat | (J) |

$\dot{\mathit{Q}}$ | thermal power | (W) |

R | thermal resistance | (K·W^{−1}) |

Re | Reynolds number | (/) |

s | specific entropy | (J·kg^{−1}·K^{−1}) |

S | spacing | (m) |

T | temperature | (K) |

t | time | (s) |

v | velocity | (m·s^{−1}) |

V | volume | (m^{3}) |

V* | the displaced fluid volume ratio | (/) |

$\dot{\mathit{W}}$ | mechanical power | (W) |

x | segment | (/) |

y | a spatial node within the segment | (/) |

Greek | ||

δ | thickness | (m) |

σ | stress | (MPa) |

ϵ | emissivity | (/) |

ε | strain | (/) |

ρ | density | (kg·m^{−3}) |

μ | dynamic viscosity | (Pa·s) |

τ | time | (s) |

Subscripts | ||

a | ambient | |

ad | adiabatic | |

A | austenite | |

baf | baffles | |

c | cold | |

ef | effective | |

f | fluid | |

h | hot | |

ht | heat transfer | |

hyst | hysteresis | |

H | housing | |

i | inlet | |

in | inside | |

iso | isothermal | |

irr | irreversibility | |

L | loading | |

mech | mechanical | |

o | outlet | |

out | outside | |

pump | pumping | |

reg | regenerator | |

s | solid | |

tot | total | |

trans | transformation | |

UL | unloading |

## Appendix A

Material properties | ||||||

eCM | HTF | housing | ||||

V | (m^{3}) | $2.18\xb7{10}^{-6}$ | $2.26\xb7{10}^{-6}$ | 0.000119 | ||

c | (J·kg^{−1}·K^{−1}) | Equation (6) | $f\left(T,p\right)$ | 726 | ||

ρ | (kg·m^{−3}) | 6450 | $f\left(T,p\right)$ | 5523 | ||

k | (W·m^{−1}·K^{−1}) | 8.6/18 | $f\left(T,p\right)$ | 10.9 | ||

μ | (Pa·s) | / | $f\left(T,p\right)$ | / | ||

Geometrical properties | ||||||

D | (m) | 0.003 | ||||

d | (m) | 0.0025 | ||||

S | (m) | 0.0003 | ||||

H | (m) | 0.008 | ||||

δ_{sup} | (m) | 0.004 | ||||

n_{t} | (/) | 18 | ||||

n_{s} | (/) | 4 | ||||

A_{ht,L} | (m^{2}) | $\pi \xb7{D}_{\mathrm{L}}\xb7H\xb7{n}_{\mathrm{t}}\xb7{n}_{\mathrm{s}}\xb7\left(1-\epsilon \right)$ for heat transfer from eCM to HTF | ||||

$0.00443\xb7\left(1-\epsilon \right)$ for heat transfer from HTF to housing | ||||||

A_{ht,UL} | (m^{2}) | $\pi \xb7{D}_{\mathrm{UL}}\xb7H\xb7{n}_{\mathrm{t}}\xb7{n}_{\mathrm{s}}$ for heat transfer from eCM to HTF | ||||

0.00443 for heat transfer from HTF to housing | ||||||

A_{ht,a} | (m^{2}) | 0.0187 | ||||

A_{ht,sup} | (m^{2}) | 0.00012 | ||||

d_{h} | (m) | $4\left(\left({S}^{2}\sqrt{3}/4\right)-\left(\pi {D}^{2}/8\right)\right)/\left(\pi D/2\right)$ | [72] | |||

L | (m) | 0.082 | ||||

Thermohydraulic properties | ||||||

Re | (/) | $v\xb7{d}_{\mathrm{h}}\xb7\rho /\mu $ | ||||

Nu | (/) | $0.051\xb7R{e}^{0.88}\xb7P{r}^{0.36}$ | [65] | |||

F | (/) | $3993\xb7R{e}^{-1.4}$ | [65] | |||

T_{a} | (°C) | 25 ± 1.5 | ||||

h_{out} | (W·m^{−2}·K^{−1}) | 90 | ||||

h_{in} | (W·m^{−2}·K^{−1}) | $Nu\xb7k/{d}_{\mathrm{h}}$ | ||||

h_{eff} | (W·m^{−2}·K^{−1}) | ${h}_{\mathrm{in}}/\left(1+Bi\xb7\chi \left(Fo\right)/4\right)$ $\mathsf{\chi}\left(Fo\right)=Fo\xb7exp\left[\begin{array}{c}0.246196-0.84878\xb7\mathrm{ln}\left(Fo\right)-\\ 0.05639\xb7{\left(\mathrm{ln}\left(Fo\right)\right)}^{2}\end{array}\right]$ | [83] [84] | |||

R_{a} | (m^{2}·K·W^{−1}) | $\left({\delta}_{\mathrm{h}}/{k}_{\mathrm{h}}+1/{h}_{\mathrm{out}}\right)\xb7{A}_{\mathrm{ht},\mathrm{a}}$ | ||||

R_{baf,f} | (m^{2}·K·W^{−1}) | $\left({\delta}_{\mathrm{sup}}/{k}_{\mathrm{Steel}}+2/{h}_{\mathrm{in}}\right)\xb7{A}_{\mathrm{ht},\mathrm{baf}}$ | ||||

R_{baf,s} | (m^{2}·K·W^{−1}) | $\left({\delta}_{\mathrm{sup}}/{k}_{\mathrm{NiTi}}\right)\xb7{A}_{\mathrm{ht},\mathrm{baf},\mathrm{NiTi}}$ |

## References

- Ozone Secretariat. Handbook for the Montreal Protocol on Substances That Deplete the Ozone Layer, 14th ed.; Ozone Secretariat: Nairobi, Kenya, 2020. [Google Scholar]
- Kauffeld, M. Current long-term alternative refrigerants and their applications. In Proceedings of the 31st Informatory Note on Refrigeration Technologies, International Institute of Refrigeration, Paris, France, 1 April 2016. [Google Scholar]
- McLinden, M.O.; Brown, J.S.; Brignoli, R.; Kazakov, A.F.; Domanski, P.A. Limited options for low-global-warming-potential refrigerants. Nat. Commun.
**2017**, 8, 14476. [Google Scholar] [CrossRef] [PubMed] [Green Version] - IEA. Average Efficiency of New Air Conditioners 2000–2020 and in the Net Zero Scenario; IEA: Paris, France; Available online: https://www.iea.org/data-and-statistics/charts/average-efficiency-of-new-air-conditioners-2000-2020-and-in-the-net-zero-scenario (accessed on 12 September 2022).
- Kitanovski, A. Energy Applications of Magnetocaloric Materials. Adv. Energy Mater.
**2020**, 10, 1903741. [Google Scholar] [CrossRef] - Ismail, M.; Yebiyo, M.; Chaer, I.A. Review of Recent Advances in Emerging Alternative Heating and Cooling Technologies. Energies
**2021**, 14, 502. [Google Scholar] [CrossRef] - Goetzler, W.; Zogg, R.; Young, J.; Johnson, C. Energy Savings Potential and RD & D Opportunities for Non-Vapor-Compression HVAC Technologies; United States Department of Energy: Washington, DC, USA, 2014; p. 3673. [CrossRef]
- OECD/IEA. The Future of Cooling Opportunities for Energy-Efficient Air Conditioning Together Secure Sustainable. 2018. Available online: www.iea.org/t&c/ (accessed on 12 September 2022).
- Bruederlin, F.; Bumke, L.; Chluba, C.; Ossmer, H.; Quandt, E.; Kohl, M. Elastocaloric Cooling on the Miniature Scale: A Review on Materials and Device Engineering. Energy Technol.
**2018**, 6, 1588–1604. [Google Scholar] [CrossRef] - Kalizan, J.; Tušek, J. Caloric Micro-Cooling: Numerical modelling and parametric investigation. Energy Convers. Manag.
**2020**, 225, 113421. [Google Scholar] [CrossRef] - Babapoor, A.; Aziz, M.; Karimi, G. Thermal management of a Li-on battery using carbon fiber-PCM composites. Appl. Therm. Eng.
**2015**, 62, 281–290. [Google Scholar] [CrossRef] - Bo, Z.; Li, Z.; Yang, H.; Li, C.; Wu, S.; Xu, C.; Xiong, G.; Mariotti, D.; Yan, J.; Cen, K.; et al. Combinatorial atomistic-to-AI prediction and experimental validation of heating effects in 350 F supercapacitor modules. Int. J. Heat Mass Transf.
**2021**, 171, 121075. [Google Scholar] [CrossRef] - Kabirifar, P.; Žerovnik, A.; Ahčin, Ž.; Porenta, L.; Brojan, M.; Tušek, J. Elastocaloric Cooling: State-of-the-art and Future Challenges in Designing Regenerative Elastocaloric Devices. J. Mech. Eng.
**2019**, 65, 615–630. [Google Scholar] [CrossRef] [Green Version] - Rodriguez, C.; Brown, L.C. The thermal effect due to stress-induced martensite formation in Β-CuAlNi single crystals. Metall. Mater. Trans. A
**1980**, 11, 147–150. [Google Scholar] [CrossRef] - Hou, H.; Qian, S.; Takeuchi, I. Materials, physics and systems for multicaloric cooling. Nat. Rev. Mater.
**2022**, 7, 633–652. [Google Scholar] [CrossRef] - Qian, S. Thermodynamics of elastocasloric cooling and heat pump cycles. Appl. Therm. Eng.
**2023**, 219, 119540. [Google Scholar] [CrossRef] - Frenzel, J.; Wieczorek, A.; Opahle, I.; Maaß, B.; Drautz, R.; Eggeler, G. On the effect of alloy composition on martensite start temperatures and latent heats in Ni–Ti-based shape memory alloys. Acta Mater.
**2015**, 90, 213–231. [Google Scholar] [CrossRef] - Wieczorek, A.; Frenzel, J.; Schmidt, M.; Maass, B.; Seelecke, S.; Schütze, A.; Eggeler, G. Optimizing Ni–Ti-based shape memory alloys for ferroic cooling. Funct. Mater. Lett.
**2017**, 10, 1740001. [Google Scholar] [CrossRef] - Chluba, C.; Ge, W.; de Miranda, R.L.; Strobel, J.; Kienle, L.; Quandt, E.; Wuttig, M. Ultralow-fatigue shape memory alloy films. Science
**2015**, 348, 1004–1007. [Google Scholar] [CrossRef] [PubMed] - Chluba, C.; Ossmer, H.; Zamponi, C.; Kohl, M.; Quandt, E. Ultra-Low Fatigue Quaternary TiNi-Based Films for Elastocaloric Cooling. Shape Mem. Superelasticity
**2016**, 2, 95–103. [Google Scholar] [CrossRef] [Green Version] - Mañosa, L.; Jarque-Farnos, S.; Vives, E.; Planes, A. Large temperature span and giant refrigerant capacity in elastocaloric Cu-Zn-Al shape memory alloys. Appl. Phys. Lett.
**2013**, 103, 211904. [Google Scholar] [CrossRef] - Qian, S.; Geng, Y.; Wang, Y.; Pillsbury, T.E.; Hada, Y.; Yamaguchi, Y.; Fujimoto, K.; Hwang, Y.; Radermacher, R.; Cui, J.; et al. Elastocaloric effect in CuAlZn and CuAlMn shape memory alloys under compression. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2016**, 374, 20150309. [Google Scholar] [CrossRef] [Green Version] - Xiao, F.; Bucsek, A.; Jin, X.; Porta, M.; Planes, A. Giant elastic response and ultra-stable elastocaloric effect in tweed textured Fe-Pd single crystals. Acta Mater.
**2022**, 223, 117486. [Google Scholar] [CrossRef] - Liu, J.; Zhao, D.; Li, Y. Exploring Magnetic Elastocaloric Materials for Solid-State Cooling. Shape Mem. Superelasticity
**2017**, 3, 192–198. [Google Scholar] [CrossRef] - Tong, W.; Liang, L.; Xu, J.; Wang, H.J.; Tian, J.; Peng, L.M. Achieving enhanced mechanical, pseudoelastic and elastocaloric properties in Ni-Mn-Ga alloys via Dy micro-alloying and isothermal mechanical cyclic training. Scr. Mater.
**2022**, 209, 114393. [Google Scholar] [CrossRef] - Zhang, S.; Yang, Q.; Li, C.; Fu, Y.; Zhang, H.; Ye, Z.; Zhou, X.; Li, Q.; Wang, T.; Wang, S.; et al. Solid-state cooling by elastocaloric polymer with uniform chain-lengths. Nat. Commun.
**2022**, 13, 9. [Google Scholar] [CrossRef] [PubMed] - Bennacer, R.; Liu, B.; Yang, M.; Chen, A. Refrigeration performance and the elastocaloric effect in natural and synthetic rubbers. Appl. Therm. Eng.
**2022**, 204, 117938. [Google Scholar] [CrossRef] - Imran, M.; Zhang, X. Recent developments on the cyclic stability in elastocaloric materials. Mater. Des.
**2020**, 195, 109030. [Google Scholar] [CrossRef] - Chen, J.; Lei, L.; Fang, G. Elastocaloric cooling of shape memory alloys: A review. Mater. Today Commun.
**2021**, 28, 102706. [Google Scholar] [CrossRef] - Imran, M.; Zhang, X. Reduced dimensions elastocaloric materials: A route towards miniaturized refrigeration. Mater. Des.
**2021**, 206, 109784. [Google Scholar] [CrossRef] - Tušek, J.; Engelbrecht, K.; Eriksen, D.; Dall’Olio, S.; Tušek, J.; Pryds, N. A regenerative elastocaloric heat pump. Nat. Energy
**2016**, 1, 16134. [Google Scholar] [CrossRef] - Kirsch, S.M.; Welsch, F.; Michaelis, N.; Schmidt, M.; Wieczorek, A.; Frenzel, J.; Eggeler, G.; Schütze, A.; Seelecke, S. NiTi-Based Elastocaloric Cooling on the Macroscale: From Basic Concepts to Realization. Energy Technol.
**2018**, 6, 1567–1587. [Google Scholar] [CrossRef] - Chen, J.; Zhang, K.; Kan, Q.; Yin, H.; Sun, Q. Ultra-high fatigue life of NiTi cylinders for compression-based elastocaloric cooling. Appl. Phys. Lett.
**2019**, 115, 93902. [Google Scholar] [CrossRef] - Snodgrass, R.; Erickson, D. A multistage elastocaloric refrigerator and heat pump with 28 K temperature span. Sci. Rep.
**2019**, 9, 18532. [Google Scholar] [CrossRef] - Ahčin, Ž.; Dall’Olio, S.; Žerovnik, A.; Baškovič, U.Ž.; Porenta, L.; Kabirifar, P.; Cerar, J.; Zupan, S.; Brojan, M.; Klemenc, J.; et al. High-performance cooling and heat pumping based on fatigue-resistant elastocaloric effect in compression. Joule
**2022**, 6, 2338–2357. [Google Scholar] [CrossRef] - Bachmann, N.; Fitger, A.; Maier, L.M.; Mahlke, A.; Schäfer-Welsen, O.; Koch, T.; Bartholomé, K. Long-term stable compressive elastocaloric cooling system with latent heat transfer. Commun. Phys.
**2021**, 4, 194. [Google Scholar] [CrossRef] - Chen, Y.; Wang, Y.; Sun, W.; Qian, S.; Liu, J. A compact elastocaloric refrigerator. Innovation
**2022**, 3, 100205. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.; Zhu, Y.; Cheng, S.; Yao, S.; Sun, Q. Enhancing cooling performance of NiTi elastocaloric tube refrigerant via internal grooving. Appl. Therm. Eng.
**2022**, 213, 118657. [Google Scholar] [CrossRef] - Emaikwu, N.; Catalini, D.; Muehlbauer, J.; Hwang, Y.; Takeuchi, I.; Radermacher, R. Experimental Investigation of a Staggered-Tube Active Elastocaloric Regenerator. Int. J. Refrig. 2022, in press. [CrossRef]
- Li, X.; Cheng, S.; Sun, Q. A compact NiTi elastocaloric air cooler with low force bending actuation. Appl. Therm. Eng.
**2022**, 215, 118942. [Google Scholar] [CrossRef] - Barclay, J.A.; Steyer, W.A. Active Magnetic Regenerator. U.S. Patent 4332135A, 1 June 1982. [Google Scholar]
- Torelló, A.; Lheritier, P.; Usui, T.; Nouchokgwe, Y.; Gérard, M.; Bouton, O.; Hirose, S.; Defay, E. Giant temperature span in electrocaloric regenerator. Science
**2020**, 370, 125–129. [Google Scholar] [CrossRef] [PubMed] - Han, Y.; Lai, C.; Li, J.; Zhang, Z.; Zhang, H.; Hou, S.; Wang, F.; Zhao, J.; Zhang, C.; Miao, H.; et al. Elastocaloric cooler for waste heat recovery from proton exchange membrane fuel cells. Energy
**2022**, 238, 121789. [Google Scholar] [CrossRef] - Al-Hamed, K.H.M.; Dincer, I.; Rosen, M.A. Investigation of elastocaloric cooling option in a solar energy-driven system. Int. J. Refrig.
**2020**, 120, 340–356. [Google Scholar] [CrossRef] - Qian, S.; Ling, J.; Hwang, Y.; Radermacher, R.; Takeuchi, I. Thermodynamics cycle analysis and numerical modeling of thermoelastic cooling systems. Int. J. Refrig.
**2015**, 56, 65–80. [Google Scholar] [CrossRef] - Luo, D.; Feng, Y.; Verma, P. Modeling and analysis of an integrated solid state elastocaloric heat pumping system. Energy
**2017**, 130, 500–514. [Google Scholar] [CrossRef] - Tušek, J.; Engelbrecht, K.; Millán-Solsona, R.; Mañosa, L.; Vives, E.; Mikkelsen, L.P.; Pryds, N. The Elastocaloric Effect: A Way to Cool Efficiently; Supporting Information. Adv. Energy Mater.
**2015**, 5, 1500361. [Google Scholar] [CrossRef] - Qian, S.; Wang, Y.; Xu, S.; Chen, Y.; Yuan, L.; Yu, J. Cascade utilization of low-grade thermal energy by coupled elastocaloric power and cooling cycle. Appl. Energy
**2021**, 298, 117269. [Google Scholar] [CrossRef] - Sebald, G.; Komiya, A.; Jay, J.; Coativy, G.; Lebrun, L. Regenerative cooling using elastocaloric rubber: Analytical model and experiments. J. Appl. Phys.
**2020**, 127, 94903. [Google Scholar] [CrossRef] - Zhu, Y.; Hur, J.; Cheng, S.; Sun, Q.; Li, W.; Yao, S. Modelling of elastocaloric regenerators with enhanced heat transfer structures. Int. J. Heat Mass Transf.
**2021**, 176, 121372. [Google Scholar] [CrossRef] - Tušek, J.; Engelbrecht, K.; Pryds, N. Elastocaloric effect of a Ni-Ti plate to be applied in a regenerator-based cooling device. Sci. Technol. Built Environ.
**2016**, 22, 489–499. [Google Scholar] [CrossRef] - Tan, J.; Wang, Y.; Xu, S.; Liu, H.; Qian, S. Thermodynamic cycle analysis of heat driven elastocaloric cooling system. Energy
**2020**, 197, 117261. [Google Scholar] [CrossRef] - Qian, S.; Wang, Y.; Yuan, L.; Yu, J. A heat driven elastocaloric cooling system. Energy
**2019**, 182, 881–899. [Google Scholar] [CrossRef] - Qian, S.; Yuan, L.; Yu, J.; Yan, G. Numerical modeling of an active elastocaloric regenerator refrigerator with phase transformation kinetics and the matching principle for materials selection. Energy
**2017**, 141, 744–756. [Google Scholar] [CrossRef] - Cirillo, L.; Rosaria Farina, A.; Greco, A.; Masselli, C. The optimization of the energy performances of a single bunch of elastocaloric elements to be employed in an experimental device. Therm. Sci. Eng. Prog.
**2022**, 27, 101152. [Google Scholar] [CrossRef] - Qian, S.; Yuan, L.; Hou, H.; Takeuchi, I. Accurate prediction of work and coefficient of performance of elastocaloric materials with phase transformation kinetics. Sci. Technol. Built Environ.
**2018**, 24, 673–684. [Google Scholar] [CrossRef] - Ulpiani, G.; Saliari, M.; Bruederlin, F.; Kohl, M.; Ranzi, G.; Santamouris, M. On the cooling potential of elastocaloric devices for building ventilation. Sol. Energy
**2021**, 230, 298–311. [Google Scholar] [CrossRef] - Bachmann, N.; Schwarz, D.; Bach, D.; Schäfer-Welsen, O.; Koch, T.; Bartholomé, K. Modeling of an Elastocaloric Cooling System for Determining Efficiency. Energies
**2022**, 15, 5089. [Google Scholar] [CrossRef] - Heintze, O.; Seelecke, S. A coupled thermomechanical model for shape memory alloys-From single crystal to polycrystal. Mater. Sci. Eng. A
**2008**, 481–482, 389–394. [Google Scholar] [CrossRef] - Yuan, L.; Wang, Y.; Yu, J.; Greco, A.; Masselli, C.; Qian, S. Numerical study of a double-effect elastocaloric cooling system powered by low-grade heat. Appl. Therm. Eng.
**2023**, 218, 119302. [Google Scholar] [CrossRef] - Bachmann, N.; Fitger, A.; Unmüßig, S.; Bach, D.; Schäfer-Welsen, O.; Koch, T.; Bartholomé, K. Phenomenological model for first-order elastocaloric materials. Int. J. Refrig.
**2022**, 136, 245–253. [Google Scholar] [CrossRef] - Hess, T.; Vogel, C.; Maier, L.M.; Barcza, A.; Vieyra, H.P.; Schäfer-Welsen, O.; Wöllenstein, J.; Bartholomé, K. Phenomenological model for a first-order magnetocaloric material. Int. J. Refrig.
**2020**, 109, 128–134. [Google Scholar] [CrossRef] - Griffith, L.D.; Alho, B.P.; Czernuszewicz, A.; Ribeiro, P.O.; Slaughter, J.; Pecharsky, V.K. Toward efficient elastocaloric systems: Predicting material thermal properties with high fidelity. Acta Mater.
**2021**, 217, 117162. [Google Scholar] [CrossRef] - de Oliveira, N.A.; von Ranke, P.J. Theoretical aspects of the magnetocaloric effect. Phys. Rep.
**2010**, 489, 89–159. [Google Scholar] [CrossRef] - Ahčin, Ž.; Liang, J.; Engelbrecht, K.; Tušek, J. Thermo-hydraulic evaluation of oscillating-flow shell-and-tube-like regenerators for (elasto)caloric cooling. Appl. Therm. Eng.
**2021**, 190, 116842. [Google Scholar] [CrossRef] - Porenta, L.; Trojer, J.; Brojan, M.; Tušek, J. Experimental investigation of buckling stability of superelastic Ni-Ti tubes under cyclic compressive loading: Towards defining functionally stable tubes for elastocaloric cooling. Int. J. Solids Struct.
**2022**, 256, 111948. [Google Scholar] [CrossRef] - Tušek, J.; Engelbrecht, K.; Mikkelsen, L.P.; Pryds, N. Elastocaloric effect of Ni-Ti wire for application in a cooling device. J. Appl. Phys.
**2015**, 117, 124901. [Google Scholar] [CrossRef] [Green Version] - Miyazaki, S.; Imai, T.; Igo, Y.; Otsuka, K. Effect of cyclic deformation on the pseudoelasticity characteristics of Ti-Ni alloys. Metall. Trans. A
**1986**, 17, 115–120. [Google Scholar] [CrossRef] - Porenta, L.; Kabirifar, P.; Žerovnik, A.; Cebron, M.; Žužek, B.; Dolenec, M.; Brojan, M.; Tušek, J. Thin-walled Ni-Ti tubes under compression: Ideal candidates for efficient and fatigue-resistant elastocaloric cooling. Appl. Mater. Today
**2020**, 20, 100712. [Google Scholar] [CrossRef] - Brey, W.; Nellis, G.; Klein, S. Thermodynamic modeling of magnetic hysteresis in AMRR cycles. Int. J. Refrig.
**2014**, 47, 85–97. [Google Scholar] [CrossRef] [Green Version] - Kitanovski, A.; Tušek, J.; Tomc, U.; Plaznik, U.; Ožbolt, M.; Poredoš, A. Magnetocaloric Energy Conversion—From Theory to Applications; Springer: Berlin, Germany, 2015. [Google Scholar] [CrossRef]
- Žukauskas, A. Heat Transfer from Tubes in Crossflow. Adv. Heat Transf.
**1987**, 18, 87–159. [Google Scholar] [CrossRef] - Trevizoli, P.V.; Barbosa, J.R. Thermal-hydraulic behavior and influence of carryover losses in oscillating-flow regenerators. Int. J. Therm. Sci.
**2017**, 113, 89–99. [Google Scholar] [CrossRef] - Masche, M.; Ianniciello, L.; Tušek, J.; Engelbrecht, K. Impact of hysteresis on caloric cooling performance. Int. J. Refrig.
**2021**, 121, 302–312. [Google Scholar] [CrossRef] - Hess, T.; Maier, L.M.; Bachmann, N.; Corhan, P.; Schäfer-Welsen, O.; Wöllenstein, J.; Bartholomé, K. Thermal hysteresis and its impact on the efficiency of first-order caloric materials. J. Appl. Phys.
**2020**, 127, 75103. [Google Scholar] [CrossRef] - Lu, B.; Song, M.; Zhou, Z.; Liu, W.; Wang, B.; Lu, S.; Wu, C.; Yang, L.; Liu, J. Reducing mechanical hysteresis via tuning the microstructural orientations in Heusler-type Ni44.8Mn36.9In13.3Co5.0 elastocaloric alloys. J. Alloys Compd.
**2019**, 785, 1023–1029. [Google Scholar] [CrossRef] - Yuan, B.; Zhu, X.; Zhang, X.; Qian, M. Elastocaloric effect with small hysteresis in bamboo-grained Cu–Al–Mn microwires. J. Mater. Sci.
**2019**, 54, 9613–9621. [Google Scholar] [CrossRef] - Ahadi, A.; Kawasaki, T.; Harjo, S.; Ko, W.S.; Sun, Q.; Tsuchiya, K. Reversible elastocaloric effect at ultra-low temperatures in nanocrystalline shape memory alloys. Acta Mater.
**2019**, 165, 109–117. [Google Scholar] [CrossRef] - Zhu, X.; Zhang, X.; Qian, M. Reversible elastocaloric effects with small hysteresis in nanocrystalline Ni-Ti microwires. AIP Adv.
**2018**, 8, 125002. [Google Scholar] [CrossRef] - Lin, H.; Hua, P.; Sun, Q. Effects of grain size and partial amorphization on elastocaloric cooling performance of nanostructured NiTi. Scr. Mater.
**2022**, 209, 114371. [Google Scholar] [CrossRef] - Kabirifar, P.; Trojer, J.; Brojan, M.; Tušek, J. From the elastocaloric effect towards an efficient thermodynamic cycle. J. Phys. Energy
**2022**, 4, 44009. [Google Scholar] [CrossRef] - Schmidt, M.; Kirsch, S.M.; Seelecke, S.; Schütze, A. Elastocaloric cooling: From fundamental thermodynamics to solid state air conditioning. Sci. Technol. Built Environ.
**2016**, 22, 475–488. [Google Scholar] [CrossRef] - Jeffreson, C.P. Prediction of breakthrough curves in packed beds: II. Experimental evidence for axial dispersion and intraparticle effects. AIChE J.
**1972**, 18, 416–420. [Google Scholar] [CrossRef] - Engelbrecht, K.L.; Nellis, G.F.; Klein, S.A. The effect of internal temperature gradients on regenerator matrix performance. J. Heat Transf.
**2006**, 128, 1060–1069. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the experimental setup for superelastic and elastocaloric characterization of the tube, showing the tube with the marked regions of interest for the thermal camera and video extensometer measurements.

**Figure 2.**The first and last training cycles and determination of E

_{A}and E

_{M}(

**a**); isothermal superelastic stress-strain behavior at 24 °C and 45 °C and determination of transformation strain and critical stress at the middle of the transformation (

**b**); the phase diagram for the forward transformation showing the martensite peak temperature and the corresponding Clausius-Clapeyron coefficient (

**c**); and the isothermal hysteresis loop area as a function of strain for both evaluated temperatures (

**d**).

**Figure 3.**Comparison of experimentally determined and modeled superelastic behavior at different temperatures (

**a**); comparison of experimentally determined and modeled positive adiabatic temperature changes during loading (

**b**); entropy-temperature diagram (

**c**); Modelled entropy irreversibilities as a function of strain at different temperatures (

**d**).

**Figure 4.**CAD model of the shell-and-tube AeCR: cross-sectional view (

**a**); longitudinal view with a schematic representation of the 1D discretization of regenerators’ domain (

**b**); and schematic representation of the heat transfer through the baffles in the 1D domain (

**c**).

**Figure 5.**Time evolution of the temperature span of HTF outlet temperatures (

**a**); presentation of the modeled loading curves at three different temperatures (

**b**); presentation of the modeled active elastocaloric cycle in the T-s diagram with detail of the inclusion of irreversibility in the case of the first cycle (

**c**) and the steady-state cycle (

**d**).

**Figure 6.**Evolution of the eCM temperature profile along the AeCR during each phase of the thermodynamic cycle.

**Figure 7.**Comparison between experimentally determined and numerically calculated maximum temperature spans established along the AeCR under different operating conditions for heat-pumping (

**a**) at a stress level of 775 MPa and cooling (

**b**) and under different mechanical loads (

**c**).

**Figure 8.**Comparison between experimentally determined and numerically calculated temperature span—cooling/heating power characteristics for heat-pumping (

**a**) and cooling (

**b**) at a stress level of 775 MPa.

**Figure 9.**Comparison between experimentally determined and numerically calculated COP—cooling/heating power characteristics for heat-pumping (

**a**) and cooling (

**b**) at the stress level of 775 MPa.

**Figure 10.**The effect of different applied strains at different operating conditions (operating frequency and displaced fluid volume ratio) on established temperature spans at no thermal load conditions for heat-pumping (

**a**) and cooling (

**b**).

**Figure 11.**The effect of temperature span on AeCR performance at a fixed strain of 2.7% for heat-pumping (

**a**,

**b**) and cooling (

**c**,

**d**).

**Figure 12.**The effect of displaced volume ratio and frequency on AeCR performance at a fixed strain of 2.7% and temperature span of 15 K (based on inlet fluid temperatures) for heat-pumping (

**a**,

**b**) and cooling (

**c**,

**d**).

**Figure 13.**The effect of reduced hysteresis losses on the performance of the AeCR at a fixed strain of 2.7%, temperature span of 15 K and displaced fluid volume ratio of 0.6 for heat-pumping (

**a**,

**b**) and cooling (

**c**,

**d**).

**Table 1.**Experimentally determined properties of NiTi tube required for further phenomenological modeling.

E_{A}(GPa) | σ_{AM,24}(MPa) | σ_{AM,45}(MPa) | ε_{trans}(/) | C_{Mp}(MPa·K ^{−1}) | M_{p}(K) | M_{s}(K) | M_{f}(K) |
---|---|---|---|---|---|---|---|

113.3 | 463 | 633 | 0.0225 | 7.83 | 239 | 315 | 163 |

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**MDPI and ACS Style**

Ahčin, Ž.; Kabirifar, P.; Porenta, L.; Brojan, M.; Tušek, J.
Numerical Modeling of Shell-and-Tube-like Elastocaloric Regenerator. *Energies* **2022**, *15*, 9253.
https://doi.org/10.3390/en15239253

**AMA Style**

Ahčin Ž, Kabirifar P, Porenta L, Brojan M, Tušek J.
Numerical Modeling of Shell-and-Tube-like Elastocaloric Regenerator. *Energies*. 2022; 15(23):9253.
https://doi.org/10.3390/en15239253

**Chicago/Turabian Style**

Ahčin, Žiga, Parham Kabirifar, Luka Porenta, Miha Brojan, and Jaka Tušek.
2022. "Numerical Modeling of Shell-and-Tube-like Elastocaloric Regenerator" *Energies* 15, no. 23: 9253.
https://doi.org/10.3390/en15239253