# CFD-Simulation Assisted Design of Elastocaloric Regenerator Geometry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{ad}, and isothermal entropy change, Δs

_{iso}. The potential for heat transfer between the active material and the heat transfer fluid is quantified by ΔT

_{ad}, while the capability of the active material to establish a high cooling power is measured by Δs

_{iso}[1]. Very recently, a (Ni

_{50}Mn

_{50−x}Ti

_{x})

_{99.8}B

_{0.2}alloy exhibiting colossal elastocaloric properties has been developed [17]. This alloy demonstrates a superior reproducible adiabatic temperature change, ΔT

_{ad}, of 31.5 K, and an isothermal entropy change, Δs

_{iso}, of 45 J kg

^{−1}K

^{−1}, compared to using thin wires of a benchmark material Ni-Ti which provides ΔT

_{ad}= 25 K and Δs

_{iso}= 35 J kg

^{−1}K

^{−1}[10]. However, (Ni

_{50}Mn

_{50−x}Ti

_{x})

_{99.8}B

_{0.2}has not been tested in an elastocaloric device and there is no available information about its functional and structural fatigue or machinability.

^{8}operational cycles [33], and provide a system temperature span of at least 40 K [20], both of which are challenging. Despite the fact that numerous laboratory setups have been reported [28,29,34,35], demonstrating the increased performance and enhanced fatigue life of the eCE regenerators, additional improvements of eCE regenerator geometries and mechanical properties are required.

^{®}, are presented. Note that the presented geometry might as well be exploited as a plate heat exchanger, which has an enhanced heat transfer surface. However, the main goal of this study is to develop a novel regenerator geometry that will be successfully exploited under compression loading in elastocaloric cooling applications, and would contribute to the development of sustainable cooling sources. Thus, not all corrugated or otherwise enhanced geometries are suitable for elastocaloric applications. Therefore, the results obtained in this study are not compared to those found in the literature on corrugated geometries, such as tubes that are spirally corrugated [36,37,38], grooved [39] or fitted with inserts [40]. The analysed cases involve the passage of the heat transfer fluid between two parallel flat plates, one flat and one double corrugated plate stacked in parallel, and two parallel double corrugated plates stacked in a mirrored manner.

## 2. Materials and Methods

_{c}, reported in [43,44] and the constant hydraulic diameter, D

_{h}, reported in [43,45]. The surface of the plates analysed in this study is mathematically described by Equation (1), which is derived from a set of equations describing tubular geometry in [43,45]. In this study, Equation (1) mimics the involute of the double corrugated tubes, disregarding the concept of constant hydraulic diameter or constant cross-sectional area, which were investigated in the case of a tube shape. A desired double corrugated plate is created by extruding the surfaces obtained with Equation (1) by the thickness z. Figure 1 shows the volumetric volume of a double corrugated plate AR1.6P02 with plate thickness z = 0.1 mm.

_{m}is mean fluid velocity, k is thermal conductivity, T

_{w}is wall temperature and ΔT

_{lm}is log-mean temperature difference, calculated as given by Equation (11).

_{h}, at the inlet would be held the same. Equation (3) defines the hydraulic diameter.

_{c}is the cross-sectional area of the inlet, and P is the perimeter of the inlet.

_{s}, strongly depends on the corrugation intensity. For example, the A

_{s}of geometry AR1.6P02 is 40% larger compared to the flat geometry, however the A

_{s}of AR1.6P04 decreases by 24% compared to AR1.6P02. This change in A

_{s}is attributed to the difference in period.

^{−3}. All models were solved using the PARDISO solver with a relative error of 10

^{−6}. The mass and energy balance for each model were solved to an accuracy of at least 10

^{−4}and 10

^{−3}, respectively.

_{p}is a specific heat capacity.

## 3. Results

_{c}, which is provided in Table 2, lead to a different mass flow rate, $\dot{m}$, at the inlet of each analysed geometry. One can also see that with the increasing corrugation intensity fluid flow is disturbed more effectively. However, as it is seen from Figure 4d,e, a fast-flowing fluid tends to bypass the very intense corrugation, i.e., corrugation with high amplitude and small period, along the flow direction, resulting in an effectively narrowed size of the flow channel. This observation was also reported in [43]. More clearly, the effect of the geometrical features on the fluid flow velocity is presented in Figure 5.

_{i}, and the plate surface temperature, T

_{s}. The model output data were mass flow rate, $\dot{m}$, average fluid flow velocity, u

_{m}, and fluid temperature at the outlet, T

_{o}. The Nusselt number was calculated using Equations (9)–(12) [50]. The net rate of outflow thermal energy q was calculated by Equation (9).

_{o,i}is the temperature difference between the temperature of a plate surface, T

_{s}, and the bulk temperature of the fluid, T, at the inlet and outlet. The average Nusselt number was calculated as given in Equation (12).

_{e}, and of the flat (reference) geometry, Nu

_{p}, and the cubic root of the ratio between the friction factor of the enhanced geometry, f

_{e}, and the flat geometry, f

_{p}. All the Nu and f must be evaluated at the same Reynolds number, Re. The friction factor is calculated using the simulation results for mean flow velocity, u

_{m}, and the specified pressure drop, Δp, as given in Equation (15) [51].

## 4. Discussion

^{®}. It must be emphasised that the models of the flat plate geometry did not reach the required convergence accuracy under the fluid flow conditions with the three highest Δp, therefore the data obtained with these models were omitted from the further analysis.

^{®}. The friction factor data were fit to the most commonly used expression given by Equation (18).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Construction of the fluid flow in the passage between two double corrugated plates AR1.1P02 in COMSOL Multiphysics

^{®}environment.

**Figure 3.**The modelled fluid flow in the passages in between (

**a**) two flat plates; (

**b**) one flat and one double corrugated plate (AR1.1P02); (

**c**) two double corrugated “mirrored” plates (AR1.1P02M) stacked in parallel. The light blue colour denotes the fluid inlet.

**Figure 4.**The temperature field at the constant pressure drop Δp = 3397 Pa of the fluid flow in the passage of (

**a**) flat (mass flow rate $\dot{m}$ = 0.21 kg/h), (

**b**) AR1.1P02 (mass flow rate $\dot{m}$ = 0.12 kg/h), (

**c**) AR1.1P02M (mass flow rate $\dot{m}$ = 0.15 kg/h), (

**d**) AR1.6P02 (mass flow rate $\dot{m}$ = 0.49 kg/h) and (

**e**) AR1.6P04 (mass flow rate $\dot{m}$ = 0.66 kg/h) plates.

**Figure 5.**Fluid flow velocity at Δp = 3397 Pa in the middle cross-section of the modelled geometry as a function of the normalised distance across the fluid flow direction.

**Figure 6.**Nusselt number as a function of Reynolds number. Symbols represent the data points and the dashed line represent the least square data fit.

Density, $\mathit{\rho}$ (kg m^{−3})
| Dynamic Viscosity, μ (Pa s) | Thermal Conductivity, k (W m^{−1} K^{−1}) | Specific Heat, c _{p} (J kg^{−1} K^{−1}) | Pressure Drop, Δp (Pa) | Inlet Temperature, T_{in} (K) | Wall Temperature, T_{s} (K) |
---|---|---|---|---|---|---|

1000 | 8 × 10^{−4} | 0.603 | 4184 | 5869; 4891; 4076; 3397; 2830; 2359; 1966; 1638; 1310.40; 1048.32; 838.66; 670.92 | 303 | 330 |

Geometry Name | Length, l (mm) | Width, w (mm) | Hydraulic Diameter, D_{h} (mm) | Inlet Cross-Section Area, A_{c} (mm^{−2}) | Heat Transfer Surface Area, A_{s} (mm^{−2}) | Corr. Period, P (mm) | Aspect Ratio, AR (−) | Corrugation Amplitude Coefficient, R (mm) |
---|---|---|---|---|---|---|---|---|

Flat | 1.6 | 1.6 | 0.1 | 0.09 | 5.1 | $\infty $ | 1.0 | 0.1 |

AR1.1P02 | 1.6 | 1.6 | 0.1 | 0.09 | 5.2 | 0.2 | 1.1 | 0.1 |

AR1.1P02M | 1.6 | 1.6 | 0.1 | 0.09 | 5.3 | 0.2 | 1.1 | 0.1 |

AR1.6P02 | 1.6 | 1.6 | 0.1 | 0.11 | 7.2 | 0.2 | 1.6 | 0.1 |

AR1.6P04 | 1.6 | 1.6 | 0.1 | 0.11 | 5.8 | 0.4 | 1.6 | 0.1 |

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**MDPI and ACS Style**

Navickaitė, K.; Penzel, M.; Bahl, C.; Engelbrecht, K.; Tušek, J.; Martin, A.; Zinecker, M.; Schubert, A. CFD-Simulation Assisted Design of Elastocaloric Regenerator Geometry. *Sustainability* **2020**, *12*, 9013.
https://doi.org/10.3390/su12219013

**AMA Style**

Navickaitė K, Penzel M, Bahl C, Engelbrecht K, Tušek J, Martin A, Zinecker M, Schubert A. CFD-Simulation Assisted Design of Elastocaloric Regenerator Geometry. *Sustainability*. 2020; 12(21):9013.
https://doi.org/10.3390/su12219013

**Chicago/Turabian Style**

Navickaitė, Kristina, Michael Penzel, Christian Bahl, Kurt Engelbrecht, Jaka Tušek, André Martin, Mike Zinecker, and Andreas Schubert. 2020. "CFD-Simulation Assisted Design of Elastocaloric Regenerator Geometry" *Sustainability* 12, no. 21: 9013.
https://doi.org/10.3390/su12219013