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Phenomenological model for a first-order magnetocaloric material
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Phenomenological model for a first-order magnetocaloric material T. Hess , C. Vogel , L.M. Maier , A. Barcza , H.P. Vieyra , ¨ ¨ O. Schafer-Welsen , J. Wollenstein , K. Bartholome´ PII: DOI: Reference:
S0140-7007(19)30423-2 https://doi.org/10.1016/j.ijrefrig.2019.10.003 JIJR 4541
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International Journal of Refrigeration
Received date: Revised date: Accepted date:
4 June 2019 23 September 2019 5 October 2019
Please cite this article as: T. Hess , C. Vogel , L.M. Maier , A. Barcza , H.P. Vieyra , ¨ ¨ O. Schafer-Welsen , J. Wollenstein , K. Bartholome´ , Phenomenological model for a first-order magnetocaloric material, International Journal of Refrigeration (2019), doi: https://doi.org/10.1016/j.ijrefrig.2019.10.003
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Highlights:
Material model for a first-order magnetocaloric material Model tailored for an easy integration into system simulations Model based only on the specific heat, providing a built-in model consistency Parameters are determined from experimental data Consistency of the model is verified with experimental data
Research paper
Phenomenological model for a first-order magnetocaloric material T. Hess1,2, C. Vogel3, L. M. Maier1,2, A. Barcza4, H. P. Vieyra4, O. Schäfer-Welsen1, J. Wöllenstein1,2, K. Bartholomé1 1. Fraunhofer Institute for Physical Measurement Techniques IPM, Heidenhofstraße 8, 79110 Freiburg, Germany 2. Laboratory for Gas Sensors, Department of Microsystems Engineering - IMTEK, University of Freiburg, Germany 3. GSI Technologies UG, Kalmistr. 30, D-67269 Gruenstadt, Germany 4. Vacuumschmelze GmbH & Co. KG, Grüner Weg 37, 63450 Hanau, Germany
Corresponding Author's main affiliation address: e-mail: [email protected]
Fraunhofer-Institute for Physical Measurement Techniques IPM, Heidenhofstraße 8, 79110 Freiburg, Germany Declaration if interest: None Comment: Figures are added in the text as pixelated graphs for better readability and are also submitted as vector graphs. Frames around the figures and equations will be removed for final version. Abstract: In order to predict the potential of magnetocaloric heating and cooling devices, system simulations are an essential instrument. These simulations, in turn, depend to a large extent on the model implemented for the magnetocaloric material. Magnetocaloric materials with a first-order phase transition, such as found in some La(Fe,Mn,Si)13-based alloys, show excellent magnetocaloric properties. The aim of this work is thus to provide a material model for a first-order La(Fe,Mn,Si)13-based alloy. The model is tailored to be used in system simulations. This includes thermodynamic consistency of the model and a relatively simple implementation. All relevant equations of the material model are determined from the specific heat capacitance as function of the temperature and the magnetic field. Since all equations are derived from the same base equation, they are consistent in terms of the first and second law of thermodynamics. As base function for the specific heat capacitance, a modified Cauchy–Lorentz function is used. The model parameters are determined from experimental data. Consistency of the model is
verified with further data. The present model enables the simulation of the exergetic efficiency of a magnetocaloric cooling or heating device based on first-order La(Fe,Mn,Si)13 alloys. Keywords: Magnetocaloric refrigeration, Material model, First-order material, System simulation
Nomenclature Roman A,B,C
-1 -1
areas in figure 2 and 3 corresponding to specific entropies [J kg K ] -1 -1 specific heat capacitance [J kg K ] -1 -1 specific baseline heat capacitance at [J kg K ] -1 -1 specific heat capacitance at constant applied magnetic field [J kg K ] -1 magnetic field [A m ] externally applied magnetic field [T] change of the externally applied magnetic field [T] -1 magnetization [A m ] -1 -1 specific entropy [J kg K ] temperature [K] temperature of the maximum specific heat capacitance with no external field[K] specified temperature as integration boundary [K]
Greek -1
model parameter corresponding to the shift of the peak per applied field [K T ] -3 density [g cm ] -1 -1 thermal conductivity [W m K ] model parameter corresponding to the peak width [-] -1 vacuum permeability [T m A ] finite difference
Subscripts ad iso max rel
Adiabatic Isotherm Maximum Relative
Acronyms La Fe Si Mn
Lanthanum Iron Silicon Manganese
1. Introduction Magnetocaloric materials with a transition temperature close to room temperature have the potential to form the basis for disruptive cooling and heating devices. Compared to compressor-based systems, magnetocaloric systems do not require hazardous coolants and are potentially even more efficient than compressor-based systems (Fähler and Pecharsky, 2018; Takeuchi and Sandeman, 2015). Magnetocaloric materials can be classified into materials with a first-order and a second-order phase transition. One of the most promising magnetocaloric material classes are La(Fe,Mn,Si)13-based alloys (Katter et al., 2008). Most of them show a first-order phase transition (Bratko et al., 2017), resulting in a large isothermal entropy as well as adiabatic temperature change. To determine the true potential of magnetocaloric cooling devices, system simulations are an essential tool. As part of those simulations, the magnetocaloric effect of the material has to be modelled. Nielsen et al. (Nielsen et al., 2011) summarize two possibilites to implement the magnetocaloric effect: The first option, the so-called »direct approach«, is the direct manipulation of the temperature by implementing the adiabatic temperature change of the magnetocaloric material during an instantaneous magnetization. This way is straightforward to implement and is used by various authors for system simulations e.g. (Aprea et al., 2013; Schroeder and Brehob, 2016; Silva et al., 2018; Teyber et al., 2016). Drawbacks of this approach are the fact that real systems do not show an instantaneous magnetization and that thermodynamic consistency, which is crucial to precisely predict the efficiency of a given sytem, is not necessarily fulfilled. The second option is the so-called »built-in approach«, where the magnetocaloric effect is modelled by the heat flux, which is released into a varying heat capacitance of the caloric material during a continuous magnetization. The heat-source-term and the varying heat capacitance are coupled thermodynamically. Therefor a numerically differentiable model of e.g. the specific heat as functions of both temperature and magnetic field is required to give a thermodynamic consistent description. Thermodynamic consistency means in this context that all material properties such as adiabatic temperature change, isothermal entropy change and specific heat are interconnected to form a clearly defined state description in the entropy-temperature-space (Gottschall et al., 2016). Engelbrecht (Engelbrecht, 2008) introduced three general approaches for material modelling of magnetocaloric materials: the interpolation of empirical data, the application of mean field theory as a kind of first principle approach, and curve fitting. The interpolation of empirical data strongly depends on the quality of the measured data. It is very challenging to compose a consistent material model based on interpolated data. Therefore, this approach is not considered further here. Mean field theory provides acceptable results for second-order materials but is no option for first-order materials. The revised model introduced by Bean and Rodbell (C. P. Bean and D. S. Rodbell, 1962) describes some properties of a first-order phase transition but as Neves Bez et al. (Neves Bez et al., 2016) point out, fails to fully describe the properties of first-order La(Fe,Mn,Si)13. Curve fitting combines the benefits of the two other approaches. It is also based on measured data but not as dependent on the data quality as data interpolation, because some of the information is already given by the choice of the fitting functions. For second-order materials, latest material models utilize for example the presence of a so-called master curve for the isothermal entropy change (Franco et al., 2006; Lei et al., 2016). This means, the curve only scales with applied magnetic field and does not change its shape. First-order materials do not show this master curve characteristic. For first-order materials, Benedict et al. introduced a set of equations to fit the adiabatic temperature change and the specific heat capacitance at zero applied magnetic field 1
(Benedict et al., 2016). These quantities can also be used to create a clearly defined state space, yet there is no analytical access to the heat capacitance as function of temperature and applied magnetic field and the change in entropy with the magnetic field δs/δµ0H, which are used for the built-in implementation of the magnetocaloric effect (Nielsen et al., 2011). The present work is based on the assumption that hysteresis can be neglected. This is obvious for second-order materials but requires some justifications for first-order materials, because first-order phase transitions are strongly associated with hysteresis (Basso et al., 2018; Gutfleisch et al., 2016). However, latest material researches show that materials can undergo a first-order phase transition with close to zero hysteresis (Zhang et al., 2015). Additionally, even if a material shows hysteresis at full phase change, it seems that some materials can be trapped into so called minor hysteresis loops when stimulated under cyclic conditions, whereby they do not undergo the full phase change (Gottschall et al., 2017; Kaeswurm et al., 2016). Moreover, even if the material shows hysteresis and it is not reduced by the presence of minor hysteresis loops, for system simulations it can still be beneficial to treat the material as if it was free of hysteresis and then add an additional source term to account for the dissipative losses due to hysteresis. In this work we present a material model for a first-order La(Fe,Mn,Si)13 based alloy. The model is based on the specific heat as function of temperature and applied magnetic field. All other quantities are derived from this base function to ensure thermodynamic consistency. The selection of the base function forms a compromise between good representation of measured data and analytical accessibility of the remaining material properties, to provide easy to implement equations for system simulations. The model parameters are determined from measured data.
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2. Material model 2.1. First- and second-order phase change Magnetocaloric materials are classified according to the Ehrenfest classification into the order of the phase change they undergo. First-order materials show a discontinuous change of the magnetization M (figure 1 top left), while second-order materials show a continuous change of the magnetization around the phase change (figure 1 top right). The simple Ehrenfest classification is conceptionally explicit, however a clear determination of real magnetocaloric probes is still matter of discussion in the scientific community (Bratko et al., 2017; Law et al., 2018). Beside the continuity or discontinuity of the magnetization, another difference between first- and second-order materials is the different deformation of the peak of the specific heat capacitance as a function of the temperature at different applied magnetic fields. In first-order materials, the peak-temperature increases with larger applied magnetic fields, whilst the shape of the peak does not change drastically (figure 1 middle right). In second-order materials, with an increasing field the peak decreases and broadens without any significant shift of the peak temperature (figure 1 middle left). As indicated in the bottom line of figure 1, the magnetocaloric response of first-order materials saturates with large magnetic fields (Zverev et al., 2010) whereas second-order materials do not saturate (Gottschall et al., 2019). At moderate magnetic fields, as created by permanent magnet field sources, first-order materials generally show a higher magnetocaloric effect than second-order materials. As downsides of first-order materials, they generally only work in a smaller temperature range and are associated with hysteresis. One current goal of material scientists is to develop magnetocaloric materials at sweet spot between first- and second-order characteristics to get the best of both worlds. In the present work, we have placed our focus on the description of first-order materials.
Figure 1: Distinctions between a first-order (left) and second-order (right) magnetocaloric material via magnetization (top), specific heat capacitance (middle) and the adiabatic temperature change (bottom) as function of the temperature and for different magnetic 3
fields. Figure is inspired by (Kitanovski et al., 2015) p. 25 and (Smith et al., 2012).
2.2. Base equations and visualization The assumption of no hysteresis allows a clearly defined description by state variables. State variables are the temperature T, the applied magnetic field H and the entropy s. With the specific heat capacitance cH at a constant applied magnetic field the entropy is (
)
∫
(
)
1
When s(T,µ0H) is known, the material is clearly defined in a state space. This is represented in a sT-space by state trajectories for different applied magnetic fields. This means modelling cH(T,µ0H) for all possible µ0H gives a uniquely defined material model. In an ideal first-order material, the specific heat capacitance shows an infinite Dirac-like function in temperature at the phase change. However, for real magnetocaloric materials this peak is broadened, resulting in a finite peak function. The two material properties, isothermal entropy change ∆siso and adiabatic temperature change ∆Tad, are derived. The isothermal entropy change is the entropy change of the caloric material whilst the material is magnetized or demagnetized at a constant temperature. For a magnetization from zero field to µ0∆H the isothermal entropy change is given by (
)
(
)
(
)
2
To visualize this equation an exemplary material is introduced. Figure 2 shows cH/T of this first-order magnetocaloric material for µ0H = 0 T (black curve) and µ0∆H (green curve). By applying a magnetic field, the peak shifts to higher temperatures as measured in various studies (Basso et al., 2015; Burriel et al., 2014; Wang, 2012). An isothermal magnetization decreases the entropy of the material, an isothermal demagnetization increases it. The area below the curve from 0 K to a certain temperature T represents the entropy of the material at this temperature (equation 1). At this temperature the entropy at zero field corresponds to the area A and at µ0∆H to A+B. The entropy change is the difference of these areas and here given by A. The entropy change has a maximum in temperature at the crossing point of the two curves (orange dot in figure2). Furthermore, when assuming a constant base line as offset for the peak functions the maximum entropy change ∆siso,max at very high fields is given by the area of the peak function without offset.
4
Figure 2: Visualization of the isothermal entropy change set free by the magnetization of a first-order magnetocaloric material by using the temperature normalized heat capacitance curves at different applied magnetic fields. Besides the isothermal entropy change, the adiabatic temperature change is a very important material property of magnetocaloric materials. An adiabatic and reversible process is always isentropic (constant entropy). Since the material is assumed to have no thermal hysteresis and therefore shows no dissipative losses, the adiabatic temperature change is also an isentropic temperature change. This means, even after magnetization the area below the cH/T curve must be the same as before magnetization. This can be expressed by ∫
(
)
∫
(
)
3
This equation is visualized in figure 3. Without external magnetic field, the entropy at a certain temperature T is given by the sum of the areas A and B. When magnetized isentropically, the entropy is given by B+C. For an isentropic process A = C must be fulfilled. In other words, the area below the black curve is squeezed below the green curve, pushing the right boundary of the integral and therefor the material temperature towards a higher temperature.
Figure 3: Visualization of the adiabatic temperature change ∆Tad of a first-order magnetocaloric material. To keep the entropy constant during magnetization, the area below the black curve is squeezed under the green curve up to a certain temperature T. This shifts the right boundary of the area to the higher temperature T+∆Tad.
5
2.3. Empirical parametrization For an empirical description, an analytical peak function for cH/T at temperatures around the phase change is introduced. It is a Cauchy–Lorentz function given by (
)
4 (
(
))
where σ defines the width of the peak and T0 the position in temperature at zero applied field. The Cauchy-Lorentz function has an offset of c0/ T0 and an area under the curve without offset of ∆siso,max. With an applied magnetic field, the function shifts linear to higher temperatures. The shift in temperature is given by β, which can be interpreted as the inverse Clausius Clapeyron coefficient (Tušek et al., 2016).
Figure 4: Parametrization of Cauchy–Lorentz function for the temperature normalized entropy change cH/T. Figure 4 shows the graphical representation of equation 4. The area of the peak function does not change with the applied field, thus leading to a convergence of the state trajectories in the sT-space at higher temperatures. Thereby, the physical boundary condition that in the paramagnetic state the entropy of the material is not a function of the applied magnetic field is fulfilled. Inserting equation 4 into equation 1 gives the following expression for the relative entropy: (
)
(
(
) ) (
(
(
)
5
) )
For most numeric applications, only the relative entropy is of interest but not the absolute entropy. With a T0 of around 300 K and a µ0H of ~1 T the second term can be simplified to be constant in μ0H and T and therefore is ignored. Inserting 5 into 2 gives the analytic expression of the isothermal entropy change in expanded and simplified form of (
)
(
(
(
( (
6
)
)
(
)
)) (
(
)
)
(
))
( )
6
This function has a global maximum of ∆siso,max. Inserting equation 4 into equation 3 results in the following expanded and simplified expression for the adiabatic temperature change: (
)
(
( ) (
( (
(
(
)
(
)
7
) ))
)
(
(
)
))
Note that this is a recursive expression in ∆Tad and can only be solved numerically. As the maximum of the isothermal entropy changes, the maximum of the adiabatic temperature change also saturates with increasing applied magnetic fields. The value of saturation is ∆siso,max T0/c0, which is the area of the zero field cH/T curve without the offset divided by the offset. As final material parameter required for a thermodynamic constant material model δs/δµ0H is derived by differentiating equation 4: (
)
(
(
) (
(
(
(
))
)
8
))
This quantity is part of the magnetocaloric source term and an analytical description enables a relatively simple implementation into system simulations.
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3. Methods to determine model parameters A LaFe11.5Mn0.24Si1.26 alloy was produced by powder metallurgy methods as described in (Katter et al., 2008). A powder blend was isostatically pressed and subsequently sintered at around 1100°C under inert conditions. The sample was milled down to a coarse powder and sieved between 125-250 µm. In order to adjust the transition temperature to around room temperature the sample was hydrogenated as described in (Barcza et al., 2011). The material belongs to the commercially available material family CALORIVAC H and has a density of = 7.2 g cm-3 and a thermal conductivity of = 6-8 W m-1 K-1. All measurements where carried out at the same sample. Quasi-adiabatic ∆Tad characterization was performed for different external magnetic field changes (0.8 / 1 / 1.25 / 1.5 T) using a bespoke device with a rotating magnet assembly. Field-dependent magnetization measurements as a function of temperature were performed with a commercial vibrating sample magnetometer. The data have been used to determine ∆siso at 0.4 / 0.8 / 1.2 and 1.6 T using the Maxwell relation. Zero-field heat capacity curves were obtained at a heating rate of 2.5 K/min in a commercial differential scanning calorimeter using the sapphire method. The parameters c0, ∆siso,max, σ and T0 are determined by fitting equation 4 to measured data of cH/T at zero applied magnetic field. Parameters c0, ∆siso,max, and σ determined in this way are taken as input parameters for the following analysis, whereas T0 is left as free parameter to account for different temperature offsets of the different measurement techniques. To determine the shift of the peak function per applied field β from ∆siso or ∆Tad a multivariable fit is used. The functions for ∆siso and ∆Tad have the variables T and H and the previously determined constants c0, ∆siso,max, and σ and the parameters β and T0. For the fits, the experimental data are weighted according to their relative quantity in cH/T, ∆Tad and ∆siso to make the fit curves more accurate close to the respective peaks.
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4. Comparison with experimental data In Figure 5, the experimental data of cH/T without external field against the temperature are shown as black dots. There is an asymmetry of the slopes of the left hand side of the peak function compared to the right hand side. Compared the Lorentzian function there is also an additional turning point at the peak characteristic. Whether this is a measurement artifact or a physical effect is unclear. The present material model cannot take into account either the asymmetry or the additional turning point. The Lorentzian fit to the experimental data is shown as red line. Model parameters of c0 = 558 J kg-1K-1, ∆siso,max = 17.6 J kg-1K-1 σ = 0.8 and T0 = 310.0 K are determined.
Figure 5: Measured data of the temperature normalized heat capacitance (black) and fitted Lorentzian curve (red) are shown. The parameters c0, ∆siso,max, σ and T0 are determined. In figure 6, the measured data and the fitted model of the adiabatic temperature change ∆Tad for different applied magnetic fields is plotted against the temperature. Plot markers represent the measured data and the solid lines the fitted model. For the fit, the parameters c0, ∆siso,max und σ are taken from the cH/T fit. T0 is not taken over from cH/T to account for an offset of the measurement of the absolute temperature. The fit parameters are β = 3.97 K T-1 and T0 = 309.9 K. The model shows good agreement with the measured data close to the maximum of the according peaks. Next to the peaks, the model underestimates the measured data, especially at temperatures lower than T0.
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Figure 6: Measured data and fitted model of the adiabatic temperature against the temperature at different applied magnetic fields. Figure 7 shows the measured data and the fitted model of the isothermal entropy change ∆siso at different applied magnetic fields against the temperature. With increasing applied magnetic field, the peak broadens and attains larger values. At higher fields, the measured data show a tilted plateau at the peak maximum. As in figure 6, β and T0 are determined as fit parameters. The parameters are β = 4.02 K T-1 and T0 = 308.9 K. T0 is about 1 K lower than the fitted values from the cH/T and ∆Tad data. This can be explained by a temperature offset of the ∆siso measurement. The determined value for β is consistent with the β-value from the ∆Tad data. All determined fit parameters from the different measurements are summarized in table 1. These results show good agreements with the values measured by Basso et al. (Basso et al., 2015).
Figure 7 Measured data and fitted model of the isothermal entropy change against the temperature at different applied magnetic fields. Table 1: Summery of determined fit parameters for the different sets of data Experimental data cH/T ∆Tad ∆Siso
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c0 [J kg-1K-1] 558 -
∆siso,max [ J kg-1K-1] 17.6 -
σ [-] 0.8 -
T0 [K] 310 309.9 308.9
β [K T-1] 4.02 3.97
5. Further perspective The present model shows a good accuracy with built in consistency. In order to simulate the system efficiency of a magnetocaloric cooling device, the consistency of the model is much more important than a highly accurate representation of the material parameters, because it prevents unphysical results. More degrees of freedom for the heat capacitance increase the accuracy but due to the various integration and differentiation steps, the complexity of the model would significantly increase. The present model gives an analytical state description of the caloric material at any field and temperature, used in system simulations. The corresponding sT-diagram (equation 5) of the material is shown in figure 8. A high-field curve and a low-field curve span up state space. Any cooling cycle for a given high and low field takes place within this state space, and the state of the material can be traced at any time step. The area enclosed by such a state trajectory corresponds to the work that has to be applied to the magnetocaloric material in order to create a certain cooling or heating power. With the sum of those path integrals for each knot the exergetic efficiency of a given system can be determined.
Figure 8: Calculated relative entropy over the temperature. This defines the sT-state space of the magnetocaloric material. When simulating a cooling system it has to be considered that the phase transition of La(Fe,Mn,Si)13 materials approaches second-order type transition towards lower temperatures (Basso et al., 2015). In order to model a machine with a cascade of caloric material ranging from e.g. 40°C to 0°C it may be necessary to adapt the material model gradually from first- to second-order. By giving the model more degrees of freedom, the agreement with measured data will improve, but with the disadvantage of increased model complexity. For a more second-order material, a peak function for cH/T also should shrink and change its shape with an applied magnetic field. The concepts presented here can also be adopted for a material model for elastocaloric and electrocaloric materials.
6. Conclusion An empirical model for the mathematical description of a first-order magnetocaloric material has been presented. The model is based on a Cauchy–Lorentz function for cH/T. From this base equation, expressions for all relevant material properties where derived, resulting in a thermodynamically consistent material model tailored for system simulations. The model parameters were determined by 11
fits to experimental data of the specific heat capacitance at zero field, the adiabatic temperature change and the isothermal entropy change. The model is in good agreement with the experimental data and can be implemented in system simulations for cooling devices with first-order magnetocaloric materials.
7. Acknowledgments This work is funded by the Fraunhofer internal project MagCon, as part of the Sustainability Center Freiburg within the project ActiPipe and by the Federal Ministry of Economics and Energy (BMWi) of Germany within the project MagMed.
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Gottschall, T., Stern-Taulats, E., Mañosa, L., Planes, A., Skokov, K.P., Gutfleisch, O., 2017. Reversibility of minor hysteresis loops in magnetocaloric Heusler alloys. Appl. Phys. Lett. 110 (22), 223904. 10.1063/1.4984797. Gutfleisch, O., Gottschall, T., Fries, M., Benke, D., Radulov, I., Skokov, K.P., Wende, H., Gruner, M., Acet, M., Entel, P., Farle, M., 2016. Mastering hysteresis in magnetocaloric materials. Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 374 (2074). 10.1098/rsta.2015.0308. Kaeswurm, B., Franco, V., Skokov, K.P., Gutfleisch, O., 2016. Assessment of the magnetocaloric effect in La,Pr(Fe,Si) under cycling. Journal of Magnetism and Magnetic Materials 406, 259–265. 10.1016/j.jmmm.2016.01.045. Katter, M., Zellmann, V., Reppel, G.W., Uestuener, K., 2008. Magnetocaloric Properties of La(Fe,Co,Si)13 Bulk Material Prepared by Powder Metallurgy. IEEE Trans. Magn. 44 (11), 3044–3047. 10.1109/TMAG.2008.2002523. Kitanovski, A., Plaznik, U., Tomc, U., Poredoš, A., 2015. Present and future caloric refrigeration and heatpump technologies. International Journal of Refrigeration 57, 288–298. 10.1016/j.ijrefrig.2015.06.008. Law, J.Y., Franco, V., Moreno-Ramírez, L.M., Conde, A., Karpenkov, D.Y., Radulov, I., Skokov, K.P., Gutfleisch, O., 2018. A quantitative criterion for determining the order of magnetic phase transitions using the magnetocaloric effect. Nature communications 9 (1), 2680. 10.1038/s41467-018-05111-w. Lei, T., Engelbrecht, K., Nielsen, K.K., Neves Bez, H., Bahl, C.R.H., 2016. Study of multi-layer active magnetic regenerators using magnetocaloric materials with first and second order phase transition. J. Phys. D: Appl. Phys. 49 (34), 345001. 10.1088/0022-3727/49/34/345001. Neves Bez, H., Bahl, C., Nielsen, K.K., Smith, A., 2016. Magnetocaloric materials and first order phase transitions. PhD Thesis. Denmark, 192 pp. Nielsen, K.K., Tusek, J., Engelbrecht, K., Schopfer, S., Kitanovski, A., Bahl, C.R.H., Smith, A., Pryds, N., Poredos, A., 2011. Review on numerical modeling of active magnetic regenerators for room temperature applications. International Journal of Refrigeration (34), 603–616. Schroeder, M.G., Brehob, E., 2016. A flexible numerical model of a multistage active magnetocaloric regenerator. International Journal of Refrigeration 65, 250–257. 10.1016/j.ijrefrig.2016.01.023. Silva, D.J., Amaral, J.S., Amaral, V.S., 2018. Heatrapy: A flexible Python framework for computing dynamic heat transfer processes involving caloric effects in 1.5D systems. SoftwareX 7, 373–382. 10.1016/j.softx.2018.09.007. Smith, A., Bahl, C.R.H., Bjørk, R., Engelbrecht, K., Nielsen, K.K., Pryds, N., 2012. Materials Challenges for High Performance Magnetocaloric Refrigeration Devices. Adv. Energy Mater. 2 (11), 1288–1318. 10.1002/aenm.201200167. Takeuchi, I., Sandeman, K., 2015. Solid-state cooling with caloric materials. Phys. Today 68 (12), 48–54. 10.1063/PT.3.3022. Teyber, R., Trevizoli, P.V., Christiaanse, T.V., Govindappa, P., Niknia, I., Rowe, A., 2016. Performance evaluation of two-layer active magnetic regenerators with second-order magnetocaloric materials. Applied Thermal Engineering 106, 405–414. 10.1016/j.applthermaleng.2016.06.029. Tušek, J., Engelbrecht, K., Mañosa, L., Vives, E., Pryds, N., 2016. Understanding the Thermodynamic Properties of the Elastocaloric Effect Through Experimentation and Modelling. Shap. Mem. Superelasticity 2 (4), 317–329. 10.1007/s40830-016-0094-8. Wang, G., 2012. Magnetic and Calorimetric Study of the Magnetocaloric Effect in Intermetallics Exhibiting First-order Magnetostructural Transitions, 277 pp. 13
Zhang, M.X., Liu, J., Zhang, Y., Dong, J.D., Yan, A.R., Skokov, K.P., Gutfleisch, O., 2015. Large entropy change, adiabatic temperature change, and small hysteresis in La(Fe,Mn)11.6Si1.4 strip-cast flakes. Journal of Magnetism and Magnetic Materials 377, 90–94. 10.1016/j.jmmm.2014.10.035. Zverev, V.I., Tishin, A.M., Kuz’min, M.D., 2010. The maximum possible magnetocaloric ΔT effect. Journal of Applied Physics 107 (4), 43907. 10.1063/1.3309769.
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Phenomenological model for a first-order magnetocaloric material T. Hess , C. Vogel , L.M. Maier , A. Barcza , H.P. Vieyra , ¨ ¨ O. Schafer-Welsen , J. Wollenstein , K. Bartholome´ PII: DOI: Reference:
S0140-7007(19)30423-2 https://doi.org/10.1016/j.ijrefrig.2019.10.003 JIJR 4541
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
4 June 2019 23 September 2019 5 October 2019
Please cite this article as: T. Hess , C. Vogel , L.M. Maier , A. Barcza , H.P. Vieyra , ¨ ¨ O. Schafer-Welsen , J. Wollenstein , K. Bartholome´ , Phenomenological model for a first-order magnetocaloric material, International Journal of Refrigeration (2019), doi: https://doi.org/10.1016/j.ijrefrig.2019.10.003
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Highlights:
Material model for a first-order magnetocaloric material Model tailored for an easy integration into system simulations Model based only on the specific heat, providing a built-in model consistency Parameters are determined from experimental data Consistency of the model is verified with experimental data
Research paper
Phenomenological model for a first-order magnetocaloric material T. Hess1,2, C. Vogel3, L. M. Maier1,2, A. Barcza4, H. P. Vieyra4, O. Schäfer-Welsen1, J. Wöllenstein1,2, K. Bartholomé1 1. Fraunhofer Institute for Physical Measurement Techniques IPM, Heidenhofstraße 8, 79110 Freiburg, Germany 2. Laboratory for Gas Sensors, Department of Microsystems Engineering - IMTEK, University of Freiburg, Germany 3. GSI Technologies UG, Kalmistr. 30, D-67269 Gruenstadt, Germany 4. Vacuumschmelze GmbH & Co. KG, Grüner Weg 37, 63450 Hanau, Germany
Corresponding Author's main affiliation address: e-mail: [email protected]
Fraunhofer-Institute for Physical Measurement Techniques IPM, Heidenhofstraße 8, 79110 Freiburg, Germany Declaration if interest: None Comment: Figures are added in the text as pixelated graphs for better readability and are also submitted as vector graphs. Frames around the figures and equations will be removed for final version. Abstract: In order to predict the potential of magnetocaloric heating and cooling devices, system simulations are an essential instrument. These simulations, in turn, depend to a large extent on the model implemented for the magnetocaloric material. Magnetocaloric materials with a first-order phase transition, such as found in some La(Fe,Mn,Si)13-based alloys, show excellent magnetocaloric properties. The aim of this work is thus to provide a material model for a first-order La(Fe,Mn,Si)13-based alloy. The model is tailored to be used in system simulations. This includes thermodynamic consistency of the model and a relatively simple implementation. All relevant equations of the material model are determined from the specific heat capacitance as function of the temperature and the magnetic field. Since all equations are derived from the same base equation, they are consistent in terms of the first and second law of thermodynamics. As base function for the specific heat capacitance, a modified Cauchy–Lorentz function is used. The model parameters are determined from experimental data. Consistency of the model is
verified with further data. The present model enables the simulation of the exergetic efficiency of a magnetocaloric cooling or heating device based on first-order La(Fe,Mn,Si)13 alloys. Keywords: Magnetocaloric refrigeration, Material model, First-order material, System simulation
Nomenclature Roman A,B,C
-1 -1
areas in figure 2 and 3 corresponding to specific entropies [J kg K ] -1 -1 specific heat capacitance [J kg K ] -1 -1 specific baseline heat capacitance at [J kg K ] -1 -1 specific heat capacitance at constant applied magnetic field [J kg K ] -1 magnetic field [A m ] externally applied magnetic field [T] change of the externally applied magnetic field [T] -1 magnetization [A m ] -1 -1 specific entropy [J kg K ] temperature [K] temperature of the maximum specific heat capacitance with no external field[K] specified temperature as integration boundary [K]
Greek -1
model parameter corresponding to the shift of the peak per applied field [K T ] -3 density [g cm ] -1 -1 thermal conductivity [W m K ] model parameter corresponding to the peak width [-] -1 vacuum permeability [T m A ] finite difference
Subscripts ad iso max rel
Adiabatic Isotherm Maximum Relative
Acronyms La Fe Si Mn
Lanthanum Iron Silicon Manganese
1. Introduction Magnetocaloric materials with a transition temperature close to room temperature have the potential to form the basis for disruptive cooling and heating devices. Compared to compressor-based systems, magnetocaloric systems do not require hazardous coolants and are potentially even more efficient than compressor-based systems (Fähler and Pecharsky, 2018; Takeuchi and Sandeman, 2015). Magnetocaloric materials can be classified into materials with a first-order and a second-order phase transition. One of the most promising magnetocaloric material classes are La(Fe,Mn,Si)13-based alloys (Katter et al., 2008). Most of them show a first-order phase transition (Bratko et al., 2017), resulting in a large isothermal entropy as well as adiabatic temperature change. To determine the true potential of magnetocaloric cooling devices, system simulations are an essential tool. As part of those simulations, the magnetocaloric effect of the material has to be modelled. Nielsen et al. (Nielsen et al., 2011) summarize two possibilites to implement the magnetocaloric effect: The first option, the so-called »direct approach«, is the direct manipulation of the temperature by implementing the adiabatic temperature change of the magnetocaloric material during an instantaneous magnetization. This way is straightforward to implement and is used by various authors for system simulations e.g. (Aprea et al., 2013; Schroeder and Brehob, 2016; Silva et al., 2018; Teyber et al., 2016). Drawbacks of this approach are the fact that real systems do not show an instantaneous magnetization and that thermodynamic consistency, which is crucial to precisely predict the efficiency of a given sytem, is not necessarily fulfilled. The second option is the so-called »built-in approach«, where the magnetocaloric effect is modelled by the heat flux, which is released into a varying heat capacitance of the caloric material during a continuous magnetization. The heat-source-term and the varying heat capacitance are coupled thermodynamically. Therefor a numerically differentiable model of e.g. the specific heat as functions of both temperature and magnetic field is required to give a thermodynamic consistent description. Thermodynamic consistency means in this context that all material properties such as adiabatic temperature change, isothermal entropy change and specific heat are interconnected to form a clearly defined state description in the entropy-temperature-space (Gottschall et al., 2016). Engelbrecht (Engelbrecht, 2008) introduced three general approaches for material modelling of magnetocaloric materials: the interpolation of empirical data, the application of mean field theory as a kind of first principle approach, and curve fitting. The interpolation of empirical data strongly depends on the quality of the measured data. It is very challenging to compose a consistent material model based on interpolated data. Therefore, this approach is not considered further here. Mean field theory provides acceptable results for second-order materials but is no option for first-order materials. The revised model introduced by Bean and Rodbell (C. P. Bean and D. S. Rodbell, 1962) describes some properties of a first-order phase transition but as Neves Bez et al. (Neves Bez et al., 2016) point out, fails to fully describe the properties of first-order La(Fe,Mn,Si)13. Curve fitting combines the benefits of the two other approaches. It is also based on measured data but not as dependent on the data quality as data interpolation, because some of the information is already given by the choice of the fitting functions. For second-order materials, latest material models utilize for example the presence of a so-called master curve for the isothermal entropy change (Franco et al., 2006; Lei et al., 2016). This means, the curve only scales with applied magnetic field and does not change its shape. First-order materials do not show this master curve characteristic. For first-order materials, Benedict et al. introduced a set of equations to fit the adiabatic temperature change and the specific heat capacitance at zero applied magnetic field 1
(Benedict et al., 2016). These quantities can also be used to create a clearly defined state space, yet there is no analytical access to the heat capacitance as function of temperature and applied magnetic field and the change in entropy with the magnetic field δs/δµ0H, which are used for the built-in implementation of the magnetocaloric effect (Nielsen et al., 2011). The present work is based on the assumption that hysteresis can be neglected. This is obvious for second-order materials but requires some justifications for first-order materials, because first-order phase transitions are strongly associated with hysteresis (Basso et al., 2018; Gutfleisch et al., 2016). However, latest material researches show that materials can undergo a first-order phase transition with close to zero hysteresis (Zhang et al., 2015). Additionally, even if a material shows hysteresis at full phase change, it seems that some materials can be trapped into so called minor hysteresis loops when stimulated under cyclic conditions, whereby they do not undergo the full phase change (Gottschall et al., 2017; Kaeswurm et al., 2016). Moreover, even if the material shows hysteresis and it is not reduced by the presence of minor hysteresis loops, for system simulations it can still be beneficial to treat the material as if it was free of hysteresis and then add an additional source term to account for the dissipative losses due to hysteresis. In this work we present a material model for a first-order La(Fe,Mn,Si)13 based alloy. The model is based on the specific heat as function of temperature and applied magnetic field. All other quantities are derived from this base function to ensure thermodynamic consistency. The selection of the base function forms a compromise between good representation of measured data and analytical accessibility of the remaining material properties, to provide easy to implement equations for system simulations. The model parameters are determined from measured data.
2
2. Material model 2.1. First- and second-order phase change Magnetocaloric materials are classified according to the Ehrenfest classification into the order of the phase change they undergo. First-order materials show a discontinuous change of the magnetization M (figure 1 top left), while second-order materials show a continuous change of the magnetization around the phase change (figure 1 top right). The simple Ehrenfest classification is conceptionally explicit, however a clear determination of real magnetocaloric probes is still matter of discussion in the scientific community (Bratko et al., 2017; Law et al., 2018). Beside the continuity or discontinuity of the magnetization, another difference between first- and second-order materials is the different deformation of the peak of the specific heat capacitance as a function of the temperature at different applied magnetic fields. In first-order materials, the peak-temperature increases with larger applied magnetic fields, whilst the shape of the peak does not change drastically (figure 1 middle right). In second-order materials, with an increasing field the peak decreases and broadens without any significant shift of the peak temperature (figure 1 middle left). As indicated in the bottom line of figure 1, the magnetocaloric response of first-order materials saturates with large magnetic fields (Zverev et al., 2010) whereas second-order materials do not saturate (Gottschall et al., 2019). At moderate magnetic fields, as created by permanent magnet field sources, first-order materials generally show a higher magnetocaloric effect than second-order materials. As downsides of first-order materials, they generally only work in a smaller temperature range and are associated with hysteresis. One current goal of material scientists is to develop magnetocaloric materials at sweet spot between first- and second-order characteristics to get the best of both worlds. In the present work, we have placed our focus on the description of first-order materials.
Figure 1: Distinctions between a first-order (left) and second-order (right) magnetocaloric material via magnetization (top), specific heat capacitance (middle) and the adiabatic temperature change (bottom) as function of the temperature and for different magnetic 3
fields. Figure is inspired by (Kitanovski et al., 2015) p. 25 and (Smith et al., 2012).
2.2. Base equations and visualization The assumption of no hysteresis allows a clearly defined description by state variables. State variables are the temperature T, the applied magnetic field H and the entropy s. With the specific heat capacitance cH at a constant applied magnetic field the entropy is (
)
∫
(
)
1
When s(T,µ0H) is known, the material is clearly defined in a state space. This is represented in a sT-space by state trajectories for different applied magnetic fields. This means modelling cH(T,µ0H) for all possible µ0H gives a uniquely defined material model. In an ideal first-order material, the specific heat capacitance shows an infinite Dirac-like function in temperature at the phase change. However, for real magnetocaloric materials this peak is broadened, resulting in a finite peak function. The two material properties, isothermal entropy change ∆siso and adiabatic temperature change ∆Tad, are derived. The isothermal entropy change is the entropy change of the caloric material whilst the material is magnetized or demagnetized at a constant temperature. For a magnetization from zero field to µ0∆H the isothermal entropy change is given by (
)
(
)
(
)
2
To visualize this equation an exemplary material is introduced. Figure 2 shows cH/T of this first-order magnetocaloric material for µ0H = 0 T (black curve) and µ0∆H (green curve). By applying a magnetic field, the peak shifts to higher temperatures as measured in various studies (Basso et al., 2015; Burriel et al., 2014; Wang, 2012). An isothermal magnetization decreases the entropy of the material, an isothermal demagnetization increases it. The area below the curve from 0 K to a certain temperature T represents the entropy of the material at this temperature (equation 1). At this temperature the entropy at zero field corresponds to the area A and at µ0∆H to A+B. The entropy change is the difference of these areas and here given by A. The entropy change has a maximum in temperature at the crossing point of the two curves (orange dot in figure2). Furthermore, when assuming a constant base line as offset for the peak functions the maximum entropy change ∆siso,max at very high fields is given by the area of the peak function without offset.
4
Figure 2: Visualization of the isothermal entropy change set free by the magnetization of a first-order magnetocaloric material by using the temperature normalized heat capacitance curves at different applied magnetic fields. Besides the isothermal entropy change, the adiabatic temperature change is a very important material property of magnetocaloric materials. An adiabatic and reversible process is always isentropic (constant entropy). Since the material is assumed to have no thermal hysteresis and therefore shows no dissipative losses, the adiabatic temperature change is also an isentropic temperature change. This means, even after magnetization the area below the cH/T curve must be the same as before magnetization. This can be expressed by ∫
(
)
∫
(
)
3
This equation is visualized in figure 3. Without external magnetic field, the entropy at a certain temperature T is given by the sum of the areas A and B. When magnetized isentropically, the entropy is given by B+C. For an isentropic process A = C must be fulfilled. In other words, the area below the black curve is squeezed below the green curve, pushing the right boundary of the integral and therefor the material temperature towards a higher temperature.
Figure 3: Visualization of the adiabatic temperature change ∆Tad of a first-order magnetocaloric material. To keep the entropy constant during magnetization, the area below the black curve is squeezed under the green curve up to a certain temperature T. This shifts the right boundary of the area to the higher temperature T+∆Tad.
5
2.3. Empirical parametrization For an empirical description, an analytical peak function for cH/T at temperatures around the phase change is introduced. It is a Cauchy–Lorentz function given by (
)
4 (
(
))
where σ defines the width of the peak and T0 the position in temperature at zero applied field. The Cauchy-Lorentz function has an offset of c0/ T0 and an area under the curve without offset of ∆siso,max. With an applied magnetic field, the function shifts linear to higher temperatures. The shift in temperature is given by β, which can be interpreted as the inverse Clausius Clapeyron coefficient (Tušek et al., 2016).
Figure 4: Parametrization of Cauchy–Lorentz function for the temperature normalized entropy change cH/T. Figure 4 shows the graphical representation of equation 4. The area of the peak function does not change with the applied field, thus leading to a convergence of the state trajectories in the sT-space at higher temperatures. Thereby, the physical boundary condition that in the paramagnetic state the entropy of the material is not a function of the applied magnetic field is fulfilled. Inserting equation 4 into equation 1 gives the following expression for the relative entropy: (
)
(
(
) ) (
(
(
)
5
) )
For most numeric applications, only the relative entropy is of interest but not the absolute entropy. With a T0 of around 300 K and a µ0H of ~1 T the second term can be simplified to be constant in μ0H and T and therefore is ignored. Inserting 5 into 2 gives the analytic expression of the isothermal entropy change in expanded and simplified form of (
)
(
(
(
( (
6
)
)
(
)
)) (
(
)
)
(
))
( )
6
This function has a global maximum of ∆siso,max. Inserting equation 4 into equation 3 results in the following expanded and simplified expression for the adiabatic temperature change: (
)
(
( ) (
( (
(
(
)
(
)
7
) ))
)
(
(
)
))
Note that this is a recursive expression in ∆Tad and can only be solved numerically. As the maximum of the isothermal entropy changes, the maximum of the adiabatic temperature change also saturates with increasing applied magnetic fields. The value of saturation is ∆siso,max T0/c0, which is the area of the zero field cH/T curve without the offset divided by the offset. As final material parameter required for a thermodynamic constant material model δs/δµ0H is derived by differentiating equation 4: (
)
(
(
) (
(
(
(
))
)
8
))
This quantity is part of the magnetocaloric source term and an analytical description enables a relatively simple implementation into system simulations.
7
3. Methods to determine model parameters A LaFe11.5Mn0.24Si1.26 alloy was produced by powder metallurgy methods as described in (Katter et al., 2008). A powder blend was isostatically pressed and subsequently sintered at around 1100°C under inert conditions. The sample was milled down to a coarse powder and sieved between 125-250 µm. In order to adjust the transition temperature to around room temperature the sample was hydrogenated as described in (Barcza et al., 2011). The material belongs to the commercially available material family CALORIVAC H and has a density of = 7.2 g cm-3 and a thermal conductivity of = 6-8 W m-1 K-1. All measurements where carried out at the same sample. Quasi-adiabatic ∆Tad characterization was performed for different external magnetic field changes (0.8 / 1 / 1.25 / 1.5 T) using a bespoke device with a rotating magnet assembly. Field-dependent magnetization measurements as a function of temperature were performed with a commercial vibrating sample magnetometer. The data have been used to determine ∆siso at 0.4 / 0.8 / 1.2 and 1.6 T using the Maxwell relation. Zero-field heat capacity curves were obtained at a heating rate of 2.5 K/min in a commercial differential scanning calorimeter using the sapphire method. The parameters c0, ∆siso,max, σ and T0 are determined by fitting equation 4 to measured data of cH/T at zero applied magnetic field. Parameters c0, ∆siso,max, and σ determined in this way are taken as input parameters for the following analysis, whereas T0 is left as free parameter to account for different temperature offsets of the different measurement techniques. To determine the shift of the peak function per applied field β from ∆siso or ∆Tad a multivariable fit is used. The functions for ∆siso and ∆Tad have the variables T and H and the previously determined constants c0, ∆siso,max, and σ and the parameters β and T0. For the fits, the experimental data are weighted according to their relative quantity in cH/T, ∆Tad and ∆siso to make the fit curves more accurate close to the respective peaks.
8
4. Comparison with experimental data In Figure 5, the experimental data of cH/T without external field against the temperature are shown as black dots. There is an asymmetry of the slopes of the left hand side of the peak function compared to the right hand side. Compared the Lorentzian function there is also an additional turning point at the peak characteristic. Whether this is a measurement artifact or a physical effect is unclear. The present material model cannot take into account either the asymmetry or the additional turning point. The Lorentzian fit to the experimental data is shown as red line. Model parameters of c0 = 558 J kg-1K-1, ∆siso,max = 17.6 J kg-1K-1 σ = 0.8 and T0 = 310.0 K are determined.
Figure 5: Measured data of the temperature normalized heat capacitance (black) and fitted Lorentzian curve (red) are shown. The parameters c0, ∆siso,max, σ and T0 are determined. In figure 6, the measured data and the fitted model of the adiabatic temperature change ∆Tad for different applied magnetic fields is plotted against the temperature. Plot markers represent the measured data and the solid lines the fitted model. For the fit, the parameters c0, ∆siso,max und σ are taken from the cH/T fit. T0 is not taken over from cH/T to account for an offset of the measurement of the absolute temperature. The fit parameters are β = 3.97 K T-1 and T0 = 309.9 K. The model shows good agreement with the measured data close to the maximum of the according peaks. Next to the peaks, the model underestimates the measured data, especially at temperatures lower than T0.
9
Figure 6: Measured data and fitted model of the adiabatic temperature against the temperature at different applied magnetic fields. Figure 7 shows the measured data and the fitted model of the isothermal entropy change ∆siso at different applied magnetic fields against the temperature. With increasing applied magnetic field, the peak broadens and attains larger values. At higher fields, the measured data show a tilted plateau at the peak maximum. As in figure 6, β and T0 are determined as fit parameters. The parameters are β = 4.02 K T-1 and T0 = 308.9 K. T0 is about 1 K lower than the fitted values from the cH/T and ∆Tad data. This can be explained by a temperature offset of the ∆siso measurement. The determined value for β is consistent with the β-value from the ∆Tad data. All determined fit parameters from the different measurements are summarized in table 1. These results show good agreements with the values measured by Basso et al. (Basso et al., 2015).
Figure 7 Measured data and fitted model of the isothermal entropy change against the temperature at different applied magnetic fields. Table 1: Summery of determined fit parameters for the different sets of data Experimental data cH/T ∆Tad ∆Siso
10
c0 [J kg-1K-1] 558 -
∆siso,max [ J kg-1K-1] 17.6 -
σ [-] 0.8 -
T0 [K] 310 309.9 308.9
β [K T-1] 4.02 3.97
5. Further perspective The present model shows a good accuracy with built in consistency. In order to simulate the system efficiency of a magnetocaloric cooling device, the consistency of the model is much more important than a highly accurate representation of the material parameters, because it prevents unphysical results. More degrees of freedom for the heat capacitance increase the accuracy but due to the various integration and differentiation steps, the complexity of the model would significantly increase. The present model gives an analytical state description of the caloric material at any field and temperature, used in system simulations. The corresponding sT-diagram (equation 5) of the material is shown in figure 8. A high-field curve and a low-field curve span up state space. Any cooling cycle for a given high and low field takes place within this state space, and the state of the material can be traced at any time step. The area enclosed by such a state trajectory corresponds to the work that has to be applied to the magnetocaloric material in order to create a certain cooling or heating power. With the sum of those path integrals for each knot the exergetic efficiency of a given system can be determined.
Figure 8: Calculated relative entropy over the temperature. This defines the sT-state space of the magnetocaloric material. When simulating a cooling system it has to be considered that the phase transition of La(Fe,Mn,Si)13 materials approaches second-order type transition towards lower temperatures (Basso et al., 2015). In order to model a machine with a cascade of caloric material ranging from e.g. 40°C to 0°C it may be necessary to adapt the material model gradually from first- to second-order. By giving the model more degrees of freedom, the agreement with measured data will improve, but with the disadvantage of increased model complexity. For a more second-order material, a peak function for cH/T also should shrink and change its shape with an applied magnetic field. The concepts presented here can also be adopted for a material model for elastocaloric and electrocaloric materials.
6. Conclusion An empirical model for the mathematical description of a first-order magnetocaloric material has been presented. The model is based on a Cauchy–Lorentz function for cH/T. From this base equation, expressions for all relevant material properties where derived, resulting in a thermodynamically consistent material model tailored for system simulations. The model parameters were determined by 11
fits to experimental data of the specific heat capacitance at zero field, the adiabatic temperature change and the isothermal entropy change. The model is in good agreement with the experimental data and can be implemented in system simulations for cooling devices with first-order magnetocaloric materials.
7. Acknowledgments This work is funded by the Fraunhofer internal project MagCon, as part of the Sustainability Center Freiburg within the project ActiPipe and by the Federal Ministry of Economics and Energy (BMWi) of Germany within the project MagMed.
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