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A flexible numerical model of a multistage active magnetocaloric regenerator Michael G. Schroeder a,b,*, Ellen Brehob b a b

Advanced Development Group, General Electric Appliances Division, Louisville, KY, USA Department of Mechanical Engineering, University of Louisville, Louisville, KY, USA

A R T I C L E

I N F O

A B S T R A C T

Article history:

A flexible one-dimensional model was created in Python to determine the periodic steady

Received 4 December 2015

state cooling power of an active magnetocaloric regenerator. Several features of the model

Received in revised form 22 January

provide advantages when performing large parametric studies. Geometry, material prop-

2016

erties, and operation parameters are read in as input. Fluid flow and magnetization profiles

Accepted 26 January 2016

are parameterized, and thermodynamic consistency is forced in magnetocaloric material

Available online 25 April 2016

properties. A dynamic time step is used, and three associated model constants are set to work over a wide range of input parameter combinations. Derivative gain is adjusted to

Keywords:

minimize convergence time resulting in an average convergence time reduction of approxi-

Magnetocaloric

mately 50%. © 2016 Elsevier Ltd and IIR. All rights reserved.

AMR Magnetic refrigeration Numerical model Convergence acceleration Time step

Un modèle numérique flexible d’un régénérateur multiétagé actif magnéto calorique Mots clés : Magnéto calorique ; AMR ; Froid magnétique ; Modèle numérique ; Accélération de convergence ; Pas de temps

1.

Introduction

Magnetic heating and cooling at near room temperature is an emerging technology, which has the potential to replace vapor compression in many applications. Continued development of magnetic refrigeration technology is driven by efficiency, which may exceed that of vapor compression (Engelbrecht, 2008).

Magnetic cooling relies fundamentally on the magnetocaloric effect (MCE); an adiabatic temperature change due to a change in magnetic field strength. The magnitude of temperature change due to the MCE is a function of temperature and magnetic field strength, and is largest near a material’s Curie temperature. An example of a magnetocaloric material (MCM) is gadolinium, which has a maximum effect near room temperature.

* Corresponding author. Advanced Development Group, General Electric Appliances Division, Louisville, KY, USA. Tel.: +1 419 615 0054; Fax: +1 502 452 0512.. E-mail address: [email protected] (M.G. Schroeder). http://dx.doi.org/10.1016/j.ijrefrig.2016.01.023 0140-7007/© 2016 Elsevier Ltd and IIR. All rights reserved.

international journal of refrigeration 65 (2016) 250–257

Nomenclature

Abbreviations MCE magnetocaloric effect MCM magnetocaloric material AMR active magnetocaloric regenerator Variables A Area [m2] α Thermal diffusivity [m2/s] C Specific heat capacity [J/kgK] D Diameter [m] E Energy [J] ε Fluid fraction of the packed bed H Magnetic field strength [H] k Thermal conductivity or thermal dispersion coefficient [W/mK] L Length [m] M Ratio of volumetric heat capacities of fluid and MCM m Mass [kg] N Number or count P Pressure [Pa] Pe Peclet number Pr Prandtl number Re Reynolds number ρ Density [kg/m3] S Entropy [J/kgK] T Temperature [K] t Time [s] U Heat transfer coefficient [W/m2K] V Volume [m3] v Velocity [m/s]

In order to harness the magnetocaloric effect, a secondary material is often introduced as a heat carrier. Because the MCE is only a few degrees per K per Tesla of applied field, most magnetic refrigeration machines employ a regenerative cycle to increase temperature span. The interface between MCM and heat carrier in regenerative machines is referred to as an active magnetocaloric regenerator (AMR). Many reviews summarize the operating principles as well as details of constructed machines (Yu et al., 2010), (Gomez et al., 2013), (Kitanovski et al., 2014). The simplified AMR cycle as described by (Brown, 1976) consists of four segments: 1. Adiabatic magnetization; MCM temperature raises to high level. 2. Constant magnetization heat transfer; fluid is displaced. Fluid enters from the cold side, and heated fluid exits the hot-side. 3. Adiabatic demagnetization; MCM temperature falls to lower level than step 1. 4. Constant magnetization heat transfer; fluid is displaced in the opposite direction. Ambient temperature fluid enters from the hot side, and cooled fluid exits the cold side. There have been many studies performed on AMR systems since 1976, and magnetocaloric technology continues to develop

251

(Gschneider and Pecharsky, 2008). A major branch of the magnetocaloric field of study is machine and cycle simulation. An accurate model can allow for much better understanding of the complex challenges in achieving cooling power and efficiency in a real machine. Many numerical models have been published, utilizing similar fundamental equations: Engelbrecht (2008), Roudaut et al. (2011), Aprea and Maiorino (2010), Li et al. (2011), Ivan(2012), Risser et al. (2013), Tagliafico et al. (2013), and Govindaraju et al. (2014). Most of these place an emphasis on modeled physics, which are extremely important for model accuracy. Previous numerical models typically utilize either a constant MCE or MCE as a function of temperature at the simulation field strength. Data for the MCE come either from direct MCE measurements or analytical material models for second order materials such as Gadolinium. Accurate mathematical models of first order materials are not readily available. There is also a lack of measured MCE at multiple field strengths. One objective of the presented model is to scale measured MCE by magnetic field strength and force thermodynamic consistency, eliminating the need for measured data at multiple field strengths. A thermodynamically consistent model helps eliminate physically impossible solutions that come about from MCE measurement error. The presented model also seeks to facilitate large design studies by reducing computation time and maximizing model robustness to large parameter ranges by using convergence acceleration and a variable time step. AMR models often break the cycle into four basic segments, while the presented model treats the cycle as a continuous process subjected to magnetic and fluid flow profiles. Because of this the model is able to represent non-ideal cycles that occur in real machines, where cycles may not be symmetric or have clearly defined segments. If a cycle is treated as separate segments with various simplifications explicit solutions can be made to further speed up a model (Torregrosa-Jaime et al., 2015). This is not the case in the presented model, as flexibility is a primary concern.

2.

One-dimensional AMR model

2.1.

Model inputs and outputs

Fluid cycles are defined in the model input file by event start times, ramp times, and peak durations and magnitudes. Magnetic profiles are defined as a square profile with a start and stop time. An example flow profile can be observed in Fig. 1. Cycles are defined this way to enable parameterization of flow and magnetic profiles. Flow can be modeled as anything from step profiles to fully triangular profiles to any custom profile input into the model via a read-in file. The decision was made to use a square magnetic profile in order to reduce overall calculations required for the simulation to complete. Nonsquare profiles can be estimated be using an average amplitude. MCM thermal conductivity and density are considered constant, as are fluid thermal conductivity, density, viscosity, and heat capacity. Geometry parameters such as cross sectional area, regenerator length, fluid fraction, and characteristic length are also included in the input file. Finally, the input file provides the location of MCE data file(s) to be read into the program.

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Fig. 1 – An example fluid velocity profile as defined by ramp start times, ramp durations, peak durations, and peak magnitudes.

Hot and cold side temperatures are treated as boundary conditions. Single cycle total rejected and accepted heat are calculated.

2.2.

Magnetocaloric material properties

Magnetocaloric properties are read into the program as arrays of adiabatic temperature rise on magnetization and heat capacity at zero magnetic field, both as a function of temperature. Adiabatic temperature rise is then scaled to the proper peak magnetic field strength using 2/3 scaling,

⎛H ⎞ ΔTadiabatic = ΔTadiabatic ⎜ 2 ⎟ ⎝ H1 ⎠ 0−H2 0 − H1

23

(1)

where H2 is the simulated field strength in Tesla, H1 is the input data field strength, ΔTadiabatic is the input adiabatic tempera0 − H1

ture rise data, and ΔTadiabatic is the scaled adiabatic temperature 0−H2

rise. 2/3 scaling has been shown to be a good thermodynamic approximation for second order materials such as Gadolinium, and can still be used under certain conditions on first order materials with minimal error (Smith et al., 2014). Entropy at each temperature point is calculated using the definition of heat capacity,

∂S =

CMCM (T ) ∂T T

multistage studies. Using multiple material files provides superior predictions specific to a single batch of materials, and also allows for material property sensitivity studies. Alternatively, a single material data file can be read into the program and temperature shifted to generate multiple material stages. The single material file method is ideal for high level machine design studies, where material to material variation is unwanted noise. This way the average impact of machine parameters can be understood independent of material batch variation.

2.3.

Assumptions

In order to maintain model simplicity the following assumptions are made: 1. 2. 3. 4.

All materials are incompressible Fluid properties are constant with temperature and pressure MCM density and thermal conductivity are constant. MCE is applied as an instantaneous change in temperature, and is isentropic 5. The ends of the regenerator are adiabatic and the only energy exchanged with the hot and cold reservoir occurs via mass transfer 6. There is no radial heat leak to the environment 7. The AMR container does not interact with the AMR fluid or MCM

2.4.

Governing equations

(2)

where CMCM is low field heat capacity, T is absolute temperature. The lowest point on the data temperature scale is initialized with S = 0. In order to force thermodynamic consistency total entropy during magnetization and demagnetization is held constant, following a method proposed by (Pecharsky et al., 2001). High field entropy is calculated by shifting the low field entropy by the adiabatic change in temperature. This method assumes that the MCE is both adiabatic and reversible. For example, at 300K a 3K adiabatic temperature rise is observed in a material on magnetization. The low field entropy at 300K must then match the high field entropy at 303K for the transition to be isentropic. Adiabatic temperature change on demagnetization can be found from low and high field entropy in a similar fashion. MCE data can be input into the model in two ways, each of which provides a different insight into performance. Multiple material stage data files can be input to perform cascaded

One MCM node and one fluid node are represented at each axial location. A basic schematic of interactions is shown in Fig. 2. Only three node locations are shown for simplicity. Not shown in the figure, but included in the model is the application of viscous heating and the MCE. In a real machine, the MCE is magnetic work coming from the magnetization system. Viscous heating is applied as a generation term in the fluid nodes during fluid flow, and corresponds in a real machine to fluid pumping power. When combined with node thermal mass a set of equations can be used to calculate rate of temperature change of each node due to all effects combined. Fluid node temperature changes are calculated using

∂Tfluid node

⎛ ⎞ ⎜ ⎛ ∂T ⎞ ⎟ ∂T ⎞ fluid ∂T ⎞ ∂T ⎞ ⎛ ⎛ ⎛ + =⎜ + ⎟ ∂t fluid + mass ⎝ ∂t ⎠ MCM − f ⎝ ∂t ⎠ axial ⎝ ∂t ⎠ viscous ⎟ ⎜ ⎝ ∂t ⎠ trans . heating heat ⎜⎝ disp. ⎠⎟ trans.

(3)

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253

Fig. 2 – Modeled interactions between nodes within the AMR, as well as interactions between the AMR and the hot and cold reservoirs.

where ΔPnode is pressure change across a node length using a version of the correlation suggested by Ergun and Orning (1949).

Solid node temperature changes are calculated using

∂TMCM node

⎛ ⎞ ⎜ ⎛ ∂T ⎞ ∂T ⎞ MCM ⎟ ⎛ =⎜ MCM + ⎟ ∂t + ∂TMCE MCM − f ⎜ ⎝ ∂t ⎠ axial ⎝ ∂t ⎠ heat ⎟ ⎜⎝ ⎟⎠ cond.

(4)

2.4.2.

trans.

2.4.1.

MCM nodal temperature change rate components

Temperature change rate of MCM due to axial conduction is defined as

Fluid nodal temperature change rate components

Next the individual components are defined. Fluid temperature change rate due to mass transfer during flow in the positive “x” direction is defined as

kMCM ⎛ ∂T ⎞ MCM = − 2TnMCM + TnMCM −1 ) (T ⎝ ∂t ⎠ axial L2node ρMCMCMCM nMCM + 1

vsf ⎛ ∂T ⎞ = − Tn f ) T ⎝ ∂t ⎠ mass Lnode ε ( n f + 1

where kMCM is MCM thermal conductivity, ρMCM is MCM density, and CMCM is MCM heat capacity. Temperature change rate of MCM due to MCM to fluid heat transfer is defined as

(5)

trans.

where vsf is superficial fluid velocity, Lnode is node length, ε is fluid fraction of the packed bed, Tn f is current node temperature, and Tn f +1 is adjacent node temperature (see Fig. 2). During flow in the negative “x” direction Tn f +1 is replaced with Tn f −1 . Temperature change rate of fluid due to MCM to fluid heat transfer is defined as

⎛ ∂T ⎞ fluid = UAMCM − f T − Tn f ) ⎝ ∂t ⎠ MCM − f ρ f VnodeC f ε ( nMCM heat trans.

(6)

where ρ f is fluid density, Vnode is total node volume, TnMCM is temperature of the co-located MCM node, UAMCM − f is absolute thermal conductance between MCM and fluid in W/K, and C f is fluid heat capacity. Temperature change rate of fluid due to axial dispersion is defined as

keff ⎛ ∂T ⎞ fluid = − 2Tn f + Tn f −1 ) T ⎝ ∂t ⎠ axial L2node ρ f C f ε ( n f + 1

(7)

(9)

cond.

UAMCM − f ⎛ ∂T ⎞ fluid = T − TnMCM ) ⎝ ∂t ⎠ MCM − f ρMCM VnodeCMCM (1 − ε ) ( n f heat trans.

2.4.3.

(10)

Correlations and supporting equations

Heat transfer coefficient multiplied by area for a given AMR volume is calculated using a relationship based on the work of Wakao et al. (1979),

UAMCM − f =

6 (1 − ε ) ⎞⎤ ⎡ ⎛ ⎢ ⎜ 1 1 ⎟⎥ 2 ⎢ Dp ⎜ + ⎟⎥ 1 6kMCM ⎟ ⎥ ⎢ ⎜ ⎛ 2 1 1 0 .6 3 ⎞ + . Re Pr k f ⎜ ⎜ ⎟ ⎢⎣ ⎝ ⎝ ⎠⎟ ⎥⎦ ⎠

(11)

where Dp is particle diameter, Re is Reynolds number based on Dp , and Pr is Prandtl number. Axial dispersion coefficient in equation 7 is calculated assuming Stokes flow (Koch and Brady, 1985),

disp.

where keff is axial dispersion coefficient. The final fluid component is viscous heating, defined as

ΔPnode vsf ⎛ ∂T ⎞ = ⎝ ∂t ⎠ viscous Lnode ρ f C f ε heating

(8)

keff = 1 +

α 3 1 1 1 2 Pe + π 2 (1 − ε ) Pe ln ( Pe) + (1 − γ )2 f Pe 4 6 15 α MCM M

(12)

where Pe is Peclet number based on particle radius, M is the ratio of volumetric heat capacities of the fluid and MCM, α is thermal diffusivity, and

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(1 − ε ) ⎛⎝ 1 −

1⎞ ⎠ M γ = 1⎞ ⎛ 1 − (1 − ε ) 1 − ⎝ M⎠

2.5.

∂t max = (13)

Time stepping

Model stability and accuracy are both closely tied to time step size when an iterative solution is calculated. Using too large of a time step will create overshoot and rebound, while using too small of a time step will increase computation time without improving accuracy. In the model presented, three criteria are used to determine time step size: temperature change rate, mass transfer rate, and minimum cycle resolution. Error is largely a function of temperature change per time step, and it accumulates as additional time steps are simulated. Using maximum permissible change in temperature across a single time step as a control for time step size,

∂t max ( x ) = ΔT

∂Tmax ∂T (x ) ∂t

(14)

where ∂Tmax is the set maximum temperature change across a single time step and ∂∂Tt ( x ) is local temperature change rate calculated by the governing equations. A second criterion for determining time step is based on fluid velocity. An implicit assumption in the application of the governing equations is that exchanged mass during a single time step is less than the mass of a fluid node. In order to ensure that this holds true, a maximum fluid exchange fraction is specified to yield another time step limit,

⎛ m ⎞L ε ∂t max = ⎜ ex ⎟ node ⎝ mnode ⎠ vsf vel

(15)

where mex is fluid mass exchanged in a single time step and mnode is the mass of a fluid node. Finally, a third criterion for determining time step is created to avoid losing cycle resolution. It is enforced in the form of a global maximum time step, and is determined by total cycle time,

res

t cycle N

(16)

where t cycle is total cycle time and N is the number of increments needed to fully define a cycle profile. Minimum N depends on the complexity of the fluid flow profile. The minimum time step specified by the three criteria is used for each time step. Time steps calculated throughout the first cycle are stored and repeated in subsequent cycles until convergence is reached in order to avoid instability.

2.6.

Convergence acceleration

During the convergence of a cyclical model, the temperature profile at the beginning of the cycle is not equal to the temperature profile at the end of the cycle. By initializing the subsequent cycle with the end-of-cycle temperature profile, a natural progression to periodic steady state is established. This represents the progression of an AMR to steady state in a real experiment with matching initial conditions. If the only output of interest is the steady state condition, the path to steady state becomes irrelevant and convergence can be accelerated. The implementation of acceleration is shown in Fig. 3. Subscripts of “i” and “i + 1” indicate the most recent and upcoming cycle iterations, respectively, and subscripts of “0” and “final” indicate initial and final states within a cycle iteration, respectively. As shown in Fig. 3, convergence acceleration is applied on every other cycle in the form of derivative gain until convergence criteria are met. Convergence is only checked when gain has not been applied. Energy exchange at both the cold and hot sides of the AMR are used as a comparison to assess convergence. If the difference in energy exchange between cycles is less than a limit value for both the hot and cold sides the model is considered converged. The method of convergence acceleration is best visualized by looking at the output of interest, normalized cooling power, versus cycles simulated, shown in Fig. 4;where normalized cooling power is

Fig. 3 – Flowchart of the convergence process with acceleration. K d represents the gain parameter used to decrease convergence time.

international journal of refrigeration 65 (2016) 250–257

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Fig. 4 – A depiction of model output power as a function of cycles simulated, with and without convergence acceleration.

E cooling current Ecooling = * Ecooling

(17)

max

In Fig. 4, the total number of cycles simulated corresponds to the cumulative length of the solid lines in the horizontal direction. Convergence occurs when the slope reduces to a set value.

3.

Results and discussion

A high resolution version of the model was created with 10000 time steps and 1000 nodes. The output of this version of the model serves as a reference point when determining the accuracy of any combination of time step control parameters. This will be considered the zero error case moving forward. For validation, the high resolution code was then benchmarked against real machine performance in a rotary machine (Engelbrecht et al., 2012). The high resolution model was compared to experimental measurements of the rotary machine (Lozano et al., 2014), resulting in less than 20% error to measured data in all cases. Twenty percent error is comparable to results obtained by Lozano et al., who also performed numerical predictions on all cases with a model based on Engelbrecht (2008). Thus, the high resolution case should be

a good comparative point for determining the error resulting from any combination of control parameters in the simulation. Gadolinium data for the simulation was taken from Dan’kov et al. (1998) at 2 Tesla field, scaled and repopulated internally via equations 1 and 2, and used by the code to produce these results. This shows the applicability of 2/3 scaling, and that the code can give good high level design guidance related to the effect of field strength. As suggested in the time stepping section, a decrease in time step size can produce a reduction in error. Further reduction in time step size yields diminishing returns; shown in Fig. 5. A minimum of 400 time steps per cycle is required for 5% or better accuracy when using a constant time step in the case shown. Minimum time step requirements will differ depending on the parameters being modeled. This provides a good starting point for setting constants for equations 14, 15, and 16. A series of gadolinium simulations was run using different levels of each constant. A full factorial design of experiments was simulated, equivalent to 6750 runs, using the following parameters. Regenerator length is held constant at 50 mm, cycle speed at 5 Hz, and fluid displacement at 20% of the contained regenerator fluid volume. Number of stages is varied to ensure that any discontinuities in temperature profile at stage interfaces are accurately simulated. Simulated span has a large impact on axial thermal effects, and also must be varied. Number of nodes or nodes per stage must be varied since

Fig. 5 – Cooling power error between modeled and high resolution cases for a single gadolinium stage at 12 K span, using a constant time step, for 300 nodes. Error axis is log-scaled.

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Table 1 – Variables with their respective minimums, maximums, and number of levels simulated. Eq.

Parameter

16 17

∂Tmax ⎛ mex ⎞ ⎝⎜ mnode ⎠⎟ N Number of nodes Span Stages Particle diameter

18 – – – –

temperature change rate due to axial effects varies with node length (Equations 7 & 9). Particle diameter is varied because of its direct effect on fluid thermal dispersion (Equation 12) and MCM to fluid heat transfer (Equation 11). Varying these 4 parameters changes the relative impact of all effects, which should exercise the control parameters effectively. The full range and number of levels simulated for all operating and time step control parameters can be seen in Table 1. As control parameters forced smaller time steps, output should approach the high resolution case. For development of a tool to use for parametric studies an error of 5% or less was selected as the acceptable range. The worst stability and highest error was consistently observed with a single stage at 12 K span, therefore span and number of stages were held constant at this point to simplify analysis. Minimum number of nodes is observed to be 300 for up to 6 stages, but more should be added with additional stages or regenerator length to maintain spatial stage definition. Additional nodes should also be added for extremely low fluid displacements which only interact directly with the ends of the regenerator. These two considerations could be used to generate new control parameters for future simulations. Applying a cutoff limit of 5% error to the remaining parameter space yields a smaller range of possible constants for equations 14, 15, and 16. A final output of interest is total simulation time, which should be minimized. Sorting the remaining parameter space by total simulation time yields a final combination of parameters, shown in Table 2. It should be noted that these parameters work well for second order material profiles similar to gadolinium. First order materials often exhibit much more dramatic changes in MCE with temperature, and will likely require a smaller time steps.

Min

Max

0.01 0.05

1 0.4

100 100 0 1 100 μm

1600 300 12 K 6 1000 μm

Levels 5 5 5 3 2 3 3

The cycle resolution limit of 400 (Equation 16) was calculated using a fluid profile containing 9 changes in velocity versus time slope. Derivative gain is optimized using the same parameter range for span and number of stages. The only output parameter of interest is total simulation time, which changes proportionally with number of cycles simulated to convergence. The effect of derivative gain on cycles to convergence is shown in Fig. 6. Increasing gain successfully decreases the number of simulated cycles up to 4.5. Beyond this point overshoot becomes too large, the model fails to converge, and the simulation is stopped by the maximum cycles limit (400 cycles). When compared with the non-accelerated model, a derivative gain of 4.5 decreases simulation time by more than 50% in many cases.

4.

Summary

A flexible one-dimensional model was created to determine periodic steady state cooling power of an AMR. Model inputs include fluid velocity profiles, magnetic field timing and magnitude, AMR geometry, and material properties. MCE data is derived from measured data at one arbitrary field strength. A variable time stepping method is implemented utilizing three criteria, which force stability. Constants are tuned to work over a wide range of input parameter combinations. Convergence acceleration is applied as a derivative gain value, and is tuned to minimize simulation time for a range of input parameters. Average observed convergence time reduction is approximately 50% with acceleration. The model can now be used for future high level screening studies, where.

Fig. 6 – Gain versus simulated cycles to convergence for a parameter space.

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Table 2 – Final parameter selection, optimized for model stability, accuracy, and computation time. Eq.

Parameter

16 17

∂Tmax ⎛ mex ⎞ ⎜⎝ ⎟ mnode ⎠ N Minimum number of nodes Minimum nodes per stage

18 –

Value or limit 0.1 0.2 400 300 50

Acknowledgements The authors would like to acknowledge the support of General Electric throughout this project. REFERENCES

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