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Validation of an active magnetic regenerator test apparatus model J. Dikeos, A. Rowe* Department of Mechanical Engineering, Institute for Integrated Energy Systems, University of Victoria, PO Box 3055 STN CSC, Victoria, B.C., Canada V8W 3P6
article info
abstract
Article history:
This work focuses on the development and validation of a transient one-dimensional
Received 10 September 2012
numerical model of an active magnetic regenerator (AMR) test apparatus. Simulation
Received in revised form
results are validated by comparison to room temperature experiments for varying hot heat
13 November 2012
sink temperature, system pressure, and applied heat load. Three different second-order
Accepted 3 December 2012
magnetocaloric materials are used. In addition to external heat leaks, parameters such
Available online 12 December 2012
as thermal conductivity, Curie temperature, and peak magnetocaloric effect are adjusted to obtain better fits to experimental results. In the case of gadolinium, where material
Keywords:
properties are well-characterized, the inclusion of parasitic heat leaks as well as an
Magnetic refrigeration
increase in diffusivity resulted in good fits across a broad range of operating conditions.
Simulation
Adjustments to Curie temperature and peak magnetocaloric effect produced good matches
Magnetocaloric effect
with experimental data for Gd0.85Er0.15. Predictive simulations of a Gd e Gd0.85Er0.15 two-
Active magnetic regenerator
layer regenerator are briefly discussed. ª 2012 Elsevier Ltd and IIR. All rights reserved.
Gadolinium
Validation d’un mode`le de banc d’essai utilise´ pour tester les re´ge´ne´rateurs actifs magne´tiques Mots cle´s : froid magne´tique ; simulation ; effet magne´tocalorique ; re´ge´ne´rateur actif magne´tique ; gadolinium
1.
Introduction
Magnetic cycles are being considered for room-temperature refrigeration, heat pumps, and gas liquefaction systems. These devices employ the magnetocaloric effect (MCE), a reversible temperature change that can be induced in a magnetic material through the application or removal of a magnetic field. The MCE is generally a strong non-linear function of temperature and is largest in magnitude when
a material is near its magnetic ordering temperature, also known as its Curie temperature. Gadolinium, Gd, a rare-earth metal, is the most thoroughly studied material for its MCE and the standard by which other materials are compared for room-temperature applications (Pecharsky and Gschneidner, 2002). However, the maximum value of the MCE is limited to about 2 K T1 at the Curie point and decreases as temperature changes. For temperatures above w20 K, temperature spans are improved by implementation of the active magnetic
* Corresponding author. Tel.: þ1 250 721 8920. E-mail address:
[email protected] (A. Rowe). 0140-7007/$ e see front matter ª 2012 Elsevier Ltd and IIR. All rights reserved. http://dx.doi.org/10.1016/j.ijrefrig.2012.12.003
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Nomenclature AMR AMRTA FEM MCE Af Ad Ap B cf cs cs,ref D Dd Dh Deq fo G h hp jH k Ks keff kstatic L lp m _ m
Active Magnetic Regenerator Active Magnetic Regenerator Test Apparatus Finite Element Modeling Magnetocaloric Effect Fluid flow area (m2) Displacer piston surface area (m2) Particle surface area (m2) Applied magnetic field intensity (T) Fluid heat capacity (J kg1 K1) Solid heat capacity (J kg1 K1) Reference solid heat capacity (J kg1 K1) Tube diameter (m) Dispersion coefficient Hydraulic diameter (m) Equivalent spherical diameter (m) Effective conductivity constant Mass velocity (kg m2 s1) Convection heat transfer coefficient (W m2 K1) Particle height (m) Colburn j-factor Conductivity (W m1 K1) Dimensionless solid conductivity parameter Effective conductivity (W m1 K1) Static conductivity (W m1 K1) Sub-domain length (m) Particle length (m) Mass (kg) Fluid mass flow rate (kg s1)
regenerator (AMR) concept. An AMR acts as both the working material (or refrigerant) and the heat transfer medium. A temperature gradient is developed along the length of the AMR bed such that each location within the bed undergoes a slightly different cycle. For high efficiency to be realized, optimum materials, regenerator design, and cycle parameters must be determined. Due to the broad range of variables impacting AMR performance, simulation tools with predictive capabilities are needed. One area where simulations can be particularly useful is with the optimization of multi-material AMRs. Varying the amount of each material experimentally can be tedious and is an area where a validated modeling tool would be useful. A reliable modeling tool will help optimize parameters such as system mass flow rate, operating frequency, AMR geometry, aspect ratio, and material composition. The purpose of this work is to develop a model that can accurately simulate the operation of an AMR test apparatus (AMRTA) developed at the University of Victoria (Rowe, 2002). An important step in developing a predictive tool is validation of simulation results with experimental results.
2.
Model background
As summarized in by Nielsen et al. (2011), a number of simulation tools to model AMR refrigeration systems have been
NTU Pe Pr Q_ load Q_ rad r Re T TC TH TN t* Vp v wp x x* a ao DTad ε k m s s f fref fs j
Number of Transfer Units Peclet number Prandtl number Rate of heat load input (W s1) Rate of radiative heat input (W s1) Radius (m) Reynolds number Temperature (K) Cold temperature (K) Hot heat sink temperature (K) Temperature of surroundings (K) Non-dimensionalized time Particle volume (m3) Velocity (m s1) Particle width (m) Space (m) Non-dimensionalized space Porosity Effective conductivity constant Adiabatic temperature change due to magnetocaloric effect (K) Emissivity Thermal capacity ratio Dynamic viscosity (kg m1 s1) StefaneBoltzmann constant Blow duration/period (s) Utilization Reference utilization Sphericity Symmetry
developed (Trevizoli et al., 2012; Li et al.,2008; Engelbrecht et al., 2005; Rowe, 2002; Smaı¨li and Chahine, 1998; Carpetis, 1994; Spearing, 1994; DeGregoria, 1992; Matsumoto and Hashimoto, 1990). To reduce computational time, simulations of this type are often one-dimensional analyses. Further, except for a small number of models (Spearing, 1994), the model domain typically encompasses only one AMR. In this case a temperature span is imposed across the AMR and the resulting cooling power is calculated. Other simulations approximate the operation of an AMR refrigerator with a simplified model such as a magnetic Brayton cycle. Stepping and ramping functions are typically used to model the application of fluid blow and magnetization. Furthermore, all models make simplifications to reduce numerical complexity and computational time. These may include neglecting axial conductivity, dispersion effects, and void space thermal mass in addition to simplified properties. Few works have been thoroughly validated with experimental results from a magnetic refrigeration apparatus using multiple alloys over a broad range of operating conditions. Although the model presented in this paper is also based on a one-dimensional analysis, it differs from previous work in several ways. The model domain encompasses two AMRs and a cold space between them. This mimics the operation of a magnetic refrigeration apparatus since it requires only the hot heat sink boundary temperature and allows for a temperature span to develop across each of the regenerators based on the magnitude of heat absorbed in the cold section. Heat loads
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can be imposed and the resulting temperature span predicted. The model also employs sinusoidal variations of magnetic field and mass flow rate which occur simultaneously and closely approximate the AMRTA. The most important difference between this work and previous models is the use of extensive experimental data from three different magnetocaloric materials to validate results. Previously, the model was used to show AMR performance using Gd and DyAl2 (Dikeos and Rowe, 2006). The AMR test apparatus is composed of a cylindrical superconducting magnet, a composite cylinder housing two regenerators, and a piston-cylinder arrangement (Arnold et al., 2011; Tura et al., 2006). The piston-cylinder configuration is responsible for driving the regenerators in and out of the magnetic field. The AMRs in the cylinder are set a distance apart from each other so that they are out of phase with regards to the magnetic field applied to them. The applied field is in the x-direction, as shown in Fig. 1. While one regenerator is being subjected to the maximum intensity field in the center of the magnet, the other is at its furthest distance from the center of the magnet and is, therefore, subject to the minimum possible magnetic field intensity. Using two regenerators in this way allows for a cold section to be created between them. Resistive heaters between the AMRs are then used to mimic the application of a particular cooling load. The movement of the cylinder containing the AMRs is coupled to a displacer that acts to pulse the fluid flow through the regenerators in alternating directions. Helium is used as the heat transfer fluid for the work presented here.
3.
Model description
The model domain is intended to represent the cylinder of the AMRTA in its entirety. As such, it is composed of three distinct sub-domains. Two of these represent AMR beds while a third, central sub-domain is used to model the area between the regenerators. A heat source can be added to this region to
Top Regenerator
simulate a cooling load. Fig. 1 shows a cross-sectional view of the cylinder as well as the one-dimensional model domain used to represent it. Sinusoidal representations of the variation of magnetic field intensities and fluid flow are used in the implementation of this model to closely approximate the actual machine dynamics. Two sets of governing equations characterize the transient temperature profiles of the heat transfer fluid and the solid structure. Several assumptions are made in the development of the governing equations in order to simplify the analysis and remove terms of lesser importance. These assumptions are: 1. A one-dimensional analysis is sufficiently representative. This assumes a fully-developed flow and uniformity in the radial direction. 2. The regenerator matrix is homogeneous, stationary, and has a uniform porosity and cross-sectional area 3. The mass flow rate is a function of time only and the amount of fluid entrained in the matrix and cold section is constant 4. Viscous dissipation is negligible in the energy balance 5. The magnetic field intensity is constant across the regenerators with negligible eddy-current dissipation 6. Internal temperature gradients in the solid are negligible 7. The heat transfer fluid is incompressible 8. Thermal diffusion between solid particles is negligible The magnetic field applied to each of the regenerators at each point in time is assumed to be constant since the length of each of the AMRs is small compared to the sweep length of the entire cylinder. Eddy-current dissipation is neglected since it is typically small in particle bed regenerators where current loops are minimized due to the point contacts between particles and small particle size. Viscous dissipation is often neglected in regenerator models since this quantity is often relatively small compared to other terms in the governing equations (Nielsen et al., 2011). Based on these assumptions, the reduced governing equations for the temperatures of the heat transfer fluid and regenerator solid in the AMR sections, respectively, are (Dikeos, 2006):
Spacer
Heater
Phenolic Insert
Bottom Regenerator G-10 Flange G-10 Bearing Pad
AMR 1
COLD SECTION
k
vTf vTf v fref vTf þ NTUf Ts Tf ¼ f þ j vt vx vx Pef vx
vTs dðDTad Þ ¼ NTUf Tf Ts þ vt dt
G-10 Cylinder
AMR 2
x
Fig. 1 e A cross-sectional view of the cylinder containing two AMR beds and the one-dimensional model domain that is used to represent it.
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(1)
(2)
where, Tf and Ts represent the fluid and regenerator temperatures, and DTad is the adiabatic temperature change incurred as a result of the magnetocaloric effect. The NTU, or Number of Transfer Units, is a well-known grouping that is a measure of the heat transfer effectiveness. The parameters f and k are thermal capacity ratios; f, also known as the utilization, is the ratio of fluid thermal capacity rate to refrigerant thermal capacity rate, while k is the ratio of refrigerant thermal capacity to void space fluid thermal capacity. Pef, a form of the Peclet number, relates energy transfer through advection to that through heat conduction. Lastly, j, is the inverse nondimensional specific heat. The dimension of the governing
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differential equations is temperature. These dimensionless parameters for the AMR sections are defined as follows: hAw NTUh _ f mc
(3)
_ fs mc ms cs
(4)
_ fs mc fref h ms cs;ref
(5)
mf cf kh ms cs
(6)
_ fL mc Pef h keff Af
(7)
cs;ref jh cs
(8)
fh
_ >¼ 0; m
where, h is the convection heat transfer coefficient, Aw is the _ is the fluid mass flow rate, cf is the total wetted area, m constant-density specific heat of the fluid, s is the blow period or the length of time of each blow, ms is the mass of the solid matrix, cs is the solid heat capacity at a constant applied magnetic field, cs,ref is a reference solid heat capacity, the peak zero-field value, fref is a proxy for a reference parameter known as utilization, mf is the mass of the fluid entrained in the regenerator matrix, L is the characteristic length or the length of each of the regenerators, and keff is the effective conductivity for the combination of the solid and the fluid. The use of the last parameter is discussed in more detail in the material properties and correlations section. Further, in order to generalize the analysis and application of the model, the time and space variables, t and x, are non-dimensionalized through the use of the following definitions: t t h ; s
x h
x L
(9)
The governing equations for the heat transfer fluid and solid in the cold section between the regenerators, are similar to Equations (1) and (2), but without the magnetocaloric effect term: k
section tubing, while cs is the heat capacity of the cold-section tubing. Unlike in the regenerator solid equation, the transient storage term in the solid cold section is neglected. By omitting this term, the rate of convergence is improved and this assumption has a negligible effect on the periodic steady state condition that is achieved. However, without the energy storage term and with the radiative term in the solid equation, sharp gradients are created at the boundaries between the cold section and the AMR sections. To correct this, the radiative load is applied to the fluid equation instead. Numerically determined temperature spans are found to be more consistent with heat load experiments when this approach is used (Dikeos, 2006). The following periodic boundary conditions are assumed:
vTf vTf v fref vTf þ NTUf Ts Tf ¼ f þ j vt vx vx Pef vx Q_ load þ Q_ rad s þ ms cs
v vTs j Ks þ NTUf Tf Ts ¼ 0 vx vx
(10)
vTf x¼0: ¼0 vx x ¼ 3L : Tf ¼ TH
(11)
(12)
where, ks is the tubing conductivity. In the cold section, As is the contact area between the heat transfer fluid and the cold
(13)
(14)
where, a positive mass flow rate indicates fluid flow in the positive x direction. These conditions are set such that the fluid temperature at the inlet boundary is equal to the hot heat sink temperature while the boundary condition at the outlet of the model domain specifies that there is no heat flux at this point. The initial temperature throughout the computational domain is the hot heat sink temperature represented by TH.
4.
Material properties and correlations
4.1.
AMR sections
The density of helium is calculated using the ideal gas equation. Other helium properties are obtained using the National Institute of Standards and Technology’s (NIST) database of thermophysical properties, which is available online. Curve fits of this data allow for interpolation of the properties to be made at any given temperature. Other than density, all helium properties are assumed to be independent of pressure. This is a reasonable assumption considering that values of specific heat, conductivity, and viscosity are found to vary by less than 0.4% over the pressure range being considered. The heat transfer coefficient within a porous bed is determined using the following relationship (Barron, 1999): h ¼ jH Gcp Pr2=3
where, Q_ load represents a distributed heat load input across the cold section used to mimic operation with a particular cooling power, Q_ rad represents radiative heat transfer between the surroundings and the outside wall of the cold section tubing. Ks is a non-dimensional conductivity defined as: ks As s Ks h ms cs;ref L
_ < 0; m
x ¼ 0 : Tf ¼ TH vTf ¼0 x ¼ 3L : vx
(15)
where, h is the convection heat transfer coefficient, jH is the Colburn j-factor, G is the mass flux or the mass flow rate per unit fluid flow area, and cp and Pr are the specific heat and Prandtl number of the fluid, respectively. The following relation for j-factor is used (Barron, 1999): jH ¼ 0:23Re0:3
(16)
The expressions for the mass flux and the Reynolds number are as follows: G¼
_ m Af
(17)
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Re ¼
GDh m
(18)
where, Af is the fluid flow area, Dh is the characteristic dimension for the analysis (the hydraulic diameter in this case), and m is the dynamic viscosity of the fluid. The hydraulic diameter can be calculated from the equivalent spherical diameter such that (Barron, 1999): Dh ¼
2aDeq 3ð1 aÞ
(19)
where, a is the porosity of the regenerator, and Deq is the equivalent spherical diameter. For a spherical particle regenerator, the average particle diameter is used. For flaked particle beds, such as those sometimes used in the AMR test apparatus, the following expression applies (Caudle, 1997): Deq ¼
6Vp Ap fs
(20)
where, Vp and Ap are the volume and surface area of an average particle, respectively, and fs is Gamson’s shape factor, also known as the sphericity. For flaked particles, the shape factor is assumed to be 0.86 (Caudle, 1997). Flaked particles can be modeled as prismatic structures such that: Vp ¼ lp wp hp Ap ¼ 2lp wp þ 2wp hp þ 2lp hp
(21)
where, lp, wp, and hp are the length, width, and height of an average particle. The effective conductivity is a local volume-averaged property and is dependent on the conductivity of each phase and the porosity of the porous media (Kaviany, 1991). Although this quantity is principally derived for the type of analysis where the fluid and solid phases are in thermal equilibrium, it can also be used when this assumption is not valid and twomedium treatment is required. For instance, the dispersionparticle-based model includes the effective thermal conductivity in the fluid phase equation and accounts for only radial conduction in the solid phase equation (Kaviany, 1991). Thus, with one-dimensional analyses, conduction can be eliminated completely from the solid energy balance. The effective conductivity can be expressed as (Engelbrecht et al., 2005): keff ¼ kstatic þ kf Dd
(22)
where, keff is the effective conductivity, kstatic is the static component of the effective conductivity, kf is the conductivity of the fluid phase, and Dd is the dispersion coefficient. The static component of the effective conductivity is essentially the effective conductivity of the packed bed when there is no flow. The correlation developed by Hadley is used (Kaviany, 1991): 2 ks afo þ 1 afo 6 kf 6 kstatic ¼ kf 6ð1 ao Þ ks 4 1 a 1 fo þ a 1 fo kf 3 2 ks ks 2 ð1 aÞ þ ð1 þ 2aÞ 7 kf kf 7 (23) þ ao 7 ks 5 ð2 þ aÞ þ 1 a kf where, ks is the conductivity of the solid and a is the porosity of the packed bed. The constants ao and fo are defined as:
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fo ¼ 0:8 þ 0:1a
0 a 0:0827 log ao ¼ 4:898a 0:0827 a 0:298 log ao ¼ 0:405 3:154ða 0:0827Þ 0:298 a 0:580 log ao ¼ 1:084 6:778ða 0:298Þ
(24)
The following expression for the dispersion coefficient, which models this quantity as an additional conduction loss, is used in this analysis (Engelbrecht et al., 2005): Dd ¼ 0:75a Pe
(25)
where, Pe is the Peclet number, which is defined as: Pe ¼ Re Pr
(26)
where, Re and Pr are the Reynolds and Prandtl numbers. It should be noted that the same definition for the Reynolds numbers is used in this case as is given in Equation (18). Material properties such as density, conductivity, heat capacity, and adiabatic temperature change are obtained from several sources for the magnetocaloric material in the AMR sections, such as Pecharsky and Gschneidner, 2005 and Dan’kov and Tishin (1998). Density and conductivity are assumed to be constant, while heat capacity data at several field intensities is used to interpolate the heat capacity for a given temperature and field intensity. With the magnetic field intensity varying sinusoidally and simultaneously with the mass flow rate, a simple step function representing the temperature change due to the magnetocaloric effect cannot be used. With the rate of change of the magnetic field, dB/dt, set in the simulation, the instantaneous variation in temperature due to the magnetocaloric effect is determined using: dðDTad Þ dðDTad Þ dB ¼ dt dB dt
(27)
A function to determine d(DTad)/dB at a specified temperature and field is created using the original MCE data. Further details are available in Dikeos (2006).
4.2.
Cold sections
The convection coefficient in the cold section is assumed to be through a simple circular cross-section. Calculations with typical flow parameters and component dimensions, as specified in Tables 1 and 2, indicate that the flow is laminar. For a circular tube with uniform surface heat flux and laminar, fully-developed flow, the convection coefficient is (Incropera and DeWitt, 2002): h ¼ 4:36
k D
(28)
where, k is the conductivity of the fluid and D is the inner diameter of the tube. The thermal properties of the solid portion of the cold section are modeled using the properties of G10-CR. This data is obtained from the National Institute of Standards and Technology (NIST) website. The radiative heat transfer in Equation (10) accounts for heat transfer between the outside of the cold section and the surrounding apparatus. This is modeled as radiation transfer between two concentric cylinders, where the rate of heat transfer is given by Incropera and DeWitt (2002):
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Table 1 e A summary of several key operating parameters used in the simulations.
Magnetic field intensity, Max (T) Magnetic field intensity, Min (T) Frequency (Hz) Hot heat sink temperature (K) Average pressure (atm) Cooling load (Heaters) (W) Ambient temperature (K)
Q_ rad
Model input
Apparatus
2.0 0.1 0.65e0.8 270e310 3.0e15.0 0e10.0 300
0e5.0 0.05*Bmax 0.2e1.2 263e313 0e10.0 0e40.0 w300
Fig. 2 e Crank and connecting rod assembly used to drive the displacer.
¼ 1 1 ε2 r1 þ ε2 r2 ε1 sA1 T42
T41
(29)
Emissivity values and area are listed in Table 2.
4.3.
A sinusoidal variation of the magnetic field intensity is assumed with the minimum field intensity set to 5% of the maximum (Rowe, 2002).
System properties
Two identical magnetically-coupled displacers are used to drive helium within the AMRTA. A representation of the piston-cylinder type assembly used to drive these displacers is shown in Fig. 2. The crank, AeB, rotates about A and is attached to the connecting rod, BeC, which is used to drive the displacer pistons, C, along the x-axis. The mass flow rate of fluid driven by the pistons is given by: _ ¼ 2rvAd m
(30)
_ is the mass flow rate, r is the density of the fluid, and where, m v and Ad are the velocity and surface area of each displacer piston. The density of the fluid is known since the displacers operate at ambient temperatures. The surface area of each of the displacer pistons is also known. The velocity of the displacer pistons, C, along the x-direction is (Dikeos, 2006): vC ¼ uo dsinðuo tÞ
(31)
Table 2 e A summary of properties and dimensions required by the simulations. Regenerator (per puck) Mass (kg) Length (cm [in]) Diameter (cm [in]) Porosity Average particle dimensions (length, width, height) (106 m) Particle Size (106 m) Cold section, tubing Material Length (cm [in]) Inner Diameter (cm [in]) Outer Diameter (cm [in]) Emissivity Cold section, apparatus inner wall Material Diameter (cm [in]) Emissivity Gas displacer Crank length (cm [in]) Connecting rod length (cm [in]) Displacer diameter (cm)
0.045 2.54 [1.00] 2.54 [1.00] 0.56 1330, 780, 120 335 G10-CR 15.24 [6.00] 3.18 [1.25] 3.81 [1.50] 0.85 Stainless steel 19.05 [7.50] 0.22 10.16 [4.00] 54.61 [21.50] 4.00
5.
Model implementation
The finite element modeling (FEM) package Comsol Multiphysics is used with MATLAB to solve the equations describing the AMRTA. The transient partial differential equation (PDE), general form modeling option is used with two dependent variables in order to represent the solid and fluid temperatures throughout the model domain. A uniform mesh consisting of 121 nodes is used and the maximum time step is limited to 0.01 s. Sinusoidal variations of the fluid mass flow rate and magnetic field intensities are implemented and cycling is repeated until the temperature profiles of the fluid and regenerator attain a periodic steady state condition. Periodic steady state is said to be achieved when the temperature difference between the hot heat sink and the average solid in the cold section changes by less than 0.01 K between cycles. Fig. 3 gives a graphical representation of simulation convergence by depicting the transient temperature profile of the solid domain at every fifth cycle using gadolinium. Snapshots of select, instantaneous temperature profiles are shown. The
300
Solid Temperature (K)
Operating variable
295 290
Time
285 280 275
AMR 1 270 0
0.5
Cold Section 1 1.5 2 Axial Location
AMR 2 2.5
Fig. 3 e Transient temperature profile of the simulated solid temperature throughout the model domain.
3
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6.
Model validation
The model results are compared to experimental results so that the simulation can be calibrated. The first set of simulations generate baseline model results where conduction and dispersion are the only loss mechanisms (i.e. radiation is neglected). With these base values, the model yields temperature spans that are significantly larger than those obtained experimentally. A number of parameters can be adjusted to help produce results that are more consistent with experiments. In general, fitting options fall into two general categories: increased losses, or altered material properties. Examples include: 1. Adding a heat leak from the surroundings to the cold section: Because the cylinder is operating in a vacuum, the external heat leak can be modeled as radiation. The net radiation can be adjusted by varying the length of the solid cylinder subjected to the radiative load, emissivity, or by modifying the environmental reference temperature. 2. Increasing the conductivity: This can be done by altering the effective static component, kstatic, or the dispersive component, Dd. 3. Adjusting the value of the peak MCE for the material being used: This accounts for impurities, phase non-uniformity, experimental uncertainties in material characterization, or demagnetizing effects. 4. Adjusting the Curie temperature of the material being used: The reported values of this quantity vary by as much as 5 K, so adjusting this parameter is not unreasonable. This also accounts for the possibility of a slightly different composition than expected for a regenerator material composed of two or more elements (e.g. Gd0.85Er0.15). 5. Including viscous dissipation in the fluid energy balance expression: This introduces an internal thermal load due to pumping power. As discussed earlier, this is not considered in this work since viscous losses are assumed to be small.
Gd0.85Er0.15 and Gd0.74Tb0.26, adjustments to material properties are made to fit experimental results.
6.1.
Heat rejection temperature variation
The first set of simulations examines the effect of hot heat sink temperature variation on the temperature span. Hot heat sink temperature is an important operating parameter because it affects where an AMR operates with respect to the Curie temperature of the material that it is composed of (or the Curie temperatures in the case of multi-material AMRs). For tests near room temperature, setting the hot heat sink temperature in the simulations is analogous to adjusting the temperature of the recirculating chiller used in experiments with the AMRTA. Here, the metric used for the performance characterization of the AMRs is the magnitude of the temperature span developed across them. The results for the temperature variation behavior for Gd, Gd0.74Tb0.26, and Gd0.85Er0.15 AMRs are shown in Figs. 4e6. Fig. 4 depicts the fitting procedure that is used for Gd. The experimental data, represented by the solid points, shows that the peak temperature span is achieved at a hot sink temperature significantly above the Gd Curie temperature of approximately 295 K. Baseline model results are significantly larger than the experimental results, with the largest deviation at higher temperatures. When radiation is considered, a drop is seen in the overall performance but the effect is most pronounced at lower temperatures. A reasonable fit to the data is obtained if radiation is included and the value of the conductivity is increased. The nominal static conductivity is calculated to be 1 W m1 K1 while, for the tests shown, the nominal dispersion is 5.5 W m1 K1. Obtaining a fit with Gd0.74Tb0.26, and Gd0.85Er0.15 is more challenging. Fig. 5 illustrates the fitting procedure for Gd0.74Tb0.26. In this case, the modifications used with Gd simulations are used as the baseline; however, model results employing these parameters are found to predict a temperature span between 2.5 and 6.0 K higher than the experimental results obtained for this material. Simulation results that fit well with the experimental data are obtained if the static conductivity is increased further.
Experiment Experiment Model Model (Rad) (Rad) Model (Rad, (Rad, kstatic+2.5) kstatic+2.5) Model (Rad, (Rad, kstatic+4) kstatic+4.0)
30
Temperature Span (K)
initial profile is uniform at 295 K and, as time passes, the curves begin to converge. The final temperature of the cold section in this case is approximately 278 K. Details regarding experimental method and results can be found in Tura (2005). Simulation results are obtained for Gd, Gd0.74Tb0.26, and Gd0.85Er0.15 single material regenerators. A broad range of simulations with varying hot heat sink temperature, system pressure, and applied heat load, are performed for Gd. However, only hot heat sink temperature was varied for the simulations with the other materials. Zero applied load simulations are conducted at an operating frequency of 0.65 Hz, while simulations with an applied heat load are at 0.8 Hz. Tables 1 and 2 summarize the key operating parameters and component dimensions used in the simulations.
25
20
15
10
In the case of Gd0.85Er0.15 and Gd0.74Tb0.26, less material data is available as compared to Gd. As such, less certainty is given to the accuracy of the data for these materials. So, for this work, Gd material properties are assumed to be correct and other physical mechanisms are adjusted. Once this is done, the loss parameters are fixed for all materials and, for
275
280
285
290
295
300
305
310
Hot Sink Temperature, TH (K)
Fig. 4 e Model results compared to experimental results for the temperature variation behavior of Gd at 9.5 atm and 0.65 Hz.
Temperature Span (K)
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25
Table 3 e Summary of parameters used to obtain most consistent results.
20
Parameter
4 Static conductivity adjustment (W m1 K1) Curie temperature 295/295 (used/reference) (K) Peak MCE 5.65/5.65 (used/reference) (K)
15
10 Experiment Experiment Model Model,(Rad, Rad,kkstatic+4 static+4.0)
5
Gd
Gd0.76Tb0.24 Gd0.85Er0.15 10
0
278/278
265/261
5.80/5.80
5.00/4.50
Model,(Rad, Rad,kkstatic+8 Model static+8.0) Model,(Rad, Rad,kkstatic+10 Model static+10.0)
0 270
275
280 285 290 295 Hot Sink Temperature, TH (K)
300
Fig. 5 e The simulated temperature variation results for a Gd0.74Tb0.26 AMR compared to the experimental results, both at 9.5 atm and 0.65 Hz.
The process used to obtain model results that are in agreement with experimental data for Gd0.85Er0.15 is depicted in Fig. 6. For this material, it is not possible to obtain a good fit to experimental results by only accounting for radiation and adjusting the value of the static conductivity, as is done with the other materials. Simulation results begin to better resemble experimental data if both the Curie temperature and the magnitude of the peak MCE are adjusted. Table 3 summarizes the final parameter modifications used to obtain simulation results that are relatively consistent with the experimental findings. The significance of these modifications is examined in the discussion.
6.2.
Pressure variation
Pressure is also an important operating parameter since it affects the density and hence the mass flow rate of the fluid. Pressure variation simulations are conducted for single puck Gd regenerators and compared to experimental results for the same conditions. The results for base case simulations with no additional losses are shown in Fig. 7. This plot shows that,
Fig. 6 e Simulation validation results (1Tcurie [ 261 K, MCEmax [ 4.5 K; 2Tcurie [ 265 K, MCEmax [ 5.0 K) for Gd0.85Er0.15 at 9.5 atm and 0.65 Hz.
for these conditions, a reduction in the simulated system pressure actually results in an increase in temperature span, contrary to the trend observed in the experimental results. The effects of varying pressure after applying radiation and the modified thermal conductivity for the single Gd puck are shown in Fig. 8. This process yields model results that are in good agreement with experimental data at all three pressures, with deviations up to 1.5 K. However, it is important to note that effect of accounting for radiation is quite different in each of the cases. Applying radiation to simulations at 9.5 atm has a small impact on the performance of the regenerators, causing the simulated temperature spans to drop by less than 3.5 K. Conversely, accounting for radiation in the 3.0 atm simulations has a dramatic effect on the temperature span across the AMRs, decreasing these values by up to 12.5 K. The effect of this modification on the performance of Gd regenerators operating at 6.0 atm is more moderate. Although not shown here, pressure variation simulations are also carried out without any radiative loading. In this case, it is not possible to obtain simulation results that are in agreement with experimental data across the range of operating pressures being considered by solely adjusting the value of the static conductivity.
6.3.
Heat load application
Simulations are performed to mimic the application of a heat load in the cold section. These tests are used to represent the operation of the AMRTA with a particular net cooling power. It is important to investigate the sensitivity of AMRs with
Fig. 7 e Experimental pressure variation results compared to baseline simulation results for Gd AMRs with an operating frequency of 0.65 Hz.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 6 ( 2 0 1 3 ) 9 2 1 e9 3 1
Fig. 8 e The effect of varying pressure on the temperature span of single Gd AMRs for a specified hot heat sink temperature and an operating frequency of 0.65 Hz.
respect to applied heat load because this demonstrates the refrigeration potential of AMR devices. However, it should be noted that the cooling power of the single puck regenerators being investigated here is relatively small. This is largely due to the small mass of magnetocaloric material and the low fluid flow rate (small utilization). Fig. 9 illustrates the results of the Gd heat load simulations in comparison to the results obtained from the AMRTA for a system pressure of 9.5 and an operating frequency of 0.80 Hz. Although the magnitudes of some of the temperature spans are found to deviate by up to 2.0 K from the experimental data, the slopes of the curves are in good agreement.
7.
Discussion
Fig. 10 provides a summary of the single puck results for hot heat sink temperature variation. This plot compares data obtained through AMRTA experiments to fitted simulation results. As noted earlier, the fit to experimental data was most easily obtained with the Gd simulations. Obtaining fitted results with the other materials was more challenging and
Fig. 9 e The effect of heat loading on the temperature span of Gd single puck AMRs at a system pressure of 9.5 atm and a frequency of 0.8 Hz.
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Fig. 10 e A summary of the hot sink temperature variation behavior of single puck regenerators operating at 0.65 Hz and 9.5 atm.
required that additional parameters be adjusted. Some of this discrepancy can be accounted for by noting that Gd is the most well characterized magnetocaloric material. Compared to the other materials being investigated, it has been studied in much more detail and there is much less uncertainty in the value of its properties. Thus, there is a greater probability that the reference data obtained for the Gd is valid. Investigating the parameter changes that were required to yield the fitted model results with the different materials also generates some interesting observations. Compared to Gd, Gd0.76Tb0.24 simulations required much larger conductivity, while Gd0.85Er0.15 simulations necessitated a shift in the Curie temperature, and an increase in the maximum MCE used for MCE curve scaling. Therefore, simulation results based on reference values and validation parameters used in the Gd modeling would have yielded a much larger temperature span for Gd0.76Tb0.24 and decreased performance for Gd0.85Er0.15. The beds used in the experiments consist of irregular crushed particles with a relatively broad variation in size and shape. This may produce variations in porosity and local hydraulic diameter resulting in an effective thermal conductivity poorly predicted by the correlation. In regards to the Curie temperature and MCE, these can differ between alloys due to impurities and phase non-uniformity. It should be noted there are other approaches to fitting model results to experimental results (Burdyny and Rowe, 2013); the choice as to which parameters to use for fitting may not be obvious and will depend on what experimental data is actually measured. Consulting the MCE curve for each of the materials (Dan’kov and Tishin, 1998; Pecharsky and Gschneidner, 2005), the peak MCE values for Gd, Gd0.76Tb0.24, and Gd0.85Er0.15, are 5.65 K, 5.80 K, and 4.50 K, respectively. Based solely on these peak MCE values, one might expect that an AMR composed of Gd0.76Tb0.24 would have the best performance while a Gd0.85Er0.15 AMR would generate the lowest no-load temperature span. This is contrary to the observed experimental results and may be explained by the material heat capacities (Tura, 2005). The comparatively low heat capacity of Gd0.85Er0.15 results in a larger utilization than with Gd0.76Tb0.24 and can increase the performance of Gd0.85Er0.15. When the apparatus is operating at
930
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 6 ( 2 0 1 3 ) 9 2 1 e9 3 1
a lower pressure, and thus lower utilization, the temperature span tends to be more sensitive to heat load. Temperature span as a function of rejection temperature, TH, is an important experimental result for characterizing AMR performance. The results of this validation work indicate the following general features for fitting models to temperature span: 1. External heat leak mechanisms should be considered. With low applied loads, utilizations, and large spans, parasitic heat leaks can significantly alter the temperature span. 2. Temperature span sensitivity to diffusion increases as the hot rejection temperature increases above the Curie temperature. 3. Reducing MCE decreases span for all rejection temperatures while adjusting the Curie temperature shifts the temperature span curve in the direction of the changed TCurie.. 4. Temperature span sensitivity to specific heat is determined by the impact on utilization. Thus, experimental spans are better understood when plotted with respect to utilization instead of displaced fluid volume or mass. Reasonable fits of single-material experimental data are realized with the model. The data used for validation is a result of tests using helium and relatively low pressure drops are seen. This is not always the case, and smaller particle sizes or longer regenerators will increase viscous losses. Also, devices that use lower applied fields and liquids as the heat transfer fluid may have more significant dissipation terms. In addition, it was found that for certain operating conditions, temperature span can be more sensitive to increased losses. With this in mind, the assumption of negligible viscous dissipation in the energy balance expressions should be reassessed in future work.
7.1.
Predictive simulations
Model results were also compared to experimental data for two-material (i.e. two layer) regenerators. In addition, the
Fig. 12 e The heat load performance of GdeGd0.85Er0.15 AMRs with varying proportions of Gd, operating at 9.5 atm and 0.8 Hz.
model was used to predict the effects of increased pressure, varying aspect ratio, and varying material proportion in two material regenerators (Dikeos, 2006). The particular case of an AMR consisting of Gd and Gd0.85Er0.15 was investigated to determine the impact of the relative amount of each alloy on temperature span. The predicted temperature span at zero applied load for a fixed volume regenerator with varying fractions of the two refrigerants is shown in Fig. 11. The results of this analysis for the assumed conditions, suggest that the optimum regenerator for large temperature spans and small thermal loads is approximately 20% Gd and 80% Gd0.85Er0.15. This result is specific to the condition of heat rejection at 290 K, and may not hold true if the heat rejection temperature is changed. With this in mind, the thermal load sensitivity for this composition was modeled and compared to experimental data for an equivalent volume of Gd only and a regenerator consisting of 50% Gd and 50% Gd0.85Er0.15. Results are shown in Fig. 12. As can be seen, while an increased span is realized at low loads, the sensitivity of span to applied load increases and performance is worse than Gd only under high load conditions. There is no simple rule to describe how the behavior of a multi-material AMR will change with rejection temperature. Results will vary with operating conditions, amount of material, matrix geometry, porosity, and separation between Curie temperatures of the two materials. Multimaterial performance is an area where further studies are needed and is one reason validated numerical models are required. Additional studies are warranted.
8.
Fig. 11 e The effect of varying the Gd proportion in a GdeGd0.85 Er0.15 AMR operating at 9.5 atm and 0.65 Hz, and with a hot heat sink temperature of 290 K.
Conclusions
A numerical model of an active magnetic test apparatus is described. Model results are compared to experimental data for three different second-order magnetic refrigerants. Loss mechanisms and material properties are employed as fitting parameters to correct model predictions. Base results show that the model tends to over-predict the realized temperature span. Increased heat leaks to the cold-section and higher internal losses due to diffusion correct the shape of the
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 6 ( 2 0 1 3 ) 9 2 1 e9 3 1
temperature span curve as a function of rejection temperature for gadolinium. Additional adjustments to magnetocaloric effect and Curie temperature are used to accurately model Gd0.76Tb0.24 and Gd0.85Er0.15. A more complete validation is recommended when lower field strengths and liquid heat transfer fluids are employed. Predictive simulations with twomaterial regenerators indicate that preferred compositions may not be to use equal amounts of material.
Acknowledgments The support of the Natural Sciences and Engineering Research Council of Canada is greatly appreciated. Partial support for this work was provided by the NSERC Hydrogen Canada (H2Can) Strategic Network.
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