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Predicting the performance of an active magnetic regenerator refrigerator used for space cooling and refrigeration.

Active magnetic regenerative refrigeration (AMRR) systems represent an environmentally attractive refrigeration alternative that does not use a fluorocarbon working fluid. Recently, families of magnetocaloric material alloys with adjustable Curie temperatures have been developed. Using these materials, it is possible to construct a layered regenerator bed that can achieve a high magnetocaloric effect across its entire operating temperature range. This paper describes a numerical model capable of predicting the practical limits of performance of this technology applied to space-conditioning and refrigeration applications. The model treats the regenerator bed as a one-dimensional matrix of magnetic material with a spatial variation in Curie temperature and, therefore, magnetic properties. The matrix is subjected to a spatially and temporally varying magnetic field and fluid mass flow rate. The numerical model is solved using a fully implicit discretization of the governing energy equations. The nonlinear aspects of the governing equations (e.g., fluid and magnetic property variations) are handled using a relaxation technique. Modeling results are presented that illustrate how an AMRR system can be optimized for a particular operating condition. The performance of layered and nonlayered AMRRs are compared to current vapor compression technology for space-conditioning and refrigeration applications.


A well-designed magnetic cooling system may be competitive with vapor compression systems for space-cooling and refrigeration applications. The metallic refrigerant has essentially zero vapor pressure and no ozone depletion potential or direct global warming potential. Active magnetic regenerative refrigeration (AMRR) systems may also have advantages relative to noise, control, and part-load efficiency. A rotary bed configuration was developed and demonstrated by Zimm et al. (2002); it achieved moderate levels of COP and utilized affordable permanent magnet materials. Concurrently, the development of magnetic materials with "tunable" Curie temperatures allowed layered magnetic regenerator beds to be fabricated that exhibit a large magnetocaloric effect over a large temperature range (Smaili and Chahine 1997).

Vapor compression is a mature technology with well-known fundamental performance limits resulting from inherent throttling, compression, and heat exchange processes. Irreversible throttling and compression losses are avoided in a magnetic system, although new dissipation mechanisms are encountered, such as pumping losses through the bed and additional heat transfer losses in the regenerator. A well-designed magnetic system may be competitive with or even more efficient than vapor compression systems. This paper uses a physics-based, numerical model to predict the practical limits of the efficiency of a magnetic cooling cycle and compares the AMRR cycle to current technology.

The Magnetocaloric Effect

The thermal and magnetic properties of magnetocaloric materials are highly coupled over a specific, typically small, temperature range, allowing them to be used within energy conversion systems. Temperature (T) and entropy (S) form a canonical conjugate property pair that, together, defines the transfer of heat. If hysteresis is ignored, the applied field ([[mu].sub.o]H) and magnetic moment (VM) describe the transfer of magnetic work (Guggenheim 1967). The fundamental property relation for a substance capable of experiencing magnetic work is shown in Equation 1.

dU = TdS + [[mu].sub.o]Hd(VM) (1)

Examination of Equation 1 reveals that the applied field is analogous to pressure and that magnetic moment is analogous to (the inverse of) volume for compressible substances.

Magnetic Refrigerator Configurations

The thermodynamic coupling between thermal and magnetic properties allows a magnetocaloric material to be used as the working fluid in a refrigeration cycle. Giauque and MacDougall (1933) used this effect to reach temperatures below 1 K, breaking the temperature barrier that was previously imposed by the properties of compressible fluids. These early "one-shot" systems consisted of a solid piece of magnetocaloric material, and they used an adiabatic demagnetization to achieve very low temperatures. More advanced, continuously operating versions of these adiabatic demagnetization refrigeration (ADR) systems have been developed, culminating in the sophisticated multi-stage ADRs used by NASA (Shirron et al. 2004) to achieve and maintain sub-Kelvin detector temperatures. The temperature lift in an ADR system is limited to the adiabatic magnetization temperature change exhibited by the material. ADR cycles require complex heat switches with limited capacities and are not practical for near room temperature commercial devices.

The technical barriers associated with the ADR cycle have been overcome by the use of a regenerator in the AMRR cycle. A porous, packed bed of magnetic material is exposed to a time-varying magnetic field and flow of heat transfer fluid. Each segment of the bed undergoes a unique refrigeration cycle and interacts with the adjacent material via the heat transfer fluid. The net result of these cascaded refrigeration cycles is a temperature lift that is much larger than can be achieved by an ADR cycle.

Practical residential AMRR systems will likely use permanent magnets to produce a magnetic field and achieve variations in applied field by physically moving the magnetic regenerator relative to the magnetic field, either linearly as demonstrated by Rowe and Barclay (2002), Hirano et al. (2002), Wu (2003), and Zimm et al. (1998) or rotationally as demonstrated by Zimm et al. (2002). A conceptual drawing illustrating the four processes that make up the operation of a rotary AMRR, such as that described by Zimm et al. (2002), is shown in Figure 1. One of the six beds within the refrigerator is considered in the following discussion. The bed is magnetized by rotating it into the field provided by the permanent magnet (Figure 1a). The magnetocaloric effect causes the temperature of the material in the bed to increase. While the bed is in the magnetic field, it experiences a flow from the cold end to the hot end, which causes heat rejection in the hot heat exchanger, (Figure 1b). The bed is demagnetized when it rotates out of the magnetic field (Figure 1c), causing the temperature of the bed to decrease. The regenerator then experiences a flow from the hot end to the cold end while it is out of the magnetic field (Figure 1d), and a cooling load is accepted in the cold heat exchanger.


Layered Regenerator Beds

Pure magnetic materials exhibit a large magnetocaloric effect only over a narrow temperature range that is centered at the Curie temperature ([T.sub.Curie]) of the material. As a result, there is only a very small temperature range (near the Curie temperature) where an AMR composed of a pure magnetic material can maintain its otherwise potentially high performance. Researchers have identified families of alloys whose Curie temperatures can be engineered, within some range, through variation in the alloying formula; examples of research on such magnetocaloric families include Pecharsky and Gschneidner (1997), Chen et al. (2003), and Hu et al. (2001). Therefore, layered AMR beds composed of several alloys may be constructed with engineered, spatial variations in the Curie temperature that is chosen to match the local, average regenerator temperature and therefore deliver a maximum magnetocaloric effect.


A detailed model of the AMRR system has been developed (Engelbrecht 2005). The conventional equipment that must be integrated with the magnetic regenerator bed, including the pumps, heat exchangers, fans, drive motor, and permanent magnets, is not explicitly modeled; its effect on the bed is included through an imposed time variation of the mass flow rate ([dot.m](t)) and the variation of the magnetic field in time and space ([[mu].sub.o]H(x, t)). When the fluid mass flow rate is positive, flow is in the positive x direction, which is defined as entering the hot end of the regenerator bed. A negative mass flow rate indicates flow entering from the cold end of the bed. The flow entering the bed is assumed to have the temperature of the adjacent thermal reservoir, [T.sub.H] or [T.sub.C], depending on whether the flow rate is positive or negative. The fluid is assumed to be incompressible ([[rho].sub.f] = constant). The remaining required fluid properties, specific heat capacity ([c.sub.f]), viscosity ([[mu].sub.f]), and thermal conductivity ([k.sub.f] ), are assumed to be functions of temperature but not pressure.

The fluid flows within a regenerator matrix that is composed of either a single magnetic material (nonlayered) or a layered magnetic material; a layered bed is modeled as having a Curie temperature that varies spatially along the matrix ([T.sub.Curie](x)). The partial derivative of entropy with respect to applied field at constant temperature,

[[partial derivative].sub.s.sub.r]/[[[partial derivative].sub.[mu].sub.o]H]|[.sub.T],

is assumed to be a function of the difference between the temperature of the material and the local Curie temperature, T - [T.sub.Curie], and of the applied magnetic field, [[mu].sub.o]H. When expressed in this manner, the magnetic effect of different alloys within a given material family can be collapsed onto a common shape by shifting the temperature scale according to the Curie temperature; this representation makes it possible to approximately model alloys with desired Curie temperatures for which detailed material property data do not currently exist. The specific heat capacity at constant applied field ([c.sub.r]) and the thermal conductivity ([k.sub.r]) of the material are assumed to be functions of the material temperature, applied field, and the local Curie temperature.

The geometry of the matrix consists of many small passages that allow the fluid to flow in intimate thermal contact with the regenerator material. The regenerator geometry is characterized by a hydraulic diameter ([d.sub.h]), porosity ([epsilon]), and specific surface area ([a.sub.s]). The Nusselt number of the matrix is assumed to be a function of the local Reynolds number and Prandtl umber of the fluid, Nu(R.sub.[e.sub.f], P[r.sub.f]). The friction factor is assumed to be a function of the local Reynolds number, f (R[e.sub.f]). The matrix/fluid combination is characterized by an effective thermal conductivity, [k.sub.eff], that accounts for conduction through the liquid-solid matrix as well as axial dispersion caused by mixing of the flowing fluid in the axial direction. The overall size of the regenerator is specified according to its length (L) and total cross-sectional area ([A.sub.c]).

Derivation of Governing Equations

The fluid and regenerator temperature variations over a periodic steady-state cycle are the primary outputs of the numerical model ([T.sub.f] (x,t) and [T.sub.r](x,t)). These variations, coupled with the prescribed mass flow rate and applied field, allow the calculation of cycle performance metrics such as refrigeration load and magnetic input power. The temperature variations are obtained by solving a set of coupled, partial differential equations in time and space that are obtained from energy balances on the fluid and the matrix. After some simplification (Engelbrecht 2005), the energy balance on the fluid is

[dot.m][c.sub.f][[[partial derivative][T.sub.f]]/[[partial derivative].sub.x]] + [[Nu[k.sub.f]]/[d.sub.h]][a.sub.s][A.sub.c]([T.sub.f] - [T.sub.r]) + [[rho].sub.f][A.sub.c][epsilon][c.sub.f][[[partial derivative][T.sub.f]]/[[partial derivative].sub.t]] = |[[partial derivative]P/[partial derivative]x][[dot.m]/[[rho].sub.f]]. (2)

The first term in Equation 2 represents the enthalpy change of the flow; the second term represents heat transfer from the fluid to the magnetic material; the third term represents the energy stored by the fluid that is entrained in the matrix structure; and the fourth term represents viscous dissipation. Axial conduction is ignored in the fluid equation and instead is applied to the matrix and modeled using the concept of effective bed conductivity. The governing equations are simplified and the numerical solution is stabilized considerably if the entrained fluid heat capacity is "lumped" with the heat capacity of the matrix itself; therefore, this term is removed from the fluid energy equation but will be included in the regenerator energy equation. Investigations are ongoing relative to the impact of this simplification, but it is a conservative assumption in the passive regenerator case as shown by Nellis and Klein (2006); that is, this simplification tends to reduce the regenerator performance when no magnetic field variation is present. An approximate technique to correct for this assumption is discussed later. After removing the entrained fluid heat capacity, moving the axial conduction term to the regenerator equation, and expressing the pressure gradient in terms of a friction factor, Equation 2 becomes

[dot.m][c.sub.f][[[partial derivative][T.sub.f]]/[partial derivative]x] + [[Nu[k.sub.f]]/[d.sub.h]][a.sub.s][A.sub.c]([T.sub.f] - [T.sub.r] = |[f[dot.m.sup.3]]/[2[[rho].sub.f.sup.2][A.sub.c.sup.2][d.sub.h]]|. (3)

The energy balance on the magnetic material in the regenerator is

[[Nu[k.sub.f]]/[d.sub.h]][a.sub.s][A.sub.c]([T.sub.f] - [T.sub.r]+[A.sub.c])(1 - [epsilon])[[mu].sub.o]H[[partial derivative]M/[partial derivative]t] + [k.sub.eff][A.sub.c][[[[partial derivative].sup.2][T.sub.r]]/[[partial derivative][x.sup.2]]] = [[rho].sub.r][A.sub.c](1 - [epsilon])[[[partial derivative][u.sub.r]]/[partial derivative]t], (4)

where the first term represents heat transfer from the fluid to the regenerator, the second term represents the magnetic work transfer, the third term is the effective axial conduction through the regenerator and fluid, and the fourth term represents energy storage. The magnetic work term is grouped with the change in the internal energy of the matrix in order to obtain an expression involving the partial derivative of entropy with respect to magnetic field at constant temperature. The fluid heat capacity that was removed from Equation 2 is added to the regenerator energy balance so that the governing equation for the regenerator becomes.

[[Nu[k.sub.f]]/[d.sub.h]][a.sub.s][A.sub.c]([T.sub.f] - [T.sub.r]) + [k.sub.eff][A.sub.c][[[[partial derivative].sup.2][T.sub.r]]/[[partial derivative][x.sup.2]]] = [A.sub.c](1 - [epsilon])[[rho].sub.r][T.sub.r][[[partial derivative][s.sub.r]]/[[partial derivative][[mu].sub.o]H]]|[[[partial derivative][[mu].sub.o]H]/[partial derivative]t] + [A.sub.c][[[rho].sub.f][epsilon][c.sub.f]+(1 - [epsilon])[[rho].sub.r][c.sub.[mu].sub.o]H][[[partial derivative][T.sub.r]]/[partial derivative]t]. (5)

The boundary conditions for Equations 3 and 5 require that the fluid enter the matrix at the temperature of the associated thermal reservoir. The ends of the regenerator are assumed to be insulated from conductive heat transfer, and the regenerator must undergo a steady-state cycle. This last boundary condition leads to the constraint that the temperature at any location in the regenerator at time t must equal the temperature at the same point at time t + [tau], where [tau] is the duration of a cycle. These boundary conditions are summarized in Table 1.

Numerical Model

The numerical solution for the fluid and regenerator temperatures is obtained over a grid that extends from 0 to L in space and from 0 to [tau] in time. Initial "guess" values for the temperatures at each node ([T*.sub.ri,j] and [T*,j]) are assigned based on a spatially linear and time-invariant assumption. The properties, local flow characteristics, and other temperature-dependent aspects of the matrix characteristics are computed for each control volume based on these "guess" temperature values. The fluid and regenerator governing equations are linearized, discretized, and solved using a sparse matrix decomposition algorithm. The absolute value of the maximum error between the "guess" values of the regenerator and fluid temperatures and the calculated values is determined--if the error is less than a specified relaxation tolerance of 5.0 mK, then the relaxation process is complete; otherwise, a new set of "guess" values is used in a subsequent iteration. These new "guess" values ([T*.sup.(+)]) are computed as the weighted average of the calculated and "guess" values. Additional details regarding the numerical model are provided in Engelbrecht (2005).

Material Properties

The specific heat, thermal conductivity, and viscosity of water and other heat transfer fluids, such as propylene glycol and ethylene glycol solutions, are represented by polynomial correlations as functions of temperature. Experimental property data for a 94% gadolinium/6% erbium alloy* were used to provide entropy over a range of temperatures and applied fields through interpolation using a two-dimensional spline technique. The interpolated entropy data are numerically differentiated to determine the required model inputs: constant field specific heat capacity and the partial derivative of entropy with respect to applied field.

Thermal/Hydraulic Performance of Matrix

Although the model is generally applicable to a number of matrix configurations, the initial analyses have focused on a packed sphere regenerator. The total axial conductivity of a regenerator bed is the sum of the dispersive conductivity, [k.sub.f][D.sub.d], and static effective thermal conductivity, [k.sub.static], where [D.sub.d] is the dimensionless dispersion coefficient.

[k.sub.eff] = [k.sub.static] + [k.sub.f][D.sub.d] (6)

The static conductivity of the fluid/regenerator matrix for packed spheres is computed using the correlation presented by Hadley (1986).

[k.sub.static] = [k.sub.f][(1 - [[alpha].sub.0])[[[epsilon][f.sub.0] + [k.sub.r]/[k.sub.f] (1 - [epsilon][f.sub.0])]/[1 - [epsilon](1 - [f.sub.0]) + [k.sub.r]/[k.sub.f][epsilon](1 - [f.sub.0])]] + [[alpha].sub.0][[2([k.sub.r]/[k.sub.f])[.sup.2](1 - [epsilon]) + (1+2[epsilon])[k.sub.r]/[k.sub.f]]/[(2 + [epsilon])[k.sub.r]/[k.sub.f] + 1 - [epsilon]]]]. (7)

[f.sub.0] = 0.8 + 0.1[epsilon] (8)

log[[alpha].sub.0] = -4.898[epsilon]

0 [less than or equal to] [epsilon] [less than or equal to] 0.0827.

log[[alpha].sub.0] = -0.405 - 3.154([epsilon] - 0.0827)

0.0827 [less than or equal to] [epsilon] [less than or equal to] 0.298

log[[alpha].sub.0] = -1.084 - 6.778([epsilon] - 0.298)

0.298 [less than or equal to] [epsilon] [less than or equal to] 0.580 (9)

Axial dispersion is calculated using a correlation presented by Kaviany (1995).

[D.sub.d] = [epsilon][3/4]P[e.sub.f], P[e.sub.f] [much greater than] 1 (10)

where P[e.sub.f] is the Peclet number of the flow, defined as P[e.sub.f] = R[e.sub.f] P[r.sub.f]. The friction factor is computed using the Ergun equation with constants suggested by Kaviany (1995).

[f.sub.f] = 360[[(1 - [epsilon])[.sup.2]]/[[[epsilon].sup.3] x R[e.sub.f]]] + 3.6[1 - [epsilon]/[[epsilon].sup.3]] (11)

The Nusselt number is calculated using a correlation for packed bed spheres given by Wakao and Kaguei (1982).

N[u.sub.f] = [2[epsilon]/3(1 - [epsilon])](2.0 + 1.1R[e.sub.f.sup.0.6]P[r.sub.f.sup.1/3]) (12)

Internal Temperature Gradients

For a practical packed sphere regenerator design, the Biot number (Bi) associated with the magnetic material is not generally much less than unity throughout the entire cycle, particularly during flow periods. Therefore, the temperature within the spheres that comprise the regenerator matrix cannot be considered spatially uniform. As a result, the heat transfer from the fluid to the magnetic material is significantly affected by conduction from the center of the spheres to the outer surface as well as by the convection from the surface of the spheres to the fluid. This effect is accounted for by calculating a modified heat transfer coefficient ([h.sub.eff]) as suggested by Jeffreson (1972) according to Equation 13.

[h.sub.eff] = h/[1+[Bi/5]] (13)

Entrained Fluid Heat Capacity

In most applications employing regenerative heat exchangers, the heat capacity of the fluid is small relative to the heat capacity associated with the regenerator matrix itself. However, AMRR systems used in space-cooling applications utilize liquid heat transfer fluids; therefore, the heat capacity of the fluid entrained in the matrix is significant. In a practical AMRR design, the capacity of the entrained fluid may be equal to the matrix heat capacity. As a result, the entrained fluid heat capacity cannot be neglected when modeling an AMRR system with a liquid heat transfer fluid. Nellis and Klein (2006) developed a method that can be used to approximately correct for the lumped capacitance assumption that was made during the development of the numerical regenerator model--that is, the combining of the fluid and regenerator heat capacity in the governing equations. The method was developed for a passive regenerator and provides a correction factor that is applied to the heat transfer coefficient; the augmented heat transfer coefficient, [h.sub.aug], is given by

[h.sub.aug] = [h.sub.eff](1 + 1.7640R + 1.0064[R.sup.2]), (14)

where R is the ratio of fluid heat capacity to regenerator heat capacity, defined in Equation 15.

R = [[[rho].sub.f][c.sub.f][epsilon]]/[[[rho].sub.r][c.sub.r](1 - [epsilon])] (15)

Model Outputs

The basic outputs of the model are the refrigeration capacity ([dot.Q.sub.refrigeration]), heat rejection ([dot.Q.sub.refection]), and the magnetic work into the regenerator ([dot.W.sub.mag]) during cyclic steady-state operation. The heat rejection and refrigeration loads are calculated by numerically carrying out the integrals:

[dot.Q.sub.rejection] = -[1/[tau]][[tau].[integral].0][dot.m][h.sub.f,x = 0]dt (16)

[dot.Q.sub.refrigeration] = -[1/[tau]][[tau].[integral].0] [dot.m][h.sub.f, x = L]dt (17)

where [h.sub.f, x = 0] and [h.sub.f, x = L] are the specific enthalpies of the fluid evaluated at the hot and cold ends of the regenerator, respectively. For cyclic steady-state operation with no external parasitics, the total power input must be the difference between the heat rejection and refrigeration loads.

[dot.W.sub.tot] = [dot.Q.sub.rejection] - [dot.Q.sub.refrigeration] (18)

Heat Exchangers and Auxiliary Equipment

The electric motor used to rotate the regenerator and the pump that provides the desired fluid mass flow rate are both modeled by assigning an overall efficiency relative to an ideal thermodynamic process for these components. Using this approach, the ideal pump power, [W.sub.pump,ideal], is

[dot.W.sub.pump, ideal] = [1/[[[rho].sub.f][tau]]][[tau].[integral].0] [L.[integral].0]|[dot.m.sub.f]|[dP/dx]dxdt, (19)

where dP/dx is calculated using the friction factor from Equation 11 and [dot.m.sub.f] is the fluid mass flow rate. The integration is performed numerically by summing the pressure drop, calculated using the local friction factor via Equation 11, for each spatial step in the regenerator. The magnetic work can be calculated when the total work and ideal pump work are known.

[dot.W.sub.mag] = [dot.W.sub.tot] - [dot.W.sub.pump, indeal] (20)

The actual pump power is calculated using the total pump efficiency, [[eat].sub.pump].

[dot.W.sub.pump] = [dot.W.sub.pump, ideal]/[[eat].sub.pump] (21)

Similarly, the power required to run the electric motor ([dot.W.sub.pump]) is based on the magnetic work input to the regenerator:

[dot.W.sub.motor] = [dot.W.sub.mag]/[[eat].sub.motor] (22)

where [[eat].sub.motor] is the overall motor efficiency. A pump efficiency of 0.7 and a motor efficiency of 0.9 were used for this analysis.

A practical AMRR space-cooling device will interact with the conditioned space and the ambient environment via conventional heat exchangers. The use of a liquid heat transfer fluid suggests that these heat exchangers will have size and performance characteristics that are comparable to conventional evaporators and condensers. The hot and cold heat exchangers are modeled as cross-flow air-to-liquid heat exchangers. The overall conductance, UA, and mass flow rate of air through each heat exchanger are specified based on values that are reasonable for an equivalent vapor compression system. The overall conductance of these heat exchangers was specified using the values reported by the DOE/ORNL heat pump design model developed by Rice (2005) for comparably sized domestic air-conditioning and by Gan (1998) for comparably sized refrigeration systems. The fan power for the domestic air-conditioning system is accounted for using values calculated by the DOE/ORNL heat pump model developed by Rice (2005). Fan power is neglected for the refrigeration application. Therefore, while neither the fan power nor the conductance of these external heat exchangers is computed by the model, the impact of these effects on the efficiency of the AMRR cycle is explicitly considered.

Unlike the evaporator and condenser in a vapor compression cycle, the heat exchangers in an AMRR system will have a finite capacity ratio; therefore, the performance, as indicated by an approach temperature difference, will not remain constant as the mass flow rate of the refrigerant (in this case the heat transfer fluid) is varied. An iterative calculation is therefore required to ensure that the reservoir temperatures that are used as the fluid temperature boundary conditions for the numerical model of the active magnetic regenerator bed ([T.sub.H] and [T.sub.C]) are consistent with the mass flow rates of the air and heat transfer fluids and the operating temperatures. Analysis of the cold heat exchanger must consider the significant amount of condensation that takes place on the cold heat exchanger. Heat transfer and moisture removal for the cold heat exchanger are modeled using a heat transfer analogy that is similar to the [epsilon]-NTU method presented by Braun et al. (1989).

There is no latent enthalpy change in the heat transfer fluid as there is with the refrigerant in a vapor compression cycle, and the temperature change in the heat transfer fluid due to the magnetocaloric effect is relatively low; therefore, the mass flow rate of heat transfer in an AMRR system will be significantly higher than the refrigerant flow in a comparable vapor compression system. Therefore, care must be taken in the design of the heat exchangers in order to avoid high pumping losses associated with flow through the heat exchangers. It may also be necessary to increase the diameter of connecting tubing to avoid high pumping losses. In this analysis, the pressure drop and fan power for the air side are considered and included in the system COP for the air-conditioning application. However, the additional pressure drop associated with the flow of the liquid through the hot and cold heat exchangers is ignored, as it has been shown that, with good design, it can be made small relative to the pressure drop through the AMRR bed itself.


The entrained heat capacity correction factor presented in the previous section was developed for a passive regenerator. It is not yet completely understood how the entrained heat capacity affects the performance of an active magnetic regenerator, although work is ongoing to develop a correction factor for an active regenerator.

The additional pumping power that is associated with the acceleration and deceleration of fluid in the regenerator is not considered. For relatively low frequency operation (e.g., operation near 5 Hz), the effect of fluid oscillation is small; however, more advanced cycles will likely operate at higher frequencies in order to allow smaller equipment and thus exploit the more favorable economics associated with these design points. Therefore, the heat transfer, dispersion, and pressure drop correlations that are currently used in the model may not be accurate for more advanced cycles operating at higher frequencies. Studies of the impact of oscillating flow on the behavior of passive regenerators have found that heat transfer is enhanced and pressure drop is increased relative to a steady flow (for example, Zhao and Cheng [1996]). The effect of oscillating flow may be accounted for using appropriate correlations for oscillating flows that are a function of the Strouhal number or the Valensi number as well as the Reynolds number and Prandtl number.

The correction for internal temperature gradients that is used in the model and was discussed previously was developed assuming steady-state conduction through the solid material. At high frequencies, the thermal penetration depth associated with the cycle time may become comparable to the spatial extent of the local matrix structure, indicating that the entire matrix does not participate in the refrigeration process. This effect can be accounted for using a correction factor that is a function of the Fourier number as well as the Biot number.


Technique for Selecting Aspect Ratio and Mass Flow Rate

Results are presented in the subsequent sections for two applications, refrigeration and space conditioning, at specified operating conditions that include the cooling load. The active magnetic regenerator bed internal geometry, material, and operating frequency are limited by the technical maturity of AMRR technology and specified for these studies. The remaining free parameters are the regenerator volume and aspect ratio (defined as the ratio of the length to the cross-sectional area) and the heat transfer fluid flow rate.

Figure 2 illustrates the behavior of the refrigeration capacity and coefficient of performance, COP (defined as the ratio of the refrigeration capacity to the sum of the motor, pump, and fan power), as a function of the fluid mass flow rate for a typical AMRR system. Note that at very low values of mass flow rate, the regenerator acts as a thermal short between the hot and cold reservoirs so that both the refrigeration capacity and COP are negative. As the mass flow rate increases, the system is capable of producing refrigeration in direct proportion to the mass flow rate so both the refrigeration capacity and the COP increase. At some point, the losses induced by the fluid flow overwhelm the conductive loss and the COP hits a maximum value, which occurs at very low mass flow rate, and subsequently drops. Eventually, the mass flow rate becomes large enough that it overwhelms the magnetocaloric effect exhibited by the bed and, thereafter, the refrigeration capacity is reduced. At extremely high mass flow rates, the refrigeration capacity and COP both become negative as the fluid entering the cold reservoir is higher than the cold reservoir temperature.


Note that a horizontal line may be drawn on Figure 2 corresponding to a specific, required refrigeration capacity. Unless the specified refrigeration capacity exceeds the maximum capability of the system, the horizontal line would intersect the capacity curve at two points, each corresponding to a different mass flow rate; it is always true that the lower mass flow rate corresponds to a higher COP and is, therefore, the more optimal selection. Therefore, the lower mass flow rate operating point corresponding to the specified refrigeration load is selected for all subsequent results.

There is also an optimal aspect ratio for the regenerator bed. Figure 3 illustrates the general behavior of the COP as a function of aspect ratio for a given regenerator volume; note that in order to generate this curve, it is necessary to select the mass flow rate at each value of aspect ratio in order to obtain a desired refrigeration capacity using the previously described procedure. Figure 3 shows that an optimal aspect ratio exists that maximizes the COP of the bed; lower aspect ratios result in a pancake-shaped bed with excessive conduction losses, while higher aspect ratios result in excessive pumping losses. The optimum mass flow rate and aspect ratio are chosen as functions of regenerator volume in the context of a specific application and AMRR configuration in the following results.


Space-Cooling Application

The predicted performance of an AMRR system is compared to a standard vapor compression system for a space-cooling application. The default conditions for the DOE/ORNL heat pump design model (Rice 2005) were used to represent the vapor compression system. The parameters used to carry out the comparison are summarized in Table 2. The active magnetic regenerator bed model coupled to the previously described heat exchanger models was operated in order to determine how system performance varies with the regenerator volume at the optimal aspect ratio and mass flow rate determined as described previously.

For the conditions summarized in Table 2, the DOE/ORNL heat pump design model predicts a COP of 3.10 for the baseline vapor compression cycle, including 0.59 kW of fan power. The AMRR system model was subsequently run at the same conditions and using the same heat exchanger sizes (as indicated by the UA values in Table 2) and airflow rates. The numerical model does not explicitly model the fan power associated with the airflow rate; therefore, the fan power predicted by the heat pump design model was added to the total power predicted by the numerical model in order to compute the system COP for the AMRR cycle. Note that the COP presented corresponds to the ratio of the refrigeration provided to the sum of the motor, pump, and fan powers.

Figure 4 illustrates the predicted COP as a function of regenerator volume for a layered and a nonlayered bed; these curves were generated for a cooling capacity of 8.76 kW and the optimal aspect ratio and mass flow rate for each value of the regenerator volume. The layered regenerator was modeled as having a Curie temperature that varies linearly along the bed from the AMRR hot reservoir temperature to the cold reservoir temperature, whereas the nonlayered regenerator contains material with a constant Curie temperature, which is set at the average of the hot and the cold reservoir temperatures. The magnetic material properties represent a 94% gadolinium/6% erbium alloy, as described previously. Figure 4 indicates that an AMRR cycle may be capable of achieving higher values of COP than an equivalent vapor compression cycle; however, the COP that can be achieved using the AMRR depends strongly on the volume of the magnetic regenerator bed-a layered bed may be much smaller than a nonlayered bed. Note that the volume indicated in Figure 4 corresponds to the total volume of all six regenerator beds but does not include any allowance for the external hardware (e.g., the magnet, regenerator housing, motor, pump, heat exchangers, etc.). As the regenerator volume increases, the operating efficiency increases.


The sensitivity of the COP of the AMRR system to the heat rejection temperature was investigated by carrying out a parametric study using a particular regenerator design (volume and aspect ratio) for a layered and a nonlayered bed. The selected regenerator designs were chosen so that the predicted COP was higher than the equivalent vapor compression system at the design condition and also so that the COPs of the layered and nonlayered regenerators were approximately equal; these choices are shown in Figure 4 as designs A (a 3 L layered bed with a 1.20 kg/s mass flow rate and aspect ratio of 0.15) and B (a 10 L nonlayered bed with a 1.43 kg/s mass flow rate and aspect ratio of 0.15). The regenerator bed material, and, therefore, the Curie temperature distribution in the layered regenerator bed, was held constant at the values used to develop Figure 4. The heat rejection temperature (the temperature of the air entering the hot heat exchanger) was varied while the cooling capacity was held constant at 8.76 kW by varying the mass flow rate of the fluid as required. The COPs for designs A and B are shown in Figure 5 as functions of the hot reservoir temperature (i.e., the temperature of the ambient air).

In Figure 5, the layered bed is no longer capable of producing 8.76 kW, regardless of mass flow rate, when the heat rejection temperature rises above approximately 320 K (116[degrees]F), while the nonlayered bed is unable to produce the desired cooling when the heat rejection temperature is above 325 K (135[degrees]F). The nonlayered bed operates at a slightly higher coefficient of performance than the layered bed as the heat rejection temperature increases from the design temperature of 308.2 K (95.1[degrees]F). This non-intuitive result occurs because the material in the layered bed is specifically chosen to maximize the magnetocaloric effect over a specific temperature range. Because the maximum change in entropy with respect to magnetic field occurs over a narrow temperature span for the materials studied here, when the temperature span changes, the decrease in magnetocaloric effect for the layered bed is relatively high. The nonlayered bed is composed of a single material with a Curie temperature chosen as the average of the hot and cold reservoir temperatures. As the hot reservoir temperature increases, the magnetocaloric effect decreases, but the maximum magnetocaloric effect is still achieved at some location in the bed, and the decrease in performance for the nonlayered bed is lower than that for the layered bed.


It is important to quantify the dominant losses in a well-designed AMRR system. This information can be used to assess risk areas in modeling, to direct additional research, and to provide insight into more advanced regenerator designs. The losses of performance that can be associated with each entropy generation mechanism have been individually quantified by "turning off" that specific loss mechanism in the model while leaving all other loss mechanisms on. The change in the performance of the system that results from the deactivation of that particular loss mechanism provides a measure of its importance. This analysis assumes that the major loss mechanisms do not interact. As each loss mechanism was deactivated, the fluid mass flow rate was adjusted so that the same refrigeration power was provided. The change in the required input power indicates the corresponding improvement in performance.

The major loss mechanisms in an AMRR system for space conditioning are pumping losses, losses due to viscous dissipation, axial conduction due to dispersion, axial conduction due to the static conduction through the regenerator bed (solid and fluid), heat transfer over a finite temperature difference in the regenerator, motor losses, and heat exchanger losses in the hot and cold heat exchangers. The nominal AMRR design point used to carry out the parametric study is design case A from Figure 4, which corresponds to a 3 L layered bed with an aspect ratio of 0.15. The results of this study are summarized in Figure 6. The Carnot COP for a cooler operating at the hot and cold reservoir temperatures specified in Table 1 with no dehumidification is 35.7, which corresponds to a work input of 0.25 kW, whereas the predicted work input for design A under these conditions is 2.31 kW. The difference between these values, 2.06 kW, can therefore be attributed to the various losses listed previously. The relative contribution of these losses is estimated in Figure 6.


Figure 6 indicates that the dominant losses directly controlled by the magnetic regenerator bed are related to pumping losses due to pressure drop across the regenerator, conduction, and imperfect heat transfer within the regenerator; these are approximately equal, which is the result of the design process described previously, which was used to specify design point A. External to the regenerator bed, there are significant heat exchange and fan losses associated with the hot and cold heat exchangers; these losses are significant in a vapor compression cycle as well. The importance of the internal regenerator losses shown in Figure 6 indicates the magnitude in the performance improvement that can be obtained through the use of more advanced regenerator configurations that provide higher values of specific area. As the losses within the bed are reduced, the amount of heat that must be rejected will be reduced, so the loss associated with the hot heat exchanger, which is also large, as shown in Figure 6, can be reduced indirectly.

Refrigeration Application

The hot and cold heat exchangers for the refrigeration application are modeled using the same method that was previously described for the space-cooling application. Condensation and frost accumulation are not considered when modeling the cold heat exchanger in the refrigeration application. The model was used to predict the performance of an AMRR operating at freezer temperatures. Gan (1998) studied the total conductance of heat exchangers in a residential 150 W (500 Btu/h) refrigerator/freezer with a split freezer and refrigerator cycle. The conductance of the AMRR hot heat exchanger is taken to be the sum of the conductances of the refrigerator and freezer condensers, and the cold heat exchanger conductance is the sum of the conductances of the refrigerator and freezer evaporators for a domestic refrigerator/freezer system. The specific heat exchanger parameters and other model inputs that were used to carry out the analysis are summarized in Table 3.

Figure 7 illustrates the predicted COPs as a function of regenerator volume for a layered and a nonlayered bed; these curves were generated using a refrigeration capacity of 150 W and an optimal aspect ratio and mass flow rate for each volume. The fan power is a smaller fraction of the input power for a refrigeration application and is ignored in this analysis. From Jaehnig (1999), a typical vapor compression system operating under these conditions has a COP of about 1.9. Comparing Figure 7 to Figure 4 shows that layering the regenerator bed of an AMRR system has a more significant impact on its refrigeration application performance than it does on its space-cooling application performance. This behavior results because the increased temperature spanned by the refrigeration application causes the regenerator material within a nonlayered bed to operate further from its Curie temperature; therefore, it exhibits a relatively low magnetocaloric effect. In contrast, the material in the layered bed is chosen such that the local temperature is always near the Curie temperature of the material at that regenerator location. For the refrigeration application, the nonlayered bed is unable to achieve a COP that is competitive with current vapor compression technology, while the layered bed is able to exceed vapor compression performance provided that a sufficiently large regenerator bed is utilized.



The numerical model presented in this paper is capable of predicting the performance of an AMRR system with a layered regenerator matrix for space-cooling and refrigeration applications. The model was used to analyze the effects of mass flow rate, aspect ratio, and regenerator volume in the context of a gadolinium and erbium alloy packed sphere matrix. The model was exercised in the context of a space-cooling and a refrigeration application and showed that the COP of an appropriately designed AMRR system may be comparable to current vapor compression technology, provided that the volume of the regenerator is sufficiently large.


Financial support for this project was provided by the Air-Conditioning and Refrigeration Technology Institute (ARTI) and by a grant from the American Society of Heating and Air-Conditioning Engineers. The technical assistance of personnel at Astronautics Corporation and the ARTI 21CR Emerging Technology subcommittee is gratefully acknowledged.


[A.sub.c] = cross-sectional area of regenerator bed, [m.sub.2]

[a.sub.s] = surface area density of regenerator bed, [m.sub.2]/[m.sub.3]

b = wall thickness between flow passages, m

Bi = Biot number

c = specific heat, J/kg x K

[c.sub.[mu]oH] = specific heat of the matrix material at constant magnetic field, J/kg x K

COP = coefficient of performance

[D.sub.d] = dispersion coefficient

[d.sub.h] = hydraulic diameter, m

[d.sub.p] = particle diameter, m

f = friction factor

k = thermal conductivity, W/m x K

[k.sub.ef]f = effective thermal conductivity of regenerator and fluid, W/m x K

h = specific enthalpy, J/kg; convection heat transfer coefficient, W/[m.sub.2] x K

[h.sub.aug] = augmented convection heat transfer coefficient, W/[m.sub.2] x K

[h.sub.eff] = effective heat transfer coefficient of the regenerator, W/[m.sup.2] x K

L = length, m

[dot.m] = mass flow rate, kg/s

M = magnetic intensity, A/m

Nu = Nusselt number

P = pressure, Pa

Pe = Peclet number

Pr = Prandtl number

[dot.Q] = heat transfer, W

R = ratio of fluid heat capacity to the regenerator heat capacity

Re = Reynolds number, Re = [rho]v[d.sub.h]/[mu]

s = entropy, J/kg x K

t = time, s

T = temperature, K; torque, N/m

tol = relaxation tolerance, K

u = internal energy, J/kg

UA = overall heat transfer coefficient, W/K

[dot.W] = mechanical work, W

x = axial position, m


[epsilon] = porosity of matrix, heat exchanger effectiveness

[eat] = efficiency

[mu] = viscosity, N x s/[m.sub.2]

[[mu].sub.o]H = applied field, Tesla

[rho] = density, kg/[m.sub.3]

[tau] = cycle duration, s


aug = augmented

C = cold or refrigeration temperature

eff = effective

f = fluid

H = hot or heat rejection temperature

mag = magnet or magnetic

r = regenerator material

tot = total


* = guess value or modified value

+ = new value of guess value


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Kurt L. Engelbrecht

Student Member ASHRAE

Greg F. Nellis, PhD


Sanford A. Klein, PhD


Received September 6, 2005; accepted April 17, 2006

Kurt L. Engelbrecht is a graduate student, Greg F. Nellis is an assistant professor, and Sanford A. Klein is a professor in the Mechanical Engineering Department, University of Wisconsin-Madison, Madison, WI.

* Data were measured at Ames Laboratory, Ames, IA, and provided by Astronautics, Inc., Madison, WI.
Table 1. Model Boundary Conditions

Cold end Adiabatic end condition
 Fluid enters at constant temperature [T.sub.C] when [dot.m]
 > 0
Hot end Adiabatic end condition
 Fluid enters at constant temperature [T.sub.H] when [dot.m]
 < 0
Cyclic Steady state, [T.sub.r](x,t = 0) = [T.sub.r](x,t = [tau])

Table 2. Parameters Used for Space-Cooling Comparative Study

Parameter Value

Heat rejection temperature 308.2 K (95[degrees]F)
Load temperature 299.8 K (80[degrees]F)
Cooling capacity 8.76 kW (2.5 ton)
Maximum applied field 1.5 Tesla
Cold airflow rate 0.57 kg/s (1000 cfm)
Hot airflow rate 1.42 kg/s (2500 cfm)
 880 W/K
Cold heat exchanger UA (1160 Btu/h x [degrees]F)
 1430 W/K
Hot heat exchanger UA (1910 Btu/h x [degrees]F)
Motor efficiency 0.9
Pump efficiency 0.7
Heat transfer fluid water
Number of beds 6
Period 0.2 s (5Hz)
Sphere size for packing 0.2 mm (0.0079 in.)

Table 3. Parameters Used for Refrigeration Analysis

Parameter Value

Heat rejection temperature 305.4 K (90[degrees]F)
Load temperature 255.4 K (0[degrees]F)
Cooling capacity 150 W (512 Btu/h)
Maximum applied field 1.5 Tesla
Cold air mass flow rate 0.061 kg/s (93.5 cfm)
Hot air mass flow rate 0.109 kg/s (199.8 cfm)
Cold heat exchanger UA 86 W/K (160 Btu/h x [degrees]F)
Hot heat exchanger UA 73 W/K (140 Btu/h x [degrees]F)
Motor efficiency 0.9
Pump efficiency 0.7
Heat transfer fluid 50% ethylene glycol, 50% water
Number of beds 6
Period 0.2 s (5 Hz)
Sphere size for packing 0.2 mm
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Author:Engelbrecht, Kurt L.; Nellis, Greg F.; Klein, Sanford A.
Publication:HVAC & R Research
Geographic Code:1USA
Date:Oct 1, 2006
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