Balancing performance of active magnetic regenerators: a comprehensive experimental study of aspect ratio, particle size, and operating conditions

Effective and, at the same time, efficient active magnetic regenerator (AMR) performance requires balanced geometry and operating conditions. Here the influence of regenerator shape, magnetocaloric material size, operating frequency, and utilization on the performance of gadolinium packed-particle bed AMRs is demonstrated experimentally. Various metrics are applied to assess effectiveness and efficiency. Observed temperature spans and cooling powers across a wide range of operating conditions are used to evaluate system performance and estimate exergetic cooling power and exergetic power quotient. A new metric combining exergetic cooling power and pump power provides an estimate of the maximum achievable second law efficiency. Five regenerator geometries with equal volumes and the aspect ratio from 1.0 to 3.8, and four different ranges of Gd spherical particles between 182 and 354 µm, are investigated. Improvements in system performance are demonstrated by a boost in specific cooling power of gadolinium from 0.85 to 1.16 W g−1 and maximum temperature span from 8.9 to 15.1 K. The optimum exergetic cooling power is observed for 1.37 utilization and 3 Hz operating frequency, exergetic power quotient exhibits a maximum at the same utilization but at 2 Hz frequency, while the highest efficiency is recorded at 1 Hz and utilization of 0.5, demonstrating that multiple performance metrics must be balanced to achieve regenerator design meeting all performance targets.


Introduction
Cutting greenhouse gas emissions is necessary to halt climate change and governments worldwide are dedicating their efforts and resources to transition to net-zero energy systems in the near future. In 2021 the United States launched the Net-Zero World Initiative [1] led by the U.S. Department of Energy focused on establishing a partnership between countries to accelerate development and implementation of strategies and technologies that make it feasible. According to the U.S. Environmental Protection Agency, total greenhouse gas emissions in the United States is composed of 80% carbon dioxide, 10% methane, 7% nitrous oxide, and 3% fluorinated gases [2]. Fluorinated gases, regardless of their small percentage, are the most powerful at trapping heat in the atmosphere with global warming potential (GWP) thousands to tens of thousands of times higher than carbon dioxide, and many remain in the atmosphere for millennia [2]. The most commonly used fluorinated gases are hydrofluorocarbons (HFCs), utilized primarily in air conditioners and refrigerators, and due to the increasing demand, especially in the developing world, HFCs are among the fastest-growing greenhouse gases, with emissions increasing at a rate of 10%-15% annually [3].
Magnetic refrigeration [4] based on the magnetocaloric effect has the potential to become a more environmentally benign replacement for vapor-compression technology commonly used in air conditioning and refrigeration. The magnetocaloric effect manifests as adiabatic temperature changes of materials upon introduction of alternating magnetic fields. Magnetic refrigeration employs zero GWP solid-state refrigerants instead of HFC gases, hence even in the case of system malfunction, there is no risk of refrigerant emission, and the refrigerant is easily reclaimable. Additionally, magnetic refrigeration is potentially 25%-30% [5] more energy efficient, lowering emission of greenhouse gases related to energy production.
Near-room temperature magnetic cooling relies on active magnetic regeneration [6] to accomplish high temperature spans and cooling powers using reasonably achievable magnetic fields. Active magnetic regeneration coordinates cyclic heat transfer between the solid refrigerant and a heat exchange fluid with application and removal of magnetic fields to move heat up a temperature gradient. Major components of a magnetic cooling device include active magnetic regenerators (AMRs) that consist of magnetocaloric material placed in regenerator beds, magnets to cyclically stimulate AMRs, a pump that oscillates heat transfer fluid in the system, heat exchangers to dissipate heat, and control equipment to coordinate magnetic fields and fluid flows [7][8][9][10]. While magnetic field sources receive a lot of attention [11,12] because they are the most costly component of magnetic refrigeration [13], thoughtful regenerator bed design ensures that magnetic fields are used most effectively and can significantly boost performance of AMR devices.
The AMR has a porous structure enabling the heat exchange fluid to flow through. Packed-sphericalparticle and parallel-plate beds are the most common regenerator configurations [5,14,15]. The packed-particle bed is easy to manufacture even with very small magnetocaloric material feature sizes. The major disadvantages of this AMR configuration are fixed porosity, nominally 0.36, and relatively large pressure drops [16], accompanied by large viscous dissipation losses. Parallel plate regenerators have controlled porosity that can be adjusted by changing spacing between plates, however, manufacturing AMRs with very thin plates necessary for efficient heat transfer is challenging. Even a few percent non-uniformity in spacing has a disastrous effect on performance [17].
For a fixed regenerator volume, the main parameter describing the regenerator shape is the aspect ratio, defined as the ratio of the regenerator length to the square root of its cross-sectional area. AMR aspect ratio should be carefully considered since it has a significant influence on both viscous and axial conduction losses inside the regenerator [15,18,19]. Viscous losses are related to the pressure drop, ∆P, which, for packed beds of spherical particles, depends on the regenerator length, L, fluid density, ρ F , fluid viscosity, µ, porosity, ε, particle diameter, D p , and superficial fluid velocity, v s , according to the Ergun equation [20]. Expressing the superficial fluid velocity of a square fluid flow profile in terms of the utilization, U, frequency, f, regenerator volume, V R , regenerator cross-sectional area, A c , pump fraction, φ p , and heat capacity, c, of the solid, s, and fluid, F, in equation (1), helps to clarify the relationship between the pressure drop and operating conditions for the AMR, shown in equation (2), demonstrating that significant viscous losses are expected in regenerators with higher aspect ratios. On the other hand, smaller aspect ratios with shorter lengths and larger cross-sectional areas increase axial conduction losses. Regenerator performance can also be affected by insufficient heat transfer between the heat transfer fluid and magnetocaloric material. Larger material dimensions and larger flow channels have higher heat transfer resistance requiring reduced operating frequency to ensure enough time for complete exchange between the solid and fluid [21], however, smaller flow channels increase the pressure drop and viscous losses [20]. Optimizing regenerator performance by finding a configuration generating minimum total losses and maximum cooling power will lead to efficient system design. Literature focused on using experimental investigation to probe the influence of aspect ratio and active material size on magnetocaloric system performance is scarce and at times contradictory. Six regenerators with the aspect ratio between 7.6 and 123.7, filled with 7.5 g of 1.6 mm gadolinium spheres, were evaluated in [22]. The largest temperature span of 6.9 K was measured for the aspect ratio of 23.2. Results also showed higher sensitivity to utilization than to operating frequency. Experimental and numerical evaluation of two regenerators with aspect ratios of 6.9 and 13.5 were presented in [23] using AMRs packed with 80 g of 0.9 mm La(FeMnSi) 13 H z irregular particles. The temperature span, cooling power, and maximum power span product [24] were evaluated. Results revealed larger power span product for the longer regenerator. Using smaller, 0.5 mm particles in the shorter regenerator resulted in nearly identical power span product for both particle sizes, however, the larger particles reached larger maximum power, while smaller particles achieved larger temperature span.
A numerical analysis of geometry influence on AMR performance was presented in [19]. The study focused on coefficient of performance (COP) and entropy production rates due to different loss mechanisms inside AMRs [18] using a one-dimensional model [25]. Aspect ratios between 0.1 and 8.0, and Gd sphere diameter from 0.14 to 0.57 mm were considered. Results showed that COP is much more sensitive to changes in aspect ratio than frequency. With the increase of particle size, optimum aspect ratio rises and the optimum frequency drops. The largest COP of 7.6 was observed when the aspect ratio, particle size, and frequency were 1.0, 0.2 mm, and 2.3 Hz, respectively. Another numerical study focused on geometry optimization via entropy generation minimization [26] used performance evaluation criteria of variable geometry and fixed face (cross-sectional) area defined in [27]. The variable geometry study indicated an advantage of long regenerators and large magnetocaloric particles for low flow rates and higher frequencies, while short, wide AMRs of smaller particles performed better at high flow rates and small frequencies. The fixed face area analyses showed quite different results suggesting optimal configurations of short regenerators and small particles for low flow rates. As the flow increases, higher aspect ratios are necessary to achieve the target effectiveness and larger particles are required to minimize entropy generation. These studies and their disparate results indicate the important influence of aspect ratio on performance of AMRs and the metrics used to define optimum operation.
All the above-discussed articles focus on either effectiveness or efficiency, however, there is still lack of a detailed report guiding regenerator design leading to minimum total losses as well as maximum performance across a wide range of operating conditions. The two above-mentioned experimental analyses [22,23] evaluated regenerators with much larger aspect ratios and particle sizes than what numerically [19] was shown to provide high COP, leading us to focus our experimental studies on AMR geometry and operating conditions where both high power and high efficiency are expected. To analyze results, metrics that combine cooling power and temperature span are applied, and a new efficiency metric is proposed that estimates the maximum possible second law efficiency. In our study, we focus on packed particle bed AMRs due to their simplicity and ease of manufacture, however, we expect that observed performance trends will aid in optimizing other AMR configurations as well [15].

Experimental methods
To investigate the impact of the aspect ratio on regenerator performance, five AMRs with the aspect ratio between 1 and 3.8 were designed. Regenerator bed shapes are based on [28] and are sized to fit in the CaloriSMART system using its 1.13 T magnetic field configuration. Despite the small size of the test system, its heat losses to the environment are quite small as exemplified by performance comparable to a wide range of devices, making the results of this study useful for predicting operation of much larger multibed systems [28]. The regenerators have the same volume, and therefore mass of gadolinium, and the same width while the length and height are varied. Note that the regenerator length is parallel to the direction of the heat exchange fluid flow through regenerator, while the height is aligned with the direction of the magnetic field. Each regenerator consists of 16.6 ± 0.3 g of spherical gadolinium powder in one of four particle size ranges (182-210 µm, 210-250 µm, 250-300 µm, 300-354 µm). The material was obtained from a single batch and sieved to different size fractions, hence the magnetocaloric properties [29] are not dependent on particle size. Previous testing with CaloriSMART [28,30] used regenerators with an aspect ratio of 3.0 and 25.0 g of 182-210 µm spherical gadolinium powder. Table 1 summarizes regenerator dimensions while figure 1 shows solid models of all AMR beds.
For each aspect ratio, temperature span is measured for different displaced fluid volumes, operating frequencies, and cooling powers. Displaced fluid volumes range from 0.7 to 1.9 ml with a step of 0.4 ml, which corresponds to utilizations [31] between 0.50 and 1.37, using the 350 J kg −1 K −1 peak heat capacity of gadolinium [29]. Operating frequencies range from 1 to 3 Hz with a 0.5 Hz step. Cooling power, provided by a resistive heater, is increased in 3 W steps until each regenerator reaches 0 K temperature span. Temperature and pressure are measured at each end of the regenerator and the temperature spans are calculated based on the regenerator inlet measurements. During all experiments, the hot inlet temperature is stabilized at around 300 K. Periodic steady state, as defined by a change of less than 0.05 K between the averages of the most recent 100 stream temperature values and the previous 100 stream temperature values, is used as a cut-off criterion. Testing is performed using a single pump fraction of 0.8 with center-aligned flow and magnetic field profiles, and constant rpm based on previous results [30].

Results and discussion
The first part of the investigation focuses on determining the aspect ratio that provides the best performance for a single size of gadolinium spheres, 182-210 µm, and is described in section 3.1. The second part, section 3.2, uses the optimum aspect ratio and explores performance of regenerators with different Gd sphere sizes. Table 2 lists all the regenerators tested with their designated identifier. The empty cells indicate configurations that are not investigated.

Aspect ratio
The performance of five regenerators with the aspect ratio between 1.0 and 3.8 consisting of 182-210 µm Gd spheres is presented in this section. The temperature span and cooling power results for the operating frequencies and utilizations are shown as envelopes in figure 2. Each subgraph represents different aspect ratio, from the lowest at the top of the graph for A1 regenerator to the highest at the bottom for regenerator E1.
Only the three highest utilizations are visible in the graph, indicating that the lowest utilization always results in smaller temperature spans and cooling powers. For all aspect ratios, temperature span and cooling power rise with the increase in frequency, reaching a maximum at 3 Hz, which is the highest value in the experiments. Across the investigated aspect ratio range, increasing the AMR length results in larger temperature spans while enlarging the cross-sectional area drives up the output power. Figure 3 compares obtained maximum temperature spans and cooling powers for all tested regenerators. Maximum temperature spans are recorded at zero load, while maximum cooling powers are estimated from the fits to the data for zero temperature span.
Maximum temperature spans show close to linear dependency for all investigated cases. The longest regenerator demonstrates a 15.1 K temperature span which is a 19% improvement over the shortest one providing 12.7 K. All maximum temperature spans are obtained for the same utilization of 0.79 and a  frequency of 3 Hz. The correct utilization, hence the volume of heat transfer fluid moved during the cycle, is important for regenerator performance. Too small amount will prevent full utilization of magnetocaloric effect, while too large will reduce the temperature span due to insufficient regeneration. Maximum cooling power results differ from temperature span with the highest values for the lowest aspect ratio. For the shortest AMRs, cooling power is relatively constant and begins to decrease rapidly at values higher than 2.2. A 36% improvement for the shortest regenerator (19.2 W of cooling power) vs. the longest regenerator (14.1 W of cooling power) is observed, which corresponds to 1.16 W g −1 and 0.85 W g −1 specific cooling power of Gd, respectively. All maximum cooling powers occurred at 3 Hz and for the utilization of 1.37 except the longest regenerator, which performs best at 1.08 utilization. No clear optimum for either cooling power or temperature span is observed, hence it can be expected that decreasing the aspect ratio could result in even higher cooling powers and increasing it could maximize the temperature span. Based on the isothermal entropy change in Gd at 300 K for a magnetic field change of 1.1 T [29], a maximum cooling power of 26.1 W could be achieved for 16.6 g of Gd operating at 3 Hz, ignoring demagnetization.
More detailed AMR comparison requires metrics that capture both the cooling power and losses inside the regenerator. The COP, a ratio of the device cooling power to the power required to run the device, and the second law efficiency that compares COP to Carnot COP are good indicators of system performance [32]. However, CaloriSMART, the system used for experimentation, was not designed to provide high efficiency but rather flexibility in single-regenerator testing [28]. Because of that its COP and second law efficiency are not meaningful metrics to use. Two metrics based on cooling power and temperature span are used to compare regenerators' performance: exergetic cooling power, P Ex [33], and exergetic power quotient, θ Ex [28]. The former (exergetic cooling power) represents the power required by a Carnot cycle to lift the measured cooling power, P c , over the measured temperature span between regenerator cold, T c , and hot side, T h , the latter normalizes P Ex by the magnetic power available from the permanent magnet assembly, P m , where µ 0 is the vacuum permeability, V MCM is the volume of active magnetocaloric material, B is the maximum magnetic field, while f is the operating frequency. Exergetic cooling power and exergetic power quotient are shown in figures 4 and 5, respectively. Results are displayed as contour plots for a frequency of 3 Hz and varying aspect ratio and utilization (left), contour plots for utilization of 1.37 and varying aspect ratio and frequency (middle), and two-dimensional (2D) plots of maximum exergetic cooling power or exergetic power quotient vs. aspect ratio (right). Both metrics show the highest values for regenerator C1 with the aspect ratio of 2.2. Regenerator B1 with the aspect ratio of 1.6 performs only slightly worse indicating the optimum aspect ratio is likely in the range between 1.6 and 2.2. The longest regenerator, E1, achieves the lowest results. The difference in exergetic cooling power and exergetic power quotient between C1 (the highest performance) and E1 (the lowest performance) is 37% and 27%, respectively. Note that maximum exergetic power and exergetic power quotient are observed for different operating parameters. All regenerators show maximum exergetic power for the highest frequency but different utilizations. The three shortest regenerators reach the maximum for  These results differ from the optimum performance found in [19] where the minimum entropy production rate related to the losses inside the AMR and maximum COP for similar size particles (200 µm) was found for the aspect ratio of 1.0. There is high similarity in a spindle shaped characteristic of our exergetic power quotient results and COP in [19], both showing higher sensitivity to aspect ratio than frequency, however, the frequency where the best performance is observed is much lower (2 Hz vs. 3.7 Hz) in our study.
As mentioned above, regenerator designs are based on [28] where aspect ratio was 3.0, the same as aspect ratio of the regenerator D1 investigated here. In [28] maximum exergetic power quotient in maximum magnetic field of 1.13 T was equal to 0.048, while D1 reaches 0.057. The improved performance of D1 can be attributed to optimized operating conditions, i.e. pump fraction of 0.8 with center-aligned flow and magnetic field profiles [30], that are applied in current experiments.
We also measured pressure drop across the regenerators since it is indicative of pump work and viscous dissipation losses. Figure 6 shows a comparison of the minimum and maximum observed pressure drops for each regenerator. Minimum pressure drop is measured at the lowest operating frequency of 1 Hz and the smallest utilization of 0.5, while maximum pressure drop is observed for the highest operating frequency of 3 Hz and the largest utilization of 1.37. The pressure drop increases with the increase in the regenerator aspect ratio as expected for packed particle beds [20], indicating more significant viscous loss in regenerators with longer aspect ratios.
Larger pressure drop results in higher pump work which lowers overall system efficiency. One way to assess influence of pressure drop and pump work on regenerator efficiency is estimation of η * [34], which relates the exergetic cooling power to the pump power, P p , where the pump power is calculated based on the pressure drop across the regenerator, cycle time, t c , and volumetric flow rate,V, Our experiments yield many results where η * is substantially higher than one, which is not a physically meaningful quantity. The two main inputs to a magnetic refrigerator are the power to run the pump and the power to spin the magnet [35]. Omission of the latter by η * , results in a significant overestimation of possible device efficiency. We suggest that, in the absence of measured magnetic work, the minimum magnetic work be estimated as the Carnot work necessary to elevate the given cooling power from the cold to the hot temperature. Adding this to the denominator of equation (5) gives a better efficiency estimation, η est , that demonstrates the reduction in achievable device efficiency owing to the pumping power, Results for η est show the large advantage of low pressure drops, and therefore low viscous losses, on system efficiency, demonstrating the best performance at the frequency of 1 Hz and the utilization of 0.5 for all investigated AMR configurations (figure 7). The highest η est is observed for the shortest bed with the lowest pressure drop and decreases significantly for regenerators with higher aspect ratio.

Particle size
Since the regenerator with the aspect ratio of 2.2 (C1) demonstrates the highest exergetic cooling power and exergetic power quotient with 182-210 µm Gd particles, we use this configuration to test the influence of magnetocaloric particle size on AMR performance. Three new regenerator beds were manufactured and filled with Gd spheres in ranges 210-250 µm, 250-300 µm, and 300-354 µm. Maximum temperature spans and cooling powers as a function of Gd sphere diameter are compared with regenerator C1 in figure 8.
Results are plotted at the centers of the size ranges.
Similar to previous results for the aspect ratio, all maximum temperature spans are observed for the utilization of 0.79 and a frequency of 3 Hz. Maximum cooling powers are also reached at the highest operating frequency of 3 Hz, however, at 1.37 utilization. Both maximum temperature spans and cooling powers decrease almost linearly with the increase in Gd sphere diameter, showing 36% and 23% drop, respectively, between the smallest (C1) and the largest (C4) particle regenerators which corresponds to 0.86 W g −1 specific cooling power of Gd for the latter. The drop in performance for the larger particles, regardless of the same regenerator mass, results from higher heat transfer irreversible losses due to smaller heat transfer area and, therefore, heat transfer coefficient. Figure 9 presents maximum exergetic cooling power results for regenerators with different particle sizes, while figure 10 shows exergetic power quotient performance. Results are displayed as contour plots for a frequency of 3 Hz and varying particle size and utilization (left), contour plots for utilization of 1.37 and varying particle size and frequency (middle), and 2D plots of maximum exergetic cooling power or exergetic power quotient vs. particle size (right).   Both metrics, exergetic cooling power, and exergetic power quotient confirm the best performance of the regenerator consisting of the smallest Gd particles. With the increase in sphere size, the performance rapidly decreases, showing the smallest magnitudes for the largest particles. For all regenerators, maximum exergetic cooling power is observed at 3 Hz, but optimum utilization decreases from 1.08 for the three smallest particle size AMRs to 0.79 for the largest. Maximum exergetic power quotient is obtained for the biggest investigated utilization, but the frequency decreases from 2 Hz for regenerators C1-C3 to 1 Hz for C4 showing that larger particles require reduced operating frequency to ensure enough time for heat transfer.
The minimum and maximum pressure drop results presented in figure 11 are obtained for the smallest and highest utilization and frequency, and show a decrease with the increase in particle size, which is expected since larger particles provide larger hydraulic diameters for the heat exchange fluid flow.
Results for η est (figure 12) demonstrate increasing efficiency with the rising size of Gd spheres due to reduced pump work regardless of decreasing exergetic cooling power. The highest η est are observed for the frequencies of 1.0-1.5 Hz and utilization of 0.5. Higher efficiency for larger particles shows that, in terms of efficiency, decrease in viscous losses related to lower pressure drop for the larger spheres is more significant than improved heat transfer for smaller particles.    Table 3 summarizes optimum operating conditions leading to the maximum performance in terms of temperature span, cooling power, exergetic cooling power, exergetic power quotient, and η est for all investigated regenerators.

Trends and observations
As expected from previous studies, high temperature span favors high aspect ratio regenerators of small particles, high operating frequencies, and moderate utilizations. Another result consistent with previous studies is that high efficiency prefers low pressure drop operation, reaching maximum for the shortest regenerators with largest particles at the lowest flow rates and operating frequencies. There are, however, some surprising results that point at different optimum regenerator designs when multiple metrics are considered. The results presented in table 3 suggest general AMR design directions and inform optimized operation as follows: • Combining both temperature span and power, maximum exergetic cooling power and maximum exergetic power quotient are observed for regenerators with an aspect ratio around 2 consisting of 197 µm or smaller particle sizes with utilizations >1. However, they differ in their optimum frequencies, with exergetic cooling power peaking at frequencies ⩾3 and power quotient at 2 Hz, suggesting possibly higher efficiency when maximizing this metric. • While larger aspect ratios tend to have lower estimated efficiency, C1 compared to A1, this reduction in efficiency can be offset completely by increasing particle size, C4. • Considering both exergetic power quotient and efficiency, larger aspect ratios may be combined with larger particle sizes, low operating frequencies, and moderate utilizations to achieve relatively high performance for both metrics.
Observed trends for Gd AMRs are expected to be valid with other magnetocaloric materials and layered regenerators, however, with different optimum geometries and operating parameters.

Conclusions
An experimental study measured 2075 temperature spans to evaluate influence of AMR geometry and operating conditions on its performance with the goal of informing future AMR designs. A broad range of aspect ratios, particle sizes, frequencies and utilizations were considered. It is clear from the results that there is not a single combination of geometry and operating conditions that optimize all relevant performance metrics for magnetic cooling systems.
Maximum temperature span and cooling power show opposite trends with aspect ratio indicating a necessary compromise for high performance in both. Exergetic cooling power and exergetic power quotient both have peaks near the same aspect ratio but under different operating conditions. Efficiency, on the other hand, always favors small aspect ratios, low frequencies, and low utilizations at the expense of useful cooling power and temperature span. In contrast with aspect ratio, the smallest particles in this study result in large temperature spans, cooling power, exergetic cooling power, and exergetic power quotient. Because the pressure drop strongly depends on particle size, the largest estimated efficiencies occur with the largest particle sizes, again at the lowest frequencies and flow rates.
Multiple performance metrics must be balanced to achieve all system performance targets in a single AMR design. For example, if the system design goal is to maximize cooling power and temperature span, the optimum regenerator design would be different than if high cooling power and minimum magnet size were emphasized. Results show that a small change in regenerator design can significantly influence the performance of magnetic cooling systems using the same amount of magnetocaloric material. While performance boost could be achieved by adding magnetocaloric material or increasing the strength of magnetic field, those changes would rise the cost and possibly the footprint of the system. Informed regenerator design requires little to no investment and can vastly improve system performance.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.