CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 70, 2018 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Timothy G. Walmsley, Petar S. Varbanov, Rongxin Su, Jiří J. Klemeš 
Copyright © 2018, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-67-9; ISSN 2283-9216 

Numerical Investigation of a Packed-Bed Active Magnetic 

Regenerator 

Haoning Shi, Ting Ma, Qiuwang Wang* 

Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, Xi’an Jiaotong University, Xi’an, Shaanxi 

710049, China 

wangqw@mail.xjtu.edu.cn 

Magnetic refrigeration at room temperature is of great potential for the new generation of refrigeration 

technology. The performance of the active magnetic regenerator is greatly influenced by many design and 

operation parameters. In this paper, a one-dimensional numerical model for active magnetic regenerator is 

presented and validated. The performances of active magnetic regenerator with different particle diameters, 

utilization factors and operating frequencies are calculated with the present model. The performance curves of 

utilization factor with different particle diameters are presented and analysed. The results indicate that the 

cooling power and COP both increase at first and then decrease with the increase of the utilization factor. And 

the deviation between the optimal utilization factor of cooling power and COP becomes larger with decrease of 

the particle diameter. Comprehensively, the optimal range of particle diameter is 0.4-0.6 mm with the 

utilization factor of 0.7-1.5 which also depends on the operating frequency based on the present data. 

1. Introduction 

Magnetic refrigeration at room temperature is an emerging and promising refrigeration technology based on 

the magnetocaloric effect (MCE). The MCE is the thermal response of magnetocaloric material (MCM) in the 

process of magnetization or demagnetization. In the application of magnetic refrigeration, the use of 

traditional refrigerants CFC and HCFC can be completely avoided, which means no damage to the ozone 

layer and no contribution to the greenhouse effect. The compressor is not required as well thus leading to a 

compact structure without much noise. Therefore, the magnetic refrigeration is of great potential for the 

replacement of the vapor compression refrigerator technology. In recent decades, a great number of studies 

on magnetic refrigeration at room temperature have been performed. In summary, the researches 

mainly focus on two aspects, i.e., the magnetic prototypes and the numerical models. Recently, a number of 

prototypes have been created. Jacobs et al. (2014) presented and tested a large-scale rotary magnetic 

refrigerator. Its cooling power achieved 2502 W over a temperature span of 11 K, with electrical COP of 1.9. 

Trevizoli et al. (2016) designed and built a single packed-bed active magnetic regenerator (AMR) with 195.5 g 

of Gd spheres. The performance of the AMR under different operating frequencies and utilization factors has 

been evaluated. Its COP achieved 4.6 with a temperature span of 5 K, at 0.25 Hz working frequency. Eriksen 

et al. (2015) built a rotary active magnetic regenerator (AMR) prototype. Its cooling power achieved 103 W 

and the COP was 3.1 under a temperature span of 10.2 K. Meanwhile, a number of numerical studies have 

been performed. Nielsen et al. (2011) reviewed the published models of AMR. Tušek et al. (2011) presented a 

one-dimensional model for the operation of AMR. The model has been used to analyse cases under different 

operating conditions. Lionte et al. (2015) numerically analysed a reciprocating AMR by a two-dimensional 

mathematical model. The regenerator performance was evaluated and the detailed data of the temperature 

and velocity fields in the microchannel was presented. Based on the previous studies, it has been widely 

accepted that the AMR performance and the heat transfer characteristics are closely related, and the heat 

transfer characteristics are directly affected by the particle diameter. However, the utilization factor should also 

be considered when optimizing the particle diameter. The interaction effect among the particle diameter, 

utilization factor and operating frequency should be discussed. In this work, a one-dimensional numerical 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1870124 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Shi H., Ma T., Wang Q., 2018, Numerical investigation of a packed-bed active magnetic regenerator , Chemical 
Engineering Transactions, 70, 739-744  DOI:10.3303/CET1870124   

739

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model for AMR is presented and validated. The performances of AMR with different particle diameters, 

utilization factors and operating frequencies are calculated and analysed.  

2. Mathematic model 

Viewed from the heat transfer process, the magnetic refrigeration system consists of a cold heat exchanger 

(CHEX), an active magnetic regenerator (AMR) and a hot heat exchanger (HHEX). As shown in Figure 1, the 

CHEX and HHEX are connected with the cold end and the hot end of AMR respectively. The AMR is the heat 

transfer channel of packed bed structure. It is made of the magnetocaloric material. In this work, 

gadolinium(Gd) is selected as the magnetocaloric material. 

 

Figure 1: Basic components of heat transfer system of magnetic refrigerator 

Based on the MCE, periodic change of the AMR’s temperature field is achieved by periodic magnetization and 

demagnetization process. The heat or cold quantity generated in the AMR is carried away by the periodic 

oscillating heat transfer fluid to the HHEX and CHEX. Finally, the heat transfer between the fluid and the cold 

source or the heat source occurs in the CHEX and the HHEX. Specifically, each complete AMR cycle consists 

of four stages successively: 

1) Magnetization: No fluid flows, the magnetic field is increased, and the magnetocaloric packed bed heats up 

due to MCE. 

2) Cold flow: Fluid flows from the cold end to the hot end, absorbs heat from the packed bed, and then 

releases heat to the hot source in the HHEX. 

3) Demagnetization: No fluid flows, the magnetic field is reduced, the magnetocaloric packed bed cools down 

due to MCE. 

4) Hot flow: Fluid flows from the hot end to the cold end, releases heat to the packed bed, and then absorbs 

heat from the cold source in the CHEX, which provides the cooling power. 

After a certain number of such magnetocaloric cycles, the temperature gradient along the length direction of 

AMR will be established and the temperature profile of the AMR will achieve a periodic steady state. For 

simplification, several important assumptions are made as follows: 

1) In the process of magnetization, the applied magnetic field is uniform. 

2) There is no heat loss to/from the surroundings in the AMR, the adiabatic boundary condition is adopted at 

both the cold end and hot end of the AMR. 

3) The fluid velocity of cold and hot flow is regarded as constant and the fluid flow is uniform. 

4) The fluid thermal properties are regarded as constants. 

5) The heat transfer process in cold or hot heat exchanger is considered sufficient enough that the fluid can be 

heated or cooled to the cold or hot source temperature. 

6) The magnetization and demagnetization are both linear process, and the time ratio of the 4 stages is kept 

as 1:4:1:4 (From magnetization to hot flow). 

The governing equations for energy of fluid and solid domains can be expressed as follows: 

2
,

2
( )



    

  
    

  

f f f eff eff hts

s f

f f f f f f

T T h AT u dp
u T T

t x c c V c dxx
 (1) 

2
, 0

2
( )

(1 )

 

 

  
   

   

s eff eff hts s s

f s

s s s s s s

h AT T T HM
T T

t c m c c T tx
 (2) 

s
T , fT , u , 

 ,
eff

h ,
ht

A , M , H represent the solid temperature, the fluid temperature, the fluid velocity, the 

porosity, the effective convective heat transfer coefficient, the heat exchange area, the magnetization 

intensity, the magnetic field intensity. The first term and the second term on the right side of Eqs(1, 2) are 

the thermal conductivity term and the convection heat transfer term. The third term on the right side of Eq(1) is 

the viscous loss source term of fluid flow, which reflects the fluid friction loss in the packed-bed structure due 

740



to fluid viscosity. The viscous loss is greatly related to the fluid velocity and the pressure drop. The 

magnetocaloric source term is coupled into the governing equation of solid domain, which is effective at the 

stage of magnetization or demagnetization. The use of the magnetocaloric source term requires the support of 

detailed data of the magnetization intensity and the specific heat capacity of magnetocaloric material under 

different temperatures and magnetic fields. In this work, the related data of Gd are calculated using the Mean 

field model (Tishin A.M. et al., 2003). The magnetic intensity and specific heat capacity of Gd as functions of 

temperature are shown in Figures 2a and b. 

 

200 220 240 260 280 300 320 340 360 380
0.0

2.0x10
5

4.0x10
5

6.0x10
5

8.0x10
5

1.0x10
6

1.2x10
6

1.4x10
6

 

 

M
 (

A
/m

)

T (K)

 0T

 1T

 2T

 

270 280 290 300 310 320
160

180

200

220

240

260

280

300

320

 

 

c
 (

J/
k

g
/K

)

T (K)

 0T

 0.5T

 1T

 1.5T

2T

 
(a) (b) 

Figure 2: The magnetocaloric properties of Gd as a function of temperature under different magnetic fields(a) 

The magnetic intensity; (b) The specific heat capacity  

The effective thermal conductivity for solid and fluid can be expressed as follows (Wakao et al., 1982): 

,
0.5 Pr Re   

f eff f f  (3) 

,
(1 )   

s eff s
 

(4) 

The pressure gradient can be calculated by the correlation as follow, which is given by Kaviany (1995): 

2 3

90 1 1
0.9

Re

 

 

 
 

dp

dx
 (5) 

The Nusselt number is calculated by the correlation for a spherical packed-bed structure given by Wakao et al. 

(1982): 

0.6 1/ 3
2 1.1Re Pr Nu  (6) 

A correction factor related to the packed-bed structure is applied to balance the temperature distribution 

perpendicular to the fluid flow in solid domain as follow (Kitanovski et al., 2015):  

3
1




eff Bi

h
h

 

(7) 

h

 

is the convective heat transfer coefficient and Bi is the Biot number. The adiabatic boundary conditions are 

adopted at both the cold end and hot end of the AMR, and the temperature of fluid that enters the AMR is 

considered as the temperature of cold source or hot source. 

At cold flow stage (Fluid flows from the cold end to the hot end): 

, 0

0

, 0, 0, 0


 

  
   

  

f s s

f x c

x x Lx L

T T T
T T

x x x
 (8) 

At hot flow stage (Fluid flows from the hot end to the cold end): 

741



,

0

, 0, 0, 0


 

  
   

  
0

f s s

f x L h

x Lx x

T T T
T T

x x x
 (9) 

As shown in Figure 3, the computational domain consists of the fluid domain and the solid domain. 

The finite volume method is used for the numerical discretization. The computation is completed with the 

MATLAB software. The unsteady state calculation is carried out with the implicit scheme. The numerical 

calculation will continue for a certain number of cycles until the periodic steady operating state is reached. The 

key performance parameters of the AMR can be obtained when the steady state is reached, specifically 

including the heating power, cooling power, and COP.  

In the cold flow stage, the heating power transferring to the hot source in the HHEX can be expressed as: 

,
( ) / 2( )

 


 



  
m f

m
h f f fh out h m f

Q m c T T dt  (10) 

In the hot flow stage, the cooling power transferring to the cold source in the CHEX can be expressed as: 

2( )

,
2

( ) / 2( )
 

 
 




  

m f

m f
c f f c fc out m f

Q m c T T dt  (11) 


m

and 
f
are the time of magnetization (demagnetization) and hot (cold) flow in one cycle respectively. COP 

is the ratio of the cooling power and the input work, which can be expressed as: 




c

h c

Q
COP

Q Q
 (12) 

 

 

Figure 3: Schematic of the discretization of AMR  

3. Results and discussion 

3.1 Validation of present model 

The present numerical model is validated by solving the temperature field of a single-blow transient passive 

regenerator. As shown in Figure 4, the result of the temperature profile is in accordance with the simplified 

Schumann solution (Shitzer et al., 1983) and Engelbrecht’s results (Engelbrecht et al., 2008) under the same 

operating condition. The validity of the numerical model for unsteady calculation has been proved. 

 

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

5

10

15

20

25

30

35

 

 

T
 (

K
)

x (m)

 Schumann solution(1983) 

 Engelbrecht's results(2008)

 Present calculation

 

Figure 4: Comparison of the regenerator temperature predicted by the simplified Schumann solution (1983), 

Engelbrecht (2008) and present results 

742



3.2 Influence of operating frequency, utilization factor, and particle diameter on AMR performance 

By numerical calculation with the present model, the performance of the AMR under different operating 

frequencies, utilization factors, and particle diameters is obtained. In the calculation, the geometric and 

operating parameters of AMR are listed in Table 1. Utilization factor (UF), a non-dimensional parameter of 

mass flow rate, is defined as below: 




f f f

s s

m c
UF

m c
 (13) 

f
m

 

is the fluid mass flow rate,

 

s
m

 

is the total mass of the magnetocaloric material, 
s

c

 

is the average 

specific heat of the magnetocaloric material. 

Table 1: Constant geometric and operating parameters of AMR 

Constant geometric and operating parameters of AMR 

Dimensions of the AMR 120 mm (length)×40 mm×20 mm 

Heat-transfer fluid Water 

Porosity 0.4 

Operating frequency 1 Hz/2 Hz 

Utilization factor 0-3 

Particle diameter 0.2/0.4/0.6/0.8/1.0 mm 

Magnetic field change 0—1 T 

Temperature span 12 K (Tc =288 K, Th=300 K) 

 

0.0 0.5 1.0 1.5 2.0 2.5
0

50

100

150

200

 

 

C
o

o
li

n
g

 p
o

w
e
r 

Q
c
/W

Utilization factor UF

 d=0.2mm

 d=0.4mm

 d=0.6mm

 d=0.8mm

 d=1.0mm

 
0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

20

 

 

C
O

P

Utilization factor UF

 d=0.2mm

 d=0.4mm

 d=0.6mm

 d=0.8mm

 d=1.0mm

 
(a) (b) 

0.0 0.5 1.0 1.5 2.0 2.5
0

50

100

150

200

250

300

 

 

C
o

o
li

n
g

 p
o

w
e
r 

Q
c
/W

Utilization factor UF

 d=0.2mm

 d=0.4mm

 d=0.6mm

 d=0.8mm

 d=1.0mm

 
0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

20

 

 

C
O

P

Utilization factor UF

 d=0.2mm

 d=0.4mm

 d=0.6mm

 d=0.8mm

 d=1.0mm

 
(c) (d) 

Figure 5: Cooling power and COP as functions of UF for different particle diameters (a) cooling power of 1 Hz; 

(b) COP of 1 Hz; (c) cooling power of 2 Hz; (d) COP of 2 Hz 

As shown in Figure 5, with increase of the UF, the cooling power and COP both increase at first and then 

decrease under different particle diameters and frequencies. Generally, with increase of the UF, the COP 

reaches its maximum earlier than the cooling power. Such results are in good agreement with the previous 

study (Tušek et al.,2011). Specifically, with decrease of the particle diameter, the deviation between the 

optimal UF of cooling power and COP becomes larger, which is not desirable. Besides, as the particle 

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diameter is reduced from 1.0 mm to 0.4 mm, the maximal cooling power increases gradually. Generally, with 

the decrease of the particle diameter, the heat transfer in AMR is more sufficient, thus increasing the 

refrigeration capacity of AMR. However, the maximal cooling power of d=0.4 mm is greater than d=0.2 mm, 

shown in Figures 5(a) and (c). This trend becomes more obvious when the operating frequency is 2 Hz in 

comparison with 1 Hz. Such result can be explained by the viscous loss source term in the energy equation. 

The viscous heating effect has negative effect on the refrigeration capacity, which cannot be ignored for the 

packed-bed AMRs. Especially, the influence of viscous losses is much more significant when the size of 

magnetocaloric particle is small. Comprehensively, the optimal range of particle diameter is 0.4-0.6 mm with 

the utilization factor of 0.7-1.5 in consideration of both cooling power and COP, which also depends on the 

operating frequency. Finally, as the operating frequency increases from 1 Hz to 2 Hz, the cooling power 

increases, however, the overall level of COP decreases as the expense. 

4. Conclusions 

In this study, a 1-D numerical model for AMR is presented and validated. By numerical calculation with the 

present model, the performances of the AMR under different operating frequencies, utilization factors, and 

particle diameters are obtained. Based on the present study, several conclusions are obtained as follows: 

1) With increase of the UF, the cooling power and COP both increase at first and then decrease under 

different particle diameters and frequencies. 

2) With decrease of the particle diameter, the deviation between the optimal UF of cooling power and COP 

becomes larger, which is not desirable. 

3) In this work, the optimal range of particle diameter is 0.4-0.6 mm, with the utilization factor of 0.7 - 1.5 which 

also depends on the operating frequency. A comprehensive optimization combing the numerical model and 

the optimization algorithm is needed for the accurate optimal operation based on this work. 

Acknowledgments 

This work is financially supported by the International Cooperation and Exchanges Project of NSFC of China 

(Grant No. 51120165002), National Natural Science Foundation of China (Grant No. 51306139) and the 

Fundamental Research Funds for the Central Universities. 

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