Ўзбекистон республикаси ахборот технологиялари ва коммуникацияларини

2 Mathematical Model of the 
Active Magnetic Regenerator 
One-dimensional 
time-
dependent models of the AMR have 
been developed in [1, 2, 5, 6, 7] based 
on the law of energy conservation.
The most detailed model, which takes 
into account the axial thermal 
conduction in the bed, thermal 
dispersion, and dissipation of heat as a 
result of friction in fluid, has been 
developed in [5] through the following 
nonlinear system of partial differential 
equations (PDE):  




3
4
3
(
)
1
(1
)
,
f
f
p
f
f
p
m
f
f
p
f
f
f
p
p
T
V
c
T
t
Ac
c
x
h a
T
T
c
T
k
d
c
x
x
V
ff
Ac
D

 
 

 




 

 

 

























 




(1)


(1
)
1
,
m
f
m
m
m
m
S
m
m
m
m
T
h a
T
T
t
c
T
dH
H
dt
T
k
c
x
x
 






 






 











(2) 
where 
)
,
(
t
x
T
T
f
f

is the fluid 
temperature;
)
,
(
t
x
T
T
m
m

is the bed 
temperature;
x
is the spatial 
coordinate 
)
0
(
L
x


;
t
is the 
chronological coordinate 
)
0
(
P
t


;
(Re)
h
h

is 
the 
heat 
transfer 
coefficient between the fluid and 
material; 
a
is the contact area of the 
fluid and material per unit of bed 
volume;
)
(x
Ac
Ac

is the cross 
section area of bed;

is the bed 
porosity 
(pore 
volume 
ratio);
)
,
(
t
x
H
H

is the magnetic induction; 
)
,
(
H
T
c
c
m
m
m

is the heat capacity of 
the material;
m

is the density of the 
material;
L
is the bed length;
P
is the 

---

11 
time period of the flow in either 
direction;
)
(
m
m
m
T
k
k

is the thermal 
conductivity 
of 
the 
material;
)
(
f
f
f
T



is the density of the fluid;
)
(
f
p
p
T
c
c

is the heat capacity of the 
fluid;
p
D
is the particle diameter; 
(Re)
ff
ff

is the fluid friction factor; 
V
is the volumetric flow rate;
)
(
f
f
f
T
k
k

is the thermal conductivity 
of the fluid;
(Re)
d
d

is the 
coefficient of thermal dispersion;
)
(
f
T



is the viscosity of the fluid; 
and
)
Re(
Re
f
T

is the Reynolds 
number.
3 Natural Cubic Interpolating 
Spline for the Heat Capacity of 
Gadolinium 
One of the difficulties in the 
development of a computer simulator 
of AMR is determination of the heat 
capacity of the magnetic material 
(gadolinium), 
)
,
(
H
T
c
c
m
m
m

, which 
depends on the temperature of the 
material, 
)
,
(
t
x
T
T
m
m

, as well as on the 
magnetic induction, 
)
(t
H
H

. To 
obtain a highly accurate approximation 
function for the heat capacity of 
gadolinium, we used the combination 
of natural cubic spline and the least 
squares curve fitting techniques. First, 
we constructed the natural cubic 
splines of the heat capacity of 
gadolinium, 
(
)
m
m
m
c
c T

, using the 
experimental measurements at the 
fixed values of the magnetic induction, 
)
(t
H
H

. Then we used the least 
squares curve fitting technique to 
obtain the approximation function of 
the heat capacity of gadolinium, 
)
,
(
H
T
c
c
m
m
m

, which depends on the 
temperature 
of 
the 
material, 
)
,
(
t
x
T
T
m
m

, as well as on the magnetic 
induction, 
)
(t
H
H

: 


,
,
, 1
(
,
)
,
m
m
m
m i
m i
m
m i
c
c T H
c
T
T
T





where 
 
 
 
2
3
,
1
2
3
4
,
m i
i
i
m
i
m
i
m
c
a
a
T
a
T
a
T







 
 
 
 
2
3
,1
,1
,1
,1
4
,1
1
1
2
3
4
5
,
i
i
i
i
i
i
a
b
b
H
b
H
b
H
b
H









 
 
 
 
2
3
,2
,2
,2
,2
4
,2
2
1
2
3
4
5
,
i
i
i
i
i
i
a
b
b
H
b
H
b
H
b
H









 
 
 
 
2
3
,3
,3
,3
,3
4
,3
3
1
2
3
4
5
,
i
i
i
i
i
i
a
b
b
H
b
H
b
H
b
H









 
 
 
 
2
3
,4
,4
,4
,4
4
,4
4
1
2
3
4
5
,
i
i
i
i
i
i
a
b
b
H
b
H
b
H
b
H









 


 


 
 
,
,
,
,
,
,
1
1
,
2
2
,
3
3
,
4
4
,
5
5
,
,
i j
i j
i j
i j
i j
m i
B
b
B
b
B
b
B
b
B
b
Temp
T






1, 2,3,..., 24,
1, 2,3, 4,
i
j


 
,
( 288.870, 289.079, 289.409, 289.446, 290.049,
290.611, 290.837, 291.132, 291.191, 291.845,
292.366, 292.616, 292.813, 292.894, 293.603,
294.153, 294.419, 294.536, 294.633, 295.363,
295.942, 296.193, 296.270, 296.3
m i
Temp
T


73 ),

---

12 
 
,
1
1
41277883.91
430197.88 1494.50
1.73
33481355.65
345637.51
1189.32 1.36
33481344.55
345637.39
1189.32 1.36
33481327.22
345637.22
1189.32 1.36
33481402.69
345638.00
1189.32 1.36
33481400.50
345637.97
1189
i j
B
b














.32 1.36
21879263.72
225410.39
774.15
0.89
21879253.51
225410.28
774.15
0.89
21879261.24
225410.36
774.15
0.89
21879303.08
225410.79
774.15
0.89
21879292.89
225410.69
774.15
0.89
34427574.32
351866.70
1198.67 1.36












34427563.40
351866.59
1198.67 1.36
34427575.01
351866.71
1198.67 1.36
34427567.79
351866.64
1198.67 1.36
34427556.02
351866.52
1198.67 1.36
4394662.67
43714.80
144.93
0.16
4394655.12
43714.72
144.93
0.16
4394670.












09
43714.88
144.93
0.16
4394633.34
43714.50
144.93
0.16
4394627.92
43714.45
144.93
0.16
11114599.04
111777.88
374.72
0.42
11114598.90
111777.88
374.72
0.42
11114566.80
111777.56
374.72
0.42





































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
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












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








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


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
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
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
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
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












,
2
2
36934992.66
384928.41
1337.19 1.55
25364719.16
261604.67
899.34
1.03
19519921.62
201017.77
689.99
0.79
2456779.77
26762.55
96.96
0.12
50967030.24
525804.19 1808.12
2.07
50017112.71
515998.12 1774.37
2.0
i j
B
b














3
3882970.00
40121.85
138.14
0.16
9223817.09
95157.16
327.18
0.37
472935.11
4743.79
15.89
0.02
30029296.32
308566.31
1056.93 1.21
25515453.85
262249.27
898.51
1.03
21407195.72
218817.89
745.51
0.85
15571819.86
159













031.84
541.33
0.61
30323660.34
310129.25 1057.21
1.20
25187270.73
257646.24
878.45
1.00
19986436.54
204604.10
698.13
0.79
12365591.79 125048.82
421.54
0.47
8280509.14
83440.13
280.27
0.31
27356313.52
277673.03
93













9.51
1.06
1341918.70
13444.98
44.92
0.05
1033975.32
10639.74
36.46
0.04
4566031.67
46080.10
155.03
0.17
4492152.01
45332.02
152.51
0.17
36073644.01
365290.37 1232.98
1.39























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







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
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



---

13 
 
,
3
3
15672405.43
163121.93
565.93
0.65
1771608.53
17908.32
60.30
0.07
2319725.28
24502.26
86.24
0.10
11842987.16
123207.32
427.25
0.49
21991992.97
226750.59
779.29
0.89
21295392.54
219559.53
754.55
0.86
8377
i j
B
b















762.50
86313.45
296.40
0.34
12116332.94 124837.93
428.73
0.49
7914395.98
81547.41
280.06
0.32
10804577.62
110872.95
379.26
0.43
7494453.23
76907.40
263.09
0.30
5643959.94
57792.14
197.24
0.22
1559220.94
15942.16
5













4.32
0.06
7951701.61
81417.87
277.87
0.32
4698664.38
48178.72
164.65
0.19
884750.15
9281.47
32.42
0.04
8173866.94
83021.85
281.09
0.32
5314325.73
53895.94
182.20
0.21
13580528.91
138063.75
467.87
0.53
2895206.01
29













280.19
98.70
0.11
4637514.05
46942.18
158.38
0.18
3069503.58
31060.53
104.76
0.12
3121219.02
31584.19
106.53
0.12
20699776.16
209521.02
706.91
0.80






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


,
4
4
2360656.0
24553.2
85.1
0.0984
367045.1
3863.9
13.6
0.0158
1185308.7 12346.0
42.9
0.0496
2503917.2
26012.9
90.1
0.1040
3194601.9
32927.4 113.1
0.1296
3042617.2
31358.4 107.7
0.1234
1566309.7
16130.2
55.4
0
i j
B
b
















.0633
2314021.0
23835.1 81.8
0.0937
1732213.0
17841.0
61.2
0.0701
1420454.1 14566.6
49.8
0.0567
698249.2
7156.0
24.4
0.0278
803290.8
8238.3
28.2
0.0321
13654.0
131.7
0.4
0.0005
871460.7
8934.2
30.5
0.0348
323581.1
3















336.1
11.5
0.0131
508540.9
5150.6
17.4
0.0196
1543816.3 15699.6
53.2
0.0601
971910.1
9874.4
33.4
0.0378
2116463.9
21528.4
73.0
0.0825
658394.8
6655.8
22.4
0.0252
1038532.6
10509.3
35.4
0.0399
859330.5
8694.2
29.3
0.















0330
869673.6
8799.0
29.7
0.0334
3303633.1
33436.4 112.8
0.1269

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
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


---

14 
 
,
5
5
112571.8
1170.4
4.1
0.0047
32827.1
342.8
1.2
0.0014
79584.9
827.5
2.9
0.0033
138189.8
1434.9
5.0
0.0057
146736.1 1512.1
5.2
0.0059
136603.8 1407.5
4.8
0.0055
77551.3
798.4
2.7
0.0031
120277.6 1238.6
4.3
0.0
i j
B
b


















049
94419.4
972.2
3.3
0.0038
63213.9
648.1
2.2
0.0025
15067.1
154.1
0.5
0.0006
44994.7
461.7
1.6
0.0018
1687.8
16.6
0.1
0.0001
37650.7
386.3
1.3
0.0015
10256.8
106.4
0.4
0.0004
45217.8
459.4
1.6
0.0018
86629.0
881.3
3.

















0
0.0034
53948.7
548.5
1.9
0.0021
104817.9
1066.4
3.6
0.0041
33925.1
342.8
1.2
0.0013
59267.5
599.7
2.0
0.0023
52099.4
527.1
1.8
0.0020
52690.4
533.1
1.8
0.0020
160866.6 1628.1
5.5
0.0062

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
In Figure 2, the fitted surface of 
the heat capacity of the magnetic 
material is illustrated, where x is the 
temperature of gadolinium in Kelvin, y 
is the magnetic induction in Tesla, and 
z is the heat capacity of gadolinium in 
J/(kg*K): 
Figure 2: The heat capacity of 
gadolinium.