Natural Cubic Interpolating Spline for the Heat Capacity of ... - wseas.us

Recent Researches in Applied Mathematics and Economics

Natural Cubic Interpolating Spline for the Heat Capacity of Gadolinium BAKHODIRZHON SIDDIKOV Mathematics Department Ferris State University 820 Campus Drive, Big Rapids, MI 49307 U.S.A. [email protected] Abstract: A time-dependent one-dimensional model of the Active Magnetic Regenerator (AMR) that takes into account most of the physical and practical design problems for the AMR is given as a highly nonlinear system of partial differential equations. The accurate approximation function for the heat capacity of the magnetic material (gadolinium) is obtained by using the natural cubic spline and the least squares curve fitting techniques. Key-Words: magnetic refrigeration, heat capacity of gadolinium, numerical simulation, natural cubic spline

1 Introduction Magnetic Refrigeration (MR) is rapidly developing and becoming competitive with conventional gas compression technology, primarily because the most inefficient component of the refrigerator – the compressor – is eliminated. In addition, MR operating near room temperature provides important environmental benefits. MR uses a solid magnetic material as the cooling source and water (perhaps with antifreeze additives) as the heat transfer medium. There is no need to use volatile chemicals with potential environmental problems. MR is based on the magnetocaloric effect, where a magnetic material changes its temperature with variations of magnetic field. One of the key components of MR is the Active Magnetic Regenerator (AMR), which produces refrigeration without gas expansion by using the magnetocaloric effect. An AMR cycle consists of four operations: bed magnetization, warming of the magnetic material; fluid flow from cold to hot reservoirs through the bed, transferring heat to the Hot Heat Exchanger, HHEX (this semi-cycle is called the Hot Blow Period); bed demagnetization, cooling of the magnetic material; fluid flow from hot to cold reservoirs through the bed, and absorption of heat at the Cold Heat Exchanger, CHEX (called the Cold Blow Period) [1, 2]. Regeneration occurs during fluid flows. A schematic of the AMR is illustrated in Figure 1.

ISBN: 978-1-61804-076-3

Figure 1: Schematic of the AMR. To predict operating characteristics and improve the design of the Active Magnetic Regenerative Refrigerators (AMRR) it is desirable to develop mathematical models for AMR and accurate, stable numerical solvers of the model. In [3, 4], we developed numerical scheme for the model [5] to obtain a computer simulator of AMR. One of the difficulties in this work is obtaining a dependable approximation function for the heat capacity of the magnetic material (gadolinium). In [3, 4], we used the least squares curve fitting technique to obtain the approximation function for the heat capacity of gadolinium. Further research in this field indicated that we need more accurate approximation function for the heat capacity of gadolinium to improve the simulation results. This paper reports on obtaining of such a highly accurate approximation function.

125

Recent Researches in Applied Mathematics and Economics

the cross section area of bed; ε is the bed porosity (pore volume ratio); H = H ( x , t ) is the magnetic

2 Mathematical Model of the Active Magnetic Regenerator

induction; cm = cm (Tm , H ) is the heat capacity of

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):

the material; ρ m is the density of the material; L is the bed length; P is the time period of the flow in either direction; k m = k m (Tm ) is the thermal conductivity of the material; ρ f = ρ f (T f ) is the density of the fluid;

capacity of the fluid; D p is the particle diameter;

ff = ff (Re) is the fluid friction factor; V& is the volumetric flow rate; k f = k f (T f ) is the thermal conductivity of the fluid; d = d (Re) is the coefficient of thermal dispersion; µ = µ (T f ) is the

∂T f

V& ∂ =− ⋅ ( ρ f ⋅ cp ⋅Tf ) ∂t Ac ⋅ ε ⋅ ρ f ⋅ c p ∂x

+

h⋅a ⋅ (T − T ) ε ⋅ ρ f ⋅ cp m f

viscosity of the fluid; and

1

ε ⋅ ρ f ⋅ cp

⋅

Re = Re(T f ) is the

Reynolds number.

(1)

+

c p = c p (T f ) is the heat

∂T  ∂  ε ⋅kf + d )⋅ f  (  ∂x  ∂x 

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), cm = cm (Tm , H ) , which depends on the

(1 − ε ) ⋅V& 3 ⋅ ff + 4 , ε ⋅ Ac3 ⋅ D p

temperature of the material, Tm = Tm ( x, t ) , as well as on the magnetic induction, H = H (t ) . 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, cm = cm (Tm ) , using the experimental measurements at the fixed values of the magnetic induction, H = H (t ) . Then we used the least squares curve fitting technique to obtain the approximation function of the heat capacity of gadolinium, cm = cm (Tm , H ) , which depends on

∂Tm h⋅a = (T f − Tm ) ∂t (1 − ε ) ⋅ ρ m ⋅ cm +

∂Tm dH ⋅ ∂H S dt

+

∂T  1 ∂  ⋅  km ⋅ m , ρ m ⋅ cm ∂x  ∂x 

(2)

the temperature of the material, Tm = Tm ( x, t ) , as well as on the magnetic induction, H = H (t ) :

where T f = T f ( x, t ) is the fluid temperature;

Tm = Tm ( x, t ) is the bed temperature; x is the spatial coordinate (0 ≤ x ≤ L) ; t is the chronological coordinate (0 ≤ t ≤ P ) ; h = h (Re)

cm = cm (Tm , H ) = {cm,i , Tm ,i ≤ Tm < Tm ,i +1} where

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; Ac = Ac(x ) is

ISBN: 978-1-61804-076-3

cm ,i = a1i + a 2i ⋅ (Tm ) + a3i ⋅ ( Tm ) + a 4i ⋅ (Tm ) , 2

126

3

Recent Researches in Applied Mathematics and Economics

a1i = b1i ,1 + b 2i ,1 ⋅ ( H ) + b3i ,1 ⋅ ( H ) + b4i ,1 ⋅ ( H ) 2

B1 = ( b1i , j ) =

3

+ b5i ,1 ⋅ ( H ) ,

 41277883.91  −33481355.65   −33481344.55   −33481327.22  −33481402.69   −33481400.50  21879263.72   21879253.51  21879261.24   21879303.08   21879292.89  −34427574.32   −34427563.40  −34427575.01   −34427567.79   −34427556.02  4394662.67   4394655.12  4394670.09   4394633.34  4394627.92  11114599.04  11114598.90 11114566.80

4

a 2i = b1i ,2 + b2i ,2 ⋅ ( H ) + b3i ,2 ⋅ ( H ) + b4i ,2 ⋅ ( H ) 2

3

+ b5i ,2 ⋅ ( H ) , 4

a3i = b1i ,3 + b2i ,3 ⋅ ( H ) + b3i ,3 ⋅ ( H ) + b4i ,3 ⋅ ( H ) 2

3

+ b5i ,3 ⋅ ( H ) , 4

a 4i = b1i ,4 + b2i ,4 ⋅ ( H ) + b3i ,4 ⋅ ( H ) + b4i ,4 ⋅ ( H ) 2

3

+ b5i ,4 ⋅ ( H ) , 4

B1 = ( b1i , j ) , B 2 = ( b 2i , j ) , B3 = ( b3i , j ) , B 4 = ( b 4i , j ) , B5 = ( b5i , j ) , Temp = (Tm ,i ) , i = 1, 2,3,..., 24,

j = 1, 2,3, 4,

Temp = (Tm,i ) = ( 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, 296.373 ),

ISBN: 978-1-61804-076-3

127

−430197.88 1494.50 −1.73 −1189.32 1.36  345637.51 −1189.32 1.36  345637.39  −1189.32 1.36  345637.22 −1189.32 1.36  345638.00  345637.97 −1189.32 1.36  −225410.39 774.15 −0.89  −225410.28 774.15 −0.89 −225410.36 774.15 −0.89 −225410.79 774.15 −0.89  −225410.69 774.15 −0.89 351866.70 −1198.67 1.36   −1198.67 1.36  351866.59 −1198.67 1.36  351866.71  −1198.67 1.36  351866.64  −1198.67 1.36  351866.52 −43714.80 144.93 −0.16  −43714.72 144.93 −0.16 −43714.88 144.93 −0.16  −43714.50 144.93 −0.16 −43714.45 144.93 −0.16  −111777.88 374.72 −0.42  −111777.88 374.72 −0.42 −111777.56 374.72 −0.42

Recent Researches in Applied Mathematics and Economics

B 2 = ( b 2i , j ) =  −36934992.66  25364719.16  19519921.62   −2456779.77 50967030.24  50017112.71 3882970.00  9223817.09  −472935.11   −30029296.32   −25515453.85  21407195.72  15571819.86 30323660.34   25187270.73  19986436.54  −12365591.79   −8280509.14  −27356313.52   −1341918.70 1033975.32   −4566031.67   −4492152.01 36073644.01

B3 = ( b3i , j ) = −1337.19 1.55  384928.41 −261604.67 899.34 −1.03  −201017.77 689.99 −0.79   −96.96 26762.55 0.12  −525804.19 1808.12 −2.07   −515998.12 1774.37 −2.03  −40121.85 138.14 −0.16   −95157.16 327.18 −0.37  −15.89 4743.79 0.02  −1056.93 1.21  308566.31  262249.27 −898.51 1.03  −218817.89 745.51 −0.85   −0.61  −159031.84 541.33 −310129.25 1057.21 −1.20   −257646.24 878.45 −1.00   −204604.10 698.13 −0.79  −421.54 0.47  125048.82  −280.27 0.31  83440.13 277673.03 −939.51 1.06   13444.98 0.05  −44.92 −10639.74 36.46 −0.04   46080.10 −155.03 0.17   45332.02 −152.51 0.17  −365290.37 1232.98 −1.39 

ISBN: 978-1-61804-076-3

15672405.43  −1771608.53   2319725.28  11842987.16  −21991992.97   −21295392.54  −8377762.50   −12116332.94  −7914395.98  10804577.62  7494453.23  −5643959.94   −1559220.94  −7951701.61   −4698664.38   −884750.15 8173866.94  5314325.73 13580528.91   −2895206.01  −4637514.05   −3069503.58   −3121219.02  −20699776.16

128

−163121.93 565.93 −60.30 17908.32 −24502.26 −123207.32 226750.59 219559.53 86313.45

86.24 427.25 −779.29 −754.55 −296.40

124837.93 81547.41 −110872.95 −76907.40

−428.73 −280.06 379.26 263.09

57792.14 15942.16 81417.87 48178.72

−197.24 −54.32 −277.87 −164.65

9281.47 −83021.85 −53895.94 −138063.75 29280.19

−32.42 281.09 182.20 467.87 −98.70

46942.18 31060.53 31584.19 209521.02

−158.38 −104.76 −106.53 −706.91

−0.65 0.07  −0.10  −0.49 0.89   0.86  0.34   0.49  0.32  −0.43  −0.30 0.22   0.06  0.32   0.19   0.04  −0.32  −0.21 −0.53  0.11  0.18   0.12   0.12  0.80 

Recent Researches in Applied Mathematics and Economics

B 4 = ( b 4i , j ) =  −2360656.0  −367045.1   −1185308.7   −2503917.2 3194601.9  3042617.2 1566309.7   2314021.0 1732213.0   −1420454.1   −698249.2 803290.8   −13654.0 871460.7  323581.1   −508540.9  −1543816.3   −971910.1  −2116463.9  658394.8 1038532.6  859330.5  869673.6 3303633.1

B5 = ( b5i , j ) = 24553.2 3863.9 12346.0 26012.9 −32927.4 −31358.4 −16130.2

−23835.1 −17841.0 14566.6 7156.0 −8238.3 131.7 −8934.2 −3336.1 5150.6 15699.6 9874.4 21528.4 −6655.8

−10509.3 −8694.2 −8799.0 −33436.4

−85.1 0.0984  −13.6 0.0158  −42.9 0.0496   −90.1 0.1040  113.1 −0.1296   107.7 −0.1234  55.4 −0.0633   −0.0937  81.8 61.2 −0.0701  −49.8 0.0567   −24.4 0.0278  28.2 −0.0321   −0.4 0.0005  −0.0348  30.5  −0.0131  11.5  −17.4 0.0196  −53.2 0.0601   −33.4 0.0378  −73.0 0.0825   22.4 −0.0252  −0.0399  35.4  29.3 −0.0330   29.7 −0.0334  112.8 −0.1269 

112571.8 32827.1  79584.9  138189.8  −146736.1   −136603.8  −77551.3   −120277.6  −94419.4  63213.9  15067.1  −44994.7  1687.8  −37650.7   −10256.8   45217.8 86629.0  53948.7 104817.9   −33925.1  −59267.5   −52099.4   −52690.4  −160866.6

−1170.4 4.1 −342.8 1.2 −827.5 −1434.9 1512.1 1407.5 798.4

2.9 5.0 −5.2 −4.8 −2.7

1238.6 972.2 −648.1 −154.1

−4.3 −3.3 2.2 0.5

461.7 −16.6 386.3 106.4

−1.6 0.1 −1.3 −0.4

−459.4 −881.3 −548.5 −1066.4 342.8

1.6 3.0 1.9 3.6 −1.2

599.7 527.1 533.1 1628.1

−2.0 −1.8 −1.8 −5.5

−0.0047  −0.0014  −0.0033   −0.0057  0.0059   0.0055  0.0031   0.0049  0.0038  −0.0025   −0.0006  0.0018   −0.0001  0.0015   0.0004   −0.0018  −0.0034   −0.0021  −0.0041   0.0013  0.0023   0.0020   0.0020  0.0062 

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):

ISBN: 978-1-61804-076-3

129

Recent Researches in Applied Mathematics and Economics

[6] K. Matsumoto and T. Hashimoto, Thermodynamic Analysis of Magnetically Active Regenerator, Proceedings of International Conference Cryogenics and Refrigeration, Hangzhou, China, 1989, pp. 110-115. [7] R. Li, O. Yoshida and T. Hashimoto, Measurement of Ineffectiveness on Regenerators Packed with Magnetic Regenerator Materials between 4 and 35K, Advances in Cryogenic Engineering, Vol. 35(B), 1990, pp. 1183-1190.

Figure 2: The heat capacity of gadolinium.

4 Conclusion We obtained a highly accurate approximation function of the heat capacity of gadolinium, cm = cm (Tm , H ) by using the natural cubic spline and the least squares curve fitting techniques. We are planning to use the obtained approximation function cm = cm (Tm , H ) in our future research work to improve the results of the computer simulator of AMR.

References: [1] A. J. DeGregoria, Modeling the Active Magnetic Regenerator, Advances in Cryogenic Engineering, Vol. 37(B), 1992, pp. 867-873. [2] A. Smaili and R. Chahine, Thermodynamic Investigation of Optimum Active Magnetic Regenerators, Cryogenics, Vol. 38, No. 2, 1998, pp. 247-252. [3] B. Siddikov, D. Schultz and B. Wade, Numerical Simulation of the Active Magnetic Regenerator, International Journal of Computers and Mathematics with Applications, Vol. 49, No. 9-10, 2005, pp. 1525-1538. [4] B. Siddikov, D. Schultz and B. Wade, Numerical Simulation of the Passive Regenerator, International Journal of Applied Science and Computations, Vol. 9, 2002, pp. 89-97. [5] A. J. DeGregoria, J. A. Barclay, P. J. Claybaker, S. R. Jaeger, S. F. Kral, R. A. Pax, J. R. Rowe and C. B. Zimm, Preliminary Design of a 100W 1.8K to 4.7K Regenerative Magnetic Refrigerator, Advances in Cryogenic Engineering, Vol. 35(B), 1990, pp. 1125-1131.

ISBN: 978-1-61804-076-3

130