Ўзбекистон республикаси ахборот технологиялари ва коммуникацияларини

Расулов Акбарали Махаматович, 
Муҳаммад ал-Хоразмий номидаги 
Тошкент ахборот технологиялари 
университети Фарғона филиали директори, 
физика-математика фанлари доктори, профессор
 
 

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8 
УДК 501 
 
 
 
 
 
 
 
 
 
ТАБИИЙ ФАНЛАРНИ ЎҚИТИШДА АХБОРОТ-
КОММУНИКАЦИЯ ТЕХНОЛОГИЯЛАРИНИНГ ЎРНИ 
 

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9 
NATURAL CUBIC INTERPOLATING SPLINE FOR THE HEAT 
CAPACITY OF GADOLINIUM 
B. Siddikov 
Ferris State University 
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.
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.

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10 
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.