International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04
5
Development of Computational Algorithms to
Control the State of Total Thermal Energy
of an Elastic Rod
Kalimoldayev M.N., Amirtayev K.B., Naizabayeva L.
Institute of Information and Computing Technologies
Almaty, Republic of Kazakhstan
A.Yesevi International Turkish-Kazakh University
Turkestan, Republic of Kazakhstan
[email protected], [email protected], [email protected]
Abstract--
The universal numerical algorithm based on
minimization to the full thermal energy functional by node
temperature values was developed. The field of temperature
distribution and of the potential energy functional of thermo
elastic deformation of the rod by the nodal values of
displacement were built; the deformation and stress field
was constructed.
Index Term — Elastic deformation, thermo elastic strain,
functional, heat isolation, heat exchange.
I. INTRODUCTION
A. Relevance
Today, the existing studies of thermo-elastic state of onedimensional elements of the construction are relatively of
private character and are not subject to all exploitation
conditions [1-4]. Also, very important tasks of technology
and chemical engineering, thermo elastic steady state of
one-dimensional construction elements with axial force,
unequal distribution of temperature field, thermal flow, heat
exchange and thermal insulation are worth careful studying.
In this paper, on the basis of laws of energy conservation
and change, combined by the finite element method based
on minimization of the energy functional, the linear system
of algebraic equations whose allows finding the values of
variables, was obtained. The task is solved separately first
on modeling and the definition of the field of temperature
distribution along the length of the rod with partial heat
isolation, heat exchange, as well as the presence of the heat
flow and temperatures. After that, while minimizing the
potential energy functional, the system of algebraic
equations relatively to node displacement was obtained.
After defining the field of removal on the basis of the
Cauchy ratio of and the Hooke's law the fields of
deformation and voltage talking into account temperature
field distribution law, were determined.
B. The scientific novelty
-On the basis of the energy principles and finite element
method with quadratic elements, the universal numerical
algorithm for solving applied tasks of thermo elastic state of
a partially heat insulated rod of limited length, in the
presence of axial force, temperature, heat flow and heat
exchange, was developed.
-The field regularities of distribution of the temperature,
displacement, deformation and strain along the length of the
rod, depending on the area of the lateral heat-isolated
surface, on the applied temperature, heat flow, heat
exchange and heat exchange coefficients with the
surrounding parts of the lateral surface, were revealed.
II. ROD INSULATED ALONG THE LATERAL
SURFACE WITH IT ’ S ENTIRE LENGTH
Earlier the authors made the fundamental justification for
the fact that the function which gives the minimum to the
functional that characterizes the full thermal energy is the
solution to of the heat-transfer equation with the appropriate
natural boundary conditions [2]. The field of temperature
distribution along the rod length, with rod lengthening and
tensile strength was defined.
h, TОC
P
q

i


j
k
Fig. 1. Rod insulated along the lateral
surface with it’s entire length
The differential equation for temperature distribution
along the full rod length of the rod was considered:
K xx
d 2T
 0 (1)
dx 2
with the following boundary conditions:
K xx
K xx
dT
 q  0, x  0
dx
dT
 h(T  TOC )  0, x   (2)
dx
If the coefficient of thermal expansion of the rod material
is  , then the lengthening of the rod due to field of
temperature distribution is

q
q 
 T   T ( x)dx   TОС  

h 2 K xx 

0

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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04
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To compare our results with those obtained by other
authors, the parameter values representing the geometric,
physical and mechanical and thermal properties of the rod
are defined as follows:
elements are under the influence of heat flow and tensile
strength.
K xx  75 (Wt / см о С ) ; h  10 (Wt / см 2 о С ) ;
Here the numerical study of lengthening of partially heatisolated rod in the presence of a heat sources and tensile
strength is given. The numerical study with the different
source data was conducted.
  125 107 (1/ оС ) ; P  1000 кG ;
q  150 (Wt / см 2 ) ;   7,5 (см) ;
III. CALCULATION SCHEME OF THE TASK
F  4 (см 2 ) ; TОС  40 (  C )
The
temperature
values
T(x=0)=T1
Ti , T j , Tk in
the
points
х1
( x  xi  0) , ( x  x j   / 2) , ( x  xk  ) of the rod
will be:
150 150  7,5


 70 (C ); 
10
75


150 150  7,5


T  x  x j    T j  40 

 62,5 (C );
2
10
2

75




150
T ( x  xk  )  Tk  40 
 55 ( C ).

10

h1 , TOC1
х2
T ( x  xi  0)  Ti  40 
These found values exactly coincide with the values found
previously. Extension of the rod due to the impact of the
tensile strength has the following form
P
 P 
EF
х3
h1 , TOC 2
х4
х5=L
h1 , TOC 3
(4)
P
Then, using (3), (4) the value of the ratio is
x
 T
EF 
q q  EF 
q q  (5)
 
TОС  

TОС  

 P
P 
h 2 K xx 
P 
h 2 K xx 
If  P  P  1000  7,5  7500  0,0009375 (см) ,
EF 2  106  4 8000000
Then

q
q 
 T   TОС  

h 2  K xx 

150 150  7,5 

 125 10 7  7,540 


10
2  75 

Fig. 2. Calculation scheme of the task
To do it, the rod with length L will be to discrete quadratic
finite elements with three nodes.
Then, for the final elements in the heat isolated lateral
surface parts, the view of the functional, expressing
conservation and change of the total thermal energy is as
follows
K  T 
I i   xx   dV (6)
2  x 
Vi
2
1125 

 125 10 7  7,540  15 

150 

 125 10 7  7,562,2  0,00585938 (см ).
For the finite element heat isolated by the lateral surface,
the functional type is as follows
Then for the rod under consideration there will be
 T
 6,25 .
 P
As it can be seen, here it is necessary to take into account
the fact that the extension of the rod due to the temperature
will be much more considerable than the extension of the
rod due to the applied tensile force, when the rod structural
K  T 
h
I n   xx   dV   3 (T  TОС 3 ) 2 dS (7)
2  x 
2
n
Vn
Slsс
2
On the lateral surface x1  x  x2 of the rod, the heat
exchange with the environment takes place.
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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04
K  T 
h
I j   xx   dV   1 (T  Tос1 ) 2 dS (8)
2  x 
2
j
Vj
Slsп
7
2
1500
1
1000
2
500
the rod, the heat
Ik 
K xx
2
Vk

h
 T 
2

 dV   2 (T  TОС 2 ) dS (9)
2
k
 x 
Slsп
76,2
72,2
68,2
64,2
60,2
56,2
52,2
48,2
44,2
40,2
36,2
32,2
28,2
24,2
20,2
16,2
-1000
-1500
3
-2000
4
-2500
Then, for the rod in general, the expression of the
corresponding functional is as follows
I  Ii  I j  I k  I n
8,2
-500
2
12,2
exchange with the environment takes place as well.
4,2
0
0,2
x3  x  x4 of
On the lateral surface
Длина стержня
Fig. 4.The law on the distribution
of displacements of nodal points along
the rod length.
(10)
0,004
79,2
74,8
66
64,2
70,4
61,6
57,2
52,8
48,4
44
39,6
35,2
30,8
26,4
22
60,2
1
-0,006
2
3
-0,01
4
-0,012
стержня
Fig. 5.The field of Длина
distribution
of along
the length of the rod
0,0006
0,0005
0,0004
0,0003
0,0002
1
3
2
0,0001
76,2
72,2
68,2
56,2
52,2
48,2
44,2
40,2
36,2
32,2
28,2
24,2
20,2
16,2
12,2
8,2
4,2
0
-0,0001
0,2
After testing of the developed computational algorithm,
the influence of local temperature and heat exchange on the
thermal-strained and deformed state of the test rod is
analyzed. To do this, the values of the compressive force
and the true stress on different rod parts are
calculated[1].The influence of the heat flow on the thermal
strained deformed and state of the test rod was analyzed [2].
The analysis of the effect of heat flow on the crosssectional area and analyses of the effect of the heat
exchange with the environment through apportion of the
lateral surface on the thermal-strained and deformed state of
the test rod were carried out. These results are shown in Fig.
and 3-6 inTable1.
17,6
-0,004
-0,008
IV. T HE ANALYSIS OF THE EFFECT
13,2
-0,002
8,8
0

0
4,4

0,002
Перемещение
The influence of temperature and tensile strength on the
lengthening of the rod was analyzed [1].
Moreover, the lengthening of considered rod is affected
the by heat transfer coefficient h, Wt см 2  С between the
rod material and the cross-sectional area of the clamped rod
ends [2].
-0,0002
-0,0003
100
4
-0,0005
80
70
Длина стержня
3
60
50
Fig. 6.The field of distribution along
the length of the rod
2
1
40
30
20
10
Длина стержня
Fig. 3.The field of temperature distribution along
the length of the rod
80
76
72
68
64
60
56
52
48
44
40
36
32
28
24
20
16
8
12
4
0
0
Температура
4
-0,0004
90
V. CONCLUSION
On the basis of the minimization of the functional
characterizing the full thermal energy using quadratic finite
elements with three nodes, the computational algorithm of
tasks of determining the temperature distribution field in the
rods of limited length in the presence of heat flow, heat
exchange and partial insulation was constructed.
The corresponding developed computational algorithms
were tested by several test problems.
The corresponding calculation algorithm of numerical
study of thermal strained and deformed state of the rod
partially thermally insulated and clamped by two ends, in
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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:14 No:04
8
the presence of the local temperature and heat exchange,
was developed.
[1]
[2]
[3]
[4]
[5]
REFERENCES
Kudaykulov A.K., Amirtayev K.B., Tuleuova R., Utebaev
U.B., Tokkuliev B.M. Steady thermal stress-strained and
deformed state of the rod in the presence of heat sources //
Proceedings of the Fifth All-Russian scientific conference with
international participation "Mathematical modeling and
boundary value problems" - Samara - 2008. - p. 161-165.
Kudaykulov A.K., Amirtayev K.B., Utebaev U.B. Numerical
study of strain-deformed state of the rod filled-rigidly at both
ends, when exposed along the length to the parabolic law of the
temperature field// Materialy IV mezinarodni vedeckoprakticka conference «Научно пространство на европа2008». – Sofia – 2008. – Т.28, – p. 21–24.
Amirtayev K.B., Utebaev U.B., Tokkuliev B.M.,
Zhumadillaeva A. Determination of the law of temperature
distribution in a partially insulated tube of limited length, with
the heat flow on the inner bounded surface of closed mid-pipe
// Materialy IV mezinarodni vedecko-prakticka conference
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Kudaykulov A.K., Arapov B.R., Kenzhegulov B.Z.,
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solution of the established thermal strained and deformed state
of the heat exchanger in the presence of internal heat flow and
external heat transfer of constant intensity// Materialy IV
mezinarodni vedecko-prakticka conference «Veda a vznik2008/2009». – Praha: Publishing house «Education and
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