The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating
For the isotropic beam elements of the technological equipment (BETE), which are used in the production of carbon-carbon composite materials (CCCM) in direct heating furnaces, the analytical solution of the temperature stress-strain state, taking into account the stiffness of the spring of the curre...
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| Zitieren: | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating / M.V. Meltyukhov, Y.V. Kravtsov // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 143-147. — Бібліогр.: 2 назв. — англ. |
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| citation_txt | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating / M.V. Meltyukhov, Y.V. Kravtsov // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 143-147. — Бібліогр.: 2 назв. — англ. |
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| description | For the isotropic beam elements of the technological equipment (BETE), which are used in the production of carbon-carbon composite materials (CCCM) in direct heating furnaces, the analytical solution of the temperature stress-strain state, taking into account the stiffness of the spring of the current collector, was obtained for the first time in the work. Formulas for calculating the critical diameters of loss of compressive strength are given. Using the example of graphite and molybdenum rods, it was established in the work that the decisive factor of the load-bearing capacity is the loss of stability of the elements under consideration.
Для ізотропних стержневих елементів технологічного оснащення (СЕТО), яке використовується при виготовленні вуглець-вуглецевих композиційних матеріалів (ВВКМ) у печах прямого нагріву, в роботі вперше отримане аналітичне рішення температурного напружено-деформованого стану з урахуванням жорсткості пружини струмовідводу. Наводяться формули для обчислення критичних діаметрів втрати міцності на стискання. На прикладі графітових і молібденових стержнів у роботі встановлено, що вирішальним фактором несучої здатності є втрата стійкості елементів, що розглядаються.
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 143
https://doi.org/10.46813/2023-144-143
UDC 621.762
THE TERMAL DEFORMATION OF ISOTROPIC BEAM ELEMENTS
OF TECHNOLOGICAL EQUIPMENT IN THE MANUFACTURING
OF CARBON-CARBON COMPOSITE MATERIALS
IN FURNACES OF DIRECT HEATING
M.V. Meltyukhov, Y.V. Kravtsov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: meltyukhov_m@kipt.kharkov.ua; tel./fax +38(057)349-10-61
For the isotropic beam elements of the technological equipment (BETE), which are used in the production of
carbon-carbon composite materials (CCCM) in direct heating furnaces, the analytical solution of the temperature
stress-strain state, taking into account the stiffness of the spring of the current collector, was obtained for the first
time in the work. Formulas for calculating the critical diameters of loss of compressive strength are given. Using the
example of graphite and molybdenum rods, it was established in the work that the decisive factor of the load-bearing
capacity is the loss of stability of the elements under consideration.
INTRODUCTION
In the production of CCCM by the method of ther-
mogradient gas-phase densification in direct heating
furnaces, beam elements are often used as equipment,
which are in conditions of high temperatures (up to
1500 °C) and compression, which occurs as a result of
thermal expansion. It is advisable to study the stress-
strain state of such elements at the stage of densification
processes planning.
Fig. 1. General view of the direct heating furnace:
1 – upper power supply; 2 – equipment element (heater
rod); 3 – lower power supply; 4 – camera body;
5 – product made of CCCM;
6 – spring of the upper current supply
The general view of the direct heating furnace is
shown in Fig. 1. It is a chamber 4 that is cooled. During
the operation of the furnace, the initial heating of the
CCCM frame for densification is often carried out using
the rod element of the equipment 2, which is heated to a
temperature of 900 °C. For reliable electrical contact,
the spring 6 is pressed using a screw mechanism. This
spring should also compensate for the thermal expan-
sion of rod 2 and CCCM. At the end of the process, the
temperature in the rod and in the center of the product
reaches 1300…1500 °С. At the same time, the central
areas of the finished product will be compressed be-
tween the current leads due to thermal expansion with a
temperature difference along the radius of about 600 °C.
The equipment element 2 can be a molybdenum rod
with a diameter of 6…8 mm or a rod made of carbon
material.
1. STRESS-STRAIN STATE (SSS)
OF THE FREE HEATER ROD WITHOUT
TAKING INTO ACCOUNT THE CURRENT
SUPPLY SPRING
First, free expansion of BETE due to heating is con-
sidered. The calculation diagram of the deformation of
rod element 2 for this case is presented in Fig. 2. It is a
rod of constant cross-section fixed from below, which is
heated to a temperature of ΔT. Let's determine the dis-
placement of the free end of the rod. If the diameter of
the BETE is small, then a hole is made in the upper cur-
rent lead, which should compensate for the elongation
of the rod, provided that electrical contact is maintained.
Then moving the free end of the rod will give us infor-
mation about what the depth of this hole should be.
Fig. 2. Calculation diagram the free extension of the
heater rod: 1 – BETE; 2 – lower current supply
The elongation of BETE can be found using the ex-
pression for temperature deformations t [1]:
l
l
T t
t
, (1)
1
2
3
4
∆Т
5
6
∆lt
1
2
∆Т
l
144 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144)
where is the coefficient of temperature linear expan-
sion of the material (CTLE);
t
l – elongation of the
free end of the rod due to its heating.
Table 1 shows the initial data and calculation results
of the movement of the free end of the heater rod
(l = 1 m), which is made of MPG-7 graphite at different
temperatures [2].
Table 1
Temperature movements of a BETE free end
of the MPG-7 graphite
Т, °C
610 ,
1/deg
Δ l, mm
1 2 3
200 9.0 1.8
400 13.0 4.8
600 16.0 9.6
800 19.0 15.2
2. SSS OF THE COMPRESSED HEATING
ROD WITHOUT TAKING INTO ACCOUNT
THE SPRING OF THE CURRENT SUPPLY
Fig. 3 shows a rod of constant cross-section, which
is located between two current leads. Its ends cannot
move vertically. The temperature has changed by ΔT.
Let's determine the axial stress σ [1]:
T
l
l
Е , (2)
where Е – modulus of elasticity of the rod material. In
(2) in the case of fixed ends Δl = 0. Thus we have:
TЕ . (3)
As can be seen from equation (3), the compressive
stresses in the rod do not depend on its diameter and
length.
Fig. 3. Calculation diagram of the heating rod
deformation without taking into account the spring
of the current supply: 1 – upper current lead;
2 – equipment element; 3 – lower current lead
Table 2 shows the initial data and the results of cal-
culations of the axial stress of BETE at different tem-
peratures, from 200 to 800 °C. The length of the rod is
1 m. Data for MPG-7 graphite [2] were taken for CLTE.
The compressive strength of MPG-7 graphite of the
highest grade is 103.00 MPa [2]. Fig. 4 shows the
graphical dependence of the axial stress on the tempera-
ture, which is non-linear due to the fact that the CLTE
and the modulus of elasticity depend on the tempera-
ture.
Table 2
Calculation of the stresses of the BETE
due to heating
Т, °C 610 , 1/deg E, GPа σ(Т), МPа
1 2 3 4
200 9.0 7.10 12.78
400 13.0 7.35 38.22
600 16.0 7.50 72.00
800 19.0 7.84 119.17
Fig. 4. Dependence of the axial stress of the rod on the
temperature without taking into account the stiffness of
the current lead spring
3. STRESSED-STRAIN STATE OF BETE,
TAKING INTO ACCOUNT THE SPRING
OF THE CURRENT SUPPLY
The calculation scheme for this case is shown in
Fig. 5. It consists of a heating rod with a diameter d
(BETE), length l1 and a spring with stiffness C. The
values characterizing the stressed-strain state of the rod
will be denoted by the subscript “1”.
Thus, we get a static nondeterministic system con-
sisting of two parts. The first section is a heating rod,
and the second is a spring with stiffness C. This system
deforms in the axial direction due to the heating of the
first section.
The generalized view of the calculation scheme in
this case is shown in Fig. 6,a.
Fig. 5. Calculation scheme of the heating rod taking
into account the spring of the current supply:
1 – upper current supply; 2 – BETE; 3 – lower current
supply; 4 – spring of the upper current supply
As can be seen from Figs. 6,b and c, the thermal
elongation of the first section will consist of the sum of
the elastic elongations of the first and second sections:
1 2.tl l l (4)
1
2
3
∆Т
l
2
3
∆Т
4
С
l1
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 145
Fig. 6. Deformation of the heater rod
together with the spring: a – the system consists of two
sections; b – the first section is heated and lengthens by
the amount tl ; c – thermoelastic deformation of the
entire system
In other words, the spring prevents the first section
from elongating, (4) is the geometric side of the prob-
lem.
The static side of the problem is that the longitudinal
force xN on all sections of the system will be the same
and will be equal to the reactions in the resistances (in
the current leads). If there is a previous longitudinal
compression of the system, then it will be added to the
chart of longitudinal forces according to the principle of
superpositions [1].
The physical side of the problem [1]:
Tllt 1 ,
11
1
1
1
1111
FE
Nl
E
lll
T
x
, (5)
C
N
l
T
x 2 .
where – cross-sectional area of BETE.
After substituting equations (5) into (4), we get the
expression for calculating the longitudinal force T
xN :
1
1
1 1
,
T T
x xl N N
l T
E F C
1
1
1 1
.
1
T
x
Tl
N
l
E F C
(6)
In the case when 0С , 0xN we have a free
end. When С , we have a rigid support,
11FTENx , the formula for the longitudinal force
turns into the one resulting from (2).
To obtain the stress in the rod, it is enough to divide
the value of the axial force by the cross-sectional area
[1]:
1F
N x
x
. (7)
In the case of preliminary compression by force 𝑁𝑥
П,
formula (6) takes the form:
П
xx N
CFE
l
Tl
N
1
11
1
1
. (8)
In order to verify that at С , 11FTENx
calculations of axial stresses were carried out at
E = 7.84 GPa; 61019 1/deg; ΔT = 800 °C for
different values of the stiffness of the spring C and the
diameter of rod. The results are given in Table 3.
Table 3
The value of BETE stresses x , MPa at different
diameters and stiffness of the spring
d, mm
С, kN/m
2 20 200 2000 20000
25 0.03 0.31 3.02 24.58 86.05
15 0.09 0.85 8.02 49.96 104.6
5 0.77 7.27 46.9 103.3 117.4
As can be seen from the data shown in Table 3, with
a spring stiffness of 20000 kN/m and a rod diameter of
5 mm, the axial stress of the BETE is close to that
obtained earlier with absolutely rigid resistances
17,119x MPa (see Table 2).
Table 4 contains the initial data for calculations
according to formula (8).
The results of calculations for BETE, which is made
of MPG-7 graphite according to formula (8), are pre-
sented in Table 5 at different values of the diameter of
the graphite rod. The following notations are accepted:
– axial ductility of the graphite rod, ductility of the
spring 51051 С m/N, η – coefficient of strength re-
serve for axial stresses due to temperature expansion;
taking into account the preliminary compression of
500 N at the compressive strength limit of MPG-7
graphite [ ] МPа.
Table 4
Input data for temperature SSS calculations
Physical quantity
Value
MPG-7
graphite
Molybdenum
Temperature drop ∆𝑇, °С 800 800
CLTE 610 , 1/grad 19 6,2
Modulus of elasticity E, GPа 7,84 160
Rod length 𝑙 , m 1 1
Spring stiffness,
410С N/m 2 2
Pre-compression
П
xN , N 500 500
С ∆Т, l1 , E1, F1
1 2
x
а
∆Т
b
c
146 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144)
Fig. 7. Comparison of the compliance of the BETE
and the current supply spring for different values of its
diameter
On the basis of the data entered in Table 5, it can be
concluded that the destruction of the BETE due to
reaching the strength limit during compression occurs at
a diameter of 3 mm.
Fig. 7 shows a graph for comparing the axial com-
pliance of the BETE for different values of its diameter
with the compliance of the current supply spring. As can
be seen from the graph, the component corresponding to
the compliance of BETE becomes significant at diame-
ters that are close to the critical ones for the loss of rod
strength.
In order to avoid selecting the critical diameter of
the BETE, we obtain the cross-sectional area using for-
mula (7). Let's write down the strength condition:
x
x
x
F
N
1
. (9)
Instead of
xN let's substitute its expression (8) in (9).
After bringing similar ones, we get a quadratic equation
with respect to 1F of the form:
01
2
1 cbFaF , (10)
where 1Ea ; 1111 ENCETlClb П
x ;
ClNc П
x 1 .
One positive root of the equation (10) was obtained
6
1
1002.7 F m
2
= 0.0702 сm
2
. It corresponds to the
value of the diameter d = 2.99 mm, which coincides
with what was obtained by selection and is given in Ta-
ble 5. With a length of 𝑙 = 0.5 m, the failure occurs at
d = 2.78 mm. For molybdenum, the critical diameter of
temperature compression at a length 11 l m is
2.72 mm, that is, it is almost no different from that ob-
tained for graphite.
4. BUCKLING OF THE ROD ELEMENT
OF THE EQUIPMENT UNDER
CONDITIONS OF THERMAL EXPANSION
The analysis of the results regarding the critical di-
ameter of graphite BETE failure due to thermal com-
pression shows that the obtained diameter values are
very small. In addition, the dangerous diameter depends
slightly on the length of the rod element. At the same
time, the elements under consideration have a sufficient-
ly large flexibility, so it is appropriate to consider the
buckling problem for them.
According to [1], the critical strength of the loss of
stability for the rod element of the structure, which has a
rigid clamping at the ends (Fig. 8), will be calculated
according to the formula:
𝑃𝑐𝑟
4𝜋2 𝐽𝑚𝑖𝑛
2
, (11)
where Е – modulus of elasticity at a given temperature;
𝐽𝑚𝑖𝑛 – moment of inertia in the plane of minimum stiff-
ness of the section, for a rod of round cross-section:
𝐽𝑚𝑖𝑛
𝜋𝑑4
64
, (12)
l – rod lenghth.
Fig. 8. Calculation scheme for determining the critical
strength of the loss of stability of the rod element
The safety margin of the structure under considera-
tion is defined as a fraction of the critical force of buck-
ling and the axial force of temperature expansion. At the
same time, we will reduce the diameter of the rod until
the safety margin approaches unity. The calculation re-
sults for MPG-7 graphite (at 11 l m) are given in Ta-
ble 6, and for molybdenum in Table 7.
The data presented in Tables 6 and 7 confirm the
general information about the structural buckling. For
example, for a graphite rod, the critical value of the di-
ameter of the loss of stability is 16 mm (for molyb-
denum, 7 mm), although this diameter corresponds to a
compressive stress of only 3.98 MPa, and the loss of
strength due to thermal expansion occurs at a diameter
of 3.0 mm (see Table 5), that is, the destruction caused
by the loss of stability occurs at stresses much lower
than the strength limit of the material.
The following values of the critical diameter of
buckling were obtained for the length of the BETE
5,01 l m: graphite MPG-7 11 mm, molybdenum
5 mm.
CONCLUSIONS
The paper investigates the temperature resistance of
beam elements of technological equipment used in the
production of CCCM in direct heating furnaces. The
calculation scheme of the problem is built taking into
account the stiffness of the furnace current supply
spring, and the deformation model contains such a type
of destruction as loss of stability. In the course of nu-
merical studies, it was proved that the buckling is deci-
sive for this case.
0,00
1,00
2,00
3,00
4,00
5,00
6,00
0 10 20 30
1/С,
m/N
d, mm
spring
BЕТE
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 147
Table 5
Axial stresses of a graphite rod at different diameters
𝑙 𝑚
d, mm F, cm
2
∙ 5,
m/N
𝑁𝑥
𝑇, N 𝑁𝑥
Σ, N 𝑥, МPа η
25 4.909 0.026 302 802 1.63 63.01
15 1.77 0.072 300 800 4.53 22.76
5 0.196 0.65 269 769 39.17 2.63
3 0.071 1.8 223 723 102.33 1.006
Table 6
Axial force and critical buckling force of a graphite rod at different diameters (l = 1m)
d, mm
9
min 10J ,
cm
4 𝑁𝑥
∑
, N 𝑥, МPа 𝑃𝑐𝑟 , N 𝜂
𝑃𝑐𝑟
𝑁𝑥
∑
20 7.85 802 2.55 2430 3.03
18 5.153 801 3.15 1594 1.99
16 3.217 800 3.98 996 1.24
15 2.485 800 4.53 769 0.962
Table 7
Axial force and critical buckling force of a molybdenum rod at different
diameters (l = 1m)
d, mm
10
min 10J ,
cm
4 𝑁𝑥
∑
, N 𝑥, МPа 𝑃𝑐𝑟 , N 𝜂
𝑃𝑐𝑟
𝑁𝑥
∑
10 4.91 599 7.63 3100 5.18
7 1.18 599 15.56 744.5 1.24
6 0.636 599 21.18 401.8 0.67
The article provides formulas for calculating the crit-
ical diameters of the loss of compressive strength due to
heating, as well as calculations for determining the
buckling critical diameters of rod elements at a tempera-
ture of 800 °C, which are made of MPG-7 graphite and
molybdenum, and have a length of 1 and 0.5 m.
REFERENCES
1. A.P. Yakovlev, V.V. Matveev. Reference book on
resistance of materials / Opening ed. Pisarenko H.S.
2nd ed., revised. and add. Kyiv: “Nauk. opinion”, 1988,
736 c.
2. Billets and products from MPG-6 and MPG-7
graphite: TU U 23.9-37494898-001:2018.
Article received 14.03.2023
ТЕМПЕРАТУРНІ ДЕФОРМАЦІЇ ІЗОТРОПНИХ СТЕРЖНЕВИХ ЕЛЕМЕНТІВ
ТЕХНОЛОГІЧНОГО ОСНАЩЕННЯ ПРИ ВИГОТОВЛЕННІ ВУГЛЕЦЬ-ВУГЛЕЦЕВИХ
КОМПОЗИТНИХ МАТЕРІАЛІВ У ПЕЧАХ ПРЯМОГО НАГРІВУ
М.В. Мельтюхов, Я.В. Кравцов
Для ізотропних стержневих елементів технологічного оснащення (СЕТО), яке використовується при ви-
готовленні вуглець-вуглецевих композиційних матеріалів (ВВКМ) у печах прямого нагріву, в роботі вперше
отримане аналітичне рішення температурного напружено-деформованого стану з урахуванням жорсткості
пружини струмовідводу. Наводяться формули для обчислення критичних діаметрів втрати міцності на стис-
кання. На прикладі графітових і молібденових стержнів у роботі встановлено, що вирішальним фактором
несучої здатності є втрата стійкості елементів, що розглядаються.
|
| id | nasplib_isofts_kiev_ua-123456789-196111 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:23:38Z |
| publishDate | 2023 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Meltyukhov, M.V. Kravtsov, Y.V. 2023-12-10T12:58:48Z 2023-12-10T12:58:48Z 2023 The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating / M.V. Meltyukhov, Y.V. Kravtsov // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 143-147. — Бібліогр.: 2 назв. — англ. 1562-6016 DOI: https://doi.org/10.46813/2023-144-143 https://nasplib.isofts.kiev.ua/handle/123456789/196111 621.762 For the isotropic beam elements of the technological equipment (BETE), which are used in the production of carbon-carbon composite materials (CCCM) in direct heating furnaces, the analytical solution of the temperature stress-strain state, taking into account the stiffness of the spring of the current collector, was obtained for the first time in the work. Formulas for calculating the critical diameters of loss of compressive strength are given. Using the example of graphite and molybdenum rods, it was established in the work that the decisive factor of the load-bearing capacity is the loss of stability of the elements under consideration. Для ізотропних стержневих елементів технологічного оснащення (СЕТО), яке використовується при виготовленні вуглець-вуглецевих композиційних матеріалів (ВВКМ) у печах прямого нагріву, в роботі вперше отримане аналітичне рішення температурного напружено-деформованого стану з урахуванням жорсткості пружини струмовідводу. Наводяться формули для обчислення критичних діаметрів втрати міцності на стискання. На прикладі графітових і молібденових стержнів у роботі встановлено, що вирішальним фактором несучої здатності є втрата стійкості елементів, що розглядаються. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Irradiation installations, diagnostic and research methods The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating Температурні деформації ізотропних стержневих елементів технологічного оснащення при виготовленні вуглець-вуглецевих композитних матеріалів у печах прямого нагріву Article published earlier |
| spellingShingle | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating Meltyukhov, M.V. Kravtsov, Y.V. Irradiation installations, diagnostic and research methods |
| title | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| title_alt | Температурні деформації ізотропних стержневих елементів технологічного оснащення при виготовленні вуглець-вуглецевих композитних матеріалів у печах прямого нагріву |
| title_full | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| title_fullStr | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| title_full_unstemmed | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| title_short | The termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| title_sort | termal deformation of isotropic beam elements of technological equipment in the manufacturing of carbon-carbon composite materials in furnaces of direct heating |
| topic | Irradiation installations, diagnostic and research methods |
| topic_facet | Irradiation installations, diagnostic and research methods |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/196111 |
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