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DOI: 10.17516/1997-1397-2022-15-3-267-272 УДК 517.96
Numerical Simulation of Temperature and Thermal Stress Fields in a Carbon Block under External Thermal Effect
Evgeniy N. Vasil'ev*
Institute of Computational Modelling of SB RAS Krasnoyarsk, Russian Federation
Received 23.11.2021, received in revised form 28.01.2022, accepted 20.03.2022 Abstract. The paper is devoted to modelling thermal and stress-strain state of a carbon block when it is partially immersed in an electrolyte. The temperature field in the block was determined from the solution of a non-stationary three-dimensional heat conduction equation. The calculation of temperature stresses was carried out on the basis of the solution of the Poisson equation for the thermoelastic displacement potential. The temperature fields in the carbon block were obtained at various points in time. The stress-strain field was also obtained. Then the location and magnitude of the maximal temperature stresses were determined. It allows one to assess the fracture of the carbon block.
Keywords: heat conduction equation, Poisson equation, temperature stresses, thermoelastic displacement potential, numerical simulation.
Citation: E.N. Vasil'ev, Numerical Simulation of Temperature and Thermal Stress Fields in a Carbon Block under External Thermal Effect, J. Sib. Fed. Univ. Math. Phys., 2022, 15(3), 267-272. DOI: 10.17516/1997-1397-2022-15-3-267-272.
Introduction
The technological process of aluminium production requires regular replacement of carbon blocks (anodes). In the industrial electrolytic cells, when the cold anode is initially immersed in a hot electrolytic solution at a temperature of about 960oC, a heat wave propagates from the contact boundary into the volume of the anode. An increase in the local temperature causes thermal expansion of the anode material. The difference in the magnitude of the expansion of different zones of the anode leads to the occurrence of thermal stresses. In a zone of the highest temperature gradients significant thermal stresses arise which can exceed the ultimate strength of the material. It leads to the formation of cracks and further fracture of the anode. The phenomena that accompanies the process of immersing a cold carbon anode into the melt is called thermal shock [1,2]. The state of the carbon block during thermal shock depends on the thermophysical (thermal conductivity, heat capacity) and mechanical (thermal expansion coefficient, shear modulus, Poisson's ratio, tensile strength) properties of graphite as well as on conditions of heat exchange with electrolyte. Numerical simulation allows one to analyse the state of the carbon block taking into account these factors.
* ven@icm.krasn.ru https://orcid.org/0000-0003-0689-2962 © Siberian Federal University. All rights reserved
The aim of the work is to calculate the temperature field and the stress-strain field of the carbon block of the electrolytic cell. To describe the formation of thermal stresses, the mathematical modelling procedure includes two consecutive stages:
1. Determination of the temperature field in the volume of the carbon block is based on the solution of 3D heat conduction problem.
2. Calculation of thermal stresses is based on the solution of the Poisson equation for the obtained temperature field at various points in time.
1. Determination of the temperature field of the carbon block
The anode block is a parallelepiped made of carbon graphite (Fig. 1). In the electrolysis cell, the anode block is mounted using a steel bracket. The geometrical dimensions of the anode along the x, y, and z axes are 1450 x 700 x 600 mm3.
Fig. 1. The anode block of the industrial electrolytic cell
The heat transfer process in a carbon block is described by non-stationary three-dimensional
heat conduction equation (
c — = \(— — —
P dt \ dx2 dy2 dz2
where c, p are specific volumetric heat capacity and density of the material; T is temperature; A is the coefficient of thermal conductivity; t is time; x, y, z are spatial coordinates. The solution of equation (1) was obtained with the use of the method of finite differences with the splitting of the problem in spatial coordinates [3,4].
(1)
Calculations were performed for the anode block shown in Fig. 1. The size of the part of the block immersed in the electrolyte is 120 mm. For graphite, the following thermophysical properties were set: A = 4.4 W/(m-K), c = 942 J/(kg-K), p= 1560 kg/m3 [5, 6].The heat exchange coefficient of the surface of the anode block with air 3 a = 10 W/(m2-K) and with the electrolyte solution 3e = 18 W/(m2-K). A homogeneous spatial grid with the number of nodes 146 x 71 x 61 was used for calculations, and the time step was 5 s.
The results of calculation of the temperature field of the anode for the moment of time At = 15 min are shown in Fig. 2. The temperature field of the lower part of the anode in the middle cross-section of the xz plane is shown in the left figure. Taking into account the symmetry of the problem, one quarter of the temperature field of the lower surface of the anode (plane xy) is shown in the right figure. The temperature values on the isolines are given in degrees Celsius. The most intense heating is observed in the zone where the unit is in contact with the electrolyte. In this area, the highest temperature is in the lower corner and and it is 467°C.
Fig. 2. Distribution of temperature in the middle xz and bottom xy planes of the anode block
The temperature distributions obtained from the solution of equation (1) at various times are the initial data for solving the problem of the stress-strain state of the carbon block.
2. Calculation of the temperature stresses in the carbon block
The temperature stresses are calculated by solving the Poisson equation for the thermoelastic displacement potential [7]
d2$ d2$ d2$ _ «0(1 + n) dx2 dy2 dz2 1 — ^ '
where $ is the thermoelastic potential of displacements; n, a are Poisson's ratio and coefficient of thermal expansion; 0 _ (T — T0) is the temperature increment with respect to the temperature of the natural state of the body T0. Equation (2) is supplemented with the conditions of the absence of externally applied normal and tangential stresses on the carbon block surface: az _ 0,
Txz 0, Tyz
The values of the thermoelastic potential $ were used to determine the stresses at the corresponding points of the difference grid
= 2*^
a, = 2* I 5-2 + V2$
\ dx2 j
>xy
d 2 $
(3)
(4)
where ax, ay, az, Txy, Tyz, Txz are normal and tangential elastic stresses; G is the shear modulus of the material at a given point and at a given moment in time, (xyz) is the symbol of cyclic permutation of x, y, z. The number of nodes of the grid in thermal stresses equations (2)-(4) corresponds to the thermal problem.
The distribution of thermal normal stresses of the anode for a time instant of 15 minutes is shown in Fig. 3. The distribution of thermal normal stresses for the middle xz plane is shown in the left figure. The magnitude of the temperature stresses in this plane reaches 3.4 MPa. The highest stresses are observed in zones of the highest temperature gradients. The maximum values of temperature gradients and stresses occur at the corners of the anode, they can be displayed in the vertical diagonal section passing along the bisector of the angle of the anode base. The right figure shows the distribution of thermal stresses in this diagonal plane. Comparison of the distributions in the middle xz and the diagonal vertical planes shows that in the second case the values of maximum stress are more than 1.5 times higher.
Fig. 3. Distribution of normal temperature stresses in the middle xz and diagonal vertical planes of the anode block
The variation of the maximum normal stresses in the anode block with time is shown in Fig. 4. The calculated maximum values of thermal stresses at various moments of time are marked with circles. The dotted line is obtained with the use of interpolation. The greatest increase in thermal stresses occurs at the initial stage of the process, and then the slope of the curve is significantly decreased. Considering results of calculation of the stress-strain state, it is possible to assess the possibility of fracture of the anode by comparing stresses with the ultimate strength of carbon. The most important from the point of view of cracking of the anode are tensile stresses that arise from the inhomogeneous thermal expansion of the material during heating. The ultimate tensile strength of the anode material is in the range of 5-15 MPa [2, 8, 9]. The scatter in the data of the limiting values of thermal stresses for graphite depends on the manufacturing technology and
composition. It follows from the results of calculations that when graphite with a low ultimate strength (less than 8 MPa) is used there is a probability of fracture of the anode block.
Fig. 4. The time dependence of the maximum normal temperature stresses in the anode block
Conclusion
Numerical modelling of thermal processes occurring when a carbon block is immersed in a hot electrolyte allows one to determine the magnitude and location of the maximal temperature gradients and stresses at various moments of time. Calculations have shown that the maximum values of temperature stresses in the corners of the anode block exceed the lower limit of the tensile strength of graphite. This points up the possible fracture of the anode block.
References
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[2] Yu.G.Mikhalev, P.V Polyakov, A.S.Yasinskiy, S.G.Shahrai, A.I.Bezrukikh, A.V.Zavadyak, Anode processes malfunctions causes. An overview, Journal of Siberian Federal University. Engineering and Technologies, 10(2017), no. 5, 593-606 (in Russian).
DOI: 10.17516/1999-494X-2017-10-5-593-606
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DOI: 10.1134/S1063784218040266
[5] J.P.Schneider, B.Coste, Thermomechanical modelling of thermal shock in anodes, Light Metals, 1993, 621-628.
[6] S.N.Akhmedov, V.V.Tikhomirov, B.S.Gromov, R.V.Pak, A.I.Ogurtsov, Specific features of the lining deformation of the cathode devices of aluminum electrolysers, Tsvetnye metally, (2004), no. 1, 48-51 (in Russian).
[7] N.I.Bezukhov, V.L.Bazhanov, I.I.Gol'denblatt, N.A.Nikolaenko, A.M.Sinyukov, The Calculations for Strength, Stability, and Oscillations in High Temperature Conditions, Mashinos-troenie, Moscow, 1965 (in Russian).
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Вычислительное моделирование полей температур и термических напряжений в угольном блоке при внешних тепловых воздействиях
Евгений Н. Васильев
Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация
Аннотация. Работа посвящена моделированию теплового режима и напряженно-деформированного состояния угольного блока при его частичном погружении в электролит. Температурное поле в блоке определялось из решения нестационарного трехмерного уравнения теплопроводности. Расчет температурных напряжений проводился на основе решения уравнения Пуассона, записанного для термоупругого потенциала перемещений. В результате моделирования теплового режима получены температурные поля в угольном блоке для разных моментов времени. Расчет напряженно-деформированного состояния определил величину и расположение наибольших температурных напряжений и позволил оценить возможность разрушения угольного блока.
Ключевые слова: уравнение теплопроводности, уравнение Пуассона, термические напряжения, термоупругий потенциал перемещений, вычислительное моделирование.
