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Numerical simulation of the heating process in a vacuum sintering electric furnace and structural optimization | Scientific Reports

Scientific Reports volume 14, Article number: 30905 (2024) Cite this article 1323 Accesses Metrics details A three-dimensional numerical model of the vacuum sintering furnace was established, combined

Scientific Reports volume 14, Article number: 30905 (2024) Cite this article

1323 Accesses

Metrics details

A three-dimensional numerical model of the vacuum sintering furnace was established, combined with the custom program of temperature-voltage feedback regulation. Through simulationand experimental validation, the heating and holding stage as well as the thermal hysteresis phenomenon of the furnace were analyzed, a dimensionless quantity of hysteresis temperature difference was proposed and calculated, the distribution of the electric field and temperature uniformity of the furnace were discussed in detail, while the structural improvement approach was proposed based on simulation. The results show that: during the heating process, the maximum of thermal hysteresis temperature difference between the graphite cylinder and the heating tube is 0.4. The relative error between the simulation and measurement is within 4%, which verifies the accuracy of the model. By optimizing the structure of the heating tube and graphite base plate, the thermal hysteresis effect of the furnace can be effectively reduced, the surface load and the temperature difference between the surface of the heater can be significantly reduced, the temperature field and uniformity of the heating zone can be improved.

Cemented carbide has been widely used in cutting tools, high-pressure tools, engineering machinery, wear-resistant parts, and has a broad development prospect. As the most critical process in the whole production process of Cemented Carbide, sintering has a significant or even decisive influence on its structural properties1,2, and temperature control in the sintering process has always been the focus of people’s research.

Theoretical study of the temperature field in an industrial furnace is essentially a problem of solving the complex heat transfer problem of convection, radiation and heat conduction in a closed cavity, Colomer et al.3 investigated the interaction of radiation and natural convection in participating media in cavities with different heating. Albanakis et al.4 studied coupled heat transfer by convection, surface radiation, and conduction in a three-dimensional asymmetrically heated square cavity by numerical analysis. Kuznetsov et al.5 numerically studied convective-radiative during localized heating at the bottom of the cavity with finite-thickness heat-conducting walls. Raisi et al.6 used the finite element method to discretize the fluid field equations, investigated the transient natural convection problem and the effect of fluid-solid interaction in a square cavity. Based on a rectangular furnace, Nevin et al.7 evaluated the Discrete Ordinates Method (DO) and the Discrete Transfer Method (DTM) of the radiation model in terms of computational accuracy and economy, respectively. Mochida et al.8used the Monte Carlo method to calculate the radiative heat transfer and developed a computer program for analyzing the three-dimensional transient radiation-heat conduction problem in a vacuum furnace to estimate the heat transfer characteristics of a large-scale heating furnace and to calculate the effective heating region. Kantor9 investigated the heat transfer through the heat shield in a vacuum through a coupled conduction-radiation method.

Based on numerical methods, Charles Manière et al.10 predicted the uniformity of the heating and compaction in large-size gear specimens under different process conditions by FDTM model of COMSOL. Devesh Iwari et al.11 simulated the specimens during the spark plasma sintering process through MATLAB and ABAQUS, the effect of power input, sintering time, and thermal conductivity of the heat transfer process in the hearth region was investigated. LEE et al.12 studied the internal structure of a vacuum furnace and found that insulators and heaters are the most important components, how to choose them will directly affect the process and production cost as well as the energy efficiency. Ma Anjun et al.13 used simulation to optimize and analyze the heater structure of an isostatic pressing furnace, the temperature uniformity of the product was improved by optimizing the graphite heater arrangement and dimensions. Hadala et al.14 conducted experimental and theoretical calculations on an industrial annealing furnace to reach the conclusion that the total heat loss was related to the thermal insulation of the furnace wall and the mode of operation. Khodabandeh et al.15 conducted numerical study of graphite insulation in an arc furnace, the use of composite insulation (graphite sheet + graphite felt) can reduce the heating time by 5–8% and save about 15% of electric energy was found. Hooks et al.16found that a multilayer form of insulation is a very promising option in a vacuum. Paramonov17 carried out a technical and economic optimization study of multilayer furnace linings of different thicknesses and materials, various combinations were given. However, there are some limitations in the above studies, few research of simulation on the temperature control and the response of the unsteady heating process in vacuum furnaces, and there is a large optimization potential for the influencing factors of the furnace structure.

In this study, we establish a three-dimensional numerical model of the vacuum sintering furnace and make the temperature control closer to the actual heating process by customizing the program. We calculated the electric heat field distribution of each part of the heating process, and on this basis, put forward the optimization scheme such as the structure of the heater, so that the uniformity of the internal heat field is improved. The structure of the heating tube is optimized through calculation of the electric-thermal field and the uniformity of the internal thermal field is improved.

A simplified physical model of the vacuum furnace is established while the medium of the solid region are insulation cylinder, a graphite cylinder (including graphite cylinder body, graphite base plate, and graphite cylinder cover), graphite heaters, graphite dewaxing tube and the furnace shell. The medium in the fluid region is a thin gas under vacuum conditions, which is nitrogen by default. To reduce the number of grids in the simulation, based on the symmetry to the YOZ plane as the symmetry plane using one-half of the model, the origin of the coordinates is located in the center of the bottom surface of the insulation layer. The established model is shown in Fig. 1, the model size is Φ800mm×766 mm, and the grid after meshing is shown in Fig. 2.

Schematic diagram of vacuum furnace model. 1-graphite cylinder; 2-graphite heating tube; 3-insulation layer; 4-graphite dewaxing tube; 5-furnace shell; 6- temperature control thermocouple; 7- electrode.

Grid at symmetric plane.

This grid produced 362,683 nodes with 1,925,414 cells. All the quality indexes of this meshing are in the better range. After refining the grid as a whole and keeping other conditions unchanged to recalculate, the calculation error of the temperature measurement nodes in the vacuum furnace is less than 2%, and it can be assumed that grid independency is established.

Heat transfer models for vacuum heat treatment furnaces include heat conduction, radiation and convection.

Convection is negligible due to the near absence of atmosphere for the heating process in a vacuum environment. The numerical model used in this simulation is described as follows.

(1) Continuity equations.

In which: v is the fluid velocity in that direction, m·s−1. Sm is the mass source term, Kg。

(2) Momentum equation.

where P is the static pressure, Pa; \(\mathop \tau \limits^{=}\) is the stress tensor; \(\rho \mathop g\limits^{ \to }\) is the gravity, N; \(\mathop F\limits^{ \to }\) is the source term due to other forces, N;

(3) Energy equation.

where \({k_{eff}}\) is the effective thermal conductivity, W·m−2·K−1; \({\overrightarrow J _j}\) is the diffusion flux of component j; the first three terms on the right-hand side represent the energy transfer due to conduction, mass diffusion, and viscous dissipation, respectively. \({S_h}\) is the heat of chemical reaction and other internal heat sources, KJ·m−3.

(4) The DO model.

It uses the direction in the radiative transfer equation (RTE) as a field function, Its computational equation is as follows:

where \(\overrightarrow r\), \(\overrightarrow s\), \(\overrightarrow s^{\prime}\) respectively for the position vector, direction vector, and scattering direction vector; a, n, \({\sigma _s}\) respectively for the absorption index, refractive index, and heat dissipation coefficient; I is the radiation intensity, related to \(\overrightarrow r\) and \(\overrightarrow s\), W·m−2; T is the local temperature, K; Ф is the phase function; \(\Omega^{\prime}\) is the solid angle, sr.

(5) Electro-thermal coupling model.

The calculation equation of the potential field is as follows:

where \(\varphi\) is the potential, V; \(\sigma\) is the conductivity, 1·Ω−1·m−1; S is the other internal heat sources, kJ·m−3.

The internal heat source is the Joule heat calculated by the ANSYS electrical module. The results obtained through the Mechanica parameters l APDL module of ANSYS have an error of only 1.67% from the manufacturer’s supplied values.

The materials are mainly insulation layer (composite rigid carbon felt), graphite inner cylinder (GSK0.8 particles), graphite heater (isostatic graphite R8340) and dilute gas under vacuum working conditions.

The default gas in the furnace during sintering is mainly nitrogen, to simulate the vacuum conditions, the gas density and thermal conductivity are set to very small (10e-7), and the specific heat capacity is determined from the literature18. Some physical properties of the composite rigid carbon felt, graphite inner cylinder and graphite heater were obtained by the fitting curve equations from the literature19,20, the material properties are shown in Table 1, where T is the temperature, K.

The water-cooled furnace shell is a constant wall temperature boundary condition; the interface of each structure is set as a no-slip boundary, the heat transfer condition is coupled heat transfer, and the internal emissivity is the surface emissivity of each structure; the SIMPLE algorithm of the solver is adopted, the first-order upwind scheme to obtain better convergence, and the time step is 1s.

The solution strategy is to use the electric module to solve the electric field from the initial voltage given by the custom program, then input the result as an internal heat source into the fluent module to achieve the electric-thermal field coupling.

In the vacuum atmosphere, the heat transfer mode of the heating tube to the graphite cylinder is dominated by radiation, convection is extremely weak, and due to the characteristics of thermal radiation, there is a heating hysteresis problem.

To make the simulation of the heating process close to the actual, through the FLUENT UDF (user-defined function), the intelligent temperature-control custom program was developed via C language. When the thermocouple temperature is different from the setting temperature, the custom program regulates the heating power by adjusting the input voltage value to minimize the effect of heating hysteresis in the simulation process, so as to obtain more stable and accurate control, and the calculation flowchart of the subroutine is shown in Fig. 3.

Flow chart of temperature-controlled custom program calculation.

The heating zone is the space surrounded by the graphite cylinder, it is 250 mm in diameter and 250 mm in height. The temperature difference between the heating tube and the graphite cylinder at the same moment is called the thermal hysteresis temperature difference; In order to describe the thermal hysteresis phenomenon more intuitively, the dimensionless quantity Xth is defined to represent the thermal hysteresis temperature difference.

where T1 and T2 are the average temperature of the heating tube and graphite cylinder, respectively, Taim is the target temperature of the heating tube, K; C1 and C2 are the specific constant pressure heat capacity for the temperature, respectively, Caim is the specific constant pressure heat capacity of the heating tube at the target temperature, J/(kg•K); ρ1 and ρ2 are the densities of the heating tube and the graphite cylinder at the corresponding temperature, respectively, ρaim is the density of the heating tube at the target temperature, kg/m3.

During the heating stage, the target temperature is preset as 1373 K. The relationship between the average temperature of the heating tube and the graphite cylinder with the heating time is shown in Fig. 4. With the increase of heating time, there has been a large temperature difference between the heating tube and the graphite cylinder, Xth shows the first increase and then decrease, Xth reaches a maximum value of 0.4 with the heating time at 3708s; it shows that in the early stage of heating, the thermal hysteresis phenomenon is serious.

The main reason for the thermal hysteresis phenomenon is that the heat is transferred from the heating tube to the graphite cylinder by radiation, which indirectly heats the internal space of the graphite cylinder, so the heating process is slow. The thermal hysteresis phenomenon is closely related to the intensity of radiation, especially the temperature of the heating tube and the area of radiation, during the low-temperature heating stage, there will be a more obvious thermal hysteresis phenomenon and it is quite a problem for engineers.

Combined with Eq. (6) and Fig. 4, it can be seen that choosing the proper material and heating curve can reduce the peak value of Xth and effectively minimize the effect of thermal hysteresis, which has guiding significance for production design.

Temperature and the Xth variation in the heating stage.

The temperature variation of the thermocouple in the holding stage is shown in Fig. 5. After the thermocouple temperature reaches 1373 K, the temperature of the thermocouple shifts up and down around the target temperature, the highest temperature is 1374.42 K and the lowest temperature is 1368.23 K, the difference of 1.27 K and − 4.92 K from the target temperature, respectively, the relative error is less than 0.36%, which can be limited in the error allowable range of the custom program and the temperature control effect is good.

Temperature variation curve of thermocouple.

The steady-state measurement method was used in the experiment, a total of four measurement points, the point arrangement is shown in the Fig. 6, point 1 is the graphite cylinder inner wall temperature, point 2 is the insulation layer inner wall temperature, point 3 is the insulation layer outer wall temperature and point 4 is the furnace shell inner wall temperature. The temperature curves of the measurement points are shown in Fig. 7.

Layout of measurement points.

The temperature variation at measurement points.

After the holding stage, the simulated values of the measurement points are compared with the experimental values, the results are shown in Table 2. The error of the simulated and experimental values is within 4%, which is within the engineering range of error.

The temperature field distribution inside the furnace is shown in Fig. 8, the high-temperature zone is mainly distributed in the hearth surrounded by the insulation layer, decreasing from the inside to the outside. The graphite electrode and the dewaxing tube have more drastic temperature variation, mainly due to a lot of heat loss from there.

Temperature distribution in the hearth zone. (a) Temperature distribution in YOZ section; (b) Temperature distribution in XOZ section.

Taking the centerline of the axial direction in the hearth (y = 0 mm, z = 332 mm), the temperature variation along the axial direction is shown in Fig. 9. Taking the inner wall of the insulation layer (x = 215 mm) as the boundary, the maximum temperature is 1397.91 K at 180 mm in the x-axis direction, which corresponds to the region where the heating tube is located; and the minimum temperature of 1370.6 K at the origin of the coordinates is the center of the corresponding heating zone. The dramatic downwards of the temperature behind the inner wall of the insulation layer, which indicates that the insulation performance is better.

X-axial temperature distribution in the hearth (z = 332 mm).

The voltage distribution of the heating tube is shown in Fig. 10. The voltage of the electrode decreases uniformly from the current input port to the output port.

Voltage distribution of the heating tube.

The temperature distribution is shown in Fig. 11, the higher temperature zone is mainly centralized on the heating tube, the temperature of the upper and lower side blocks is low. The highest temperature is 1487.36 K, which is located in the center of the heating tube; the lowest temperature is 1358.83 K, located at the interface of the electrode. The temperature difference is up to 128.53 K. In the horizontal direction, the average temperatures of the four heating tubes are 1466.35 K, 1470.92 K, 1470.96 K, and 1466.33 K, respectively. The temperatures of the second and third tubes are slightly higher than the first and fourth, which may be due to the effect of the different structures of the series-parallel connection.

Temperature distribution of the heater.

The heat flow density distribution is displayed in Fig. 12, The region of high heat flow density is localized on the heating tube, which has a high surface load with an average value of 43905.94 W/m2. The upper and lower side blocks are 5977.06 W/m2 and 4656.42 W/m2, respectively. This shows that the main source of radiant heat flow is the heating tube, the upper and lower blocks as the linkage mainly play the role of supporting and supplementing heat. On the one hand, this feature of the heater is conducive to the centralization of heat radiation to the graphite cylinder and then heating the product to be sintered, on the other hand, it will cause heat loss of the heater tube, and the material requirements are higher, at the same time, it will reduce the service life of the heater.

Heat flow density distribution of the heater.

The temperature distribution of the heating zone is depicted in Fig. 13, it can be seen from Fig. 13(a) that the boundary is the inner wall surface of the graphite cylinder, the high-temperature zone is close to the heating tube, and the temperature can reach to 1405.53 K. According to Fig. 13(b), the temperature is increasing from the center outward along the axial axis because the heat from the heater is received by the graphite cylinder firstly, and then transfer the heat to the heating zone after the temperature of the cylinder rises. The lowest temperature is 1334.06 K near the graphite base plate, because it is connected to the dewaxing tube, the heat will lost through the graphite dewaxing tube.

Temperature distribution of heating zone boundary and axial section. (a) Boundary temperature distribution of the heating zone; (b) Axial temperature distribution in the heating zone.

The effective heating zone is the maximum working zone in meeting the heat treatment requirement, the temperature uniformity of the experimental furnace requires ± 5 K. The maximum axial and radial temperature variation in the heating zone are shown in Fig. 14 (a) and Fig. 14 (b), respectively. The maximum temperature is 1377.75 K in (a), the minimum temperature is 1377.74 K in (b), considering the temperature uniformity requirement of 5 K, the minimum temperature is 1372.74 K in the axial direction, the maximum temperature in the radial direction is 1382.75 K, corresponding to the effective heating zone is the height of 100 mm to 250 mm, the diameter of 240 mm or less. The actual production process can be considered by adding a 100 mm high graphite bottom bracket as an auxiliary part, which can obtain a better sintering effect.

Temperature variation in the heating zone. (a) Axial temperature variation in the heating zone; (b) Radial temperature variation in the heating zone.

The temperature distribution of the insulation layer is shown in Fig. 15. The temperature from the inside to the outside is constantly attenuating. It can be seen from Fig. 15(a) that the temperature is higher in the region near the electrode holes in the outer wall, thus the thermal shielding structure can be added appropriately without the affect of airflow.

It is shown in Fig. 15 that the temperature is constantly attenuated along the axis direction, the temperature of the inner and outer walls are 1365.73 K and 458.97 K, respectively.

Temperature distribution results along the insulation layer. (a) Temperature distribution of insulation layer in YOZ section; (b) Temperature distribution of insulation layer in XOZ section.

Localized overheating of the heater will lead to a shortened service life. Adopting a new plate heater to replace the tube heater will improve the high loading of the heater, at the same time, which has a favorable effect on the uniformity of the temperature field. The structure of the plate heater is schematically shown in Fig. 16. The diameter and height of the heater are not changed.

Schematic diagram of the plate heater structure.

After structural optimization, the temperature variation curves of the graphite heater and heating zone in the heating stage are shown in Fig. 17. With the increase of heating time, the temperature difference between the heater and the graphite cylinder has been very small. When the heating time is 4214s, Xth reaches the maximum value of 0.29. Compared with the heating tube, the Xth decrease by 27.5%. Therefore, the plate heater to replace the tube heater is beneficial to alleviate the thermal hysteresis phenomenon at the low-temperature stage in the furnace.

Temperature and the Xth variation in the heating stage.

The temperature distribution of the plate heater after holding stage is shown in Fig. 18. As can be seen from the figure that the highest temperature is 1389.94 K, which is located in the middle region. Compared with the heating tube, the maximum temperature is reduced by nearly 100 K; the minimum temperature of 1341.50 K, which is at the interface of the electrodes. The temperature difference is 48.43 K. The overall temperature distribution is more uniform compared to the tube heater.

Temperature field distribution of the plate heater.

The heat flow density distribution is shown in Fig. 19. The overall distribution is more uniform and the average heat flow density is 6682.23 W/m2, which is about 84.78% lower than the tube heater. When the surface load is reduced, it is favorable to improve the service life of the heater.

Heat flow density distribution of the plate heater.

The temperature field distribution before and after the optimization of the heater is shown in Fig. 20, The average temperature before and after the change of the heater are 1371.99 K and 1369.21 K, respectively, with a decrease of 2.18 K. The variation in axial temperature distribution is not obvious, the variation in radial temperature distribution is displayed in Fig. 21, the maximum radial temperature difference under the tube heater is 3.5 K, while the plate heater is 2.36 K. Therefore, the temperature field in the radial direction of the plate heater is more uniform.

Temperature distribution in heating zone before and after optimization. (a) Temperature distribution before optimization; (b) Temperature distribution after optimization.

Radial temperature distribution in the heating zone before and after optimization.

Due to the heat is dissipated by the dewaxing tube, the material of the graphite base plate is changed to composite hard carbon felt, which can reduce the negative effect of the graphite dewaxing tube on the temperature field.

The temperature distribution before and after the graphite base plate material improvement is shown in Fig. 22 When the thermal conductivity of the base plate is very small (0.28), the effect of the dewaxing tube is minimized and the temperature uniformity is improved.

The temperature distribution along the axial and radial direction is shown in Figs. 23 and 24. As can be seen from Fig. 23, the average temperature along the axial direction after the improvement is 1382.76 K, which is about 14 K higher than the original structure, the variance is reduced from 149.23 to 0.34. According to Fig. 24, the radial average temperature after the improvement is 1383.56 K, increasing about 5 K. The variance is decreased from 2.05 to 1.05. Therefore, when the graphite base plate is changed to composite hard carton felt and the thermal conductivity is small, the uniformity of the temperature field can be greatly improved.

Temperature distribution in the heating zone before and after the graphite base plate optimization. (a) Temperature field distribution before the optimization; (b) Temperature field distribution after the optimization.

Axial temperature distribution in the heating zone before and after the graphite base plate optimization.

Radial temperature distribution in the heating zone before and after the graphite base plate optimization.

The heat loss through the insulation layer is related to the structural dimensions, the thermal properties and the temperature of the inner wall surface in the furnace shell,. It is preferable to explore the effect of the thermal conductivity, surface emissivity, and water-cooled wall temperature on the insulation performance, which has a certain guiding significance for the optimization of the process in vacuum sintering furnace.

When the thermal conductivity of the insulation material varies between 0.36 W m−1 K−1 and 0.2 W m−1 K−1. The performance of the insulation layer sidewall is shown in Table 3, as the thermal conductivity decreases, the radiant heat fluxes on both the inner and outer walls are reduced, indicating the heat loss decreases; whereas the temperature of the inner wall increases and the outer wall decreases, showing that the insulation performance is improved. When the material with lower thermal conductivity is selected as the insulation felt, a better thermal insulation effect can be obtained.

When the surface emissivity of the insulation material changes to 0.6 and 1.0, the effect on the insulation performance of the sidewall is shown in Table 4, with the increase of the surface emissivity, the radiant heat fluxes on both the inner and outer wall surfaces are increased, indicating an increase in the heat loss; The temperature of the inner and outer wall surfaces decreases, which shows that the performance is improved, with the increase in surface emissivity, the insulation effect will be enhanced, but the heat loss through the insulation layer will increase.

When the temperature of the water-cooled furnace shell inner wall changes to 325 K and 365 K, the effect on the insulation performance of the side wall is shown in Table 5, with the decrease of the wall temperature, the radiant heat flux of the inner and outer wall surfaces increases, indicating that the heat loss is increased; while the temperature of the inner and outer wall surfaces decreases, showing that the insulation performance has been improved. When the water-cooled wall temperature decreases, the heat loss increases.

A three-dimensional numerical model of the heating process is established and analyzed in detail. A dimensionless quantity Xth of hysteresis temperature difference was proposed and calculated while combining the temperature-voltage feedback regulation with the custom program and optimizing the electric-thermal distribution, the results show that:

In the heating stage, the maximum of the Xth reaches 0.4. In the holding stage, the relative error of temperature is lower than 0.36%, the temperature control achieves the expected effect, the error between the simulated temperature and the experimental value is within 4%, the accuracy of the simulation is verified.

After the holding stage, considering the temperature distribution differences between the surface of the heating tube and the heating zone, the workpiece can be located higher than 100 mm heating zone to meet the uniformity requirement of 5 K.

After optimizing the structure of the heater, the maximum of the Xth is reduced by 27.5%; The temperature difference on the surface of the heater is reduced by 62.32%, and the heat load is reduced by 84.78%. The maximum radial temperature difference in the heating zone is reduced by 32.57%.

After optimizing the graphite base plate material, the axial temperature variance is reduced from 149.23 to 0.34, the radial temperature variance is decreased from 2.05 to 1.05, the temperature and uniformity of the heating zone are improved.

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

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School of Energy Science and Engineering, Central South University, Changsha, 410083, China

Mao Li, Jishun Huang, Ting Hu, Benjun Cheng & Hesong Li

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“Mao Li conceived and designed the study, reviewed the manuscript, supervised and provided expert guidance. Jishun Huang wrote the first drafted manuscript, analyzed data, and edited the manuscript. Ting Hu performed experiments, collected data. Benjun Cheng and Hesong Li contributed to writing the manuscript, revised the manuscript critically, and supervised and provided expert guidance.”

Correspondence to Mao Li or Benjun Cheng.

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Li, M., Huang, J., Hu, T. et al. Numerical simulation of the heating process in a vacuum sintering electric furnace and structural optimization. Sci Rep 14, 30905 (2024). https://doi.org/10.1038/s41598-024-81843-8

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Received: 07 July 2024

Accepted: 29 November 2024

Published: 28 December 2024

DOI: https://doi.org/10.1038/s41598-024-81843-8

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