Study of heat transfer model and buried thermocouple test of bell-type annealing furnace based on thermal equilibrium

Abstract

The bell-type annealing furnace is a key equipment in the cold rolling process, and the heat transfer process inside the furnace has a significant impact on the annealing process and product quality. At present, there are shortcomings in the calculation accuracy of model research based on heat transfer theory and practical experience. In addition, existing research has not fully considered the influence of gas flow distribution and corresponding heat transfer coefficient in each channel of multi tube steel coils on the temperature field. The aim of this study is to develop a more accurate heat transfer model for bell type annealing furnaces and verify its reliability. In this work, a heat transfer model for a bell jar furnace was established based on the thermal equilibrium method, and its accuracy was verified through industrial buried thermocouple tests. The results indicate that the heat transfer model has high accuracy due to considering the distribution of protective gas flow rate, convective heat transfer coefficient, and radial thermal conductivity of the steel coil. The average relative error of the cold spot temperature calculated by the model is less than 0.5%, and the hot spot temperature does not exceed 5%, which proves the accuracy and reliability of the model, especially for cold spot temperature. When the relative error of the cold spot temperature is within 2.5%, the accuracy of the coil is above 95%. This study provides an accurate heat transfer model for fitting the heat transfer process of bell type annealing furnaces.

Introduction

Cold-rolled sheet, having a thickness of less than 4 mm, is obtained from a hot-rolled steel strip made of ordinary carbon structural steel. The surface quality of the cold-rolled sheet is good, and the dimensional accuracy is high due to rolling at room temperature, and no iron oxide skin is produced. In addition, the mechanical properties and process properties of the cold-rolled sheet are superior to those of the hot-rolled steel sheet because of its annealing treatment. In many fields, especially in the field of home appliance manufacturing, it has gradually replaced hot-rolled steel sheets with it. With the development of various industries such as mobile manufacturing, electrical products, locomotives and vehicles, aerospace, precision instruments, food appliances, architectural decoration, etc., the demand for quality and high-value-added cold-rolled sheets has increased from 22.5 million tons in 2010 to 40 million tons in 2017 in China. The cold-rolled steel coils must be recrystallized and annealed before completion to achieve the required microstructure structure, mechanical properties, and process performance indicators. This annealing process needs to be conducted in the annealing furnace which is one of the important furnace types for cold rolling annealing. Therefore, it is of great significance to study the heat transfer process in a bell-type furnace and establish a reasonable steel coil heat transfer model for optimizing the annealing control parameters, enhancing product quality, and improving production efficiency.

Many scholars have carried out research work on the bell-type furnace and the steel coil heat transfer model from different perspectives on the heat transfer process. Ding Fei and Yang Haijiang et al.^1,2,3,4 established the mathematical models for the heating cover, furnace space, inner cover, hydrogen, cooling cover, and cooling medium through the analysis of the heat transfer characteristics of the bell-type annealing furnace and solved it. That provides correct convection and radiation boundary conditions for the calculation of the steel coil temperature field. Yi Zuo et al.⁶ proved that the convective heat transfer coefficient and the equivalent radial thermal conductivity great influence on the overall heat transfer in the furnace, the calculated temperature curve and the test data are consistent with the temperature error of less than 25 °C. Pei-pei et al.⁶ used the Monte Carlo method to solve the radiation heat transfer angle coefficient in the furnace, and established the heat transfer model of the steel coil considering the radiation heat transfer. Through the on-site buried thermocouple test, the temperature calculated by the simulation was consistent with the measured value. The relative error of the model falls within 3% with a probability of more than 70%. Da-jun⁷ established a digital simulation platform for the annealing process of the full-hydrogen bell-type furnace and corrected the correction coefficient by using the actual measured data of the buried thermocouple test. The simulation platform matched with the actual production conditions on the site, the calculated temperature difference between the steel coil temperature and the measured temperature does not exceed 10 °C, and the error is within 4%. An online model was explored to predict and control the steel coil cold spot temperature⁸, and the average cold spot temperature prediction error of the two coils and three coils were − 0.45 °C and 2.16 °C, respectively. At present, there is still a lack of calculation accuracy in the model study based on heat transfer theory and practical experience, especially for the simulation of the temperature field of multiple coils in the same furnace. Besides, some scholars have applied intelligent algorithms to it. For example, Dong and Quan-li et al.^9,10 used neural networks, genetic algorithms, and fuzzy C-means clustering methods for hybrid modeling. Moreover, Wei-jie et al.¹¹ developed an online adaptive control (OSAC) method to predict and optimize the annealing process system, and used the self-learning correction module to improve the precision of annealing calculation. The results obtained by this method agree well with the test values. The cold spot temperature’s calculation error is within 10 °C, the relative error is within 1.1%. Although these models have many advantages, the computational accuracy of these models depends on the testing and training of large-scale production data, and the controlling temperature of the annealing process for the development of new steel products is not applicable.

Domestic and foreign scholars have reached a consensus on the heat transfer model, thermal resistance, and radial equivalent thermal conductivity of steel coil, and they have studied it very thoroughly. However, the current study does not fully consider the effect of gas flow distribution in each flow channel of the multi-barrel steel coil and the corresponding heat transfer coefficient on the temperature field, and there are few reports on the heat transfer model and the related industrial buried thermocouple test. In addition, the accuracy, stability, and reliability of the currently established model need to be further improved. The thermal equilibrium analysis method is an important means of studying the laws of heat transfer and conversion. By analyzing the heat balance of each part of the system, an energy conservation equation was established to describe the thermal state of the system. In the research field of bell type annealing furnaces, the thermal balance analysis method can clearly display the heat transfer path and conversion relationship inside the furnace from the perspective of energy, providing a theoretical basis for establishing accurate heat transfer models. Hence, in this study, a heat transfer model for bell-type annealing furnaces was established based on the method of thermal equilibrium, and its accuracy was analyzed via the comparison of industrial buried thermocouple test and model calculation.

Mathematical model of heat transfer

This article conducts a heat transfer analysis of the heating, insulation, and cooling steps of a bell type annealing furnace, and summarizes the heat transfer methods of the thermal process inside the furnace. Finally, a heat transfer model for the steel coil annealing process in a fully hydrogen bell type furnace is established based on the thermal equilibrium method. The technical roadmap for establishing a mathematical model of heat transfer is shown in Fig. 1.

Fig. 1

Technical route.

Heat transfer analysis of steel coil annealing process

The thermal process of the bell-type annealing furnace mainly includes heating, heat preservation, and cooling. During the heating process, the ignition of the burner is started until the protection gas reaches the prescribed phase transition point of the annealing process after the heating cover is hoisted. The heat preservation process starts from the end of the heating process. The steel coil is kept at the prescribed annealing process temperature until the steel coil reaches the required temperature and the difference in temperature between the inside and outside of the steel coil meets the requirements. A heating cover was used for cooling after the end of the heat preservation process. The gas supply is stopped first, and then the combustion air is sprayed out of the burner to scour and cool the inner cover. The air cooling process is finished when the temperature of the control thermocouple in the furnace reaches the required value, and the water cooling process is started. During the water cooling process, the inner cover is forced to cool by spraying the normal temperature water at the upper part of the cooling cover, thereby cooling the steel coil in the furnace. The entire annealing process is completed with the end of water cooling when the temperature of the control thermocouple reaches the set tapping temperature. The thermal process in the bell-type furnace is shown in Fig. 2.

Fig. 2

The diagram of thermal process heat transfer in the heating process of bell-type annealing furnace.

In Fig. 2, for the heating and heat preservation process, heat is transferred from the high-temperature flue gas to the steel coil in the combustion chamber through the protection gas in the furnace. Firstly, the high-temperature flue gas between the heating cover and the inner cover heats the inner cover through convection and radiation. Then the inner cover transmits the heat from its outer wall to the inner wall through heat conduction. On the one hand, the inner wall of the inner cover directly transmits heat in the form of heat radiation to the coil, and the inner wall of the inner cover heats the protection gas by the convection heat transfer. The gradually warmer protection gas contacts the steel coil and heats the steel coil by convective heat transfer. Finally, the steel coil internally transfers heat through heat conduction. For the cooling process, the direction of heat transfer is exactly the opposite of the heating process, which is transmitted from the steel coil in the furnace to the cooling medium.

The temperature change of the steel coil in the furnace has two reasons: one is that the protection gas heats and cools the steel coil in the form of convection heat transfer; the other is the radiation heat transfer of the inner cover to the coil. Studies have shown that the heat flux of the convective heat transfer is over 250 times that of the radiation heat transfer heat flux in the furnace. During the heat preservation process, the ratio is also maintained at more than 20 times^12,13. Therefore, the convective heat transfer of the protection gas and the coil plays a decisive role in the heat exchange in the furnace. The radiation heat transfer of the inner cover and the steel coil is negligible under normal working conditions¹⁴. The composite heat transfer coefficients can be used to replace the convective heat transfer coefficients when the proportion of radiation heat transfer in a special working condition is large.

Model establishment

Establishing a heat transfer model for bell furnace steel coils can predict hot spots, cold spots, heating time, and annealing time by simulating the temperature field, thereby controlling the temperature of steel coils during annealing, improving their performance and quality, shortening production time, and reducing heat treatment defects.

The thermal equilibrium method analyzes the heat change of the unit cell represented by each node in the divided grid by the Fourier heat conduction law, Newton cooling formula Stephen Boltzmann’s law, etc., and directly establishes its energy conservation expression. Then, under the given initial conditions and boundary conditions, the algebraic equations formed by the energy equations of the unit cells are solved. This model uses the thermal equilibrium method, and the model assumptions are as follows:

• (1) The heat conduction of circumferential direction can be ignored due to the symmetry of the coil structure and the even distribution along the circumference;

• (2) There is no internal heat source of the steel coil;

• (3) Ignoring the effects of cover changing during the cooling process;

• (4) The temperature of the protection gas in the furnace is considered to be uniform when modeling because of the high-speed flow of protection gas under the action of the circulating fan;

• (5) The coil heat exchange is mainly convection heat transfer. The radiation heat transfer is ignored when the protection gas temperature is lower than 800 K. When the protection gas temperature is higher than 800 K, the composite heat exchange coefficient is used instead of the convection heat transfer coefficient.

Based on the above assumptions, a heat transfer model for a steel coil in the furnace was established. First, the steel coil section was meshed, and the thermal equilibrium method was used to establish the thermal equilibrium equations for each node. The thermal equilibrium equations for the steel coil interior, the steel coil boundary, and the corner part of the steel coil are given here, as shown in Fig. 3.

Fig. 3

The diagram of mesh division and unit body heat transfer.

The steel coil interior (point 1 in the Fig. 3):

$$\begin{gathered} \quad \rho c\frac{{T_{m,n}^{i + 1} - T_{m,n}^{i + 1} }}{\Delta \tau } = \lambda_{r} \Delta z\frac{{T_{m + 1,n}^{i} - T_{m,n}^{i} }}{\Delta r} + \lambda_{r} \Delta z\frac{{T_{m - 1,n}^{i} - T_{m,n}^{i} }}{\Delta r} + \lambda_{z} \Delta r\frac{{T_{m,n + 1}^{i} - T_{m,n}^{i} }}{\Delta z} + \lambda_{z} \Delta r\frac{{T_{m,n - 1}^{i} - T_{m,n}^{i} }}{\Delta z} \hfill \\ \Rightarrow T_{m,n}^{i + 1} = \left( {1 - \frac{{2\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{2\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z}} \right)T_{m,n}^{i} + \frac{\Delta \tau }{{\rho c}}\left[ {\frac{{\lambda_{r} \Delta z}}{\Delta r}\left( {T_{m + 1,n}^{i} + T_{m - 1,n}^{i} } \right) + \frac{{\lambda_{z} \Delta r}}{\Delta z}\left( {T_{m,n + 1}^{i} + T_{m,n - 1}^{i} } \right)} \right] \hfill \\ \end{gathered}$$

(1)

The steel coil boundary (point 2 in the Fig. 3):

$$\begin{gathered} \quad { }\rho c\frac{{T_{m,N}^{i + 1} - T_{m,N}^{i + 1} }}{\Delta \tau } = \lambda_{r} \Delta z\frac{{T_{m + 1,N}^{i} - T_{m,N}^{i} }}{\Delta r} + \lambda_{r} \Delta z\frac{{T_{m - 1,N}^{i} - T_{m,N}^{i} }}{\Delta r} + \lambda_{z} \Delta r\frac{{T_{m,N - 1}^{i} - T_{m,N}^{i} }}{\Delta z} + h_{N} \Delta r\left( {T_{g}^{i} - T_{m,N}^{i} } \right) \hfill \\ \Rightarrow T_{m,N}^{i + 1} = \left( {1 - \frac{{2\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z} - \frac{{h_{N} \Delta r\Delta \tau }}{\rho c}} \right)T_{m,N}^{i} + \frac{\Delta \tau }{{\rho c}}\left[ {\frac{{\lambda_{r} \Delta z}}{\Delta r}\left( {T_{m + 1,N}^{i} + T_{m - 1,N}^{i} } \right) + \frac{{\lambda_{z} \Delta r}}{\Delta z}T_{m,N - 1}^{i} + h_{N} \Delta rT_{g}^{i} } \right] \hfill \\ \end{gathered}$$

(2)

The corner part of the steel coil (point 3 in the Fig. 3):

$$\begin{gathered} \quad \rho c\frac{{T_{M,N}^{i + 1} - T_{M,N}^{i + 1} }}{\Delta \tau } = h_{E} \Delta z\left( {T_{g}^{i} - T_{M,N}^{i} } \right) + \lambda_{r} \Delta z\frac{{T_{M - 1,N}^{i} - T_{M,N}^{i} }}{\Delta r} + h_{N} \Delta r\left( {T_{g}^{i} - T_{M,N}^{i} } \right) + \lambda_{z} \Delta r\frac{{T_{M,N - 1}^{i} - T_{M,N}^{i} }}{\Delta z} \hfill \\ \Rightarrow T_{M,N}^{i + 1} = \left[ {1 - \frac{{\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z} - \frac{{\left( {h_{E} \Delta z + h_{N} \Delta r} \right)\Delta \tau }}{\rho c}} \right]T_{M,N}^{i} + \frac{\Delta \tau }{{\rho c}}\left[ {\frac{{\lambda_{r} \Delta z}}{\Delta r}T_{M - 1,N}^{i} + \frac{{\lambda_{z} \Delta r}}{\Delta z}T_{M,N - 1}^{i} + \left( {h_{E} \Delta z + h_{N} \Delta r} \right)T_{g}^{i} } \right] \hfill \\ \end{gathered}$$

(3)

To control the steel coil temperature in real-time during the actual production process of the bell-type annealing furnace, a control thermocouple for monitoring the temperature of the protection gas is provided in the furnace. According to the actual process and assumptions, the temperature of the control thermocouple (T_R) can be completely considered to be the temperature of the protection gas, since the outer surface of the steel coil is directly in contact with the protection gas. Therefore, the boundary conditions for the heat transfer model of the steel coil annealing process can be set, and it’s considered that the temperature of the protection gas in the furnace is known, i. e. it’s equal to control thermocouple temperature, the temperature can be obtained through actual production data or manual settings.

For non-steady-state heat transfer numerical calculations, the influence of two adjacent moments’ nodes should remain unchanged. The reasonable situation is: the temperature of point n after i increases or decreases with the temperature at i. Hence the expression before the coil temperature in the heat balance equation established by each unit body should be greater than 0, as obtained by Eqs. (1),  (2), and (3):

$$\left. \begin{gathered} \left( {1 - \frac{{2\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{2\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z}} \right) > 0 \hfill \\ \left( {1 - \frac{{2\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z} - \frac{{h_{N} \Delta r\Delta \tau }}{\rho c}} \right) > 0 \hfill \\ \left[ {1 - \frac{{\lambda_{r} \Delta z\Delta \tau }}{\rho c\Delta r} - \frac{{\lambda_{z} \Delta r\Delta \tau }}{\rho c\Delta z} - \frac{{\left( {h_{E} \Delta z + h_{N} \Delta r} \right)\Delta \tau }}{\rho c}} \right] > 0 \hfill \\ \end{gathered} \right\}$$

(4)

Because the boundary conditions of the nodes on the coil are different, $\Delta \tau$ the required to meet the conditions at each node is different. To ensure that the solution of the equation converges at any nodes, the minimum value is chosen as the calculated time step. For the calculation conditions in this paper, this value was taken as 30 s.

Protection gas flow distribution

The protection gas is an important medium for the heat transfer process of the hood furnace. The flow distribution of the protection gas in the furnace directly affects the heat transfer speed of the steel coil. Therefore, determining the reasonable distribution of protection gas flow in the heat transfer model is of great significance to calculation with accuracy. Flow distribution refers to the flow rate of protection gas in the furnace through the various flow channels. The flow path in the furnace mainly includes the outer surface of the steel coil to the inner cover area, the area enclosed by the inner surface of the steel coil, the bottom furnace diffuser, the top space area, and the convection plate, etc., as shown in Fig. 4.

Fig. 4

The diagram of furnace protection gas distribution.

At present, domestic and foreign scholars mainly determine the flow channels in the bell-type annealing furnace through empirical assumptions¹⁵, cold tests¹⁶, flow resistance analysis and prediction correction¹⁷, and numerical simulation^18,19. There are some differences in the distribution of gas flow, while the overall rules are the same, showing that the flow of protection gas is the greatest at the bottom of the convection plate and the top of the furnace, and the flow at other convection plates decreases from bottom to top. In this work, the continuity equation and flow resistance equation of the protection gas circulation process in the furnace were established based on the theoretical analysis and guidance of previous research. Besides, a reasonable resistance coefficient was selected according to the distribution interval of the flow at each location in the experience. Moreover, the flow distribution coefficient and the flow attenuation coefficient were introduced, and the relationship between the flow distribution coefficient and the position of the flow path and the number of steel coils was obtained through trial calculation of a large number of experiments and simulations. Relational expressions such as formula (5) and formula (6) can be used to obtain the flow distribution at each flow channel of 4 layers of steel coils (Table 1).

$$\left\{ \begin{gathered} fd3 = {1 \mathord{\left/ {\vphantom {1 {\left( {2 + coNum} \right)}}} \right. \kern-0pt} {\left( {2 + coNum} \right)}} \hfill \\ fd2 = \zeta \cdot fd3 \hfill \\ fd1 = \zeta \cdot fd2 \hfill \\ \end{gathered} \right.$$

(5)

$$\Rightarrow \left\{ \begin{gathered} Q_{cp} 1 = Q \cdot fd1 \hfill \\ Q_{cp} 2 = Q \cdot \left( {1 - fd1} \right) \cdot fd2 \hfill \\ Q_{cp} 3 = Q \cdot \left( {1 - fd1} \right) \cdot \left( {1 - fd2} \right) \cdot fd3 \hfill \\ Q_{top} = Q \cdot \left( {1 - fd1} \right) \cdot \left( {1 - fd2} \right) \cdot \left( {1 - fd3} \right) \hfill \\ \end{gathered} \right.$$

(6)

Table 1 Flow distribution coefficient and flow ratio of each flow channel.

Protection gas convection heat transfer coefficient

The flow of the protection gas in the furnace can be considered as turbulence in the pipe slot, so the Dittus-Boelter forced convection heat transfer correlation in the pipe slot can be used. And the convective heat transfer coefficient of the protection gas is calculated using its correction formula after considering the effect of convection plate and diffuser²⁰:

$$h = 0.023 \cdot \psi \cdot \frac{{u_{g}^{0.8} }}{{D^{0.2} }} \cdot \lambda_{g} \cdot \left( {\frac{{\mu_{g} }}{{\rho_{g} }}} \right)^{ - 0.8} \left[ {1 + \left( \frac{D}{L} \right)^{0.7} } \right]$$

(7)

The following points should be noted when we calculate the convection heat transfer coefficient: Firstly, the flow distribution coefficient of the corresponding flow channel should be considered for the convection heat transfer coefficient at different flow channels. Secondly, the circulating medium in the full-hydrogen bell-type furnace is hydrogen, and its viscosity and thermal conductivity all change with temperature. Therefore, the actual convective heat transfer coefficient changes with the heating time in the calculation process.

To calculate the heat transfer coefficient, only the convection heat transfer was taken into consideration first and then the composite heat transfer coefficient was obtained after employing the radiation-induced enhancement coefficient (ξ) by comparing the calculated results with the production data, as shown in the formula (8):

$$k = \left\{ {\begin{array}{*{20}c} h & {T_{g} < 800\;{\text{K}}} \\ {\xi \cdot h,} & {T_{g} \ge 800\;{\text{K}}} \\ \end{array} } \right.$$

(8)

Steel coil radial thermal conductivity

The radial thermal resistance caused by the overlapping of thousands of thin steel sheets is the main problem that restricts the steel coil’s annealing heat exchange. The radial heat transfer of the steel coil is the combined effect of heat transfer within the air gap between two adjacent layers of steel coils, including the thermal conductivity of the protection gas in the air gap, the radiation heat transfer between the steel coils, and the heat conduction through the contact points of the two adjacent layers of steel coils. In recent years, many scholars, including Seong-Jun PARK, Saboonchi, and Witeket al.^21,22,23, have considered the thermal conductivity of the oxide layer on the coil when calculating the radial thermal conductivity. Some scholars, only consider the thermal resistance of the emulsion in the total radial thermal resistance because the emulsion on the surface of the coil will prevent the coil from being oxidized to a certain extent²⁴.

Some results were concluded in previous research on the mechanism of various physical and chemical changes of emulsion and grease in the surface of steel coils during the annealing of a full hydrogen furnace^{24,25,26,27,28}. The emulsion and grease on the coil surface will affect the radial heat transfer of the coil in the annealing process. And the series of changes will affect the atmosphere between the steel coils. However, most emulsions will be evaporated or reacted during the high-temperature process. Therefore, this paper only considers the effect of emulsion on radial thermal resistance in the heating process below 800 K, it’s considered that the surface of the steel coil is mainly an oxide layer and the content of the emulsion is negligibly small above 800 K, and the cooling process.

To simplify the calculation, it is assumed that the physical properties involved are only functions of the average temperature of the unit body, the cylindrical surface is equivalent to a plane, and the thermal resistance problem is discussed by the concept of steady-state heat transfer, so the simulation of the internal thermal resistance of the coil is shown in Fig. 5.

Fig. 5

Radial Equivalent Thermal Resistance Network Simulation.

The expression of each thermal resistance is:

$$R_{{{S \mathord{\left/ {\vphantom {S 2}} \right. \kern-0pt} 2}}} = \frac{S}{{2\lambda_{S} }}$$

(9)

$$R_{O} = \frac{{b_{O} }}{{\lambda_{O} }}$$

(10)

$$R_{R} = \frac{2 - \varepsilon }{{4\left( {1 - \varphi } \right)\varepsilon \sigma T_{m}^{3} }}$$

(11)

$$R_{G} = \frac{b}{{\left( {1 - \varphi } \right)\lambda_{g} }}$$

(12)

$$R_{D} = \frac{{\sigma_{P} }}{{1.13\lambda_{S} \tan \theta }}\varphi^{ - 0.94}$$

(13)

The thickness b_O of the oxide layer can be measured from the weight reduction during the pickling process, for different coils the thickness is approximately 10 μm²²; and the values of b and φ can be given by the following formulas^6,22:

$$\varphi { = }\frac{{\text{P}}}{{{\text{H}} + {\text{P}}}}$$

(14)

$$b = 42.7 \times 10^{ - 6} \exp \left( { - 5 \times 10^{ - 2} {\text{P}}} \right)$$

(15)

The total thermal resistance $R_{\Sigma }$ of the unit body can be expressed as:

$$R_{\Sigma } = \left\{ \begin{gathered} 2R_{{{S \mathord{\left/ {\vphantom {S 2}} \right. \kern-0pt} 2}}} + R_{O} + \left( {\frac{1}{{R_{R} }} + \frac{1}{{R_{G} }} + \frac{1}{{R_{D} }}} \right)^{ - 1} ,T_{g} \ge 800{\text{K}} \hfill \\ 2R_{{{S \mathord{\left/ {\vphantom {S 2}} \right. \kern-0pt} 2}}} + R_{E} + \left( {\frac{1}{{R_{R} }} + \frac{1}{{R_{G} }} + \frac{1}{{R_{D} }}} \right)^{ - 1} ,T_{g} < 800{\text{K}} \hfill \\ \end{gathered} \right.$$

(16)

Equations (9)–(16) are brought into the definition of the thermal resistance, and the formula for calculating the radial equivalent thermal conductivity of the steel coil can be obtained:

$$\lambda_{r} = \left\{ \begin{gathered} \frac{{S + b + b_{O} }}{{R_{\Sigma } }} = \frac{{S + b + b_{O} }}{{\frac{S}{{\lambda_{S} }}{ + }\frac{{b_{O} }}{{\lambda_{O} }} + \left[ {\frac{{4\left( {1 - \varphi } \right)\varepsilon \sigma T_{m}^{4} }}{2 - \varepsilon } + \left( {1 - \varphi } \right)\frac{{\lambda_{g} }}{b} + \frac{{1.13\lambda_{S} \tan \theta }}{{\sigma_{P} }}\varphi^{0.94} } \right]^{ - 1} }},T_{g} \ge 800{\text{K}} \hfill \\ \eta \cdot \frac{S + b}{{R_{\Sigma } }} = \eta \cdot \frac{S + b}{{\frac{S}{{\lambda_{S} }} + \left[ {\frac{{4\left( {1 - \varphi } \right)\varepsilon \sigma T_{m}^{4} }}{2 - \varepsilon } + \left( {1 - \varphi } \right)\frac{{\lambda_{g} }}{b} + \frac{{1.13\lambda_{S} \tan \theta }}{{\sigma_{P} }}\varphi^{0.94} } \right]^{ - 1} }},T_{g} < 800{\text{K}} \hfill \\ \end{gathered} \right.$$

(17)

Solution of heat transfer model for Bell-type annealing furnace

Using MATLAB to program and solve the heat transfer model of steel coils in the furnace, the specific programming solution is shown in Fig. 6.

Fig. 6

Flow chart of solving heat transfer model.

Buried thermocouple test

To verify the results of the model and ensure that the optimization results are consistent with the on-site data, it is necessary to test the trend of the internal temperature of the steel coil over time, analyze the important factors affecting the temperature distribution, and adjust reasonable model parameters to provide data support for model modification and verification. The temperature of the steel coil needs to be obtained by buried thermocouple test, i. e. inserting a thermocouple at a specific position of the coil. And the temperature change of the coil at different positions during the entire annealing process will be directly measured. The required equipment for this buried thermocouple test is shown in Table 2.

Table 2 The table of buried thermocouple test equipment.

Thin steel was inserted in the middle of the coiling process of the steel coil to form a certain gap in the radial direction, and the thermocouple can be inserted in the thin steel’s slot when lifting the steel coil (Fig. 7). The thermocouple can make good contact with the coil due to the coiling tension of the steel coil. The asbestos was stuffed into the other gaps which were formed by the thermocouples and the thin steel inserts, and compacted to prevent the hydrogen gas from being blown out during the test and convection heat transfer in the air gaps of the steel coils. The above measures can ensure that the temperature measured by the thermocouple can truly reflect the temperature of the steel coil at the measuring point.

Fig. 7

The diagram of inserted sheets and steel coil after inserting (The red circle in the picture is the insert). (a) Inserted sheets; (b) inserted steel coil; (c) sealing apparatus; (d) K-type armored thermocouple; (e) computer and temperature collector).

In the test, four representative steel coil layers were selected, and steel coils numbered 1# to 4# from bottom to top. The coil information is shown in Table 3. After confirming the position of the measuring points, the on-site worker calculated the specific welding position of the inserted sheets in the flat state of the steel coil according to the control system before welding the inserted sheets. The coil information and location of the specific measurement point are shown in Table 3 and Fig. 8, respectively.

Table 3 Bell-type furnace heat treatment steel coil data information sheet.

Fig. 8

The diagram of measuring points distribution.

Figure 9 shows the on-site installation process. The prepared thermocouple was inserted into the inserted sheet slot, and asbestos was plugged at the gap to prevent the hydrogen from being blown away before the steel coils were hoisted on the furnace table. When the convection plate was hoisted, the gap was aligned to avoid the thermocouple being broken. The thermocouple used in this test was a K-type armored thermocouple with a diameter of 2 mm. The temperature signal was recorded by a temperature collector, the collection frequency was 2 times/min, and the test results were recorded in the computer through the interface. After all the coils on the furnace table and the thermocouples in the steel coils were installed, the 12 thermocouples were numbered according to the measuring point number and corresponded with each other. The thermocouples were then passed through 12 perforations in the upper flange in sequence and tightened. Finally, the refractory clay was used to seal the interface of the lower flange. The seal apparatus was completely connected with the entire bell-type annealing furnace table after the lower flange of the sealing apparatus and the flange at the bottom of the furnace table. During the annealing process, the temperature change of each measuring point can be monitored in real-time through the temperature collector, as shown in Fig. 9.

Fig. 9

The diagram of on-site operation, coil stacking, and temperature collector interface. (a): thermocouple installation; (b) steel coil lifting; (c) steel coil stack; (d) temperature collector interface).

Results and discussion

As mentioned above, the heat transfer model was calculated by taking the temperature of the protection gas as a known number. The changes in gas flow velocity, flow rate, etc., with temperature need to be considered when the steel coil temperature field was calculated cyclically. The calculation was terminated and the data was saved when the calculation jumped out of the loop before being visualized. To achieve a precise comparison between the model and the test, the model calculation initial conditions, the size of the coil, type, and other information are shown in Table 3.

The calculation results of the model under the test conditions are shown in Fig. 10. From the temperature curves, it can be seen that the annealing process is divided into three stages heating, heat preservation, and cooling.

Fig. 10

Model calculation results under test conditions.

Production practice shows that the temperature of the measuring point varies with time in different positions of the same steel coil and the different layers of steel coil in the bell-type annealing furnace. However, each steel coil has a highest temperature point and a lowest temperature point during the heating and cooling process and is usually referred to as a hot spot and a cold spot. The hot spot measured by the test is usually at the edge of the steel coil, and the cold spot is usually near the center of the steel coil. As can be seen from Fig. 8, the temperature lags behind the hot spot in time, and the maximum temperature of the cold spot was also lower than the hot spot at the end of the heating period. The probable reason is the direct convection heat transfer happened between the steel coil hot spot and the protection gas. Moreover, the steel coil temperature and gas temperature are also close because the heating and cooling speeds are fast, and the heating and cooling of the cold spot must rely on its adjacent point’s heat conduction. It is necessary to set the insulation phase in the annealing process to give the cold spot a sufficient heating time reduce the temperature difference between the cold spot and the hot spot and keep the steel coil heated evenly. And the ideal product performance will be obtained.

From Fig. 10, it can be judged that the temperature curve of the characteristic points (hot and cold spots) on the steel coil calculated by the heat transfer model conforms to the actual production state. However, the accuracy of the calculation results further needs to be verified by tests.

The results of the buried thermocouple test are shown in Fig. 11. Obviously, the trend of the curve in the figure is divided into three stages heating, heat preservation, and cooling. At the beginning of the heating process, the temperature rises rapidly in the furnace, and the heating rate is significantly slowed down in the late period of the heating process. This is because the internal crystal phase structure of the steel coil changes abnormally leading to the heating rate preventing from being too fast, which in turn causes the mechanical properties of the steel coil to fail to meet customer requirements. About 3 h after heating (marked (a) in Fig. 11), the protective gas and the hot spot temperature decreased significantly due to a short-term failure of the combustion system. On the one hand, it was found that fluctuations in protection gas and hot spot temperature did not have a significant effect on the rising trend of the cold spot temperature. During the coil annealing process from the 23rd to the 24th hour (marked (b) in Fig. 11), the protection gas and the hot spot temperature increase significantly. This reason is that this stage turns from air cooling to water cooling and the cooling mode was changed from the original forced convection cooling to natural convection cooling, leading to the greatly reduction of cooling intensity. It caused the protection gas to be cooled at a lower rate by the outer side of the inner cover and the air than by the inner side of the steel coil so that the temperature of the protection gas rose. In addition, although the protection gas is in the actual process circulated in the furnace, there is still a temperature difference in the protection gas in all places, and there is also a temperature difference in the surface of the inner cover. Therefore, uneven cooling will cause its temperature to rise, which will further cause the steel coil temperature to rise slightly.

Fig. 11

The buried thermocouple test results.

Figure 12 shows the cold spot temperature curves and the error curves calculated by the model and the buried thermocouple test. It can be seen that the model calculation results and the test results almost exactly coincide with each other regardless of whether they were 1# steel coil or 2# steel coil. Specifically, the average error of the model temperature calculation was |± 5| °C; the maximum cold spot temperature error did not exceed |± 2| °C; and the temperature error at the end of the heating process did not exceed |± 3| °C, either. Therefore, the cold spot temperature average relative error calculated by the model was less than 0.5%. When the cold spot reaches the maximum temperature, the highest cold spot temperature of the 1# steel coil measured by the test was 667.2 °C, and the appeared moment was18.5 h; the maximum cold spot temperature of the 1# steel coil calculated by the model was: 666.8 °C, and the appeared moment was18.3 h. The time error was only 0.2 h, and the relative error of the model calculation was 1.1%. The highest cold spot temperature of the 2# steel coil measured by the test was671.2 °C, and the appeared moment was18.5 h; the maximum cold spot temperature of the 2# steel coil calculated by the model was 673.5 °C, and the appeared moment was18.2 h. The time error was only 0.3 h, and the relative error of the model calculation was 1.6%. From this, we can see that the accuracy of the model has been greatly improved compared with the previous models^[5,7,11] of some scholars.

Fig. 12

The comparison of the cold spot temperature by model and test and model calculation error curve. (a) 1# steel coil; (b) 2# steel coil).

The hot spot temperature curves and the error curves calculated from the model and the buried thermocouple test are presented in Fig. 13. The model calculation results and the test results almost exactly agree with each other regardless of whether they were 3# steel coil or 4# steel coil. Especially, the average error calculated by the model did not exceed 25 °C, and the temperature error at the end of the heating process did not exceed 5 °C. Therefore, the hot spot temperature average relative error calculated by the model was less than 5%. The test measured the hotspot temperature of 3# steel coil at the end of the heating process was 707.3 °C, and the appeared moment was17.30 h; The model calculated the hot spot temperature of the 3# steel coil at the end of the heating processwas702.9 °C, and the appeared momentwas17.27 h. The time error was only 0.03 h. The relative error of the model calculation was 0.17%. The test measured the hot-spot temperature of the 4# steel coil at the end of the heating process was 711.6 °C, and the appeared moment was17.28 h; the model calculated the hot spot temperature of the 4# steel coil at the end of the heating process was 706.9 °C, and the appearance moment was 17.24 h. The time error was only 0.04 h. The relative error of the model calculation was 0.23%.

Fig. 13

The comparison of the hot spot temperature by model and test and model calculation error curve. (a) 3# steel coil; (b) 4# steel coil).

The error of the hot spot temperature calculated by the model is greater than that of the cold spot error. The reason is mainly because the existence of the convection plate will cause turbulence and negative pressure in some areas of the furnace, thus affecting the heat transfer of the steel coil surface. On the other hand, the non-uniformity of the temperature of the protection gas and the inner cover will also increase the calculation error of the hot spot temperature.

From the error comparison and analysis of the model calculation results and the buried thermocouple test results, the heat transfer model has small calculation errors for different layers of steel coils’ temperature and time, and the reliability is very high. Especially for the calculation of the cold spot temperature, the accuracy was above 95% when the relative error of cold spot temperature is within 2.5%, which proves the accuracy of the model (Fig. 14), indicating the accurate calculation of the flow distribution at different flow channels in the furnace and the convection heat transfer coefficient of the protection gas. The flow distribution of each flow channel will directly affect the convection heat transfer coefficient of the protection gas and steel coil, and then affect the cold and hot spot temperature of different steel coils. Therefore, it is very important to accurately predict the change of the cold spot and the hot spot temperature to judge whether the temperature difference between cold spots and hot spots of all the steel coils in the annealing process meets the requirements.

Fig. 14

Model calculated cold spot temperature relative error curve.

The cross-sectional isotherm distribution of steel coils during the initial heating and cooling stages is shown in Fig. 15. After heating for 0.5 h, as shown in Fig. 15a, we can see that the temperature of the steel coil gradually increases. From the distribution characteristics of isotherms, it can be analyzed in depth that the number of isothermal lines in the axial direction is significantly denser than in the radial direction per unit length. The density difference of this isotherm actually reflects the severity of temperature changes, that is to say, the temperature difference in the axial direction is greater than that in the radial direction per unit length. The magnitude of the temperature difference is closely related to the heat transfer rate, and it can be inferred that the axial heat transfer rate is greater than the radial heat transfer rate. The main reason for this situation is that the thermal conductivity of the steel coil in the radial direction is much smaller than that in the axial direction. Thermal conductivity is an important parameter for measuring the thermal conductivity of materials, and a smaller thermal conductivity means that heat transfer is relatively difficult. Therefore, in this case, the axial heat transfer depth will be greater, and heat will be more easily transferred along the axial direction. Looking at Fig. 15b again, this is the isotherm distribution of the steel coil after cooling for 0.5 h. A similar pattern can also be observed, where the heat transfer rate in the axial direction is greater than that in the radial direction. The same conclusion can be drawn from the density of isotherms and the trend of temperature changes as during the heating process. Taking into account both heating and cooling processes comprehensively, although the heat transfer rate in the axial direction is locally higher than that in the radial direction, the radial thermal conductivity coefficient is smaller, which has a more significant hindering effect on the overall heat transfer. Therefore, the heat transfer rate of the steel coil mainly depends on the radial heat transfer rate. Based on the above analysis, it is particularly important to accurately calculate the radial thermal conductivity of steel coils. In the actual annealing process of steel coils, the precise value of radial thermal conductivity is directly related to the efficiency and uniformity of heat transfer inside the steel coil. If this parameter can be accurately grasped, the annealing process can be optimized to reduce the annealing time of steel coils, thereby improving production efficiency and better ensuring the quality of steel coils, reducing quality problems caused by uneven temperature.

Fig. 15

Steel coil cross-section isotherm distribution. (a) heating 0.5 h; (b) cooling 0.5 h. (Matlab R2022b, https://www.mathworks.com)

Figure 16 shows the cross-section temperature field at various times during the annealing process of steel coils. The steel coil temperature field gradually changes from external hot and internal cold to external cold and internal hot, indicating that the steel coil is first heated and then cooled. Moreover, there is a temperature difference between the inside and outside of the steel coil, and the internal temperature change always lags behind the surface temperature change.

Fig. 16

Steel coil cross-section temperature field. (a): 5 h; (b) 15 h; (c) 18 h; (d) 20 h; (e) 25 h. (Matlab R2022b, https://www.mathworks.com).

In addition, it can be seen from Figs. 16 and 10 that the best heat transfer condition for 4 layers of steel coils is the 2nd layer of steel coils (2#). This is because during the actual annealing process, although the protection gas passing through the flow path at the upper surface of the 1st layer coil is much (20.2%), see Table 1, no convection plate is placed on the lower surface of the 1st layer coil. Therefore, the steel coil has only 3 heat exchange surfaces and the heat transfer conditions are relatively poor. However, the 2nd layer of steel coils has 4 heat exchange surfaces, and the protection gas passing through the inner and outer surfaces of the coil is also relatively high, accounting for 79.8% of the total, which is only lower than 100% of the first layer of coils (Table 1). Therefore, the heat transfer condition of the second layer coil is better than that of the first layer. The aforementioned results imply that the heat exchange of the steel coil depends mainly on the radial heat transfer, so the protection gas passing through the inner and outer surfaces of the coil plays a major role in the heat transfer of the coil. The protection gas passed through the inner and outer surfaces of the 2nd layer coil is higher than the 3rd and 4th layers (65.2% and 54.3%, respectively). Therefore, the heat transfer conditions of the second-layer steel coil are better than those of the third and fourth-layer steel coils. The worst heat transfer condition in the 4 steel coils is the 3rd layer coil (3#). This is because the 3rd layer coils have the least protection gas flowing through the upper and lower convection plates, only 10.9% and 14.6% (Table 1). Moreover, the protection gas passing through the inner and outer surfaces of the coil is also very small, only 65.2% is higher than that of the 4th layer (54.3%, Table 1). However, the heat transfer condition of the 4th layer coil is also better than that of the 3rd layer because the protection gas passed through the top channel is the most (54.3%, Table 1), which causes the convection heat transfer coefficient of the 4th layer of coil top and protection gas is greater than the other three coils. On the other hand, heat transfer is affected by the heat radiation of the inner cover because the upper surface of the 4th layer steel coil does not stack the convection plate. Hence, the heat transfer condition of the 4th layer coil is not the worst. In addition, from Figs. 10 and 16, it can be seen that there is a large temperature difference in the cold spot temperature of 4 stacks of steel coils in the same furnace, and the maximum value is nearly 100 °C. Therefore, it is very important to accurately calculate the temperature of different coils in the furnace by using the model and determine whether the cold spot temperature of the coil with poor heat transfer conditions meets the required value for the production of high-performance cold-rolled coils.

Conclusions

• 1. A heat transfer model with high accuracy for a bell-type annealing furnace based on the method of Thermal Equilibrium was established successfully. The cold spot temperature average relative error calculated by the model is less than 0.5% and no more than 5% for a hot spot, which proves the accuracy and reliability of the model, especially for the cold spot temperature. The relative error of the model calculation is less than 2% and 0.5% at the maximum temperature and the end of the heating process of the cold spot, respectively.

• 2. The heat transfer condition of the furnace can be well described by model. The best heat transfer conditions for the 4 layers of steel coils is the 2nd steel coil (2#), while the worst is the 3rd steel coil (3#).

• 3. This work provides an accurate heat transfer model for fitting the heating transfer process of bell-type annealing, which is significant for adjusting the parameters of the furnace to get better heat transfer and enhance the quality of the product.