J. Cent. South Univ. (2016) 23: 1719-1728

DOI: 10.1007/s11771-016-3226-6

Temperature control for thermal treatment of aluminum alloy in a large-scale vertical quench furnace

SHEN Ling(沈玲), HE Jian-jun(贺建军), YU Shou-yi(喻寿益), GUI Wei-hua(桂卫华)

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

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: The temperature control of the large-scale vertical quench furnace is very difficult due to its huge volume and complex thermal exchanges. To meet the technical requirement of the quenching process, a temperature control system which integrates temperature calibration and temperature uniformity control is developed for the thermal treatment of aluminum alloy workpieces in the large-scale vertical quench furnace. To obtain the aluminum alloy workpiece temperature, an air heat transfer model is newly established to describe the temperature gradient distribution so that the immeasurable workpiece temperature can be calibrated from the available thermocouple temperature. To satisfy the uniformity control of the furnace temperature, a second order partial differential equation (PDE) is derived to describe the thermal dynamics inside the vertical quench furnace. Based on the PDE, a decoupling matrix is constructed to solve the coupling issue and decouple the heating process into multiple independent heating subsystems. Then, using the expert control rule to find a compromise of temperature rising time and overshoot during the quenching process. The developed temperature control system has been successfully applied to a 31 m large-scale vertical quench furnace, and the industrial running results show the significant improvement of the temperature uniformity, lower overshoot and shortened processing time.

Key words: large-scale vertical quench furnace; temperature calibration; thermal dynamic model; decoupling control

1 Introduction

Large-scale vertical quench furnaces are the key equipment for the thermal treatment of aluminum alloy workpieces to improve their hardness, strength and anticorrosion properties. The temperature fluctuation during the temperature holding period is often strictly confined to a very narrow range about ±3 °C. To obtain the qualified symmetrical high-strength workpiece, the temperature control of the vertical quench furnace requires high-precision, stability and uniformity. However, the large-scale vertical quench furnace is with height of 31 m and diameter of 2.8 m. The furnace consists of multiple heating zones in the axial direction and two chambers in the radial direction. The temperature control system of the large-scale vertical quench furnace, which is characterized by distributed parameters, long time delay, nonlinear and strongly coupled variables, is thus challengeable to be developed to achieve the desired control precision.

Only a few publications concerned with the temperature control of the large-scale vertical quench furnace. Based on the assumption that the heat exchange inside the vertical quench furnace has reached a balance, ZHOU et al [1] established five thermal equilibrium equations to describe the radial temperature gradient distribution. Therefore, the equations were only suitable for the steady state of the heating process. JIANG et al [2] presented a Hammerstein fault calibrating model based on least squares support vector machine to predict the temperature of the quench furnace. Although these researches about the quench furnace are out of date, they are still valuable. Considering the temperature control systems of the reheating furnaces and other large-scale furnaces have been widely studied, researches about the reheating furnace can be taken as references [3-4]. ORZYLOWSKI et al [5] considered the temperature control problem as a lumped parameter dynamic optimiz- ation system for horizontal furnaces. With the increasing demand of the control precision, conventional lumped parameter based temperature control systems which disregard the space distribution of the temperature, are hardly accomplish optimal control for the large-scale vertical furnace. AGGELOGIANNAKI et al [6] proposed a nonlinear model calibrative control for hyperbolic distributed parameter systems (DPS) which was applied in the flow-based temperature control in a long duct. The system used a radial basis function neural network to estimate the temperature distribution. CAO et al [7] developed a new method for modeling DPS using a B-spline neural network and a linear recurrent neural network. A hybrid intelligent control method is proposed by CHAI et al [8] to control the technical indices into the desired range. The proposed approach has efficiency on the roasting process of shaft furnace. More recently, STEINBOECK et al [9] designed a nonlinear model predictive controller which is suitable for non-steady-state operating situations.

For large-scale heating furnaces, an early assumption considered that the heat transfer effects among heating zones were relatively small, and the dynamics of the furnace can be modeled by multiple independent single-input/single-output (SISO) models. However, when there are a large number of zones, coupling effects caused by heat transfer between two neighboring zones become strong. In this case, the independent SISO models are no longer accurate enough to describe the heating system. Hence, the entire furnace dynamics have to be identified as a multi-input/multi- output (MIMO) model. ZHOU et al [10] extended SISO techniques to design MIMO calibrative temperature control systems. HE et al [11] focused on a multi- stage control algorithm to improve control performance in the large-scale quench furnace. There are papers regarding the coupling issue in the reheating furnace [12-14]. LI and MENG [15] proposed a PID decoupling control method based on DRNN neural network setting for electrical heater furnace. REDDY et al [16] used the decoupling technique and the Smith compensation method to solve the strong coupling in a reheating furnace. All of the above works are enlightening to our search.

This work contributes to introduce and summarize our research results on temperature control of the large-scale vertical quench furnace thoroughly, including workpiece temperature calibration, temperature uniformity control, and its application in large-scale vertical quench furnaces. First, an air dynamic mathematical model is established to describe the temperature gradient distribution, which is used to calibrate the workpiece temperature according to the available sensor temperature. Then, a MIMO second- order partial differential equation (PDE) model describing the dynamics of thermal treatment is derived. Based on the proposed dynamic model, a feedback gain matrix, also known as a decoupling matrix, is designed to decouple the heating process into multiple independent heating zones. The temperature control system (TCS), integrating the temperature calibration of the workpiece and the decoupling control of multi-heating zones, is developed for thermal treatment of aluminum alloy workpieces in the large-scale vertical quench furnace. The running of TCS increases the eligible rate of the products, improves the efficiency of thermal processing, and decreases the electrical power consumption.

2 Heating process analysis

Large-scale workpieces made from aluminum alloy are basic components of rockets, missiles, airplanes, submarine, etc. Thermal treatment is an essential process to improve the mechanical and physical properties of the aluminum alloy, including strength, hardness and corrosion resistance. To reach the desired mechanical and physical properties, the temperature control in vertical quench furnace used for thermal treatment of aluminum alloy workpieces requires exceptionally high precision, stability and uniformity.

As shown in Fig. 1, the large-scale vertical quench furnace is composed of a heating chamber and a working chamber in the radial direction, while the outer diameter of the working chamber is 1.5 m and the external diameter of the heating chamber is 2.8 m, with height of 31.64 m. The heat insulation material between the heating chamber and the furnace wall is made of silicate cotton to minimize heat loss. The working chamber wall is made of stainless steel to increase the radiation uniformity. The alloy workpiece is suspended in the central area of the furnace via a hook. The electric heaters are firmly installed on the heating chamber wall in the axial direction, responsible for the regulation of furnace temperature by means of changing the electric power. The heaters divide the furnace into multiple heating zones from top to bottom. For each zone, temperature is measured by a thermocouple. Two ventilators are installed at the bottom of the furnace to circulate airflow between the heating and working chambers and to accelerate heat convection.

During the quenching process, the workpiece is first loaded through the furnace door within 3 to 5 minutes. After the furnace door is closed, the ventilators are switched on to circulate airflow. Then, the electricheaters will heat up the workpiece through heat convection and radiation for about three hours, including the temperature rising period and holding period. During the temperature rising period, the workpiece temperature increases rapidly because the air in the furnace is heated up really fast. While in the temperature holding period, the workpiece temperature tends to fluctuate within the allowed temperature range. When the measured temperature profile satisfies the expected temperature distribution, the workpiece is at last discharged through the furnace door. Generally, the allowed maximum overshoot is about 10 °C in the temperature rising period, while the axial temperature distribution uniformity inside the furnace should be within ±5 °C, sometimes even ±3 °C [17], so that the mechanical and physical properties of aluminum alloy workpieces can be guaranteed in the holding period.

Fig. 1 Sectional view of large-scale vertical quench furnace

Based on the above analysis, it can be found that the workpiece temperature cannot be measured directly.

Moreover, the heating air keeps circulating in the furnace resulting in strong coupling effects among multi-heating zones. To solve these two issues, a temperature control system (TCS) is developed as shown in Fig. 2. The TCS consists of parameter setting module, temperature calibration module, and temperature uniformity control module. Once the batch number of workpiece is determined, parameter setting module presets the temperature setpoint, dimension and alloy content of corresponding workpiece. Temperature calibration module calibrates temperature measurements from thermocouples based on the air thermal mathematical model. As a result, the workpiece temperature can be accurately obtained. Temperature uniformity control module implements the decoupling between multiple heating zones, and provides real-time control signal of each heating zone to regulate electric power.

3 Workpiece temperature calibration

The workpiece temperature cannot be measured directly. Because the workpiece is suspended in the center of the furnace, and sometimes it swings during the heating process because of the circular air flow. Therefore, the thermocouples are evenly installed along the working chamber wall with only 2-3 cm extending to the working chamber to avoid collision. Apparently, there inevitably exists a temperature gap between the workpiece and the working chamber wall. The deviation between the measured temperature and the real temperature of the workpiece easily leads to overburning, which results in unsuitable thermal treatment and even produces lots of wasted workpieces. Thus, accurate workpiece temperature calibration is the premise of qualified thermal treatment.

Because the air is the main heat transfer medium between the working chamber wall and the workpiece by heat convection and heat conduction. The temperature distribution of the heating air is able to represent the temperature gap. If the temperature distribution of the air is obtained, the surface temperature of the workpiece can be estimated. Based on the above analysis, the heat transfer model of the air is developed as the workpiece temperature calibration model. The main assumptions and features of the model are as follows:

1) The radiation heat exchange is neglected because the air is a completely transparent body;

2) The flow rate of the air is constant;

3) Heat convection of the air is neglected in the radial direction, that is the flow rate of the air in the radial direction is assumed to zero;

4) Two-dimensional temperature distribution is calculated for the air.

Based on the above assumptions, the unsteady thermal exchange model is calculated as follows:

  (1)

where r, φ and z are shown in Fig. 3.

Fig. 2 Schematic diagram of temperature control

Fig. 3 Cylindrical coordinate of air dynamic model

Because the furnace geometry is axisymmetric, the variable T is independent of the circumferential axis f, the Eq. (1) becomes

           (2)

The initial conditions and boundary conditions for the PDE are:

, in Ω

on Γ_D

The finite element method [18] which is widely used to solve the PDE is adopted.

First, introducing the weighted function w. According to the weighted residual method, the integral becomes

        (3)

Because the geometry of the air in the furnace is axisymmetric, the domain integral is demonstrated as

                (4)

Herein, T(r, z) is any function which is independent of φ. Combining Eq. (4) and Eq. (3), and using integration by parts gives the weak formulation of Eq. (2).

            (5)

Second, using the finite element discretization in domain Ω. The considered temperature field is represented by a group of finite triangular elements. Hence, Ω is divided into n_e three-noded triangular

elements, then

Temperature T^e(r, z) within a finite element is given by the variable interpolation of nodal temperature :

              (6)

where H_i(r, z) is the shape function of the linear triangular element; is the node temperature; [H] and [T^e] are the shape function matrix, and the node temperature matrix, respectively.

Due to Galerkin method, the weight function is chosen as w=T^e. Substituting Eq. (6) into Eq. (5), each of the sub-element matrix [K^e], [M^e] can be obtained.

       (7)

where

Third, the system element K, M are yielded by assembling each K^e, M^e. Combining the given boundary conditions F, the ordinary differential equations for Eq. (2) is obtained.

                      (8)

The PDE has transformed into a set of ordinary differential equations (ODEs) using the finite element method. To solve the equations, the forward difference method is adopted for the time derivative and Eq. (8) becomes

        (9)

Solving Eq. (9) can get the temperature distribution of the working chamber air and the temperature gap between the workpiece and the thermocouple can be thus obtained.

4 Temperature uniformity control

4.1 Thermal dynamics model

Once obtain the workpiece temperature, the thermal dynamic model of the quenching process needs to be built to reach the uniformity control. For the thermal treatment process of the aluminum alloy workpiece, heat conduction, heat convection and heat radiation coexist in the furnace. The accurate analysis of the heat exchange during the quenching process is difficult. Although thermal dynamics in the quench furnace is complex, reasonable assumptions can simplify the model structure and reduce the model complexity. Based on the facts that the aluminum alloy workpiece in the vertical quench furnace has good thermal conductivity, and its diameter (less than 0.8 m) is far shorter than its length (about 31 m), therefore, it can be supposed that the radial temperature distribution of the workpiece is uniform. And the dynamic model only concerns about the temperature distribution of the workpiece in the axial direction. The working chamber wall made of stainless steel is use to enhance the temperature uniformity and is irrelevent to the control problem, therefore, it can be neglected. The heat transfer inside the furnace can be simplified as shown in Fig. 4. The real line arrowheads represent the heat radiation from the electric heater to the workpiece, and the dash line arrowheads represent the direction of air flow and the heat convection.

Fig. 4 Simplified heat exchange process

The temperature distribution problem is distributed and nonlinear. Considering the thermal radiation and convection coexist inside the furnace, the nonlinear thermal dynamic system model is derived as

         (10)

where θ₁(T), θ₂(T) and θ₃(T) vary nonlinearly with temperature during the entire heating process. But in the temperature holding period, furnace temperature rarely fluctuates, which is regarded as a stable working point. In the vicinity of the steady-state working point, nonlinear model (Eq. (10)) is linearized as

     (11)

When the large-scale vertical quench furnace is heated at discrete points then whereis Dirac function.

Neglecting heat loss at two ends of the furnace. Assuming the initial workpiece temperature is equal to the ambient temperature T₀, then the boundary conditions are simplified as

In Eq. (11), θ₁, θ₂ and θ₃ are unknown model parameters, which are needed to be identified.

There are 11 heating zones in the large-scale vertical quench furnace, and the workpiece to be heated is usually between zone 3 and zone 10. Because zone 11 closes to the bottom of the furnace and it is easily influenced by ambient temperature, therefore, the heating effect of zone 11 is neglected. So does the zone 1. The samples from zone 2 and zone 10 are regarded as boundary condition. The length of 8 zones is the efficient height (0≤z≤23 m). The height of each zone is h=23/8= 2.857 m, and the efficient time is 0≤t≤480 s.

Once the temperatures of zone 2 to zone 9 fall in the range of ±5 °C of the initial temperature set-point, the heating process goes into the holding period. Samples are collected from zone 2 to zone 10 per minute, lasting for 9 min. Its model parameters are identified by means of iterative least square method:

In industry application, distributed parameter system can be controlled by discrete controllers. In this paper, finite difference method is adopted to realize finite-dimensional approximation. In Eq. (11), x replaces T for readable presentation, and Let h=L/m is the space step length, dividing furnace axially to m zones. For the i-th zone, the discrete form of Eq. (11) is:

 (12)

Letand The state equation and the output equation is described as

                   (13)

                             (14)

where

,

,

,

C₁=1.

Then, the convergence property of finite difference approximation for discrete dynamic model should be simply analyzed below [19].

Firstly, the step-size τ is used for the discretization of the time.

Secondly, according to the Euler method, the conclusion is derived that ifand the finite-dimensional discrete model is convergent.

4.2 Expert control

Because of the special quenching technique of the aluminum alloy workpiece, heat convection and heat radiation cause strong coupling effects between multiple heating zones. To eliminate the coupling effects, a commonly used decoupling control algorithm is adopted for the heating process. From Eqs. (13)-(14), the state equation is demonstrated as

                               (15)

To simplify the controller design, the coupling between non-adjacent zones is neglected because the temperature of each zone is influenced mostly by the adjacent zones. Hence, only coupling between the adjacent zones is taken into account, and the output of controller in each zone u(t, z_i) is decided by state variables x_i-1, x_i, x_i+1.

Suppose the state feedback gain matrix is

and the state space equation of closed loop system is

                        (16)

After calculation yields the K,

                                (17)

                           (18)

                           (19)

After decoupling, the state equation becomes

                              (20)

The system is decoupled into eleven independent heating zones under the state feedback matrix K, also called as the decoupling matrix. The off-diagonal elements in matrix are called decoupling coefficients. Because the heating condition has a little fluctuation, the decoupling matrix needs on-line adjustment.

Suppose and (i=1, 2, …, m) are decoupling terms of adjacent zones, where denotes the decoupling coefficient between i-th zone and (i-1)-th zone, denotes the decoupling coefficient between i-th zone and (i+1)-th zone, and are temperature differences of adjacent zones at time k, given as where is the temperature of i-th zone at time k. and are calculated based on decoupling coefficients in the decoupling matrix K. Supposeis the compensation output generated by the effect of (i-1)-th zone on i-th zone at time k, and is the compensation output generated by the effect of (i+1)-th zone on ith zone at time k. Then,

         (21)

        (22)

After obtaining the decoupling matrix, the coupling between zones is simplified as the influence of temperature gradient between adjacent zones. The temperature control in the ith zone is illustrated by the diagram shown in Fig. 5. R_i(k) and Y_i(k) denote the temperature setpoint and the calibrated temperature of workpiece in ith zone. Y_i+1(k) denotes the calibrated temperature of workpiece in (i+1)-th zone. G_i(z) is the transfer function of thermal dynamics in ith heating zone . And the output control U_i(k) at time k is defined as

               (23)

where u_i(k) is the output of expert controller at time k. Suppose t_on denotes the time of pulse current output, t_off denotes the turn-off time, t_p denotes the on-off period, and the ratio of t_on to t_off is called on-off ratio of electric current. Then

                              (24)

In the rising period, a compromise must be found between the rising time and overshoot. If the rising time is shortened, the overshoot will consequently be large, resulting in over-burn effect. If the overshoot is carefully maintained, the long rising time will affect the production efficiency. In holding period, the objective is to diminish temperature fluctuation. To solve the dilemmas, multi-stages approach is needed. In different heating stage, u_i(k) is determined empirically. A typical expert rule is exemplified as follows:

IF THEN

ELSE IF THEN

ELSE IF THEN

ELSE IF THEN

ELSE

where

Parameters ET1, ET2, ET3, p₁, p₂, p₃ are determined by the forged workpiece state, including its dimension, alloy content, etc. All parameters are stored as a form of prescriptions. When one workpiece state comes to work, the corresponding prescription is selected. For example, the batch number of workpiece is B478, its alloy state is LY11CZ, the on-off control period t_p=120 s, ET1=3 °C, ET2=10 °C, ET3=15 °C, p₁=4 s/°C, p₂=6 s/°C, p₃=8 s/°C.

To prevent harmonic wave pollution, high-power solid-state-relay has replaced thyristor, which has advantages like simple structure, convenient control and low fault rate. Pulse-width-modulation (PWM) is used to control electric power. And zero trigger replaces phase trigger to eliminate harmonic wave pollution on electric power grid. Once U_i(k) is calculated, on-off ratio of electric current is calculated to control the heaters by PWM. Consequently in rising period, the temperature of each zone is rising at an optimal rate and the overshoot is confined to an acceptable range. In holding period the temperature of each zone can be kept within allowable fluctuation range of set-points.

Fig. 5 Control principle diagram in i-th zone

6 System implementation results

The developed system has been running for 31 m large-scale vertical quench furnace in Southwest Aluminium Co. Ltd., China.

6.1 Calibration results

To testify the calibration accuracy of the developed model, industrial experiments were conducted to acquire temperature measurements, including the measured temperature from thermocouples installed along the working chamber wall and the workpiece temperature measured by tying the thermocouple wire around the workpiece. Because the thermocouple wire costs a lot and is easy to be broken, it is only used to measure workpiece temperature in industrial experiments. Three thermocouples were assembled in the upper surface, middle surface, and lower surface of the workpiece which are represented as Sensor 1, Sensor 2, and Sensor 3, respectively. To obtain the air temperature, in the same way, the other three thermocouples were fixed on an iron shelf which was hung in the working chamber just for the experiment requirement.

Figure 6 shows the comparison of the calibrated and the measured workpiece temperature when the temperature set-point is 470 °C. In the temperature rising period, all of the electric heaters are switched on. Therefore, the heat exchange in the furnace during this period is drastic. The heating air primarily accumulates on the top of the furnace, and resulting in the higher temperature of sensor 1. It can be seen from Fig. 6 that the quick air flow speed makes the furnace temperature rising rapidly, and the temperature profiles thus appear gradient distribution. Only in half an hour, the workpiece temperature arrives at the maximum 470 °C and minimum 460 °C. During this period, the error in a certain range is reasonable, because there is rarely overburing accident in this period. The quality of the workpiece mainly depends on the temperature distribution uniformity of the holding period. Therefore, although the calibrated temperature is slightly higher than the measured temperature, there exists a reasonable agreement between the calibrated and measured temperature profiles during the temperature rising period. But in the temperature holding period, especially in thefinal period, the heat exchange reaches at the thermal equilibrium, and the calibrated temperature becomes in good agreement with the measured temperature. This means that the heat transfer model can be successfully applied to the large scale vertical quench furnace for the calibration of the workpiece surface temperature.

Fig. 6 Comparison of predicted and measured workpiece temperature during temperature holding period

Figure 7 illustrates the relative error of the temperature calibration of the three sensors. Note that the calibration error in the temperature rising period is a little larger than that in the temperature holding period because of the rapid temperature rising trend. However, the relative error is still in rational range and it completely satisfy the calibration accuracy. The agreement between the measured temperature and the calculated temperature shows the established model is valid for estimating the temperature gap and obtaining the workpiece temperature.

Fig. 7 Relative error of calibration model

6.2 Uniformity control results

The batch number of the trial aluminum workpiece is BX-29, and its alloy state is LC52CS. The temperature data of each of the heating zones were stored in a real time database per minute.

Figure 8 is a result from the actual running of TCS during the heating process when the temperature set-point is 470 °C. Zone 7 is not considered because it closes to the furnace door, and the temperature of Zone 7 is influenced by the external temperature. Note that during the temperature rising period, the temperature distribution of six zones appears gradient distribution, there are mainly two reasons. Firstly, in the rising period, all of the heaters are switched on and the temperature rising is rapid which makes the heated air mostly accumulating in the upper zones of the furnace, thereby, peak temperature appears in the upper zone. Secondly, the furnace door lies in the bottom of the furnace, there inevitably exists heat loss, and seriously affects the ambient temperature. From the temperature profiles of zone 6, we can also see that the temperature fluctuation is larger than other zones. But during the temperature holding period, the six temperature profiles are more smooth and stable because of the suitable control strategy of the proposed TCS. In Fig. 9, the control effect is clearer. As shown in Fig. 9, before 1.15×10⁴ s, the temperature distribution is not uniform, and there exists a large temperature difference between each of the zones. However, when the temperature nearly arrives at the set-point of 470 °C, the TCS begins to work. Figure 9 shows in the holding period (after about 1.15×10⁴ s), the temperature gradient becomes smaller, and the temperature fluctuation of the six zones are all confined to 467 °C-473 °C. It is clear that the temperature control precision in the axial direction in the holding period is within ±3 °C, which guarantees the temperature uniformity of thermal processing for alumina alloy workpieces.

Fig. 8 Temperature profiles during entire quenching process

Fig. 9 Local enlargement of temperature profiles during temperature holding period

The long-term actual running in the 31 m vertical quench furnace shows that the temperature uniformity inside the furnace in the holding period has been improved from more than ±6 °C to within ±3 °C. The temperature rising time has been shortened from more than 40 min to less than 30 min, which obviously increases the efficiency of thermal processing and also decreases the electrical power consumption.

The developed TCS has also been applied to the 24m large-scale vertical quench furnace in the Southwest Aluminium Co. Ltd., China. Running results show that the eligible rate has been improved by 6.3%, the temperature uniformity inside the furnace in the holding period has been confined to within ±3 °C, and the electrical power consumption is decreased by 5.5%.

7 Conclusions

The design and implementation of the temperature control system for thermal treatment of aluminum alloy workpieces in the large-scale vertical quench furnace is described and illustrated by the practical industrial application. There are two main contributions in this work. The first is an air dynamic mathematical model is built to calibrate the temperature measurement, and the calibrated temperature reflects the temperature gap, which provides accurate feedback information for precise temperature control of the furnace. The second is the temperature uniformity control strategy. Based on a derived MIMO second order PDE, which is the basis of the decoupling matrix to decouple the heating process into multiple independent zones, and combining with the expert control, the temperature uniformity control realizes to control the furnace temperature within the desired range at the temperature set-point. The proposed system increases the eligible rate and shortens the rising time to improve the efficiency of thermal processing when applying to both 31 m and 24 m quench furnaces. Such research results will be helpful to similar large-scale furnaces for thermal treatment.

Nomenclature

r   Radial axe

φ  Circumferential axe

z   Axial axe

u_z   Air flow rate in axial direction

λ  Air heat conductivity

ρ  Air density

c_p   Air specific heat capacity

T   Temperature

Γ_D  Whole boundary

T_W   Measured temperature

m   Number of element nodes

 Node temperature

θ₁(T) Radiation coefficient

θ₂(T) Convection coefficient

θ₃(T) Heating coefficient

U(z,t) Distribution of heat generation rate

t   Time

z   Axial space variable of furnace

L   Furnace height

τ  Step-size

  Diagonal matrix

K   Decoupling matrix

R_i(k) Temperature setpoint

Y_i(k) Calibrated temperature of workpiece

G_i(z) Transfer function

t_on   Time of pulse current output

t_off  Turn-off time

t_p    On-off period

References

[1] JIANG Shao-hua, GUI Wei-hua, YANG Chun-hua. Vertical quench furnace Hammerstein fault calibrating model based on least squares support vector machine and its application [C]// In Proc Int CCDC. Guilin: IEEE, 2009: 203-207. (in Chinese)

[2] ZHOU Xuan, YU Shou-yi, LIANG Lie-quan. Multivariable temperature measurement and control system research of 31m vertical quench furnace [J]. Journal of Mini-Micro Systems, 2005, 26(2): 310-314. (in Chinese)

[3] HAN S H, CHUANG D, HUH C. Efficiency analysis of radiative slab heating in a walking-beam-type reheating furnace [J]. Journal of Energy, 2011, 36(2): 1265-1272.

[4] TAN C, JENKINS J, WARD J. Zone modelling of the thermal performances of a large-scale bloom reheating furnace [J]. Journal of Applied Thermal Engineering, 2013, 50(1): 1111-1118.

[5] ORZYLOWSKI M, KALUZNIACKI T, RUDOLF Z, NOWICKI G. Precise temperature control for measurement purpose [C]// Proc of the 16th IEEE Instrumentation and Measurement Technology Conference. Venice, Italy: IEEE, 1999: 16-21.

[6] AGGELOGIANNAKI E, SARIMVEIS H, KOUBOGIANNIS D. Model calibrative temperature control in long ducts by means of a neural network approximation tool [J]. Applied Thermal Engineering, 2007, 27(14/15): 2363-2369.

[7] CAO Liu-lin, LI Da-zi, ZHANG Chun-yu, WU Hai-yan. Control and modeling of temperature distribution in a tubular polymerization process [J]. Computers and Chemical Engineering, 2007, 31(11): 1516-1524.

[8] CHAI Tian-you, DING Jin-liang, WU Feng-hua. Hybrid intelligent control for optimal operation of shaft furnace roasting process [J]. Control Engineering Practice, 2011, 19(3): 264-275.

[9] STEINBOECK A, WILD S, KUGI A. Nonlinear model predictive control of a continuous slab reheating furnace [J]. Control Engineering Practice, 2013, 21(4): 495-508.

[10] ZHOU Xuan, YU Shou-yi, LIANG Lie-quan. Temperature measurement of large-scaled vertical quench furnace based on temperature field [J]. Journal of Control Theory and Applications, 2004, 2(4): 401-405.

[11] HE Jian-jun, LIU Yu-qiao, YU Shou-yi, GUI Wei-hua. Neural adaptive PSD decoupling controller and its application in three-phase electrode adjusting system of submerged arc furnace [J]. Journal of Central South University Technology, 2013, 20(2): 405-412.

[12] DING Yomg-sheng, LIU Bao. An intelligent bi-cooperative decoupling control approach based on modulation mechanism of internal environment in body [J]. IEEE Trans on Control Systems Technology, 2011, 19(3): 692-698.

[13] GHOSH A, DAS S K. Open-loop decoupling of MIMO plants [J]. IEEE Trans on Automatic Control, 2009, 54(8): 1977-1981.

[14] WEN Ding-du. Decoupling control of electric heating furnace temperature based on DRNN neural network [C]// Int Conference on Electronic Computer Technology. Kuala Lumpur: ECT, 2010: 261-264.

[15] LI Jin-yang, MENG Xiao-feng. Temperature decoupling control of double-level air flow field dynamic vacuum system based on neural network and calibration principle [J]. Applications of Artificial Intelligence, 2013, 26(4): 1237-1245.

[16] REDDY J N, GARTLING D K. The finite element method in heat transfer and fluid dynamics [M]. Florida: CRC, 2010.

[17] SHEN Ling, HE Jiang-jun, YANG Chun-hua, GUI Wei-hua, XU Hong-lei. Temperature uniformity control of large-scale vertical quench furnaces for aluminum alloy thermal treatment [J]. IEEE Trans on Control Systems Technology, 2015, (99): 1-10.

[18] KWON Y W, BANG H. The finite element method using matlab [M]. New York: Taylor & Francis Group, 2000.

[19] ZHOU Xuan, YU Shou-yi, HE Jian-jun. Dynamically decoupling control algorithm of temperature DPS in large-scale vertical quench furnace [J]. Journal of Central South University Technology: Scieace and Technolagy, 2007, 38(3): 533-539. (in Chinese)

(Edited by DENG Lü-xiang)

Foundation item: Project(61174132) supported by the National Natural Science Foundation of China; Project(2015zzts047) supported by the Fundamental Research Funds for the Central Universities, China; Project(20130162110067) supported by the Research Fund for the Doctoral Program of Higher Education of China

Received date: 2015-03-18; Accepted date: 2016-01-18

Corresponding author: HE Jian-jun, Professor; Tel: +86-13507465056; E-mail: jjhe@csu.edu.cn