# Research on Modeling and Dynamic Characteristics Analysis of Alkali Recovery Furnace

^{1}; Qian Chen

^{2}

^{, †}; Fei Wei

^{3}; Qing-yu Dai

^{4}

2Department of Electrical & Control Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China, Student

3Department of Electrical & Control Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China, Student

4Department of Electrical & Control Engineering, Shaanxi University of Science & Technology, Weiyang District, Xi’an, 710021, China, Lecturer

^{†}E-mail: 876397837@qq.com (Addresse: Department of Electrical & Control Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China, Student)

## Abstract

In order to analyze the relationship between the temperature of the furnace, the oxygen content of the flue gas, the pressure of the furnace, and the flow of black liquor, the amount of air supply, and the amount of induced air during the combustion of the alkali recovery furnace, this paper designs an experiment of alkali recovery furnace combustion process by using the control variable method in the actual field. The analysis of the experimental results shows that there is a certain coupling relationship between variables. On this basis, using the principle of material balance and energy balance, a dynamic control oriented mathematical model of alkali recovery combustion process is established, and the model is verified. The verification results show that the model is consistent with the experimental results, which can reflect the on-site operation of the alkali recovery furnace. Finally, the relative gain matrix is used to analyze the correlation degree of the system. The three-input and three-output control system can be divided into a two-input and two-output coupling system and a single-input and single-output control system, which can simplify the design of the control system of the alkali recovery furnace and provide a theoretical basis for the formulation of the control strategy and control scheme of the alkali recovery furnace.

## Keywords:

Alkali recovery furnace, mechanism modeling, coupling degree analysis## 1. Introduction

Alkali recovery is the most effective method for completely eradicating the pollution of paper making black liquor. The alkali recovery furnace is the key equipment for alkali recovery, its operation condition has a great influence on the evaluation indexes of alkali recovery efficiency, pollutant emissions in flue gas and thermal efficiency. Therefore, it is crucial to deeply analyze the operation mechanism of the combustion process of the alkali recovery furnace and establish its dynamic mathematical model, and then analyze the dynamic characteristics based on it, and then lay the foundation for the design of the control strategy and control scheme of the alkali recovery furnace important.

The purpose of burning black liquor mainly includes: recovering the heat of organic matter in black liquor and alkali in inorganic matter. The process flow of the alkali recovery combustion section is shown in Fig. 1. The concentrated black liquor is first pumped into the thick black liquor tank, further concentrated by the disc evaporator, and then sent into the furnace black liquor tank, where the black liquor is mixed with the alkali ash and glauber’s salt and then sent to the concentrated black liquor heater, and then the black liquor spray gun is sprayed into the furnace with high pressure. In the alkali recovery furnace, the black liquid droplets are atomized, dried and burned. Part of the burnt black ash falls on the cushion of the alkali recovery furnace, the black ash continues to burn, inorganic substances continue to melt, and some organic substances are carbonized into elemental carbon for combustion and reduction of sodium sulfate. After the reaction is completed, glauber’s salt is reduced to sodium sulfide, and the resulting melt flows out of the chute. The air required for the combustion of black liquor is fed by the air supply system of the alkali recovery furnace. The primary and secondary air are sent to the air preheater through the blower for heating, after the air is heated to about 150℃, it is sent to the combustion furnace through the primary and secondary air nozzles. The tertiary air does not need to be preheated, and is directly sent into the combustion furnace by the blower. The proportion of the first, second and third air is generally: 45%, 40%, 15%. The high-temperature flue gas generated by the combustion of black liquor passes through the economizer, disc evaporator and other equipment to absorb the residual heat, and then is removed by the electrostatic precipitator, and then extracted by the induced fan and discharged through the chimney.^{1,2)} The melt produced by the burning of black liquor is discharged to the dissolution tank through the melt chute, and then sent to the caustic chemical section through the green liquor filter for causticization, thereby recovering industrial alkali.

The key indicators to measure the operation effect of the alkali recovery furnace are combustion efficiency, heat utilization rate and alkali recovery rate, and the key parameters that affect the above indicators are furnace temperature, flue gas oxygen content, and furnace pressure. The furnace temperature has a certain influence on the loss of sulfur-containing gas and alkali ash, thus affecting the efficiency of alkali recovery. The oxygen content of flue gas is an important indicator reflecting the excess air supply to the furnace, and its size affects the combustion efficiency of the alkali recovery furnace. Furnace pressure is an important parameter that reflects the stability of the combustion conditions, the size of the furnace negative pressure affects the black liquor combustion efficiency, heat utilization rate, and the load of the induced draft fan.

Therefore, during the operation of the alkali recovery furnace, the furnace temperature, the oxygen content of the flue gas and the furnace pressure need to be strictly controlled. The in-depth analysis of the operation mechanism of the alkali recovery furnace combustion process and the establishment of its dynamic mathematical model, based on the analysis of the mathematical model to propose a suitable control scheme is the basis for precise control of the above three parameters. However, there are few studies on the operation mechanism and mathematical model of alkali recovery furnace. Reference 3 put forward the idea of using the principle of material and energy balance to establish a steady-state, dynamic mathematical model of the alkali recovery furnace, and made sufficient preparations for the modeling of the alkali recovery furnace, but did not get a specific mathematical model expression. Reference 4 estimated the specific composition of black liquor particles and the construction of a black liquor particle combustion model from a micro perspective based on the characteristics of combustion in an alkali furnace, but did not point out a mathematical model of combustion from a macro perspective of the control of an alkali recovery furnace. Reference 5 used Fluent and other computational fluid dynamics software to divide the furnace into several units through mesh division, and established the material and energy equations of each unit. Although this model can well reflect the actual operating conditions of the furnace However, the mechanism modeling process is computationally intensive and the model is complex, which makes it difficult to meet the requirements of controller design.

The above studies have contributed to the establishment of the mathematical model of the alkali recovery furnace, but none of them has reached the control-oriented dynamic mathematical model of the alkali recovery furnace, and did not consider the coupling relationship among the variables such as furnace temperature, flue gas oxygen content, and furnace pressure. And its non-linear relationship with variables such as black liquor flow, supply air volume, and induced air volume, so it cannot provide a theoretical basis for the formulation of control strategies and control plans for alkali recovery furnaces.

In order to establish a dynamic mathematical model that better reflects the real-time operating status of the alkali recovery furnace, this paper designed and completed the experiment of the combustion process of the alkali recovery furnace in a paper-making enterprise in Henan, China. Based on the analysis of the experimental results, the principle of material balance and energy balance is established a dynamic mathematical model with black liquor flow rate, supply air volume and induced air volume as input volume, and furnace temperature, furnace pressure and flue gas oxygen content as output volume is introduced. The relative gain matrix is used to analyze the established mathematical model, and the three-input and three-output control system of the alkali recovery furnace is decomposed into a two-input and two-output coupling system of furnace temperature and flue gas oxygen content, and a single-input and single-output control system of the furnace pressure. Therefore, the design of the alkali recovery furnace control system is simplified, which provides a certain theoretical basis for the design of the alkali recovery furnace control scheme.

## 2. Materials and Methods

### 2.1 Materials

The entire experiment includes online measurement and engineering operations. The control of the combustion section of the alkali recovery furnace used a multi-level computer distributed control system (DCS). With the help of DCS, the on-site monitoring value can be transmitted to the host computer for display through the communication system in DCS, and the instructions issued by the operator can also be sent to the control device for execution through the communication system in DCS. Therefore, the operation instructions of this experiment are issued on the man-machine interaction interface of the operation station, and the execution of the instructions is performed by the actuators of each sub-control system. The hardware structure of the system is shown in Fig. 2.

This experiment was completed in a paper-making enterprise in Henan, China. The main experimental equipment includes an alkali recovery furnace, a black liquid pump, and air supply and air induction equipment to control the air volume. The alkali recovery equipment is shown in Fig. 3. Among them, the flow rate of black liquor is mainly adjusted by changing the rotation speed of the black liquor pump. The black liquor flow adjustment device is shown in Fig. 4. The control of flue gas oxygen content in the experiment is mainly achieved by adjusting the black liquor flow rate and the air supply volume. Generally speaking, the air supply volume is divided into three times through the air blower into the furnace, and the air supply volume is mainly adjusted by changing the frequency of the inverter to change the speed of the air blower. Air supply equipment is shown in Fig. 5. The adjustment of the induced air is mainly achieved by finely adjusting the frequency of the inverter. The induced air equipment is shown in Fig. 6.

### 2.2 Design of experimental control scheme

Through the foregoing analysis, it can be known that the system is a three-input and three-output control system. According to the theoretical analysis, it can be known that when the black liquor flow increases, the black liquor consumes oxygen, and the oxygen content of the flue gas will decrease; the black liquor combustion releases a large amount of heat and generates various gases at the same time, which causes the furnace temperature to increase and the furnace pressure to increase. When the supply air flow increases, the temperature of the furnace will decrease because the supply air temperature is much lower than the furnace temperature; the oxygen content in the furnace will increase significantly, and the furnace pressure will increase slightly. When the induced air flow increases, the heat taken away increases, so the temperature of the furnace will be reduced, and the negative pressure of the furnace will increase, and the oxygen content of the flue gas will also decrease. In order to study the specific change relationship between the variables under actual working conditions, the research team designed and completed the experiment of the combustion process of the alkali recovery furnace. This test was conducted under the condition that the alkali recovery furnace is operating normally and each output has reached a stable output. During the collection process, the research team applied the control variable method to the three control quantities of the black liquor flow rate, the air supply volume, and the air induced volume to ensure that the other two quantities remained unchanged, and applied a stable value of 5% short-term and small disturbances, collecting data of three outputs of furnace temperature, furnace pressure, and flue gas oxygen content, to analyze the changes of these three outputs with a certain input. The time interval for collecting data is designed to be 1 minute, and the change data of each output variable is obtained within 70 minutes. Using the pauta criterion (also known as the 3*δ* criterion), it is assumed that a set of test data contains only random errors. Through the calculation and processing of the data, the standard deviation is obtained, and then a certain interval is determined according to a certain probability. Finally, it is considered that any error beyond this interval is not a random error and should be eliminated. After eliminating the bad data, the change data collected twice are divided into two groups, one group is used to establish the mathematical model, the other group is used to verify the accuracy of the model.

## 3. Results and Discussion

### 3.1 Results

Use MATLAB to draw a set of data obtained in section 2.2 into the curve shown in Fig. 7.

It can be seen from the figures that under the condition of keeping the supply air volume and the induced air volume constant, when the flow of black liquor increases, the heat generated by the combustion of the black liquor will cause the furnace temperature to rise and the furnace pressure to increase; oxygen needs to be consumed and the oxygen content of the flue gas will decrease. In the case of maintaining the black liquid flow and the induced air volume, when the air supply volume is increased, the air supply temperature lower than the furnace temperature will reduce the furnace temperature; the air supply volume will also bring oxygen, which will increase the oxygen content of the flue gas. The furnace pressure also increases. Under the condition that the black liquid flow rate and the air supply volume are kept unchanged, when the induced air volume is increased, a part of the heat will be taken away, which will cause the furnace temperature to decrease and the furnace pressure to decrease accordingly. The increase of the air supply will increase the oxygen content of the furnace flue gas to a certain extent, but the change is small and can be ignored, that is, the oxygen content of the flue gas is approximately unchanged.

In summary, when any one of the variables of the black liquor flow rate, air supply volume, and induced air volume is changed, the furnace temperature, furnace pressure, and oxygen content of the flue gas will change, which indicates that there is a certain coupling relationship between them. However, the above is only a qualitative analysis of the relationship between the input and output. To analyze the relationship between the variables more clearly and provide a practical basis for the design of the control system, it is necessary to theoretically analyze and establish Mathematical model of alkali recovery furnace.

### 3.2 Modeling of alkali recovery combustion section

In this paper, the input of the black liquor flow rate, the supply air volume, and the induced draft volume are selected, and the furnace temperature, the oxygen content of the flue gas, and the furnace pressure are used as outputs. The dynamic mathematical model of the alkali recovery furnace is established by using the relationship between energy balance and material balance. Because the furnace temperature cannot be measured in actual engineering, the actual furnace temperature is generally replaced by the furnace outlet smoke temperature. Therefore, the furnace outlet smoke temperature is used as the output during the establishment of the mathematical model. In addition, this mathematical model was established under the following assumptions:

**Assumption 1**: Consider the furnace interior as an area of uniform density.

**Assumption 2**: The flue gas in the furnace of alkali recovery furnace is the ideal gas meeting the ideal gas state equation *PV*=*NRT*. (Note: According to the definition of ideal gas, the gas with temperature greater than 500K or pressure not higher than 1.01×10^{5} Pa is generally considered as the ideal gas, while the furnace temperature is generally around 1,000℃ and the pressure is generally -20--50 Pa, that is, the pressure is below Atmospheric pressure 20-50 Pa, meeting the requirements of ideal gas.)

3.2.1.1 Conservation of quality

The material in the hearth of the alkali recovery furnace is conserved, the materials that enter the furnace are black liquor and air, and the gases that exit the furnace are smoke and smelt. During the combustion of the black liquor, the concentration of solid matter in the black liquor is about 60%, and about 70% of the organic matter in the solid matter participates in the combustion, after a series of chemical reactions, various flues gases such as CO, CO_{2} and TRS are generated. About 30% of the inorganic substances are discharged from the furnace as a molten stream after a chemical reaction.^{6,7)} The change in the quality of the flue gas in the hearth is related to the substances involved in the combustion, the amount of air, and the amount of flue gas flowing out of the hearth. The relationship between the substances involved in combustion in the black liquor and the black liquor flow is

$$${W}_{f}-{W}_{f}\times 60\%\times 30\%={W}_{f}\times 82\%$$$ |

The change in the quality of the furnace flue gas is related to the substances involved in combustion, the amount of air, and the amount of flue gas flowing out of the furnace. According to the conservation of mass, there is the following relationship

$$${W}_{f}\times 82\%+{W}_{a}-{W}_{g}=\frac{d{M}_{g}}{dt}$$$ | [1] |

3.2.1.2 Conservation of energy

From the perspective of the energy flow inside the alkali recovery furnace, most of the energy brought by the blower and the energy released by the combustion of black liquor are converted into radiant energy. A large part of the radiant energy is absorbed by the steam drum and water-cooled wall and converted into steam for use. The remaining small part of the radiant energy is dissipated from the furnace wall to the surroundings. And the other part of the energy is taken out by the flue gas through the furnace outlet.^{8)} It can be seen that the change of energy in the furnace is related to the heat released by the combustion of black liquor, the enthalpy values of the supply and induced air, the radiant energy, and the heat carried by the flue gas at the outlet of the furnace. According to the conservation of energy

$$$\begin{array}{l}{W}_{f}{Q}_{d}{W}_{0}+0.85{W}_{a}{C}_{pa}{t}_{0}+0.15{W}_{a}{C}_{pa}{t}_{0}-\\ {W}_{g}{C}_{pa}{t}_{g}-{Q}_{r}={C}_{pg}\frac{d\left({M}_{g}{t}_{g}\right)}{dt}\end{array}$$$ | [2] |

In Eqs. 1-2, *W _{f}* is the black liquor flow rate (kg·s

^{-1}) entering the furnace,

*W*is the supply air flow rate (kg·s

_{a}^{-1}), and

*W*is the induced air flow rate (kg·s

_{g}^{-1}),

*M*is the mass of flue gas (kg),

_{g}*W*

_{0}is the percentage of solid in black liquor (%),

*Q*is the low calorific value of black liquor combustion (J·kg

_{d}^{-1}),

*C*is the specific heat capacity of the air (J·kg

_{pa}^{-1}·℃

^{-1}),

*t*is the temperature of the primary and secondary air (℃),

_{a}*t*

_{0}is the temperature of the tertiary air, that is, the ambient temperature (℃), and

*C*is the specific heat capacity of the flue gas (J·kg

_{pg}^{-1}·℃

^{-1}),

*t*is the furnace temperature (℃), and

_{g}*Q*is the radiant energy (J).

_{r}For the right side of Eq. 2, the partial derivation is

$$$\begin{array}{l}{W}_{f}{Q}_{d}{W}_{0}+0.85{W}_{a}{C}_{pa}{t}_{a}+0.15{W}_{a}{C}_{pa}{t}_{0}-{W}_{g}{C}_{pi}{t}_{g}-{Q}_{r}=\\ {C}_{pg}\left({M}_{g}\frac{d{t}_{g}}{dt}+{t}_{g}\frac{d{M}_{g}}{dt}\right)\end{array}$$$ | [3] |

If the volume of molten material in the alkali recovery furnace is ignored, there are

$$${M}_{g}={V}_{l}{\rho}_{g}$$$ | [4] |

The transformation of the ideal gas equation of state *PV*=*NRT*can be obtained

$$${\rho}_{g}=k\frac{{P}_{g}}{{t}_{g}}$$$ | [5] |

In Eqs. 4-5, *V _{l}* is the furnace volume,

*ρ*is the flue gas density,

_{g}*P*is the furnace pressure, and

_{g}*k*is the ideal gas constant.

Comprehensive Eqs. 1-5 can be obtained

$$$\begin{array}{l}{C}_{pg}{V}_{l}k\frac{{P}_{g}}{{t}_{g}}\frac{d\left({t}_{g}\right)}{dt}+\\ \left(82\%{C}_{pg}{W}_{f}+{C}_{pg}{W}_{a}+\left({C}_{pi}-{C}_{pg}\right){W}_{g}\right){t}_{g}=\\ 0.85{W}_{a}{C}_{pa}{t}_{a}+0.15{W}_{a}{C}_{pa}{t}_{0}+{W}_{f}{Q}_{d}{W}_{0}-{Q}_{r}\end{array}$$$ | [6] |

Where radiant energy

$$${Q}_{\mathrm{r}}={F}_{1}\psi {a}_{1}{\mathrm{\sigma}}_{0}\left({t}_{g}^{4}-{t}_{0}^{4}\right)$$$ | [7] |

Among them, *F*_{1} is the furnace area (m^{2}), *ψ* is the thermal effective coefficient, *a*_{1} is the furnace blackness, *σ*_{0} is Boltzmann constant, *t*_{0} is the ambient temperature (℃), which is small compared to the furnace temperature, and *t*_{0} can be ignored excluding, the radiant energy can be approximated as

$$${Q}_{\mathrm{r}}={F}_{1}\psi {a}_{1}{\mathrm{\sigma}}_{0}{t}_{g}^{4}$$$ | [8] |

Eq. 6 shows that there is a coupling relationship between the furnace temperature and the furnace pressure, and there is a non-linear relationship between them and the variables such as the black liquor flow, the supply air volume, and the induced air volume. In order to facilitate the subsequent control research, the non-linear function is expanded by Taylor series near the equilibrium point, and the higher-order derivative term is ignored, so that

$$$\begin{array}{l}\frac{{P}_{g}}{{t}_{g}}\frac{d\left({\mathrm{t}}_{g}\right)}{dt}=\frac{{P}_{g0}}{{t}_{g0}}{\dot{t}}_{g0}+\\ \left\{{\left(\frac{\partial \left({P}_{g}/{t}_{g}\right)}{\partial {P}_{g}}\right)}_{0}{\mathrm{\Delta}{P}_{g}+\left(\frac{\partial \left({P}_{g}/{t}_{g}\right)}{\partial {\mathrm{t}}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}\right\}{\dot{t}}_{g0}+\frac{{P}_{g0}}{{t}_{g0}}{\mathrm{\Delta}\dot{t}}_{g}\end{array}$$$ | [9] |

$$${W}_{a}{t}_{g}={W}_{a0}{t}_{g0}+{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {W}_{a}}\right)}_{0}\mathrm{\Delta}{W}_{a}{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}$$$ | [10] |

$$${W}_{f}{t}_{g}={W}_{f0}{t}_{g0}+{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {W}_{f}}\right)}_{0}\mathrm{\Delta}{W}_{f}{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}$$$ | [11] |

$$${W}_{g}{t}_{g}={W}_{g0}{t}_{g0}+{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{g}}\right)}_{0}\mathrm{\Delta}{W}_{g}{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}$$$ | [12] |

Substituting the above linearization processing result Eqs. 8-12 into Eq. 6, we get

$$$\begin{array}{l}{C}_{pg}{V}_{l}k\left[\frac{{P}_{g0}}{{t}_{g0}}{\dot{t}}_{g0}+{\left(\frac{\partial \left({P}_{g0}/{t}_{g0}\right)}{\partial {P}_{g}}\right)}_{0}{\mathrm{\Delta}{P}_{g}\mathrm{\Delta}\dot{t}}_{g0}+\right.\\ \left.{\left(\frac{\partial \left({P}_{g0}/{t}_{g0}\right)}{\partial {t}_{g}}\right)}_{0}{\mathrm{\Delta}{t}_{g}\mathrm{\Delta}\dot{t}}_{g0}+\frac{{P}_{g0}}{{t}_{g0}}\mathrm{\Delta}{\dot{t}}_{g0}\right]+\\ {C}_{pg}\left[{W}_{a0}{t}_{g0}+\right.{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{a}}\right)}_{0}\mathrm{\Delta}{W}_{a}{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}+\\ 82\%\left.\left({W}_{f0}{t}_{g0}+{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {W}_{f}}\right)}_{0}\mathrm{\Delta}{W}_{f}\right)+{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}\right]+\\ \left({C}_{pi}-{C}_{pg}\right)\left({W}_{g0}{t}_{g0}+{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{g}}\right)}_{0}\mathrm{\Delta}{W}_{g}+{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}\right)\\ =0.85\left({W}_{a0}+\mathrm{\Delta}{W}_{a}\right){C}_{pa}{t}_{a}+0.15\left({W}_{a0}+\mathrm{\Delta}{W}_{a}\right){C}_{pa}{t}_{0}+\\ \left({W}_{f0}+\mathrm{\Delta}{W}_{f}\right){Q}_{d}{W}_{0}-{Q}_{r0}-4{F}_{1}\psi {a}_{1}{t}_{g0}^{3}\mathrm{\Delta}{t}_{g}\end{array}$$$ | [13] |

The equilibrium equation of the Eq. 6 at the equilibrium point is

$$$\begin{array}{l}{C}_{pg}{V}_{l}k\frac{{P}_{g0}}{{t}_{g0}}\frac{d\left({t}_{g0}\right)}{dt}+\\ \left(82\%{C}_{pg}{W}_{f0}+{C}_{pg}{W}_{a0}+\left({C}_{pi}-{C}_{pg}\right){W}_{g0}\right){t}_{g0}=\\ 0.85{W}_{a0}{C}_{pa}{t}_{a}+0.15{W}_{a0}{C}_{pa}{t}_{a}+{W}_{f0}{Q}_{d}{W}_{0}-{Q}_{r0}\end{array}$$$ | [14] |

$$$\begin{array}{l}{C}_{pg}{V}_{l}k\left[{\left(\frac{\partial \left({P}_{g}/{t}_{g}\right)}{\partial {P}_{g}}\right)}_{0}{\mathrm{\Delta}{P}_{g}\mathrm{\Delta}\dot{t}}_{g0}+\right.\\ \left.{\left(\frac{\partial \left({P}_{g}/{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}{\mathrm{\Delta}{t}_{g}\mathrm{\Delta}\dot{t}}_{g0}+\frac{{P}_{g0}}{{t}_{g0}}{\mathrm{\Delta}\dot{t}}_{g0}\right]+\\ {C}_{pg}\left[{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{a}}\right)}_{0}\mathrm{\Delta}{W}_{a}{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}+\right.\\ 82\%\left.\left({\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {W}_{f}}\right)}_{0}\mathrm{\Delta}{W}_{f}\right)+{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}\right]+\\ \left({C}_{pi}-{C}_{pg}\right)\left({\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{g}}\right)}_{0}\mathrm{\Delta}{W}_{g}+{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}\mathrm{\Delta}{t}_{g}\right)=\\ 0.85\mathrm{\Delta}{W}_{a}{C}_{pa}{t}_{a}+0.15\mathrm{\Delta}{W}_{a}{C}_{pa}{t}_{0}+\mathrm{\Delta}{W}_{f}{Q}_{d}{W}_{0}-4{F}_{1}\psi {a}_{1}{\mathrm{\sigma}}_{0}{t}_{g0}^{3}\mathrm{\Delta}{t}_{g}\end{array}$$$ | [15] |

Sorted out

$$${k}_{1}{\dot{t}}_{g}+{k}_{2}{t}_{g}+{k}_{3}{P}_{g}={k}_{4}{W}_{a}+{k}_{5}{W}_{f}-{k}_{6}{W}_{g}$$$ | [16] |

Among them,

$$$\begin{array}{l}{k}_{1}={C}_{pg}{V}_{l}k\frac{{P}_{g0}}{{t}_{g0}}\\ {k}_{2}=\begin{array}{l}{C}_{pg}{V}_{l}k\left(\frac{\partial \left({P}_{g0}/{t}_{g0}\right)}{\partial {t}_{g}}\right){\mathrm{\Delta}\dot{t}}_{g0}+{C}_{pg}\left[{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}+\right.\\ \left.82\%{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial \left({t}_{g}\right)}\right)}_{0}\right]+\left({C}_{pi}-{C}_{pg}\right){\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}+\\ 4{F}_{1}\psi {a}_{1}{\mathrm{\sigma}}_{0}{t}_{g0}^{3}\end{array}\\ {k}_{3}={C}_{pg}{V}_{l}k{\left(\frac{\partial \left({P}_{g}/{t}_{g}\right)}{\partial {P}_{g}}\right)}_{0}{\dot{t}}_{g0}\\ {k}_{4}=0.85{C}_{pa}{t}_{a}{-C}_{pg}{\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {W}_{a}}\right)}_{0}+0.15{C}_{pa}{t}_{0}\\ {k}_{5}={Q}_{d}{W}_{0}-82\%{C}_{pg}{\left(\frac{\partial \left({W}_{f}{t}_{g}\right)}{\partial {W}_{f}}\right)}_{0}\\ {k}_{6}=\left({C}_{pi}-{C}_{pg}\right){\left(\frac{\partial \left({W}_{a}{t}_{g}\right)}{\partial {W}_{g}}\right)}_{0}\end{array}$$$ |

In view of the expression of the coefficient in Eq. 16, where the rate of change of the furnace temperature near the equilibrium point is approximately 0, that is $$ {\dot{t}}_{g0}\approx 0,\mathrm{s}\mathrm{o}{k}_{3}\approx 0$$. Eq. 16 becomes

$$${k}_{1}{\dot{t}}_{g}+{k}_{2}{t}_{g}={k}_{4}{W}_{a}+{k}_{5}{W}_{f}-{k}_{6}{W}_{g}$$$ | [17] |

The Laplace transform of Eq. 17 under zero initial conditions yields

$$${k}_{1}s{t}_{g}\left(s\right)+{k}_{2}{t}_{g}\left(s\right)={k}_{4}{W}_{a}\left(s\right)+{k}_{5}{W}_{f}\left(s\right)-{k}_{6}{W}_{g}\left(s\right)$$$ | [18] |

The expression of the dynamic mathematical model of the furnace temperature obtained by Eq. 18 is

$$${t}_{g}\left(s\right)=\frac{{k}_{4}{W}_{a}\left(s\right)+{k}_{5}{W}_{f}\left(s\right)-{k}_{6}{W}_{g}\left(s\right)}{{k}_{1}s+{k}_{2}}$$$ | [19] |

During the combustion of black liquor, the rate of change in the mass of oxygen in the alkali recovery furnace is equal to the difference between the amount of oxygen taken in by the supply air volume per unit time and the oxygen consumed by the combustion of the black liquor and the amount of oxygen taken away by the induced air.^{9,10)} According to the mass conservation of oxygen in the alkali furnace, we can get

$$${W}_{a}\times 23\%-{W}_{g}{\omega}_{{O}_{2}}-{W}_{f}{W}_{0}{\omega}_{0}\times 23\%=\frac{d\left({M}_{g}{\omega}_{{O}_{2}}\right)}{dt}$$$ | [20] |

Among them, *ω*_{O2} is the mass fraction (%) of oxygen element in the flue gas, and *ω*_{0} is the theoretical air mass (Kg) of black liquor combustion.

Partial differentiation on the right side of the above formula gives

$$$\begin{array}{l}{W}_{a}\times 23\%-{W}_{g}{\omega}_{{O}_{2}}-{W}_{f}{W}_{0}{\omega}_{0}\times 23\%=\\ {M}_{g}\frac{d{\omega}_{{O}_{2}}}{dt}+{\omega}_{{O}_{2}}\frac{d{M}_{g}}{dt}\end{array}$$$ | [21] |

Where

$$${\omega}_{0}=\frac{2.67\omega \left(C\right)+8\omega \left(H\right)+\omega \left(S\right)+2.286\omega \left(N\right)-\omega \left(O\right)}{0.23\times 100}$$$ |

*ω*(*C*), *ω*(*H*), *ω*(*S*), *ω*(*O*), *ω*(*N*) are the mass fractions of C, H, S, O, N in the black liquid solids.

Combining Eqs. 1, 4, and 5 with Eq. 21 gives

$$$\begin{array}{l}{v}_{1}k\frac{{P}_{g}}{{t}_{g}}\frac{d{\omega}_{{O}_{2}}}{\mathrm{d}t}+\left({W}_{a}+82\%{W}_{f}\right){\omega}_{{O}_{2}}=\\ 23\%{W}_{a}-23\%{W}_{f}{W}_{0}{\omega}_{0}\end{array}$$$ | [22] |

From Eq. 22, it can be seen that there is also a coupling relationship between the oxygen content of the flue gas, the furnace pressure, and the furnace temperature, and there is a non-linear relationship with the black liquor flow and air supply. The method of linearizing it is the same as that in the dynamic model of furnace temperature.

The nonlinear function $$ \frac{{P}_{g}}{{t}_{g}}\frac{d{\omega}_{{O}_{2}}}{dt},{W}_{a}{\omega}_{{O}_{2}},{W}_{f}{\omega}_{{O}_{2}}$$ is expanded using Taylor series near the equilibrium point and subtracted from the equilibrium point state equation.

$$${b}_{1}{\dot{\omega}}_{{O}_{2}}+{b}_{2}{\omega}_{{O}_{2}}+{b}_{3}{P}_{g}+{b}_{4}{t}_{g}={b}_{5}{W}_{a}-{b}_{6}{W}_{f}$$$ | [23] |

Among them

$$$\begin{array}{l}{b}_{1}={V}_{l}k\frac{{P}_{g0}}{{t}_{g0}}\\ \begin{array}{cc}{\left.{b}_{2}=\frac{\partial \left({W}_{a}\omega \left({O}_{2}\right)\right)}{\partial \left(\omega \left({O}_{2}\right)\right)}\right|}_{\begin{array}{c}{W}_{a}={W}_{a}0\\ \omega \left({O}_{2}\right)={\omega}_{0}\left({O}_{2}\right)\end{array}}& {\left.+82\%\frac{\partial \left({W}_{f}\omega \left({O}_{2}\right)\right)}{\partial \left(\omega \left({O}_{2}\right)\right)}\right|}_{\begin{array}{c}{W}_{f}={W}_{f}0\\ \omega \left({O}_{2}\right)={\omega}_{0}\left({O}_{2}\right)\end{array}}\end{array}\\ {b}_{3}={\left.\frac{\partial f}{\partial {P}_{g}}\right|}_{\left({P}_{g0},{t}_{g0}\right)}{\dot{\omega}}_{0}\left({O}_{2}\right)\\ {b}_{4}={\left.\frac{\partial f}{\partial {t}_{g}}\right|}_{\left({P}_{g0},{t}_{g0}\right)}{\dot{\omega}}_{0}\left({O}_{2}\right)\\ {b}_{5}=23\%-{\left.\frac{\partial \left({W}_{a}\omega \left({O}_{2}\right)\right)}{\partial \left({W}_{a}\right)}\right|}_{\begin{array}{c}{W}_{a}={W}_{a}0\\ \omega \left({O}_{2}\right)={\omega}_{0}\left({O}_{2}\right)\end{array}}\\ {b}_{6}={W}_{0}{\omega}_{0}\times 23\%+82\%{\left.\frac{\partial \left({W}_{f}\omega \left({O}_{2}\right)\right)}{\partial \left({W}_{f}\right)}\right|}_{\begin{array}{c}{W}_{f}={W}_{f}0\\ \omega \left({O}_{2}\right)={\omega}_{0}\left({O}_{2}\right)\end{array}}\end{array}$$$ |

For the same reason, the rate of change of the oxygen content of the flue gas near the equilibrium point is basically 0, that is $$ {\dot{\omega}}_{0}\left({O}_{2}\right)\approx 0$$, that is *b*_{3} ≈ 0, *b*_{4} ≈ 0, the Eq. 23 becomes

$$${b}_{1}{\dot{\omega}}_{{O}_{2}}+{b}_{2}{\omega}_{{O}_{2}}={b}_{5}{W}_{a}-{b}_{6}{W}_{f}$$$ | [24] |

By performing Laplace transform on Eq. 24 under zero initial conditions, the dynamic mathematical model expression of the oxygen content of the flue gas is

$$${\omega}_{{O}_{2}}\left(s\right)=\frac{{b}_{5}{W}_{a}\left(s\right)-{b}_{6}{W}_{f}\left(s\right)}{{b}_{1}s+{b}_{2}}$$$ | [25] |

According to the ideal gas state equation: *PV*=*NRT* , the following relationship is obtained

$$${P}_{g}=k{\rho}_{g}{t}_{g}$$$ | [26] |

In Eq. 26, $$ k=\frac{M}{R}$$, *M* is a relative molecular mass of air, and *R* is a gas constant. Derivative on both sides of Eq. 26,^{9)} the incremental expression is

$$$\frac{d{P}_{g}}{dt}=k{\rho}_{g}\frac{d{t}_{g}}{dt}+{kt}_{g}\frac{d{p}_{g}}{dt}$$$ | [27] |

Combining Eqs. 1-2, Eq. 27 can be summarized as

$$$\frac{d{P}_{g}}{dt}=\begin{array}{l}\frac{k}{{V}_{l}{C}_{pg}}\left[0.85{W}_{a}{C}_{pa}{t}_{a}+0.15{W}_{a}{C}_{pa}{t}_{0}+\right.\\ \left.{W}_{f}{Q}_{d}{W}_{0}-{W}_{g}{C}_{pi}{T}_{g}-{Q}_{r}\right]\end{array}$$$ | [28] |

Linearize the Eq. 28 (the method is the same as above), and get

$$${c}_{1}{\dot{P}}_{g}+{\mathrm{c}}_{2}{t}_{g}={c}_{3}{W}_{a}+{c}_{4}{W}_{f}-{c}_{5}{W}_{g}$$$ | [29] |

Among them

$$$\begin{array}{l}{c}_{1}={V}_{l}{C}_{pg}\\ {c}_{2}=k{C}_{pi}{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {t}_{g}}\right)}_{0}+4k{F}_{1}\psi {a}_{1}{\sigma}_{0}{t}_{g0}^{3}\\ {c}_{3}=0.85k{C}_{pa}{t}_{a}+0.15k{C}_{pa}{t}_{0}\\ {c}_{4}=k{Q}_{d}{W}_{0}\\ {c}_{5}=k{C}_{pi}{\left(\frac{\partial \left({W}_{g}{t}_{g}\right)}{\partial {W}_{g}}\right)}_{0}\end{array}$$$ |

A Laplace transform is performed on Eq. 29 under zero initial conditions, and the dynamic mathematical model expression of the furnace pressure obtained by combining Eq. 19 is

$$${p}_{g}\left(s\right)=\frac{{a}_{1}{W}_{a}\left(s\right)+{a}_{2}{W}_{f}\left(s\right)-{a}_{3}{W}_{g}\left(s\right)}{{c}_{1}s\left({k}_{1}s+{k}_{2}\right)}$$$ | [30] |

Among them

$$$\begin{array}{l}{a}_{1}={k}_{1}{c}_{3}s+{k}_{2}{c}_{3}-{c}_{2}{k}_{4}\\ {a}_{2}={k}_{1}{c}_{4}s+{k}_{2}{c}_{4}-{c}_{2}{k}_{5}\\ {a}_{3}={k}_{1}{c}_{5}s+{k}_{2}{c}_{5}-{c}_{2}{k}_{6}\end{array}$$$ |

The research group selected representative 150 sets of data from the data obtained during the experiment. The crippled data was removed using 3*δ* criteria to obtain 141 sets of data. The remaining 141 data of each variable were averaged to obtain Table 1 data.

The data in Table 1 is substituted into the expressions of the correlation coefficients, and the values of the coefficients are shown in Table 2.

The parameters in Table 2 are substituted into their respective mathematical model expressions, and the mathematical model expressions of the furnace temperature, the oxygen content of the flue gas, and the furnace pressure of the alkali recovery furnace are simplified to Eqs. 31-33.

$$${t}_{g}\left(s\right)=\frac{-17.61{W}_{a}\left(s\right)+66.9{W}_{f}\left(s\right)-0.23{W}_{g}\left(s\right)}{\mathrm{4.617.4}s+1}$$$ | [31] |

$$${\omega}_{{O}_{2}}\left(s\right)=\frac{0.071{W}_{a}\left(s\right)-0.02{W}_{f}\left(s\right)}{9.15s+1}$$$ | [32] |

$$${P}_{g}\left(s\right)=\begin{array}{l}\frac{\left(2655.11s+5.07\right){W}_{a}\left(s\right)}{\left(4617.4s+1\right)s}\\ \frac{\left(9335.96s+20.28\right){W}_{f}\left(s\right)}{\left(4617.4s+1\right)s}\\ \frac{\left(21692.89s+47.6\right){W}_{g}\left(s\right)}{\left(4617.4s+1\right)s}\end{array}$$$ | [33] |

From the Eqs. 31-33, the transfer function matrix between the input and output variables of the combustion section of the alkali recovery furnace is

$$$\begin{array}{l}\left[\begin{array}{c}{t}_{g}\left(s\right)\\ {\omega}_{{O}_{2}}\left(s\right)\\ {P}_{g}\left(s\right)\end{array}\right]=\left[\begin{array}{ccc}{G}_{11}& {G}_{12}& {G}_{13}\\ {G}_{21}& {G}_{22}& {G}_{23}\\ {G}_{31}& {G}_{32}& {G}_{33}\end{array}\right]\left[\begin{array}{c}{W}_{f}\left(s\right)\\ {W}_{a}\left(s\right)\\ {W}_{g}\left(s\right)\end{array}\right]=\\ \left[\begin{array}{ccc}\frac{66.9}{4617.4s+1}& \frac{-17.61}{4617.4s+1}& \frac{-0.23}{4617.4s+1}\\ \frac{-0.02}{9.15s+1}& \frac{0.071}{9.15s+1}& 0\\ \frac{20.28\left(460.35s+1\right)}{\left(4617.4s+1\right)s}& \frac{5.07\left(523.69s+1\right)}{\left(4617.4s+1\right)s}& \frac{-47.6\left(455.7s+1\right)}{\left(4617.4s+1\right)s}\end{array}\right]\\ \times \left[\begin{array}{c}{W}_{f}\left(s\right)\\ {W}_{a}\left(s\right)\\ {W}_{g}\left(s\right)\end{array}\right]\end{array}$$$ | [34] |

In Eq. 34, the transfer functions *G*_{12}, *G*_{22}, *G*_{31}, *G*_{32} are negative, indicating that an increase in the input amount will cause a decrease in the output amount. In actual engineering, when the supply air flow increases, the temperature of the furnace will decrease because the supply air temperature is much lower than the furnace temperature. When the induced air flow increases, the heat taken away increases, so the furnace temperature will be reduced, and the furnace pressure will be reduced. *G*_{11}, *G*_{22}, *G*_{31}, *G*_{32} are positive, indicating that the output volume increases as the input volume increases. During the combustion of the black liquor, when the flow of the black liquor increases, a large amount of heat is released during combustion, and various gases are generated at the same time, which causes the furnace temperature to increase and the furnace pressure to increase. When the amount of air supply is increased, the oxygen content in the furnace is significantly increased, and the furnace stress will also increase slightly. In addition, the induced air flow increases, and the proportion of oxygen in the furnace flue gas basically does not change, that is *G*_{23}=0, the oxygen content of the flue gas does not change. From the above analysis, it can be known that the obtained mathematical model can reflect the change law between the input and output of the actual industrial site, which is consistent with the actual production situation.

### 3.3 Model verification

In order to verify the accuracy of the model, the research group using MATLAB to simulate and verify the control model. The simulation conditions are the same as the experimental conditions, that is: when the alkali recovery furnace operates normally, that is, when each output reaches a stable level, for the three control quantities of black liquor flow rate, supply air quantity and induced air quantity entering the furnace, while ensuring that the other two quantities remain unchanged, apply a small disturbance with a stable value of 5% to one of them, and the data collection interval is set to 1min to obtain the change data of each output variable within 70 min, and compared with another set of experimental data obtained in Section 2.2, the comparison results as shown in Fig. 8.

In order to quantitatively explain the accuracy of the mathematical model established in this paper, the research group calculates the mean square error (MSE) between the actual output value of each output and the predicted value of the model, and the calculation formula is

$$$MSE=\frac{1}{n}\sum _{t=1}^{n}{\left(observe{d}_{t}-predicte{d}_{t}\right)}^{2}$$$ |

Among them, *observed* and *predicted* are actual output data and model prediction data respectively. The calculation results are shown in Table 3.

From the three groups of comparison curves in Fig. 8 and the mean square error values of each group of data in Table 3, it can be seen that the simulation curve obtained by the mathematical model established in this paper under the same input disturbance has the same change trend with the field test curve, and the maximum mean square error is 0.004237, which shows that the mathematical model can reflect the influence of each control quantity on each controlled quantity. In addition, it should be noted that in Fig. 8(c), in the initial stage of induced air disturbance, the oxygen content of on-site flue gas fluctuates by a small amount of 0.12%. This is because in the initial stage of the increase of induced air volume, the furnace pressure decreases, and the air supply flow will increase a small amount, so that the oxygen content of flue gas increases a small amount. Later, when the air supply volume is controlled unchanged, the oxygen content of flue gas almost does not change, which is similar to the simulation results of *G*_{23}=0 in the transfer function matrix are consistent.

### 3.4 Discussion

The transfer function matrix in Eq. 34 is a non-diagonal matrix, so there is a mutual coupling between the furnace temperature, the furnace pressure, and the oxygen content of the flue gas. However, further analysis of the proportion coefficient of each transfer function shows that: *K*_{23}=0, the change of induced air flow will not affect the change of oxygen content; and *K*_{13} is much smaller than *K*_{11} and *K*_{12} (*K*_{13} is only $$ \frac{\text{1}}{\text{290}}$$ of *K*_{11}, and $$ \frac{1}{76.6}$$ of *K*_{12}), that is, the influence of induced air flow on the furnace temperature is small; *K*_{11} and *K*_{12} are not much different, and *K*_{21} and *K*_{22} are not much different, it shows that the coupling relationship between furnace temperature, oxygen content, black liquor flow rate and supply air flow rate cannot be ignored. In this paper, the relative gain matrix is used to quantitatively analyze the correlation between the input and output of the alkali recovery furnace to determine the degree of coupling and the input-output variable pairing relationship.

The relative gain indicates the degree of influence of an operation amount on a controlled amount (relative to the influence of other control amounts on the controlled amount in the system). Assume that a multivariable control system contains *n* control loops, the controlled quantity of the *i*-th (1≤*i*≤*n*) control loop and the *j*-th manipulated quantity are *y _{i}*,

*m*respectively. The relative gain between the controlled quantity of the

_{j}*i*-th control loop and the

*j*-th manipulated variable is defined as

$$$\lambda {i}_{j}=\frac{\left.\frac{\partial {y}_{i}}{\partial {m}_{j}}\right|{m}_{k}=c\left(1\le k\le n,k\ne j\right)}{\left.\frac{\partial {y}_{i}}{\partial {m}_{j}}\right|{y}_{k}=c\left(1\le k\le n,k\ne j\right)}$$$ | [35] |

According to the mechanism model of the alkali recovery furnace, the control system of the alkali recovery furnace is a three-input three-output control system, so the relative gain matrix is defined as

$$$\mathrm{\Lambda}=\left[\begin{array}{ccc}{\lambda}_{11}& {\lambda}_{12}& {\lambda}_{13}\\ {\lambda}_{21}& {\lambda}_{22}& {\lambda}_{23}\\ {\lambda}_{31}& {\lambda}_{32}& {\lambda}_{33}\end{array}\right]$$$ | [36] |

According to the relative gain calculation method shown in Eq. 35, the relative gain matrix is obtained as

$$$\mathrm{\Lambda}=\left[\begin{array}{ccc}0.55& 0.37& 0.08\\ 0.35& 0.65& 0\\ 0.23& 0.06& 0.71\end{array}\right]$$$ | [37] |

From Eq. 37, it can be seen that *λ*_{12}=0.37, *λ*_{21}=0.35 are both greater than 0.3 and less than 0.7, which indicates that the correlation between the furnace temperature and the oxygen content of the flue gas is strong and needs to be decoupled. *λ*_{13}=0.08, *λ*_{23}=0, it shows that the amount of induced air has little effect on furnace temperature and flue gas oxygen content; because *λ*_{11}=0.55, *λ*_{22}=0.65 is greater than 0.5, it shows that the temperature of the furnace can be controlled by the flow rate of black liquor in the coupling system, and the oxygen content of the flue gas can be controlled by the supply air volume. *λ*_{31}=0.23, *λ*_{32}=0.06 is less than 0.3, it shows that the flow of black liquor and the amount of air supply have little effect on the furnace pressure. *λ*_{33}=0.71 means that the pressure of the furnace is mainly affected by the amount of induced air and can be controlled independently. Therefore, the three-input three-output coupling system of the alkali recovery furnace can be decomposed into a two-input two-output coupling system of black liquor flow-furnace temperature and supply air flow-flue gas oxygen content and furnace pressure single-input single-output system. ^{11)} The simplified control block diagram is shown in Fig. 9.

For comparison, the simplified control block diagram is shown in Fig. 10.

Comparing Fig. 9 and Fig. 10, it can be seen that 8 simplified decoupling controllers need to be designed before the simplification, and only 4 decoupled controllers need to be designed after the simplification. Before the simplification, due to the large number of variables, there are many interaction factors between the controllers. Tuning the parameters is also cumbersome. After the simplification, the number of variables that affect each other is reduced, and the tuning of the controller parameters is relatively simple, providing a basis for the formulation of control strategies and schemes.

## 4. Conclusions

This article focuses on the analysis of the operating mechanism of the alkali recovery combustion process and the relationship between the furnace temperature, the oxygen content of the flue gas, the pressure in the furnace, and the alkali recovery rate, thermal efficiency, and pollutant emissions in the flue gas. In the actual site, a combustion process experiment of an alkali recovery furnace was designed by using the control variable method. Analysis of the experimental results showed that there was a certain coupling relationship between the variables. Based on this, a control-oriented control method was established with the input of black liquor flow rate, air supply volume, and induced air volume as inputs, and furnace temperature, flue gas oxygen content, and furnace pressure as output quantities. The dynamic nonlinear model of the alkali recovery furnace was linearized. The accuracy of the model was verified using the experimental data, and the correlation between the furnace temperature, the oxygen content of the flue gas, and the furnace pressure was analyzed using the relative gain matrix. The conclusions are as follows:

- (1) Through the qualitative analysis of the established mathematical model and the analysis of the data obtained from the experiment, it is shown that the flow rate of black liquor, the amount of air supply, and the amount of induced air have a great influence on the furnace temperature, flue gas oxygen content, and furnace pressure, so the black liquid flow rate Control the furnace temperature, supply air volume to control the oxygen content of flue gas, and induce air volume to control the furnace pressure. Through model verification, the results show that: The maximum mean square error between the output value of the model and the actual output is 0.004237, which indicates that the mathematical model has higher accuracy and can simulate the actual dynamic characteristics of the alkali recovery combustion process.
- (2) The quantitative analysis of the relative gain matrix shows that there is a strong coupling relationship between the furnace temperature and the flue gas oxygen content, and decoupling control is carried out; while the influence of the supply air volume and the black liquor flow rate relative to the induced air volume on the furnace pressure is small Too much, so the air supply volume and black liquor flow can be regarded as interference, and the single-loop control of the furnace pressure can be implemented.
- (3) The three-input and three-output control system of the alkali recovery furnace can be decomposed into a two-input and two-output coupling system of furnace temperature and flue gas oxygen content and a single-input and single-output control system of furnace pressure, thereby simplifying the design of the control system of the alkali recovery furnace.

## Acknowledgments

This work was supported by the Key R&D Plan in Shaanxi Province of China (2018GY-042) and Light Industry Intelligent Detection and Control Innovation Team (AI2019005). We sincerely thank for the funding of the project.

## Literature Cited

- Wessel, R. A. and Baxter, L. L., Comprehensive model of alkali-salt deposition in recovery boilers, Tappi J. 2(2):19-24 (2003).
- Zhang, R., Zheng, S., and Ma, S., Recovery of alumina and alkali in Bayer red mud by the formation of andradite-grossular hydrogarnet in hydrothermal process, Journal of Hazardous Materials 189(3):827-835 (2011). [https://doi.org/10.1016/j.jhazmat.2011.03.004]
- Zhang, L. J., Li, J., and Li, P., Research on modeling and optimal control strategy of alkali recovery furnace in large pulp mill, Mechanical and Electrical Engineering 17(3):96-99 (2000).
- Zhang, J. X., Modeling and numerical simulation of black liquor combustion process, Master’s Thesis, Harbin Institute of Technology, China (2015).
- Wang, X. J., Jin, B. S., and Yong, Z., Three dimensional modeling of a coal-fired chemical looping combustion process in the circulating fluidized bed fuel reactor, Energy&Fuels 27(4):2173-2184 (2013). [https://doi.org/10.1021/ef302075n]
- Liu, F., Optimal design of alkali recovery workshop of paper mill based on distributed control system, Master’s Thesis, Hebei University of Technology, China (2014).
- Jin, F. M., Application research of low lignin black liquor recovery engineering technology, Master’s Thesis. South China University of Technology, China (2018).
- Gan, H. B., Ren, G., and Zhang, J. D., Modeling and dynamic system simulation of fuel boiler for large oil tanker, Journal of System Simulation 21(3):913-917 (2009).
- Liu, J. Z., Lu, Y., and Yang, T. T., Least squares support vector machine modeling of boiler NO_x emissions based on variable selection, China Electrical Engineering News 32(20):102-107+146 (2012).
- Ni, H., Cheng, G., and Sun, F. R., Modular modeling and simulation of marine natural water circulation boiler, Computer Simulation 30(05):9-14+18 (2013).
- Yuan, Z. G., Su, Z., and Zhang, Q., Modeling T-S for outlet temperature of cement decomposition furnace, Control Engineering 23(2): 211-217 (2016).