Optimal operation of alumina proportioning and mixing process based on stochastic optimization approach

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Abstract

The quality of raw slurry and hence the alumina production is determined by its proportioning and mixing process. However, the existence of uncertainty elements in this process is unavoidable. These uncertainty elements are regarded as stochastic variables and the problem is formulated as a stochastic programming problem. Then, a second-order cone programming optimization approach is applied to find the optimal mixing proportioning strategy, with which the productivity of qualified raw slurry is maximized. A practical on-site experiment is carried out using the proposed approach and the results obtained show that this approach is effective in the maximization of the productivity.

Introduction

Alumina is an important raw material for many products in industrial production. Over the past 20 years, China’s monthly output of alumina products has ranked first in the world. During the production of alumina, bauxite and limestone, waste carbonic liquor (SCL) and other auxiliary material recovered from downstream steps are mixed into raw slurry, which is then delivered to the next steps for the production of alumina. The quality of raw slurry will affect the quality of alumina and also the productivity. However, it is a challenge to control and regulate the indices of various ingredients in the raw slurry Kong, Yin, Yang, Gui, and Teo (2020) and Yang, Gui, Kong, and Wang (2009). In the process of producing alumina, several indices, such as Fe–Al ratio (F/A), Al–Si ratio (A/S), Alkali ratio (A/R) and Calcium ratio (C/S), need to be focused on. In general, the adjustment of the last two indices is usually by proportioning and mixing raw slurry amongst several tanks many times between the higher and lower indices. The procedure is detailed as follows: Several tanks are used to fill the raw slurry with different composition ratios by pumps, and then re-mix them so that the required quality of raw slurry is achieved, that is, mixing several times between tanks until the relevant indices are within their respective required intervals.

This method depends on the measurement results of the indices. However, there is a long time delay before these results can be obtained. Furthermore, the sampled indices do not represent the overall characteristics of the entire tank, and any interference that causes the indices to change will make the results obtained unreliable. Thus, it is required to repeat the remixing process. As a consequence, it is difficult to achieve high productivity because the required specifications of the indices are to be met. In many alumina production plants, although more than two tanks can be accommodated, only two tanks are used for mixing due to the difficulty involved in the calculation of various indices between multiple tanks. Thus, only two tanks are normally used for remixing. For the interference in the measurement of the sampling and the uncertain factors in the operation of each tank, they are regarded together as one noisy variable. Clearly, if more tanks are used for mixing, better qualified raw slurry will be obtained with less times of mixing and proportioning processes. However, as mentioned earlier, the most difficult part in the proportioning and mixing process is how to accurately describe the various indices of each tank based on available information and how to calculate the appropriate proportioning of each tank.

Much work has been done in the past in the control and optimization of industrial processes, see Adetola and Guay, 2010, Huang et al., 2020, Jia et al., 2020 and Li, Ding, Chai, and Lewis (2020). Also, some attention has been given to the development of metallurgical industry (Ding et al., 2012, Zhou et al., 2020). In addition, some methods have been proposed and some results have been obtained for the blending process of alumina production (Duan, Yang, Li, Gui, & Deng, 2008). Although significant advancement has been made for the improvement of the quality of raw slurry when compared with non-optimal or experience-based operation, it is still a challenge to achieve high productivity due to insufficient information of the indices. It is, indeed, difficult to obtain satisfactory indices through proportioning and mixing. To obtain optimal proportioning such that maximal productivity is achieved, it is important to have a proper representation of the existing uncertainty. To date, some stochastic based results have been obtained and they have been successfully used in industrial applications (Salvendy, 2001). In many engineering applications, their uncertainty is regarded as a Poisson process such as in Basin and Maldonado (2014). Robust, estimation, optimization and learning methods are discussed in Alvarez and Odloak, 2010, Chen et al., 2020, Yin et al., 2015 and Zhang, Monder, and Fraser Forbes (2002) to deal with different uncertainties in different processes.

However, the challenge for optimal set points decision or optimal operation in industrial process under uncertain environment is how to solve the decision related programming. By now, it is known that deterministic optimization problems are constructed to model optimal set points decision or optimal operation problems, in many real world applications. It is worth mentioning that stochastic programs, which have been studied since 1950s as a tool to deal with uncertainties in data or models, belong to a class of optimization problems in the form of linear and quadratic programs. Second-order cone programming (SOCP), which is also named as conic quadratic programming Luo, Sturm, and Zhang (2000) and Miao, Fan, Aghamolki, and Zeng (2018), is a type of convex programming problems where a linear function is optimized subject to a set of second order cone constraints. SOCP is a nice structured convex programming problems which is solvable by efficient interior-point solvers (Nesterov & Nemirovskii, 1994). Many interesting results and successful applications are now available in the literature, see, for example, Ben-Tal and Nemirovski (2001) and Lobo, Vandenberghe, Boyd, and Lebret (1998).

In this paper, by making use of the nature of uncertainty in the mixing process of alumina production, the content of each component in each tank can be assumed to follow a certain random distribution based on the information obtained from measurement. Then, appropriate proportion for each tank is obtained through solving a stochastic programming problem such that the maximal qualified raw slurry is achieved, where only one re-mixing operation is needed but more than two tanks are involved. It is well-known that optimization has been successfully applied in many complex industrial and management systems Kim et al., 2017, Liu et al., 2018 and Yin, Liu, Teo, and Wang (2018). However, it remains a challenge to find an effective computational method to deal with the complex problem considered in this paper because the relationship between contents and indices varies stochastically. This problem will be dealt with using the stochastic programming approach. It shows that the problem here can be formulated in the form of a second-order cone programming, which can be solved effectively. Thus, the optimal proportions are obtained such that the maximal productivity is achieved and the indices are within the required respective intervals with a high confidence.

The remainder of the paper is organized as follows. The alumina operation process is introduced and some challenges in finding the proportioning and mixing are explained in Section 2. In Section 3, the formulation as a stochastic optimization problem is presented. In Section 4, a solvable stochastic algorithm is proposed to find the optimum solutions. Section 4 shows the experimental results, where real production data are used. Section 5 concludes the paper.

Section snippets

Mixing process of alumina production

Alumina production is a complex metallurgical production process. The first step is to feed multiple raw materials, which include bauxite, limestone, auxiliary materials and carbonic acid waste liquid (SCL) recycled from subsequent steps, to grinders, to form a mixture, called raw slurry, which is the main material for the production of alumina. For alumina production, the indices between various compositions should be strictly controlled. Due to the uncertainty involved in raw slurry, many

Formulation of stochastic programming problem

Obviously, the random parameter δ will affect the actual content of relevant components in each tank. Thus C(t) is expressed as a random matrix c̃(t). Therefore, system (2) is rewritten as: R(t)=f(X(t),c̃(t)),where f()={f1,,fq}, c̃(t)={c̃ls(t)},l=1,,q,s=1,,m, and c̃ls are independent Gaussian random parameters, with mean μls and variance σls. They are obtained based on measurement results and statistical historical data.

The mixing model can now be expressed by the following equations: fl(X(t

Industrial experiment

As mentioned earlier, for the proportioning and mixing operation problem in alumina production, there are uncertainty factors that will affect the content of compositions in each tank before proportioning and mixing. For each tank, there are 5 types of components to consider and monitor. They are CaO, Na2O, SiO2, Fe2O3 and Al2O3, the content of each chemical composition can be obtained from laboratory through assay being carried out in the laboratory. These chemical composition contents are

Conclusions

The quality of raw slurry, which is determined by its proportioning and mixing process, is critically important for alumina production. The unavoidable uncertainty factors, that exist, are regarded as a stochastic random variable with its mean and variant being assumed known from statistical historical data obtained from measurement. On this basis, the alumina production process was formulated as a stochastic programming problem, and the second-order cone programming optimization approach is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under grant Nos. 61773011, 61773183, Hunan Provincial Natural Science Foundation of China (2018JJ2100) and a Discovery Grant from Australian Research Council .

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