CONSTRUCTION OF A MODEL OF THE KAOLIN ENRICHMENT PROCESS

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 15 CONSTRUCTION OF A MODEL OF THE KAOLIN ENRICHMENT PROCESS E.R. Kоmolov Senior Lecturer, Department of Mathematics and Physics, Alfraganus unversity https://doi.org/10.5281/zenodo.17606402 Abstract. The article develops a method for selecting scaling fuzzy concepts and forming membership functions on universal sets for the values of fuzzy variables that make up term sets; fuzzy modeling of the process of obtaining enriched kaolin is carried out. Keywords: kaolin enrichment process, modeling methods, fuzzy modeling, membership function, term set, syntax rules, oxidation of sulfide minerals, iron-oxidizing bacteria. Introduction: The kaolin beneficiation control system contains parameters described in terms of natural or formal language, i.e., they have no quantitative measurement. Fuzzy set theory, the foundations of which were proposed by L. Zadeh, allows for the operation of such concepts. Fuzzy modeling is currently one of the most successful ways to adequately represent objective and subjective knowledge about an object, especially when the equation contains parameters with various types of uncertainty. In this case, the equation describing the object, 𝑦 =𝑓(𝑥,𝑧) (1) Along with the vector of point parameters x, there is also a vector of fuzzy variables z, each of which is represented by its own fuzzy set Z. Accordingly, the vector of estimated parameters y in this case will be a fuzzy set Y. Then the membership function for Y, subject to the independence of fuzzy variables z1, z2,...,zn, can be written based on the use of the generalized concept of a function [11]: 𝜇𝑌= max 𝑦−𝑓(𝑧)[min(𝜇1(𝑧1),𝜇2(𝑧2),...,𝜇𝑛(𝑧𝑛))], (2) where 𝑌 =𝑓(𝑍1,𝑍2,...,𝑍𝑛),𝜇𝑖(𝑍𝑖) – membership functions for the parameters zi. Before calculation, it is necessary to evaluate the coefficient vector 𝛼 in the model. 𝑌 =𝐹(𝑥,𝛼)+𝐾𝐻, (3) where KH – parameters that introduce some uncertainty into the estimate 𝛼; x – vector. This model can be static and dynamic (when past values of y are included in the vector x). The estimation problem can be solved using various methods: least squares, maximum likelihood, adaptation, etc. In this case, various assumptions are made about the type of fuzziness (additive, random, Gaussian, with zero mathematical expectation, multiplicative, etc.), the nature of the a priori information about the process is taken into account and based on the measured values (yi, xi), 𝐼 =1,𝑁 an assessment is being made for research 𝛼. Let us consider two methods of evaluation based on fuzzy modeling, which is one of the most successful ways of adequately representing our objective and subjective knowledge about an object. In this case, we will consider the coefficients 𝛼 as fuzzy quantities in their domains of definition. The obtained membership functions can be used further directly in fuzzy models of type (6), and point estimates can also be obtained on their basis. One of the ways to construct a fuzzy set L, which represents the coefficient 𝛼, can be the method described in [6]. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 16 Research methods: Let y=(y1,y2,…,yn)T is a column vector, and the positive definite weight matrix W is chosen such that 𝛥𝑦𝑇𝑊𝛥𝑦 represents the cost associated with the error 𝛥𝑦. Let us define the function 𝐹 ~(𝛼) as 𝐹 ~(𝛼)=∑[𝑦𝑖−𝐹(𝑥𝑖,𝛼)]𝑇𝑊[𝑦𝑖−𝐹(𝑥𝑖,𝛼)] 𝑁 𝑖=1 . (4) A vector of crisp values for which the function 𝐹 ~(𝛼) reaches a minimum value, we denote 𝛼0. Then the membership function of any 𝛼 subset L can be defined as follows: 𝜇𝐿(𝛼)=exp{(𝑁−1)(1− 𝐹 ~(𝛼) 𝐹 ~(𝛼0))}. (5) Using the generalized concept of function (4) and taking into account the possibility of the presence of fuzzy constraints 𝐹 ~ on the set X, we write the membership function for the fuzzy set of solutions A: 𝜇𝐴(𝑦)= max 𝑦=𝐹(𝑥,𝛼)[min(𝜇𝐿(𝛼),𝜇𝐹 ~(𝑥))]. (6) If necessary, a vector of clear solutions y0 can be found for which the membership function 𝜇𝐴(𝑦) reaches its maximum value. For equation (5), an adaptive recurrent procedure for estimating the coefficients can be proposed 𝛼. Moreover, for the convenience of using the information available in practice about the errors in parameter measurements in this case, the uncertainty in the assessment 𝛼 can be attributed to the inaccuracy of measuring the input and output parameters at time t, i.e. 𝑍𝑡=𝑋𝑡+𝑉𝑡;𝑍𝑡=𝑌𝑡+𝑊𝑡, 𝑡= 1,2,...; , (7) where Yt, Xt are the exact values of the output and input parameters; Vt and Wt are fuzzy noise with known membership functions 𝜇(𝑣) and 𝜇(𝑤). Let the biotechnological system be represented as a sequence of vectors S of the form: S={S1,…,Se,…, Sn}, Sn={x1,…,𝑥𝑘 𝑒,…, 𝑥𝑁 (𝑛)}, where 𝑥𝑘 𝑒 means that the vector Sn indicates a specific value of the K-th parameter n= 1, 2,..., i.e. Sn={𝑥1 𝛼,𝑥2 𝛽,...,𝑥𝑘 𝑓,...,𝑥𝑁 𝜏}, 𝛼,𝛽,...,𝜏 ∈[1,𝑀𝑘], where Mk is the maximum number of terms in the term set T(Xk) of the fuzzy parameter Xk. The value of the parameter in each specific vector is a fuzzy set of the type 𝑥𝑘 𝑚=, where (𝜇𝑥𝑘 𝑚(𝑙𝑘𝜂) – the degree of membership of an element of the universal set Lk to a fuzzy set (term) 𝑋𝑘 𝑚. Each fuzzy parameter is assigned a universal set Lk = {𝑙𝑘 1,𝑙𝑘 2,...,𝑙𝑘 𝑁}, which is the Cartesian product of N universal sets: L = L1  L2 … Lk … LN. To estimate the distance between fuzzy sets, it is necessary to find the corresponding distances. Several definitions of distances between fuzzy sets are given in [13]. All of them have the feature that the distances between any disjoint sets are equal. The distance between fuzzy vectors S1 and S2 can be found using the formula. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 17 𝑑(𝑆1,𝑆2)={1 𝑅∑ ∑ [𝜇𝑥𝐾 (1)(𝑙𝐾 (1))−𝜇𝑥𝐾 (2)(𝑙𝐾 (2))]2 𝑅 𝑟=1 𝑁 𝐾=1 }1 2 ⁄. To classify fuzzy vectors, the idea proposed by A.A. Dorofeyuk [12] is used, which is as follows. Vector S1 is taken and the one closest to it in terms of minimum distance d is sought from the remaining ones. Denoting the found vector by 𝑆2, we are looking for a fuzzy vector 𝑆3 of the remaining ones, the closest to the group 𝑆= {𝑆1,𝑆2} etc. The result will be a sequence of vectors and a corresponding sequence of distances dn. When solving the problem of classifying real processes, questions arise regarding the choice of a method for scaling fuzzy concepts and the formation of a membership function on universal sets for the values of fuzzy variables that make up the term set in the Sn vector. If the formation of the desired number Mk of membership functions with the involvement of experts is difficult, it is necessary to limit the initial 𝑀𝐾 a reasonable value (3–5 terms). When solving problems, the number of terms used by experts in real-life situations is significantly greater. Therefore, the problem of generating intermediate terms using the initial (basic) ones and choosing certain syntactic rules associated with simple semantics arises. For example, having 𝜇𝑥𝑘 𝑚(𝐿𝑘) term "large", it is necessary to describe membership functions for the terms "close to large", "very close to large", etc. However, questions arise related to the construction of membership functions. Let us consider a method for describing membership functions given by experimental data using an analytical expression of a certain type. The results of experiments conducted under various conditions show that Th ferroxidans quickly oxidizes ferrous iron, while the chemical process is very slow. The rate of bacterial oxidation of ferrous iron by spontaneous microflora was increased by aerating the solutions and adding phosphorus salts, as can be seen from the data obtained with additional aeration with compressed air (2.8–3.0 m3/min) and adding bacterial inoculations from vats at a temperature of 20–31 °C under stationary cultivation conditions and with an excess of oxygen, carbon dioxide and other components. Keenan found that Th ferroxidans can oxidize Fe2+ at a rate 200,000 times greater than the rate without bacteria. Thus, Th ferroxidans is highly effective in oxidizing ferrous iron and can be used to regenerate leach solutions. Solubility of the final oxidation products of sulfide minerals, the interaction between the oxidation rate of the metal 2 dM dt  and solubility of the product sm K sulfide mineral can be expressed by the equation 222 sm dM V AK A S M dt         , (8) where A is the proportionality factor. As bacteria consume the substrate, new cells are formed. This process can be described by the following equation: i dx x dt   . (9) The specific growth rate of bacteria characterizes their physiological properties. Ironoxidizing bacteria obtain energy for their development by oxidizing ferrous iron, consume carbon for their development by oxidizing ferrous iron, and obtain carbon for their development from the carbonate ion dissolved in the nutrient medium. Bacteria also require other biogenic elements: SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 18 nitrogen, phosphorus, potassium, and magnesium. This depends on the substrate concentration, the presence of inhibitors and activators in the medium, temperature, and pH. The optimal pH in each case is determined experimentally. For sulfide minerals, the highest oxidation rate has been found to occur in the pH range of 2.2–2.6, which is optimal for the growth of Th ferroxidans bacteria. The specific growth rate of bacteria depending on pH with constant maximum values is affected by the oxidation rate of Fe2 with the growth optimum for bacteria and corresponds to the pH zone of 2.0–3.0 (optimum Ph 2.3–2.5) parameters. The degree of influence of pH on the specific growth rate of bacteria, the influence of T optimum temperature on the oxidation of sulfide minerals is within the range of 28–35 °C. When selecting microorganism cultures for biotechnology, the generation time of bacteria (cell doubling time) is of great importance, since this factor determines the efficiency of the process as a whole. Taking into account the kinetic dependencies, we will compose an equation describing the process of culturing bacteria in continuous mode: i dx x dt   ,   Fe i i i i dM mx dt   , 2 2 m Fe Fe MFe M KM   , (13) 222 sm dM V AK A S M dt       . The rate of bacterial oxidation of ferrous iron was increased by aerating the solutions and adding phosphorus salts. Additional aeration was achieved with compressed air (2.8–3.0 m/min) and the addition of bacterial inoculants from vats at a temperature of 25–35°C under stationary cultivation conditions and with excess oxygen, carbon dioxide, and other components. The oxidation rate in the cultivator reached 50 g/L. The oxidation rate is determined by the bacterial growth rate, which, in turn, depends on the method and conditions of their cultivation. The generation time of bacteria in a medium at a temperature of 25–35°C ranges from 3.6 to 10 g. Conclusion. Thus, the application of fuzzy set theory for the evaluation of process parameters in the presence of imprecisely specified values in the equations makes it possible to obtain a quantitative characteristic of the uncertainty of the estimated parameters using membership functions and creates a basis for decision-making in the biotechnological process control system. In the process of biotechnological deironing of kaolin, it is necessary to take into account the amount of iron contained in the raw material, and the S:L ratio and pH must be maintained such that there is no re-precipitation of iron in the form of hydroxide or to create conditions under which 3-valent iron will remain in solution. However, increasing the concentration of cells in the parent material by 10 times reduces the enrichment time not by 70– 80 hours, but only by 25–30 hours. Numerous conditions have shown that the oxidation rate is directly proportional to the enzyme concentration. Table 1 SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 19 The effect of the amount of mother material introduced into the kaolin suspension of the bacterium Thiobacillus ferrox on the culture titer Volume of mother material per 1 liter of suspension, ml Titer of suspension culture, cells/ml The amount of iron introduced into suspensions with parent material, g/l Note 20 2·10 0,016 The cell titer in the suspension is below optimal 25 2,5·10 0,02 30 3·10 0,024 The culture titer is sufficient for the development of the enrichment process 35 3,5·10 0,028 40 4·10 0,032 The amount of iron introduced exceeds the optimal norms Table 2 The number of removals of 1/2 the volume of the liquid phase of the reaction mixture at the maximum achievable concentration of iron in the solution depending on the T:L ratio T:F ratio Massvolume ratio, g:ml Removable volume of liquid phase, ml Total mass of iron removed, g One-time removal of iron mass, g Number of times liquid phase is removed, times 1:1 1:2 1:5 1:4 1:5 500:500 333:666 250:750 200:800 166:830 250 333 375 400 415 2,5 1,66 1,25 1,0 0,83 0,2 0,264 0,3 0,32 0,332 12 6 4 3 2,6 During the bubbling of the reaction mixture, starting from the 2nd day, a portion of the suspension was collected and analyzed for iron oxide content. Once the suspension concentration reaches 0.75 g/L, bubbling is stopped for 3–4 hours, then the upper supernatant is pumped out or drained, taking care not to disturb the sediment. Part of the liquid phase is drained periodically until the iron oxide concentration in it stops increasing and settles below 0.75 g/L. The goal of parametric identification is to experimentally determine the characteristics of an object. The object's parameters are estimated within the framework of a mathematical model of a specific class. The difference between the data obtained on the actual object and the corresponding mathematical model should be as minimal as possible. REFERENCES 1. Бочкарев В.В. Оптимизация химико-технологических процессов. Серия Университеты России. – М.: Изд-во Юрайт, 2016. – 263 с. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 20 2. Черносвитов Ю.Л. Требования промышленности к качеству минерального сырья. М.: Госгеолтехиздат. 1982.123 с. 3. Шакиров Ш.Ю., Салимов З.С, Шакиров Т. И., Мкртчян Р.В. Обогащение Ангренских каолинов с помощью электромагнитного сепаратора. Узбекский химический журнал. 1993.№1,с. 52-58. 4. Семенов В.С. и др. способ обогащения каолина. А.С.№412164.1974. 5. Ахметов К.А., Исмаилов М.А. Математическое моделирование и управление технологическими процессами биохимического производства. Ташкент,: Фан 1988.96 с. 6. Биестур У.Э., Шмите И.А., Жилевич А.В. Биотехнология. Биологические агенты, технология, аппаратура. Рига: Зинатне.1987. 263 с. 7. Кафаров В.В. Методы кибернетики в химии и химической технологии. М.: химия. 1985. 448 с. 8. Кафаров В.В., Винаров А.Ю., Гордеев Л.С. Моделирование биохимических реакторов. М.: Химия. 1983.342 с. 9. Аверкин А.Н. и др. Нечеткие множества в моделях управления искусственного интеллекта. -М.: Наука, 1986 10. Халин В.Г. Теория принятия решений. М Наука 2018. 11. Камолов Э.Р. Математическая модель технологического процесса обагащения каолина. Молодой ученый. № 27(161). С.3-7.2017г. 12. Камолов Э.Р. Основные виды и типы неопределенности информации, характерные для слогинных биотехнологических систем. Молодой ученый. № 27(161). С.33-37. 2017г 13. Камолов.Э.Р “Нечеткое моделирования процесса получения обогощенного каолина. ”, научный журнал “Интернаука”, Москва, 2019 г, №11(93). 14. КамоловЭ.Р. Мет одика идентификации математической м одели. Международный научный журнал, № 3 (79), 2020, Том.