Available online at www.HighTechJournal.org HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 663 ISSN: 2723-9535 Permissible Extrapolation Justification of the Multiplicative Multifactorial Model and Its Application to the White Soot Production Technology Lyutsiya Karimova 1, 2* , Guldana Makasheva 1, 3 , Vitaliy Malyshev 4, Yelena Kharchenko 1, 5 , Yerlan Kairalapov 1, 2 1 Metallurgy Laboratory of LLP “Innovation”, Karaganda 100024, Kazakhstan. 2 LLP “KazHydroMed”, Karaganda 100000, Kazakhstan. 3 Satbayev University, Almaty 050013, Kazakhstan. 4 Chemical-Metallurgical Institute named after Zh. Abisheva, Karaganda 100009, Kazakhstan. 5 Non-Profit Joint Stock Company “Karaganda Industrial University”, Temirtau City 101400, Kazakhstan. Received 25 May 2024; Revised 23 August 2024; Accepted 27 August 2024; Published 01 September 2024 Abstract The solution to a specific technological problem is combined with the methodological development of a nonlinear multifactorial relationship to justify the boundaries of its extrapolation beyond the experimental range used. White soot was produced through two-stage carbonization of a silicate solution (composition, g/l: Na2O = 126.5, SiO2 = 107.7, Al2O3 = 3.1), obtained after processing waste tailings with carbon dioxide in a recirculation system. The influence of the deposition duration, temperature, and final pH value of the pulp on the formation of the specific surface area (Ssp, m2/g) of white soot was studied. The specific surface area was calculated from the average diameter of the white soot particles measured using an electron microscope. A multifactorial experiment was designed, and the experimental results were processed using a probabilistic deterministic method for experiment design (PDED) to obtain a nonlinear multiplicative combined model. A new interpretation of the subordination of the nonlinear multiple correlation coefficient R and the R2 value was given as relating to the structural and adaptive components of complex self-organizing systems. This determines their use for assessing the ratio of the basic R and extrapolated R2 ranges of variation for each factor and any combinations thereof and multifactorial dependence in general. The results are presented in the form of multifactor tabular nomograms measured by the number of multifactor cells in localized areas of optimal sets that allow isolation by one or another combination of factors. The technological object of extrapolation of the ‘white soot’ production is linked to the solution of emerging methodological problems and illustrates the accessibility of the engineering application of the proposed method for combining nonlinear and linear approaches to mathematical experimental design. Keywords: White Soot; Specific Area; Multifactor Model; Correlation; Permissible Extrapolation. 1. Introduction Recently, mathematical planning of experiments has been developed to improve the accuracy of displaying the relationships of an existing sample using methods for combining linear and nonlinear additive and multiplicative models [1-6]. Moreover, combinatorial enumeration generates an optimal model structure, which is the individual for * Corresponding author: lyuciya_karimova@bk.ru http://dx.doi.org/10.28991/HIJ-2024-05-03-08 ➢ This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). © Authors retain all copyrights. https://creativecommons.org/licenses/by/4.0/ https://orcid.org/0000-0001-6205-6585 https://orcid.org/0000-0003-2875-9433 https://orcid.org/0000-0002-5206-2620 https://orcid.org/0000-0003-4616-5436 HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 664 each sample each time. For example, when using the neural network method, it is directly indicated that an identical generalized mathematical model of an object simply does not exist [1]. In this case, there is no actual planning for the experiment, and it formally belongs to the category of passive experiments, which use a randomly generated set of data without observing any conditions. In the absence of a mathematical model, questions about the possibility of extrapolation, let alone permissible ones, are inappropriate. Meanwhile, the problem of increasing the accuracy and reliability of the results of a multifactorial experiment can be solved by active intervention in the initial data to identify an unknown space of optimal results. In the proposed method, this is achieved by revealing the relationship between the statistical criteria R and R2 as general systemic criteria of stability, achieved with a ratio equal to the proportion of the golden section. In this case, the intrasystemic limitation of the extrapolation procedure is the adaptive component R2 ≤ R. The production of multifactor linear and nonlinear models of technological processes is well-developed [7-13]; however, the procedure for multifactor extrapolation into unexplored areas of the multifactor technological space and optimization of results remain ununified. Therefore, it is necessary to accumulate such data when solving production problems and to find rational methods for processing them. Over the past decades, there has been intensive growth in the research areas of science and technology based on the use of various forms of silica [14]. Quartz sand and tailings are promising raw materials for the production of amorphous silica, which is used in various industries and is in great demand in the market. Therefore, the study of processes for processing quartz-containing raw materials to obtain pure silicate products (white soot) is an urgent task. Figure 1 presents a flowchart of the research process. Figure 1. Flowchart of the research process Formulating the Problem Obtaining BS-100 white soot with a specific surface area of 100-150 m2/g from waste tailings to justify the boundaries of its extrapolation beyond the experimental range used. Selecting the Research Methodology Based on a combination of a nonlinear multifactor model with the known procedure of the steepest (linear) ascent to the extremum region like in the Box-Wilson method. Method and Novelty • for partial functions, only linear equations were used combining dependencies in the form of their product with the normalization of the multifactorial function by the geometric mean value; • for the first time, the square of the correlation coefficient R2 is used as a measure of limiting the extrapolation of a multifactorial dependence into an unstudied area for all factors, regarding the value of the R2 criterion for each particular dependence, which ensures uniform adjustment of the factor variation step to the area of optimal values. Conclusion A nonlinear multiplicative equation was obtained in compliance with the linearity of partial dependencies, which are most clearly expressed by the influence of temperature, duration and pH of the solution on the specific surface area of microscopic grains of this product according to GOST in the range of 100-150 m2/g. Results Formation of multifactor tabular nomograms, which include the full number of combinations of all factors and levels among them. The results that meet the requirements of GOST in compliance with the interval of 100-150 m2/g are isolated from this set. HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 665 2. Literature Review Alkali sintering methods were studied in previous studies [15, 16], and the enrichment of the materials was confirmed. The extraction of silicon from metallurgical waste into a solution can be used for the deposition of white soot. White soot is finely dispersed hydrated silicon oxide containing 85-95% SiO2 and admixtures of iron, aluminum, magnesium, and sodium oxides. It is in demand as an active mineral filler in tire, rubber, chemical, cosmetic, and other industries. Silicon dioxide is deposited from a solution of sodium silicate (liquid glass) with acid or carbon dioxide, followed by filtration, washing, and drying. The main characteristics of various white soot brands are presented in Table 1 (GOST 18307-78) [17]. Table 1. Main characteristics of white soot Indicator BS-30 BS-50 BS-100 BS-120 Mass fraction of silicon dioxide, %, at least 85 76 86 87 Mass fraction of moisture, %, at most 6.5 6.0 6.5 6.5 Weight loss on ignition, % 4.5-7.5 7.0-10.0 5.0-7.0 3.5-7.0 Mass fraction of iron in terms of iron oxide, % at most Not standardized 0.03 0.15 0.17 Mass fraction of aluminum in terms of aluminum oxide, % at most Not standardized 0.10 0.15 0.10 pH of aqueous extract • For powder white soot • For granular white soot 8.0-10.0 - 9.0-10.5 - 7.0-8.5 - 8.0-9.5 7.0-8.5 Specific surface area, m2/g 35±10 45±10 100±20 120±20 By agreement with the consumer, it is allowed to produce white soot BS-100 with a specific surface area of 100-150 m2/g. Depending on the method for producing white soot, the final property of the product is determined by the size and shape of the particles, presence or absence of pores, etc. [18, 19]. The obtained experimental values were processed using the standard methods of probability theory and mathematical statistics. The main approach to solving many problems is the least squares method, which works for linear dependencies, but in practice, nonlinear ones are more common. In this case, approximate nonlinear processing methods were used [20]. The main tasks of regression analysis include establishing the form of dependence, determining the regression function, and estimating unknown values of the dependent variable. To obtain the experimental and statistical functions of objects, mathematical designs of complete factorial experiments (CFE) and fractional factorial experiments (FFE) have been developed [20-23]. Due to the significantly smaller number of experiments compared to CFE designs, FFE designs are widely used in industrial experiments and when it is necessary to study a sufficiently large number of factors with a small number of experiments and determine the factors that have the strongest influence on the property of the factor. If necessary, in an experiment compared to the CFE design, FFE mathematical designs are drawn up depending on the preferred method of statistical analysis of the results. Mathematical experimental design is a constantly improving method; important indicators of the quality of compiled mathematical experimental designs are their orthogonality and optimality [24, 25]. There are different types of compositional designs: three-level Box designs (3k), Box-Wilson design, Box-Hunter design, and Kono design. These designs make it possible to find the regression equation in the following family of polynomials, for example, for coded values of the input factors: To obtain a quadratic regression equation, it is necessary to have at least three levels for each factor (ml ≥ 3). Three- level complete factorial experimental designs are known as Box designs. The advantage of Box designs is their high accuracy in determining factor effects. Box designs are not orthogonal and D-optimal. The disadvantages of Box designs also include the large number of experiments, which far exceeds the maximum possible number of coefficients in the quadratic polynomial L (Table 2) [24, 25]. Table 2. Parameters of Box designs Parameter Parameter value with the number of input factors k 2 3 4 5 6 Number of experiments N=3k 9 27 81 243 729 L 6 10 15 21 28 Next, the remaining linear regression coefficients are adjusted to simultaneously and uniformly move the expected optimum, for which the existing largest regression coefficient should be reduced and the existing smallest value should be increased. This can be achieved in various ways. Since there is no theoretical justification for limiting the regression HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 666 coefficient, further continuation of the Box-Wilson method is provided only in an additional experimental implementation. Special extrapolation methods other than the Box-Wilson method are unknown, including the neural- network-based experimental design method [26]. Therefore, each time it is necessary to adjust the known methods for designing nonlinear experiments, such as the well-known multiplicative [27] and above-mentioned research methods. In contrast to similar methods, this method is developed only for linear terms of a multifactor regression equation and with the obligatory elimination of regression terms in the form of a product of the first-degree factors; otherwise, the procedure for simultaneously reaching the global extremum region will not be reliable. This research aims to study the possibility of obtaining a commercial product, BS-100 white soot, in the range with a specific surface area of 100-150 m2/g using a new method by determining the optimal values of this surface based on combining the known nonlinear multiplicative multifactor model [27] with the known steepest (linear) ascent procedure to the extremum region using the Box-Wilson method. 3. Research Method To converge with the Box-Wilson method (BWM), by steeply ascending to the region of the global optimum based on a linear regression equation, the following modification of PDED is proposed: for partial functions, only linear equations are used, and their arithmetic mean values are introduced into the multiplicative model to normalize their partial dependencies in dimensionless form. For the first time, the square of the correlation coefficient R2 i was used as a measure of extrapolation of multifactorial dependence into an unstudied area for all factors simultaneously. This is achieved, as in the BWM, by adjusting the rate of entry into the global extremum region using the value R2 for each particular dependence rather than by averaging the linear regression coefficients. Extrapolation and optimization processes are detailed in multidimensional tabular nomograms and measured by the number of multifactor cells in localized areas of optimal (acceptable) values. These nomograms can be used to monitor and control the technological process, as illustrated by the example of white-soot production. Our research is characterized by the possibility of combining linear and nonlinear mappings in multifactorial dependencies, if it is necessary to extrapolate beyond the studied range to obtain separate areas of optimal values of a multidimensional function. The combination method includes the coupling of additive and multiplicative fragments, of which the former are represented by rectilinear partial dependencies normalized by arithmetic mean values, and the latter are represented by the products of partial functions and geometric mean values. The research objective (in contrast to the known ones) is to establish the degree of extrapolation of multifactorial dependence based on the R2 value, which eliminates the possibility of unjustified extrapolation and unreliable results. This research was conducted by conducting technological operations with multifactor dependencies within optimal areas and using them, for example, in the event of a random exit from the optimal mode of a technological object and an accelerated return to the optimal mode based on four-factor tabular nomograms. Carbon dioxide was used as a neutralizing agent to separate white soot from a silicate solution. White soot (mSiO2‧nН2О) was obtained through the two-stage carbonization of a silicate solution (liquid glass) with carbon dioxide in a recirculation system, bringing the pH to 9-10 within 30 min, and then within 60 min, until the residual alkali content in the solution was 90 g/l. The main reaction for producing white soot with carbon dioxide is as follows: Na2SiO3+CO2 = Na2CO3 + SiO2↓ with sedimentation in the form of mSiO2‧nН2О. At the stage of preliminary desiliconization of the rough concentrate into a solution under arbitrary search conditions, a silicate solution was obtained with the following composition: g/l: Na2O = 126.5, SiO2 = 107.7, Al2O3 = 3.1. In a series of experiments, carbon dioxide was purged through the solution volume for such a time that the required final pH value (9.5 - 9.8 units) of the pulp was achieved within a technologically acceptable duration. When conducting experiments according to a special design, the influence of deposition duration (τ, min), temperature (t, °C) and the final pH value of the pulp on the formation of the specific surface area (Ssp, m2/g) of white soot was studied. The resulting sediment was separated by filtration, washed, and dried at 105°C. The specific surface area was calculated from the average diameter of the white soot particles measured using an electron microscope. The experimental design for the sequential study of the operating factors was implemented using a method that involved setting up a generalizing central experiment with a single experimental point for all factors. This is indicated on the graph of partial functions and is considered when constructing them and determining the correlation coefficient [27, 28]. The resulting partial dependencies regarding the significant functions to describe the set of operating factors were generalized according to Malyshev et al. [27] in the form of their product with normalization by the arithmetic mean experimental value of each function. In all cases, when deriving the equation, to check its adequacy, we used the nonlinear multiple correlation coefficient R and its significance tR, which are expressed by the following formulas [27, 28]: HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 667 𝑅 = √1 − (𝑛−1)∑ (𝑦𝑒,𝑖−𝑦𝑑,𝑖) 2𝑛 𝑖=1 (𝑛−𝑘−1)∑ (𝑦𝑒,𝑖−𝑦𝑒,𝑎𝑣) 2𝑛 𝑖=1 (1) 𝑡𝑅 = 𝑅√𝑛−𝑘−1 1−𝑅2 > 2 (2) here уe,i is an experimental value; уd,i is a design value; уe,av is an average experimental value; n is the number of independent (non-repeated) experimental data; k is the number of operative factors; (n-1) is the number of degrees of freedom for reproducibility variance; (n-k-1) – the number of degrees of freedom for adequacy variance. The geometric mean of all experimental values 𝑦 𝑒,𝑔 was introduced into the multifactor equation as a normalizing divisor of the generalized function 𝑦/𝑦 𝑒,𝑔 : 𝑦 = 𝑦 𝑒,𝑔 ∏ 𝑦𝑖 𝑦𝑖,𝑒,а 𝑖=𝑛 𝑖=1 (3) In its most general form, this equation expresses a nonlinear multiplicative multifactor function and fundamentally differs from a multifactor linear regression equation in the form of a sum of terms. The regression equation has two major drawbacks. • It is not reset to zero at zero values for any factor, that is, a generalizing function is not identically transformed into a particular function. • The regression equation does not allow the inversion of variables because of the representation of each variable, not only separately, but also as a product with other factors. The nonlinear multiplicative multifactor Equation 3 is deprived of these shortcomings because of the obvious algebraic operations involving zero and the transfer of variables on the left or right sides of equality during ordinary algebraic procedures. The mandatory normalization of partial dependencies by their arithmetic mean values is a significant feature of Equation 3. This results in the reduction of the natural dimensions of partial dependencies and their consideration in the form of shares in a unit. Moreover, the generalized multiplicative function y is strictly equal to unity if all the partial dependencies used in the multifactor equation are equal to their arithmetic mean values. This ensures the possibility of reducing it to unity at the average values of all partial functions 𝑦 𝑦𝑖,э = ∏ 𝑦𝑖,р 𝑦𝑖,э,а = 1𝑖=𝑛 𝑖=1 . The multiplicative structure of the multifactor Equation 3 considers the influence of each factor according to its deviation from unity: when deviating upward from unity, the influence of this factor increases; when deviating from unity downward, it decreases; if it is equal to unity, the influence of each factor is neutralized. Determining the optimal conditions for obtaining technological products using multifactor models usually involves searching for extreme values in each partial dependence, with further substitution of the corresponding largest (or smallest) partial optima into the generalizing dependence and obtaining a single multifactor extremum. This practice has taken root since the time when there were no multifactorial mathematical models and the multifactorial process was studied in a sequential transition from the best indicators in the previous partial dependence to the initial conditions in the subsequent one, etc., to the last one. In fact, it turned out that in this way, it is possible to enter the region of a particular extremum, located arbitrarily far from the global extremum. Special methods have emerged for using multifactor models to determine the shortest paths to reach the region of optimal values of a multifactor function. Most of these methods are based on simplex-lattice progression to an extremum, moving in the opposite direction to the worst conditions (rather than in the continuation of the best). Another method is the so-called “steepest ascent” to the optimum area, based on a multifactorial regression equation with the correlation of the step of progress toward the target result, while simultaneously “stepping” in a general order, implementing the last stage of the “ascent” by staging a step-by-step experimental completion of the process when the indicators begin to deteriorate. In the new experimental design methods, including those based on neural networks, the possibility and conditions for limiting extrapolation were not considered. The main task of the BWM is to provide simultaneous and uniform step-by-step movements toward the optimum region. This is achieved by adjusting the regression coefficients for each factor after obtaining an incomplete quadratic regression model and removing all the products of the factors as non-linear terms. Thereafter, the remaining linear part of the equation was tested for the significance of each factor by its regression coefficient and the removal of all insignificant factors. Then, the most essential procedure for adjusting the regression coefficient is conducted, as they determine the rate (by an absolute value) and direction of the effects of each factor (increase, plus, decrease, minus) with the code designation of the dimensionality of the variables (вi ·xi=1). Vinarsky and Lurie consider several methods of correlating вi ·xi to smooth out the impact of strong and weak factors, and among them, normalization of вi by вmax, averaged в𝑖 = в,̅,taking the reciprocal value 1/вi, change of sign в𝑖 ⟹в𝑖 2, and normalization by sum в𝑖 = в𝑖/∑ в𝑖 𝑖=𝑛 𝑖=1 . The ambiguity of preparing for the steepest ascent to the region of the global optimum forces to use crushing the adjusted steps to the smallest overall step, especially because the ascent is recommended to be conducted experimentally HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 668 before the results begin to deteriorate and repeat the entire procedure, starting with obtaining a new regression equation under the conditions of stopping the ascent to the top and so on until the region of global extremum transforms to a limit that cycles without deterioration. Thus, it is not easy to implement process control with the linear display of a complex object when a change in linearly acting factors is linearly transmitted to a linearly perceiving multifactorial response function. Our attempt to apply a similar procedure to nonlinear multifactor objects was based on the complete replacement of the main tool for setting up a uniform and simultaneous ascent to the region of optimal results of regression coefficients with correlation coefficients of particular and generalized dependencies. Compared to regression, correlation seems to be a more general concept, including probabilistic and information aspects and those related to the stability of complex systems. This requires expanding the concept of the optimum and is represented by its multidimensional mapping. At the same time, we tried to adhere to the conditions of subordination of the requirements for linear systems and the basic idea of the steepest ascent – uniform and rectilinear ones (as in Newton’s first law), according to the Box-Wilson method. To combine the steepest (linear) ascent to the optimum region (in this case, for the best Ssp values in the GOST interval) with the Box-Wilson method, basic nonlinear model (3) was supplemented with the following restrictions: • The original partial-point experimental dependencies are approximated only by the equation of straight lines, which are typically used for linear multifactor models in the form of regression equations. • The basic nonlinear (multiplicative) model is expressed as a product of linear partial functions with normalization by their arithmetic mean values, while the dimensionalities of the partial dependencies for each factor are reduced during normalization and become dimensionless (unit fractions, u.f.). • To normalize a nonlinear multifactor function using the general average, which is defined as the geometric mean. These limitations relate to the experimental design to obtain a single multifactor nonlinear equation with fragments of a linear response in combination with nonlinear ones for better adaptability of the combined model to the object of display, in this case, to the technological process for a wider scope of optimal conditions for obtaining a product of a given quality. In addition to these limitations, an entropy-information justification for the structure of the correlation coefficient (1) and its significance (2) has been added to contain structural i and adaptive h components normalized by the maximum value of information entropy Hmax in the form of the law of conservation of the sum of information and entropy [29, 30]: 𝑖 + ℎ = 1 (4) This supplement to the meaning of nonlinear correlation makes it possible to isolate the additive influence of the structural and adaptive components on the limit of extrapolation and optimization in an expanded multifactor space equal to the proportion of the golden section [31, 32] and in normalized form, not leaving the equality: 𝑖 + ℎ = 0.618 + 0.382 = 1 (5) This equality is valid for the first level of self-organization of a complex system (n=2) [28]. The values of i=R are obtained from the dependence of the structural component on the degree of coherent (integer) correspondence of the measures of the information (i) and entropy (h) components as a result of the analytical or numerical solution of the equation [28]. 𝑖𝑛 + 𝑖 − 1 = 0 (6) When n=2, the superiority of i>h is achieved for the first time, and in all cases 𝑅 ≥ 𝑅2, so that neither within the framework of general system laws nor according to purely mathematical properties of the value 0 ≤ х ≤ 1⟹ 𝑥2 < 𝑥. In this case, this made it possible to more strictly substantiate the need for double consideration of the correlation coefficient, not only to express the real dependencies of the multifactorial empirical connection of data, but also for reliable extrapolation to the same extent in a broader process control mode. In addition, the nonlinear multifactor model acquires additional reliability and degrees of freedom, which can be used for linear models, while simultaneously eliminating the approximate and contradictory interpretation of R2 in relation to the determinism and functionality of any model. To verify these equations, it is necessary: • To use the value R2 as the permissible fraction σ of the expansion of the studied interval for each factor Δxi by multiplying it by R2 i. 𝜎𝑖 = ∆𝑥𝑖𝑅𝑖 2. HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 669 The greater Ri, the greater R2 i, but it is always within Δxi. When 𝑡𝑅 ≤ 2, the correlation coefficient was considered insignificant; that is, when Rmin=0. 𝜎𝑖,𝑚𝑖𝑛 = ∆𝑥𝑖𝑅𝑖,𝑚𝑖𝑛 2 = 0. • With Ri>2, the share of σi is distributed equally between the left and right boundaries of the range of values x by adding or decreasing x by Δxi‧R2 i/2, depending on the direction of extrapolation 𝜎𝑖 = ±∆𝑥𝑖𝑅𝑖 2/2, toward increasing or decreasing yi in proportion to xi, which is a consequence of the primary processing of dependencies on the equation of straight lines y=ax+b. For nonlinear functions, their behavior during extrapolation can be greatly distorted. • Extrapolation for all factors is not conducted immediately over the entire range but with an adjustment for an increase from 50 to 100% of the full share. To prepare particular dependencies and a multifactor model together for extrapolation procedures, the initial data should be tabulated in the order in which they are sequentially filled out. The results of extrapolation and optimization of acceptable options for quality indicators (Ssp = 100-150 m2/g of white soot) were in a wider range of combinations of operating parameters of the white soot production process. 4. Results and Discussions The results of the experiments using the sequential study of factors after conducting a central experiment are presented in Table 3. Table 3. Experimental data on the influence of pH, leaching duration and temperature on the specific surface area of white soot Factor under study Experimental conditions Ssp., m 2/g exp. Ssp., m 2/g for partial functions Ssp., m 2/g by Eq. 7 τ, min. (CNa2O– 126.5 g/l, t– 40°С) 50 340 338.9 357.7 70 320 314.9 332.4 90 280 291.0 307.1 120 260 255.1 269.3 t, °C (τ–60 min., CNa2O– 126.5 g/l) 25 455 436.7 383.7 40 330 368.0 323.3 60 290 276.3 242.8 рН, units (t–40°С, τ – 60 min., CNa2O– 126.5 g/l) 9.5 125 168.5 178.6 9.7 316.1 255.3 270.6 10.2 455 472.4 500.7 The processing of the experimental results on the equation of a straight line (using the least-squares method) to identify significant linear functions is presented in Figure 2. Crosses represent the coordinates of the central experiment. In all experiments, the alkalinity was adjusted at the final stage to a level of 126.5 g/l. The obtained linear partial dependences of the specific surface area with the determination of the nonlinear multiple correlation coefficient R and its significance tR are listed in Table 4. (a) (b) 150 200 250 300 350 400 0 20 40 60 80 100 120 S sp ., m 2 /g τ, min 150 250 350 450 0 10 20 30 40 50 60 S sp ., m 2 /g t, °C HighTech and Innovation Journal Vol. 5, No. 3, September, 2024 670 (c) Figure 2. Influence of various carbonization factors on the specific surface area of white soot: (a) duration, min; (b) temperature, °С; (c) solution рН (developed by the authors). Note: ○ – experimental data; × – average value for all partial functions; ∆ – according to Equation 7. Table 4. Particular functions of the specific surface area (Ssp., m2/g) of white soot for τ, t and рН of the solution. Functions R tR Ss𝑝. = (398.7 − 1.196 ∙ 𝜏) 0.967 21.21 Ss𝑝. = (551.3 − 4.58 ∙ t) 0.887 4.166 S𝑠𝑝. = (434.2 ∙ pH − 3956) 0.958 11.73 Based on experimental data, partial equations were obtained (Table 4), which were used to derive a mathematical model [27] for the specific surface area of white soot. Partial equations were generalized in the form of their product with normalization to the average experimental value (in this case, m2/g: 300.0 for  =60 min; 360.3 for t=40°C; 298.8 - pH=9.7 units). The generalized equation for the specific surface area is expressed as: 𝑆𝑠𝑝 = 9.335 ∙ 10−6 ∙ (398.7 − 1.196 ∙ 𝜏) ∙ (551.3 − 4.58 ∙ 𝑡) ∙ (434.2 ∙ 𝑝𝐻 − 3956) (7) The nonlinear multiple correlation coefficient was R=0.843, its significance was tR =6.001>2 (calculated by equations 1 and 2), and based on the geometric mean value 𝑦 𝑒,𝑔 R=0.847, tR=6.249> 2. A comparison of the experimental data on the specific surface area of white soot and those obtained from Equation 7 is shown in Figure 2 (triangles). Generalized Equation 7 makes it possible to identify the joint influence of the existing factors. Moreover, not exceeding the permissible values for the influence of factors can be ensured by many combinations of specified levels rather than by what was used during the central experiment under average conditions of the recorded factors. This can be demonstrated using a multifactor tabular nomogram, which is presented in Tables 5 and 6. Each mini-rectangle (cell) displays a multifactor nonlinear function with a flat multifactor approximation for each of the common coordinate systems. Isolating groups of cells according to the condition of being in the permissible (allowed) limits of the objective function with varying accuracy creates an image of optimal behavior and control of the object. Table 5 shows the initial data for an allowable extrapolation of the multifactor model into the unexplored area of the factor space regarding R2 for partial functions (Table 4). Moreover, R2