Open AccessArticle Modeling of Gibbsite Solubility in Na +-K +-OH −-Al(OH) 4−-SO 42−-CO 32− System Applied to Precipitation at Sintering Alumina Refineries Tatiana E. Litvinova Tatiana E. Litvinova Nickolai V. Tuleshov Nickolai V. Tuleshov * The Department of General and Physical Chemistry, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia * Author to whom correspondence should be addressed. Metals 2026, 16(6), 633; https://doi.org/10.3390/met16060633 (registering DOI) Submission received: 5 May 2026 / Revised: 5 June 2026 / Accepted: 5 June 2026 / Published: 9 June 2026 Abstract The quantitative prediction of the equilibrium gibbsite content in alkaline liquors of alumina production is a key parameter for controlling precipitation kinetics and ensuring product quality. Unlike Bayer alumina refineries, sintering alumina refineries use different alkali and impurity content ranges; moreover, they are characterized by the presence of significant amounts of potassium in aluminate liquors that are not considered in the existing gibbsite equilibrium models. This paper presents an extended mathematical model, which is based on the Rosenberg–Healey methodology and incorporates sodium (Na 2O) and potassium (K 2O) components into the total alkaline system. Model coefficients were optimized using 18 tests designed by the D-optimal method at temperatures of 50–85 °C and a total alkali content of [R 2O k] = 40–80 g/L, which contains up to 30% of potassium alkali K 2O, as well as the impurities in the form of soda [Na 2CO 3] = 0–58 g/L and sodium sulfate [Na 2SO 4 ]= 0–25 g/L. The fine-tuned model was verified using the composition of actual refinery liquors and provides RMSE = 0.71 g/L Al 2O 3, thus demonstrating satisfactory solubility prediction accuracy. The presented model can be used to calculate aluminate liquor productivity at existing sintering refineries with a sodium–potassium system. Keywords: gibbsite equilibrium; mathematical modeling; thermodynamic model; sintering method; aluminate liquor; gibbsite precipitation; alumina production 1. Introduction Models of gibbsite solubility in alkaline aluminate liquors are an important tool for controlling the precipitation process in alumina production. An important variable in various complex population balance models of aluminum hydroxide particles [ 1], which explain the dynamics of particle formation and growth in the liquors, is the driving force ( A*) of the aluminate ion decomposition (1), i.e., the difference [ 2] between the equilibrium and effective concentrations of dissolved alumina (2). A l ( O H ) 4 − ↔ A l ( O H ) 3 c r . + O H − (1) A * = A − A e q (2) where A is the effective content of alumina in the liquor in terms of Al 2O 3, and A e q is the calculated equilibrium content of alumina in the liquor in terms of Al 2O 3. One of the first industrially applied models of gibbsite solubility in aluminate liquors is considered to be the empirical Misra model [ 3], which had sufficient convergence for narrow ranges of concentrations and temperatures; however, it did not make allowance for the content of impurities in the precipitation process [ 4], i.e., sulfate, carbonate, chloride, and oxalate anions (3). The model was validated for the liquors containing [NaOH] from 0.57 to 6.14 mol/L in the temperature range from 25 to 100 °C. ln A l = 5.7128 − 2486.7 T + 33.702 [ N a O H ] T + l n ( N a O H ) (3) where A l is the content of dissolved gibbsite in terms of aluminum, mol/L; T is the liquor temperature, K; and [ N a O H ] is the content of alkali in terms of NaOH, mol/L. The main limitation of Misra’s model is the absence of various impurity contents implemented as variables to calculate gibbsite solubility, which applies to real plant-case scenarios. Bennett applied [ 5] the group method of data handling (GMDH), i.e., a neural network, which uses the hierarchical self-organizing approach to evaluate the impurities’ effects on the equilibrium alumina content [ 6]. The model was validated for use in the ranges indicated in Table 1. Thus, a compact analytical model (4) was developed, which mathematically relates input factors to Al 2O 3 equilibrium concentration without thermodynamic assumptions. A l 2 O 3 = − 2.62 z 19 + 25.7 z 18 − 1.33 z 19 2 + 86.5 (4) where A l 2 O 3 is the content of dissolved gibbsite in terms of Al 2O 3, g/L; and z 19 z 18 are partial descriptors of the model. The aforementioned semi-empirical Rosenberg–Healey model [ 7] is considered a significant accomplishment in determining the equilibrium content of dissolved gibbsite. Based on the results of studies of semi-synthetic solutions, model (5) was validated on the actual liquors of the Worsley refinery and demonstrated sufficient convergence over wide ranges of concentrations and temperatures ( Table 2). Unlike the previously mentioned approaches, this model had a physical and chemical basis, i.e., the Debye–Hückel thermodynamic model [ 8] with a Bromley correction, as well as took into account the influence of impurity anions through empirically obtained coefficients for each anion. A = 0.96197 C 1 + 10 a 0 I 1 + I − a 3 I − a 4 I 3 e d G R T (5) where A is the content of dissolved gibbsite in terms of Al 2O 3, g/L; C is the content of sodium alkali in terms of Na 2CO 3, g/L; a 0 , a 3 , a n d a 4 are the coefficients obtained by the optimization method; and I is a modified expression for ionic strength (6). I = 0.011887 C + k 1 [ N a C l ] 58.44 + k 2 [ N a 2 C O 3 ] 105.99 + k 3 [ N a 2 S O 4 ] 142.01 + k 4 · 0.01887 [ T O C ] (6) where C is the content of sodium hydroxide in terms of Na 2CO 3, g/L; k 1 , k 2 , k 3 , and k 4 are coefficients obtained by the optimization method; [ N a C l ] , [ N a 2 C O 3 ] , and N a 2 S O 4 are the contents of chlorine in terms of NaCl, inorganic carbon in terms of Na 2CO 3, and sulfur in terms of Na 2SO 4, respectively, g/L; and TOC is total organic carbon in terms of a salt of a monacid, g/L. More complex models of gibbsite solubility have also been developed, which consider complex interionic interactions [ 9]. In the paper on gibbsite solubility in pure alkaline liquors, Li [ 10] determined the temperature-dependent parameters of the Pitzer model for N a A l ( O H ) 4 ( β 0 , β 1 , C ϕ ), as well as the binary ( θ O H − – A l ( O H ) 4 − ) and ternary ( ψ N a + – O H − – A l ( O H ) 4 − ) parameters of ion interactions [ 11], and obtained an analytical expression for the equilibrium constant (7). l g K = − 161.1495 + 4629.7868 T + 26.6959 l n T − 0.0256 T (7) Most of the above-mentioned models were developed based on the results of studies of aged synthetic/semi-synthetic liquors [ 12] from unsaturated zones. Tuned models of the equilibrium content of dissolved hydrate in alkaline liquors are used at most Bayer alumina refineries [ 13]. For alumina refineries that produce alumina by nepheline sintering, due to the specifics of their raw materials, the above-mentioned hydrate solubility models are mostly not applicable [ 14]. Thus, the ranges of sodium hydroxide concentrations, dissolved impurities, and the absence of the potassium hydroxide used as variables in the above-mentioned models make said models applicable only for a rough estimate of the equilibrium alumina content for various sintering methods [ 15] applied, for example, at the Achinsk Alumina Refinery [ 16] and the Pikalevo Alumina Refinery [ 17] ( Table 3). Some dependencies, for example, Wesolowski [ 18] and Brichkin–Fedorov [ 19], consider the effect of the potassium ion on the solubility of gibbsite in alkaline liquors. For example, Wesolowski experimentally determined the equilibrium content of gibbsite in the Na–K–Cl–OH–Al(OH) 4 system in the temperature range of 6.4–80 °C and ionic strength of 0.01–5.0 mol/kg, as well as analyzed Russell’s [ 20] data at higher concentrations. Based on Pitzer’s approach, the authors derived a parametric equation that allows estimating the equilibrium content of alumina with an accuracy of ±0.02 logarithmic units. The authors found that gibbsite solubility is virtually independent of the form of the alkaline cation (Na + or K +), while the impact of the anionic composition is only evident at ionic strengths above 0.1 mol/kg. The Brichkin–Fedorov paper proposes an original method of thermodynamic modeling of gibbsite solubility in potassium-containing liquors, based on the analysis of the nonlinearity of solubility isotherms as an indicator of the complication of the ionic composition of aluminate liquors during the formation of dimers of tetrahydroxo complexes [Al 2O(OH) 6]. It was experimentally established that the equilibrium compositions, in particular, sections of the Na 2O–K 2O–Al 2O 3–H 2O system with an accuracy of ±5%, can be determined based on the additivity principle using the data for the equilibrium in the particular systems of Na 2O–Al 2O 3–H 2O and K 2O–Al 2O 3–H 2O. At the mole fraction of K 2O = 0.33, the authors determined a change in the mechanism of decomposition of the aluminate ion due to a change in the ionic composition of the liquor. The developed model enables the calculation of the ionic composition of equilibrium liquors and predicts phase equilibria in the technologically significant region of the above-mentioned system. The above-mentioned studies make a significant contribution to understanding the impact of potassium on the equilibrium of gibbsite specifically and on the precipitation of aluminate liquors in general; however, they have limitations that hinder their application for sintering refineries; namely, they do not consider the influence of sulfate and carbonate anions on the equilibrium content of gibbsite [ 21]. Wesolowski studied the influence of chloride anions, which undoubtedly are present in some quantities in the Bayer precipitation; however, it does not play a significant role in the sintering process [ 22]. Brichkin–Fedorov’s paper makes some conclusions on the complexity of the influence of impurity anions on the equilibrium of gibbsite; however, it does not present any data on the influence of impurities on the content of dissolved gibbsite. This paper is aimed at eliminating the above-mentioned shortfalls in the understanding of the solubility of gibbsite in sodium–potassium liquors, taking into account the influence of impurities, as well as the concentration and temperature conditions of the precipitation at sintering refineries. 2. Materials and Methods Actual operating parameters of the precipitation process were used to generate the concentration and temperature ranges for further mathematical planning of the experiment ( Table 4). These variables were selected [ 23] due to the need to reduce the experiment factoring while maintaining a sufficiently descriptive ability for the model, i.e., the sodium hydroxide content is not specified directly, but is determined from a combination of the total alkali content and potassium alkali content parameters. 2.1. Planning an Experiment The Rosenberg–Healey model, as the most widely applicable industrial model, was chosen as an example for a preliminary assessment [ 24] of the linearity of the influence of the selected variables on alumina solubility. Gibbsite solubility was analyzed at various impurity and alkali concentrations, as well as the temperature: (5,6) were used to calculate the 10-point sets of Al 2O 3 equilibrium concentrations by varying various studied parameters. Figure 1a shows the dependence of the calculated equilibrium alumina content on the temperature at fixed caustic alkali contents of [Na 2O c] = 30, 50 and 70 g/L with no impurities. Figure 1b contains similar dependences in the presence of [Na 2CO 3] = 70 g/L and [Na 2SO 4] = 25 g/L as impurities. Figure 2a shows the dependences of the calculated equilibrium alumina content on the sum of dissolved impurities at the set temperatures of 50, 70, 90°C, with a sodium alkali content of [Na 2O c] = 30 g/L, and Figure 2b shows the same but with a content of [Na 2O c] = 70 g/L. A preliminary assessment of the influence of temperature, sodium hydroxide and impurities on the gibbsite solubility showed that in the selected ranges, the impact of the selected factors can be taken as linear: the calculated R 2 values exceed 0.9 in all cases. The rank order of factors Nos. 3–5 was taken as 2 in planning [ 25] the experiment. The influence of the K 2O c/R 2O c ratio in combination with different sodium hydroxide contents on the solubility of gibbsite is nonlinear according [ 19] to Brichkin–Fedorov’s paper: the rank order of factors Nos. 1 and 2 was taken as 3. A full factorial experiment (72 tests) under the above-said conditions with five selected factors was redundant [ 26]. To adjust the number of experiments while maintaining a high descriptive ability of the developed model, the D-optimal experimental design method was used [ 27], implemented in DoE module (Design of Experiments) of the STATISTICA v13.3 software (Statsoft). The licensed software is the property of the Empress Catherine II Saint Petersburg Mining University. The design was generated using the Fedorov exchange algorithm, which iteratively optimizes the design matrix by exchanging candidate points to maximize the determinant of the information (Fisher’s) matrix. The design was constructed to support the regression model (8). C A l 2 O 3 = β 0 + ∑ i = 1 5 β i x i + β 11 x 1 2 + β 22 x 2 2 + ε (8) where β 0 is the intercept, β i are linear coefficients, β 11 β 22 are quadratic coefficients for three-level factors Nos. 1–2, and ε represents random experimental error. The overall regression model comprised 10 parameters (1 intercept + 5 linear terms for all factors + 2 quadratic terms for three-level factors + an intercept for error estimation). The resulting 18-run D-optimal design achieved a 75% reduction in experimental effort compared to the full factorial design while preserving statistical efficiency for parameter estimation. The complete design matrix in coded units L1–L2 for two-level factors and L1–L3 for three-level factors is presented in Table 5. Design Matrix Efficiency Evaluation The statistical quality of the generated D-optimal design was evaluated using various criteria: D-efficiency, condition number and degrees of freedom analysis. The D-efficiency quantifies how close the design is to the theoretical D-optimal benchmark and is calculated using (9): D eff = 100 % · d e t X T X d e t X r e f T X r e f 1 p (9) where X is an 18 × 10 model matrix for the actual design, Xref represents the reference (theoretically optimal) design matrix, and p is the number of model parameters. Xref was obtained via numerical optimization ( Table 6). The condition number k of the information matrix M = X TX assesses numerical stability and multicollinearity (10). k M = λ m a x λ m i n (10) where λ m a x λ m i n are the largest and smallest eigenvalues of M = X TX, respectively. Values below 20 indicate low multicollinearity and well-conditioned parameter estimation [ 28]. The obtained value k = 8.17 confirms that the design provides stable and reliable coefficient estimates. The degree of freedom calculation based on a given methodology is (11): D e g r e e s o f f r e e d o m = n − p (11) where n is the total number of runs, and p is the number of model parameters, resulting in the residual degree of freedom value to be 8, which allows for adequate estimation of experimental error. The D-optimal design demonstrated balanced coverage of the experimental domain ( Figure 3a–c). All nine combinations of the three-level factors ( x 1 , x 2 ) were represented, while the two-level factors ( x 3 – x 5 ) showed near-equal distribution between low and high levels [ 29]. 2.2. Experiment Procedure Most researchers studying the gibbsite solubility in alkaline liquors noted [ 30] the unstable results when obtaining the concentration equilibrium from the supersaturation region by cooling a liquor supersaturated with alumina (by a process similar to precipitation). Therefore, a method for obtaining the equilibrium from the undersaturation region was selected by excessively adding dry aluminum hydroxide to a model solution containing no aluminum. Therefore, the experiments to determine the effect of chemical composition and temperature on gibbsite solubility were carried out as follows. 2.2.1. General Procedure Excessive seed was added to the prepared model solution containing no aluminum at a given temperature, i.e., 100 g Al(OH) 3 was added to 200 mL liquor. The resulting slurry was then held for 4 days (96 h) in parallel-operating steel vessels at reflux while stirring at the given temperature to achieve equilibrium. The hot slurry was filtered to separate the seed residue and analyzed by the ICP-MS (Inductively coupled plasma mass spectroscopy) method with selective duplication of the results using the volumetric method. The experiments were carried out in duplicate; the final result of the experiment was the arithmetic mean value if the duplicate results differed by no more than 5% rel. 2.2.2. Preparation of Reaction Slurry The required weighed samples of chemically pure sodium hydroxide, chemically pure potassium hydroxide, chemically pure sodium sulfate, and chemically pure sodium carbonate were calculated to achieve the desired results for the selected parameters. It was experimentally established that the selected seed hydrate did not show significant sorption capacity; to adjust the solution composition, the weighed samples were increased by 2–3%. Eighteen model solutions were prepared; the solutions were analyzed for total potassium, sodium, carbonate, and sulfate anion content using ICP-MS to ensure accuracy. 2.2.3. Experimental Setup The H.E.L. Auto-MATE laboratory setup (H.E.L Group Limited, London, UK) was used to perform the experiments, which allows for 4 experiments to be carried out simultaneously at a given temperature, with stirring and vapor condensation to avoid slurry concentrating [ 31] ( Figure 4). 3. Results Table 7 presents the obtained results of the chemical composition of the studied liquors. Due to the model liquor preparation in a proper manner, the discrepancies between the planned and actual contents of the model liquors, studied by ICP-MS analysis, were negligibly low; values marked as <0.01 were treated as zero. The mathematical expression for the alumina solubility model in the alkaline sodium–potassium liquors containing carbonate and sulfate anions as impurities is presented below (12). The model parameters have been adapted to be presented in a form familiar to alumina refineries operating in the CIS (Commonwealth of Independent States). [ A l 2 O 3 e q ] = 1.6451 · [ N a 2 O c ] + 1.9863 · [ K 2 O c ] 1 + 10 a 1 I 1 + I − a 2 I − a 3 I 1.5 e d G R T (12) where N a 2 O c is sodium alkali content, g/L; K 2 O c is potassium alkali content, g/L; a 1 , a 2 , and a 3 are the coefficients obtained by the optimization method; I is a modified expression of the ionic strength of the liquor (9), dG is dissolution free energy, J/mol; R is universal gas constant, J/K·mol; and T is temperature, K. The Gibbs energy dG value was taken as 30.96 kJ/mol in accordance with the Rosenberg–Healy calculations. No attempts were made to calculate or fit the dG value due to the use of only two different temperature points as the test set. The coefficients for sodium [Na 2O c] and potassium [K 2O c] oxides were calculated by converting to (OH −) in (12) and (Al(OH) 4−) in (13) using molar masses, assuming the complete dissociation of the aforementioned oxides in aqueous liquor, similar to the work by Rosenberg and Healy. The coefficients a 1–a 3 describe the behavior of ion activity: a 1 is a composite constant created by combining the parameters for aluminate (Al(OH) 4−)and hydroxyl (OH −) ions; it effectively represents the slope of the activity coefficient ratio between these two species; a 2 is a linear correction term, analogous to the Bromley adjustment, used to account for specific ion interactions at moderate to high concentrations; a 3 is an empirical virial coefficient for the term I 1.5 , necessary to maintain model accuracy at extreme ionic strengths. The expression for ionic strength was modified to account for alkalis in the form of oxides (13). The influence of impurity carbonates and sulfates was taken into account through a correction factor before the sum, and when calculating the ionic strength, also similar to the work by Rosenberg and Healy. The correction factors b1, b2 before sodium carbonate and sulfate are reduced contributions of these salts to the ionic strength of the solution, reflecting the assumption that only partial dissociation of them takes place in complex aluminate liquors [ 4]. I = 0.03225 · [ N a 2 O c ] + 0.02128 · [ K 2 O c ] + b 1 · N a 2 C O 3 106 + b 2 · [ N a 2 S O 4 ] 142 (13) where b 1 b 2 are the correction factors obtained by the optimization method; [ N a 2 C O 3 ] is the content of carbonates in terms of sodium carbonate, g/L; and [ N a 2 S O 4 ] is sulfate content in terms of sodium sulfate, g/L. To obtain the coefficients a 1 , a 2 , a 3 b 1 , b 2 , the Levenberg–Marquardt method was applied. The objective function for optimization [ 32] is formulated as (14): f = ∑ i = 1 n A m e a s i − A p r e d i 2 (14) where A m e a s i is the experimentally determined alumina content, g/L; and A p r e d i is the alumina content calculated by the developed model, g/L. Table 8 presents the calculated coefficients. 4. Discussion 4.1. Verification of the Model Using Actual Plant Liquor Compositions To verify the developed model, actual compositions and temperatures of the refinery’s spent liquors from the precipitation area were requested from the RUSAL Achinsk alumina refinery. The liquor compositions were reproduced under laboratory conditions and adjusted by evaporation [ 36] and the addition of various additives [ 37] to simulate various process operating conditions in the precipitation area. The presence of impurities, which are not presented in the table, in the refinery are typically considered negligible (<1 g/L); therefore, they are not considered [ 38]. Further, similar to the previous procedure, the experiments were conducted to determine the equilibrium content of dissolved alumina. Table 9 presents the results of the analysis of the spent liquor samples. Figure 7 shows the model errors for the test and verification sample sets. The root mean square error (RMSE) for the verification sample set is 0.71 g/L Al 2O 3 and is close to that for the test sample set, thus proving the sufficient descriptive ability of the model within the studied ranges [ 39] ( Table 4). Table 10 shows both test and validation sample sets and actual-predicted model errors. A study of the model’s reliability outside the studied ranges was not conducted due to the low technological value of such a study: all the process parameters were within the studied ranges. Due to the absence of statistically significant correlations between the relative error and the parameters, the authors postulate that these discrepancies occur from random analytical uncertainties in the determination of dissolved alumina and the limitations of the model, which does not fully account for the solution-phase speciation and physicochemical interactions of impurities. 4.2. Developed Model Results Comparison to Existing Models Given the previously described lack of existing gibbsite solubility models that account for the full ionic composition and temperatures of liquors used in alumina sintering processes, it was not possible to carry out a comprehensive comparison of the entire test and validation data sets across the existing models. Given the semi-empirical nature and relatively narrow applicable ranges of this work’s model, the overall RMSE of the test data set points compared across various models indicated a good convergence over the test models ( Table 11). 5. Conclusions The analysis of the available studies on gibbsite solubility in alkaline liquors revealed that a suitable mathematical model does not exist for describing the equilibrium aluminum content in the precipitation processes at sintering alumina refineries that consider the concentration and temperature conditions of the process. The experimental study of gibbsite equilibrium in sodium–potassium aluminate liquors was preceded by the study of the influence of various liquor parameters on the gibbsite solubility using the Rosenberg–Healey model, so that the D-optimal method could be later used to develop the design matrix. Based on the Rosenberg–Healey methodology, a mathematical model was developed that considers the presence of potassium alkali and sodium sulfate and carbonate impurities in amounts typical for sintering precipitation. The developed model demonstrated a low root mean square error with a verification data set, which is based on the refinery’s liquor compositions (RMSE = 0.71 g/L), making the model suitable for process calculations. A cross-model comparison of the selected data points showed no significant discrepancies between this work and existing gibbsite solubility models. Author Contributions Conceptualization, T.E.L. and N.V.T.; Methodology, T.E.L.; Software, N.V.T.; Validation, T.E.L., N.V.T.; Formal Analysis, T.E.L.; Investigation, N.V.T.; Resources, T.E.L.; Data Curation, N.V.T.; Writing—Original Draft Preparation, N.V.T.; Writing—Review & Editing, T.E.L.; Visualization, N.V.T.; Supervision, T.E.L.; Project Administration, N.V.T.; Funding Acquisition, T.E.L. All authors have read and agreed to the published version of the manuscript. Funding This research received no external funding. Data Availability Statement The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Conflicts of Interest The authors declare no conflicts of interest. Abbreviations The following abbreviations are used in this manuscript: ICP-MS Inductively coupled plasma mass spectroscopy CIS Commonwealth of Independent States RMSE Root mean square error R 2O Sum of sodium and potassium oxides References Golubev, V.O.; Chistyakov, D.G.; Brichkin, V.N.; Postika, M.F. Population balance of aluminate solution decomposition: Physical modelling and model setup. Tsvetnye Met. 2019, 2019, 75–81. [] [ CrossRef] Li, T.S.; Livk, I.; Ilievski, D. Supersaturation and Temperature Dependency of Gibbsite Growth in Laminar and Turbulent Flows. J. Cryst. Growth 2003, 258, 409–419. [] [ CrossRef] Misra, C. Solubility of Alumina Trihydrate in Sodium Hydroxide Solution. Chem. Ind. 1970, 9, 619–623. [] Nortier, P.; Chagnon, P.; Lewis, A.E. Modelling the Solubility in Bayer Liquors: A Critical Review and New Models. Chem. Eng. Sci. 2011, 66, 2596–2605. [] [ CrossRef] Bennett, F.R.; Crew, P.; Muller, J.K. A GMDH Approach to Modelling Gibbsite Solubility in Bayer Process Liquors. Int. J. Mol. Sci. 2004, 5, 101–109. [] [ CrossRef] Mustafaev, A.S.; Grabovskiy, A.Y.; Sukhomlinov, V.S.; Shtoda, E.V. Technology for Monitoring the Surface Emission Inhomogeneity in Plasma Electronics Devices. J. Appl. Phys. 2024, 136, 203303. [] [ CrossRef] Rosenberg, S.P.; Healy, S.J. A Thermodynamic Model for Gibbsite Solubility in Bayer Liquors. In Proceedings of the 4th International Alumina Quality Workshop, Darwin, Australia, 2–7 June 1996. [] Costa Reis, M. Ion Activity Models: The Debye-Hückel Equation and Its Extensions. ChemTexts 2021, 7, 9. [] [ CrossRef] Königsberger, E.; May, P.M.; Hefter, G. Comprehensive Model of Synthetic Bayer Liquors. Part 3. Sodium Aluminate Solutions and the Solubility of Gibbsite and Boehmite. Monatshefte Für Chem.-Chem. Mon. 2006, 137, 1139–1149. [] [ CrossRef] Li, X.B.; Yan, L.; Zhou, Q.S.; Liu, G.H.; Peng, Z.H. Thermodynamic Model for Equilibrium Solubility of Gibbsite in Concentrated NaOH Solutions. Trans. Nonferrous Met. Soc. China Engl. Ed. 2012, 22, 447–455. [] [ CrossRef] Povarov, V.G.; Cheremisina, O.V.; Alferova, D.A.; Fedorov, A.T. Determination of the Activity Coefficients of Components in a Di-2-Ethylhexylphosphoric Acid–n-Hexane Binary System Using Gas Chromatography. Chemistry 2025, 7, 92. [] [ CrossRef] Povarov, V.G.; Cheremisina, O.V.; Alferova, D.A. Determination of Energy Interaction Parameters for the UNIFAC Model Based on Solvent Activity Coefficients in Benzene–D2EHPA and Toluene–D2EHPA Systems. Chemistry 2025, 8, 2. [] [ CrossRef] Brichkin, V.N.; Fedorov, A.T. Thermodynamic Modelling of Ion Equilibria in the Na 2O–Al 2O 3–H 2O System with Gibbsite. Tsvetnye Met. 2022, 74–81. [] [ CrossRef] Chistyakov, D.G.; Golubev, V.O.; Sizyakov, V.M.; Brichkin, V.N. Raw Material Composition at Rusal Achinsk and Its Impact on the Production Indicators. Tsvetnye Met. 2020, 10, 27–34. [] [ CrossRef] ElDeeb, A.B.; Brichkin, V.N.; Bertau, M.; Savinova, Y.A.; Kurtenkov, R.V. Solid State and Phase Transformation Mechanism of Kaolin Sintered with Limestone for Alumina Extraction. Appl. Clay Sci. 2020, 196, 105771. [] [ CrossRef] Sizyakov, V.M.; Brichkin, V.N.; Kurtenkov, R.V.; Maksimova, R.I. Synthesis, Properties and Applications of Complex Calcium Aluminates and Hydroaluminates. Non-Ferr. Met. 2025, 1, 10–17. [] [ CrossRef] Sizyakov, V.M. Chemical and Technological Mechanisms of a Alkaline Aluminum Silicates Sintering and a Hydrochemical Sinter Processing. J. Min. Inst. 2016, 217, 102–112. [] Wesolowski, D.J. Aluminum Speciation and Equilibria in Aqueous Solution: I. The Solubility of Gibbsite in the System Na-K-CI-OH-Al (OH) 4 from 0 to 100°C. Geochim. Cosmochim. Acta 1991, 56, 1065–1091. [] [ CrossRef] Brichkin, V.N.; Fedorov, A.T. Calculation and Experimental Determination of Equilibrium Composition of Liquors in Certain Instances of the Na 2O–K 2O–Al 2O 3–H 2O System. Tsvetnye Met. 2022, 33–38. [] [ CrossRef] Russell, A.S.; Edwards, J.D.; Taylor, C.S. Solubility and Density of Hydrated Aluminas in NaOH Solutions. JOM 1955, 7, 1123–1128. [] [ CrossRef] Healy, S.J. Bayer Process Impurities and Their Management. In Smelter Grade Alumina from Bauxite; Raahauge, B.E., Williams, F.S., Eds.; Springer Series in Materials Science; Springer International Publishing: Cham, Switzerland, 2022; Volume 320, pp. 375–426. [] Bai, G.; Teng, W.; Wang, X.; Qin, J.; Xu, P.; Li, P. Alkali Desilicated Coal Fly Ash as Substitute of Bauxite in Lime-Soda Sintering Process for Aluminum Production. Trans. Nonferrous Met. Soc. China 2010, 20, s169–s175. [] [ CrossRef] Özdemir, A.; Turkoz, M. Development of a D-Optimal Design-Based 0–1 Mixed-Integer Nonlinear Robust Parameter Design Optimization Model for Finding Optimum Design Factor Level Settings. Comput. Ind. Eng. 2020, 149, 106742. [] [ CrossRef] Koshevoy, N.D.; Dergachov, V.A.; Pavlyk, H.V.; Zabolotnyi, O.V.; Koshevaya, I.I.; Kostenko, Е.М. The Method of Building Plans of Multifactorial Experiments with Minimal Number of Factor Levels Measurements and Optimal by Cost (Time) Expenses. Radio Electron. Comput. Sci. Control 2020, 4, 55–64. [] [ CrossRef] Pryakhin, E.I.; Dranova, A.Y. Application of Direct Laser Marking of Products Made of Different Types of Alloys Using Ultra-Dense Matrix Nanobar-Code. CIS Iron Steel Rev. 2025, 30, 92–98. [] [ CrossRef] Maksarov, V.V.; Sinyukov, M.S. Methods of Ensuring the Quality of Assembly of Non-Removable Joints from Dissimilar Materials. Int. J. Eng. 2026, 39, 1191–1199. [] [ CrossRef] Akra, U.P.; Isaac, A.A.; Francis, R.E.; Tim, I.M. D-Optimality-Based Approach for Solving Second-Order Response Surface Methodology Problems. Sci. Technol. Sci. Soc. 2025, 2, 94–104. [] [ CrossRef] Myers, R.H.; Montgomery, D.C.; Anderson-Cook, C. Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2016. [] Wang, J.; Xie, W.; Ryzhov, I.O. Algorithms for Budget-Constrained D-Optimal Design. Math. Oper. Res. 2024, 1–61. [] [ CrossRef] Pan, H.B.; Chen, Z.F.; Darvell, B.W. Solubility of Sparingly-Soluble Electrolytes—A New Approach. Anal. Methods 2010, 2, 973. [] [ CrossRef] Kuzmin, M.I.; Ponomareva, A.I.; Gerasimov, R.V. Approach to Adaptive Management of Well Stock with Electric Vane Pump Installation. Oil Ind. 2025, 130–134. [ CrossRef] Brigadnov, I.A. Numerical Method for Solving of Ill-Conditioned Boundary Value Problems in Nonlinear Elasticity. Acta Mech. 2023, 234, 1293–1303. [] [ CrossRef] Silvanovich, O.V.; Shirokov, N.A. Inverse Theorem of Approximation by Entire Functions of Exponential Type. J. Math. Sci. 2026, 295, 440–448. [] [ CrossRef] Brichkin, V.N.; Gumenyuk, A.M.; Panov, A.V.; Suss, A.G. Determination of the Optimal Technological Conditions of Processing of the Alkali Alumosilicate. In Proceedings of the Scientific-Practical Conference “Research and Development—2016”; Anisimov, K.V., Dub, A.V., Kolpakov, S.K., Lisitsa, A.V., Petrov, A.N., Polukarov, V.P., Popel, O.S., Vinokurov, V.A., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 639–647. [] Kostoglotov, A.A.; Kalienko, I.V.; Kornev, A.S.; Lazarenko, S.V. Synthesis of Algorithms for Compensating Systematic Errors Based on the Construction of Polynomial Mathematical Models of Radar Measurements. Meas. Tech. 2022, 65, 290–296. [] [ CrossRef] Dembowski, M.; Graham, T.R.; Reynolds, J.G.; Clark, S.B.; Rosso, K.M.; Pearce, C.I. Influence of Soluble Oligomeric Aluminum on Precipitation in the Al–KOH–H 2O System. Phys. Chem. Chem. Phys. 2020, 22, 24677–24685. [] [ CrossRef] [ PubMed] Delegard, C.; Pearce, C.; Dembowski, M.; Snyder, M.; Leavy, I.; Baum, S.; Fountain, M. Aluminum Hydroxide Solubility in Sodium Hydroxide Solutions Containing Nitrite/Nitrate of Relevance to Hanford Tank Waste; Pacific Northwest National Laboratory (PNNL): Richland, WA, USA, 2018; PNNL-27969; RPT-OSIF-003; 1660940. [] Mikhnev, A.D.; Tregubova, E.V. Estimation of Errors of Sampling, Sample Preparation, and Sample Analysis in the Monitoring Scheme of Hydrochemistry Shop of the Achinsk Alumina Refinery. Russ. J. Non-Ferr. Met. 2007, 48, 6–9. [] [ CrossRef] Westad, F.; Marini, F. Validation of Chemometric Models—A Tutorial. Anal. Chim. Acta 2015, 893, 14–24. [] [ CrossRef] [ PubMed] Figure 1. Dependences of the equilibrium content of alumina on the temperature for different caustic concentrations, calculated based on the Rosenberg–Healy model: ( a)—without impurities, ( b)—containing [Na 2CO 3] = 70 g/L and [Na 2SO 4] = 25 g/L as impurities. Figure 1. Dependences of the equilibrium content of alumina on the temperature for different caustic concentrations, calculated based on the Rosenberg–Healy model: ( a)—without impurities, ( b)—containing [Na 2CO 3] = 70 g/L and [Na 2SO 4] = 25 g/L as impurities. Figure 2. Dependences of the equilibrium content of alumina on the total impurities at different temperatures, calculated based on the Rosenberg–Healy model: ( a)—with caustic content [Na 2O c] = 30 g/L, ( b)—with [Na 2O c] = 70 g/L. Figure 2. Dependences of the equilibrium content of alumina on the total impurities at different temperatures, calculated based on the Rosenberg–Healy model: ( a)—with caustic content [Na 2O c] = 30 g/L, ( b)—with [Na 2O c] = 70 g/L. Figure 3. The 3d-projections of obtained D-optimal plan: ( a)—for factors Nos. 1–3, ( b)—for factors Nos. 3–4, ( c)—for factors Nos. 1,2,4, the numbers are the order Nos. of runs. Figure 3. The 3d-projections of obtained D-optimal plan: ( a)—for factors Nos. 1–3, ( b)—for factors Nos. 3–4, ( c)—for factors Nos. 1,2,4, the numbers are the order Nos. of runs. Figure 4. Diagram of the HEL laboratory setup: 1—300 cm 3 stainless steel vessel, 2—fluoroplastic magnetic stirrer, 3—reflux condenser for vapor condensation, 4—setup body with separate heating devices for each vessel. Figure 4. Diagram of the HEL laboratory setup: 1—300 cm 3 stainless steel vessel, 2—fluoroplastic magnetic stirrer, 3—reflux condenser for vapor condensation, 4—setup body with separate heating devices for each vessel. Figure 5. Convergence of the model and measured (actual) values for dissolved alumina. Figure 5. Convergence of the model and measured (actual) values for dissolved alumina. Figure 6. Distribution of the model residues from the measured dissolved alumina for the test data set. Figure 6. Distribution of the model residues from the measured dissolved alumina for the test data set. Figure 7. Convergence of the model and actual values of alumina solubility using test and verification sample sets. Figure 7. Convergence of the model and actual values of alumina solubility using test and verification sample sets. Figure 8. Convergence of potassium-free models, adapted from [ 7, 10] across selected data points of the test data set. Figure 8. Convergence of potassium-free models, adapted from [ 7, 10] across selected data points of the test data set. Figure 9. Convergence of potassium-included impurities-free models, adapted from [ 18, 19] across selected data points of the test data set. Figure 9. Convergence of potassium-included impurities-free models, adapted from [ 18, 19] across selected data points of the test data set. Table 1. Ranges of applicability of the Bennett model. Table 1. Ranges of applicability of the Bennett model. Parameter Temp, °C C (Na 2O), g/L Na 2CO 3, g/L NaCl, g/L Na 2SO 4, g/L NaF, g/L TOC, g/L Range 55–95 110–160 0–70 0–13 0–4 0–7 0–10 Table 2. Range of applicability of the Rosenberg–Healey model. Table 2. Range of applicability of the Rosenberg–Healey model. Parameter Temp., °C C (Na 2CO 3), g/L Na 2CO 3, g/L NaCl, g/L Na 2SO 4, g/L TOC, g/L Range 55–95 °C 100–300 0–50 0–70 0–55 0–33 Table 3. Comparison of average process parameters of aluminate liquors in the precipitation area at the Bayer refineries and sintering refineries processing nepheline-limestone mixture. Table 3. Comparison of average process parameters of aluminate liquors in the precipitation area at the Bayer refineries and sintering refineries processing nepheline-limestone mixture. Parameter, g/L Bayer Sintering Al 2O 3120–130 60–65 Na 2O total 155–165 50–55 Na 2O caustic 135–145 40–45 K 2O total <3 15–20 K 2O caustic <1 10–15 Carbonate as Na 2CO 325–30 30–35 Sulfate as Na 2SO 46–8 10–15 Table 4. Range of parameters for developing a model based on actual data of the sintering refinery. Table 4. Range of parameters for developing a model based on actual data of the sintering refinery. No. Factor Designation Minimum Maximum 1 Total alkali content: Na 2Oc + K 2Oc R 2Oc, g/L 40 80 2 Mass fraction of potassium alkali K 2Oc, wt.% 0 30 3 Liquor temperature T, °C 50 85 4 Sulfate in terms of sodium sulfate Na 2SO 4, g/L 0 25 5 Carbonate in terms of carbonate anion Na 2CO 3, g/L 0 58 Table 5. Design matrix obtained by the D-optimal experiment planning method. Table 5. Design matrix obtained by the D-optimal experiment planning method. No. of Run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1. R 2O c, g/L L2 L1 L2 L2 L3 L1 L3 L3 L3 L1 L1 L1 L2 L3 L1 L2 L2 L3 2. K 2O c, % L2 L1 L1 L3 L2 L3 L1 L2 L1 L3 L2 L2 L2 L3 L1 L1 L3 L3 3. T, °C L2 L1 L2 L1 L1 L2 L1 L1 L2 L1 L1 L2 L2 L2 L2 L1 L1 L2 4. SO 42−, g/L L2 L2 L2 L1 L1 L1 L2 L2 L1 L2 L1 L2 L1 L2 L1 L1 L2 L1 5. CO 3−2, g/L L1 L1 L2 L2 L1 L1 L2 L1 L2 L2 L2 L2 L2 L1 L1 L1 L1 L2 Table 6. D-efficiency calculation results. Table 6. D-efficiency calculation results. Parameter det X TX det X refTX refD-Efficiency Value ୮.୯୨·୍ଠ ୧୦ 6୧.୧୨ ୍ଠ·୧୦ 797.8% Table 7. Chemical composition of model slurries after the experiment to determine the equilibrium concentration. Table 7. Chemical composition of model slurries after the experiment to determine the equilibrium concentration. No. Temp, C Na 2O c, g/L K 2O c, g/L Na 2CO 3, g/L Na 2SO 4, g/L Al 2O 3, g/L 1 50 41.57 <0.01 <0.01 23.59 8.81 2 50 70.66 12.47 57.55 <0.01 25.15 3 50 43.66 18.71 <0.01 <0.01 15.94 4 50 29.07 12.46 57.49 23.59 12.49 5 50 43.68 18.72 <0.01 23.62 17.37 6 50 83.09 <0.01 57.55 23.62 24.50 7 50 62.30 <0.01 57.49 <0.01 17.08 8 50 35.36 6.24 <0.01 <0.01 7.43 9 50 70.70 12.47 <0.01 23.62 24.21 10 85 52.12 9.19 <0.01 23.60 36.54 11 85 34.72 6.13 57.50 23.60 27.07 12 85 81.88 <0.01 <0.01 <0.01 51.14 13 85 28.60 12.25 57.50 <0.01 27.12 14 85 61.30 <0.01 57.55 23.60 40.38 15 85 52.08 9.19 57.50 <0.01 39.86 16 85 57.31 24.55 <0.01 23.60 53.20 17 85 40.92 <0.01 <0.01 <0.01 18.51 18 85 57.27 24.55 57.55 <0.01 55.54 Table 8. Coefficients obtained by the optimization method. Table 8. Coefficients obtained by the optimization method. Parameter a 1a 2a 3b 1b 2Value −9.2406 −0.8908 0.2174 2.4977 1.9512 Table 9. Results of determining the compositions of the samples of adjusted liquors for the model verification after the experiment. Table 9. Results of determining the compositions of the samples of adjusted liquors for the model verification after the experiment. No. Temp, C Na 2O c, g/L K 2O c, g/L Na 2CO 3, g/L Na 2SO 4, g/L Al 2O 3, g/L 1 51.6 36.61 12.18 35.34 8.4 15.12 2 51.6 42.09 12.07 35.11 11.89 16.44 3 59.2 36.49 16.74 43.34 8.56 19.70 4 59.2 54.74 25.11 65.01 12.84 30.76 5 67.6 36.61 12.18 35.34 8.4 23.09 6 67.6 42.09 12.07 35.11 11.89 24.03 7 77.6 36.49 16.74 43.34 8.56 31.24 8 77.6 54.74 25.11 65.01 12.84 48.12 Table 10. Model residues (errors): letter t in caption stands for test data set, v for validation data set. Table 10. Model residues (errors): letter t in caption stands for test data set, v for validation data set. No. Measured, g/L Predicted, g/L Model Error, g/L Relative Error, % 1t 8.81 9.31 −0.50 −5.7 2t 25.15 25.68 −0.53 −2.1 3t 15.94 15.45 0.49 3.1 4t 12.49 12.34 0.15 1.2 5t 17.37 17.12 0.25 1.4 6t 24.50 25.51 −1.01 −4.1 7t 17.08 18.12 −1.04 −6.1 8t 7.43 7.47 −0.04 −0.5 9t 24.21 24.54 −0.33 −1.4 10t 36.54 38.35 −1.81 −5.0 11t 27.07 26.93 0.14 0.5 12t 51.14 52.00 −0.86 −1.7 13t 27.12 26.55 0.57 2.1 14t 40.38 40.75 −0.37 −0.9 15t 39.86 41.11 −1.25 −3.1 16t 53.20 55.63 −2.43 −4.6 17t 18.51 18.53 −0.02 −0.1 18t 55.54 57.84 −2.30 −4.1 1v 15.12 14.19 0.93 6.2 2v 16.44 16.09 0.35 2.1 3v 19.70 20.03 −0.33 −1.7 4v 30.76 31.93 −1.17 −3.8 5v 23.09 21.71 1.38 6.0 6v 24.03 24.55 −0.52 −2.2 7v 31.24 30.77 0.47 1.5 8v 48.12 48.60 −0.48 −1.0 Table 11. Combined cross-model RMSE for selected test data set points. Table 11. Combined cross-model RMSE for selected test data set points. Potassium-Free Impurities-Included Solubility Models This Work Rosenberg–Healy Li 0.48 0.71 0.74 Impurities-free potassium-includedsolubilitymodelsThis Work Brichkin–Fedorov Wesolowski 0.35 0.68 0.54 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Share and Cite MDPI and ACS Style Litvinova, T.E.; Tuleshov, N.V. Modeling of Gibbsite Solubility in Na +-K +-OH −-Al(OH) 4−-SO 42−-CO 32− System Applied to Precipitation at Sintering Alumina Refineries. Metals 2026, 16, 633. https://doi.org/10.3390/met16060633 AMA Style Litvinova TE, Tuleshov NV. Modeling of Gibbsite Solubility in Na +-K +-OH −-Al(OH) 4−-SO 42−-CO 32− System Applied to Precipitation at Sintering Alumina Refineries. Metals. 2026; 16(6):633. https://doi.org/10.3390/met16060633 Chicago/Turabian Style Litvinova, Tatiana E., and Nickolai V. Tuleshov. 2026. "Modeling of Gibbsite Solubility in Na +-K +-OH −-Al(OH) 4−-SO 42−-CO 32− System Applied to Precipitation at Sintering Alumina Refineries" Metals 16, no. 6: 633. https://doi.org/10.3390/met16060633 APA Style Litvinova, T. E., & Tuleshov, N. V. (2026). Modeling of Gibbsite Solubility in Na +-K +-OH −-Al(OH) 4−-SO 42−-CO 32− System Applied to Precipitation at Sintering Alumina Refineries. Metals, 16(6), 633. https://doi.org/10.3390/met16060633 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details . 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Modeling of Gibbsite Solubility in Na+-K+-OH−-Al(OH)4−-SO42−-CO32− System Applied to Precipitation at Sintering Alumina Refineries
