519.873 Modeling and optimization the process of liquid mechanical mixing in a vertical vessel
Karpushkin S. V. (Tambov State Technical University)
VERTICAL VESSEL, MECHANICAL MIXING OF LIQUID, NAVIER-STOKES EQUATIONS, TURBULENCE MODEL, LABORATORY AND INDUSTRIAL EXPERIMENT, MIXING EFFICIENCY CRITERION, MULTIFACTOR COMPUTATIONAL EXPERIMENT
doi: 10.18698/2309-3684-2025-4-148172
A mathematical model of the mechanical mixing process (MMP) of a homogeneous liquid in a vertical capacitive apparatus is proposed, including the Reynolds-averaged Navier-Stokes equations and equations of semiempirical RNG k-ε model of turbulence in a cylindrical coordinate system. The conclusion about the adequacy of the proposed model is justified by comparing the mixing power consumption values, determined by calculations' results of the velocity field in stirred liquid and the results of laboratory and industrial experiments: measurements supply voltage for electric motor of a mechanical mixing device (MMD) and the current consumption when stirring a homogeneous liquid and rotating the stirrer in an empty apparatus. The problem's formulation of optimizing the MMD's parameters provides for the choice of values the diameter and height of the agitator blade, the height of its installation above the bottom of the apparatus and the rotation frequency of its shaft, minimizing the dispersion of the velocity vector's medium length of the mixed liquid. It is proposed to determine the values of MMD design's additional parameters by iteration at the optimal values of the above-mentioned basic parameters. The basis for the development of a numerical algorithm for solving the problem was the method of planning a multifactorial computational experiment. Examples of solving problems of optimizing design parameters and operating mode of MMDs of laboratory and industrial devices are given. The optimal blade widths of the most common mechanical agitators exceed those recommended by the NIIHIMMASH Guidance Document by 1.7-2.5 times. Based on the results of solving the problem of optimizing the MMD's parameters of the industrial apparatus of JSC Pigment, Tambov, modifications of its design were proposed, which made it possible to eliminate the formation of deposits on the walls of the apparatus, reduce by 10% the duration of the repulpation stage of the copper phthalocyanine paste, and reduce by 16% the cost of mixing power.
[1] Garbaruk A.V. Lektsii po techeniyu vyazkoy zhidkosti i modelyam turbu-lentnosti: metody rascheta turbulentnykh techeniy [Lectures on viscous fluid flow and turbulence models: methods for calculating turbulent flows]. St. Petersburg, SPbPU, 2010, 127 p.
[2] Braginskiy L.N., Begachev V.I., Barabash V.M. Peremeshivaniye v zhidkikh sredakh. Fizicheskiye osnovy i inzhenernyye metody rascheta [Mixing in liquid media. Physical foundations and engineering methods of calculation]. Leningrad, Khimiya, 1984, 336 p.
[3] RD 26-01-90-85. Mekhanicheskiye peremeshivayushchiye ustroystva. Metod rascheta. Rukovodyashchiy normativnyy dokument. Vved [Mechanical mixing devices. Calculation method. Guiding normative document.Introduction.]. 1986-01-01. Leningrad, LenNIIkhimmash, 1985, 257 p.
[4] Korkodinov YA.A. The review of set of k–ε models for modeling turbulence. Vestnik Permskogo nauchno-issledovatel'skogo politekhnicheskogo universiteta [Bulletin of the Perm Scientific Research Polytechnic University], 2013, no. 2, pp. 5-16.
[5] Minibayeva L.R., Mukhametzyanova A.G., Klinov A.V. Chislennoye modeliro-vaniye gidrodinamicheskoy struktury potoka v apparatakh s peremeshivayushchimi ustroystvami [Numerical modeling of the hydrodynamic structure of flow in devices with mixing devices]. Vestnik Kazanskogo tekhnologicheskogo universiteta [Bulletin of Kazan Technological University], 2008, no. 6, pp. 191-198.
[6] Minibayeva L.R., Mukhametzyanova A.G., Klinov A.V. Modeli turbulentnosti dlya adekvatnogo opisaniya polya skorosti v apparatakh s peremeshivayushchimi [Turbulence models for an adequate description of the velocity field in apparatus with stirring]. Vestnik Kazanskogo tekhnologicheskogo universiteta [Bulletin of Kazan Technological University], 2010, no. 9, pp. 469-477.
[7] Yang F., Zhou S. Free surface turbulent flow in an unbaffled stirred tank: Detached eddy simulation and VOF Study. Chemical and Biochemical Engineering Quarterly, 2015, vol. 29, iss. 3, рp. 395-403.
[8] Liangchao L., Bin X. Numerical simulation of hydrodynamics in an uncovered unbaffled stirred tank. Chemical Papers, 2017, vol. 71, iss. 10, pр. 1863-1875.
[9] Hirsch C. Numerical computation of internal and external flows. Journal of Fluid Mechanics, 1991, 225, рp. 691-695.
[10] Voytovich R., Lipin A.A., Lipin A.G. Matematicheskoye modelirovaniye gidrodinamiki smesitelya s ekstsentricheski raspolozhennoy meshalkoy [Mathematical modeling of the hydrodynamics of a mixer with an eccentrically located stirrer]. Izvestiya vysshikh uchebnykh zavedeniy. Seriya: Khimiya i khimicheskaya tekhnologiya [News of higher educational institutions. Series: Chemistry and chemical technology], 2015, vol. 58, no. 11, pp. 83-86.
[11] Minibayeva L.R., Mukhametzyanova A.G., Klinov A.V. O vliyanii konstruktsii peremeshivayushchego ustroystva na kharakteristiki potoka v apparatakh s mnogoyarusnymi meshalkami [On the influence of the design of the mixing device on the flow characteristics in devices with multi-tier mixers]. Vestnik Kazanskogo tekhnologicheskogo universiteta [Bulletin of Kazan Technological University], 2010, no. 11, pp. 201-210.
[12] Ostrovskaya E.N., PolyakovaT.V. Raschetikonstruirovaniyekhimicheskikhapparatovsmeshalkami: uchebnoyeposobiye [Calculation and design of chemical apparatuses with stirrers: a textbook]. Kazan', KSTU, 2006, 119 p.
[13] OpenFOAM – otkrytaya integriruyemaya platforma dlya chislennogo modelirovaniya zadach mekhaniki sploshnykh sred [Electronic resource]. Access mode: http://fsweb.info/caecad/openfoam.html. (Date of access: 11/15/2024).
[14] Kurakhtina G.S. Obshchaya elektrotekhnika: uchebnoye posobiye dlya studentov tekhnicheskikh spetsial'nostey vuzov regiona [General electrical engineering: a textbook for students of technical specialties at universities in the region]. Petropavlovsk-Kamchatskiy, KamchatGTU, 2007, 144 p.
[15] Panteleyev, A.V., Letova T.A. Metody optimizatsii v primerakh i zadachakh [Optimization methods in examples and problems]. St. Petersburg, Lan Publishing House, 2015, 511 p.
[16] Sidnyayev N.I. Teoriya planirovaniya eksperimenta i analiz statisticheskikh dannykh: uchebnoye posobiye [The theory of experimental planning and statistical data analysis: a textbook]. Moscow, Publishing URAIT, 2012, 399 p.
Карпушкин С.В. Моделирование и оптимизация процесса механического перемешивания жидкости в вертикальной емкости. Математическое моделирование и численные методы, 2025, № 4, с. 148–172.
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