Mathematical modelling of crystallization of polymer solutions




homogenization, integral transformations, iterative method, crystallization, mathematical model, nonlinear boundary value problem


The processes of homogenization and crystallization of polymer solutions in cylindrical pipes are considered, which are described by the convective-diffusion equation with respect to the solution temperature and kinetic equations with respect to homogenization and crystallization of the polymer known as the thermokinetic nonlinear boundary value problem. A numerical-analytical iterative method for solving this problem is proposed, which consists of stepwise obtaining solutions of kinetic equations with respect to homogenization and crystallization of polymer solutions depending on the solution temperature and obtaining a solution of the convective-diffusion problem with respect to melt temperature. The accuracy of the obtained solution is determined by the norm of the difference between two adjacent iterations. The value of the crystallization coefficient, which is close to unity, determines the length of the dosing zone and the transition to the next zone – the flow of homogenized polymer into the distribution head of the extruder. The results of mathematical modelling are given.

Author Biography

Kyryl Zelensky, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv

Associate professor, Doctor of Technical Sciences, a professor at the Department of Biomedical Cybernetics of the Faculty of Biomedical Engineering of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine.


N.A. Belyaeva, Mathematical modeling of deformation of viscoelastic structured polymer (composite) systems: Author. dr. diss. Chelyabinsk, 2008, 35 p.

K.Kh. Zelensky, “A numerical-analytical method for solving space-time problems with moving boundaries,” Interv. science and technology coll. “Adaptive automatic control systems”, no. 14 (34), pp. 107–117, 2009.

K.Kh. Zelensky, Mathematical modeling of nonlinear polymer materials in extruders: Author.diss... of Doctor of Technical Sciences. Kyiv, 2021, 43 p.

R.R. Zinatullin and N.M. Trufanova, “Numerical modeling of technological stresses during the production of plastic wire insulation,” Computational Mechanics of Solid Media, vol. 2, no. 1, pp. 38–53, 2009.

T.G. Kulikova and N.A. Trufanov, “Numerical solution of the boundary value problem of thermomechanics for a crystallizing viscoelastic polymer,” Ex. Mechanics of continuous media, vol. 1, no. 2, pp. 38–52, 2008.

V.N. Mitroshin, “Structural modeling of the cooling process of an insulated cable core when it is produced on an extrusion line,” Vestn. Samar Mr. technical Univ. Ser.: “Technical Sciences”, no. 40, pp. 22–33, 2006.

O.S. Sakharov, V.I. Sivetskyi, and O.L. Sokolskyi, Modeling of processes of melting and homogenization of polymer compositions in worm equipment: monograph. Kyiv: VP “Edelweiss”, 2012, 120 p.

I.N. Shardakov and N.A. Trufanov, “Modeling of thermomechanics of crystallizing polymers,” Izv. Tula State University. Natural sciences, vol. 2, pp. 117–123, 2008.

D. Andreucci, A. Fasano, and M. Primicerio, “Numerical simulation of polymer crystallization,”Mathematical Models and Methods in Applied Sciences, November 2011, 11 p.

V. Capasso, H. Engl, J. Periaux (Eds.), “Computational Mathematics Driven by Industrial Problems,” in Mathematical models for polymer crystallization processes, Springer, Berlin/Heidelberg, 2000, pp. 39–67.

V. Capasso, R. Escobedo, and C. Salani, “Moving Bands and Moving Boundaries in a Hybrid Model for the Crystallization of Polymers,” in Free Boundary Problems Theory and Applications, vol. 147, ed. by P. Colli, C. Verdi, A. Visintin, Birkhauser, 2004, pp. 75–86.

R. Escobedo and L. Fernandez, “Free boundary problems (FBP) and optimal control of axisymmetric polymer crystallization processes,” Computers and Mathematics with applications, 68, pp. 27–43, 2014.

R. Escobedo and L. Fernandez, “Optimal control of chemical birth and growth processes in a deterministic model,” J. Math. Chem., 48, pp. 118–127, 2010.

A. Friedman and A. Velazquez, “A free boundary problem associated with crystallization of polymers in a temperature field,” Indiana Univ. Math. J., 50, pp. 1609–1649, 2001.

J. Ramos, “Propagation and interaction of moving fronts in polymer crystallization,” Appl. Math. Comput., 189, pp. 780–795, 2007.

O. Trofymchuk, K. Zelensky, V. Pavlov, and K. Bovsunovska, “Modeling of heat and mass transfer processes in the melting zone of polymer,” System Research & Information Technologies, no. 4, pp. 68–80, 2021.

N. Zelenskaya and K. Zelensky, “Approximation of Bessel functions by rational functions,” Electronics and control systems, no. 2 (44), pp. 123–129, 2015.






Mathematical methods, models, problems and technologies for complex systems research