najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-6-749-754najo-1613Research ArticleMATHEMATICSМАТЕМАТИКАMathematical modeling of industrial ammonia synthesis using nonlinear reactiondiffusion equationsМатематическое моделирование промышленного синтеза аммиака с использованием нелинейных уравнений реакции-диффузииhttps://orcid.org/0000-0002-3712-4635ХасановЖ.KhasanovJ.

Жамшид Хасанов

Jamshid Khasanov 

1A Gurlan str., Urgench 220100

jamshid2425@mail.ruhttps://orcid.org/0000-0003-2471-4836МуминовС.MuminovS.

Сохибжан Муминов

Sokhibjan Muminov 

2 Bolkhovuz Street, Khiva 220900

sokhibjan.muminov@gmail.comhttps://orcid.org/0009-0001-4998-5141ИскандаровС.IskandarovS.

Сарвар Искандаров

Sarvar Iskandarov 

4 Kh. Alimdjan str., Urgench 220100

iskandarovsb1993@gmail.comUrgench State Pedagogical InstituteUzbekistanMamun UniversityUzbekistanUrgench State University named after Abu Rayhan BiruniUzbekistan202506012026166749754Copyright © Khasanov J., Muminov S., Iskandarov S., 20262026Хасанов Ж., Муминов С., Искандаров С.Khasanov J., Muminov S., Iskandarov S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1613

This study proposes a mathematical model for ammonia synthesis based on nonlinear reactiondiffusion equations. The model integrates degenerate gas diffusion in the reactor with Haber-Bosch reaction kinetics to explore efficiency and environmental sustainability. A theoretical analysis is conducted to establish the existence and stability of global solutions for the underlying degenerate parabolic system. Numerical simulations were validated against industrial data from Navoiyazot facility in Uzbekistan, demonstrating 98.2% accuracy in concentration profiles and outperforming constant-diffusivity models by 12–15% in low-concentration regions.

В данном исследовании предлагается математическая модель синтеза аммиака, основанная на нелинейных уравнениях реакции-диффузии. Модель объединяет вырожденную диффузию газа в реакторе с кинетикой реакции Габера-Боша для исследования эффективности и экологической устойчивости. Проведен теоретический анализ для установления существования и устойчивости глобальных решений для базовой вырожденной параболической системы. Численное моделирование было проверено на основе промышленных данных завода «Навоиазот» в Узбекистане, продемонстрировав 98,2% точность определения профилей концентрации и превосходя модели с постоянной диффузией на 12–15% в областях с низкой концентрацией.

нелинейная реакция-диффузиявырожденная диффузиясинтез аммиакакинетика Габера-Бошаглобальная устойчивостьnonlinear reaction-diffusiondegenerate diffusionammonia synthesisHaber-Bosch kineticsglobal stabilityThe authors are grateful to Navoiyazot JSC for providing access to operational data from the industrial ammonia synthesis unit A-15.ReferencesTemkin M., Pyzhev V. Kinetics of ammonia synthesis on iron catalysts. Acta Physicochim. URSS, 1941, 12, P. 327–356.Temkin M., Pyzhev V. Kinetics of ammonia synthesis on iron catalysts. Acta Physicochim. URSS, 1941, 12, P. 327–356.Smith C., Hill A.K., Torrente-Murciano L. Current and future role of Haber–Bosch ammonia in a carbon-free energy landscape. Energy Environ. Sci., 2020, 13, P. 331–344.Smith C., Hill A.K., Torrente-Murciano L. Current and future role of Haber–Bosch ammonia in a carbon-free energy landscape. Energy Environ. Sci., 2020, 13, P. 331–344.Nielsen A. (Ed.). Ammonia: Catalysis and Manufacture. Springer, Berlin Heidelberg, 1995.Nielsen A. (Ed.). Ammonia: Catalysis and Manufacture. Springer, Berlin Heidelberg, 1995.Nadiri A., Perez-Ram ´ ´ırez J., et al. Ammonia synthesis rate over a wide operating range: From experiments to validated kinetic models. ChemCatChem, 2024, 16, e202301462.Nadiri A., Perez-Ram ´ ´ırez J., et al. Ammonia synthesis rate over a wide operating range: From experiments to validated kinetic models. ChemCatChem, 2024, 16, e202301462.Burdyny T., van Haveren J., Smith W.A. Entropy production minimization in a tubular ammonia synthesis reactor with variable geometry. Front. Energy Res., 2025, 13, 1515577.Burdyny T., van Haveren J., Smith W.A. Entropy production minimization in a tubular ammonia synthesis reactor with variable geometry. Front. Energy Res., 2025, 13, 1515577.Fragnelli G., Mugnai D. Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations. Mem. Amer. Math. Soc., 2016, 242(1146).Fragnelli G., Mugnai D. Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations. Mem. Amer. Math. Soc., 2016, 242(1146).Boutaayamou I., Fragnelli G., Maniar L. Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions. J. Anal. Math., 2018, 135, P. 1–35.Boutaayamou I., Fragnelli G., Maniar L. Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions. J. Anal. Math., 2018, 135, P. 1–35.Erhardt A.H. Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving cross-diffusion. J. Elliptic Parabol. Equ., 2020, 6, P. 735–772.Erhardt A.H. Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving cross-diffusion. J. Elliptic Parabol. Equ., 2020, 6, P. 735–772.Camasta S., Fragnelli G. Degenerate fourth order parabolic equations with Neumann boundary conditions. Nonlinear Anal. Real World Appl., 2023, 67, 103573.Camasta S., Fragnelli G. Degenerate fourth order parabolic equations with Neumann boundary conditions. Nonlinear Anal. Real World Appl., 2023, 67, 103573.Topayev T.N., Popov A.I., Popov I.Y. On Keller-Rubinow model for Liesegang structure formation. Nanosystems: Phys. Chem. Math., 2022, 13(4), P. 365–371.Topayev T.N., Popov A.I., Popov I.Y. On Keller-Rubinow model for Liesegang structure formation. Nanosystems: Phys. Chem. Math., 2022, 13(4), P. 365–371.Maksimova M.A., Polyakov E.V., Volkov I.V., Tyutyunnik A.P., Ioshin A.A. Kinetic of colloidal-chemical transformations during the decomposition of ammonia complexes of Zn(II) in alkaline solutions. Nanosystems: Phys. Chem. Math., 2024, 15(4), P. 498–509.Maksimova M.A., Polyakov E.V., Volkov I.V., Tyutyunnik A.P., Ioshin A.A. Kinetic of colloidal-chemical transformations during the decomposition of ammonia complexes of Zn(II) in alkaline solutions. Nanosystems: Phys. Chem. Math., 2024, 15(4), P. 498–509.Borisov V.A., Fedorova Z.A., Ichetovkin Z.N., et al. La and Co-based materials for ammonia decomposition: activity, stability and structural changes. Nanosystems: Phys. Chem. Math., 2025, 16(4), P. 498–509.Borisov V.A., Fedorova Z.A., Ichetovkin Z.N., et al. La and Co-based materials for ammonia decomposition: activity, stability and structural changes. Nanosystems: Phys. Chem. Math., 2025, 16(4), P. 498–509.Navoiyazot JSC. Annual Technical Report: Ammonia Production Unit A-15. Internal Report, Navoi, Uzbekistan, 2022.Navoiyazot JSC. Annual Technical Report: Ammonia Production Unit A-15. Internal Report, Navoi, Uzbekistan, 2022.Aripov M.M. Nonlinear Degenerate and Singular Parabolic Equations. Fan Publishers, Tashkent, 1988.Aripov M.M. Nonlinear Degenerate and Singular Parabolic Equations. Fan Publishers, Tashkent, 1988.Khasanov J.O. Mathematical modeling of processes described by cross-diffusion source and variable density. Problems of Computational and Applied Mathematics, 2022, 6(45), P. 39–47.Khasanov J.O. Mathematical modeling of processes described by cross-diffusion source and variable density. Problems of Computational and Applied Mathematics, 2022, 6(45), P. 39–47.Aripov M., Matyakubov A.S., Khasanov J.O. Global solvability and explicit estimation of solutions of a cross-diffusion parabolic system in non-divergent form with a source and variable density. Bulletin of the Institute of Mathematics, 2022, 5(4), P. 22–31.Aripov M., Matyakubov A.S., Khasanov J.O. Global solvability and explicit estimation of solutions of a cross-diffusion parabolic system in non-divergent form with a source and variable density. Bulletin of the Institute of Mathematics, 2022, 5(4), P. 22–31.Aripov M., Matyakubov A.S., Khasanov J.O. To the qualitative properties of self-similar solutions of a cross-diffusion parabolic system not in divergence form with a source. AIP Conference Proceedings, 2023, 2781, 020005.Aripov M., Matyakubov A.S., Khasanov J.O. To the qualitative properties of self-similar solutions of a cross-diffusion parabolic system not in divergence form with a source. AIP Conference Proceedings, 2023, 2781, 020005.Muminov S., Agarwal P., Muhamediyeva D. Qualitative properties of the mathematical model of nonlinear cross-diffusion processes. Nanosystems: Phys. Chem. Math., 2024, 15(6), P. 742–748.Muminov S., Agarwal P., Muhamediyeva D. Qualitative properties of the mathematical model of nonlinear cross-diffusion processes. Nanosystems: Phys. Chem. Math., 2024, 15(6), P. 742–748.Cholewa T., Steinbach B., Heim C., Nestler F., Nanba T., Guttel R., Salem O. Reaction kinetics for ammonia synthesis using ruthenium and iron ¨ based catalysts under low temperature and pressure conditions. Sustainable Energy & Fuels, 2024, 8(10), P. 2245–2255.Cholewa T., Steinbach B., Heim C., Nestler F., Nanba T., Guttel R., Salem O. Reaction kinetics for ammonia synthesis using ruthenium and iron ¨ based catalysts under low temperature and pressure conditions. Sustainable Energy & Fuels, 2024, 8(10), P. 2245–2255.Zecevic N. Multiscale catalyst model for ammonia synthesis: coupling kinetics, diffusion and deactivation. Reaction Kinetics, Mechanisms and Catalysis, 2025, 138, P. 3645–3664.Zecevic N. Multiscale catalyst model for ammonia synthesis: coupling kinetics, diffusion and deactivation. Reaction Kinetics, Mechanisms and Catalysis, 2025, 138, P. 3645–3664.Baltaeva U.I., Alikulov Y., Baltaeva I.I., Ashirova A.I Analog of the darboux problem for a loaded integro-differential equation involving the caputo fractional derivative. Nanosystems: Phys. Chem. Math., 2021, 12(4), P. 418–424.Baltaeva U.I., Alikulov Y., Baltaeva I.I., Ashirova A.I Analog of the darboux problem for a loaded integro-differential equation involving the caputo fractional derivative. Nanosystems: Phys. Chem. Math., 2021, 12(4), P. 418–424.Agarwal P., Baltaeva U.I., Madrakhimov U., Baltaev J.I. The Cauchy problem for a high-order wave equation with a loaded convolution type. Nanosystems: Phys. Chem. Math., 2024, 15(4), P. 448–456.Agarwal P., Baltaeva U.I., Madrakhimov U., Baltaev J.I. The Cauchy problem for a high-order wave equation with a loaded convolution type. Nanosystems: Phys. Chem. Math., 2024, 15(4), P. 448–456.

The authors declare that there are no conflicts of interest present.