English
Русский
Objectives: development of a mathematical model to describe the phenomena occurring in a medium-temperature solid oxide fuel cell. Solving the model equations requires finding a number of kinetic parameters, including the rate constants of electrochemical reactions.
Methods. To determine the viscosity of polymer solutions, their molecular weight and to study the adsorption of k-carrageenan on CX, the method of capillary viscometry was used. The assessment of the stability of the zones over time was carried out photometrically.
Results. The mathematical model is based on a system of partial differential equations and includes material and heat balance equations, as well as charge balance relationships. An algorithm for numerically solving the mathematical model equations and a corresponding software module for calculating the equations implemented in the Python programming language have been developed.
Conclusions. The developed mathematical model adequately describes the processes occurring at the electrodes of a medium-temperature fuel cell. The optimal fuel-to-oxidizer ratio was determined to be 1:10.
Methods. To determine the viscosity of polymer solutions, their molecular weight and to study the adsorption of k-carrageenan on CX, the method of capillary viscometry was used. The assessment of the stability of the zones over time was carried out photometrically.
Results. The mathematical model is based on a system of partial differential equations and includes material and heat balance equations, as well as charge balance relationships. An algorithm for numerically solving the mathematical model equations and a corresponding software module for calculating the equations implemented in the Python programming language have been developed.
Conclusions. The developed mathematical model adequately describes the processes occurring at the electrodes of a medium-temperature fuel cell. The optimal fuel-to-oxidizer ratio was determined to be 1:10.
1. Hu Y., Li D., Guo H., Liu S.-H., Meng Y., Ding S., Li C.-X. Recent progress of high-performance intercon-nectors for SOFC: From materials, protective coatings, optimizing strategies, towards the real stack applications. Chemical Engineering Journal. 2025. Vol. 505. P. 159321. https://doi.org/10.1016/j.cej.2025.159321
2. Zeng Z., Qian Y., Zhang Y., Hao C., Dan D., Zhuge W. A review of heat transfer and thermal management methods for temperature gradient reduction in solid oxide fuel cell (SOFC) stacks. Applied Energy. 2020. Vol. 280. P. 115899. https://doi.org/10.1016/j.apenergy.2020.115899
3. Zakaria Z., Mat Z.A., Abu Hassan S.H., Kar Y.B. A review of solid oxide fuel cell component fabrication methods toward lowering temperature. International Journal of Energy Research. 2020. Vol. 44. No. 5. P. 5946 – 5961. https://doi.org/10.1002/er.4907
4. Leah R. T., Brandon N. P., Aguiar P. Modeling of cells, stacks and systems based around metal-supported planar IT-SOFC cells with CGO electrolytes operating at 500-600° C. Journal of Power Sources. 2005. Vol. 145. No. 2. P. 336 – 352. https://doi.org/10.1016/j.jpowsour.2004.12.067
5. Blum L., Buchkremer H.-P., Gross S., Gubner A., (Bert) de Haart L.G.J., Nabielek H., Quadakkers W.J., Reisgen U., Smith M. J., Steinberger-Wilckens R., Steinbrech R.W., Tietz F., Vinke I. C. Solid oxide fuel cell de-velopment at Forschungszentrum Juelich. Fuel Cells. 2007. Vol. 7. No. 3. P. 204 – 210. https://doi.org/10.1002/fuce.200600039
6. Peters Ro., Tiedemann W., Hoven I., Deja R., Kruse N., Kunz F., Blum L., Peters R., Eichel R.-A. Experi-mental results of a 10/40 kW-class reversible solid oxide cell demonstration system at Forschungszentrum Jülich. Journal of the Electrochemical Society. 2023. Vol. 170. No. 4. P. 044509. https://doi.org/10.1149/1945-7111/accbf0.
7. Hajimolana S.A., Hussain M.A., Wan Daud W.M.A., Soroush M., Shamiri A. Mathematical modeling of solid oxide fuel cells: A review. Renewable and Sustainable Energy Reviews. 2011. Vol.15. P. 1893 – 1917. https://doi.org/10.1016/j.rser.2010.12.011
8. Khazaee I., Rava A. Numerical simulation of the performance of solid oxide fuel cell with different flow channel geometries. Energy. 2017. Vol. 119. P. 235-244. https://doi.org/10.1016/j.energy.2016.12.074
9. Zazhigalov S.V., Zagoruiko A.N., Popov M.P., Nemudry A.P., Belotserkovsky V.A. Mathematical modeling and experimental studies of microtubular solid oxide fuel cells. Theoretical Foundations of Chemical Engineering. 2020. Vol. 54. No. 4. P. 647 – 654. https://doi.org/10.1134/S0040579520040284
10. Sarkar B., Hariharan D., Gundlapally S.R. Hybrid 1+1D model for direct ammonia-fed planar SOFCs: Sim-ulation and Analysis. Fuel. 2025. Vol. 392. P. 134780. https://doi.org/10.1016/j.fuel.2025.134780
11. Fu Q., Li Z., Wei W., Liu F., Xu X., Liu Z. Performance enhancement of planar solid oxide fuel cell using a novel interconnector design. International Journal of Hydrogen Energy. 2021. Vol. 46. No. 41. P. 21634 – 21656. https://doi.org/10.1016/j.ijhydene.2021.04.001
12. Nguyen X.-V. Three-dimensional simulation of the operating characteristics of cell layers in solid oxide fuel cells // Applied Sciences. 2025. Vol. 15. P. 4462. https://doi.org/10.3390/app15084462
13. Tikiz I., Taymaz I., Pehlivan H. CFD modeling and experimental validation of cell performance in a 3-D planar SOFC. International Journal of Hydrogen Energy. 2019. Vol. 44. No. 29. P. 15441 – 15455. https://doi.org/10.1016/j.ijhydene.2019.04.152
14. Zhang Z., Chen J., Yue D., Yang G., Ye S., He C., Wang W., Yuan J., Huang N. Three-dimensional CFD modeling of transport phenomena in a cross-flow anode-supported planar SOFC. Energies. 2014. Vol. 7. P. 80 – 98. https://doi.org/10.3390/en7010080
15. Yakubu A.U., Qingsheng L., Kai M., Jinwei C., Abbaker O. Mohammed A., Zhao J., Jiang Q., Ye X., Liu J., Yu Q., Aurangzeb M., Xiong S. Modeling, optimization, and thermal management strategies of hydrogen fuel cell systems. Results in Engineering. 2025. Vol. 27. P. 105924. https://doi.org/10.1016/j.rineng.2025.105924
16. Bedi U. Recent advances in fuel cell design and modeling: A comprehensive review. Next Energy. 2026. Vol. 11. P. 100517. https://doi.org/10.1016/j.nxener.2026.100517
17. Zhang Z., Liu H., Lan G., Wang Y., Liu C., Yin Z., Lu K. Single-cell modeling, simulation and optimization of high-temperature proton exchange membrane fuel cells: current status, challenges, and perspectives. Journal of Power Sources. 2026. Vol. 664. P. 238943. https://doi.org/10.1016/j.jpowsour.2025.238943
18. Ba S., Xia D., Gibbons E.M. Model identification and strategy application for solid oxide fuel cell using ro-tor Hopfield neural network based on a novel optimization method. International Journal of Hydrogen Energy. 2020. Vol. 45. No. 51. P. 27694 – 27704. https://doi.org/10.1016/j.ijhydene.2020.07.12
19. Guo H., Gu W., Khayatnezhad M., Ghadimi N. Parameter extraction of the SOFC mathematical mod-el based on fractional order version of dragonfly algorithm // International Journal of Hydrogen Energy. 2022. Vol.47. No. 57. P. 24059 – 24068. https://doi.org/10.1016/j.ijhydene.2022.05.190
20. Xiong X., Liang K., Ma G., Ba L. Three-dimensional multi-physics modeling and structural optimization of SOFC large-scale stack and stack tower. International Journal of Hydrogen Energy. 2023. Vol. 48. No. 7. P. 2742 – 2761. https://doi.org/10.1016/j.ijhydene.2022.10.146
21. Beloev H.I., Saitov S.R., Filimonova A.A., Chichirova N.D., Mayorov E.S., Babikov O.E., Iliev I.K. Solid oxide fuel cell voltage prediction by a data-driven approach. Energies. 2025. Vol. 18. P. 2174. https://doi.org/10.3390/en18092174.
22. Testasecca T., Maniscalco M.P., Brunaccini G., Airò Farulla G., Ciulla G., Beccali M., Ferraro M. Toward a digital twin of a solid oxide fuel cell microcogenerator: data-driven modeling. Energies. 2024. Vol. 17. P. 4140. https://doi.org/10.3390/en17164140.
23. Kulikovsky A.A. Thermal waves in SOFC stacks. Journal of the Electrochemical Society. 2008. Vol. 155. No. 9. P. 693-698. https://doi.org/10.1149/1.2953579
2. Zeng Z., Qian Y., Zhang Y., Hao C., Dan D., Zhuge W. A review of heat transfer and thermal management methods for temperature gradient reduction in solid oxide fuel cell (SOFC) stacks. Applied Energy. 2020. Vol. 280. P. 115899. https://doi.org/10.1016/j.apenergy.2020.115899
3. Zakaria Z., Mat Z.A., Abu Hassan S.H., Kar Y.B. A review of solid oxide fuel cell component fabrication methods toward lowering temperature. International Journal of Energy Research. 2020. Vol. 44. No. 5. P. 5946 – 5961. https://doi.org/10.1002/er.4907
4. Leah R. T., Brandon N. P., Aguiar P. Modeling of cells, stacks and systems based around metal-supported planar IT-SOFC cells with CGO electrolytes operating at 500-600° C. Journal of Power Sources. 2005. Vol. 145. No. 2. P. 336 – 352. https://doi.org/10.1016/j.jpowsour.2004.12.067
5. Blum L., Buchkremer H.-P., Gross S., Gubner A., (Bert) de Haart L.G.J., Nabielek H., Quadakkers W.J., Reisgen U., Smith M. J., Steinberger-Wilckens R., Steinbrech R.W., Tietz F., Vinke I. C. Solid oxide fuel cell de-velopment at Forschungszentrum Juelich. Fuel Cells. 2007. Vol. 7. No. 3. P. 204 – 210. https://doi.org/10.1002/fuce.200600039
6. Peters Ro., Tiedemann W., Hoven I., Deja R., Kruse N., Kunz F., Blum L., Peters R., Eichel R.-A. Experi-mental results of a 10/40 kW-class reversible solid oxide cell demonstration system at Forschungszentrum Jülich. Journal of the Electrochemical Society. 2023. Vol. 170. No. 4. P. 044509. https://doi.org/10.1149/1945-7111/accbf0.
7. Hajimolana S.A., Hussain M.A., Wan Daud W.M.A., Soroush M., Shamiri A. Mathematical modeling of solid oxide fuel cells: A review. Renewable and Sustainable Energy Reviews. 2011. Vol.15. P. 1893 – 1917. https://doi.org/10.1016/j.rser.2010.12.011
8. Khazaee I., Rava A. Numerical simulation of the performance of solid oxide fuel cell with different flow channel geometries. Energy. 2017. Vol. 119. P. 235-244. https://doi.org/10.1016/j.energy.2016.12.074
9. Zazhigalov S.V., Zagoruiko A.N., Popov M.P., Nemudry A.P., Belotserkovsky V.A. Mathematical modeling and experimental studies of microtubular solid oxide fuel cells. Theoretical Foundations of Chemical Engineering. 2020. Vol. 54. No. 4. P. 647 – 654. https://doi.org/10.1134/S0040579520040284
10. Sarkar B., Hariharan D., Gundlapally S.R. Hybrid 1+1D model for direct ammonia-fed planar SOFCs: Sim-ulation and Analysis. Fuel. 2025. Vol. 392. P. 134780. https://doi.org/10.1016/j.fuel.2025.134780
11. Fu Q., Li Z., Wei W., Liu F., Xu X., Liu Z. Performance enhancement of planar solid oxide fuel cell using a novel interconnector design. International Journal of Hydrogen Energy. 2021. Vol. 46. No. 41. P. 21634 – 21656. https://doi.org/10.1016/j.ijhydene.2021.04.001
12. Nguyen X.-V. Three-dimensional simulation of the operating characteristics of cell layers in solid oxide fuel cells // Applied Sciences. 2025. Vol. 15. P. 4462. https://doi.org/10.3390/app15084462
13. Tikiz I., Taymaz I., Pehlivan H. CFD modeling and experimental validation of cell performance in a 3-D planar SOFC. International Journal of Hydrogen Energy. 2019. Vol. 44. No. 29. P. 15441 – 15455. https://doi.org/10.1016/j.ijhydene.2019.04.152
14. Zhang Z., Chen J., Yue D., Yang G., Ye S., He C., Wang W., Yuan J., Huang N. Three-dimensional CFD modeling of transport phenomena in a cross-flow anode-supported planar SOFC. Energies. 2014. Vol. 7. P. 80 – 98. https://doi.org/10.3390/en7010080
15. Yakubu A.U., Qingsheng L., Kai M., Jinwei C., Abbaker O. Mohammed A., Zhao J., Jiang Q., Ye X., Liu J., Yu Q., Aurangzeb M., Xiong S. Modeling, optimization, and thermal management strategies of hydrogen fuel cell systems. Results in Engineering. 2025. Vol. 27. P. 105924. https://doi.org/10.1016/j.rineng.2025.105924
16. Bedi U. Recent advances in fuel cell design and modeling: A comprehensive review. Next Energy. 2026. Vol. 11. P. 100517. https://doi.org/10.1016/j.nxener.2026.100517
17. Zhang Z., Liu H., Lan G., Wang Y., Liu C., Yin Z., Lu K. Single-cell modeling, simulation and optimization of high-temperature proton exchange membrane fuel cells: current status, challenges, and perspectives. Journal of Power Sources. 2026. Vol. 664. P. 238943. https://doi.org/10.1016/j.jpowsour.2025.238943
18. Ba S., Xia D., Gibbons E.M. Model identification and strategy application for solid oxide fuel cell using ro-tor Hopfield neural network based on a novel optimization method. International Journal of Hydrogen Energy. 2020. Vol. 45. No. 51. P. 27694 – 27704. https://doi.org/10.1016/j.ijhydene.2020.07.12
19. Guo H., Gu W., Khayatnezhad M., Ghadimi N. Parameter extraction of the SOFC mathematical mod-el based on fractional order version of dragonfly algorithm // International Journal of Hydrogen Energy. 2022. Vol.47. No. 57. P. 24059 – 24068. https://doi.org/10.1016/j.ijhydene.2022.05.190
20. Xiong X., Liang K., Ma G., Ba L. Three-dimensional multi-physics modeling and structural optimization of SOFC large-scale stack and stack tower. International Journal of Hydrogen Energy. 2023. Vol. 48. No. 7. P. 2742 – 2761. https://doi.org/10.1016/j.ijhydene.2022.10.146
21. Beloev H.I., Saitov S.R., Filimonova A.A., Chichirova N.D., Mayorov E.S., Babikov O.E., Iliev I.K. Solid oxide fuel cell voltage prediction by a data-driven approach. Energies. 2025. Vol. 18. P. 2174. https://doi.org/10.3390/en18092174.
22. Testasecca T., Maniscalco M.P., Brunaccini G., Airò Farulla G., Ciulla G., Beccali M., Ferraro M. Toward a digital twin of a solid oxide fuel cell microcogenerator: data-driven modeling. Energies. 2024. Vol. 17. P. 4140. https://doi.org/10.3390/en17164140.
23. Kulikovsky A.A. Thermal waves in SOFC stacks. Journal of the Electrochemical Society. 2008. Vol. 155. No. 9. P. 693-698. https://doi.org/10.1149/1.2953579
Vasilenko V.A., Lebedev I.D., Makarenkov D.A., Koltsova E.M. Mathematical modeling of a solid oxide fuel cell. Chemical Bulletin. 2026. 9 (1). 3. https://doi.org/10.58224/2619-0575-2026-9-1-3