Theoretical analysis of hydrodynamics of the cream churning process in a batch butter churn equipped with a Möbius strip working body

The study investigates the hydrodynamic processes occurring during the whipping of cream in a device with a working element shaped like a double Möbius strip. The relevance of the work stems from the need to improve the efficiency of butter production, particularly for small-scale farms, where traditional churning methods are often energy-intensive and insufficiently productive. The aim of the research is to analyze the flow characteristics generated by the working element and their impact on the cream-whipping process. The object of the study was a butter churn with a double Möbius strip working element, located at the «Agromechanics» Research Institute (Ganja, Azerbaijan). Methods included mathematical modeling to analyze centrifugal and vortex flows, as well as the determination of key parameters such as rotational speed (n=700→900 rpm) and strip width (b=0.03→0.05 m). Formulas for calculating centrifugal velocity, force, and turbulent flows were applied. The results showed that increasing the rotational speed to (n=900 rpm) and the strip width to (b=0.05 m) leads to maximum values of centrifugal velocity (v=12.5 m/s) and force (F=1408.5 N), significantly intensifying the churning process. The turbulent flows generated by the Möbius strip promote uniform mixing and the disruption of fat globule membranes. The theoretical investigations show that the highest hydrodynamic activity is formed in the peripheral zone of the working chamber, where according to the calculated data, the most favorable conditions are created for coalescence of fat globules and formation of butter grains. The results obtained make it possible to predict that realization of the indicated hydrodynamic regimes can facilitate reduction of the duration of the churning process and specific energy consumption. The experimental assessment of these effects requires additional studies and is the subject of the further work.

1. Kuliev, Z.V., Yakubov, K. G., Aliev, E. M. (2024). Effect of churning parameters on certain properties of buttermilk and butter developed from yogurt. Molochnokhozayistvenny Vestnik, 3(55), 156–171. (In Russian)] https://doi.org/10.52231/2225-4269_2024_3_156

2. Arsenieva, T. P. (2013). Butter technology. St. Petersburg: National Research University; Institute of Refrigeration and Biotechnology, 2013. (In Russian)]

3. Gogaev, O. K., Karaeva, Z. A., Kadieva, T. A., Morgoeva, D. G. (2024). Technology of milk and dairy products. St. Petersburg: Lan, 2024. (In Russian)]

4. Biryukova, E. A., Lazutkina, S. A. (February 7–8, 2017). Analysis of existing provisions of the theories of cream churning and butter grain formation. Agrarian science and education at the modern stage of the development: Experience, problems and ways of their solution: Proceedings of the 8th International scientific-practical conference. Ulyanovsk: Ulyanovsk State Agricultural Academy named after P. A. Stolypin, 2017. (In Russian)]

5. Tverdokhleb, G. V., Sazhinov, G. Yu., Ramanauskas, R. I. (2006). Technology of milk and dairy products. Moscow: DeLi print, 2006. (In Russian)]

6. Yashin, A. V., Polyvyany, Yu. V. (2019). The results of laboratory studies of the butter maker with a flexible vibration drive. Niva Povolzhya, (3(52)), 177–183. (In Russian)]

7. Yashin, A., Polyvyannii, Y. (2020). Results of research on justification the device for producing ecologically pure butter. Scientific Papers. Series A. Agronomy, 63(1),632–636.

8. Aljaafreh, A., Al-Tahiri, R., Abadleh, A., Mansour, A. M., Alaqtash, M. (2019). Design and development of an automated and quality controlled system for traditional butter and ghee production. Wseas Transactions on Environment and Development, 15, 478–484.

9. Kalla, A. M., Sahu, C., Agrawal, A. K., Bisen, P., Chavhan, B. B., Sinha, G. (2016). Development and performance evaluation of frustum cone shaped churn for small scale production of butter. Journal of Food Science and Tech‑ nology, 53(5), 2219–2226. https://doi.org/10.1007/s13197-015-2110‑y

10. Lazutkina, S. A., Minnibaev, M. R. (June 20–21, 2018). Batchoilmanufacturer. Agrarian science and education at the modern stage of the development: Experience, problems and ways of their solution: Proceedings of the 9th International scientific-practical conference dedicated to the 75th anniversary of the Ulyanovsk State Agrarian University named after P. A. Stolypin. Ulyanovsk: Ulyanovsk State Agrarian University, 2018. (In Russian)]

11. Ebrahimi, M., Khoshnam, F., Kamandar, M. R., Namjoo, M. (2024). Development and performance evaluation of machine-type reciprocating churner. Biomechanism and Bioenergy Research, 3(2), 108–116. https://doi.org/10.22103/BBR.2024.22972.1080

12. Wang, M., Lu, J., Li, Z., Gu, F., Chen, Z. (2023). Study on the unsteady characteristics and radial force of a single-channel centrifugal pump. ACS Omega, 8(2), 2291–2305. https://doi.org/10.1021/acsomega.2c06614

13. Cao, W., Mao, J., Li, W. (2020). Analysis of unsteady pressure fluctuation in a semi-open cutting pump. Energies, 13(14), Article 3657. https://doi.org/10.3390/en13143657

14. Kang, Y., Liu, S., Zou, W., Hu, X. (2019). Numerical investigation on pressure pulsation characteristics and radial force of a deep-sea electric lifting pump at off-design conditions. Shock and Vibration, 2019(1), Article 4707039. https://doi.org/10.1155/2019/4707039

15. Diehl, E. (2021). Non-rectangular coordinate systems: Polar coordinates. Chapter in a book: The Engineering Dynamics Course Companion, Part 1: Particles. Synthesis Lectures on Mechanical Engineering, Springer, Cham, 2021. https://doi.org/10.1007/978-3-031-79677-7_6

16. Lee, G., Kim, W. (2024). Image processing in L1‑norm-based discrete Cartesian and polar coordinates. Electronics, 13, Article 1088. http:// doi.org/10.3390/electronics1301088

17. Zhu, J., Wang, Y., Gao, F., Zhang, Z. (2023). Optical differentiation in a polar coordinate system. Applied Physics Letters, 122(9), Article 091107. https://doi.org/10.1063/5.0140272

18. Walstra, P., Wouters, J. T. M., Geurts, T. J. (2005). Centrifugation. Chapter in a book: Dairy science and technology. CRC Press, 2005.

19. Panchal, B., Truong, T., Prakash, S., Bansal, N., Bhandari, B. (2021). Influence of emulsifiers and dairy ingredients on manufacturing, microstructure, and physical properties of butter. Foods, 10(5), Article 1140. https://doi.org/10.3390/foods10051140

20. Nozawa, E., Deguchi, T. (2025). Simulating phase inversion processes by coupled map lattice: Towards the theoretical design of food texture and quality in dairy processing from fresh cream to butter via whipped cream. Journal of Chemical Physics, 162, Article 064902. https://doi.org/10.1063/5.0251375

21. Nozawa, E. (2026). Theoretical relationship between the macro texture and micro structure in dairy processing revealed by the multi scale simulation of coupled map lattice. arXiv, 1, Article 15051. https://arxiv.org/abs/2601.15051

22. Lars, W., De Graef, V., Rasmussen, M., Dewettinck, K. (2009). Relations between crystallisation mechanism and microstructure of milk fat. International Dairy Journal, 19(8), 424–430. https://doi.org/10.1016/j.idairyj.2009.03.003

23. Buldo, P., Kirkensgaard, J. J. K., Wiking, L. (2013). Crystallization mechanisms in cream during ripening and initial butter churning. Journal of Dairy Science, 96(11), 6782–6791. https://doi.org/10.3168/jds.2012-6066

24. Bronstein, I. N., Semendyayev, K. A. (1986). Handbook of mathematics for engineers and students of technical universities. Moscow: Nauka, 1986. (In Russian)]

25. Serway, R. A., Jewett, J. W. (2018). Physics for scientists and engineers with Modern Physics (10th ed.). Cengage Learning, 2018.

26. Hibbeler, R. C. (2016). Engineering mechanics: Dynamics (14th ed.). Pearson, 2015.

27. Landau, L. D., Lifshitz, E. M. (1988). Theoretical physics: Volume 1. Mechanics. Moscow: Nauka, 1988. (In Russian)]