Mass-Flow Balance Modeling in the Production of Freeze-Stable Cream Cheese

Introduction. The study is motivated by the need to justify the efficiency of localizing the production of high-protein dairy products capable of withstanding a single exposure to sub-zero temperatures during transport to the consumer.

Purpose. To develop a balanced mathematical model of raw material flows for the production of cryostable curd cheese from milk, enabling the scientifically grounded selection of formulation and processing parameters for production in the sub-Arctic regions of the Russian Far East.

Materials and Methods. The study employs classical material balance equations for milk separation and established patterns of protein transfer during curd production. An invariant mathematical model was developed and implemented as software in the Wolfram Mathematica 10.2 environment using principle of mass conservation, Raoult's second law (with the van't Hoff correction), linear programming, the Ostwald - de Waele power law, and the Atwater system.

Results. Two scenarios for utilizing the cream obtained during separation were analyzed: partial (Scenario A) and complete (Scenario B) utilization. Variables included the proportion of curd in the binary "curd–cream" system (T = 65…85 %) and the cream fat content ( =10…85 %). Non-linear relationships were established regarding the impact of factors T and  on the yields of curd cheese, whey, and secondary cream; on protein and fat mass fractions; and on the product's cryoscopic depression and dynamic viscosity. A fundamental transition boundary between the scenarios was identified (T = 65 % при = 44,48 %), along with a terminal region where solutions for Scenario B do not exist without the addition of milk protein concentrate.

Conclusion. The proportion of curd in the binary mixture is the key limiting factor of the technology, whereas cream fat content acts as a supplementary factor. The developed model and Powersoftware enable the targeted design of the formulation and production parameters for curd cheese with specified yield, composition, and cryostability.

1. Aleksandrovsky, S. A. (2012). Materials and raw materials calculations for food production: a textbook. Kazan: KNRTU.

2. Gorbatova, K. K., Gunkova, P. I. (2021). Biochemistry of milk and dairy products: textbook (5th ed., revised and additional). St. Petersburg: GIORD.

3. Lipatov, N. N. (1990). Principles and methods for designing food formulations that balance diets. News of higher educational institutions. Food Technology, (6), 5–10.

4. Order of the State Agricultural Industry of the USSR dated December 31, 1987 No. 1025 “On approval of standards for consumption and losses of raw materials in the production of whole milk products at dairy industry enterprises and organization of work on standardization of consumption of raw materials.” (1987). URL: https://www.consultant.ru/document/cons_doc_LAW_55974/ (date of access: 05.28.2026).

5. Serba, E. V., Barkovskaya, I. A. (2025). The influence of quality indicators of whole milk from Jersey cows on the technological aspects of cottage cheese production. Food Industry, (11), 36–40. https://doi.org/10.52653/PPI.2025.11.11.005

6. Agarwal, S., Beausire, R. L. W., Patel, S., & Patil, S. (2015). Innovative uses of milk protein concentrates in product development. Journal of Food Science, 80(S1), A23–A29. https://doi.org/10.1111/1750-3841.12807

7. Belsito, P. C., Ferreira, M. V. S., Cappato, L. P., et al. (2017). Manufacture of Requeijão cremoso processed cheese with galactooligosaccharide. Carbohydrate Polymers, 174, 869–875. https://doi.org/10.1016/j.carbpol.2017.07.021

8. Cardello, A. V., Ball, D. H., Alabran, D. M., et al. (1984). The taste, odor and hedonic quality of polyglycerols. Chemical Senses, 9(3), 285–301. https://doi.org/10.1093/chemse/9.3.285

9. Chen, C. S. (1986). Effective molecular weight of aqueous solutions and liquid foods calculated from the freezing point depression. Journal of Food Science, 51(6), 1537–1539. https://doi.org/10.1111/j.1365-2621.1986.tb13853.x

10. Chen, P., Chen, X. D., & Free, K. W. (1996). Measurement and data interpretation of the freezing point depression of milks. Journal of Food Engineering, 30(1–2), 239–253.

11. Dave, P. J. (2012). Rheological properties of low-fat processed cheese spread made with inulin as a fat replacer (Dissertation). University of Wisconsin–Stout.

12. Guinee, T. P., O'Kennedy, B. T., & Kelly, P. M. (2006). Effect of milk protein standardization using different methods on the composition and yields of Cheddar cheese. Journal of Dairy Science, 89(2), 468–482.

13. Guinee, T. P., Pudja, P. D., & Mulholland, E. O. (1994). Effect of milk protein standardization, by ultrafiltration, on the manufacture, composition and maturation of Cheddar cheese. Journal of Dairy Research, 61(1), 117–131.

14. Huang, L., Abdel-Hamid, M., Romeih, E., et al. (2020). Textural and organoleptic properties of fat-free buffalo yogurt as affected by polydextrose. International Journal of Food Properties, 23(1), 1–8. https://doi.org/10.1080/10942912.2019.1682010

15. Hyvönen, L., & Slotte, M. (1983). Alternative sweetening of yoghurt. International Journal of Food Science and Technology, 18(1), 97–112. https://doi.org/10.1111/j.1365-2621.1983.tb00248.x

16. Lawless, H. T. (1977). The pleasantness of mixtures in taste and olfaction. Sensory Processes, 1(3), 227–237.

17. Lee, S. K., Anema, S. G., & Klostermeyer, H. (2004). The influence of moisture content on the rheological properties of processed cheese spreads. International Journal of Food Science and Technology, 39(7), 763–771. https://doi.org/10.1111/j.1365-2621.2004.00842.x

18. Mistry, V. V. (2002). Manufacture and application of high milk protein powder. Le Lait, 82(4), 515–522.

19. Moura, R. S., Guimarães, J. T., Scudino, H., et al. (2024). Thermosonication as an effective substitution for fusion in Brazilian cheese spread (Requeijão Cremoso) manufacturing: The effect of ultrasonic power on technological properties. Ultrasonics Sonochemistry, 105, 106867. https://doi.org/10.1016/j.ultsonch.2024.106867

20. Paul, A. A., & Southgate, D. A. T. (1978). McCance and Widdowson's the composition of foods (4th ed.). Amsterdam: Elsevier North, Holland Biomedical Press.

21. Power law (Ostwald–deWaele model). (2007). In J. W. Gooch (Ed.), Encyclopedic dictionary of polymers (p. 781). New York: Springer. https://doi.org/10.1007/978-0-387-30160-0_9193

22. Rahman, M. S. (Ed.). (2009). Food properties handbook (2nd ed.). CRC Press. https://doi.org/10.1201/9781420003093

23. Regulation (EU) No 1169/2011 of the European Parliament and of the Council of 25 October 2011 on the provision of food information to consumers. (2011). Official Journal of the European Union, L304, 18–63. URL: https://eur-lex.europa.eu/eli/reg/2011/1169/oj (дата обращения: 28.05.2026).

24. Santomauro, R. A., Papi, L. L., de Almeida, A. H. P., et al. (2025). Effect of lactose-free UHT milk macromolecules on the colligative property of cryometry. Food Science and Technology, 45, e00510. https://doi.org/10.5327/fst.00510

25. Zhang, J. (2024). The logic and application of greedy algorithms. Applied and Computational Engineering, 82, 154–160. https://doi.org/10.54254/2755-2721/82/20241110