Heat Transfer Modeling in Sweetened Condensed Milk

Introduction: Sweetened condensed whole milk (SCM) is a highly demanded product among consumers and processing plants. The study of temperature profiles is essential for optimizing its logistics under extreme temperature conditions to prevent quality defects and minimize costs associated with specialized transport when delivering to regions with hot climates and the Far North, considering the absolute temperature range in Russia, which is approximately 90°C. Previously, this aspect of transportation had not been studied. The maximum allowable storage temperature for the product was set at 25°C, while the minimum temperature was not regulated.

Purpose: To investigate the temperature profiles of SCM in transport packaging under various ambient conditions.

Materials and Methods: A simulation of the heating and cooling processes of SCM in transport packaging, modeled as a one-dimensional multilayer system, was conducted. To describe heat transfer within the temperature ranges of 5°C to 35°C and 5°C to -35°C, a system of differential equations was formulated, with specified initial and boundary conditions.

Results: According to the proposed model, the duration of heating SCM from 5°C to 35°C is 36.7 hours, while cooling from 5°C to -35°C takes 41.1 hours. Based on the study results, software was developed to calculate the duration of SCM temperature changes depending on the initial and final ambient temperatures.

Conclusion: A new approach has been developed for theoretically predicting the duration of temperature changes in SCM within transport packaging during storage and transportation. This approach can be utilized in specialized business software solutions for logistics route planning, transportation cost estimation, and consideration of ambient conditions during shipping. Additionally, the proposed solution can be adapted for other food products.

1. Antonopoulos, K. A., & Tzivanidis, C. (1996). Analytical solution of boundary value problems of heat conduction in composite regions with arbitrary convection boundary conditions. Acta Mechanica, 118, 65–78. https://doi.org/10.1007/BF01176344

2. Biswas, P., & Singh, S. (2015, December). Analytical solution of 1-D multilayer heat conduction problem with time varying ambients. In 23rd National and 1st International ISHMT-ASTFE Heat and Mass Transfer Conference, Trivandrum, Kerala, India.

3. Churchill, S. W., & Chu, H. H. S. (1975). Correlating equations for laminar and turbulent free convection from a vertical plate. International Journal of Heat and Mass Transfer, 18, 1323–1329. https://doi.org/10.1016/0017-9310(75)90235-2

4. Derossi, A., de Pilli, T., La Penna, M. P., & Severini, C. (2012). Prediction of heating length to obtain a definite F value during pasteurization of canned food. Journal of Food Process Engineering, 36, 211–219. https://doi.org/10.1111/j.1745-4530.2012.00686.x

5. Dairy News. (2023). Vologda region exported more than 90 tons of condensed milk. DairyNews Today. Retrieved from https://dairynews.today/news/vologodskaya-oblast-otpravila-na-eksport-bolee-90-.html

6. Fujii, T., & Imura, H. (1972). Natural-convection heat transfer from a plate with arbitrary inclination. International Journal of Heat and Mass Transfer, 15, 755–767. https://doi.org/10.1016/0017-9310(72)90063-1

7. Galstyan, A. G., Petrov, A. N., Illarionova, E. E., Semipyatniy, V. K., Turovskaya, S. N., Ryabova, A. E., et al. (2019). Effects of critical fluctuations of storage temperature on the quality of dry dairy product. Journal of Dairy Science, 102, 10779–10789. https://doi.org/10.3168/jds.2019-16979

8. Illarionova, E. E., Turovskaya, S. N., & Radaeva, I. A. (2020). To the question of increasing canned milk storage life. In Actual Issues of the Dairy Industry: Intersectoral Technologies and Quality Management Systems (pp. 225–230). All-Russian Dairy Research Institute.

9. Isachenko, V. P., Osipova, V. A., & Sukomel, A. S. (1981). Heat exchange. Energoizdat.

10. Kızıltaş, S., Erdoğdu, F., & Palazoğlu, T. K. (2010). Simulation of heat transfer for solid–liquid food mixtures in cans and model validation under pasteurization conditions. Journal of Food Engineering, 97, 449–456. https://doi.org/10.1016/j.jfoodeng.2009.11.024

11. Ledneva, I., Patsai, L., & Karmyzov, A. (2023). Transformation of logistics supply chains in modern economic conditions. Scientific Works of the Belarusian State University of Economics, 210–216.

12. Lienhard, J. H. (2019). A heat transfer textbook (5th ed.). Courier Dover Publications.

13. Micheev, M. A., & Micheeva, I. M. (1977). Basics of heat exchange (2nd ed.). Energiya.

14. Patel, A. A., Gandhi, H., Singh, S., & Patil, G. R. (1996). Shelf-life modeling of sweetened condensed milk based on kinetics of Maillard browning. Journal of Food Processing and Preservation, 20, 431–451.

15. Paul, D. A., Anishaparvin, A., & Anandharamakrishnan, C. (2011). Computational fluid dynamics studies on pasteurisation of canned milk. International Journal of Dairy Technology, 64, 305–313. https://doi.org/10.1111/j.1471-0307.2011.00687.x

16. Rohsenow, W. M., Hartnett, J. P., & Cho, Y. I. (1998). Handbook of heat transfer (3rd ed.). McGraw-Hill.

17. Roif Expert. (2023). Condensed milk market in Russia - 2023 exports amounted to + 6.5 million $ in the Kazakh direction. VC.ru. Retrieved from https://vc.ru/u/406653-roif-expert/636061-rynok-sgushchennogo-moloka-v-rossii-2023-eksport-sostavil-6-5-mln-po-kazahskomu-napravleniyu

18. Ryabova, A., Tolmachev, V., & Galstyan, A. (2022). Phase transitions of sweetened condensed milk in extended storage temperature ranges. Food Processing: Techniques and Technology, 52, 526–535. https://doi.org/10.21603/2074-9414-2022-3-2379

19. Ryabova, A. E. (2023). Study of the thermophysical properties of sweetened condensed milk. Food Processing Industry, 5(2), 52–55. https://doi:10.52653/ppi.2023.2.2.012

20. Ryabova, A. E., Burkov, I. A., Semipyatny, V. K., et al. (2023). The program for calculating the cooling time of a tin of condensed milk (Certificate of state registration No. 2023662843). Federal State Autonomous Scientific Institution "All-Russian Scientific Research Institute of the Dairy Industry". Application filed June 20, 2023, and published June 27, 2023.

21. Sharma, P., Patel, H., & Patel, A. (2015). Evaporated and sweetened condensed milks. In Dairy Processing and Quality Assurance (pp. 310–332). https://doi:10.1002/9781118810279.ch13

22. Turovskaya, S. N., Kruchinina, G., Illarionova, E., & Bolshakova, E. I. (2024). Study of the critical ambient temperatures' impact on sweetened condensed milk. Bulletin of KrasGAU, 189–200. https://doi:10.36718/1819-4036-2024-1-189-200

23. Zhu, S., Li, B., & Chen, G. (2022). Improving prediction of temperature profiles of packaged food during retort processing. Journal of Food Engineering, 313, 110758. https://doi.org/10.1016/j.jfoodeng.2021.110758