Abstract

A stochastic model studying the formation and destruction of a dispersed protein gas–liquid system (foam) is proposed. The regularities governing the formation of dispersed systems strongly depend on the conditions of a chemical engineering or engineering process, and both the formation of a foam and the destruction of the obtained foam layer occur simultaneously in the process of foam generation. Since a necessary condition for the construction of a stochastic model is the availability of statistical data, which provide the estimation of the number of both forming and bursting bubbles, the method of such a calculation is of topical interest. The model enables the description of the process state at every time moment of the first cycle. One of the characteristics of a foam is its dispersion, so the random variable characterizing the number of bubble per unit volume is introduced to study the processes of foam formation. The mathematical expectation, dispersion, and also the foam destruction rate function are proposed as a basis for the calculation of foaming efficiency characteristics. Since the model is formalized by a set of differential equations, it can also be used in the simulation modeling of the foaming process. The first cycle of the formation and destruction of a protein foam has systematically been studied. The constructed stochastic model has allowed the mathematical expectation and dispersion of the number of protein foam bubbles per unit volume to be calculated at any time moment of gas saturation within the first cycle. It has been shown that the applied numerical solutions of the differential equations are in good agreement with the analytical solutions given by simple formulas convenient for engineering calculations. A method of estimating the model parameters has been developed. The proposed model has allowed the quantitative description of the foaming process both on average and by states. It has been established that the time of the formation of a protein foam in a rotor-stator device at specified process parameters is advisable to be limited by the moment, at which the highest foam destruction rate is attained.

Keywords

dispersed protein based gas–liquid systems, stability, stochastic model, probability, random variable moments, differential equations, numerical and analytical solution

REFERENCES

1. Fridrikhsberg, N.N., Kurs kolloidnoi khimii (Course in Colloid Chemistry) (Khimiya, St. Petersburg, 1995).
2. Gel’fman, M.I., Kovalevich, O.V., and Yustratov, V.P., Kolloidnaya khimiya (Colloid Chemistry) (Lan’, St. Petersburg, 2003).
3. Kruglyakov, P.M. and Ekserova, D.R., Pena i pennye plenki (Foam and Foam Films) (Khimiya, Moscow, 1990).
4. Tikhomirov, V.K., Peny. Teoriya i praktika ikh polucheniya i razrusheniya (Foams. Theory and Practice of Their Formation and Destruction) (Khimiya, Moscow, 1983).
5. Kann, K.B., Kapillyarnaya gidrodinamika pen (Capillary Hydrodynamics of Foams) (Nauka, Sib. Otd., Novosibirsk, 1989).
6. Bikerman, J., Foams (Springer, Berlin, 1973).
7. Schramm, L.L., Emulsions, Foams, and Suspensions: Fundamentals and Applications (Wiley-VCH, Weinheim, 2005).
8. Merkin, A.P. and Traube, P.R., Neprochnoe chudo (Fragile Miracle) (Khimiya, Moscow, 1983).
9. Walstra, P., Physical Chemistry of Foods (Marcel Dekker, New York, 2003).
10. Dickinson, E., Food Emulsions and Foams (Royal Soc. Chem., London, 1987).
11. Balerin, C., Aymard, P, Ducept, F., Vaslin, S., and Cuvelier, G., Effect of formulation and processing factors on the properties of liquid food foams, J. Food Eng., 2007. V.78. №3. P. 802–809.
12. Narchi, I., Vial, C., Labbafi, M., and Djelveh, G., Comparative study of the design of continuous aeration equipment for the production of food foams, J. Food Eng., 2011. V.102. №2. P. 105–114.
13. Indrawati, L., Wang, Z., Narsimhan, G., and Gonzalez, J., Effect of processing parameters on foam formation using a continuous system with a mechanical whipper, J. Food Eng., 2008. V.88. №1. P. 65–74.
14. Müller-Fischer, N., Suppiger, D., and Windhab, E.J., Impact of static pressure and volumetric energy input on the microstructure of food foam whipped in a rotor-stator device, J. Food Eng., 2007. V.80. №1. P. 306–316.
15. Meshalkin, V.P. and Boyarinov, Yu.G., Semi-markovian models of the functioning of complex chemical engineering systems, Theor. Found. Chem. Eng., 2010. V.44. №2. P. 186–191.
16. Hanselmann, W. and Windhab, E., Flow characteristics and modelling of foam generation in a continuous rotor/stator mixer, J. Food Eng., 1998. V.38. №4. P. 393–405.
17. Ho, Q.T., Carmeliet, J., Datta, A.K., Defraeye, T., Delele, M.A., Herremans, E., Opara, L., Ramon, H., Tijskens, E., van der Sman, R., Liedekerke, P.V., Verboven, P., and Nicolaï, B.M., Multiscale modeling in food engineering, J. Food Eng., 2013. V.114. №3. P. 279–291.
18. Vetoshkin, A.G. Hydromechanics of separation of gas–liquid systems with foam structure, Theor. Found. Chem. Eng., 2004. V.38. №6. P. 569–574.
19. Liszka-Skoczylas, M., Ptaszek, A., and Żmudziński, D., The effect of hydrocolloids on producing stable foams based on the whey protein concentrate (WPC), J. Food Eng., 2014. V.129. P. 1–11.
20. Oboroceanu, D., Wang, L., Magner, E., and Auty, M.A.E., Fibrillization of whey proteins improves foaming capacity and foam stability at low protein concentrations, J. Food Eng., 2014. V. 121. P. 102–111.
21. Indrawati, L. and Narsimhan, G., Characterization of protein stabilized foam formed in a continuous shear mixing apparatus, J. Food Eng., 2008, V. 88. №4. P. 456–465.
22. Britten, M. and Lavoie, L., Foaming properties of proteins as affected by concentration, J. Food Sci., 1992. V. 57. №5. P. 1219–1241.
23. Damodaran, S., Protein stabilization of emulsions and foams, J. Food Sci., 2005. V.70. №3. P. 54–66.
24. Germick, R.J., Rehill, A.S., and Narsimhan, G., Experimental investigation of static drainage of protein stabilized foams―Comparison with model, J. Food Eng., 1994. V.23. №4. P. 555–578.
25. Saaty, T., Elements of Queueing Theory with Application (Dover Publications, New York, 1961).
26. Ventsel’, E.S. and Ovcharov, L.A., Teoriya sluchainykh protsessov i ee inzhenernye prilozheniya (Theory of Stochastic Processes and Its Engineering Applications) (Vysshaya shkola, Moscow, 2000).
27. Kleinrock, L., Queueing Systems. Volume I: Theory (Wiley, New York, 1975).
28. Feller, W., An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968).
29. Pavsky, V.A., Pavsky, K.V., and Khoroshevsky, V.G., Vychisleniye pokazatelei zhivuchesti raspredelennykh vychislitel’nykh sistem i osushchestvimosti resheniya zadachi (Calculating the characteristics of the robustness of distributed computational systems and the feasibility of problem solutions), Iskusstvennyi intellekt (Artificial Intelligence), 2006. №4. P. 28–34.
30. Khoroshevsky, B.G., Pavsky, V.A., and Pavsky, K.V., Raschet pokazatelei zhivuchesti raspredelennykh vychislitel’nykh sistem (Calculation of the robustness characteristics of distributed computational systems), Vestnik Tomskogo Gosudarstvennogo Universiteta. Upravlenie, vychislitel’naya tekhnika i informatika (Tomsk State University Journal of Control and Computer Science), 2011. №2. P. 81.
31. Khoroshevsky, V.G. and Pavsky, V.A., Calculating the efficiency indices of distributed computer system functioning, Optoelectron. Instrum. Data Process., 2008. V. 44. №2. P. 95–104.
32. Yustratov, V.P., Pavskii, V.A., Krasnova, T.A., and Ivanova, S.A., Mathematical modeling of electrodialysis demineralization using a stochastic model, Theor. Found. Chem. Eng., 2005. V.39. №3. P. 259–262.
33. Ivanova, S.A, Stokhasticheskie modeli tekhnologicheskikh protsessov pererabotki dispersnykh sistem obezzhirennogo moloka (Stochastic Models of the Processing of Dispersed Skim Milk Systems) (KemTIPP, Kemerovo, 2010).
34. Ivanova, S.A. and Prosekov, A.Yu., Intensifikatsiya tekhnologii aerirovaniya molochnykh productov (Intensification of Dairy Product Aeration Technologies) (KemTIPP, Kemerovo, 2011).