Foods and Raw Materials Foods and Raw Materials Foods and Raw Materials 2308-4057 2310-9599 3357 10.12737/5472 BIOTECHNOLOGY BIOTECHNOLOGY BIOTECHNOLOGY Studing the Foaming of Protein Solutions by Stochastic Methods Studing the Foaming of Protein Solutions by Stochastic Methods https://orcid.org/0000-0002-1252-9572 Иванова Светлана Анатольевна Ivanova Svetlana A. pavvm2000@mail.ru

доктор технических наук;

doctor of technical sciences;

 ФГБОУ ВО «Кемеровский государственный университет» Кемерово Россия Kemerovo State University Kemerovo Russian Federation 01 09 2014 01 09 2014 2 2 140 146 https://jstrategizing.kemsu.ru/en/nauka/article/3357/view

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.

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.

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

INTRODUCTIONDispersed gas–liquid systems (foams) in both the liquid and solid form find wide application in different industries (oil-and-gas, food, and metallurgical industries, firefighting, etc.). The mechanism of the foam formation process is complicated due to the combined effect of numerous physicochemical, physicotechnical, and other factors. The regularities governing the formation of dispersed systems strongly depend on the conditions of a chemical engineering or engineering process, and both the formation and destruction of an obtained gas–liquid layer occur simultaneously in the process of foam generation [1–14]. As a consequence, these features complicate to a great extent the mathematical description of the foaming process [3–5, 15–18].Among the principal characteristics of a foam are the expansion factor (foam-to-solution volumetric ratio), the dispersion (air bubble size), and the stability (time period from the formation of a foam to its partial or complete destruction) [3–8]. The foam stability applicable to any foam independently of its purpose may be considered as a basic characteristic.It is known that foams based on protein solutions, an increase in the concentration of which improves the foaming properties of a system as a whole, are highly stable [9, 10, 19–24]. The formation of bubbles generally depends on the composition of a foamed solution (foaming agent) and the intensity of a mechanical action, whereas their destruction proceeds under the action of both internal and external forces. For this reason, the entire foam generation process representing a process flow may be considered as a dynamic system of flows or a queueing system. This queueing system will be studied by the methods of stochastic processes and queueing theory [25–27].The objective of this study is to create a stochastic model, which would systematically describe the processes of foaming in protein solutions and determine the time of the formation of a foam of specified quality.OBJECTS AND METHODS OF STUDYFoam formation regularities were studied via the gas saturation of a protein solution (skim milk protein concentrate; protein mass fraction, 4.4%) in a rotor-stator device (GID-100/1 hydrodynamic disperser, which was developed, manufactured, and mounted in the All-Russian Research Institute of Dairy Industry) at a rotor revolution speed from 1750 to 3000 rpm, a working chamber filling coefficient of 0.3, a rotor-stator gap of 0.1 mm, and a processed solution temperature of 13 ± 2°C. Control measurements were performed each three minutes after the freezing of samples in a nitrogen atmosphere and their transmitted-light microscopy on an AxioVert.A1 microscope with an AxioCamERc5 camera and a photo recording block. The number of bubbles was calculated from digital images with the use of corresponding software (the comparison of automatic and manual calculation results for the number of bubbles in the frozen samples shows that the former were underestimated by 18% on average). 

Fridrikhsberg, N.N., Kurs kolloidnoi khimii (Course in Colloid Chemistry) (Khimiya, St. Petersburg, 1995). Fridrikhsberg, N.N., Kurs kolloidnoi khimii (Course in Colloid Chemistry) (Khimiya, St. Petersburg, 1995). Gel’fman, M.I., Kovalevich, O.V., and Yustratov, V.P., Kolloidnaya khimiya (Colloid Chemistry) (Lan’, St. Petersburg, 2003). Gel’fman, M.I., Kovalevich, O.V., and Yustratov, V.P., Kolloidnaya khimiya (Colloid Chemistry) (Lan’, St. Petersburg, 2003). Kruglyakov, P.M. and Ekserova, D.R., Pena i pennye plenki (Foam and Foam Films) (Khimiya, Moscow, 1990). Kruglyakov, P.M. and Ekserova, D.R., Pena i pennye plenki (Foam and Foam Films) (Khimiya, Moscow, 1990). Tikhomirov, V.K., Peny. Teoriya i praktika ikh polucheniya i razrusheniya (Foams. Theory and Practice of Their Formation and Destruction) (Khimiya, Moscow, 1983). Tikhomirov, V.K., Peny. Teoriya i praktika ikh polucheniya i razrusheniya (Foams. Theory and Practice of Their Formation and Destruction) (Khimiya, Moscow, 1983). Kann, K.B., Kapillyarnaya gidrodinamika pen (Capillary Hydrodynamics of Foams) (Nauka, Sib. Otd., Novosibirsk, 1989). Kann, K.B., Kapillyarnaya gidrodinamika pen (Capillary Hydrodynamics of Foams) (Nauka, Sib. Otd., Novosibirsk, 1989). Bikerman, J., Foams (Springer, Berlin, 1973). Bikerman, J., Foams (Springer, Berlin, 1973). Schramm, L.L., Emulsions, Foams, and Suspensions: Fundamentals and Applications (Wiley-VCH, Weinheim, 2005). Schramm, L.L., Emulsions, Foams, and Suspensions: Fundamentals and Applications (Wiley-VCH, Weinheim, 2005). Merkin, A.P. and Traube, P.R., Neprochnoe chudo (Fragile Miracle) (Khimiya, Moscow, 1983). Merkin, A.P. and Traube, P.R., Neprochnoe chudo (Fragile Miracle) (Khimiya, Moscow, 1983). Walstra, P., Physical Chemistry of Foods (Marcel Dekker, New York, 2003). Walstra, P., Physical Chemistry of Foods (Marcel Dekker, New York, 2003). Dickinson, E., Food Emulsions and Foams (Royal Soc. Chem., London, 1987). Dickinson, E., Food Emulsions and Foams (Royal Soc. Chem., London, 1987). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Vetoshkin, A.G. Hydromechanics of separation of gas-liquid systems with foam structure, Theor. Found. Chem. Eng., 2004. V.38. №6. P. 569-574. Vetoshkin, A.G. Hydromechanics of separation of gas-liquid systems with foam structure, Theor. Found. Chem. Eng., 2004. V.38. №6. P. 569-574. 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. 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. 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. 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. 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. 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. Britten, M. and Lavoie, L., Foaming properties of proteins as affected by concentration, J. Food Sci., 1992. V. 57. №5. P. 1219-1241. Britten, M. and Lavoie, L., Foaming properties of proteins as affected by concentration, J. Food Sci., 1992. V. 57. №5. P. 1219-1241. Damodaran, S., Protein stabilization of emulsions and foams, J. Food Sci., 2005. V.70. №3. P. 54-66. Damodaran, S., Protein stabilization of emulsions and foams, J. Food Sci., 2005. V.70. №3. P. 54-66. 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. 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. Saaty, T., Elements of Queueing Theory with Application (Dover Publications, New York, 1961). Saaty, T., Elements of Queueing Theory with Application (Dover Publications, New York, 1961). 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). 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). Kleinrock, L., Queueing Systems. Volume I: Theory (Wiley, New York, 1975). Kleinrock, L., Queueing Systems. Volume I: Theory (Wiley, New York, 1975). Feller, W., An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968). Feller, W., An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968). 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). Ivanova, S.A. and Prosekov, A.Yu., Intensifikatsiya tekhnologii aerirovaniya molochnykh productov (Intensification of Dairy Product Aeration Technologies) (KemTIPP, Kemerovo, 2011). Ivanova, S.A. and Prosekov, A.Yu., Intensifikatsiya tekhnologii aerirovaniya molochnykh productov (Intensification of Dairy Product Aeration Technologies) (KemTIPP, Kemerovo, 2011).