Abstract and keywords
Abstract (English):
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


Dispersed 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.


Foam 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).



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