Studing the Foaming of Protein Solutions by Stochastic Methods

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.