The ensemble-averaged characteristics of the bubble system during cavitation in the separator

The object of research is the process of formation of vapor-gas bubbles at an early stage of hydrodynamic cavitation during throttling of the liquid in the axial valve separator. Designing a separator under conditions of decreasing bubble formation intensity is an urgent problem. Based on the energy approach, a stochastic model of the formation of an energetically closed system of bubbles in the separator according to the Gibbs ensemble is proposed. Analytical dependences are obtained for the density function and the bubble geometric characteristics averaged over the ensemble, depending on the design parameters, operating parameters, and physic-mechanical properties of the working medium, including the ratio of the gas and vapor contents in the inner cavity of the bubble. The expressions for the indicated distribution function over the size of the bubbles and for the ensemble-averaged characteristics of the system of bubbles formed (diameter and surface area) can be used in the formation of an engineering methodology for calculating control equipment.


Introduction
The phenomenon of bubble cavitation is of particular interest for multiphase flow processes in axial valve separators. An active study of this phenomenon is associated with the need to design equipment with an increased durability due to erosion of the inner surfaces of the valves with a guaranteed safe limit of noise effects and vibrations of structural elements. Studying the mechanism of the appearance of vapor-gas bubbles in the indicated flows at the stage of initial hydrodynamic cavitation before their growth and collapse helps to further predict the behavior of the bubble system at the developed stage of cavitation. The problem of describing this mechanism was raised in classical works [1][2][3].
Evaluation of the size of the bubble in the valve separator depends on many factors, including the physical state of the liquid medium, a set of design and operational parameters of the device. The identification of valve operating modes with a reduced intensity of cavitation bubble formation is one of the most important tasks in the design of pipeline valves. However, the development of the theoretical foundations of the bubble cavitation process at its early stage remains an urgent problem [4,5]. Known deterministic approaches to the description of the mechanism of bubble cavitation associated with solving systems of equations of mechanics of heterogeneous media have recently become dependent on the implementation of numerical simulation methods [6][7][8].
Stochastic approaches [9] in solving this problem, as a rule, are based on kinetic equations obtained by analogy with equations from the theory of Brownian particle motion for the nucleation process [10,11] in media with different degrees of viscosity taking into account surface effects. The variety of problems in modeling the nucleation process is determined by a homogeneous [10,11] (or modified homogeneous [12]) and heterogeneous [13] mechanisms. Moreover, in the latter case, features of the formation of vapor bubbles on the boundary surfaces of various nature are revealed [14], including in electrochemical systems [15]. In addition, in the simulation of bubble cavitation, the Euler-Lagrange approach is used, when it is proposed to use various variables to describe the behavior of the phases of the hydrodynamic system taking into account the influence of interphase transfers [16], including when combined with the process of liquid degassing [17]. The density functions used in the models for bubble cavitation are often semi-empirical in nature, and the desired solutions are numerically represented.
The well-known stochastic approach presented in [18] allows one to describe random processes from the standpoint of both the energy closure of the system by the Gibbs ensemble and the energy openness of the system in the presence of macroscale fluctuations of its states. In particular, the application of the energy method [19] for the latter case was tested by the authors on the example of modeling the motion of rarefied particles during their mixing [19,20]. The aim of this work is to obtain analytical dependences for the ensemble-averaged characteristics of the system of generated bubbles in the axial valve separator, taking into account the main parameters of its design [21] and the safe operation mode. The practical application of the results relates to the design of regulatory equipment in conditions of decreasing intensity of bubble cavitation. To achieve this goal, it is necessary to identify the most significant factors affecting the appearance of bubbles in the axial valve separator, set a sufficient set of input parameters for the stochastic model based on the approach [18], and form a differential function for the distribution of bubbles by their size based on the works of the authors [22][23][24].

Methods
Suppose that when a fluid moves with a density ρ L in the axial valve separator [21], a system of bubbles is formed at an early stage of hydrodynamic cavitation and is a macro-system of spherical cavities of radius r filled with steam and gas in the ratio : α α According to the stochastic approach [18], the construction of a stationary solution of the Fokker-Planck kinetic equation in the energy representation for the equilibrium bubble distribution function is performed depending on the indicated phase variables [25]. It is indicated here: is the energy of the bubble during its stochastic movement; 0 E is the energy parameter equal to the stationary value of the bubble energy at the time of its stochastization.
Then the differential distribution function of the bubbles by their specific sizes and the number of bubbles in the phase volume element du are determined by the expressions In accordance with the method of constructing the bubble energy during its stochastic motion ( , ) λ µ E , proposed in [22], we have where the dependences 1 2 ( ), ( ) λ λ q q on the specific radius of the bubble are indicated as follows The coefficients , 1, 5 = j k j in (4), (5) are given by the expressions: b is the number of rows. Expression (5) includes a parameter M as a random component of the angular momentum of the vapor-gas system inside the bubble, the value of which is determined using the calculation method proposed in [22,24] together with the energy parameter 0 E .In this case, the thermodynamic relation is used for the adiabatic process for the vapor-gas system inside the bubble [26], as well as the balance equation between the energy of the bubble system during their stochastic movement and the energy that is spent on the hydrodynamic disruption of the fluid flow over a given period of time [22,24].
The substitution of expression (3) for the bubble energy during its stochastic movement ( , ) λ µ E with auxiliary dependences 1 2 ( ), ( ) λ λ q q from (4), (5) into expression (2) leads to the desired expression for the differential function of the distribution of bubbles by their specific sizes according to definition (1).
In addition, the limiting value max vy K for the conditional valve capacity vy K (m 3 /h) was taken into account at its maximum degree of opening z = 1 according to the simulation results of the differential distribution function of the bubbles with respect to the parameter z from [23,24,26]. Therefore, from (1) 3/ 2 2 2 1/ 2 2 max max 3/ 2 4 5/ 2 2 max max max 1 [ ] 2 − γ = θ . Obtained in expression (6) of the differential function of the distribution of bubbles by their specific sizes 1 ( ) ϕ λ , taking into account (7) -(13), can be used to calculate the ensemble average characteristics of the radius, diameter and surface for the macroscopic system of bubbles in the region of the valve separator using the expressions As shown by practical calculations, when performing integration on the right-hand sides of expressions (14), (15) according to (1), (2) for satisfactory convergence with experimental data [26], it is sufficient to take into account linear approximations for the error functions  (14), (15) for the values of the ensemble average characteristics of the diameter and surface for the macroscopic system of bubbles at the early stage of hydrodynamic cavitation in the axial valve separator take the form where the coefficients 1 2 , , , 1, 3 ψ ψ ε = j j are given by expressions (7), (8), (10) and correspond to the design parameters of the separators, the operational characteristics of the liquid throttling process, and the physical and mechanical properties of the system; the constants are respectively equal 3 4 4 ε = π , 5 5 / 4 ε =π .

Results and Discussion
Set the input parameters of the model. Design parameters for the axial valve separator: hole diameter is ; number of rows is 2 5 = b . Performance parameters for the process of throttling the liquid in the separator (for calculating the parameters M and 0 E according to [22,24]): the maximum achievable liquid flow through the separator is max Using the specified input data, the values of the ensemble mean diameter and surface characteristics , sb sb D S for the macro-system of bubbles in the separator were calculated according to expressions (17), (18). The obtained numerical values of the integral characteristics for the bubble, depending on the degree of opening of the valve opening z and conditional values of the Reynolds criterion Re y , are contained in Table 1. In addition, the corresponding graphs were constructed for the dependences of the indicated bubble characteristics on the main indicators of the liquid throttling process in the axial valve separator, shown in Fig. 1, 2. Endpoints of graphs 1-4 in Fig. 1a 1 and 4, Fig. 1a), but less 1.1 times than with z = 0/8 (see endpoints of graphs 3 and 4, Fig. 1a).  1 and 3, Fig. 1a), and then -a decrease in the characteristic sb D to almost its initial value 1 4 ≈ sb sb D D with an increase in throughput