In this paper, an improved probability model is introduced to provide a more comprehensive prediction of annular flow. Unlike previous work, which did not consider atomization and deposition, as well as breakup and coalescence, simultaneously, the improved model integrates all four processes into its framework. The mechanisms of these processes are described in detail by the present model. When annular flow is fully developed, the four processes reach dynamic equilibrium. A numerical program is compiled based on the influence of the four processes on the droplets. Five important parameters, including the droplet-diameter probability density distribution, characteristic droplet diameters, entrainment ratio, liquid film thickness, and interfacial shear-stress coefficient, are calculated when the annular flow is in dynamic equilibrium. The validity and accuracy of the improved model are assessed by comparison with 377 cases from 12 experiments. For most cases used to validate the predicted droplet-diameter probability density distribution, the prediction curves closely match the experimental data. The mean absolute percentage errors (MAPEs) for the two experiments used to validate the predicted characteristic droplet diameters are 20.74% and 24.04%, respectively. Additionally, the mean MAPEs of the entrainment ratio, liquid film thickness, and interfacial shear-stress coefficient are found to be 34.11%, 20.60%, and 27.28%, respectively. This demonstrates the effectiveness and reliability of the improved probability model in predicting annular flow. 

Probability model for gas–liquid annular flow in dynamic equilibrium among the processes of atomization, deposition, breakup, and coalescence


 The equivalent Markov motion model (EMMM) is used to improve the probability model for solid particle erosion and droplet deposition. When calculating the solid particle erosion, the probability density distribution of particle velocity developed by EMMM is used to improve the solution of the impact parameters of solid particles. When calculating the droplet deposition rate of annular flow, the conversion coefficient proposed by EMMM is used to correct the variance of the particle displacement. The proportion of the particles deposited is calculated by solving the probability of a single particle reaching the pipe wall or liquid film. The improved model involving the EMMM is well verified by comparison with experimental data. Compared with the previous probability model, the present model can provide more accurate predictions. In addition, the self-compatibility of the EMMM is discussed in detail.

Application of the Equivalent Markov Motion Model in Solving Solid Particle Erosion and Droplet Deposition

A novel model is proposed for analyzing high-concentration granular flow systems comprising equally sized spherical particles within vertical, long straight pipelines. This model is specifically tailored for simulating the vertical hydraulic transport of ore particles in marine mining projects. The proposed model treats the granular system akin to a pseudo-fluid and operates through three mechanisms. First, fluid characteristics of the granular system are derived from particle–particle collisions. Second, the resistance exerted by the pipe wall on the granular system is calculated based on the momentum loss of particles during particle–wall collisions. Third, the interaction between individual particles and the surrounding fluid is transformed into an interaction between the carrier fluid and the pseudo-fluid. Additionally, the present work develops a dedicated numerical format and iterative method for solving the one-dimensional two-fluid governing equations. The one-dimensional (1D) model notably enhances computational efficiency and facilitates accurate tracking of high-concentration particles over extended distances within straight pipelines. Notably, the proposed 1D model demonstrates a high degree of predictive accuracy when compared against experimental data as well as results from computational fluid dynamics and discrete element method simulations. 

One-dimensional model for vertical hydraulic transport of high-concentration mineral particles

Steady-state modeling and analysis of solid–liquid two-phase flow in vertical and flexible pipelines for deep-sea mining

Abstract Droplet deposition on a turbine cascade is important for the turbine system performance but its experimental analysis is difficult. Thus, the simulation of a droplet liquid phase in a gas flow field was studied to optimize turbine cascade design. However, the computational fluid dynamics (CFD)-based approach for such droplet problems requires enormous costs. Thus, the application of CFD simulation in the turbine blade's early-stage design is challenging, requiring iterative optimization for adjusting design with performance prediction. Therefore, this study proposed an analytical prediction method, having a reasonable cost and moderate accuracy, as an alternative to the whole multi-phase numerical simulation approach. The proposed method predicts droplet motion using the outline of the turbine blade and gas–liquid physical properties. Further, the approach was validated by performing a three-dimensional Eulerian–Lagrangian simulation with low-pressure turbine blade T106. It was found that the droplet trajectories in the turbine cascade are governed by Stokes number. Furthermore, the streamlines of the gas flow were characterized by the shape of the turbine blade. The proposed model reproduced droplet trajectories obtained using CFD within an error margin of 10%. Consequently, it was concluded that the proposed analytical model is a promising approach to predict the droplet trajectory in a turbine cascade, obtained by the three-dimensional numerical simulation at a low cost. 

Pragmatic Prediction Model of Droplet Trajectory in a Turbine Cascade

\n Two methods are used to study the process of particle deposition in a turbulent pipe flow. The Monte Carlo method tracks 10,000 particles in the turbulent pipe flow to reproduce the deposition process of the particles. The deposition velocity of the particles is determined by calculating the proportion of particles passing through the test section. The simplified deposition model uses an equivalent Markov motion instead of the radial movement of the particle in the turbulent core. The probability that a particle leaves the turbulent core depends on the radial particle position and the probability density distribution of the random vortex. The probability that a particle penetrates the boundary layer can be solved by integrating the probability density distribution of radial particle velocity. The deposition velocity of particles can be obtained by calculating the probability of an individual particle leaving the turbulent core and penetrating the boundary layer. Five experimental data series from the literature are applied to examine the predictive abilities of the two methods. The results demonstrate that the Monte Carlo method can be properly used to track the particle deposition process in the diffusion–impaction and inertia-moderated regimes. The simplified model is suitable for high-inertia particles.

Deposition Process and Equivalent Markov Motion of High-Inertia Particles in a Long Straight Pipeline

Hydraulic gradient analysis in vertical hydraulic transport for coarse particles

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Application of the Equivalent Markov Motion Model in Solving Solid Particle Erosion and Droplet Deposition

Deposition Process and Equivalent Markov Motion of High-Inertia Particles in a Long Straight Pipeline

Hydraulic gradient analysis in vertical hydraulic transport for coarse particles