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In this paper, the complex flow evaporation process of droplet impact on the liquid film in a horizontal falling film evaporator is numerically studied based on smoothed particle hydrodynamics (SPH) method. The purpose of this paper is to present the mechanism of the water treatment problem of the falling film evaporation for the high salinity mine water in Xinjiang region of China.

To effectively characterize the phase transition problem, the particle splitting and merging techniques are introduced. And the particle absorbing layer is proposed to improve the nonphysical aggregation phenomenon caused by the continuous splitting of gas phase particles. The multiresolution model and the artificial viscosity are adopted.

The SPH model is validated qualitatively with experiment results and then applied to the evaporation of the droplet impact on the liquid film. It is shown that the larger single droplet initial velocity and the smaller single droplet initial temperature difference between the droplet and liquid film improve the liquid film evaporation. The heat transfer effect of a single droplet is preferable to that of multiple droplets.

A multiphase SPH model for evaporation after the droplet impact on the liquid film is developed and validated. The effects of different factors on liquid film evaporation, including single droplet initial velocity, single droplet initial temperature and multiple droplets are investigated.

This study highlights the impact of mental accounts on a user's decision-making regarding payment schemes and aims to determine the pricing strategy for the first-enjoy-after-pay service offered by the two-sided media platforms.

This study establishes a game-theoretic model and utilizes backward induction to derive the equilibrium price by maximizing the monopolist's profit.

The findings indicate that the conditions for a two-sided media platform to offer the first-enjoy-after-pay service depend on the trade-off between pleasure attenuation and pain buffering and the effect of time discounts. Moreover, the authors found that the time discount is a critical factor in determining pricing strategies under various payment schemes offered by the platform.

This work adopts a uniform pricing strategy for users who opt for either immediate or post-payment schemes. Nevertheless, it is important to note that this approach has limitations in terms of offering discriminatory pricing for those who choose both payment schemes.

This analytical work provides valuable insights for two-sided media platforms to optimize their payment scheme strategies and pricing considering the influence of a user's mental account.

In a two-sided media platform, the authors provide applicable conditions for the platform to offer first-enjoy-after-pay service considering the effect of mental accounts. Further, the authors show the optimal pricing strategy under different payment schemes provided by the platform.