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An equivalent analytical model of sloshing in a two-dimensional (2-D) rigid rectangular container equipped with multiple vertical baffles is presented. Firstly, according to the subdomain partition approach, the total liquid domain is partitioned into subdomains with the pure interface and boundary conditions. The separation of variables is utilized to achieve the velocity potential for subdomains. Then, sloshing characteristics are solved according to continuity and free surface conditions. According to the mode orthogonality of sloshing, the governing motion equation for sloshing under horizontal excitation is given by introducing generalized time coordinates. Besides, by producing the same hydrodynamic shear and overturning moment as those from the original container-liquid-baffle system, a mass-spring analytical model of the continuous liquid sloshing is established. The equivalent masses and corresponding locations are presented in the model. The feasibility of the present approach is verified by conducting comparative investigations. Finally, by utilizing normalized equivalent model parameters, the sloshing behaviors of the baffled container are investigated regarding baffle positions and heights as well as the liquid height, respectively.

Sloshing in liquid storage tanks is a critical phenomenon that affects the stability, performance, and safety of various engineering systems, including fuel tanks, offshore structures, and industrial storage units. The presence of internal structures, such as vertical baffles, significantly influences the natural sloshing frequencies and fluid motion. However, existing theoretical models often rely on simplified assumptions that restrict their capacity to capture the complexities of fluid–structure interactions in baffled tanks. This study integrates the coupled Eulerian–Lagrangian method with the impulse excitation technique to predict natural sloshing frequencies in a rectangular tank with vertical baffles. By analyzing the system’s response to an impulse excitation, we extracted the dominant sloshing frequencies while considering the impact of baffles on fluid dynamics. This computational approach provides a more realistic representation of sloshing phenomena and enables a parametric analysis of how various tank dimensions, fluid properties, and baffle configurations influence sloshing behavior. The findings of this study contribute to the improved design and optimization of liquid storage tanks, ensuring enhanced stability and performance in practical engineering applications. The integration of impulse excitation with the coupled Eulerian–Lagrangian method marks a significant advancement in sloshing analysis, offering a robust framework for understanding and mitigating the effects of sloshing in baffled tanks.

Violent sloshing induced by excitation with large amplitudes or resonant frequencies may result in structural damage of the liquid-tank or even the overturning of the liquid cargo transport system. Therefore, impermeable and permeable vertical baffles were investigated numerically to suppress sloshing. The numerical simulations were based on the finite element method and arbitrary Lagrangian–Eulerian (ALE) method. The numerical model was verified by the available experimental data, numerical results and linear theoretical results. Based on the study of the effects of impermeable baffle height, amplitude and frequency of excitation on sloshing, the effects of baffle permeability on sloshing were investigated. Importantly, a critical permeability coefficient that was most effective to suppress sloshing was found. In addition, the maximum flow velocities in the tank with a baffle of small permeability coefficient were smaller than those in the tank with an impermeable baffle. While, the maximum flow velocities under a baffle of large permeability coefficient were larger than those in the tank with an impermeable baffle. Vortices were observed in the whole region of the baffle, tank bottom, tank walls and the free surface in the tank with a permeable baffle.

Rectangular aqueducts are critical building structures in large-scale water conveyance systems used worldwide. Liquid sloshing can produce hydrodynamic forces that threaten structural safety and long-term performance. This study analytically investigates the vibration characteristics of two-dimensional rectangular cantilever aqueduct systems while accounting for soil–structure interaction (SSI). To reduce sloshing and enhance the performance of the mechanical system, a bottom-mounted vertical baffle is proposed as a hydrodynamic damping solution. Through subdomain analysis, mathematical expressions for liquid potential fields are derived. The continuous liquid is represented through discrete mass–spring elements for dynamic analysis. Horizontal soil impedance is characterized by using Chebyshev orthogonal polynomial approximations with optimized least squares fitting techniques. A dynamic mechanical model for the soil–aqueduct–liquid–baffle coupling system is developed by using the substructure method. Convergence and comparative studies are conducted to validate the reliability of the proposed method. Between the current results and those reported previously, the variation in the first-order sloshing frequency is less than 1.10%. Parametric analyses evaluate how baffle size, baffle position, and soil properties influence sloshing behavior. The presentation of an equivalent analytical model is the novelty of this research. The results can provide the theoretical basis for optimizing anti-sloshing designs in hydraulic building structures, thereby supporting safer and more sustainable engineering practices.

The hydrodynamics and heat transfer in a reactor with a fluidized bed of catalyst particles and an inert material were simulated. The particle bed (the particle density was 2350 kg/m3, and the particle diameter was 1.5 to 4 mm) was located in a distribution device which was a grid of 90 × 90 × 60 mm vertical baffles. The behavior of the liquefying medium (air) was modeled using a realizable k-ε turbulence model. The behavior of particles was modeled using the discrete element method (DEM). In order to reduce the slugging effect, the particles were divided into four separate horizontal layers. It was determined that with the velocity of the liquefying medium close to the minimum fluidization velocity (1 m/s), slugging fluidization is observed. At a velocity of the liquefying medium of 3 m/s, turbulent fluidization in the lowest particle layer and bubbling fluidization on subsequent particle layers are observed. With an increase in the velocity of the liquefying medium over 3 m/s, entrainment of particles is observed. It was shown that a decrease in the density of the liquefying medium from 1.205 kg/m3 to 0.383 kg/m3 when it is heated from 298 K to 923 K would not significantly affect the hydraulic resistance of the bed. Based on the obtained results, it can be stated that the obtained model is optimal for such problems and is suitable for the further description of experimental data.

Understanding the damping mechanism of baffles is helpful to make more reasonable use of them in suppressing liquid sloshing. In this study, the damping effect and mechanism of vertical baffles in shallow liquid sloshing under a rotational excitation are investigated by an improved particle method. By incorporation of a background mesh scheme and a modified pressure gradient model, the accuracy of impact pressure during sloshing is significantly enhanced. Combined with the advantages of the particle method, the present numerical method is a wonderful tool for the investigation of liquid sloshing issues. Through the analysis of impact pressure, the influences of baffle height and baffle position on the damping mechanism are discussed. The results show that the damping effect of vertical baffles increases with the increase of the elevation of baffle top and decreases with the increase of the elevation of the baffle bottom. Moreover, the resonance characteristics of sloshing are altered when static water is divided into two parts by the vertical baffle. The dominant damping mechanism of vertical baffles depends on the configurations.

The influence of the impeller type, impeller speed, volumetric gas flow rates, type of sugar and type of yeast on the hydrodynamics (gas hold-up and residence time of gas bubbles) in a vessel with 24 vertical tubular baffles has been presented in this paper. The measurement of hydrodynamics was conducted in the vessel with inner diameter D = 0:288 m and liquid height of H = 0:288 m. Three different agitators were used in the experimental study. Seven different series, three of which refer to the two-phase system and four to the three-phase system were mixed in the vessel. The influence of gas flow number, Weber number, the mass fraction of aqueous sugar solution ci , and mass fraction of yeast suspension yss for twoand three-phase systems on the gas hold-up φ were described mathematically. These equations do not have equivalents in the literature.

A finite difference model for solving Navier Stokes viscous liquid sloshing-wave interaction with baffles in a tank. equations with turbulence taken into accotmt is used to investigate The volume-of-fluid and virtual boundary force methods are employed to simulate free surface flow interaction with structures. A liquid sloshing experimental apparatus was established to evaluate the accuracy of the proposed model, as well as to study nonlinear sloshing in a prismatic tank with the baffles. Damping effects of sloshing in a rectangular tank with bottom-mounted vertical baffles and vertical baffles touching the free surface are studied numerically and experimentally. Good agreement is obtained between the present numerical results and experimental data. The numerical results match well with the current experimental data for strong nonlinear sloshing with large free surface slopes. The reduction in sloshing-wave elevation and impact pressure induced by the bottom-mounted vertical baffle and the vertical baffle touching the free surface is estimated by varying the external excitation frequency and the location and height of the vertical baffle under horizontal excitation.

The influence of the agitator type, agitator speed, superficial gas velocity, type of sugar (glucose or sucrose) and the presence of yeast in the system on the gas hold-up in an agitated vessel with 24 vertical tubular baffles (located on the circuit in the vessel) has been presented in this paper. The measurement of gas hold-up was conducted in an agitated vessel with inner diameter of D = 0.288 m and liquid height of H = 0.288 m. Three different agitators were used in the experimental study. Five gas-liquid and two biophase-gas-liquid systems were agitated in an agitated vessel. Air was used as gas. The influence of gas flow number, Weber number, the mass fraction of aqueous sugar solution ci, and mass fraction of yeast suspension ys for gas-liquid and biophase-gas-liquid systems on the gas hold-up ϕ was described mathematically. These equations do not have equivalents in the literature.

The influence of impellers system and type of liquid on the gas hold-up in the vessel has been presented in this paper. The analysis of gas hold-up was conducted on the basis of the data obtained in the vessel of the diameter D = 0.288 m, where the vessel was filled by a liquid up to the height H = 2D. The vessel was equipped in 24 vertical tubular baffles located on the circuit and two high-speed impellers situated on a shaft. Five different configurations of high-speed impellers were employed. The experiments in the gas-liquid system were conducted for setups which differed in capability of gas bubbles coalescence. The results of the experiment of the gas holdup for the five impellers configurations and four gas-liquid systems were presented in the graphic form and they were described mathematically.