Based on the concept of discontinuity displacement, an analytical solution for cracked thin plates has been derived in which displacements and stresses in a solid can be expressed by the linear distributed discontinuity displacements on the whole boundary. By way of the potential variational principle and the analytical solution newly developed, a boundary element model for 2D multiple crack problems has been presented and applied to fracture and damage analysis of thin plates with many cracks. Two numerical examples are considered to illustrate applications of the proposed element model. #
Liquid-liquid extraction is an important separation technology in the chemical industry, and its separation efficiency depends on thermodynamics (two-phase equilibrium), hydrodynamics (two-phase mixing and contact), and mass transfer (molecular diffusion). For hydrodynamics, the dispersion size of droplets reflects the mixing of two phases and determines the mass transfer contact area of the two phases. Therefore, a deep understanding of the droplet dispersion mechanism can help guide process intensification. The mass transfer and droplet behaviors in the liquid-liquid extraction process are reviewed based on three scales: equipment, droplets, and the interface between two liquids. Studies on the interaction between mass transfer and other performance parameters in extraction equipment as well as liquid-liquid two-phase flow models are reviewed at the equipment scale. The behaviors of droplet breakage and coalescence and the kernel function of the population balance equation are reviewed at the droplet scale. Studies on dynamic interfacial tension and interaction between interfaces are reviewed at the interface scale. Finally, the connection among each scale is summarized, the existing problems are analyzed, and some future research directions are proposed in the last section.
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