A Numerical Solution for the Coexisting Field of Surface and Internal Solitary Waves

The numerical solutions for the coexisting fields of surface and internal solitary waves have been obtained, where the set of nonlinear equations based on the variational principle for steady waves are solved using the Newton- Raphson method. The relative phase velocity of surface-mode solitary waves is smaller in the coexisting fields of surface and internal solitary waves than in the cases without the coexistence of internal waves. The relative phase velocity of internal-mode solitary waves is also smaller in the coexisting fields of surface and internal solitary waves than in the cases without surface waves. The interfacial position of an internal mode internal solitary wave in a coexisting field of surface and internal waves can exceed the critical level determined in the corresponding case without a surface wave. The wave height ratio between internal-mode surface and internal solitary waves is smaller than the corresponding linear shallow water wave solution, and the difference increases, as the relative wave height of internal-mode internal solitary waves is increased.

Hold-up and flooding in a vibrating plate extractor

The mean volumetric hold-up of the dispersed phase and the limiting flow rates of phases in five sizes of experimental vibrating plate extractors VPE were measured and the results compared with the relations of Richardson and Zaki, Míšek and Ishii and Zuber for the relative phase velocity. The experiments were performed with the system toluene-water, toluene dispersed. All the relations were found to describe the dependence of hold-up on flow rates very well. They also allowed a satisfactory estimation of the limiting phase velocities. Within the range of the experimental conditions the characteristic velocity as well as the limiting phase velocities could be approximated by linear functions of the frequency of vibrations. For the columns studied no influence of column size on limiting velocities was detected. The deviations of individual columns from geometrical similarity, however, brought about an appreciable effect both on the hold-up and on the flooding rates. This phenomenon has been explained in terms of longitudinal hold-up profiles, the shape of which depends on the boundary conditions in the particular column. This explanation was corroborated by case studies using a polydisperse mathematical model which show that even in the presence of hold-up profiles the effect of phase velocities on the mean hold-up can be correlated by means of the investigated relations with constant parameters and that the parameters depend on the hold-up profiles form.