We present a modeling approach to determine the permeability-selectivity tradeoff for microfiltration and ultrafiltration membranes with a distribution of pore sizes and pore shapes. Using the formulated permeability-selectivity model, the effect of pore aspect ratio and pore size distribution on the permeability-selectivity tradeoff of the membrane is analyzed. A finite element model is developed to study the effect of membrane stretching on the distribution of pore sizes and shapes in the stretched membrane. The effect of membrane stretching on the permeability-selectivity tradeoff of membranes is also analyzed. The results show that increasing pore aspect ratio improves membrane performance while increasing the width of pore size distribution deteriorates the performance. It was also found that the effect of membrane stretching on the permeability-selectivity tradeoff is greatly affected by the uniformity of pore distribution in the membrane. Stretching showed a positive shift in the permeability-selectivity tradeoff curve of membranes with well-dispersed pores while in the case of pore clustering, a negative shift in the permeability-selectivity tradeoff curve was observed.
Drillstring vibration is one of the major causes for a deteriorated drilling performance, and if left untreated may result in a complete failure of the drilling process. Although the variations in the drilling load, stick-slip, and whirling are known to be the primary causes of severe vibrations, they often give rise to excessive flexural deformations and subsequent string—borehole interaction. Drillstring contact-impact interaction with borehole results in high-frequency excitations, which further deteriorate the drilling performance, and may cause damage to the bottomhole assembly. Modelling of impact is crucial to understanding the associated dynamic response, and to providing means for controlling the collision phenomenon. A continuous force—displacement law is introduced to model the impulsive force during the short-lived interval of impact, wherein the material compliance and damping coefficients are determined from energy balance relations. The impact model is integrated to the dynamic model of the whole drillstring; including drillpipes and drillcollars. The dynamic model of the rotating drillstring is formulated using a Lagrangean approach in conjunction with the finite-element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric non-linear coupling. In addition, the model accounts for the gyroscopic effect and the effect of the gravitational force field. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of string—borehole collisions.
This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson's ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.
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