Conservation equations mathematical characteristics

Mathematical modeling and determination of the characteristic parameters to predict the performance of membrane solvent extraction processes has been studied widely in the literature. The analysis of mass transfer in hollow fiber modules has been carried out using two different approaches. The first approach to the modeling of solvent extraction in hollow fiber modules consists of considering the velocity and concentration profiles developed along the hollow fibers by means of the mass conservation equation and the associated boundary conditions for the solute in the inner fluid. The second approach consists of considering that the mass flux of a solute can be related to a mass transfer coefficient that gathers both mass transport properties and hydrodynamic conditions of the systan (fluid flow and hydrodynamic characteristics of the manbrane module). 

If we consider the set of six equations on mass and momentum conservation on the one hand, and the characteristic energy extrema for stability of the three broad regimes of operation on the other, mathematical modeling for local hydrodynamics of particle-fluid two-phase flow beyond minimum fluidization needs therefore to satisfy the following constraints ... 

The mathematical model, which makes it possible to consider the influence of the hydrodynamic conditions of flow on the processes of mixing and chemical transformations of reacting substances in a liquid phase, assumes that the average flow characteristics of a multicomponent system can be described by the equations of continuum mechanics and will satisfy conservation laws. 

Thus, the mathematical model (2), (6), (7), (8) describes the change in the velocity field in the formation of thrombus in the vessel. To simulate the obstacles of arbitrary shape (in this problem blood clot) is introduced by a discretetime artificial power. This force is applied only on the surface and within the constraints of the body. Force application point disposed in a spaced, similar velocity components defined on a staggered grid. When the point of application of force coincides with a virtual border, an artificial force is applied so as to satisfy the boundary conditions on the obstacle. The cell containing the virtual boundary, does not satisfy the equation of conservation of mass. Therefore, we introduce the source / drain weight to the cell that contains the virtual border. Discrete in time force is used to meet the conditions of adhesion on a virtual border, while the source / drain weight, to meet the conservation of mass for the cell that contains the virtual boundary. Procedure nondimensionalization this system involves choosing the characteristic scales the concentrations Oq and 4>q, lines size L, the characteristic scale of velocity V. In view of the above equations (1) - (2) takes the form ... 

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