Local fluid flow hydrodynamics

In this category, one may classify all the attempts in describing theoretically the fluid-solid interactions. The main characteristic of these models is a local description of the fluid flow hydrodynamics. Some models for the prediction of the liquid holdup are worth noting in this category. 

The mass flux of a solute can be related to a mass transfer coefficient which gathers both mass transport properties and hydrodynamic conditions of the system (fluid flow and hydrodynamic characteristics of the membrane module). The total amount transferred of a given solute from the feed to the receiving phase can be assumed to be proportional to the concentration difference between both phases and to the interfacial area, defining the proportionality ratio by a mass transfer coefficient. Several types of mass transfer coefficients can be distinguished as a function of the definition of the concentration differences involved. When local concentration differences at a particular position of the membrane module are considered the local mass transfer coefficient is obtained, in contrast to the average mass transfer coefficient [37]. 

Studies on the effect of hydrodynamics on localized corrosion and electrochemical etching processes have been reviewed by West et al. Much of the work has been performed by Alkire and co-workers." They have used FIDAP, a commercial FEM code, to investigate the influence of fluid flow on geometries relevant to etching and to pitting corrosion. In most cases, Stokes flow was considered. The Stokes flow approximation is frequently valid inside the cavity because its characteristic dimension is small. However, the flow outside the cavity may not be in the Stokes flow regime. Since it is the external fluid motion that induces flow inside the cavity, under many (especially unsteady) situations, the use of the Stokes flow approximation may be problematic. Some of the work of Alkire and co-workers has been extended hy Shin and Economou, " who simulated the shape evolution of corrosion pits. Natural convection was also considered in their study. 

A reactor model is also available which uses actual local fluid velocity and turbulence measurements throughout a vessel to describe the flow pattern (simplifled to a rectangular grid of cells ). This is getting even nearer the true hydrodynamics, but still with coarse resolution. It is now superceded by the following CFD techniques. 

Pulsatile flow in an elastic vessel is very complex, since the tube is able to undergo local deformations in both longitudinal and circumferential directions. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classical model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the tube thin wall (Womersley, 1955b Atabek and Lew, 1966). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for the wall of an elastic tube, which are coupled together by the boundary conditions at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system (f , 6, f), and the dynamic... 

The fact that a local equilibrium assumption is an essential part of the hydrodynamic description of fluid flow suggests that we look for solutions of the Boltzmann equation where the distribution function is close to a local Maxwellian distribution /ie(r, v, t) given by... 

Fouling may be defined as the formation of deposits on heat transfer surfaces which impede the transfer of heat and increase the resistance to fluid flow. The growth of these deposits causes the thermai and hydrodynamic performance of heat transfer equipment to decline with time. Fouling affects the energy consumption of industrial processes and it can also decide the amount of material employed in the construction of heat transfer equipment. In addition, where the heat flux is high, fouling can lead to local hot spots and ultimately it may result in mechanical failure, and hence an unscheduled shutdown of the equipment. 

However, an exact solution to the problem of convective diffusion to a solid surface requires first the solution of the hydrodynamic equations of motion of the fluid (the Navier-Stokes equations) for boundary conditions appropriate to the mainstream velocity of flow and the geometry of the system. This solution specifies the velocity of the flrrid at any point and at any time in both tube and yam assembly. It is then necessary to substitute the appropriate values for the local fluid velocities in the convective diffusion equation, which must be solved for boundary cortditiorts related to the shape of the package, the mainstream concentration of dye and the adsorptions at the solid surface. This is a very difficrrlt procedure even for steady flow through a package of simple shape. " ... 

Two methods dissipative particle dynamics (DPD) was initially devised by Hoogerbrugge and Koelman as a particle-based off-lattice simulation method for the flow of complex fluids and to tackle hydrodynamic time and space scales beyond those available with MD. Since DPD is a coarse-grained model and individual atoms or molecules are not represented directly by the particles but they are grouped together into beads, these beads represent local fluid packages able to move independently. 

The mobility tensor can be derived from Stokes-flow hydrodynamics. Consider a set of spherical particles, located at positions r, with radius a, surrounded by a fluid with shear viscosity rj. Each of the particles has a velocity v which, as a result of stick boundary conditions, is identical to the local fluid velocity on the particle surface. The resulting fluid motions generate hydrodynamic drag forces Ff, which at steady state are balanced by the conservative forces, Ff + F- = 0. The commonly used approximation scheme is a systematic multipole expansion, similar to the analogous expansion in electtostatics [17-21]. For details, we refer the reader to the original literature [17], where the contributions from rotational motion of the beads are also considered. As a result of the linearity of Stokes flow, the particle velocities and drag forces are linearly related,... 

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