Hydrodynamic eddy

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

First of all, the increased computer power makes it possible to switch to transient simulations and to increase spatial resolution. One no longer has to be content with steady flow simulations on relatively coarse grids comprising 104-105 nodes. Full-scale Large Eddy Simulations (LES) on fine grids of 106—107 nodes currently belong to the possibilities and deliver realistic reproductions of transient flow and transport phenomena. Comparisons with quantitative experimental data have increased the confidence in LES. The present review stresses that this does not only apply to the hydrodynamics but relates to the physical operations and chemical processes carried out in stirred vessels as well. Examples of LES-based simulations of such operations and processes are due to Flollander et al. (2001a,b, 2003), Venneker et al. (2002), Van Vliet et al. (2005, 2006), and Flartmann et al. (2006). 

A further option is to forget about simulating the flow and the processes in the whole vessel and to zoom into local processes by carrying out a DNS for a small box. The idea is to focus on the flow and transport phenomena within such a small box, such as mass transport and chemical reactions in or around a few eddies or bubbles, or the hydrodynamic interaction of a limited number of bubbles, drops, and particles including their readiness to collisions and coalescence. Examples of such detailed studies by means of DNS are due to Ten Cate et al. (2004) and Derksen (2006b). 

Bakker, A., Oshinowo, L. M., and Marshall, E. M., The Use of Large Eddy Simulation to Study Stirred Vessel Hydrodynamics . Proceedings of the 10th European Conference on Mixing, Delft, Netherlands, 247-254 (2000). 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

We will note how the shadow is in a state of continual movement. The patterns are caused by eddy currents around the heater as the air warms and then rises. After just a quick glance, it s clear that the movement of the warmed air is essentially random. By extension, we see that, as an electroanalytical tool, electrode heating is not a good form of convection, because of this randomness. Conversely, a hydrodynamic electrode gives a more precisely controlled flow of solution. In consequence, the rate of mass transport is both reproducible and predictable. 

Yoke, P. R., and M. W. Collins. 1983. Large eddy simulation Retrospect and prospects. Physico-Chemical Hydrodynamics 4(2) 119-61. 

Dobetti and Pantaleo (38) investigated the influence of hydrodynamic parameters per se on the efficiency of a coacervation process for microcapsule formation. They based their work on that of Armenante and Kirwan (39) who described the size of the smallest eddies or vortices generated in a turbulent regime on a microscopic scale in the vicinity of the agitation source, i.e., microeddies, as... 

Equation (51) assumes that the eddy properties are isotropic. In addition, no effect of other gradients such as temperature or gravity upon the molecular transport is taken into account. The expression was written for a single component and it is necessary to solve a set of such expressions one for each component, simultaneously if the interrelation of the material transport upon the hydrodynamic velocity is to be taken into account. 

Surface active substances (surfactants) are chemicals which accumulate at the water surface and reduce the air-water interfacial tension. The influence of such films on air-water exchange is twofold (1) they create an additional transport barrier, and (2) they change the hydrodynamics at the water surface such that the transport of solutes by eddies approaching the water surface is reduced (hydrodynamic damping). 

DESIGNER also contains different hydrodynamic models (e.g., completely mixed liquid-completely mixed vapor, completely mixed liquid-vapor plug flow, mixed pool model, eddy diffusion model) and a model library of hydrodynamic correlations for the mass transfer coefficients, interfacial area, pressure drop, holdup, weeping, and entrainment that cover a number of different column internals and flow conditions. 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

A comparative analysis of the hydrodynamical situations that occurred in different years has made it evident that anticydonic eddies may represent a typical element of the circulation in the eastern deep basin at least during the warm season (April-December). This fact contradicts the traditionally accepted concept (see [1-3]). The appearance and existence of these kinds of anticydonic eddies in the deep basin is related to the separation of coastal anticyclones from the coast. Some events of this kind together with the subsequent movement of the deep-sea anticyclones were registered with the use of satellite information of high spatio-temporal resolution and derived from the hydrographic surveys of different years. 

Macro-mixing concerns the state of flow produced by the stirrer in the vessel. The stirrer generates primary eddies whose size is of the same order of magnitude as the stirrer diameter d. The macro-scale, A, is therefore given by A d. It is described by the hydrodynamical pi-numbers such as Re = n d2/v, Fr = n2 d/g, and the like. 

The positive effect of velocity on the permeate flux is a result of enhanced hydrodynamic effects at the membrane surface, since high velocities lead to high shear and turbulent flow, which results in the formation of vortices and eddies that minimize the concentration polarization effects and the development of a fouling layer. The bigger the thickness of this layer, the higher its flow resistance and the smaller the permeate flux through the membrane becomes. Under turbulent flow conditions, shear effects induce hydrodynamic diffusion of the particles from the boundary layer back into the bulk, with a positive effect on the permeate flux. 

Interception capture refers to the situation where drops are captured on the surface of the rock, in vugs, and in recirculation eddies. Re-entrainment of the captured droplets can occur when a repulsive hydrodynamic force on the droplet is much larger than the van der Waals electrostatic attraction between the droplet and the rock surface. Droplet breakup will occur when interfacial tension is low and hydrodynamic forces are high. 

The issue of the role of hydrodynamic phenomena in the propagation of small cavities is far from settled, even for ambient-temperature systems. Any calculations of this type must be checked by carefully conceived experiments that are capable of providing quantitative data on the growth of pits as a function of aspect ratio and flow velocity. The key question appears to be to what depth do flow-induced eddies penetrate in a crevice, and how do these eddies affect the aggressive conditions that are established within the crevice due to the differential aeration ... 

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