Fluid dynamics velocity profile

Computer Models, The actual residence time for waste destmction can be quite different from the superficial value calculated by dividing the chamber volume by the volumetric flow rate. The large activation energies for chemical reaction, and the sensitivity of reaction rates to oxidant concentration, mean that the presence of cold spots or oxidant deficient zones render such subvolumes ineffective. Poor flow patterns, ie, dead zones and bypassing, can also contribute to loss of effective volume. The tools of computational fluid dynamics (qv) are useful in assessing the extent to which the actual profiles of velocity, temperature, and oxidant concentration deviate from the ideal (40). 

This transition has profound effects in all fluid dynamics, and certainly so in aerodynamics. The velocity profile in (he boundary layer becomes fuller neat the surface on account of Ihe higher average kinetic energy of the layer created by turbulent energy exchange from layer lo layer. The effective viscosity is therefore larger in turbulent than laminar flow, ihe turbulent boundary layer thickens more rapidly downstream, the skin friction increases. 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

Dynamic segmented fluid packets in 200 Compute Hone l sim ule tion o velocity profile contiguous segmented fluid... 

The first step consists in deducing the velocity profile. This fluid dynamics problem was solved by von Karman14 and Cochran15 and gives... 

In Chap. 2, the concept of the diffusion layer was established. It is a thickness, within which a large fraction of diffusional changes take place, and at a distance of several times this thickness, practically no more diffusional changes are observed. This layer will here be given the symbol 8b (D for diffusion). In fluid dynamics, there is a similar layer, within which most of the velocity changes occur. This is the hydrodynamic layer 8. It turns out that for diffusive mass transfer, 8b is usually much smaller than <5/,. This is fortunate, because it justifies to some extent the linearised velocity profiles often assumed near walls, making analysis easier. These relations are very lucidly discussed in a classic paper by Vielstich [560]. 

For laminar flow (ReD < 2100) that is fully developed, both hydro-dynamically and thermally, the Nusselt number has a constant value. For a uniform wall temperature, NuD = 3.66. For a uniform heat flux through the tube wall, NuD = 4.36. In both cases, the thermal conductivity of the fluid in NuD is evaluated at Tb. The distance x required for a fully developed laminar velocity profile is given by [(x/D)/ReD] 0.05. The distance x required for fully developed velocity and thermal profiles is obtained from [(x/D)/(ReD Pr)] = 0.05. 

Thermal similarity is achieved in the ACR by providing a temperature profile which can be held geometrically similar when scaled. The temperature profile drives the ACR chemical kinetics and is a combined result of the heat transfer attributable to cracking and the heat effects caused by the bulk fluid movement. Thus, true thermal similarity in the ACR can only be achieved in conjunction with chemical and kinematic similarity. Kinematic similarity in the ACR is made possible during scale-up by forcing geometrically similar velocity profiles. The ACR temperature, pressure, and velocity profiles are governed by compressible gas dynamics so that an additional key scale parameter is the Mach number. 

In developing the arguments that are presented later in this review, it is necessary to keep in mind the relative scales (dimensions) at which each phase occurs. This is important because the effect of flow on localized corrosion is largely (though not totally) a question of the relative dimensions of the nucleus and the velocity profile in the fluid close to the surface. However, the velocity profile is a sensitive function of the kinematic viscosity, which in turn depends on the density and the dynamic viscosity. Because the kinematic viscosity of water drops by a factor of more than 100 on increasing the temperature from 25 °C to 300 °C, the conclusions drawn from ambient temperature studies of the effect of flow on localized corrosion must be used with great care when describing flow effects at elevated temperatures. 

Under plug flow conditions the convective transport is completely dominant over the diffusive mass transport term. The fluid moves like a plug and the diffusive term can be neglected. The conditions for plug flow are closely satisfied for narrow and long tubular reactors when the viscosity is low. However, this approximation is clearly best for fully developed turbulent flow, for which the velocity profiles are relatively fiat. For dynamic conditions, the species mass balance is a PDF with z and t as the independent variables. The Eulerian species mass balance (1.301) reduces further to ... 

E/N, f(u) as well as of the fluid dynamics of the chemical discharge reactor i.e., type of flow (plug, laminar, turbulent) and related dimensionless quantities relevant to the definition of (a) velocity profiles and corresponding effective residence times,... 

The both considered limit situations can be encountered in numerous problems of convective heat transfer they are schematically shown in Figure 3.1. One can see that in the case Pr — 0, which approximately takes place for liquid metals (e.g., mercury), one can neglect the dynamic boundary layer in the calculation of the temperature boundary layer and replace the velocity profile v(x, y) by the velocity v<, (x) of the inviscid outer flow. As Pr-)- oo, which corresponds to the case of strongly viscous fluids (e.g., glycerin), the temperature boundary layer is very thin and lies inside the dynamic boundary layer, where the velocity increases linearly with the distance from the plate surface. 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Analytical expressions for the RTDs have been derived for noncircular ducts, non-Newtonian fluids, and helically coiled tubes. Computational fluid dynamics has been used for really complicated geometries such as motionless mixers. 

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