Diffusion tangential

A tangential electric field VE acting on these charges produces a relative motion between the interface and the solution just outside the diffuse layer. In view of the thinness of the diffuse layer, a balance of the tangential viscous and electrical forces can be written... 

For many particles, the diffuse-charge layer can be characterized adequately by the value of the zeta potential. For a spherical particle of radius / o which is large compared with the thickness of the diffuse-charge layer, an electric field uniform at a distance from the particle will produce a tangential electric field which varies with position on the particle. Laplace s equation [Eq. (22-22)] governs the distribution... 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

When the fractions of molecules reflected specularly and diffusively are known, the slip length can be determined, as shovm by Maxwell. Maxwell introduced a tangential momentum accommodation coefficient defined as... 

The simplest and most commonly used convection apparatus consists of a disc electrode rotating with a constant angular velocity u [1-5]. The disc sucks the solution toward its surface, much in the way a propeller would as the solution approaches the disc, it is swept away radially and tangentially (see Fig. 14.1). The transport of the reacting species to the disc occurs both by convection and diffusion. Though the mathematics are complicated, the rate of transport can be calculated exactly for an infinite disc. A particularly nice feature of this setup is the fact that the transport is uniform so that the surface concentration of any reacting species is constant over the surface of the electrode. 

Carlos and Latif both fluidised glass particles in dimethyl phthalate. Data on the movement of the tracer particle, in the form of spatial co-ordinates as a function of time, were used as direct input to a computer programmed to calculate vertical, radial, tangential and radial velocities of the particle as a function of location. When plotted as a histogram, the total velocity distribution was found to be of the same form as that predicted by the kinetic theory for the molecules in a gas. A typical result is shown in Figure 6.11(41 Effective diffusion or mixing coefficients for the particles were then calculated from the product of the mean velocity and mean free path of the particles, using the simple kinetic theory. 

When the voltage is critical, regime b), there is no concentration polarization because the electrophoretic transport is equal to the convective transport. Any build up of species on the membrane will be dissipated due to diffusion driven by the concentration difference. In this regime, increasing the tangential velocity is expected to have no influence on the flux because fluid shear can only improve the transport of particles down a concentration gradient. In this case, there is no concentration gradient. 

Filtration separates components according to their size. Efficiency depends on the shape and compressibility of the particles, the viscosity of the liquid phase and the driving force, which is the pressure created by overpressure or by vacuum. Filtration can be performed either as dead-end filtration, where the feed stream flows perpendicular to the filter surface (Lee, 1989) or as tangential flow filtration, where the feed stream flows parallel to the filter and the filtrate diffuses across it. Examples of the former are the continuous rotaiy vacuum dram filter, where a rotaiy vacuum filter has a filter medium covering the surface of a rotating drum and the filtrate is drawn through the dram by an... 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

Alternatively, multivane triangular diffusers (Fig. 14-7lb. c) such as the Schoepentoeter have been successful for high-velocity vapor-rich feeds. These are used with radial (as distinct from tangential) nozzles. The vanes knock out the liquid and direct it downward while the vapor expands to the tower diameter. 

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