Hydrodynamic boundary layer, rotating

In contrast to the RDE, the range of rotation speeds used in the RDC is rather limited. The upper limit is around 6-8 Hz, while Eq. (8) breaks down below approximately 1-2 Hz, where the hydrodynamic boundary layer. 

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

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... 

It was shown above that the limiting c.d. increases with velocity raised to the 0.8 power and the pipe diameter raised to the -0.2 power for piping corrosion rates that are controlled by mass transport. In contrast, it is evident that the shear stress increases with the fluid velocity raised to the 1.75 power and the pipe diameter raised to the -0.2 power. Thus equality of shear stress does not give equality of mass transfer rates. In both cases corrosion is enhanced in pipes of smaller diameter for the same solution velocity. Such a relationship can be rationalized based on the effect of pipe diameter on the thickness of the mass transport and hydrodynamic boundary layers for a given fixed geometry. Cameron and Chiu (19) have derived similar expressions for defining the rotating cylinder rotation rate required to match the shear stress in a pipe for the case of velocity-... 

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

Consider one side of a disk whose diameter is large compared to 6, rotating in a large vessel of liquid. Since liquid rotates with the disk and acquires a radial as well as an angular motion, it must also flow toward the face of the disk. The velocity of the perpendicular flow can be considered independent of distance except very near the disk. The distance at which flow becomes essentially parallel to the disk can be called the hydrodynamic boundary layer thickness. The distance at which a concentration gradient begins was calculated by Levich (7) to be... 

Both qualitative and quantitative insight can be garnered from transient X -i, i-t and r -t measurements in quiescent or stirred solutions, while measurements of steady-state behavior are best performed under well-defined hydrodynamic conditions. Typically, a rotating disc electrode (RDE), or a related method, is used to specify and/or modulate the hydrodynamic boundary layer thickness, 8. With an RDE the boundary layer is specified by... 

How does axial circulation, a precondition for large-space mixing, take place in an unbaffled tank A purely rotational motion would be useless. Answer A boundary layer is formed at the wall due to the van der Waals and the viscosity forces, which is kept there by the shear stress. The tank contents therefore rotates more slowly than the stirrer, and the centrifugal forces of the stirrer convey the liquid radially outwards. (It has been found that axial rotation immediately almost completely fails, when a friction-reducing tenside is added [572].) The deceleration of the tank contents is much more effective with baffles (stream crossways to the baffle surfaces) than by hydrodynamic boundary layers (stream parallel past the wall surface). 

The rotating disc method may be used to study the separate roles of diffusion and integration in crystal growth since it enables the mass transfer (diffusion) step to be isolated. A uniform hydrodynamic boundary layer of thickness... 

As with chronoamperometry, there are limits to the range of experimental conditions under which equation (19.12) applies, but with the RDE the important parameter is the rotation rate co. At low values of co, the hydrodynamic boundary layer becomes large and the assumptions used to derive the Levich equation no longer hold. The lower limit for co can be found from... 

We cannot deal here with the details of the hydrodynamic Navier-Stokes equations and their consequences. For dimensional reasons one can derive the following expression [150] for the thickness of the boundary layer when the crystal rotates with angular frequency u>... 

Hull and Kitchener (2) measured the rate of deposition of 0.3- an-diameter polystyrene latex particles onto a rotating disk coated with a film of polyvinyl formaldehyde. In electrolytes of high ionic strength (where the double-layer repulsion is negligible), they found close agreement between experiments and the prediction of Levich s boundary-layer analysis (Eq. 3]), indicating that a diffusion boundary layer exists and that its thickness is large compared to the domain of van der Waais and hydrodynamic interactions. These are neces-... 

This section describes selected mass transport correlations for laboratory devices such as the rotating disk and cylinder. These mass transport correlations may be used in order to establish the same mass transport conditions (diffusional boundary layer thicknesses) as those obtained in a pipe or under impinging flow. Essentially, the experimenter may vary the rotation rate and geometry of the cylinder or disk to dial in the same mass transport conditions as obtained in the field for pipes or impinging jets. The user should also verify that the same hydrodynamic conditions also exist through use of Reynolds numbers, as shown above. 

In a real electrochemical system, convection is usually introduced by such means as rotating electrode, stirring, or other forced circulation. In any case, the electrolyte moves relative to electrode surfaces. Due to the mechanical friction between electrolyte solution and electrode surface, a velocity v(x) variation exists. The velocity of solution flow is generally a constant (vqo) in bulk solution (far from the electrode surface and the wall of solution container) and decreases while approaching the solid surfaces [6]. The solution flow velocity v(x) = 0 at solid surface (x = 0). A hydrodynamic (or Prandtl) boundary layer is defined as [6]... 

At low pH, rate depends significantly on the hydrodynamic transport constant for H" which is not well defined. For example, at 25°C, our calculations from observed rates show that ki may vary from > 0.007 cm sec under approximately laminar boundary layer conditions at the end of a rotating disk (10) to about 0.23 cm sec at the impact of a jet (at v 35 m sec ) on the calcite surface ( ). Under the turbulent conditions of the stirred batch experiments of Plummer al. (1 ), ki is near 0.05 cm sec . ... 

The problem of transferring corrosion rate data from one hydrodynamic system to another has also been considered in some depth by Chen et al. [18], by using the corrosion of 90 10 Cu Ni alloy in aerated 1 m NaCl solution at 25 °C in pipe-flow, annular-flow, and rotating-cylinder systems. The authors recognized that two mass-transfer processes should be distinguished transfer through the diffusion boundary layer in e solution (mass-transfer coefficient, h), and transfer through the corrosion product film ( f). The overall mass-transfer coefficient was defined as... 

Both Sq and S have the same value over the entire electrode surface, which has given rise to the description of the electrode as a uniformly accessible surface. The concepts of a hydrodynamic and a diffusion boundary layer have no theoretical significance as such but serve mainly to provide a suitable model for the hydrodynamic conditions related to the rotating electrode. 

The question at hand is whether circumstances exist for this rather simple situation in which the conditions (1) and (2) are satisfied so that boundary-layer analysis can be applied. So far as the first condition is concerned, the only flows of (9-266) that have open streamlines are those with X > 0 (which includes simple shear flow). On the other hand, there is a nonzero hydrodynamic torque on the sphere that causes it to rotate for all flows in this subgroup except X = 1. Thus, for a sphere in the general linear 2D flow, given by (9-266), there are only two cases that satisfy the conditions for applicability of boundary-layer theory ... 

Recently, Hynes et al. [221, 222] have pointed out that continuum models of rotational relaxation become unreliable when the molecule of interest rotates in a solvent comprising molecules of similar size. To improve on the model, they considered a sphere to be surrounded by a first co-ordination shell of solvent molecules. All these were taken as rough spheres, that is hard spheres which reverse their relative velocity (normal and tangential components) on impulsive collision. Of specific interest are CCI4 and SF. The test sphere and its boundary layer is surrounded by a hydrodynamic continuum. To model this, Hynes et al. used linearised hydrodynamic equations for the solvent with a modified boundary condition between solvent and test molecule, which relates the rotational stress on the test sphere to the angular velocity of the sphere. A coefficient of proportionality, 3, is introduced as a slip coefficient (j3 0... 

A schematic of the RDE is presented in Fig. 2.2. The hydrodynamics of this electrode system are well-known, and they have been discussed in detail elsewhere. In simple terms the rotating electrode acts as a pump drawing fresh solution from bulk regions of the fluid toward the electrode surface, then spinning it around, and subsequently flinging it sideways. This flow pattern is illustrated in Fig. 2.5. The electrode action establishes a stationary boundary layer, called the diffusion layer, at the electrode... 

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