Convection hydrodynamic layer

Lionbashevski et al. (2007) proposed a quantitative model that accounts for the magnetic held effect on electrochemical reactions at planar electrode surfaces, with the uniform or nonuniform held being perpendicular to the surface. The model couples the thickness of the diffusion boundary layer, resulting from the electrochemical process, with the convective hydrodynamic flow of the solution at the electrode interface induced by the magnetic held as a result of the magnetic force action. The model can serve as a background for future development of the problem. 

M. EISENBERG (Lockheed) I think that this was an important paper I have felt that many electrochemists did not appreciate the necessity of incorporating the dynamics of the diffusion layer and convection hydrodynamics in their studies. 

Taking into account that Sand s equation is valid only as long as the change of concentration occurs within a stagnant layer undisturbed by convection and introducing the Nemst diffusion layer boundary 8 and hydrodynamic layer boundary Ah, the minimum current density that must be applied in the first pulse for electrodeposition of the second layer to take place is given by... 

A similar situation occurs in the case of free convection and exothermic chemical reaction phenomena where the hydrodynamic boundary layer is separated from the reaction front. 

In chemical micro process technology with porous catalyst layers attached to the channel walls, convection through the porous medium can often be neglected. When the reactor geometry allows the flow to bypass the porous medium it will follow the path of smaller hydrodynamic resistance and will not penetrate the pore space. Thus, in micro reactors with channels coated with a catalyst medium, the flow velocity inside the medium is usually zero and heat and mass transfer occur by diffusion alone. 

Diffusion in a convective flow is called convective diffusion. The layer within which diffnsional transport is effective (the diffnsion iayer) does not coincide with the hydrodynamic bonndary layer. It is an important theoretical problem to calcnlate the diffnsion-layer thickness 5. Since the transition from convection to diffnsion is gradnal, the concept of diffusion-layer thickness is somewhat vagne. In practice, this thickness is defined so that Acjl8 = (dCj/ff) Q. This calcniated distance 5 (or the valne of k ) can then be used to And the relation between cnrrent density and concentration difference. 

In free-convection mass transfer at electrodes, as well as in forced convection, the concentration (diffusion) boundary layer (5d extends only over a very small part of the hydrodynamic boundary layer <5h. In laminar free convection, the ratio of the thicknesses is... 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

The dissolution rate of a solid from a rotating disc is governed by the controlled hydrodynamics of the system, and it has been treated theoretically by Levich [104]. This theory considers only forced convection due to rotation and ignores natural convection, which may occur at low speeds of rotation. Figure 16 shows the solvent flow held near the surface of the rotating disc. The apparent thickness, h, of the diffusion layer next to the surface of the disc is given by... 

Ary given catalytic material can be abstracted based on the same underlying similar architecture — for ease of comparison, we describe the catalytic material as a porous network with the active centers responsible for the conversion of educts to products distributed on the internal surface of the pores and the external surface area. Generally, the conversion of any given educt by the aid of the catalytic material is divided into a number of consecutive steps. Figure 11.13 illustrates these different steps. The governing transport phenomenon outside the catalyst responsible for mass transport is the convective fluid flow. This changes dramatically close to the catalyst surface from a certain boundary onwards, named the hydrodynamic boundary layer, mass transport toward and from the catalyst surface only takes place... 

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

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... 

At a given stirring rate, the effective hydrodynamic boundary layer is expected to be independent of particle size beyond a maximal particle size range, since the particle surface cannot bind the surrounding fluid to an infinite distance into the bulk. As a matter of course, dissolution still depends on convection. 

Convection (of the electrolyte liquid phase as a whole) can be natural (due to thermal effects or density gradients) or forced (principal mass transport mode in hydrodynamic techniques). Still, however, close to the electrode surface a diffusion layer develops. 

The diffusion-layer concept is an artifice for handling the flux arising from what would be, if treated in a proper hydrodynamic way, a complicated space variation of concentration at the interface. There is always some gradient of concentration at the interface there is an initial region in which the concentration changes linearly with distance, but there is, in the real case, no sharply defined layer of definite thickness, even when convection (natural or forced) produces a steady-state concentration... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

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