Diffusional boundary layer

The limitation of using such a model is the assumption that the diffusional boundary layer, as defined by the effective diffusivity, is the same for both the solute and the micelle [45], This is a good approximation when the diffusivities of all species are similar. However, if the micelle is much larger than the free solute, then the difference between the diffusional boundary layer of the two species, as defined by Eq. (24), is significant since 8 is directly proportional to the diffusion coefficient. If known, the thickness of the diffusional boundary layer for each species can be included directly in the definition of the effective diffusivity. This approach is similar to the reaction plane model which has been used to describe acid-base reactions. 

From Eq. (28), the diffusional boundary layer for the rotating disk, Eq. (29), can be substituted to give... 

The potential production of sulfide depends on the biofilm thickness. If the flow velocity in a pressure main is over 0.8-1 ms-1, the corresponding biofilm is rather thin, typically 100-300 pm. However, high velocities also reduce the thickness of the diffusional boundary layer and the resistance against transport of substrates and products across the biofilm/water interphase. Totally, a high flow velocity will normally reduce the potential for sulfide formation. Furthermore, the flow conditions affect the air-water exchange processes, e.g., the emission of hydrogen sulfide (cf. Chapter 4). 

All modern pictures and models of hydrate crystal growth include mass transfer from the bulk phases to the hydrate. Unfortunately, some confusion arises due to the fact that two interfaces are usually considered, and the driving forces may not be intuitive for those not familiar with the area. In order to provide a basis for the modeling section, a brief overview of the diffusional boundary layer is given. 

Equation (3.58) and Equation (3.61) are the Hixson and Crowell cube-root and the Higuchi and Hiestand two-thirds-root expressions, respectively. The cube-root and the two-thirds-root expressions are approximate solutions to the diffusional boundary layer model. The cube-root expression is valid for a system where the thickness of the diffusional boundary layer is much less than the particle radius whereas the two-thirds-root expression is useful when the thickness of the boundary layer is much larger than the particle radius. In general, Equation (3.57) is more accurate when the thickness of the boundary layer and the particle size are comparable. 

The precise transition from laminar to turbulent flow occurs at different values of Re depending on geometry. Even in turbulent flow there exists a thin laminar hydrodynamic sublayer of thickness 8h near the metal surface. If mass transport is also occurring at the surface, there will be a diffusional boundary layer of thickness 8d. 8h is a function of v while 8d is a function of D. The Schmidt number quantifies a relationship between these two parameters ... 

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. 

Data is shown in Figs. 7a and 7b for oxygen reduction on carbon steel in room temperature 0.6M NaCl. iL increases with co0 5 as predicted. Hence if the corrosion rate is determined by the mass transport of oxygen to the disk surface to support oxygen reduction, then the corrosion rate will increase as a function of the rotation rate, co, raised to the 0.5 power and linearly with dissolved oxygen concentration. The diffusion boundary layer thickness, 8d, may be calculated from Fick s first law after iL is determined. Recall that 8 = nFDCJiL for one dimensional diffusion at the steady state. This leads to the following expression for the diffusional boundary layer thickness ... 

Generally, the highest flow rate that produces a film of the desired properties on a particular substrate should be used. Increasing the gas flow also reduces the diffusional boundary layer resulting in more uniform coating coverage at a higher deposition rate. 

The degree of saturation in diffusional boundary layer next to the predominant face of a growing... 

A similar variation in yields results from changing the rate of reaction varying the speed of rotation of a reacting disk of Mg. A faster rotation results in a thinner diffusional boundary layer and a faster reaction. For the reaction of evelopeniyl bromide in DKI at 25 t the yield of RMgBr varies from -86V< for slow rotation to .W// at 7000 rpnt. with products of v varying in the opposite way [28[. 

For quiescent donor and acceptor solutions, the equilibration time ranges from minutes to several hours and is dependent on the thickness and other geometrical parameters of the corresponding chambers, the membrane permeability, and the temperature. The equilibration is dominated by diffusion. The equilibration can be accelerated by convective mass transport according to Figure 1 by achieving thin diffusional boundary layers on the membrane. 

Mass transfer coefficients are the basis for models where the dissolved species are transported by a combination of diffusive and advective processes. The diffusive mass transfer coefficient ko, m/sec) is based on boundary layer theory. The basic premise of boundary layer theory is that, for laminar ffow, the ffuid velocity adjacent to a solid surface is zero (the no slip condition ) and the velocity increases as a parabolic function of distance away from the surface until it matches the velocity of the bulk fluid (Figure 7.5). This means that there is a thin layer of fluid with a thickness of 5d (m) adjacent to the surface that is effectively static. The rate of mass transport through this layer is limited by the diffusion rate of the dissolved species. The diffusional boundary layer is much thinner than the velocity boundary layer. For laminar flow past a flat surface, the thickness of the diffusional boundary layer is related to the thickness of the velocity boundary layer (Sy) by the Schmidt number, which compares the fluid viscosity to the diffusivity (Probstein, 1989). 

Figure 7.5. The velocity boundary layer thickness (S)/) snd the diffusional boundary layer thickness So) for laminar flow past a flat surface.The diffusional boundary layer is approximated as a thin layer of static fluid at the solid surface where only diffusional mass transport occurs. See discussions in Probstein (1989), Denny (1993), and Vogel (1994) for more details.

The bulk concentration of anolyte, CA,Na. is almost unchanged under normal operations. However, 5a is a function of the flow rate and of the surface roughness of the membrane. Chlorine bubbles, generated at the anode, agitate the anolyte solution near the membrane. As bubble action becomes more intense, the diffusional boundary layers become thinner. The membrane surface itself is not flat. Industrial membranes are normally reinforced with PTFE fiber or cloth. A metal oxide coating on zero-g membranes improves their hydrophilicity and allows easier detachment of chlorine bubbles. These coatings also affect the thickness of the diffusion layer and the limiting current density. 

Water vapor diffusion coefficients as functions of time are shown in Fig, 8. Values obtained from (2), (3), and (4) are all shown for comparison. Due to the scatter obtained, coefficients resulting from (2) are shown as a band of values rather than a single curve. Such a coefficient should be a function of the partial pressure gradient across the diffusional boundary layer. However, under the ambient conditions existing during this study, this quantity was essentially constant, and no partial pressure relationship was determined. However, the effect of boundary layer condensation without subsequent adherence to the container surface is illustrated by the data shown in Fig. 8. Equations (2), (3) and (4) pro-... 

Many authors (Nienow, 1975 Nienow and Miles, 1978 Chaudhari, 1980 Conti and Sicardi, 1982) have reported the effect of agitation on the diffusional mass transfer coefficient, ksLUp. It is sufficient to say that the diffusional mass transfer rate is affected primarily by the impact of agitation on the hydrodynamic environment near the surface of the particle, in particular the thickness of the diffusional boundary layer surrounding the solid. The hydrodynamic environment near the particle surface depends on the properties of the fluid properties as well as those of the particles. The specific variables were introduced in Section 10-2.1.1. In addition to these, the diffusivity. Da, also influences the diffusional mass transfer. 

McGregor R., Peters R.H. (1965), The Effect of Rate of Flow on Rate of Dyeing I -The Diffusional Boundary Layer in Dyeing Journal of the Society of Dyers and Colourists, 81, 393 00. 

Key words dye transport, sorption, adsorption, adsorption isotherms, standard affinity of dyes, diffusion, diffusional boundary layer. Pick s laws. 

Etters J.N. (1991), The Influence of the Diffusional Boundary Layer on Dye Sorption from Finite Baths Journal of the Society of Dyers and Colourists, 107, 114-16. 

McGregor s work showed the full complexity of some of the problems inherent in rigorously predicting the effect of rate of flow on rate of dyeing. The diffusional boundary layer model outlined in his paper is as simple and useful a treatment of the problem as can be found. 

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