Diffusive sublayer

Boudreau, B.P. and Guinasso, N.L. 1982, The influence of a diffusive sublayer on accretion, dissolution, and diagenisis at flie sea floor. In The Dynamic Environment of the Ocean Floor. Fanning, K.A., Manheim, FT., Eds. Lexington Books Toronto. 

Interfacial transfer of chemicals provides an interesting twist to our chemical fate and transport investigations. Even though the flow is generally turbulent in both phases, there is no turbulence across the interface in the diffusive sublayer, and the problem becomes one of the rate of diffusion. In addition, temporal mean turbulence quantities, such as eddy diffusion coefficient, are less helpful to us now. The unsteady character of turbulence near the diffusive sublayer is crucial to understanding and characterizing interfacial transport processes. 

We will now adjust our position to within a few microns of the interface and examine the flux. Flux is molecular at the interface between two phases, as illustrated in Figure 8.3. This means that is zero and the only active process is diffusion. Note the difference in the order of magnitudes of D and s given in Figure 8.3, and recall the resistance concepts that we discussed in Chapter 2. The resistance to mass transfer is much larger near the interface than away from the interface, so the controlling process would tend to be transfer in the diffusive sublayer. 

In the diffusive sublayer, flux is due purely to diffusion. Then, we can utilize the fact that mean flux is constant with distance ... 

As noted previously, most environmental flows are turbulent. The diffusive sublayer, where only diffusion acts to transport mass and the concentration profile is linear, is typically between 10 /xm and 1 mm thick. Measurements within this sublayer are not usually feasible. Thus, the interfacial flux is typically expresses as a bulk transfer... 

Equations (8.4), (8.8), and (8.9) can be combined with the assumption that all resistance to transfer occurs in the diffusive sublayer, as illustrated in Figure 8.5, to show that... 

It may seem as though we have abandoned our statement that the unsteady aspects of the interaction of the diffusive sublayer and turbulence are paramount, because Kb, Kl, and Kg are all bulk quantities. However, the unsteady relationships that exist will still be brought into the analysis of equation (8.10) and (8.11) (i.e., 5 = 8 D, turbulance)). This relatively simple characterization provides for most of the research regarding interfacial transport rates. 

Archer D., Emerson S., and Smith C. R. (1989b) Direct measurement of the diffusive sublayer at the deep sea floor using oxygen microelectrodes. Nature 340, 623-626. 

The border diffusion layer model was introduced as an amendment to the film model to present a more realistic description. It accounts for an undefined film thickness, turbulence effects, and the role of molecular diffusion. When the flow is turbulent, the flow around the bubble is split into four sections the main turbulent stream, the turbulent boundary layer, the viscous sublayer, and the diffusion sublayer. Eddy turbulence accounts for mass transfer in the main turbulent stream and the turbulent boundary layer. The viscous sublayer limits eddy turbulence effects so that the flow is laminar and mass transfer is controlled by both molecular diffusion and eddy turbulence. Microscale eddy turbulence is assumed to be dominant in the viscous sublayer. Mass transfer in the diffusion sublayer is controlled almost completely by molecular diffusion (Azbel, 1981). 

This model can be used as a rough estimate. It is still plagued by the steady-state assumption which is oftentimes an inadequate description of mass transfer. Tlie diffusion sublayer is a function of viscosity (v), diffusivity (Z)), and viscous sublayer thickness (Sq), bnt realistic measurements are difficult, if not impossible, to obtain. An empirical correlation has been suggested for S (Azbel, 1981) ... 

Santschi, P.H., R.F. Anderson, M.Q. Fleisher, and W. Bowles. 1991. Measurementsof diffusivity sublayer thicknesses in the ocean by alabaster dissolution and their implications for the measurement of benthic fluxes. Journal of Geophysical Research, 96(10), 641-657. 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

Eddy diffusion as a transport mechanism dominates turbulent flow at a planar electrode ia a duct. Close to the electrode, however, transport is by diffusion across a laminar sublayer. Because this sublayer is much thinner than the layer under laminar flow, higher mass-transfer rates under turbulent conditions result. Assuming an essentially constant reactant concentration, the limiting current under turbulent flow is expected to be iadependent of distance ia the direction of electrolyte flow. 

I0-38Z ) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38Z ) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. ln.st. Chem. Eng. J., 10, 269(1964)]. 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

In the limit of extreme turbulence, when eddies of fresh solution are rapidly swept into the immediate vicinity of the interface, neither the laminar sublayer nor a stationary surface can exist the diffusion path may, according to Kishinevskii, become so short that diffusion is no longer rate-controlling, and consequently for such liquid-phase transfer (14)... 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

In Illustrative Example 19.1, we calculated the vertical exchange of water across the thermocline in a lake by assuming that transport from the epilimnion into the hypolimnion is controlled by a bottleneck layer with thickness 5 = 4m. From experimental data the vertical diffusivity was estimated to lie between 0.01 and 0.04 cm2s 1. Closer inspection of the temperature profiles (see figure in Illustrative Example 19.1) suggests that it would be more adequate to subdivide the bottleneck boundary in two or more sublayers, each with its own diffusivity. 

For a substance to be absorbed into the body following dermal exposure, it must initially dissolve in the stratum corneum sublayer, then diffuse through the remaining sublayers of the epidermis and into the dermis, where it will eventually diffuse into the blood capillaries. This absorption barrier ranges in thickness from 100 to 1000 pm, depending on area of the body (Klaassen and Rozman, 1991). 

Using the results of the preceding section, it is easy to show that if only component A is diffusing, then the ratio, yi/y2, of the thicknesses of these sublayers in the ApBq-B reaction couple is... 

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