Diffusive boundary exchange velocity

When comparing Eqs. 19-1 and 19-3, the reader may remember the discussion in Chapter 18 on the two models of random motion. In fact, these equations have their counterparts in Eqs. 18-6 and 18-4. If the exact nature of the physical processes acting at the bottleneck boundary is not known, the transfer model (Eqs. 18-4 or 19-3) which is characterized by a single parameter, that is, the transfer velocity vb, is the more appropriate (or more honest ) one. In contrast, the model which started from Fick s first law (Eq. 19-1) contains more information since Eq. 19-4 lets us conclude that the ratio of the exchange velocities of two different substances at the same boundary is equal to the ratio of the diffusivities in the bottleneck since both substances encounter the same thickness 5. Obviously, the bottleneck model will serve as one candidate for describing the air-water interface (see Chapter 20). However, it will turn out that observed transfer velocities are usually not proportional to molecular diffusivity. This demonstrates that sometimes the simpler and less ambitious model is more appropriate. 

To understand the principal idea of Deacon s model we have to remember the key assumption of the film model according to which a bottleneck boundary is described by an abrupt drop of diffusivity, for instance, from turbulent to molecular conditions (see Fig. 19.3a). Yet, theories on turbulence at a boundary derived from fluid dynamics show that this drop is gradual and that the thickness of the transition zone from fully turbulent to molecular conditions depends on the viscosity of the fluid. In Whitman s film model this effect is incorporated in the film thicknesses, 8a and 8W (Eq. 20-17). In addition, the film thickness depends on the intensity of turbulent kinetic energy production at the interface as, for instance, demonstrated by the relationship between wind velocity and exchange velocity (Figs. 20.2 and 20.3). 

In the film model of Whitman the water-phase exchange velocity, v,w, is a function of the molecular diffusion coefficient of the chemical, while in Deacon s boundary layer model v[W depends on the Schmidt Number Sc W. Explain the reason for this difference. 

To understand this point, we must recall that the sediment surface acts as a wall boundary with respect to diffusion (Chapter 19). According to Eq. 19-33, the exchange flux, Fsed, and thus the corresponding apparent exchange velocity, vsedex, decreases with elapsed time t. From Eq. 19-33, with C p replaced by/wCt, we get ... 

The solution of the corresponding mass exchange problem for a circular cylinder and an arbitrary shear flow was obtained in [353] in the diffusion boundary layer approximation. It was shown that an increase in the absolute value of the angular velocity Cl of the shear flow results in a small decrease in the intensity of mass and heat transfer between the cylinder and the ambient... 

In the seventies, the growing interest in global geochemical cycles and in the fate of man-made pollutants in the environment triggered numerous studies of air-water exchange in natural systems, especially between the ocean and the atmosphere. In micrometeorology the study of heat and momentum transfer at water surfaces led to the development of detailed models of the structure of turbulence and momentum transfer close to the interface. The best-known outcome of these efforts, Deacon s (1977) boundary layer model, is similar to Whitman s film model. Yet, Deacon replaced the step-like drop in diffusivity (see Fig. 19.8a) by a continuous profile as shown in Fig. 19.8 b. As a result the transfer velocity loses the simple form of Eq. 19-4. Since the turbulence structure close to the interface also depends on the viscosity of the fluid, the model becomes more complex but also more powerful (see below). 

The equations (3.109), (3.117) or (3.118) and (3.120) for the velocity, thermal and concentration boundary layers show some noticeable similarities. On the left hand side they contain convective terms , which describe the momentum, heat or mass exchange by convection, whilst on the right hand side a diffusive term for the momentum, heat and mass exchange exists. In addition to this the energy equation for multicomponent mixtures (3.118) and the component continuity equation (3.25) also contain terms for the influence of chemical reactions. The remaining expressions for pressure drop in the momentum equation and mass transport in the energy equation for multicomponent mixtures cannot be compared with each other because they describe two completely different physical phenomena. 

The airborne flight of a spore draws to a close when various factors tend to accelerate the downward velocity of spores. Rain is perhaps the most important factor in this process. Another mechanism of deposition is sedimentation in association with boundary layer exchange, a process by which spores from a cloud of particles overhead diffuse into the boundary layer of air in which settling is mainly gravitational. Deposition of spores is also achieved by their impaction against solid objects. The relative importance of these various deposition processes varies with the circumstances [1],... 

Here D is the molecular diffusivity of CO2, z is the film thickness, a, is the solubility of i, V and A are the volume and surface area of the ocean, and X is the decay coefficient. Use of pre-industrial mean concentrations gave a global boundary layer thickness of 30pm (D/z 1800m y = piston velocity). The film thickness is then used to estimate gas residence times either in the atmosphere or in the mixed layer of the ocean. For CO2 special consideration must be made for the chemical speciation in the ocean, and for " C02 further modification is necessary for isotopic effects. The equilibration times for CO2 with respect to gas exchange, chemistry, and isotopics are approximately 1 month, 1 year, and 10 years, respectively. 

Movement of a soluble chemical throughout a water body such as a lake or river is governed by thermal, gravitational, or wind-induced convection currents that set up laminar, or nearly frictionless, flows, and also by turbulent effects caused by inhomogeneities at the boundaries of the aqueous phase. In a river, for example, convective flows transport solutes in a nearly uniform, constant-velocity manner near the center of the stream due to the mass motion of the current, but the friction between the water and the bottom also sets up eddies that move parcels of water about in more randomized and less precisely describable patterns where the instantaneous velocity of the fluid fluctuates rapidly over a relatively short spatial distance. The dissolved constituents of the water parcel move with them in a process called eddy diffusion, or eddy dispersion. Horizontal eddy diffusion is often many times faster than vertical diffusion, so that chemicals spread sideways from a point of discharge much faster than perpendicular to it (Thomas, 1990). In a temperature- and density-stratified water body such as a lake or the ocean, movement of water parcels and their associated solutes will be restricted by currents confined to the stratified layers, and rates of exchange of materials between the layers will be slow. 

Next we consider the mechanism of internal friction (viscosity) of flowing liquid or gas. It consists in the following the neighboring layers which move with different velocities exchange impulses these impulses are transferred by particles which diffuse through the boundary between layers. Besides, the intermolecular bonds of structural elements (normally - i.e., perpendicularly - orientated to the sliding surface) deform and tear up this is the second factor of viscosity inherent to liquid only. 

This is known as perfect slip condition. It may be noted that the velocity slip boundary condition for u - u, i becomes obsolete, since the Euler equations require only the no-penetration boundary condition in this limit. For diffuse reflection, the boundary condition is (T = 1 (Figure 3.2(c)). Here, the molecules are reflected from the walls with zero average tangential velocity for the stationary wall case. Therefore, diffuse reflection leads to tangential momentum exchange (thus friction) of the gas with the walls. 

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