Bottleneck Boundaries

Simple Bottleneck Boundaries A Simple Noninterface Bottleneck Boundary Illustrative Example 19.1 Vertical Exchange of Water in a Lake Two- and Multilayer Bottleneck Boundaries Bottleneck Boundary Between Different Media Illustrative Example 19.2 Diffusion of a Volatile Compound from the Groundwater Through the Unsaturated Zone into the Atmosphere... 

Bottleneck Boundary Around a Spherical Structure Sorption Kinetics for Porous Particles Surrounded by Water Box 19.3 Spherical Wall Boundary with Boundary Layer Finite Bath Sorption... 

A different situation is encountered at the bottom of a water body. The sediment-water interface is characterized by, on one side, a water column which is mostly turbulent (although usually less intensive than at the water surface), and, on the other side, by the pore space of the sediment column in which transport occurs by molecular diffusion. Thus, the turbulent water body meets a wall into which transport is slow, hence the term wall boundary (Fig. 19.3b). A wall boundary is like a one-sided bottleneck boundary, that is, like a freeway leading into a narrow winding road. 

Before dealing with this and other examples, let us derive the mathematical tools which we need to describe the flux of a chemical across a simple bottleneck boundary. First, we recognize that for a conservative substance at steady-state, the flux, F(x), along the boundary coordinate x orthogonal to the boundary must be constant. According to Fick s first law (Eq. 18-6) the flux is given by ... 

Figure 19.4 Transfer across a simple bottleneck boundary of thickness 8 connecting two zones A and B. The solid line is concentration C(x) and the dashed line is diffu-sivity D x).

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. 

Figure 19.6 Transfer across a two-layer bottleneck boundary. Diffu-sivity in the bottleneck zone A is smaller than in zone B, but diffu-sivity in both zones is much smaller than in the adjacent bulk zones. The solid line is concentration C(x), the dashed line is diffusivity D(x). Note that here we assume that the phases in both zones are identical. In Fig. 19.7 the model is extended to a boundary with phase change.

Figure 19.7 Transfer across a two-layer bottleneck boundary between two phases. The situation is analogous to Fig. 19.6 except for the fact that the equilibrium condition between the two layers is now expressed by the relation KB/A = (CB/A / CA/B),.q. The dashed line in zone A gives the concentration in zone B expressed as the cor-responding A-phase equilibrium concentration.

Eqs. 19-19 and 19-20 represent a powerful tool for the description of multilayer bottleneck boundaries. In fact, the validity of the result extends beyond the special picture of a series of films across which transport occurs by molecular diffusion. Since the transfer velocities, vA and vBKB/A, can be interpreted as inverse resistances, Eq. 19-20 states that the total resistance of a multilayer bottleneck boundary is equal to the sum of the individual resistances. Note that the resistance of the nonreference phase includes the additional factor KBIA. In Problem 19.3, the above result shall be extended to three and more layers. 

The above results will be useful for the two-film model of air-water exchange (Chapter 20). A very different bottleneck boundary, that is, the unsaturated zone of a soil, is discussed in Illustrative Example 19.2. 

The unsaturated zone can be modeled as a bottleneck boundary of thickness 8 = 4 m. The TCE concentration at the lower end of the boundary layer is given by the equilibrium with the aquifer and at the upper end by the atmospheric concentration of TCE, which is approximately zero. Thus, you need to calculate the nondimensional Henry coefficient of TCE at 10°C, KTCB a/w(10°C). 

In these equations we recognize expressions which by now should have become familiar to us. During the initial phase of the exchange process (t Zcm), boundary concentration and flux at the interface remind us of a (B-side controlled) bottleneck boundary with transfer velocity vbl = Db,/S (see Eq. 19-19). The concentrations on either side are C and CBq=CA/FA/B, where the latter is the B-side concentration in equilibrium with the initial A-side concentration CA. 

A diffusive boundary connects two systems in which diffusivity is of similar size or equal (Fig. 19.3c). In contrast to a bottleneck boundary, which is characterized by one or several zones with significantly reduced diffusivity, or to a wall boundary, which exhibits an asymmetric drop in diffusivity, transport at a diffusive boundary is not very different from the inner part of the systems involved. What makes it a boundary is either a phase change (thus, the boundary is also an interface) or an abrupt change of one or several properties. By property we mean, for instance, the concentration of some chemical compound or of temperature. 

Due to the spherical geometry of the surface, the concentration profile across the boundary layer is no longer a straight line as was the case for the flat bottleneck boundary (Fig. 19.4). We can calculate the steady-state profile by assuming that CF and CFq = Cs/Fs/F are constant. Then, the integrated flux, ZF, across all concentric shells with radius r inside the boundary layer (r0 < r < r0 + 8) must be equal ... 

At first sight Eq. 19-63 does not resemble the type of equation that we found earlier for bottleneck boundaries (e.g., Eq. 19-19). That is not surprising since XF is an integrated flux (mass per unit time), while in the case of flat boundaries we have always dealt with specific fluxes (mass per unit area and time). If we divide XF by the surface of the sphere, 4n r, after some algebraic rearrangements we get ... 

The flux across a bottleneck boundary can be expressed either in terms of Fick s first law or by a transfer velocity. Explain how the two views are related. 

In the expression for the total exchange velocity across a two-layer bottleneck boundary between different media (Eq. 19-20) the transfer velocity vB is multiplied by the extra factor Kpj. What is AiA/B Can you imagine a scheme in which vA carries an extra factor instead How are the two schemes related ... 

Assume that the concentrations on either side of a boundary, CA and CB, are constant. You calculate the flux across this boundary by treating it (a) as a bottleneck boundary and (b) as a wall boundary, respectively. How does the flux evolve as a function of time in these two models ... 

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. 

All the necessary tools to develop kinetic models for air-water exchange have been derived already in Chapters 18 and 19. However, we don t yet understand in detail the physical processes which act at the water surface and which are relevant for the exchange of chemicals between air and water. In fact, we are not even able to clearly classify the air-water interface either as a bottleneck boundary, a diffusive boundary, or even something else. Therefore, for a quantitative description of mass fluxes at this interface, we have to make use of a mixture of theoretical concepts and empirical knowledge. Fortunately, the former provide us with information which is independent of the exact nature of the exchange process. As it turned out, the different flux equations which we have derived so far (see Eqs. 19-3, 19-12, 19-57) are all of the form ... 

The first model, the film model by Whitman (1923), depicted the interface as a (single- or two-layer) bottleneck boundary. Although many aspects of this model are outdated in light of our improved knowledge of the physical processes occurring at the interface, its mathematical simplicity keeps the model popular. 

In the film model the air-water interface is described as a one- or two-layer bottleneck boundary of thicknesses 5a and 8W, respectively. Thus, according to Eq. 19-9 ... 

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). 

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