Air-Water Exchange Models

In the preceding discussion, we presented experimental information on the singlephase air-water exchange velocities. Water vapor served as the test substance for the air-phase velocity v,a, while 02, C02 or other compounds yielded information on v,w. Now, we need to develop a model with which these data can be extrapolated to other chemicals which either belong also to the single-phase group or are intermediate cases in which both via and vlw affect the overall exchange velocity v,a/w (Eq. 20-3). 

To this end we use the boundary models derived in Chapter 19. Since each model has its own characteristic dependence on substance-specific properties (primarily molecular diffusivity in air or water), the experimental data from different compounds help us recognize the strengths and limitations of the various theoretical concepts. 

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 Although many aspects of this model are outdated in light of our improved knowledge of the physical processes occurring at the interface, its mathematical simplicity keeps the model popular. 

An alternative approach, developed by chemical engineers as well, is the surface renewal model by Higbie (1935) and Danckwerts (1951). It applies to highly turbulent conditions in which new surfaces are continuously formed by breaking waves, by air bubbles entrapped in the water, and by water droplets ejected into the air. Here the interface is described as a diffusive boundary. 

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). 

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In this example the subdivision of the system is obvious since the air-water interface clearly defines the boundary between the two boxes. In Chapter 20 we learned that independently of the specific air-water exchange model used, the exchange across the interface is always described mathematically by Eq. 20-1. This expression can be separated into two unidirectional fluxes with the form ... 

Calculate the excess ratio, Rt = C(Up/ C,down, for all five compounds (1) from the concentration measured above and below the river section (/ , is then called R 3S), and (2) from the linear air-water exchange model, Eqs. 24-21 and 24-22 (Rt is then called R odd). Compare R,meas with R,model. 

To assess the relative importance of the volatilisation removal process of APs from estuarine water, Van Ry et al. constructed a box model to estimate the input and removal fluxes for the Hudson estuary. Inputs of NPs to the bay are advection by the Hudson river and air-water exchange (atmospheric deposition, absorption). Removal processes are advection out, volatilisation, sedimentation and biodegradation. Most of these processes could be estimated only the biodegradation rate was obtained indirectly by closing the mass balance. The calculations reveal that volatilisation is the most important removal process from the estuary, accounting for 37% of the removal. Degradation and advection out of the estuary account for 24 and 29% of the total removal. However, the actual importance of degradation is quite uncertain, as no real environmental data were used to quantify this process. The residence time of NP in the Hudson estuary, as calculated from the box model, is 9 days, while the residence time of the water in the estuary is 35 days [16]. 

The approach pursued in this and the next chapter is focused on the common mathematical characteristics of boundary processes. Most of the necessary mathematics has been developed in Chapter 18. Yet, from a physical point of view, many different driving forces are responsible for the transfer of mass. For instance, air-water exchange (Chapter 20), described as either bottleneck or diffusive boundary, is controlled by the turbulent energy flux produced by wind and water currents. The nature of these and other phenomena will be discussed once the mathematical structure of the models has been developed. 

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. 

In Chapter 20, the diffusive boundary scheme serves as one of several models to describe air-water exchange. 

Diffusive boundaries also exist between different phases. The best known example is the so-called surface renewal (or surface replacement) model of air-water exchange, an alternative to the stagnant two-film model. It will be discussed in Chapter 20.3. 

Box 20.2 Temperature Dependence of Air-Water Exchange Velocity v(w of Volatile Compounds Calculated with Different Models Overall Air-Water Exchange Velocities... 

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 ... 

Independent of the model that is used to describe air-water exchange at the sea surface, the flux Ft is proportional to (Eqs. 20-1,20-2, where subscript w is replaced by sw for seawater) ... 

What makes these data difficult to interpret is that they originate from studies in which different substances, mainly C02 and 02, but also radon and sulfur hexafluoride (SF6), have been used. If the exact nature of the exchange process is not known, it is not immediately evident how data from different gases should be compared. A thorough discussion has to be postponed to Section 20.3 where models of air-water exchange are presented. Then we will also tackle the question of how water temperature affects the transfer velocity. 

In the last 20 years, considerable efforts have been made to measure air-water exchange rates either in the laboratory or in the field. One central goal of these investigations was to check the validity of Eq. 20-19 or of alternative expressions. Thus let us see how corresponding forms of Eq. 20-19 look for other models. 

Figure 20.8 Depending on the roughness of the river bed, the production of turbulence leads either to (a) eddies which are much smaller than the river depth h, or to (b) large eddies which are able to transport dissolved chemicals fast from and to the water surface. Both situations can be described by two different models for air-water exchange (a) the small-eddy model by Lamont and Scott (1970), and (b) the large-eddy model by O Connor and Dobbins (1958). See Box 20.3 for details.

Note that the inverse of -Kha /ha is identical with aa which was introduced in Eq. 8-21. Here we choose the Annotation to indicate that the ratio is like a partition coefficient which appears in the flux (Eq. 20-1) if different phases or different chemical species are involved (see section 19.2 and Eq. 19-20). In order to show how the combination of both partitioning relationships, one between air and water (Eq. 20-42), the other between neutral and total concentration (Eq. 20-43), affect the air-water exchange of [HA], we choose the simplest air-water transfer model, the film or bottleneck model. Figure 20.11 helps to understand the following derivation. 

Whatever the detailed physicochemical model of the interface, most models of the air-water exchange flux are written as the product of two factors, one describing the physics, the other the chemistry. What are these factors ... 

Hint The process of heat exchange across an interface can be treated in the same way as the exchange of a chemical at the interface. To do so, we must express the molecular thermal heat conductivity by a molecular diffusivity of heat in water and air, Z)thw and Z)tha, respectively. At 20°C, we have (see Appendix B) flthw = 1.43 xl(T3 cm2 s-1, Dlh a = 0.216 cm2 s 1. Use the film model for air-water exchange with the typical film thicknesses of Eq. 20-18a. 

The mass balance equations for the epilimnion and hypolimnion look like Eq. 21-38, except for the air-water exchange fluxes which are replaced by the vertical fluxes across the thermocline, 7) EH and 7) HE. According to the general form of mass transfer models (Eq.18-4), we can express these fluxes as ... 

Figure21.10 Two-box model for stratified lake. The numbered processes are (1) input by inlets (rj is relative fraction of input going to the hypolimnion), (2) air-water exchange, (3) loss at the outlet, (4) loss by in situ chemical transformation (chemical, photochemical, biological), (5) flux on settling solid matter, (6) exchange across the thermocline. See text for definition of parameters. Note that the substance subscript i is omitted for brevity.

The question arose whether contaminants in the fairly dirty city air could pollute the drinking water by air-water exchange. You remember the two-box model shown in Fig. 21.9 and decide to make a first assessment by using the steady-state solution of this model. As an example you use the case of benzene, which can reach a partial pressure in air of up to p = 10 ppbv in polluted areas. You use a water temperature of 10°C and the corresponding Henry s law constant K, H = 3.1 L bar mol-1. The air-water exchange velocity of benzene under these conditions is estimated as vi a/w = 5 x 10 4 cm s 1. 

Figure 23.1 General view of a linear one-box model of a well-mixed pond, a lake, or part of a lake or ocean. See Box 23.1 for definitions. If the box represents the complete water body of a lake, the terms with vex (water exchange with adjacent boxes) do not exist. Similarly, if the box is not in contact with the atmosphere, the air-water exchange flux (va,w) is absent.

In this section we treat the exchange at the sediment-water interface in the same manner as the air-water exchange. That is, we assume that the concentration in the sediments is a given quantity (an external force, to use the terminology of Box 21.1). In Section 23.3 we will discuss the lake/sediment system as a two-box model in which both the concentration in the water and in the sediments are model variables. 

In Part 2 of the PCB story, we introduced the exchange between the water column and the surface sediments in exactly the same way as we describe air/water exchange. That is, we used an exchange velocity, vsedex, or the corresponding exchange rate, ksedex (Table 23.6). Since at this stage the sediment concentration was treated as an external parameter (like the concentration in the air, Ca), this model refinement is not meant to produce new concentrations. Rather we wanted to find out how much the sediment-water interaction would contribute to the total elimination rate of the PCBs from the lake and how it would affect the time to steady-state of the system. As shown in Table 23.6, the contribution of sedex to the total rate is about 20% for both congeners. Furthermore, it turned out that diffusion between the lake and the sediment pore water was much more important than sediment resuspension and reequilibration, at least for the specific assumptions made to describe the physics and sorption equilibria at the sediment surface. 

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