1 The Global Perspective on Environmental Transport and Fate

Estimating the transport and fate of chemicals released into the environment is an interesting and challenging task. The environment can rarely be approximated as well mixed, and the chemicals in the environment often are not close to equilibrium. Thus, chemical transport and fate in the environment require a background in the physics of fluid flow and transport, chemical thermodynamics, chemical kinetics, and the biology that interacts with all of these processes. We will be following chemicals as they move, diffuse, and disperse through the environment. These chemicals will inevitably react to form other chemicals in a manner that approaches – but rarely achieves – a local equilibrium. Many times these reactions are biologically mediated, with a rate of reaction that more closely relates to an organism being hungry, or not hungry, than to the first- and second-order type of kinetics that we were taught in our chemistry courses.

   To which environmental systems will these basic principles be applied? The global environment is large, on the chemical transport and fate scale. We will attempt to apply the mathematics of diffusion techniques that we learn to the atmosphere, lakes, rivers, groundwater, and oceans, depending on the system for which the material we are learning is most applicable. To a limited extent, we will also be applying our mathematics of diffusion techniques to transfer between these media. Volatilization of a compound from a water body, condensation of a compound from the air, and adsorption of a compound from a fluid onto a solid are all interfacial transport processes. Thus, the transport and fate of chemicals in the environmental media of earth, water, and atmosphere will be the topic. In this text, we will attempt to formulate transport and fate problems such that they can be solved, regardless of the media or the transport process, through the mathematics of diffusion.

A. Transport Processes

A transport process, as used herein, is one that moves chemicals and other properties of the fluid through the environment. Diffusion of chemicals is one transport process, which is always present. It is a spreading process, which cannot be reversed (without the involvement of another media such as in reverse osmosis). Convection or advection is the transport of chemicals from one place to another by fluid flow. The convection and diffusion of a chemical cloud, as represented in Figure 1.1, are the movements of the cloud and spreading of the cloud over time.

   Turbulent diffusion is actually a form of advection, but the turbulent eddies tend to mix fluid with a random characteristic similar to that of the diffusion process, when viewed from enough distance. The representation given in Figure 1.1 could also be used to represent convection and turbulent diffusion, except that the pace of turbulent diffusion is normally more than one order of magnitude greater than diffusion. This higher pace of turbulent diffusion means that diffusion and turbulent diffusion do not normally need to be considered together, because they can be seen as parallel rate processes, and one has a much different time and distance scale from the other. If two parallel processes occur simultaneously, and one is much faster than the other, we normally can ignore the second process. This is discussed further in Section 1.D.

   Dispersion is the combination of a nonuniform velocity profile and either diffusion or turbulent diffusion to spread the chemical longitudinally or laterally. Dispersion is something very different from either diffusion or turbulent diffusion, because the velocity profile must be nonuniform for dispersion to occur. The longitudinal dispersion of a pipe flow is illustrated in Figure 1.2. While there is diffusion of the chemical, the nonuniform velocity profile creates a dispersion that is much greater than would occur with diffusion alone. The other important difference is that dispersion reflects the spreading of a cross-sectional mean concentration, while diffusion represents the spreading of a local concentration. In some contexts, typically in atmospheric applications, turbulent diffusion is also considered to be a form of dispersion. This is only a semantic difference, and herein we will continue to distinguish between turbulent diffusion and the dispersion of a mean concentration.

   Interfacial transfer is the transport of a chemical across an interface. The most studied form of interfacial transfer is absorption and volatilization, or condensation and evaporation, which is the transport of a chemical across the air–water interface. Another form of interfacial transfer would be adsorption and desorption, generally from water or air to the surface of a particle of soil, sediment, or dust. Illustration of both of these forms of interfacial transfer will be given in Section 1.D.

   Finally, there is multiphase transport, which is the transport of more than one phase, usually partially mixed in some fashion. The settling of particles in water or air, the fall of drops, and the rise of bubbles in water are all examples of multiphase transport. Figure 1.3 illustrates three flow fields that represent multiphase transport.

   Mass transport problems are solved with the diffusion equation, often represented as where C is the concentration of a chemical; t is time; u, v, and w represent the temporal mean velocity in the x, y, and z directions, respectively; and D represents a diffusion coefficient. The first term (1) on the far left of equation (1.1) represents the rate of accumulation of chemical concentration. The second terms (2) represent the mean convection of the chemical. The third terms (3), to the right of the equal sign, represent either diffusion or turbulent diffusion of the chemical. The fourth term (4) represents the multitude of reactions that are possible in a fluid in environmental media.

   We will be solving equation (1.1), or a similar equation, for various applications.

B. Chemical Fate

Chemical fate is the eventual short-term or long-term disposition of chemicals, usually to another chemical or storage. Some examples that fit the concept of short-term and long-term fate are given in Table 1.1. If a polychlorinated biphenol (PCB) compound is in groundwater, the media are soil and water. The short-term fate will be that the PCB will primarily adsorb to the soil. The long-term fate is that the chemical will desorb, when the PCB-laden water has left, and eventually be bioremediated by microbacteria looking for carbon sources. If this PCB is in the atmosphere, it will be adsorbed primarily to aerosols and particles in the short term, whereas its long-term fate will probably be photocatalyzed degradation.

   There are as many or more examples of short-term and long-term fate as there are chemical–media combinations. An important consideration for this topic is whether we are interested in short-term or long-term fate. This is often a question to be answered by toxicologists. We will, for example, take the results of their computations and experiments and track the more toxic forms of a spill. Sometimes this involves a short-term fate, and sometimes this involves a long-term fate. The time scale of the calculations is important in determining how we deal with the problem or how we set up our solution.

Table 1.1: Examples of short-term and long-term fates

PCB, polychlorinated biphenyl.

C. The Importance of Mixing

Mixing is a rate-related parameter, in that most rates of reaction or transport are dependent on mixing in environmental systems. When mixing is dominant (the slowest process), the first-order rate equation can be described as

Thus, we need two items to compute the rate of the process: the equilibrium concentrations for all species involved and the mixing rate parameter. A common example would be dissolved oxygen concentration in aquatic ecosystems.

   One of the most common chemicals of concern in water bodies is oxygen. Without sufficient oxygen, the biota would be changed because many of the “desirable” organisms in the water body require oxygen to live. The rate of oxygen transfer between the atmosphere and a water body is therefore important to the health of the aquatic biota. For air–water oxygen transfer, equation (1.2) can be formulated as

where dM/dt is the rate of mass transfer into the water, K_L is a bulk oxygen transfer coefficient, A is the surface area for transfer, C_a is the concentration of oxygen in the air, H is a coefficient that partitions oxygen between the air and water at equilibrium (called Henry’s law constant for liquid and gas equilibrium), and C is the concentration of oxygen in the water. Air is approximately 20.8% oxygen, so the concentration of oxygen in the atmosphere is determined primarily by atmospheric pressure. Henry’s law constant for oxygen is a function of pressure as well as temperature. Thus, the equilibrium concentration of oxygen is influenced by the thermodynamic variables: pressure and temperature. The rate parameter is K _L A, which has units of volume per second. The difference from equilibrium partitioning is represented by C_a /H - C. It is C that we typically need to bring as close to equilibrium with the atmosphere as possible, and the means to do it is by having a large dM/dt This usually means a large K_LA because it would be difficult to alter either C_a or H is determined by turbulence and diffusion coefficient (i.e., mixing) close to the water surface and represents the rate of mixing per unit surface area. Thus, the primary variable that can be changed in order to increase dM/dt is the mixing parameter represented by K_L. Some further examples of mixing rate and equilibrium parameters in environmental processes are given in Table 1.2.

D. Resistance to Transport

An important concept for environmental transport is resistances. The inverse of a rate parameter is a resistance to chemical transport. Or, in equation form:

Table 1.2 Examples of important mixing rate and equilibrium parameters in environmental process

   Figure 1.4 gives an example of the adsorption of a compound to suspended sediment, modeled as two resistances in series. At first, the compound is dissolved in water. For successful adsorption, the compound must be transported to the sorption sites on the surface of the sediment. The inverse of this transport rate can also be considered as a resistance to transport, R₁. Then, the compound, upon reaching the surface of the suspended sediment, must find a sorption site. This second rate parameter is more related to surface chemistry than to diffusive transport and is considered a second resistance, R₂, that acts in series to the first resistance. The second resistance cannot occur without crossing the first resistance of transport to the sorption site; so, they must occur in series.

   Now, if R₁ is much greater than R₂, we can assume that R₂ is zero without compromising the accuracy of the rate calculation. In electric circuits, two resistances applied in series are simply added together in calculating the line resistance. The same is true for resistance to chemical transport. If R₁ is 1,000 resistance units and R₂ is 1 resistance unit, we can ignore R₂ and still be within 99.9% of the correct answer. For most environmental transport and fate computations, it is sufficient to be within 99.9% of the correct answer.

   Another example is the air–water transfer of a compound, illustrated in Figure 1.5. This example will be used to explain volatile and nonvolatile compounds. There is resistance to transport on both sides of the interface, regardless of whether the compound is classified as volatile or nonvolatile. The resistance to transport in the liquid phase is given as R_L = 1/K_L. If we are describing chemical transfer through an equation like (1.3), the resistance to transfer in the gas phase is given as R_G = 1/(HK_G). The equilibrium constant is in the R_G equation because we are using the equivalent water side concentrations to represent the concentration difference from equilibrium, and the gas phase resistance needs to be a resistance to an equivalent water concentration.

   The gas phase and the liquid phase resistances are applied in series. In general, gas film coefficients are roughly two orders of magnitude greater than liquid film coefficients. It is also true that Henry’s law constant, H, varies over many orders of magnitude as the transported compounds are varied. Nitrogen gas, for example, has a Henry’s law constant of approximately 15, using mass concentrations. The herbicide atrazine has a Henry’s law constant of 3 × 10 ^{- 6}. Thus, the ratio R_G/R_L would vary by seven orders of magnitude between nitrogen gas and atrazine (see Appendix A–5).

   If we put these orders of magnitude into a series resistance equation

Because of Henry’s law constants, we can see that, for nitrogen gas,R≅R_L, and for atrazine, R ≅R_G. If the ratio of K_G /K_L ~ 100 is applied (Mackay and Yuen, 1983), R_G = R_L when H = 0.01.

   Now, the mass transfer between phases is given as

Nitrogen gas would be a volatile compound, because the equilibrium is strongly to the gas phase, and there is little gas phase resistance to its transfer (that is, 1/K_L＞＞ 1/(HK_G)). For that reason, nitrogen is generally called a gas, as are many other volatile compounds, such as methane, oxygen, and propane.

   Atrazine, however, would be a nonvolatile compound − 1/(HK_G)＞＞1/K_L − because equilibrium is strongly to the liquid phase due to the small Henry’s law constant. There is also a strong gas phase resistance to the transfer. Atrazine was manufactured to remain in the liquid phase, where it will act as a herbicide, rather than in the gas phase, where farm personnel will be breathing this toxic chemical. If you were going to pick a compound that is not made by humans from the list of those that are a gas or liquid in our environment, a good guess is that it would be a volatile or semivolatile compound. There are only a few nonionic environmental compounds that are nonvolatile. Remarkably, one of them is water. While the atmosphere may be as much as 3% water, the water bodies in the world are very close to 100% water. The equilibrium is strongly to the liquid side because of the small Henry’s law constant.

   One theme of this discussion can now be stated as follows: when transport processes occur in series, it is the slower transport processes that are important for chemical transport calculations, because the resistance to transport is large, just as the large resistors of a series in an electronic circuit are the most important.

   Now we are ready for the second theme: when transport processes occur in parallel, the fast transport process with the low resistance dominates. The result is the opposite of resistances in series. Figure 1.6 illustrates this concept with the transport of a compound from the water body to a sorption site on a solid. In the bulk solution, there is diffusion and turbulent diffusion occurring simultaneously. Transport can occur from either process, so there are two different paths that may be followed, without the need of the other path. These transport processes are operating in parallel, and the faster transport process will transport most of the compound. The analogy to electronic circuits applies in this case as well. Beginning with a compound in solution in Figure 1.6, there are two parallel transport paths, each with a resistance to transfer. Most of the compound will be transported through the path with the least resistance. Many times, we can ignore the path with the greater resistance because the quantity of compound transported through this path is very small. When the compound comes close to the solid, however, the turbulent diffusion dissipates, because eddies become so small that they are dissipated by viscous action of the water. Now, we are back to one transport path, with the act of sorption and diffusion acting in series. Thus, the slowest transport path once again becomes the important process.

   The overall resistance to the sorption process illustrated in Figure 1.6 can be written as follows:

where R_T, R_D1, R_D2, and R_S are the resistance to turbulent transport, diffusive transport in the bulk of the fluid, diffusive transport near the solid surface, and adsorption, respectively. We can see that, in Figure 1.6 and in equation (1.8), the resistance from diffusion in the bulk of the fluid can be neglected because turbulent diffusion is a parallel path. The resistance from diffusion only needs to be considered when there is no parallel path for turbulent diffusion, such as very near the surface of the solid. Thus, we can ignore R_D1, but not R_D2.

   In this chapter, we have discussed some of the topics in the bulk of the text, where the physics of mass transport – rather than the mathematics of the diffusion equation – are essential. We will return to these and similar engineering concepts throughout the text in an attempt to develop models in the environmental transport and fate of chemicals that are realistic but can be solved, even if that solution is approximate.

E. Terminology of Chemical Transport

There is some terminology of chemical transport that needs to be defined. Some terminology has already been implied, and some may seem obvious, but these are so important to this text that we need to define them in one place and at one time.

   Diffusion – In this text, we will define diffusion (and most other processes) from an engineering perspective, in that we will go to the level of detail that suits our objective. Diffusion can then be defined as the mixing of chemicals by random molecular motion. Diffusion coefficients in dilute solutions will be discussed in detail in Chapter 3.

   Convection (or advection) – The transport of a chemical or other quantity by an imposed flow.

   Turbulent diffusion – The mixing of chemicals by turbulence, such that a turbulent diffusion coefficient can be defined separately from the temporal mean convection.

   Dilution – The mixing of a more concentrated solution with one that is less concentrated. The adage “The solution to pollution is dilution” is still used, sometimes appropriately, for many pollution and mitigation processes.

   Density – Total mass per unit volume.

   Concentration – The quantity of a compound or chemical per unit volume, unit mass, or unit moles, where 1 mole = 6.02 × 10²³ molecules of the chemical or compound. In this text, we will typically be discussing concentration in mass or moles per volume of water, mass per mass of solid, and moles per mole of gas, depending on the media of interest.

   The conversion between concentration units and the expression of the units themselves can be confusing. We will now review the typical concentration units used in various environmental media. Concentration in water is usually given as mass per unit volume or moles per unit volume. The conversion between them is a straightforward application of molecular weights. For example, we have 2.0 g/m³ of CO₂ dissolved in water. The molecular weight of carbon dioxide is 44 g/mole. Then the concentration in moles/m³ is

The concentration in air, however, is typically given in units that are different from those of water, because mass per unit volume can be misleading in a media that can be significantly compressed. Thus, concentration in the atmosphere is often given as a partial pressure at one atmosphere of total pressure. Because the pressure of a gas at a given temperature is proportional to the number of molecules in a given volume, the following relations are applied: