Two-Box Models

Let us define a two-box model for a steady-stafe ocean as shown in Fig. 10-22. The two well mixed reservoirs correspond to the surface ocean and deep oceans. We assume that rivers are the only source and sediments are the only sink. Elements are also removed from the surface box by biogenic particles (B). We also assume there is mixing between the two boxes that can be expressed as a velocity Vmix = 2 m/yr and that rivers input water to the surface box at a rate of Vnv = 0.1 m/yr. The resulting ratio of F mix/V riv is 20. 

The inadequacy of the two-box model of the ocean led to the box-diffusion model (Oeschger et al, 1975). Instead of simulating the role of the deep sea with a well-mixed reservoir in exchange with the surface layer by first-order exchange processes, the transfer into the deep sea is maintained by vertical eddy diffusion. In... 

Two box models are used for the parameterization of water as conservative material and salts, including sulfur, as non-conservative material. For example, the first... 

The remaining two chapters of Part IV set the basis for the more advanced environmental models discussed in Part V. Chapter 21 starts with the simple one-box model already discussed at the end of Chapter 12. One- and two-box models are combined with the different boundary processes discussed before. Special emphasis is put on linear models, since they can be solved analytically. Conceptually, there is only a small step from multibox models to die models that describe the spatial dimensions as continuous variables, although the step mathematically is expensive as the model equations become partial differential equations, which, unfortunately, are more complex than the simple differential equations used for the box models. Here we will not move very far, but just open a window into this fascinating world. 

Linear Two-Box Model with One Variable Linear Two-Box Model of a Stratified Lake Box 21.7 Linear Two-Box Model for Stratified Lake Illustrative Example 21.5 Tetrachloroethene (PCE) in Greifensee From the One-Box to the Two-Box Model Linear Two-Box Models with Two and More Variables Nonlinear Two-Box Models... 

This is the first of several chapters which deal with the construction of models of environmental systems. Rather than focusing on the physical and chemical processes themselves, we will show how these processes can be combined. The importance of modeling has been repeatedly mentioned before, for instance, in Chapter 1 and in the introduction to Part IV. The rationale of modeling in environmental sciences will be discussed in more detail in Section 21.1. Section 21.2 deals with both linear and nonlinear one-box models. They will be further developed into two-box models in Section 21.3. A systematic discussion of the properties and the behavior of linear multibox models will be given in Section 21.4. This section leads to Chapter 22, in which variation in space is described by continuous functions rather than by a series of homogeneous boxes. In a sense the continuous models can be envisioned as box models with an infinite number of boxes. 

One-box/ two-box/ multibox models Model consisting of one or several boxes. Each box is characterized by one or several state variables Example Two-box model of a lake consisting of the boxes epilimnion and hypolim- nion... 

The derivation of the two-box model follows naturally from the one-box model. It is useful for describing systems consisting of two spatial subsystems which are connected by one or several transport processes. The mass balance equations for the individual boxes look like Eq. 21-1 with the addition of terms describing mass fluxes between the boxes. Each box can be characterized by one or several state variables. Thus, the dimension of the system of coupled differential equations is the product of the number of boxes and the number of variables per box. 

Figure 21.8 Two-box model of trwo completely mixed environmental compartments. Definition of subscripts first subscript (7 or j) designs the compound, the second (1 or 2) the box. Transfer fluxes Tcarry three subscripts. For instance, T ll describes the interbox flux of variable i from box 1 to box 2. X and Y denote other chemicals which are not state variables.

From a mathematical point of view, the distinction between boxes and chemicals is not relevant. In fact, a one-box model with two chemicals and a two-box model with one chemical lead to the same system of differential equations. Therefore, Box 21.6 also helps in solving a linear two-box model with one variable. 

Figure 21.9 Linear two-box model for a volatile compound in a coupled air-water system. Box 1 is the air volume (1 = a), box 2 the water volume (2 = w). Both volumes (Va and Vb) are flushed by the volumetric rates, Qa and Qb, respectively. The system is described by Eq. 21-38. See text for further explanations.

As a second example of a two-box model we discuss the case of a stratified lake which is divided into the surface layer (epilimnion E, box 1) and the deep-water layer (hypolimnion H, box 2). The model and its parameters are shown in Fig. 21.10. It includes the following processes (numbers as in the figure) ... 

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.

If the size of vex cannot be determined by fitting the two-box model to observed concentration profiles, then Eq. 21-42 is the appropriate expression to estimate this model parameter (see Illustrative Example 19.1). 

Illustrative Example 21.5 Tetrachloroethene (PCE) in Greifensee From the One-Box to the Two-Box Model... 

Hence, the one-box and two-box models yield the same result. There is a simple reason for that. Since the only removal processes of PCE act at the lake surface, at steady-state the surface concentration in both models (C°°for the one-box model, ClE for the two-box model) must attain the same value to compensate for the input /, tot. Furthermore, since the hypolimnion has neither source nor sink, the net exchange flux across the thermocline must be zero, and this requires C(E= C,H. 

As discussed below (Illustrative Example 21.6), the real advantage of the two-box model is the description of the transient behavior of the concentration. That is, such models allow us to examine the dynamic change from the initial value to the steady-state, rather than the computation of the steady-state itself. 

In order to understand the physical meaning of the eigenvalues we compare them with the rates characterizing the two-box model (see Illustrative Example 21.5). 

In the epilimnion/hypolimnion two-box model the vertical concentration profile of a chemical adopts the shape of two zones with constant values separated by a thin zone with an abrupt concentration gradient. Often vertical profiles in lakes and oceans exhibit a smoother and more complex structure (see, e.g., Figs. 19.1a and 19.2). Obviously, the two-box model can be refined by separating the water body into three or more horizontal layers which are connected by vertical exchange rates. 

You have constructed a linear two-box model for tetrachloroethene (PCE) in a lake in which the only input of PCE is from the outlet of a sewage treatment plant. The atmospheric PCE concentration is assumed to be zero in your model. How will the steady-state of the model be altered if the PCE input from sewage is reduced by... 

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. 

In Illustrative Examples 21.1 and 21.2 we have studied the fate of nitrilotriacetic acid (NTA) in Greifensee, especially its in situ degradation rate. Now we want to refine our analysis using the two-box model developed in Illustrative Example 21.5. 

Two-Box Model for Lake/Sediment System Box 23.3 Solution of Linear Water-Sediment Model PCBs in Lake Superior (Part 3)... 

In Section 23.1, this procedure will be applied to just one completely mixed water body. This control volume may represent the lake as a whole or some part of it (e.g., the mixed surface layer). Section 23.2 deals with the dynamics of particles in lakes and their influence on the behavior of organic chemicals. Particles to which chemicals are sorbed may be suspended in the water column and eventually settle to the lake bottom. In addition, particles already lying at the sediment-water interface may act as source or sink for the dissolved chemical. In Section 23.3, two-box models of lakes are discussed, particularly a model consisting of the water body as one box and the sediment bed as the other. Finally, in Section 23.4, one-dimensional vertical models of lakes and oceans are discussed. 

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 Chapter 21 the model of a stratified lake served as a prototype of a linear two-box model (Fig. 21.10). The necessary mathematics were developed in Boxes 21.6 and 21.7. In Illustrative Example 21.5 the fate of tetrachloroethene (PCE) in Greifensee was used to demonstrate that for the case of a two-box model it is still possible to carry out back-of-the-envelope calculations. Further examples are given in Problems 23.2 and 23.3, where the behavior of anthracene in a mixed as well as in a stratified lake is assessed. 

Figure 23.5 Processes considered for the combined sediment-water two-box model to describe the fate of PCB congeners in lakes.

Note that the only remaining variables are Ctop and Cssc, that is, the chosen state variables of the two-box model. 

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