One-box models

Given that the long residence time of uranium should place limits on how much the marine value could change over Late Quaternary time scales, several workers have used models to determine what these limits should be (see Henderson and Anderson 2003). Chen et al. (1986) and Edwards (1988) used a simple one-box model and assumed steady state conditions. They showed that ... 

Richter and Turekian (1993) also assumed a simple one-box model for the ocean. They derived a set of equations with the assumption that 1) the volume of the ocean does not change, 2) mass is conserved, i.e., the change in the marine U concentration with time equals the flux in from rivers minus the flux out, and 3) that the marine U concentration does not change with time, which implies that the flux in equals the flux out. This results in the equation ... 

Indeed Kim et al. (1999) have argued that better agreement is obtained from a one-box model approach, in which the particulate Th residence time is calculated as the difference between the residence times of total Th and dissolved Th (effectively the difference in the l/k values calculated from Eqns. 5 and 6). However, Buesseler and Charette (2000) argued in a response to Kim et al. (1999) that there is abundant evidence to support the notion that residence times of POC and Th are different in the euphotic zone. 

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. 

It is useful to briefly consider a simple conceptional model that considers simultaneous inputs and outputs of a compound in an organism, the one-box approach (see Section 12.4 for a general discussion of one-box models). In this approach we assume that the organism (i.e., the fish) is a well-mixed reactor (which, of course, it is not), and we define all processes as first-order reactions. The temporal change in concentration of a given compound i in the fish, Clfish, can then be described simply by ... 

Finally, in the last section of this chapter, we will introduce the simplest approach for modeling the dynamic behavior of organic compounds in laboratory and field systems the one-box model or well-mixed reactor. In this model we assume that all system properties and species concentrations are the same throughout a given volume of interest. This first encounter with dynamic modeling will serve several pur-... 

We conclude this chapter by introducing a simple tool with which we will be able to put the reactivities of organic compounds into an environmental context the well-mixed reactor or one-box model. 

Figure 12.5 Schematic representation of a simple one-box model including constant input (/), output (O), and transformation reactions in the bulk phase. M is the total mass of the compound in the constant volume V, C is the total concentration, and Q is the flow rate.

This equation can be very helpful for assessing the behavior of a substance in a natural system. If input and system concentration at steady-state are equal, ktot must be zero. In turn, for any reactive substance must be smaller than Cjn. A simple application of the one-box model is given in Illustrative Example 12.3. 

Consider the linear one-box model. What can we learn from a comparison of Cin and regarding the behavior of the chemical in the system ... 

In order to fully appreciate the consequences of the rather simple mathematical rules which describe the random walk, we move one step further and combine Fick s first law with the principle of mass balance which we used in Section 12.4 when deriving the one-box model. For simplicity, here we just consider diffusion along one spatial dimension (e.g., along the x-axis.)... 

Calculate the half-life of lindane in the pond (days), assuming that the reservoir volume remains constant. The one-box model introduced in Chapter 12 may help. 

Box 21.3 Time-Dependent External Forcing of Linear One-Box-Model Box 21.4 Temporal Variability of PCE-Input and Response of Concentration in Greifensee... 

Box 21.5 One-Box Model with Water Throughflow and Second-Order Reaction (Advanced Topic)... 

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. 

In this chapter we will keep the description of transport simpler than Fick s law, which would eventually lead to partial differential equations and thus to rather complex models. Instead of allowing the concentration of a chemical to change continuously in space, we assume that the concentration distribution exhibits some coarse structure. As an extreme, but often sufficient, approximation we go back to the example of phenanthrene in a lake and ask whether it would be adequate to describe the mass balance of phenanthrene by using just the average concentration in the lake, a value calculated by dividing the total phenanthrene mass in the lake by the lake volume. If the measured concentration in the lake at any location or depth would not deviate too much from the mean (say, less than 20%), then it may be reasonable to replace the complex three-dimensional concentration distribution of phenanthrene (which would never be adequately known anyway) by just one value, the average lake concentration. In other words, in this approach we would describe the lake as a well-mixed reactor and could then use the fairly simple mathematical equations which we have introduced in Section 12.4 (see Fig 12.7). The model which results from such an approach is called a one-box model. 

In case the actual concentration of phenanthrene would exhibit significant spatial variations, the one-box model would not be the ideal description. Instead, it may be adequate to subdivide the lake into two or more boxes in such a way that within the defined subvolumes, phenanthrene concentration would be fairly homogeneous. So we would end up with a two- or multi-box model. In certain situations this box model approach may still not be sufficient. We may need a model which allows for a continuous concentration variation in time and in space. Such models will be discussed in Chapter 22. 

The simplest and often most suitable modeling tool is the one-box model. One-box models describe the system as a single spatially homogeneous entity. Homogeneous means that no further spatial variation is considered. However, one-box models can have one or several state variables, for instance, the mean concentration of one or several compounds i which are influenced both by external forces (or inputs) and by internal processes (removal or transformation). A particular example, the model of the well-mixed reactor with one state variable, has been discussed in Section 12.4 (see Fig. 12.7). The mathematical solution of the model has been given for the special case that the model equation is linear (Box 12.1). It will be the starting point for our discussion on box models. 

Figure 21.3 One-box model or well-mixed reactor model. State variables are the concentrations, Ch Cpof chemicals ij,... They are influenced by inputs (/,-, Ip. ..), outputs (Oj, Op. ..), and internal transformations between the state variables or between other chemicals X, Y,... which do not appear as state variables in the model.

In this box we demonstrate the construction and application of a simple one-box model to a small lake like Greifensee to analyze the dynamic behavior of a chemical such as PCE. Characteristic data of Greifensee are given in Table 21.1. Measurements of PCE in the water column of the lake yield the following information ... 

Time-Dependent External Forcing of Linear One-Box Model... 

Some important sources of uncertainty for the indirect determination of the degradation rate, NXA, from the linear one-box model include ... 

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