Well-Mixed Reactor or One-Box Model

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. 

In Chapter 21 we will show that the construction of any environmental model first consists of the appropriate choice of a boundary between the system and the outside world (see Fig. 21.1). Here we choose the simplest possible system [i.e., a homogeneous (completely mixed) box that is connected to the outside world by an input I and an output O, Fig. 12.5]. We consider one single chemical compound that shall be described either by the total amount in the system, M, or by its mean concentration, C=IM/V, where V is the volume of the system. Note that for simplicity we omit the subscript i in the following derivation. Several reaction processes, Rj, act on the compound in the inteior of the system we characterize them by the total rate Rlal  

Then the mass balance for the compound in the well-mixed box is simply  

All quantities on the right-hand side have the dimension mass per time  

If the volume V is constant, we can rewrite this equation using M = VC and dividing both sides by V  

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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. 

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-... 

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. 

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 Part IV we repeatedly used box models for describing the dynamics of chemicals in lakes. In this chapter we will summarize this information. As a first step, Fig. 23.1 illustrates the one-box model approach for the average total concentration of a chemical, Ct, in a well-mixed water body such as a pond, a shallow lake, a subcompartment of a deep lake or ocean (e.g., the mixed surface layer), or even an engineered system like a completely stirred reactor. 

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