Attainable Regions for Variable Density Systems

All of the systems and associated AR constructions provided thus far have been computed, primarily, in concentration space. Understanding concepts related to concentration is more intuitive, we believe. Thus, it is simpler to base discussions in concentration space, as opposed to other process variables that also obey linear mixing laws. The AR has also been developed historically with isothermal constant density systems in mind, although this constraint is relaxed in Chapter 7. Yet, concepts such as those from Chapters 6 to 8 demonstrate that even when these constraints are relaxed, the resulting theory might still be quite complex. 

Many systems of interest typically occur under nonconstant density conditions, and heterogeneous reactions are possibly more common in industry than constant density systems. It is hence beneficial to understand how such systems might be improved via AR analysis. 

In this chapter, we wish to demonstrate how the existing set of AR theory, developed in previous chapters, may be adapted to include variable density systems as well. Certainly, many of the founding ideas, involving reaction concentration and mixing, may be adapted for use with mass fractions instead. The use of mass fractions is an important concept in nonconstant density systems, and thus an adequate understanding of the idea is required. 

We will begin by discussing a number of important formulae for converting common process variables involving moles to equivalent quantities involving mass fraction. These concepts are not difficult to understand, however, they are fundamental to how the computation of ARs in mass fraction space must be organized. Discussion of how the stoichiometric subspace may be computed and how residence time may be incorporated in mass fraction space is also provided. From this, a number of examples are provided that demonstrate the theory. In particular, isothermal and nonisothermal unbounded gas phase systems shall be investigated. 

In the following discussions, it is assumed that an appropriate equation of state is available. This is necessary for describing how the system temperature, pressure, and volume are related to each other. It is also assumed that molar masses for all components participating in the system are available, which will be used to convert between species mass and moles. 

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