Variable-Density System

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

For variable density systems it is more convenient to work in terms of the fraction conversion (equation 8.2.9), but for constant density systems either equation 8.2.9 or equation 8.2.10 is appropriate. 

This example may also be solved using the E-Z Solve software (see file exl2-2.msp). 12.3.2.2 Variable-Density System... 

A material balance for A around the ith tank of volume Vi in the N-tank series (all tanks of equal size), in the case of unsteady-state behavior of a variable-density system,... 

Space-time is commonly referred to as mean residence time, holding time, or simply residence time. However, for a system with expansion (variable density system), these quantities are not equal and the residence time is a variable (Levenspiel, 1972) ... 

In the case of A as limiting reactant and a variable-density system, the solution of the model is the same as for first-order irreversible homogeneous reactions of the form A - products (Levenspiel, 1972) ... 

To a first approximation, the density in the variable-density system varies in the neighborhood of r as... 

DSR sidestream mixing parameter for constant density systems DSR sidestream mixing parameter for variable density systems... 

It shall always be assumed that density remains constant for mixtures, unless specified otherwise. This assumption is generally acceptable for most liquid mixtures. In Chapter 9, variable density systems will be discussed. 

The given description of residence time is the most natural choice for many reactor problems, although, this may not be the sole choice available. Take, for example, if reaction rate is expressed with respect to catalyst mass. An appropriate set of units for the rate might then incorporate the mass of the catalyst contained within the reactor body over reactor volume. In Chapter 9, we will define r in terms of mass fractions for variable density systems. 

This chapter marks the end of Section I of the book. All of the information discussed up to this point provides a firm foundation of the basics of AR theory. In the following chapters, we shall extend on these ideas, and relate them to higher dimensional constructions. Automated AR construction methods and variable density systems will also be discussed, which will allow us to tackle even more realistic problems. 

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. 

Figure 9.1 Reactor network involving a variable density system.

It might not be apparent why mass fractions are used over mol fractions in variable density systems. Mole fractions behave in a manner similar to mass fractions (for example, both sum to unity), and mole fractions are commonly used in rate expressions for gas phase reactions, such as when species partial pressures are used. Although it is possible to construct the AR in mole fraction space (see Zhou and Manousiouthakis, 2007), mass fractions are often easier to handle from a computational perspective. We therefore prefer to use mass fractions when the AR for a nonconstant density system is to be determined. 

Observe that the units of a are no longer dimensionally consistent with the units of time. Instead, the units of a are given by [volume of reactor x time/total mass]. Even though the units of a are different to t, the underlying mathematical and geometric behavior of a is equivalent to residence time. Hence, a is still a useful measure of the volume of a reactor network, and thus it is an important variable when determining reactor structures with minimum total volume in variable density systems. 

Understanding the bounds imposed by the reaction stoichiometry in mass fraction space plays an important role in helping to deploy standard AR construction schemes (in concentration space) for use in variable density systems. 

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