Mass fraction space

To view the data in a graphical manner, we shall borrow an idea from Section 2.1.2 and plot the mass fraction of NaCl and H2O, each on an axis in two-dimensional mass fraction space. For this particular example, we choose the x-axis to represent the mass fraction of NaCl in solution the mass fraction of water in solution is then placed on the y-axis. 

For any amount of NaCl and H2O added or removed from the system, it is possible to visualize the solution mixture as a point in NaCl-H20 mass fraction space. That is, any ratio of NaCl and H2O produces a point in space. For... 

Note that by plotting the species concentrations in mass fraction space, we can visualize each scenario as a distinct point on the graph. Provided that only combinations of water and salt are added to the system, it is clear that the resulting mixtures are associated with points that lie on a line in mass fraction space. This is easy to interpret once we notice that the species mass fractions must sum to unity ... 

When only two components are present, mass balance requires that species mass fractions lie on a straight line in mass fraction space. This result is also clear by observing that the equation... 

Much of the content in this chapter is taken from important contributions by Martin Feinberg (Feinberg, 1999, 2(X)0a, 2000b Feinberg and Hildebrandt, 1997). As will be shown, results from four papers by Feinberg, in particular, broadly define the major findings of AR theory in concentration space—at the time of writing, these results have yet to be expanded to wider state spaces, such as mass fraction space. It is for this reason that primary focus will be placed on AR constructions in concentration space alone wherein density is assumed constant. 

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. 

SIDE NOTE Anal< between Q in concen-1 1 tration space and G in mass fraction space... 

That is, for any mixture containing n species, we may associate the mixture with a mass fraction vector z. From a geometric viewpoint, z is the vector in K" mass fraction space associated with a unique magnitude and direction. 

Hence, the geometric interpretation of this is that a mixture resulting from combinations of mixtures Zj and produces a mass fraction vector z that lies on a straight line between Zj and in z -Zb space. Linear mixing is therefore maintained in mass fraction space. 

To show that residence time in mass fraction space obeys a linear mixing law, the traditional definition for t written in terms of volumetric flow rate is no longer suitable. Instead, we define, in an analogous fashion, the equivalent mass fraction residence time for a reactor i as follows ... 

CSTR The CSTR expression operating in mass fraction space is derived in a manner similar to the traditional CSTR expression in molar concentration space. A species molar balance around the CSTR operating at steady state... 

Observe that the geometric behavior of the mass fraction version of the CSTR expression shares identical traits to the constant density molar CSTR expression. In the case of mass fractions, the vector (z - Zj) must be collinear with the vector Wr(z) for z to be a feasible CSTR state in mass fraction space (provided that a > 0). Hence, the same geometric interpretations for CSTRs in mass fraction space apply. 

DSR A DSR operating in mass fraction space is best derived in a manner similar to that given in Chapter 4. However unlike the PFR and CSTR, it is preferable to begin with an expression in terms of species mass flow rates, g, instead of species molar flow rates n,-. Hence, consider a component mass balance over a small differential slice of the DSR, operated at steady state. The sidestream concentration is written in terms of a sidestream mass fraction z° and... 

Note that this is analogous to the DSR expression in concentration space. The variable p used in Equation 9.18 is the mass fraction equivalent of a. Note the geometric interpretation of a DSR operating in mass fraction space is also equivalent—a DSR solution trajectory in mass fraction space is resultant from the linear combination of a reactive component Wr(C) and a mixing vector (z° — z). 

Determining the stoichiometric subspace of a system of reactions in mass fraction space is carried out in a manner similar to that performed in concentration space. In fact, the stoichiometric subspace represented in mass fraction space is slightly more convenient, for all species mass fractions are necessarily bounded by 0 < z, < 1. 

Similar to the procedure carried out in Chapter 8, computation of the stoichiometric subspace S begins with the stoichiometric coefficient matrix A. The dimension of S in mass fraction space is equivalent to that in concentration space, and it is found by computing the rank of A. This is determined by the number of independent reactions present in the system. For n components in d reactions, the size of A is (nxd). 

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. 

Here, Wr(C) is the equivalent rate vector in mass fraction space. Assuming that we are working in residence time space,... 

EXAMPLE 6 Stoichiometric Subspace for CH4 Steam Reforming in Mass Fraction Space... 

In Chapter 8, we showed how the stoichiometric subspace for the methane steam reforming reaction can be computed in concentration space. Since the reaction occurs in the gas phase, it is more appropriate to determine the stoichiometric bounds in mass fraction space. This approach is preferable as the density of the mixture is no longer required to be constant. Compute the stoichiometric subspace for the CH4 steam reforming reaction and compare it to the answer obtained in Chapter 8. Assume that a feed molar vector of Uf = [1,1, l,0,0] kmol/s is available, and that the gas mixture obeys the ideal gas assumption to simplify calculations. Assume a constant pressure and temperature of P = 101 325 Pa and T = 500 K, respectively. 

Since rank(A) = 2, the stoichiometric subspace in this instance is two-dimensional residing in R mass fraction space. Due to the dimension of the component space, it is not possible to view the entire region in a single plot however, two-dimensional projections onto different component spaces may be performed. This is shown in Figure 9.4. 

Figure 9.4 Stoichiometric subspace for methane steam reforming for different component pairs in mass fraction space (i) CH -HjO,...

The system involves three independent reactions with four components. It follows that the AR is a three-dimensional subspace in Later on, it will be useM to provide a comparison of the AR generated in this chapter in mass fraction space, to that produced in Chapter 7 originally in concentration space. For this reason, the AR shall be generated in z -Zb-Zjj space. Components C may be found by mass balance. The mass fraction and rate vectors are then defined as z = [z, Zg, Zjj] and r(z) = [r (z), rg(z), rjj(z)] . It is assumed that the feed available is pure in component A. The feed molar flow rate vector is hence given as = [1, 0, 0]. Since the feed is pure in A, it follows that the mass fraction feed vector be given as Zj = [1,0,0]. ... 

Using this equation, the rate expressions given in the previous section may be utilized to describe the kinetics in mass fraction space. 

From these values. Equations 9.7, 9.17, and 9.18 may be employed to generate PFR, CSTR, and DSR reactor stmc-tures with corresponding output states in mass fraction space. The AR is expected to exist as a convex poly tope in... 

Observe that the region is represented by a tetrahedron in mass fraction space. The benefit of generating regions in mass fraction space is that the results are always scaled between 0 and 1. 

Figure 9.6 Stoichiometric subspace for the Van de Vusse system, generated in mass fraction space.

Figure 9.8 (a) Full AR for the Van de Vusse system in mass fraction space, (b) AR for the Van de Vusse kinetics, converted to concentration space. The transparent region is the AR obtained for an identical feed point when constant density is assumed. (See color plate section for the color representation of this figure.)... 

The stoichiometric subspace for this system has already been established, both in concentration space (Chapter 6) and mass fraction space (Section 9.2.7). The same numbering convention shall be adopted as used for the stoichiometric subspace calculation. Hence, rows 1-5 correspond to CH4 to CO2, respectively. 

Conversion to Mass Fraction Space Sinee the rate expressions for CH4 reforming and the water-gas shift reaction are given in terms of species coneentration, the Peng-Robinson equation of state must be employed to express the rate expressions in terms of species mass fractions instead. 

Moreover, although the result may not appear significant, the fact that the AR can be computed at all bears special meaning the AR for a non-ideal system involving temperature dependence and minimum reactor volume has been found. From the results of the construction, the AR in mass fraction space can be converted to an equivalent AR in concentration space. An appropriate objective function may then be overlaid to determine its intersection with the boundary. Optimization of the system may then follow. From this, deeper insights into the limits of the system can be obtained. 

A number of simple transformations were explained in this chapter that help to express common quantities used in AR theory in terms of species mass fractions. Mass fractions in AR computations are useful for describing industrial systems, as the conservation of mass guarantees that mixing in mass fraction space always obeys a linear mixing law. This result, in turn, allows for the use of AR theory in variable density systems (when molar concentration does not mix linearly). 

Conversion of systems to mass fraction space often requires an appropriate equation of state to relate the temperature, pressure, and composition of a mixture. Use of... 

Figure 9.9 (a) AR for the methane steam reforming reaction in mass fraction space, (b) Comparison of the stoichiometric subspace and... 

Chapter 9 discussed the construction of candidate ARs in variable density systems. Mass fractions are hence useful in broadening the use of AR theory to a wider class of systems (i.e., gas phase reactions). Mass fractions always obey a linear mixing law, and thus these constructions are valid even when the system does not obey constant density. We described a number of useful relations for converting common process variables used in AR theory, such as concentration and mole fraction, in terms of species mass fractions. These conversions are often assisted by use of an appropriate equation of state that relates the system volume to the process variables of interest. Rate expressions involving concentration are then easily viewed in mass fraction space, and the AR may be computed appropriately. 

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