Transformed compositions

The transformed variables describe the system composition with or without reaction and sum to unity as do Xi and yi. The condition for azeotropy becomes X, = Y,. Barbosa and Doherty have shown that phase and distillation diagrams constructed using the transformed composition coordinates have the same properties as phase and distillation region diagrams for nonreactive systems and similarly can be used to assist in design feasibility and operability studies [Chem Eng Sci, 43, 529, 1523, and 2377 (1988a,b,c)]. A residue curve map in transformed coordinates for the reactive system methanol-acetic acid-methyl acetate-water is shown in Fig. 13-76. Note that the nonreactive azeotrope between water and methyl acetate has disappeared, while the methyl acetate-methanol azeotrope remains intact. Only... 

Transformation toughening ceramics, 5 621-622 Transformed composition space, 22 330-331 Transforming particles ceramic—matrix composite reinforcement, 5 571—572 Transgene expression, controllable, 12 453 Transgenes, 12 452, 453... 

Due to the conservation of elements, the rank of J will lie less than or equal to K — E 1 In general, rank(J) = Ny < K - E, which implies that V = K — T eigenvalues of J are null. Moreover, since M is a similarity transformation, (5.51) implies that the eigenvalues of J and those of J are identical. We can thus limit the definition of the chemical time scales to include only the Nr finite ra found from (5.50). The other N components of the transformed composition vector correspond to conserved scalars for which no chemical-source-term closure is required. The same comments would apply if the Nr non-zero singular values of J were used to define the chemical time scales. 

If AW AW the process of finding a linear-mixture basis can be tedious. Fortunately, however, in practical applications Nm is usually not greater than 2 or 3, and thus it is rarely necessary to search for more than one or two combinations of linearly independent columns for each reference vector. In the rare cases where A m > 3, the linear mixtures are often easy to identify. For example, in a tubular reactor with multiple side-injection streams, the side streams might all have the same inlet concentrations so that c(2) = = c(iVin). The stationary flow calculation would then require only AW = 1 mixture-fraction components to describe mixing between inlet 1 and the Nm — I side streams. In summary, as illustrated in Fig. 5.7, a turbulent reacting flow for which a linear-mixture basis exists can be completely described in terms of a transformed composition vector ipm( defined by... 

The surfaces described by Eqs. (27a) and (27b) in the three-dimensional composition space intersect with each other and yield the PSPS as curves given in Fig. 4.10(a). The PSPS contain several branches, three of which pass through the pure components HOAc, IPOAc and water, and are located outside the composition space but are not depicted. The branch passing through the I PA-vertex locates four nonre-active azeotropes - that is, IPA-IPOAc, IPOAc-Water, IPA-IPOAc-Water, and IPA-Water. This branch also contains the reactive azeotrope. The PSPS is also displayed in the transformed composition space (Fig. 4.10(b)). 

Fig. 4.10. Potential singular point surface for isopropyl acetate (IPOAc) reaction system at 1.01 X 105 Pa. (a) Liquid phase composition space in mole fractions x, (b) representation in transformed composition space.

A singular point of reactive membrane separation should be denoted as kinetic arheotrope because it is a process phenomenon rather than a thermodynamic phenomenon. The condition for arheotropy can be elegantly expressed in terms of new transformed variables, which are a generalized formulation of the transformed composition variables first introduced to analyze reactive azeotropes. 

Figure A.l Conventional and transformed composition variables in a ternary diagram.

For example, the AB mixture expressed in Figure A.1 by XA and XB mole fractions on the AB edge leads at equilibrium to a mixture (xA, xB, xc) obtained by intersecting the equilibrium curve with the stoichiometric line passing through the initial mixture. Conversely, a ternary mixture where a chemical reaction at equilibrium takes place may be described only by two transformed composition variables. 

Corresponding relations hold for multicomponent mass transfer with transformed compositions Xio and Xib-... 

It will be convenient to determine the left rather than the right characteristic vectors. Except for a possible discrepancy in length, which determines the size of unit amounts of the various species these vectors form the rows of the inverse matrix used to transform compositions from the A to the B system of coordinates (see footnote Section IV,A,4,a). These ... 

These matrices 2 and 2 transform compositions between the natural and characteristic systems in the two dimensional spaces ... 

The composition of the metastable structural transition in the experimental studies lies between about 0.63 and 0.69 mol fraction AIN. The calculated transformation composition is therefore in good agreement with these observations. 

Effect of grinding time on the Form-H-to-Form-I phase transformation composition of pure fostedil (O), and also for fostedil/microcrystalline cellulose mixtures having ratios of 3 1 ( ), 1 1 ( ), and 1 3 ( ). (The figure is adapted from data presented in Ref. 6.)... 

Solutions of (6.14) and (6.15), the rectifying and stripping cascade flash trajectories, can be represented in mole fraction space (three dimensional for the IPOAc system). However, we represent the solutions in transformed composition space, which is two dimensional for IPOAc system (for a derivation and properties of these transformed variables [46]). This transformed composition space is a projection of a three dimension mole fraction space onto a two dimensional transformed composition subspace for the IPOAc system. Even though the correspondence between real compositions and transformed compositions is not one-to-one in the kinetic regime, we will make use of these transforms because of ease of visualization of the trajectories, and because overall mass balance for reactive systems (kinetically or equilibrium limited) can be represented with a lever rule in transformed compositions. We use this property to assess feasible splits for continuous RD. 

Parametric column simulations for the I POAc system were performed with different Damkohler numbers, reflux ratios, reboil ratios as well as total number of stages, (N-I-) and feed tray location, (/). The distillate and bottoms compositions obtained were recorded in transformed composition space. Fig. 6.9 compares the products obtained from column simulations with 30 stages and using different values of r and s at D = 0.25 and D = 0.75. The column feed specification is the same as that to the co-current flash cascade. The flash trajectories provide a good estimate of the product compositions from a continuous column. We also compared the product compositions from column simulations with the flash trajectories in mole fraction space. We found that product compositions from column simulations surrounded the flash trajectories, in agreement with the hypothesis that the flash trajectories lie in the feasible product regions for continuous RD. 

The transformed composition variables have two convenient properties they have the same numerical value before and after reaction, and their sum is unity ... 

S. Banegee, R. Ghosh, and B. Bagchi, Structural transformations, composition anoma-hes and a dramatic collapse of linear polymer chains in dilute eflianol-water mixtures. J. Phys. Chem. B, 116 (2012), 3713-3722. 

Figure 7.40 Transformed compositions for isobutene, methanol, and MTBE in chemical an) phase equilibrium. (Reprinted from Doherty and Buzad, 1992).

These necessary and sufficient conditions for reactive azeotropes have been generalized and theoretically established for the case of multicomponent mixtures undergoing multiple equilibrium chemical reactions by Ung and Doherty (19956). The starting point for their analysis is the introduction of transformed compositions. It is widely recognized that mole fractions are not the most convenient measures of composition for equilibrium reactive mixtures, as they might lead to distortions in the equilibrium surfaces (Barbosa and Doherty, 1988a Doherty and Buzad, 1992). In order to visualize in a much more... 

For this system, the transformed compositions are given by the following generalized expressions. 

These azeotropy expressions 2.19 state that in the space of transformed composition variables the bubble-point and dew-point surfaces are tangent at an azeotropic state (Barbosa and Doherty, 1988a), allowing the azeotropes to be found easily by visual inspection in the reactive phase diagram for the case of Uc — rirx < 3 (figure 2.4). For systems beyond this space, a graphical determination of azeotropes might not be feasible. 

Figure 2.4. Phase diagram for methanol in the synthesis of MTBE expressed in terms of transformed compositions (equation 2.18). Remrirks the location of the reactive eizeotrope is tracked down at the intersection of the residue curve and the line X =Y reacting mixtures of various compositions are depicted. System features operating pressure is 11-10 Pa inert nC4 is present in the mixtme.

Figure 3.3. Dimension reduction through transformed compositions. Systems features chemical reaction A + B + I C + 7 where I is an inert component (adapted from Prey and Stichhnair (19996)).

These definitions present convenient and simplifying properties ( ) the dimensions of the system are reduced, simplifying the depiction of equilibrium (figure 3.3) (Frey and Stichlmair, 19996 Barbosa and Doherty, 1987a) (ii) they have the same numerical values before and after reaction (m) they sum up to unity (iv) they clearly indicate the presence of reactive azeotropes when Xi =Yf, (v) the nonreactive hmits are well defined (vi) the number of linear independent transformed composition variables coincides with the number of independent variables that describe the chemical equihbrium problem and vii) the lever rule is valid as the chemical reaction no longer impacts the material balance. 

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