Mixtures concentration space

Concentration space is the number of points representing all possible compositions of an n-component mixture. Concentration space of a binary mixture C2 is a segment of unit length the ends correspond to pure components, and the inner points correspond to mixtures of various compositions (Fig. 1.1a)... 

A bond, bond chain, distillation subregion, andregion are the nonlocal structural elements of the azeotropic mixture concentration space. 

By the structural matrix of the azeotropic mixture concentration space, we will name a square matrix, the columns and lines of which correspond to the stationary points and the elements of which aij = 1, if there is a bond directed from stationary point i to stationary point (a, = 0, if such a bond is missing). For the purpose of obviousness, some examples of three-component mixture structural matrices are shown in Fig. 1.8. 

We examine separation of the mixtures, concentration space of which contains region of existence of two hquid phases and points of heteroazeotropes. It is considerably easier to separate such mixtures into pure components because one can use for separation the combination of distillation columns and decanters (i.e., heteroazeotropic and heteroextractive complexes). Such complexes are widely used now for separation of binary azeotropic mixtures (e.g., of ethanol and water) and of mixtures that form a tangential azeotrope (e.g., acetic acid and water), adding an entrainer that forms two liquid phases with one or both components of the separated azeotropic mixture. In a number of cases, the initial mixture itself contains a component that forms two liquid phases with one or several components of this mixture. Such a component is an autoentrainer, and it is the easiest to separate the mixture under consideration with the help of heteroazeotropic or heteroextractive complex. The example can be the mixture of acetone, butanol, and water, where butanol is autoentrainer. First, heteroazeotropic distillation of the mixture of ethanol and water with the help of benzene as an entrainer was offered in the work (Young, 1902) in the form of a periodical process and then in the form of a continuous process in the work (Kubierschky, 1915). 

Figure 9.9 Oxygen-fuel mass concentrations in mixture fraction space by Equation (9.45b), and at y = 6,...

Figure 5.16. The four-environment model divides a homogeneous flow into four environments, each with its own local concentration vector. The joint concentration PDF is thus represented by four delta functions. The area under each delta function, as well as its location in concentration space, depends on the rates of exchange between environments. Since the same model can be employed for the mixture fraction, the exchange rates can be partially determined by forcing the four-environment model to predict the correct decay rate for the mixture-fraction variance. The extension to inhomogeneous flows is discussed in Section 5.10.

A map is the deshed result, for example a clean and noise free spectrum, or the concentration of several compounds in a mixture. Map space exists in a similar fashion to data space. 

The calculation of the minimum reflux and reboil ratios of nonpreferred separations is based on the fact that, in ideal mixtures, the states of constant reflux ratio = const, constitute a straight line in the triangular concentration space of Fig. 5.2-30. Their endpoints on the side lines of the triangle can easily be determined from the McCabe-Thiele diagram showing the equilibrium curves of the binary mixtures a-b and a-c. From a first estimation of the reflux ratio the operating line is drawn. Its points of intersection with the equilibrium lines dehver the endpoints... 

Consider, for example, the original 5g of NaCl and 10 g of KOH described before. The mixture corresponds to the coordinate (0.085 0.178) in NaCl-KOH molar concentration space, and is represented by point 1 in Figure 2.7. When an additional 5 g NaCl is added, the mixture moves 0.085 mol/L to the right. The new coordinate is hence given by (0.171 0.178), which is shown as point 2 in Figure 2.7. 

When density is assumed constant, then mixing has a special geometric property. Mixtures lie on a straight line joining the two concentrations being mixed in concentration space. Mixing is therefore a linear process. This has important consequences in AR theory, as will be seen later. 

Although it is still possible to visualize these points, as well as any possible mixture concentrations, it is noteably more cumbersome and time consuming to identify the points that lie on the perimeter/boundary. Figure 2.24 illustrates this for 30 points in a three-dimensional space (R ). 

Extents can take on negative values Extent of reaction is positive for products and negative for reactants. Concentrations, by comparison, may never take on negative values. This property provides a convenient set of bounds for all species present in a mixture. If concentration is used over extent for a given reactive system, the associated AR for the system must then also always lie in the positive orthant of concentration space." Additionally, using reaction stoichiometry and mass balance constraints, it is possible to compute a definite bound (called the stoichiometric subspace) that the AR must reside in. Similar bounds are rather less well-defined when extent is employed, and thus greater care must be taken. 

Description In the same manner that one might define a mixture concentration compactly as a vector of species concentrations, it is also possible to form a vector, of equal dimension to the concentration vector, associated with the rate of reaction. If benzene and toluene are the components of interest to the problem, the concentration vector is C = [Cb, c-p]. It is natural to express the corresponding rates of reaction for benzene and toluene as a two-component column vector as well. The rate vector, r(C), in Ca-Cp space is hence defined as follows ... 

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. 

Concentration space of an n-component mixture C is a space in which every point corresponds to a mixture of definite composition. Dimensionality of concentration space corresponds to the number of concentrations of components that can be fixed independently. 

The (n - 1) concentration for an n-component mixture can be fixed independently because concentration of the nth component can be found from Eq. (1.2). That is why the dimensionality of the concentration space of binary mixture Ci is one, of ternary mixture C3 - two, of four-component mixture C4 - tree, etc. 

Equation (1.5) represents the so-called lever rule points xip, Xio, and xib are located on one straight line, and the lengths of the segments [xif, xid and [xib, Xip] are inversely proportional to the flow rates D and B (Fig. 1.1b). Mixture with a component number n > 5 cannot be represented clearly. However, we wiU apply the terms simplex of dimensionality (n -1) for a concentration space of n-component mixture C , hyperfaces C i of this simplex for (n - l)-component constituents of this mixture, etc. 

The boundaries separating one bundle from another are specific residue curves that are called the separatrixes of saddle stationary points. In contrast to the other residue curves, the separatrixes begin or come to an end, not in the node points but in the saddle points. A characteristic feature of a separatrix is that in any vicinity of its every point, no matter how small it is, there are points belonging to two different bundles of residue curves. The concentration space for ideal mixtures is filled with one bundle of residue curves. Various types of azeotropic mixtures differ from each other by a set of stationary points of various types and by the various sequence of boihng temperatures in the stationary points. 

Figure 2.13b shows a structure of concentration space of mixture benzene (entrainer) (1) -isopropyl alcohol(2)-water(3). The mixture has a ternary azeotrope and three binary azeotropes. 

The important advantage of mixture separability analysis for each sharp split consists of the fact that this analysis, as is shown in this and three later chapters, can be realized with the help of simple formalistic rules without calculation of distillation. A spht is feasible if in the concentration space there is trajectory of distillation satisfying the distillation equations for each stage and if this trajectory connects product points. That is why to deduct conditions (rules) of separability it is necessary to study regularities of distillation trajectories location in concentration space. 

Because sharp separation is not always feasible for azeotropic mixtures, we also consider the best semisharp splits, when one of the products contains a smaller number of components than the feeding and when the possible product point of the second product is the farthest from the product point of the first product in the concentration space. 

Fill up structural matrix for four-component azeotropic mixture the structure of concentration space of which is shown at Fig. 3.15. 

Petlyuk, F. B. (1979). Structure of Concentration Space and Synthesis of Schemes for Separating Azeotropic Mixtures. Theor. Found. Chem. Eng, 683-9. 

Therefore, the stated algorithm of calculation of minimum reflux mode, based on the geometry of the trajectory bundles in concentration space, are potentially as one likes precise and most general, because they embrace any spUts on mixtures with any components number and any degree of nonideahty. 

However, in some cases, even usage of nonadiabatic columns does not maintain separability. These are the cases, when reversible distillation trajectories for both product points do not have node points. In these cases, section trajectory bundles not only of adiabatic, but also of nonadiabatic columns, are limited because reversible distillation trajectories at which section trajectory bundles stationary point lie are located in limited parts of concentration space, adjacent to product boundary elements (see Fig. 5.17b for xd). To check whether it is possible to separate the mixture of this kind into a set product, it is necessary to examine the bundles 5 - N/r and 5 - A(+ for points 5 and S, the most remote along... 

First, we consider the variation of concentration of a species in a solution/mixture with space and time inside a separator. An equation that considers such variations in a phase/region will be illustrated. Since such a concentration distribution in time and separator location is influenced... 

An illustrative example generates a 2 x 2 calibration matrix from which we can determine the concentrations xi and X2 of dichromate and permanganate ions simultaneously by making spectrophotometric measurements yi and j2 at different wavelengths on an aqueous mixture of the unknowns. The advantage of this simple two-component analytical problem in 3-space is that one can envision the plane representing absorbance A as a linear function of two concentration variables A =f xuX2). 

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