Region attainable

CSTR (point O), and a CSTR followed by a PFR (curve D). Within the region bounded by the three arcs and the horizontal base line Cg = 0), product compositions can be achieved with some combination of these reactor configurations. The appropriate reactor conhguradoo along the boundary of the attainable region depends on the desired effluent concentration of A. When 1 0.38 kmol/m, a CSTR with bypass (curve C) provides the maximum 

A systematic method for the construction of the attainable region using CSTRs and ITRs, with or without mixing and bypass, for a system of chemical reactions, as presented by Hildebrandt and Biegler (1995), is demonstrated for van de Vusse kinetics  

Step 1 Begin by constructing a trajectory for a PFR from the feed point, continuing to the complete conversion of A or chemical equilibrium. In this case, the PFR trajectory is computed by solving simultaneously the kinetic equations for A and B  

Step 3 The PFR trajectory is expanded by linear arcs, representing mixing betweenh PFR effluent and the feed stream, extending the candidate attainable region. Note that a linear arc connecting two points on a composition trajectory is expressed by the equation 

Step 5 A PFR trajectory is drawn from the position where the mixing line meets the CSTR trajectory. If this PFR trajectory is convex, it extends the previous AR to form an 

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Glasser, D., Hildebrandt, D., and Crowe, C. (1987). A geometric approach to steady flow reactors The attainable region and optimization in concentration space. Ind. Eng. Chem. Res., 26, 1803-1810. 

Horn, F. (1965). Attainable and non-attainable regions in chemical reaction technique, in Third European Symposium on Chemical Reaction Engineering Pergammon Press London, 293-302. 

The effects of deviations from the Born-Oppenheimer approximation (BOA) due to the interaction of the electron in the sub-barrier region with the local vibrations of the donor or the acceptor were considered for electron transfer processes in Ref. 68. It was shown that these effects are of importance for long-distance electron transfer since in this case the time when the electron is in the sub-barrier region may be long as compared to the period of the local vibration.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes. However, nonadiabatic effects in the classically attainable region may also be of importance in atom transfer processes. In the harmonic approximation, when these effects are taken into account exactly, they manifest themselves in the noncoincidence of the... 

Feinberg, M. and D. Hildebrandt. Optimal Reactor Design from a Geometric Viewpoint I. Universal Properties of the Attainable Region. Chem Eng Sci 52(10) 1637-1666 (1997). 

Remark 1 Note that the borderlines between the three main approaches are not necessarily distinct. For instance, the targets in (ii) can be viewed as heuristics or rales that simplify the combinatorial problem and allow for its decomposition into smaller, more tractable problems (see chapter on heat exchanger network synthesis via decomposition approaches). The optimization approach (iii) can formulate thermodynamic targets, or targets on the attainable region of reaction mechanisms as optimization models, and can either utilize them so as to decompose the large-scale problem or follow a simultaneous approach that treats the full-scale mathematical model. The first... 

D. Hildebrandt and D. Glasser. The attainable region and optimal reactor structures. Chem. Eng.Sci., 45 2161, 1990. 

D. Hildebrandt, D. Glasser, and C. M. Crowe. Geometry of the attainable region generated by reaction and mixing with and without constraints. / EC Res., 29 49, 1990. 

Nicol W, Hemier M, Hildebrandt D, Glasser D. The attainable region and process synthesis reaction systems with external cooling and heating. Chem Eng Sci 2001 56 173-191. 

Around the same time, Glasser et al. (17) retrieved and extended the insightful methods of Horn (18) and presented graphical procedures known as the attainable region (AR) method. Their approach requires the graphical construction of the convex hull of the problem and helps to exemplify the need for a systematic and general methodology. In principle, the reactor network with maximum performance in terms of yield, selectivity, or conversion can be located on the boundary of the AR in the form of DSR and CSTR cascades with... 

Feinberg M, Hildebrandt D. Optimal reactor design from a geometric viewpoint. I. Universal properties of the attainable region. Chem Eng Sci 1997 52 1637. 

Glasser D, Hildebrandt D, Godorr S. The attainable region for segregated maximum mixed and other reactor models. Ind Eng Chem Res 1994 33 1136. 

Fig. 2.4. The conversion diagram, stoichiometrically attainable region and reaction rate vector.

Nisoli et al. [22] have studied the attainable regions for this reaction system with simultaneous distillative separation. These authors found an immiscible region between water and DME, which shrinks as the pressure increases. The PSPS of this system was calculated based on the VLE parameters given in the studies of Nisoli et al. [22]. Water was chosen as the reference component and DME as independent variable to represent the system ... 

From the above said, it may be concluded that a detailed kinetic model of coal combustion process that combines all three basic processes can not virtually be constructed, as it is impossible to do for each process separately. Therefore, the empirical models based on separation and experimental study of the limiting stages are extensively used. Such models separately do not reveal general regularities and do not allow the generalized conclusions to be drawn. The thermodynamic model makes it possible to study the whole attainability region and hence to consider states of the considered system as a whole and to keep track of the variation in the amounts of any component as a function of some or other kinetic constraints. The latter are written, as was shown above, easily enough even for such complex processes as coal combustion. 

Gorban, A. N., Kaganovich, B. M. and Filippov, S. P., "Thermodynamic Equilibria and Extrema Analysis of Attainability Regions and Partial Equilibria in Physicochemical and Technical systems", 296 p. Nauka, Novosibirsk (2001). (in Russian). 

The WWW shows how to use these plots along with the attainable region technique to maximize the amount of B produced. 

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