The Attainable Region

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. 

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. 

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 ... 

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

The Fun Part 315 The Attainable Region CD-ROM 316 Summary 318 Questions and Problems 320... 

For chemical reactor networks, the attainable region concept was first defined by Horn (1964), who noted that... 

V]ariables such as recycle flow rate and composition of the product form a space which in general can be divided into an attainable region and a non-attainable region. The attainable region corresponds to the totality of physically possible reactors.. . . Once the border is known the optimum reactor corresponding to a certain environment can be found by simple geometric considerations. 

To illustrate this concept, consider the attainable region for the following series reaction ... 

We construct the attainable region by noting that the concentration space is a vector field with a rate vector (e.g., in Fig. 1, dC /dC/ = RB/R ) defined at each point. Moreover, we are not restricted to concentration space, but can consider any other variable that satisfies a linear conservation law (e.g., mass fractions, residence time, energy, and temperature—for constant heat capacity and density). The attainable region is an especially powerful concept once it is known, performance of the network can often be determined without the network itself. 

More recently, Glasser et al. (1987) developed geometric properties of the attainable region along with a constructive approach for determining this region. They defined the necessary conditions for the attainable region as follows ... 

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