Attainable region analysis

Cautions For rate laws with unusual concentration functionalities or for nonisothermal operation, these bounds may not be accurate and one must use attainable region analysis (ARA) discussed in Chapter 13. 

CD Solved Problems — Blood Coagulation FAQ [Frequently Asked Questions] — In Updates/FAQ icon section Professional Reference Shelf R6.I Attainable Region Analysis (ARA)... 

P6-27. This problem requires the use of Attainable Region Analysis (ARA) to solve it and in fact is solved on the web. 

First, the focus is on chemical reactors in Chapter 6, in which reactor models are reviewed, followed by a discussion of configurations for heat transfer in exothermic and endothermic reactors. Then, attainable region analysis is introduced for the optimal design of reactor networks. 

Kauchali, S., Rooney, W.C., Biegler, L.T., Glasser, D., Hilde-hrandt, D., 2002. Linear programming formulations for attainable region analysis. Chem. Eng. Sci. 57, 2015-2028. 

Seodigeng, T., Hausberger, B., Hildebrandt, D., Glasser, D., 2009. Recursive constant control policy algorithm for attainable regions analysis. Comput. Chem. Eng. 33, 309-320. 

Scott, R, Conejeros, R., Aroca, G., 2013. Attainable region analysis for continuous production of second generation bioethanol. Biotechnol. Biofuels 6, 171. 

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). 

As it has been shown, in AR approach the construction of the boundary is essential. Feinberg and Hidebrandt (1997) demonstrated that the boundary of attainable region could be assembled only by means of combinations of PFRs, CSTRs, and DSRs (differential side-stream reactors). The calculation of these elementary reactors is simple. The construction of an AR is not complicated, if the analysis is restricted at two dimensions. Moreover, 3-D representation could be visualised relatively easy with an appropriate computer tool. In the case of systems of higher dimensions the method relies rather on numerical techniques, and is not easy to apply. 

Thus far, the attainable region has been shown for the analysis of systems with two key compositions to be tracked. In the following, the principle of reaction invariants is used to reduce the composition space in systems of larger dimension. 

Table 3.2. Qualitative fingerprint of the design methods used in reactive distillation. Nomenclature. Gi Statics analysis G2 RGM G3 Attainable region G4 Fixed-points G5 Reactive cascade ...

Table 3.2. Qualitative fingerprint of the design methods used in reactive distillation (continuation). Nomenclature. Gi Statics analysis G2 ROM G3 Attainable region G4 Fixed-points G5 Reactive cascade Ge Thermodynamic-based G7 Conventional graphic techniques Gg Phenomena-based Gg Difference points Mi MINLP M2 OCFE M3 MIDO Hi Heuristics. Key. applicable nonapplica ble/nonmentioned A original assumption relaxed by later contributions...

Gorban, A.N., Kaganovich, B.M., Filippov, S.I., et al., 2006. Thermodynamic Equilibria and Extrenra. Analysis of Attainability Regions and Partial Equilibrium. Springer, New York, NY. 

This paper describes an integrated MINLP synthesis of overall process schemes using a combined synthesis / analysis approach. The synthesis is carried out by a multilevel-hierarchical MINLP optimization of the flexible superstructure, whilst the analysis is performed in an economic attainable region (EAR). The role of the MINLP synthesis step is to obtain a feasible and optimal process structure, and the role of the subsequent EAR analysis step is to verify the MINLP solution and to propose in the feedback loop, any profitable superstructure modifications for the next MINLP. The main objective of the integrated synthesis is to exploit interactions between the reactor network, separator network and the remaining part of the heat/energy integrated process scheme. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

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