The attainable region for systems with mixing and multiple-rate processes: finding optimal reactor structures

Godorr, Sven; Hildebrandt, Diane; Glasser, David

doi:10.1016/0923-0467(94)00212-6

Cited by 15 publications

(9 citation statements)

References 20 publications

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“…We have not included it as a variable in this paper as including this variable increases the dimensionality of the problem by one. We have rather taken the approach as used by Godorr et al (1994) where we consider E as a control variable that we have at our disposal. This will be discussed below in detail when we define the breakage vector.…”

Section: Representing the Psd In Mass Fraction Spacementioning

confidence: 99%

The application of the attainable region analysis to comminution

Khumalo

Glasser

Hildebrandt

et al. 2006

Chemical Engineering Science

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Abstractiii ABSTRACT This work applies the concepts of the attainable region for process synthesis in comminution. The attainable region analysis has been successfully applied for process synthesis of reactor networks. The Attainable Region is defined as the set of all possible output states for a constrained or unconstrained system of fundamental processes . A basic procedure for constructing the attainable region for the fundamental processes of reaction and mixing has been postulated in reaction engineering . This procedure has been followed in this work to construct the candidate attainable region for size reduction processes as found in a size reduction environment.A population balance model has been used to characterise the evolution of particle size distributions from a comminution event. postulated the dependency of grinding on the specific energy. A specific energy dependent population balance model was used for the theoretical simulations and for the fitting of experimental data.A new method of presenting particle size distributions as points in the Euclidian space was postulated in place of the traditional cumulative distribution. This allows successive product particle size distributions to be connected forming a trajectory over which the objective function can be evaluated. The curve connects products from successive batch grinding stages forming a pseudo-continuous process.Breakage, mixing and classification were identified as the fundamental processes of interest for comminution. Agglomeration was not considered in any of the examples. Mathematical models were used to describe each fundamental process, i.e. breakage, mixing and classification, and anThe application of the attainable region analysis in comminution Abstract iv algorithm developed that could calculate the evolution of product particle size distributions. A convex candidate attainable region was found from which process synthesis and optimisation solutions could be drawn in two dimensional Euclidian space. As required from Attainable Region Theory, the interior of the bounded region is filled by trajectories of higher energy requirements or mixing between two boundary optimal points.Experimental validation of the proposed application of the attainable region analysis results in comminution was performed. Mono-sized feed particles were broken in a laboratory ball mill and the products were successfully fitted using a population balance model. It was shown that the breakage process trajectories were convex and they follow first order grinding kinetics at long grind times. The candidate attainable region was determined for an objective function to maximise the mass fraction in the median size class 2. It was proved that the same specific energy input produces identical products. The kinematic and loading conditions are supposed to be chosen as a subsequent event after the required specific energy is identified.Finally the fundamental process of classification was added to the system of breakage and mixing. The attainable regions analysi...

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Section: Representing the Psd In Mass Fraction Spacementioning

confidence: 99%

The application of the attainable region analysis to comminution

Khumalo

Glasser

Hildebrandt

et al. 2006

Chemical Engineering Science

Self Cite

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“…The structure of the boundary is interpreted as a process layout by following various trajectories from one process surface to another via the intersectors, so tracing the sequence of fundamental processes used. Although it may not be entirely clear yet how the interpretation of the boundary is achieved, the easiest way to understand the procedure is to look at problems that have been solved. − ,, …”

Section: Interpreting the Boundarymentioning

confidence: 99%

“…Over the past decade the attainable region method has been used to solve a large variety of optimization problems. − Some of these problems had not been or could not be solved by the classical methods. It is a therefore instructive to compare the AR method with these other methods.…”

Section: Introductionmentioning

confidence: 99%

The Attainable Region and Pontryagin's Maximum Principle

McGregor

Glasser

Hildebrandt

1999

Ind. Eng. Chem. Res.

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Attainable region analysis has been used to solve a large number of previously unsolved optimization problems. This paper examines its relationship to Pontryagin's maximum principle and highlights the similarities and differences between the methods. It is shown that not all problems that can be solved by attainable region analysis are readily formulated as maximum principle problems. The fundamental process of mixing seems to be the main reason for this difference and the consequences of this are highlighted. The class of problems that can be formulated as maximum principle problems are then examined and the relationship between the two methods highlighted. Here, the maximum principle gives rise to a set of results that are very useful for finding an attainable region. In fact, from these results and the experience gained by solving AR problems, postulates on the nature of the boundary of the attainable region are proposed. Previous to this work the construction of the attainable region required a trial and error approach and the region thus generated was tested using the necessary conditions. These postulates should allow a more constructive approach to finding the attainable region boundary.

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“…Previously, Glasser et al (1987) described how to generate the attainable region for two quantities which follow linear mixing laws (i.e., two additive quantities, such as concentrations). and Godorr et al (1994) described how to construct the attainable region in higher dimensions. A short review of the relevant steps in the construction is presented here.…”

Section: Introductionmentioning

confidence: 99%

“…The CSTR and PFR reactors represent two extremes of mixing and are the only reactors needed in the construction of the attainable region for two additive quantities (Glasser et al, 1987). For constructing the attainable region in higher dimensions one must also consider the DSR reactor, which may or may not be needed depending on the detailed kinetics (Godorr et al, 1994).…”

Section: Introductionmentioning

confidence: 99%

Attainable Regions for Polymerization Reaction Systems

and

Malone

1997

Ind. Eng. Chem. Res.

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A method is described for determining the limits of number-average molecular weights, polydispersities, monomer conversions, residual initiator concentrations, and reactor network residence times that are possible in isothermal polymerizations. The general approach provides a visual representation of the “attainable region” and reactor networks which achieve the attainable limits. The approach is based on an extension of the work of Glasser et al. (1987) to polymerization systems and is illustrated with a specific example of the free-radical polymerization of poly(methyl methacrylate). The results show how design conditions affect the range of attainable number-average molecular weights; the results indicate that much higher molecular weights than those typically produced in industry are feasible. The results also demonstrate that while many reactor networks can produce a desired molecular weight, only a few carefully selected networks can yield the narrowest molecular weight distributions. These include a CSTR reactor, a CSTR reactor with a bypass feed stream, and a CSTR reactor followed by a PFR reactor.

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The attainable region for systems with mixing and multiple-rate processes: finding optimal reactor structures

Cited by 15 publications

References 20 publications

The application of the attainable region analysis to comminution

The application of the attainable region analysis to comminution

The Attainable Region and Pontryagin's Maximum Principle

Attainable Regions for Polymerization Reaction Systems

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