Diane Hildebrandt

Optimal reactor design from a geometric viewpoint—I. Universal properties of the attainable region

Abstract A geometric framework for studying optimal reactor design is developed. For a given feed and a prescribed kinetics (perhaps involving many reactions), focus is on the full set of product composition vectors that can be produced in principle by means of all possible steady-state designs that employ only reaction and mixing (including designs that transcend current imagination). This set, called the attainable region by F. J. M. Horn, carries the full range of outcomes available to the designer. Of special importance are its extreme points , for these determine the region completely, and reactor optima are often realized there. Although the attainable region is not generally discernible in advance, one can nevertheless prove that it has certain universal properties, which, in turn, provide information about qualitative designs that provide access to the extreme points. Despite the vast spectrum of designs the attainable region is intended to embrace, two theorems suggest that its extreme points will always be accessible by means of classical elementary reactor types taken in simple combination. These results suggest that any reactor product that is realizable can, in fact, be realized by parallel operation of those canonical reactor building blocks that give rise to the extreme points. This paper lays the groundwork for additional theory, in which special properties of reactors that access the extreme points will be studied in some detail.

The attainable region and optimal reactor structures

Abstract Given a system of reactions and associated kinetics and assuming only the processes of reaction and mixing, one may derive, for variables which obey linear mixing laws, geometrically based constructive necessary conditions. Using these necessary conditions, global optimization problems with algebraic objective functions of the basis variables may be solved generating both optimal reactor configurations as well as operating parameters. The technique is used to solve the well known three dimensional problems of van de Vusse, Denbigh and Westerterp. It is further shown in general for all dimensions that it would need to be a very unusual situation where the optimal reactor network was not a series-parallel arrangement of CSTRs, plug flow reactors and bypasses.

Linear programming formulations for attainable region analysis

We propose linear programming (LP) models for attainable region (AR) analysis by considering a rate vector field in concentration space with an arbitrarily large number of points. One model provides a method to construct candidate ARs using a fully connected network of continuously stirred tank reactors (CSTRs) of arbitrary volume. More importantly, these methods are extended to derive linear programming conditions that are stronger necessary conditions than have proposed previously by Glasser and Hildebrandt. We state the LP condition as: No combination of nonzero volume CSTRs, operating at discretized points in the complement of the candidate AR, can extend the region. We demonstrate these proposed linear programming techniques on several two-dimensional reaction mechanisms and then apply the LP methods to verify extensions for a previously published three-dimensional candidate AR.

Reactor and process synthesis

Abstract It is shown how the Attainable Region method may be used to synthesize both chemical reactors and more general systems. This is done by using geometric ideas to first define the allowable processes as vectors in a space of the basic variables of the problem. A set of results is used to construct a region which satisfies the necessary conditions for the Attainable Region, that is the set of all possible outcomes using the allowable processes in a steady state system. The properties of the boundary of this region are of particular importance. In particular, the boundary is the union of surfaces on which single processes occur. Smooth intersections between single process surfaces may represent curves along which two or more processes occur simultaneously. Paths are traced along which one can move from a feed point(s) to other points on the boundary of the region; the paths can then be interpreted in terms of a structure, which we call the optimal structures which will allow us to achieve any point on the boundary of the region. As these optimal structures arise from the solution of the problem they have actually been synthesized. It is further shown how these ideas might be used even for situations where the dimensionality of the problem makes it unlikely that one could find the full Attainable Region.

Producing Transportation Fuels with Less Work

New reaction chemistry may reduce the energy input and carbon dioxide emissions from processes that convert coal into liquid fuels.

Optimal reactor structures for exothermic reversible reactions with complex kinetics

Abstract Exothermic reversible reactions are industrially very significant. Previous work has looked at finding the Attainable Region for elementary kinetics and in this paper we extend the work to find optimal reactors for more complex kinetics. Using these results we show how the complexity of the kinetics affects the nature of the optimal reactor. In particular it is found that the optimal reactor structure includes a novel element, namely two parallel structures, the first of which has preheating followed by the following reactors connected in series: a plug flow reactor, CSTR, plug flow reactor and finally differential side stream reactor, while the second structure is a plug flow reactor. The side streams in the DSR are taken from various points along the plug flow reactor making up the second parallel structure. Furthermore, it is also shown that in order to satisfy tangency conditions on the boundary of the Attainable Region, the rate of reaction of the material in the side stream to the DSR and that in the DSR at the point where the side stream is added must be equal. The equations describing the DSR operation as well as the conditions describing the operation of the other reactors in the optimal reactor structure are given.

An annual and seasonal characterisation of winery effluent in South Africa.

Winery effluent is known to have a high chemical oxygen demand (COD) and a low pH. In this study, we extensively analysed effluent from two cellars and studied the temporal changes over the duration of a harvest and the duration of a year. We found that ethanol contributes approximately 85% to 90% of the COD of raw winery effluent, with acetic acid being the next significant contributor. The pH showed some dependence on the concentration of acetic acid. The concentration of sodium in the effluent is strongly dependent on the cleaning regime in place at the cellar, and the concentration of potassium has been shown to be linked to the spillage of juice, wine or lees. The data and correlations presented here could allow for an artificial effluent to be prepared easily for research purposes.

Process synthesis for reaction systems with cooling via finding the attainable region

The attainable region (AR) technique is used to find the optimum process design for an exothermic reversible reaction system where allowance is made for external cooling using two different constant temperature utilities. Heat exchange with the two utilities has different associated costs. The general optimum process structure (equipment) for the specified system was obtained via finding the attainable region. The temperature progression required for the exothermic reversible reaction was obtained by a sequence of structures. The hot reaction mixture was adiabatically reacted in a CSTR followed by a plug flow reactor then cooled down in a differential cold shot converter. This was followed with external cooling of the reactor, firstly with the hotter utility and then with the colder and more expensive utility. The optimum control of the cold shot cooling as well as the external cooling was determined using the AR technique.

Novel separation system design using “moving triangles”

Abstract Shortcut design techniques have been employed in the initial design of traditional distillation systems. Current techniques are not useful in the design of novel or complex configurations however. We will show that by using column profile mapping “moving triangles” to model the behaviour of column sections (CS), any distillation configuration, no matter how complex, can be modelled and its behaviour more thoroughly understood. As an example, a thermally coupled column will be modelled using column profile maps. It is suggested that by gaining an understanding of the behaviour of the configuration quickly and easily, using column profile maps, time and money can be saved by avoiding poor initial decisions and designs.

DRIFT spectroscopy and optical reflectance of heat-treated coal from a quenched gasifier

Coal samples collected from a quenched coal gasifier were characterized for (thermal) heterogeneity by optical reflectance measurements. DRIFT spectroscopy served to estimate the final temperature to which the samples were exposed. Particle size information on the coal samples was superimposed on the results for the thermal history of the char samples collected from the gasifier bed. A good match was found between these different sets of data. The implications for gasifier operation are discussed.