The application of the attainable region analysis to comminution

The aim of any comminution circuit is to produce material of a desired particle size distribution (PSD) at a minimum operational cost. Currently, the comminution process is energy intensive and operates at very low efficiency when the input energy is compared to the breakage achieved. The attainable region (AR) technique has been successfully used to solve optimization problems simultaneously with the process synthesis formulation of reactor systems. The AR looks at the fundamental processes of a given system and determines all the possible outputs to which the objective function can be applied and an optimal process solution selected.Particle breakage, separation (classification) and mixing are identified as the three fundamental processes of interest taking place during comminution. Breakage and mixing processes are used in this paper to illustrate the applicability of the AR theory in comminution. We develop a fundamentally based model which is equipment independent to describe breakage. Specific energy is the independent variable and the production of particles with a certain PSD is the objective function. We use geometric construction to represent this PSD as a point in an n-dimensional space in relation to an input specific energy. Output PSDs are dependent on the input PSDs, allowing connectivity of the batch grinding stages to form a pseudo-continuous process.Specific energy is used as the control variable to obtain sharper product PSDs. It is shown that the same net energy consumed in the system can produce different product PSDs. Therefore, this implies that the design of comminution circuits should achieve better control of the specific energy. Once the candidate AR is constructed, operational process targets can be defined more accurately. This establishment of targets permits a measure of the actual process efficiency against a theoretical target. The advantage of the AR method lies in its ability to develop not only the performance of the optimal circuit but also the operational conditions to be used in the optimal process circuit. This also answers the process synthesis question of the type of equipment to be used which is a function of the specific energy.     

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.     

An experimental simulation of distillation column concentration profiles using a batch apparatus

There has been a growing interest in the use of residue curves for the preliminary design and sequencing of distillation columns. Residue curves are used not only to predict composition changes in separation processes, but also to determine the feasibility of proposed separations, and flowsheet development (Chem. Eng. Sci. 33 (1977) 281).An experimental technique has been developed for the measurement of these residue curves. (Distillation & absorption ’97, Inst. Chem. Eng. 1 (1997) 187). It can be shown that the time-dependent composition profiles obtained in a modified form of this apparatus are mathematically equivalent to the position-dependent profiles in a continuous distillation column. Hence, it is possible to experimentally simulate a distillation column profile in a small batch apparatus using only small quantities of material.The modified apparatus consists of a still immersed in a heated oil bath so that a liquid feed is continuously supplied to the still. Samples of liquid are then analysed over time using a gas chromatograph. The results from an experimental system have been compared to available information and simulations to determine the accuracy of the apparatus.This technique has several advantages over working with distillation columns, firstly in the sample size required, and secondly in the ease of operation. The method allows quick and low-cost measurements of the concentration variables that model a distillation column. The information obtained this way could prove useful for the selection of feasible systems and for finding minimum reflux requirements. It could also be very valuable for screening of complex systems where only small amounts of material are available and simulations may be very difficult.