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.