The application of the attainable region analysis in comminution.
Date
2008-06-09T10:03:28Z
Authors
Khumalo, Ngangezwe
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Abstract
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 (Horn, 1964). A basic procedure for
constructing the attainable region for the fundamental processes of reaction
and mixing has been postulated in reaction engineering (Glasser et al., 1987).
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. Herbst and Fuerstenau
(1973) 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 an
The application of the attainable region analysis in comminution Abstract
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 analysis affords the opportunity
to quantify exactly the reduction in energy consumption due to classification
in a comminution circuit, thus giving optimal targets. Classification showed the
potential to extend the candidate attainable region for a fixed specific energy
input. The boundary of the attainable region is interpreted as pieces of
equipment and optimum process conditions. This solves both the original
process synthesis and successive optimisation problems.
Description
Keywords
Attainable region, comminution, particle size distribution, population balance model, specific energy, classification, energy consumption, targets