In the context of this paper a simulation model for a size reduction device is an equation or equation set which allows the calculation of the product size distribution from the device for a specified size distribution of feed. The model might be expressed as a simple algebraic expression, as a matrix equation, or as a finite difference formulation. The model proposed here for once-through crushers was originally derived and developed by Austin, et al for the treatment of smooth double roll crushers. The model is based on the assumption that if the particle size range is split into geometric size intervals (i.e. the √2 sieve series) the breakage of each size interval occurs independently of other sizes.
That is, if fi and pi give the weight fraction of material in size interval i in the crusher feed and product, respectively, the relationship between the pi and fi, can be expressed as the transfer parameter equation (3)
where size 1 is the largest size, size 2 the second size, etc. In this equation, the d give the weight fraction of size j feed which is “transferred” to size i upon passage through the crusher. The formulation of these transfer parameters is based on an analysis of the breakage process which is schematically illustrated in Figure 1 and can be described as follows.
When material of size i is passed through the crusher, a fraction d is left within the starting size i. Defining breakage as leading to material smaller than the starting size range, dii can be considered as material which has bypassed through the crusher. Thus, a fraction dii = (1 – ai) bypasses through the crusher without breakage and a fraction ai is broken. The fraction ai for a given size i is referred to as the primary bypass parameter for size i. It is also assumed that the first fracture leads to a set of primary daughter fragments. These primary daughter fragments are given as the primary breakage distribution parameter set bij. The bij give the weight fraction of size j material which is broken to size i as a result of primary breakage. Daughter fragment material can in turn bypass through the crusher or be selected for another fracture, with a fraction ai’ broken. The ai’ do not necessarily equal the ai because a fragment of size i produced by fracture in the crushing zone might be expected to be in a better position to rebreak than one just entering the crusher, so that l>ai’>ai. The fraction ai’ for a given size i is erred to as the secondary bypass parameter for size i. The ai and ai’ are assumed to be constant and independent of the size distribution of material being subjected to the bypass mechanisms.
The equations given by Austin et al for this repeated breakage process were
This formulation was put as equation (1) by Rogers (3) with
The cumulative crusher product weight size distribution is given by
where n is the smallest or sink size interval and P(xi) is the weight fraction of crusher product smaller than size xi, where x is the upper size of size interval i in millimetres.
It is interesting to compare this model to the one developed by Whiten and utilized by others for simulating cone crusher performance. Adopting the present nomenclature, the Whiten model is illustrated schematically in Figure 2. As seen, this model employs the primary breakage distribution and material bypass concepts. Furthermore, this model gives a relationship between fi and pi that is simply a special case of the more general once-through model formulation, that is Equations (1) and (3) but with ai = ai’. Thus the only essential difference between these models is in the treatment of the bypass for particles produced by fracture.
Model Parameter Estimation
It is possible, in principle, to estimate the parameters of the model for once-through crushers using the same approach that Whiten, et al and Karra have taken for the cone crusher model. That is, one can assume that model parameters can be described by equations which are a function of particle size and which contain a few descriptive parameters. Then by incorporating these equations into the model, the descriptive parameters can be estimated by fitting the model to experimental data obtained from crusher tests. For the cone crusher model Karra notes that this approach can result in parameter estimates that are reasonably consistent from one set of test data to another but can also lead to questionable parameter values. Another approach, and one that can also be used for the Whiten model, calls for performing a series of crusher tests on the feed material of interest and results in direct measurements of parameters. Hits preferred approach, which has been demonstrated for both laboratory and full-scale once-through crushers. Involves the experimental and calculational procedures described as follows.