**Distributed Component Comminution Model**

The general form of the comprehensive distributed component comminution model has been described as follows

d/dt(m(t)) = – [I – B]S m(t) = A m(t)…………………………………………..(1)

Where A is the fundamental matrix/state companion matrix or the so-called mill matrix in comminution literature. The reason being that A as a general real matrix may have complex eigen-values and complex eigenvectors but more importantly, there is no guarantee that a set of eigenvectors of A will turn out to be independent.

Unfortunately e1+u = el . eu is true only for scalars but not true for matrices i.e. eL + U ≠ eL . eU unless they commute. In other words eAt = e(L + U)t = eLt. eUt ⇔ LU = UL. Since this is not the case, this decomposition scheme must be replaced by a mathematically acceptable method known as the “splitting method” (Trotter, 1959). The splitting method uses the “Trotter product formula” to approximate eAt as follows:

Where:

L is strictly lower triangular matrix

U is strictly upper triangular matrix and m is a large number which is known as Trotter parameter

For a general splitting A = B + C the approximation is:

**Results and Discussion**

In order to demonstrate the validity of the computational procedure, an arbitrary real matrix having essentially the same characteristics as that of mill matrix A was chosen.

**Comparison of Old and New Methodology in Terms of Model Fitting**

The novelty of the new method was demonstrated by considering a simple case i.e. binary model having two composite classes. The two methods (old and new) were tried independently on the same data.

**Parameter Estimation Utilizing Model Solution Computed by The New Method**

A binary model with 2 composite classes was considered and estimation was carried out for S parameters. A set of true parameters (ideal) was used to simulate the multi-component system for t = 1, 2, and 4 minutes.