(220h) Analytical Solutions to Linear Breakage Population Balance Models of Various Process Configurations

For linear breakage, it is well known that one obtains an analytical solution to the resulting sectional population balance for batch milling in terms of the time trajectories of the size-class mass occupancies. The model can be expressed in vector-matrix form as m’ = A ´ m(t) in which m(t) is the time-dependent vector of the size-class mass occupancies, m’ is the vector of their time derivatives, and A is a triangular matrix, the diagonal of which is comprised of the negative of the selection functions for each size class, and the remainder of the non-zero triangle is comprised of products of these selection functions and fragment mass allocations showing how mass broken in one section is distributed to the sections at smaller sizes. The resulting system is a set of linear, coupled ODEs.

The system is decoupled by the classical approach of finding a similar diagonal matrix D. The diagonal elements of D are the eigenvalues of A and similarity is expressed via A = E ´ D ´ E^-1. The columns of the matrix E are the eigenvectors of A while E^-1 denotes the inverse of E. Given an initial condition of mass occupancy in terms of the vector m_o, the analytical solution for the trajectories is simply:

m(t) = E ´ J(t) ´ E^-1´ m_o with J(t) = exp(Dt)

To be clear, J(t) is a diagonal matrix with elements that are the exponentials of the product of the diagonal elements of D and time.

It is not so well known that this decoupling through similarity to a diagonal matrix allows the solution of the sectional breakage population balance for a myriad of other process configurations besides batch. The solution for a continuous plug-flow mill is identical to the batch solution with time replaced by axial position divided by axial velocity so this will not be explored. Instead, we will illustrate employment of this approach for a perfectly mixed continuous mill (either a single mill or mills in series), and for a continuous plug-flow mill with recycle as a function of recycle ratio. These configurations are often employed commercially. For example, perfectly-mixed mills in series is analogous to the practice of “pendulum” milling.