Even though it is tried to keep this chapter on Parameter Identification of Flocculated Suspensions as self-comprehensive as possible, preliminar knowledge on numerical Methods and the Modeling of suspensions are useful. In particular, the Newton-Raphson scheme to solve nonlinear systems of equations for the optimization and Finite-Volume-Methods for the solution of partial differential equations are applied.
Modeling of flocculated supensions
The batch settling process of flocculated suspensions is modeled as an intitial value problem
denotes the volume fraction of the dispersed solids phase.
For the closure, the convective flux function is given by the Kynch batch settling function with Richardson-Zaki hindrance function
and the diffusive flux is given by
which results from the insertion of the power law
In the closure, the constants
are partly known.
The numerical scheme for the solution of the direct problem is written in conservative form as a marching formula for the interior points ("interior scheme") as
and at the bondaries ("boundary scheme") as
For a first-order scheme, the numerical flux function becomes
If the flux function has one single maximum, denoted by
the Engquist-Osher numerical flux function can be stated as
For linearization, the Taylor formulae
Abbreviating the time evolution step as
the linearized marching formula for the interior scheme becomes
Rearrangement leads to a block-triangular linear system
which is of the form
or, in more compact notation,
Parameter identification as Optimization
The goal of the parameter identification by optimization is to minimize the cost function over the parameter space
where h(e) denotes the interface that is computed by simulations and H is the measured interface.
Without loss of generality we consider a parameter set e=(e_1, e_2) consiting of two parameters.
The optimization can be iteratively done by employing the Newton method as
is the Hessian of q. The Newton method is motivated by the Taylor expansion
where is the optimal parameter choice.