Sedimentation/Parameter Identification

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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[edit]

The batch settling process of flocculated suspensions is modeled as an intitial value problem

where 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.

Numerical scheme[edit]

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


are inserted. 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[edit]

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.