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.

Contents

IntroductionEdit

Modeling of flocculated supensionsEdit

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

 

into

 

In the closure, the constants   are partly known.

Numerical schemeEdit

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

 

and

 

are inserted. Abbreviating the time evolution step as

 

the linearized marching formula for the interior scheme becomes

   

where

 

Rearrangement leads to a block-triangular linear system

     

which is of the form

 

or, in more compact notation,

 


Parameter identification as OptimizationEdit

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

 

where

 

is the Hessian of q. The Newton method is motivated by the Taylor expansion

 

where   is the optimal parameter choice.

WeblinksEdit