Sedimentation/Parameter Identification

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. Introduction Modeling of flocculated suspensions The batch settling process of flocculated suspensions is modeled as an initial 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 scheme 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 boundaries ("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 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) consisting 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. Weblinks http://de.wikipedia.org/wiki/Wikipedia:TeX Wikipedia:TeX http://en.wikipedia.org/wiki/Matlab