No abstract

The increasing complexity in particle science and technology requires the ability to deal with multidimensional property distributions. We present the theoretical background for multidimensional fractionations by transferring the concepts known from one dimensional to higher dimensional separations. Particles in fluids are separated by acting forces or velocities, which are commonly induces by external fields, e.g., gravitational, centrifugal or electro-magnetic fields. In addition, short-range force fields induced by particle interactions can be employed for fractionation. In this special case, nanoparticle chromatography is a recent example. The framework for handling and characterizing multidimensional separation processes acting on multidimensional particle size distributions is presented. Illustrative examples for technical realizations are given for shape-selective separation in a hydrocyclone and for density-selective separation in a disc separator.

In this work, we present algorithmically tractable reformulations of distributionally robust optimization (DRO) problems. The considered ambiguity sets can exploit information on moments as well as confidence sets. Typically, reformulation approaches using duality theory need to make strong assumptions on the structure of the underlying constraints, such as convexity in the decisions or concavity in the uncertainty. In contrast, here we present a very general duality-based reformulation approach for distributionally robust problems that are allowed to depend on elementary functions, which renders the problem nonlinear and nonconvex. In order to be able to reformulate the semiinfinite constraints nevertheless, a safe approximation is presented that is realized by a discretized counterpart. Its reformulation leads to a mixed-integer positive semidefinite problem that yields sufficient conditions for distributional robustness of the original problem. For specific models with uncertainties that are only one-dimensional, it is proven that with increasingly fine discretizations, the discretized reformulation converges to the robust counterpart of the original distributionally robust problem. The approach is made concrete for a one-dimensional robust chance-constrained problem, where the discretized counterpart results in a mixed-integer linear problem. We study a challenging, fundamental task in particle separation that appears in material design. Computational results for a realistic setting show that the safe approximation yields robust solutions of high-quality within short time.

No abstract