In this work, we present algorithmically tractable safe approxima-tions of distributionally robust optimization (DRO) problems. The consideredambiguity sets can exploit information on moments as well as confidence sets.Typically, reformulation approaches using duality theory need to make strongassumptions on the structure of the underlying constraints, such as convexityin the decisions or concavity in the uncertainty. In contrast, here we present aduality-based reformulation approach for DRO problems, where the objective ofthe adverserial is allowed to depend on univariate indicator functions. This ren-ders the problem nonlinear and nonconvex. In order to be able to reformulatethe semiinfinite constraints nevertheless, an exact reformulation is presentedthat is approximated by a discretized counterpart. The approximation is re-alized as a mixed-integer linear problem that yields sufficient conditions fordistributional robustness of the original problem. Furthermore, it is proven thatwith increasingly fine discretizations, the discretized reformulation convergesto the original distributionally robust problem. The approach is made concretefor a challenging, fundamental task in particle separation that appears inmaterial design. Computational results for realistic settings show that the safeapproximation yields robust solutions of high-quality and can be computedwithin short time.