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An approach for the numerical topology optimization of baffled stirred tanks is presented and demonstrated exemplarily on impeller blades. The methodology is based on the well-known Brinkman penalization method, which is used to alter the impeller blade geometry without having to adapt the computational grid. Since the simulation of this time-dependent problem is computationally very expensive, non-uniform rational basis splines are employed to describe the interface between solid and fluid. This method allows topological changes of complex geometries to be realized with a minimum number of design variables, thus speeding up the optimization process. Two test cases are investigated to demonstrate the capabilities of the approach and serve as a proof-of-concept. In the first case, the power requirement of the stirrer is minimized, while the second case aims to maximize the degree of mixing. As the result of the second test case is at first useless from a practical point of view, an additional constraint is introduced to limit the maximum allowable displacement of the impeller blade, which is computed using a finite element analysis.