Optimal sensor network design for multi-scale, time-varying differential algebraic equation systems: Application to an entrained-flow gasifier refractory brick

•A sensor placement algorithm for multi-scale systems is proposed.•The algorithm can be applied to time-varying differential algebraic systems.•A multi-scale filtering framework with adaptive time steps is used.•The algorithm is used for estimating temperature and slag penetration in gasifier.•The algorithm can facilitate health monitoring of gasifier wall.