Flotation is a separation process where particles or droplets are removed from a suspension with the aid of floating gas bubbles. Applications include dissolved air flotation (DAF) in industrial wastewater treatment and column froth flotation (CFF) in wastewater treatment and mineral processing. One-dimensional models of flotation have been limited to steady-state situations for half a century by means of the drift-flux theory. A newly developed dynamic one-dimensional model formulated in terms of partial differential equations can be used to predict the process of simultaneous flotation of bubbles and sedimentation of particles that are not attached to bubbles. The governing model is a pair of first-order conservation laws for the aggregate and solids volume fractions as functions of height and time. An analysis of nonlinear ingredients of the governing equations helps to identify desired steady-state operating conditions. These can be chosen by means of operating charts, which are diagrams that visualize regions of admissible values of the volumetric flows of the feed input and underflow outlet. This is detailed for the DAF thickening process. Dynamic simulations are obtained with a recently developed numerical method. Responses to control actions are demonstrated with scenarios in CFF and DAF.