A heat exchange system with at least two heat exchange chambers is provided. Each of the heat exchange chambers includes heat exchange chamber boundaries which surround at least one heat exchange chamber interior of the heat exchange chamber. The heat exchange chamber boundaries include at least one first opening for guiding in of an inflow of at least one heat transfer fluid into the heat exchange chamber interior and at least one second opening for guiding out of an outflow of the heat transfer fluid out of the heat exchange chamber interior. At least one heat storage material is arranged in the heat exchange chamber interior such that a heat exchange flow of the heat transfer fluid through the heat exchange chamber interior causes a heat exchange between the heat storage material and the heat transfer fluid.