An apparatus for the extraction of heat energy from hot off-gases (2) from productive or service processes and for the transfer of the energy to an external user device (3) is positioned in contact with the off-gases (2), at least partly inside a containing chamber (22) or one or more pipes of a plant performing the productive or service processes. The apparatus comprises at least two sections (4), each of which is defined by at least a heat exchange module (5) having a container/exchanger (6), located at least partly in contact with the off-gases (2), in order to exchange heat energy with them. A first phase change material (7) having high heat diffusivity is located inside the container/exchanger (6). An extraction exchanger (8), disposed in contact with the first phase change material (7), has delivery and return pipes (9) in which a heat-carrying fluid (10) circulates, to extract and transfer the heat energy accumulated by the first phase change material (7). Furthermore, movement means (14, 15) move the heat-carrying fluid (10) and convey and manage the flow rate of the latter to selectively determine, for each section (4), a variation in the values of the heat flow extracted from the off-gases (2) and exchanged with said first phase change material (7).
