Air-conditioning system for an aircraft cabin, comprising a turbomachine provided with a radial turbine distributor

The invention relates to an air conditioning system for an aircraft comprising a turbine engine comprising at least one radial turbine and a compressor, said radial turbine extending along a central axis (10) and comprising a wheel (20) equipped with vanes (21) mounted so as to be able to rotate about said central axis (10), a nozzle (30) arranged at the periphery of said wheel (20) comprising a plurality of variable-pitch blades (31) arranged around said central axis (10), and a volute (12) comprising an air inlet (12a) and an air outlet (12b) opening onto said plurality of variable-pitch blades (31), characterized in that said variable-pitch blades (31) are arranged around said central axis (10) between two nozzle casing walls (11a, 11b) delimiting an air passage section of said nozzle between them, each variable-pitch blade (31) further comprising means for axial displacement of the blade between the two casing walls (11a, 11b).