Multi-chamber Compressor (6, 206, 506) of Mechanical Vapor re-Compression (MVC) and water treatment methods, the compressor bearing independent compression chambers of positive displacement, for heat-pumps, of two main variants: a) reciprocating-rotary motion (6, 206) wherein the compression chambers (7V) are radially arranged cylindrical sectors based on concentric circular sectors of the same angle, with, pistons of radially arranged vanes (20, 220) of respective surface and with the plane of the vanes passing through the axis of the common rotor (14) and the shaft (16) and b) reciprocating-linear motion (506) wherein the compression chambers (52v) are in series arranged cylinders with pistons/vanes (50v) of corresponding circular surfaces and with the plane of the vanes perpendicular to the common shaft (51). In both cases, the shaft (16, 51) and the motor are common to all the vanes (20v, 50v), which follow identical strokes. The surfaces of the vanes (20v, 50v), as well as of the compression chambers (7V, 52v), differ from each other, since each compression chamber (7V, 52v) has its own and independent pair of evaporation (ev, dv, Lv, by) and heat-exchanger chambers/areas (Cv/eCv, 32v/33v, 132v, 54v/53v), said compression chamber exclusively sucks from, compresses and discharges to, and the fluids/vapors being dispensed, are under different thermodynamic state conditions. The stages are independent from each other, the medium-vapor providing the energy of evaporation is produced in the stage itself, and flow rate and compression ratio CR are independently controlled and adjusted in each stage.