exchange in human lungs and waste removal in kidneys (dialysis) are complex mass-transfer processes [14, 16].

: Techniques like freeze-drying, microencapsulation, and controlled-atmosphere packaging rely heavily on mass-transfer rates to maintain quality [25].

: Assumes the streams leaving a stage (like a tray in a tower) are in thermodynamic equilibrium. The number of "ideal stages" is calculated and then adjusted for efficiency [1, 2, 32].

Mass-transfer operations are physical processes in chemical engineering that involve the movement of chemical species from one phase to another or within a single phase, primarily driven by a or chemical potential [1.1, 1.4, 4]. These operations are essential for the purification of raw materials and the final separation of products in industries like petroleum, pharmaceuticals, and environmental engineering [3, 8, 32]. Fundamental Principles