Explain the principles and applications of distillation in chemical engineering, highlighting its significance in separating liquid mixtures. Begin by elucidating the fundamental theory behind distillation, encompassing concepts such as vapor$-$liquid equilibrium, Raoult's Law, and the ideal stages of distillation. Subsequently, delve into the various types of distillation processes, including simple distillation, fractional distillation, and azeotropic distillation, elucidating their respective mechanisms and distinguishing factors. Discuss the key parameters that influence the efficiency of distillation, such as temperature, pressure, reflux ratio, and tray efficiency. Furthermore, explore the industrial applications of distillation across diverse sectors$,$ elucidating its role in refining petroleum, producing alcoholic beverages, and purifying chemicals. Additionally, discuss the advancements in distillation technology, including innovations in column design, process optimization, and energy efficiency. Finally, evaluate the sustainability aspects of distillation processes, considering factors like energy consumption, waste generation, and environmental impact, and propose potential strategies for enhancing the eco$-$efficiency of distillation operations.