Discuss the fundamental principles governing the design and operation of distillation columns for the separation of binary and multicomponent mixtures in chemical engineering processes. Explore the theoretical basis of distillation, including vapor$-$liquid equilibrium, relative volatility, and mass transfer phenomena. Analyze the factors influencing column efficiency, such as tray or packing design, reflux ratio, and operating conditions. Evaluate the application of distillation in various industries, highlighting its significance in petrochemical refining, pharmaceutical production, and beverage distillation processes.