"How does distillation separate components in a mixture based on differences in volatility? Discuss the principles underlying fractional distillation and its applications in chemical engineering processes such as crude oil refining and ethanol purification. Explain the role of vapor$-$liquid equilibrium and the concept of relative volatility in designing efficient distillation columns. Compare the efficiency of batch distillation versus continuous distillation techniques. Provide examples of industries where distillation plays a critical role in product purification and separation processes."