"What are the key principles governing distillation processes in chemical engineering? Explain the role of vapor$-$liquid equilibrium and the concept of relative volatility in designing efficient distillation columns. Discuss how operating conditions such as temperature and pressure influence separation efficiency. Additionally, elucidate the significance of tray efficiency and reflux ratio in optimizing distillation performance. Provide examples illustrating the application of these principles in industrial$-$scale distillation units."