"How does the design of a distillation column affect the separation efficiency in chemical processes? Discuss the role of column diameter, packing material, and reflux ratio in achieving desired separation outcomes. Explain the concept of vapor$-$liquid equilibrium and its significance in distillation processes. How do changes in temperature and pressure influence the equilibrium composition of a binary mixture? Illustrate the operating principles of a continuous distillation process and compare it with batch distillation in terms of efficiency and throughput."