$9.$ The Fenske, Underwood, and Gilliland equations are critical in the design and analysis of distillation columns. Describe the purpose of each equation and how they contribute to determining the minimum number of stages, minimum reflux ratio, and the actual number of stages needed for separation.

$10.$ Reflux ratio plays a pivotal role in the efficiency and cost of distillation operations. Discuss the concepts of total reflux, minimum reflux, and optimum reflux ratios. How do these parameters influence the design and economic aspects of a distillation column, especially in the context of multiple feeds and side streams?