$9.$ To determine the optimal feed stage in a distillation column, engineers use the $\_\_\_\_$ method, which takes into account the feed $\_\_,$ operating $\_\_,$ and relative $\_\_\_\_$ of the components.

$10.\_\_\_\_$ distillation is advantageous for separating compounds with $\_\_$ boiling points or those that degrade at high temperatures, by using $\_\_$ to lower the overall system $\_\_\_\_.$

$11.$ The $\_\_\_\_-$Savarit method differs from the $\_\_$ method by incorporating both mass and $\_\_$ balances, making it suitable for systems where $\_\_\_\_$ effects cannot be ignored.

$12.$ The $\_\_\_\_$ equation helps in estimating the $\_\_$ number of stages in a distillation column under $\_\_$ reflux conditions, which is crucial for $\_\_\_\_$ cost$-$effective separation.