Problem $3[60$ points$]$

Liquid extraction is an operation used to separate the components of a liquid mixture of two or more species. In the simplest case, a mixture contains two components: a solute $($A$)$ and a liquid solvent $($B$).$ The mixture is contacted in an agitated vessel with a second liquid solvent $($C$)$ that has two key properties: A dissolves in it$,$ and B is immiscible or nearly immiscible with it$.$ Some of the A transfers from $B$ to $C,$ and then the B$-$rich phase $($called the raffinate$)$ and the C$-$rich phase $($called the extract$)$ separate from each other in a settling tank. The extraction can be improved by adding more mixing with fresh $\frac{C}{?}$ settling steps.

Shown below is a flowchart of a process in which acetic acid $($A$)$ is extracted from a mixture of acetic acid and water $($B$)$ using $1-$hexanol $($C$).$

$($a$)$ What is the maximum number of independent material balances that can be written for the overall process? Write down one possible set of balances.

$($b$)$ Prepare a stream table with the information provided in the flowchart.

$($c$)$ Calculate the mass flow rates of $C$ in stream $1,$ the mass flow rate of stream $4,$ and the mass flow rate of stream $5.$

$($d$)$ Calculate the difference between the amount of acetic acid in the feed mixture and that in stream $5.$ Show that it equals the amount of acetic acid leaving in stream $4.$

$($e$)$ Calculate the $\%$ recovery of acetic acid of this process.

$($f$)$ Acetic acid is relatively easily to separate from $1-$hexanol. Draw a flowchart that might be used to recover nearly pure acetic acid from the initial acetic acid$-$water mixture.S