A contimuous extraction of benzoic acid from toluene is carried out in a two stage cross$-$current mode

of operation under steady$-$state conditions using water as the solvent. Each stage consists of $(1)$ a

mixer where vigorous stirring of the contents takes place, and $(2)$ a settler where the mixture is

pumped from the mixer and allowed to settle into two layers. The upper toluene layer and the lower

water layer are removed separately.

$($a$)$ Develop a mathematical model to find the total fraction of benzoic acid extracted by clearly

mentioning the assumptions involved in the analysis.

$($b$)$ Identify the dimensionless groups in the final expression and explain their physicat significance.