Water $($solvent$)$ is to be used in a $2-$stage countercurrent cascade to dissolve Li$2$CO$3$ from

insoluble calcium carbonate $($CaCO$3).$ The mixture is in a solid form. The following information is

given:

$$ The flow rate of calcium carbonate, FA$,$ is $400$ kg$/$hour$.$

$$ The mass flow rate ratio between Li$2$CO$3$ and CaCO$3$ is $1$:$4$

$$ The flow rate of fresh water $($solvent$),$ S$,$ is $1,000$ kg$/$hour

$$ The solvent$-$to$-$insoluble solids mass ratio $($R$)$ is $0\mathrm{.}25$

$$ Li$2$CO$3$ is highly soluble in water

a$)$ Calculate Li$2$CO$3$ mass flow rate $($FB$)$ and washing factor $($W$).$

b$)$ Calculate the percentage of Li$2$CO$3$ recovered in the overflow stream.

c$)$ Provided that only $900$ kg$/$hour of fresh water is introduced into the second stage and $100$

kg$/$hour of additional fresh water is introduced into the first stage $($the stage where solids

enter$),$ calculate the recovery rate under such a configuration.