A raw material consisting of freshly produced orange juice containing $12wt\%$ solids, undergoes $3-$step

unit operations to obtain a concentrated juice at $98\mathrm{.}5\%$ solids yield $($in the final product$).$ Instead of using

a single step evaporation to concentrate the juice, a bypass is used $($stream B$)$ to reduce the volatile

constituents' loss which results in juice product with a flat taste. So$,$ the juice is first strained to give a

strained stream $S($flow rate $=85$ and a solid fraction $x)$ and a bypass $($B$)$ with a solid fraction $($Y$).$ The

stream $S$ entering the evaporator is concentrated to $65wt\%$ solids $($stream E$),$ and the evaporator product

stream is mixed with the bypassed fresh juice to achieve the desired final concentration.

$(1)$ Calculate the product flow rate $($C$)$ and the evaporation rate $($W$)($use overall system boundary!$)$

$(2)$ Make a MB on the evaporator to obtain stream E and the solid composition $($X$)$ of stream S $!$

$(3)$ Calculate the bypass stream B and its solid composition!