As shown in this flow diagram a countercurrent thickener consisting of three stages is used in a Kraft paper mill to wash a "white mud" consisting of $35\%$ solids $(CaC{O}_3)$ and $17\%$NaOH in water. Two wash streams are used, the first $($stream $5)$ contains $4\%$ suspended solids, $6\%$NaOH, and the rest $H2O$; the second $($stream $8)$ contains no solids, $2\%$NaOH, and the rest $H_{2}O.$ The clear liquids from stage I and stage II contain $0\mathrm{.}5\%$ suspended solids; the clear liquid from stage III contains $0\mathrm{.}4\%$ solids. The flow of wash liquid to stage II $($stream $5)$ is $1\mathrm{.}5$ times the flow of the feed sludge $($stream $1),$ and the washed sludge from stage II contains one third solids. The washed sludge from the third stage $($stream $9)$ contains $32\mathrm{.}5\%$ solids and $2\mathrm{.}5\%$ NaOH, and the flow rates of streams $7$ and $9$ are equal. All compositions are given on a mass basis. Assume that in each stage the clear solution and the solution contained in the washed sludge have the same concentration.

$($a$)$ Construct a degree$-$of$-$freedom table and determine whether the problem is correctly specified.

$($b$)$ Calculate the concentration og NaOH in stream $4$