A chemical company synthesizing catalysts for the petrochemical industry is generating $180$ m$3/$d of a

concentrated solution of K$2$SO$4(80$ g$/$l$)$ as by$-$product. One potential valorisation route could be the production of

KOH and H$2$SO$4$ from K$2$SO$4$ by integration of a$)$ a concentration stage with ED followed by; b$)$ an EDBP stage

for the production of both chemicals.

P$1$a$)$ For the concentration stage with ED $(180$ m$3/$d $80$ g$/$l K$2$SO$4)$ and taking into account that:

i$)$ the water recovery objective in the ED stage is $70\%,$

ii$)$ the salt $($K$2$SO$4)$ recovery objective is $91\%$;

iii$)$ the initial proposal from the stack provider is to use a $3-$stage design, as the maximum desalting ratio that is

recommended for each stage is $55\%,$ where a concentration factor $($CF$)$ of $3\mathrm{.}1$ will be achieved.

Check if the ED treatment scheme proposed by the manufacturer is suitable to achieve a salt recovery ratio of

$91\%.$ Please remember that the flow$-$rates of the diluate and concentrate streams along the ED stacks are equal

and the same along the stages. $($follow the ED scheme from class notes for inspiration$)$

Determine:

a$1)$ the flow$-$rates of the product diluate, and the product concentrate stream, as well as the recirculation product

stream flow$-$rate and the total feed flow$-$rate of the $80$ g$/$L K$2$SO$4$ that will be needed to feed the ED stage number

$1($diluate and concentrate circuits$).(4$p$)$

a$2)$ the concentration values of K$2$SO$4$ along the different stack stages $(1$p$)$

a$3)$ membrane area requirements $($m$2)$ at each stage $(3$ p$)$ and the electrical power $($kW$)$ in each stage $(2$p$).$

Notes: The current efficiency in the selected cells is estimated at $85\%,$ the recommended current density is $80$

A$/$m$2$ and the potential drop between membranes is $1\mathrm{.}2$ V$.$ The membranes used have an effective area of $1$ m$2$

$.$

Assume that the different circuit flow$-$rates along the different stages of ED scheme remain constant.

Atomic weight $($g$/$mol$)$: K $(39),$ S $(32),$ O $(16)$

P$1.$b$)$ For the production of KOH and H$2$SO$4$ from K$2$SO$4$ concentrate $($ED stage$)$ by EDBP stage, consider: i$)$ the

inlet flow$-$rate will be the concentrate product flow$-$rate generated in the ED stage; ii$)$ the total conversion ratio

of K$2$SO$4$ to KOH and H$2$SO$4$ will be $80\%.$ The provider has recommended to use stacks with $60$ cells and a cell

length $($Y$)$ of $0\mathrm{.}5$ and cell width $($X$)$ of $0\mathrm{.}5$ m$.$

b$1)$ Propose the type of stack cell that will be needed, and provide a scheme of the system indicating the flow$-$

rates of the different circuits and the total concentrations $($eq$/$l$)$ of the main components in each stream $(3$p$)$

b$2)$ Determine the required membrane area Am $($m$2),$ number of cells and the number of stacks $(4$ p$)$

b$3)$ Calculate the process path length $($Lpp$)(1$ p$)$

b$4)$ Indicate the total power needed $($kW$)$ and the specific energy consumption per m$3$ EDBP product $($kWh$/$m$3)$

$(2$ p$)$

Notes: The current efficiency in the selected cells is estimated at $80\%,$ the recommended current density is $1100$

A$/$m$2$ and the potential drop per stack and electrodes is $110$ V$.$ Cell dimensions and flow velocities of the

different solutions in the stack are assumed to be identical.