Question $2.$

There is a $10m^3$ pure water with a density of $1000k\frac{g}{m^3}$ in a perfectly mixed tank. At $t=0,$ a $10wt\%$ salt solution is started to be fed into the tank with a flowrate of $1\frac{m^3}{h}(F_i).$ The outlet stream has also a flowrate of $1\frac{m^3}{h}(F).$

a$)$ Derive a mathematical model defining the variation of the salt concentration with time.

b$)$ Determine the time necessary for the new steady$-$state conditions $(10\%$ salt concentration$)$ in the tank.

After the tank content settles down into the new steady$-$state you obtained in part $($b$),$ the inlet flowrate and salt concentration are increased to $2\frac{m^3}{h}$ and $15wt\%,$ respectively because of a pump malfunction. However, the outlet flowrate is kept constant at its nominal value $(1\frac{m^3}{h}).$ For this case,

c$)$ What will be the salt concentration in the tank after $4$ hours.

d$)$ If the volume of the tank is $20m^3,$ what do you expect to see after $10$ hours.