Wash alumina solids to remove \(\mathrm{NaOH}\) from the entrained liquid. Underflow from the settler tank is \(20.0 \mathrm{vol} \%\) solid and \(80.0 \mathrm{vol} \%\) liquid. Two feeds to the system are also \(20.0 \mathrm{vol} \%\) solids. In one of these feeds, the \(\mathrm{NaOH}\) concentration in the liquid is \(5.0 \mathrm{wt} \%\). This feed's solid flow rate (on a dry basis) is \(1000.0 \mathrm{~kg} / \mathrm{h}\). The second feed has a \(\mathrm{NaOH}\) concentration in the liquid of \(2.0 \mathrm{wt} \%\), and its solids flow rate (on a dry basis) is 2000.0 \(\mathrm{kg} / \mathrm{h}\). We desire the final \(\mathrm{NaOH}\) concentration in underflow liquid to be 0.6 \(\mathrm{wt} \% \mathrm{NaOH}\). A countercurrent operation is used. Inlet washing water is pure and flows at \(4000.0 \mathrm{~kg} / \mathrm{h}\). Find the optimum feed location for the intermediate feed and the number of equilibrium stages required.

D4. ata: \(ho_{\mathrm{w}}=1.0 \mathrm{~kg} /\) liter (constant), \(ho_{\text {alumina }}=2.5 \mathrm{~kg} /\) liter (dry crushed).