How can you separate alumina from silica in bauxite ore associated with silica? Give equations,if any.

(N/A) To separate alumina from silica in bauxite ore,the powdered ore is digested with a concentrated $NaOH$ solution at $473-523 \ K$ and $35-36 \ bar$ pressure.
This process leaches out alumina $(Al_2O_3)$ as sodium aluminate and silica $(SiO_2)$ as sodium silicate,leaving behind other impurities.
$Al_2O_{3(s)} + 2NaOH_{(aq)} + 3H_2O_{(l)} \to 2Na[Al(OH)_4]_{(aq)}$
$SiO_2 + 2NaOH_{(aq)} \to Na_2SiO_{3(aq)} + H_2O_{(l)}$
Next,$CO_2$ gas is passed through the resulting solution to neutralize the aluminate,which causes the precipitation of hydrated alumina. The solution is seeded with freshly prepared samples of hydrated alumina to induce precipitation.
$2Na[Al(OH)_4]_{(aq)} + 2CO_{2(g)} \to Al_2O_3 \cdot xH_2O_{(s)} + 2NaHCO_{3(aq)}$
During this step,sodium silicate remains in the solution. The hydrated alumina is filtered,washed,dried,and heated to obtain pure alumina.
$Al_2O_3 \cdot xH_2O_{(s)} \xrightarrow{1470 \ K} Al_2O_{3(s)} + xH_2O_{(g)}$