Titanium dioxide, the most common white pigment in paint, can be produced from the titanium mineral rutile by chlorination to TiCl₄, followed by oxidation to TiO₂. To purify insoluble TiO₂, it is washed free of soluble impurities in a continuous, countercurrent system of thickeners with water. Two hundred thousand kg/h of 99.9 wt% TiO₂ pigment is to be produced by washing, followed by filtering and drying. The feed contains 50 wt% TiO₂, 20 wt% soluble salts, and 30 wt% water. Wash liquid is pure water at a flow rate equal to the feed on a mass-flow basis. 

(a) Determine the number of washing stages required if the underflow from each stage is 0.4 kg solution/kg TiO₂. 

(b) Determine the number of washing stages required if the underflow varies as follows:

Derive (16-20), assuming that (Yi)b, kc, m, and a are constants and that (Xi)o is uniform through the solid.

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Concentration of Solute, kg solute/kg solution 0.0 0.2 0.4 0.6 Retention of Solution, kg solution/kg TiO₂ 0.30 0.34 0.38 0.42