Question

what is the appropriate method for separating coloured dyes from blue ink

Answer 1

chromatography  is the appropriate method for separating coloured dyes from blue ink

Answer 2

The blue ink used in fountain pens is a liquid mixture. It is a mixture of dye with water . This dye the colored component of the ink . We can separate the color component of ink by the process of evaporation.
Precaution that should be taken is that ink should not be heated directly as it is decomposed by high temperature. So we use a water bath for evaporating ink .

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