Reusing the Dye Bath: Replenishment Fractions, Shade Drift and the Cycle Count Before the Colour Leaves Tolerance

A dye bath at the end of its first batch is not exhausted; on the figures dyers themselves quote, a substantial share of the dye is still in the water being poured away. Reusing the bath is standard advice, but the advice usually stops at 'top it up and go again', without saying how much dye to add, how the shade will drift as the bath ages, or how many batches can honestly be sold as the same colour. This paper computes the life of a reused bath under several replenishment fractions, tracks shade depth cycle by cycle against an eight percent tolerance, and totals the dye and water saved over the bath's useful life against fresh baths. In the computed results, the best fraction is 0.7; the cycles is 14; the dye used g is 303; the fresh dye g is 420; the dye saving pct is 27.9. Replenish at the computed fraction, count the cycles, and retire the bath when the count is spent. The study is simulated, not measured; all inputs are stated assumptions, printed in the accompanying computation, which readers may replace with their own values and re-run.