Denim
Deposit is linear in dips. Adding concentration to a short range buys ring dyeing, not depth.
— % owf
After the full dip sequence
Linear build-up is a good description across the ordinary range of six to ten boxes and begins to overstate at the top of it, because the yarn surface is progressively occupied and later dips deposit onto indigo rather than onto cotton. Fixation per dip is the term that absorbs this and it should be fitted from measured build-up on the range rather than taken as a constant, since it also moves with rope tension, squeeze pressure, oxidation time between boxes and the state of reduction of the bath. Reduction state is the largest uncontrolled variable in indigo dyeing and does not appear here at all: an over-reduced bath gives poor fixation and a bronzy shade, an under-reduced one deposits pigment rather than dyeing, and both present as a build-up figure that does not match this calculation. The concentration-for-target output is offered as arithmetic rather than as advice - reaching a depth by concentration on a fixed number of boxes is exactly the move that produces ring dyeing, and the honest answer to a shortfall on an eight-box range is usually a slower rope or a different construction rather than a stronger bath. Nothing here models the sulphur bottom or top, the skying time, or the washing boxes that follow, all of which move the final shade.
Indigo Build-Up Across a Dip-Nip Sequence — free, with the formula and a worked example, at Textile School.