More dye does not mean proportionally deeper colour

A common household assumption is that doubling the dye in a bath doubles the depth of colour on the fibre. For madder on alum-mordanted wool, a saturation model shows that this assumption fails, and by how much. Dye uptake is modelled with a Langmuir-type isotherm, saturation capacity 90 g of dyestuff equivalent per kilogram of fibre and affinity constant 0.45 L/g, combined with a bath mass balance at a liquor ratio of 20 L per kilogram. Uptake rises from 28.94 g/kg at a dose of 50 g/kg to 48.37 g/kg at 100 g/kg, 67.41 g/kg at 200 g/kg, and only 79.05 g/kg at 400 g/kg. Doubling the dose from 200 g/kg to 400 g/kg buys a depth gain of 17.3%; doubling from 400 g/kg to 800 g/kg buys 7.2%, while bath exhaustion falls to 10.6%. A linear prediction calibrated at 50 g/kg overstates uptake at 800 g/kg by a factor of 5.46. Liquor ratio matters at fixed dose: at 100 g/kg, uptake is 58.58 g/kg at a liquor ratio of 10, 48.37 g/kg at 20, and 37.24 g/kg at 40. Mordant arithmetic is set out: alum at 20% on weight of fibre is 200 g per kilogram of wool, and raising alum from 20% to 25% raises modelled uptake at a 100 g/kg dose only from 48.37 g/kg to 49.11 g/kg. Colour yield per kilogram of dyestuff is tabulated so a household can price a shade; at an illustrative root price of 18 per kilogram, dyeing 1 kg of wool costs 1.8 in dyestuff at the 100 g/kg dose and 7.2 at the 400 g/kg dose, for a depth gain of less than two thirds. All results are model calculations. SIMULATED, NOT MEASURED.