Dyebath Exhaustion Kinetics & Cycle Time Fit

The formula

This is the expression the tool evaluates. Every term is named underneath, with the unit it must be supplied in.

How the result is derived

Step by step, from the values you type to the figure on screen.

1. The 4 inputs are read from the form on every keystroke: Sampling Time, Exhaustion at that Time, Equilibrium Exhaustion and Dyeing Time at Temperature.
2. Each value is checked against the accepted range in the input table below. A value outside its range stops the calculation rather than producing a misleading figure — the results blank out and a message appears.
3. The validated values are substituted into the expression above, which resolves Exhaustion at End of Cycle together with every supporting figure in one pass — no value is carried over from a previous entry.
4. The supporting outputs — Rate Constant K, Sample as Share of Equilibrium, Half-Dyeing Time, Time to 95% of Equilibrium, Time to 99% of Equilibrium, Dye Left in Bath, Extra Time Needed for 99% and Dye Discharged per Tonne at 1% owf — come from the same pass, so they always describe the same case as the headline figure.
5. Results are rounded for display only. The full-precision value is used throughout the chain, so reading a rounded intermediate figure back into the tool by hand can shift the last digit.

What each input means

Where to read each value on the floor, the unit it must be in, and the range the tool accepts.

What the tool returns

The headline figure and every supporting value it is built from.

Worked example

The tool loads with this case already solved — the Exhaustion at End of Cycle shown above is its answer. Change one value and the difference from this baseline is the sensitivity of the result to that variable.

How to use it

1. Work through the input groups in order — Measured Point and Programmed Cycle. The defaults are a realistic case, so you can change one value at a time and watch what moves.
2. There is no calculate button. Every figure recalculates as you type or drag, which is what makes this usable for a what-if sweep rather than a single answer.
3. Read Exhaustion at End of Cycle in the dark results panel — that is the headline figure, expressed in %.
4. Check the supporting rows underneath (Rate Constant K, Sample as Share of Equilibrium, Half-Dyeing Time, Time to 95% of Equilibrium, Time to 99% of Equilibrium, Dye Left in Bath, Extra Time Needed for 99% and Dye Discharged per Tonne at 1% owf) before acting on the headline — they are where an implausible input usually shows itself first.
5. Reset to defaults returns every field to the reference case, which is the quickest way to check whether a surprising result came from the tool or from an input you had changed earlier.

Where this is used

• Process planning — establishing Exhaustion at End of Cycle before a trial is booked, so machine time and material in Dyeing, Printing, Color Management & Chemical Control are committed against a calculated figure rather than an estimate.
• Costing and quotation — Exhaustion at End of Cycle is an input to the cost sheet, and quoting from a worked number rather than a remembered one is what keeps a margin intact.
• Troubleshooting — when the floor result drifts from plan, entering the measured values (starting with Sampling Time) shows how much of the gap in Exhaustion at End of Cycle each variable explains.
• Teaching and study — the accepted ranges bracket normal Dyeing, Printing, Color Management & Chemical Control practice, so moving one variable at a time shows the shape of the relationship rather than a single answer.

Assumptions and limits

• The fit uses one point, so it inherits everything that point carries: a sample drawn while the bath is still climbing to temperature describes a system that was never at the temperature the constant is supposed to belong to, and the resulting K will flatter the cycle. Draw the sample during the isothermal hold and take the equilibrium figure from a deliberately over-run laboratory dyeing rather than from the recipe sheet. Equilibrium exhaustion is a property of the whole system and moves with liquor ratio, electrolyte and temperature, so a constant fitted at one liquor ratio does not transfer to a machine running a different one - which is the usual reason a laboratory cycle fails to reproduce in bulk. The model describes exhaustion, not fixation: for reactive dyes a bath can be handsomely exhausted and still fix badly, and the hydrolysed fraction that follows is a wash-off problem rather than a kinetic one. Time to 99 percent is reported because it is a useful bound, not because it is a target; the last percent of equilibrium is almost never worth the steam, and the honest reading of a large extra-time figure is usually that the recipe should change rather than that the cycle should lengthen.
• Every input is bounded to the range normal practice occupies (Sampling Time 1 to 300 min, Exhaustion at that Time 1 to 99 % and Equilibrium Exhaustion 5 to 100 %, and so on for the rest). Those bounds are guard rails against typing errors, not a claim that the formula fails one unit outside them.
• The calculation is deterministic: the same inputs always give the same result. It carries no allowance for machine condition, operator skill, ambient conditions or lot-to-lot material variation unless an input above explicitly represents one.
• Nothing is sent anywhere. The maths runs in your browser, so the numbers you type never leave the page.

Questions people ask