Left: washing-machine drum, stain blobs shrink with C(t). Right: live stain-concentration curve for the current cycle.
This simulation models the macroscopic kinetics of a laundry wash cycle — not the molecular-scale self-assembly of surfactant micelles (that's covered separately), but how quickly a stain's overall concentration on a fabric falls as the wash proceeds. Stain concentration C(t) is treated as decaying by a first-order-style law, dC/dt = −k·C, numerically integrated every frame so the curve reshapes the instant you move a slider. The rate constant k is not a single knob: it is built from three independent, physically motivated factors that you control directly.
A drum of garments whose stain blobs shrink and fade as concentration falls, paired with a live C(t) vs time chart. Water temperature contributes an Arrhenius-style exponential speed-up, detergent dose contributes a saturating (Michaelis-Menten) term with diminishing returns past a point, and agitation acts as a simple multiplier on the whole rate.
Drag Water Temperature, Detergent Dose, Agitation Intensity and Cycle Duration to see the rate constant k, the time to 50% removal (t½ = ln2/k), and the remaining stain percentage update live. The cycle loops continuously; use Restart Cycle to reset the concentration to 100% and clear the chart.
Because the detergent term saturates as D/(D+K_D), doubling the dose rarely doubles the cleaning speed once you're past the saturation point — the extra surfactant has nowhere useful left to bind. This mirrors why real detergent guidance settles on a recommended dose range rather than "more is always better".
Treating dC/dt as proportional to the remaining concentration C is the simplest model that captures the observed pattern of real cleaning processes: removal is fastest when the most stain is present and slows as less remains, producing the characteristic exponential-shaped decay curve shown in the chart.
Through an Arrhenius-style factor, exp(−(Ea/R)·(1/T−1/Tref)), the same mathematical form used for chemical reaction rates generally: raising the absolute temperature lowers 1/T, which increases the exponent and speeds up the effective rate constant k, consistent with hotter water cleaning faster.
The dose term follows a Michaelis-Menten-style saturation curve, D/(D+K_D), which rises steeply at low dose but flattens out as it approaches 1. This represents a finite capacity for surfactant to usefully engage with the stain — beyond that capacity, extra detergent contributes little extra rate.
Agitation is modelled as a linear multiplier on k, from a floor at zero agitation (representing slow diffusive removal) up to roughly triple that rate at full agitation, reflecting how mechanical action helps carry dissolved surfactant and loosened soil away from the fabric surface.
Not necessarily — while this model's Arrhenius factor makes higher temperature accelerate removal without limit, real fabrics and certain stains (proteins in particular) can be damaged or set by excessive heat. The simulation notes this trade-off but does not encode it as a numeric penalty, since it depends heavily on the specific fabric and stain.