The Cleaning Equation: formulae and derivation

The basic idea in three sentences

Everything that temperature, chemistry and flow achieve is held in a single number: the cleaning rate k. How much work has to be done is held in a second number: the cleaning task A. And the whole calculation consists of comparing the two.

k · t ≥ A

Cleaning rate times time must be at least as large as the task. That is the whole cleaning equation. Everything else is its components.

Component 1: grease does not disappear evenly

The assumption: at every instant a fixed proportion of the grease still present is removed, not a fixed amount. A decay curve follows from this:

m(t) = m₀ · e^{−k · t}

m₀: grease on the part before cleaning, in mg/m²
m(t): what is still left after time t
k: cleaning rate, in 1/min

The practical consequence is the one nobody on the shop floor believes: the first ninety per cent goes quickly, the last one per cent takes the longest. And: what counts is not the absolute amount of grease, but the ratio of start to finish.

This assumption is testable and can fail. With very thick, closed grease films the removal is even at first rather than proportional. The model is then too optimistic. To check it, measure the residual soil after 1, 2, 4, 8 and 16 minutes and plot it logarithmically against time: the result must be a straight line.

Component 2: the cleaning task A

Rearranging the decay curve for the condition that at most the permissible residue may remain at the end puts the time on one side and everything else on the other:

A = ln ( m₀ / m_zul )

A is a pure number without a unit. It combines two things that are usually discussed separately: how dirty the part arrives and how clean it has to leave. Those two statements are exactly what the drop-down lists in the calculator ask for.

Case	Requirement	m₀ → m_perm	A
Light oil film, drained	Interim cleaning, transport protection	200 → 20	2.30
Normal machining, oiled	Before powder coating, painting	500 → 1	6.21
Heavily oiled, drawn part	Before powder coating, painting	1,000 → 1	6.91
Grease applied, preserved	Bonding, soldering	2,000 → 1	8.29

Between the lightest and the heaviest task there is a factor of 3.6. That is how much more cleaning work a preserved drawn part before bonding demands than a lightly oiled sheet before an interim clean, at an identical bath setting.

What is remarkable about this: twice as much oil raises the task only by ln 2 = 0.69, at A = 6.2 that is about 11 %, not 100 %. The requirement, by contrast, weighs heavily: lowering the limit from 20 to 1 mg/m² raises the task from 3.22 to 6.21, that is, by 93 %.

Component 3: the three laws

Each of the three factors follows its own law, known from the literature. They differ not only in strength but in shape, which is why they cannot be set off against one another like for like.

Temperature: Arrhenius

f_T = exp [ − E_A/R · ( 1/T − 1/T₀ ) ]

T is the temperature in kelvin (°C + 273.15), R the gas constant (8.314 J/mol·K), E_A the so-called activation energy, a measure of how temperature-sensitive the soil is. In practice this is known as the rule of thumb: roughly twice the rate for every 10 °C. That is the same statement, only rounded.

The calculator uses true Arrhenius, not the rule of thumb. Over a span of 45 °C the rule of thumb is about 18 % too optimistic at the top end.

Chemistry: diminishing returns

f_C = κ₀ + (1 − κ₀) · ( c / c₀ )^a

The exponent a lies well below one: typically 0.2 to 0.6. At a = 0.5, twice the concentration buys only the square root of 2, that is 41 % more rate rather than 100 %.

Mechanics: wall shear stress

g(τ) = μ₀ + (1 − μ₀) · ( τ / τ₀ )^b
f_M = g(τ_actual) / g(τ_spec)

τ is the force with which the flow drags at the surface. It can hardly be measured in a plant; flow velocity, nozzle pressure or, for ultrasound, the power density serve as substitutes.

Two steps rather than one: the base effect μ₀ is what still comes off with no flow at all. It therefore belongs at τ = 0 and not at the reference point. Only afterwards is the value divided by the one at the specification. That is the only way the same step, taken there and back, comes out as one again.

The reference point is usually the still immersion bath: chemical supplier data sheets are written for it. Where it says “ten minutes at 60 °C”, immersion without part movement is what is meant.

Time: the only linear factor

Time sits outside k and therefore acts directly: twice the time, twice the output. Related to a relative adjustment it is thus the strongest of the four, stronger than chemistry and mechanics, which both enter damped.

Why κ₀ and μ₀? Without these base effects the power law predicts a rate of zero at zero cleaner or zero flow, that is, “never clean”. That is wrong: hot water alone cleans too. κ₀ is that remainder without cleaner, μ₀ the remainder in a still bath.

The complete equation

k = k₀ · f_C(c) · f_T(T) · f_M(τ)

k₀ is the rate at the reference point. In the calculator the reference point is the chemical supplier's specification: at exactly those values all three factors equal 1, and k equals k₀.

And where is time? It deliberately sits outside k. k is a rate (unit 1/min), time is a duration. Only their product k·t is a pure number and therefore comparable with A, which carries no unit either. The three factors inside k say how fast cleaning proceeds; time says how long. That is precisely why time is the only linear lever.

Together with the task this gives the condition for a successful step:

k(T, c, τ) · t ≥ A

Why multiplied and not added

Sinner's circle, four sectors in a circle of constant area, suggests that the four shares always add up to the same sum. As a picture of interchangeability that is right. As a calculation rule it is wrong: the effects do not add, they reinforce one another. Raising temperature and flow at the same time gains the product, not the sum.

Take the logarithm of the product and it does become a sum, and then the picture of the circle is correct:

ln k₀ + a·ln(c/c₀) − E_A/R·(1/T − 1/T₀) + b·ln(τ/τ₀) + ln t ≥ ln(m₀/m_zul)

That is Sinner's circle as an equation. He was right, but only in the logarithm, and only with the weights a, b and E_A/R that he never named himself.

The exchange rates

This finally answers what the circle leaves open: at what rate one may trade. Both tables are calculated from the formulae above.

What a doubling buys

Adjustment	Change	Factor on k · t
Temperature	+10 °C from 50 °C	2.00
Time	twice as long	2.00
Mechanics	twice the flow	1.59
Chemistry	twice the dosing	1.35

The temperature row is not directly comparable with the others: it shows an absolute change of 10 °C, the others a doubling. “Twice the temperature” is not a meaningful quantity.

The mechanics row applies to a spray plant. It is the only one that depends on the starting point: in a quietly run immersion bath the same doubling gives only 1.20. There, up to three quarters of the cleaning happens without any flow anyway. Only the remainder gets doubled.

What halving the time costs

Adjustment	Change required	Assessment
Temperature	+10.0 °C from 50 °C	usually the cheapest, often forbidden by the substrate
Mechanics	flow × 2.8	rebuilding the plant, but permanent
Chemistry	dosing × 4.7	usually impossible or uneconomic

Calculated with a = 0.5 · b = 0.8 · κ₀ = 0.15 · μ₀ = 0.2 · E_A = 62 kJ/mol, reference point spraying. Out of a quiet immersion bath the same halving would require seven times the flow instead of 2.8 times. Temperature is the strongest lever, not because its functional form is special, but because 10 °C is usually cheap to come by in this application.

What the calculator makes of it

The calculator compares two states: the supplier's specification and today's actual condition. Both pass through the same equation, and the ratio is the degree of fulfilment, the percentage shown on the scale:

Fulfilment = ( k_act·t_act ) / ( k_spec·t_spec ) · A_spec / A_act

Because everything except the ratios cancels in both fractions, the calculator needs no measured plant value. From this follows the statement that counts on the shop floor: how far the residual soil sits above the target value.

Residue / target = e^{(1 − fulfilment) · A}

Here too the masses cancel out. That is why a figure in mg/m² is nowhere required, only their ratio, and that is contained in A.

Where the numbers come from

The six parameters of the equation, k₀, a, b, E_A, κ₀, μ₀, are not natural constants. They depend on the combination of grease, cleaner and surface. The values stored in the calculator are guide values from the literature. They carry the direction and the order of magnitude reliably; which lever gives most in a given case is worked out afresh each time. There is no general ranking: whether chemistry or mechanics comes first depends on the type of grease and on how strongly the plant already flushes the parts.

Determining them for your own plant takes two blocks of trials and about one working day:

Block 1 · 7 samples: Fixed conditions, residual soil after 1, 2, 4, 8, 16 minutes, plus two repeats at 4 minutes. Shows whether the basic model holds at all. The repeat points give the measurement scatter: only a deviation larger than that counts as refutation.
Block 2 · 11 samples: Do not vary each factor on its own; run all eight combinations of two levels for temperature, concentration and flow, plus three repeats at the centre. The regression yields a, E_A and b.

The water-break test is not sufficient for this. There a breaking water film only shows that grease is still present, not how much. What is needed is weighing after extraction (gravimetry) or a fluorescence measurement. That always includes the blank value, the reading from a clean sample, subtracted as the zero point, and the detection limit, the smallest amount the method can still distinguish from zero at all. At a few mg/m² and 100 cm² of sample area we are talking about hundredths of a milligram.

Building block 5: etching on aluminium

On aluminium the degreasing bath usually removes metal at the same time. That is not a refinement of the cleaning equation but a second condition alongside it, and it follows a different logic.

The cleaning equation knows only one direction: more is better, it only costs money. Etching has a window. Too little removal leaves the deformation layer in place, the microcrystalline near-surface zone left by forming, holding metal oxides and intermetallic phases. Adhesion and filiform corrosion then become the problem. Too much costs metal, dimensional accuracy and appearance.

What is required

Unlike residual grease, there is a real measured value here. QUALICOAT requires it as a weight difference on a test panel of AA6060 or AA6063: weigh it, run it through the process, strip it, weigh it again. The specifications call the quantity Beizgrad (degree of etching), the trade literature mostly Beizabtrag (etch removal), the same thing.

Standard: at least 1.0 g/m² across all etching steps together.
SEASIDE: at least 2.0 g/m²; with a double etching step the last step must contribute at least 0.5 g/m².
There is no standard for the upper end. Neither QUALICOAT nor GSB names one; the only published window that could be found is 0.5 to 4.0 g/m². Where your own ceiling sits is an in-house decision. The calculator suggests 3.0 g/m² and lets you change it.

Because this measured value exists, the etching part of the calculator needs no calibration out of thin air: it shifts a measured point, exactly as the rest of the calculator shifts the supplier specification.

The formula

Removal is linear in time, unlike the grease, which decays exponentially, and depends on temperature by Arrhenius, the same equation as above:

removal [g/m²] = r₀ · f_T(T) · (c/c₀)ⁿ · f_pH · t

r₀ is the measured removal at the specified point. For the activation energy the calculator uses 50 kJ/mol on the alkaline route (literature for aluminium in caustic soda between 15 and 45 °C, 45-51 kJ/mol is measured; across alloys, concentrations and temperature ranges, though, the literature scatters widely, from around 40 to over 90 kJ/mol). The acid route uses the same value. More on that below. The concentration exponent n is 1.0, linear: half the etchant concentration, half the etch rate. That is the most striking difference from cleaning, where the yield falls off as dosage rises.

On temperature sensitivity, honestly: a lower value stood here for the acid route at first, on the widespread grounds that acid etching responds less to temperature. That claim could not be substantiated. The only direct comparison found, on the same alloy, shows the opposite trend. So both routes now carry the same value, and the field is editable in the calculator. It is the least certain number in the whole building block.

The acid route: why pH does count here

Acid etching runs below pH 4, with fluoride: hydrofluoric acid or complexed fluorides. The shop-floor rule is "the lower the pH, the better", and behind it lies not experience but a textbook number: hydrofluoric acid is a weak acid with pK_a 3.17. The attacking species is undissociated hydrogen fluoride, and how much of it is free depends on pH alone.

pH	Share of free hydrogen fluoride
2.00	93.7 %
2.50	82.4 %
3.00	59.7 %
3.17	50.0 %
4.00	12.9 %
5.00	1.5 %

That explains both ends: why the work is done below pH 4, and why going below pH 2 gains almost nothing, because nearly everything is active there already. A pH of 1 instead of 2 costs acid and plant wear for six per cent more effect.

The lever reverses

The linear concentration term leads to something that directly contradicts the main statement of the degreasing part, which is why it belongs on record. Compared here: ten degrees more against twice the dosage.

Effect on …	+10 °C	twice the dosage
etch removal	1.69	2.00
cleaning: Light oil, cutting fluid	1.49	1.35
cleaning: Grease, wax, drawing compound (solid at room temperature)	1.98	1.29
cleaning: Baked-on oil (carbonised residue)	2.20	1.21
cleaning: Polishing paste (grease with solids)	1.69	1.29
cleaning: Corrosion protection oil, preservative	1.61	1.32
cleaning: Dried-on emulsion, machining residue	1.78	1.33

For cleaning, temperature beats dosage, for every soil type; that is the core statement of the first part. For etch removal it is the other way round. So economising on the etchant costs twice as much at the etching as at the degreasing: half the dosage means half the removal, but only about a quarter less cleaning effect.

The reason is in the formula: concentration enters removal linearly, and cleaning with a falling yield plus a base effect that remains even without any detergent.

The rule you will not read anywhere else

Anyone wanting a shorter cycle runs hotter. For plain degreasing that is always a good idea. With etch cleaning two quantities have to hold at once. So let us work the question through properly: we go up by 10 °C and cut the time just far enough that the etch removal stays the same (to 58 %). What happens to the cleaning?

Soil	Ea in kJ/mol	Cleaning effect
Light oil, cutting fluid	38	−12 %
Corrosion protection oil, preservative	45	−5 %
Polishing paste (grease with solids)	50	±0 %
Dried-on emulsion, machining residue	55	+6 %
Grease, wax, drawing compound (solid at room temperature)	65	+18 %
Baked-on oil (carbonised residue)	75	+31 %

Running hotter and shorter only wins if the soil is more temperature-sensitive than the etching. The switching point sits exactly where the two activation energies are equal, with the default value, at 50 kJ/mol. With carbonised oil the trade is a real gain; with thin oil it makes matters worse.

Etching helps the cleaning: as mechanics

When the metal dissolves underneath the soil, the soil loses its grip, regardless of what the detergent does. That is not a chemical but a mechanical effect; the calculator therefore counts it towards mechanics and not towards chemistry.

It is added, not multiplied: undercutting works in a still bath too, where there is no flow at all. At the reference point it cancels completely, so like everything else here it needs no calibration. It only becomes visible on deviation, and then in an uncomfortable direction:

An etching bath run too cold loses twice over. The detergent slows down and the etch attack weakens, and the second loss does not appear in any usual assessment.

Where this building block ends

Dissolved aluminium slows it down. A bath in service etches more slowly than a freshly made-up one at the same caustic level. After a bath change, measure again.
Alloy and condition. AA6060 etches differently from AA5754, extruded differently from rolled. The measured value holds for what was measured.
Inhibited etch cleaners contain additives that deliberately limit removal. With those, removal is less temperature-dependent than the formula assumes - the calculation then overstates the gain from more temperature. This is the most uncomfortable limitation, because it weakens precisely the rule above.
Removal is an area average. Edges and cavities etch differently. A figure in g/m² says nothing about uniformity.
The share attributed to loosening from underneath (40 % alkaline, 25 % acid) is an estimate. There is no dependable series of measurements for it.
Temperature sensitivity scatters widely. 50 kJ/mol is a mid-range value, not a constant of nature. Markedly different figures appear in the literature for the same alloy family. If you have measured removal at two temperatures, enter your own value.
The link between etch removal and filiform corrosion is debated in the trade. That too little removal leaves the deformation layer in place and therefore encourages it is well arguable. As a chain of cause and effect it is not conclusively established.

Where the model ends

These points belong to the equation. Anyone who does not know them overestimates it:

Thick, closed films. There removal is governed by area rather than by amount, and the decay curve no longer holds.
Immersion and withdrawal. In laboratory trials about 90 % of the removal was attributed to the air-liquid interface travelling across the part as it passes through the bath surface. Dipping several times can therefore be more effective than dwelling longer. The equation does not know this lever.
Thresholds. The melting point of the grease, the cavitation threshold in ultrasound, a critical wall shear stress, an optimum concentration. These are jumps, not curves. With waxes, 5 °C can decide between good parts and scrap.
The age of the soil. Dried-on oil can have a rate an order of magnitude lower. In many plants the time parts are left standing is the single largest source of scattered results.
pH, water hardness, foam, choice of cleaner, rinsing and drying do not appear. pH is contained silently in the exponent a.
Particulate cleanliness to VDA 19.1 / ISO 16232: there the particle count matters, not the mass.

Sources

The full derivation, with trial plan and review of the literature, runs to 32 pages. The principal references:

Sinner's circle, Herbert Sinner (1900–1988), Henkel: the four factors and their interchangeability.
E. Kissa, Textile Research Journal 1978: measures a reaction order of 1.1 for particulate soil, that is, close to first order. Also notes, however, that the rate falls with washing time.
H. Schott, Textile Research Journal 1976, the critical counter-argument: bond strengths are distributed, so a single k is always an average.
JAOCS, Kinetic studies of detergency I and II: source for the Dupré mechanism accounting for about 90 % of removal.
Food and Bioproducts Processing: CIP modelling with removal and redeposition; evidence for the bath load.
Journal of Food Engineering: evidence for the optimum concentration, above which more chemistry cleans worse again.
VDA 19.1 / ISO 16232: technical cleanliness by particle count; the area this model explicitly does not cover.

The paper behind this

This page is the short form. The full derivation is in a separate foundation paper: every equation with its origin, the material and soil tables, etch removal on aluminium, a test plan of 18 samples, and a section on what the model cannot do.

The Cleaning Equation, Edition 1.3 PDF · 36 pages · 0.6 MB · September 2026

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Said plainly: the paper is a derivation, not a test series. The parameters in its tables are reasoned starting values for your own trials, not design data, and the one experiment that could refute the model has yet to be run. The paper states at every point where the calculation breaks down.