Pendulum hardness of a 2K PU clearcoat.

1. Runs used in total

31 runs: 12 + 16 + 3. No run failed.

2. Suggested optimum

About 89 s König hardness. That is the mean of the three confirmation runs (88.7, 92.2, 86.4 s). The model predicted 97 s, so it overestimates at this corner.

3. Recommended settings

X02, X04 to X07 and X09 to X11 had no clear effect. Their levels follow the sign of their small screening effects, so any level in range will do.

4. Final model

Fitted on design 2, coded units. A = X01, B = X03, C = X08, D = X12.

Hardness = 80.08 + 6.49 A + 1.49 B + 9.77 C + 1.71 D + 0.17 AB + 3.82 AC − 1.23 BC + 3.62 A² − 18.01 B² − 9.09 C²

Residual standard deviation 2.5 s, R² = 0.99 (5 residual degrees of freedom).

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Hardness depends on three factors: cure temperature, NCO:OH index and catalyst level. Raise the index to 1.3, cure at about 95 °C and add about 200 ppm DBTDL. The confirmation runs gave about 89 s, double the 42 s of the all-low run.

I screened 11 of the 12 factors with a 12-run Plackett–Burman design. A PB-12 design has only 11 columns, so substrate temperature (X12) stayed at its centre point. Cure temperature (+9.6 s) and NCO:OH index (+5.9 s) stood out. The other effects were under 2 s, within the noise.

The second design was a face-centred central composite design in index, catalyst and cure temperature. I picked the catalyst because it interacts with binder, curing agent and cure temperature. I aliased X12 with the three-factor interaction so it got a test too; its effect was small (+1.7 s). The design showed strong curvature in the catalyst: too little and too much both soften the film. It also showed a ceiling in cure temperature and an index × temperature interaction. I held the inactive factors at their better screening levels.

Three confirmation runs at the model optimum averaged 89 s, against a prediction of 97 s. The model is too optimistic at the corner it extrapolates to. One design-2 run at index 1.3 and 80 °C gave 94 s, so the true best cure temperature may lie between 80 and 95 °C. If the customer needs more than 90 s, a few runs along cure temperature at index 1.3 would settle this.

Only two of the 12 factors control hardness: the NCO:OH index and the cure temperature. Set the index to its maximum and cure at 95 °C, and the film reaches about 89 s König hardness. Before the study, typical settings gave about 60 s.

1. Runs used in total

40 runs: 25 + 11 + 4. None failed.

2. Suggested optimum

The model predicts 91 s König hardness. The four confirmation runs averaged 88.6 s (86.7 to 90.6 s), which is within the noise of the prediction.

3. Recommended settings

Factors X02 to X07 and X09 to X12 had no clear effect, so their midpoints are a safe choice. You can move them for cost or for other properties.

4. Final model

Quadratic model in X01 and X08, fitted to all 36 runs from designs 1 and 2. A block term absorbs the 12 s offset between the two designs. The coefficients below are for the design 2 block, where the other ten factors sit at their midpoints.

Residual standard deviation: 5.8 s. R²: 0.86, including the block term.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I chose a definitive screening design because it tests all 12 factors in 25 runs. It also detects curvature and keeps main effects free of two-factor interactions. It found two strong factors: cure temperature (+9.8 s per coded unit) and NCO:OH index (+6.3 s). The other ten effects were all below 1 s. The center point measured 85 s while the average was 60 s, which pointed to a hardness maximum inside the range.

Next I ran a face-centered CCD in the two active factors and held the others at their midpoints. This mapped the curvature. Hardness peaks at about 95 °C cure temperature, and higher index always helps. Index and temperature also reinforce each other. The CCD runs came out about 12 s harder than the DSD runs at the same X01 and X08 settings. In the DSD, the other ten factors sat at their extremes, so pushing several of them to the edge of their range together probably costs some hardness. That is another reason to keep them near their midpoints.

Four confirmation runs at the recommended settings averaged 88.6 s, close to the 91 s prediction. Of the 40 runs, 36 built the model and 4 confirmed it.

1. Runs used in total

42 runs: 29 + 10 + 3. None failed.

2. Suggested optimum

The model predicts 87 s König hardness at the recommended settings. The three confirmation runs measured 91.1, 89.5 and 92.6 s, a mean of 91 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. Set the other nine factors anywhere in their ranges to suit cost or processing. The table puts them at mid-range.

4. Final model

Hardness (s), coded units, fitted to designs 1 and 2 (39 runs):

Residual standard deviation 2.8 s, R² = 0.96.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I started with a definitive screening design. With 12 factors, it tests all main effects in 29 runs, keeps them clear of two-factor interactions, and detects curvature. That mattered here: the center point measured 76 s, while the design average was 58 s. Two factors stood out: cure temperature (X08) and NCO:OH index (X01). Catalyst (X03) had a smaller effect. A second-order model on these factors showed a strong curvature in catalyst and in cure temperature, plus an index × temperature interaction. The other nine factors did nothing that the noise could not explain.

In a DSD, squared terms are partly correlated with each other. So I added the axial and center points of a face-centered CCD in X01, X03, X08 and cure time (X09), with the other factors at center. That took 10 runs. They confirmed the model. Cure time had no effect. Catalyst has a clear optimum near 220 ppm. Hardness rises with the index up to the top of its range. Cure temperature peaks near 95 °C, and the peak moves higher as the index rises.

Three confirmation runs at the optimum gave a mean of 91 s, which is 3 to 4 s above the prediction. That fits the noise, plus a slight underestimate at the corner of the index range. Compared with the old center recipe at about 77 s, the gain is about 14 s.

1. Runs used in total

44 runs (25 + 16 + 3). None failed.

2. Suggested optimum

About 90 s König hardness (95 % prediction interval for one run: 86 to 94 s). The three confirmation runs gave 88.8, 90.1 and 96.1 s, a mean of 91.7 s.

3. Recommended settings

• X01 NCO:OH index: 1.3 (+1)
• X08 Cure temperature: 92 °C (+0.6)
• X03 Catalyst (DBTDL): 210 ppm (+0.05)
• All other factors have no clear effect. Keep them at the centre of their range: X02 160 mg KOH/g, X04 0.55 %, X05 2 %, X06 17.5 %, X07 12.5 min, X09 40 min, X10 45 µm, X11 50 % r.h., X12 24 °C (all 0)

(X08: 80 °C + 0.6 × 20 °C = 92 °C.)

4. Final model

Fitted to the 16 CCD runs, in coded units:

hardness = 80.32 + 5.79·X01 + 7.81·X08 + 1.51·X03 + 4.30·X01·X08 − 9.74·X08² − 13.57·X03²

Residual standard deviation 1.53 s, R² = 0.992.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Three factors control hardness: the NCO:OH index, the cure temperature and the catalyst level. The other nine don’t matter inside their ranges.

I started with a definitive screening design because it tests all 12 factors in 25 runs and also shows whether the response curves. Only the NCO:OH index and the cure temperature had clear main effects. But the centre point scored 85 s, while the factorial runs averaged 61 s, so at least one factor has a strong curve. The DSD could not tell which one.

Next I ran a face-centred central composite design in the NCO:OH index, the cure temperature and the catalyst level, with all other factors held at their centre. I added the catalyst because it usually works together with the curing agent and the cure temperature. The CCD showed where the curve comes from: hardness peaks at a middle catalyst level and just above the middle of the cure temperature range. The index raises hardness steadily, and more so at higher cure temperature.

Three confirmation runs at the predicted optimum averaged 91.7 s, inside the prediction interval. Compared with the old design centre (about 80 s), hardness goes up by about 10 s.

1. Runs used in total

45 runs (25 + 16 + 4). None failed.

2. Suggested optimum

König hardness of about 90 s (model: 89.6 s; 95 % range for one run: 82 to 97 s). The four confirmation runs averaged 89.5 s.

3. Recommended settings

X02, X04 to X07 and X09 to X12 showed no effect on hardness. Set them for cost or for other properties.

4. Final model

Fitted to designs 1 and 2 (41 runs), coded units:

hardness = 79.46 + 5.51·X01 + 0.31·X03 + 10.09·X08 + 4.00·X01·X08 − 11.98·X03² − 10.70·X08²

Residual standard deviation 3.4 s, R² = 0.947.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Three of the twelve factors control hardness: NCO:OH index, catalyst level and cure temperature. The other nine do nothing you can measure.

I started with a definitive screening design, because with 25 runs it screens all 12 factors and still catches curvature. It showed that the index and cure temperature raise hardness. It also showed that hardness at the centre point is far above the average, so some factors have an optimum inside their range. Cure temperature and catalyst showed this curvature.

Next I ran a face-centred central composite design on the three active factors, with the rest at their centre, to fit a full quadratic model. Hardness rises with the index all the way to 1.3. Cure temperature has an optimum near 93 °C, and the gain from heat is larger at high index. Catalyst has an optimum near 200 ppm: both no catalyst and 400 ppm give softer films.

Four confirmation runs at the optimum averaged 89.5 s, which matches the prediction of 89.6 s. Recipes near the old centre gave about 78 s, so this is a gain of about 12 s.

1. Runs used in total

46 runs (25 + 17 + 4). None failed.

2. Suggested optimum

About 89 s König hardness, by the model. The four confirmation runs averaged 93.5 s (89.9 to 96.0 s).

3. Recommended settings

None of the factors set to 0 had a measurable effect, so they can go wherever cost or processing suits best. For example, a shorter cure time or no catalyst costs no hardness.

4. Final model

Fitted to the CCD (design 2), in coded units:

hardness = 80.99 + 3.95·X01 + 8.45·X08 + 3.39·X01·X08 − 9.41·X08²

Residual standard deviation 2.99 s, R² = 0.925.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Two factors control hardness: the NCO:OH index and the cure temperature. The other ten made no difference we could measure. Hardness rises with the index all the way to the top of its range. It also rises with cure temperature, but levels off and drops again near 100 °C. A higher index moves that peak a little higher.

I started with a definitive screening design because it tests all 12 factors in 25 runs, keeps main effects clear of two-factor interactions, and catches curvature. It showed strong effects for the index (+6 s) and cure temperature (+11 s). The centre run came out 21 s above the average, so the response clearly curves.

Next I ran a face-centred CCD in the index, cure temperature and cure time, with the other factors at their centre. I included cure time because chemistry suggested it could interact with temperature, and the screening residuals hinted at it. The CCD confirmed the index and temperature effects, their interaction, and a strong quadratic term for temperature. Cure time had no effect. The model puts the peak at index 1.3 and about 93 °C.

Four confirmation runs at that point gave 93.5 s on average. That is a little above the predicted 89 s but within the noise, and well above anything seen in screening.

1. Runs used in total

48 runs, none failed.

2. Suggested optimum

About 91 s König hardness (95 % range for one new panel: 85 to 97 s). The four confirmation panels averaged 89.9 s.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine can sit anywhere in their ranges; the centre is shown.

4. Final model

Coded units, fitted to all 48 runs:

hardness = 79.36 + 5.76·X01 + 0.94·X03 + 9.95·X08 + 1.54·X01² − 13.54·X03² − 10.08·X08² − 1.61·X01·X03 + 3.70·X01·X08 − 1.17·X03·X08

Residual standard deviation 2.7 s, R² = 0.976.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I screened first, then modelled the few factors that mattered, then checked the answer.

A definitive screening design came first. With 12 factors it needs only 27 runs, gives clean main effects, and also shows curvature, which a two-level screen would miss. It found three active factors. Cure temperature (X08) and NCO:OH index (X01) raise hardness. Catalyst (X03) has a strong optimum in the middle of its range. Index and cure temperature also work together. The other nine factors showed no effect.

A face-centred CCD on those three factors, with the rest at their centre, then gave a full quadratic model. It confirmed the screening picture, and the block shift between the two designs was negligible (−0.2 s). The model puts the maximum at the highest index, catalyst near 200 ppm, and cure at about 94 °C. Hotter than that, hardness falls again.

Four confirmation runs at those settings averaged 89.9 s, against 92.5 s predicted. That is well inside the prediction interval, so the model holds. To fix the soft film, raise the NCO:OH index to 1.3, bake at about 94 °C, and keep the catalyst at 200 ppm.

1. Runs used in total

50 runs, none failed.

2. Suggested optimum

About 92 s König hardness (predicted 92.3 s; three confirmation runs gave 90.3, 92.1 and 95.0 s, mean 92.5 s).

3. Recommended settings

Only X01, X03 and X08 matter. The other nine can sit anywhere in their ranges; the mid values are given as a default.

4. Final model

Fitted on all 47 runs from designs 1 to 3, coded units:

Residual standard deviation 2.9 s, R² = 0.97.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-CCD-axial.csv, design-3-CCD-axial-results.csv
• design-4-confirmation.csv, design-4-confirmation-results.csv

7. Summary

Hardness depends on three factors: cure temperature, NCO:OH index and catalyst level. Cure temperature has the strongest effect. A higher index raises hardness, and it raises it more when the cure is hot. Catalyst has a clear peak near 200 ppm: too little or too much softens the film. Film build, humidity, flash-off and the other process settings had no effect we could detect.

I started with a definitive screening design because it tests all 12 factors in 25 runs. It also keeps main effects free of interactions and can spot curvature. The centre run scored far above the average, so the response clearly curved. A three-factor quadratic fit on the screening data alone already explained 98 % of the variation.

I then ran a face-centred central composite design in the three active factors, with the other nine at mid-range, to estimate the curvature properly. A scripting error placed all six axial runs at the same point (high cure temperature, the other two at centre). Those runs still served as replicates. I sent the six correct axial points as design 3. That cost six extra runs.

The combined model puts the optimum at index 1.3, about 210 ppm catalyst and about 94 °C cure. Three confirmation runs there matched the prediction within noise.

1. Runs used in total

60 runs (27 + 29 + 4). None failed.

2. Suggested optimum

König hardness of about 90 s. The model predicts 89.7 s, and four confirmation runs averaged 89.8 s.

3. Recommended settings

Only X01, X03 and X08 matter. You can set the other nine factors anywhere in their ranges, for example to save cost or time.

4. Final model

Fitted to design 2, in coded units:

hardness = 80.26 + 5.46·X01 + 0.93·X03 + 8.86·X08 + 4.01·X01·X08 − 13.31·X03² − 10.49·X08²

Residual standard deviation 3.3 s, R² = 0.958.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I started with a definitive screening design because it handles 12 factors in 27 runs and also shows curvature. NCO:OH index (X01) and cure temperature (X08) had clear effects. The three centre points came in near 81 s, against a design mean of 62 s. That showed strong curvature, but this design could not tell which factor caused it.

Next I ran a face-centred CCD on the five likely candidates: X01, X08, catalyst (X03), cure time (X09) and OH number (X02). The other seven factors stayed at their centre. Three things drive hardness: X01, X08 and their interaction. X03 and X08 both have an optimum inside their range. The curvature from screening came from catalyst level and cure temperature. Cure time and OH number had no real effect.

The model puts the best film at the highest index, about 92 °C cure and about 200 ppm DBTDL. Four confirmation runs at these settings averaged 89.8 s, against a prediction of 89.7 s. The model holds there. Compared with the centre of the old window, the gain is about 10 s.

1. Runs used in total

72 runs. None failed.

2. Suggested optimum

About 91.5 s König hardness (95 % prediction interval for one run: 86 to 97 s). The four confirmation runs averaged 89.9 s.

3. Recommended settings

Only X01, X03 and X08 matter. The rest have no measurable effect, so set them wherever cost or handling suits you. X09 has only a small effect: 20 min is about 1 s harder than 60 min at 95 °C.

4. Final model

Fitted to designs 1 to 3 (68 runs), coded units:

hardness = 81.04 + 5.42·X01 + 1.05·X03 + 9.33·X08 + 0.19·X09 + 4.16·X01·X08 − 1.09·X08·X09 − 14.22·X03² − 10.11·X08²

Residual standard deviation 2.66 s, R² = 0.976.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-axial-curvature.csv, design-3-axial-curvature-results.csv
• design-4-confirmation.csv, design-4-confirmation-results.csv

7. Summary

Hardness depends on three of the twelve factors: cure temperature, NCO:OH index and catalyst level. A high index, 95 °C and about 200 ppm DBTDL give roughly 90 s, against 45 to 60 s for much of the old window.

I started with a definitive screening design because it handles 12 factors in 27 runs and shows curvature. It found two strong main effects, cure temperature and NCO:OH index. The centre runs came out far above the rest, so some factor had to have a peak inside the range.

A CCD in X01, X08 and X09 then mapped the curvature and the interactions. Cure temperature peaks inside the range, and the index helps more at high temperature. But the whole CCD block came out about 10 s higher than the DSD. The DSD could not tell which of the other factors caused that, because their square terms were confounded with each other.

So design 3 moved each remaining factor alone to −1 and +1, at a good X01/X08 setting, with four centre runs. Only the catalyst stood out: both 0 and 400 ppm cost about 19 s, so the best level sits in the middle. With the catalyst’s square term in the model, the block offset vanished. Four confirmation runs at the predicted optimum averaged 89.9 s, inside the prediction interval.

1. Runs used in total

28 runs: 25 screening runs and 3 confirmation runs. None failed.

2. Suggested optimum

92.5 s König pendulum hardness (standard error of the prediction 1.1 s). The three confirmation runs at these settings averaged 91.3 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. The other nine sit in the middle of their ranges, as in the confirmation runs.

4. Final model

Fitted to all 28 runs, coded units:

Residual standard deviation 2.53 s, R² 0.982 (adjusted R² 0.977, 21 residual degrees of freedom). X03 on its own is not significant (p = 0.14). I kept it because X03² is the strongest curvature in the model.

5. Designs in order

6. Run sheets

• design-1-DSD.csv
• design-1-DSD-results.csv
• design-2-confirmation.csv
• design-2-confirmation-results.csv

7. Summary

I used two designs: a definitive screening design to find the active factors and their curvature, then three confirmation runs at the predicted best point.

With 12 factors and a maximum to find, the first design had to do two jobs. It had to pick out the few factors that matter and show whether any of them peak inside the range. A two-level design can’t see a peak. A 16-run fractional factorial would also mix main effects with the interactions you’d expect between index, catalyst and cure temperature. A definitive screening design does both jobs in 25 runs. Its main effects are free of two-factor interactions and curvature, and if three or fewer factors are active, the same runs fit a full quadratic model in them.

Three factors were active. Cure temperature had the largest effect: hardness rises with it and, at index 1.3, levels off at about 93 °C. A higher NCO:OH index also raises hardness, more so at high cure temperature. Catalyst has almost no straight-line effect but a strong peak in the middle. At 0 or 400 ppm, hardness is about 14 s lower than at 200 ppm. The centre run gave 85 s, the best screening result and the first sign of this peak. None of the other nine factors changed hardness by more than 1.5 s from low to high setting. A six-term model fitted the 25 runs with R² 0.98 and a residual standard deviation of 2.3 s, so I skipped a response surface design and went straight to confirmation.

The model predicted 94 s at index 1.3, 200 ppm catalyst and 93 °C. Three repeat runs there gave 87.1, 91.8 and 95.1 s. Their average of 91.3 s is inside the 95 % prediction interval of 90 to 99 s, so the model holds. With these runs added, it predicts 92.5 s at the recommended settings: 7 s above the best screening run and 51 s above the worst.

Hardness was still rising at index 1.3, the top of the allowed range, so a higher index might give more if the recipe allows it. There is also a weak hint (p = 0.05) that cure time works best at 40 min, the middle of its range. You can set the other inactive factors for cost or appearance, but only their middle settings have been confirmed.

1. Runs used in total

28 runs: 25 screening runs and 3 confirmation runs. None failed.

2. Suggested optimum

König hardness of 91 s at the recommended settings. The final model predicts 91.3 s (95 % confidence interval 88.6 to 94.0 s). The three confirmation runs gave 92.6, 90.8 and 91.7 s.

3. Recommended settings

The nine factors at 0 showed no clear effect, so I left them at the centre, where the confirmation runs were made.

4. Final model

Fitted to all 28 runs, in coded units, with hardness in seconds:

Residual standard deviation 2.86 s, R² 0.974 (adjusted 0.966).

Coding: X01 = (index - 1.1) / 0.2, X03 = (ppm - 200) / 200, X08 = (°C - 80) / 20.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, results in design-1-DSD-results.csv
• design-2-confirmation.csv, results in design-2-confirmation-results.csv

7. Summary

Three of the 12 factors control hardness. Cure temperature matters most: hardness rises up to about 98 °C, then levels off. A higher NCO:OH index helps across its whole range, and more so at high cure temperature. Catalyst has a sharp optimum near 200 ppm: none costs about 16 s, and 400 ppm about 13 s. Moving any of the other nine factors from low to high changed hardness by 2 s or less, which is within the noise.

I used two steps: a screening design that could also see curvature, then confirmation runs at the predicted optimum. A definitive screening design (DSD) needs 25 runs for 12 factors. Its main effects are free of interactions and squared terms, and each factor runs at three levels, so it shows curvature. I expected curvature, since cure chemistry often has an optimum. A Plackett-Burman design costs fewer runs, but it can’t see curvature and partly mixes main effects with interactions. And if three or fewer factors turn out active, the DSD fits a full quadratic model in them, so it doubles as the optimization design.

That is what happened. Cure temperature and NCO:OH index had large main effects. The centre run gave 80.9 s, the best of all 25 against an average of 60 s, so the curvature was strong. A search over squared terms and interactions placed it on catalyst and cure temperature, plus an NCO:OH × cure temperature interaction. This model predicted 90.8 s at NCO:OH 1.3, 210 ppm catalyst and 98 °C. I didn’t need a separate optimization design: NCO:OH sits at its upper limit, catalyst has a clear peak, and any cure temperature from 94 to 100 °C costs less than 0.5 s.

The confirmation runs tested a point the screening never ran. Their mean of 91.7 s was within 1 s of the prediction, and 31 s above the screening average. The three repeats spread less (standard deviation 0.9 s) than the model residual (2.9 s), so some of the other nine factors may have small effects that 28 runs can’t separate from noise.

1. Runs used in total

31 runs: 25 in design 1 and 6 in design 2. No run failed.

2. Suggested optimum

93 s König pendulum hardness. The 95 % confidence interval for the mean is 90 to 96 s. Single panels will scatter more widely, from about 85 to 101 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. The other nine sit at the midpoint, where I tested the optimum, and you can move them within their ranges to suit cost or other properties. The model’s exact best point is X03 = +0.05 and X08 = +0.62. I rounded it to the tested point, which costs less than 0.1 s.

4. Final model

Fitted to all 31 runs, in coded units:

Residual standard deviation 3.47 s (24 degrees of freedom). R² = 0.963, adjusted R² = 0.954.

5. Designs in order

6. Run sheets

• design-1-DSD.csv and design-1-DSD-results.csv
• design-2-I-optimal.csv and design-2-I-optimal-results.csv

7. Summary

I used two steps. A screening design found the active factors and a first model. A few more runs then pinned down the optimum, and two of them sat on the optimum itself, so they also served as the confirmation.

I started with a definitive screening design because nobody knew which of the 12 factors mattered. Hardness in a 2K PU also tends to peak inside the window, since too little or too much crosslinker or heat both cost hardness. A DSD tests every factor at three levels in 25 runs (2k + 1). It keeps main effects clear of two-factor interactions and can fit a full quadratic in any three factors. So if only a few factors matter, the same runs give both the screen and the first model.

The DSD found three active factors. Cure temperature had the biggest effect, with a peak inside its range. A higher NCO:OH index raised hardness, and more so at higher cure temperature (the X01·X08 term). Catalyst had almost no straight-line effect but a strong peak: 0 ppm and 400 ppm DBTDL lost 15 s and 13 s against 200 ppm. The other nine factors all had effects of 1.4 s or less, within the noise. The best run was the centre point at 84 s. The model put the optimum at index 1.3, 200 ppm and about 92 °C, where no run had been.

Because the optimum lay in an untested spot, I wanted the next runs to sharpen the predictions there rather than across the whole space. An I-optimal search over that region kept catalyst at 200 ppm and spread index and cure temperature around the optimum. I rounded its picks to four clean points and added two repeats at the predicted optimum.

Design 2 bore the model out. Its six runs came in 2 s below the predictions on average, which is within the noise. At index 1.3, cures at 80 °C and 100 °C gave 89.7 s and 93.3 s, against predicted 90.4 s and 92.7 s. The two repeats at the optimum gave 98.7 s and 87.3 s, a mean of 93.0 s. After the refit the optimum moved by less than 0.05 coded units, so I stopped. Those two repeats also show how much single panels scatter, so judge the clearcoat on the mean of several panels.

Two smaller points remain. Flash-off time showed a weak effect (-1.4 ± 0.7 s per coded unit), too small to call real, so I left it at the midpoint. Cure time had no effect between 20 and 60 min, so a shorter bake may keep the same hardness, but I haven’t tested that at the new settings.

1. Runs used in total

35 runs (25 + 7 + 3). None failed.

2. Suggested optimum

91.4 s König pendulum hardness (95% confidence interval 89.0 to 93.9 s). The three confirmation runs averaged 92.0 s.

3. Recommended settings

The eight factors at 0 had no detectable effect. I held them at mid-range, where the confirmation ran, but you can set them for cost or handling.

4. Final model

Hardness in seconds, coded units, fitted to all 35 runs.

Residual standard deviation 2.60 s (26 df). R² = 0.977, adjusted R² = 0.970.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I worked in three steps: screen all 12 factors, model the few that matter, then confirm. Each step reused the runs before it, so the final model rests on all 35 runs.

Design 1 was a definitive screening design. I chose it because it keeps main effects clear of two-factor interactions and shows curvature in the same 25 runs. Interactions were likely, since index, polyol, catalyst and cure temperature tend to act together in a 2K PU. Four factors mattered: cure temperature and NCO:OH index strongly, catalyst and film thickness weakly. The other eight showed no effect. The centre point gave the hardest film of all 25 runs (82.6 s), so the optimum lay inside the ranges. The best fit put that curvature on catalyst and cure temperature, plus an interaction between index and cure temperature.

Design 2 checked that picture. In a DSD the squared terms are correlated, so the curvature could also belong to a factor I had dropped. I varied only index, catalyst and cure temperature and held the rest at mid-range, with film thickness at 60 µm. Six runs were D-optimal for the quadratic model. The seventh sat at the predicted optimum and gave 84.9 s. The runs confirmed the curvature in catalyst and cure temperature and showed none in the index, so the index belongs at the top of its range. The refit also found a peak in film thickness near 47 µm (p = 0.03). Only three runs at 45 µm supported it, so the confirmation tested it.

Design 3 repeated the recommended settings three times. The runs averaged 92.0 s against a prediction of 90.4 s, and beat the 60 µm run from design 2 by 7 s, which backs the thinner film.

1. Runs used in total

36 runs: 25 + 8 + 3. None failed.

2. Suggested optimum

91.7 s König hardness predicted by the model. Three confirmation runs at these settings gave 93.7, 90.1 and 90.9 s, mean 91.6 s. The 95% prediction interval for a mean of three runs was 87.1 to 96.3 s.

3. Recommended settings

X01, X03 and X08 set the hardness. The other nine sit at mid-range, where the confirmation runs were made. Within their ranges you can set them for cost or for other film properties.

4. Final model

Fitted to the 33 runs of designs 1 and 2, coded units, response in s:

Residual standard deviation 3.16 s (26 df), R² 0.969, adjusted R² 0.961. Refitting with the three confirmation runs moves no coefficient by more than 0.05 and leaves the optimum where it is.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-I-optimal.csv, design-2-I-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Only three factors control hardness: NCO:OH index, catalyst and cure temperature. I found them in three steps: screen all 12 factors, model the region around the best settings, then confirm. Hardness often peaks inside a process window, so the screening design had to detect curvature. A two-level design can’t.

The DSD screens 12 factors in 25 runs and keeps every main effect clear of interactions and curvature. Higher NCO:OH and higher cure temperature both raised hardness strongly. The center run gave 83.0 s, more than any other run, so the optimum lies inside the ranges. Catalyst had a small linear effect but strong curvature: runs at 200 ppm were harder than comparable runs at 0 or 400 ppm. The other nine factors all had main effects under 1 s, inside the noise. Second-order terms overlap in a DSD, so several models fitted. I kept the one with second-order terms only among the three active factors (effect heredity). It fitted as well as any other model of that size.

Design 2 added eight runs where that model put the optimum: NCO:OH mid or high, catalyst 100 to 300 ppm, cure temperature 80 to 100 °C, everything else at mid-range. I chose them for the best predictions in that region. Together with design 1, catalyst then had five levels and cure temperature four, so a curve that isn’t quadratic would show. More than eight runs improved the predictions only a little. The best run gave 96.7 s. Cubic terms added nothing, and no interaction with the nine other factors survived a correction for the 81 terms tested. Dropping the terms that showed no effect left the six-term model above.

The model puts the optimum at the top of the NCO:OH range, 220 ppm catalyst and 94 °C. Three repeats there averaged 91.6 s against 91.7 s predicted, with a standard deviation of 1.9 s. NCO:OH sits at its upper limit, so a higher index might give harder films, but that is outside the allowed range.

1. Runs used in total

36 runs: 25 + 8 + 3. None failed.

2. Suggested optimum

90.7 s König pendulum hardness. The 95 % confidence interval for the mean is 88.0 to 93.4 s. A single panel will read between about 83 and 98 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. Set the other nine for cost, appearance or other properties. The values above are mid-range, where the runs near the optimum were made.

4. Final model

Coded units, fitted to all 36 runs:

Residual standard deviation 3.57 s, R² = 0.956, 29 residual degrees of freedom.

5. Designs in order

6. Run sheets

• design-1-DSD.csv and design-1-DSD-results.csv
• design-2-D-optimal.csv and design-2-D-optimal-results.csv
• design-3-confirmation.csv and design-3-confirmation-results.csv

7. Summary

I worked in three steps: screen all 12 factors, model the ones that matter near the best corner, then repeat the best points.

A definitive screening design tests 12 factors in 25 runs, keeps main effects clear of two-factor interactions and shows curvature. Three factors stood out. Hardness rises with the NCO:OH index (+6.6 s per coded unit) and the cure temperature (+9.6 s), and the two reinforce each other (+3.5 s). The catalyst has a sharp optimum near the middle of its range (−13 s on the squared term). The other nine factors showed nothing. No effect was above 0.7 s, and adding all nine to the final model explains nothing more (F-test, p = 0.99). The centre run gave 79 s against a design average of 60 s, the first sign of the catalyst curvature.

The screening model put the best point at high index and high temperature, where the design had few runs. I added eight D-optimal runs in that corner, with five catalyst levels and the other nine factors at mid-range. I chose them so the model could also fit a catalyst curve that isn’t a plain parabola. They confirmed the curvature on both sides of the optimum. But two neighbouring points disagreed: 82.6 s at 200 ppm, where the model predicted 95 s, and 95.3 s at 300 ppm.

The three confirmation runs repeated those two points and added 250 ppm. They gave 84.0, 86.1 and 91.4 s. The low value at 200 ppm came back, the high one at 300 ppm didn’t. The repeats put the noise at about 4.7 s, close to the model’s 3.6 s. Fitted alone, the corner runs put the catalyst peak at 210 ppm, as the full model does, and extra terms for the curve’s shape didn’t improve the fit. The five runs near the optimum average 87.9 s, within noise of the 90.7 s prediction.

The top of the response is flat. Any catalyst level from 155 to 265 ppm, or any bake from 91 to 100 °C, costs less than 1 s. The NCO:OH index has no slack: each 0.02 below 1.3 costs about 1 s.

1. Runs used in total

37 runs. None failed.

2. Suggested optimum

90.5 s König hardness. The three confirmation runs averaged 90.0 s.

3. Recommended settings

Only X01, X03, X08 and X10 change the hardness. The other eight had no effect we could measure, so you can set them for cost or other film properties. We kept them at the centre, where we tested the model.

4. Final model

Fitted to the 34 runs of designs 1 and 2, in coded units:

Residual standard deviation 2.93 s, R² = 0.973 (adjusted 0.964). The prediction uses block = 1, because design 2 ran at the recommended settings of the eight minor factors. Adding the confirmation runs moves no coefficient by more than 0.1 and leaves the optimum at 93 °C and 200 ppm.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

The scripts that built the designs and fitted the models are in scripts/.

7. Summary

I screened all 12 factors, modelled the ones that mattered, then checked the prediction. The screen was a definitive screening design (DSD). It handles 12 factors in 25 runs, keeps main effects clear of two-factor interactions and shows curvature. Curvature turned out to matter: the catalyst has no linear effect, but hardness drops sharply at both 0 and 400 ppm. A two-level screen such as Plackett-Burman would have missed it. The screen also showed that hardness rises strongly with the NCO:OH index and with cure temperature, and that the temperature effect flattens towards 100 °C. Going from 30 to 60 µm film adds about 3 s. The other eight factors showed nothing.

Design 2 looked closer at the two factors with curvature. The NCO:OH index has a strong straight-line effect, so I fixed it at its limit of 1.3, with film thickness at 60 µm and the minor factors at their centre. I used a face-centred CCD in catalyst and cure temperature, but swapped the two 60 °C corners, which lie far from the optimum, for two repeat runs at 90 °C. The 9 runs confirmed both curves. Catalyst is best at 200 ppm. Cure temperature is best at 93 °C. The two factors interact, so the high NCO:OH index moves the best temperature up a little. Between 90 and 100 °C the predicted hardness changes by about 1 s, less than the noise, so the lower temperature also saves oven energy.

The three confirmation runs gave 92.2, 88.2 and 89.7 s, a mean of 90.0 s. The model predicted 90.5 s, with a 95 % interval of 86 to 95 s for the mean of three runs. The prediction held, so I stopped.

1. Runs used in total

37 runs, none failed.

2. Suggested optimum

About 90 s König pendulum hardness. The final model predicts 91.1 s (95 % confidence interval 88.9 to 93.4 s). The three confirmation runs averaged 89.1 s. For comparison, the middle of the window gave 78.6 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. The other nine stay mid-range, where the confirmation ran. In the screening, none of them changed hardness by more than about 1 s.

4. Final model

Fitted to the 23 runs at NCO:OH index 1.3 (X01 = +1). The nine factors without an effect are not in the model.

Residual standard deviation 2.91 s, R² 0.971, 17 residual degrees of freedom. Repeated runs give a pure-error standard deviation of 2.05 s.

The model’s peak is 92.2 s at 100 °C and 150 ppm. It stays above 90 s over roughly 92 to 100 °C and 90 to 220 ppm catalyst. The recommended point is 1.1 s below the peak, which is within the noise. I kept it because the confirmation tested it directly and it sits 5 °C below the oven limit.

5. Designs in order

6. Run sheets

All files are in projects/E1-hardness/.

7. Summary

I screened all 12 factors in one design, then spent every later run near the hardest films. A definitive screening design needs only 25 runs for 12 factors. It keeps main effects free of two-factor interactions and runs each factor at three levels, so it also shows curvature.

The screening found three active factors. Cure temperature (+10.3 s per coded unit) and NCO:OH index (+6.5 s) raise hardness, and they reinforce each other (+4.7 s). Catalyst has almost no linear effect but a strong peak mid-range. Each of the other nine factors moved hardness by less than 1 s, inside the noise. That includes cure time: a 20-minute cure gives the same hardness as 60 minutes. A quadratic model in the three active factors (R² 0.987) predicted 98 s at index 1.3, 100 °C and 200 ppm.

Design 2 tested that prediction and missed it: the two runs at the predicted optimum gave 86.5 and 88.5 s. At high temperature the catalyst peak is lower and sits at less catalyst, which a quadratic through three catalyst levels cannot show. Design 3 added five D-optimal runs near the hardest corner. It showed that temperature and catalyst trade off: the more catalyst, the lower the best temperature. Its local model put the peak at 94.5 °C and 140 ppm, with 94.7 s.

Design 4 tested 95 °C and 140 ppm three times and gave 91.7, 87.5 and 88.0 s. The two best earlier runs (94.4 and 93.3 s) had been lucky, so the top is flatter and lower than the local model said. Refitting all 23 runs at index 1.3 gives a plateau of 90 to 92 s. Further runs could only move the choice by about 1 s inside that plateau, less than the 2 s noise, so I stopped there.

1. Runs used in total

42 runs (25 + 14 + 3). None failed.

2. Suggested optimum

93.6 s König pendulum hardness (95 % confidence interval for the mean: 91.2 to 96.0 s). The three confirmation runs at these settings gave 92.5, 87.9 and 92.9 s, a mean of 91.1 s.

3. Recommended settings

4. Final model

Coded units, fitted on all 42 runs:

Residual standard deviation: 3.01 s (34 degrees of freedom). R² = 0.972, adjusted R² = 0.966. Six repeat runs (three centre points, three confirmation runs) give a pure-error standard deviation of 2.3 s. The lack-of-fit test is not significant (p = 0.31).

5. Designs in order

6. Run sheets

7. Summary

Three of the twelve factors control hardness: NCO:OH index, catalyst and cure temperature. Raise the index to 1.3 and cure at about 98 °C. The two work together, so the higher index pays off most in a hot oven. Catalyst is best at 200 ppm, and both 0 and 400 ppm cost about 15 s. Going from 10 to 25 % solvent may add about 2 s, but that evidence is weak. The other eight factors showed no effect in their ranges.

I worked in three steps: screen, find the source of the curvature, confirm. A definitive screening design came first, because it tests all twelve factors at three levels in 25 runs and keeps main effects clear of interactions and curvature. It found two main effects: cure temperature (+11 s per coded unit) and NCO:OH index (+6 s). The centre run gave 82.5 s, against an average of 60 s. So some factors were best inside their range, but one centre run can’t show which ones.

Next I added 14 runs. Twelve D-optimal runs estimate the curvature of every factor and the interactions among index, OH number, catalyst, cure temperature and cure time. Two repeat centre points came back at 81.3 and 79.0 s, so the high centre was real and the noise is only 2 to 3 s. The curvature came from catalyst and cure temperature, and the only interaction was index × cure temperature. The model predicted 95.8 s at index 1.3, catalyst 200 ppm and 98 °C.

Three confirmation runs there averaged 91.1 s, just below the lower 95 % limit for a three-run mean (91.2 to 100.5 s). Fitted models usually promise a little too much at their own optimum, so I refitted on all 42 runs. The new model predicts 93.6 s, and the settings stay the same. Its best cure temperature is 97 °C, but 98 °C gives less than 0.1 s less.

The index is at its upper limit of 1.3, and hardness is still rising there. A higher index would probably give a harder film, but it’s outside the allowed range.

1. Runs used in total

45 runs in three designs. None failed.

2. Suggested optimum

92 s König hardness (95 % confidence interval for the mean: 90.5 to 93.7 s).

3. Recommended settings

Only X01, X03, X08 and X10 change hardness. The other eight had no effect I could measure, so you can set them for cost or other film properties. I confirmed the optimum with them at their centre values.

4. Final model

Coded units, fitted to all 45 runs:

Residual standard deviation 2.94 s with 37 degrees of freedom. R² = 0.977 (adjusted 0.972). X03 stays in only because X03² is in the model.

5. Designs in order

6. Run sheets

• design-1-DSD.csv and design-1-DSD-results.csv
• design-2-I-optimal.csv and design-2-I-optimal-results.csv
• design-3-L9.csv and design-3-L9-results.csv

The analysis scripts are in analysis/.

7. Summary

Four of the 12 factors control hardness: cure temperature (X08) most, then NCO:OH index (X01), then catalyst level (X03). A thicker film (X10) adds about 2 s. The other eight showed no effect. I worked in three steps: screen all 12 factors, model the four that matter, then confirm the optimum and check the factors I had set aside.

Design 1 was a definitive screening design. It screens 12 factors in 25 runs and keeps main effects clear of two-factor interactions. Each factor runs at three levels, so it also shows curvature, which a two-level screen can’t. It found strong effects for cure temperature and NCO:OH index and a weak one for film thickness. The centre run gave 79 s against an average of 61 s, so the response clearly bends, but 25 runs couldn’t say which factors cause the bend.

Design 2 added 10 runs to fit a full quadratic model in X01, X03, X08 and X10, with the other eight factors at centre. An I-optimal search picked runs that work together with the 25 DSD runs and give the best average prediction, which is what an optimum search needs. Catalyst was in the set because the DSD hinted at catalyst curvature and catalyst often interacts with cure temperature. It has no linear effect but a sharp optimum at 200 ppm: 0 or 400 ppm both cost about 12 s. A two-level screen would have missed it. Cure temperature peaks at 93 to 94 °C, and it pays off more at a high NCO:OH index.

The fit still hinted at curvature in booth humidity (X11) and OH number (X02). Design 3 tested this at the predicted optimum, where it matters. An L9 array set X02, X06, X11 and X12 to three levels each in 9 runs, and one more run repeated the optimum. None of the four had an effect. Designs 1 and 2 predicted 92.1 s at the optimum, and the 10 runs averaged 92.1 s. Across all 45 runs, no main effect of the eight held factors is significant, and their curvature terms aren’t significant together (p = 0.08). So the final model leaves them out.

1. Runs used in total

33 runs sent to the lab. None failed.

2. Suggested optimum

About 92 s König hardness. The confirmation runs near these settings gave 84 to 92 s, so expect 88 to 92 s in practice.

3. Recommended settings

“Any” means the factor had no measurable effect on hardness. Pick whatever suits cost or other properties.

4. Final model

Hardness (s) in coded units, fitted on all 33 runs:

• Intercept: 80.7
• X01: +6.97
• X08: +10.67
• X03²: −13.9
• X08²: −11.51
• X01·X08: +3.32

Residual standard deviation 3.09 s, R² 0.977.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

7. Summary

I treated 12 factors with a definitive screening design (25 runs). It finds the few factors that matter and also shows curvature, which a plain two-level screen would miss.

The screen showed that only NCO index (X01), cure temperature (X08) and catalyst (X03) matter. The other nine factors had no effect. The centre run was far above the average of the others, so X03 and X08 have a peak inside their ranges. X01 and X08 also interact: a higher index gains more at higher cure temperature.

The second design had 8 runs near the predicted best point, with three repeats of it. It checked the model and gave a clean estimate of noise. Results fell 2 to 4 s below the first prediction, so I refitted with all 33 runs. The catalyst peak sits at about 200 ppm. The hardness gain from the NCO index continues to the upper limit, so I set it to 1.3. Cure temperature peaks at about 92 °C.

1. Runs used in total

36 runs sent to the lab. None failed.

2. Suggested optimum

About 92 s König hardness. The final model predicts 94.7 s. Three confirmation runs measured 91.7 s on average (91.7, 93.0, 90.5).

3. Recommended settings

4. Final model

Fitted on the 17 runs of design 2, coded units. The factors not listed were held at the settings above.

hardness = 81.87 + 6.52·X01 + 9.19·X08 − 1.94·X11 + 5.12·X01·X08 − 11.78·X08²

Residual standard deviation 2.64 s, R² 0.965.

5. Designs in order

6. Run sheets

• design-1-fractional-factorial.csv, design-1-fractional-factorial-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

With 12 factors and a noisy response, I first ran a 16-run screening design. It asks for the fewest runs that still give every main effect. It showed that cure temperature (X08) and NCO:OH index (X01) matter most. Humidity (X11) came a distant third. The other nine factors had small effects, at or below the noise level. I did not chase them.

The screening design mixes main effects with two-factor interactions, so the two big effects could have hidden an interaction. I gave the three leading factors a face-centred CCD to settle this and to look for curvature. The other factors were held at the levels the screening favoured. The CCD showed curvature in cure temperature (hardness peaks near 92 °C, not at the top of the range) and a real X01·X08 interaction. A higher index pays off most when the cure is hot. Humidity helps slightly when low.

Three confirmation runs at the predicted optimum gave about 92 s, close to the predicted 94.7 s. The catalyst, cure time and the weak factors were not studied in detail. If more hardness is needed, test cure time and catalyst next, around this optimum.

1. Runs used in total

43 runs (20 + 20 + 3). None failed.

2. Suggested optimum

About 91 s König hardness at the recommended settings. The three confirmation runs gave 85.4, 89.6 and 95.5 s (mean 90.2 s).

3. Recommended settings

Only X01 and X08 matter. The other ten factors are at mid-range because they showed no clear effect.

4. Final model

Coded units, fitted on designs 2 and 3 (23 runs):

hardness = 79.02 + 5.80·X01 + 9.73·X08 + 4.47·X01·X08 − 7.97·X08²

Residual standard deviation 3.02 s, R² 0.954.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I screened all 12 factors with a 20-run Plackett-Burman design. It gives clean main effects for 12 factors and leaves degrees of freedom to estimate noise. Cure temperature (X08) and NCO:OH index (X01) stood out. Film thickness (X10) and polyol OH number (X02) were weak candidates.

Next I ran a 20-run D-optimal design for a quadratic model in X01, X08, X10 and X02, with the other factors at centre. It showed a strong X01×X08 interaction and a curved X08 effect, with a peak near the top of the cure temperature range. X10 and X02 had no significant effect.

I refitted a model with only X01 and X08 terms and set X01 at its high level. The curve is flat near X08 = 0.8 to 1.0, so I chose 0.9. Three confirmation runs there averaged 90.2 s, close to the 91 s predicted.

The other ten factors were not tested beyond screening, so interactions among them are not ruled out. The confirmation runs were at centre values for them.

1. Runs used in total

45 runs (25 + 17 + 3). No run failed.

2. Suggested optimum

About 91 s König hardness (model prediction). The three confirmation runs gave 86.7, 93.0 and 89.6 s, mean 89.7 s.

3. Recommended settings

4. Final model

Fitted on the 17 central composite runs (other factors at centre), coded units:

hardness = 80.11 + 5.49·X01 + 1.95·X03 + 9.56·X08 − 12.73·X03² − 8.98·X08² + 4.66·X01·X08

Residual standard deviation 3.4 s, R² = 0.96.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors, noisy data and likely curvature led me to a definitive screening design. It needs 25 runs and estimates main effects and quadratic effects.

The screen showed that cure temperature (X08) and NCO:OH index (X01) raise hardness. The centre run (79 s) was far above the design average (60 s), so some factor had a curved effect. Catalyst (X03) showed the strongest quadratic term. The other nine factors had effects well inside the noise, so I fixed them at their centre values.

A face-centred CCD on X01, X03 and X08 (17 runs, three centre points) fitted well. Catalyst has an optimum near the middle of its range. Cure temperature rises to about 96 °C and then levels off. The X01·X08 interaction means a high index pays off most at high temperature. The optimum sits at X01 at its upper limit, catalyst near centre and cure temperature just below the upper limit.

Three confirmation runs at that point averaged 89.7 s, close to the 91 s predicted. I stopped there.

1. Runs used in total

45 runs, none failed (25 + 17 + 3).

2. Suggested optimum

About 92 s König hardness. The 3 confirmation runs gave 85.4, 94.2 and 90.0 s (mean 89.9), in line with this value given a noise of about 3 s.

3. Recommended settings

Factors X02, X04 to X07 and X09 to X12 had no effect, so any value in range works. Use the centre values or whatever is cheapest.

4. Final model

Coded units, fitted to all 45 runs:

hardness = 79.37 + 6.39·X01 + 1.50·X03 + 10.99·X08 − 13.08·X03² − 8.93·X08² + 3.89·X01·X08

Residual standard deviation 2.98 s, R² = 0.970.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I started with a 25-run definitive screening design for all 12 factors. It tests every factor at three levels, so it can show curvature as well as main effects. The centre run was far above the average of the others, which meant strong curvature. Only NCO:OH index, cure temperature and catalyst mattered. The catalyst had a sharp optimum near the middle of its range, and cure temperature was curved too.

The other nine factors showed nothing, so I dropped them. I ran a 17-run face-centred central composite design on the three active factors. It confirmed the curvature in catalyst and cure temperature, and it found an interaction: a higher NCO:OH index helps more at a high cure temperature. The best point sits at the upper limit for index and cure temperature, and near the middle for catalyst.

Three runs at the predicted optimum closed the study. They agreed with the model.

1. Runs used in total

45 runs (22 + 19 + 4). No run failed.

2. Suggested optimum

Hardness of about 91.5 s at the recommended settings. The four confirmation runs measured 92.3, 90.4, 91.0 and 94.3 s.

3. Recommended settings

4. Final model

Coded units, fitted to the 19 CCD runs and the 4 confirmation runs:

hardness = 81.09 + 5.95·X01 + 9.14·X08 + 4.88·X01·X08 − 11.01·X08²

Residual standard deviation 2.46 s, R² 0.963.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I screened all 12 factors with a 20-run Plackett-Burman design plus two centre runs. That is the cheapest way to see main effects for this many factors. Only X01 and X08 were clearly active. The centre runs averaged about 80 s against 56 s for the corners, so the response is strongly curved.

I then ran a face-centred CCD (a half-fraction cube, axial and centre runs) on X01, X08, X06 and X10, the factors that looked most active. The other factors stayed at centre. X06 and X10 did not matter. The X06·X10 term in the full fit is aliased with X01·X08 in the half fraction, so I credited it to X01·X08, which has strong main effects. X08 has a clear peak inside the range, near +0.65, while X01 is best at its upper limit.

I fitted the reduced model, set X01 to +1 and X08 to +0.65, and ran four confirmation runs there. They agreed with the prediction. Catalyst, flow additive and the other factors were not tested beyond the screening, so they sit at mid-range, and small gains from X03 or X12 may remain.

1. Runs used in total

46 runs sent to the lab. None failed.

2. Suggested optimum

About 91.7 s König hardness. The three confirmation runs averaged 90.7 s.

3. Recommended settings

4. Final model

Coded units, fitted on the 18 CCD runs (X02, X04–X07, X09–X12 held at 0):

hardness = 78.89 + 5.93·X01 + 10.30·X08 + 2.92·X01·X08 − 13.85·X03² − 6.37·X08²

Residual standard deviation 3.44 s, R² = 0.954.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

With 12 factors and noisy data, I first ran a 25-run definitive screening design. It finds main effects and also shows curvature and the strongest interactions. Cure temperature (X08) was clearly the biggest effect. NCO:OH index (X01) came next. Catalyst (X03) had no linear effect but strong curvature. The other nine factors showed no effect above the noise.

I then ran an 18-run face-centred CCD on X01, X03 and X08, with the other nine at their centres. The face-centred form keeps every run inside the allowed range. It confirmed the three effects and showed a hardness peak at mid catalyst. Hardness rises with the index and with cure temperature. The two interact a little, and the cure-temperature gain bends over near the top.

The model puts the optimum at index 1.3, 100 °C and 200 ppm catalyst. Three runs at those settings gave 90.7 s on average, close to the 91.7 s predicted. Cure time and film thickness had no clear effect in screening, so I left them at the centre.

1. Runs used in total

54 runs (25 + 26 + 3). No run failed.

2. Suggested optimum

About 89 s König hardness (model prediction 89.4 s). The three confirmation runs gave 87.2, 90.9 and 86.8 s, mean 88.3 s.

3. Recommended settings

Factors that do not matter can sit anywhere in their range. Mid values are the safe choice.

4. Final model

Coded units, fitted to the 26 D-optimal runs and the 3 confirmation runs (29 runs):

hardness = 78.99 + 5.52·X01 + 0.73·X03 + 9.33·X08 + 3.34·X01·X08 − 13.61·X03² − 8.28·X08²

Residual standard deviation 3.35 s, R² 0.954.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

I used a definitive screening design first, because 12 factors are many and hardness probably bends. It needs only 25 runs and can show curvature. It showed that cure temperature (X08) and NCO:OH index (X01) matter, and that the response curves with temperature. The centre run (77 s) sat far above the average of the other runs (60 s), which meant strong curvature somewhere. The main-effects fit left a large residual (9 s), so the screening result was not the whole story.

I then ran a 26-run D-optimal design with a quadratic model in five factors: X01, X08, and X03, X09, X10 (catalyst, cure time, film thickness, kept as the most plausible partners for binder index and cure temperature). It found a large catalyst effect that the screening design had missed: hardness peaks near mid catalyst level and falls at both ends, so a linear screening could not see it. Cure time and film thickness did not matter once the other terms were in. X01 and X08 also interact: a higher index moves the best cure temperature up a little.

The optimum from the model is X01 at its top (1.3), catalyst near the middle, cure temperature at 95 °C. Three confirmation runs there averaged 88.3 s, in line with the 89.4 s prediction given noise of about 3 s. The other seven factors showed no effect in screening. I did not test them again, so a small effect on them could still be hidden in the noise.

1. Runs used in total

54 runs (25 + 26 + 3). No run failed.

2. Suggested optimum

About 92 s König hardness (model prediction 91.7 s). The three confirmation runs gave 92.6, 97.4 and 90.9 s, mean 93.7 s.

3. Recommended settings

4. Final model

Coded units, fitted on the 26 CCD runs:

hardness = 80.64 + 7.26·X01 + 1.34·X03 + 8.78·X08 + 3.90·X01·X08 + 1.35·X03·X08 − 12.77·X03² − 10.86·X08²

Residual standard deviation 2.26 s, R² 0.983.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

With 12 factors and curvature likely, I started with a 25-run definitive screening design. It sorts out main effects, quadratic effects and some interactions in few runs. It showed that cure temperature (X08) and NCO:OH index (X01) matter, and that catalyst (X03) and cure temperature act in a curved way. The other eight factors showed nothing.

I then ran a 26-run face-centered CCD on X01, X03, X08 and X10, with the other eight factors at their middle. X10 was borderline in screening, and the CCD showed it has no effect. Hardness rises with the index and with cure temperature, the two reinforce each other, and both catalyst and cure temperature have a peak inside the range. The index peak lies at its upper limit, so the optimum sits on that edge.

Three confirmation runs at the predicted optimum averaged 93.7 s against 91.7 s predicted, so the model holds. The eight inactive factors were left at their middle; any value in range should do.

1. Runs used in total

57 runs (25 + 29 + 3). No run failed.

2. Suggested optimum

About 91 s König hardness. The three confirmation runs gave 88.8, 85.7 and 90.0 s (mean 88.2 s).

3. Recommended settings

Factors X02, X04, X05, X06, X07, X09, X10, X11 and X12 had no effect I could separate from noise, so they stay at mid-range.

4. Final model

Coded units, fitted to the 29 CCD runs and the 3 confirmation runs.

hardness = 80.88 + 5.57·X01 + 1.18·X03 + 9.34·X08 − 12.92·X03² − 10.70·X08² + 4.03·X01·X08 + 1.74·X03·X08

Residual standard deviation 3.03 s, R² = 0.971.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors and a noisy response called for a screening design that also shows curvature. I chose a definitive screening design: 25 runs, with main effects clear of two-factor interactions and every factor at three levels.

The DSD showed that cure temperature (X08) and NCO:OH index (X01) raise hardness. The centre run (87 s) sat far above the linear fit (60 s), so strong curvature was present. Catalyst (X03) and X08 curved most. Flow additive (X04) and film thickness (X10) also looked active, and X09 and X06 were borderline.

I then ran a face-centered CCD on X01, X03, X04, X08 and X10, with the other seven factors at mid-range. It dropped X04 and X10. The model that remained had a hardness peak in catalyst near the middle of its range, a cure-temperature optimum inside the range, and a gain from raising the NCO:OH index that grows with temperature (the X01·X08 term). X09 and X06 were not carried forward. Their DSD effects were small, and checking them would have cost further runs. That is the main risk.

The confirmation runs at the predicted optimum averaged 88 s against 92 s predicted. That is within the noise. I refitted with them and report 91 s.

1. Runs used in total

28 runs, none failed: 25 in the screening design and 3 in the confirmation.

2. Suggested optimum

About 92 s König pendulum hardness at the recommended settings (95% interval for the mean: 89 to 95 s). The three confirmation runs at these settings averaged 89.5 s.

3. Recommended settings

• X01 NCO:OH index: 1.3 (+1)
• X02 Polyol OH number: 160 mg KOH/g (0)
• X03 Catalyst (DBTDL): 200 ppm (0)
• X04 Flow additive: 0.55 % (0)
• X05 Matting agent: 2 % (0)
• X06 Solvent dilution: 17.5 % (0)
• X07 Flash-off time: 12.5 min (0)
• X08 Cure temperature: 91 °C (+0.55)
• X09 Cure time: 40 min (0)
• X10 Dry film thickness: 45 µm (0)
• X11 Booth humidity: 30 % r.h. (-1)
• X12 Substrate temperature: 24 °C (0)

Catalyst sits at 0 because that is its peak. The other factors at 0 showed no effect, so they stay at the midpoint, where the confirmation runs measured them.

4. Final model

Coded units, fitted to all 28 runs.

Residual standard deviation: 3.11 s (20 degrees of freedom). R²: 0.974 (adjusted 0.964).

X03 stays in the model because its square is large, but its own coefficient is not different from zero.

5. Designs in order

6. Run sheets

• design-1-DSD.csv
• design-1-DSD-results.csv
• design-2-confirmation.csv
• design-2-confirmation-results.csv

7. Summary

Twelve factors is too many for a full factorial, and I expected only a few to matter. So I ran a 25-run definitive screening design (DSD). It tests every factor at three levels and estimates all 12 main effects free of curvature and interactions. If only three or four factors matter, it also holds a full quadratic model for them, so I needed no separate optimization design.

Three factors are active. Cure temperature is the biggest: hardness rises with temperature, peaks at about 91 to 92 °C and falls above that. A higher NCO:OH index helps more the hotter the cure, so 1.3 is best. Catalyst has no straight-line effect but a clear peak at 200 ppm, and 0 or 400 ppm each cost about 10 s. These terms explain about 97% of the variation. Booth humidity may lower hardness by about 1 s per coded unit (p = 0.12). That evidence is weak, but low humidity costs nothing, so I set it low. The other eight factors (polyol OH number, flow additive, matting agent, solvent dilution, flash-off time, cure time, film thickness and substrate temperature) showed no effect above noise. Each was estimated at 1.5 s or less over its whole range.

The DSD alone predicted 94 to 96 s at the optimum. Three confirmation runs there gave 92.6, 85.6 and 90.4 s. That fits the noise, since a single run varies by about 3 s. Refitting on all 28 runs lowered the prediction to 92 s and left the settings the same. I stopped there. Every model I tried put the optimum at the same index and catalyst and at nearly the same temperature, and the response is flat near the top, so more runs would not change the advice. No run failed, so the study says nothing about where failures occur.

1. Runs used in total

32 runs went to the lab (25 + 7). None failed.

2. Suggested optimum

93.4 s König pendulum hardness (95% confidence interval for the mean: 91.2 to 95.5 s).

3. Recommended settings

• X01 NCO:OH index: 1.3 (coded +1)
• X02 Polyol OH number: 160 mg KOH/g (coded 0)
• X03 Catalyst (DBTDL): 200 ppm (coded 0)
• X04 Flow additive: 0.55 % (coded 0)
• X05 Matting agent: 2 % (coded 0)
• X06 Solvent dilution: 17.5 % (coded 0)
• X07 Flash-off time: 12.5 min (coded 0)
• X08 Cure temperature: 95.5 °C (coded +0.77)
• X09 Cure time: 40 min (coded 0)
• X10 Dry film thickness: 45 µm (coded 0)
• X11 Booth humidity: 50 % r.h. (coded 0)
• X12 Substrate temperature: 24 °C (coded 0)

4. Final model

Coded units, fitted to all 32 runs:

Residual standard deviation 2.95 s (26 degrees of freedom), R² 0.977.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-augmentation.csv, design-2-augmentation-results.csv

7. Summary

With 12 factors, noisy measurements and likely curvature, I started with a 25-run definitive screening design. It takes only 2 × 12 + 1 runs, keeps the main effects clear of two-factor interactions and shows curvature. All 25 runs were measured.

Three factors control hardness. Cure temperature (X08) and NCO:OH index (X01) both raise it, and they work together: at 60 °C the index does nothing, at 100 °C the high index is worth 21 s over the low one. Cure temperature also levels off, with a peak near 95 °C at the high index. Catalyst (X03) has no straight-line effect but a sharp peak in the middle: 0 and 400 ppm both cost about 17 s against 200 ppm. These five terms explain 96% of the variation in the screening runs.

None of the other nine factors stood out. Their main effects ran from -1.0 to +1.3 s per coded unit with a standard error of 0.6 s, so I kept them at mid-range. The largest, substrate temperature, is about what noise alone gives when nine factors are tested.

The second design had 7 runs near the predicted optimum, with the nine quiet factors at mid-range. It doubled as the confirmation. The screening model predicted 93.7 s at NCO:OH +1, catalyst 0 and cure temperature +0.8, and the two runs there gave 91.4 and 99.9 s. That gap is large, but noise of 3 s produces it about one time in twenty. The other five runs fell within 2.4 s of the model, so it needed no new terms. I stopped at 32 runs.

The optimum sits at the upper limit of the NCO:OH range, so the data cannot say whether a higher index would help.

1. Runs used in total

36 runs, none failed (25 + 8 + 3).

2. Suggested optimum

About 93 s König hardness. The final model predicts 93.2 s at the recommended settings. Three confirmation runs measured 92.7, 92.7 and 92.4 s (mean 92.6 s).

3. Recommended settings

The eight factors at 0 had no effect on hardness. They stay at the middle of their range, as in all runs after the screening.

4. Final model

Coded units, fitted to all 36 runs:

hardness = 80.37 + 5.63 X01 + 1.49 X03 + 9.07 X08 - 2.04 X11 + 2.40 X01² - 15.19 X03² - 9.53 X08² - 2.29 X11² + 2.16 X01·X03 + 3.42 X01·X08

Residual standard deviation 2.59 s (25 degrees of freedom), R² = 0.984.

5. Designs in order

6. Run sheets

7. Summary

I screened all 12 factors, refined around the best region, then confirmed. The screening was a 25-run definitive screening design. It keeps the main effects clear of two-factor interactions and tests for curvature, and both mattered here because index, catalyst and cure temperature are known to interact.

Four factors matter. Cure temperature (X08) and NCO:OH index (X01) have the largest effects. Catalyst (X03) has a sharp peak near the middle of its range, and both ends of the range cost a lot of hardness. Booth humidity (X11) has a small negative effect. The other eight factors showed no effect. The centre run (81.1 s) sat far above the average of all screening runs (58.7 s), which points to this curvature.

The two best screening models disagreed on how low humidity should go. So design 2 added 8 D-optimal runs in the four active factors, near the predicted optimum, with the other eight factors at the middle. They showed the index is best at its upper limit, the cure temperature peaks inside the range (about 93 °C), and humidity is best at a low to middle level.

Design 3 repeated the recommended settings three times. The mean of 92.6 s agrees with the model’s 93.2 s, so I stopped there.

1. Runs used in total

36 runs, and none failed: 25 in design 1, 8 in design 2 and 3 in design 3.

2. Suggested optimum

About 91 s König pendulum hardness (model prediction 91.1 s, standard error 0.7 s). The three confirmation runs at these settings gave 90.95, 88.09 and 93.30 s, mean 90.8 s.

3. Recommended settings

The nine factors not in the model are held at mid level, where all runs of designs 2 and 3 were made.

4. Final model

Fitted to all 36 runs, coded units, hardness in seconds:

Residual standard deviation 1.90 s (26 degrees of freedom), R² 0.991.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Three factors control hardness: cure temperature, catalyst level and NCO:OH index. Flash-off time and cure time may matter a little, and the other seven factors show no effect we can detect. Cure temperature and catalyst both peak inside their range. At the recommended settings, a 60 °C cure instead of 97 °C costs about 27 s, and moving the catalyst 200 ppm from the middle costs about 14 s. Hardness keeps rising with the NCO:OH index up to its limit of 1.3, and more so at a high cure temperature.

With 12 factors and no prior knowledge, I started with a 25-run definitive screening design. It estimates every main effect free of two-factor interactions and it shows curvature, which I expected because cure temperature and catalyst usually have an optimum. The center run was the second highest of the 25, which pointed the same way. A full quadratic model in the three drivers explained 98.5 % of the variation (residual standard deviation 2.1 s). Flash-off time and cure time came out weak, about 1 s per coded unit, and the other seven factors at 0.4 s or less.

Design 2 added 8 runs, chosen by a D-optimal search, near the expected optimum. They varied the three drivers plus flash-off and cure time, with the other factors at mid level. The design 1 model predicted them with a root mean square error of 2.5 s (largest miss 5.9 s), close to the noise level. The two weak effects shrank to about 0.7 s per coded unit with the same sign, still borderline (p 0.04 to 0.08). I set both at their low level. If they are real, that adds about 1.3 s over mid level. If not, it costs nothing and shortens the process.

Design 3 repeated the predicted optimum three times. The mean of 90.8 s matches the prediction of 91.1 s, and the spread (standard deviation 2.6 s) is in line with the noise in the fits. I stopped there. The open question is the two weak factors, and testing a gain of about 1 s against 2 s of noise would take many more runs.

1. Runs used in total

36 runs (25 + 8 + 3). None failed.

2. Suggested optimum

About 90 s König pendulum hardness. The model predicts 89.9 s at the recommended settings (standard error 1.0 s). Three confirmation runs at these settings gave 90.4, 93.0 and 91.4 s, a mean of 91.6 s.

3. Recommended settings

Only X01, X03 and X08 influence hardness. The other nine are held at mid-range, where the confirmation runs were made.

• X01 NCO:OH index: 1.3 (coded +1)
• X02 Polyol OH number: 160 mg KOH/g (coded 0)
• X03 Catalyst (DBTDL): 200 ppm (coded 0)
• X04 Flow additive: 0.55 % (coded 0)
• X05 Matting agent: 2 % (coded 0)
• X06 Solvent dilution: 17.5 % (coded 0)
• X07 Flash-off time: 12.5 min (coded 0)
• X08 Cure temperature: 98 °C (coded +0.9)
• X09 Cure time: 40 min (coded 0)
• X10 Dry film thickness: 45 µm (coded 0)
• X11 Booth humidity: 50 % r.h. (coded 0)
• X12 Substrate temperature: 24 °C (coded 0)

4. Final model

Fitted to the 33 runs of designs 1 and 2, in coded units. The nine inactive factors are left out.

Residual standard deviation 2.47 s (25 degrees of freedom). R² 0.982.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors and a noisy response, so I screened first, then fitted a small response surface on the factors that mattered, then confirmed the optimum with replicate runs.

Design 1 was a 25-run definitive screening design. It keeps the main effects clear of two-factor interactions and also shows curvature. A main-effects fit alone found cure temperature (+10.4 s per coded unit) and NCO:OH index (+5.8 s), but left most of the variation unexplained (R² 0.67). The centre run gave 80.4 s while the other 24 runs averaged 59.1 s, which pointed to curvature. Adding squared and interaction terms for the three leading factors raised R² to 0.99. Catalyst turned out to matter through curvature: hardness peaks near 200 ppm and falls by about 12 s at either end. Cure temperature also peaks inside the range, and NCO:OH works best at its upper limit, more so at high cure temperature. The other nine factors moved hardness by less than 1 s per coded unit. X02 (+0.8) and X06 (+0.7) were the largest, and neither was significant.

Design 2 was eight D-optimal runs. Only the three active factors varied, near the predicted peak. The design 1 model had over-predicted some of these runs by up to 7 s, so the refit moved the peak from 91.8 s at coded cure temperature +0.76 to 90.0 s at +0.92.

Design 3 was three replicates at the recommended settings. They averaged 91.6 s, within error of the model’s 89.9 s. The all-mid recipe gave 80.4 s in design 1, so the new settings add about 10 s. Refitting with the confirmation runs changed no coefficient by more than 0.4.

1. Runs used in total

36 runs, and none failed.

2. Suggested optimum

König pendulum hardness of about 92 s (model: 91.7 s, 95% interval for the mean 89.9 to 93.6 s). The three confirmation runs at these settings gave 93.7, 89.8 and 93.9 s, a mean of 92.4 s.

3. Recommended settings

4. Final model

Coded units, fitted to all 36 runs:

hardness = 81.33 + 5.21 X01 + 0.98 X03 + 9.66 X08 - 15.18 X03² - 9.44 X08² + 4.34 X01·X08

Residual standard deviation 2.92 s (29 degrees of freedom), R² 0.975.

5. Designs in order

6. Run sheets

7. Summary

Three of the 12 factors move hardness: cure temperature (X08), NCO:OH index (X01) and catalyst level (X03). The other nine showed nothing I could tell from the noise, which is about 3 s, so I left them at their mid-points.

I began with a 25-run definitive screening design on all 12 factors. It tests each factor at three levels, so it shows curvature as well as main effects, and it keeps the main effects clear of interactions. I chose it because binder, catalyst and cure temperature often interact, and I could not rule out curvature. Every run was measured. Cure temperature and NCO:OH index raised hardness. The center point gave 82 s, higher than most corner runs, so some factors peak in the middle of their range. A model search then picked catalyst as the third factor, with a sharp peak near 200 ppm, and showed that a higher index and a higher cure temperature help each other.

The second design was eight runs chosen by an I-optimal search, placed near the predicted best region. I varied cure time and film thickness in them too, because both had borderline signals in design 1. The design 1 model predicted these eight runs to within about 3 s, which is the noise level, so the model held. Cure time and film thickness each moved hardness by about 1 s, too little to act on, so I kept them at the center.

The last design was three repeat runs at the predicted optimum. Their mean of 92.4 s matches the 91.4 s the model predicted before they were added. The final model uses all 36 runs.

The NCO:OH index sits at its upper limit of 1.3 and hardness was still rising there, so the allowed range sets this setting. Cure temperature and catalyst both peak inside their ranges, near 95 °C and 200 ppm. Moving away from either one lowers hardness.

1. Runs used in total

37 runs, none failed (25 + 9 + 3).

2. Suggested optimum

König hardness of about 92.4 s (95% interval 90.7 to 94.1 s). The three confirmation runs at the recommended settings gave 91.6, 92.5 and 92.3 s, a mean of 92.1 s. The screening center point gave 79.6 s.

3. Recommended settings

Only X01, X03 and X08 affect hardness. The other nine can stay anywhere in their range, and I left them at the midpoint.

4. Final model

Fitted to all 37 runs, in coded units, response in seconds:

Residual standard deviation 2.84 s (30 degrees of freedom), R² 0.980. The model’s maximum inside the ranges is X01 = +1, X03 = +0.02, X08 = +0.94. I used X03 = 0 and X08 = +0.9 because the curve is flat there: anything from X08 = +0.7 to +1.0 (94 to 100 °C) predicts 92.0 to 92.4 s.

5. Designs in order

6. Run sheets

7. Summary

I screened all 12 factors in one design, then spent a small second design on the few that matter, then confirmed the result.

Design 1 was a 25-run definitive screening design. It estimates every main effect free of interactions and curvature, and it puts each factor on three levels, so strong curvature shows up in the same runs. It found three active factors. Cure temperature (X08) raises hardness most, about 10.5 s per coded unit, with a flattening curve. NCO:OH index (X01) raises it by about 6.5 s, and the two interact: a high index pays off more at a high cure temperature. Catalyst (X03) has no linear effect but a sharp peak in the middle. Both 0 ppm and 400 ppm lose about 15 s against 200 ppm. None of the other nine factors moved hardness by more than 1 s per coded unit, and no interaction with them stood out.

Design 2 had 9 runs in the region where the screening model put the optimum: index 0.5 to 1.0, cure temperature 0.5 to 1.0, catalyst at 0 and ±0.5. The other nine factors stayed at their midpoint. The screening model predicted these runs to within 5.6 s. The runs showed that the catalyst peak sits at the midpoint, that no lower index beat an index of 1.0, and that hardness changes by only about 1.5 s between cure temperatures of +0.5 and +1.0.

Design 3 repeated the recommended settings three times. The mean of 92.1 s matches the predicted 92.4 s, and the three runs agree within 1 s.

The model’s residual standard deviation (2.8 s) is larger than the spread of the three repeats (0.5 s). I did not chase this, because the prediction held at the optimum. The exact cure temperature matters little: any value from 94 to 100 °C works, with the index at 1.3 and the catalyst at 200 ppm.

1. Runs used in total

37 runs, all measured. No run failed.

2. Suggested optimum

89.8 s König pendulum hardness (model prediction, standard error 0.9 s). Two confirmation runs near these settings gave 88.9 s and 90.0 s.

3. Recommended settings

X02 and X04 to X07, X09 to X12 have no detectable effect, so they stay at mid-range.

4. Final model

Fitted to all 37 runs, coded units, hardness in seconds.

Residual standard deviation 2.68 s (30 degrees of freedom), R² = 0.977.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

With 12 factors and no idea which mattered, I started with a 25-run definitive screening design. It ranks the main effects and shows curvature in the same runs, and curvature was likely for catalyst and cure conditions. Three factors stood out: cure temperature, NCO:OH index and catalyst. Catalyst peaks near the middle of its range, and both ends lose about 14 s. A higher cure temperature helps with diminishing returns, and helps more at a high NCO:OH index. The other nine factors moved hardness by less than 0.8 s per coded unit (p above 0.2), which is within noise.

A model in the three active factors fit the 25 runs well. It put the optimum at the top of the NCO:OH range, near mid catalyst, at a cure temperature around 94 °C. For the second step I added 10 D-optimal runs around that point to check the curve shapes and the peak position. Three replicated pairs gave a noise standard deviation of 2.9 s, close to the model’s residual of 2.7 s, so the model shows no lack of fit. Cubic terms were not significant.

Two confirmation runs at NCO:OH 1.3, 220 ppm and 93 °C gave 88.9 s and 90.0 s, against a prediction of 89.7 s. I then refit all 37 runs and dropped the terms that were not significant (X01², X01·X03, X03·X08). The optimum sits at the upper limit of the NCO:OH range, so a higher index might harden the film more, but it is outside the allowed window.

1. Runs used in total

41 runs went to the lab (25 + 10 + 6). None failed.

2. Suggested optimum

König pendulum hardness of 89.6 s at the recommended settings (95% confidence interval for the mean: 87.6 to 91.5 s). Three repeat runs at these settings measured 89.4, 88.2 and 90.3 s, mean 89.3 s.

3. Recommended settings

X02, X05, X06, X07, X09, X10, X11 and X12 had no effect I could detect, so they sit at mid-range. Any value in their range should do.

4. Final model

Coded units, fitted to all 41 runs:

Residual standard deviation 2.83 s (33 degrees of freedom), R² 0.972.

5. Designs in order

6. Run sheets

7. Summary

Hardness depends on four of the twelve factors: cure temperature, NCO:OH index, catalyst and flow additive. I screened all twelve, fitted a curved model on the four, then checked the predicted best point in the lab.

Design 1 was a 25-run definitive screening design. Twelve factors are too many for a full model, and this design estimates main effects and curvature together. Cure temperature and NCO:OH index stood out. Hardness peaks inside the cure temperature range, so a hotter cure is not a harder one. Catalyst also peaks near the middle of its range. Flow additive had a small negative effect. The other eight factors showed nothing, alone or combined with the four.

Design 2 added 10 D-optimal runs. The screening runs could not pin down a full curved model in four factors, so these runs filled the gaps. They varied the four active factors and substrate temperature, which turned out to have no effect, and held the rest at mid-range. The fit put the best point at the highest NCO:OH index, the lowest flow additive, about 210 ppm catalyst and about 96 °C.

Design 3 had six runs: three repeats at that point and three neighbours. The repeats averaged 89.3 s against 90.2 s predicted before the runs, with a standard deviation of 1.0 s. A lower cure temperature (0.4 coded, 85.9 s) and less catalyst (100 ppm, 83.6 s) gave softer films, as the model says. More catalyst (300 ppm, 90.9 s) was as hard as the optimum, so the catalyst peak may be broad.

The residual standard deviation is 2.8 s, so differences under about 3 s between settings are not reliable. The results hold only inside the tested ranges.

1. Runs used in total

48 runs. None failed.

2. Suggested optimum

About 92 s König pendulum hardness. The final model gives 92.1 s at the recommended settings (95% interval for the mean: 90.4 to 93.8 s). Three confirmation runs at these settings measured 94.2, 87.8 and 90.5 s (mean 90.8 s).

3. Recommended settings

4. Final model

Coded units, fitted to all 48 runs. Factors not listed have no term.

Residual standard deviation: 2.93 s (40 degrees of freedom). R² = 0.977 (adjusted 0.972).

5. Designs in order

6. Run sheets

7. Summary

Twelve factors and noisy measurements called for three stages: screen, zoom in, confirm. I screened with a definitive screening design. It covers 12 factors in 25 runs, keeps the main effects free of two-factor interactions, and shows curvature, which a two-level screen would miss. The centre run gave 86 s, far above the 59 s average of the 25 screening runs, so curvature was likely from the start.

The screening pointed to four active factors: cure temperature, NCO:OH index, catalyst and film thickness. Catalyst and cure temperature both peak inside their ranges, and a high NCO:OH index makes a high cure temperature pay off more. The other eight factors (OH number, flow additive, matting agent, solvent dilution, flash-off time, cure time, humidity and substrate temperature) showed nothing above the noise in the final fit.

Designs 2 and 3 were D-optimal add-ons around the predicted optimum. They showed that the model fitted to the screening data was too optimistic there: nine of the ten design 3 runs came in 3 to 10 s below its prediction. I refitted on all runs. Index and film thickness sit at their upper limits and catalyst near 210 ppm. Cure temperature has a flat top: anything from 94 to 100 °C predicts the same hardness within 0.3 s.

The three confirmation runs averaged 90.8 s against 92.1 s predicted. That gap is within the noise, which has a standard deviation of about 3 s (3.2 s from the three replicates, 2.9 s from the model residuals). The model shows no lack of fit (p = 0.69), though the replicate estimate rests on only 2 degrees of freedom.

Hardness may rise further with index and film thickness beyond the allowed range. Humidity and substrate temperature are set low because early data hinted that drier air helps, but the final model finds no effect for either. The six factors with no effect are at mid-range and can move within their ranges for cost or other reasons.

1. Runs used in total

39 runs, all measured successfully (no failures): 20 in the screen, 14 in the optimization, 5 in the confirmation.

2. Suggested optimum

König pendulum hardness ≈ 91 seconds at the recommended settings (predicted 89.5 s; confirmed by 5 replicate runs averaging 91.3 s, sd 3.2 s).

3. Recommended settings

4. Final model

Coded units, X01 = NCO:OH index, X08 = cure temperature:

hardness = 80.78 + 6.37·X01 + 8.94·X08 − 18.27·X08² + 4.17·X01·X08

Residual standard deviation: 7.05 s. R² = 0.795.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

The screen used a 16-run Plackett-Burman design across all 12 factors, plus 4 center points to check for curvature: the minimum arrangement that covers 12 factors in one block. It separated three factors with real main effects from nine with none. NCO:OH index (X01), cure temperature (X08), and substrate temperature (X12) all came back significant, while the rest did not move hardness beyond noise. The center points also came back far higher than the average factorial corner, a clear sign that the optimum sits inside the box rather than at a corner. That ruled out simply picking the best corner and called for a response-surface design next.

The second design was a face-centered CCD on the three active factors, reusing the four center-point runs already measured in the screen rather than measuring fresh ones. Face-centering keeps every axial point within the ±1 equipment limits, which a rotatable CCD would have violated. Fitting the combined screen and CCD data in a full quadratic model showed that substrate temperature (X12) was not actually significant once curvature and the X01×X08 interaction were accounted for. Its apparent effect in the screen was most likely confounded with that interaction, a known risk of a resolution-III Plackett-Burman design. Dropping X12 left a clean model: linear effects of X01 and X08, a strong quadratic term in X08 (hardness peaks partway through the cure-temperature range, not at the edge), and a positive X01×X08 interaction.

Optimizing that model over the ±1 box places the optimum at the top of the NCO:OH range (more crosslinker keeps raising hardness, so the model pushes to the boundary) and at a cure temperature of 87 °C, comfortably inside the window. The remaining nine factors can be run anywhere in their allowed range without materially affecting hardness; recommending their center values keeps the process close to current practice. Five confirmation runs at the recommended point measured a mean hardness of 91.3 s against a model prediction of 89.5 s, well within the model’s noise band, closing the study.

1. Runs used in total

43 runs, all measured successfully (0 failed).

2. Suggested optimum

Predicted König pendulum hardness: 95 s (roughly 89-101 s, based on the residual standard deviation). Five confirmation runs at these settings averaged 91.8 s, which matches the prediction.

3. Recommended settings

Only NCO:OH index and cure temperature are set away from their mid-point; the other eight factors had no measurable effect on hardness and can stay at any convenient setting inside their ranges.

4. Final model

Coded units, fit on the two active factors (X01 = NCO:OH index, X08 = cure temperature):

hardness = 80.97 + 6.39·X01 + 10.61·X08 + 2.57·X01² − 11.35·X08² + 4.79·X01·X08

Residual standard deviation: 2.59 s. R²: 0.955.

Substrate temperature (X12) showed a significant main effect in screening but dropped to a near-zero, non-significant effect once tested properly in the response-surface design, so it was removed from the final model along with the other seven inactive factors.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

The brief listed twelve factors with no clue which ones mattered, so the first step was a screening design: a 16-run Plackett-Burman on all twelve, plus four center points to check for curvature. A Plackett-Burman design finds main effects in the fewest runs by confounding them with higher-order interactions, a fair trade when the goal is to separate the few factors that matter from the many that don’t. The center points caught something a corner-only design would have missed: hardness at the midpoint averaged 80.5 s, far above the 57.2 s average of the sixteen corner runs. That gap meant the true response curves, so a model built only from straight lines would have sent the study in the wrong direction.

The screening fit turned up three factors with effects clearly bigger than noise: NCO:OH index, cure temperature, and substrate temperature. The other nine, including catalyst level, dry film thickness, and cure time, had effects no bigger than noise and were dropped.

Since the screening data already showed curvature, the second design skipped straight to a response surface instead of a confirming factorial: a face-centered central composite on the three active factors, with the other nine held at their midpoint. This design estimates quadratic and two-factor effects directly, which a screening design can’t. It showed that substrate temperature’s apparent effect in screening wasn’t real. It was an artifact of confounding with other factors: its main effect, quadratic term, and interactions all came back within noise. NCO:OH index and cure temperature, by contrast, had a strong positive interaction and a sharp quadratic curve in cure temperature, so hardness rises with cure temperature only up to a point, then falls.

Solving the fitted quadratic model for its maximum inside the allowed range landed on NCO:OH index at its upper bound (1.3) and cure temperature at 93.6 °C, predicting about 95 s of hardness. Five confirmation runs at that setting averaged 91.8 s, which matches the prediction once you account for the ±2.6 s measurement noise. The recommendation: raise the NCO:OH index to 1.3 and cure at 93.6 °C. The remaining formulation and process settings can sit wherever is convenient, since none of them measurably affects hardness.

1. Runs used in total

43 runs, no failures:

• Design 1 (PB screening): 19 runs
• Design 2 (CCD): 19 runs
• Design 3 (confirmation): 5 runs

2. Suggested optimum

König pendulum hardness ≈ 93 s (model prediction 92.1 s; confirmation runs averaged 93.1 s, sd 2.4 s, n=5).

3. Recommended settings

All other factors showed no detectable effect on hardness; set them anywhere in their allowed range (midpoint recommended):

4. Final model

Coded units, X01 = NCO:OH index, X08 = cure temperature:

hardness = 80.11 + 6.43·X01 + 10.72·X08 + 3.17·X01·X08 − 8.61·X08²

Residual standard deviation: 2.23. R² = 0.966 (14 residual df).

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

With 12 factors to check, the study started with a Plackett-Burman screen: 16 runs plus 3 center points to test for curvature. Two main effects stood out well above the noise set by the center-point replicates (sd about 1.1): NCO:OH index (X01, effect +11.1) and cure temperature (X08, effect +20.1). Booth humidity (X11, effect -2.6) was a weaker third candidate, carried forward out of caution. The gap between the factorial runs and the center points (+21.9) also pointed to strong curvature, so a plain linear screen would not have been enough.

The second design was a face-centered CCD in the three candidate factors (X01, X08, X11), with every other factor held at its midpoint. The face-centered form (alpha = 1) keeps every run within the lab’s +-1 range. Fitting the full quadratic model showed that X11, and every term built from it, had no real effect (t below 1 in each case), so it was dropped. The remaining model in X01 and X08 fit well (R squared 0.966): both main effects are strong and positive, there is a genuine X01-X08 interaction (a higher NCO:OH excess helps more at higher cure temperatures, which fits how isocyanate curing works), and cure temperature has a strong negative quadratic term, so hardness peaks partway through its range rather than at the edge. NCO:OH, by contrast, stayed roughly linear across the tested range.

Optimizing the fitted model over the allowed square puts the best settings at the top of the NCO:OH range (X01 = +1, index 1.3) and at X08 = +0.81 (cure temperature about 96 degrees C), an interior peak on temperature that fits with over-cure or heat damage lowering hardness above that point. Five confirmation runs at these settings averaged 93.1 s (sd 2.4), close to the model’s prediction of 92.1 s.

1. Runs used in total

45 runs, none failed: 20 in the screening design, 20 in the optimization design, and 5 in the confirmation run.

2. Suggested optimum

König pendulum hardness of about 90 seconds. The confirmation run measured a mean of 92.8 seconds across five replicates, close to this prediction.

3. Recommended settings

Only X01, X08 and X09 moved from their midpoints. The rest stayed at the centre of their allowed range because neither design found an effect on hardness.

4. Final model

Fitted in coded units, over X01 and X08 only (X09 dropped after the optimization stage, see summary):

hardness = 80.42 + 5.03 · X01 + 9.15 · X08 − 8.83 · X08² + 4.03 · X01 · X08

Residual standard deviation: 2.45. R²: 0.947.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

The study ran in three stages, to spend as few runs as possible while still finding the real optimum.

The first stage screened all 12 factors with a 16-run Plackett-Burman design, the smallest resolution-III design that covers 12 factors. We added four centre points to catch curvature, since the design itself only tests the corners of the box. Three factors came out significant: the NCO:OH index (X01), cure temperature (X08), and cure time (X09). The centre points’ average hardness was 23 seconds above the average of the corner runs, a gap far too large to be noise. The true optimum had to sit inside the box, not at a corner, so a model built only from straight lines would miss it.

The second stage built a response surface around those three factors, holding the other nine at the centre of their range, where screening found no effect. We used a face-centered central composite design, not a standard rotatable one, because a rotatable design’s star points fall outside the lab’s -1 to +1 limits. A full quadratic model, with the weakest terms removed, showed that cure time (X09) had no detectable effect once NCO:OH index and cure temperature were fixed; the earlier screening signal for X09 likely reflected curvature from cure temperature instead. The final model kept only X01, X08, and their square and interaction terms, and explained 95% of the variation in hardness.

Solving that model within the allowed range gives the high end of the NCO:OH index (1.3) and a cure temperature just below the maximum, around 95°C, where the quadratic term starts pulling the gain back down. We set cure time to its shortest allowed value, 20 minutes, since it made no measurable difference and a shorter cure saves time on the line.

The third stage ran five replicates at these settings to check the prediction against real measurements. The average came out at 92.8 seconds against a predicted 90.4, well within the measurement noise seen throughout the study: the recommended settings hold up.

1. Runs used in total

45 runs, all measured. None failed.

2. Suggested optimum

93 seconds König pendulum hardness.

3. Recommended settings

Hold every other factor at the middle of its range: polyol OH number 160 mg KOH/g, catalyst 200 ppm, flow additive 0.55%, matting agent 2%, solvent dilution 17.5%, flash-off time 12.5 min, dry film thickness 45 µm, booth humidity 50% r.h., substrate temperature 24°C. None of them moved hardness, so any value in range should do, and the middle is the safest default.

4. Final model

In coded units, over NCO:OH index (X01), cure temperature (X08) and cure time (X09):

hardness = 80.9 + 5.67·X01 + 8.78·X08 + 1.29·X09 − 10.96·X08² + 5.16·X01·X08

Residual standard deviation: 2.2 seconds. R²: 0.97.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

We started with 12 suspect factors and no idea which of them set hardness, so the first step was to screen all of them at once rather than guess. A 20-run Plackett-Burman design, with four runs added at the center of every factor, let us estimate all 12 main effects from a single batch and check for curvature at the same time.

Three factors stood out: NCO:OH index, cure temperature and cure time. The other nine, including catalyst level, film thickness and booth humidity, showed no effect worth trusting. The center points also told us something important: hardness there averaged 76 seconds, well above the 57-second average at the corners of the design. That gap is curvature, a sign that the best settings lie inside the tested range rather than at an edge, and it meant a simple main-effects model would not be enough to find the optimum.

To map that curvature, we ran a face-centered central composite design on the three active factors, holding the other nine at their midpoint. This design adds points at the center and along each factor’s axis to the corners of a factorial design, so it can estimate curved effects, not just straight-line ones. We kept the axial points at the equipment’s own limits (coded ±1) instead of pushing past them, since the lab rejects settings outside that range. The resulting model showed a strong bend in cure temperature and a real interaction between NCO:OH index and cure temperature: raising both together beats raising either one alone.

Solving that model for the best combination pointed to the top of the range for NCO:OH index and cure time, and a cure temperature partway up the range rather than at its ceiling. Five confirmation batches at that setting averaged 90 seconds, close to the model’s prediction of 92 and well inside the run-to-run noise. A higher NCO:OH index, cured longer at a moderate cure temperature, fixes the soft film the customer rejected.

1. Runs used in total

54 runs, all measured. None failed.

2. Suggested optimum

91.3 seconds König pendulum hardness.

3. Recommended settings

4. Final model

In coded units, fit to the 29 runs of the optimization design:

hardness = 80.63 + 5.20·X01 - 0.47·X03 + 10.26·X08 + 0.25·X09 　- 14.42·X03² - 9.39·X08² 　+ 2.07·X01·X08 - 1.17·X03·X08 + 1.61·X08·X09

Residual standard deviation: 2.15 s. R²: 0.985.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors were on the table, with no idea which ones drive hardness. The study ran in three steps: screen all twelve, model the ones that matter, then confirm the predicted optimum in the lab.

The first design screened all twelve factors in 20 runs: a Hadamard-based resolution III design (equivalent to a Plackett-Burman screen), plus four center points to check for curvature. Center points are cheap and catch something a pure two-level design would miss: if the true relationship bends, the corners alone say nothing about where the peak sits. That is what happened here. The center points averaged 81.0 seconds against a factorial average of 57.3, a gap too large to be noise, so the optimum sits inside the tested range, not at a corner.

The screen also showed which factors matter. Cure temperature (X08) and the NCO:OH index (X01) stood out, with catalyst level (X03) and cure time (X09) close behind. Flash-off time (X07) and matting agent (X05) showed smaller, negative effects. The other six factors, including film thickness, booth humidity and substrate temperature, showed no effect worth chasing. Given the curvature and the cost of fitting a quadratic term for every factor, the second design built a response surface around the four strongest factors and fixed flash-off time and matting agent at their better extreme, saving the runs a full twelve-factor response surface would have needed.

The second design was a face-centered central composite design on the index, catalyst, cure temperature and cure time: 29 runs covering the factorial corners, the face centers, and five center-point replicates. The fitted model explains 98.5% of the variance in hardness, with a residual spread of about 2.1 seconds. The index and cure temperature both have strong linear effects, and the catalyst and cure temperature both curve sharply, which is why the catalyst optimum sits near the middle of its range instead of at an end. Cure temperature also interacts with the index and with cure time: at a higher cure temperature, both pay off more than they would alone.

Solving the fitted model gives a corner for the index (highest allowed) and cure time (longest allowed), but an interior optimum for catalyst and cure temperature, at a predicted 91.3 seconds. The third design confirmed this: five replicate runs at the recommended settings measured 95.0, 88.2, 91.6, 93.5 and 89.5 seconds, averaging 91.5 seconds, close to the model’s prediction and well within the spread expected from five noisy measurements.

The clearcoat gets harder with a higher NCO:OH index, a longer cure, and a cure temperature and catalyst level tuned to a middle ground rather than pushed to an extreme. Formulation and cure decide hardness; the application conditions tested here do not.

1. Runs used in total

55 runs: 20 in the screening stage, 30 in the optimization stage, 5 as confirmation. None failed.

2. Suggested optimum

König pendulum hardness of about 93 seconds, with repeat measurements at that setting averaging 91.3 seconds.

3. Recommended settings

4. Final model

Coded units, fit to the 30-run optimization stage:

hardness = 78.92 + 6.70·X01 + 0.42·X03 + 9.64·X08 - 1.23·X09

• 14.26·X03² - 7.67·X08²
• 1.98·X01·X03 + 2.58·X01·X08 + 1.25·X03·X08 - 1.19·X08·X09

Residual standard deviation: 2.59. R²: 0.976.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-D-optimal.csv, design-2-D-optimal-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors were on the table, so the first step was to find out which ones actually move hardness. A 16-run Plackett-Burman screen, plus 4 center points to check for curvature, covered all 12 factors in 20 runs. Six factors came out active: NCO:OH index, catalyst level, solvent dilution, cure temperature, cure time and booth humidity, with cure temperature and NCO:OH index by far the largest effects. The center points also came back much higher than the corner runs averaged, a sign that the response bends somewhere inside the region instead of running straight to an edge.

That curvature called for a response-surface design next. A full central-composite design on six factors would have needed around 50 runs to fit linear, quadratic and interaction terms. A D-optimal design built for a narrower model, six main effects, six quadratic terms, and the interactions the catalyst-cure chemistry would predict between NCO:OH index, catalyst and cure temperature, covered the same ground in 30 runs plus 4 center points for pure error.

Fitting that model and dropping the terms that did not hold up under a t-test left nine terms plus the intercept, with an R² of 0.976. Hardness rises with NCO:OH index and cure temperature, then bends over at high catalyst and high cure temperature, where both quadratic terms are negative, and it drops slightly with longer cure time, more so at high temperature. That fits the chemistry: more catalyst speeds the reaction, but past a point the film is already fully cross-linked, and a long bake at high temperature can start working against the network rather than for it. Booth humidity and solvent dilution, both active in the screen, had no significant main, quadratic or interaction effect once the response surface was fit, so the final recommendation holds them at mid-range.

Solving the fitted model for its maximum put NCO:OH index and cure temperature high, catalyst near the middle of its range, and cure time at the low end. Five confirmation runs at that setting measured 86 to 96 seconds, averaging 91.3, close to the model’s prediction of about 93. That holds up the model and the setting it points to.

1. Runs used in total

55 runs, all measured (no failures): 20 for screening, 31 for the response surface design, and 4 for confirmation.

2. Suggested optimum

96 seconds König pendulum hardness.

3. Recommended settings

The other seven factors (flow additive, matting agent, solvent dilution, flash-off time, dry film thickness, booth humidity, substrate temperature) had no measurable effect on hardness. Run them at their current mid-range settings.

4. Final model

In coded units, with hardness as the response:

hardness = 80.3
           + 5.26 * X01          (NCO:OH index)
           - 1.21 * X02          (polyol OH number)
           + 10.21 * X08         (cure temperature)
           + 2.31 * X09          (cure time)
           + 3.86 * X01 * X08
           + 1.21 * X01 * X09
           - 14.50 * X03^2       (catalyst)
           - 8.62 * X08^2

Residual standard deviation: 2.33. R²: 0.981.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

The brief named twelve factors and no leads on which of them drove hardness, so the study started with a screen. A 16-run Plackett-Burman design on all twelve factors, with four centre points added to check for curvature, used the fewest runs that could still separate main effects from noise. The centre points came back far higher than the average of the factorial runs, which meant the response bent sharply somewhere inside the box rather than climbing toward a corner. Cure temperature, NCO:OH index and cure time stood out as the strongest effects, with polyol OH number and catalyst a step behind. The other seven factors, including dry film thickness and booth humidity, showed effects no larger than the noise.

The curvature ruled out a plain factorial follow-up. The second design was a response surface study on the five factors that mattered: a face-centred central composite design, built from a resolution V half-fraction so two-factor interactions would not hide behind main effects, with axial points on the faces of the cube rather than beyond it. Equipment limits in this process run from -1 to +1 in coded units, so a rotatable design with axial points outside the cube was never an option. Centre points gave a check on pure error. Fitting a full quadratic model and dropping terms with a t-value under 2 left cure temperature and NCO:OH index as the two strongest drivers, each curving on its own and interacting with each other. Catalyst had a sharp optimum near the middle of its range. Polyol OH number and cure time were smaller, mostly linear effects.

Solving that model inside the allowed ranges pointed to the high end for NCO:OH index and cure time, the low end for polyol OH number, catalyst near the centre of its range, and cure temperature at 96 °C, short of the 100 °C ceiling, because the fitted curve turns over there. Four confirmation runs at these settings averaged 91 seconds against a predicted 96, close enough given the model’s residual spread to trust the settings. The seven factors dropped after screening can stay wherever is convenient in production; none of them affected hardness.

1. Runs used in total

56 runs, none failed: 23 for screening, 28 for optimization, 5 to confirm the result.

2. Suggested optimum

The model predicts a König pendulum hardness of 77.2 s at the recommended settings. Five confirmation runs there averaged 78.8 s, close to the prediction.

3. Recommended settings

X01 and X08 drive hardness and sit where the model says to put them. The rest showed no real effect, so they sit at mid-range or at the side the screening data leaned toward.

4. Final model

Coded units, fit on the 28 optimization runs:

hardness = 66.861 + 5.269 X01 + 10.220 X08 + 3.173 (X01 x X08) - 8.804 X08^2

Residual standard deviation: 2.83 s. R^2: 0.943.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

With 12 factors and no idea which ones mattered, the first step was to screen all of them in as few runs as possible. A Plackett-Burman design does this in 20 runs instead of the 4096 a full factorial would need. We added 3 center points to check whether the system behaved linearly inside the tested range. Two factors stood out clearly: the NCO:OH index (X01) and the cure temperature (X08). Two more, the matting agent (X05) and the cure time (X09), showed a weaker signal worth checking further. The center points came back much harder than the corners of the design, so the best settings for at least one factor lie inside the tested range, not at its edges.

That curvature ruled out another screening-style design for the next step; we needed a model that could bend. So we ran a face-centered central composite design on the four candidate factors, with the other eight held at the settings the screening data favored. Face-centered, because the equipment cannot run outside the -1 to +1 range, so the usual CCD arms that reach past it were not an option. This design let us fit curves and interactions, not just straight lines.

The matting agent and cure time, it turned out, do not move hardness enough to matter here. Hardness comes down to the NCO:OH index, the cure temperature, and how they work together: a higher cure temperature helps more when the index is also high. The cure temperature has a best setting inside the tested range; the index does not, it keeps helping all the way to the top of its range. That gave a four-term model with a good fit (R^2 = 0.94) and a clear recommendation: push the NCO:OH index to its allowed maximum, set the cure temperature about three-quarters of the way to its allowed maximum, and leave everything else at a comfortable mid-point or at the better side from screening.

Five confirmation runs at these settings averaged 78.8 s against a model prediction of 77.2 s, well within the noise seen throughout the study. The result holds.

1. Runs used in total

56 runs, no failures: 20 for screening, 31 for optimization, 5 for confirmation.

2. Suggested optimum

93.0 seconds König pendulum hardness (measured mean at the recommended settings; the model predicts 90.3).

3. Recommended settings

4. Final model

Coded units, fit on the 31-run optimization design:

hardness = 80.4 + 5.3·X01 + 1.1·X03 + 9.1·X08 − 14.4·X03² − 9.8·X08² + 4.3·X01·X08

Residual standard deviation: 4.2. R²: 0.94 (adjusted R²: 0.92).

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Nobody knew which of the twelve factors still controlled hardness, so the first step was to screen all of them at once. A 16-run Plackett-Burman design covers twelve factors in the fewest runs that still separate main effects from noise. Four center points came with it to catch curvature early, and they paid off: the center points averaged 82 seconds against a corner average of 57, a gap too large to be noise. Something in the process pushed hardness up near the middle of the ranges, not at an edge.

The screening data also showed which factors drove that curve. Cure temperature (X08) and the NCO:OH index (X01) had by far the largest effects, with catalyst (X03) and booth humidity (X11) next. The other eight factors, including solvent dilution, film thickness, flash-off time and cure time, fell inside the noise band set by the center-point replicates and were dropped.

The second design built a response surface on the five remaining factors, using a half-fraction central composite design at resolution V. That fraction keeps every main effect and quadratic term clear of the others while cutting the run count roughly in half against a full design, which is what matters once curvature is confirmed and the question becomes where it peaks. Center points went in again, this time to estimate pure error for testing the model. The fit showed humidity and solvent dilution had no reliable effect after all, so they and their interactions dropped out. What remained was a model built on NCO:OH index, catalyst and cure temperature, with strong quadratic curvature on catalyst and cure temperature and an interaction between index and temperature. It explains 94% of the variance in the data.

Solving that model inside the allowed ranges put the optimum at the high end of the NCO:OH index, a catalyst level close to the design’s center, and a cure temperature well above the original process midpoint. Five confirmation runs at those settings averaged 93.0 seconds, close to the model’s prediction of 90.3 and clearly above anything seen during screening. The clearcoat was soft because the process ran colder and leaner on isocyanate than hardness needed, not because of any of the other nine factors on the original list.

1. Runs used in total

41 runs, all measured successfully, none failed.

2. Suggested optimum

About 89 seconds (95% CI: 86.6 to 91.8 s), based on five confirmation runs at the recommended settings.

3. Recommended settings

NCO:OH index and cure temperature drive hardness. The other ten factors showed no measurable effect, so we left them at the midpoint of their range.

4. Final model

In coded units, fit on the 12-run face-centered CCD on X01 and X08:

hardness = 78.84 + 7.27·X01 + 12.32·X08 + 3.70·X01² − 11.32·X08² + 0.91·X01·X08

Residual standard deviation: 3.52 s. R²: 0.955.

5. Designs in order

6. Run sheets

• design-1-PB.csv, results in design-1-PB-results.csv
• design-2-CCD.csv, results in design-2-CCD-results.csv
• design-3-confirmation.csv, results in design-3-confirmation-results.csv

7. Summary

Twelve factors were on the table, so the first job was to find out which few actually matter. We screened all of them in a 24-run Plackett-Burman design: 20 runs at the corners of the factor space, plus 4 at the midpoint. We chose the 20-run version over the smaller 16-run one because the 16-run design fully confounds each main effect with a two-factor interaction, while the 20-run design spreads that risk thinly across many interactions. That mattered here, since catalyst and cure conditions in a 2K PU system are known to interact. The midpoint runs let us test for curvature at the same time.

Two factors stood out: NCO:OH index and cure temperature, both with effects several times larger than any other factor and clearly significant. The midpoint runs also pointed to strong curvature: they averaged close to 79 seconds, well above the roughly 57 seconds typical of the corner runs, which meant the best settings were likely to sit inside the tested range rather than at its edges.

With the field narrowed to two factors and curvature confirmed, the next design was a face-centered central composite on NCO:OH index and cure temperature, holding the other ten factors at their midpoint. We used a face-centered design, with its star points set at +1 and -1 rather than further out, because the lab rejects any setting outside that range. It’s the smallest design that still fits a full quadratic model for two factors.

That design showed cure temperature has a real peak partway through its range: raising it helps up to a point, then starts to hurt, which fits with over-cure breaking down the crosslink network in a PU film. NCO:OH index behaved differently: hardness kept rising all the way to the top of its allowed range, so the best setting there is the edge of the window, not the middle. The two factors showed no detectable interaction with each other.

Finally, we ran the recommended settings five times to check the model against fresh samples. The measured average, about 89 seconds, came in a bit below the model’s point estimate of about 94 seconds. That gap is normal for a quadratic model fit from only 12 runs, since the fitted peak sits at the edge of the design, where predictions are least certain. Even so, every other setting tried across all three designs scored lower, so this region is the best one found, and we recommend moving ahead with it.

1. Runs used in total

42 runs, all measured. No runs failed.

2. Suggested optimum

The model predicts a König pendulum hardness of about 93 seconds at the recommended settings (95% prediction interval: 88 to 98 seconds). Four confirmation runs at these settings averaged 89.3 seconds (87.8, 87.8, 89.2, 92.4), inside that interval.

3. Recommended settings

X01 and X08 drive hardness and sit at or near the top of their allowed range. X09 showed no measurable effect, so it’s set to the short, cheaper end. The other nine factors showed no effect either; they’re set to the middle of their range, the safest choice when a factor’s effect is unknown.

4. Final model

Fitted in coded units, on X01 (NCO:OH index) and X08 (cure temperature) only. Cure time (X09) and the nine other factors dropped out as not significant:

hardness = 80.6 + 7.37·X01 + 9.48·X08 + 4.28·X01·X08 - 9.50·X08²

Residual standard deviation: 2.15 seconds. R²: 0.971.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

Twelve possible drivers of hardness were on the table, from resin stoichiometry to booth humidity, with no idea which ones mattered. The cheapest way to find out was a 16-run Plackett-Burman screen, which tests all twelve main effects at once at the cost of some risk of mixing up a main effect with a two-factor interaction. Four center runs (all factors at their midpoint) came along to catch overall curvature a two-level design can’t see, and to give a direct read on measurement noise, needed to judge whether an effect is real.

The screen pointed at three candidates, all raising hardness: cure temperature (X08), the NCO:OH index (X01), and cure time (X09), with cure temperature mattering most. It also turned up something the screen alone couldn’t explain: the four center runs averaged 81, well above the 58 average of the sixteen factorial runs. That gap was too big to be noise, so the response had to be curved, not a straight line.

The next step was a response-surface design on just those three factors, holding the other nine at their midpoint: a central composite design, which fits curves and interactions as well as straight lines, in far fewer runs than testing all twelve factors at three levels each would take. Its corner, face, and axial points all stayed within the -1 to +1 equipment limit by using face-centered axial points instead of the usual ones further out.

That design traced the curvature to one factor, not two. Cure time had no real effect and dropped out. Cure temperature and the NCO:OH index both help, and they reinforce each other: pushing both up at once helps more than pushing either one up alone. Cure temperature also levels off and turns back down past a point, so its best setting isn’t at the top of the range but just past three-quarters of the way up it, 94°C of the 60 to 100°C range. The NCO:OH index keeps helping all the way to the top of its range, so it’s set to the maximum, 1.3.

Four confirmation runs at these settings came in at 87.8 to 92.4 seconds, averaging 89.3, inside the model’s 95% prediction interval, enough to trust the recommendation: raise the NCO:OH index to 1.3, hold cure temperature near 94°C, and cut cure time to the cheaper 20 minutes, since shortening it costs nothing. The other nine factors, including the catalyst and the application settings, can stay wherever is most convenient; none of them moved hardness enough to tell apart from noise.

1. Runs used in total

43 runs, sent to the lab in three batches. None failed.

2. Suggested optimum

König pendulum hardness of about 91 seconds at the recommended settings. Five confirmation runs there averaged 90.6 s, so the number holds up.

3. Recommended settings

Only X01 and X08 move the result. The other ten factors sit at the middle of their allowed range, since none of them changed hardness by more than noise.

4. Final model

Coded units, fit to the 18-run central composite design:

hardness = 81.18 + 5.28·X01 + 10.69·X08 + 3.50·X01·X08 - 11.63·X08²

Residual standard deviation: 3.44 s. R²: 0.93.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors and no idea which ones mattered, so the first step was a screen, not a guess. A 16-run Plackett-Burman design covers all twelve at two levels each in the fewest runs that still keep every factor’s effect uncorrelated with the others. Four center points came along to check whether the real optimum might sit inside the tested range instead of at a corner.

It did, by a wide margin: the center points averaged 24 s higher than the corners. Three factors stood out above the noise: cure temperature, the NCO:OH index, and booth humidity. But a saturated 16-run screen can’t always tell a factor’s own effect from an interaction between two other factors, and the humidity signal sat on exactly that kind of pair: the index and the cure temperature. With both of those already looking active, their interaction was a plausible explanation for what the screen had actually picked up.

The second design settled it. A face-centered central composite design in just those three factors, with the other nine held at mid-range, fits every two-way interaction cleanly, because its core is a full factorial. The flat faces replace the usual rounded ones, so no run asks for a setting outside the equipment’s range. Once the index times cure-temperature interaction was in the model, humidity’s own effect vanished. It was never a driver of hardness: the screen had mistaken that interaction for it. Cure temperature’s effect curves, rising with temperature up to a point and then falling, while the index pushed hardness up in close to a straight line across the tested range.

The index showed no interior peak, so pushing it to its highest allowed value, 1.3, always helped. Cure temperature did have an interior peak, at 92.2°C. Five confirmation runs there measured 90.6 s on average, close to the model’s prediction of 91 s. The fix for the soft film: raise the index to 1.3 and hold cure at 92.2°C. Nothing else in the recipe or the process needs to move.

1. Runs used in total

43 runs, none failed: 25 for the screening design, 13 for the optimization design, 5 confirmation runs.

2. Suggested optimum

The model predicts a König pendulum hardness of about 91.9 seconds at the recommended settings. Five confirmation runs at those settings measured a mean of 92.4 seconds, so the prediction holds up.

3. Recommended settings

Only X01 and X08 change hardness. The study found no effect from the other ten factors, so they sit at the midpoint of their allowed range. You can move them for cost, flow, or other process reasons without changing hardness.

4. Final model

In coded units:

hardness = 80.59 + 7.03 X01 + 7.72 X08 - 9.12 X08² + 4.72 X01 X08

Residual standard deviation: 3.43 seconds R-squared: 0.92 (adjusted: 0.87)

The model uses only X01 and X08. No other factor or interaction showed an effect in the screening data.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

The customer’s complaint was simple: the clearcoat cures too soft, and nobody knew which of the 12 factors to blame. With that many factors and a noisy measurement, the plan was to screen first, optimize second, and confirm last, so that no run went to a factor that turned out not to matter.

For the screening step, a definitive screening design (DSD) fit the job better than a classic Plackett-Burman design. A 12-factor DSD needs only 25 runs, keeps every main effect clear of two-factor interactions, and can also catch curvature in a factor’s own effect. That mattered here, because coatings cure chemistry often hides interactions, such as between a catalyst and the cure temperature, that a resolution III design would have blurred into the main effects.

The screening design pointed at two factors: the NCO:OH index (X01) and the cure temperature (X08). Both had large main effects, cure temperature also showed strong curvature, and the two factors interacted with each other. None of the other ten factors, including the catalyst, moved hardness by more than noise. That’s a clean case of a few active factors among many that don’t matter.

With two active factors and real curvature, the next step called for a response surface design rather than another screening round. A central composite design (CCD) fits a full quadratic model in X01 and X08 and needed only 13 runs. Because the lab’s equipment range stops at coded ±1, the design used face-centered axial points at ±1 instead of the usual ±1.41, so every run stayed inside the allowed range. This design gave a tight fit: R-squared of 0.92 on a residual standard deviation of about 3.4 seconds, clearly better than squeezing the same model out of the screening data, where the ten inactive factors added noise.

The fitted surface has no peak inside the studied region. Hardness keeps rising with the NCO:OH index right up to its allowed limit, while the best cure temperature sits partway up its range, near 94°C, because the curvature in that factor pulls hardness back down at the very top. The recommended settings sit at that point: NCO:OH index at its high limit, cure temperature at about 94°C, and every other factor left at the midpoint of its range. Five confirmation runs at these settings measured a mean hardness of 92.4 seconds against a predicted 91.9, well within the run-to-run noise seen throughout the study. That closes the case: the softness came from running the index and the cure temperature too low, not from the other ten factors in the recipe or the process.

1. Runs used in total

43 runs sent to the lab. None failed.

2. Suggested optimum

About 93 seconds König pendulum hardness. The model predicts 93.4 s at the recommended settings. Five confirmation runs at those settings measured 92.9 s on average (SD 2.3 s), which matches the prediction well.

3. Recommended settings

4. Final model

Coded units, fit on the 38 runs from designs 1 and 2:

hardness = 82.00 + 6.31·X01 + 9.28·X08 − 9.25·X08² + 4.47·X01·X08 − 1.53·R

R is the summed square of the coded settings of the other ten factors (X02 to X07, X09 to X12). It is zero at the recommended settings.

Residual standard deviation: 3.65. R²: 0.946.

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Of the twelve factors, only two moved the needle: the NCO:OH index and the cure temperature. They also interact, so raising the index helps more when the cure runs hot.

We started with a definitive screening design (DSD) rather than a plain Plackett-Burman design. With twelve factors and a cure step involved, curvature was a real possibility, and a DSD estimates main effects and quadratic effects in a single design, at the same run count a screening design would need anyway. It also keeps main effects free of two-factor interactions, so the screening result didn’t depend on guessing which interactions might be hiding in it. Twenty-five runs pointed straight at the NCO:OH index and cure temperature; every other factor looked like noise on its own. But the other ten factors still mattered as a group: moving any of them away from its nominal setting cost hardness, even though no single one of them showed a clear effect by itself. So we fixed all ten at their mid-range setting for the rest of the study, which turned out to be where they perform best anyway.

To map the response surface in the two active factors, we ran a face-centered central composite design (CCD). The equipment can’t go outside the coded range of -1 to +1, so a face-centered design was the only way to add axial points without leaving that range. It confirmed a strong, downward-bending curve in cure temperature, and the interaction between the two factors held up as well.

Pooling both designs gave a model with an R² of 0.95. The optimum sits at the highest allowed NCO:OH index. Cure temperature stops short of its own upper limit: hardness keeps rising with heat, then turns over near 95 °C, so curing all the way up at 100 °C would give less hardness, not more. Five confirmation runs at the recommended settings measured 92.9 s on average, close to the model’s prediction of 93.4 s, confirming the recommended settings.

1. Runs used in total

48 runs sent to the lab. All 48 came back measured; none failed.

2. Suggested optimum

About 92 seconds of König pendulum hardness. Five confirmation runs at the recommended settings averaged 92.1 s (standard error 0.9 s), close to the 94.9 s the final model predicted.

3. Recommended settings

Only the NCO:OH index, the catalyst, and the cure temperature move hardness. The other nine factors sit at the midpoint of their allowed range, because none of them showed an effect worth acting on.

4. Final model

Coded units, centred at 0:

hardness = 81.23 + 6.01·X01 + 2.35·X03 + 9.73·X08
           − 15.00·X03² − 6.72·X08²
           + 4.68·X01·X08 − 2.48·X03·X08

Residual standard deviation: 2.82 s. R²: 0.976 (adjusted R²: 0.960).

5. Designs in order

6. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

7. Summary

Twelve factors were on the table, with no record of which ones drove hardness. The first design was a 25-run definitive screening design (DSD), which tests every factor at three levels while keeping all twelve main effects independent of each other. With twelve factors, a few real effects can easily hide behind eleven that do nothing, so each main effect has to read cleanly on its own. Two effects came through clearly: the NCO:OH index and the cure temperature, the latter with a hint of curvature, meaning an interior optimum rather than “turn it up as far as it goes”. The catalyst (DBTDL) fell just short of significance, but since it is the compound that drives the cure reaction, it carried into the next design anyway. The other nine factors (polyol OH number, flow additive, matting agent, solvent dilution, flash-off time, cure time, film thickness, booth humidity, substrate temperature) showed nothing worth separating from noise.

The second design followed up on those three factors with an 18-run face-centred central composite design, holding the other nine at the midpoint of their range. With the field narrowed down, this design could fit a full quadratic surface: every main effect, every two-way interaction, and every curvature term among the three. The catalyst turned out to matter after all, with a curved effect: hardness rises then falls as DBTDL increases, peaking near the middle of the tested range rather than at either extreme. Cure temperature and the NCO:OH index also interact, so pushing both up together gives more hardness than either one alone. The resulting model explains 96% of the variance in hardness (adjusted R²) with a residual error of 2.8 seconds, after dropping the terms that added nothing.

Five confirmation runs at the recommended settings averaged 92.1 seconds, close to the model’s prediction of 94.9 and well above the roughly 80 seconds seen at the midpoint of the design space. The result rests on three dials: push the NCO:OH index to the top of its allowed range, hold the catalyst near 200 ppm rather than at either extreme, and cure at the highest allowed temperature. Every other setting in the recipe (the solvent cut, the flow and matting additives, the application and booth conditions) can stay wherever is convenient. None of them affect hardness.

1. Runs used in total

53 runs, all measured. No runs failed.

2. Suggested optimum

92.0 seconds König pendulum hardness (95% CI 88.5 to 95.5, from six confirmation runs). The fitted model predicted 89.3 seconds at these settings, so the confirmation runs match the model.

3. Recommended settings

X02, X04 to X07, and X09 to X12 did not show an effect on hardness. The settings above hold them at the middle of their range.

4. Final model

Coded units, fitted to the 19 CCD runs plus the 4 center runs from the screening design (23 runs, all other factors at 0):

hardness = 79.03 + 6.02·X01 + 1.88·X03 + 9.30·X08 - 1.23·X01² - 10.78·X03² - 9.91·X08² - 1.14·X01·X03 + 5.38·X01·X08 - 1.38·X03·X08

Residual standard deviation: 2.99. R²: 0.970 (adjusted 0.949).

5. Designs in order

6. Run sheets

• design-1-DSD.csv / design-1-DSD-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

7. Summary

Twelve factors is too many to test by trial and error, so the study ran in three stages: screen all of them at once, model the few that mattered, then confirm the recommended settings in the lab.

The screen (design 1) was a definitive screening design, 28 runs. It reads every factor’s straight-line effect and its curvature without one hiding the other. A plain two-level design would have used fewer runs, but can’t see curvature at all, and coatings chemistry is full of factors with a sweet spot rather than a direction, so that risk wasn’t worth taking. Two factors came out as clear drivers of hardness: the NCO:OH index and the cure temperature. The catalyst load was weaker and could have been noise, but it chemically belongs with the other two, since stoichiometry, catalyst, and cure temperature together set how far the crosslinking reaction runs, so it stayed in for the next stage. The other nine factors showed no effect. The screen also found strong overall curvature: the best runs sat near the middle of the design, not at a corner. But the design carries only one shared curvature term for all twelve factors, so it couldn’t say which factor the curvature belonged to.

Design 2, a face-centered central composite design on the three candidate factors with everything else held at its middle setting, answered that in 19 runs. It gave each factor its own curvature term and all three two-way interactions. The catalyst turned out to matter mainly through curvature, not a straight-line effect, and the NCO:OH index and cure temperature interact strongly: raising both together helps more than raising either alone. The index is best pushed to the top of its range, cure temperature has a real sweet spot inside its range, and the catalyst is best left near the middle.

The six confirmation runs (design 3) measured 92.0 seconds on average, against a model prediction of 89.3. The match is good, so the three-factor model is a reliable guide to where hardness is highest within the allowed ranges.

1. Runs used in total

58 runs, all measured. None failed.

2. Suggested optimum

König pendulum hardness around 89 seconds. Five confirmation runs at the recommended settings averaged 88.7 s (std 1.6 s), close to the model’s prediction of 94.7 s and far above the roughly 56 to 81 s we saw across the earlier designs.

3. Recommended settings

Only NCO:OH index and cure temperature move from their mid-range value; both sit at the top of their allowed range. Every other factor stays at the centre of its range, where the data gave no reason to move it.

4. Final model

Coded units, fit on the central composite design (design 3):

hardness = 78.67 + 5.14·X01 + 10.60·X08 + 0.58·X01² − 6.48·X08² + 6.16·X01·X08

Residual standard deviation: 3.22. R²: 0.939.

5. Designs in order

6. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-fractional-factorial.csv / design-2-fractional-factorial-results.csv
• design-3-CCD.csv / design-3-CCD-results.csv
• design-4-confirmation.csv / design-4-confirmation-results.csv

7. Summary

The brief named twelve factors and asked which ones control hardness. With that many candidates, a full factorial was out of reach, so the study started with a 16-run Plackett-Burman screen plus four centre points. That screen pointed to three factors with a real effect: NCO:OH index, cure temperature, and cure time. It also showed strong curvature: the four centre points averaged 83, well above the 57 average at the corners of the design.

Plackett-Burman designs alias main effects with each other’s interactions, so the next step was a cleaner 16-run fractional factorial on the three candidates plus three more that had shown a weak signal, again with centre points. Cure time dropped out here: its earlier signal had come from aliasing, not a real effect. NCO:OH index and cure temperature were the only two factors that mattered, together with an interaction between them.

With two active factors and clear curvature, the study moved to a response-surface design: a face-centred central composite design on just NCO:OH index and cure temperature, with the other ten factors held at the centre of their ranges. Thirteen runs were enough to fit a full quadratic model. It showed a concave bend in cure temperature, a positive interaction between the two factors, and the best fit at the top corner of the box: NCO:OH index at +1, cure temperature at +1. There was no sign of a true interior peak, so the recommended settings sit at that corner, the highest values the brief allows for both factors.

Five confirmation runs at these settings came back at 88.7 s on average, close to the model’s prediction and consistent run to run. Push NCO:OH index and cure temperature to the top of their allowed ranges, keep every other factor where it already sat, and expect hardness close to 89 seconds.

1. Runs used in total

69 runs across five designs. None failed.

2. Suggested optimum

About 90 seconds of pendulum hardness. Five confirmation runs at the recommended settings averaged 90.3 seconds (95% CI: 86.6 to 93.9). The fitted model predicts 95.8 seconds at that point, but that point sits at a corner of the design space, where the model’s curvature terms are least reliable. Trust the confirmed 90 seconds over the model’s 95.8.

3. Recommended settings

X02, X04 to X07, and X10 to X12 never showed an effect, so their settings are just the middle of the allowed range.

4. Final model

Fitted on the 40 runs of the factorial and central-composite designs, coded units:

hardness = 81.35
  + 7.36 X01 + 0.90 X03 + 10.41 X08 + 1.13 X09
  - 3.00 X01^2 - 15.44 X03^2 - 7.67 X08^2
  + 6.18 X01 X08

Residual standard deviation: 2.55 seconds. R²: 0.980 (adjusted 0.975).

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-full-factorial.csv, design-2-full-factorial-results.csv
• design-3-CCD.csv, design-3-CCD-results.csv
• design-4-CCD.csv, design-4-CCD-results.csv
• design-5-confirmation.csv, design-5-confirmation-results.csv

7. Summary

Twelve factors were too many to test one at a time, so the study began with a screening design: a 20-run Plackett-Burman layout with four center points added, 24 runs in total. Only two factors showed a real main effect, the NCO:OH index and the cure temperature. But the center points showed something the main effects could not: they averaged 79 seconds, well above the 57-second average of the corner runs. That gap is curvature, and a two-level screen can flag it without saying which factor is behind it.

The next design narrowed to four factors: the index and cure temperature, already shown to matter, plus the catalyst and cure time, which the underlying chemistry ties to the same reaction even though neither had shown an effect yet. A full factorial on these four factors, 16 runs plus five center points, found a strong interaction between the index and cure temperature. Catalyst and cure time still showed no effect on their own.

A two-factor central composite design on the index and cure temperature came next, to map the curvature between them. It found the best result so far, 92.5 seconds at the highest index and highest temperature. But its center points averaged 80, well below the 82 the earlier full factorial had measured at the same mid-point for all four factors. Catalyst and cure time had to be carrying curvature of their own, the kind a two-level design cannot see.

Six more runs, axial points on catalyst and cure time with everything else held at center, confirmed it. Catalyst had a sharp interior optimum: both low and high loadings cut hardness by 15 to 20 seconds compared with the mid-point. Cure time behaved almost linearly across its range, with no such penalty at either end.

Pooling all 40 runs from the factorial and composite designs gave the model in part 4: an interior optimum for catalyst, mild curvature for the index and cure temperature, and a strong positive interaction between index and cure temperature that pushes their joint optimum to the high end of both ranges. Confirmation runs at those settings measured 90 seconds on average, short of the model’s predicted 95.8. That gap fits a model whose curvature terms are extrapolated to a corner of the design rather than interpolated between tested points.

1. Runs used in total

72 runs went to the lab. All 72 came back measured; none failed.

2. Suggested optimum

About 90 seconds König pendulum hardness. Five confirmation runs at the recommended settings averaged 89.8 s (84 to 93 s); the fitted model predicts 94.0 s at the same point.

3. Recommended settings

Only the NCO:OH index and the cure temperature move off center, both to the top of their range. The other ten factors showed no real effect on hardness, so they sit at mid-range; pick whichever setting among those is cheapest or easiest to run.

4. Final model

Fitted to the 13-run response surface design, in coded units (X01 and X08, each -1 to +1):

hardness = 78.69 + 7.73 X01 + 9.59 X08 + 0.38 X01² - 6.34 X08² + 3.96 X01·X08

Residual standard deviation: 4.9 s. R² = 0.87.

5. Designs in order

6. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-fractional-factorial.csv, design-2-fractional-factorial-results.csv
• design-3-CCD.csv, design-3-CCD-results.csv
• design-4-confirmation.csv, design-4-confirmation-results.csv

7. Summary

Two of the twelve factors control hardness: the NCO:OH index and the cure temperature. Push both to the top of their allowed range and hardness comes out around 90 s, well above the roughly 56 s average seen across the tested region. The other ten factors, including the catalyst, cure time, flash-off time, and booth humidity, made no measurable difference.

The study ran in four stages. First, a 16-run Plackett-Burman design screened all twelve factors at once, the fewest runs that resolution allows for a factor list this size. Four extra runs at the midpoint of every factor checked for curvature. That design pointed to six candidates: the NCO:OH index, the catalyst, flash-off time, cure temperature, cure time, and booth humidity. It also showed the midpoint runs averaging about 80 s against a factorial average near 56 s, a strong sign that the response bends sharply somewhere inside the box.

Second, a 34-run resolution VI half-fraction on those six candidates freed every two-factor interaction from every main effect, so each candidate could be retested with more power. Only the NCO:OH index and the cure temperature held up as main effects, together with a strong interaction between them. The catalyst, flash-off time, and booth humidity all dropped out. So did cure time, which had looked significant in the screening design: a Plackett-Burman design confounds each main effect with a mix of interactions, and that first result was most likely a stand-in for the NCO:OH-by-cure-temperature interaction, not a real effect of its own.

Third, a 13-run face-centered central composite design mapped the curve in just those two factors, with five runs at the center to pin down the model and check lack of fit. The fitted surface climbs toward both the NCO:OH index and the cure temperature across their full range, with no interior peak: the best point in the box sits at the top corner, NCO:OH at 1.3 and cure temperature at 100 °C.

Fourth, five confirmation runs at that corner averaged 89.8 s, close to the model’s prediction of 94.0 s and clearly above anything else measured in the study.

1. Runs used in total

50 runs. None failed.

2. Suggested optimum

About 92 s König hardness. The six confirmation runs averaged 92.5 s, against a model prediction of 90.5 s.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine sit at the middle of their range because hardness does not depend on them; set them to whatever else the job needs.

4. Final model

Coded units, fitted to the 17 runs of design 2:

hardness = 80.21
         +  4.97 * X01
         -  1.21 * X03
         + 10.86 * X08
         - 14.72 * X03^2
         -  7.67 * X08^2
         +  1.87 * X01*X08

Residual standard deviation 2.98 s, R² = 0.972.

5. Designs in order

6. Run sheets

7. Summary

Twelve factors and no idea which ones count: that calls for screening first, then a close look at the few survivors. I used three steps.

The first was a definitive screening design, 25 runs plus 2 extra runs at the center of the range. A design of this kind separates main effects from two-factor interactions and still shows curvature, which a Plackett-Burman design of the same size cannot do. The extra center runs measured the noise: about 1.2 s. The result was blunt. Three factors moved hardness: the NCO:OH index (X01), the catalyst level (X03) and the cure temperature (X08). The other nine did nothing a lab could see. The three center runs came back near 78 s while the average of the outer runs was 59 s, so the surface bends strongly downward at the edges and the best settings had to lie inside the range, not at a corner.

The second step put those three factors in a face-centered central composite design, 17 runs, with the other nine held at the middle of their range. This gives a full quadratic model. It confirmed the picture: hardness rises with the NCO:OH index and with the cure temperature, but the catalyst has a sharp peak in the middle of its range. Too little catalyst and the film does not cross-link enough; too much and hardness drops away again just as fast. Cure temperature peaks as well, a little above the middle. The index wants to be as high as the range allows, and it helps more at a high cure temperature than at a low one.

The third step was confirmation: six runs at the predicted best settings, three of them with the minor factors pushed to their apparently better levels. The two groups measured the same, 92.1 s and 92.8 s, which settles it. Those minor factors are noise. The predicted 90.5 s and the measured 92.5 s agree inside the scatter of the measurement.

So the answer to the customer is short. Run the index at 1.3, the catalyst at about 200 ppm, and bake at 95 °C. Everything else in the recipe and the booth can stay where it is.

1. Runs used in total

56 runs. None failed.

2. Suggested optimum

König pendulum hardness of about 89 seconds. The three confirmation runs at these settings measured 85.9, 96.5 and 88.3 seconds, a mean of 90.2.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine sit at the middle of their range, so you can keep them wherever production finds them convenient.

4. Final model

Coded units, fitted on all 56 runs:

hardness = 79.14
         +  9.96 * X08
         -  9.72 * X08^2
         +  5.01 * X01
         +  1.34 * X03
         - 11.89 * X03^2
         +  3.37 * X01 * X08

Residual standard deviation 2.97 s, R² = 0.966.

5. Designs in order

6. Run sheets

7. Summary

Twelve factors and no idea which ones matter is a screening problem, so I started with a definitive screening design: 29 runs, main effects free of two-factor interactions, and middle levels that show curvature. The curvature showed up at once. Corner runs came back between 42 and 61 seconds, the centre run at 83. Hardness peaks inside the range, not at a corner, which is why nobody found it by turning one knob at a time.

The screening design pointed at three factors. Cure temperature (X08) has the largest effect and a clear peak. Catalyst (X03) has a strong negative quadratic term, so too little and too much both cost hardness. The NCO:OH index (X01) rises steadily towards the top of its range and works together with cure temperature. Nothing else came out of the noise. That fits the chemistry: index, catalyst and cure temperature set the crosslink density, and the rest of the recipe is there for look and handling.

The second design was a face-centred central composite on those three factors plus film thickness (X10), which the screening run had hinted at. Nineteen runs gave the full quadratic model, and film thickness dropped out as noise. The surface is a broad dome. Hardness stays above 85 seconds over a wide band of cure temperature and catalyst, so the process can drift a little without the customer noticing.

Eight confirmation runs closed it out: three repeats at the predicted optimum and five points just off it. All landed between 84 and 96 seconds, and the repeats averaged 90.2 against a prediction of 88.8. The spread of those repeats, about 5 seconds, is the measurement noise, and it sets the limit on how much further tuning is worth.

Run the clearcoat at index 1.3, 210 ppm DBTDL and 94 °C. That should take you from the 60 seconds the current corner settings give to around 89.

1. Runs used in total

61 runs. None failed.

2. Suggested optimum

About 93 seconds König hardness. Four confirmation runs at the recommended settings gave 102.2, 91.7, 94.0 and 92.6 seconds, a mean of 95.1.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine are set to the middle of their range because hardness does not depend on them.

4. Final model

Coded units, fitted on all 61 runs:

Residual standard deviation 2.33 seconds. R² = 0.985.

5. Designs in order

6. Run sheets

7. Summary

I worked in four steps: screen all twelve factors, look for curvature in the few that survived, map the surface around the best region, then confirm.

The first design was a 20-run Plackett-Burman with three centre points. It separates twelve main effects and still leaves room to judge the noise. Two factors stood out at once: the NCO:OH index and the cure temperature. The centre points told me more than the effects did. They averaged 79.9 seconds against 56.7 for the corners, so the response bends sharply and the best settings lie inside the box, not at a corner.

Curvature means squared terms, and a Plackett-Burman cannot give them. So the second design was a definitive screening design on the six most promising factors, 13 runs plus two extra centre points. It estimates every main effect and every squared term without interactions getting in the way. Fitted together with the first design, it showed what is going on. Hardness rises with the NCO:OH index and the cure temperature, the two work together, and the catalyst has a strong peak in the middle of its range. Every other factor came out near zero, well inside the noise.

The third design was a face-centred central composite on those three factors, squeezed into the region the model pointed at: index 1.1 to 1.3, catalyst 100 to 300 ppm, cure 88 to 100 °C. Seventeen runs put the peak at the top of the index range, at 200 ppm catalyst and 96 °C.

The last six runs checked it. Four replicates at the recommended settings averaged 95.1 seconds, close to the predicted 93. The other two kept those settings but pushed the nine quiet factors to the levels the first design had weakly favoured. They gave 94.6 and 91.3, no better than the rest, which is good evidence those nine do nothing.

Two things the customer should know. The index wants to go as high as the window allows, so hardness may still be climbing above 1.3. And the catalyst has a real optimum near 200 ppm: both no catalyst and 400 ppm cost about 14 seconds, which is the most likely reason the recipe drifted soft when somebody changed the catalyst level.

1. Runs used in total

65 runs. None failed.

2. Suggested optimum

König pendulum hardness of about 90 seconds. Three confirmation runs at the recommended settings gave 90.3, 89.1 and 91.3 seconds, so the model holds.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine factors had no measurable effect, so set them wherever production prefers. We put them at the middle of their range.

4. Final model

Coded units, fitted to all 65 runs:

Residual standard deviation 2.90 seconds. R² = 0.972.

5. Designs in order

6. Run sheets

7. Summary

Twelve factors, and nobody knew which ones counted. So we screened first and mapped the hill afterwards, in four stages: screen, narrow, optimize, confirm.

Design 1 was a 20-run Plackett-Burman. It separates twelve main effects in twenty runs, and because it is not a regular fraction, no main effect sits entirely on top of one two-factor interaction. Two effects stood out well above the Lenth noise margin: cure temperature X08 at 21 seconds and the NCO:OH index X01 at 12. Catalyst X03, cure time X09 and flash-off X07 were borderline. But the two centre points told us more than the effects did. They came back at 80.6 and 77.9 seconds while the twenty corner runs averaged 58. A gap that big means the response is a hill and the corners sit on its slopes.

Design 2 chased the hill. A 13-run definitive screening design on the five live candidates estimates main effects, two-factor interactions and pure quadratic terms, which a Plackett-Burman cannot do. It found the curvature. Catalyst X03 and cure temperature X08 both have strong negative quadratic terms, and catalyst is the clearest case: hardness drops at 0 ppm and at 400 ppm, and peaks in between. Cure time and flash-off were flat, so we dropped them.

Design 3 mapped the surface. A face-centred central composite on X01, X03 and X08, with the other nine factors held at their midpoint, gave a full quadratic model from 16 runs. It fit well (R² = 0.99, residual standard deviation 2.0 seconds) and showed one interaction worth keeping: a higher NCO:OH index pays off more at a high cure temperature than at a low one. The peak sat at X01 = +1, X03 near 0 and X08 near +0.5.

Design 4 was insurance. Four factors had looked marginal in the screen, so we varied X02, X07, X09 and X11 in an 8-run resolution IV fraction while holding the three real drivers at their best settings. All four came back inside the noise. We had missed nothing.

The last five runs confirmed the recommendation and checked how flat the peak is in cure temperature. Hardness changed by less than a second between 88 and 96 °C, which is good news for the plant: the window is wide.

For the recipe, that means three moves. Push the NCO:OH index to 1.3, the top of the allowed range. The model still slopes upward there, so a wider range would be worth asking for. Keep the catalyst near 200 ppm, because both ends cost about 13 seconds of hardness. Bake at about 93 °C. Everything else, film thickness and booth humidity included, is free to set for other reasons. Together these take the clearcoat from the 58 seconds the corner settings averaged to about 90.

1. Runs used in total

65 runs. None failed.

2. Suggested optimum

König pendulum hardness of about 90 seconds. Four confirmation runs at the recommended settings gave 89.7, 93.1, 90.7 and 93.7 seconds, a mean of 91.8.

3. Recommended settings

Only X01, X03 and X08 need to be set deliberately. The other nine factors do not change hardness, so pick them on cost, looks or line speed. The midpoints above are what we measured.

4. Final model

Hardness in seconds, factors in coded units:

hardness = 79.19
         +  6.74 * X01
         +  0.83 * X03
         +  8.61 * X08
         +  3.05 * X01*X08
         - 14.44 * X03^2
         -  8.00 * X08^2

Residual standard deviation 2.06 s, R² = 0.976, fitted on the 26 runs of design 2.

5. Designs in order

6. Run sheets

7. Summary

Twelve factors and no idea which ones matter is a screening problem, so I started with a definitive screening design. It costs 25 runs for 12 factors, estimates every main effect free of two-factor interactions, and, unlike a Plackett-Burman design, it can see curvature. I added two extra centre points to get a read on the noise. That paid off at once: the centre runs came back near 81 seconds while the corner runs averaged 61, so the surface bends a long way down at the edges. Two main effects stood out, the NCO:OH index and the cure temperature, and the curvature loaded onto the catalyst and the cure temperature.

A screening design cannot separate interactions, so design 2 was a response surface on the five candidates the screening left standing: X01, X03, X08, X09 and X12. A central composite design would have cost 30 runs or more, so I used a D-optimal design and got the full quadratic model in 26. The NCO:OH index, the cure temperature, their interaction and the squared terms of catalyst and cure temperature all came out with large t-values. Everything else, including the two factors I had carried along as insurance, sits in the noise.

The catalyst behaves the way the screening hinted. Its squared coefficient of -14.4 is the biggest number in the model and the linear term is almost zero, so hardness peaks in the middle of the range and drops off sharply either way. Too little catalyst leaves the film undercured, too much ruins it. Cure temperature also has an interior optimum, but the positive interaction with the NCO:OH index pushes it up the range: the more isocyanate you run, the hotter you want to cure. At X01 = +1 the model puts the best cure temperature at 94 °C.

Design 3 checked the factors I had parked at their midpoints. The screening had only seen them in a linear model, so I ran an 8-run resolution III fractional factorial on the seven that were not in design 2, with X01, X03 and X08 already at their best settings. No effect came out above 1.6 seconds against a noise level of about 3, so those factors really are inert in the region that matters. That block doubles as a second read on the optimum: its mean was 92.3 seconds.

Design 4 was four straight repeats at the recommended settings. They averaged 91.8 seconds against the model’s prediction of 90.2, well inside the measurement spread. The customer’s soft film is most likely a catalyst level near one end of the range together with a cure that is too cool for the index being run.

1. Runs used in total

67 runs. None failed.

2. Suggested optimum

About 94 s König hardness. The four confirmation runs at the recommended settings gave 96.7, 98.2, 93.1 and 89.9 s, a mean of 94.5 s. The model predicted 91 s.

3. Recommended settings

Only X01, X03 and X08 matter. The other nine are free; the levels above are where we measured them, so keep them there unless something else argues against it.

4. Final model

Coded units, fitted to the 27 runs of designs 2 and 3:

hardness = 78.70
         +  5.85 * X01
         +  2.00 * X03
         +  9.32 * X08
         +  5.80 * X01*X08
         +  1.81 * X03*X09
         - 12.42 * X03^2
         -  8.71 * X08^2

Residual standard deviation 3.14 s, R² = 0.965.

5. Designs in order