S2 - Process product optimization using design experiments and response surface methodolgy
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Transcript of S2 - Process product optimization using design experiments and response surface methodolgy
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Process/product optimization using design of experiments and response surface methodology
M. Mäkelä
Sveriges landbruksuniversitetSwedish University of Agricultural Sciences
Department of Forest Biomaterials and TechnologyDivision of Biomass Technology and ChemistryUmeå, Sweden
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Contents
Practical course, arranged in 4 individual sessions:
Session 1 – Introduction, factorial design, first order models
Session 2 – Matlab exercise: factorial design
Session 3 – Central composite designs, second order models, ANOVA,
blocking, qualitative factors
Session 4 – Matlab exercise: practical optimization example on given data
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Session 1Introduction
Why experimental design
Factorial design
Design matrix
Model equation = coefficients
Residual
Response contour
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Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
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Research problem
A chemist is interested on the effect of temperature (A), catalyst concentration (B)
and time (C) on the molecular weight of polymer produced
She performed a 23 factorial design
Molecular weight (in order): 2400, 2410, 2315, 2510, 2615, 2625, 2400, 2750
Parameter Low High
Temperature, A (°C) 100 120
Catalyst conc., B (%) 4 8
Time, C (min) 20 30
Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 131.
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Research problem
Empirical model:
, ,
⋯
In matrix notation:
→
yy⋮y
1 ⋯1 ⋯1 ⋮ ⋮ ⋱ ⋮1 ⋯
bb⋮b
ee⋮e
Measure ChooseUnknown!
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Design matrixN:o A B C Resp.
1 -1 -1 -1 2400
2 1 -1 -1 2410
3 -1 1 -1 2315
4 1 1 -1 2510
5 -1 -1 1 2615
6 1 -1 1 2625
7 -1 1 1 2400
8 1 1 1 2750
Build a design matrix with interactions in Matlab
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Coefficients
Model coefficients
Least squares calculation
Coefficient significance?
8 model terms
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Degrees of freedom
In statistics, degrees of freedom (df) are lost by imposing linear constraints on
a sample
E.g. sample variance:
∑
where ∑ 0. Hence 1 residuals can be used to completely
determine the other.
In RSM, dfs are lost due to the constraints imposed by the coefficients
Residual df with observations and 1 model terms
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Coefficients
Remove unimportant model terms
and calculate new coefficients
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Predicted response
Calculate predicted response,
Based on X and b
Graphical tools are useful
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Residuals
Calculate model residuals
A common way is to scale the residuals
E.g. standardized residual
→ Need an error approximation
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Error estimation
Estimated error of predicted response
∑ , where p = k + 1
Standardized residuals
>│3│ a potential outlier
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ANOVA
Analysis of variance (ANOVA) allows to statistically test the model
Fisher’s F test
H0: ⋯ 0
H1: 0 for at least one j
Parameter df Sum of squares (SS)
Meansquare (MS)
F-value p-value
Total corrected n-1 SStot MStot
Regression k SSmod MSmod MSmod/MSres
<0.05>0.05
Residual n-k-1 SSres MSres
Lack of fit
Pure error
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R2 statistic
Data variability explained by the
model
Compares model and total sum of
squares
R2 = 95.2%
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Response contour
For a k=3 design, 2D contours require
one constant factor
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Response contour
Use of the model
Optimization within specific coordinates
Prediction of an optimum
Verification
A = 115 and B = 7
1.5.51.25
2645 90
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Research problem
The chemist has a possibility to perform three
additional center-points in her factorial design
Molecular weight values: 2800, 2750, 2810
Can this change our view on the behaviour of
the system?
A
BC
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Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
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Nomenclature
Design matrix
Coefficient
Degree of freedom
Prediction
Residual
Outlier
ANOVA
Sum of squares
Mean square
Contour
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Contents
Practical course, arranged in 4 individual sessions:
Session 1 – Introduction, factorial design, first order models
Session 2 – Matlab exercise: factorial design
Session 3 – Central composite designs, second order models, ANOVA,
blocking, qualitative factors
Session 4 – Matlab exercise: practical optimization example on given data
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Thank you for listening!