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 landbruksuniversitet Swedish University of Agricultural Sciences Department of Forest Biomaterials and Technology Division of Biomass Technology and Chemistry Umeå, Sweden
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An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system. Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.”
Transcript of S2 - Process product optimization using design experiments and response surface methodolgy
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
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
Session 1Introduction
Why experimental design
Factorial design
Design matrix
Model equation = coefficients
Residual
Response contour
Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
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.
Research problem
Empirical model:
, ,
⋯
In matrix notation:
→
yy⋮y
1 ⋯1 ⋯1 ⋮ ⋮ ⋱ ⋮1 ⋯
bb⋮b
ee⋮e
Measure ChooseUnknown!
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
Coefficients
Model coefficients
Least squares calculation
Coefficient significance?
8 model terms
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
Coefficients
Remove unimportant model terms
and calculate new coefficients
Predicted response
Calculate predicted response,
Based on X and b
Graphical tools are useful
Residuals
Calculate model residuals
A common way is to scale the residuals
E.g. standardized residual
→ Need an error approximation
Error estimation
Estimated error of predicted response
∑ , where p = k + 1
Standardized residuals
>│3│ a potential outlier
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
R2 statistic
Data variability explained by the
model
Compares model and total sum of
squares
R2 = 95.2%
Response contour
For a k=3 design, 2D contours require
one constant factor
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
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
Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
Nomenclature
Design matrix
Coefficient
Degree of freedom
Prediction
Residual
Outlier
ANOVA
Sum of squares
Mean square
Contour
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
Thank you for listening!
