S2 - Process product optimization using design experiments and response surface methodolgy
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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.”
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S2 - Process product optimization using design experiments and response surface methodolgy
- 1. 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
- 2. 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
- 3. Session 1 Introduction Why experimental design Factorial design Design matrix Model equation = coefficients Residual Response contour
- 4. Session 2 Factorial design Research problem Design matrix Model equation = coefficients Degrees of freedom Predicted response Residual ANOVA R2 Response contour
- 5. 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 Parameter Low High Temperature, A (°C) 100 120 Catalyst conc., B (%) 4 8 Time, C (min) 20 30 Molecular weight (in order): 2400, 2410, 2315, 2510, 2615, 2625, 2400, 2750 Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 131.
- 6. Research problem Empirical model: ݕ ൌ ݂ ۯ, ۰, ۱ ߝ ݕ ൌ ߚ ߚଵݔଵ ߚଶݔଶ ⋯ ߚݔݔ ߝ In matrix notation: ܡ ൌ ܆܊ ܍ → yଵ yଶ ⋮ y୬ ൌ 1 ݔଵଵ ݔଶଵ ⋯ ݔଵ 1 ݔଵଶ ݔଶଶ ⋯ ݔଶ 1 ⋮ ⋮ ⋱ ⋮ 1 ݔଵ ݔଶ ⋯ ݔ b bଵ ⋮ b୩ eଵ eଶ ⋮ e୬ Measure Choose Unknown!
- 7. Design matrix N: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
- 8. Coefficients Model coefficients Least squares calculation Coefficient significance? 8 model terms
- 9. Degrees of freedom In statistics, degrees of freedom (df) are lost by imposing linear constraints on a sample E.g. sample variance: ݏଶ ൌ Σ ሺ௬ି௬ത ሻమ సభ ିଵ where Σ ݕ െ ݕത ൌ 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
- 10. Coefficients Remove unimportant model terms and calculate new coefficients
- 11. Predicted response Calculate predicted response, ࢟ෝ Based on X and b Graphical tools are useful
- 12. Residuals Calculate model residuals A common way is to scale the residuals E.g. standardized residual → Need an error approximation
- 13. Error estimation Estimated error of predicted response ߪොଶ ൌ Σ ሺ௬ି௬ොሻమ ୀଵ , where p = k + 1 ି Standardized residuals >│3│ a potential outlier
- 14. ANOVA Analysis of variance (ANOVA) allows to statistically test the model Fisher’s F test H0: ߚ ൌ ߚଵ ൌ ⋯ ൌ ߚ ൌ 0 H1: ߚ ് 0 for at least one j Parameter df Sum of squares (SS) Mean square (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
- 15. R2 statistic ݕො ݕ ݁ Data variability explained by the model Compares model and total sum of squares R2 = 95.2%
- 16. Response contour For a k=3 design, 2D contours require one constant factor
- 17. Response contour Use of the model Optimization within specific coordinates Prediction of an optimum Verification A = 115 and B = 7 ܠܕ ൌ ሾ1 .5 .5 1 .25ሿ ݕො ൌ 2645 േ 90
- 18. 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 B C
- 19. Session 2 Factorial design Research problem Design matrix Model equation = coefficients Degrees of freedom Predicted response Residual ANOVA R2 Response contour
- 20. Nomenclature Design matrix Coefficient Degree of freedom Prediction Residual Outlier ANOVA Sum of squares Mean square Contour
- 21. 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
- 22. Thank you for listening!