Process/product optimization using design of experiments and response surface methodology

Transcription

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 DOE and RSM You DOE RSM Design of experiments (DOE) Planning experiments Maximum information from minimized number of experiments Response Surface Methodology (RSM) Identifying and fitting an appropriate response surface model Statistics, regression modelling & optimization

3 What to expect? Background and philosophy Theory Nomenclature Practical demonstrations and exercises (Matlab) What not? Matrix algebra Detailed equation studies Statistical basics Detailed listing of possible designs

4 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

5 Session 1 Introduction Why experimental design Factorial design Design matrix Model equation = coefficients Residual Response contour

6 Response surfaces If the current location is known, a response surface provides information on: - Where to go - How to get there - Local maxima/minima

7 Is there a difference? vs.? Mäkelä et al., Appl. Energ. 131 (2014) 490.

8 Research problem, A and B constant reagents C reaction product (response), to be maximized T and P reaction conditions (continuous factors), can be regulated

9 Response as a contour plot What kind of equation could describe C behaviour as a function of T and P? C = f(t,p)

10 What else do we want to know? Which factors and interactions are important Positions of local optima (if they exist) Surface and surface function around an optimum Direction towards an optimum Statistical significance

11 How can we do it? The expert method

12 How can we do it? The shotgun method

13 How can we do it? The Soviet method x k possibilities with k factors on x levels 2 factors on 4 levels = 16 experiments

14 How can we do it? The classical method P fixed x T fixed

15 How can we do it? Factorial design T, P Factor interaction (diagonal)

16 Why experimental design? Reduce the number of experiments Cost, time Extract maximal information Understand what happens Predict future behaviour

17 Challenges Multiple factors on multiple levels 6 factors on 3 levels, 3 6 experiments Reduce number of factors Only 2 levels Discard factors = SCREENING 2 3 1

18 Factorial design 3 N:o T P P T

19 Factorial design 1 In coded levels N:o T T coded P P coded P T The smallest possible full factorial design!

20 Factorial design Design matrix: N:o T P C P T

21 Factorial design Average T effect: T = 20 P Average P effect: T 1 P = 30 Interaction (TxP) effect: TxP = 10

22 Research problem,, A and B constant reagents C reaction product (response), to be maximized T, P and K reaction conditions (continuous factors) at two different levels Number of experiments 2 3 = 9 ([levels] [factors] ) How to select proper factor levels?

23 Research problem Empirical model:,, In matrix notation: y y y b b b e e e Measure Choose Unknown!

24 Factorial design First step Selection and coding of factor levels Design matrix K 1 T = [80, 120] P P = [2, 3] K = [0.5, 1] T

25 Factorial design N:o Order T P K Factorial design matrix Notice symmetry in diffent colums Inner product of two colums is zero E.g. T P = 0 This property is called orthogonality Randomize!

26 Orthogonality For a first-order orthogonal design, X X is a diagonal matrix: , 2x x x If two columns are orthogonal, corresponding variables are linearly independent, i.e., assessed independent of each other.

27 Factorial design Design matrix: N:o T P K Resp. (C) P K T

28 Factorial design N:o T P K Resp. (C) Model equation, main terms: where denotes response factor (T, P or K) coefficient residual mean term (average level)

29 Factorial design Equation = coefficients b b b b b o average value (mean term) Large coefficient important factor Interactions usually present Due to coding, the coefficients are comparable!

30 Factorial design Model equation with interactions: N:o T P K TxP TxK PxK TxPxK Resp. (C)

31 Factorial design Main effects and interactions: T P + K + - TxP TxK PxK

32 Factorial design Equation = coefficients b b b b b b b b Large interaction b 13 (TxK) Important interaction, main effects cannot be removed Which coefficients to include?

33 Factorial design An estimate of model error needed Center-points Duplicated experiments Model residual

34 Factorial design Error estimation allows significant testing Remove insignificant coefficients Leave main effects Important interaction, main effect cannot be removed

35 Factorial design Error estimation allows significant testing Remove insignificant coefficients Leave main effects Important interaction, main effect cannot be removed Recalculate significance upon removal!

36 Factorial design Model residuals Checking model adequacy Finding outliers Normally distributed Random error Several ways to present residuals Possibility for response transformation

37 Factorial design R 2 statistic Explained variability of measured response R 2 = % explained

38 Factorial design More things to look at Normal distribution of coefficients Residual Standardized residual Residual histogram Residual vs. time ANOVA

39 Factorial design

40 Factorial design Prediction: T = 110 K = 0.9 P = 2 (min. level) Coded location: Predicted response:

41 Session 1 Introduction Why experimental design Factorial design Design matrix Model equation = coefficients Residual Response contour

42 Nomenclature Factorial design Screening Design matrix Model equation Response Effect (main/interaction) Coefficient Significance Contour Residual

43 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

44 Thank you for listening! Please send me an that you are attending the course