What methodolgy can be used to create an-ethic-of-empowerment?
S1 - Process product optimization using design experiments and response surface methodolgy
44
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
-
Upload
cacheme -
Category
Engineering
-
view
478 -
download
0
description
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 S1 - 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
DOE and RSM
DOE RSM
You
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
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
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
If the current location is
known, a response surface
provides information on:
- Where to go
- How to get there
- Local maxima/minima
Response surfaces
Is there a difference?
vs. ?
Mäkelä et al., Appl. Energ. 131 (2014) 490.
Research problem
,
A and B constant reagents
C reaction product (response), to be maximized
T and P reaction conditions (continuous factors), can be regulated
Response as a contour plot
What kind of equation could
describe C behaviour as a
function of T and P?
C = f(T,P)
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
How can we do it?
The expert method
How can we do it?
The shotgun method
How can we do it?
The ”Soviet” method
xk possibilities with k
factors on x levels
2 factors on 4 levels = 16
experiments
How can we do it?
The classical method
P fixed
T fixed
x
How can we do it?
Factorial design
∆T, ∆P
Factor interaction (diagonal)
Why experimental design?
Reduce the number of experiments
→ Cost, time
Extract maximal information
Understand what happens
Predict future behaviour
Challenges
Multiple factors on multiple levels
6 factors on 3 levels, 36 experiments
Reduce number of factors
Only 2 levels
→ Discard factors
= SCREENING
1
23
1
2
Factorial design
T
P
N:o T P
1 80 2
2 120 2
3 80 3
4 120 32
3
12080
Factorial design
T
P
1-1-1
1In coded levels
The smallest possible full factorial design!
N:o T T coded
P P coded
1 80 -1 2 -1
2 120 1 2 -1
3 80 -1 3 1
4 120 1 3 1
Factorial design
T
P
25 35
45 75
1-1-1
1Design matrix:
N:o T P C
1 -1 -1 25
2 1 -1 35
3 -1 1 45
4 1 1 75
Factorial design
T
P
25 35
45 75
1-1-1
1Average T effect:
T = 20
Average P effect:
P = 30
Interaction (TxP) effect:
TxP = 10
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 23 = 9 ([levels][factors])
How to select proper factor levels?
Research problem
Empirical model:
, ,
⋯
In matrix notation:
→
yy⋮y
1 ⋯1 ⋯1 ⋮ ⋮ ⋱ ⋮1 ⋯
bb⋮b
ee⋮e
Measure ChooseUnknown!
Factorial design
First step
Selection and coding of factor levels
→ Design matrix
T = [80, 120]
P = [2, 3]
K = [0.5, 1]
0.5
280 120
3
1
P
T
K
Factorial design
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
N:o Order T P K
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 1
Randomize!
Orthogonality
For a first-order orthogonal design, X’X is a diagonal matrix:
If two columns are orthogonal, corresponding variables are linearly independent, i.e., assessed independent of each other.
1 11 11 11 1
, 1 1 1 11 1 1 1
1 1 1 11 1 1 1
1 11 11 11 1
4 00 4
2x4
4x22x2
Factorial design
N:o T P K Resp. (C)
1 -1 -1 -1 60
2 1 -1 -1 72
3 -1 1 -1 54
4 1 1 -1 68
5 -1 -1 1 52
6 1 -1 1 83
7 -1 1 1 45
8 1 1 1 80
Design matrix:
-1
-1-1 1
1
1
60 72
52 83
6854
45 80
T
PK
Factorial design
Model equation, main terms:
where
denotes response
factor (T, P or K)
coefficient
residual
mean term (average level)
N:o T P K Resp. (C)
1 -1 -1 -1 60
2 1 -1 -1 72
3 -1 1 -1 54
4 1 1 -1 68
5 -1 -1 1 52
6 1 -1 1 83
7 -1 1 1 45
8 1 1 1 80
Factorial design
Equation = coefficients
bbbb
64.211.52.50.8
bo average value (mean term) Large coefficient → important factor
Interactions usually present
Due to coding, the coefficients are comparable!
Factorial designModel equation with interactions:
N:o T P K TxP TxK PxK TxPxK Resp. (C)
1 -1 -1 -1 1 60
2 1 -1 -1 -1 72
3 -1 1 -1 1 54
4 1 1 -1 -1 68
5 -1 -1 1 -1 52
6 1 -1 1 1 83
7 -1 1 1 -1 45
8 1 1 1 1 80
Factorial design
+-
T
+
-P
+-
K
TxP
- +
TxK
+-
PxK
+
-
Main effects and interactions:
Factorial designEquation = coefficients
bbbbbbbb
64.211.52.50.80.85.000.3
Large interaction b13 (TxK)
Important interaction, main effects cannot be removed
→ Which coefficients to include?
Factorial design
An estimate of model error needed
Center-points
Duplicated experiments
Model residual
Factorial design
Error estimation allows significant testing
Remove insignificant coefficients
Leave main effects
Important interaction, main effect
cannot be removed
Factorial design
Error estimation allows significant testing
Remove insignificant coefficients
Leave main effects
Important interaction, main effect
cannot be removed
Recalculate significance upon removal!
Factorial design
Model residuals
Checking model adequacy
Finding outliers
Normally distributed
→ Random error
Several ways to present residuals
Possibility for response transformation
Factorial design
R2 statistic
Explained variability of
measured response
R2 = 0.9962
99.6% explained
Factorial design
More things to look at
Normal distribution of coefficients
Residual Standardized residual
Residual histogram
Residual vs. time
ANOVA
Factorial design
Factorial design
Prediction:
T = 110
K = 0.9
P = 2 (min. level)
Coded location:
1 0.5 1 0.6 0.3
Predicted response:
74.5 2.4
Session 1Introduction
Why experimental design
Factorial design
Design matrix
Model equation = coefficients
Residual
Response contour
Nomenclature
Factorial design
Screening
Design matrix
Model equation
Response
Effect (main/interaction)
Coefficient
Significance
Contour
Residual
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!
Please send me an email that you are attending the course
