doe review torren carlson. goals review of experimental design -we can use this for real...
TRANSCRIPT
DOE Review
Torren Carlson
Goals
Review of experimental design -we can use this for real experiments?
Review/Learn useful Matlab functions Homework problem
An Example
Experimental Considerations Operating objectives
• Maximize productivity• Achieve target polymer properties
Input variables• Catalyst & co-catalyst concentrations• Monomer and co-monomer concentrations• Reactor temperature
Output variables• Polymer production rate• Copolymer composition• Two molecular weight measures
Experimental Design Problem
• Determine optimal input values Brute force approach
• Select values for the five inputs• Conduct semi-batch experiment• Calculate polymerization rate from on-line data• Obtain polymer properties from lab analysis• Repeat until best inputs are found
Statistical techniques (work smarter not harder)• Allow efficient search of input space• Handle nonlinear variable interactions• Account for experimental error
The Experimental Design Problem
The Experimental Design Problem cont.
Design objectives• Information to be gained from experiments
Input variables (factors)• Independent variables• Varied to explore process operating space• Typically subject to known limits
Output variables (responses)• Dependent variables• Chosen to reflect design objectives• Must be measured
Statistical design of experiments• Maximize information with minimal experimental
effort• Complete experimental plan determined in advance
Design Objectives
Comparative experiments• Determine the best alternative
Screening experiments• Determine the most important factors• Preliminary step for more detailed analysis
Response surface modeling• Achieve a specified output target• Minimize or maximize a particular output• Reduce output variability• Achieve robustness to operating conditions• Satisfy multiple & competing objectives
Regression modeling• Determine accurate model over large operating
regime
Incre
asin
g
Com
ple
xity
Input Levels
Input level selection• Low & high limits define operating regime• Must be chosen carefully to ensure feasibility
Two-level designs• Two possible values for each input (low, high)• Most efficient & economical• Ideal for screening designs
Three-level designs• Three possible values for each input (low,
normal, high)• Less efficient but yield more information• Well suited for response surface designs
Empirical Models
Scope• Three factors (x1, x2, x3) & one response (y)
Linear model
• Accounts only for main effects• Requires at least four experiments
Linear model with interactions
• Includes binary interactions• Requires at least seven experiments
3322110 xxxy
3223311321123322110 xxxxxxxxxy
Empirical Models cont. Quadratic model
• Accounts for response curvature• Requires at least ten experiments
Number of parameters/response
2333
2222
21113223
11321123322110
xxxxx
xxxxxxxy
Factors 2 3 4 5 6
Linear 3 4 5 6 7
Interaction 4 7 11 16 22
Quadratic 6 10 15 21 28
Linear vs. Quadratic Effects
Linear function
Two levels sufficient Theoretical basis for
all two-level designs
Quadratic function
Three levels needed to quantify quadratic effect
Two-level design with center points confounds quadratic effects
Two levels adequate to detect quadratic effect
General Design Procedure
1. Determine objectives2. Select output variables 3. Select input variables & their levels4. Perform experimental design5. Execute designed experiments6. Perform data consistency checks7. Statistically analyze the results8. Modify the design as necessary
Response Surface with Matlab>> rstool(x,y,model) x: vector or matrix of input values y: vector or matrix of output values model: ‘linear’ (constant and linear terms), ‘interaction’
(linear model plus interaction terms), ‘quadratic’ (interaction model plus quadratic terms), ‘pure quadratic’ (quadratic model minus interaction terms)
Creates graphical user interface for model analysis VLE data – liquid composition held constant
x = [300 1; 275 1; 250 1; 300 0.75; 275 0.75; 250 0.75; 300 1.25; 275 1.25; 250 1.25]
y = [0.75; 0.77; 0.73; 0.81; 0.80; 0.76; 0.72; 0.74; 0.71]
Experiment 1 2 3 4 5 6 7 8 9
Temperature 300 275 250 300 275 250 300 275 250
Pressure 1.0 1.0 1.0 0.75 0.75 0.75 1.25 1.25 1.25
Vapor Composition
0.75 0.77 0.73 0.81 0.80 0.76 0.72 0.74 0.71
Response Surface Model Example cont.>> rstool(x,y,'linear')>> beta =0.7411 (bias)
0.0005 (T)-0.1333 (P)
>> rstool(x,y,'interaction')>> beta2 = 0.3011 (bias)
0.0021 (T)0.3067 (P)-0.0016 (T*P)
>> rstool(x,y,'quadratic')>> beta3 = -2.4044 (bias)
0.0227 (T)0.0933 (P)-0.0016 (T*P)-0.0000 (T*T)0.1067 (P*P)
Plotting the Main EffectsSyntax maineffectsplot(Y,GROUP)>> load carsmall;>> maineffectsplot(Weight,{Model_Year,Cylinders}, ... 'varnames',
{'Model Year','# of Cylinders'})
Plotting Interaction EffectsSyntax interactionplot(Y,GROUP)>> y = randn(1000,1); % response >> group = ceil(3*rand(1000,4)); % four 3-level factors >> interactionplot(y,group,'varnames',{'A','B','C','D'})
Homework Problem Data file from polymerization experiment
-five factors, four responses (32 runs + CP) Comment on the effects of the factors on the responses
-i.e. does temperature effect the polymerization rate? How? Estimate quadratic effects.
-what are the pros and cons on varying the factors? Relate to the goals.
-Do we need all of the data? Could we do a fractional design?
Work as a group No more than two pages
-Use graphs to illustrate your conclusions