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DOE Review Torren Carlson. Goals  Review of experimental design -we can use this for real experiments?  Review/Learn useful Matlab functions  Homework.

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Presentation on theme: "DOE Review Torren Carlson. Goals  Review of experimental design -we can use this for real experiments?  Review/Learn useful Matlab functions  Homework."— Presentation transcript:

1 DOE Review Torren Carlson

2 Goals  Review of experimental design -we can use this for real experiments?  Review/Learn useful Matlab functions  Homework problem

3 An Example

4 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

5 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

6 The Experimental Design Problem

7 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

8 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 Increasing Complexity

9 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

10 Empirical Models  Scope Three factors (x 1, x 2, x 3 ) & 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

11 Empirical Models cont.  Quadratic model Accounts for response curvature Requires at least ten experiments  Number of parameters/response Factors23456 Linear34567 Interaction47111622 Quadratic610152128

12 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

13 General Design Procedure 1.Determine objectives 2.Select output variables 3.Select input variables & their levels 4.Perform experimental design 5.Execute designed experiments 6.Perform data consistency checks 7.Statistically analyze the results 8.Modify the design as necessary

14 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] Experiment123456789 Temperature300275250300275250300275250 Pressure1.0 0.75 1.25 Vapor Composition 0.750.770.730.810.800.760.720.740.71

15 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)

16 Plotting the Main Effects Syntax maineffectsplot(Y,GROUP) >> load carsmall; >> maineffectsplot(Weight,{Model_Year,Cylinders},... 'varnames',{'Model Year','# of Cylinders'})

17 Plotting Interaction Effects Syntax 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'})

18 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

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