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DOE Review Torren Carlson

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

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An Example

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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

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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

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The Experimental Design Problem

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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

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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

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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

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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

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

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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

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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

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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

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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)

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

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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'})

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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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