DOE ReviewGoalsAn ExampleExperimental ConsiderationsExperimental DesignThe Experimental Design ProblemThe Experimental Design Problem cont.Design ObjectivesInput LevelsEmpirical ModelsEmpirical Models cont.Linear vs. Quadratic EffectsGeneral Design ProcedureResponse Surface with MatlabResponse Surface Model Example cont.Plotting the Main EffectsPlotting Interaction EffectsHomework ProblemDOE ReviewTorren CarlsonGoalsReview of experimental design -we can use this for real experiments?Review/Learn useful Matlab functionsHomework problemAn ExampleExperimental ConsiderationsOperating objectives•Maximize productivity•Achieve target polymer propertiesInput variables•Catalyst & co-catalyst concentrations•Monomer and co-monomer concentrations•Reactor temperatureOutput variables•Polymer production rate•Copolymer composition•Two molecular weight measuresExperimental DesignProblem•Determine optimal input valuesBrute 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 foundStatistical techniques (work smarter not harder)•Allow efficient search of input space•Handle nonlinear variable interactions•Account for experimental errorThe Experimental Design ProblemThe Experimental Design Problem cont.Design objectives•Information to be gained from experimentsInput variables (factors)•Independent variables•Varied to explore process operating space•Typically subject to known limitsOutput variables (responses)•Dependent variables•Chosen to reflect design objectives•Must be measuredStatistical design of experiments•Maximize information with minimal experimental effort•Complete experimental plan determined in advanceDesign ObjectivesComparative experiments•Determine the best alternativeScreening experiments•Determine the most important factors•Preliminary step for more detailed analysisResponse surface modeling•Achieve a specified output target•Minimize or maximize a particular output•Reduce output variability•Achieve robustness to operating conditions•Satisfy multiple & competing objectivesRegression modeling•Determine accurate model over large operating regimeIncreasing ComplexityInput Levels Input level selection•Low & high limits define operating regime•Must be chosen carefully to ensure feasibilityTwo-level designs•Two possible values for each input (low, high)•Most efficient & economical•Ideal for screening designsThree-level designs•Three possible values for each input (low, normal, high)•Less efficient but yield more information•Well suited for response surface designsEmpirical ModelsScope• Three factors (x1, x2, x3) & one response (y)Linear model•Accounts only for main effects•Requires at least four experimentsLinear model with interactions•Includes binary interactions•Requires at least seven experiments3322110xxxy3223311321123322110xxxxxxxxxyEmpirical Models cont.Quadratic model•Accounts for response curvature•Requires at least ten experimentsNumber of parameters/response233322222111322311321123322110xxxxxxxxxxxxyFactors 2 3 4 5 6Linear 3 4 5 6 7Interaction 4 7 11 16 22Quadratic 6 10 15 21 28Linear vs. Quadratic EffectsLinear functionTwo levels sufficientTheoretical basis for all two-level designsQuadratic functionThree levels needed to quantify quadratic effectTwo-level design with center points confounds quadratic effectsTwo levels adequate to detect quadratic effectGeneral Design Procedure1. 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 necessaryResponse Surface with Matlab>> rstool(x,y,model)x: vector or matrix of input valuesy: vector or matrix of output valuesmodel: ‘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 analysisVLE data – liquid composition held constantx = [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 9Temperature 300 275 250 300 275 250 300 275 250Pressure 1.0 1.0 1.0 0.75 0.75 0.75 1.25 1.25 1.25Vapor Composition0.75 0.77 0.73 0.81 0.80 0.76 0.72 0.74 0.71Response 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 ProblemData 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 groupNo more than two pages -Use graphs to illustrate your
View Full Document