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UMass Amherst CHE 401 - CHE 401 Lecture Notes

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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 CarlsonGoalsReview of experimental design -we can use this for real experiments?Review/Learn useful Matlab functionsHomework problemAn ExampleExperimental 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 measuresExperimental 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 errorThe Experimental Design ProblemThe 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 advanceDesign 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 regimeIncreasing ComplexityInput 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 designsEmpirical 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 experiments3322110xxxy3223311321123322110xxxxxxxxxyEmpirical Models cont.Quadratic model•Accounts for response curvature•Requires at least ten experimentsNumber of parameters/response233322222111322311321123322110xxxxxxxxxxxxyFactors 2 3 4 5 6Linear 3 4 5 6 7Interaction 4 7 11 16 22Quadratic 6 10 15 21 28Linear vs. Quadratic EffectsLinear functionTwo levels sufficientTheoretical basis for all two-level designsQuadratic functionThree levels needed to quantify quadratic effectTwo-level design with center points confounds quadratic effectsTwo 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 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 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 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


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UMass Amherst CHE 401 - CHE 401 Lecture Notes

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