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 Increasing Complexity

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 (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

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

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 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

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; ; ; ; ; ; ]  y = [0.75; 0.77; 0.73; 0.81; 0.80; 0.76; 0.72; 0.74; 0.71] Experiment Temperature Pressure Vapor Composition

Response Surface Model Example cont. >> rstool(x,y,'linear') >> beta =0.7411(bias) (T) (P) >> rstool(x,y,'interaction') >> beta2 = (bias) (T) (P) (T*P) >> rstool(x,y,'quadratic') >> beta3 = (bias) (T) (P) (T*P) (T*T) (P*P)

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

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'})

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