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

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g

Com

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xity

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 (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 experiments

3322110 xxxy

3223311321123322110 xxxxxxxxxy

Empirical Models cont. Quadratic model

• Accounts for response curvature• Requires at least ten experiments

Number of parameters/response

2333

2222

21113223

11321123322110

xxxxx

xxxxxxxy

Factors 2 3 4 5 6

Linear 3 4 5 6 7

Interaction 4 7 11 16 22

Quadratic 6 10 15 21 28

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 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 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; 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 9

Temperature 300 275 250 300 275 250 300 275 250

Pressure 1.0 1.0 1.0 0.75 0.75 0.75 1.25 1.25 1.25

Vapor Composition

0.75 0.77 0.73 0.81 0.80 0.76 0.72 0.74 0.71

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)

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 conclusions