CHEN 4860 Unit Operations Lab Design of Experiments (DOE) With excerpts from “Strategy of Experiments” from Experimental Strategies, Inc.

DOE Lab Schedule

DOE Lab Schedule Details  Lecture 2 Limitations of Factorial Design  Centerpoint Design  Screening Designs  Response Surface Designs  Formal Report

Limitations of Factorial Design Circumventing Shortcomings

Limitations of 2 k Factorials  Optimum number of trials? “Signal-to-Noise” ratio  Nonlinearity? 3 k factorial or center point factorial  Inoperable regions? Tuck method  Too many variables? Screening designs  Fractional Factorial  Plackett-Burman  Need detailed understanding? Response Surface Plots

Number of Runs vs. Signal/Noise Ratio  Confidence Interval or Signal FEavg + t*SeffFEavg - t*Seff   FEavg + t*SeffFEavg - t*Seff

Number of Runs vs. Signal/Noise Ratio  Avg + t*Seff   = 2*t*Seff  Seff = 2*Se/sqrt(N)   = 2*2*t*Se/sqrt(N)  Rearrange, N (total number of trials) is:  N=[2*2*t/(/Se)]^2  Estimate t as approximately 2  N=[(7 or 8)/(/Se)]^2

Number of Runs vs. Signal/Noise Ratio  (/Se) is the signal to noise ratio. Ratio (/Se) Number of Runs to to to to 256

Number of Runs vs. Signal/Noise Ratio

Factorial Design (2 k )  2 is number of levels (low, high)  What about non-linearity? A B C LO, HI, LO LO, LO, LO Pts (A, B, C) HI, LO, LO HI, HI, LO HI, LO, HI HI, HI, HI LO, HI, HI LO, HI, LO

Centerpoint Test for Nonlinearity  Additional pts. located at midpoints of factor levels. (No longer 8 runs, Now 20) A B C LO, HI, LO LO, LO, LO Pts (A, B, C) HI, LO, LO HI, HI, LO HI, LO, HI HI, HI, HI LO, HI, HI LO, HI, LO

Centerpoint Test for Non-linearity  Effect(nonlinearity) =Ynoncpavg-Ycavg  What about significance? Calculate variance of non-centerpoint (cp) tests as normal (S^2) Calculate variances of cp (Sc^2) Degrees of Freedom (df) for base design  (#noncp runs)*(reps/run-1) DF for cp (dfc)  (#cp runs-1) Calculate weighted avg variance  Se^2 = [(df*S^2)+(dfc*Sc^2)]/(dfc+df) Snonlin=Se*sqrt(1/Nnoncp+1/Ncp) dftot=dfc+df Lookup t from table using dftot Calculate DL = + t*Snonlin

Better Way to Test Non-Linearity  Use response surface plots with Face Centered Cubes, Box-Behnken Designs, and others. Face-Centered Cube (15 runs)Box-Behnken Design (13 runs)

Inoperable Regions  Don’t shrink design, pull corner inward X1 X2 X1 X2 BADGOOD

Diagnosing the Environment  Too many variables, use screening designs to pick best candidates for factorial design Screening Designs Full Factorial Designs Response Surface Designs Many Independent Variables Fewer independent variables (<5) “Crude” Information Quality Linear Prediction Quality non-linear Prediction

Screening Designs  Benefits: Only few more runs than factors needed Used for 6 or more factors  Limitations: Can’t measure any interactions or non- linearity. Assume effects are independent of each other

Screening Designs  # of runs needed # of FactorsFull FactorialScreening Design , , , ,388, ,217,72828

Screening Designs  Fractional Factorial Interactions are totally confounded with each other in identifiable sets called “aliases”. Available in sizes that are powers of 2.  Plackett-Burman Interactions are partially correlated with other effects in identifiable patterns Available in sizes that are multiples of 4.

Fractional Factorial (1/2-Factorial)  Suppose we want to study 4 factors, but don’t want to run the 16 experiments (or 32 with replication). ABCABACBCABC Typical Full Factorial

Fractional Factorial  What happens if we replace the unlikely ABC interaction with a new variable D? The other 2 factor interactions become confounded with one another to form “aliases”  AB=CD, AC=BD, AD=BC The other 3 factor interactions become confounded with the main factor to also form “aliases”  A=BCD, B=ACD, C=ABD

Fractional Factorial  Ignoring the unlikely 3 factor interaction, we have… ABCDAB=CDAC=BDAD=BC

Fractional Factorial  Calculations performed the same  If the effects of interactions prove to be significant, perform a full factorial with the main effects to determine which interaction is most important.

Plackett-Burman  Benefits: Can study more factors in less experiments  Costs: Main factor in confounded with all 2 factor interactions.  Suppose we want to study 7 factors, but only want to run 8 experiments (or 16 with replication).

Plackett-Burman A=BD= CG=EF B=AD= CE=FG C=AG= BE=DF D=AB= CF=EG E=AF= BC=DG F=AE= BG=CD G=AC= BF=DE

Plackett-Burman  Calculations performed the same  How do you handle confounding of main affects? Use General Rules:  Heredity: Large main effects have interactions  Sparsity: Interactions are of a lower magnitude than main effects  Process Knowledge Use Reflection

Reflection of Plackett-Burman  Reruns the same experiment with the opposite signs. A=BD= CG=EF B=AD= CE=FG C=AG= BE=DF D=AB= CF=EG E=AF=B C=DG F=AE=B G=CD G=AC= BF=DE

Reflection of Plackett-Burman  Treats 2 factor responses as noise  Average the effects from each run to determine the true main effect Normal  E(A)calc=E(A)act-Noise Reflected  E(A)calcr=E(A)actr+Noise Combined  E(A)est=(E(A)calc+E(A)calcr)/2

Response Surface Plots  Need detail for more than 1 response variable and related interactions  Types 3 level factorial Face-Centered Cube Design Box-Behnken Design  Many experiments required

Size of Response Surface Design Number of Factors 3-level Factorial Face- Centered Cube* Box- Behnken* *extra space left for multiple center points due to blocking

Summary  Diagnose your problem  Use one of the many different methods outlined to circumvent it  Many more options and designs listed on the web

Formal Memo  Follow outline presented for formal memo presented on Dr. Placek’s website. Executive Summary Discussion and Results Appendix with Data, Calcs, References, etc.  **GOAL IS PLANNING**

Formal Memo Report Questions  What are your objectives?  How did you minimize random and bias error?  What variables did you control and why?  What variables did you measure and why?  What were the results of your experiment?  Which factors were most important and why?  What is your theory (based on chem-eng knowledge) on why the experiment turned out the way it did?  Was there any codependence?  What will be your next experiment?  What would you do differently the next time?