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CHEN 4860 Unit Operations Lab

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

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CHEN 4860 Unit Operations Lab

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

  2. DOE Lab Schedule

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

  4. Limitations of Factorial Design Circumventing Shortcomings

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

  6. Number of Runs vs. Signal/Noise Ratio • Confidence Interval or Signal D FEavg - t*Seff FEavg + t*Seff FEavg - t*Seff D FEavg + t*Seff

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

  8. Number of Runs vs. Signal/Noise Ratio • (D/Se) is the signal to noise ratio.

  9. Number of Runs vs. Signal/Noise Ratio

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

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

  12. 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

  13. 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)

  14. Inoperable Regions • Don’t shrink design, pull corner inward BAD GOOD X2 X2 X1 X1

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

  16. 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

  17. Screening Designs • # of runs needed

  18. 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.

  19. 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). Typical Full Factorial

  20. 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

  21. Fractional Factorial • Ignoring the unlikely 3 factor interaction, we have…

  22. 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.

  23. 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).

  24. Plackett-Burman

  25. 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

  26. Reflection of Plackett-Burman • Reruns the same experiment with the opposite signs.

  27. 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

  28. 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

  29. Size of Response Surface Design *extra space left for multiple center points due to blocking

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

  31. 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**

  32. 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?

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