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CHEN 4860 Unit Operations Lab

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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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