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CHEN 4860 Unit Operations Lab

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3k factorial or center point factorial. Inoperable regions? Tuck method. Too many variables? Screening designs. Fractional Factorial. Plackett-Burman ... – PowerPoint PPT presentation

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Title: CHEN 4860 Unit Operations Lab


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 tSeff
FEavg - tSeff
FEavg tSeff
FEavg - tSeff
D
7
Number of Runs vs. Signal/Noise Ratio
  • Avg tSeff
  • D 2tSeff
  • Seff 2Se/sqrt(N)
  • D 22tSe/sqrt(N)
  • Rearrange, N (total number of trials) is
  • N22t/(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
A
LO, LO, LO
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
A
LO, LO, LO
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 (S2)
  • Calculate variances of cp (Sc2)
  • Degrees of Freedom (df) for base design
  • (noncp runs)(reps/run-1)
  • DF for cp (dfc)
  • (cp runs-1)
  • Calculate weighted avg variance
  • Se2 (dfS2)(dfcSc2)/(dfcdf)
  • SnonlinSesqrt(1/Nnoncp1/Ncp)
  • dftotdfcdf
  • Lookup t from table using dftot
  • Calculate DL tSnonlin

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
  • Dont shrink design, pull corner inward

BAD
GOOD
X2
X2
X1
X1
15
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
  • Cant 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 dont
    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
  • ABCD, ACBD, ADBC
  • The other 3 factor interactions become confounded
    with the main factor to also form aliases
  • ABCD, BACD, CABD

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)calcE(A)act-Noise
  • Reflected
  • E(A)calcrE(A)actrNoise
  • Combined
  • E(A)est(E(A)calcE(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. Placeks 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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