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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)

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
• 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

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
• 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
• 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