The document describes a 23 factorial design used to optimize chromatographic conditions. Three factors (temperature, ethanol concentration, and mobile phase flow rate) were each tested at two levels in a 23 factorial design. Resolution was used as the response. Regression analysis was performed on the results to develop a polynomial equation relating the factors and their interactions to resolution. This allowed determination of optimum conditions for chromatographic separation.

2. •In factorial design, levels of factors are independently
varied, each factor at two or more levels.
•The effects that can e attributed to the factor and their
interactions are assed with maximum efficiency in
factorial design. So predictions based on results of an
undersigned experiment will be less reliable than those
which would be obtained in a factorial design.
•The optimization procedure is facilitated by costruction
of an equation that describes the experimental results
as a function of the factorial design. Here in case of a
factorial , a polynomial equation can be constructed
where the coefficients in the equation are related to
effects and interations of the factors.

3. •Now factorial design with fators at only two
level is called as 2n factorial design where n is
the no. of factors. these designs are simplest
and often adequate to achieve the experimental
objectives.
•The optimization procedure is facilitated by
fitting of an empirical polynomial equation to the
experimental results. The equation from for 2n
factorial experiment is of the following form:
• Y= b0 + b1X1 + b2X2 + b3X3 +………+
b12X1 X2 + b13X1 X3
+ b23X2 X3+……+ b123X1 X2 X3

4.  Optimization of chromatographic conditions for
both c8 and c18 columns carried out by a factorial
design which evaluates temperature, ethanol
concentration and mobile phase flow rate.
 So design matrix would be 23 factorial design for c
8
column.

5. NO. FACTORS LOW LEVEL HIGH LEVEL
1 TEMP (X1) 30 50
2 %ETHANOL (X2) 55 60
FLOW RATE OF M.
3 PHASE (X3) 0.1 0.2

6.  In chromatographic condition responses can
be
1. Efficiency
2. Retention factor
3. Assymetry
4. Retention time
5. Resolution
 In this example resolution is considered as
response

7. NO. X1 X2 X3
1 -1 -1 -1
2 -1 1 -1
3 1 -1 -1
4 1 1 -1
5 -1 -1 1
6 -1 1 1
7 1 -1 1
8 1 1 1

8. Data analysis for 23 factorial design
resolution/res
temp %ethanol flow rate ponse
30 55 0.1 2.4
50 55 0.1 1.8
30 60 0.1 1.9
50 60 0.1 1.4
30 55 0.2 2.4
50 55 0.2 1.8
30 60 0.2 1.6
50 60 0.2 1.3

9.  The formula for transformation is
X-the average of the two levels
one half the difference of the levels

10. RESPO
X1 X2 NSE
NO. X1 X2 X3 X1 X2 X1 X3 X2 X3 X3 (Y)
1 -1 -1 -1 1 1 1 -1 2.4
2 -1 1 -1 -1 1 -1 1 1.8
3 1 -1 -1 -1 -1 1 1 1.9
4 1 1 -1 1 -1 -1 -1 1.4
5 -1 -1 1 1 -1 -1 1 2.4
6 -1 1 1 -1 -1 1 -1 1.8
7 1 -1 1 -1 1 -1 -1 1.6
8 1 1 1 1 1 1 1 1.3

11.  The coefficients for polynomial equation are
calculated as
Σ XY/2n
Where X is the value (+1 or -1) in the column
appropriate for the coefficient being calculated,
Y is the response.

12. X1Y X2 Y X3Y X1X2Y X1X3Y X2X3Y X1X2X3Y Y
-2.4 -2.4 -2.4 2.4 2.4 2.4 -2.4 2.4
-1.8 1.8 -1.8 -1.8 1.8 -1.8 1.8 1.8
1.9 -1.9 -1.9 -1.9 -1.9 1.9 1.9 1.9
1.4 1.4 -1.4 1.4 -1.4 -1.4 -1.4 1.4
-2.4 -2.4 2.4 2.4 -2.4 -2.4 2.4 2.4
-1.8 1.8 1.8 -1.8 -1.8 1.8 -1.8 1.6
1.6 -1.6 1.6 -1.6 1.6 -1.6 -1.6 1.6
1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3
average
B1 b2 b3 b12 b13 b23 b123 b0
-0.275 -0.25 -0.05 0.05 -0.05 0.025 0.025 1.825

13. Summary output
Regression Statistics
Multiple R 1
R Square 1
Adjusted R
Square 65535
Standard
Error 0
Observations 8
ANOVA
Significance
Df SS MS F F
Regression 7 1.175 0.167857 0 #NUM!
Residual 0 6.9E-31 65535
Total 7 1.175

14. Lower Upper
Coefficie Standard Lower Upper 95.0 95.0
nts Error t Stat P-value 95% 95% % %
Intercept 1.825 0 65535 #NUM! 1.825 1.825 1.825 1.825
X1 -0.275 0 65535 #NUM! -0.275 -0.275 -0.275 -0.275
X2 -0.25 0 65535 #NUM! -0.25 -0.25 -0.25 -0.25
X3 -0.05 0 65535 #NUM! -0.05 -0.05 -0.05 -0.05
X1 X2 0.05 0 65535 #NUM! 0.05 0.05 0.05 0.05
X1 X3 -0.05 0 65535 #NUM! -0.05 -0.05 -0.05 -0.05
X2 X3 0.025 0 65535 #NUM! 0.025 0.025 0.025 0.025
X1 X2 X3 0.025 0 65535 #NUM! 0.025 0.025 0.025 0.025

15. RESIDUAL PROBABILITY
OUTPUT OUTPUT
Predicted Standard RESPO
Observation RESPONSE (Y) Residuals Residuals Percentile NSE (Y)
1 2.4 0 0 6.25 1.3
2 1.8 2.22E-16 0.797724 18.75 1.4
3 1.9 -4.4E-16 -1.59545 31.25 1.6
4 1.4 -4.4E-16 -1.59545 43.75 1.8
5 2.4 0 0 56.25 1.8
6 1.8 -2.2E-16 -0.79772 68.75 1.9
7 1.6 0 0 81.25 2.4
8 1.3 -2.2E-16 -0.79772 93.75 2.4

16. SUMMARY OUTPUT
Regression Statistics
Multiple R 0.99787
R Square 0.995745
Adjusted R Square 0.970213
Standard Error 0.070711
Observations 8
ANOVA
Significanc
df SS MS F eF
Regression 6 1.17 0.195 39 0.121965
Residual 1 0.005 0.005
Total 7 1.175

17. Standard Lower Upper Lower Upper
Coefficients t Stat P-value
Error 95% 95% 95.0% 95.0%
Intercept 1.825 0.025 73 0.00872 1.507346 2.142654 1.507346 2.142654
X1 -0.275 0.025 -11 0.057716 -0.59265 0.042654 -0.59265 0.042654
X2 -0.25 0.025 -10 0.063451 -0.56765 0.067654 -0.56765 0.067654
X3 -0.05 0.025 -2 0.295167 -0.36765 0.267654 -0.36765 0.267654
X1 X2 0.05 0.025 2 0.295167 -0.26765 0.367654 -0.26765 0.367654
X1 X3 -0.05 0.025 -2 0.295167 -0.36765 0.267654 -0.36765 0.267654
X2 X3 0.025 0.025 1 0.5 -0.29265 0.342654 -0.29265 0.342654

18. PROBABILITY
RESIDUAL OUTPUT OUTPUT
Predicted
Observati RESPON Standard RESPON
on SE (Y) Residuals Residuals Percentile SE (Y)
1 2.425 -0.025 -0.93541 6.25 1.3
2 1.775 0.025 0.935414 18.75 1.4
3 1.875 0.025 0.935414 31.25 1.6
4 1.425 -0.025 -0.93541 43.75 1.8
5 2.375 0.025 0.935414 56.25 1.8
6 1.825 -0.025 -0.93541 68.75 1.9
7 1.625 -0.025 -0.93541 81.25 2.4
8 1.275 0.025 0.935414 93.75 2.4

19. Regression Statistics
Multiple R 0.969755
R Square 0.940426 Normal Probability Plot
Adjusted R
0.916596
Square 3
RESPONSE (Y)
2.5
Standard
0.118322 2
Error
1.5
y = 0.013x + 1.1774
Observations 8 1
R2 = 0.937
0.5
ANOVA 0
Significanc 0 20 40 60 80 100
df SS MS F
eF Sample Percentile
39.464
Regression 2 1.105 0.5525 2 0.000866
9
Residual 5 0.07 0.014
Total 7 1.175

20. Lower Upper
Std Lower Upper
Coefficients t Stat P-value 99.0 99.0
Error 95% 95%
% %
Intercept 1.825 0.04183 43.625 1.19E-07 1.717465 1.932535 1.656324 1.993676
X1 -0.275 0.04183 -6.5737 0.00122 -0.38253 -0.16747 -0.44368 -0.10632
X2 -0.25 0.04183 -5.9761 0.00187 -0.35753 -0.14247 -0.41868 -0.08132

21. PROBABILITY
RESIDUAL OUTPUT
OUTPUT
Predicted
Observatio Standard RESPON
RESPON Residuals Percentile
n Residuals SE (Y)
SE (Y)
1 2.35 0.05 0.5 6.25 1.3
2 1.85 -0.05 -0.5 18.75 1.4
3 1.8 0.1 1 31.25 1.6
4 1.3 0.1 1 43.75 1.8
5 2.35 0.05 0.5 56.25 1.8
6 1.85 -0.05 -0.5 68.75 1.9
7 1.8 -0.2 -2 81.25 2.4
8 1.3 -2.2E-16 -2.2E-15 93.75 2.4

22. Residuals
0.2
0
0 1 2 3 4 5 6 7 8 9
-0.2
Residuals
-0.4

23. X1 LOW X1 HIGH
1 2.4 2 1.8 prediction of interaction from graph
X2
LOW 5 2.4 6 1.8
2.4 1.8 3
2.5 2.4
3 1.9 4 1.4
X2
HIGH 7 1.6 8 1.3 2
1.75 1.8
1.75 1.35 low
1.5
average
1.35 Series2
1
0.5
0
low high
value of factor
Interaction plot showing (by the parallel lines) that factors A and B do not
influence each other.

24.  Diagnostic Checking: Adjusted 2 R
Rule of Thumb: Values > 0.8 typically indicate that the
regression model is a good fit.
 Otherwise, a second order model is required because
the linear regression is not fit for our experiment.
 Final equation for this final reduced model will be y =
1.825-0.275*temp-0.25*(%ethanol).

25. Prediction from equation
Coefficients of both temperature and %ethanol are having (-) negative
value. So if we put lesser the value for both we will get good/ highest
response / resolution.
Now, batch 5 is good , so we can say that batch 5 is best which give good
resolution.