Applied Statistics and Probability for Engineers

EXAMPLE S14-3 In Example S14-2 we described an experiment on a chemical process in which two factors,

reaction time (x 1 ) and reaction temperature (x 2 ), affect the percent conversion or yield (Y).

Figure S14-3 shows the 2^2 design plus five center points used in this study. The engineer

found that both factors were important, there was no interaction, and there was no curvature

in the response surface. Therefore, the first-order model

should be appropriate. Now the effect estimate of time is 1.55 hours and the effect estimate of

temperature is 0.65F, and since the regression coefficients and are one-half of the

corresponding effect estimates, the fitted first-order model is

Figure S14-8(a) and (b) show the contour plot and three-dimensional surface plot of this

model. Figure S14-8 also shows the relationship between the coded variablesx 1 and x 2 (that

defined the high and low levels of the factors) and the original variables, time (in minutes) and

temperature (in F).

From examining these plots (or the fitted model), we see that to move away from the de-

sign center—the point (x 1 0, x 2 0)—along the path of steepest ascent, we would move

0.775 unit in the x 1 direction for every 0.325 unit in the x 2 direction. Thus, the path of steepest

ascent passes through the point (x 1 0, x 2 0) and has a slope 0.3250.775. The engineer

decides to use 5 minutes of reaction time as the basic step size. Now, 5 minutes of reaction

time is equivalent to a step in the codedvariable x 1 of x 1 1. Therefore, the steps along the

path of steepest ascent are x 1 1.0000 and x 2 (0.3250.775)x 1 0.42. A change of

x 2 0.42 in the coded variable x 2 is equivalent to about 2F in the original variable temper-

ature. Therefore, the engineer will move along the path of steepest ascent by increasing

reaction time by 5 minutes and temperature by 2F. An actual observation on yield will be

determined at each point.

Figure S14-9 shows several points along this path of steepest ascent and the yields actually

observed from the process at those points. At points A–Dthe observed yield increases steadily, but

beyond point D, the yield decreases. Therefore, steepest ascent would terminate in the vicinity of

55 minutes of reaction time and 163F with an observed percent conversion of 67%.

yˆ40.440.775x 1 0.325x 2

ˆ 1 ˆ 2

Y 0  1 x 1  2 x 2 

14-15

40

160

50 60 70

150

Original fitted

contours

Original region

of experimentation

Path of

steepest ascent

30

A

B

C

D

E

F

Time

170

Temperature

Point A: 40 minutes, 157°F, y = 40.5

Point B: 45 minutes, 159°F, y = 51.3

Point C: 50 minutes, 161°F, y = 59.6

Point D: 55 minutes, 163°F, y = 67.1

Point E: 60 minutes, 165°F, y = 63.6

Point F: 65 minutes, 167°F, y = 60.7

Figure S14-9 Steepest ascent experiment for Example S14-3.

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