11 Process Yield Steepest Ascent

EXAMPLE 14-11 Process Yield Steepest Ascent

  In Example 14-6 we described an experiment on a chemical

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

  process in which two factors, reaction time (x 1 ) and reaction

  hours and the effect estimate of temperature is 0.65 ⬚F, and

  temperature (x 2 ), affect the percent conversion or yield (Y ).

  since the regression coefficients ␤ ˆ 1 and ␤ ˆ 2 are one-half of the

  Figure 14-27 shows the 2 2 design plus five center points used

  corresponding effect estimates, the fitted first-order model is

  in this study. The engineer found that both factors were important, there was no interaction, and there was no curva-

  y ˆ ⫽ 40.44 ⫹ 0.775x 1 ⫹ 0.325x 2

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

  Figure 14-45(a) and (b) show the contour plot and three-

  Y ⫽␤ 0 ⫹␤ 1 x 1 ⫹␤ 2 x 2 ⫹⑀

  dimensional surface plot of this model. Figure 14-45 also

  x 2 (temperature)

  32.00 150.0 30.00 x 1 (time)

  x 1 (time)

  –1

  Contour plot

  Three-dimensional surface plot

  (a)

  (b)

  Figure 14-45 Response surface plots for the first-order model in Example 14-11.

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  CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

  F

  Original region of experimentation

  C Point

  A: 40 minutes, 157°F, y = 40.5

  B Path of

  Point

  B: 45 minutes, 159°F, y = 51.3

  A steepest ascent

  Point

  C: 50 minutes, 161°F, y = 59.6

  Point

  D: 55 minutes, 163°F, y = 67.1

  Temperature

  Point

  E: 60 minutes, 165°F, y = 63.6

  Point

  F: 65 minutes, 167°F, y = 60.7

  Original fitted contours

  30 40 50 60 70 Time

  Figure 14-46 Steepest ascent experiment for Example 14-11.

  shows the relationship between the coded variables x 1 and x 2 (0.325 兾0.775) x 1 0.42. A change of x 2 0.42 in the

  (that defined the high and low levels of the factors) and the

  coded variable x 2 is equivalent to about 2

  F in the original

  original variables, time (in minutes) and temperature (in F).

  variable temperature. Therefore, the engineer will move along

  From examining these plots (or the fitted model), we see

  the path of steepest ascent by increasing reaction time by

  that to move away from the design center—the point (x 1 0,

  5 minutes and temperature by 2

  F. An actual observation on

  x 2 0)—along the path of steepest ascent, we would move

  yield will be determined at each point. 0.775 unit in the x 1 direction for every 0.325 unit in the x 2 Next Steps: Figure 14-46 shows several points along this

  direction. Thus, the path of steepest ascent passes through the

  path of steepest ascent and the yields actually observed from

  point (x 1 0, x 2 0) and has a slope 0.325兾0.775. The engi-

  the process at those points. At points A–D the observed yield

  neer decides to use 5 minutes of reaction time as the basic step

  increases steadily, but beyond point D, the yield decreases.

  size. Now, 5 minutes of reaction time is equivalent to a step in

  Therefore, steepest ascent would terminate in the vicinity of

  the coded variable x 1 of x 1 1. Therefore, the steps along

  55 minutes of reaction time and 163

  F with an observed per-

  the path of steepest ascent are x 1 1.0000 and x 2 cent conversion of 67.