Applied Statistics and Probability for Engineers

Analysis of a Second-Order Response Surface

When the experimenter is relatively close to the optimum, a second-order model is usually

required to approximate the response because of curvature in the true response surface. The

fitted second-order model is

where denotes the least squares estimate of . In this section we show how to use this fitted

model to find the optimum set of operating conditions for the x’s and to characterize the nature

of the response surface.

EXAMPLE S14-4 Continuation of Example S14-3

Consider the chemical process from Example S14-3, where the method of steepest ascent

terminated at a reaction time of 55 minutes and a temperature of 163F. The experimenter decides

to fit a second-order model in this region. Table S14-7 and Fig. S14-10 show the experimental

design, which consists of a 2^2 design centered at 55 minutes and 165F, five center points, and four

runs along the coordinate axes called axial runs. This type of design is called a central compos-

ite design,and it is a very popular design for fitting second-order response surfaces.

Two response variables were measured during this phase of the experiment: percentage

conversion (yield) and viscosity. The least-squares quadratic model for the yield response is

The analysis of variance for this model is shown in Table S14-8.

Figure S14-11 shows the response surface contour plot and the three-dimensional surface

plot for this model. From examination of these plots, the maximum yield is about 70%,

obtained at approximately 60 minutes of reaction time and 167F.

The viscosity response is adequately described by the first-order model

yˆ 2 37.083.85x 1 3.10x 2

yˆ 1 69.11.633x 1 1.083x 2

 0.969x^21 1.219x^22 0.225x 1 x 2

ˆ

yˆˆ 0 a

k

i 1

ˆixia

k

i 1

ˆiix^2 ib

ij

ˆijxixj

14-16

Table S14-7 Central Composite Design for Example S14-4

Conversion Viscosity

Observation Time Temperature Coded Variables (percent) (mPa-sec)

Number (minutes) (F) x 1 x 2 Response 1 Response 2

1 50 160 

1 

1 65.3 35

2 60 160 1 

1 68.2 39

3 50 170 

1 1 66 36

4 60 170 1 1 69.8 43

5 48 165 1.414 0 64.5 30

6 62 165 1.414 0 69 44

7 55 158 0 1.414 64 31

8 55 172 0 1.414 68.5 45

9 55 165 0 0 68.9 37

10 55 165 0 0 69.7 34

11 55 165 0 0 68.5 35

12 55 165 0 0 69.4 36

13 55 165 0 0 69 37

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