Response Surface Methodology

Abstract

Response surface methodology or in short RSM is a collection of mathematical and statistical tools and techniques that are useful in developing, understanding, and optimizing processes and products. Using this methodology, the responses that are influenced by several variables can be modeled, analyzed, and optimized.

Problems

9.1

An article entitled “Fermentation of molasses by Zymomonas mobilis: Effects on temperature and sugar concentration on ethanol production” published by M. L. Cazetta et al. in Bioresource Technology, 98, 2824–2828, 2007, described the experimental results shown in Table 9.10.

Table 9.10 Data for Problem 9.1

• (a) Name the experimental design used in this study.

• (b) Construct ANOVA for ethanol concentration.

• (c) Do you think that a quadratic model can be fit to the data? If yes, fit the model and if not, why not?

9.2

Find the path of steepest ascent for the following first-order model

$$\hat{y}=1000+100 x_1+50x_2$$

where the variables are coded as $-1\le x_i\le 1$.

9.3

In a certain experiment, the two factors are temperature and contact pressure. Two central composite designs were constructed using following ranges on the two factors.

$$\begin{aligned} \text {Temperature}&: 500\,^{\circ }\text {F}-1000\,^{\circ }\text {F}\\ \text {Contact pressure}&: 15~\text {psi}-21~\text {psi} \end{aligned}$$

The designs listed in Table 9.11 are in coded factor levels.

Table 9.11 Data for Problem 9.3

• (a) Replace the coded levels with actual factor level.

• (b) Is either design rotatable? If not why?

• (c) Construct a rotatable central composite design for this.

9.4

A disk-type test rig is designed and fabricated to measure the wear of a textile composite under specified test condition. The ranges of the three factors chosen are as follows:

$$\begin{aligned} \text {Temperature}&: 500-1000\,^{\circ }\text {F}\\ \text {Contact pressure}&: 15-21~\text {psi}\\ \text {Sliding speed}&: 54-60~\text {ft/sec} \end{aligned}$$

Construct a rotatable $3^3$ central composite design for this experiment.

9.5

Construct a Box Behnken design for the experiment stated in Problem 9.5.

9.6

In a study to determine the nature of response system that relates yield of electrochemical polymerization (y) with monomer concentration $(x_1)$ and polymerization temperature $x_2$, the following response surface equation is determined

$$\hat{y}=79.75+10.18x_1+4.22x_2-8.50x_1^\mathrm{2}-5.25x_2^\mathrm{2}-7.75x_1x_2.$$

Find the stationary point. Determine the nature of the stationary point. Estimate the response at the stationary point.

9.7

Consider the following model

$$\hat{y}=1.665-32\times 10^{-5}x_1+372\times 10^{-5}x_2+1\times 10^{-5}x_1^\mathrm{2}+68\times 10^{-5}x_2^\mathrm{2}-1\times 10^{-5}x_1x_2$$

Find the stationary point. Determine the nature of the stationary point. Estimate the response at the stationary point.

9.8

An article entitled “Central composite design optimization and artificial neural network modeling of copper removal by chemically modified orange peel” published by A. Ghosh et al. in Desalination and Water Treatment, 51, 7791–7799, 2013, described the experimental results shown in Table 9.12.

Table 9.12 Data for Problem 9.8

• (a) Name the design of experiments used here.

• (b) Develop a suitable response surface model and construct ANOVA.

• (c) Find out the stationary point and comment on the nature of stationary point.

• (d) State the optimum process factors that maximize the percentage removal of copper.

9.9

An article entitled “Optimization of the conductivity and yield of chemically synthesized polyaniline using a design of experiments” published by E.J. Jelmy et al. in Journal of Applied Polymer Science, 1047–1057, 2013, described a three-factor Box Behnken design with the results shown in Table 9.13.

Table 9.13 Data for Problem 9.9

• (a) Develop an appropriate model and construct ANOVA for conductivity.

• (b) Find out the stationary point and comment on the nature of stationary point.

• (c) What operating conditions would you recommend if it is important to obtain a conductivity as close as 0.2 S/cm?

• (d) Develop an appropriate model and construct ANOVA for yield.

• (e) Find out the stationary point and comment on the nature of stationary point.

• (f) What operating conditions would you recommend if it is important to obtain a yield as close as 90%?

• (g) What operating conditions would you recommend if you wish to maximize both conductivity and yield simultaneously?