RESPONSE SURFACE METHODOLOGY

1.7 RESPONSE SURFACE METHODOLOGY:

1.7.1 INTRODUCTION:

A common goal in many types of experiments is to characterize the relationship between a response and a set of factors or variables of interest to the experimenter. This can be accomplished by constructing model that describes the response over a wide range of factors of interest. The fitted model is called as the response surface if plotted in 2 – D and a contour if plotted in 1 – D. When the response function is simple, it can be solved for one of the factors as a function of the response and the other factor but if the function is complicated then we can plot the numerical values of the response on a graph as

function of two factors. The different types of contours available for two factor responses are mound shaped, stationary ridge, rising ridge and saddle type[3,7].

1.7.2 DESIGN SELECTION FOR FILLER OPTIMIZATION:

The following broad criteria are to be satisfied by the design for a rubber-compounding problem[10,19].

• The first requirement for a statistical design is that the number of experiments should be greater than or equal to the number of co – efficients in the regressed model.
• It should allow a second-degree polynomial equation to be estimated with satisfactory accuracy within the region of interest of the experimenter.
• It should allow a check to be made on the representational accuracy of the fitted equation with replicate points.
• It should not contain an excessively large number of points.
• It should form a nucleus from which a satisfactory design of higher degree can be built if the second-degree polynomial is found inadequate.
• It should be rotatable.

Therefore based upon the above guidelines a CCD was found to be the most suitable one since:

• The first requirement is to involve more than the number of co – efficients of the regressed model is met. Since we are tentatively assuming a second – degree polynomial equation to be adequate which contains 6 co – efficient. Therefore, a CCD with 8 points on its side and 1 point on the centre and 1 replicate at the centre is chosen.
• CCD will form an effective nucleus and can include 4 points from which a satisfactory design of three factors can be easily conceived.
• CCD is rotatable since all the points are at equal distance from centre. So, if there are any errors in any design they will be distributed uniformly over all the points.

1.7.3 CENTRAL COMPOSITE DESIGN:

CCD is an efficient and proven rotatable design especially for two factors. The number of design points in CCD based upon a complete 2k factorial is

N = 2^k + 2k + m

Where k is the number of factors and m is the number of observations. The factor space would be like the one shown in the fig.

To begin with, experimentation the following scheme is followed:

Step 1: Factorial and fraction of the centre points are run. After obtaining the experimental data, a linear model is fitted and tested for representational accuracy.

Step 2: Star points and remainder of central points is run.

After obtaining the experimental data, a quadratic response is fitted and this quadratic equation is tested for representational accuracy.

Fig.1.12 Factor space of a CCD

1.8 SCOPE AND OBJECTIVES:

As discussed above , fillers play an important role in polymer compounding. In an industry environment, filler masterbatches are prepared and subsequently mixed with virgin material and other additives. The properties of the polymer compound are dependent on the type and loading of the filler and processing conditions. In this project report, an attempt has been made to optimize the filler loading in HDPE matrix, which is to be used in the blow molding applications.

The importance of design of experiments (DoE) in this context has been discussed in the earlier section. By using Design of Experiments, maximum information can be obtained from minimum number of trials. Since the responses do not have a linear relationship with the factors ( the filler content and the let down ratio) the factorial design can be eliminated. The Central Composite Design (CCD) for two factors is used for the project.

Calcium carbonate – HDPE samples are prepared as per the design and the samples are characterized for Mechanical properties such as Tensile Strength, Youngs Modulus, Percentage Elongation, Flexural strength, Flexural Modulus, Hardness and

Impact strength. Based on the results the linear regression analysis is performed using MINITAB and MATLAB and the regression equations are generated. From the response equations, the Contour Diagrams and Response Surfaces are generated for various properties. Response Surface Methodology (RSM) is applied to provide model equations using which , an optimum CaCO₃ loading can be selected to obtain the desired properties. The equations are used to make the prediction of properties for a number of CaCO₃ loadings by overlaying of individual contour plots.