The graphical method was applied as an optimization technique (Carrillo et al., 2002). For LLDPE and HDPE composites, contour plots of the response variables (refer to appendix G) were utilized by applying superposition surface methodology to obtain three contour plots for observation and selection of the best (optimum) combination of fiber content (FC), temperature (T) and screw speed (SS) for the production of flax-fiber-reinforced polyethylene composites.
Contour plots were generated using Tecplot 9.2. The plots of LLDPE were separated from the plots of HDPE. LLDPE plots were again subdivided into three groups according to the two process variables involved: FC-SS (fiber content – screw speed) plots (1), FC-T (fiber content – temperature) plots (2) and T-SS (Temperature – Screw speed) plots (3). The same was done for the HDPE plots. Superposition of the plots within a group was carried out to determine the best combination of process variables (FC-SS, FC-T, and T-SS). Since there were two values for each variable, the two values were averaged to obtain the optimum value for each of the variables FC, T, and SS. This was done both for the LLDPE and HDPE treatments.
A sample computation follows. The following are LLDPE plots grouped according to the two variables [fiber content (FC) and screw speed (SS)] involved.
a b
c d
Figure 5.20 Contour plots (LLDPE) for the effects of fiber content (FC) and screw speed (SS) on the different response variables (extrudate color, extrudate density, extrudate melt flow index, and composite color).
a b
c
Figure 5.21 Contour plots (LLDPE) for the effects of fiber content (FC) and screw speed (SS) on the different response variables (composite density, composite tensile strength, and composite water absorption).
Since the superposition could not be shown here, the contour plots were analyzed one by one.
The first plot (Figure 5.20a) shows the effect of fiber content (FC) and screw speed (SS) on extrudate color. The desired value would be the lowest value since it signifies proximity to the pure plastic color. The lowest value can be found on the FC level of 0% and the SS level of 110 rpm.
The second plot (Figure 5.20b) shows the effect of FC and SS on extrudate density. The desired value would be the highest value (most dense). The highest value lies on the FC level of 12.5% and the SS level of 110 rpm.
The third plot (Figure 5.20c) shows the effect of FC and SS on melt flow index. The desired value would be the highest value since this signifies the greatest flowability. The highest value lies on the FC level of 0% and the SS levels of 110 rpm and 150 rpm.
The fourth plot (Figure 5.20d) shows the effect of FC and SS on composite color. The desired value is the lowest value which can be found on the FC level of 0% and the SS levels of 110 rpm and 150 rpm.
The fifth plot (Figure 5.21a) shows the effect of FC and SS on composite density. The desired value (highest value) lies on the FC level of 12.5% and the SS level of 110 rpm.
The sixth plot (Figure 5.21b) shows the effect of FC and SS on composite tensile strength. The desired value is the highest value which lies on the FC level of 0% and SS levels of 110 rpm and 150 rpm.
The seventh plot (Figure 5.21c) shows the effect of FC and SS on the composite water absorption. The desired value would be the lowest value (least absorbent) and this can be found on the FC levels of 0% and 12.5% and the SS levels of 110 rpm and 150 rpm.
Table 5.15 summarizes these results.
Table 5.15 Optimum levels of fiber content (FC) and screw speed (SS) for the different response variables.
Response Variable Levels with desired values
Fiber content (%) Screw speed (rpm)
Extrudate color 0% 110 rpm
Extrudate density 12.5% 110 rpm
Extrudate melt index 0% 110 rpm and 150 rpm
Composite color 0% 110 rpm and 150 rpm
Composite density 12.5% 110 rpm
Composite tensile strength 0% 110 rpm and 150 rpm Composite water absorption 0% and 12.5% 110 rpm and 150 rpm
To obtain the optimum levels for fiber content (FC) and screw speed (SS), the values were averaged:
% 69 . 4 8 ) 3 %( 5 . 12 ) 5 %( 0 + = = FC 55 . 124 11 ) 4 ( 150 ) 7 ( 110 = + = SS rpm
The preceding values were the optimum levels derived from the subgroup LLDPE FC-SS contour plots. The same procedure was done for the subgroup LLDPE FC-T plots and the LLDPE T-SS plots, generating two values each for FC, T and SS variables. The two values were averaged to finally arrive at the optimum values of FC, T and SS for LLDPE. The same procedure was applied to the HDPE contour plots.
Table 5.16 presents the computed optimum values for both LLDPE and HDPE.
Table 5.16 Optimum values of process variables fiber content (FC), temperature (T), and screw speed (SS) for LLDPE and HDPE composites based on extrudate color, extrudate density, extrudate melt flow index, composite color, composite density, composite tensile strength and composite water absorption.
Process Variable Optimum Values
LLDPE HDPE Fiber content (FC) 4.69% 3.96% Temperatures (T) 146.7°C or 75-116.7-126.7-136.7-146.7°C 147.7°C or 75-117.7-127.7-137.7-147.7°C Screw speed (SS) 120.13 rpm 125 rpm
The preceding table presents the optimum values of the three process variables FC, T and SS based on the seven response variables, including
The purpose of including the response variables extrudate color and composite color was to measure the degree of degradation of the flax fibers. However, it was observed that the fibers never underwent degradation during the whole process since process temperatures used for both extrusion and rotational molding did not go beyond the degradation temperature of flax fibers (213.9°C – refer to appendix A). Hence, we can exclude the two response variables in the optimization process.
Table 5.17 presents the optimum values of the three process variables FC, T and SS based on the remaining five response variables: extrudate density, extrudate melt flow index, composite density, composite tensile strength and composite water absorption.
Table 5.17 Optimum values of process variables fiber content (FC), temperature (T), and screw speed (SS) for LLDPE and HDPE composites based on extrudate density, extrudate melt flow index, composite density, composite tensile strength and composite water absorption.
Process Variable Optimum Values
LLDPE HDPE Fiber content (FC) 6.25% 5.02% Temperatures (T) 147.3°C or 75-117.3-127.3-137.3-147.3°C 148.1°C or 75-118.1-128.1-138.1-148.1°C Screw speed (SS) 117.5 rpm 125.56 rpm