TAGUCHI ANOVA ANALYSIS

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1 CHAPTER 10 TAGUCHI ANOVA ANALYSIS Studies by varying the fin Material, Size of Perforation and Heat Input using Taguchi ANOVA Analysis 10.1 Introduction The data used in this Taguchi analysis were obtained experimentally for various materials like mild steel, copper, and aluminum, by different heat inputs and varying the size of the perforation (Solid, 10mm and 20mm perforations having Porosity ratio 0, 0.38 and 0.5) Fig 10.1 shows the copper fin arrays being tested. In these experiments, the four fins were mounted on the base plate of size mm. Fin dimensions were (H L), mm, fin thickness 3mm and the fin spacing S=20mm. Using the Taguchi experimental design method, the influence of the parameters like material, porosity ratio and heat input on the rate of heat transfer were investigated. Heat transfer coefficient was considered as performance parameters. An L 9 (3 3 ) orthogonal array was selected as an experimental plan. Fig 10.1 Perforated fin arrays of material Copper with 20mm size of perforation (Experimentation for Taguchi ANOVA analysis) 170

2 10.2 Taguchi Method For all types of experimental findings, it require to design the experiment and take the enough actual readings. To support the outcomes of the reaserch. That can be done by different ways: i) trial-and-error approach : By conducting the series of actual experiments those which gives diierent understanding. This requires taking actual readings after each experiments. so that analysis of data of readings will give idea to decide what to do next - "Which parameters can be varied and at what extent". In many cases it may not work may lead to negative results it will discourage, and or may not lead to selection of exact influencing parameter. Such experimentation will not reach to end or number of experiments will be more. Thus data may be insufficient to reach any significant conclusions and the main problem still remains unsolved. ii) Design of experiments: In a systematically planned set of experiments, having included parameters of interest are changed for a specified range, this is good approach to gain systematic data. In Mathematical language, such a full set of experiments is set to give desired results. In such cases number of experiments and resources (materials and time) required are possibly large. Many times researchers decides to carry out a subset of the total set of experiments to save efforts and money. Even though, it does not simply provide itself to understanding of art behind the phenomenon. The investigation is not so simple and obvious effects of different parameters on the practical data are not readily noticeable. ii) Taguchi Method The Taguchi method involves reducing the variation in a process through robust design of experiments. The overall objective of the method is to produce high quality product at low cost to the manufacturer. The Taguchi method was developed by Dr. Genichi Taguchi, a method for designing experiments to investigate how different parameters affect the mean and variance of process performance characteristics that defines how well the process is functioning. The Taguchi method gives the S/N ratio as 171

3 the performance index to evaluate the characteristics of the product or process. It can be easily defined as the ratio of the mean (signal) to the standard deviation (noise) by S/N ratio. The S/N ratios may be depended on the particular type of performance characteristics, including smaller-is-better (Z S ) or larger-is-better (Z L ) z s = 10 log 1 n Y i n i = z L = n 10 log 2 n i =1 Y i 172 (10.1) (10.2) Where n is the number of tests in trial Y i is the performance value of i th experiment. In this study, Orthogonal array L 9 (3 3 ) [50] experimental design method was chosen to determine the experimental plan. In this study the control parameters like porosity, heat input, and thermal conductivity of material were set as a level as shown in Table 9.5. In order to observe the effect of noise to source ratio on the heat transfer coefficient each experiment was repeated three times under the same condition as per L 9 (3 3 ) table. Values were determined by comparing the standard method and analysis of variance (ANOVA) which is based on the Taguchi method. The objective was to obtain performance characteristics (maximum heat transfer coefficient) hence, larger the better was chosen. Table 10.1 Control Parameters and Levels for Maximum Heat Transfer Coefficient Control parameters Level I Level II Level III A Porosity ratio B Heat input (watt) C Material (Thermal Cond.) Cu Al MS 10.3 Taguchi and ANOVA Analysis Analysis of variance (ANOVA) is one of the statistical models used to study the difference among group means plus their connected procedures like differences between groups. In ANOVA, the variance observed in a prescribed parameter is divided into parts

4 attributable to various sources of deviation. ANOVA provides a statistical test of means for several groups are equal or not and accordingly generalizes the t-test for more than 173

5 two groups. By many t-tests of two-samples will increase chance of a type I error. Because of which, ANOVAs are useful in comparing three or more means for statistical significance. As explained above the parameters chosen are: Heat input, Porosity and Fin material and the heat transfer coefficient was the measure of the outcomes of varying these parameters. In this study, from experimental readings, the average heat transfer coefficient (h a ) was calculated using Eq. (10.1). Eq. (10.2) the larger- is the better, used to calculate the S/N ratio. Table 10.2.S/N Response Table for Maximum Heat Transfer Coefficient Experiment no. A B C h a (W/m 2 k) S/N Ratio Both the values of h a and S/N ratio, are presented in Table After calculating the S/N ratio for each experiment, the average S/N value is calculated for each factor and level. For example, the mean S/N ratio for the heat input level II can be calculated by averaging the S/N ratios for experiment no. 1, 4, 7, and for level II experiment no. 2, 5, 8 and for level III experiment no 3, 6, 9. The mean S/N ratio for each level of the other parameters can be computed in similar manners that are presented in the response Table The main effect of each parameter is nothing but difference of highest and lowest value among the levels. 174

6 Table 10.3 Orthogonal Array L 9 (3 3 ) of the Experimental Results And Corresponding S/N Ratio Parameters Level I Level II Level III Main effects (A) Porosity a 2.1 (B) Heat Input a 1.4 (C) Thermal Conductivity a Performance Characteristics Cu Al MS Porosity Heat Input Mterial Fig.10.2 the Effect of Parameters on Heat Transfer Coefficient (ha) The plot in Fig.10.2 shows the degree of effect of the each parameter on the performance characteristics. The procedure can be explained with an example, for an instance, Fig 10.2 shows the variation of the performance characteristics with the porosity ratio. Now, let us try to determine the experimental condition for the first data point. The porosity ratio for this point is 0, which is the level I, the performance characteristics value is which is tabulated in Table Similarly for second data point, the performance characteristics value is under level II. Similarly for third data point and so on. The numerical value of the maximum point in each graph shows the best value of that particular parameter. They also indicate the optimum conditions in the range of the experimental conditions. The most effective parameter to enhance the heat transfer rate is the porosity ratio. Fig.10.2 shows that the design parameter combination A3 B3 C1, and 175

7 the corresponding values of each factor for the maximum heat transfer coefficient i.e. A3 porosity ratio (0.5), B3 heat input (80), material (Copper). Table 10.4 summarizes the ANOVA results for maximum heat transfer coefficient and shows percentage contribution and variation of factors A B and C. Table 10.4 ANOVA to Maximize Heat Transfer Coefficient Factors SS DOF MS F actual % Contribution A B C Error Total % Contribution Porosity Heat Input Thermal Cond. Fig.10.3 Percentage Contribution of Each Control Parameter to Enhance the Heat Transfer Fig 10.3 indicates that porosity ratio having % contribution and more significant, material (thermal conductivity) having 29.38% contribution and heat input having 20.39%contribution and less significant influence upon the maximum heat transfer coefficient in our study. 176

8 10.4 Confirmation Test In this study ANOVA was used to analyze the effects on the heat transfer coefficient of porosity, heat input and thermal conductivity of the material. ANOVA is the statistical technique used to find the influence of all control parameters. In the analysis, the percentage distributions of each control factor were used to measure the corresponding effects on the performance characteristics. The significance level of 5%, i.e. for 95% level of confidence was considered in this analysis. ANOVAs values belonging to the experimental results for the heat transfer coefficient and S/N ratios are shown in Table 10.1 the optimal heat transfer coefficient was obtained by taking into account the influential parameter within the evaluated optimum combination. The predicated optimum heat transfer coefficient was calculated by considering the individual effect of the parameters A 3, B 3 and C 1 and their levels using Eq. (10.3) (ha) p =T ha +(A 3 -)+(B 3 - T ha )+(C 1 - T ha ) (10.3) Where T ha is the total mean value of the heat transfer coefficient. A 3, B3, and C 1 are the (11.56,10.85 and 10.73) ha of experimental trials at the corresponding parameter. The optimal heat transfer coefficient is computed as (w/m 2 k). The confidence interval for the predicated optimal values is calculated as follows CI = Ö ËØ Ø» ÌÀ g 8 Ù aúú 8 Ê, h (10.4) Where ËØ Ø is the F-ratio required for (α=0.05 with a confidence of 95%)» 8,» < are the number of degrees of freedom of the mean and the number of degrees of freedom of error,» ÌÀ is the error of variance, r is the number confirmation experiments. h eff is no. of effective measured results defined as [51] h eff = ÛÜC$ ÝÞÑÝFßàÝáÜC$ ÜFßC$ 8 H ÜÛÜC$ âýãfýý Ûä äfýýâûà Ûä äcåüûf æ Ýâ äûf ÑFÝâßåÜßÛá (10.5) The total number of experimental trials are 9, with error of variance 0.134, number of confirmation experiments 3, h eff calculated using Eqs.(10.5) which is 1.28.Therefore, the CI is computed to be CI = ±0.820.The confirmation test result are presented in Table The optimal levels of corresponding parameters are Porosity at level III (i.e. 0.5), Heat Input at Level III (i.e. 80W) and Material at Level I (i.e. Cu) for 177

9 this combination real experimental value of ha is 13.54, which falls between the predicted confidence interval. Table 10.5 Results of confirmation experiment Average heat transfer coefficient (ha) Prediction Confidence Interval CI Real ± The most important parameter affecting heat transfer enhancement is the porosity ratio (50.23% contribution). A perforation in the fin enhances the heat transfer and enhancement increases with an increase in the porosity ratio for tested range. Secondly, thermal conductivity also plays an important role in enhancing the heat transfer. Average heat transfer coefficients of perforated fin arrays of both 0.38 and 0.5 porosity ratio s are higher than solid fin arrays (0 porosity ratio) for materials (Cu, Al, MS). Finally one of the most important benefits of the utilization of perforated fins (increases porosity ratio) is reduction of fin s weight. Low weight certifies saving material of fins and related equipments such as heat sinks. From the experimental trials and Taguchi results, new design of the fin structure (copper fin with perforation) can maximize the rate of heat transfer Effect of Heat Input The results of the perforated fin arrays with and without cross fin, non perforated fin arrays with and without cross fin at the center shows that the average heat transfer coefficient increases with increasing with heat input which has proved by both experimental as well as CFD analysis. This is explained by the increase in local fin surface temperature when fin base Watt heat input is increased, which results in an increase in the local and average heat transfer coefficient. And also from Taguchi analysis; it is proved that heat input has least contribution in enhancing the heat transfer. 178

10 10.6 Closure Taguchi analysis is carried by varying the fin material, porosity and heat input as a parameters of heat transfer coefficient. This reveals that the factor most affecting the enhancement of heat transfer is the porosity i.e. size of the perforation and secondly the thermal conductivity of the material. 179

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