Optimization of Muffler and Silencer

Transcription

1 Chapter 5 Optimization of Muffler and Silencer In the earlier chapter though various numerical methods are presented, they are not meant to optimize the performance of muffler/silencer for space constraint problems. The trial and error method in the enhancement of the muffler/silencer design is considered tedious and expensive. Therefore the interest to optimize muffler performance under space constraint is increasing in the field. So this chapter emphasizes on shape optimization of mufflers. Also it was necessary to do pilot experiments to identify key design levels with which future experiments were to be designed. Shape optimization of mufflers within a limited space is the need of the industry where equipment layout is occasionally tight and the available space for a muffler is limited for maintenance and operation purposes [96]. Because of the necessity of operation and maintenance within an enclosed machine room or an automobile, a space-constrained problem within a noise abatement facility will occur; therefore, there is a growing need to optimize the acoustical performance within a fixed space. Yet, the need to investigate the optimal muffler design under space constraints is rarely tackled. To proficiently enhance the acoustical performance within a constrained space, the selection of an appropriate acoustical mechanism and optimizer becomes crucial. The four-pole system matrix for evaluating the acoustic performance of muffler viz, sound transmission loss (STL) is derived by using transfer matrix method. To recognize the correctness of model for silencer performance using four-pole matrix method, the three-dimensional finite element analysis is adopted [97]. Moreover, the maximal acoustic performance of a silencer can be precisely and quickly achieved by using the grid search method. 66

2 By using Taguchi and ANOVA methods several better designs can be obtained. Consequently, optimization technique is used for the optimal design of a silencer for the noise reduction by a silencer. 5.1 Optimization by using Grid Search Method This method involves setting up a suitable grid in the design space, evaluating the objective function at all the grid points, and finding the grid point corresponding to the highest function value [98]. It can be seen that the grid search method requires prohibitively large number of function evaluations in most practical problems. However, for problems with a small number of design variables, the grid search method can be used conveniently to find an approximate maximum. Objective Function As indicated in Figure 5.1, the silencer includes two fixed compartments; and three straight ducts along with central inlet and side outlet. Fixed dimensions of two chambers and variable dimensions L 1, L 2 and L 3 are shown. Figure 5.1: Central extended inlet side outlet double chamber muffler L = 150 mm, Diameter of chamber = 125 mm, Diameter of pipe = 35 mm, Length of first chamber = 125 mm, Length of second chamber = 375 mm 67

3 The transfer matrices of individual segments are known. These are used to calculate overall TM of the silencer as follows: T = [T 1 2 ][T 4 ][T 3 5 ][T 6 ][T 7 8 ][T 10 ][T 9 11 ][T ][T ] T = A 11 A 12 A 21 A 22 [ ( 1 T L = 20 log 10 A 11 + A )] 12 + A 21 Y outlet + A 22 2 Y inlet where Y = Volume velocity = ρc S [ ( 1 Objective function = T L = 20 log 10 A 11 + A )] 12 + A 21 Y outlet + A 22 2 Y inlet 5.2 Design of Experiments Industry has become increasingly aware of the importance of quality. It is being used as a business strategy to increase market share. Organizations are achieving world-class quality by using proper design of experiments. Experimental design is one of the most powerful techniques for improving quality and productivity [99, 100] with least expenditure. Through experimentation changes are intentionally introduced into the process or the system in order to observe their effect on performance characteristics or response of the system or process. A statistical approach is the most efficient method for optimizing these changes. An engineer can do investigation without statistics using ad hoc and heuristic approaches. However, an engineer can be much more efficient in his investigation if armed with statistical tools. Basic Statistics For any group data two parameters are of greater interest, the mean and variance. For a group of data, X 1, X 2,,,,,,,,, X n where n is the number of observations in the group, the mean or average is a measure of the central tendency of the group of data; that is, X = 1 n n X i (5.1) i=1 68

4 The variance is a measure of the dispersion of the group of data about the mean; that is, S = n i=1 (X i X) 2 n 1 (5.2) The standard deviation is often stated as the square root of the variance. The variance is also referred to as the mean square, MS, which is the sum of squares, SS, divided by the number of degrees of freedom, ν; that is, S = MS = SS v The variance consists of n quantities (X i X) in the numerator; however, there are only n 1 independent quantities, because a sample statistics is used rather than the population parameters. In this case, X is used rather than µ. Thus, the degree of freedom, ν, is given by, ν = n 1 As the sample size of a population increases, the variance of the sample approaches the variance of the population; that is, n i=1 MS = lim (X i X) 2 n n 1 = n i=1 (X i µ) 2 n One Factor at a Time The conventional approach for optimizing analytical methods in the laboratory is the onefactor-at-a-time approach, where each experimental factor or parameter is optimized separately and independently of other factors. Consider an experimental design with N f factors with two levels for each factor, as shown in Table 5.1. At treatment condition 1, all factors are run at level 1 to obtain the benchmark or reference value. The next treatment condition is to run factor A at level 2 while the other factors are kept at level 1. For the third treatment condition, factor B is run at level 2 while the other factors are kept at level 1. This sequence is continued until all factors are run at level of the factors at level 1. The change in the response due to a change in the level of the factor is known as the effect of a factor. The main effect of factor is defined as the difference between the average responses at the two levels. In order to determine the effect of each factor, the 69

5 Table 5.1: One-Factor-at-a-time Experimental Design Treatment Factors/Levels Response Condition A B C D... N f (Results) Y Y Y Y Y n T C Y N responses at level 2 are compared to the reference value (level 1); that is, E A = Y A Y 0 E B = Y B Y 0 E C = Y C Y 0 and so on, for each factor. In this manner the factors that have a strong effect on the response can be determined. In addition, the sign of the effect indicates whether an increase in the factor level increases the response, or decreases the response. One-Factor-at-a-time Experimental Design can be more effective if number of runs is limited, primary goal is to attain improvements in the system, and experimental error is not large compared to factor effects, which must be additive and independent of each other. It has several attractive features such as run size economy, fewer level changes and providing protection against the risk of premature termination of experiments. It is neither balanced nor efficient. Two-Factor Factorial Design A two-factor factorial design, also called as two-factor design, is an experimental design that simultaneously evaluates the effects of two factors (Xs) on a dependent variable (Y ). Factors can have any numbers of levels. However, only factors with two levels, or treatments, such as high and low, or on and off are considered. 70

6 5.2.1 Analysis of Variance CHAPTER 5. OPTIMIZATION OF MUFFLER AND SILENCER Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups), in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. Doing multiple twosample t-tests would result in an increased chance of committing an error. For this reason, ANOVAs are useful in comparing (testing) three or more means (groups or variables) for statistical significance [101]. The terminology of ANOVA is largely from the statistical design of experiments. The experimenter adjusts factors and measures responses in an attempt to determine an effect [102]. It is a method to find the significant factors using the sum of squares. In other words, the total dispersion of a characteristic value is represented by the sum of squares and this dispersion is divided into the sum of squares of each factor and the error. Then, the mean square of each factor, that is the value of the sum of squares of each factor divided by the corresponding degree of freedom can be obtained. Thus the ratio of the mean square of a factor and the mean square of error leads to the percentage contribution of each factor and decision whether a factor is significant or not using the F test. Classical ANOVA for balanced data does three things at once: As exploratory data analysis, an ANOVA is an organization of additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition (or, equivalently, each set of terms of a linear model). Comparisons of mean squares, along with F-tests allow testing of a nested sequence of models. Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors. 71

7 ANOVA for a single factor The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available [101]. ANOVA for multiple factors ANOVA generalizes the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used. Degree of Freedom The number of degrees of freedom is very important value because it determines the minimum number of treatment conditions. In a statistical sense it is associated with each piece of information that is estimated from the data. Sum of Squares The sum of squares is the measure of the deviation of the experimental data from the mean value of data. Summing each squared deviation emphasizes the total deviation. S t = n (Y i Y ) 2 i=1 where Y is the average value of Y i. Mean Sum of Squares Let T 2 = n i=1 (Y i Y 0 ) 2 /n the sum of all deviations from the target value. Then, the mean sum of squares of the deviation is: S = T 2 n 72

8 Percent Contribution CHAPTER 5. OPTIMIZATION OF MUFFLER AND SILENCER The percentage contribution for any factor is obtained by dividing the pure sum of squares for that factor by S t and multiplying result by 100. Percentage contribution is denoted by P and can be calculated by using following equation: Confidence Interval P m = S m 100 S t In statistics, a confidence interval (CI) is a type of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient. Variance Ratio The ratio obtained by dividing the mean square representing the association with a parameter in the model divided by residual mean square is known as variance ratio. Number of Trials F m = V m V e Many factors can be used in a screening experiment (pilot experiment) for sensitivity analysis to determine which factors are the main drivers of the response variable. However, as the number of factors increases, the total number of combinations increases exponentially. Thus, screening studies often use a fractional factorial design, which produces high confidence in the sensitivity results using a feasible number of runs [103]. In an experiment designed to determine the effect of factor A on response Y, factor A is to be tested at L levels. Assume N 1 repetitions of each trial that includes A 1. Similarly at level A 2 the trial is to be repeated N 2 times. The total no. of trials is the sum of the numbers of trials at each level i.e. N = N 1 + N 2 + N 3 + N N n 73

9 Pure sum of squares When (DOF σ 2 ) term is subtracted from the sum of the squares expression, the reminder is called the pure sum of squares. Thus pure sum of squares Sm is: S m = S m V e Orthogonal Design Orthogonality means that the experimental design is balanced. A balanced experiment leads to a more efficient one. Table 5.2 shows an orthogonal experimental design for three factors with two levels each. Although there are more treatment conditions, the design is balanced and efficient. Treatment conditions 1, 2, 3, and 5 are the same as treatment conditions 1, 2, 3, and 4 for the one-factor-at-a-time experiment in the previous section. A review of this particular design shows that there are four level 1s and four level 2s in each column. Although different designs have a different number of columns, rows, and levels, there will be an equal number of occurrences for each level. The idea of balance produces statistically independent results. Table 5.2: Orthogonal Experimental Design Treatment Factors/Levels Response Condition A B C (Results) Y Y Y Y Y Y Y Y 222 Looking at the relationship between one column and another, we find that for each level within one column, each level in any other column will occur an equal numbers of times. Therefore, the four level 1s for factor A have two level 1s and two level 2s for factor B, and two level 1s and two level 2s for factor C. The effect of factor A at level is 74

10 statistically independent, because the effects of factor B and factor C for each level are averaged into both levels of the factor A. Statistical independence is also true for factor B and factor C. In order to obtain the main effect of each factor/level, the results need to be averaged. The average of the response for the factor A due to level i (i = 1, 2) is given by, A = 1 2 Y ijk...m Similarly, the average of the response for factor B due to level j (j = 1, 2) is B = 1 2 Y ijk...m And so on, for all factors. While the number of treatment conditions increased by a factor levels of two, the number of values that were used to calculate the factor/level average increased by a factor of four. This improvement in the number of values used to determine the average makes the orthogonal design very efficient in addition to being balanced. The main effect of factors A and B are given by, E A = A 2 A 1 E B = B 2 B Taguchi Principle Most of the body of knowledge associated with the quality sciences was developed in the United Kingdom as design of experiments and in the United States as statistical quality control. More recently, Taguchi [104] has added to this body of knowledge. The application of DOE requires careful planning, prudent layout of the experiment, and expert analysis of results. Based on years of research and applications Dr. Genechi Taguchi has standardized the methods for each of these DOE application steps described below. Thus, DOE using the Taguchi approach has become a much more attractive tool to practicing engineers and scientists. The DOE using Taguchi approach can economically 75

11 satisfy the needs of problem solving and product/process design optimization projects. By learning and applying this technique, engineers, scientists, and researchers can significantly reduce the time required for experimental investigations. In particular, he introduced the loss function concept, which combines cost, target, and variation into one metric with specifications being of secondary importance. Furthermore, he developed the concept of robustness, which means that noise factors are taken into account to ensure that the system functions correctly. Noise factors are uncontrollable variables that can cause significant variability in the process or the product. DOE using Taguchi approach attempts to improve quality which is defined as the consistency of performance. Consistency is achieved when variation is reduced. This can be done by moving the mean performance to the target as well as by reducing variations around the target. The prime motivation behind the Taguchi experiment design technique is to achieve reduced variation (also known as Robust Design). This technique, therefore, is focused to attain the desired quality objectives in all steps. The classical DOE does not specifically address quality. Taguchi s work includes three principles: A specific loss function The philosophy of off-line quality control and Innovations in the design of experiments. 8-Steps in Taguchi Methodology [105] 1. Identify the main function, side effects and failure modes; 2. Identify the noise factors, testing conditions and quality characteristics; 3. Identify the objective function to be optimized; 4. Identify the control factors and their levels; 5. Select the orthogonal array matrix experiment; 6. Conduct the matrix experiment; 76

12 7. Analyze the data, predict the optimum levels and performance; 8. Perform the verification experiment and plan the future action; 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. Taguchi developed a method for designing experiments to investigate how different parameters affect the mean and variance of a process performance characteristic that defines how well the process is functioning. Loss Function Taguchi loss function is the loss curve which expresses the loss of deviating from nominal value within specifications. Under this, the continuous reduction of unit to unit variation around the nominal value is the most economical course of action [106]. Taguchi followed the statistical theory mainly from the followers of Fisher [107], who had avoided loss function. Reacting to Fisher s methods in the design of experiment, Taguchi interpreted Fisher s methods as being adapted for seeking to improve the mean outcome of a process. Indeed, Fisher s work had been largely motivated by programmers to compare agricultural yields under different treatments and blocks, and such experiments were done as part of a long-term programmes to improve harvests. However, Taguchi realized that in industrial production, there is a need to produce an outcome on target, for example, to machine a hole to a specified diameter, or to manufacture a cell to produce a given voltage. He also realized, as had Walter A. Shewhart [108] and others before him, that excessive variation laid at the root of poor manufactured quality and that reacting to individual items inside and outside specification was counterproductive. Orthogonal Arrays Orthogonal arrays (OA) are simplified method of putting together an experiment. The effect of many different parameters on the performance characteristic in a condensed set of experiments can be examined by using the orthogonal array experimental design proposed by Taguchi [101]. Once the parameters affecting a process that can be controlled have been determined, the levels at which these parameters should be varied must be determined. Determining what levels of a variable to test requires an in-depth understanding of the process, including the minimum, maximum, and current value of the parameter. If 77

13 the difference between the minimum and maximum value of a parameter is large, the values being tested can be further apart or more values can be tested. If the range of a parameter is small, then less values can be tested or the values tested can be closer together. Typically, the number of levels for all parameters in the experimental design is chosen to be the same to aid in the selection of the proper orthogonal array. Knowing the number of parameters and the number of levels, the proper orthogonal array can be selected. These standard arrays stipulate the way of conducting the minimum number of experiments that could give the full information of all the factors that affect the performance of the parameters. The crux of the orthogonal method lies in choosing the level combinations of the input design variables [109]. An orthogonal property of the OA is shown in Table 5.3. Table 5.3: Taguchi Table (L9) Treatment Factors/Levels Response Condition A B C D (Results) Y Y Y Y Y Y Y Y Y 3321 The 9 in the designation L9 represents the number of rows, which is also the number of treatment conditions (TC) and the degree of freedom. Across the top of orthogonal array is the maximum number of factors that can be used, which in our case are four. The levels are designated by 1 and 2. If more levels occur in the array, then 3, 4, 5, etc., are used. Other schemes such as -, 0, and + can be used. 78

14 The orthogonal property of the OA is not compromised by changing the rows or the columns. Taguchi changed the rows from traditional design so that TC 1 was composed of all level 1s and, if the team desired, could thereby represent the existing conditions. Also, the columns were switched so that the least amount of change occurs in the assigned factors with long set-up times to those columns. An advantage of the Taguchi method is that it emphasizes a mean performance characteristic value close to the target value rather than a value within certain specification limits, thus improving the product quality. Additionally, Taguchi s method for experimental design is straightforward and easy to apply to many engineering situations, making it a powerful yet simple tool. It can be used to quickly narrow down the scope of a research project or to identify problems in a manufacturing process from data already in existence. Also, the Taguchi method allows for the analysis of many different parameters without a prohibitively high amount of experimentation. The main disadvantage of the Taguchi method is that the results obtained are only relative and do not exactly indicate what parameter has the highest effect on the performance characteristic value. Also, since orthogonal arrays do not test all variable combinations, this method should not be used if all relationships between all variables are needed Signal-To-Noise (S/N) Ratio Another of Taguchi s contributions is the signal-to-noise (S/N) ratio. It was developed as a proactive equivalent to the relative loss function. Signal-to-noise ratio (often abbreviated SNR or (S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise [106]. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise. While SNR is commonly quoted for electrical signals, it can be applied to any form of signal. To analyse the results of experiments involving multiple runs, use of the (S/N) ratio over the standard analysis (use average of results) is preferred. Analysis using (S/N) ratio offers following two main advantages It provides guidance to a selection of the optimum level based on the least variation around the target and also on the average value closest to the target. 79

15 It offers objective comparison of two sets of experimental data with respect to variation around the target and the deviation of the average from the target value. There are 3 Signal-to-Noise ratios of common interest for optimization of static problems: Nominal the Best n = 10 log 10 ( Square of mean ) variance It is used wherever there is a nominal or target value and the target is finite but not zero. Smaller the Best n = 10 log 10 (mean of sum of squares of measured data) This is usually the chosen (S/N) ratio for all undesirable characteristics like defects etc. for which the ideal value is zero. Also, when an ideal value is finite and its maximum or minimum value is defined then the difference between measured data and ideal value is expected to be as small as possible. The generic form of (S/N) ratio then becomes, n = 10 log 10 [mean of sum of squares of measured - ideal] Larger the Best n = 10 log 10 (mean of sum squares of reciprocal of measured data) It is used where the largest value is desired, such as weld strength, gasoline mileage. From a mathematical viewpoint, the target value is infinity. Like loss function, it is the reciprocal of smaller the best. 80

16 5.2.5 Taguchi Principle for Muffler Optimization For muffler optimization the factorial design can be used. However, such a design requires excessive number of experiments. Therefore, fractional factorial design has been used in this case, since it takes less time than factorial design. This design is not concerned about interactions with more than two factors, which are not significant in general. This thesis proposes an optimal design scheme to improve the muffler s capacity of noise reduction of the exhaust system by combining the Taguchi method and a fractional factorial design. The change in the quality characteristics of a product under investigation, in response to a factor introduced in the experimental design is termed as signal of the desired effect. However, when an experiment is conducted, there are numerous external factors which are not considered into actual experimentation. However, these factors also influence the outcome (response). These external factors are termed as noise factors and their effect on the outcome of the quality characteristics under test is termed the noise. The signal to noise ratio (S/N) measures the sensitivity of the quality characteristics being investigated in a controlled manner, to those external influencing factors (noise factor) not under control. For present work Larger the Best was used. This is because the best transmission loss is one with higher value. Thus (S/N) ratio of this TL needs to be maximum. The accuracy of optimization technique depends upon many factors. Though they predict results with 90% confidence level there is need to validate the results with experimentation procedure. So general practice is to optimize the model by numerical methods and validate the result by experimental methods. For the experimental validation experimental set-up must be developed with due care and precaution. So next chapter covers experimental set-up and procedure for determining transmission loss. 81