OPTIMISATION OF PIN FIN HEAT SINK USING TAGUCHI METHOD

CHAPTER - 5 OPTIMISATION OF PIN FIN HEAT SINK USING TAGUCHI METHOD The ever-increasing demand to lower the production costs due to increased competition has prompted engineers to look for rigorous methods of decision making such as optimization. As a result engineering optimization was developed to help engineers design systems that are both more efficient and less expensive and to develop innovative methods to improve the performance of the existing systems. Engineering optimization can best be classified as a rigorous mathematical approach to identify and select a best candidate from a set of probable design alternatives (Rao, 1996). Optimization in its broad sense can be applied to solve any engineering problem. Optimization techniques are currently being used in a wide variety of industries, including aerospace, automotive, MEMS, chemical, electrical and manufacturing industries. With the development of computer technology, complexity of problems being solved using optimization methods is no longer an issue. Optimization methods coupled with modern tools of Computer Aided Design (CAD) are also being used to enhance the creative process of conceptual and detailed design of engineering systems. There is no single method or technique for solving all optimization problems efficiently. Hence a number of optimization methods have been developed for solving different types of optimization problems. It is in the entire discretion of the engineer to choose a method which is computationally efficient, accurate and appropriate for his design problem. Because a good design point is often the result of a trade-off between various objectives, the exploration of a given design cannot be performed by using only direct optimization algorithms leading to a single design point. It is important to gather enough information about the current design so as to be able to answer the so-called what-if questions quantifying the influence of design variables on the performance of the product in an exhaustive manner. By doing so, the right decisions can be made based on accurate information - even in the event of an unexpected change in the design constraints. The optimal performance is obtained when the heat sink fin geometry produces the best turbulent flow conditions that 131

generate the largest heat transfer coefficient, and heat sink fins are large enough to provide minimum internal thermal resistance and maximum convective surface area. In this chapter optimum design of heat sink under multi-jet impingement condition is investigated using a novel technique Taguchi method. 5.1 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 of Japan who maintained that variation. 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. The experimental design proposed by Taguchi involves using orthogonal arrays to organize the parameters affecting the process and the levels at which they should be varied. Instead of having to test all possible combinations like the factorial design, the Taguchi method tests pairs of combinations. This allows for the collection of the necessary data to determine which factors most affect product quality with a minimum amount of experimentation, thus saving time and resources. The Taguchi method is best used when there are an intermediate number of variables (3 to 50), few interactions between variables, and when only a few variables contribute significantly. The Taguchi arrays can be drawn out manually and large arrays can be derived from deterministic algorithms. The arrays are selected by the number of parameters (variables) and the number of levels (states). Analysis of variance on the collected data from the Taguchi design of experiments can be used to select new parameter values to optimize the performance characteristic. The data from the arrays can be analyzed by plotting the data and performing a visual analysis, ANOVA, bin yield and Fisher's exact test, or Chi-squared test to test significance. The various phases and steps involved in optimization using Taguchi method is shown in the Fig. 5.1. The general steps involved in the Taguchi Method are as follows: 1. Define the process objective, or more specifically, a target value for a performance measure of the process. This may be a flow rate, temperature, etc. The target of a process may also be a minimum or maximum; for example, the 132

Determine the factors Identify the test conditions Identify control and noise factors Phase-1 Design the matrix experiments (OAs) Define the Data analysis procedure Conduct Designed experiments Phase-2 Analyse the Data using software Phase-3 Predict the performance at these Individual factor contribution Relative factor interaction ANOVA and S/N analysis Performance under optimal condition Phase-4 Validation of experiment Fig.5.1: Flow Chart of Taguchi Method 133

goal may be to maximize the output flow rate. The deviation in the performance characteristic from the target value is used to define the loss function for the process. 2. Determine the design parameters affecting the process. Parameters are variables within the process that affect the performance measure such as temperatures, pressures, etc. that can be easily controlled. The number of levels that the parameters should be varied at must be specified. For example, a temperature might be varied to a low and high value of 40 0 C and 80 0 C increasing the number of levels to vary a parameter at increases the number of experiments to be conducted. 3. Create orthogonal arrays for the parameter design indicating the number of and conditions for each experiment. The selection of orthogonal arrays is based on the number of parameters and the levels of variation for each parameter, and will be expounded below. 4. Conduct the experiments indicated in the completed array to collect data on the effect on the performance measure. 5. Complete data analysis to determine the effect of the different parameters on the performance measure. 5.2 ORTHOGONAL ARRAY 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. 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 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 value can be tested or the values tested can be closer together. For example, if the temperature of a reactor jacket can be varied between 20 and 80 o C and it is known that the current operating jacket temperature is 50 C; three levels might be chosen at 20, 50, and 80 C. Also, the cost of conducting experiments must be considered when determining the number of levels of a parameter to include in the experimental design. In the previous 134

example of jacket temperature, it would be cost prohibitive to do 60 levels at 1 degree intervals. 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. Using the array selector Table 5.1 shown below, the name of the appropriate array can be found by looking at the column and row corresponding to the number of parameters and number of levels. Once the name has been determined (the subscript represents the number of experiments that must be completed), the predefined array can be looked up. Links are provided to many of the predefined arrays given in the array selector table. Table 5.1 shows Orthogonal Array selector details. These arrays were created using an algorithm Taguchi developed, and allows for each variable and setting to be tested equally. For example, if we have three parameters (voltage, temperature, and pressure) and three levels (high, medium, low), it can be seen the proper array is L9. It can be seen nine different experiments are given in the L9 array as shown in Table 5.2. The levels designated as 1, 2, 3 etc. should be replaced in the array with the actual level values to be varied and parameters P1, P2, P3 should be replaced with the actual parameters (i.e. voltage, temperature, etc.). 5.3 OPTIMISATION OF PIN FIN HEAT SINK In this section the details procedure of optimization is explained for the case of multi-jet impingement on pin fin heat sink by exiting air through effusion slots provided on the nozzle plate as shown in the Fig. 5.2. 5.3.1 Design of Experiments ( DOE ) From the detailed experimentation studied so far it has been observed that Reynolds number, ratio Z/d, total area inlet through nozzle and total area of convection (base area of heat sink and pin fin area) are the major controlling parameters for the optimization of pin fin heat sink. Total convective surface depend upon the geometric parameters of heat sink such as array size, height, pin diameter and pitch. Total nozzle area depends up on the nozzle diameter, array size and nozzle pitches. For the optimization it is considered that for each heat sink configuration total area of nozzles must be same and thereby quantity of air impinging will be automatically same. 135

Nozzles Inlet Spent air exit Effusion Slots Pin fin heat sink Adiabatic walls Fig.5.2: Geometry of Computational Domain Considered for Optimization 136

Table 5.1: Orthogonal Array Selectors Number of parameters 2 3 4 5 6 7 8 9 10 No of levels 2 L4 L4 L8 L8 L8 L8 L12 L12 L12 3 L9 L9 L9 L18 L18 L18 L18 L27 L27 4 L 16 L 16 L 16 L 16 L32 L32 L32 L32 L32 5 L25 L25 L25 L25 L50 L50 L50 L50 L50 Table 5.2: L 9 Array for 3 Parameters with 3 Levels L9 Orthogonal Array Parameter 1 Parameter 2 Parameter 3 1 Level 1 Level 1 Level 1 2 Level 1 Level 2 Level 2 3 Level 1 Level 3 Level 3 4 Level 2 Level 1 Level 2 5 Level 2 Level 2 Level 3 6 Level 2 Level 3 Level 1 7 Level 3 Level 1 Level 3 8 Level 3 Level 2 Level 1 9 Level 3 Level 3 Level 2 137

Thus under same input condition (same amount of heat flux supplied and same amount of air has been impinged) optimization can be done. To find such combinations, pin diameter (d) and array size (N N) are varied from 1 to 5 mm and 4 4 to 11 11 respectively. For all such cases total nozzle area is calculated. It is observed that only three configurations have same total nozzle area (176.8 mm 2 ) at array size 4 4, 6 6 and11 11 with fin diameter 5, 3 and 1.5 mm respectively. Therefore three heat sinks shown in Table 5.3 (Fig.1.9) are considered as baseline for the optimization pin fin heat sink. Thus for the considered problem, the three parameters and three levels are used for Taguchi method with very careful understanding of the levels taken for the factors. The factors to be studied are mentioned in Table 5.4. Before selecting an orthogonal array, the minimum number of experiments to be conducted can be fixed by using the following relation, = 1 + ( 1). (5.1) where N Taguchi is the number of experiments to be conducted, NV is the number of variables and NL is the number of levels. In this analysis, NV = 3 and NL = 3. Hence a minimum of seven experiments are to be conducted. The standard orthogonal arrays available are L4, L8, L9, L12, L16, L18 etc. According to the Taguchi design concept L9 orthogonal array is chosen for the experiments as shown in Table 5.5 and Experimental design in Table 5.6. In this study, the observed values of Z/d ratio, CFM and Pin Fin diameter will be set at maximum level. The impinging velocity at the nozzle plate and the total area of the nozzle holes are identical for these selected nozzle plates when the airflow rate is the same. The important parameter associated with correlating the convective heat transfer for a fin bundle is found to be fin bundle void fraction α (Ko-Ta et al., 2006). The void fraction of the fin- bundle is that fraction of a cross-sectional area of the fin-bundle that is occupied by air. It can be expressed as- = 1. For all the cases it is kept constant (0.913). 5.3.2 Data Analysis Each experimental trail was performed as per L9 Table (Table 5.6) and the optimization of the observed values was determined by comparing the standard method and analysis of variance (ANOVA) which is based on the Taguchi method. 138

Table 5.3: Heat Sinks Considered as Baseline for the Optimization Sr. No. Width of fin distribution area Fin Array Fin Height Fin diameter Fin Spacing Fin Pitch One Fin surface Area (π.dh+ π.d 2 ) Total Fin Surface Area (π DH+ π D 2 ) N 2 W mm N N H mm D mm S mm X p mm A s mm 2 A t mm 2 A f Total fin area Cross section reduction ratio (A b -A f ) /A b No. of nozzle Total Area of Nozzle Ratio of total convective area to total nozzle area (N-1) 2 A n A f / A n 1 50 11 7.5 1.5 3.35 4.85 35.3 4276.1 213.7 0.91 10 176.8 1.208 2 50 6 15 3 6.4 9.4 141.4 5089.3 254.3 0.9 5 176.8 1.438 3 50 4 25 5 10 15 392.7 6283.0 314.0 0.87 3 176.8 1.776 139

Table 5.4: Control Parameters and Levels Control Parameters Level 1 Level 2 Level 3 Pin Fin diameter, d (mm) 1.5 (11 11 fin array) 3 (6 6 fin array) 5 (4 4 fin array) Z/d Ratio 6 8 10 Air flow rate (m/s) 26.59 39.88 53.17 Table 5.5: Orthogonal Array for L9 Design Experiment no. P1 P2 P3 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 2 5 2 2 3 6 2 3 1 7 3 1 3 8 3 2 1 9 3 3 2 Table 5.6: Experimental Design for Orthogonal Array of L9 Array and Finding Heat Transfer Coefficient L9 Orthogonal Array Pin Fin diameter, d (mm) Z/d Ratio Air flow rate (m/s) Average heat transfer coefficient h, (W/m 2 K) 1 1.5 6 26.59 245.674 2 1.5 8 39.88 247.782 3 1.5 10 53.17 242.181 4 3 6 39.88 239.37 5 3 8 53.17 250.84 6 3 10 26.59 150.44 7 5 6 53.17 283.19 8 5 8 26.59 189.25 9 5 10 39.88 216.43 140

Table 5.7 shows the S/N ratio for L9 orthogonal array. In the Taguchi method, all the observed values are calculated based on the concept that higher the better and smaller the better. In this analysis, the observed values of heat transfer coefficient or Nusselt number will be set to the maximum. Analysis of Variance (ANOVA) is used to analyze the experimental data as follows. 1. Sound to Noise ratio (S/N ratio) was calculated using formula log where Y i was the output or performance characteristic. The performance of the heat sink (ANOVA-significant factor) was calculated for each experiment of the L9 by using the observed values of the heat transfer coefficient from Table 5.7. 2. After calculating S/N ratio for L9 array, correction factor (C.F) was calculated using following formula (Table 5.7) ( ) = ( ) Where N was the total number of experiments 3. Sum of Squares (SS) is found by using the equation, (Table 5.8) (5.2) = 1 2 + + (5.3) 4. The % of contribution of each parameter on heat transfer coefficient (Table 5.9) was calculated using formula: % = ( ) (5.4) It was observed that the contributions of all the working parameters (Fin diameter, Z/d and CFM) performance had equal importance (Fig. 5.3). 5. The optimum working parameters of heat sink (ANOVA-optimum condition) were determined using Taguchi methodology (Table 5.10). From Fig.5.4 it was observed that optimum working parameters were, pin fin diameter = 1.5 mm (11 11array heat sink), Z/d = 6 and air flow rate is 53.17 m/s. Thus, the optimum solution for the given problem was obtained for the above parameters with heat transfer coefficient as 343.38 W/m 2 K and this value was found to be better than the previous observations. (Table 5.11). 141

Table 5.7: S/N Ratio for L9 Orthogonal Array L9 Orthogonal Array Pin Fin diameter, d (mm) Z/d Ratio Air flow rate (m/s) Average heat transfer coefficient h, (W/m 2 K) S/N Ratio 1 1.5 6 26.59 245.674 47.80718 2 1.5 8 39.88 247.782 47.8814 3 1.5 10 53.17 242.181 47.6828 4 3 6 39.88 5 3 8 53.17 6 3 10 26.59 7 5 6 53.17 8 5 8 26.59 9 5 10 39.88 239.37 250.84 150.44 283.19 189.25 216.43 47.58139 47.98794 43.54727 49.04156 45.54072 46.70635 CF 19954.06 Table: 5.8: Sigma S/N Values for Heat Transfer Coefficient CF >> 19954.06 Diameter Z/d Air flow rate Sigma S/N - I 143.3714 144.4301 136.8952 Sigma S/N - II 139.1166 141.4100 142.1691 Sigma S/N - III 141.2886 137.9364 144.7123 142

Table 5.9: Analysis of Variance for Heat Transfer Coefficient (h) Diameter Z/d Air flow rate 1 Mean 47.8000 48.15 45.64 2 Mean 46.3800 47.13668 47.39 3 Mean 47.1000 45.98 48.24 SS 19958.5941 19964.62687 19969.96595 % CONTRIBUTION 33.33 % 33.34 % 33.34% Table 5.10: Optimum Working Parameters Control Parameters Parameter1 Parameter 2 Parameter 3 Heat transfer coefficient 1.5 mm (11 11 array heat sink) Z/d = 6 53.17 m/s. Table 5.11: Optimum Solution of Heat Sink Parameter Result Heat transfer coefficient 343.38 W/m 2 K 143

Air CFM Flow Rate 34% Diameter 33% Z/d 33% (a) Percentage Contribution 48.5 48.0 47.5 Optimum working parameters Diameter Z/d Air CFMFlow Rate Vairance 47.0 46.5 46.0 45.5 45.0 1 2 3 Level of design variables (b) Effect Fig. 5.3: (a) Percentage Contribution and (b) Effect of Each Factor on Heat Transfer Coefficient 144