Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3): 493-500 © Scholarlink Research Institute Journals, 2013 (ISSN: 2141-7016) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016)
Prediction and Optimization of Design Parameters of Microelectronic Heat Sinks Mathias Ekpu, Raj Bhatti, Michael I. Okereke, Sabuj Mallik, Kenny C. Otiaba Manufacturing Engineering Research Group, School of Engineering University of Greenwich at Medway, Chatham, ME4 4TB, Kent, U.K. Tel: +441634883873, e-mail:
[email protected] Corresponding Author: Mathias Ekpu __________________________________________________________________________________________ Abstract This research presents an approach for predicting and optimizing the thermal resistance and mass of microelectronics heat sinks parameters. The Minitab software used the Taguchi design of experiments method to generate L27 Orthogonal array combinations for the heat sink geometry. The combinations consist of 6 factors (fin height, base thickness, fin thickness, length, number of fins, and width of heat sink) and 3 levels (low, medium, high). Ansys finite element software was used in the thermal analysis of the heat sink to determine the thermal resistance while the mass was calculated. The thermal resistance and mass responses were used to develop a multiple linear regression analysis model for predicting both thermal resistance and mass of heat sink. The developed regression model showed over 90% accuracy in predicting the thermal resistance and mass of heat sinks. In addition, the optimized heat sink geometry showed a 50.75% reduction in mass and 5.26% reduction in thermal resistance. This model could be used as an alternative to simulation software and could save cost and time in selecting an optimal heat sink for microelectronic applications. __________________________________________________________________________________________ Keywords: heat sink, multiple linear regression analysis, Taguchi design of experiment, mass, thermal resistance. INTRODUCTION The use of high performance chips and large scale integrated (LSI) circuits becomes more widespread due to recent development of electronics and communications technologies (Yin et al 2008; Black et al 1998). The demand for high power electronic chips grows continually, but their volumes are steadily reduced. In recent years, laptop computer microprocessors could produce a power loss up to 50W or more on chip during operation (Yin et al 2008). These chips generate heat and when the heat becomes excess it could lead to thermal failures. Therefore, it is important that heat is dissipated from the chip to avoid such failures. Heat sink is a device that could be used to dissipate such heat from the chip level device (Ekpu et al 2011a). Fig. 1 shows a picture of an Intel Pentium II flat plate heat sink with rectangular fins. The geometry of the heat sink is designed by mechanical engineers with the main aim of reducing the thermal resistance and the mass of the heat sink for better cooling of the microelectronic device. Aluminium flat plate heat sink with rectangular fins is a commonly used design of a heat sink because of its good reliability and relatively low production cost (Ekpu et al 2011b).
112W LED street lamp. The heat dissipation from exposed base area and fin array, and optimization and realization was investigated. Dogruoz and Arik (2010) studied the conduction and convection heat transfer from lightweight advanced heat sinks. The research was motivated by the lack of understanding of the use of anisotropic advanced materials in natural convection environments. The research found that the weight of heat sinks manufactured with advanced materials may be reduced drastically compared to relatively used materials. Kulkarni and Das (2005) studied analytical and numerical studies on microscale heat sinks for electronic applications. The research presented thermal models for cooling microscale electronic processor chips through forced and natural convection heat sinks. The thermal models prediction equations were enhanced by the performance of a Pentium III heat sink. A novel hybrid heat sink using phase change materials for transient thermal management of electronics was proposed by Krishnan et al (2004). The research aimed at improving thermal performance of electronic heat sinks. In another study, Yin et al (2008) carried out experimental research on heat transfer mechanism of heat sink with composite phase change materials. The research suggest that the application of composite phase change material to an electronic device’s heat sink could effectively improve the performance of
Luo et al (2009) studied the design and optimization of horizontally located plate fin heat sink for high power LED street lamps. The research was aimed at improving thermal management of a heat sink for 493
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) resisting the shock of high heat flux and ensure the reliability and operating stability of electronic and electrical equipment. Kreutz et al (2000) investigated the simulation of micro-channel heat sinks for optoelectronic Microsystems. The research implied that the designed micro-channel heat sinks results in a decrease of their thermal resistance and the pressure drop of the coolant. This allowed an increased heat load of an optoelectronic Microsystems. Wan et al (2011) studied the flow and heat transfer in porous micro heat sink for thermal management of high power LEDs. The research proposed a novel porous micro heat sink system for thermal management of high power LEDs. The research result suggested that the heated surface temperature of porous micro heat sink is low at high heat fluxes and is much less than the bearable temperature level of LED chips.
model comprising of a heat sink, thermal interface material, and chip was designed as shown in Fig. 3.
Fig. 1 Intel Pentium II Heat Sink.
Fig. 3 Schematic of a Microelectronic Package
In this paper, the focus is on developing a Multiple Linear Regression Analysis (MLRA) model for heat sink performance factors such as thermal resistance and mass. A flat plate heat sink with geometric factors such as fin height (a), base thickness of heat sink (b), fin thickness (d), length of heat sink (L), number of fins (N), and width of heat sink (W) were considered. A design of experiment was carried out using the Taguchi design method. The geometry combinations of the heat sink generated by the Taguchi design method were created using the Ansys finite element software. The software was used to find out the various performance characteristics of the different combinations of the heat sink geometries. The importance of this research is to improve the selection of heat sinks and thermal performance of microelectronic devices. The developed MLRA model could be a good replacement for the use of simulation software, thereby saving cost and time of modelling and simulation.
The chip has a 15 x 15mm dimension and a thickness of 1mm while the thermal interface material has a dimension of 15 x 15mm with a thickness of 0.035mm. The microelectronic package materials and properties used in this study are presented in Table 1
Fig. 2 Schematic of a Flat Plate Heat Sink
Table 1 Material and Properties Material
Thermal Conductivity (W/mK)
Density (K/m3 )
Aluminium Heat Sink Silicon Chip Solder (Thermal Interface Material)
237.5
2689
Specific Heat Capacity (J/kgC) 951
148 48
2330 8000
712 167
Boundary Conditions The overall boundary conditions for the simplified model include a uniform heat flow ‘Q’ (Watts) on the surfaces of the chip. A heat generating surface area (active area) of 80W is applied on the surface of the silicon chip. A natural convection coefficient for stagnant air simplified case is applied to the exposed area of the heat sink. The ambient temperature is assumed to be 22oC and the system is assumed to be adiabatic therefore, the heat transfer due to radiation and convection are neglected. Conduction is the only mode of heat transfer considered in this research
MATERIALS AND METHODS Model Description A schematic of a flat plate heat sink with ‘N’ number of fins is shown in Fig. 2. The geometry model was created with Ansys workbench version 13 software. In other to represent the microelectronic package, a
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) Where: ΔT = Tb – Ta (2) Where Tb is the temperature of the heat sink base, Ta is the ambient temperature, and ‘Q’ is the heat dissipation.
because heat is dissipated from the chip through the bonding layers to the heat sink primarily by conduction. METHODOLOGY In order to effectively analyse the microelectronic model created in Ansys finite element software, the model was symmetrically divided into four equal parts. This was done in other to reduce computational time and storage space of the system used for the simulation. The part of interest of the microelectronic model was subdivided into assembly of finite elements (mesh) as shown in Fig. 4. The mesh consists of a total of 2102017 nodes and 1144583 elements. The transient thermal analysis was used to analyse the thermal conduction of the microelectronic model. The maximum temperatures of the heat sinks over time were recorded and used for the thermal resistance calculation
The mass of the heat sink is given in Equation 3 as: Mass = [(bLW) + (adLN)] x 10-6 x ρAl (3) Where ρAl is the density of the Aluminium heat sink. The L27 Orthogonal array combinations with the corresponding responses of thermal resistance and mass generated by the Minitab software are given in Table 3 Table 2 Heat Sink Design Factors and Levels Design Parameters Fin Height (mm) Heat Sink Base Thickness (mm) Fin Thickness (mm) Length of Heat Sink (mm) No of Fins Width of heat Sink (mm)
Level 1 (low) 10
Level 2 (medium) 20
Level 3 (high) 30
3 1.3
5 1.5
7 1.7
60 10
70 20
80 30
60
70
80
Linear Regression Analysis Problems in most cases have two or more related variables, and it is of importance to model such relationship. A mathematical model called regression model is used to characterized the relationship between the variables (Montgomery 2009). A regression model could be used to predict a dependant variable from independent variable(s). Therefore, Multiple Linear Regression Analysis (MLRA) is used to find the relationships between a dependent variable and several independent variables. The multiple linear regressions model with response variable ‘y’ may be related to the ‘k’ regressor variable as given in Equation 4 (Montgomery 2009). y = β0 + β1 x1 + β2x2 + ... + βkxk (4)
Fig. 4 Mesh of the Microelectronic Package Taguchi Design of Experiments The generation of the orthogonal array of the heat sink geometry factors comprised of 6 factors (fin height (a), heat sink base thickness (b), fin thickness (d), length of heat sink (L), number of fins (N), and width of heat sink (W)) with 3 levels (low, medium, high) as shown in Table 2. Applying the full factorial design of experiment method will yield a Minimum Number of Experimental (MNE) combinations equivalent to ‘kn’, where ‘k’ is the number of levels and ‘n’ the number of factors. Therefore, the full factorial design method will give 36 (729) simulation combinations. Performing this large amount of simulations will be tasking and will take a lot of time and resources. Hence, the choice of Taguchi design method of experiment over the full factorial designs method. The Taguchi Orthogonal array method allows fewer simulations and gives similar results as the full factorial design method. The Minitab software used the 6 factors and 3 levels to generate a combination of L27 Orthogonal array. The responses for each of the L27 combinations are thermal resistance (K/W) and mass (g). For each thermal simulation carried out, the temperature of the heat sink base were recorded and use to calculate the thermal resistance of the heat sink as shown in Equation 1. R = ΔT/Q (1)
The parameters βj, for j = 0, 1, . . . , k, are the regression coefficient and xj, for j = 1, 2, . . . , k, are the independent variables. The regression analysis technique is used to determine the model’s regression coefficients from the available data. A new standard model for predicting the thermal resistance and mass of heat sink is suggested in this research work. The Taguchi design approach is used for the suggested regression model. The response ‘y’ is either thermal resistance or mass of heat sink depending on the response data used. While the independent variables include: fin height (a), heat sink base thickness (b), fin thickness (d), length of heat sink (L), number of fins (N), and width of heat sink (W). The regression coefficient is assumed to be β0, β1, β2, β3, β4, β5, and β6 respectively. This new standard model is given in Equation (5). y = β0 + β1a + β2b + β3d + β4L + β5N + β6W (5)
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) Table 3 L27 Taguchi Design of Experiment with Corresponding Responses L27 Combination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Fin Height 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30
Heat Sink Base Thickness 3 3 3 5 5 5 7 7 7 3 3 3 5 5 5 7 7 7 3 3 3 5 5 5 7 7 7
Fin Thickness 1.3 1.3 1.3 1.5 1.5 1.5 1.7 1.7 1.7 1.5 1.5 1.5 1.7 1.7 1.7 1.3 1.3 1.3 1.7 1.7 1.7 1.3 1.3 1.3 1.5 1.5 1.5
Length of Heat Sink 60 60 60 70 70 70 80 80 80 80 80 80 60 60 60 70 70 70 70 70 70 80 80 80 60 60 60
RESULTS AND DISCUSSIONS Prediction of Thermal Resistance of Aluminium Heat Sink using Linear Regression Analysis Table 4 presents the linear regression analysis of the thermal resistance response carried out using Minitab software and the corresponding values of coefficients of the heat sink independent variables. The analysis of variance is presented in Table 5. The regression equation for the thermal resistance of the heat sink is given as:
No of Fins 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30
Width of Heat Sink 60 70 80 60 70 80 60 70 80 70 80 60 70 80 60 70 80 60 80 60 70 80 60 70 80 60 70
Resistance (K/W) 0.7735750 0.6830875 0.6580250 0.6103625 0.5755750 0.5562875 0.5457125 0.5298125 0.5225250 0.7665500 0.6834625 0.5712125 0.6057125 0.5711375 0.5215125 0.5545125 0.5404625 0.5173625 0.7695625 0.6174875 0.5748625 0.6255250 0.5702000 0.5520375 0.5539500 0.5267500 0.5183000
Mass (g) 50.0154 75.8298 101.6442 84.7035 122.3495 159.9955 126.9208 178.5496 230.1784 109.7112 180.7008 232.3296 111.3246 174.2472 212.9688 141.1725 203.2884 225.876 141.1725 225.876 327.5202 169.9448 232.3296 326.9824 162.9534 212.9688 296.8656
Resistance (K/W) = 0.901 - 0.000813 Fin Height 0.0358 Heat Sink Base Thickness - 0.0601 Fin Thickness -0.000250 Length of Heat Sink - 0.00452 No of Fins + 0.00126 Width of heat Sink (6) This could be re-written as: Resistance (K/W) = 0.901 - 0.000813a - 0.0358b 0.0601d - 0.000250L - 0.00452N + 0.00126W (7)
Table 4; Linear Regression Analysis for Predicting Thermal resistance of a Heat Sink Predictor
Coefficient Values
SE Coefficient
T
P
Constant, β0
0.9012000
0.1049000
8.59
0.000
VIF
Fin Height, β1
-0.0008127
0.0008069
-1.01
0.326
1.000
Heat Sink Base Thickness, β2
-0.0357900
0.0040350
-8.87
0.000
1.000
Fin Thickness, β3
-0.0601300
0.0403500
-1.49
0.152
1.000
Length of Heat Sink, β4
-0.0002501
0.0008069
-0.31
0.760
1.000
No of Fins, β5
-0.0045185
0.0008069
-5.60
0.000
1.000
Width of heat Sink, β6
0.0012598
0.0008069
1.56
0.134
1.000
S = 0.0342340 R-Sq = 85.3% R-Sq(adj) = 80.9% PRESS = 0.0451130 R-Sq(pred) = 71.66%
Table 5 Analysis of Variance for Resistance of Al Heat Sink Source
DF
SS
MS
F
P
Regression
6
0.135738
0.022623
19.3
0.000
Residual Error
20
0.023439
0.001172
Total
26
0.159178
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) The following could be deduced from Tables 4 and 5. At an α-level of 0.05 the p-value in the Analysis of Variance table (0.000) shows that the model estimated by the regression procedure is significant. This also indicates that at least one coefficient is different from zero. The p-values for the estimated coefficients of the factors, indicates that the factors are significantly related to the thermal resistance of the aluminium heat sink. The R-sq value indicates that the predictors explain 85.3% of the variance in the thermal resistance of the aluminium heat sink. The adjusted R-sq is 80.9%, which accounts for the number of predictors in the model. Both values indicate that the model fits the data well. The predicted R-sq value is 71.66%. The predicted R-sq value is close to the R-sq and adjusted R-sq values; therefore the model has adequate predictive ability. The multiple linear regression model developed in Equation (7) were used to test the L27 combinations generated using Taguchi design of experiments. The comparison graph of the MLRA prediction of thermal resistance against the simulated results of thermal resistance is shown in Fig. 5. The standard residual of the thermal resistance against the observation run is shown in Fig. 6. The standard residual ranges between – 1.3 to 2.3 therefore implying that the predicted model could serve as a good alternative for the simulation software in predicting thermal
Fig. 5 Comparison of Simulation results vs. MLRA Prediction of Thermal Resistance for Aluminium Heat Sink Versus Order (response is Resistance) 2.5
Standardized Residual
2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 2
4
6
8
10
12 14 16 18 Observation Order
20
22
24
26
Fig. 6 Residual Graph for Thermal Resistance Prediction of the Mass of Aluminium Heat Sink using Linear Regression Analysis The linear regression analysis of the calculated mass response using Minitab software and the corresponding values of coefficients of the heat sink independent variables are presented in Table 6. The analysis of variance is presented in Table 7. The regression equation for the mass of the heat sink is given as: Mass of Al Heat Sink (g) = -292 + 5.37 Fin Height + 9.28 Heat Sink Base Thickness + 56.0 Fin Thickness + 2.16 Length of Heat Sink + 5.65 No of Fins – 0.444 Width of heat Sink (8) This could be re-written as: Mass (g) = -292 + 5.37a + 9.28b + 56.0d + 2.16L + 5.65N – 0.444W (9) Table 6 Linear Regression Analysis for P Predictor
Coefficient Values
SE Coefficient
T
P
VIF
Constant, β0
-292.43
61.29
-4.77
0.000
Fin Height, β1
5.3690
0.4712
11.39
0.000
1.000
Heat Sink Base Thickness, β2
9.277
2.356
3.94
0.001
1.000
Fin Thickness, β3
56.02
23.56
2.38
0.028
1.000
Length of Heat Sink, β4
2.1602
0.4712
4.58
0.000
1.000
No of Fins, β5
5.6469
0.4712
11.98
0.000
1.000
Width of heat Sink, β6 S = 19.9919 R-Sq = 94.1% R-Sq(adj) = 92.3% PRESS = 15407.6 R-Sq(pred) = 88.54%
-0.4437
0.4712
-0.94
0.358
1.000
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) Table 7 Analysis of Variance for Mass of Al Heat Sink Source
DF
SS
MS
F
P
Regression
6
126495
21083
52.75
0.000
Residual Error
20
7994
400
Total
26
134489
The following could be inferred from Tables 6 and 7. The p-value in the Analysis of Variance table (0.000) shows that the model estimated by the regression procedure is significant at an α-level of 0.05. This also indicates that at least one coefficient is different from zero. The p-values for the estimated coefficients
Versus Order (response is Mass of Al Heat Sink)
Standardized Residual
3
of the heat sink factors, indicates that the factors are significantly related to the mass of Al heat sink. The R-sq value indicates that the predictors explain 94.1% of the variance in the mass of Al heat sink. The adjusted R-sq is 92.3%, which accounts for the number of predictors in the model. Both values indicate that the model fits the data well. The predicted R-sq value is 88.54%. Since the predicted R-sq value is close to the R-sq and adjusted R-sq values, the model does not appear to be over-fit and has adequate predictive ability.
2
1
0
-1
-2 2
4
6
8
10
12 14 16 18 Observation Order
20
22
24
26
Fig. 8 Residual Graph for Mass of Al Heat Sink Validation of Simulation and MLRA Prediction Results Performing experiments for this particular research work would prove difficult because of the number of heat sinks that will be fabricated due to the L27 geometric combinations generated by Taguchi design of experiments. However, in order to validate the simulation results and the MLRA equations for both thermal resistance and mass of aluminium heat sinks, a Pentium III flat plate heat sink was investigated (Intel 2001). The dimensions of the heat sink are presented in Table 8.
The L27 combinations generated using Taguchi design of experiments was used to test the multiple linear regression model developed in Equation (9). The comparison graph of the MLRA prediction of the mass against the results calculated for mass is shown in Fig. 7. The standard residual of the mass against the observation run is shown in Fig. 8. The standard residual of the mass response is between – 1.5 to 1.9. However, one of the observation run was above 1.9. Therefore, the predicted model could serve as a good alternative for the calculation software in predicting mass of a heat sink. This could save cost and time.
Table 8 Pentium III Heat Sink Dimensions Design Parameters
Dimension
Fin Height (mm)
19.939
Heat Sink Base Thickness (mm)
3.175
Fin Thickness (mm)
0.762
Length of Heat Sink (mm)
63.5
No of Fins Width of heat Sink (mm)
20 79.502
Applying the MLRA for predicting thermal resistance in Equation (7), the resulting thermal resistance is 0.722775 K/W, while the simulated thermal resistance is 0.805462 K/W. The difference between both thermal resistances is about 10%, which infer that the developed MLRA model equation for predicting thermal resistance of flat plate aluminium heat sink is about 90% accurate. Fig. 7 Comparison of Simulation results vs. MLRA Prediction of Thermal Resistance for Aluminium Heat Sink
The mass of the heat sink is calculated to give 94.99g while the MLRA for predicting mass in Equation (9) gives a mass of 102.07g. The difference between both 498
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3):493-500 (ISSN: 2141-7016) masses is about 6.9%. Therefore, the developed MLRA model equation for predicting mass of flat plate heat sink with rectangular fins is about 93% accurate.
Table 9 Results of Genetic Algorithm Heat Sink Response Mass (g) Thermal Resistance (K/W) Optimal Geometry
Optimization of Pentium III Heat Sink Using Genetic Algorithm (GA) The MLRA model is employed to optimize the Pentium III heat sink parameters in Section 4.3 base on the thermal resistance and mass. Since this is a combination of two factors, the evolutionary algorithms technique is employed. These techniques converges the solutions to globally accepted optimal solutions. The Genetic Algorithm (GA) approach is used in this research to predict the optimal heat sink settings. The objective of the combination approach is to minimize error function (EF) of the two factors. The objective function is given as: EF = [((M – Md)/M) + ((R – Rd )/R] (10)
Desired Output 102.07 0.722775
Predicted Optimal Output 50.27 0.68473
a = 19.939 mm b = 3.175 mm d = 0.762 mm L = 63.5 mm N = 20 W = 79.502 mm
a = 10.39 mm b = 3.38 mm d = 1.7 mm L = 60.21 mm N = 10 W = 60 mm
From Table 9, it is observed that the Pentium III heat sink predicted mass is 50.75% less than that of the desired mass. While the predicted thermal resistance of the heat sink is 5.26% less than the desired thermal resistance. This showed that the developed MLRA model could be effectively applied in the optimization of heat sink geometry. CONCLUSIONS This research has focused on developing multiple linear regression models for predicting thermal resistance and mass of aluminium flat plate heat sinks used in microelectronic applications. In addition, Genetic Algorithm was applied to optimize Pentium III heat sink geometry. The Taguchi design of experiment was used to generate L27 Orthogonal array combinations with the following factors: fin height, heat sink base thickness, fin thickness, length of heat sink, number of fins, and width of heat sink. Ansys finite element software was used for the simulation to determine the thermal resistance and the mass was calculated. The MLRA models developed for predicting thermal resistance and mass of heat sink showed over 90% accuracy. This could substitute the simulation software and save cost and time. In addition, the optimized heat sink geometry showed a 50.75% reduction in mass and 5.26% reduction in thermal resistance. These results would aid in the selection of appropriate heat sinks for microelectronic applications. However, conducting experiments to validate the results will further strengthen the findings of this research work.
Where ‘M’ and ‘R’ are the mass and resistance given by Equations (9) and (7) respectively. While ‘Md’ and ‘Rd’ are the desired mass and resistance of the Pentium III heat sink respectively. Matlab is used to solve the objective function for finding optimal heat sink parameter. The target values of the mass and thermal resistance of the Pentium III heat sink are calculated from the developed MLRA model. Hence, Equations (7), (9), and (10) are embedded into the genetic algorithm. In order to avoid the generation of negative values from the minimization problem, it is necessary to map the objective function to fitness function form. The change from minimization objective f(x) to equivalent maximization problem is given by the fitness mapping F(x) as: F(x) = 1/(1 + f(x)) (11) The Matlab inputs for the objective function and constraints used for the optimization of the Pentium III heat sink are given as: Objective function: function f = fun(x) y(1) = -292 + 5.37*x(1) + 9.28*x(2) + 56.0*x(3) + 2.16*x(4) + 5.65*x(5) – 0.444*x(6) (12) y(2) = 0.901 – 0,000813*x(1) – 0.0358*x(2) – 0.0601*x(3) – 0.00025*x(4) – 0.00452*x(5) + 0.00126*x(6) (13) f = 1/(1 + (((y(1) – 102.07)/y(1)) + (y(2) – 0.722775)/y(2))) (14)
ACKNOWLEDGEMENT The authors would like to thank University of Greenwich for providing the bursary for this research work and also members of Electronics Manufacturing Research Group (EMERG). REFERENCES Black W.Z, Glezer A, Hartley JG. (1998). Thermal Management of a Laptop Computer with Synthetic Air Microjets. IEEE Inter-Society Conference on Thermal Phenomena. pp: 43-50.
Linearity constraints of the variables: Lower bounds: [10 3 0.762 60 10 60] Upper bounds: [30 7 1.7 80 30 80] The optimal parameters of the Pentium III heat sink gotten from the genetic algorithm iterations are presented in Table 9.
Dogruoz M.B, Arik M. (2010). On the Conduction and Convection Heat Transfer from Lightweight Advanced Heat Sinks. IEEE Transaction on Components and Packaging Technologies. pp: 1-8. 499
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