Passive Enhancement of Fin Performance Using Fractal- Like Geometries

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Passive Enhancement of Fin Performance Using Fractal- Like Geometries Daniel Dannelley * and John Baker. Department of Mechanical, University of Alabama, Tuscaloosa, AL, Results of a study into the use of fractal geometries for extended surface heat transfer enhancement are presented. Certain fractal geometries are shown to increase surface area while significantly decreasing the mass of a fin. Two fractal geometries were selected, the modified Koch snowflake and the Sierpinski carpet. This study examines fin performance for the baseline cases (a triangular fin and a square fin) relative to three fractal iterations for a total of eight different fin geometries. Constant heat rate conditions were applied to the fin base and the temperature distribution across the fins was observed using an infrared camera. Fin effectiveness and fin efficiency were calculated, for each fin geometric configuration, to quantify the value of using fractal geometries to improve fin performance. Based upon the observed results, fractal geometries can be used to improve fin performance, especially when the decrease in fin mass is a performance criteria. As fins are used for passive thermal management in many industrial and electronic devices, the use of fractallike geometries has wide reaching potential. Nomenclature A b = base area, m 2 A s = surface area, m 2 b = fin base length, m h = heat transfer coefficient, W/m 2 -K I = current, A m = mass, kg n = iteration index Q = total heat rate, W Qc = corrected heat rate, W t = fin thickness, m T avg = average fin temperature, K T b = fin base temperature, K T = ambient temperature, K T t = fin tip temperature, K U = uncertainty V = voltage, V T = average fin to ambient temperature difference, K ϵ = fin effectiveness ε = emissivity η = fin efficiency θ b = base to ambient temperature difference, K ρ = fin density, kg/m 3 σ = Stefan-boltzmann constant, W/m 2 -K 4 * Graduate Research Assistant, Mechanical Engineering, 290 Hardaway Hall, 7 th Ave. Tuscaloosa, AL, 35487, Student Member. Professor, Mechanical Engineering, 290 Hardaway Hall, 7 th Ave. Tuscaloosa, AL, 35487, Senior Member. 1 Copyright 2012 by the, Inc. All rights reserved.

2 I. Introduction Fins are used to increase heat transfer by increasing the available surface area. They can be found in most electronics, engines, industrial equipment, and a variety of other mechanical devices. The ability of an object to dissipate heat has a direct correlation to the performance, and if not accomplished sufficiently, can result in device malfunction or even failure. In most situations it is ideal to utilize the least amount of space while still obtaining a high level of performance. This can often be challenging because the heat transfer for an object is directly proportional to the available surface area. When fractal geometries are considered, significant gains in available surface areas can be achieved without large increases in fin size or mass. For certain fractal geometries, surface area can even be increased while reducing the mass of the fin. Fractal geometries could be used to increase natural convection for commercial and industrial applications. Also, minimized cost for spacecraft thermal management could be achieved due to a decrease in weight. The potential for using fractals to enhance heat transfer is not a new idea. Lee 1 noted that a major problem in precision instruments is temperature induced error. With the desire to decrease the size of precision instruments while still maintaining the required heat transfer, a fractal Hilbert Curve pattern was used and measurement accuracy was increased. Murali 2 used grooves and threads to increase the heat transfer through radiating pin fins. Groove angles and thread count were varied to find optimal geometries. Plawsky 3,4 used branching systems to increase heat transfer and diffusion properties. It was noted that although infinite branching provided infinite surface area, after a finite number of branch generations the maximum heat transfer was reached. Plawsky also noted that contracting fins provided better response than expanding fins. Xu 5 found optimal branching angles, length ratios, and diameter ratios for branching networks. Mahmoud 6 studied micro scale heat sinks to show that macro scale could not be used to accurately predict behavior at micro scale. Coppens 7 discussed scaling for processes using fractal techniques to preserve micro scale properties. Gao 8 noted the fractal nature of goose down structure used for insulation techniques. This can show that fractals can lead to an increase in heat transfer as well as to a decrease in heat transfer. Kukulka 9 looked at the combined effects of surface texture and arrangements to show that heat transfer can be increased due to surface parameters. Van der Vyver 10,11 studied the effects of a quadratic Koch island tube-inshell heat exchanger. The pattern was found to increase heat transfer while reducing the pressure drop. Sahin 12 showed that circular perforations in a pin fin can result in an increase the Nusselt number with forced convection. Al Essa also showed that enhanced natural convection heat transfer can be achieved by introducing perforations into fins. Square, rectangular, and triangular perforations were used with spacing, orientation, and size of perforations varied. Adrover 16 studied forced convection across fractal boundaries and found that the thickness of the thermal boundary layer corresponds to the boundary fractal dimension. The previously described approaches are just a few of the methods that have been used by other researchers to try to improve fin performance or utilize fractal techniques. In this study, fin surface area was increased by using a fractal pattern. Since heat transfer is proportional to the surface area of the device dissipating thermal energy, fractal patterns are predicted to increase the fin effectiveness. The two fractal patterns used in this study are the Sierpinski carpet and a modified version of the Koch snowflake. The baseline (0th iteration) and the first 3 iterations of both sets can be found in Fig. 1. The Sierpinski carpet removes material with each iteration while the modified Koch snowflake adds material. Figure 2 shows the change in area of each set Figure 1. Sierpinski Carpet (left) and Modified Koch Snowflake (right) for the first ten iterations of fractals and corresponds to the dimensions of the manufactured fractals for experimentation. The fractals fins were manufactured with a width of 4 inches and a thickness of 1/8 inch. The material chosen was Al 5052 because of its ability to be easily machined. The surface area of Sierpinski carpet can be found using Eq. (1) and the surface area of the modified Koch snowflake can be found using Eq. (2). Although the Sierpinski Carpet initially reduces the area with the first two iterations, the surface area increases with subsequent iterations and results in an increase in surface area of 117% after only five iterations. The modified Koch Snowflake increases area with each iteration, but only results in an increase in surface area of 58% after five iterations. 2

3 ( ) ( ) ( ) (1) ( ) ( ) ( ) ( ) ( ) (2) Figure 2. Area/Initial Area vs. Iteration Figure 3 shows the change in mass for each of the fin geometries represented in Fig. 2. The mass of the Sierpinski carpet can be found using Eq. (3) and the mass of the modified Koch snowflake can be found using Eq. (4). The modified Koch snowflake initially increases in mass but converges to an asymptotic value of mass increase of approximately 40%. The Sierpinski carpet, however, decreases in mass and converges to an asymptotic value of zero mass. After only five iterations, the Sierpinski carpet results in an increase in surface area of 117% and a reduction in mass of 45%. Although the Sierpinski carpet seems to be more effective relative to the modified Koch snowflake, both were examined to determine how the fractal-like fins would perform experimentally. ( ) ( ) (3) ( ) ( ) ( ) (4) Figure 3. Mass/Initial Mass vs. Iteration 3

4 II. Experimentation A. Experimental Methods Figure 4 provides a simplified system drawing of the experimental test apparatus. A known heat rate was applied by a set of ceramic resistors. The resistors were clamped to the base of the fins, with a high conductivity thermal paste used to minimize contact resistance. The base of the fins, with the attached ceramic resistors, was insulated to minimize thermal energy losses from the resistance heaters. The fins were sandblasted and painted with a flat black paint to try to provide a diffuse surface for imaging. Once the fins were painted the emissivity of the fins were calibrated using an infrared thermometer and a Figure 4. Experimental Schematic thermocouple. The thermocouple was used to find the base temperature of the fin and the infrared thermometer was used to correlate the emissivity to the temperature. The emissivity was found to be approximately With the emissivity known, the fins were then heated by providing a voltage across the resistors and allowing sufficient time for the system to achieve steady state before any measurements were taken. A FLIR A325 infrared camera was then used to find an average base temperature and average tip temperature as well as to record the temperature profile across the fin. A thermocouple was used to measure ambient temperature. Each fin geometry was tested five times to find data at various ambient temperatures and conditions. B. Calculations Using the average base and average tip temperatures gathered from the infrared camera, an average fin temperature was found. The average fin to ambient temperature difference was found by subtracting the ambient temperature from the average fin temperature. The base temperature to ambient temperature difference was found by subtracting the ambient temperature from the average base temperature. The total heat rate was calculated from the product of the measured voltage and current. A corrected heat rate was calculated using Eq. (5) to take into account losses due to thermal radiation. The emissivity used was the emissivity correlated as described earlier in the experimental methods. The heat transfer coefficient was determined from Eq. (6) using the corrected heat rate instead of the total heat rate. ( ) (5) ( ) (6) The fin efficiency was calculated using Eq. (7) and refers to the ratio of heat transfer achieved to the heat transfer that would occur if the entire fin was at the base temperature. The fin effectiveness was calculated using Eq. (8) and refers to the ratio of heat transfer achieved to the heat transfer that would occur if no fin was present. (7) (8) C. Experimental Accuracy An experimental uncertainty analysis was performed for all measured parameters using the methods described by Wheeler 17. The uncertainty in the heat rate was found by using the uncertainty values of the voltage and current supplied by the power supply. The uncertainty for the fin surface temperatures were found using the data provided by the manufacturer for the FLIR A325. With the uncertainty of the fin surface temperatures known, the uncertainty of the average fin surface temperature was found by adding the uncertainty of the fin tip temperature and fin base temperature. The uncertainty of the ambient temperature was that corresponding to the manufacturer of the thermocouple. The uncertainty of the base area and surface area of the fins was minimal as the measurements were made using a set of calipers, with a measurement uncertainty of 0.1 mm. However, the uncertainty was assumed to 4

5 be one percent, a value larger than the actual uncertainty, for simplicity. The uncertainty of the corrected heat rate was found using the uncertainty of the average fin surface temperature and ambient temperature. The emissivity and Stefan-Boltzmann constant were assumed to have negligible uncertainty and therefore did not contribute to the uncertainty of the corrected heat rate. The maximum values of the uncertainty of the values discussed can be found Table 1. Maximum Uncertainty Values of Measured Parameters V, V I, A Q, W Q c, W T, K T b, K T t, K T avg, K A b, m2 A s, m E E-04 in Table 1. The heat transfer coefficient uncertainty was calculated using the previously determined values with Eq. (9) and was found to be a maximum of 1.12 W/m 2 -K. With these uncertainties available, the uncertainty analysis could be completed for both the fin effectiveness and fin efficiency using Eq. (10) and Eq. (11). The maximum uncertainty for the fin efficiency was calculated to be 0.14 and for the fin effectiveness Although these values are high, the standard deviation was found and used to get 95% confidence intervals based on sample size. Using these values the uncertainty can be found as for the efficiency and 0.56 for the effectiveness. With the close proximity of calculated effectiveness and efficiency values, the confidence intervals appear to be a more accurate reflection of the their uncertainty. ( ) ( ) ( ) (9) ( ) ( ) ( ) ( ) (10) ( ) ( ) ( ) ( ) (11) D. Results The fin efficiency and fin effectiveness are the main characteristics used to quantify the performance of fins. These values were calculated for all sets of data for each iteration and are found tabulated in Table 2. These characteristics can also be found plotted in Fig. (5) and Fig. (6). For the Sierpinski carpet, the fin effectiveness follows the same trend as the surface area. Although it decreases for the first two iterations due to a reduction in surface area, the third iteration results in an increase from the second iteration as predicted. The efficiency of the Sierpinski carpet decreases with each iteration due to a larger temperature differential across the fin profile. The modified Koch snowflake also follows the same trend as the surface area. It results in an increase in effectiveness with each iteration. The efficiency of the modified Koch snowflake remains fairly constant due to a similar temperature differential across the fin profile. Table 2. Calculated Fin Performance Characteristics Sierpinski Carpet Although the fin effectiveness and fin efficiency are key parameters, a reason for selecting the fractal fin profile is the potential for achieving high performance per mass. In order to compare the performance in this light, the effectiveness per mass was calculated and can be found plotted in Fig. (7). As can be seen, the Sierpinski carpet results in increases in effectiveness per mass with each iteration. This increase becomes larger with each iteration. For the modified Koch snowflake, the effectiveness per mass initially decreases due to the large increase in mass from the 0 th to 1 st iteration. However, the effectiveness per mass increases with subsequent iterations. While the Sierpinski carpet can be found to be more effective on a mass basis with each iteration, the modified Koch snowflake would require at least a 2 nd iteration to be more effective. 5 Modified Koch Snowflake Iteration Average Fin Efficiency Average Fin Effectiveness Effectiveness/Mass, kg

6 Figure 5. Average Fin Efficiency vs. Iteration Figure 6. Average Fin Effectiveness vs. Iteration III. Conclusion An investigation into the use of fractal geometries to enhance fin performance was conducted. Based upon the observed results of the fractal patterns the following statements can be made: Effectiveness of the tested geometries behaves proportional to the surface area available as predicted. Effectiveness per unit mass gains increase by larger steps with higher order iterations. Efficiency of the Sierpinski carpet decreases with each iteration. Efficiency of the modified Koch snowflake maintains a relatively constant value with each iteration. This could be attributed to the large increases in surface area near the base where the temperature is more constant than at the tip. Although an increase in inter-surface radiation is produced due to the increase in perforations, it is believed that the performance is helped by an onset of turbulence due to the perforations and edges. For the Sierpinski carpet, the potential for increased performance is available with only a few more iterations while utilizing less mass. For the modified Koch snowflake, the potential for increased performance is available after the first iteration if mass is not a concern. Although the trends predict that a fourth and fifth iteration would result in greater effectiveness than the 0 th iteration for the Sierpinski carpet, more testing is need to correlate this. Due to cost constraints of manufacturing a fourth and fifth iteration, future iterations are likely to be modeled computationally so that the behavior of higher order iterations can be obtained. 6

7 Figure 7. Average Fin Effectiveness/Mass vs. Iteration Potential areas for future study are the scaled behavior of fins. For instance, a comparison between the fin performance at macroscale and microscale could be examined. Radiation only heat transfer is another potential study case. The fins could be tested in a vacuum to see how they would perform in spacecraft applications. Another case study could be finned arrays for application in a heat sink. These areas for future study could also be applied to alternative fractal patterns such as the Sierpinski gasket or the Koch island curve. Acknowledgments This work was supported by the von Braun Center for Science and Innovation and the UA GK-12 Program. The UA GK-12 Program is supported by the US Department of Education under NSF Award The authors would like to acknowledge the assistance of Hunter Corum, who assisted with the fabrication of the experimental test cell and with data collection. References 1 Lee, S.R., Z.G. Li, B.G. Wang, and Chiou, H. S., "An Application of the Fractal Theory in the Design of Heat Sink for Precision Measurement Instrument," Key Engineering Materials, Vol. 295, Oct. 2005, pp Murali, J. Ganesh, and Katte, Subrahmanya S., "Experimental Investigation of Heat Transfer Enhancement in Radiating Pin Fin," Jordan Journal of Mechanical and Industrial Engineering, Vol 2, No. 3, 2008, pp Plawsky, Joel L. "Transport In Branched Systems I: Steady-State Response," Chemical Engineering Communications, Vol. 123, No. 1, 1993, pp Plawsky, Joel L., "Transport In Branched Systems I: Transient Response," Chemical Engineering Communications, Vol. 123, No. 1, 1993, pp Xu, P., B. Yu, M. Yun, and Zou, M., "Heat Conduction in Fractal Tree-like Branched Networks," International Journal of Heat and Mass Transfer, Vol 49, No. 19, 2006, pp Mahmoud, S., R. Al-Dadah, D. K. Aspinwall, S. L. Soo, and Hemida, H., "Effect of Micro Fin Geometry on Natural Convection Heat Transfer of Horizontal Microstructures," Applied Thermal Engineering, Vol. 31, No. 5, 2011, pp Coppens, Marc-Olivier, "Scaling-up and -down in a Nature-Inspired Way," Industrial & Engineering Chemistry Research, Vol. 44, No. 14, 2005, pp Gao, Jing, Ning Pan, and Yu, Weidong, "A Fractal Approach to Goose Down Structure,"International Journal of Nonlinear Sciences and Numerical Simulations, Vol. 7, No. 1, 2006, pp Kukulka, David J., and Fuller, Kevin G., Development of an Enhanced Heat Transfer Surface 20 th European Sysmposium on Computer Aided Process Engineering, Escape Meyer, Josua, and Vyver, Hilde Van Der, "Heat Transfer Characteristics of a Quadratic Koch Island Fractal Heat Exchanger," Heat Transfer Engineering, Vol. 26, No. 9, 2005, pp Van Der Vyver, Hilde, Validation of a CFD Model of a Three-Dimensional Tube-In-Tube Heat Exchanger, Proc. of Third International Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia, 2003, pp Sahin, Bayram, and Demir, Alparslan, "Performance Analysis of a Heat Exchanger Having Perforated Square Fins," Applied Thermal Engineering, Vol. 28, No.5, 2008, pp Al-Essa, Abdullah H., and Al-Hussien, Fayez M.S., "The Effect of Orientation of Square Perforations on the Heat Transfer Enhancement from a Fin Subjected to Natural Convection," Heat and Mass Transfer Vol 40, No. 6, 2004, pp

8 14 AlEssa, Abdullah H., Ayman M. Maqableh, and Ammourah, Shatha, "Enhancement of Natural Convection Heat Transfer from a Fin by Rectangular Perforations with Aspect Ratio of Two," International Journal of Physical Sciences, Vol. 4, No. 10, 2009, pp AlEssa, Abdullah H., and Al-Widyan, Mohamad I., "Enhancement of Natural Convection Heat Transfer from a Fin by Triangular Perforation of Bases Parallel and toward Its Tip," Applied Mathematics and Mechanics, Vol. 29, No.8, 2008, pp Adrover, A., "Laminar Convective Heat Transfer across Fractal Boundaries," Europhysics Letters, Vol. 90, April, 2010, pp Wheeler, Anthony J., and Ganji, Ahmad R., Introduction to Engineering Experimentation, 2 nd ed., Pearson Education, Upper Saddle River, New Jersey, 2004, Chap. 7. 8