Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 3468246 | https://doi.org/10.1155/2016/3468246

Jianxin Zhu, Lixia Sun, "Modelling and Numerical Simulations of Heat Distribution for LED Heat Sink", Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3468246, 8 pages, 2016. https://doi.org/10.1155/2016/3468246

Modelling and Numerical Simulations of Heat Distribution for LED Heat Sink

Academic Editor: Luisa Di Paola
Received24 Jan 2016
Accepted12 Oct 2016
Published10 Nov 2016

Abstract

Light-emitting diode (LED) has higher efficiency and longer lifetime when compared with the conventional lighting. However, the efficiency and lifetime will be degraded greatly when it is operated at a high temperature. Now, both previous simulation and experimental results have already indicated that the heat transfer in vertical direction of the LED lamp by conduction is the most critical component. In this paper, a simplified numerical simulation model is built to estimate the heat distribution of the LED heat sink in the spherical coordinate system, which would be useful for its shape optimization design. With this model, some mathematical treatments are provided to a heat conduction equation, in order to rapidly compute the static heat distribution and the temperature of different designs of LED heat sinks. The built rapid heat sink evaluation method, implicit finite difference method (IFDM), is unconditionally stable. Several heat distribution simulations could demonstrate that our built mathematical model conforms well to the reality and our method is full of feasibility and effectiveness.

1. Introduction

Recently, light-emitting diode (LED) lamp has played an important role in the illumination market, mainly due to the advantages of more compact package in size, higher efficiency, and longer lifetime than the conventional lighting types. It has become the fourth-generation light source in the auto industry [1, 2]. Moreover, it has been considered as the ultimate light source. However, when we want to produce more light output, the higher power needs to be applied to LED lamp. Subsequently, the amount of heat generated from the LED lamp will be greatly increased at the same time, which could reduce both the lifespan and the luminous efficiency. Researchers have found that both the efficiency and lifetime will be degraded pronouncedly when the LED is operated at a temperature higher than 70C. Thus, the design of heat sink used to cool LED lamp is full of importance to ensure their long lifetime and high efficiency.

Numerous heat sinks in different shapes, sizes, and structures are currently used in the commercial LED light bulbs. However, most of these heat sinks were not optimized in cooling efficiency, dimension, and material cost. Lots of researchers have already studied the cooling of LED lamps [320], which are mainly operated under natural or forced convection. More specifically, a numerical and experimental research considering the whole set constituted by the vapour chamber and a finned heat sink was conducted in [3]. A 3D finite element simulation was presented for an array of high power LED lamp with a heat sink in [4], where the thermal resistance network was performed to estimate different contributions for the heat management. In [5], a radial heat sink with pin fins was optimized while keeping a similar cooling performance as in the previous study. In the related further work [17], a three-dimensional fin-height profile heat sink was optimized. In [11], the enhanced model with an aluminum nitride (AIN) insulation plate was more effectively conducted than the conventional chip on board (COB) model. In [12], LED effectively produced a thermal effect by the changes of the fin shape and intervals of various heat sink designs. In [13], the thermal performance of LED illumination systems could be improved by a detailed CFD analysis. In [14], the numerical simulation and optimization of a radial heat sink were performed, and the impossibility of optimizing both thermal performance and heat sink mass at the same time was concluded. In [15], some experimental results were shown for a similar heat sink configuration. In [16], the effect of both radiation and natural convection on the thermal performance of a radial heat sink was studied similar to [14, 15]. In [20], numerical simulation was conducted to investigate the impingement and film composite cooling on blade leading region. The luminance of LED lamp is greatly related to junction temperature, so evaluating the junction temperature is one of the methods to determine its heat dissipation [610]. In [6], the junction temperature was assessed by the heat dissipation simulation of a single chip high power LED package, and an electric-heat-optical system dynamics model for LED luminance control was provided in [7]. A 3 W high power LED array system with an in-line pin fin heat sink was presented in [8]. In [9], a tandem 12-chip module with three types of heat sinks was discussed, while the impact factors of the junction temperature were studied in [10] through the thermal performance of conventional plate-fin heat sinks and novel cooling device of heat conductive plates. Recently, the related research can also be found in [18, 19].

From the above research, we could find that it is major importance to get the heat distribution when we want to optimize the designs of LED heat sinks. To the LED heat sink, recent simulation and experimental results have already proved that the heat transfer in vertical direction of the LED lamp by conduction is the most critical component. Then we mainly consider the heat conduction equation concerning the LED heat sink in this paper. Many methods have been developed to the heat transfer problems in the Cartesian coordinate system, such as the technique of alternating directional implicit method (ADIM) [21], the finite volume method (FVM) [22], and the finite element method [23]. However, for the LED heat sink, its model of heat distribution can be much simpler in the spherical coordinate system than in the Cartesian coordinate system, though there is a little research about 3D heat transfer problems in the spherical coordinate system. The objective of our present work is building a simplified mathematical model and developing a rapid heat sink evaluation method by making some mathematical treatments, that is, the implicit finite difference method (IFDM), to a heat conduction equation in the spherical coordinate system for the LED heat sink. With this method, we can estimate the static heat distribution and the temperature in different designs of LED heat sink. It can be more quickly achieved than the explicit finite difference method (EFDM) because it is unconditionally stable.

The paper is organized as follows. In Section 2, modelling of LED heat sink is introduced. In Section 3, we propose the discretization schemes and give some mathematical treatments. Some numerical simulations of heat distribution for the LED heat sink are provided in Section 4, and the conclusions are made in Section 5.

2. Modelling for LED Heat Sink

To the commonly used LED lamp (Figure 1), regardless of its top cover and base, heat conduction equation for the heat sink can be expressed in the Cartesian coordinate system as follows:whereWe suppose that the LED lamp is continuously homogeneous and isotropic, while all variables are with the international standard unit system, whose domains are also supposed to be workable. Heat source lies near the top of LED lamp. For simplicity, we suppose that there is only one heat source within the LED lamp in this paper, which lies at . Besides the top heat source, the inside longitudinal direction figure of LED heat sink can be described in Figure 2. The blank place is air, while in general the red is aluminum.

Concerning the influence of heat source to the LED lamp, the popular treatment is to restrict the influence at slender grids, which closely surround in very limited grids. Obviously, it does not so well conform to the reality. The influence of heat source on LED should be continuous instead of being truncated by the limited grids. More reasonably, we can describe the influence with the pulse function; then the last part of (1) can be written as represents the height of LED lamp. is a constant which is decided by the specific LED shape and power, which represents the influence of heat source to each part of LED lamp. We let simplify (3), where the unit of is . Then the heat conduction equation (1) becomesIn our following work, with convenience, we consider the top and the base of heat sink as spherical instead of reality planar, and the position of heat source is limited at (). All segmentation is supposed equally, so the number of surrounding fins of the lamp body can be chosen as an even number. In considering spherical coordinate system, the heat conduction equation (4) becomeswhere , , , and , or , , , and . is the location of heat source. represents the total number of surrounding fins of the lamp body ( is even). , , , and are all piecewise constants, which are defined as follows.

(I) When or , , ; , , , we have , , , .

(II) When , , , we have , , , .

(III) When , , ; , , , we have , , , .

We consider the relevant boundary conditions as follows.

(IV) When , , , or , , , we havewhere is a constant.

(V) When , , , or , , , we have

(VI) When , , or , , , we have

(VII) When , , , we have

(VIII) When , , , we havewhere and .

(IX) When , , , , we have

(X) When or , , , , we have

(XI) When , , , , we have

Specifically, in the following work, we take as an example. Then the longitudinal and crosscutting map of the LED heat sink considered in this paper could be shown in Figures 3 and 4. The color bar which lies in the right side of each map represents temperature. That is to say that we only consider four surrounding fins in our work. Other numerical values of will be able to be finished similarly.

3. Discretization Schemes and Mathematical Treatments

Now we process the discretization at the points for the heat equation (5), where

When (i.e., ), we let

When (i.e., ), we letwhere .

For simplicity, we denote as afterwards. For the IFDM used in our paper, the partial derivative about is approximated with the forward difference scheme, while the others are approximated with the central difference schemes. Particularly, we only indicate the treatments about and .

For the periodicity of , when (i.e., ), we let the related central differences be the following forms.

When , , , and , we let

When , , , and , we let

For the central differences about , we only show the following case.

When , and , or , , , we letwhere is the virtual symmetrical point of about . With (7), we can getWe make the approximationSo can be deduced, and and in this case are solved.

Similarly, with (8)–(13), other partial derivatives can be finished. With these discretization schemes, the truncation error of (5) is , where and . With denser discretization, the result will turn better.

Now, after the discretization of all variables, we can arrange the discretization of (5) aswhere is matrix, whose elements are , , and , , or , The point sequences can be ordered as , , , , , where , , , and , . All of them are sorted in ascending order. is matrix, whose every element is . is block matrix, whose every block is similar to a band matrix. But the band characteristic is destroyed by the disordered elements in the first row and last row. is also matrix, which greatly relies on .

To solve the heat equation (5), we should run iteration (22) step by step. This treatment is unconditionally stable for the implicitness [24]. Our method is independent of the discretization step about heat transfer time (), so it can be rapidly achieved. It is very convenient and fast.

4. Numerical Simulations of Heat Distribution

For all heat distribution simulations here, we take a 0.5 w LED lamp as an example, where only a quarter of height is considered. For its shape structure, we let , , , , , , , , , , and . The ambient temperature is 25.00C. The numerical value of is fixed at . The radius and height of heat source are both 2 mm, so . To obtain , we just consider . It should satisfy when , while and at the same time. After a short Matlab program, approximately. The heat source influence is shown in Figure 5.

Numerical Simulation 1. To prove the validity of our presented schemes, we let as verification. For the discretization of all variables, we let , , , , , and , and the heat transfer time is . When , after a few seconds of Matlab program, the stable solution is represented that the final temperature of LED keeps 25.00C. When we prolong the heat transfer time, the result is still the same. It has no change since the heat transfer does not happen under this certain circumstance. This simulation indicates that our presented schemes are correct.

Numerical Simulation 2. The discretization is the same as numerical simulation 1; that is, , , , , , , and the heat transfer time is  s. When (), the final temperature of LED ranges from 25.21C to 171.37C. This result is reasonable. That is the result for ignoring the heat dissipation through convection; in addition our built mathematical model is different from the actual model. We let Figure 6 describe the temperature distribution of the LED heat sink under this circumstance. In this paper, we only show figures of the temperature distributions at for the symmetrical shape. When (), the final temperature of LED ranges from 25.19C to 171.36C. The temperature distribution is shown in Figure 7. From the two simulations, we can find that the calculation about time step has a little influence on the final result. Thus, in the next numerical simulations, we fix  s. In contrast, with the same discretization, the related explicit difference schemes need  s when we want to obtain the same result, so our present method is very fast and convenient.

Numerical Simulation 3. When the discretization is , , , , , and , which is like the above two numerical simulations, we prolong the heat transfer time to ; then we get the stable solution representing the final temperature ranges from 25.93C to 172.10C. The temperature distribution is shown in Figure 8. When the heat transfer time is , the stable solution represents the final temperature ranges from 26.88C to 173.06C. Its distribution of temperature is shown in Figure 9. From Figures 7, 8, and 9, we can find that the heat transfer becomes much clearer and better conformed to the reality when the heat transfer time becomes larger.

Numerical Simulation 4. The discretization is denser than the above three numerical simulations, , , , , , and , when the heat transfer time is ; then the stable solution represents the final temperature ranges from 26.10C to 275.49C, whose distribution of the temperature is shown in Figure 10, while the discretization becomes sparser than the above three numerical simulations, , , , , , and ; the heat transfer time is still ; then we get the stable solution representing the final temperature ranges from 25.12C to 34.77C. In this case, the related distribution of temperature is shown in Figure 11. Obviously, Figures 10 and 11 are far away from the reality for the too much dense or sparse mesh dissections.

5. Conclusions

From the above numerical simulations of heat distribution, based on our fairly realistic mathematical model and treatments, we can design the unconditionally stable scheme of the heat distribution for LED heat sink. It is effective and convenient to evaluate the heat distribution in the spherical coordinate system. But the mesh dissections about , , and should be chosen properly, because the too dense or sparse mesh dissections will become far away from the reality result. To the technique of computer, the large size of iterative matrix which is a multiplication of mesh nodes about , , , , and is a great challenge when we want much denser mesh dissections. In addition, when the size of becomes larger, the evaluation speed becomes slower as a result. In spite of these shortcomings, our built mathematical model and unconditionally stable treatments still have advantage on the optimization of LED heat sink. With our treatments, the heat distribution and the temperature in different designs of LED lamp can be rapidly estimated. Furthermore, the built spherical coordinate system is better for the shape optimization design than in the Cartesian coordinate system, for the variable in the spherical coordinate system is in great accordance with actual angle shape. It is promising for the optimization design and actual engineering application of LED heat sink.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was partially supported by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010002).

References

  1. T. Acikalin, S. V. Garimella, J. Petroski et al., “Optimal design of miniature piezoelectric fans for cooling light emitting diodes,” in Proceedings of the IEEE Conference on Thermal and Thermomechanical Phenomena in Electronic System ( ITHERM '04), vol. 1, pp. 663–671, Las Vegas, Nev, USA, June 2004. View at: Publisher Site | Google Scholar
  2. D. A. Steigerwald, J. C. Bhat, D. Collins et al., “Illumination with solid state lighting technology,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 8, no. 2, pp. 310–320, 2002. View at: Publisher Site | Google Scholar
  3. X. Luo, R. Hu, T. Guo et al., “Low thermal resistance LED light source with vapor chamber coupled fin heat sink,” in Proceedings of the 60th Electronic Components and Technology Conference (ECTC '10), pp. 1347–1352, Las Vegas, Nev, USA, June 2010. View at: Publisher Site | Google Scholar
  4. A. Christensen and S. Graham, “Thermal effects in packaging high power light emitting diode arrays,” Applied Thermal Engineering, vol. 29, no. 2-3, pp. 364–371, 2009. View at: Publisher Site | Google Scholar
  5. D. Jang, S.-H. Yu, and K.-S. Lee, “Multidisciplinary optimization of a pin-fin radial heat sink for LED lighting applications,” International Journal of Heat and Mass Transfer, vol. 55, no. 4, pp. 515–521, 2012. View at: Publisher Site | Google Scholar
  6. R. Vairavan, Z. Sauli, and V. Retnasamy, “High power LED heat dissipation analysis using cylindrical Al based slug using Ansys,” in Proceedings of the IEEE Regional Symposium on Micro and Nano Electronics (RSM '13), pp. 186–189, Langkawi, Malaysia, September 2013. View at: Publisher Site | Google Scholar
  7. B.-J. Huang, C.-W. Tang, and M.-S. Wu, “System dynamics model of high-power LED luminaire,” Applied Thermal Engineering, vol. 29, no. 4, pp. 609–616, 2009. View at: Publisher Site | Google Scholar
  8. F. Z. Hou, D. G. Yang, G. Q. Zhang, Y. Hai, D. Liu, and L. Liu, “Thermal transient analysis of LED array system with in-line pin fin heat sink,” in Proceedings of the 12th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE '11 ), pp. 1–5, Linz, Austria, April 2011. View at: Publisher Site | Google Scholar
  9. S.-J. Wu, H.-C. Hsu, S.-L. Fu, and J.-N. Yeh, “Numerical simulation of high power LED heat-dissipating system,” Electronic Materials Letters, vol. 10, no. 2, pp. 497–502, 2014. View at: Publisher Site | Google Scholar
  10. X.-J. Zhao, Y.-X. Cai, J. Wang, X.-H. Li, and C. Zhang, “Thermal model design and analysis of the high-power LED automotive headlight cooling device,” Applied Thermal Engineering, vol. 75, pp. 248–258, 2015. View at: Publisher Site | Google Scholar
  11. M. W. Jeong, S. W. Jeon, S. H. Lee, and Y. Kim, “Effective heat dissipation and geometric optimization in an LED module with aluminum nitride (AlN) insulation plate,” Applied Thermal Engineering, vol. 76, pp. 212–219, 2015. View at: Publisher Site | Google Scholar
  12. D. W. Lee, S.-W. Cho, and Y.-J. Kim, “Numerical study on the heat dissipation characteristics of high-power LED module,” Science China Technological Sciences, vol. 56, no. 9, pp. 2150–2155, 2013. View at: Publisher Site | Google Scholar
  13. C.-J. Weng, “Advanced thermal enhancement and management of LED packages,” International Communications in Heat and Mass Transfer, vol. 36, no. 3, pp. 245–248, 2009. View at: Publisher Site | Google Scholar
  14. S.-H. Yu, K.-S. Lee, and S.-J. Yook, “Optimum design of a radial heat sink under natural convection,” International Journal of Heat and Mass Transfer, vol. 54, no. 11-12, pp. 2499–2505, 2011. View at: Publisher Site | Google Scholar
  15. S.-H. Yu, K.-S. Lee, and S.-J. Yook, “Natural convection around a radial heat sink,” International Journal of Heat and Mass Transfer, vol. 53, no. 13-14, pp. 2935–2938, 2010. View at: Publisher Site | Google Scholar
  16. S.-H. Yu, D. Jang, and K.-S. Lee, “Effect of radiation in a radial heat sink under natural convection,” International Journal of Heat and Mass Transfer, vol. 55, no. 1–3, pp. 505–509, 2012. View at: Publisher Site | Google Scholar
  17. D. Jang, S.-J. Yook, and K.-S. Lee, “Optimum design of a radial heat sink with a fin-height profile for high-power LED lighting applications,” Applied Energy, vol. 116, pp. 260–268, 2014. View at: Publisher Site | Google Scholar
  18. K. C. Yung, H. Liem, H. S. Choy, and Z. X. Cai, “Thermal investigation of a high brightness LED array package assembly for various placement algorithms,” Applied Thermal Engineering, vol. 63, no. 1, pp. 105–118, 2014. View at: Publisher Site | Google Scholar
  19. Y. Tang, X. Ding, B. Yu, Z. Li, and B. Liu, “A high power LED device with chips directly mounted on heat pipes,” Applied Thermal Engineering, vol. 66, no. 1-2, pp. 632–639, 2014. View at: Publisher Site | Google Scholar
  20. Z. Liu, L. Ye, C. Wang, and Z. Feng, “Numerical simulation on impingement and film composite cooling of blade leading edge model for gas turbine,” Applied Thermal Engineering, vol. 73, no. 2, pp. 1432–1443, 2014. View at: Publisher Site | Google Scholar
  21. C.-H. Huang and S.-C. Chin, “A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium,” International Journal of Heat and Mass Transfer, vol. 43, no. 22, pp. 4061–4071, 2000. View at: Publisher Site | Google Scholar
  22. C.-L. Chang and M. Chang, “Non-iteration estimation of thermal conductivity using finite volume method,” International Communications in Heat and Mass Transfer, vol. 33, no. 8, pp. 1013–1020, 2006. View at: Publisher Site | Google Scholar
  23. H. H. Cheng, D.-S. Huang, and M.-T. Lin, “Heat dissipation design and analysis of high power LED array using the finite element method,” Microelectronics Reliability, vol. 52, no. 5, pp. 905–911, 2012. View at: Publisher Site | Google Scholar
  24. G. I. Marchuk, Methods of Numerical Mathematics, Springer, New York, NY, USA, 1982. View at: MathSciNet

Copyright © 2016 Jianxin Zhu and Lixia Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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