İNCE TABAKADA DİFÜZYON, TERMAL DALGA VE ÇİFT FAZLI-LAG ISI İLETİMİNİN KARŞILAŞTIRMALI BİR ÇALIŞMASI

Öz

Bu çalışmada, sabit bir sıcaklığa ve yalıtılmış sınır koşullarına maruz kalan ince bir katman boyunca üç farklı ısı iletimi, difüzyon, termal dalga ve çift fazlı gecikme modu, sonlu bir eleman çözümü kullanılarak karşılaştırılmıştır. Sonlu eleman modeli, tek bir eleman için akıyı ısıtmak için gevşeme süresi ve sıcaklık gradyanı için gevşeme süresi dikkate alınarak geliştirilmiştir. Tüm elemanları birleştirdikten sonra, elde edilen cebirsel denklemlerin sayısı Python kullanılarak ince katman boyunca sıcaklık dağılımını tahmin etmek için çözülür. Çift fazlı gecikme ile tahmin edilen çözüm, tek fazlı Cattaneo – Vernotte modeli ve difüzyon Fourier modeli ile elde edilen çözüm ile karşılaştırılır. Geliştirilen model, analitik, sayısal ve deneysel çözümlerle iyi bir uyum içinde doğrulanmıştır. Her üç koşul için sıcaklık sınırları çizilmiştir ve ince katman boyunca farklı şekilde yayılma şekli açıkça gösterilmiştir. Ayrıca, çarpışmanın meydana geldiği katmanın merkezindeki sıcaklık değişimi tahmin edilir ve Fourier difüzyon modelinde sonsuz ve hem tek hem de çift fazlı gecikmede sonlu olan termal dalganın hızı, geçici olarak kararlı durum koşulu

Anahtar Kelimeler

• çift fazlı gecikme
• CV modeli
• gevşeme süresi
• sonlu eleman modeli
• python

A COMPARATIVE STUDY OF DIFFUSION, THERMAL WAVE AND DUAL-PHASE-LAG HEAT CONDUCTION IN THIN LAYER

Öz

In the present work, three different modes of heat conduction, diffusion, thermal wave, and dual-phase lag, across a thin layer subjected to a constant temperature and insulated boundary conditions are compared by using a finite element solution. The finite element model is developed by considering relaxation time to heat flux and relaxation time to temperature gradient for a single element. After assembling all the elements, the number of algebraic equations obtained is solved to predict the temperature distribution across the thin layer using Python. The solution predicted by the dual-phase lag is compared with that obtained by the single-phase Cattaneo–Vernotte’s model and diffusion Fourier model. The developed model is validated with analytical, numerical, and experimental solutions with good agreement. The temperature contours are plotted for all three conditions and the way it propagates differently through the thin layer is clearly shown. Further, the temperature variation at the center of the layer, at which collision occurred, is predicted and the speed of the thermal wave, infinite in the Fourier diffusion model and finite in both single and dual-phase lag, is examined under transient to steady-state condition.

Anahtar Kelimeler

• dual-phase lag
• CV model
• relaxation time
• finite element model
• python

Kaynakça

1. Al-Nimr M. A., Naser S. Al-Huniti, 2000, Transient Thermal Stresses In A Thin Elastic Plate Due To A Rapid Dual-Phase-Lag Heating, Journal of Thermal Stresses, 23, 8, 731-746.
2. Antaki P. J., 1998, Solution for non-Fourier dual phase lag heat conduction in a semiinfinite slab with surface heat flux, International Journal of Heat and Mass Transfer, 41, 14, 2253-2258.
3. Cattaneo C., 1958, Sur une forme de l'equation de la chaleur eliminant la paradoxe d'une propagation instantantee, Compt. Rendu, 247, 431-433.
4. Dhanaraj S. N., Karthikeya Sharma T., Amba Prasad Rao G., and Madhu Murthy K., 2019, Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal, Heat Transfer Engineering, 41, 6-7, 665-675.
5. Elsayed-Ali H. E., Juhasz T., Smith G. O., and Bron W. E., 1991, Femtosecond thermoreflectivity and thermotransmissivity of polycrystalline and single-crystalline gold films, Physical Review B, 43, 5, 4488-4491.
6. Fong E. and Lam T. T., 2014, Asymmetrical collision of thermal waves in thin films: An analytical solution, International Journal of Thermal Sciences, 77, 55-65.
7. Fujimoto J. G., Liu J. M., Ippen, E. P. and Bloembergen, N., 1984, Femtosecond Laser Interaction with Metallic Tungsten and Nonequilibrium Electron and Lattice Temperatures, Physical Review Letters, 53, 19, 1837-1840.
8. Hector L. G., Kim W. S. and Özisik M. N., 1992, Hyperbolic heat conduction due to a mode locked laser pulse train, International Journal of Engineering Science, 30, 12, 1731-1744.

9. Körner C. and Bergmann H. W., 1998, The physical defects of the hyperbolic heat conduction equation, Applied Physics A, 67, 4, 397-401.
10. Lam T. T. and Fong E., 2011, Heat diffusion vs. wave propagation in solids subjected to exponentially-decaying heat source: Analytical solution, International Journal of Thermal Sciences, 50, 11, 2104-2116.
11. Lewandowska M. and Malinowski L., 2006, An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides, International Communications in Heat and Mass Transfer, 33, 1, 61-69.
12. Li J., Cheng P., Peterson G. P. and Xu J. Z., 2005, Rapid Transient Heat Conduction In Multilayer Materials With Pulsed Heating Boundary, Numerical Heat Transfer, Part A: Applications, 47, 7, 633-652.
13. Liu K. C., and Cheng P. J., 2006, Numerical Analysis for Dual-Phase-Lag Heat Conduction in Layered Films, Numerical Heat Transfer, Part A: Applications, 49, 6, 589-606.
14. Majumdar A., 1993, Microscale Heat Conduction in Dielectric Thin Films, Journal of Heat Transfer, 115, 1, 7-16.
15. Mitra K., Kumar S., Vedevarz A., and Moallemi M. K., 1995, Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat, Journal of Heat Transfer, 117, 3, 568-573.
16. Python 3.6.3, https://www.python.org/
17. Qiu T. Q., and Tien C. L., 1992, Short-pulse laser heating on metals, International Journal of Heat and Mass Transfer, 35, 3, 719-726.
18. Reddy J. N., 2015, An Introduction to the Finite Element Method, McGraw Hill Education (India) Private Limited, New Delhi.
19. Siva Prakash G., Sreekanth Reddy S., Sarit K. Das, Sundararajan T., and Seetharamu K. N., 2000, Numerical Modelling Of Microscale Effects In Conduction For Different Thermal Boundary Conditions, Numerical Heat Transfer, Part A: Applications, 38, 5, 513-532.
20. Tan Z. M., and Yang W. J., 1997, Heat transfer during asymmetrical collision of thermal waves in a thin film, International Journal of Heat and Mass Transfer, 40, 17, 3999-4006.
21. Tan Z. M., and Yang, W. J., 1997, Non-Fourier Heat Conduction in a Thin Film Subjected to a Sudden Temperature Change on Two Sides, Journal of Non-Equilibrium Thermodynamics, 22, 1, 75.
22. Tang D., Araki N., and Yamagishi N., 2007, Transient temperature responses in biological materials under pulsed IR irradiation, Heat and Mass Transfer, 43, 6, 579-585.
23. Tang D. W., and Araki N., 2000, Non-fourier heat condution behavior in finite mediums under pulse surface heating, Materials Science and Engineering: A, 292, 2, 173-178.
24. Torii S., and Yang W. J., 2005, Heat transfer mechanisms in thin film with laser heat source, International Journal of Heat and Mass Transfer, 48, 3, 537-544.
25. Tzou D. Y., 1995, Experimental support for the lagging behavior in heat propagation, Journal of Thermophysics and Heat Transfer, 9, 4, 686-693.
26. Tzou D. Y., 1995, The generalized lagging response in small-scale and high-rate heating, International Journal of Heat and Mass Transfer, 38, 17, 3231-3240.
27. Tzou D. Y., 1995, A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales, Journal of Heat Transfer, 117, 1, 8-16.
28. Tzou D. Y., 2014, Macro- to Microscale Heat Transfer: The Lagging Behavior, pp. 25-56, Wiley Online Library.
29. Vernotte P., 1958, Les paradoxes de la theorie continue de l'equation de la chaleur, Compt. Rendu, 246, 3154-3155.
30. Yuvaraj R., and Senthil Kumar D., 2020, Numerical simulation of thermal wave propagation and collision in thin film using finite element solution, Journal of Thermal Analysis and Calorimetry, 142, 6, 2351-2369.