منابع را میتونی پیدا کنی زیر مقاله
منابع را میتونی پیدا کنی زیر مقاله
من به یک مقاله انگلیسی در مورد مکانیک احتیاج دارم که موضوعش فرق نمیکنه و اگر هم متن فارسیش کنارش باشه که خیلی عالی میشه.
Implementation of dual-phase lag model at different Knudsen numbers within slab heat transfer
J. Ghazanfarian, P. Forooghi& H. Basirat Tabrizi
1Graduated Engineer
2Associate professor Amirkabir University of Technology, Iran.
Abstract
One dimensional problem of heat conduction for the Knudsen numbers more than 0.1 corresponding to nano structures is implemented utilizing the Dual-Phase-Lag (DPL) model and its special case, Cattaneo and Vernotte (CV) model as well as the Fourier law. The results are compared with the results obtained from the Ballistic Diffusive Equations (BDE) model, which considered being an accurate approximation of the Boltzmann equation. It is observed that in Knudsen numbers of order one, the DPL model, with considering the real value
of the ratio of temperature to heat flux time lag, leads to better results than the CV model in which this ratio is considered to be equal to zero whereas in the greater values of Knudsen number there is no advantage between two approaches. In the smaller values of Knudsen number the curves obtained from the DPL and CV models match to each other, and lay very close to the curves obtained from the Fourier law.
Keywords: dual-phase lag model, ballistic-diffusive equations, Cattaneo and Vernotte model, nano-scale slap
1 Introduction
The problem of self-heating in microelectronic devices or for situationsinvolving very low temperature near absolute zero, heat source, such as laser,heat is found to propagate at a finite speed. Solving the Boltzmann equation is the most correct option to model heat transfer in such problems. However the Boltzmann equation is too difficult to solve in general, so many other models have been proposed so far to take the finite speed of heat propagation and effects of boundaries into account. One of the best approximations of Boltzmann equation is Ballistic-Diffusive Equations (BDE) derived by Chen [1] in which the heat transfer is divided at any point into two parts, one represents the ballistic nature of heat conduction originating from boundary scattering of heat carriers, and the other characterizes the diffusive behavior with heat flux time-lag phenomenon taken into account only. The BDE approximation shows a good agreement with Boltzmann equation in both one and two dimensional problems as shown by Chen et. al.[2,3].
Another well-known approximation of non-Fourier heat conduction is the Dual Phase Lag (DPL) model firstproposed by Tzou [4,5]. This model considers only effect of finite relaxation time by using heat flux and temperature phase lags where the former is caused by microstructural interactions such as phonon scattering and the latter is interpreted as the relaxation time due to fast-transient effects of thermal inertia [6]. However in many problems the hyperbolic equation in which the temperature phase lag is omitted is utilized. So far the advantages of the DPL model over the CV [7,8] model have been proven in the field of heat transfer in processed meat [9].The present study develops heat transfer regime maps by using different valuesfor Knudsen number which makes a comparison between results of the DPL andthe BDE model, for a thin slab. The results are developed for relaxation timeratios between 0.0, the case of CV, to 0.05, and for Knudsen numbers from 0.1 to 10 to find how accurate the prediction of DPL model response in the heat
conduction of nano-structure.
References
[1] Chen, G., Ballistic-Diffusive Heat-Conduction Equations: Physical Review Letters, 86, 11, pp. 2297-2300, 2001
[2] Chen, G., Ballistic-Diffusive Equations for Transient Heat Conduction from Nano to Macroscales: ASME Journal of HeatTransfer, 124, pp. 320- 328, 2002
[3] Yang, R., Chen, G., Laroche, M.& Taur, Y., Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation: ASME Journal of Heat Transfer, 127, pp. 298-306, 2005
[4] Tzou, D.Y., A Unified Approach for Heat Conduction From Macro- to Micro- Scale: ASME Journal of Heat Transfer, 117, pp.8-16, 1995
[5] Tzou, D. Y., Macro- to Microscale Heat Transfer, The Lagging Behaviour: Taylor & Francis,1997
[6] Quintnilla, R., Racke, R., Qualitative Aspects in Dual-phase-lag Thermoelasticity: Konstanzer Schriften in Mathematik undInformatic, 206, pp. 1-28, 2005
[7] Cattaneo, C., A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation: Compte Rendus, 247, pp. 431-433,1958
[8] Vernotte, P., Les Paradoxes de la Theorie Continue de lequation DE La Chleur: Compte Rendus, 252, pp. 3154-3155,1958
[9] Antaki, P.J., New Interpretation of Non-Fourier Heat Conduction inProcessed Meat: Journal of Heat Transfer, 127, pp. 189-193, 2005
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