平板在任意周期表面热扰动作用下的非Fourier热传导的求解与分析

赵伟涛; 吴九汇

doi:10.7498/aps.62.184401

摘要

分析了平板前表面遭受任意周期热扰动这类非Fourier传热情形下的温度响应. 采用双曲型热传导方程描述平板表面温度急速变化时的热传导问题. 为求解此类方程, 首先利用分离变量法和Duhamel积分原理, 得到了平板前表面遭受突变热流和简谐热流两种情况下的解析解.然后, 在此基础上应用Fourier级数展开法和叠加原理, 获得了平板前表面热流任意周期变化时非Fourier热传导下温度场的解析表达式. 利用得到的解析表达式进行数值模拟, 分析了不同热松弛时间、不同时刻和不同位置对温度响应的影响, 讨论了非Fourier热传导模型所给出的温度响应与Fourier热传导模型的差别. 这种方法能够处理许多在生产实际中具有周期边界条件的非Fourier热传导问题.

关键词:

Abstract

In this paper, the non-Fourier heat conduction in a plane slab under arbitrary periodic surface thermal disturbance is solved analytically. Hyperbolic heat conduction equation is employed to describe this problem involving high-rate change of temperature. Firstly, when the plane slab surface is subjected to a sudden heat flux change or a harmonic heat flux change, the analytic solution of this problem is found by using the separation of variables method and Duhamel’s principle. On this basis, when the plane slab surface is subjected to an arbitrary periodic heat flux change, the analytic solution of temperature field is obtained by using the Fourier series and the principle of superposition. Using the obtained analytical solution, the temperature profiles of the plane slab are analyzed, and the differences between the temperature response obtained by using non-Fourier heat conduction model and that obtained by using Fourier model are discussed. This solution can be applied to more realistic periodic boundary conditions in technology.

Keywords:

作者及机构信息

赵伟涛^{1 2},
吴九汇^{1 2}

1.
西安交通大学机械学院, 西安 710049;

2.
西安交通大学, 机械结构强度与振动国家重点实验室, 西安 710049

基金项目: 教育部新世纪优秀人才支持计划(批准号:NCET-09-0644);高等学校博士学科点专项科研基金(批准号:20090201120047)和教育部长江学者和创新团队发展计划(批准号:IRT1172)资助的课题.

Authors and contacts

Zhao Wei-Tao^{1 2},
Wu Jiu-Hui^{1 2}

1.
School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China;

2.
State Key Laboratory for Strength and Vibration of Mechanical Structres, Xi’an Jiaotong University, Xi’an 710049, China

Funds: Project supported by the Program for New Century Excellent Talents in University of Education of China (Grant No. NCET-09-0644), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090201120047), and the Program for Changjiang Scholars and Innovation Research Team in University of Ministry of Education of China (Grant No. IRT1172).

参考文献

[1]	Wang Y Z, Song X N 2012 Acta Phys. Sin. 61 234601 (in Chinese) [王颖泽, 宋新南 2012 物理学报 61 234601]
[2]	Guo Z Y, Cao B Y 2008 Acta Phys. Sin. 57 4273 (in Chinese) [过增元, 曹炳阳 2008 物理学报 57 4273]
[3]	Tian X G, Shen Y P 2012 Adv. Mech. 42 18 (in Chinese) [田晓耕, 沈亚鹏 2012 力学进展 42 18]
[4]	Tung T L, Fong E 2011 Int. J. Heat Mass Transfer 54 4796
[5]	Cattaneo C 1948 Atti. Sem. Mat. Fis. Univ. Modena 3 83
[6]	Vernotte P 1958 C. R. Acad. Sci. 246 3154
[7]	Tao Y J, Huai X L, Li Z G 2006 Chin. Phys. Lett. 23 2487
[8]	Li S R, Zhou F X, Wu H M 2007 Engineer. Mech. 24 48 (in Chinese) [李世荣, 周凤玺, 吴红梅 2007工程力学 24 48]
[9]	Sarkar D, Haji-Sheikh A 2012 Int. Commun. Heat Mass Transfer 39 1009
[10]	Tang D W, Araki N 1996 Int. J. Heat Mass Transfer 39 1585
[11]	Moosaie A 2007 Int.Commun. Heat Mass Transfer 34 996
[12]	Barletta A, Zanchini E 1997 Heat and Mass Transfer 32 285
[13]	Atefi G, Talaee M R 2011 Arch. Appl. Mech. 81 569
[14]	Mishra S C, Sahai H 2012 Int. J. Heat Mass Transfer 55 7015
[15]	Jiang F M 2006 Heat and Mass Transfer 42 1083
[16]	Shirmohammadi R, Moosaie A 2009 Int. Commun. Heat Mass Transfer 36 827

施引文献

[1]	Wang Y Z, Song X N 2012 Acta Phys. Sin. 61 234601 (in Chinese) [王颖泽, 宋新南 2012 物理学报 61 234601]
[2]	Guo Z Y, Cao B Y 2008 Acta Phys. Sin. 57 4273 (in Chinese) [过增元, 曹炳阳 2008 物理学报 57 4273]
[3]	Tian X G, Shen Y P 2012 Adv. Mech. 42 18 (in Chinese) [田晓耕, 沈亚鹏 2012 力学进展 42 18]
[4]	Tung T L, Fong E 2011 Int. J. Heat Mass Transfer 54 4796
[5]	Cattaneo C 1948 Atti. Sem. Mat. Fis. Univ. Modena 3 83
[6]	Vernotte P 1958 C. R. Acad. Sci. 246 3154
[7]	Tao Y J, Huai X L, Li Z G 2006 Chin. Phys. Lett. 23 2487
[8]	Li S R, Zhou F X, Wu H M 2007 Engineer. Mech. 24 48 (in Chinese) [李世荣, 周凤玺, 吴红梅 2007工程力学 24 48]
[9]	Sarkar D, Haji-Sheikh A 2012 Int. Commun. Heat Mass Transfer 39 1009
[10]	Tang D W, Araki N 1996 Int. J. Heat Mass Transfer 39 1585
[11]	Moosaie A 2007 Int.Commun. Heat Mass Transfer 34 996
[12]	Barletta A, Zanchini E 1997 Heat and Mass Transfer 32 285
[13]	Atefi G, Talaee M R 2011 Arch. Appl. Mech. 81 569
[14]	Mishra S C, Sahai H 2012 Int. J. Heat Mass Transfer 55 7015
[15]	Jiang F M 2006 Heat and Mass Transfer 42 1083
[16]	Shirmohammadi R, Moosaie A 2009 Int. Commun. Heat Mass Transfer 36 827

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