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Hamilton’s principle based on thermomass theory

Wu Jing Song Bai Guo Zeng-Yuan

Hamilton’s principle based on thermomass theory

Wu Jing, Song Bai, Guo Zeng-Yuan
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  • Based on thermomass theory, the Hamilton's principle as well as the Lagrangian equations governing the motion of thermomass were established by methods analogous to those of classical mechanics. With the kinetic energy of thermomass taken into consideration, the Hamiltons principle for thermomass is expected to be capable of dealing with non-Fourier phenomena. When the kinetic energy is small enough to be ignored, the principle gets back to Fourier transfer. The application of Lagrangian equations was illustrated by the approximate solution of a 1D transient heat conduction problem with heat source. The unification of thermal and mechanical theories was demonstrated from the perspective of analytical mechanics, the drawbacks of existing theory are discussed, a new way to the approximate solution of heat transfer problem was suggested, and in the meantime the concepts of thermomass and energy of thermomass were to some extent justified.
    • Funds:
    [1]

    Goldstine H H 1980 A History of the Calculus of Variations from the 17th through the 19th Century (New York: Springer-Verlag)

    [2]

    Finlayson B 1972 The Method of Weighted Residuals and Variational Principles: with Application in Fluid Mechanics,Heat and Mass Transfer (New York: Academic Press) pp335—351

    [3]

    Qian W C 1985 Generalized Variational Principles (Shanghai: Knowledge Press) (in Chinese) [钱伟长 1985 广义变分原理 (上海: 知识出版社)]

    [4]

    Onsager L 1931 Phys. Rev. 37 405

    [5]

    Prigogine I 1947 Etude Thermodynamique des Processus Irréversibles (Liege: Desoer)

    [6]

    Rosen P 1953 J. Chem. Phys. 21 1220

    [7]

    Glansdorff P, Prigogine I 1964 Physica 30 351

    [8]

    Finlayson B A, Scriven L E 1967 Int. J. Heat Mass Transfer 10 799

    [9]

    Biot M A 1954 Appl. Phys. 25 1385

    [10]

    Biot M A 1957 J. Aeronaut. Sci. 24 857

    [11]

    Biot M A 1970 Variational Principles in Heat Transfer (Oxford: Oxford University Press) pp3—20

    [12]

    Vujanovic B, Strauss A M 1971 Am. Inst. Aeron. Astron. J. 9 327

    [13]

    Guo Z Y, Zhu H Y, Liang X G 2007 Int. J. Heat Mass Transfer 50 2545

    [14]

    Chen L, Chen Q, Li Z, Guo Z Y 2009 Int. J. Heat Mass Transfer 52 4778

    [15]

    Chen L G, Wei S H, Sun F R 2009 J. Appl. Phys. 105 094906

    [16]

    Chen Q, Ren J X, Guo Z Y 2009 Chin. Sci. Bull. 54 2862

    [17]

    Chen Q, Wang M R, Pan N, Guo Z Y 2009 Energy 34 1199

    [18]

    Liu X B, Guo Z Y 2009 Acta Phys. Sin. 58 4766 (in Chinese) [柳雄斌、 过增元 2009 物理学报 58 4766]

    [19]

    Liu X B, Meng J A, Guo Z Y 2009 Chin. Sci. Bull. 54 943

    [20]

    Wu J, Liang X G 2008 Sci. China Ser. E 51 1306

    [21]

    Fourier J B J (Translated by Gui Z L) 1993 The Analytical Theory of Heat (Wuhan: Wuhan Press) (in Chinese) 中译本[傅立叶著, 桂质亮译 1993 (武汉: 武汉出版社)]

    [22]

    Cao B Y, Guo Z Y 2007 J. Appl. Phys. 102 053503

    [23]

    Guo Z Y, Cao B Y, Zhu H Y, Zhang Q G 2007 Acta Phys. Sin. 56 3306 (in Chinese) [过增元、 曹炳阳、 朱宏晔、 张清光 2007 物理学报 56 3306]

    [24]

    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]

    [25]

    Guo Z Y, Wu J, Cao B Y 2009 Chin. J. Mech. Engng. 45 10 (in Chinese) [过增元、 吴 晶、 曹炳阳 2009 机械工程学报 45 10]

    [26]

    Wu J, Guo Z Y, Song B 2009 Tsinghua Sci. Technol. 14 12

    [27]

    Morse P M, Feshbach H 1953 Methods of Theoretical Physics (Part 1)(New York: McGraw-Hill) pp280—281

    [28]

    Aharoni J 1985 The Special Theory of Relativity (New York: Dover Publications)

    [29]

    Hu R F, Cao B Y 2009 Sci. China Ser. E 52 1786

    [30]

    Luo Y X, Guan F, Guan J H, Li P 1982 Theoretical Mechanics (Vol. 3)(3rd Ed. )(Beijing: People's Education Press) pp38—44 (in Chinese) [罗远祥、 官 飞、 关冀华、 李 苹 1982 理论力学 (下册) (第3版) (北京: 人民教育出版社) 第38—44页]

    [31]

    Lardner T J 1963 Am. Inst. Aeron. Astron. J. 1 196

    [32]

    Richardson P D 1964 J. Heat Transfer 86 298

    [33]

    Chu H N 1964 J. Spacecraft Rockets 1 686

    [34]

    Carslaw H S, Jaeger J C 1959 Conduction of Heat in Solids (2nd ed.) (Oxford: Calarendon Press) p130

  • [1]

    Goldstine H H 1980 A History of the Calculus of Variations from the 17th through the 19th Century (New York: Springer-Verlag)

    [2]

    Finlayson B 1972 The Method of Weighted Residuals and Variational Principles: with Application in Fluid Mechanics,Heat and Mass Transfer (New York: Academic Press) pp335—351

    [3]

    Qian W C 1985 Generalized Variational Principles (Shanghai: Knowledge Press) (in Chinese) [钱伟长 1985 广义变分原理 (上海: 知识出版社)]

    [4]

    Onsager L 1931 Phys. Rev. 37 405

    [5]

    Prigogine I 1947 Etude Thermodynamique des Processus Irréversibles (Liege: Desoer)

    [6]

    Rosen P 1953 J. Chem. Phys. 21 1220

    [7]

    Glansdorff P, Prigogine I 1964 Physica 30 351

    [8]

    Finlayson B A, Scriven L E 1967 Int. J. Heat Mass Transfer 10 799

    [9]

    Biot M A 1954 Appl. Phys. 25 1385

    [10]

    Biot M A 1957 J. Aeronaut. Sci. 24 857

    [11]

    Biot M A 1970 Variational Principles in Heat Transfer (Oxford: Oxford University Press) pp3—20

    [12]

    Vujanovic B, Strauss A M 1971 Am. Inst. Aeron. Astron. J. 9 327

    [13]

    Guo Z Y, Zhu H Y, Liang X G 2007 Int. J. Heat Mass Transfer 50 2545

    [14]

    Chen L, Chen Q, Li Z, Guo Z Y 2009 Int. J. Heat Mass Transfer 52 4778

    [15]

    Chen L G, Wei S H, Sun F R 2009 J. Appl. Phys. 105 094906

    [16]

    Chen Q, Ren J X, Guo Z Y 2009 Chin. Sci. Bull. 54 2862

    [17]

    Chen Q, Wang M R, Pan N, Guo Z Y 2009 Energy 34 1199

    [18]

    Liu X B, Guo Z Y 2009 Acta Phys. Sin. 58 4766 (in Chinese) [柳雄斌、 过增元 2009 物理学报 58 4766]

    [19]

    Liu X B, Meng J A, Guo Z Y 2009 Chin. Sci. Bull. 54 943

    [20]

    Wu J, Liang X G 2008 Sci. China Ser. E 51 1306

    [21]

    Fourier J B J (Translated by Gui Z L) 1993 The Analytical Theory of Heat (Wuhan: Wuhan Press) (in Chinese) 中译本[傅立叶著, 桂质亮译 1993 (武汉: 武汉出版社)]

    [22]

    Cao B Y, Guo Z Y 2007 J. Appl. Phys. 102 053503

    [23]

    Guo Z Y, Cao B Y, Zhu H Y, Zhang Q G 2007 Acta Phys. Sin. 56 3306 (in Chinese) [过增元、 曹炳阳、 朱宏晔、 张清光 2007 物理学报 56 3306]

    [24]

    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]

    [25]

    Guo Z Y, Wu J, Cao B Y 2009 Chin. J. Mech. Engng. 45 10 (in Chinese) [过增元、 吴 晶、 曹炳阳 2009 机械工程学报 45 10]

    [26]

    Wu J, Guo Z Y, Song B 2009 Tsinghua Sci. Technol. 14 12

    [27]

    Morse P M, Feshbach H 1953 Methods of Theoretical Physics (Part 1)(New York: McGraw-Hill) pp280—281

    [28]

    Aharoni J 1985 The Special Theory of Relativity (New York: Dover Publications)

    [29]

    Hu R F, Cao B Y 2009 Sci. China Ser. E 52 1786

    [30]

    Luo Y X, Guan F, Guan J H, Li P 1982 Theoretical Mechanics (Vol. 3)(3rd Ed. )(Beijing: People's Education Press) pp38—44 (in Chinese) [罗远祥、 官 飞、 关冀华、 李 苹 1982 理论力学 (下册) (第3版) (北京: 人民教育出版社) 第38—44页]

    [31]

    Lardner T J 1963 Am. Inst. Aeron. Astron. J. 1 196

    [32]

    Richardson P D 1964 J. Heat Transfer 86 298

    [33]

    Chu H N 1964 J. Spacecraft Rockets 1 686

    [34]

    Carslaw H S, Jaeger J C 1959 Conduction of Heat in Solids (2nd ed.) (Oxford: Calarendon Press) p130

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  • Received Date:  24 September 2009
  • Accepted Date:  20 November 2009
  • Published Online:  15 October 2010

Hamilton’s principle based on thermomass theory

  • 1. (1)College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; (2)Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, School of Aerospace, Tsinghua University, Beijing 100084, China

Abstract: Based on thermomass theory, the Hamilton's principle as well as the Lagrangian equations governing the motion of thermomass were established by methods analogous to those of classical mechanics. With the kinetic energy of thermomass taken into consideration, the Hamiltons principle for thermomass is expected to be capable of dealing with non-Fourier phenomena. When the kinetic energy is small enough to be ignored, the principle gets back to Fourier transfer. The application of Lagrangian equations was illustrated by the approximate solution of a 1D transient heat conduction problem with heat source. The unification of thermal and mechanical theories was demonstrated from the perspective of analytical mechanics, the drawbacks of existing theory are discussed, a new way to the approximate solution of heat transfer problem was suggested, and in the meantime the concepts of thermomass and energy of thermomass were to some extent justified.

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