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Research Papers: Conduction

Thermodynamic Extremum Principles for Nonequilibrium Stationary State in Heat Conduction

[+] Author and Article Information
Yangyu Guo, Ziyan Wang

Key Laboratory for Thermal Science and
Power Engineering of Ministry of Education,
Department of Engineering Mechanics
and CNMM,
Tsinghua University,
Beijing 100084, China

Moran Wang

Key Laboratory for Thermal Science and
Power Engineering of Ministry of Education,
Department of Engineering Mechanics
and CNMM,
Tsinghua University,
Beijing 100084, China
e-mail: mrwang@tsinghua.edu.cn

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 26, 2016; final manuscript received February 21, 2017; published online April 4, 2017. Assoc. Editor: Robert D. Tzou.

J. Heat Transfer 139(7), 071303 (Apr 04, 2017) (7 pages) Paper No: HT-16-1307; doi: 10.1115/1.4036086 History: Received May 26, 2016; Revised February 21, 2017

Minimum entropy production principle (MEPP) is an important variational principle for the evolution of systems to nonequilibrium stationary state. However, its restricted validity in the domain of Onsager's linear theory requires an inverse temperature square-dependent thermal conductivity for heat conduction problems. A previous derivative principle of MEPP still limits to constant thermal conductivity case. Therefore, the present work aims to generalize the MEPP to remove these nonphysical limitations. A new dissipation potential is proposed, the minimum of which thus corresponds to the stationary state with no restriction on thermal conductivity. We give both rigorous theoretical verification of the new extremum principle and systematic numerical demonstration through 1D transient heat conduction with different kinds of temperature dependence of the thermal conductivity. The results show that the new principle remains always valid while MEPP and its derivative principle fail beyond their scopes of validity. The present work promotes a clear understanding of the existing thermodynamic extremum principles and proposes a new one for stationary state in nonlinear heat transport.

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Figures

Grahic Jump Location
Fig. 1

Schematic of 1D transient heat conduction across an infinite plate with thickness L: (a) case I, (b) case II, and (c) case III. The dashed and solid lines denote the initial and final temperature distributions, respectively. The arrow line means the direction of temporal evolution.

Grahic Jump Location
Fig. 2

Temporal evolution of temperature distributions in 1D transient heat conduction: (a) case I, an inverse temperature square-dependent thermal conductivity k=krTr2/T2 and volumetric heat capacity CV=CVrTr2/T2; (b) case II, constant thermal conductivity k=kr and volumetric heat capacity CV=CVr; and (c) case III, the temperature square-dependent thermal conductivity k=krT2/Tr2 and volumetric heat capacity CV=CVrT2/Tr2. For all the three cases, the Fourier number (Fo) is defined based on the thermal diffusivity at the reference temperature (Tr = 300 K): αr=kr/CVr. Temperature distributions at four sequential (Fo) have been displayed: Fo = 0.02, 0.05, 0.15, and 1. The arrow lines signify the direction of temporal evolution.

Grahic Jump Location
Fig. 3

Dimensionless dissipation potential versus time in 1D transient heat conduction: (a) case I, the inverse temperature square-dependent thermal conductivity k=krTr2/T2 and volumetric heat capacity CV=CVrTr2/T2; (b) case II, constant thermal conductivity k=kr and volumetric heat capacity CV=CVr; (c) case III, the temperature square-dependent thermal conductivity k=krT2/Tr2 and volumetric heat capacity CV=CVrT2/Tr2. For all the three cases, the Fourier number (Fo) is defined based on the thermal diffusivity at the reference temperature (Tr = 300 K): αr=kr/CVr. Solid line-square represents entropy production, Eq. (35), and solid line-circle represents weighted entropy production, Eq. (36), whereas the solid line-star represents the present new dissipation potential, Eq. (37).

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