Entropy production rate in the one-dimensional heat conduction for systems whose thermal conductivity or its derivative are piecewise continuous with respect to temperature

1990 ◽  
Vol 25 (2) ◽  
pp. 117-127 ◽  
Author(s):  
G. Bisio
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Entropy Production ◽  
Production Rate ◽  
Entropy Production Rate ◽  
One Dimensional ◽  
Piecewise Continuous ◽  
The One

scholarly journals Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 881 ◽  
Author(s):  
Karl Hoffmann ◽  
Kathrin Kulmus ◽  
Christopher Essex ◽  
Janett Prehl
Keyword(s):  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Irreversible Processes ◽  
Entropy Production Rate ◽  
Continuous Space ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
And Diffusion

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

scholarly journals Fractional-order heat conduction models from generalized Boltzmann transport equation

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190280 ◽  
Author(s):  
Shu-Nan Li ◽  
Bing-Yang Cao
Keyword(s):  
Heat Conduction ◽  
Entropy Production ◽  
Production Rate ◽  
Fractional Order ◽  
Mean Square Displacement ◽  
Linear Response Theory ◽  
Entropy Density ◽  
Entropy Production Rate ◽  
Boltzmann Transport ◽  
Entropy Flux

The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κ eff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δ x | 2 〉 ∼  t γ is established, namely, κ eff  ∼  t γ −1 , which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

scholarly journals Simplified Grinding Temperature Model Study

2019 ◽  
Vol 6 (2) ◽  
pp. a1-a7
Author(s):  
N. V. Lishchenko ◽  
V. P. Larshin ◽  
H. Krachunov
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Grinding Temperature ◽  
Temperature Model ◽  
One Dimensional ◽  
Heating And Cooling ◽  
Equation Of Heat Conduction ◽  
The One

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

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