An Entropy Adjoint Approach to Mesh Refinement

In the present study, steady state heat transfer in a slab is analysed by applying the principle of variation calculus to the entropy generation minimization. The governing equation of the phenomena is obtained by minimizing the total entropy generation over the slab by considering the irreversibility and variation of thermal conductivity as a function of spatial co-ordinates. The governing equation is solved to obtain the temperature distribution, internal heat generation due to irreversibility, entropy generation number and entropy transport into system. The apparent heat sources that come into existence because of the irreversibility in heat diffusion have made the minimization of entropy generation feasible.

This paper presents a method for estimating the drag error in two-dimensional computational fluid dynamics simulations using a method that does not require auxiliary adjoint solutions. The error of interest is that caused by the numerical discretization, including effects of finite mesh size and approximation order. The target output is a drag calculation based on a farfield integration of the entropy. The error estimate is motivated by an interpretation of entropy variables as adjoint solutions to an entropy balance output. As entropy variables are obtained by a direct transformation of the state, no separate adjoint solution is required. The method is shown to be applicable not only to inviscid and laminar flows, but also to turbulent Reynolds-averaged Navier-Stokes flows, for which entropy variables generally lose their symmetrization property. Since the proposed drag error estimate is based on an integral of a weighted residual over the computational domain, it can be localized to provide an adaptive indicator over each element. Adaptive results for several flows of aerodynamic interest show that the error estimate is effective and that the method performs on par with adjoint-based methods. ∗Assistant Professor, AIAA Senior Member †Graduate Research Assistant ‡Professor, AIAA Fellow

‘Stochastic thermodynamics as a conceptual framework combines the stochastic energetics approach introduced a decade ago by Sekimoto [1] with the idea that entropy can consistently be assigned to a single fluctuating trajectory [2]’. This quote, taken from Udo Seifert's [3] 2008 review, nicely summarizes the basic ideas behind stochastic thermodynamics: for small systems, driven by external forces and in contact with a heat bath at a well-defined temperature, stochastic energetics [4] defines the exchanged work and heat along a single fluctuating trajectory and connects them to changes in the internal (system) energy by an energy balance analogous to the first law of thermodynamics. Additionally, providing a consistent definition of trajectory-wise entropy production gives rise to second-law-like relations and forms the basis for a ‘stochastic thermodynamics’ along individual fluctuating trajectories.In order to construct meaningful concepts of work, heat and entropy production for single trajectories, their definitions are based on the stochastic equations of motion modeling the physical system of interest. Because of this, they are valid even for systems that are prevented from equilibrating with the thermal environment by external driving forces (or other sources of non-equilibrium). In that way, the central notions of equilibrium thermodynamics, such as heat, work and entropy, are consistently extended to the non-equilibrium realm. In the (non-equilibrium) ensemble, the trajectory-wise quantities acquire distributions. General statements derived within stochastic thermodynamics typically refer to properties of these distributions, and are valid in the non-equilibrium regime even beyond the linear response.The extension of statistical mechanics and of exact thermodynamic statements to the non-equilibrium realm has been discussed from the early days of statistical mechanics more than 100 years ago. This debate culminated in the development of linear response theory for small deviations from equilibrium, in which a general framework is constructed from the analysis of non-equilibrium states close to equilibrium. In a next step, Prigogine and others developed linear irreversible thermodynamics, which establishes relations between transport coefficients and entropy production on a phenomenological level in terms of thermodynamic forces and fluxes. However, beyond the realm of linear response no general theoretical results were available for quite a long time.This situation has changed drastically over the last 20 years with the development of stochastic thermodynamics, revealing that the range of validity of thermodynamic statements can indeed be extended deep into the non-equilibrium regime. Early developments in that direction trace back to the observations of symmetry relations between the probabilities for entropy production and entropy annihilation in non-equilibrium steady states [5–8] (nowadays categorized in the class of so-called detailed fluctuation theorems), and the derivations of the Bochkov–Kuzovlev [9, 10] and Jarzynski relations [11] (which are now classified as so-called integral fluctuation theorems). Apart from its fundamental theoretical interest, the developments in stochastic thermodynamics have experienced an additional boost from the recent experimental progress in fabricating, manipulating, controlling and observing systems on the micro- and nano-scale. These advances are not only of formidable use for probing and monitoring biological processes on the cellular, sub-cellular and molecular level, but even include the realization of a microscopic thermodynamic heat engine [12] or the experimental verification of Landauer's principle in a colloidal system [13].The scientific program Stochastic Thermodynamics held between 4 and 15 March 2013, and hosted by The Nordic Institute for Theoretical Physics (Nordita), was attended by more than 50 scientists from the Nordic countries and elsewhere, amongst them many leading experts in the field. During the program, the most recent developments, open questions and new ideas in stochastic thermodynamics were presented and discussed. From the talks and debates, the notion of information in stochastic thermodynamics, the fundamental properties of entropy production (rate) in non-equilibrium, the efficiency of small thermodynamic machines and the characteristics of optimal protocols for the applied (cyclic) forces were crystallizing as main themes. Surprisingly, the long-studied adiabatic piston, its peculiarities and its relation to stochastic thermodynamics were also the subject of intense discussions. The comment on the Nordita program Stochastic Thermodynamics published in this issue of Physica Scripta exploits the Jarzynski relation for determining free energy differences in the adiabatic piston.This scientific program and the contribution presented here were made possible by the financial and administrative support of The Nordic Institute for Theoretical Physics.

Entropy generation in the blade-tip gap layer of a radial impeller is studied by means of simple physical models solved numerically. It is shown that entropy is generated in two regions, i.e., near the shroud wall and at the fluid boundary between the gap layer and the impeller, in comparable amounts. Entropy generated near the shroud wall is carried by the flow in the gap, either upstream to the impeller inlet or downstream to the discharge. Most of the entropy generated at the fluid boundary is incorporated into the bulk flow in the channel. Operating conditions for minimal entropy generation within the tip gap are identified.

Double-diffusive laminar natural convection and entropy generation of Carreau fluid in a heated enclosure with an inner circular cold cylinder (Part II: Entropy generation)

Local entropy generation analysis on passive high-concentration DMFCs (direct methanol fuel cell) with different cell structures

6 - Entropy and transport

The shaftless ducted turbine (abbreviated as SDT), as an extraordinary innovation in tidal current power generation applications, has many advantages, and a wide application prospect. The structure of an SDT resembles a ducted turbine (abbreviated as DT), as both contain blades and a duct. However, there are some structural differences between a DT and a SDT, which can cause significant discrepancy in the hydrodynamic characteristics and flow features. The present work compares the detailed hydrodynamic-energy loss characteristics of a DT and a SDT by means of computational fluid dynamics (CFD), performed by solving the 3D steady incompressible Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Menter’s Shear Stress Transport (SST k−ω) turbulence model and entropy production model. The results show the SDT features a higher power level at low tip speed ratio (TSR) and a potential reduction in potential flow resistance and disturbance with respect to the DT. Moreover, a detail entropy production analysis shows the energy loss is closely related to the flow separation and the reverse flow, and other negative flow factors. The entropy production of the SDT is lessened than that of the DT at different TSR. Unlike the DT, the SDT allows a large mass flow of water to leak through the open-center structure, which plays an important role in improving the wake structure and avoiding the negative flow along the central axis.

<p indent="0mm">In this work, the entropy generation analysis method was applied to calculate the entropy generation rate distribution and exergy equilibrium of a typical heating unit in a regenerative reheating furnace. The entropy generation rate of the heat conduction, convective heat transfer, viscous dissipation, mass diffusion, combustion, and radiative processes were numerically determined. The transient change of exergy efficiency of the heating unit and the distribution of exergy loss and exergy efficiency along the furnace length during the billet heating process were investigated. The effects of gas jet velocity, oxygen-enriched concentration, and blast furnace gas combustion with natural gas on the furnace efficiency were investigated. The results showed that when the billet temperature is low, the radiative entropy generation accounts for 70.5% of the total entropy generation of the heating unit. Improving the billet loading temperature can effectively reduce the entropy generation rate. When the billet temperature is high, entropy generation during combustion is dominant, accounting for 59.9%, and the optimization of combustion is the main measure to reduce entropy generation. The new flame takes approximately <sc>15–20 s</sc> to stabilize after burner reversal. Because the original flame is pushed out, the furnace temperature fluctuates, and mass diffusion intensifies, which have serious impacts on the stabilization of the furnace exergy efficiency; the regenerators need to have a sufficiently high recovery efficiency to ensure that the overall exergy efficiency of the heating unit can be improved. For the entire furnace, the first type of furnace exergy loss is approximately 33.4%, mainly in the soaking zone and caused by the high-temperature flue gas leaving the furnace, accounting for 45.6% of the input exergy in the soaking zone. The second type of exergy loss is approximately 24.4%, mainly in the preheating section and caused by radiative entropy generation, accounting for 17.4% of the input energy in the preheating zone. Keeping other parameters unchanged, when the jet velocity is <sc>19 m/s,</sc> the heating unit has the highest exergy and thermal efficiency. When the oxygen concentration is increased to 35%, the exergy and thermal efficiency of the heating unit are increased by 4.1% and 5.5%, respectively. Blast furnace gas combustion with 10% natural gas can increase the furnace temperature and surface heat flux of billet, but the exergy and thermal efficiency are reduced by 1.2% and 4.4%, respectively, which is unfavorable to the energy saving of the reheating furnace. This study provides some new insights into the energy saving of reheating furnaces.

This paper presents the entropic damage indicators for metallic material fatigue processes obtained from three associated energy dissipation sources. Since its inception, reliability engineering has employed statistical and probabilistic models to assess the reliability and integrity of components and systems. To supplement the traditional techniques, an empirically-based approach, called physics of failure (PoF), has recently become popular. The prerequisite for a PoF analysis is an understanding of the mechanics of the failure process. Entropy, the measure of disorder and uncertainty, introduced from the second law of thermodynamics, has emerged as a fundamental and promising metric to characterize all mechanistic degradation phenomena and their interactions. Entropy has already been used as a fundamental and scale-independent metric to predict damage and failure. In this paper, three entropic-based metrics are examined and demonstrated for application to fatigue damage. We collected experimental data on energy dissipations associated with fatigue damage, in the forms of mechanical, thermal, and acoustic emission (AE) energies, and estimated and correlated the corresponding entropy generations with the observed fatigue damages in metallic materials. Three entropic theorems—thermodynamics, information, and statistical mechanics—support approaches used to estimate the entropic-based fatigue damage. Classical thermodynamic entropy provided a reasonably constant level of entropic endurance to fatigue failure. Jeffreys divergence in statistical mechanics and AE information entropy also correlated well with fatigue damage. Finally, an extension of the relationship between thermodynamic entropy and Jeffreys divergence from molecular-scale to macro-scale applications in fatigue failure resulted in an empirically-based pseudo-Boltzmann constant equivalent to the Boltzmann constant.

The equation of local exergy loss rate in forced convective heat transfer has been derived in present work. The obtained exergy loss equation involves exergy loss due to heat transfer and pressure drop as well as exergy generation on account of viscous heat dissipation. The dimensionless form of local exergy loss rate and entropy generation different from the previous can also be obtained. Moreover, the calculation results of two convective heat transfer problems about local exergy loss and local entropy generation proposed by Bejan are given. The effect of temperature difference, Prandtl number and Eckert number on exergy loss and entropy generation has been analysed and the comparison between exergy loss and entropy generation under various conditions has been made. Results shows that for Poiseuille flow the local exergy loss greatly differs from the entropy generation due to the presence of pressure drop. However, viscous dissipation takes an important role in exergy loss for Couette flow on account of no pressure drop.

This paper provides a review of different contributions dedicated thus far to entropy generation analysis (EGA) in turbulent combustion systems. We account for various parametric studies that include wall boundedness, flow operating conditions, combustion regimes, fuels/alternative fuels and application geometries. Special attention is paid to experimental and numerical modeling works along with selected applications. First, the difficulties of performing comprehensive experiments that may support the understanding of entropy generation phenomena are outlined. Together with practical applications, the lumped approach to calculate the total entropy generation rate is presented. Apart from direct numerical simulation, numerical modeling approaches are described within the continuum formulation in the framework of non-equilibrium thermodynamics. Considering the entropy transport equations in both Reynolds-averaged Navier–Stokes and large eddy simulation modeling, different modeling degrees of the entropy production terms are presented and discussed. Finally, exemplary investigations and validation cases going from generic or/and canonical configurations to practical configurations, such as internal combustion engines, gas turbines and power plants, are reported. Thereby, the areas for future research in the development of EGA for enabling efficient combustion systems are highlighted. Since EGA is known as a promising tool for optimization of combustion systems, this aspect is highlighted in this work.

Thermodynamics of solar cell efficiency

Direct numerical simulations with multi-step chemistry were performed for one- and two-dimensional freely propagating laminar premixed flames of methane–air and hydrogen–air mixtures with a matching density ratio to isolate the effects of hydrodynamic instability while allowing for a variable effective Lewis number, with the methane (hydrogen) flame being thermodiffusively stable (unstable). Entropy diffusion and generation mechanisms were analysed based on contributions from heat conduction, viscous dissipation, mass diffusion, and chemical reactions. Across both flames, chemical reactions were identified as the dominant source of entropy generation, with viscous dissipation contributing negligibly compared to other mechanisms. Significant differences were found in the structure of entropy generation rates across both flames, with varying degrees of correlation with curvature. Stronger correlations were found between the irreversible entropy generation rates and the heat release rate in both flames, suggesting the former as a potential marker for thermodiffusive instability. Analysis of the entropy generation profiles at representative locations across a flame front further revealed possible origins of the entropy behaviour under thermodiffusively stable and unstable conditions.

Entropy generation is tied to the exergy destroyed. Hence, the amount of entropy generation is of primary concern as it is related to unavailable work. Viscous dissipation is a form of heat generation due to work done by viscous forces. Its effect on the velocity and temperature profiles would have affected the entropy generation. In this work, second law analysis is carried out on a microchannel between parallel plates for a power-law fluid. The governing energy equation for a rectangular microchannel is first solved analytically. Analytical expression is obtained for the dimensionless entropy generation and Bejan number. Dimensionless entropy generation due to fluid flow irreversibility and heat transfer irreversibility are also computed and compared. The distribution of entropy generation due to heat transfer irreversibility and fluid friction irreversibility changes as Brinkman number increases. A comparison with a previous literature on a circular pipe for the same Brinkman number reveals that the total dimensionless entropy generation in parallel plate is more than the corresponding value in circular pipe. However, the Bejan number for a parallel plate is lower than a circular pipe.