Entropy

Research

20 pages, 407 KiB

Open AccessArticle

Thermodynamics-like Formalism for Immiscible and Incompressible Two-Phase Flow in Porous Media

by Alex Hansen and Santanu Sinha

Entropy 2025, 27(2), 121; https://doi.org/10.3390/e27020121 - 24 Jan 2025

Viewed by 387

It is possible to formulate an immiscible and incompressible two-phase flow in porous media in a mathematical framework resembling thermodynamics based on the Jaynes generalization of statistical mechanics. We review this approach and discuss the meaning of the emergent variables that appear, agiture, [...] Read more.

It is possible to formulate an immiscible and incompressible two-phase flow in porous media in a mathematical framework resembling thermodynamics based on the Jaynes generalization of statistical mechanics. We review this approach and discuss the meaning of the emergent variables that appear, agiture, flow derivative, and flow pressure, which are conjugate to the configurational entropy, the saturation, and the porosity, respectively. We conjecture that the agiture, the temperature-like variable, is directly related to the pressure gradient. This has as a consequence that the configurational entropy, a measure of how the fluids are distributed within the porous media and the accompanying velocity field, and the differential mobility of the fluids are related. We also develop elements of another version of the thermodynamics-like formalism where fractional flow rather than saturation is the control variable, since this is typically the natural control variable in experiments. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

13 pages, 13005 KiB

Open AccessArticle

Entropy Production in a System of Janus Particles

by Andrés Arango-Restrepo, Juan David Torrenegra-Rico and J. Miguel Rubi

Entropy 2025, 27(2), 112; https://doi.org/10.3390/e27020112 - 23 Jan 2025

Viewed by 485

Abstract

Entropy production is a key descriptor of out-of-equilibrium behavior in active matter systems, providing insights into both single-particle dynamics and emergent collective phenomena. It helps determine transport coefficients and phoretic velocities and serves as a crucial tool for understanding collective phenomena such as [...] Read more.

Entropy production is a key descriptor of out-of-equilibrium behavior in active matter systems, providing insights into both single-particle dynamics and emergent collective phenomena. It helps determine transport coefficients and phoretic velocities and serves as a crucial tool for understanding collective phenomena such as structural transitions, regime shifts, clustering, and self-organization. This study investigates the role of entropy production for individual active (catalytic Janus) particles and in systems of active particles interacting with one another and their environment. We employ a multiscale framework to bridge microscopic particle dynamics and macroscopic behavior, offering a thermodynamic perspective on active matter. These findings enhance our understanding of the fundamental principles governing active particle systems and create new opportunities for addressing unresolved questions in non-equilibrium thermodynamics. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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8 pages, 209 KiB

Open AccessArticle

Local Equilibrium in Transient Heat Conduction

by Kirill Glavatskiy

Entropy 2025, 27(2), 100; https://doi.org/10.3390/e27020100 - 21 Jan 2025

Viewed by 418

Abstract

Extended irreversible thermodynamics (EIT) has been widely used to overcome the deficiencies of classical irreversible thermodynamics in describing fast transport phenomena. By employing fluxes as additional independent variables and rejecting local equilibrium hypothesis, EIT may provide a thermodynamically consistent framework for high-frequency and [...] Read more.

Extended irreversible thermodynamics (EIT) has been widely used to overcome the deficiencies of classical irreversible thermodynamics in describing fast transport phenomena. By employing fluxes as additional independent variables and rejecting local equilibrium hypothesis, EIT may provide a thermodynamically consistent framework for high-frequency and non-local processes. Here, we propose an alternative approach to EIT that shares the same objective but does not reject local equilibrium hypothesis. Using the rates of change of the energy density as the additional independent variable, we illustrate this approach for two typical problems of transient heat conduction: the Cattaneo-type flux model with thermodynamic inertia and the two-temperature model of energy transfer in a phonon–electron system. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

15 pages, 2287 KiB

Open AccessArticle

Transport Numbers and Electroosmosis in Cation-Exchange Membranes with Aqueous Electrolyte Solutions of HCl, LiCl, NaCl, KCl, MgCl₂, CaCl₂ and NH₄Cl

by Simon B. B. Solberg, Zelalem B. Deress, Marte H. Hvamstad and Odne S. Burheim

Entropy 2025, 27(1), 75; https://doi.org/10.3390/e27010075 - 15 Jan 2025

Viewed by 553

Abstract

Electroosmosis reduces the available energy from ion transport arising due to concentration gradients across ion-exchange membranes. This work builds on previous efforts to describe the electroosmosis, the permselectivity and the apparent transport number of a membrane, and we show new measurements of concentration [...] Read more.

Electroosmosis reduces the available energy from ion transport arising due to concentration gradients across ion-exchange membranes. This work builds on previous efforts to describe the electroosmosis, the permselectivity and the apparent transport number of a membrane, and we show new measurements of concentration cells with the Selemion CMVN cation-exchange membrane and single-salt solutions of HCl, LiCl, NaCl, MgCl₂, CaCl₂ and NH₄Cl. Ionic transport numbers and electroosmotic water transport relative to the membrane are efficiently obtained from a relatively new permselectivity analysis method. We find that the membrane can be described as perfectly selective towards the migration of the cation, and that

{Cl}^{-}

does not contribute to the net electric current. For the investigated salts, we obtained water transference coefficients,

t_{w}

, of 1.1 ± 0.8 for HCl, 9.2 ± 0.8 for LiCl, 4.9 ± 0.2 for NaCl, 3.7 ± 0.4 for KCl, 8.5 ± 0.5 for MgCl₂, 6.2 ± 0.6 for CaCl₂ and 3.8 ± 0.5 for NH₄Cl. However, as the test compartment concentrations of LiCl, MgCl₂ and CaCl₂ increased past 3.5, 1.3 and 1.4 mol kg⁻¹, respectively, the water transference coefficients appeared to decrease. The presented methods are generally useful for characterising concentration polarisation phenomena in electrochemistry, and may aid in the design of more efficient electrochemical cells. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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17 pages, 2161 KiB

Open AccessArticle

Entropy Production in an Electro-Membrane Process at Underlimiting Currents—Influence of Temperature

by Juan Carlos Maroto, Sagrario Muñoz and Vicenta María Barragán

Entropy 2025, 27(1), 3; https://doi.org/10.3390/e27010003 - 25 Dec 2024

Viewed by 477

Abstract

The entropy production in the polarization phenomena occurring in the underlimiting regime, when an electric current circulates through a single cation-exchange membrane system, has been investigated in the 3–40 °C temperature range. From the analysis of the current–voltage curves and considering the electro-membrane [...] Read more.

The entropy production in the polarization phenomena occurring in the underlimiting regime, when an electric current circulates through a single cation-exchange membrane system, has been investigated in the 3–40 °C temperature range. From the analysis of the current–voltage curves and considering the electro-membrane system as a unidimensional heterogeneous system, the total entropy generation in the system has been estimated from the contribution of each part of the system. Classical polarization theory and the irreversible thermodynamics approach have been used to determine the total electric potential drop and the entropy generation, respectively, associated with the different transport mechanisms in each part of the system. The results show that part of the electric power input is dissipated as heat due to both electric migration and diffusion ion transports, while another part is converted into chemical energy stored in the saline concentration gradient. Considering the electro-membrane process as an energy conversion process, an efficiency has been defined as the ratio between stored power and electric power input. This efficiency increases as both applied electric current and temperature increase. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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11 pages, 809 KiB

Open AccessArticle

Computing Entropy for Long-Chain Alkanes Using Linear Regression: Application to Hydroisomerization

by Shrinjay Sharma, Richard Baur, Marcello Rigutto, Erik Zuidema, Umang Agarwal, Sofia Calero, David Dubbeldam and Thijs J. H. Vlugt

Entropy 2024, 26(12), 1120; https://doi.org/10.3390/e26121120 - 21 Dec 2024

Viewed by 540

Abstract

Entropies for alkane isomers longer than C₁₀ are computed using our recently developed linear regression model for thermochemical properties which is based on second-order group contributions. The computed entropies show excellent agreement with experimental data and data from Scott’s tables which are [...] Read more.

Entropies for alkane isomers longer than C₁₀ are computed using our recently developed linear regression model for thermochemical properties which is based on second-order group contributions. The computed entropies show excellent agreement with experimental data and data from Scott’s tables which are obtained from a statistical mechanics-based correlation. Entropy production and heat input are calculated for the hydroisomerization of C₇ isomers in various zeolites (FAU-, ITQ-29-, BEA-, MEL-, MFI-, MTW-, and MRE-types) at 500 K at chemical equilibrium. Small variations in these properties are observed because of the differences in reaction equilibrium distributions for these zeolites. The effect of chain length on heat input and entropy production is also studied for the hydroisomerization of C₇, C₈, C₁₀, and C₁₄ isomers in MTW-type zeolite at 500 K. For longer chains, both heat input and entropy production increase. Enthalpies and absolute entropies of C₇ hydroisomerization reaction products in MTW-type zeolite increase with higher temperatures. These findings highlight the accuracy of our linear regression model in computing entropies for alkanes and provide insight for designing and optimizing zeolite-catalyzed hydroisomerization processes. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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9 pages, 297 KiB

Open AccessArticle

Internal Energy, Fundamental Thermodynamic Relation, and Gibbs’ Ensemble Theory as Emergent Laws of Statistical Counting

by Hong Qian

Entropy 2024, 26(12), 1091; https://doi.org/10.3390/e26121091 - 13 Dec 2024

Cited by 1 | Viewed by 703

Abstract

Statistical counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent and identically distributed N sampling. Entropy provides the infinitesimal probability for an observed empirical frequency

\hat{ν}

with respect to a probability prior

p

, when [...] Read more.

Statistical counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent and identically distributed N sampling. Entropy provides the infinitesimal probability for an observed empirical frequency

\hat{ν}

with respect to a probability prior

p

, when

\hat{ν} \neq p

as

N \to \infty

. Following Callen’s postulate and through Legendre–Fenchel transform, without help from mechanics, we show that an internal energy

u

emerges; it provides a linear representation of real-valued observables with full or partial information. Gibbs’ fundamental thermodynamic relation and theory of ensembles follow mathematically.

u

is to

\hat{ν}

what chemical potential

μ

is to particle number N in Gibbs’ chemical thermodynamics, what

β = T^{- 1}

is to internal energy U in classical thermodynamics, and what

ω

is to t in Fourier analysis. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

24 pages, 916 KiB

Open AccessArticle

An Instructive CO₂ Adsorption Model for DAC: Wave Solutions and Optimal Processes

by Emily Kay-Leighton and Henning Struchtrup

Entropy 2024, 26(11), 972; https://doi.org/10.3390/e26110972 - 13 Nov 2024

Viewed by 910

Abstract

We present and investigate a simple yet instructive model for the adsorption of CO₂ from air in porous media as used in direct air capture (DAC) processes. Mathematical analysis and non-dimensionalization reveal that the sorbent is characterized by the sorption timescale and [...] Read more.

We present and investigate a simple yet instructive model for the adsorption of CO₂ from air in porous media as used in direct air capture (DAC) processes. Mathematical analysis and non-dimensionalization reveal that the sorbent is characterized by the sorption timescale and capacity, while the adsorption process is effectively wavelike. The systematic evaluation shows that the overall adsorption rate and the recommended charging duration depend only on the wave parameter that is found as the ratio of capacity and dimensionless air flow velocity. Specifically, smaller wave parameters yield a larger overall charging rate, while larger wave parameters reduce the work required to move air through the sorbent. Thus, optimal process conditions must compromise between a large overall adsorption rate and low work requirements. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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20 pages, 1519 KiB

Open AccessArticle

Transported Entropy of Ions and Peltier Coefficients in 8YSZ and 10Sc1CeSZ Electrolytes for Solid Oxide Cells

by Aydan Gedik and Stephan Kabelac

Entropy 2024, 26(10), 872; https://doi.org/10.3390/e26100872 - 17 Oct 2024

Viewed by 760

Abstract

In this study, the transported entropy of ions for 8YSZ and 10Sc1CeSZ electrolytes was experimentally determined to enable precise modeling of heat transport in solid oxide cells (SOCs). The Peltier coefficient, crucial for thermal management, was directly calculated, highlighting reversible heat transport effects [...] Read more.

In this study, the transported entropy of ions for 8YSZ and 10Sc1CeSZ electrolytes was experimentally determined to enable precise modeling of heat transport in solid oxide cells (SOCs). The Peltier coefficient, crucial for thermal management, was directly calculated, highlighting reversible heat transport effects in the cell. While data for 8YSZ are available in the literature, providing a basis for comparison, the results for 10Sc1CeSZ show slightly smaller Seebeck coefficients but higher transported ion entropies. Specifically, at

700 ° C

and an oxygen partial pressure of

p_{O_{2}} = 0.21

bar, values of

S_{O^{2 -}}^{*} = 52 \pm 10

J/K·F for 10Sc1CeSZ and

S_{O^{2 -}}^{*} = 48 \pm 9

J/K·F for 8YSZ were obtained. The transported entropy was also validated through theoretical calculations and showed minimal deviations when comparing different cell operation modes (O₂||O²⁻||O₂ and H₂, H₂O||O²⁻||O₂). The influence of the transported entropy of the ions on the total heat generation and the partial heat generation at the electrodes is shown. The temperature has the greatest influence on heat generation, whereby the ion entropy also plays a role. Finally, the Peltier coefficients of 8YSZ for all homogeneous phases agree with the literature values. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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13 pages, 278 KiB

Open AccessArticle

Enhanced Model for the Analysis of Thermoelectric Effects at Nanoscale: Onsager’s Method and Liu’s Technique in Comparison

by Maria Di Domenico and Antonio Sellitto

Entropy 2024, 26(10), 852; https://doi.org/10.3390/e26100852 - 9 Oct 2024

Viewed by 727

Abstract

The aim of this paper is twofold. From the practical point of view, an enhanced model for the description of thermoelectric effects at nanoscale is proposed. From the theoretical point of view, instead, in the particular case of the proposed model, the equivalence [...] Read more.

The aim of this paper is twofold. From the practical point of view, an enhanced model for the description of thermoelectric effects at nanoscale is proposed. From the theoretical point of view, instead, in the particular case of the proposed model, the equivalence between two classical techniques for the exploitation of the second law of thermodynamics is shown, i.e., Onsager’s method and Liu’s technique. An analysis of the heat-wave propagation is performed as well. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

8 pages, 225 KiB

Open AccessArticle

On the Elimination of Fast Variables from the Langevin Equation

by Dick Bedeaux

Entropy 2024, 26(10), 821; https://doi.org/10.3390/e26100821 - 26 Sep 2024

Viewed by 640

Abstract

In a multivariable system, there are usually a number of relaxation times. When some of the relaxation times are shorter than others, the corresponding variables will decay to their equilibrium value faster than the others. After the fast variables have decayed, the system [...] Read more.

In a multivariable system, there are usually a number of relaxation times. When some of the relaxation times are shorter than others, the corresponding variables will decay to their equilibrium value faster than the others. After the fast variables have decayed, the system can be described with a smaller number of variables. From the theory of nonequilibrium thermodynamics, as formulated by Onsager, we know that the coefficients in the linear flux–force relations satisfy the so-called Onsager symmetry relations. The question we will address in this paper is how to eliminate the fast variables in such a way that the coefficients in the reduced description for the slow variables still satisfy the Onsager relations. As the proof that Onsager gave of the symmetry relations does not depend on the choice of the variables, it is equally valid for the subset of slow variables. Elimination procedures that lead to symmetry breaking are possible, but do not consider systems that satisfy the laws of nonequilibrium thermodynamics. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

15 pages, 1126 KiB

Open AccessArticle

Entropy Production and Filling Time in Hydrogen Refueling Stations: An Economic Assessment

by Bruno F. Santoro, David Rincón and Diego F. Mendoza

Entropy 2024, 26(9), 735; https://doi.org/10.3390/e26090735 - 29 Aug 2024

Viewed by 611

Abstract

A multi-objective optimization is performed to obtain fueling conditions in hydrogen stations leading to improved filling times and thermodynamic efficiency (entropy production) of the de facto standard of operation, which is defined by the protocol SAE J2601. After finding the Pareto frontier between [...] Read more.

A multi-objective optimization is performed to obtain fueling conditions in hydrogen stations leading to improved filling times and thermodynamic efficiency (entropy production) of the de facto standard of operation, which is defined by the protocol SAE J2601. After finding the Pareto frontier between filling time and total entropy production, it was found that SAE J2601 is suboptimal in terms of these process variables. Specifically, reductions of filling time from 47 to 77% are possible in the analyzed range of ambient temperatures (from 10 to 40 °C) with higher saving potential the hotter the weather conditions. Maximum entropy production savings with respect to SAE J2601 (7% for 10 °C, 1% for 40 °C) demand a longer filling time that increases with ambient temperature (264% for 10 °C, 350% for 40 °C). Considering average electricity prices in California, USA, the operating cost of the filling process can be reduced between 8 and 28% without increasing the expected filling time. Full article

(This article belongs to the Special Issue The Entropy Production—as Cornerstone in Applied Nonequilibrium Thermodynamics—Dedicated to Professor Signe Kjelstrup on the Occasion of Her 75th Birthday)

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