Search Results (15)

Diffusion against a concentration gradient, also called uphill transport, has been reported for magnesium and sulfate ions in solutions of NaCl and Mg${\mathrm{SO}}_4$ in reverse electrodialysis (RED). Here we derive transport equations for such systems and explain the observed uphill transport using non-equilibrium thermodynamics (NET). A set of Nernst–Planck equations was reformulated into a set of flux equations with neutral salt driving forces. By applying the condition of entropy production invariance to the sets of variables, we show how an ideal ion selectivity model provides Onsager coupling coefficients for the ion transport. These coupling coefficients can predict and explain the observed uphill transport of magnesium and sulfate. A linear fitting scheme is developed to fit the transport coefficients to experimental data for the ideal ion selectivity model and for a more general model. The results show that even the simplest ideal ion selectivity model captures the uphill transport, and the phenomenon occurs due to diffusional ion exchange, as in Donnan dialysis. Under open-circuit conditions, deviations are large, and the simplest model is no longer sufficient; co-ion leakage corrections are essential. The results provide a basis for future modelling of RED processes; in particular, one can determine and optimise the process efficiency, as the model gives direct access to the process’s local entropy production. Osmosis is identified as a potential source of error. Full article

The article considers the description of a macrosystem in terms that do not depend on the nature of the macrosystem. The results obtained can be used to describe macrosystem models of thermodynamic processes, and to create interdisciplinary models that take into account interactions of various natures. The macrosystem model is based on its representation in the form of a self-similar oriented weighted graph where the equation of state is fulfilled for each node, which connects extensive variables. One of the extensive variables is entropy, the maximum of which corresponds to the state of equilibrium. For processes in which fluxes are linearly dependent on driving forces, Onsager’s relations are shown to be true, which makes it possible to prove that in the space of stationary processes, entropy production in a closed macrosystem is a metric similar to the Mahalanobis metric, which determines the distance between processes. Zero in such a space indicates reversible processes, and thus the production of entropy shows the degree of irreversibility as the distance from a researched process to a reversible one. Full article

This letter aims to investigate thermodynamic processes in small systems in the Onsager region by showing that fundamental quantities such as total entropy production can be discretized on the mesoscopic scale. Even thermodynamic variables can conjugate to thermodynamic forces, and thus, Glansdorff–Prigogine’s dissipative variable may be discretized. The canonical commutation rules (CCRs) valid at the mesoscopic scale are postulated, and the measurement process consists of determining the eigenvalues of the operators associated with the thermodynamic quantities. The nature of the quantized quantity β , entering the CCRs, is investigated by a heuristic model for nano-gas and analyzed through the tools of classical statistical physics. We conclude that according to our model, the constant β does not appear to be a new fundamental constant but corresponds to the minimum value. Full article

The Second Law of Thermodynamics states that entropy S increases in a spontaneous process in an ideal isothermal and isolated system. Real systems are influenced by external forces and fields, including the temperature field. In this case, only entropy is not enough, and we suggest using a new function, $L_s$, which is analogous to the Lagrangian in classical mechanics. It includes total potential energy but instead of mechanical kinetic energy, $L_s$ includes the product ST, and the system always evolves towards increasing this modified Lagrangian. It reaches an equilibrium when total potential force is balanced by both entropic and thermal forces. All forces have the same units, Newton/mol, and may be added or subtracted. For condensed systems with friction forces, it is a molecular transport velocity, and not acceleration, which is proportional to the acting force. Our approach has several advantages compared to Onsager’s non-equilibrium thermodynamics with its thermodynamic forces, which may have different units, including 1/T for energy transport. For isolated systems, the description is reduced to Second Law and Clausius inequality. It easily explains diffusion, Dufour effect, and Soret thermodiffusion. The combination of electric, thermal, and entropic forces explains thermoelectric phenomena, including Peltier–Seebeck and Thomson (Lord Kelvin) effects. Gravitational and entropic forces together inside a black hole may lead to a steady state or the black hole evaporation. They are also involved in and influenced by solar atmospheric processes. Full article

One of the most important formalisms used to describe membrane transport is Onsager–Peusner thermodynamics (TOP). Within the TOP framework, a procedure has been developed for the transformation of the Kedem–Katchalsky (K–K) equations for the transport of binary electrolytic solutions across a membrane into the Kedem–Katchalsky–Peusner (K–K–P) equations. The membrane system with an Ultra Flo 145 Dialyser membrane used for hemodialysis and aqueous NaCl solutions was used as experimental setup. The H version of K–K–P formalism for binary electrolyte solutions was used to evaluate theoretical coefficients characterizing fluxes of energies and efficiencies for membrane transport processes. The coupling coefficients of membrane processes and the dissipative energy flux were calculated on the basis of the Peusner coefficients obtained from transformation of K–K coefficients. The knowledge of dissipative energy flux, which is a function of thermodynamic forces, allows for the determination of the energy conversions during transport processes in a membrane system. In addition, a frictional interpretation of the obtained coefficients is presented. Full article

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

We develop an action principle for producing a single-fluid two-constituent system with dissipation in general relativity. The two constituents in the model are particles and entropy. The particle flux creation rate is taken to be zero, while the entropy creation rate is non-zero. Building on previous work, it is demonstrated that a new term (the proper time derivative of the matter space “metric”) is required in the Lagrangian in order to produce terms typically associated with bulk and shear viscosity. Equations of motion, entropy creation rate, and energy–momentum–stress tensor are derived. Using an Onsager approach of identifying thermodynamic “forces” and “fluxes”, a model is produced which delivers the same entropy creation rate as the standard, relativistic Navier–Stokes equations. This result is then contrasted with a model generated in the spirit of the action principle, which takes as its starting point a specific Lagrangian and then produces the equations of motion, entropy creation rate, and energy–momentum–stress tensor. Unlike the equations derived from Onsager reasoning, where the analogs of the bulk and shear viscosity coefficients are prescribed “externally”, we find that the forms of the coefficients in the second example are a direct result of the specified Lagrangian. Furthermore, the coefficients are shown to satisfy evolution equations along the fluid worldline, also a product of the specific Lagrangian. Full article

Under typical marine conditions of about 80% relative humidity, evaporation of water from the ocean is an irreversible process accompanied by entropy production. In this article, equations are derived for the latent heat of irreversible evaporation and the related nonequilibrium entropy balance at the sea surface. To achieve this, linear irreversible thermodynamics is considered in a conceptual ocean evaporation model. The equilibrium thermodynamic standard TEOS-10, the International Thermodynamic Equation of Seawater—2010, is applied to irreversible evaporation under the assumption of local thermodynamic equilibrium. The relevance of local equilibrium conditions for irreversible thermodynamics is briefly explained. New equations are derived for the mass flux of evaporation and for the associated nonequilibrium enthalpies and entropies. The estimated entropy production rate of ocean evaporation amounts to $0.004 \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-1}$ as compared with the average terrestrial global entropy production of about $1 \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-1}$. Full article

A thermodynamic description of porous media must handle the size- and shape-dependence of media properties, in particular on the nano-scale. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course-grained scale, by applying Hill’s thermodynamics of small systems to the subsystems of the medium. We argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we then derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation-dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of the Onsager conductivity matrix for transport through porous media. Practical possibilities are discussed. Full article

This article is focused on the derivation of constitutive equations for magnetic liquids. The results can be used for both ferromagnetic and magnetorheological fluids after the introduced simplifications. The formulation of constitutive equations is based on two approaches. The intuitive approach is based on experimental experience of non-Newtonian fluids, which exhibit a generally non-linear dependence of mechanical stress on shear rate; this is consistent with experimental experience with magnetic liquids. In these general equations, it is necessary to determine the viscosity of a liquid as a function of magnetic induction; however, these equations only apply to the symmetric stress tensor and can only be used for an incompressible fluid. As a result of this limitation, in the next part of the work, this approach is extended by the asymmetry of the stress tensor, depending on the angular velocity tensor. All constitutive equations are formulated in Cartesian coordinates in 3D space. The second approach to determining constitutive equations is more general: it takes the basis of non-equilibrium thermodynamics and is based on the physical approach, using the definition of density of the entropy production. The production of entropy is expressed by irreversible thermodynamic flows, which are caused by the effect of generalized thermodynamic forces after disturbance of the thermodynamic equilibrium. The dependence between fluxes and forces determines the constitutive equations between stress tensors, depending on the strain rate tensor and the magnetization vector, which depends on the intensity of the magnetic field. Their interdependencies are described in this article on the basis of the Curie principle and on the Onsager conditions of symmetry. Full article

We report the preparation of a highly-oriented graphene-based film prepared from liquid crystal dispersion of graphene oxides (GOs). We observed that the liquid crystal behavior of GOs is highly affected by the lateral size of GO flakes: the critical concentration for liquid crystal formation decreased with the increase of the lateral size of GO flakes, which is in a good agreement with Onsager’s theory. As a result, we were able to obtain highly-ordered graphene assemblies with large-sized GO flakes (150 ± 29 μm) at relatively low concentrations. By applying the shear force, we were able to obtain highly-oriented films from liquid crystal GO flakes. After hydrogen iodide (HI) reduction, GO films showed excellent mechanical strength and electrical conductivity, which were 278% and 283% higher, respectively, than those of films made from smaller sized GO flakes (28 ± 24 μm). Full article

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed. Full article

We extend Onsager’s minimum dissipation principle to stationary states that are only subject to local equilibrium constraints, even when the transport coefficients depend on the thermodynamic forces. Crucial to this generalization is a decomposition of the thermodynamic forces into those that are held fixed by the boundary conditions and the subspace that is orthogonal with respect to the metric defined by the transport coefficients. We are then able to apply Onsager and Machlup’s proof to the second set of forces. As an example, we consider two-dimensional nonlinear diffusion coupled to two reservoirs at different temperatures. Our extension differs from that of Bertini et al. in that we assume microscopic irreversibility, and we allow a nonlinear dependence of the fluxes on the forces. Full article

In this work, the irreversible processes in light heating of Silicon (Si) and Germanium (Ge) thin films are examined. Each film is exposed to light irradiation with radiative and convective boundary conditions. Heat, electron and hole transport and generation-recombination processes of electron-hole pairs are studied in terms of a phenomenological model obtained from basic principles of irreversible thermodynamics. We present an analysis of the contributions to the entropy production in the stationary state due to the dissipative effects associated with electron and hole transport, generation-recombination of electron-hole pairs as well as heat transport. The most significant contribution to the entropy production comes from the interaction of light with the medium in both Si and Ge. This interaction includes two processes, namely, the generation of electron-hole pairs and the transferring of energy from the absorbed light to the lattice. In Si the following contribution in magnitude comes from the heat transport. In Ge all the remaining contributions to entropy production have nearly the same order of magnitude. The results are compared and explained addressing the differences in the magnitude of the thermodynamic forces, Onsager’s coefficients and transport properties of Si and Ge. Full article

This paper investigates not fully explained voltage offsets observed by several researchers during the measurement of the Seebeck coefficient of high Z materials. These offsets, traditionally attributed to faulty laboratory procedures, have proven to have an irreducible component that cannot be fully eliminated in spite of careful laboratory procedures. In fact, these offsets are commonly observed and routinely subtracted out of commercially available Seebeck measurement systems. This paper offers a possible explanation based on the spontaneous formation of an adiabatic temperature gradient in the presence of a force field. The diffusion-diffusion heat transport mechanism is formulated and applied to predict two new thermoelectric effects. The first is the existence of a temperature gradient across a potential barrier in a semiconductor and the second is the Onsager reciprocal of the first, that is, the presence of a measureable voltage that arises across a junction when the temperature gradient is forced to zero by a thermal clamp. Suggested future research includes strategies for utilizing the new thermoelectric effects. Full article