State Of Minimum Entropy Production Research Articles

Minimum entropy production in a distillation column for air separation described by a continuous non-equilibrium model

In this work, we apply the rate-based model of Taylor and Krishna to describe the separation of air in a low-pressure, packed distillation column. By use of numerical optimization, we identify the temperature profile for heat exchange with the column and its surroundings that minimizes the total entropy production. Optimal operation of the column reduces the entropy production by nearly 50%, and the total heating- and cooling duties by 30% and 50%, respectively. We find that the local entropy production is more uniform for the optimal solution than in the adiabatic column, a property that may be helpful for new designs. Using the equilibrium stage model, the state of minimum entropy production has higher cooling/heating duties than in the rate-based model case. This shows that more sophisticated models can be beneficial for the development of reliable strategies to improve the energy efficiency of distillation columns.

Nature-inspired geometrical design of a chemical reactor

We explore the possibility to actively use the system geometry to search for states of minimum entropy production in a chemical reactor. This idea is inspired by the energy-efficient mass and energy transfer that takes place in the reindeer nose thanks to its complex geometry. A cylindrical plug-flow reactor for oxidation of sulfur dioxide is used as example, while optimal control theory is used to formulate the problem. We hypothesize that the nasal anatomy of the reindeer has evolved to its present shape to help reducing energy dissipation during respiration in extreme ambient temperatures.A comparable optimal diameter-profile in the plug-flow reactor resulted in 11% reduction of the total entropy production, compared to a cylindrical reference reactor. With, in addition, an optimal reactor length, the reduction is 16%. These reductions are largely due to reductions in viscous dissipation. In practice, this translates into smaller pressure drops across the system, which reduce the loads of upstream/downstream compressors. Moreover, the peak in the temperature profile was reduced with respect to that obtained by controlling the temperature of the cooling medium.With today's technological solutions, the optimal diameter profile might be easier to realize than other optimal control strategies. The possible gains from this first example are encouraging, and may serve as inspiration for further applications.

Minimum entropy generation in a heat exchanger in the cryogenic part of the hydrogen liquefaction process: On the validity of equipartition and disappearance of the highway

Liquefaction of hydrogen is a promising technology for transporting large quantities of hydrogen across long distances. A key challenge is the high power consumption. In this work, we discuss refrigeration strategies that give minimum entropy production/exergy destruction in a plate-fin heat exchanger that cools the hydrogen from 47.8 K to 29.3 K. Two reference cases are studied; one where the feed stream enters at 20 bar, and one where it enters at 80 bar. Catalyst in the hot layers speeds up the conversion of ortho-to para-hydrogen. Optimal control theory is used to formulate a minimization problem where the objective function is the total entropy production, the control variable is the temperature of the refrigerant and the constrains are the balance equations for energy, mass and momentum in the hot layers. The optimal refrigeration strategies give a reduction of the total entropy production of 8.7% in the 20-bar case and 4.3% in the 80-bar case. The overall heat transfer coefficient and duty is higher in the 20 bar case, which compensates for the increase in entropy production due to a thermal mismatch that is avoided in the 80 bar case. This leads the second law efficiency of the 20 bar case (91%) to be similar to the 80 bar case (89%). We demonstrate that equipartition of the entropy production and equipartition of the thermal driving force are both excellent design principles for the process unit considered, with total entropy productions deviating only 0.2% and 0.5% from the state of minimum entropy production. Equipartition of the thermal driving force i.e. a constant difference between the inverse temperatures of the hot and cold layers represents a particularly simple guideline that works remarkably well. We find that both heat transfer and the spin-isomer reaction contribute significantly to the entropy production throughout the length of the process unit. Unlike previous examples in the literature, the process unit considered in this work is not characterized by a “reaction mode” at the inlet followed by a “heat transfer mode”. Therefore, it does not follow a highway in state space, i.e. a band that is particularly dense with energy efficient solutions. By artificially increasing the spin-isomer conversion rate, the highway appears when the conversion rate becomes sufficiently high.

Coupled Thermoelectric Devices: Theory and Experiment

In this paper, we address theoretically and experimentally the optimization problem of the heat transfer occurring in two coupled thermoelectric devices. A simple experimental set up is used. The optimization parameters are the applied electric currents. When one thermoelectric is analysed, the temperature difference Δ T between the thermoelectric boundaries shows a parabolic profile with respect to the applied electric current. This behaviour agrees qualitatively with the corresponding experimental measurement. The global entropy generation shows a monotonous increase with the electric current. In the case of two coupled thermoelectric devices, elliptic isocontours for Δ T are obtained in applying an electric current through each of the thermoelectrics. The isocontours also fit well with measurements. Optimal figure of merit is found for a specific set of values of the applied electric currents. The entropy generation-thermal figure of merit relationship is studied. It is shown that, given a value of the thermal figure of merit, the device can be operated in a state of minimum entropy production.

Sustainable cities as organisms, a circular thermodynamics perspective

More than 50% of us now live in cities while world population growth and urbanization continues unabated. There is an urgent need for policies that can address both the challenges and opportunities for sustainable development in urban settings. Within the past two decades, the idea of cities as fractal structures resulting from social and economic processes led to a transition in thinking of cities as machines to cities as organisms. Independently, the concept of ‘sustainable cities’ had emerged earlier, with many successful implementations already underway. I propose to define a sustainable city as one able to feed and power itself as much as possible using land and energy efficiently with the least environmental pollution and impact on climate change, minimizing wastes and recycling and reconverting wastes into resources. Coherent fractal space-time structures are the key to a circular thermodynamics of organisms and sustainable systems that achieves the state of minimum entropy production and dissipation that Ilya Prigogine had proposed as a conjecture. In this article, I explain how the principles of circular thermodynamics can be applied to sustainable cities.

Equipartition of entropy production as an approximation to the state of minimum entropy production in diabatic distillation

We show that the theorem of equipartition of entropy production is important for the understanding of the state of minimum entropy production in diabatic distillation. The theorem is not valid in a strictly mathematical sense. We explain why, when and in what sense this theorem is a good approximation to the optimal state in diabatic distillation. In order to make these predictions, we use a hypothesis for the state of minimum entropy production of an optimally controlled system, which was formulated on the basis of results of entropy production minimisation in chemical reactors. The hypothesis says that the state of minimum entropy production is characterised by approximately constant local entropy production and thermodynamic forces, given that there is sufficient freedom in the system. We present numerical results which are in agreement with the predictions. The results show that a column with constant tray entropy production in the stripping section and in the rectifying section is a good approximation to the optimal column, except when the total heat transfer area is low. The agreement between the two columns becomes better and better as the total heat transfer area and the number of trays increase. The fact that the predictions and the numerical results agree very well gives support to the validity of the hypothesis.

Molecular theory of irreversibility

A generalization of the Gibbs entropy postulate is proposed based on the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy of equations as the nonequilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law. By using a generalization of the Liouville equation describing the evolution of the distribution vector, it is demonstrated that the entropy production is a non-negative quantity. Moreover, following the procedure of nonequilibrium thermodynamics a transport matrix is introduced and a microscopic expression for this is derived. This framework allows one to perform the thermodynamic analysis of nonequilibrium steady states with smooth phase-space distribution functions which, as proven here, constitute the states of minimum entropy production when one considers small departures from stationarity.

Nonlinear flux-force relations and equipartition theorems for the state of minimum entropy production

We explain that the path of minimum entropy production in a system with nonlinear flux-force relations, which do not depend explicitly on the state variables of the system, can be characterised by equipartition of driving forces and equipartition of entropy production. The explanation is based on an optimal control theory formulation of the optimisation problem. The finding helps explaining recent numerical observations on the second law optimal state of chemical reactions.

Numerical evidence for a “highway in state space” for reactors with minimum entropy production

We report on the nature of the state of minimum entropy production. Thousands of solutions for this state in chemical reactors controlled by the temperature of the coolant medium, show that the solutions fall more or less on what we have called a “highway in state space”. The solutions were found using optimal control theory for reactors that produce a fixed amount of product. A subset of the solutions, for the oxidation of sulphur dioxide, is presented. Each solution gives the most energy-efficient way of operating a reactor for the given boundary conditions. For reasonable process intensities, the highway is characterised by approximately constant entropy production and driving forces, but not necessarily by linear flux–force relations. Knowledge about the highway has implications for energy-efficient reactor design. The results give theoretical support to engineering practices.

The principle of minimum entropy production and snow structure

AbstractAn essential problem in snow science is to predict the changing form of ice grains within a snow layer. Present theories are based on the idea that form changes are driven by mass diffusion induced by temperature gradients within the snow cover. This leads to the well-established theory of isothermal- and temperature-gradient metamorphism. Although diffusion theory treats mass transfer, it does not treat the influence of this mass transfer on the form — the curvature radius of the grains and bonds — directly. Empirical relations, based on observations, are additionally required to predict flat or rounded surfaces. In the following, we postulate that metamorphism, the change of ice surface curvature and size, is a process of thermodynamic optimization in which entropy production is minimized. That is, there exists an optimal surface curvature of the ice grains for a given thermodynamic state at which entropy production is stationary. This state is defined by differences in ice and air temperature and vapor pressure across the interfacial boundary layer. The optimal form corresponds to the state of least wasted work, the state of minimum entropy production. We show that temperature gradients produce a thermal non-equilibrium between the ice and air such that, depending on the temperature, flat surfaces are required to mimimize entropy production. When the temperatures of the ice and air are equal, larger curvature radii are found at low temperatures than at high temperatures. Thus, what is known as isothermal metamorphism corresponds to minimum entropy production at equilibrium temperatures, and so-called temperature-gradient metamorphism corresponds to minimum entropy production at none-quilibrium temperatures. The theory is in good agreement with general observations of crystal form development in dry seasonal alpine snow.

Greenhouse gases and climatic states of minimum entropy production

The hypothesis of minimum entropy production is applied to a simple one‐dimensional energy balance model and is analyzed for different values of the radiative forcing due to greenhouse gases. The extremum principle is used to determine the planetary “conductivity” and to avoid the “diffusive” approximation, which is commonly assumed in this type of model. For present conditions the result at minimum radiative entropy production is similar to that obtained by applying the classical model. Other climatic scenarios show visible differences, with better behavior for the extremal case.

Deviations from minimum entropy production at steady states of reacting chemical systems arbitrarily close to equilibrium

Our analysis of reacting systems displaced from equilibrium by a matter flux across the boundaries has shown that the state of minimum entropy production differs from the steady state, even in the near equilibrium regime. When the displacement δ from equilibrium is small, the derivative of the dissipation at the steady state and the dissipation itself are both of order δ2. The state of least dissipation is displaced from the steady state by terms of order δ2 in the species concentrations. The theorem of minimum entropy production may be derived by first truncating the series expansions for the reaction rates, affinities and dissipation assuming that δ is small, and then differentiating to locate the minimum of the dissipation within the resulting idealized model. For chemical systems with an arbitrarily small but macroscopic displacement from equilibrium, this truncation procedure establishes that the dissipation is comparatively small in a neighborhood of the steady state; but it causes large relative errors in the values of the concentration derivatives and time derivatives of the dissipation within that neighborhood, because the operations of series truncation and differentiation do not commute near the steady state, when δ is small but nonzero. Near the steady state, the concentration derivative of the term of order δ3 in the dissipation is comparable to or larger than the derivative of the δ2 term.

Dissipation in steady states of chemical systems and deviations from minimum entropy production

According to the linear thermodynamic theory of irreversible processes, entropy production is minimized at the steady states of nonequilibrium systems. The principle of minimum entropy production is obtained if the dissipation is expanded in the deviation δ from equilibrium, truncated to lowest order (δ 2), and then differentiated. At this level of apprximation, the derivative of the dissipation is linear in δ and vanishes at the steady state. For a simple chemical reaction mechanism in a well-defined model system, we have derived the corrections to this approximation, by first evaluating the dissipation and its derivative exactly, and then expanding as a series in δ. To leading order in δ, the ratio of the derivative to the dissipation at the steady state is 1 2 (in dimensionless units), and this ratio does not decrease as equilibrium is approached. The steady state coincides with the state of minimum entropy production only at equilibrium. Given sufficient accuracy in measurements of species concentrations, the breakdown of the principle of minimum entropy production can be detected experimentally when the relative standard deviation in measurements of the dissipation is less than δ. Our example shows that the dissipation in a reaching chemical system may increase in time, in the later stages of relaxation toward a near-equilibrium steady state.

On the nature of the diffusion potential

The 'diffusion' potential concept is examined critically within the conditions imposed by Maxwell's equations and the Thermodynamics of Irreversible Processes. It is demonstrated that under the zero current condition during non-steady state diffusion the self-field associated with the diffusion potential gradient is due entirely to a diffusion-induced distribution of dipoles. The general failure to recognize that the charge associated with the field is due to dipoles has led to an improper use of Poisson's equation and to certain oft-remarked-upon paradoxes. The dipoles appear in systems where counter-current ion mobilities differ greatly. In seeking electroneutrality the ion trajectories become 'curved' in such a way as to generate an instantaneous dipole current and field. The magnitudes have nothing to do with the dielectric properties of the medium.It is demonstrated that the zero current condition represents a quasi-steady state of minimum entropy production. Such a state of stability can be sustained provided the process of relaxation of the field to the pure dipole state by real charge neutralization is fast compared with the relaxation process for the chemical potentials of the neutral salts in solution. The energetics of ionic solutions are usually such as to abet the neutrality condition during diffusion processes.

Thermodynamics of biological growth.

ACCORDING to some recent opinions, living organisms are nearer to open systems in the thermodynamic sense than to classical closed systems, although an organism plus its environment constitute a system which must be regarded as closed. While closed systems eventually reach equilibria characterized by minimum Gibbs free energy and maximum entropy, Prigogine1 and de Groot2 have predicted that open systems will approach steady states of minimum entropy production. On this basis, Prigogine and Wiame3 have inferred that living organisms grow towards a state of minimum metabolism per unit mass, but this view and other discussions4 on this theme are subject to two criticisms. On one hand, the theorem of minimum entropy production holds only for open systems very near to the steady state5 and, on the other, it has never been tested experimentally for living organisms so that conclusions derived from it are weakened to some extent.