Modeling a non-equilibrium distillation stage using irreversible thermodynamics

We compare transport equations derived from non-equilibrium thermodynamics to a classical rate model developed over the last 20 years, in terms of their ability to calculate the heat and mass fluxes by modeling a segment of a packed distillation column. We show, using water and ethanol separation as an example, that there is a significant coupling between heat and mass transfer. Neglect of this transport coefficient leads to variations in the magnitude, even the sign of the calculated heat flux, while the mass flux is less affected.     

Phenomenological description of the transport of isotopes in electrolytic systems

The equations of irreversible thermodynamics are applied to the isothermal—isobaric transport of isotopes in multicomponent electrolytic systems. The resulting flux—force relations can be written entirely in terms of conventional transport coefficients: transference numbers, electrical conductivity, diffusion coefficients of electroneutral electrolytes and intra-diffusion coefficients. Some applications are discussed.     

Energetics of glucose metabolism: a phenomenological approach to metabolic network modeling

A new formalism to describe metabolic fluxes as well as membrane transport processes was developed. The new flux equations are comparable to other phenomenological laws. Michaelis-Menten like expressions, as well as flux equations of nonequilibrium thermodynamics, can be regarded as special cases of these new equations. For metabolic network modeling, variable conductances and driving forces are required to enable pathway control and to allow a rapid response to perturbations. When applied to oxidative phosphorylation, results of simulations show that whole oxidative phosphorylation cannot be described as a two-flux-system according to nonequilibrium thermodynamics, although all coupled reactions per se fulfill the equations of this theory. Simulations show that activation of ATP-coupled load reactions plus glucose oxidation is brought about by an increase of only two different conductances: a [Ca(2+)] dependent increase of cytosolic load conductances, and an increase of phosphofructokinase conductance by [AMP], which in turn becomes increased through [ADP] generation by those load reactions. In ventricular myocytes, this feedback mechanism is sufficient to increase cellular power output and O(2) consumption several fold, without any appreciable impairment of energetic parameters. Glucose oxidation proceeds near maximal power output, since transformed input and output conductances are nearly equal, yielding an efficiency of about 0.5. This conductance matching is fulfilled also by glucose oxidation of β-cells. But, as a price for the metabolic mechanism of glucose recognition, β-cells have only a limited capability to increase their power output.     

Non-equilibrium thermodynamic analysis of coupled heat and moisture transfer across a membrane

Non-equilibrium thermodynamics theory is used to analyze the transmembrane heat and moisture transfer process, which can be observed in a membrane-type total heat exchanger (THX). A theoretical model is developed to simulate the coupled heat and mass transfer across a membrane, total coupling equations and the expressions for the four characteristic parameters including the heat transfer coefficient, molar-driven heat transfer coefficient, thermal-driven mass transfer coefficient, and mass transfer coefficient are derived and provided, with the Onsager’s reciprocal relation being confirmed to verify the rationality of the model. Calculations are conducted to investigate the effects of the membrane property and air state on the coupling transport process. The results show that the four characteristic parameters directly affect the transmembrane heat and mass fluxes: the heat and mass transfer coefficients are both positive, meaning that the temperature difference has a positive contribution to the heat transfer and the humidity ratio difference has a positive contribution to the mass transfer. The molar-driven heat transfer and thermal-driven mass transfer coefficients are both negative, implying that the humidity ratio difference acts to reduce the heat transfer and the temperature difference works to diminish the mass transfer. The mass transfer affects the heat transfer by 1%–2% while the heat transfer influences the mass transfer by 7%–14%. The entropy generation caused by the temperature difference-induced heat transfer is much larger than that by the humidity difference-induced mass transfer.     

Consolidated theory of fluid thermodiffusion

We present the Onsager–Stefan–Maxwell thermodiffusion equations, which account for the Soret and Dufour effects in multicomponent fluids. Unlike transport laws derived from kinetic theory, this framework preserves the structure of the isothermal Stefan–Maxwell equations, separating the thermodynamic forces that drive diffusion from the force that drives heat flow. The Onsager–Stefan–Maxwell transport-coefficient matrix is symmetric, and the second law of thermodynamics imbues it with simple spectral characteristics. This new approach allows for heat to be considered as a pseudo-species and proves equivalent to both the intuitive extension of Fick's law and the generalized Stefan–Maxwell equations popularized by Bird, Stewart, and Lightfoot. A general inversion process facilitates the unique formulation of flux-explicit transport equations relative to any choice of convective reference velocity. Stefan–Maxwell diffusivities and thermal diffusion factors are tabulated for gaseous mixtures containing helium, argon, neon, krypton, and xenon. The framework is deployed to perform numerical simulations of steady three-dimensional thermodiffusion in a ternary gas.     

Transport of organic penetrants through films of cellulose esters in transient regimes: a non-equilibrium thermodynamics analysis

Transport equations are derived in the framework of the rational thermodynamics approach for membrane processes involving a dense (active) polymer layer. The influence of the deformation of the polymer matrix, formation of interacting species, changes in polymer configuration, etc., due to mixing of the penetrant and the polymer can be accounted for in the equations. Studies of permeation fluxes in the transient regime in which the membrane is suddenly put into contact with a solvent can give useful information on the behaviour of polymer-penetrant systems. Experimental results obtained for the transient permeation of alcohols and heptane through different cellulose ester membranes are reported and discussed in the framework of this approach.     

Convective Drying Analysis of a Single Wheat Kernel Based on an Irreversible Thermodynamics Model

The application of irreversible thermodynamics offers a formal treatment for drying analysis that allows the evaluation of intra-particle or intra-medium temperature and moisture profiles, and enthalpy, liquid, and vapor fluxes. However, researchers have claimed that its implementation is complex. This work presents a simple methodology for modeling, solving, and validating the drying equations, as applied to wheat kernels, and for obtaining the inherent and usually unavailable transport coefficients. To clarify and simplify the ensuing physical analysis, a spherical shape and isotropy were assumed. Additionally, solutions obtained with both Dirichlet and convective boundary conditions were analyzed and compared against experimental data. The thermal and hydro-stresses depend heavily on internal vapor and liquid fluxes and on the respective drying evaporation fronts, all of which were evaluated and compared.     

REVIEW OF REVERSE OSMOSIS MEMBRANES AND TRANSPORT MODELS

After a brief introduction to membrane processes in general, and the reverse osmosis process in particular, the structure and properties of membranes and membrane transport theory are described. The mechanism of salt rejection and transport properties of membranes are discussed in detail. Solubility, diffusivity, and permeability of membranes to solutes and solvents are reviewed critically and compared with each other. Special attention is given to two particular types of membranes, cellulose acetate (CA) and aromatic polyamide (AP) membranes, which are often used for water desalination.

The major portion of this article is devoted to the review and discussion of membrane transport theory with application to the reverse osmosis and ultrafiltralion processes. It is shown that the solvent flux can be represented reasonably well by linear models such as the solution-diffusion model (Lonsdale, et al., 1965). The contribution of pore flow to the solvent flux is small. The solute flux, however, is not linearly dependent on the driving forces and one has to solve the differential equation of transport within the membrane which results in models such as the Spiegler-Kedem (1966) or the finely-porous (Merten, 1966) models. When the wall Peclet number is small, Pe_w =uτδ/D_sw ?1, (D_sw = bD_e one can linearize the nonlinear models. This requirement is not satisfied in most practical cases. Furthermore, the pore flow has significant effect on the solute flux equation and thus it can not be neglected.

The ambiguities that exist in the literature concerning the types of fluxes are discussed. The fluxes used in models derived from irreversible thermodynamics are purely diffusive (concentration and pressure diffusion) and they do not contain any convective effects; whereas the experimentally observed fluxes are the total fluxes with respect to the membrane which consist of a diffusive flux and a convective flux. A new model, based on irreversible thermodynamics, is derived which includes a convective term.

A membrane model is especially useful when the transport coefficients which define the model are not functions of the driving forces, i.e., pressure and concentration gradients. The coefficients in the solution diffusion and sotution-diffusion-imperfection (Sherwood, et al., 1967) models are functions of both pressure and concentration, while the coefficients in the Kedem-Katchalsky (1958) model are relatively insensitive to pressure and concentration. The nonlinear model of Spiegler-Kedem (1966) further improves the Kedem-Katchalsky model.     

ON THE RELATIONSHIP BETWEEN THE EXACT AND LINEARISED SOLUTIONS OF THE MAXWELL-STEFAN EQUATIONS FOR THE MULTICOMPONENT FILM MODEL

The exact solution of the Maxwell-Stefan equations for multicomponent mass transfer based on a film model is compared with the solution of the linearised equations. It is first shown that the formulation of the mass transfer coefficients for the two solutions can be written in identical form as the product of a square matrix of composition dependent low flux mass transfer coefficients and a square matrix of correction factors which accounts for the presence of finite rates of transfer. In the exact solution the low flux mass transfer coefficients are evaluated at a film boundary composition while in the linearised theory they are evaluated at the mean film composition. The comparison clearly shows the influence of the rates of transfer in modifying these mass transfer coefficients. It is the constituent molar fluxes which are important in the exact solution whereas it is the convective contribution to these fluxes alone which appear in the linearised theory.

Despite these fundamental differences the molar transfer rates predicted by the two methods are usually in excellent agreement relative to the average absolute rate of transfer. It is argued that this will be the case in distillation examples where composition differences are small (the matrices of mass transfer coefficients are therefore almost equal) and the matrices of correction factors are closely approximated by the diagonal unit matrix. Further, in processes involving unidirectional mass transfer, such as condensation, the large differences between the respective matrices of mass transfer coefficients and correction factors frequently compensate for each other: As a result the errors introduced by linearising the equations are usually low, even in mixtures of high concentration and with high rates of mass transfer.     

Modelling the coupled transfer of mass and thermal energy in the vapour–liquid region of a nitrogen–oxygen mixture

Current non-equilibrium distillation models do not explicitly include the coupling between thermal and mass fluxes. We present a calculation model for the coupled transfer of mass and thermal energy in the vapour–liquid region of a binary mixture. The region is modelled as a vapour–liquid interface in between two homogeneous films. The entropy production in the vapour–liquid region can be calculated using both irreversible thermodynamics and the entropy balance. The film thickness ratio is found by requiring the entropy production calculated with the two methods to be equal, while keeping the vapour film thickness fixed. Using a nitrogen–oxygen mixture as example, we show that neglecting the coupling between thermal and mass fluxes can have a large impact on the magnitude and direction of the theoretical (net) fluxes. The size of the impact depends on the vapour film thickness, but it is significant for all thicknesses. By increasing the number of control volumes that is used to represent the liquid and vapour films, we also show that the fluxes depend highly on the resistivity profiles in the films. They depend slightly on the interface resistance. A sensitivity analysis of the transport properties shows that accurate values of the Maxwell–Stefan diffusion coefficients in both homogeneous phases and of the liquid phase heat of transfer are most important. Especially the measurable heat flux at the liquid boundary of the system is sensitive to neglect of coupling, to neglect of the interface resistance and to uncertainties in the transfer properties.     

ON THE RELATIONSHIP BETWEEN THE EXACT AND LINEARISED SOLUTIONS OF THE MAXWELL-STEFAN EQUATIONS FOR THE MULTICOMPONENT FILM MODEL

The exact solution of the Maxwell-Stefan equations for multicomponent mass transfer based on a film model is compared with the solution of the linearised equations. It is first shown that the formulation of the mass transfer coefficients for the two solutions can be written in identical form as the product of a square matrix of composition dependent low flux mass transfer coefficients and a square matrix of correction factors which accounts for the presence of finite rates of transfer. In the exact solution the low flux mass transfer coefficients are evaluated at a film boundary composition while in the linearised theory they are evaluated at the mean film composition. The comparison clearly shows the influence of the rates of transfer in modifying these mass transfer coefficients. It is the constituent molar fluxes which are important in the exact solution whereas it is the convective contribution to these fluxes alone which appear in the linearised theory.

Despite these fundamental differences the molar transfer rates predicted by the two methods are usually in excellent agreement relative to the average absolute rate of transfer. It is argued that this will be the case in distillation examples where composition differences are small (the matrices of mass transfer coefficients are therefore almost equal) and the matrices of correction factors are closely approximated by the diagonal unit matrix. Further, in processes involving unidirectional mass transfer, such as condensation, the large differences between the respective matrices of mass transfer coefficients and correction factors frequently compensate for each other: As a result the errors introduced by linearising the equations are usually low, even in mixtures of high concentration and with high rates of mass transfer.     

Nomenclature for transport phenomena in electrolytic systems

The document deals with the transport phenomena of importance to electrochemists and electrochemical engineers. In the absence of a report on mass transport at large, the first section presents general definitions of mass flux, of flux density and of phenomonological coefficients, as appearing in the relationships of irreversible thermodynamics, which express the proportionality between the driving forces and the fluxes. the main part is concerned with electrolytic systems, in particular with ideal dilute solutions. Mass transport by diffusion, by convection and by migration of ions under the influence of an electric field is considered. The diffusion coefficient of a species is distinguished from that of an electrolyte. A section is devoted to the transport of charges, including transport numbers, the flow of current through the solution and through the electrode, current efficiency, and current distribution. The report also discusses the precise meaning of concepts related to mass transport, such as the Nernst diffusion layer, mass transport control, interfacial concentrations, concentration overpotential, mass transport coefficients and diffusion potentials. However, the diffusion in solid electrolytes and surface diffusion are not treated.Recommendations for symbols and definitions are given. In many cases, it is endeavoured to clarify by a brief discussion the concept itself.     

Drying of Wheat Based on the Non-Equilibrium Thermodynamics: A Numerical Study

A new mathematical model to describe simultaneous heat and mass (liquid and vapor) transfer and shrinkage during drying of capillary-porous bodies with particular reference to prolate spheroid solid is presented. As an application, the methodology was used to predict drying of soft red winter wheat (Arthur). The mathematical model was based on the nonequilibrium thermodynamics considering variable transport coefficients and convective boundary conditions at the surface of the solid. All the partial differential equations presented in the model have been written in prolate spheroidal coordinates and solved numerically by a finite-volume method using implicit fully formulation. Results of the drying and heating kinetics and moisture content and temperature distributions in a wheat kernel during drying process are presented and analyzed. The methodology allows verification of the heat, liquid, and vapor fluxes, taking into account the thermal and hydrical gradients inside the grain.     

Ambiguity and non-uniqueness in nonequilibrium thermodynamics

The ambiguity and non-uniqueness of splitting fluxes and forces from the entropy generation equation raise confusion in nonequilibrium thermodynamics and misunderstanding of the Onsager reciprocal relationships. However, they provide an opportunity to select different sets of fluxes and forces that represent a given nonequilibrium process. By symmetrization of the phenomenological coefficient matrix, one can always find a proper set of fluxes and forces. This paper shows how the implementation of the transformation theory can produce several different sets of fluxes and forces through many engineering examples such as ideal gas permeation through a membrane, reverse osmosis, nanofiltration, ultrafiltration, and simultaneous heat and mass transfer. Also, guidance is presented on how to study nonequilibrium thermodynamics for a given irreversible process together with a short summary of the principles of nonequilibrium thermodynamics. These contain the entropy generation equation, linear relations of fluxes and all generalized forces including the Curie theorem, and the Onsager reciprocal relationships.     

Learning analytics system to aid students in engineering thermodynamics: Impact of pre-requisite course attainment

Engineering thermodynamics is the core course of many majors, especially mechanical engineering and chemical engineering. Two groups of students from different majors and with different coefficients of difficulty of engineering thermodynamics examinations were selected for investigation. The students’ achievements for the three courses of mathematics, physics, and engineering thermodynamics were analysed, and the relationships between them were concluded. Investigation shows that college-level physics, especially physics 1, plays an important role in improving the study of engineering thermodynamics, and students with a poor physics knowledge foundation have difficulty obtaining scores above the average level in engineering thermodynamics. A strong advanced mathematics foundation is also important to get a good score in engineering thermodynamics. The relationships between the prerequisite courses investigated in this study and engineering thermodynamics are more evident when the engineering thermodynamics examination is difficult and are weak when the engineering thermodynamics examination is easy. Finally, based on the findings, a student learning advising system is proposed, and a feasible implementation method is presented.     

A New Theoretical Approach To Electroosmotic Dewatering (Eod) Based On Non-Equilibrium Thermodynamics☆

ABSTRACT

A novel theoretical approach to electroosmotic dewatering (EOD), with or without a pressure gradient, of clays, sludges and other colloidal suspensions is proposed. The treatment is based on nan-equilibrium thermodynamics as developed in the work of Overbeek, De Groot and others. The interpretation of electrokinetic phenomena in terms of the cancepts of irreversible thermodynamics when combined with Onsager's relations, it has been shown by Overbeek, provides a complete framework for understanding all electrokinetic phenomena.

We have applied this approach here to the electroosmotic dewatering. both in the presence and absence of applied hydrostatic pressure.

The approach provides much clarification on the nature and significance of currents and fluxes observed during EOD: these are composed of three components, during combined pressure electroosmotic dewatering: (i) electrochemicavelectrical current; (ii) hydrodynamic flux: (iii) electroosmotic current.

We have also shown the manner in which the proposed new approach to EOD based on irreversible thermodynamics can be connected to the conventional approach based on the Helmholzu-Smoluchowski equation.