Irreversible Thermodynamics Model

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Irreversible perturbation reactions, 14 617 Irreversible thermodynamics models, 21 638, 661... 

As with the finely-porous model, (Chapter 4.1.3), the mathematical representation of solvent and solute fluxes for the irreversible thermodynamic model is quite complex and beyond the scope of this work. However, it is recommended that readers consider references1 and8 for details on this transport model. 

A variety of RO membrane models exist that describe the transport properties of the skin layer. The solution-diffusion model( ) is widely accepted in desalination where the feed solution is relatively dilute on a mole-fraction basis. However, models based on irreversible thermodynamics usually describe membrane behavior more accurately where concentrated solutions are involved.( ) Since high concentrations will be encountered in ethanol enrichment, our present treatment adopts the irreversible thermodynamics model introduced by Kedem and Katchalsky.(7.)... 

The Irreversible Thermodynamics Model (Kedem and Katchalsky (1958)) is founded on coupled transport between solute and solvent and between the different driving forces. The entropy of the system increases and free energy is dissipated, where the free energy dissipation function may be written as a sum of solute and solvent fluxes multiplied by drivir forces. Lv is the hydrodynamic permeability of the membrane, AII v the osmotic pressure difference between membrane wall and permeate, Ls the solute permeability and cms the average solute concentration across the membrane. 

The Solution Diffusion Model assumes that solute and solvent dissolve in the membrane, which is imagined as a dense, non-porous layer. The membrane also has a layer of bound water at the surface, due to its low dielectric constant. The solute and solvent have different solubility and diffusion coeffieients in the membrane, and rejection of solute depends on its ability to diffuse through structured water inside the membrane (Staude (1992)). All solutes diffuse independendy, driven by their chemical potential across the membrane. It is the same as the irreversible thermodynamics model for the case where no coupling occurs. This model has lost credibility in the past due to neglected membrane imperfections, membrane-solute interactions, and solute-molecule interactions (no convection, no external forces, no coupling of flow) (Braghetta (1995)). 

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model (the fourth) is needed for some situations. The alternative Maxwell-Stefan model of mass transfer and diffusivity is explored in Section 15.7. The Maxwell-Stefan model has advantages for nonideal systems and multicomponent mass transfer but is more difficult to couple to the mass balances when designing separators. The fifth model of mass transfer, the irreversible thermodynamics model fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2QQQ Haase. 1990T is useful in regions where phases are unstable and can split into two phases, but it is beyond the scope of this introductory treatment. The... 

Irreversible thermodynamics model. This model is useful in regions where phases are unstable and can split into two phases fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2000 Haase. 19901. However, this model is beyond the scope of this introductory treatment. 

Although experimental results could be fitted well with irreversible rate models, ignoring thermodynamic facts could be disastrous. Although reversibility moderated the maximum temperature at runaway, it was not the most important qualitative result. In fact, the one dimensional (directional, or irreversible, correctly) model was not realistic at these conditions. For the prediction of incipient runaway and the AT ax permissible before runaway, the reversibility was obviously important. 

Irreversible thermodynamics has also been used sometimes to explain reverse osmosis [14,15]. If it can be assumed that the thermodynamic forces responsible for reverse osmosis are sufficiently small, then a linear relationship will exist between the forces and the fluxes in the system, with the coefficients of proportionality then referred to as the phenomenological coefficients. These coefficients are generally notoriously difficult to obtain, although some progress has been made recently using approaches such as cell models [15]. 

Another theoretical basis of the superheated liquid-film concept lies on the irreversible thermodynamics developed by Prigogine [43]. According to this theory, irreversible chemical processes would be described (Equation 13.17) by extending the equation of De Donder, provided that simultaneous reactions were coupled in a certain thermodynamic model, as follows ... 

Dependent on the model of membrane used, the systems of irreversible thermodynamics can be divided in two groups ... 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Multidisciplinary analytical and numerical models require development. These models should involve considerations of equilibrium and irreversible thermodynamics and kinetics of carbonate mineral-organic matter-water interactions within a sound hydrodynamic and basin evolution framework. 

Transport models fall into three basic classifications models based on solution/diffusion of solvents (nonporous transport models), models based on irreversible thermodynamics, and models based on porous membranes. Highlights of some of these models are discussed below. 

T. L. Hill and I. W. Plesner. Studies in irreversible thermodynamics II. A simple class of lattice models for open systems. J. Chem. Phys., 43 267-285, 1965. 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

A typical recent example of the application of the model is the irreversible thermodynamics of Cox (36). A recent paper of interest to the readers of this article is that of Thomsen (37) in which he attempts to establish the convention that microscopic reversibility is to mean that the matrix K is symmetric, and that detailed balancing is to mean that the matrix KD is symmetric. 

We have emphasized the proper modeling of thermodynamic nonideality both with regard to molecular diffusion and interphase mass transfer. The benefits of adopting the irreversible thermodynamic approach are particularly apparent here it would not be possible otherwise to explain the peculiar behavior of the Fick diffusivities. The practical implications of this behavior in the design of separation equipment operating close to the phase transition or critical point (e.g., crystallization, supercritical extraction, and zone refining) are yet to be explored. In any case, the theoretical tools are available to us. 

Kjelstrup and Bedeaux [81] speculate that the need in mechanical and chemical engineering for more accurate modeling tools to enable process equipment designs that waste less work wiU increase the use of irreversible thermodynamics in the near future. Better and more efficient use of energy resources is also a central future requirement. It may then no longer be sufficient to optimize the first law efficiency solely. The second law may have to be taken into account as well. Under future UN environmental protection conventions and protocols the process industry may be forced to report on their aimual entropy production. As a political tool, economical benefits could also be given to those industries that limit or reduce their entropy production. 

The model, based on Hnear, irreversible thermodynamics, constitutes a more general phenomenological approach, appHcable to systems with either class of membranes, multiple solutes, and driving forces involved. It includes component and overall mass balances, mass-transfer rates, local equilibrium relations, and electroneutrahty constraints. 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

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