Force-flux relationship thermodynamics

Some of the elements of thermodynamics of irreversible processes were described in Sections 2.1 and 2.3. Consider the system represented by n fluxes of thermodynamic quantities and n driving forces it follows from Eqs (2.1.3) and (2.1.4) that n(n +1) independent experiments are needed for determination of all phenomenological coefficients (e.g. by gradual elimination of all the driving forces except one, by gradual elimination of all the fluxes except one, etc.). Suitable selection of the driving forces restricted by relationship (2.3.4) leads to considerable simplification in the determination of the phenomenological coefficients and thus to a complete description of the transport process. 

THE RELATIONSHIP BETWEEN THE VALUES OF FLUX AND THE THERMODYNAMIC FORCE CLOSE TO THERMODYNAMIC EQUILIBRIUM... 

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]. 

The power (work by the system per unit time) is thus W = —Fx = —JiXiT. The work is performed under the influence of a heat flux Q leaving the hot reservoir at temperature Ti. The cold reservoir is at temperature T2 (where T > T2). The corresponding thermodynamic force is X2 = I/T2 — 1/Ti, and the flux is J2 = Q. The temperature difference Ti —T2 = AT is assumed to be small compared to T2 T kT, so one can also write X2 = AT/T. Linear irreversible thermodynamics is based on the assumption of local equilibrium with the following linear relationship between the fluxes and forces ... 

For a given reaction, the expression used to model the flux must be constrained based on the apparent equilibrium constant and overall transformed thermodynamic driving force. Specifically it is required that the flux goes to zero when the reaction reaches equilibrium and that the forward and reverse fluxes satisfy the relationship AG = —RT n(J+/J ) introduced in Section 3.1.2. 

D. A. Beard and H. Qian. Relationship between thermodynamic driving force and one-way fluxes in reversible processes. PLoS One, 2 el44, 2007. 

It is important to emphasize that thermodynamic force Xq is a vector, whereas Xq is its Cartesian component corresponding to the Cartesian coordinate i of heat flux Jq. The centuries old practice states the well known relationships between heat fluxes and temperature gradients, which are expressed by the Fourier law of heat conduction... 

Themiodynamic fluxes J are generally functions of thermodynamic forces X that induce the fluxes. Near the thermodynamic equihbrium, both the thermodynamic driving forces and fluxes of the processes are rather small. In such situations, the values of thermodynamic forces X and the conju gate fluxes J are in a simple linear relationship... 

The validity of linear relationships of type (2.1) is supported, for exam pie, by Ohm s law where the magnitude of current I—that is, the electric ity carrier flux Je—is proportional to the thermodynamic driving force that is the difference (gradient) of electric potentials AU between the part of the electric circuit with the electrical resistance R ... 

This relationship shows that if the flux of an irreversible process i is affected by thermodynamic force Xj of another irreversible process j through the mediation of coefficient Ly, then the flux of process j is also influenced by thermodynamic force Xj through the mediation of the same coefficient Lj = Ly In the case of interacting chemical processes, this statement reflects the principles of detailed (intimate) equihbrium and mass balance of the reactants that underlie the concepts of chemical kinetics. 

In accordance with the Onsager relationships, each of the fluxes under consideration is conjugate with both thermodynamic forces Xj = Ap and X2 = An. Then... 

Consider the reason for the appearance of the thermomechanical effect and its expected value. Let us say that two vessels, 1 and 2, are filled with some identical fluid (hquid or gas) and connected by a capillary, the fluids being held at preset constant temperatures T and T + dT. Let Jq desig nate the heat flux that passes through the capillary between the vessels, while Jg designates a potential fluid flux that diffuses through the same cap iUary (Figure 2.3). In accordance to the preceding deduced relationships (also see Section 1.5), the thermodynamic forces that initiate the fluxes are determined by the formula... 

Table 3-2. Number of coefficients needed to describe relationships between forces (represented by differences in chemical potential) and fluxes in irreversible thermodynamics for systems with one, two, or three components.

The relationship between ionic conductivity and Onsager s theory can now be presented in terms of the electrochemical potential. By expressing the force leading to the transport of ions in terms of the gradient of jr,-, one finds important relationships between the diffusion coefficients of the ions, and the molar conductivity and mobility. Furthermore, when the force has the correct Newtonian units, one is also in a position to calculate the rate of entropy production. On the basis of the thermodynamics of irreversible processes, the relationship between the flux of ion i and the force Vp,- is... 

When there are several different fluxes and forces brought into play in a system, then the thermodynamics of linear irreversible processes postulates that there is a matrix relationship between the different fluxes and forces (with a symmetrical matrix ONSAGER relations). The non-zero non-diagonal terms in the matrix signify that a coupling of the phenomena is taking place. 

From the angle of predictability, formulation of quantitative relationship is the primary task. At this stage, formalism developed in non-equilibrium. Thermodynamics can serve as a good guide. If cause (forces) and (fluxes) effects can be identified, linear relations can be postulated in the following form for two coupled processes ... 

Using linear relationships between the fluxes and forces in accordance with the concept of irreversible thermodynamics and assuming isothermal conditions the forces can be described as the gradient of the chemical potential, i.e. 

As we have emphasized in the preceding section, the stability of equilibrium states crucially depends on the validity of dL/dt 0 which is a direct consequence of the second law of thermodynamics only within the range of the linear relationships between the fluxes I and the forces F. Since in general this linearity will not be valid in the vicinity of steady states arbitrarily far from equilibrium, we cannot transfer the above stability proof to such states. 

As mentioned before, nonequilibrium thermodynamics could be used to study the entropy generated by an irreversible process (Prigogine, 1945, 1947). The concept ofhnear nonequilibrium thermodynamics is that when the system is close to equilibrium, the hnear relationship can be obtained between the flux and the driving force (Demirel and Sandler, 2004 Lu et al, 2011). Based on our previous linear nonequihbrium thermodynamic studies on the dissolution and crystallization kinetics of potassium inorganic compounds (Ji et al, 2010 Liu et al, 2009 Lu et al, 2011), the nonequihbrium thermodynamic model of CO2 absorption and desorption kinetics by ILs could be studied. Figure 17 shows the schematic diagram of CO2 absorption kinetic process by ILs. In our work, the surface reaction mass transport rate and diffusion mass transport rate were described using the Hnear nonequihbrium thermodynamic theory. 

If the steady state concentrations of the components are shifted, but not too far from their equilibrium values, the interconnection between the fluxes and chemical forces (chemical affinities, in our case) should satisfy the well-known linear relationships that are usually postulated in the linear thermodynamics of irreversible processes [15-18]. We do not consider here the phenomenological equations of nonequilibrium thermodynamics. For details the reader can refer to numerous excellent monographs and review articles devoted to the applications of nonequilibrium thermodynamics in the description of chemical reactions and biological processes (see, for instance, [22-30]). In many cases, the conventional phenomenological approaches of linear and nonlinear nonequilibrium thermodynamics appear to be useful tools for the... 

WesterhofT and Chen [77] were able to demonstrate that the rate of ATP synthesis (output flux) is not a unique function of the average concentration of the intermediate when the latter occurs in small number. As the result of numerical simulation, under certain conditions the system displays the behavior (positive flux Jp generated by the negative driving force AG, Fig. 3.18) that contradicts the Second Law of Thermodynamics. This is the case when the volume of the domain in which the intermediate appears, F, is so small that an average number of intermediate iV < 1. Of course, for the macroscopic system the flux-force relationship follows the Second Law. 

Haraux, 1981 de Kouchkovsky et al., 1982) or in localization by variable PSIl/PSI contribution to the proton transport (Haraux et al., 1983). In the present work, the relationships between steady-state thermodynamic fluxes and forces were varied by the membrane H" " conductivity. The results support a microchimiosmotic view where the AyH across the coupling factors, different from the bulk one, does not necessarily equalize that locally generated by the H" " redox carriers, at variance with Van Dam et al. 

These models consider either the thermodynamic or mechanical non-equilibrium between the phases. The number of conservation equations in this case are either four or five. One of the most popular models which considers the mechanical non-equilibrium is the drift flux model. If thermal non-equilibrium between the phases is considered, constitutive laws for interfacial area and evaporation/condensation at the interface must be included. In this case, the number of conservation equations is five, and if thermodynamic equilibrium is assumed the number of equations can be four. Well-assessed models for drift velocity and distribution parameter depending on the flow regimes are required for this model in addition to the heat transfer and pressure drop relationships. The main advantage of the drift flux model is that it simplifies the numerical computation of the momentum equation in comparison to the multi-fluid models. Computer codes based on the four or five equation models are still used for safety and accident analyses in many countries. These models are also found to be useful in the analysis of the stability behaviour of BWRs belonging to both forced and natural circulation type. 

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