Fluxes and Forces

While there are many possible types of flux, a single common principle underpins all transport phenomena. This is the idea that a driving force must be present to cause the transport process to occur. If there is no driving force, there is no reason to move This applies equally well to the transport of matter as it does to the transport of heat or charge. The governing equation for transport can be generalized (in one dimension) as 

As an example of the application of Equation 4.7, consider the coupling between electrical conduction and heat conduction, which is at the heart of the well-known thermoelectric effect. In a system subjected to both a temperature gradient (dT/dx) and a voltage gradient dV/dx), the resulting fluxes of heat Jq) and charge (7 ) are given as  

3 COMMON TRANSPORT MODES (FORCE/FLUX PAIRS) 

In many situations only one driving force is present or significant (e.g., when considering heat transport in many systems, there is often only a temperature gradient present, while pressure, voltage, and chemical potential gradients are not present). 

You are likely already familiar with many of the simple direct force/flux pair relationships that are used to describe mass, charge, and heat transport—they include Pick s first law (diffusion). Ohm s law (electrical conduction), Fourier s law (heat conduction), and Poiseuille s law (convection). These transport processes are summarized in Table 4.1 using molar flux quantities. As this table demonstrates. Pick s first law of diffusion is really nothing more than a simplification of Equation 4.7 for 

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Wherein the definition of the thennodynamic fluxes and forces of (A3.2,131 and (A3.2.141 have been used. Onsager defined [5] the analogue of the Rayleigh dissipation fiinction by... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

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 ... 

We can say, following these experiments, that the structural permeability loses its physical meaning. In fact, such nonlinearities between fluxes and forces and permeability variations establish the very existence of a chemical effect, leading us to the concept of the dynamical membrane permeability. 

The remaining fluxes and forces are independent and thus the Onsager relations Lki = 4 hold. The number of independent transport coefficients is l/2- -(n — 1), With the help of the above conditions, it is possible to verify the symmetry of both matrices L and L [M. Martin, et at. (1988)]... 

Note that the fluxes and forces are written as scalars, consistent with the assumption that the material is isotropic. Otherwise, terms like Jq = (djQ/dFQ)FQ must be written as rank-two tensors multiplying vectors, and the equations that result can be written as linear relations (see Section 4.5 for further discussion). 

The fluxes and forces must so be chosen that the entropy production per unit pf time can be written as ... 

The forces Fk involve gradients of intensive properties (temperature, electrochemical potential). The Ljk are called phenomenological coefficients and the fundamental theorem of the thermodynamics of irreversible processes, due originally to Onsager (1931a, b), is that when the fluxes and forces are chosen to satisfy the equation... 

In the macroscopic theory, a state of a physical (chemical) system is described by a set of thermodynamic parameters ,-,/= 1,..., n. These parameters and their derivatives determine the values of the thermodynamic fluxes/,- and forces Xj, i= 1,.. 

The general equation that relates forces and fluxes is given by Onsager s theory. For a single force X, it establishes a linear dependence between fluxes and forces that can be written as... 

As expressed by equation (3) the induced streaming potentials counteract the hydraulic driven water flow. From the fluxes and forces in table 2 the flow... 

Table 3. Flow parameters derived from the fluxes and forces in table 2...

Expression for production of entropy (8.18) can be now compared with the general results of non-equilibrium thermodynamics, which are known for both non-stationary and stationary cases. It is obvious, that last term in the right-hand side of relation (8.18) corresponds to a non-stationary case and includes the equation of change of internal variables that is relaxation equation. The first two terms in formula (8.18) correspond to a stationary case and should be considered as the products of thermodynamic fluxes and thermodynamic forces (it is possible with any multipliers). When the internal variables are absent, we should write a relation between the fluxes and forces in the form... 

Familiar examples of the relation between generalized fluxes and forces are Fick s first law of diffusion, Fourier s law of heat transfer, Ohm s law of electricity conduction, and Newton s law of momentum transfer in a viscous flow. 

In other words, under realistic conditions (/ 0), entropy is produced, with the positive entropy production being given by fluxes and forces related to the process j. Equation (5) assumes that all these processes are in series, which is mostly correct. The most obvious contributions are transport resistances, due to the finite conductivities of Li+ and e in electrolyte and electrodes. For usual geometries, these resistances are constant to a good approximation, while for resistances stemming from impeded charge transfer and phase boundaries, the dependence on current can be severe. 

This equation is independent of the type of phenomenological relations between fluxes and forces. In contrast, linear phenomenological equations and the Onsager reciprocal relations yield... 

These reciprocity relations are derived in Section 6.5. It should be clearly recognized that Eq. (6.3.6) holds only if the phenomenological relations involve conjugate fluxes and forces. If nonconjugate quantities are used, and Lji are... 

Steady—state conditions are characterized by the requirement that the fluxes and forces characterizing irreversible phenomena in a system be independent of time. Some of these forces and/or fluxes may vanish in the extreme case where all JL and X are zero the system is necessarily in an equilibrium state. 

According to Section 6.2 an appropriate choice of fluxes and forces for this problem is found by examining the dissipation function - - T 1JS VT - T 1Jn Vfn. Now, as the discussion after Eq. (6.2.12) shows, the specific quantities fk, Jtk, Nk may be converted to molecular quantities hence we replace the last term by - T-1Jn Vfn, where now Jn is a particle flux vector and is the electrochemical potential per... 

For situations not far removed from equilibrium (what this implies will be fully documented later), one postulates the usual linear relations between fluxes and forces. In the present context (A, should not be confused with A)... 

Far from thermodynamic equiHbrium we find nonfinear interdependence of thermodynamic fluxes and forces. In this case, the Onsager reciprocal relations are generally not satisfied, and the formafism developed in Chapter 2 is not fuUy applicable for analysis of the state of open systems. Analysis of systems that are far from thermodynamic equilibrium is the subject of nonlinear nonequilibrium thermodynamics. 

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

A typical problem in thermodynamics of systems that are far from their equilibrium is the analysis of the stability of stationary states of the system. Thermodynamic criteria of the stability of stationary states are found the same way as for systems that are far from and close to thermodynamic equilibrium (see Section 2.4) by analyzing signs of thermodynamic fluxes and forces arising upon infinitesimal deviation of the system from the inspected stationary state. If the system is in the stable stationary state, then any infinitesimal deviation from this state must induce the forces that push it to return to the initial position. 

A quantitative description of interdependent fluxes and forces is given by irreversible thermodynamics, a subject that treats nonequilibrium situations such as those actually occurring under biological conditions. (The concepts of nonequilibrium and irreversibility are related, because a system in a nonequilibrium situation left isolated from external influences will spontaneously and irreversibly move toward equilibrium.) In this brief introduction to irreversible thermodynamics, we will emphasize certain underlying principles and then introduce the reflection coefficient. To simplify the analysis, we will restrict our attention to constant temperature (isothermal) conditions, which approximate many biological situations in which fluxes of water and solutes are considered. 

With the equations of secs. 4.6a-c the problem of computing f from mobilities at given Ka is in principle soluble. But not in practice The mathematics are very complicated and require a number of finesses, whereas simplifications is dangerous because the various fluxes and forces are coupled, so that approximating only one of these may offset the balance and misrepresent characteristic features. Moreover, the boundary conditions are in part determined by the composition of, and the mobilities of ions in the various parts of the double layer, for which model assumptions must be made. 

The complete set of equations, relating fluxes and forces can be written in a matrix form and is called the set of phenomenological equations. It was shown by Onsager [1] that the matrix of the phenomenological proportionality constants is symmetrical ... 

Steady state conditions obtain when the fluxes and forces giving rise to irreversible phenomena in a system remain time-invariant, whereas the properties of the surroundings change. We now render this idea more precise by introducing Prigogine s Theorem Let irreversible processes take place through imposition of n forces X, X2,. ,X that result in n fluxes J, Ji,---, Jn- Let the first k forces remain fixed at values Xj, X ,..., then it is claimed that the rate of entropy production 0 is minimized when the fluxes Jk+i Jk 2, Jn ah vanish. We first prove the theorem and then discuss its relevance to steady state conditions. As before, we set 9 = Jj Xj, to construct the phenomenological... 

Entropy production is important for a proper definition of fluxes and forces in non-equilibrium systems that are relevant to the industry. A higher accuracy in the fluxes can be obtained using NET. 

The integral consists of contributions of flux-force pairs and then has the habitual form of the entropy production in non-equilibrium thermodynamics. Also assuming linear dependence between fluxes and forces, one establishes the relations... 

In Onsagerian thermodynamics, the constitutive (kinetic) equations between fluxes and forces are linear... 

The entropy production term has been written as a sum of products of fluxes and forces. However, there are only q—1 independent mass fluxes js, and the Gibbs-Duhem equation (2.451) tells us that there are only q — 1 independent forces as well [5]. 

Onsager showed in 1931 that with proper choice of fluxes and forces, = L. These relations sure now called the Onsager reciprocity relations. 

One of the fundamental tasks required to achieve the ultimate goal of hydrogeology is to understand the controls on energy distribution and transformation within an aquifer system. If this is accepted, it then becomes the hydrologists role to bring together into one concept the fluxes and forces of the chemical reactions, of the hydrodynamic flow paths, and of heat. This idea was clearly articulated and developed by G. B. Maxey (7, p. 145), who stated in part ... 

The equilibrium of the surface, the state without any transport processes, is characterised in the framework of irreversible thermodynamics by the complete absence of fluxes and forces acting at the interface... 

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