Equation phenomenological

We relate the dissipation function to the rate of local entropy production using Eqs. (3.151)—(3.153) 

If the dissipation function identifies the independent forces and flows, then we have 

As shown by Prigogine, for diffusion in mechanical equilibrium, any other average velocity may replace the center-of-mass velocity, and the dissipation function does not change. When diffusion flows are considered relative to various velocities, the thermodynamic forces remain the same and only the values of the phenomenological coefficients change. 

The formulation of linear nonequilibrium thermodynamics is based on the combination of the first and second laws of thermodynamics with the balance equations including the entropy balance. These equations allow additional effects and processes to be taken into account. The linear nonequilibrium thermodynamics approach is widely recognized as a useful phenomenological theory that describes the coupled transport without the need for the examination of the detailed coupling mechanisms of complex processes. 

In nonequilibrium systems, spontaneous decaying phenomena toward equilibrium take place. When systems are in the vicinity of global equilibrium, linear relations exist between flows Jt and thermodynamic driving forces A). 

Placing a solid under an external field (chemical, electrical, thermal, etc.) results in the appearance ofthe corresponding flows of mass, electric charge, or heat. In linear non-equilibrium thermodynamics [37], these flows can be expressed as 

Note also that the diagonal coefficients (Lu, L22 Lit) are always positive, in contrast to the cross-coefficients Ljk- For example, the heat conductivity or electrical conductivity coefficients have always a positive sign, whereas no sign maybe compulsorily ascribed to the thermodiffusion or thermoelectrical coefficients without consideration of a particular system. 

Another important relationship between the kinetic coefficients is the so-called principle of symmetry , as formulated by P. Curie and introduced to nonlinear thermodynamics by Kondepudi and Prigogine [37]. As applied to thermodynamics, this postulates that a scalar quantity could not evoke a vector effect. For example, a scalar thermodynamic force - chemical affinity (driving the process of chemical reaction) that has very high isotropy symmetry - could not cause heat flow, which has a particular direction and is therefore anisotropic. Taking into account the reciprocal relationships, this can be formulated as 

These relationships allow the description of key processes associated with the mass and charge transfer in electrochemical systems, particularly in ionic crystals. 

The methods of irreversible thermodynamics are useful in providing a quantitative approach to the phenomenon of electro-osmotic dewatering and its connection to other electrokinetic effects. The main ideas were developed by Overbeeki and reviewed by DeGroot these ideas were applied by many workers to a number of problems, following the earlier papers of Overbeek and co-workers - on the treatment of electrokinetic phenomena in terms of irreversible thermodynamics. Recently we have shown that this approach can also be applied to EOD, as follows. 

Consider the clay or sludge with the trapped water (actually an electrolyte since it contains dissolved ions) as a sort of porous diaphragm (Fig. 1). When an electric field or a pressure, or both, are applied across this diaphragm, dewatering occurs. We examine here the case of simultaneous application of pressure and the electrical field. 

Nonequilibrium treatment of EOD under these conditions yields the following rate equations for the simultaneous transport of matter (i.e., water) and electricity (i.e., current), assuming that the diaphragm is uniform  

Similarly, a plot of (I) against AV can give whereas a plot of ( Av=o against AP should give L 2- Thus all four coefficients namely, L22 L21, 111 can be determined. 

The coefficient L21 can be interpreted in the usual way in terms of the zeta potential as  

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The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Onsager postulates [4, 5] the phenomenological equations for irreversible processes given by... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The above phenomenological equations are assumed to hold in our system as well (after appropriate averaging). Below we derive formulas for P[Aq B, t), which start from a microscopic model and therefore makes it possible to compare the same quantity with the above phenomenological equa tioii. We also note that the formulas below are, in principle, exact. Therefore tests of the existence of a rate constant and the validity of the above model can be made. We rewrite the state conditional probability with the help of a step function - Hb(X). Hb X) is zero when X is in A and is one when X is ill B. 

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

It is necessary to substitute a phenomenological equation for which in this case involves only spin variables. 

Kedem-Katchalsky equations Phenomenological equations for combined convection and diffusion, derived from nonequilibrium thermodynamics. See Eqs. (19) and (20). 

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

The phenomenological equations proposed by Felix Bloch in 19462 have had a profound effect on the development of magnetic resonance, both ESR and NMR, on the ways in which the experiments are described (particularly in NMR), and on the analysis of line widths and saturation behavior. Here we will describe the phenomenological model, derive the Bloch equations and solve them for steady-state conditions. We will also show how the Bloch equations can be extended to treat inter- and intramolecular exchange phenomena and give examples of applications. 

All these theories provide the basis for using, as first approximation, the simple phenomenological equations to describe the ET rate constant k in D-B-A systems as k = k0 z d and the current flow 7 as 7 = / e /ld in molecular junctions, where d is the length of the molecule, and [1 is a decay factor. Although the decay... 

The linearity of drug release was assessed by fitting the release data to the phenomenological equation [10] ... 

In the literature, there are many transport theories describing both salt and water movement across a reverse osmosis membrane. Many theories require specific models but only a few deal with phenomenological equations. Here a brief summary of various theories will be presented showing the relationships between the salt rejection and the volume flux. 

A more rigorous way to generalize Pick s law is to use phenomenological equations based on linear irreversible thermodynamics. In this treatment of an N-component system, the diffusive flux of component i is (De Groot and Mazur,... 

Figure 1.15 Relative complex viscosity ( f/ / i/0l) versus calculated conversion for polymerization at 130, 140, and 160 C. Phenomenological equation to fit the data prior to gelation is also shown...

The rheokinetics of polycaprolactam polymerizing in the monomer shows that below 50 percent conversion, the relative complex viscosity versus conversion of the nylon 6 homopolymerization is defined by the phenomenological equation ri / t]Q = exp(19.6 X), where // is the complex viscosity of nylon 6 anionically polymerizing in its monomer, 0 is the viscosity of caprolactam monomer, and X is fractional conversion. 

In order to quantify diffiisional effects on curing reactions, kinetic models are proposed in the literature [7,54,88,95,99,127-133]. Special techniques, such as dielectric permittivity, dielectric loss factor, ionic conductivity, and dipole relaxation time, are employed because spectroscopic techniques (e.g., FT i.r. or n.m.r.) are ineffective because of the insolubility of the reaction mixture at high conversions. A simple model, Equation 2.23, is presented by Chem and Poehlein [3], where a diffiisional factor,//, is introduced in the phenomenological equation, Equation 2.1. 

II. The Strain Energy Density Function and the Phenomenologic Equations for Elasticity... 

What has been done so far is to take experimental laws and express them in the form of phenomenological equations, i.e., Eqs. (6.300) and (6.301). Just as the phenomenological equations describing the equilibrium properties of material systems constitute the subject matter of equilibrium thermodynamics, the above phenomenological equations describing the flow properties fall within the purview of nonequilibrium thermodynamics. In this latter subject, the Onsager reciprocity relation occupies a fundamental place (see Section 4.5.7). 

The phenomenological equation that relates the flux of material across the boundary to the concentration gradient at that location is given by Fick s first law. 

This is the macroscopic or phenomenological equation for the macroscopic observable Q. 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

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