Reduced phenomenological coefficients

Xi = generalised force 0j = flux of species i Llk = phenomenological coefficient E = electrical potential difference A P= pressure difference Le = electrical conductivity of one square cm of membrane n = number of components of a system ut = velocity of component i i3j j = friction coefficients 7, - = electrical transport number lt = reduced transport number or transference number z( = charge of ion i vt = partial volume of ion i per gmol vD — partial volume of the solvent per gmol... 

We may determine each phenomenological coefficient experimentally. The Onsager reciprocal relations reduce the number of coefficients to be determined. If we substitute Eq. (7.42) into Eq. (7.44), we find that the coefficients Liq and Ly obey the following relations ... 

Rate equations of product formation usually contain additive terms in the denominator if the pathway or network includes reverse steps. The number of phenomenological coefficients can then be reduced by one if numerator and denominator are divided by one of the terms. The result is a "one-plus" rate equation, with a "1" as the leading term in the denominator. (Exception This procedure is superfluous if all terms in the denominator consist only of coefficients, or of coefficients multiplied with the same concentration or concentrations, so that they can be combined to give a true power-law rate equation.)... 

Phenomenological coefficients in reduced rate equations are combinations of individual rate coefficients of steps. If a phenomenological coefficient consists entirely of a product, a ratio, or a ratio of products of individual rate coefficients, its activation energy is essentially temperature-independent. In contrast, if the phenomenological coefficient involves additive terms, its activation energy varies with temperature unless the terms have similar activation energies or one of them dominates. 

This reduces from four to three the number of independent phenomenological coefficients required to describe the intrinsic resistance of the porous medium. Inasmuch as c Vq = — V x q, the constitutive equations thus become... 

In this framework, the intensity of an entropy source is represented by a quadratic form of thermodynamic forces. The corresponding phenomenological coefficients form a matrix with remarkable properties. These properties, formulated as the Onsager reciprocity theorem, allow to reduce the number of independent quantities and to find relations between various physical effects. 

Now, the condition dL/dt 0 as required for the Liapunov-stability of the equilibrium is reduced to the condition that the matrix of the linear phenomenological coefficients is positive definite. This latter property, however, is a direct consequence of the second law of thermodynamics as we have shown in (3.82). With this conclusion we have reconfirmed our preliminary result of the preceding section in a formally precise way a sufficient condition for the stability of the equilibrium is a) a positive-definite capacitance matrix such that L 0 and b) the second law of thermodynamics such that dL/dt 0. Let us emphasize once more the significance of the equivalence between dL/dt < 0 and the second law in the form of (3.82). This equivalence, however, is valid only in the range of validity of the linear relations in (3.81). If the fluxes I were some nonlinear functions of the forces F as will be the case in situations far from the thermodynamic equilibrium, dL/dt 0 is no longer guaranteed by the second law and possibly may no longer be valid at all. 

The first term describes the dependence of the coercivity on the anisotropy field. In an ideal system, the phenomenological coefficient c would be one and is reduced to about 0.1 in a real system. The second term describes the thermal activation effects. denotes the anisotropy energy, is the molecular field and c is a phenomenological coefficient, which gives an account for the decrease of anisotropy and/or exchange interactions at defect position. The experimental data show that - has a quadratic-like behaviour which means... 

In (7) and (8) the independent variables have been decomposed into the irreducible trace ( 0 and v ), deviator (denoted by ) and skew (denoted by square brackets on subscripts) components. In the application below, each of the phenomenological coefficients is assumed to be proportional to the appropriate volume distribution function (t> . Thus = (l) etc., where are constants. This has the appealing property that the theory automatically reduces to micropolar or Navier Stokes when the concentration of the solid constituents goes to zero. As will be seen shortly, it has the nasty property of making even the simplest flow problem inherently nonlinear. 

Another key issue for the application of this theory is the determination of the phenomenological coefficients appearing in the constitutive equations. Of course, the ultimate determination must come from careful viscometric experiments. As these are apparently not yet available, we have had to resort to experience and separate theoretical analysis to provide estimates of these parameters for our calculations. For the viscous type coefficients, i.e., those appearing in (7) and (8), we have relied on prior experience with micropolar flows [7,9,14]. In particular, we have focused on those values which will produce plug-shaped profiles with appropriate boundary conditions. The values selected here reduce to the molecular value of the viscosity of water, under standard conditions, when the solid volume fraction goes to zero. As to the... 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

Experimentally, it is often found that the anodic and cathodic charge transfer coefficients are about 1/2. This is typically the case for outer-sphere electron transfer. Values between zero and one are found for several more complex reactions. We now consider whether this behavior is reasonable in the framework of the phenomenological model presented here. In an outer-sphere process, the oxidized and reduced species are outside the electrochemical double layer. The chemical potential of these species is then not influenced by the electrode potential, and the following is valid ... 

With the consideration of new forces and the asymmetry coefficients, the phenomenological equations given by Eqns (11.82)-( 11.84) reduce to... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

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