Equation phenomenological cross

Values are computed from phenomenological cross-sections using Equation 5. 

Helfferich [2,3,30] states that in addition to the mutual interference of substances i and j, characterized by the phenomenological cross coefficients of the type L,j, one should take into account the presence of a coion in the ion exchanger as well. As a result, the simplified solution is inappropriate, even to the problem of ordinary IE. By use of only one diffusion mass-transfer equation, as in this case, account for the presence of co-ion has been neglected. It is, as a consequence, necessary to consider the Nemst-Planck relation for the co-ion also. 

Straight lines are obtained in conformity with the above equation. The cross-phenomenological coefficient Lj2 values estimated from the slopes are also given in the Table 3.1. 

For inter diffusion between same-valence ions (ionic exchange) in an aqueous solution, or a melt, or a solid solution such as olivine (Fe +, Mg +)2Si04, an equation similar to Equation 3-135c has been derived from the Nemst-Planck equations first by Helfferich and Plesset (1958) and then with refinement by Barter et al. (1963) with the assumption that (i) the matrix (or solvent) concentration does not vary and (ii) cross-coefficient Lab (phenomenological coefficient in Equation 3-96a) is negligible, which is similar to the activity-based effective binary diffusion treatment. The equation takes the following form ... 

The Darken-type equation (D = [NB-DA + NA-DB,]-flh, see Section4.3.3) is obtained only if cross coefficients are zero. In order to evaluate these cross coefficients, kinetic theory beyond the phenomenological approach is needed [A. R. Allnatt, A. B. Lidiard (1993)]. 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

In Bloch s original treatment of NMR,23 he postulated a set of phenomenological equations that accounted successfully for the behavior of the macroscopic magnetization M in the presence of an rf field. These relations are based on Eq. 2.41, where M replaces X, and B is any magnetic field—static (B0) or rotating (B,). By expanding the vector cross product, we can write a separate equation for the time derivative of each component of AT ... 

These equations are called the phenomenological equations, which are capable of describing multiflow systems and the induced effects of the nonconjugate forces on a flow. Generally, any force Xt can produce any flow./, when the cross coefficients are nonzero. Equation (3.175) assumes that the induced flows are also a linear function of non-conjugated forces. For example, ionic diffusion in an aqueous solution may be related to concentration, temperature, and the imposed electromotive force. 

For a binary fluid at mechanical equilibrium and for diffusion based on the mass-average velocity, we can now establish a set of phenomenological equations (Eqs. 7.6 and 7.7) with nonvanishing cross coefficients, and hence represent the coupled heat and mass flows... 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

The matter discussed in sec. 2.3 concerned the phenomenology of adsorption from solution. To make further progress, model assumptions have to be made to arrive at isotherm equations for the individual components. These assumptions are similar to those for gas adsorption secs. 1.4-1.7) and Include issues such as is the adsorption mono- or multlmolecular. localized or mobile is the surface homogeneous or heterogeneous, porous or non-porous is the adsorbate ideal or non-ideal and is the molecular cross-section constant over the entire composition range In addition to all of this the solution can be ideal or nonideal, the molecules may be monomers or oligomers and their interactions simple (as in liquid krypton) or strongly associative (as in water). 

Here Ca is the concentration of isotopically labelled species at a point where the concentration of unlabelled species is Ca. La a and La a are the straight and cross phenomenological coefficients of the irreversible thermodynamic formulation of diffusion. The original relation, Equation 1, assumes a zero cross coefficient, which in dense intracrystalline fluids certainly is not likely to be true. 

The functions M (t) were determined from the complete unsteady axially symmetric convective diffusion equation (Eq. 4.6.7), and M (f) were obtained from the Taylor dispersion equation, which was used as the model equation. The phenomenological coefficients U and in the equation were determined by matching the first three moments of the infinite sequence M (t) to M (t) for asymptotically large times [t>a lD). Applying his scheme to the circular capillary problem, Aris showed that D fj, where axial molecular diffusion is not neglected, is given by Eq. (4.6.35). Fried Combarnous (1971) later showed that the satisfaction of the first three moments for t—implies that c x, t), obtained as a solution of the Taylor dispersion equation with = D + Pe /48), is asymptotically the solution of the complete, unsteady, axially symmetric convective diffusion equation averaged over the cross section. 

As we noted above, the phenomenological relations (4.514), (4.515) are starting equations for obtaining useful results for transport phenomena as diffusion, heat conduction and cross effects. This will be discussed in the remaining part of this Sect.4.10 for details see [1-5]. 

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