Onsager cross-coefficients

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

The Onsager reciprocity relation, when applied to the present context, predicts that the cross coefficients a2 and a, which determine the rate of flow ofliquid due to the applied electric field and the current passing due to a hydrostatic pressure difference, respectively, are equal, Le.,... 

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). 

According to the Onsager s relations, three coefficients are to be determined. They are the passive permeability to sodium ZNa, the metabolic reaction coefficient if there is no sodium transport Zr, and the cross-coefficient between the chemical reaction and the sodium flow ZNar. The linear nonequilibrium thermodynamics formulation for the active transport of sodium and the associated oxygen consumption in frog skin and toad urinary bladders are studied experimentally. Sodium flow JNa is taken as positive in the direction from the outer to the inner surface of the tissue. The term JT is the rate of suprabasal oxygen consumption assumed to be independent of the oxygen consumption associated with the metabolic functions. 

Lu and L1 are the straight and cross-coefficients, respectively. By Onsager s reciprocal rules, we have LtJ = L. The electrochemical potential differences between internal i and external e regions are defined by... 

Equations of type (2.17) for the interrelation of the rates of conjugate stepwise reactions are valid for any intermediate linear transformation pathways (including catalytic reactions). The value of A may be expressed by relations that are much more complicated than (2.15) and depends not only on parameters Sy but also on thermodynamic rushes of some external reactants of the stepwise reactions (see Section 2.3.5 for exam pies). At the same time. A > 0 always. However, the relationship between the cross coefficients Ay and Aj may be more intricate than that in the traditional Onsager equations. 

D] predicted by the above method agrees reasonably well with the experimental values except for the coefficient 21- Since cross-coefficients reflect differences in pair diffusivities, it is not uncommon to find that small cross-coefficients are not predicted accurately, even in sign. It may be noted from Example 3.3.1 that the measured do not satisfy the Onsager relations accurately. This could be one reason for the deviation between the predicted and experimental values the predicted set of coefficients implicitly satisfy the ORR. 

Again the same convention as described earlier is used, and flow into the cell is considered to be in the positive direction. The osmotic pressure due to the permeant solute is denoted by which is defined as = RT(Cf - Att has units of dyne/cm. In these equations we speak of differences in concentrations in bulk phases since the partition coefficient which relates the concentration in the membrane phase to that of the bulk phase is incorporated in the permeability coefficients. The subscripts i and s refer to impermeant and permeant solute respectively. Lpj is the cross-coefficient for the volume flow arising from differences in the osmotic pressure of the permeant solute, Aw, when there is no difference of either hydrostatic or osmotic pressure produced by impermeant solutes ( Attj = 0). L p is the relative diffusional solute mobility per unit hydrostatic (or impermeant solute) pressure difference when Asr = 0. Although is always positive, and L p are both negative and have the same units as L. If the Onsager reciprocal relation holds, then is the diffusional flow and is a measure of the relative... 

J represent the flows, F the forces, and L the phenomenological coefficients, i and j denote the different flows and forces. Thus, concentration difference may be one force, temperature difference another force, momentum difference another force, etc., and the flows can be heat transfer, mass transfer, and momentum transfer. Onsager showed that from analysis of a positive definite matrix, the cross-coefficients in Equation (B.17) have to be equal. Thus ... 

In general, the diagonal elements of a positive definite matrix must be positive. In addition, a necessary and sufficient condition for a matrix Lg to be positive definite is that its determinant and all the determinants of lower dimension obtained by deleting one or more rows and columns must be positive. Thus, according to the Second Law, the proper coefficients L k should be positive the cross coefficients, (i 7 k), can have either sign. Furthermore, as we shall see in the next section, the elements Ljk also obey the Onsager reciprocal relations Ljk = Lkj. The positivity of entropy production and the Onsager relations form the foundation for linear nonequilibrium thermodynamics. 

The phenomenological particle flow equations are given by system (1). Flows are connected linearly to generalize forces, coupling effects are expressed by way of the cross coefficients Lu and use of the Onsager reciprocal relationship is made ... 

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

The matrix of the phenomenological coefficients must be positive definite for example, for a two-flow system, we have L0 > 0, Ip >0, and Z/.p Z,pZpo > 0.1,0 shows the influence of substrate availability on oxygen consumption (flow), and Ip is the feedback of the phosphate potential on ATP production (flow). The cross-coupling coefficient Iop shows the phosphate influence on oxygen flow, while Zpo shows the substrate dependency of ATP production. Experiments show that Onsagers s reciprocal relations hold for oxidative phosphorylation, and we have Iop = Zpo. 

The coefficients rj, fj[ and 773 are shear viscosities. The twist viscosity is denoted by 7[. The symmetric traceless pressure tensor cross couples with the trace of the strain rate and the two angular velocities (l/2)Vxu-Q and (l/2)Vxu- . The corresponding cross coupling coefficients are 772 According to the Onsager reciprocity relations, they must be equal to 72/2 and 74/2. They couple the symmetric traceless strain rate to ((l/3)7 r(P)-Pg and to the two torque densities (1) and (4)- The coeffi-... 

When the fluxes are taken to be and q, — (9kr/2m )j , as suggested by the equation of change for entropy, then the numerical coefficients in the cross terms are the same, in agreement with the Onsager reciprocal relations. This provides one check on the evaluation of the flux expressions for the Hookean dumbbell model. 

Onsager showed theoretically that the coefficients Ly must be symmetrical. To ensure the non-nej tivity of the dissipation, it suffices to require Ly to be definite positive, other than being symmetiicaL The off-diagonal coefficients allow to account for cross-couphngs. This formulation seems to be better suited to moderately non-hnear problems. For example, it cannot lead to the classical plastic flow rule in solids. 

When various driving forces and fluxes become important, then, according to Jk = Sk//8kk Xk, mixed terms also appear, w hose coefficients are symmetrical according to the Onsager relation (Atk = / k k) [330]. The latter follows from the principle of microscopic reversibility [326]. We shall mostly ignore such cross effects in the treatment (however, see Sections 6.6.1, 6.10.1). 

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