Main coefficient

The parameter U consists of the two main coefficients of heat transfer shown in Equation 8.8 ... 

Similar equations can be written for components J2 and J3. The coefficients Dx x and D22 are the main coefficients they are not self-diffusion coefficients. Du and D2l are the cross-coefficients and assumed to be equal to each other for binary gas mixtures. 

For the uniform grid, the coefficients of the three nodal values involved in the interpolation become 3/8 for the downstream point, 6/8 for the first upstream node and —1/8 for the second upstream node. This scheme is more complex than CDS and it extends the computational molecule by one more node in each direction (the conventional tri-diagonal methods are, therefore, not directly applicable. See the discussion in the following subsection). The scheme has a third-order truncation error and was made popular by Leonard (1979). The transportiveness property is built into the scheme by considering two upstream and one downstream node. However, the main coefficients of the discretized equations are not guaranteed to be positive. This may lead to instability and may lead to unbounded (wiggles) solutions under certain conditions. 

The ratio of cross-coefficient to the main coefficient, B 2/Bn, is less than 5% if the salt concentration is less than 10 mol/m. So the Nernst-Planck equation can be used safely below = 10 mol/m. ... 

It is interesting to note that thermodynamic stability considerations do not require the diagonal elements and D22 to be individually positive. If recourse is made to the kinetic theory of gases, it can be shown that the main coefficients are individually positive, that is. 

Note that [G]/7 r is dimensionless and symmetric also, the cross-coefficients G12 and G21 are a large fraction of the main coefficients G and G22. Multiplying each element of the above matrix by RT gives... 

If D = 0, the cross-coefficients must match the main coefficients in magnitude—implying large coupling effects. 

For two components infinitely diluted in a third (xj and X2 close to zero), we find the cross-coefficients >12 21 vanish and the main coefficients Du and D22 are given... 

It should now be clear that diffusional interaction effects cannot occur in binary mixtures or in multicomponent systems where all the cross-coefficients, (/ = 7), vanish. This will be the case in mixtures in which all the binary diffusion coefficients are alike (the system oxygen-nitrogen-carbon monoxide discussed in Example 4.2.1 is a case in point), as well as in mixtures where one component is present in very large excess (see Example 4.2.2). Thus, the first requirement for significant interaction effects is that the cross-coefficients be large compared to the main coefficients That is. 

The advantage of this approach is that the main coefficients are positive and satisfy the requirement for conservativeness, boundedness and transportive-ness. 

Coefficients quantitatively describe main effects. In the strict sense, such coefficients are differential quotients whose numerators and denominators contain conjugated main quantities. In order to refer to them more easily, we will caU them and their offspring (which only differ by certain factors) main coefficients. If we allow only small changes, we can use difference quotients instead of differential quotients, which simplifies things. To give an example, the following coefficients, which we were introduced to in Sect. 3.9 as entropy capacities, can characterize the main effect of an increase of temperature on the De Donder system above ... 

The cross relation in question, which is raie of Maxwell s relations, is here obtained directly through flipping. If a main coefficient is flipped, it is generally reproduced there is no new informatiOTi as we can easily see ... 

The first term on the right hand side of eq. V> 15 corresponds to the flux of component 1 under its own gradient, while the second term gives the contribution of the gr ent of component 2 to the flux of component 1. Ljj is a coupling coefficient and represents the coupling effect. Lj is called the main coefficient... 

For example, from our cross coefficients and values different from zero at finite concentrations, and having in mind that main coefficients Z) and D aiQ not identical to the binary diffusion coefficients of aqueous CaClj and caffeine, we can conclude that the diffusion of CaCl2 in aqueous solutions at both temperatures, 25 C and 37 C, may be affected by the eventual presence of new different species resulting from various equilibria. 

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