Effective diffusion coefficient relations from different equations

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

In this equation, represents the effective lateral diffusion/dispersion constant for laminar flow a value on the order of is suitable, and for turbulent flow the molecular diffusion coefficient in this expression should be replaced by the turbulent diffusion coefficient. Based on these simple relations it can be calculated that under typical conditions, lateral reactant transport takes place only over distances of a few subchannels. In other words, if the gas velocity through the wall channels differs much from the velocity through the central subchannels, the nonuniform flow profile can have a significant effect on the overall reactant conversion. In these situations the CBS model can be expected to give a better estimate of the reactor performance than the CB model. 

In this chapter, we have discussed three important aspects of thermodynamics in the presence of flow. By considering different points of view ranging from kinetic and stochastic theories to thermod5mamic theories at mesoscopic and macroscopic levels, we addressed the effects of flow on transport coefficients like diffusion and viscosity, constitutive relations and equations of state. In particular, we focused on how some of these effects may be derived from the framework of mesoscopic nonequilibrium thermod5mamics. 

The observed rate coefficient for exchange (L = H, D, or T) is fe bs = k k2 /(kh. ] + k2)- If the primary isotope effect on k2 is different from that on k1 and k... j it is argued that the experimental isotope effects feob s tklb s and feobs/ ob s will not be related by the Swain—Schaad relation, kH/kT = (feH/feD)1442 which is derived with reference to a single-step proton transfer [115, 128]. The size of the discrepancy will depend upon the value of /e, /fe , the amount of internal return. In the analysis of isotope effects for triphenylmethane exchange it is assumed that k2=k2 = k2 since this represents a diffusion step. By introducing aT = k- i /k2 and Kl = k /k j eqns. (82) and (83) are obtained. A third equation (84)... 

Equations 46 have been directly derived from the full model in [19]. On the other hand, they are almost identical with the relations obtained from the so-called two-compartment model (the only difference is that the numerical coefficient is a little bit lower). The two-compartment model was first developed for sensors with receptors placed on small spheres [23]. In [24-26] it was adapted for the SPR flow cell and in [ 18] it was approved and verified by comparison of munerical results with those obtained from the full model. The two-compartment model approximates the analyte distribution in the vicinity of the receptors by considering two distinct regions. The first is a thin layer around the active receptor zone of effective thickness fiiayer> and the second is the remaining volume with the analyte concentration equal to the injected one, i.e., a. While the analyte concentration in the bulk is constant (within a given compartment), analyte transport to the inner compartment is controlled by diffusion. The actual analyte concentration at the sensor surface is then given by the difference between the diffusion flow and the consump-tion/production of the analyte via interaction with receptors. For the simple pseudo first-order interaction model we obtain ... 

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