Linear mass transport

Minimum mass transport occurs even in the absence of water filtration. We will review uni dimensional (linear) mass transport. If V = 0 and q, = 0, advective-dispersive equation acquires the format of a linear equation Fick s second law (equation 3.12). 

At linear mass-transport of nonpolar solution components, coefficient of numerical dispersion coefficient may be determined from the equation... 

Approximately how long would it take for a 20-cm-long ingot with a 3-cm x 6-cm square cross section to solidify at 1200 K assuming nucleation only occurs along the mold walls followed by linear mass-transport-limited growth inward to the center ... 

Growth will proceed inward from the mold walls toward the center. The shortest dimension of the ingot cross section determines the time required. The time required for the growth to proceed 1.5 cm (from the closest wall to the center of the ingot) under the linear mass-transport-limited growth regime is therefore... 

The response of the immobilized enzyme electrode can be made independent of the enzyme concentration by using a large excess of enzyme at the electrode surface. The electrode response is limited by the mass transport of the substrate. Using an excess of enzyme often results in longer electrode lifetimes, increased linear range, reduced susceptibiUty to pH, temperature, and interfering species (58,59). At low enzyme concentrations the electrode response is governed by the kinetics of the enzyme reaction. 

In the A sector (lower right), the deposition is controlled by surface-reaction kinetics as the rate-limiting step. In the B sector (upper left), the deposition is controlled by the mass-transport process and the growth rate is related linearly to the partial pressure of the silicon reactant in the carrier gas. Transition from one rate-control regime to the other is not sharp, but involves a transition zone where both are significant. The presence of a maximum in the curves in Area B would indicate the onset of gas-phase precipitation, where the substrate has become starved and the deposition rate decreased. 

Yet another way to detect mass transport problems is with a newly developed poisoning technique.24,26,49,50 This technique works for liquid-phase hydrogenations and possibly for other reactions that are poisoned by CS2. It takes advantage of the fact that CS2 poisons Pd and Pt linearly until all reaction stops. If mass transfer problems exist, the initial linear decrease in rate occurs at a slope less steep than the slope of the chemically controlled rate (Fig. 1.7). If no mass transport problems exist, the rate decreases linearly from the start with no change in slope. Therefore a plot of rate versus amount of CS2 reveals the existence or absence of mass transport problems 49... 

Figure 3.98 Comparison of a reversible conventional cyclic voltammogram (linear diffusion) and reversible steady-state voltammogram obtained at a single microelectrode disc where mass transport is solely by radial diffusion. Current axis not drawn to scale. From A.M. Bond and H.A.O. Hill, Metal Inns in Biological Systems, 27 (1991) 431. Reprinted by courtesy of Marcel...

Hence, the macroscopic model of Bond and Hill reinterpreted the data in Figure 3.96 in terms of a cross-over between the limiting forms of mass transport, i.e. linear to radial diffusion, and not in terms of a slowing down in the heterogeneous kinetics. 

The metal ion in electroless solutions may be significantly complexed as discussed earlier. Not all of the metal ion species in solution will be active for electroless deposition, possibly only the uncomplexed, or aquo-ions hexaquo in the case of Ni2+, and perhaps the ML or M2L2 type complexes. Hence, the concentration of active metal ions may be much less than the overall concentration of metal ions. This raises the possibility that diffusion of metal ions active for the reduction reaction could be a significant factor in the electroless reaction in cases where the patterned elements undergoing deposition are smaller than the linear, or planar, diffusion layer thickness of these ions. In such instances, due to nonlinear diffusion, there is more efficient mass transport of metal ion to the smaller features than to large area (relative to the diffusion layer thickness) features. Thus, neglecting for the moment the opposite effects of additives and dissolved 02, the deposit thickness will tend to be greater on the smaller features, and deposit composition may be nonuniform in the case of alloy deposition. 

Remarkably, the use of a fluorous biphasic solvent system in combination with a [Rh(NBD)(DPPE)]+-type catalyst (NBD = norbornadiene) copolymerized into a porous nonfluorous ethylene dimethacrylate polymer, resulted in an increased activity of the catalyst relative to a situation when only toluene was used as solvent [30]. The results were explained by assuming that fluorophobicity of the substrate (methyl-trans-cinnamate) leads to a relatively higher local substrate concentration inside the cavities of the polymer when the fluorous solvent is used. That is, the polymer could be viewed as a better solvent than the fluorous solvent system. This interpretation was supported by the observations that (i) the increase in activity correlates linearly with the volume fraction of fluorous solvent (PFMCH) and (ii) the porous ethylene dimethacrylate polymer by itself lowers the concentration of decane in PFMCH from 75 mM to 50 mM, corresponding to a 600 mM local concentration of decane in the polymer. Gas to liquid mass transport limitation of dihydrogen could be mled out as a possible cause. 

Galceran, J., Monne, J., Puy, J. and van Leeuwen, H. P. (2003). The impact of the transient uptake flux on bioaccumulation. Linear adsorption and first-order internalisation coupled with spherical semi-infinite mass transport, Mar. Chem., in press. 

In the current-voltage curve in Fig. 14.15, three different regions can be discerned. At low current densities, the performance is kinetically limited. In the linear part, ohmic losses are significant. At high current densities, mass transport losses dominate. 

If only linear diffusion is the operating mass transport process, it has been shown that the current (i) in a planar electrode is related to the concentration (c) gradient at the surface of the electrode (x = 0) by [332]... 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

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