Model convection-diffusion

B. Acock and Y. A. Pachep.sky, Convective-diffusive model of two-dimensional root growth and proliferation. Plant Soil I80 23 (1996). 

S Neervannan, LS Dias, MZ Southard, VJ Stella. A convective-diffusion model for dissolution of two non-interacting drug mixtures from co-compressed slabs under laminar hydrodynamic conditions. Pharm Res 11 1228-1295, 1994. 

JR Crison, VP Shah, JP Skelly, GL Amidon. Drug dissolution into micellar solutions Development of a convective diffusion model and comparison to the film equilibrium model with application to surfactant-facilitated dissolution of carbama-zepine. J Pharm Sci 85 1005-1011, 1996. 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

Fig. 13. Concentration profiles within an ATRllow-tiougncen convection-diffusion model (dj). The models are descri m...

Fig. 14. Comparison of the convection-diffusion behavior of acetonitrile and hemoglobin within an ATR flow-through cell as calculated by the convection-diffusion model described in the text. The concentrations of the two molecules were periodically varied between zero and a non-zero value with a frequency of 67 mHz the flow rate was 1.5mL/min. Dark areas represent high concentrations of the solute molecules (65).

Fig. 15. Response comparison between experiment and prediction of the convection-diffusion model described in the text for acetonitrile and hemoglobin at three different modulation frequencies. Solid line, simulated response dotted line, experimental response (65).

Figure 5.3-11. Scheme of the two-compartment convective-diffusion model and equations for the end zones in BSCR (tested with a liquid tracer) [47],... 

Figure 20.11 Normalized residual errors for the fit of the convective-diffusion models presented in Figure 20.12 to impedance data obtained for reduction of ferricyanide on a Pt rotating disk electrode a) real and b) imaginary.

The one-term convective-diffusion model consisting of the first term in equation (11.97)... 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response. ... 

It follows from the above discussion and numerical results that even a simple convective-diffusive model of concentration behaviour mechanism gives realistic results and yields a satisfactory description of the formation of the gaseous layer under the anode surface. The model may be improved by adding the electrolyte circulation and electromagnetic forces yet we hope that it will not change the main conclusions. The finite volume method proves to be a flexible and sufficiently accurate numerical technique for solving both the equations for the Galvani potential and the reactant concentrations. The marker-and-cell approach makes it possible to outline the electrode surfaces easily. 

The convective—diffusive model underwent further refinements, as discussed in 5.2.2. It permits a quantitative description of sample dispersion in unsegmented flow analysis and provides a good simulation to assist system design and method implementation. Other quantitative models have also been proposed for specific applications, and in this regard, the tanks-in-series model should be highlighted. 

The convective—diffusive model has been continuously expanded and applied to specific situations. The influences of reactor coiling and packing a reactor with beads were investigated in the early 1980s [38,39]. Moreover, Eq. 3.4 was expanded in order to evaluate the travel time and baseline-to-baseline time [40] the predicted results were in good agreement with experimental values but some correction factors were needed. 

Because of discrepancies between experimental tests and predicted tissue oxygen response by convection-diffusion models, the influence of physiological control mechanisms was considered. It is proposed that a minimum of two autoregulatory actions (at least conceptually) are functional in helping prevent neuron damage when low blood oxygen tensions are encountered. 

Protein Adsorption and Desorption Rates and Kinetics. The TIRF flow cell was designed to investigate protein adsorption under well-defined hydrodynamic conditions. Therefore, the adsorption process in this apparatus can be described by a mathematical convection-diffusion model (17). The rate of protein adsorption is determined by both transport of protein to the surface and intrinsic kinetics of adsorption at the surface. In general, where transport and kinetics are comparable, the model must be solved numerically to yield protein adsorption kinetics. The solution can be simplified in two limiting cases 1) In the kinetic limit, the initial rate of protein adsorption is equal to the intrinsic kinetic adsorption rate. 2) In the transport limit, the initial protein adsorption rate, as predicted by Ldveque s analysis (23), is proportional to the wall shear rate raised to the 1/3 power. In the transport-limited adsorption case, intrinsic protein adsorption kinetics are unobservable. 

Assumptions of the Convective Diffusion Model in Regular Polygon Ducts... 

Figure 14 shows a series of concentration profiles within an ATR flow-through cell as calculated by a convection-diffusion model that has been described elsewhere (65) for a small, rapidly diffusing molecule (acetonitrile) and a large, slowly diffusing molecule (hemoglobin). At time 1 = 0, the concentrations of the molecules at the inlet were switched from zero to non-zero values. The laminar flow profile is established due to relatively low flow rates (low Reynolds numbers), which is clearly... 

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