Concentration profile calculations

II. The change of the sign between these two models may be also important for clustering and the surface growth mechanism. In the selfconsistent theory, the silver atoms tend to establish bonds with palladium atoms, while in the non-selfconsistent theory, the Ag-Ag bonds are favored. This explains the differences between the concentration profiles calculated within the models I and II and the fact that they do not simply follow the variations of the on-site terms. 

Fig. 2.17 Evolution of concentration profiles, calculated from Figs. 2.15 and 2.16 (adapted from Caras et al., 1985b, p. 1922)...

Concentration changes observed between mother liquor in the flash zone and liquid product in the melt zone of an experimental triple-point crystallizer have been dramatic. A qualitative concentration profile typical of those observed in the experimental unit is shown in Figure 8. The mother liquor concentration is relatively uniform above the packed bed, but a sharp drop in contaminant concentration occurs within the top several inches of the loosely packed crystal bed. Concentration changes of the order 500 to 5000 have been observed for representative sulfurous compounds and trace contaminants, including hydrogen sulfide, carbonyl sulfide, methyl mercaptan, ethane, and ethylene. Concentration profiles calculated for the packed bed of solid carbon dioxide using a conventional packed bed axial dispersion model agree very well with the observed experimental profiles. 

In order to get an estimate of the adhesion energy between a pseudo-brush and an elastomer (cross-linked in the dry state), we can use the concentration profile calculated [65] for the case of a pseudo-brush exposed to a melt (Nc monomers per chain)... 

Figure 7.10 shows the internal concentration profiles calculated according to Eqn. 7.101 for different values of the Thiele modulus. Knowing the exact internal concentration profile, Eqn. 7.101, allows us to calculate the molar flux through the external surface of the slab from Eqn. 7.80) ... 

But how far is one unit of dimensionless distance in real terms That depends on the diffusion coefficient and on time. After 1 ms it is about 1.6 im from the electrode surface, but after 1 s it corresponds to 50 pm. Figure lD(b) shows concentration profiles calculated for the same system, but plotted versus the distance in centimeters Each curve corresponds to a different time after application of the potential pulse, and the evolution of the concentration profile with time is presented. All the information contained in the curves in Fig. lD(b) exists, of course, in the single curve shown in Fig. lD(a). This is both the advantage and the disadvantage of presenting the data in dimensionless form ... 

Figure 12. The loading and dispensing of a focused fluorescein sample a.) FVocessed images (b) Iso-concentration profiles at 0.1C 0.3C , 0.5C 0.7C and 0.9C calculated from the images and (c) Corresponding Iso-concentration profiles calculated through numerical simulation.

Figure 8.19 shows the dimensionless internal concentration profiles calculated using Eqn. (8.66) for different values of the Thiele modulus (C = C/C z - z/L). 

Unlike the solution of the diffusion equation, which is positive everywhere, the solution of Eq. 6.134a is equal to 0 for > (m t). Figure 6.16 shows some concentration profiles calculated as solution of Eq. 6.134a. When time increases, the asymmetry of the concentration profiles and the sharpness of the cutoffs at the endpoints, z = ut, increase. Eor sufficiently long times, however, their solution approaches the shifted Gaussian profile predicted by the solution of Eq. 6.22. 

Figure 4.7 compares the concentration profiles calculated for TPY+ Y in contact with different MeCN electrolytes using chronocoulometric data, as described in Section 2.5. It was assumed that electron diffusion acts as a rate-determining step and Dg values listed in Table 4.3 were taken for estimating depths. 

Figure 35 shows concentration profiles calculated from these equations (S — maltose, Z — glucose, P — hydrogen peroxide). 

When the electrode resistance is very large, e.g., with an ion-selective microelectrode, the method described above does not work. An alternative approach consists of fitting the experimental concentration profile calculated from the tip potential to the theoretical profile predicted for the substrate geometry and activity. When the system investigated is at steady state, there is a unique relationship between tip potential and tip-substrate distance and an absolute distance scale can be determined. However, the procedure fails when the system is not at steady state because most ISE are unable to follow rapid changes in concentration. 

Of the three kinds of problem, this is the easiest theoretically but its main practical use is as a stepping stone in the solution of the other two problems. The solution in discrete form is given directly by Ecp (17). Sample results are shown in Figure 2, which shows the concentration profiles calculated from Eq. (17) (with D-,j specified using Ecps. (24) to (27)) for three assumed source profiles. A sharp, strongly peaked elevated source profile produces a peak in the concentration profile and hence a countergradient flux in the layer just below the peak where the flux is upward but dc/dz is positive, because of the strongly localized near-fleld contribution to the concentration held. By contrast, a more uniform source profile, in-... 

Fig. 7.3 illustrates concentration profiles, calculated with the aid of three different models of FFF [16,19,20], and thus, substantially, a comparison of the non-equilibrium model with the dispersion model. The dimensionless mean concentration, C , in Fig. 7.3 is defined by the relationship... 

Steady-state concentration profiles calculated from (12.104) are shown in Figure 12.12. Note that as q — 0, the profiles become flat, while for larger q values significant concentration gradients develop inside the drop. 

Figure 10.6 shows concentration profiles calculated using Equations 10-20 and 10-22. Using reasonable values for all of the parameters, steady state is reached approximately 1 h after implantation of the delivery device. The time required to achieve steady state depends on the rate of diffusion and ehmination, as previously described [25], but will be significantly less than 24 h for most drug molecules. 

Fig. 33 Schematics of interference microscopy, a Two light beams, one passing through the crystal and the other through the simrounding atmosphere, b The interference microscope. c Interference patterns generated due to different optical properties of the media passed hy the two beams, d Concentration profiles calculated from the changes in interference patterns with time...

Fig. 3.4. Conceptual design of an RD column for the decomposition of MTBE and the corresponding concentration profile calculated from thermodynamic considerations [18], reprinted from Chem. Eng. Sci., Vol 57, Beckmann et al.. Pages 1525-1530, Copyright 2002, with permission from Elsevier Science)...

Figure 8.18 Initial 2D concentration profile (C/Co) of [60]PCBM in bilayer P3HT [60]PCBM OFET with P3HT and [60JPCBM thicknesses of 25 and 31 nm, respectively, and [60JPCBM concentration profile calculated using Eq. (8.1),...

Response curves and concentration profiles calculated according to the simple model with an overall effective rate coefficient defined by Eq. (8.42) provide a reasonably good approximation to the exact solutions, calculated from model 46. This is illustrated in Figure 8.10. The dimensionless parameters are defined by... 

Gottifredi JC, Gonzo EE. Approximate expression for the effectiveness factor estimation and a simple numerical method for concentration profile calculation in porous catalyst. Chemical Engineering Journal 2005 109 83-87. 

Figure 10.6 Theoretical concentration profiles calculated in the conformal space of a douhle-hand assembly (Figure 10.3C) operating in generator/collector mode in near-steady-state conditions. wig = 1 and g/2 Dt) = 0.1. C is the hulk concentration of the reactant in solution.

Table 12.3 Calculated parameters versus the degree of perturbation of the initial concentration profile calculated from the dependence D(c) with parameters a(l) = 1 x 10- a 2)° = 6 X 10 a(3) = -4 x IQ-h a is the regularization parameter, rj is the...

Figure 12.16 Concentration dependencies of the diffusion coefficient. Here curve 1 illustrates the initial concentration profile used in calculations curves 2-7 show the profiles calculated by solving the inverse problem with the initial profile distorted with amplitudes of 0, 0.1, 1, 3, 5, and 10%, respectively (a) the concentration profiles calculated for the initial profiles with perturbations <5 = 1 (b), 5 (c), and 10% (d) (curve 1 shows the initial profile, curve 2 shows the distorted profile, and curve 3 shows the profile calculated on the basis of the obtained dependencies b c)).

FIGURE 5.3 Modeling of transient water flux data for Nation 117. (a) The relaxation of the experimental outlet vapor pressure (open circle) for Nafion 117 in LE mode at 50°C, flow chamber volume V = 0.125 L, flow rate V = 0.1 L min membrane area A = 2 cm, and saturation vapor pressure = 12336.7 Pa. Plotted for comparison are model simulations for a slow transport coefficient (dash dot), fast transport coefficient (dash), and a concentration-dependent transport coefficient (gray), (b) Water concentration profiles calculated in the model at different time. (Reprinted from Electrochem. Commun. 13, Rinaldo, S. G. et al. Vaporization exchange model for dynamic water sorption in Nafion Transient solution, 5-7, Figures 1 and 2, Copyright (2011) Elsevier. With permission.)... 

Simply note that the general hydrodynamic problem of depleted layer flow including shear effects is a highly non-linear problem which must be solved using an iterative numerical scheme. Using a Carreau A curve based on experimental data for xanthan and an Auvray-type model of the tube concentration profile, calculations have been performed of rj ff as a function of 7n by Sorbie (1989, 1990), where is the Newtonian shear rate in a capillary and is given by... 

Figure 3.45 shows the critical length and concentration profile calculated by the approximate analytical model, as well as their values numerically calculated according to the full model (3.57)-(3.64), i.e., (3.117) do (3.129). 

This equation was used to predict diffusion of erucamide in polyamide-12, PA-12, at three temperatures. Film containing 15 wt% erucamide was placed between erucamide-free films and diffusion was carried out under isothermal conditions. Figures 7.1 and 7.2 show some data. Experimental points (Figure 7.1) obtained from gas chromatographic determination of concentration of erucamide confirm that the concentration profile calculated from a simple Fickian diffusion (equation 7.1) is correct. 

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