Concentration profiles from dispersion

The Higuchi model is an approximate solution in that it assumes a pseudosteady state , in which the concentration profile from the dispersed drug front to the outer surface is linear. Paul and McSpadden [24] have shown that the correct expression can be written as ... 

The experimental dependence M = /(V), i.e. the classical SEC calibration curve usually obtained by using narrow standards, in such a case can be obtained directly without calibration from the on-line LS detector. By combining the experimental function M = /(V) and the concentration profile (from DRI), one can construct the complete MMD of the HA sample. The differential and cumulative MMD of a high molar mass HA sample (Mw = 652 kDa, D = 2.1) are shown in Fig. (10). Starting from the initial MMD, the molecular weight averages and dispersity index (Mn, Mw, Mz, and D) could be easily calculated using the appropriate definitions. 

Results from the previous section in this chapter illustrate how and when interpellet axial dispersion plays an important role in the design of packed catalytic tubular reactors. When diffusion is important, more sophisticated numerical techniques are required to solve second-order ODEs with split boundary conditions to predict non-ideal reactor performance. Tubular reactor performance is nonideal when the mass transfer Peclet number is small enough such that interpellet axial dispersion cannot be neglected. The objectives of this section are to understand the correlations for effective axial dispersion coefficients in packed beds and porous media and calculate the mass transfer Peclet number based on axial dispersion. Before one can make predictions about the ideal vs. non-ideal performance of tubular reactors, steady-state mass balances with and without axial dispersion must be solved and the reactant concentration profiles from both solutions must be compared. If the difference between these profiles with and without interpellet axial dispersion is indistinguishable, then the reactor operates ideally. 

Figure 22.6. (a) Surfactant concentration profile from an experiment where a surfactant solution is flowed though a serum replacement cell containing a dispersion, and (b) the corresponding isotherm... 

The earliest and still widely used dispersion model to compute pollutant concentration profiles is the Gaussian plume model for single or multiple source pollution problems. Box-type model techniques, which can take into account nonlinear interactions among different species arising from chemical reactions, have been used in longer-range dispersion computations. 

If the concentration profile can be determined the moduli can be evaluated. In principle there is no reason why this should be a non-linear measurement, it depends upon the magnitude of the gravitational Peclet number. Buscall35 suggested that a low speed centrifuge could be used to apply different acceleration gradients to the dispersion. If the angular velocity of the rotor is cor and if X is the distance from the centre of the rotor to the top of the sediment then the pressure balance equation becomes... 

However, the two mechanisms interact and molecular diffusion can reduce the effects of convective dispersion. This can be explained by the fact that with streamline flow in a tube molecular diffusion will tend to smooth out the concentration profile arising from the velocity distribution over the cross-section. Similarly radial dispersion can give rise to lower values of longitudinal dispersion than predicted by equation 4.39. As a result the curves of Peclet versus Reynolds number tend to pass through a maximum as shown in Figure 4.6. 

Fig. 9.18 The influence of flow dispersion on the concentration profiles of the two phases (mass transfer from the continuous phase to the dispersed phase). 

Concentration profiles for compound 67 are given in Figure 7. Although the relative effect of dispersion due to tidal flow and downstream movement due to the net river flow are not precisely known, it is clear from these data that compound 67 comes from a point source located near river mile 100. In fact, the company which produces the precursor alcohols (no. 68-70), the chemically related chlorinated ethylene glycol (no. 64) and the Cg-phenol (no. 38) discharges its effluent at river mile 104. 

Another method has been proposed by Blackwell (B16) and by Hiby and Schiimmer (H8) that avoids the necessity of measuring the complete concentration profile. A pipe with a diameter smaller than the system, thus forming an annular region, is used at the sampling point. A mixed mean sample from the annular region is now sufficient to enable one to determine the radial dispersion coefficient. From Eq. (55) this concentration will be, for an annular region of dimensionless radius a,... 

Packed Beds. Data on liquid systems using a steady point source of tracer and measurement of a concentration profile have been obtained by Bernard and Wilhelm (B6), Jacques and Vermeulen (Jl), Latinen (L4), and Prausnitz (P9). Blackwell (B16) used the method of sampling from an annular region with the use of Eq. (62). Hartman et al. (H6) used a bed of ion-exchange resin through which a solution of one kind of ion flowed and another was steadily injected at a point source. After steady state conditions were attained, the flows were stopped and the total amount of injected ion determined. The radial dispersion coefficients can be determined from this information without having to measure detailed concentration profiles. 

Longitudinal concentration gradients of a chemical that is introduced into a river at a constant rate are small (except for a chemical with a very large in-situ reaction rate). Thus, according to Eq. 24-44 the effect of dispersion on C(x,t) is small. In contrast, the concentration profile resulting from a nonstationary input as caused, for instance, by an accidental spill is strongly affected by dispersion. In fact, often dispersion is the most important mechanism to reduce the maximum concentration of a concentration patch that moves along the river. 

Fig. 2.14. Response of the dispersed plug-flow model to a pulse input of tracer. Three snapshot concentration profiles at different times tx, t2, h are shown. The inset shows the C-curve derived from measurements at a fixed position, z = L. If axial dispersion occurs to only a small extent (small DJuL) the C-curve is almost symmetrical...

Konjac flour gum is reported to be pH- and cation- (sodium, potassium, and calcium ions) insensitive this is consistent with its nonionic character (FMC, 1989), but an isolate did not show a characteristic linear viscosity-concentration profile in water it did show linearity in the presence of electrolytes (Jacon et al., 1993). The apparent partial specific volume of a 0.2-0.4% dispersion was constant over a wide pH range and increased with increasing temperature from 5 to 50°C it then remained constant (Kohyama and Nishinari, 1993). 

ZS were calculated from the same Qk experimental runs as the solid settling velocity results. Many of the solid concentration profiles for the 0.5-5 ym size particles were uniform to within 0.2 weight percent. As uniform profiles suggest an infinite dispersion coefficient, scatter for the smallest size particles was too large to be included in the analysis. For the iron oxide... 

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