Dispersion coefficient determination

The design engineer can use the dispersion coefficients determined in this way for the calculation of the real course of concentrations, c, of any component in the dispersed d) and continuous (c) phases along the countercurrent column. If the mass transfer between the two phases, the actual task of an extractor, is included in the balance, the balance equations for an element of height dh of the extractor for stationary conditions is ... 

The Sutton equation belongs to a class known as Gaussian dispersion models in which the concentration of odor along any axis perpendicular to the downwind (x) direction is assumed to follow a normal or Gaussian distribution (Fig. 3.1). The dispersion coefficients determine the width of the plume and thus are related to the standard deviation of the concentration along the cross-wind and vertical axes. 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

The virtual distances, y, and z, determined using Eq. (26-69) are added to the actual downwind distance x to determine tne dispersion coefficients Cy and O, for subsequent computations. 

SCREEN allows for the selection of urban or rural dispersion coefficients. The urban dispersion option is selected by entering a U (lower or upper case) in column 1, while the rural dispersion option is selected by entering an R (upper or lower case) in column 1. Determination of the applicability of urban or rural dispersion is based upon land use or population density. In general, if 50 percent or more of an area 3 km around the source satisfies the urban criteria (Auer, 1978), the site is deemed in an urban setting. Of the two methods, the land use procedure is considered more definitive. 

The criterion for the validity of Equation 8-141 is Npg 1.0. A rough rule-of-thumb is Npg > 10. If this condition is not satisfied, the correct equation depends on the boundary conditions at the inlet and outlet. A procedure for determining dispersion coefficient Dg [ is as follows ... 

The CC2 model performes very different for static hyperpolarizabilities and for their dispersion. For methane, CC2 overestimates 70 by a similar amount as it is underestimated by CCS, thus giving no improvement in accuracy relative to the uncorrelated methods CCS and SCF. In contrast to this, the CC2 dispersion coefficients listed in Table 3 are by a factor of 3 - 8 closer to the CCSD values than the respective CCS results. The dispersion coefficients should be sensitive to the lowest dipole-allowed excitation energy, which determines the position of the first pole in the dispersion curve. The substantial improvements in accuracy for the dispersion coefficients are thus consistent with the good performance of CC2 for excitation energies [35,37,50]. 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

The standard Rodbard-Ogston-Morris-Killander [326,327] model of electrophoresis which assumes that u alua = D nlDa is obtained only for special circumstances. See also Locke and Trinh [219] for further discussion of this relationship. With low electric fields the effective mobility equals the volume fraction. However, the dispersion coefficient reduces to the effective diffusion coefficient, as determined by Ryan et al. [337], which reduces to the volume fraction at low gel concentration but is not, in general, equal to the porosity for high gel concentrations. If no electrophoresis occurs, i.e., and Mp equal zero, the results reduce to the analysis of Nozad [264]. If the electrophoretic mobility is assumed to be much larger than the diffusion coefficients, the results reduce to that given by Locke and Carbonell [218]. 

Bubble size in the circulating beds increases with Ug, but decreases with Ul or solid circulation rate (Gs) bubble rising velocity increases with Ug or Ul but decreases with Gs the ffequeney of bubbles increases with Ug, Ul or Gs. The axial or radial dispersion coefficient of liquid phase (Dz or Dr) has been determined by using steady or unsteady state dispersion model. The values of Dz and D, increase with increasing Ug or Gs, but decrease (slightly) with increasing Ul- The values of Dz and Dr can be predicted by Eqs.(9) and (10) with a correlation coefficient of 0.93 and 0.95, respectively[10]. 

J. Gotz, K. Zick, C. Heinen, T. Konig 2002, (Visualisation of flow processes in packed beds with NMR imaging Determination of the local porosity, velocity vector and local dispersion coefficients), Chem. Eng. Process. 41 (7), 611-630. 

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

Assuming the source concentration Cs is known, the virtual distance is found by using the known source concentration to find the virtual source distance. For a plume, solve Eq. (23-78) for the product GyGz, then determine the virtual source distance by iterative solution (or trial and error) using E and Cs. For a puff, solve Eq. (23-79) for the product GxGyGz (or GyGz) then determine the virtual source distance xv by iterative solution (or trial and error) using Et and Cs. The dispersion coefficients at distance xe will now represent a distance from the real source of xe — x. ... 

Parameter estimation problems result when we attempt to match a model of known form to experimental data by an optimal determination of unknown model parameters. The exact nature of the parameter estimation problem will depend on the mathematical model. An important distinction has to be made at this point. A model will contain both state variables (concentrations, temperatures, pressures, etc.) and parameters (rate constants, dispersion coefficients, activation energies, etc.). 

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

In determining how the dispersion coefficients depend on travel time one may employ atmospheric diffusion theory or the results of experiments. Because of the difficulty of performing puff experiments, however, the coefficients are usually inferred not from instantaneous releases but from continuous releases. Thus, the dispersion coefficients derived from such experiments are essentially a measure of the size of the plume envelope formed by sampling a real meandering plume emitted from a... 

The results just obtained for < y) are, however, rarely used in applications because (v ) and T are generally not known. The Gaussian dispersion parameters aj and al are, in a sense, generalizations of (Cj) and particle displacement variances o-y and a-] are not calculated by Eq. (8.8). Rather, they are treated as empirical dispersion coefficients the functional forms of which are determined by matching the Gaussian solution to data. In that way, the empirically determined a-y and deviations from stationary, homogeneous conditions which are inherent in the assumed Gaussian distribution. 

The Gaussian plume model estimates the average pheromone flux by multiplying the measured odor concentration by mean wind speed, using the following formula (Elkinton etal, 1984). Everything is the same as in the Sutton model, except that ay and az, respectively, replace the terms Cy and Cz of the Sutton model. Dispersion coefficients are determined for each experiment separately. 

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,... 

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