Dispersion coefficient estimation

The dispersion coefficient for interactions Q between molecules A and B can be estimated to an average... 

Thakkar A J 1988 Higher dispersion coefficients accurate values for hydrogen atoms and simple estimates for other systems J. Chem. Phys. 89 2092... 

Estimate die dispersion coefficient Dg[ from the definition of the Peclet number. 

A water body is considered to be a one-diiuensional estuary when it is subjected to tidal reversals (i.e., reversals in direction of tlie water quality parameter are dominant). Since the describing (differential) equations for the distribution of eitlier reactive or conserv ative (nomciictive) pollutants are linear, second-order equations, tlie principle of superposition discussed previously also applies to estuaries. The principal additional parameter introduced in the describing equation is a tid il dispersion coefficient E. Methods for estimating this tidiil coefficient are provided by Thomaim and Mueller... 

Table 4 Comparison of various ab initio results and experimental estimates for the dispersion coefficients of the electronic hyperpolarizabilities 7jj and 7 of methane. (All results in atomic units. Results for the dispersion coefficients refer to single point calculations at the equilibrium geometry. Where available, dispersion coefficients for the vibrational average are given in parentheses.)...

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 distinction here is that the kK calculated from Eq. (9.19) would be used in a linear driving force model for the actual uptake rate expression and an axial dispersion coefficient would be substituted into the pde. If however one simply desires to match the adsorption response or breakthrough curves then the kK calculated according to Eq. (9.20) would provide very satisfactory results for estimation of the length of the mass transfer zone. 

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. 

Here, is the distance between atoms i andj, C(/ is a dispersion coefficient for atoms i andj, which can be calculated directly from tabulated properties of the individual atoms, and /dampF y) is a damping function to avoid unphysical behavior of the dispersion term for small distances. The only empirical parameter in this expression is S, a scaling factor that is applied uniformly to all pairs of atoms. In applications of DFT-D, this scaling factor has been estimated separately for each functional of interest by optimizing its value with respect to collections of molecular complexes in which dispersion interactions are important. There are no fundamental barriers to applying the ideas of DFT-D within plane-wave DFT calculations. In the work by Neumann and Perrin mentioned above, they showed that adding dispersion corrections to forces... 

Rough estimates for axial dispersion coefficients can be made using random walk techniques, and these will be discussed in Section II,E. Also, a theory can be developed for predicting axial dispersion coefficients from radial dispersion coefficients which is the source of the dotted line of Figure 8. This will be discussed in Section II,D. Bischoff (B13), Fro-ment (F9), and Hofmann (Hll) have presented summaries of packed-bed data. 

Bischoff and Levenspiel (B14) considered this problem, and have presented design charts which allow estimation of errors in the calculated dispersion coefficients for various conditions. It was found that when the ratio of injection to tube diameter is less than 20%, or e < 0.2, then the assumption of a point source, or e-> 0, was good to within 5%. Thus for many cases, the neglect of the finite size of injection tube is justified. 

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

Prausnitz (PIO) has devised an approximate mixing length model for estimating the axial dispersion coefficient that allows for the interaction of the velocity profile. He used... 

Further details may be found in the above quoted references. In particular, de Josselin de Jong (D14) and Saffman (SI, S2, S3) give relatively rigorous developments that take into account the anistropy caused by the flow. Thus different estimates are obtained for the dispersion coefficients depending on whether or not the direction considered is perpendicular or parallel to the mean flow. 

As described in equation (6.59), longitudinal dispersion coefficient has a 67% confidence interval that is a factor of 1.7 times the best estimate. If the distribution of multiplicative uncertainty is normal, the 95% confidence interval would be at a factor of 3.4 times the best estimate. The reaeration coefficient has are MME of 1.8 for the Thackston and Krenkel equation (equation (9.7)). Again, if the multiplicative distribution is normal, the MME is 0.4 times the 95% confidence interval. Then the 95% confidence interval is a multiplicative factor of 4.5. 

The liquid-phase dispersion coefficient can be estimated using the Deckwer et al. (1974) correlation (Ramachandran and Chaudhari, 1980) ... 

As has been analyzed, the basic model for bubble column assumes complete mixed flow for the liquid phase and plug flow for the gas phase. The Deckwer el al. correlation (3.202) for the liquid phase and the Field and Davidson equation (3.206) for the gas phase can be used for the estimation of the dispersion coefficient. The resulting coefficients are Dll = 0.09 m2/s and DLG = 0.49 m2/s. 

Figure 24.8 Concentration of Xo estimate the dispersion coefficient Edjs we need the lateral turbulent diffusivity Ey measure t a)3 s a"ion°A (26 Tm (see Ecl- 24 45) which in tum is calculated from the friction velocity, u. Problem downstream of spill) and (b) 24.4 deals with the calculation of Edis. As it turns out, a realistic value which agrees...

There are no direct correlations of the variance (or the corresponding parameter n) in terms of the geometry and operating conditions of a vessel. For this reason the RTD is not yet a design tool, but it does have value as a diagnostic tool for the performance of existing equipment on which tracer tests can be made. RTDs obtained from tracer tests or perhaps estimated from dispersion coefficient data or correlations sometimes are applicable to the prediction of the limits between which a chemical conversion can take place in the vessel. 

The profiles of TCE concentration of the control and test columns were used to determine the effective dispersion/diffusion coefficients. As the TCE concentrations closed to the cathode were near zero during four weeks without EO (Figure 5), the variance of the four TCE concentration profiles in the control columns were used to determine the values of D/Rof TCE. The estimated Dd/R for TCE is 6.9 x 10 6 cm2/sec (equation (10)) with r-square of (0.99). Only the TCE profiles of the first week of the test column was used to determine the value of D as the peak of the TCE plume had advanced out of the boundary after two weeks of testing (Figure 6). The estimated D is 1.4 x 10 s cm2/sec. The results indicate that the effective dispersion coefficient is twice of the effective diffusion coefficient (D/R). 

In support of the WVDP, eight column tests were conducted at the University at Buffalo using WVDP groundwater spiked with nonradioactive Sr2+, over four durations 10, 20, 40, and 60 days. A single Kdof 2045 mL/g was calibrated from data from one of the 60-day columns, then used to successively predict the results for the other columns (Figure 5, 10-day data omitted for brevity). The importance of the specified boundary condition was highlighted by comparing results from various calibration schemes. For example, specification of a constant-concentration entrance boundary led to similar model fits but estimated Kd values that were 50% lower. Even when the recommended third-type BC was applied, efforts to simultaneously calibrate both the sorption and dispersion coefficient yielded similar fits for several combinations of parameters. Specification of the dispersion coefficient to a value obtained from an independent tracer test was necessary to obtain a robust estimate of the sorption coefficient. 

Knowing the viscosity and density of the reaction mixture, the flow channel diameter, void fraction of the bed, and the superficial fluid velocity, it is possible to determine the Reynolds number, estimate the intensity of dispersion from the appropriate correlation, and use the resulting value to determine the effective dispersion coefficient Del or I). Figures 8-32 and 8-33 illustrate the correlations for flow of fluids in empty tubes and through pipes in the laminar flow region, respectively. The dimensionless group De l/udt = De l/2uR depends on the Reynolds number (NRe) and on the molecular diffusivity as measured by the Schmidt number (NSc). For laminar flow region, DeJ is expressed by ... 

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