Dispersion error functions

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

The dispersion relation of (2.94) can be extracted if either or n is eliminated from (2.95). To obtain coefficients and, the maximum dispersion error will be initially expanded in a rapidly convergent series as a function of propagation angles 9 and ip and then its dominant terms will be set to zero. Hence, the coefficients, so computed, minimize the maximum dispersion error at all angles in completely adjustable way. 

According to the initial optimization concept, this technique launches a set of parametric expressions for the differential operators, whose evaluation is based on proper error estimators [59]. What is actually desirable — yet analytically not feasible — for these estimators is that their dependence on frequency and propagation angle should resemble that of the true dispersion error. Hence, an expansion of these expressions in terms of basis angular functions is performed, which enables the accuracy improvement regardless the direction of propagation. Depending... 

FIGURE 2.12 Dispersion error as a function of grid resolution for different FDTD configurations... 

FIGURE 6.4 (a) Maximum dispersion error as a function of CFLN and lattice resolution and (b) normalized phase velocity versus frequency for diverse ADI-FDTD configurations... 

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

The main consequences are twice. First, it results in contrast degradations as a function of the differential dispersion. This feature can be calibrated in order to correct this bias. The only limit concerns the degradation of the signal to noise ratio associated with the fringe modulation decay. The second drawback is an error on the phase closure acquisition. It results from the superposition of the phasor corresponding to the spectral channels. The wrapping and the nonlinearity of this process lead to a phase shift that is not compensated in the phase closure process. This effect depends on the three differential dispersions and on the spectral distribution. These effects have been demonstrated for the first time in the ISTROG experiment (Huss et al., 2001) at IRCOM as shown in Fig. 14. 

We now compare the PM3-D method with previous uncorrected DFT calculations on the S22 complexes [130], For the dispersion-bonded complexes the errors in the interaction distances for the PBE, B3LYP and TPSS functionals are reported to be 0.63, 1.16 and 0.69 A which are reduced to 0.17, 0.00 and 0.02 A when appropriate dispersive corrections are included. We see in Table 5-9 that the PM3-D method is capable of predicting the structures of dispersion-bonded complexes with greater accuracy than some uncorrected DFT functionals and with an accuracy comparable to that for the dispersion corrected PBE functional [130],... 

For small values of the dispersion parameter one may take advantage of the fact that equation 11.1.37 takes the shape of a normal error curve. This implies that for a step function input a plot of (C — Cq)/(Cq — Co) or F(t)... 

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

Pollock et al.(12) have also exploited the fact that poly dispersity index is a function of C2 only in a study utilizing a Monte-Carlo simulation technique to compare error propagation in the method of Balke and Hamielec to a revised method (GPCV2) proposed by Yau et al. (13) which incorporated correction for axial dispersion. 

Figure 1 2 10. The reduced Lifshitz parameter"z" - (ET - EF)/(EA- ET), where (EA- Er) is the full energy band dispersion in the c-axis direction, as a function of the number of holes in the G subband in A1 doped MgB2. The quantum uncertainty in the z value is indicated by the error bars that are given by D (

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