Dispersion peak value

Conservation of mass suggests a relation between the product of the peak values of mean plume concentrations and the corresponding dispersion parameters (o-j., cr ) for all averaging times... 

We will use the means and 95% confidence intervals for both and K2 to determine the time, and distance through s = Ut, when Cpeak = 0.0005 g/m. These conditions are listed in Table E9.2.1, with the times determined through iteration on equation (E9.2.3). Table E9.2.1 shows that the peak value of concentration is no longer sensitive to longitudinal dispersion coefficient after roughly 3 days, because the peak is widely spread. The time when the water treatment plants downstream of the spill could turn on the water intake, however, would likely be sensitive to longitudinal dispersion coefficient. 

This shows again that when X is small (or equivalently, Per 1), we can combine the small axial dispersion term with the mixed derivative term and simplify the general hyperbolic model [Eq. (72)] to the simpler model [Eq. (57)]. However, for X values of order unity or larger, this cannot be justified. The inverse transform of Eq. (73) can be found by integrating around the branch points but we will not pursue it here. Instead, we show in Figs. 7 and 8 the numerically determined dispersion curves for r — 0.1,1 and various values of X. As can be expected, the qualitative behavior of the full hyperbolic model [Eq. (72)] is similar to that of the simpler case of X — 0. Only for X 1, the peak value changes and shifts to lower times. 

The activation energy (E ) was calcnlated nsing the Arrhenius equation. Figure 8.17 shows the relationships between the reciprocal temperature of the tanS peaks shown in Fignre 8.15 and the logarithm of the measured frequencies. From the gradient of lines shown in Fignre 8.17, Ti s were calculated 512 kJ/mol for a dispersion and 59 kJ/mol for p dispersion. The value of the a dispersion was consistent with the main chain motion of polymers [64]. The E of the P dispersion was almost the same as that of the relaxation of water sorbed on wood, as estimated by the dielectric measurement [36]. 

During the first hour, when the entire slug of copper is introduced, dispersion, dilution, and first-order kinetic reactions result in a two-dimensional pattern of labile copper of the shape illustrated in Figure 5. The pattern shows a distribution of concentration about the peak value as a result of initial mixing and subsequent dispersion. The peak is translated downcoast by the advective process and the distribution is skewed somewhat in this same direction as a result of the kinetic transformation of labile copper to bound and sorbed states. A peak value of 18.4 /xg/L is indicated which may be compared to a maximum possible value of about 40 /xg/L if the entire slug were simply mixed with the entire volume of the dispersion zone (the diamond-shaped area around the outfall). 

In contrast, for the undrawn PVA film, there were three different peaks from the high-temperature side termed the a, p, and y dispersion peaks. The E value of the PVA film decreased drastically in the temperature range from 5°C to 50°C. This temperature range is much lower than the endothermic peak of the PVA film (see Figure 9.4). The analysis of this peak shall be discussed in relation to the positron annihilation results later. 

MPa when 10 wt% functionalized CNTs was added. With the addition of 1 wt% reinforcement, a peak value of 4.1 kJ energy absorption was obtained. The homogenous dispersion of CNT-Phenol is thought to be responsible for the considerable enhancement in the reported properties [127]. 

The relaxation time i can be correlated with the dielectric dispersion peak frequency, fmax, at which the dielectric loss tangent shows a maximum value,... 

Figure 4.4 shows graphically the effect of the standard deviation A and the mean p on the probability density function p(x) for a normal distribution. As it can be seen, the standard deviation is a measure of data dispersion. Increasing A is representative of increasing data dispersion, while the peak value decreases. At variance, small A is associated with a narrow scatter and high peak value. But standard deviation has also a very precise meaning. In fact, on the distribution curve it identifies several particular points. This is graphically shown in Fig. 4.5 in which the entire bell shaped normal distribution curve is divided into sectors each of which has a width of a standard deviation. The characteristic feature is that ... 

In practice, experimental peaks can be affected by extracolumn retention and dispersion factors associated with the injector, connections, and any detector. For hnear chromatography conditions, the apparent response parameters are related to their corresponding true column value by... 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

Thus, the variance of the peak is inversely proportional to the number of theoretical plates in the column. Consequently, the greater the value of (n), the more narrow the peak, and the more efficiently has the column constrained peak dispersion. As a result, the number of theoretical plates in a column has been given the term Column Efficiency. From the above equations, a fairly simple procedure for measuring the efficiency of any column can be derived. 

The standard deviation of the extra-column dispersion is given as opposed to the variance because, as it represents one-quarter of the peak width, it is easier to visualize from a practical point of view. It is seen the values vary widely with the type of column that is used, (ag) values for GC capillary columns range from about 12 pi for a relatively short, wide, macrobore column to 1.1 pi for a long, narrow, high efficiency column. 

To determine the band dispersion that results from a significant, but moderate, sample volume overload the summation of variances can be used. However, when the sample volume becomes excessive, the band dispersion that results becomes equivalent to the sample volume itself. In figure 10, two solutes are depicted that are eluted from a column under conditions of no overload. If the dispersion from the excessive sample volume just allows the peaks to touch at the base, the peak separation in milliliters of mobile phase passed through the column will be equivalent to the sample volume (Vi) plus half the base width of both peaks. It is assumed in figure 10 that the efficiency of each peak is the same and in most cases this will be true. If there is some significant difference, an average value of the efficiencies of the two peaks can be taken. 

Although this eliminates negative contributions, since the imaginary part of the spectrum is also incorporated in the absolute-value mode, it produces broad dispersive components. This leads to the broadening of the base of the peaks ( tailing ), so lines recorded in the absolute-value mode are usually broader and show more tailing than those recorded in the pure absorption mode. 

Peaks in homonuclear 2D /-resolved spectra have a phase-twisted line shape with equal 2D absorptive and dispersive contributions. If a 45° projection is performed on them, the overlap of positive and negative contributions will mutually cancel and the peaks will disappear. The spectra are therefore presented in the absolute-value mode. 

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