Dispersion deviation

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

Kesner et al. [36] indicated a discrepancy between the theoretical retention and experimental values for some proteins in a channel with flexible membrane walls. These deviations were probably caused by the flexible membrane walls as such effects were not observed for a channel with rigid walls [256]. Nevertheless, the experimental data of Kesner et al. [36] was in reasonable agreement with the El-FFF theory [87,263] with respect to both retention and dispersion. Deviations were attributed to an electrical field gradient in the vicinity of the membrane interface [263]. Calculation of the dependence of K on 1/E from literature data on electrophoretic mobilities and diffusion coefficients confirmed the validity of the retention theory in El-FFF. 

The earlier types of optical glasses which were melted from the traditionally employed oxides, exhibited approximately direct proportionality between refractive index and dispersion. Deviation from this rule has arisen by the introduction of new components (B2O3, BaO, ZnO, LajOj, etc.)and of quite new types of glasses. The main types and their optical properties are shown in Fig. 145. The current optical crowns and flints occupy a comparatively small region compared with that corresponding to the attainable properties of special optical glasses. 

This is the second most important metric as it is the measure of dispersion deviation from mean in root squared form (see equation (21.3)). In other words, the standard deviation describes the spread of the data. 

The problem of material type determination was decided on the base of data obtained, that is the value of deviation dispersion characteristics from analogous ultrasonic field characteristics in the model heterogeneous flawless medium with given type of structure. 

Several ways may be used to characterize the spread or dispersion in the originai data. The range is the difference between the iargest vaiue and the smaiiest vaiue in a set of observations. However, aimost aiways the most efficient quantity for characterizing variabiiity is the standard deviation (aiso caiied the root mean square). 

Another measure of dispersion is the coefficient of variation, which is merely the standard deviation expressed as a fraction of the arithmetic mean, viz., s/x. It is useful mainly to show whether the relative or the absolute spread of values is constant as the values are changed. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

Some measure of dispersion of the subgroup data should also be plotted as a parallel control chart. The most reliable measure of scatter is the standard deviation. For small groups, the range becomes increasingly significant as a measure of scatter, and it is usually a simple matter to plot the range as a vertical line and the mean as a point on this line for each group of observations. 

Another parameter is called the standard deviation, which is designated as O. The square of the standard deviation is used frequently and is called the popular variance, O". Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean [L. If the spread is broad, then the standard deviation will be larger than if it were more constrained. 

Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for ( ) < 0.06 [Ref. 156] a = 6( )/<, where d p is Sauter mean diameter when 33% mass transfer has occurred. dp = particle or drop diameter <3 = iuterfacial tension, N/m ( )= volume fraction dispersed phase a = iuterfacial volume, 1/m and k OiDf implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 153. ... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

Emulsions Almost eveiy shear rate parameter affects liquid-liquid emulsion formation. Some of the efrecds are dependent upon whether the emulsion is both dispersing and coalescing in the tank, or whether there are sufficient stabilizers present to maintain the smallest droplet size produced for long periods of time. Blend time and the standard deviation of circulation times affect the length of time it takes for a particle to be exposed to the various levels of shear work and thus the time it takes to achieve the ultimate small paiTicle size desired. 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

A related measure of efficiency is the equivalent number of stages erkngof CSTR battery with the same variance as the measured RTD. Practically, in some cases 5 or 6 stages may be taken to approximate plug flow. The dispersion coefficient also is a measure of deviation... 

Axial Dispersion and the Peclet Number Peclet numbers are measures or deviation from phig flow. They may be calculated from residence time distributions found by tracer tests. Their values in trickle beds are fA to Ve, those of flow of liquid alone at the same Reynolds numbers. A correlation by Michell and Furzer (Chem. Eng. /., 4, 53 [1972]) is... 

Sutton Micrometeorology, McGraw-Hill, 1953, p, 286) developed a solution to the above difficulty by defining dispersion coefficients, O, Gy, and O, defined as the standard deviation of the concentrations in the downwind, crosswind, and vertical x, y, z) directions, respectively, The dispersion coefficients are a function of atmospheric conditions and the distance downwind from the release. The atmospheric conditions are classified into six stability classes (A through F) for continuous releases and three stability classes (unstable, neutral, and stable) for instantaneous releases. The stability classes depend on wind speed and the amount of sunlight, as shown in Table 26-28,... 

The random nature of most physieal properties, sueh as dimensions, strength and loads, is well known to statistieians. Engineers too are familiar with the typieal appearanee of sets of tensile strength data in whieh most of the individuals eongregate around mid-range and fewer further out to either side. Statistieians use the mean to identify the loeation of a set of data on the seale of measurement and the variance (or standard deviation) to measure the dispersion about the mean. In a variable x , the symbols used to represent the mean are /i and i for a population and sample respeetively. The symbol for varianee is V. The symbols for standard deviation are cr and. V respeetively, although a is often used for both. In this book we will always use the notation /i for mean and cr for the standard deviation. 

These equations are ealled the moment equations, beeause we are effeetively taking moments of the data about a point to measure the dispersion over the whole set of data. Note that in the varianee, the positive and negative deviates when squared do not eaneel eaeh other out but provide a powerful measure of dispersion whieh... 

To express the measure of dispersion in the original scale of measurement, it is usual to take the square root of the variance to give the standard deviation ... 

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. 

The curve in Figure IB is probably more useful from a practical point of view. Although the standard deviations of any dispersion process are not additive, they do give a better impression of the actual dispersion that a connecting tube alone can cause. It is clear that a tube 10 cm long and 0.012 cm I.D. can cause dispersion resulting in a peak with a standard deviation of 4 pi. Now, a peak with a standard deviation of 4 pi would have a base width of 16 pi and, in practice, many short... 

Intensity of turbulence These factors, represented by the standard deviations of the horizontal wind direction, Og, the standard deviation of the vertical wind component, a, and the gustiness as measured by the standard deviation of the wind speed, all have significant bearing on the dispersion of emissions from a stack. 

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