Typical distributions

In the applications of gas-solid flows, there are three typical distributions in particle size, namely, Gaussian distribution or normal distribution, log-normal distribution, and Rosin-Rammler distribution. These three size distribution functions are mostly used in the curve fitting of experimental data. 

The Gaussian distribution, also known as the normal distribution, has the density function 

For a given sample, the particle size range is bounded by d and di as shown in Fig. 1.8. Thus, Eq. (1.16) becomes 

Given the number density function of Eq. (1.24), the corresponding mass density function becomes 

There is no simple, exact, and explicit expression for AM. However, for the case of a very narrow size distribution where 7d/Jo C 1, AN and Am are given by 

---

The only method utilized commercially is vapor-phase nitration of propane, although methane (70), ethane, and butane also can be nitrated quite readily. The data in Table 5 show the typical distribution of nitroparaffins obtained from the nitration of propane with nitric acid at different temperatures (71). Nitrogen dioxide can be used for nitration, but its low boiling point (21°C) limits its effectiveness, except at increased pressure. Nitrogen pentoxide is a powerful nitrating agent for alkanes however, it is expensive and often gives polynitrated products. 

Typical distributions of four commercial processes are compared in Figure 1. 

Wax Cracking. One or more wax-cracked a-olefin plants were operated from 1962 to 1985 Chevron had two such plants at Richmond, California, and Shell had three in Europe. The wax-cracked olefins were of limited commercial value because they contained internal olefins, branched olefins, diolefins, aromatics, and paraffins. These were satisfactory for feed to alkyl benzene plants and for certain markets, but unsatisfactory for polyethylene comonomers and several other markets. Typical distributions were C 33% C q, 7% 25% and 35%. Since both odd and... 

Refer to a typical distribution network shown in Figure 2.5..5 I. The capacitor is of maximum use when located as near to the load-point as possible, especially in induction motors, because ... 

Individuals differ in their sensitivity to odor. Figure 14-7 shows a typical distribution of sensitivities to ethylsulfide vapor (17). There are currently no guidelines on inclusion or exclusion of individuals with abnormally high or low sensitivity. This variability of response complicates the data treatment procedure. In many instances, the goal is to determine some mean value for the threshold representative of the panel as a whole. The small size of panels (generally fewer than 10 people) and the distribution of individual sensitivities require sophisticated statistical procedures to find the threshold from the responses. 

The primary cause of efficiency losses in an axial-flow turbine is the buildup of boundary layer on the blade and end walls. The losses associated with a boundary layer are viscous losses, mixing losses, and trailing edge losses. To calculate these losses, the growth of the boundary layer on a blade must be known so that the displacement thickness and momentum thickness can be computed. A typical distribution of the displacement and momentum thickness is shown in Figure 9-26. The profile loss from this type of bound-ary-layer build-up is due to a loss of stagnation pressure, which in turn is... 

The resin acids found in rosins are generally of the abietic- and pimaric-type. Rosins of various pine species differ in their content of abietic vs. pimaric-type acids. Rosins from species exhibiting high abietic-type acid compositions are preferred for production of rosin derivatives. However, the differences in properties of rosins are often associated with their non-resin acid content instead of their chemical compositions. On the other hand, the compositions of rosins from different sources greatly differ [22]. Table 8 shows a typical distribution of resin acids in rosins obtained from gum, tall oil and wood sources. 

Table I summarizes some typical distribution coefficients. Sediments become enriched in plutonium with respect to water, usually with a factor of vlO5. Also living organisms enrich plutonium from natural waters, but usually less than sediments a factor of 103 - 101 is common. This indicates that the Kd-value for sediment (and soil) is probably governed by surface sorption phenomena. From the simplest organisms (plankton and plants) to man there is clear evidence of metabolic discrimination against transfer of plutonium. In general, the higher the species is on the trophic level, the smaller is the Kd-value. One may deduce from the Table that the concentration of plutonium accumulated in man in equilibrium with the environment, will not exceed the concentration of plutonium in the ground water, independent of the mode of ingestion.

TABLE 10 Typical Distribution of the Alcohols in the Alfol and Epal Processes... 

Fig. 14-5 Typical distribution of P and temperature in a temperate lake in summer. Thermal stratification restricts exchange between surface and deep wafers. Phosphorus is depleted in the surface waters by the sinking of biologically produced particles.

Figure 1.14 shows a typical distribution for the geochemically abundant elements in crustal rocks. It could be seen that the proportion of the volume of material available for exploitation increases in geometrical progression as grade falls in arithmetical progression. In a sense, therefore, there is no finite limit to the availability of such elements, however, dilution with host rock implies that revenue would be insufficient to cover the fixed cost of extraction. 

Figure 1.14 Curve showing the typical distribution of the geochemically abundant elements.

A typical distribution for the geochemically scarce elements is shown in Figure 1.15. These are usually present as atomic substitutes in refractory silicate minerals. However, geological processes lead to their preferential concentration in sulfides. 

The above catalyst system is long-lived and like the Miller catalyst, it formed primarily 1,4-hexadiene, 3-methyl- 1,4-pentadiene, and 2,4-hex-adiene from ethylene and butadiene. A typical distribution of products formed by this catalyst and by the (Bu3P)2NiCl2/i-Bu2AlCl catalyst is shown in Table IX. The improved conversion and yield can be attributed to a better cocatalyst system, as shall be discussed later. Su and Collette s studies are summarized in the following discussions. 

Each phospholipid class in a given tissue has a characteristic fatty acid composition. Though the same fatty acid may be present in a number of lipids, the quantitative fatty acid composition is different for each class of lipids and remains fairly constant during the growth and development of the brain. A typical distribution profile of the major fatty acids in rat brain phospholipids is given in Table 3.1. Not only do the phosphoglycerides differ in the structure of the polar head groups, or phospholipid... 

J. Stockmann, 1999). The main achievement of this field is the establishment of universal statistics of energy levels the typical distribution of the spacing of neighbouring levels is Poisson or Gaussian ensembles for integrable or chaotic quantum systems. This statistics is well described by random-matrix theory (RMT). It was first introduced by... 

It should be noted that the plant results are variable and depend on the type of ore being treated. Typical distributions of REO in the Mountain Pass concentrate are shown in Table 24.5. 

A typical distribution is portrayed in Table 11 for a configuration of fifty cells arranged in five ten-cell banks, where for the sake of safety, a short-circuited cell will disconnect an entire bank. This arrangement is essentially identical to a five-cell bank with single short-circuits. The more efficient (i.e., the faster) the repair service (i.e., the smaller X/p), the smaller the probability that short-circuited cells are not repaired at any arbitrary time. Conversely, if repair takes a long time, a certain number of cells will always be short-circuited, and if kip is extremely large, a complete shutdown of cell operation will be highly probable (the Xlp = 10, ps 0.9 case is a case in point, albeit not overly realistic.)... 

FIGURE 10.9 Typical distribution pattern for oxygen and carbon dioxide in and around a hydrocarbon plume in porous unsaturated soil. 

A histogram of the consumption of carbonated soft drinks by UK adults and pre-school children is provided in Fig. 4.2. This shows a typical distribution with a wide range of values. In this case only about one half of respondents reported soft drink consumption during the survey and those that did consume show a... 

Fig. 5a-c. A typical distribution of polycyclic aromatic hydrocarbons in a atmospheric fallout sample, Alexandria City - Egypt b bottom incineration ash leachate of municipal solid waste - USA c hydrothermal petroleum, Escanaba Trough, NE Pacific Ocean. PAH Compound identifications N = naphthalene, MN = methylnaphthalene, DMN = dimethylnaphthalenes, P = phenanthrene, MP = methylphenanthrene, Fl = fluoranthene, Py = pyrene, BaAN = benzol anthracene, DH-Py = dihydropyrene, 2,3-BF = 2,3-benzofluorene, BFL = benzo[fc,/c]fluoranthene, BeP = benzo[e]pyrene, BaP = benzo[a]pyrene, Per = perylene, Cx-228 = methyl-228 series, Indeno = indeno[ l,2,3-c,d]pyrene, DBAN = dibenz[a,/z]anthracene, BPer = benzo[g,/z,z] perylene, AAN = anthanthrene, DBTH = dibenzothiophene, Cor = coronene, DBP = dibenzo [a,e]pyrene, DBPer = dibenzo [g,h,i] perylene... 

Fig. 11 a-d. A GC-MS trace showing a typical distribution of a pesticide polluted sample from the Los Angeles Bight... 

Fig. 2.21 Typical distribution of fiber diameters in continuous filament materials and insulation wools. (A) Continuous filament glass. The mean diameter is 14.1 micrometers, with standard deviation of 1.9 micrometers. (B) Glass fiber wool. Mean diameter is 7.0 micrometers, with a standard deviation of 4.0 micrometers.

In developing some of the relationships, it is helpful to use a four-quadrant diagram in which each quadrant represents a species in a lipid or water phase. The diagram below shows a typical distribution of an acid, AH, between two phases where ion partitioning is assumed to be negligible. The partition coefficent, P, is the ratio of the concentration of AH in the octanol to the concentration of AH in the aqueous phase. The distribution coefficient, D, is the ratio of the concentration in the octanol to that of all forms in the water. This is also called the apparent partition coefficient. 

In fines removal, both the cut size and the grade efficiency are difficult to assess because of the limited accuracy of the sieve analysis technique and the problems Involved in the determination of the solids concentration in the overflow. For a. 65 m cyclone, whilst using a 20 mm vortexfinder diameter, an apex diameter of 16 mm and a feed flow of 1.6 1/s, solids recovery is over 99 % This recovery corresponds to a cut size between 50 - 100 pm. Typical distributions of size by weight, for the feed flow as well as the overflow are shown in Figure 5 Results are summarised in Table 2. 

Fig. 4. Typical distribution of residence times of fluid flowing through a vessel (L13).

Figure 11-4 Typical distribution of polymer sizes determined on molecule and weight bases.

The overall workflow of ADME/Tox characterization of lead compounds is typically distributed across multiple departments or functional groups within pharmaceutical companies, often with specialized groups for different assays, analysis and interpretation. A representation of the overall workflow is provided in Figure 1.1. 

A typical distribution of pore radii in the hardened cement paste of concrete was shown in Fig. 1.40 which indicated that the majority of pores lie in the region of 0.05 and 1.0 pm diameter and it is through these pores that water passes by applied pressure or capillary rise, as shown in Fig. 4.5(a). 

---