Diameter dynamically equivalent

The seas may also act as a receptor for depositing aerosol. Deposition velocities of particles to the sea are a function of particle size, density, and shape, as well as the state of the sea. Experimental determination of aerosol deposition velocities to the sea is almost impossible and has to rely upon data derived from wind tunnel studies and theoretical models. The results from two such models appear in Figure 4, in which particle size is expressed as aerodynamic diameter, or the diameter of an aero-dynamically equivalent sphere of unit specific gravity.If the airborne concentration in size fraction of diameter d is c then... 

Very central to cyclone technology is the dynamically equivalent particle diameter. This is the diameter of an equi-dense sphere that has the same terminal velocity as the actual particle. Calculating this can be difficult in the range of intermediate Reynolds numbers, or when the Cunningham correction is significant. In the region where Stokes drag law applies, we call it the Stokesian diameter. 

Figure 2.3.2 from Kaye (1995) shows silhouettes of dynamically equivalent particles. The more nonspherical the actual particle, the larger it needs to be in order for it to settle with the same terminal velocity. The spheres to the right are Stokes diameters, those to the left aerodjmamic diameters. Since uranium dioxide is far denser than 1000 kg/m, the two diameters differ the most for this type of particle. 

The particle sphericity mainly enters the analysis because it influences the particle terminal velocity. We can account for its effect if we use the Stokesian diameter as a measure of particle size x rather than, for instance, a volume or mass equivalent diameter. We recall from Chap. 2 that the Stokesian (or dynamically equivalent ) diameter is the diameter of a sphere having the same terminal settling velocity and density as the particle under consideration. 

The cascade impactor and cyclone train also give us the appropriate distribution each captured fraction represents the volume (or actually mass) fraction of solids in the band between the cut diameters of two successive stages. This method also provides us with the dynamically equivalent particle size. 

Example 2-3 Scale-Up of Pipe Flow. We would like to know the total pressure driving force (AP) required to pump oil (/z = 30 cP, p = 0.85 g/cm3) through a horizontal pipeline with a diameter (D) of 48 in. and a length (L) of 700 mi, at a flow rate (Q) of 1 million barrels per day. The pipe is to be of commercial steel, which has an equivalent roughness (e) of 0.0018 in. To get this information, we want to design a laboratory experiment in which the laboratory model (m) and the full-scale field pipeline (f) are operating under dynamically similar conditions so that measurements of AP in the model can be scaled up directly to find AP in the field. The necessary conditions for dynamic similarity for this system are... 

Calculate the available net positive section head NPSH in a pumping system if the liquid density p = 1200 kg/m3, the liquid dynamic viscosity p = 0.4 Pa s, the mean velocity u = 1 m/s, the static head on the suction side 2, = 3m, the inside pipe diameter di = 0.0526 m, the gravitational acceleration g = 9.81 m/s2, and the equivalent length on the suction side SLes = 5.0 m. 

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

Particles used in practice for gas-solid flows are usually nonspherical and polydispersed. For a nonspherical particle, several equivalent diameters, which are usually based on equivalences either in geometric parameters (e.g., volume) or in flow dynamic characteristics (e.g., terminal velocity), are defined. Thus, for a given nonspherical particle, more than one equivalent diameter can be defined, as exemplified by the particle shown in Fig. 1.2, in which three different equivalent diameters are defined for the given nonspherical particle. The selection of a desired definition is often based on the specific process application intended. 

Example 1.1 One of the applications of using Stokes s law to determine the particle size is the Sedigraph particle analyzer. Table El.l shows the relationship between the cumulative weight percentage of particles and the corresponding particle terminal velocities for a powder sample. The densities of the particle and the dispersing liquid are 2,200 and 745 kg/m3, respectively. The liquid viscosity is 1.156 x 10-3 kg/m s. Find out the relationship of the mass fraction distribution to the equivalent dynamic diameter. 

The alloy is failed during some cycles of hydrogen sorption - desorption and turns into powder with particles 3-4 microns. The specific surface of such powder can be estimated with assumption of their spherical shape a (1.5-2.0 microns) with equivalent diameter of a particle ded=4s/[ 0(l- )]=1.3-1.6 microns. These values can be used in calculations of gas dynamics of hydrogen flow and heat exchange in a layer. 

Fig. 22, Hydrodynamic thickness vs. chain length plotted on a log-log scale. The points (squares and crosses) are data for PEO/PS latex/water measured through dynamic lightscattering (squares, Cohen Stuart et al., 1984c crosses, Kato et al., 1981). The curves are calculated by Ploehn and Russel (1989) using the SCF of Eqs. (71) and (72). Curves A and B denote frictions per segment equivalent to Stokes spheres of diameter l and 21, respectively. Curve C is the thickness based only on segments contained in loops (Stokes sphere diameter = /).

In order to develop the criterion more quantitatively, consider the sequence of phase-space portraits shown in Figs. 5.4(a) - (d). This sequence suggests that, as the control parameter K increases, the diameter of the resonance islands at Z = 0 mod 27t grows in action. In order to predict the touching point of the resonances, we need the widths of the resonances as a function of K. The width of the resonances is derived on the basis of the Hamiltonian (5.2.1). Since the dynamics induced by H is equivalent to the chaotic mapping (5.1.6), the Hamiltonian H itself cannot be treated analytically and has to be simplified. One way is to consider only the average effect of the periodic 6 kicks in (5.2.1). The average perturbation... 

The dynamic adsorption capacity of activated carbon containing monoliths has been shown to be equivalent to the micropore volume. However, this condition can only be met when the external area is above c. 100 m g" and the threshold diameter wide. In systems with no micropore volume or poor internal diffusion due to a low external surface area and narrow threshold diameter the breakthrough point is reached when c. 9% of the external area is covered. Future work will concentrate on using higher linear velocities and adsorption temperatures md different monolith geometries (wall thickness and channel width) in order to study the internal diffusion limitations of these types of adsorption units. 

Figure 4 shows the impact of the number of primary particles per aggregate and the coefficient of penetration on the result of a dynamic light-scattering particle sizing of those aggregates. The impact of /is very low. Therefore, it is now very easy to give the number of primary patticles as a result of a PSD measurement rather than the equivalent spherical diameter 2 n,yd. 

The differential head of the circulation water pump is relatively small, since dynamic losses are modest (short vertical pipe and a low AP spray nozzle) and the hydrauhc head is small, only about 6 m (20 ft) from the basin to the elevation of the spray header. Combined, the pumping energy demand is about 35 percent that for an equivalent CT application. The capital cost for this complete water system is also relatively small. The pumps and motors are smaller, the piping has a smaller diameter and is much shorter, and the required piping structural support is almost negligible, compared to an equivalent CT application. WSAC fan horsepower is typically about 25 percent less than that for an equivalent CT. 

Fig. 22 Mean diameter of vesicles measured by dynamic light scattering as a function of added molar equivalents of Triton X-100. Polymerized bis-SorbPC/EGDMA (filled triangles), un-polymerized bis-SorbPC/EGDMA (filled squares), and polymerized bis-SorbPC (inverted filled triangles). Reprinted with permission from [113]. Copyright 2006, American Chemical Society...

In the above, <7C is the equivalent volume diameter, i.e. the diameter of a sphere having the same volume as an irregular particle. As Hinds (1982) points out The equivalent volume diameter can be thought of as the diameter of the sphere that would result if an irregular particle were melted to form a droplet <7C is calculated from microscopic measurement of the actual particles being considered, while x is the dynamic shape factor which is included to allow for the effects of shape on terminal velocity. For example, talc dust is characterized by a dynamic shape factor (/ ) of 1.88, sand particles by 1.57, etc. Spheres have a dynamic shape factor of 1.0 while cubes have a dynamic shape factor of 1.08. 

As with most questions on particle size, the answer is very dependent on the definition used and the experimental technique. For a dynamic aerosol cloud, the correct definition is the aerodynamic particle size, which is the diameter of an equivalent sphere of unit density. An equivalent sphere is a conventional assumption in particle sizing, but for the aerodynamic size, the density is included to account for the momentum of the particle, i.e., both mass and velocity are important. The technique chosen for measurement must include these parameters, and impaction is the normally chosen technique, which also reflects the major deposition mechanism in the lung. A schematic of an impaction plate is given in Figure 10.3. 

Stokes Diameter What is the relationship connecting the volume equivalent diameter and the Stokes diameter of a nonspherical particle with dynamic shape factor X for Re < 0.1 ... 

Calculate the Stokes diameter of the NaCl particle of the previous example. The two approaches (dynamic shape factor combined with the volume equivalent diameter and the Stokes diameter) are different ways to describe the drag force and terminal settling velocity of a nonspherical particle. The terminal velocity of a nonspherical particle with a volume equivalent diameter Dve is given by (9.104),... 

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