Measurement sphericity

The estimation of dilute solutions, it is possible by (113) to determine average particle radius with great reliability. 

The laser diffraction particle sizer techniques have been used to measure spherical solid particle and droplet size distributions in dilute dispersions located in chemical reactor and separation equipment. 

The radiation and temperature dependent mechanical properties of viscoelastic materials (modulus and loss) are of great interest throughout the plastics, polymer, and rubber from initial design to routine production. There are a number of laboratory research instruments are available to determine these properties. All these hardness tests conducted on polymeric materials involve the penetration of the sample under consideration by loaded spheres or other geometric shapes [1]. Most of these tests are to some extent arbitrary because the penetration of an indenter into viscoelastic material increases with time. For example, standard durometer test (the "Shore A") is widely used to measure the static "hardness" or resistance to indentation. However, it does not measure basic material properties, and its results depend on the specimen geometry (it is difficult to make available the identity of the initial position of the devices on cylinder or spherical surfaces while measuring) and test conditions, and some arbitrary time must be selected to compare different materials. 

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

Small drops or bubbles will tend to be spherical because surface forces depend on the area, which decreases as the square of the linear dimension, whereas distortions due to gravitational effects depend on the volume, which decreases as the cube of the linear dimension. Likewise, too, a drop of liquid in a second liquid of equal density will be spherical. However, when gravitational and surface tensional effects are comparable, then one can determine in principle the surface tension from measurements of the shape of the drop or bubble. The variations situations to which Eq. 11-16 applies are shown in Fig. 11-16. 

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

An interesting historical application of the Boltzmann equation involves examination of the number density of very small spherical globules of latex suspended in water. The particles are dishibuted in the potential gradient of the gravitational field. If an arbitrary point in the suspension is selected, the number of particles N at height h pm (1 pm= 10 m) above the reference point can be counted with a magnifying lens. In one series of measurements, the number of particles per unit volume of the suspension as a function of h was as shown in Table 3-3. 

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

Hsieh and Jorgenson prepared 12-33- 4m HPFC columns packed with 5.44-pm spherical stationary phase particles. To evaluate these columns they measured reduced plate height, h,... 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

True Density or Specific Gravity. The average mass per unit volume of the individual particles is called the tme density or specific gravity. This property is most important when volume or mass of the filled composition is a key performance variable. The tme density of fillers composed of relatively large, nonporous, spherical particles is usually determined by a simple Hquid displacement method. Finely divided, porous, or irregular fillers should be measured using a gas pycnometer to assure that all pores, cracks, and crevices are penetrated. 

Emulsification is the process by which a hydrophobic monomer, such as styrene, is dispersed into micelles and monomer droplets. A measure of a surfactant s abiUty to solubilize a monomer is its critical micelle concentration (CMC). Below the CMC the surfactant is dissolved ia the aqueous phase and does not serve to solubilize monomer. At and above the CMC the surfactant forms spherical micelles, usually 50 to 200 soap molecules per micelle. Many... 

Size. The precise determination of particle size, usually referred to as the particle diameter, can actually be made only for spherical particles. For any other particle shape, a precise determination is practically impossible and particle size represents an approximation only, based on an agreement between producer and consumer with respect to the testing methods (see Size measurement of particles). 

Specific Surface. The total surface area of 1 g of powder measured ia cm /g is called its specific surface. The specific surface area is an excellent iadicator for the conditions under which a reaction is initiated and also for the rate of the reaction. It correlates in general with the average particle size. The great difference in surface area between 6-p.m reduced iron powder and 7-p.m carbonyl iron powder (Table 3) cannot be explained in terms of particle size, but mainly by the difference between the very inregular-shaped reduced and the spherical carbonyl iron powders. 

Deterrnination of the specific surface area can be made by a variety of adsorption measurements or by air-permeability deterrninations. It is customary to calculate average particle size from the values of specific surface by making assumptions regarding particle size distribution and particle shape, ie, assume it is spherical. 

Measurement of single particle settling velocity in a turbulent field is not easy. However, it is known to be a function of free settling velocity which for spherical particles can be estimated from the following ... 

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