Assumptions, particle size measurement

The theory and design of all particle size measurement devices are based on the assumption that particles are spherical. Most materials submitted for size analysis are not spherical. When these materials are analyzed by instruments employing different physical principles, the results are different. Figure 8 illustrates this point by comparing the cumulative (percent passing) distribution of three identical samples analyzed on three different instruments, each employing a different measurement technique. The absolute values of the distributions are different, but their shapes are quite similar. Because these results usually exhibit differences which are consistent, a correlation can be determined between one or more parameters of the distribution. 

Calculations of average particle size, specific surface area, or particle population of a mixture may be based on either a differential or a cumulative analysis. In principle, methods based on the cumulative analysis are more precise than those based on the differential analysis, since when the cumulative analysis is used, the assumption that all particles in a single fraction are equal in size is not needed. The accuracy of particle-size measurements, however, is rarely great enough to warrant the use of the cumulative analysis, and calculations are nearly always based on the differential analysis. 

Although particle size measurement methods mentioned in 3.2 earher provide information on surface area, those methods are based on the assumption that the particles are smooth sohd spheres without taking particle morphology into consideration. Unlike those methods, BET measurements are based on monolayer adsorption of gas molecules on the surface of sohd materials. These methods can also be used to reflect the surface morphology of particles (e.g., porosity). The BET theory was first pubhshed in 1938 by Brunauer et al. [67]. The BET theory is expressed in the equation in the succeeding text ... 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

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. 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

To measure the strength of the forces exerted on particles, various analytical techniques have been developed [6, 7]. Unfortunately, since most of these techniques are based on hydrodynamics, assumption of the potential profiles is required and the viscosities of the fiuid and the particle sizes must be precisely determined in separate experiments, for example, using the viscous flow technique [8,9] and power spectrum analysis of position fluctuation [10]. Furthermore, these methods provide information on ensemble averages for a mass of many particles. The sizes, shapes, and physical and chemical properties of individual particles may be different from each other, which will result in a variety of force strengths. Thus, single-particle... 

The separation and identification of flavanol-anthocyanin adducts in wine and in model solutions were performed with RP-HPLC coupled to DAD or ESI-MS. The investigation was motivated by the assumption that the formation of flavanol-anthocyanin complexes may influence the organoleptic characteristics of wine during ageing. Measurements were carried out in an ODS column (250 X 2 mm i.d. particle size 5 pm) at 30°C. The flow rate was 0.25 ml/min. Solvent A was water-formic acid (95 5), solvent B consisted of ACN— solvent A (80 20, v/v). The Gradient elution began with 3 per cent B for 7min to 20 per... 

To reconstruct an initial particle size distribution from observations at a later time, one must either have particle size distribution measurements at many different space-time points or make some assumption about the initial spatial distribution. The simplest assumption is that the initial spatial and particle size distributions are independent. 

If observations are made at two points in space, it is possible to check the assumption g(z) = constant by comparing the initial distributions obtained from the two measured values. In general, if observations are made at n different space (or time) points, and an initial distribution g(z) is assumed with m arbitrary parameters, these parameters may be determined from consistency relations among the various observations, and there will be an additional n-m consistency relation to test the validity of the assumptions. Note that each test is actually multiplied by the number of particle size divisions, so that the initial distribution function q n be determined quite accurately if multiple observations are available. 

Only then a correlation of the LII results with the iodine number is to be expected if different specific surfaces are caused by variations in the primary particle size and not in porosity. The assumption that LII determines a measure for the enveloping surface without consideration of small porosities has been examined in further measurements at a test reactor of the Degussa AG with finer parameter variations. The aim was to manufacture carbon blacks of same primary particle size, but different surface structure. 

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