Chord length

Figure Bl.9.7. Diagram to illustrate the relationship between chord length (i) and r in a particle.

If we consider the chord length distribution, we can express the alternating chords and I2 as ... 

Several diameter definitions are used in particle image measurements (Fig. 7). Martin diameter, the chord length which divides the projected particle into two equal areas with respect to a fixed dkection (29) Feret diameter, the projected length with respect to a reference dkection (30) and the diameter of equivalent surface area, the diameter of a ckcle the area of which is equivalent to the projected area of the particle in question (3). 

In the Lasentec instruments a chord-length distribution is generated, from a rotating infrared beam, and this is converted to a size distribution. Since highly concentrated systems can be interrogated this system can be used for on-hne size analysis. 

With segmental downcomers the length of the weir fixes the area of the downcomer. The chord length will normally be between 0.6 to 0.85 of the column diameter. A good initial value to use is 0.77, equivalent to a downcomer area of 12 per cent. 

The unperforated area can be calculated from the plate geometry. The relationship between the weir chord length, chord height and the angle subtended by the chord is given in Figure 11.32. 

Figure 11.32. Relation between angle subtended by chord, chord height and chord length...

Cylindrical Filaments. If cylindrical filaments are studied, the last-mentioned assumption is not be fulfilled. In this case it is more suitable to cross an elongated24 primary beam and the fiber in such a way that the irradiated height of the beam, Bgo, is constant. Then the irradiated volume becomes V = Bbo nrj, with /y being the radius of the fiber. In many practical applications the studied fiber is very thin, and the effective absorption coefficient can be approximated by exp (—fiteff) 1. If this approximation is not feasible, the absorption factor exp (—ji ) must be integrated for the chord length distribution gc ( ) of a circle with radius ry (cf. p 168) in order to yield the effective absorption. 

Let us introduce the number-average chord length ( P) hy For one crystal (of convex shape) it is defined by its volume divided by the shade it throws when illuminated in direction (h). Then a series expansion... 

Determination of the Average Chord Length. The average chord length p can be determined even from scattering data that are not calibrated. For this purpose Ap is determined from Porod s law in relative units and k is computed by integration of the scattering curve. Finally ip is found from Eq. (8.43). 

The isotropic chord length distribution (CLD) is of limited practical value if soft matter with only short-range order is studied. Nevertheless, the related notions have been fruitful for the development of new methods for topology visualization from SAXS data. 

Figure 8.23. The chord length distributions g (r) and g (—r) found in the 2nd derivative y" (r) of the radial correlation function. The example shows g(r) of a suspension of 10 wt.-% of silica (reproduced from a handout of DENISE TCHOUBAR)...

Figure 8.24. Demonstration of the edge-enhancement principle built into the chord length distribution, (a) Two-phase structure intersected by a straight line, (b) The density along the line, (c) The derivative of the density is a sequence of 5-functions which are marking the positions of the domain edges...

Figure 8.35. The homogeneous sphere of radius R. Radial correlation function, ys (r), distance distribution function (DDF) ps (r) and chord length distribution (CLD) gs (r)...

Check. Do the moments reflect the requirement that gsi is normalized to 1 and that the average chord length of a sphere with unit diameter is... 

Figure 14. Variation of main chord length with wall spacing. lmf = 76 cm, probe height = 46 cm, X = 1.3 Ump O = 1.5Umy, V = Tdenotes with tubes. (From Glicksman and McAndrews, 1985.)...

Feret s diameter, l.) (the mean value of the distance between pairs of parallel tangents to the projected outline of the particle), and Martin s diameter, Du (the mean chord length of the projected outline of the particle[96] D DM=I 7.1. S A//b) are used in automated analysis of microscopic images [49,50],... 

FBRM provides a real time chord length distribution within the crystallizer. The technique is excellent for determining the onset of nucleation and in following general growth rate and rate transitions associated with polymorphic transformation, reaching equilibrium and attrition effects. 

A second quantity (physically equivalent to specific energy) is the lineal energy, which is the quotient of s by /, where 7 is the mean chord length in that volume ... 

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