Diameter of a circle

The numerical values of and a, for a particular sample, which will depend on the kind of linear dimension chosen, cannot be calculated a priori except in the very simplest of cases. In practice one nearly always has to be satisfied with an approximate estimate of their values. For this purpose X is best taken as the mean projected diameter d, i.e. the diameter of a circle having the same area as the projected image of the particle, when viewed in a direction normal to the plane of greatest stability is determined microscopically, and it includes no contributions from the thickness of the particle, i.e. from the dimension normal to the plane of greatest stability. For perfect cubes and spheres, the value of the ratio x,/a ( = K, say) is of course equal to 6. For sand. Fair and Hatch found, with rounded particles 6T, with worn particles 6-4, and with sharp particles 7-7. For crushed quartz, Cartwright reports values of K ranging from 14 to 18, but since the specific surface was determined by nitrogen adsorption (p. 61) some internal surface was probably included. f... 

A chord of a circle (or sphere) is a line segment whose end points lie on the circle (or sphere). A line which intersects the circle (or sphere) in two points is a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord containing the center and a radius is a line segment from the center to a point on the circle (or sphere). 

To obtain a value for the dimensions of an irregular particle, several measurement approaches can be used Martin s diameter (defined as the length of a line that bisects the particle image), Feret s diameter (or end-to-end measurement, defined as the distance between two tangents on opposite sides of the particle parallel to some fixed direction), and the projected area diameter (defined as the diameter of a circle having the same area as that of the particle observed perpendicular to the surface on which the particle rests). With any technique, a sufficiently large number of particles is required in order to obtain a statistically valid conclusion. This is best accomplished by using a... 

D =(4SA/n)112 is the projected area diameter, equal to the diameter of a circle having the same area SA as the projected area of the particle resting in a stable position. For particles with size anisotropy, SA corresponds to the mean value derived from all possible orientations. 

Dp=PA/n is the perimeter diameter, equal to the diameter of a circle having the same perimeterPA as the projected outline of the particle. 

DIAMETER of a circle is a line segment that passes through the center of the circle whose endpoints are on the circle. The diameter is twice the radius of the circle d= 2 r. 

A mean projected diameter of the particle dp is defined as the diameter of a circle having the same area as the particle when viewed from above and lying in its most stable position. Heywood selected this particular dimension because it is easily measured by microscopic examination. 

Use the formula for the volume of a cone, replacing the Pwith 12% and the height, h, with r. Why replace the height measure with r The diameter of a circle is twice the radius, so the diameter is equal to 2r. If the diameter is twice the height, then the diameter, 2r = 2h. The length of the radius is equal to the height. 

In the preceding example and point injection the flow front will develop as a circular front, starting at the inlet, until it meets the closest side. From then on the front will tend to move unidirectionally in both directions toward the far sides (if there is no leakage at the sides). A reasonable estimate of the fill time is somewhere between the time to fill radially and the time to fill unidirectionally to the far side (flow distance 1.5 m). For the radial flow case L in Equation. 12.10 is 3 m (the diameter of a circle touching the wall farthest away). With the assumed alternative with unidirectional flow from the center toward both shorter sides, L in Equation. 12.10 is 1.5 m. It is also necessary to estimate an effective inlet radius, in this case an inlet radius of 5 mm is chosen yielding e = 3.3 10-3 that was used in the formula in Table 12.2. The constant C in Equation 12.10 is 0.5 for the unidirectional flow case and 0.65 for the radial flow case. The upper and lower limit for the fill time in this case are then ... 

Another method of characterizing irregular particles consists of reporting the diameter of a circle that projects the same cross section as the particle in question. This is done by... 

This measurement problem can be simplified somewhat by using the projected area diameter instead of Feret s or Martin s diameter. This is defined as the diameter of a circle having the same projected area as the particle in question. Figure 1.1 illustrates these three definitions. In general, Feret s diameter will be larger than the projected area diameter which will be larger than Martin s diameter. 

Numbers that are definitions and not measurements, such as the number of centimeters in a meter (100) or the number of radii in the diameter of a circle (2), are exact numbers. They do not limit the number of significant digits in a... 

T geometric constant relating circumference and diameter of a circle, 3.14159 dimensionless... 

Projected area diameter (d ) takes into account both dimensions of the particle in the measurement plane, being the diameter of a circle having the same projected area as the particle. It is necessary to differentiate between this diameter and the projected area diameter for a particle in random orientation (d ) since, in this case, the third and smallest dimension of the particle is also included. 

As with optical lenses, electromagnetic lenses have aberrations (chromatic aberration, spherical aberration, electron diffraction limit, astigmatism, etc.) each entailing an enlargement of the electron probe expressed by the diameter of a circle of least confusion. Under standard operating conditions, when the astigmatism has been adjusted, only spherical aberration plays a significant role. The expression of the probe diameter becomes ... 

The diameter of a circle with same projected area as the particle (in selected or random orientation)... 

An irregular particle can be described by a number of sizes. There are three groups of definitions the equivalent sphere diameters, the equivalent circle diameters and the statistical diameters. In the first group are the diameters of a sphere which would have the same property as the particle itself (e.g. the same volume, the same settling velocity, etc.) in the second group are the diameters of a circle that would have the same property as the projected outline of the particles (e.g. projected area or perimeter). The third group of sizes are obtained when a linear dimension is measured (usually by microscopy) parallel to a fixed direction. 

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