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NP1 - the number of grains in the sample cross-section (30 mm diameter test area).
NP2 - the number of grains in the sample centre (15 mm diameter test area).
In the second case, the number of grains was counted by a modified Jeffries–Saltykov method, where full-size grains and intersected grains are counted under the same conditions.
Changes in the shape and number of austenite grains were observed depending on the type of the inoculants applied.
The measurements taken in this area (including the total number of grains) showed an increased number of the primary austenite grains in casting inoculated with Fe-Si and iron powder.

The number of grains however depends on the gauge volume size chosen and the average size of the grains (and hence number of diffracting grains being detected) within the gauge volume.
The influence of grain size statistics depends upon the number of diffracting grains within the sampling or gauge volume.
Using Eq. 2 enough times (using random number generation) one can estimate the standard deviation in 2θ versus the number of grains 'seen' on the detector
Therefore the grain size can be estimated to be ≈(gauge volume/number of grains) 0.333.
The number of grains within SD=±0.2 can be found by diving the actua1 number by 1291 for the hkl 222 and 430 for the hkl 311 as calculated above.

Irregular grains were classified topologically by the number of their contacting neighbors, N.
An entire grain network could be sorted into its component polyhedra, each represented by its unique topological proxy, viz., the "average N-hedron" (ANH) that possesses the same number of faces, N.
In other words, imposition of network topology and the global geometry of a polyhedral grain actually combine to require that a grain's face curvature is, implicitly, a function of both the total number of faces and the grain boundary energy.
(2) Here, da/dt is the time rate of change of the grain area; M is the grain boundary mobility; γ is the grain boundary energy per unity length; κ is the in-plane curvature of the boundary; and l is the arc length; n the number of sides or borders enclosing the two-dimensional polygonal grain.
Also, increasing the exterior dihedral angle, ε=π/2, drastically reduces the number of faces for a grain to grow to just N=7.

The coarsening and grain growth were affected by the ratio of grain boundary energy to surface energy, the ratio of grain boundary mobility to surface mobility, the size of a particle, and its coordination number.
Effect of Particle Coordination Number.
The coarsening and grain growth of a particle is affected by the number of neighbor particles, or coordination number.
The slope of curves is approximately proportional to the number of necks that is the coordination number n .
The von Neumann-Mullins law [10] states that the growth rate is a function of the coordination number in grain growth.

The topological relationships between number of grain faces, grain size, number of corners and edges and how these can be calculated in a real microstructure with a statistical approach are discussed.
The relevant result of the Eq.1 is that grain with the number of sides exceeding 6 will grow whereas grains with number of sides below 6 will shrink.
The probability factor wij In a statistical ensemble of grains of size i=1, 2, …, nc (where nc is the number of size classes in the system) a simple contact symmetry law must be valid, namely the number of contacts that the grains of size i have with grains of size j must be equal to the number of contacts that the grains j have with grains of size i.
The special relationship (number of faces-grain size) in 3-D Eq. (26) and Eq. (32) can be used to calculate the average number of faces of a grain of size i by the following expression: (33) Namely this is a general expression for the number of faces of a generic grain i as a function of its geometrical parameters scaling with the grain surface Ai but, due to the dependency of Pij from i, in a not linear way.
An approximation for the special relationship (number of faces- grain size) in 3-D In the same way one can get a simplified equation for the number of faces of a grain in 3-D given by Eq. (33).

In addition to grain size, a broad range of other metrics such as the number of sides and the average side class of nearest neighbors is used to compare the experimental results with the results of two dimensional simulations of grain growth with isotropic boundary energy.
Measurement of film grain size, number of sides and side class of nearest neighbors was done from hand traced boundary networks of images obtained by transmission electron microscopy.
The metrics used for comparison of experiments and simulation were grain size (as grain area or as the diameter of a circle with area equal to the mean grain area), number of sides, side class of nearest neighbors (i.e., average number of sides of the nearest neighbors of grains with a given number of sides), and combined geometry-topology metrics such as the size of grains with a given number of sides.
This narrowing is manifested as a large drop in the number fraction of four-sided grains in the stagnant structure and a large increase in the fraction of six sided grains when compared with experiments.
Using the class of neighbors metric (i.e., the average number of sides of the nearest neighbors of grains with given number of sides) for comparison, experiments and two dimensional simulations of normal growth show a clustering in which few-sided grains are neighbored by grains with larger number of sides and vice-versa, i.e., a downward trend as a function of the number of sides.

In this paper, length of grain-to-grain contact, number of contacts per grain and ratios between the length of contacts and geometric characteristics of grains in heterogeneous Berea sandstone and Indiana limestone are studied.
Most rocks, however, are heterogeneous with complicated grain/cement configurations, various grain contact types (grain contact lengths and number of contacts per grain), different grain shapes and degree of interlocking.
However, such a coefficient does not take into account the grain boundary parameters (length of ‘grain-to–grain’ contact and number of contacts per grain).
Granular rocks are stiffer when the contact length and number of contacts per grain are larger [11, 12].
As a result, such grain boundary parameters as grain diameter (D), length of contacts (L = L1 +L2 +L3) between neighbouring grains, number of contacts per grain (n), and perimeter (P) and area (S) of a grain were measured.

For a material to have better magnetic properties after secondary recrystallization, its primary recrystallized texture should have not only larger number of ideal Goss grains, but also lower frequency of low angle grain boundary around those Goss grains.
According to CSL theory, growing Goss grains meet large numbers of CSL grain boundaries.
There are many big Goss grains, island grains inside big Goss grains and small grains neighboring big Goss grains.
Based on above experimental results, the primary recrystallized texture for better magnetic properties should have not only larger number of ideal Goss grains, but also lower frequency of low-angle grain boundary around those Goss grains.
For a material to have better magnetic properties after secondary recrystallization, its primary recrystallized texture should have not only larger number of ideal Goss grains, but also lower frequency of low angle grain boundary around those Goss grains.

A comparison of the resulting microstructure and grain boundary characteristics from the two specimens revealed that the brass specimen had approximately the same number fraction of Σ3s as the copper specimen (38%), but a lower number fraction of Σ9s and Σ27s and a markedly different microstructure.
Care must be taken to extract spurious grains (e.g. those with only one neighbour or below a certain size limit) from the maps and to eliminate noise in them in order to obtain an accurate number fraction.
The proportions of Σ3, Σ9, Σ27, Σ81 and Σ243 are shown in figure 2, where the grain boundary statistics are compiled according to both length and number.
Proportions of Σ3n boundaries in GBE copper and brass, calculated according to grain boundary length (L) and number (N). 020 40 60 3 9 27 81 243 Sigma Proportion (%) L N L N Copper Brass Conclusions GBE processing of copper specimens has resulted in 41% number fraction of Σ3s and 12% Σ9.
Data from GBE brass have approximately the same number fraction of Σ3s (38%) than the GBE Cu, a lower number fraction of Σ9s and Σ27s but a markedly different microstructure, where twins are not incorporated into the grain boundary network and instead the network is modified indirectly via the modification to the boundary crystallography that results from twinning.

The study on the establishment of the variety of forms of grains for abrasives Russian and German production made standard way - by ebb abrasive ingots, their crushing and screening a number of particle size fractions.
A pilot batch of grinding wheels with controlled grain shape, are prepared by separating the initial mass of abrasive on a number of fractions with the same shape of grains using the vibratory separator.
Full range of aspect ratios from Kf =1.0 to Kf > 3.0 is divided into six intervals , each of which is put down as a percentage of the number of grains form in relation to the total number of grains examined.
Sort conducted using a vibratory separator, which allows you to split the initial mass of abrasive number of factions, where grains have approximately the same shape [6].
Moreover, along with sorting initial weight abrasive number of factions with the same shape of grains can also realize the way of manufacturing a predetermined shape grains (like manufacturing technology sphere- and - formokorund).