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Equation (2) depends solely on the number of faces of the growing grain.
(See M-curve in Fig. 3.) and increases with the number of grain faces.
It increases with an increase in the number of grain faces.
Grains with a large number of faces tend to have a large metric contribution whereas grains with a small number of faces tend to have a small metric contribution.
Distribution of the number of faces per grain Published papers on analytical grain size distribution[9] are rarely concerned with calculating the distribution of number of faces per grain.

The increased number and reduced size of TiB2 particles provided an enhanced grain refining capability.
The final grains per unit volume �V was defined as the number of growing grains when recalescence (temperature rise) occurred i.e. when the rate of latent heat evolution due to growth exceeded the rate of heat extraction, which could then be converted to a mean linear grain intercept distance l using the relationship [3]: 3 5.0 l�V =
At 1wt.% Al5Ti1B, the number of TiB2 particles in Al5Ti1B-2 was ~100 times more than that in Al5Ti1B-1, as shown in Table 2, resulting in a final Al grain diameter reduced by ~50%.
The number fraction of the "active" TiB2 particles with diameter larger than a critical value and that are initiating new grains is listed in Table 2.
Table 2 Total number of TiB2 particles in 1wt.% Al5Ti1B, and the active nucleating particles Alloy Total number of TiB2 particles Active nucleant diameter [µm] Active nucleant number fraction Al5Ti1B-1 1.71×1013 ≥3.6 0.13 Al5Ti1B-2 1.86×1015 1.48 1.68×10-3 However, there was a discrepancy between the calculated and measured grain sizes for Al5Ti1B-1.

For the purpose of obtaining mathematical force model in grinding process, the contacting grain numbers and the single grain forces should be taken into considered.
Thus in many previous researches of theoretical grinding force model, there are two parts commonly included: one is to obtain the dynamic grain numbers in grinding contact zone and the other part is to develop single grain force model.
The composite force is: (5) Where τ is shear stress and (6) (7) Fig.2 Top view of a cutting grain Fig.3 Force analysis The relationship between shear stress and shear strain is where is the shear yield strength of workpiece material, n is the enhancement coefficient (for GCr15, ), Combining Eq.4 and Eq.6-7, get: (6) (7) Decomposite at cutting direction and integrate them on , the tangetial and normal forces of a cutting grain are: (8) Results and discussion The total grinding forces can be obtained by single grain forces (Eq.3 and Eq.8) multiplying grain number of each type of grains.
The method of calculating grain numbers is referring to reference [6] based on probability statistics.
The grain diameter can be obtained from the partical size number of a grinding wheel, for example, for 60# wheel, the grain diameter is dgx=0.211~0.255mm and for 80# wheel, the grain diameter is dgx=0.152~0.178mm.

The number of boundary nucleated in each grain is less than three.
The nucleus number of every each grain internal or defects such as dislocation is less than three.
In addition, the number of the latter three nucleation position is variable. (2)The nucleus number Ne was determined or randomly determined in accordance with the simulation steps in whole grain nucleation process, the nucleus number Nn at simulation step and the grain nucleation position distribution B1:B2:B3.
Select a new grain from all the new grains Ne, find out all possible directions number ND which new grain will grow to next step.
New grain nucleus number is limited in transformation process.

The average orientation deviation and grain number is relatively insensitive to volume nucleation and thermal boundary conditions around grain selector, and the thermal boundary conditions in the top of grain selector being of lesser importance.
Grain Number in Spiral Grain Selector.
Grain number at different heights from the grain selector with different boundary conditions and nucleation parameters are compared in Fig. 4.
It appears that the grain number in spiral grains selector is relatively insensitive to changes in the volume nucleation and thermal boundary conditions around grain selector, and the thermal boundary conditions at the top of spiral grains selector being of lesser importance.
The grain number in spiral grains selector is relatively insensitive to changes in the volume nucleation and thermal boundary conditions around grain selector, and the thermal boundary conditions at the top of spiral grains selector being of lesser importance.

On the condition of the number of the primary particles is large enough and the size is relatively small, grain refining through semisolid processing can be achieved [6, 7].
Semisolid grain refining methods The potential semisolid grain refining methods majorly focus on the size and the number of the primary particles because grain refining aims to a finer grain size rather than the shape of particles or the viscosity of the slurry.
Semisolid grain refining method is one of typical physical approach, which controls liquid temperature, the size and number of primary solid particle size and size distribution.
To achieve grain refining effect, semisolid grain refining process should focus on the size and number of the particles rather than the particle shape.
It was emphasized that the size and the number of primary solid particles are important factors for grain refining rather than the particle shape or the viscosity of the slurry.

A filling-and-elimination counting method is also introduced to count the number of grains in the digital steel microscopic image.
The grain size is related to the number of grains in a unit area.
According to the close feature of the binary steel microscopic image, we introduced a filling-and-elimination method to count the number of grains [3].
Counting the number of grains in Fig. 3 by filling-and-elimination counting algorithm, the result is N=46.
The filling-and-elimination counting algorithm can count the number of grains in the image accurately.

Fig.2 shows that the number of grain at different rows changes with time.
The grain number in every row decreased with the evolution time increased.
But the grain numbers under the different field strengths were obviously different.
Fig.3 is the grain growing curves under the different numbers of the field variables.
The grain numbers under the different field strengths have been different.

There have been number of works to data connected with the studies of evolution of such ultrafine grain microstructures in Al - based alloys at low- to moderate deformation temperatures [e.g.1-3].
At the same time, only a limited number of studies were dealt with the evolution process during severe deformation of Al alloys at elevated temperatures.
This suggests that a main mechanism of grain refinement in the present Al alloy can be directly associated with grain splitting due to formation of microshear bands followed by increase in their number and misorientation.
Various shearing directions appearing during MDF and so the number of repeated compression passes can be more useful for mutual intersection of layered boundaries evolved and formation of equiaxed grains.
Further deformation leads to increase in the number and misorientation of these boundaries and finally almost full development of fine equiaxed grains in high strain.

Although these particles contributed little in terms of total volume to the preform, they were critical in determining the number density of embryonic grains.
However, Eq (5) indicates that the total number of embryonic grains was relatively insensitive - a cube route dependency - to the value of Gi chosen [2], and essentially there is always a very large number of embryonic generated.
Given the number of simplifications required in the procedure described here, calculated final grain diameters gave excellent agreement for both the mean and the distribution of final grain diameters in IN718 rings.
The model was used to calculate the embryonic grain diameters at deposition, accounting for the number and size of the solid fragments available for grain formation.
The very large number of small diameter and solid droplets in the droplet spray was dominant in controlling final grain sizes.