MCE311_HW02_sol_fall2011

MCE311_HW02_sol_fall2011 - 14.10 Given a large volume of...

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14.10 Given a large volume of metallic powders, all of which are perfectly spherical and having the same exact diameter, what is the maximum possible packing factor that the powders can take? Solution : The maximum packing factor is achieved when the spherical particles are arranged as a face-centered cubic unit cell, similar to the atomic structure of FCC metals; see Figure 2.8(b). The unit cell of the FCC structure contains 8 spheres at the corners of the cube and 6 spheres on each face. Our approach to determine the packing factor will consist of: (1) finding the volume of the spheres and portions thereof that are contained in the cell, and (2) finding the volume of the unit cell cube. The ratio of (1) over (2) is the packing factor. (1) Volume of whole and/or partial spheres contained in the unit cell. The unit cell contains 6 half spheres in the faces of the cube and 8 one-eighth spheres in corners. The equivalent number of whole spheres = 6(.5) + 8(.125) = 4 spheres. Volume of 4 spheres = 4 π D 3 /6 = 2.0944 D 3 where D = diameter of a sphere. (2) Volume of the cube of one unit cell. Consider that the diagonal of any face of the unit cell contains one full diameter (the sphere in the center of the cube face) and two half diameters (the spheres at the corners of the face). Thus, the diagonal of the cube face = 2
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MCE311_HW02_sol_fall2011 - 14.10 Given a large volume of...

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