Series of mathematical relationships predicts compact densities of spherical shapes in a cylinder as a function of particle dimension, and compact density of angular shapes as a function of particle shape and absolute size

Ayer, J. E.

Soppet, F. E.

English

NASA Technical Reports Server

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December 1967	 Brief 67-10592 
ICY C
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Mathematical Relation Predicts Achievable Densities of Compacted Particles 
F_ 
6_1
The problem: 
To develop a method of predicting achievable den-
sities of compacted particles in a cylindrical container. 
Until the present time, no simple method has been 
suggested for predicting the uniform densities of 
compacted particles. 
The solution: 
A series of mathematical relationships for pre-
dicting (1) compact densities of spherical shapes in 
a cylinder as a function of particle dimension, and 
(2) compact density of angular shapes as a function 
of particle shape and absolute size. 
How it's done: 
An investigation was carried out to determine the 
mathematical relation of the diameter of spherical 
particles and their density when compacted. A second 
study was carried out to relate the packing char-
acteristics of angular particles to the model case of 
spheres. 
The relation for the packing efficiency of spherical 
particles in a cylinder was developed as: 
Pe = 0.635 - 0.219e_0313 D/d1 
Further, the relation for the packing efficiency of 
spherical particles within a matrix of spherical par-
ticles is:
Pe = 0.635 - 0.737e0201 d/d 
where Pe is the packing efficiency 
D is the diameter of the cylinder 
d1 is the diameter of the spherical particles 
d/d is the ratio of diameters of larger to smaller 
spherical particles 
The predicted densities show agreement within 
±0.015 with densities achieved in practice. Prediction 
of achievable densities of spherical particles or aggre-
gates of angular particles by these methods reduces 
the time and inaccuracies involved in graphical solu-
tions or extrapolations from similar systems.
Notes: 
1. The compaction of spherical shapes was done using 
a vibrator, an oscillator, a bank of amplifiers, and 
an accelerometer. Various sizes of steel shot and 
bearings were compacted together in a cylinder; 
first compacting the largest size, thçn_ihe next 
largest, and then the smallesr size. The results were 
interpreted from graphs plotting void packing effi-
ciency versus sphere-to-sphere and container-to-
sphere diameter ratios. 
2. The compaction of angular shapes was done simi-
larly, but using various sizes of malleable grit as 
the particles. Two types, ball-milled (tetragonal) 
and angular grit, were used in this study. The re-
suits were interpreted from graphs plotting void 
packing efficiency of angular grit versus ratio of 
diameter of container to particle dimension, and 
plotting void packing efficiency of ball-milled grit 
versus ratio of diameter of container to particle 
dimension. 
3. Additional details may be found in the following 
issues of the Journal of the American Ceramic 
Society, (a) Vol.. 49, No. 4, April 1966, pp. 207-
210; (b) Vol. 48, No. 4, April 1965, pp. 180-183. 
Information is also contained in Patent No. 
3,261,378, which is available from the U.S. Patent 
Office $0.50 each. 
4. Inquiries concerning this innovation may be di-
rected to:
Office of Industrial Cooperation 
Argonne National Laboratory 
9700 South Cass Avenue 
Argonne, Illinois 60439 
Reference: B67-10592 
Source: J. E. Ayer and F. E. Soppet 
Metallurgy Division 
(A RG- 10082) 
(continued overleaf) 
Now
This document was prepared under the sponsorship of the Atomic
	 liability resulting from the use of the information contained in this 
Energy Commission and/or the National Aeronautics and Space document, or warrants that the use of any information, apparatus, 
Administration. Neither the United States Government nor any per-
	 method, or process disclosed in this document may not infringe pri-
son acting on behalf of the United States Government assumes any vately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19670000591 2020-03-11T21:31:13+00:00Z
Patent status: 
Inquiries about obtaining rights for commercial 
use of this innovation may be made to: 
Mr. George H. Lee, Chief 
Chicago Patent Group 
U.S. Atomic Energy Commission 
Chicago Operations Office 
9800 South Cass Avenue 
Argonne, Illinois 60439
C] 
.. 
Brief 67-10592	 Category 03


Mathematical relation predicts achievable densities of compacted particles

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