Equal Spheres Research Articles

The cylindrically ordered packing of equal spheres

A homogeneous ordered packing of equal spheres can be formed by vibration in vertical cylindrical vessels with certain base shapes. The shapes are described. The packing consists of a concentric array of cylindrical “shells” of spheres which fills the vessel. The spheres in a shell are arrayed hexagonally, and their fitting against those in an adjacent shell is cyclic and undulatory with respect to a circle concentric with the shells. A theory based on the observed structure yields infinite-bed values for some parameters of the packing which are consistent with extrapolated experimental values. In particular, the infinite-bed void fraction is about 0.29. The number of shells in a bed with given vessel-to-sphere diameter ratio is predictable over certain ranges of the ratio.

Pressure drag measurements for turbulent air flow through a packed bed

AbstractPressure drag coefficients of spheres in a cubic packing of equal spheres have been determined at Reynolds numbers of 27,000, 10,000, 5,000 and 2,500, where NRe is based on superficial air velocity and a sphere diameter of 7 cm. Two cubic arrangements were used: in the regular arrangement, the mean flow was parallel to one of the principal axes, while in the skewed arrangement, the mean flow made equal angles with the three principal axes of the packing.The pressure drag coefficient of the central sphere has been measured for each of the 12 banks in the regular arrangement, and the effect of bed length has been determined for lengths varying between 1 and 12 banks. The pressure drag coefficient for the central sphere and the overall pressure drop in the skewed arrangement were found to be lower than for the regular arrangement at the same Reynolds number. From the distribution of local pressure measurements, the separation and reattachment points on the central sphere in each bank were determined, indicating that the boundary‐layer behavior on a sphere in a packing is similar to that over a single sphere, when allowing for the effects of turbulence and of pressure gradient.

A model of the pore space in a random packing of equal spheres

A random sphere packing is described in terms of tetrahedral subunits and a method of generating these tetrahedra is given. Analyses of the properties of an assembly of these tetrahedra give results which agree well with known properties of random sphere packings. Some capillary properties of the assembly are derived using an approximation for the draining and filling curvatures of the pore spaces.

Random Heap Structure of Molten Salts

BERNAL1 has proposed that the geometrical structure of simple monatomic liquids is similar to that of a random heap of hard spheres. From studies of random heaps of ball bearings, Bernal has shown that the maximum volume packing density, ρ, of “random close-packed” heaps of equal spheres is 0.637. (ρ is 0.741 for crystalline close-packing.)

Random packings and the structure of simple liquids. I. The geometry of random close packing

In his Bakerian Lecture, Bernal (1964) discussed those ideas of restricted irregularity which are physically realized in random packings of equal hard spheres, with particular reference to the structure of simple liquids. He stressed the need for a science of ‘statistical geometry ’, and took the first steps himself by proposing possible ways of describing such arrays. In this paper, these and other associated ideas are briefly described and extended by deriving an equivalent set of polyhedral subunits essentially inverse to the packing in real space. Examination of two independent high density arrays demonstrates the repro-ducibility of certain metrical and topological properties of these polyhedra, and their correlations over larger elements of volume. As a result, several possible ‘descriptive parameters’ are proposed. Although these essentially ‘numerical’ characteristics facilitate sensitive structural descriptions of any assembly of micro- and macroscopic subunits, we are still unable to characterize an irregular array in formal mathematical terms. Such a formulation of statistical geometry could be a powerful tool for tackling important problems in many branches of science and engineering.

Numerical Calculation of Mutual Capacitance between Two Equal Metal Spheres

The alternative method of solution by image charges is presented to the electrostatic problem of finding mutual capacitance between two metal eccentrical spheres. Numerical results and graphs for calculation of mutual capacitance between two equal metal spheres are presented for a number of relative sphere sizes and separations.

A NEW CONSTRUCTION IN THE THEORY OF LATTICE COVERINGS OF ANn-DIMENSIONAL SPACE BY EQUAL SPHERES

This article furnishes a new construction principle in the theory of lattice coverings of an n-dimensional space by equal spheres. The principle is based on the convexity of a body V in the parameter space. The whole theory is deduced from association of this convexity with strict convexity of the discriminant body.

Statistical Geometry of Random Heaps of Equal Hard Spheres

Using the porosity and average number of contacts to characterize a bed of randomly packed particles, the numbers of contacts and near neighbours can be predicted by a set of statistical equations.

A photoelastic investigation of contact stresses in equal spheres rolling together with spin

Photoelastic tests have been performed to investigate the effect of spin on the contact stresses between rolling bodies. All the tests employed equal spheres, resulting in nominally flat, circular contact areas. The shear-stress distributions on such contact areas have been experimentally determined by the ‘frozen stress’ technique. One set of tests examined the case where no resultant surface force was transmitted between the spheres while they were rolling with spin. Another set of tests allowed a resultant force perpendicular to the rolling direction to be transmitted. The test results are compared with several theories.

A minimum value for the density of random close-packing of equal spheres

A minimum value for the density of random close-packing of equal spheres

Six and Seven Dimensional Non-Lattice Sphere Packings

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

On asymmetrical slow viscous flows caused by the motion of two equal spheres almost in contact

An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.

Exact solutions of the equations of slow viscous flow generated by the asymmetrical motion of two equal spheres

Exact solutions are presented for the asymmetrical slow viscous flows of an infinite fluid caused by either the rotation of two spheres each of radius a with equal uniform angular velocities about diameters perpendicular to their line of centres or by the translation of the spheres with equal and opposite velocities along directions perpendicular to their line of centres. The technique of solution is applicable for any value of the distance 2d between the centres of the spheres. The asymptotic forms of the solutions are discussed for the cases when the ratio d/a is large or small.

On the hydrodynamics of pairs of spheres falling along their line of centres in a viscous medium

The velocities, accelerations and drag forces experienced by two equal spheres falling along their line of centres in a viscous fluid were determined for three groups of Reynolds numbers R in the range where it is commonly assumed that Stokes's approximation applies. For all groups, with R ranging between 0·060 and 0·216, both spheres continually acclerated as they fell, and the upper sphere fell faster and accelerated more than the lower one. In contrast to Stimson & Jeffery's (1926) theory, which is based on the Stokes approximation, and to most earlier experimenters, the drag-force coefficients of the upper sphere computed from the experiments were significantly smaller than those for the lower sphere. Oseen's theory for this case agreed with the experiments in some respects, but contrary to it the drag-force coefficient varied with R for the upper sphere as well as the lower sphere.

Anisotropy in Simulated Random Packing of Equal Spheres

WE have used an IBM 1130 computer to simulate the very slow settling of rigid equal spheres from a dilute slurry into a randomly packed bed. Settling took place in a box defined by {x,y,z∣0≤x≤20, 0≤y≤20, z≥0}, where the dimensions are given in sphere radii. Initial x and y coordinates of the spheres were chosen by pseudorandom numbers. Once an incoming sphere made its first contact, the process was completely deterministic: the sphere rolled and sometimes fell until it reached a stable position. Thus some spheres rolled all the way from a peak to a valley. Gross wall and end effects were avoided by varying, by means of pseudo-random numbers, the position of walls and floor for each sphere. The final packed bed consisted of 1,561 spheres, but measurements were made on interior spheres only.

A sufficient condition for an extreme covering of n-space by spheres

The problem of finding the most economical coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice, which is equivalent to a problem concerning the inhomogeneous minima of positive definite quadratic forms, has been discussed recently by Barnes and Dickson [1]. The reader is referred to [1] for a complete background on the problem. Terms and notations used will be as in that paper.

Tensile Strength of Granular Materials

RUMFF1 has calculated the tensile strength of an ideal granular material; he suggests that: where σ is the tensile strength, B is the interparticle force and D is the diameter of the particles. The ideal material is a completely random pack of equal spheres with the bonding force concentrated at the point of contact. Some of the postulated relationships in the early part of Rumpf's calculation may appear to be dimensionally anomalous, and the strength has been re-calculated to investigate the effect of these anomalies. The whole aggregate has a volume Va; a thin section of thickness t and volume Vs, perpendicular to the tensile stress direction, is considered. The average number of particles contributing to this section: where Vp is the mean crossed volume of one particle and ρ is the fractional packing density. The section is very thin and it is assumed that : where Ap is the mean crossed area for an average particle. If the cross-sectional area of the whole aggregate is Aa: Substitution in equation 4 gives:

The extreme coverings of 4-space by spheres

The object of this paper is to apply, in the case n = 4, the results of Barnes and Dickson [1] concerning extreme coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice.

Lower Bounds to the Critical Rayleigh Number in Completely Confined Regions

A method is presented for estimating lower bounds to the minimum Rayleigh number that will induce a state of convective motion in a quasi-incompressible (Boussinesq) fluid where the temperature gradient is in the direction of the body force. The fluid is completely confined by fixed-temperature, rigid bounding walls. For any arbitrary region, the critical Rayleigh number is greater than 1558(h/D)4 where h is the maximum dimension of the given region in the direction of the body force and D is the diameter of an equal volume sphere. In certain simple geometrical configurations, improved lower-bound estimates are calculated.

Random Packing of Spheres in Non-rigid Containers

AN article by Susskind and Becker1 deals with randomly packed beds of equal rigid spheres. Without questioning the value of the work with respect to reactor beds, we feel the interpretation of their results in terms of random sphere packings as such needs further examination.