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Topology of compressed pebble beds
Fusion Engineering and Design 81 (2006) 653–658
Topology of compressed pebble beds J. Reimann a,∗ , R.A. Pieritz b , R. Rolli a b
a Forschungszentrum Karlsruhe, P.O. Box 3640, D-76021 Karlsruhe, Germany Applied Research Solutions, 444–Rua General Osorio, CEP 89041-000, Blumenau, SC, Brazil
Received 7 February 2005; received in revised form 1 June 2005; accepted 1 June 2005 Available online 10 January 2006
Abstract Thermal stresses occurring during operation of helium-cooled pebble bed blankets will cause pebble deformations which increase the pebble bed thermal conductivity due to an increase of contact surfaces between the pebbles. The dependence of the thermal conductivity from pebble bed strain is generally determined in test set-ups where only global quantities (uniaxial stress, strain and temperature) are measured. For the understanding of heat transfer mechanisms, the knowledge of the topology of the pebble beds (number of pebble contacts with other pebbles or walls, corresponding contact zones, angular distribution of contacts on the pebbles) is of great interest. Experiments were performed where, first, pebble beds were uniaxially compressed to different strain levels. For better measurement accuracy, spherical 3.5 or 5 mm aluminium pebbles were used instead of blanket relevant 1 mm beryllium pebbles. Then, the topological quantities were determined by two methods: A) In the European Synchrotron Radiation Facility (ESRF) Grenoble, a special microtomography experimental set-up was used allowing the computer aided reconstruction of 3D images of pebbles within the pebble beds. By post-processing the data, both radial and axial void fraction distributions were determined as well as the topological quantities. B) In the Forschungszentrum Karlsruhe (FZK), the pebbles were chemically coloured in the compressed state in order to increase the optical contrast. Optical microscopy was then used to determine pebble contact numbers and contact surfaces. Results from both methods complement one another very well due to specific experimental limitations of both techniques. © 2005 Elsevier B.V. All rights reserved. Keywords: Pebble bed; Thermal-mechanical behaviour; Microtomography; Co-ordination number; Contact surfaces
1. Introduction
∗ Corresponding author. Tel.: +49 7247 82 3498; fax: +49 7247 82 4837. E-mail address: [email protected] (J. Reimann).
0920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2005.06.375
In helium-cooled pebble bed blankets, thermal stresses during operation will cause deformation (compression) of the breeder pebble beds which will increase the pebble bed thermal conductivity due to
654
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
an increase of contact surfaces between the pebbles [1]. This holds especially for beryllium pebble beds because of the large ratio of the beryllium conductivity to the helium conductivity [2]. The dependence of thermal conductivity on strain is determined in test set-ups where only global quantities (uniaxial stress, strain and temperature) are measured. For the understanding of heat transfer mechanisms, the knowledge of the topology of the pebble beds (number of pebble contacts with other pebbles or walls, corresponding contact zones, angular distribution of contacts on the pebbles) is of great interest. Experiments were performed where first pebble beds were uniaxially compressed to different strain levels. Then, the topological quantities were determined by: A) Computer aided microtomography (CMT) at the European Synchrotron Radiation Facility (ESRF) Grenoble. B) Optical microscopy (OM) at the Forschungszentrum Karlsruhe (FZK). A part of the CMT results were presented previously [3]; in this paper, results from both methods are compared for the case of similar pebble bed compressions. 2. Experimental 2.1. Uniaxial compression tests (UCTs) In order to determine with a higher accuracy the topological quantities, aluminium spheres with larger diameters were used instead of the blanket relevant 1 mm beryllium pebbles. For the CMT investigations, cylindrical aluminium containers (“cans”, bed height: H = 60 mm, inner diameter: D = 49 mm, wall thickness: 0.5 mm) were filled with Al spheres (diameter d = 3.5 mm). For the UCTs, these cans were placed inside of a thick-walled cylindrical steel container. For the OM investigations, 5 mm Al spheres were compressed in a stainless steel container (H = 100 mm, D = 100 mm). After filling, the pebble beds were vibrated in order to obtain dense beds. Packing factors of γ ≈ 60 and 60.5% were obtained for the 3.5 and 5 mm pebble beds, respectively. These values are smaller than the value γ ≈ 63%, relevant for densified beds in large containers [4], because of the relatively small ratios of D/d.
Fig. 1. Uniaxial compression of pebble beds.
Fig. 1 shows the measured stress–strain curves with a maximum uniaxial stress σ (identical to the piston pressure) of ≈16 MPa and strains ε up to ≈10%. The maximum deformations of Sample S1 and the pebble bed used for the OM analyses do not differ much; therefore, mainly results from these two pebble beds will be presented. During stress decrease, strain changes only marginally, indicating that elastic stresses are negligible compared to plastic pebble deformations. The figure contains also thermal conductivity measurements with a 2.3 mm aluminium pebble bed, obtained in the HECOP facility, compare [2]. These data demonstrate the strong increase of thermal conductivity with increasing compression caused by the increase of the contact surfaces Ac between the pebbles. The CMT cans were closed under maximum piston pressures in such a way that after pressure release, the pebble configuration did not change. For the CMT investigations, besides samples S1 and S6, a sample without pre-compression (S0) was used. 2.2. Microtomogaphic set-up A special experimental set-up (Fig. 2) was used at the ESRF (ID15 beamline at ≈80 keV) allowing the computer aided reconstruction of 3D images of pebbles within the pebble beds. By post-processing the data, both radial and axial void fraction distributions in the samples were determined as well as the topological quantities defined above; for details, see [3,5]. Because of beam time constraints, the microtomography scans (core samples) were limited to heights considerably smaller than the total bed height. For S1, this height was about 11 mm, and the bottom of the core
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 2. Scheme of microtomography set-up.
sample was about one sphere radius above the bottom plate of the can.
655
pebble bed regions (bulk region, zone close to lower bottom plate and close to cylindrical wall). For topological analyses, the individual spheres were glued onto a rod which was rotated by 45◦ steps. At each position, five digital photographic pictures were taken at different focus depths. A software programme was used to calculate a picture which has an increased sharpness over the total hemisphere; an example is shown in Fig. 3. After one turn, the sphere was picked up by another rod with an axis perpendicular to that of the first rod in order to visualise those zones which were previously hidden.
3. Results 2.3. Optical microscopy 3.1. Configuration of spheres In order to obtain an increased optical contrast between contact surfaces and residual pebble surfaces, the residual surfaces were coloured by yellow chrome plating. This was achieved by pouring an aqueous solution of Alodine 6105 (Henkel KGaA) into the compressed pebble bed. After a reaction time of ≈2 min, the solution was drained through an opening in the container bottom. This procedure was repeated several times. Then, before reducing the piston pressure, the pebbles were repeatedly washed with water. After drying, about 30 pebbles were taken from characteristic
As characteristic CMT result, Fig. 4 shows the horizontal and vertical cross section as well the 3D view for Sample S1. The centres of the spheres in the lowest layer are all located at the same vertical height, see Fig. 4b, indicating that these spheres are in contact with the bottom plate below. Fig. 5 displays the positions of the sphere centres in a map with vertical and radial coordinates (for mathematical reasons, both the spheres in the bottom layer and those of the first row contacting the cylindrical wall were not taken into account). The vertical structure is clearly seen; to a lesser content also the more regular packing close to the cylindrical wall. 3.2. Coordination numbers Nc The numbers of contacts per sphere Nc (Fig. 6) do not differ significantly for the different CMT samples and the OM pebble bed. The maximum values are between 6 and 7 which are in good agreement with previous results [6]. 3.3. Distribution of contact surface ratio, Ac /A, on the spheres
Fig. 3. Five millimeter Al sphere with contact zones.
Fig. 7 shows the ratio of contact surface Ac to the sphere cross section A for the S1 and OM pebble beds using pebbles from the third bottom layer. The largest value of each sphere is attributed to contact number 1; the second largest value to contact number
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J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 5. Position of sphere centres (Sample S1).
Fig. 6. Distribution of coordination numbers Nc .
Fig. 4. View of pebble bed (Sample S1).
2, etc. The trends are quite similar, the absolute values are higher for S1 due to the stronger deformation, see Fig. 1. 3.4. Poloidal distribution of contact surfaces on the spheres In UCTs, the pebble beds are subjected to a pressure in the vertical direction. The question is, if this is
Fig. 7. Distribution of contact surfaces (third layer above bottom).
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 8. Poloidal contact surface distribution; Sample S1: second bottom layer.
reflected also in the angular distribution of the contact surfaces Ac . Fig. 8 shows for the sample S1 the ratio Ac /A as a function of the poloidal angle δ (starting at the “North Pole”). The tendency is clearly seen that the largest contact surfaces occur preferentially in zones with δ < 45◦ and δ > 145◦ , that is in zones with large fractions of the contact surfaces perpendicular to the uniaxial stress. With the OM technique, the orientation of the sphere arrangement in the bed in respect to the vertical direction gets lost by removing the spheres (it was tried to obtain this information by marking the upper apex by a pen; however, this technique was not accurate enough). An exception is the spheres in the bottom layer, where in the lower hemisphere only contact occurs at the position of the “South Pole”. Fig. 9 shows the corresponding results: at δ ≈ 90◦ , primarily small contacts are observed which are of minor interest because they do not contribute to heat transfer. In the upper hemisphere, the largest deformations occur in the zone 25◦ < δ < 45◦ ; between 25 and 60◦ as mean value 3.3 contacts per sphere are observed which is quite close to the value of 3 characteristic for the hexagonal packing. The mean value of the bottom contact area agrees well (difference about 10%) with the expected value from special calibration experiments where individual spheres were compressed between two plates and
657
Fig. 9. Poloidal contact surface distribution; OM pebble bed: bottom layer.
the contact areas were determined as a function of the loading.
4. Conclusions The topology of pebble beds was determined by two methods which were complimentary to each other. Microtomography has proven to be a very powerful technique for handling large amounts of pebbles in order to provide statistical quantities. Topological quantities are obtained without disturbing the pebble configuration; this is mandatory for the determination of angular dependencies of the contacts. There are some questions in respect to the accuracy of the calculated contact sizes, and it must be still demonstrated that this technique is capable to determine pebble contact sizes with walls. With the optical microscopy technique, the information on pebble orientation in the bed, in general, cannot be obtained. However, it was shown that the measurement accuracy of contact areas is very satisfactory. Both techniques have been applied for the first time to detect coordination numbers, contact surfaces and their positions on spherical pebbles. It has been demonstrated that in uniaxial compression tests, which are standard tests for the characterisation of thermomechanical properties of pebble beds, the poloidal
658
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
distribution of contact surfaces is non-homogeneous. This should be especially considered when extrapolating results for thermal conductivities determined in uniaxial compression facilities to blanket pebble beds where different stress conditions will exist. Motivated by the encouraging first microtomography results, further very detailed experimental investigations were performed end of 2004 (ESRF Exp. Me-898); at present, these data are analysed.
Acknowledgements The authors are grateful to T. Lautenschl¨ager and K. R¨uhter from FZK and C. Ferrero and M. di Michiel from ESRF for their extensive technical support.
References [1] J. Reimann, L. Boccaccini, M. Enoeda, A. Ying, Thermomechanics of solid breeder and Be pebble bed materials, Fus. Eng. Des. 61–62 (2002) 319–331. [2] J. Reimann, G. Piazza, H. Harsch, Thermal conductivity of compressed beryllium pebble beds, this conference. [3] J. Reimann, R.A. Pieritz, M. di Michiel, C. Ferrero, Inner structures of compressed pebble beds determined by X-ray tomography, in: 23rd Symposium on Fusion Technology, Venice, Italy, 20–24 September, 2004. [4] R.K. McGeary, Mechanical packing of spherical particles, J. Am. Ceram. Soc. 44 (10) (1961) 513. [5] R. Pieritz, Mod´elisation et simulation de milieux poreux par r´eseaux topologiques, PhD Thesis, Universit´e Joseph Fourier, Grenoble, France, 1998. [6] J. Bernal, J. Mason, Co-ordination of randomly packed spheres, Nature 188 (1960) 910–911.
Topology of compressed pebble beds J. Reimann a,∗ , R.A. Pieritz b , R. Rolli a b
a Forschungszentrum Karlsruhe, P.O. Box 3640, D-76021 Karlsruhe, Germany Applied Research Solutions, 444–Rua General Osorio, CEP 89041-000, Blumenau, SC, Brazil
Received 7 February 2005; received in revised form 1 June 2005; accepted 1 June 2005 Available online 10 January 2006
Abstract Thermal stresses occurring during operation of helium-cooled pebble bed blankets will cause pebble deformations which increase the pebble bed thermal conductivity due to an increase of contact surfaces between the pebbles. The dependence of the thermal conductivity from pebble bed strain is generally determined in test set-ups where only global quantities (uniaxial stress, strain and temperature) are measured. For the understanding of heat transfer mechanisms, the knowledge of the topology of the pebble beds (number of pebble contacts with other pebbles or walls, corresponding contact zones, angular distribution of contacts on the pebbles) is of great interest. Experiments were performed where, first, pebble beds were uniaxially compressed to different strain levels. For better measurement accuracy, spherical 3.5 or 5 mm aluminium pebbles were used instead of blanket relevant 1 mm beryllium pebbles. Then, the topological quantities were determined by two methods: A) In the European Synchrotron Radiation Facility (ESRF) Grenoble, a special microtomography experimental set-up was used allowing the computer aided reconstruction of 3D images of pebbles within the pebble beds. By post-processing the data, both radial and axial void fraction distributions were determined as well as the topological quantities. B) In the Forschungszentrum Karlsruhe (FZK), the pebbles were chemically coloured in the compressed state in order to increase the optical contrast. Optical microscopy was then used to determine pebble contact numbers and contact surfaces. Results from both methods complement one another very well due to specific experimental limitations of both techniques. © 2005 Elsevier B.V. All rights reserved. Keywords: Pebble bed; Thermal-mechanical behaviour; Microtomography; Co-ordination number; Contact surfaces
1. Introduction
∗ Corresponding author. Tel.: +49 7247 82 3498; fax: +49 7247 82 4837. E-mail address: [email protected] (J. Reimann).
0920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2005.06.375
In helium-cooled pebble bed blankets, thermal stresses during operation will cause deformation (compression) of the breeder pebble beds which will increase the pebble bed thermal conductivity due to
654
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
an increase of contact surfaces between the pebbles [1]. This holds especially for beryllium pebble beds because of the large ratio of the beryllium conductivity to the helium conductivity [2]. The dependence of thermal conductivity on strain is determined in test set-ups where only global quantities (uniaxial stress, strain and temperature) are measured. For the understanding of heat transfer mechanisms, the knowledge of the topology of the pebble beds (number of pebble contacts with other pebbles or walls, corresponding contact zones, angular distribution of contacts on the pebbles) is of great interest. Experiments were performed where first pebble beds were uniaxially compressed to different strain levels. Then, the topological quantities were determined by: A) Computer aided microtomography (CMT) at the European Synchrotron Radiation Facility (ESRF) Grenoble. B) Optical microscopy (OM) at the Forschungszentrum Karlsruhe (FZK). A part of the CMT results were presented previously [3]; in this paper, results from both methods are compared for the case of similar pebble bed compressions. 2. Experimental 2.1. Uniaxial compression tests (UCTs) In order to determine with a higher accuracy the topological quantities, aluminium spheres with larger diameters were used instead of the blanket relevant 1 mm beryllium pebbles. For the CMT investigations, cylindrical aluminium containers (“cans”, bed height: H = 60 mm, inner diameter: D = 49 mm, wall thickness: 0.5 mm) were filled with Al spheres (diameter d = 3.5 mm). For the UCTs, these cans were placed inside of a thick-walled cylindrical steel container. For the OM investigations, 5 mm Al spheres were compressed in a stainless steel container (H = 100 mm, D = 100 mm). After filling, the pebble beds were vibrated in order to obtain dense beds. Packing factors of γ ≈ 60 and 60.5% were obtained for the 3.5 and 5 mm pebble beds, respectively. These values are smaller than the value γ ≈ 63%, relevant for densified beds in large containers [4], because of the relatively small ratios of D/d.
Fig. 1. Uniaxial compression of pebble beds.
Fig. 1 shows the measured stress–strain curves with a maximum uniaxial stress σ (identical to the piston pressure) of ≈16 MPa and strains ε up to ≈10%. The maximum deformations of Sample S1 and the pebble bed used for the OM analyses do not differ much; therefore, mainly results from these two pebble beds will be presented. During stress decrease, strain changes only marginally, indicating that elastic stresses are negligible compared to plastic pebble deformations. The figure contains also thermal conductivity measurements with a 2.3 mm aluminium pebble bed, obtained in the HECOP facility, compare [2]. These data demonstrate the strong increase of thermal conductivity with increasing compression caused by the increase of the contact surfaces Ac between the pebbles. The CMT cans were closed under maximum piston pressures in such a way that after pressure release, the pebble configuration did not change. For the CMT investigations, besides samples S1 and S6, a sample without pre-compression (S0) was used. 2.2. Microtomogaphic set-up A special experimental set-up (Fig. 2) was used at the ESRF (ID15 beamline at ≈80 keV) allowing the computer aided reconstruction of 3D images of pebbles within the pebble beds. By post-processing the data, both radial and axial void fraction distributions in the samples were determined as well as the topological quantities defined above; for details, see [3,5]. Because of beam time constraints, the microtomography scans (core samples) were limited to heights considerably smaller than the total bed height. For S1, this height was about 11 mm, and the bottom of the core
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 2. Scheme of microtomography set-up.
sample was about one sphere radius above the bottom plate of the can.
655
pebble bed regions (bulk region, zone close to lower bottom plate and close to cylindrical wall). For topological analyses, the individual spheres were glued onto a rod which was rotated by 45◦ steps. At each position, five digital photographic pictures were taken at different focus depths. A software programme was used to calculate a picture which has an increased sharpness over the total hemisphere; an example is shown in Fig. 3. After one turn, the sphere was picked up by another rod with an axis perpendicular to that of the first rod in order to visualise those zones which were previously hidden.
3. Results 2.3. Optical microscopy 3.1. Configuration of spheres In order to obtain an increased optical contrast between contact surfaces and residual pebble surfaces, the residual surfaces were coloured by yellow chrome plating. This was achieved by pouring an aqueous solution of Alodine 6105 (Henkel KGaA) into the compressed pebble bed. After a reaction time of ≈2 min, the solution was drained through an opening in the container bottom. This procedure was repeated several times. Then, before reducing the piston pressure, the pebbles were repeatedly washed with water. After drying, about 30 pebbles were taken from characteristic
As characteristic CMT result, Fig. 4 shows the horizontal and vertical cross section as well the 3D view for Sample S1. The centres of the spheres in the lowest layer are all located at the same vertical height, see Fig. 4b, indicating that these spheres are in contact with the bottom plate below. Fig. 5 displays the positions of the sphere centres in a map with vertical and radial coordinates (for mathematical reasons, both the spheres in the bottom layer and those of the first row contacting the cylindrical wall were not taken into account). The vertical structure is clearly seen; to a lesser content also the more regular packing close to the cylindrical wall. 3.2. Coordination numbers Nc The numbers of contacts per sphere Nc (Fig. 6) do not differ significantly for the different CMT samples and the OM pebble bed. The maximum values are between 6 and 7 which are in good agreement with previous results [6]. 3.3. Distribution of contact surface ratio, Ac /A, on the spheres
Fig. 3. Five millimeter Al sphere with contact zones.
Fig. 7 shows the ratio of contact surface Ac to the sphere cross section A for the S1 and OM pebble beds using pebbles from the third bottom layer. The largest value of each sphere is attributed to contact number 1; the second largest value to contact number
656
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 5. Position of sphere centres (Sample S1).
Fig. 6. Distribution of coordination numbers Nc .
Fig. 4. View of pebble bed (Sample S1).
2, etc. The trends are quite similar, the absolute values are higher for S1 due to the stronger deformation, see Fig. 1. 3.4. Poloidal distribution of contact surfaces on the spheres In UCTs, the pebble beds are subjected to a pressure in the vertical direction. The question is, if this is
Fig. 7. Distribution of contact surfaces (third layer above bottom).
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
Fig. 8. Poloidal contact surface distribution; Sample S1: second bottom layer.
reflected also in the angular distribution of the contact surfaces Ac . Fig. 8 shows for the sample S1 the ratio Ac /A as a function of the poloidal angle δ (starting at the “North Pole”). The tendency is clearly seen that the largest contact surfaces occur preferentially in zones with δ < 45◦ and δ > 145◦ , that is in zones with large fractions of the contact surfaces perpendicular to the uniaxial stress. With the OM technique, the orientation of the sphere arrangement in the bed in respect to the vertical direction gets lost by removing the spheres (it was tried to obtain this information by marking the upper apex by a pen; however, this technique was not accurate enough). An exception is the spheres in the bottom layer, where in the lower hemisphere only contact occurs at the position of the “South Pole”. Fig. 9 shows the corresponding results: at δ ≈ 90◦ , primarily small contacts are observed which are of minor interest because they do not contribute to heat transfer. In the upper hemisphere, the largest deformations occur in the zone 25◦ < δ < 45◦ ; between 25 and 60◦ as mean value 3.3 contacts per sphere are observed which is quite close to the value of 3 characteristic for the hexagonal packing. The mean value of the bottom contact area agrees well (difference about 10%) with the expected value from special calibration experiments where individual spheres were compressed between two plates and
657
Fig. 9. Poloidal contact surface distribution; OM pebble bed: bottom layer.
the contact areas were determined as a function of the loading.
4. Conclusions The topology of pebble beds was determined by two methods which were complimentary to each other. Microtomography has proven to be a very powerful technique for handling large amounts of pebbles in order to provide statistical quantities. Topological quantities are obtained without disturbing the pebble configuration; this is mandatory for the determination of angular dependencies of the contacts. There are some questions in respect to the accuracy of the calculated contact sizes, and it must be still demonstrated that this technique is capable to determine pebble contact sizes with walls. With the optical microscopy technique, the information on pebble orientation in the bed, in general, cannot be obtained. However, it was shown that the measurement accuracy of contact areas is very satisfactory. Both techniques have been applied for the first time to detect coordination numbers, contact surfaces and their positions on spherical pebbles. It has been demonstrated that in uniaxial compression tests, which are standard tests for the characterisation of thermomechanical properties of pebble beds, the poloidal
658
J. Reimann et al. / Fusion Engineering and Design 81 (2006) 653–658
distribution of contact surfaces is non-homogeneous. This should be especially considered when extrapolating results for thermal conductivities determined in uniaxial compression facilities to blanket pebble beds where different stress conditions will exist. Motivated by the encouraging first microtomography results, further very detailed experimental investigations were performed end of 2004 (ESRF Exp. Me-898); at present, these data are analysed.
Acknowledgements The authors are grateful to T. Lautenschl¨ager and K. R¨uhter from FZK and C. Ferrero and M. di Michiel from ESRF for their extensive technical support.
References [1] J. Reimann, L. Boccaccini, M. Enoeda, A. Ying, Thermomechanics of solid breeder and Be pebble bed materials, Fus. Eng. Des. 61–62 (2002) 319–331. [2] J. Reimann, G. Piazza, H. Harsch, Thermal conductivity of compressed beryllium pebble beds, this conference. [3] J. Reimann, R.A. Pieritz, M. di Michiel, C. Ferrero, Inner structures of compressed pebble beds determined by X-ray tomography, in: 23rd Symposium on Fusion Technology, Venice, Italy, 20–24 September, 2004. [4] R.K. McGeary, Mechanical packing of spherical particles, J. Am. Ceram. Soc. 44 (10) (1961) 513. [5] R. Pieritz, Mod´elisation et simulation de milieux poreux par r´eseaux topologiques, PhD Thesis, Universit´e Joseph Fourier, Grenoble, France, 1998. [6] J. Bernal, J. Mason, Co-ordination of randomly packed spheres, Nature 188 (1960) 910–911.