Pebble-bed density and packing factor

The following definitions are used:

• ps = density of the pebble base material

• pp = pebble density

• Pp = total pebble porosity = average fraction of porosity within (macroscopic, single) pebble volume

• Ppb = pebble-bed density = ratio of pebble-bed mass mpb to pebble-bed volume Vpb

• g = packing factor = ratio of pebble volume to pebble-bed volume

The quantities are connected by

ppb mpb/Vpb (1 Pp)psg [1 ]

In literature, the term ‘fraction of theoretical density’ is also used. This term is identical to (1—Pp). The pebble-bed density ppb is often named ‘apparent’ or ‘tap’ density. The pebble-bed density ppb is an impor­tant quantity for nuclear calculations as these require the specific density of lithium and other constituents as inputs. In pebble-bed engineering, the packing factor g (generally given in percentage) is the charac­teristic quantity.

The packing factor is influenced by the following parameters: filling procedure, pebble shape, surface roughness diameter d, diameter distribution, and con­tainer (cavity) dimensions: height H, diameter D.

Fillingprocedure. The selection of an optimum pro­cedure is not trivial, especially for larger mock-ups: filling should be assisted by vibration in such a way that a particle flow within the cavity is prevented or, in case of too small vibration energies, friction forces between pebbles and walls can be overcome.

Pebble shape and surface roughness: ceramic breeder pebble shapes are almost spherical, partly with in­dentations (melting, spraying processes) or egg-like (extrusion, spheronization, sintering processes). The pebbles originating from melting have a smoother surface than the others. The packing factor is influ­enced by the surface roughness but negligibly by the pebble shape.

Diameter distribution: The diameter d is of influence with respect to the cavity dimensions, discussed later. Ideally, the packing factor increases with increasing diameter distribution; see, for example, McGeary.87

However, sedimentation during the filling is a critical issue.

For blanket pebble beds, two groups are of inter­est: (1) Pebble beds with a relatively small d variation: for example, Li2TiO3 pebbles with a nominal diame­ter of 1 mm ranging from 0.8 to 1.2 mm and Li4SiO4 pebbles ranging from 0.25 to 0.63 mm. Production costs generally favor a wider spread, which is not detrimental in respect to the packing factor, at least for the diameter ranges cited earlier. Packing factors in the range of 63-66% are reached; (2) Binary beds: here, large pebbles, dl, and small pebbles, ds, are used, with ds« 0.1dl. First, the large pebbles are vibrated in the cavity. Then the pebbles are fixed in their posi­tion, for example, by a sieve in the filling pipe, to avoid sedimentation during the following pouring in ofthe small pebbles, which fill the spaces between the large pebbles. Packing factors of about 82% were obtained for beryllium beds with dl« 2 mm, ds« 0.2 mm.94 Critical issues are (1) the stability of such pebble packing structure during thermal cycles in the blanket and (2) the buildup of large stresses due to irradiation-induced swelling causing pebble deg­radation, leading to fracture.

Diameter d, bed height H, diameter D: For finite cavities, there exist two characteristic ratios: H/d and D/d. The reason why g depends on H/d and D/d is the existence of two different characteris­tic pebble packing structures as demonstrated by

tomography experiments101,105,106: in the pebble

bulk, there is no preferential direction of contacts between pebbles, whereas at the wall, a zone with a thickness of about 4d exists, where regular packing structures are observed and contact zones are no longer homogeneously distributed.

Figure 17 shows tomography results106 for a cylindrical container (D = 49 mm, H= 50 mm) filled with spherical pebbles (d = 2.3 mm). Distinct pebble layers close to the walls can be clearly seen in Figure 17(a) where the positions of the pebble cen­ters are plotted in a horizontal plane (left) and a vertical plane (right).

Figure 17(a) shows axial void fraction distribu­tions. The degree of regularity is largest for the pebbles in contact with the wall and decreases with increasing wall distance. The regularity is most expressed for pebbles in contact with plane walls (D = 1); here, large isles with a dense hexagonal pebble arrangement are observed; see Reimann et al}05 The average packing factor of a plane wall zone is about equal to the bulk packing factor if the bed height is sufficiently large for the two wall zones

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Figure 17 Visualization of pebble-wall layers by tomography: (a) Positions of sphere centers in region of interest (ROI) and b) Void fraction distributions. Reproduced from Pieritz, R. A.; Reimann, J.; Ferrero, C. Adv. Eng. Mater. 2011,13, Nr.3, 145-155.

 

to interfere. This is an important result because it demonstrates that the often cited statement that wall zones are characterized by reduced packing factors is not generally valid. However, with decreasing D, both the regularity of the packing and the packing factor decrease, as seen clearly in Figure 17(b).

Corners in pebble-bed cavities and inserts, for example, thermocouples, generally decrease the packing factor. This is also a reason why in small experimental setups the achieved packing factor is often distinctively below the maximum value.

Pebble beds in HCPB blanket concepts are typi­cally ‘thin’ pebble beds, that is, one dimension; the bed height H, is small compared with the other dimensions. For the ceramic breeder pebble beds, H« 10 mm; for beryllium, H« 30 mm. For a height of H« 10 mm and a pebble diameter of ~1 mm, the maximum packing factor is already difficult to achieve.

 

Best location for filling

 

X

 

Figure 18 Pebble-bed filling experiments relevant for ITER Test Blanket Modules. Reproduced from Abou-Sena, A.; Neuberger, H.; Ihli, T. Fusion Eng. Des. 2009, 84, 355-358.

 

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A disadvantage of a large pebble diameter is also the smaller heat transfer coefficient, as outlined below.

The blanket modules will probably be filled with pebbles through small pipes, as schematically shown in Figure 18. For simple cubic cavities, a homogeneous filling was achieved only by carefully vibrating the container in different tilted positions.97,100

The filling of a TBM mock-up using glass balls for the beryllium pebble beds was investigated by Abou-Sena eta/.,107 by varying the number and posi­tions of the filling pipes. The best results were obtained by using a single filling pipe close to the box corner and, keeping the filling pipe at the highest position, tilting the box around two axes. Here, a packing factor of 63.6% was achieved, which is considered to be very close to the maximum obtain­able value.

In the HELICHETTA experiment (see Dell’Orco et a/.108,109) where elongated pebble-bed cavities (10 x 100 x 480 mm3) were filled through a pipe at the top (largest dimension orientated vertically), packing factors of 60% and 62% were achieved for Li2TiO3 pebbles with diameters between 0.8 and

1.2 mm, and 0.6 and 0.8 mm, respectively. For Li4SiO4 pebbles with d between 0.25 and 0.63, the packing factor was more than 64%.