4.15.4.4.1 Thermal conductivity

There are several advanced models to describe the thermal conductivity of pebble beds, which take into account the relevant parameters such as the thermal conductivity of the pebble material ks as a function of temperature T, the thermal conductivities of the surrounding gas kg as a function of temperature Tand pressure p, the pebble diameter d, packing factor g, contact surface ratio pk2, and several other second — order effect parameters. In the following section the Schlunder Bauer Zehner model (SBZ model)114 is used to demonstrate the influence of some para­meters. Figure 23 shows ks as a function of tempera­ture for both Li4SiO4 and Li2TiO3 and kg for helium and other gases at 0.1 MPa.99 For most ceramic breeder materials, ks decreases first with increasing T, reaches at high temperatures a plateau, or increases again slightly; for details, see Abou-Sena eta/.116 The helium conductivity increases strongly with increasing T. Figure 24(a) and 24(b) from Reimann et a/.99 shows the influence of Tand pk2 on the pebble-bed conduc­tivity k of an Li4SiO4 pebble bed with mean diameter of d = 0.4 mm and g = 64%. For a noncompressed bed, pk = 0, k increases moderately with T because kg increases with T. For a moderately compressed bed,

pk2 = 0.02, k is larger than that for pk2 = 0, but the temperature dependence is quite small.

The thermal conductivity of noncompressed Li4SiO4 and Li2TiO3 pebble beds was measured by several authors. Figure 25 from Enoeda eta/.86 shows that there is a good agreement among different authors. The Li4SiO4 pebble-bed data are best fitted with a correlation established by Dalle Donne204; the Li2TiO3 data were well predicted by the SBZ model using a value pk2 = 0.0049.117 Li2TiO3 pebble-bed data for 2 mm pebbles from Abou-Sena116 are char­acterized by the tendency of a decrease in k with increasing T

Results for noncompressed beds including further ceramic breeder pebble materials were summarized by Abou-Sena eta/.116: k should increase with increas­ing ks in the sequence Li2ZrO3, Li4SiO4, Li2TiO3, Li2O for equal values of the other parameters. Because the other parameters differed, this tendency was masked.

For compressed pebble beds, the SBZ model con­tains the parameter pk, which is a priori not known. If the pebble-bed strain is measured, it is easier to use this quantity as the relevant parameter. Figures 26(a) and 26(b) from Reimann and Hermsmeyer99 show results for Li4SiO4 and Li2TiO3 pebble beds, differ­ent gas conditions, and strain values e. For noncom­pressed beds, pk = 0, the measured data agree fairly well with SBZ model predictions. With increasing strain, k increases; however, only very moderately compared with beryllium pebble bed. Even for a strain of about 4%, obtained in air at 800 °C because

of significant thermal creep, k is only increased by about 20% for both types of pebble beds. For helium atmosphere, this difference becomes even smaller.

Figure 26(b) also contains some results for binary Japanese Li2TiO3 pebble beds (0.2 and 2 mm peb­bles, g = 81.5%) in air atmosphere at ambient tem­perature. Compared with the monodisperse pebble bed with d = 2 mm pebbles and g = 64.3%, k is increased by a factor of 2. For blanket relevant con­ditions, this factor reduces to ~1.3 for T = 600 °C and helium atmosphere.

Experiments with compressed Li2TiO3 pebble beds (d = 2 mm, g = 65-67%) were also performed by Tanigawa eta/.113 For a strain of about 1%, T = 600 ° C, and helium at 0.1 MPa, k increased only by 3% compared with the noncompressed pebble bed. After annealing the pebble bed at 700 °C without com­pression for 1 day, larger bed strains (factor 2) were obtained in the subsequent cycles and with this an additional increase in conductivity.

4.15.4.4.2 Heat transfer

Close to the wall, the pebble packing differs signifi­cantly from that in the bulk, as demonstrated in Figure 17. For noncompressed pebbles, the void fraction at the wall surface is close to 100%. Heat transfer characteristics in the wall zone are, therefore, different from those in the bulk. This fact is taken into account by using the heat transfer coefficient h,
which is defined with the temperature difference (Tew-Tw), where Tew is obtained by extrapolating the bulk pebble-bed thermal conductivity k up to the wall, and Tw is the undisturbed wall temperature. In measurements, (T;w-TO is very small and is sensitively dependent on extrapolated temperature profiles. Accurate measurements are, therefore, extremely difficult, and the discrepancies in experi­mental data are significant. The smaller the pebble diameter, the thinner the wall zone, and with this the difficulty to obtain accurate data increases. Again, different models exist to predict h exist, which have been validated in better suited experiments.

Figure 27 shows h = f(T) for Li4SiO4 and Li2TiO3 pebble beds for noncompressed beds calcu­lated with the model from Yagi and Kunii118: the strong increase of h with decreasing pebble diameter d is obvious. With progressing compression, the wall

contact surfaces increase and with this h. Again, this increase is expected to be much smaller than that in beryllium pebble beds because of the small thermal conductivity of ceramic breeder materials.

At present, mechanistic models are developed taking into account the pebble arrangements close to the wall as determined by tomography101,105,106,199 or by discrete-element modeling (DEM), outlined later. A typical example is the work of Gan eta/.133