336
5 Numerical Models for Pebble-Bed Heat Transfer
the coolant will increase significantly, which may also bring trouble to the control
systems, or even make the nuclear reactor unsafe.
Decay Heat Removal
Unlike traditional coal-fired power plants and hydropower stations, decay heat
removal after shut-down is a critical issue to consider in the design of hightemperature gas-cooled reactors. In a conservative model for nuclear safety, it is
assumed that there is no coolant convection heat transport and only particle–particle
conduction and thermal radiation are available for heat removal. The effective thermal conductivity for conduction (k c ) and radiation (k r ) becomes critical parameters
to represent the ability for the decay heat removal in an HTGR [10, 38]. Although
the solid conductivity of particle and thermal conductivity of gas is functions of
temperature, the effective thermal conductivity for particle conduction is almost the
same at high temperatures [47]. Thus, it is approximated that the heat conduction
between particles is not affected by temperature. The conduction part (k c ) can be
obtained from experimental data under normal or elevated temperatures.
In the current radiation model, a packed pebble bed of the HTR-10 without coolant
convection at a very low thermal power was investigated to compute the radiation
contribution (k r ). Here, radiation is considered in the way of effective thermal conduction. Hence, the top and bottom walls were assumed to be adiabatic, and the
temperature at the outer wall was assumed to be constant. In a pseudo-porous media
model, the radial temperature distribution with constant conductivity and the uniform
heat source can be written as
T (x) = T w +
P s
4V b k eff
(R
2
o − x
2
),
(5.187)
where T w is the wall temperature at the outer wall and x is the distance to the center
line. P o , P s , and V b are the outer diameter, total heat source power, and volume of
the packed pebble bed, respectively. k eff is the effective thermal conductivity which
includes conduction and radiation and k r = k eff − k c . There is a linear relationship
between T (u) and x
2 , which is proportional to the effective thermal conductivity.
This trend is also observed in the simulation results shown in Fig. 5.81.
The values of k r (see Fig. 5.82) can be obtained using the least-squares method for
the post-processing of the simulation data. When it is within the super-high temperature region (T > 1, 262
◦ C(Λ > 1)), k r obtained from the SRM model without modification is slightly higher than the others. However, even for T > 1, 262
◦ C, it can
be seen in Fig. 5.82 that k r under different wall temperatures predicted by the SRM+
model is in good agreement with the predicted values obtained using the empirical
correlations, which are described in [5]. Particle radiation increases markedly with
an increase in the temperature. It can be seen in Fig. 5.82 that k r increases from
0.4 W/(m·
◦ C) at 100
◦ C to 10.5 W/(m·
◦ C) at 800
◦ C and 44.6 W/(m·
◦ C) at 1,600
◦ C,
which enhances the ability of the pebble bed to remove the decay heat at high temperatures significantly.
The decay heat power (P d ) with time is shown in Fig. 5.83a [1]. It can be seen
that the decay heat decreases from about 5.6% full power at shut-down to 1% at
5 Numerical Models for Pebble-Bed Heat Transfer
the coolant will increase significantly, which may also bring trouble to the control
systems, or even make the nuclear reactor unsafe.
Decay Heat Removal
Unlike traditional coal-fired power plants and hydropower stations, decay heat
removal after shut-down is a critical issue to consider in the design of hightemperature gas-cooled reactors. In a conservative model for nuclear safety, it is
assumed that there is no coolant convection heat transport and only particle–particle
conduction and thermal radiation are available for heat removal. The effective thermal conductivity for conduction (k c ) and radiation (k r ) becomes critical parameters
to represent the ability for the decay heat removal in an HTGR [10, 38]. Although
the solid conductivity of particle and thermal conductivity of gas is functions of
temperature, the effective thermal conductivity for particle conduction is almost the
same at high temperatures [47]. Thus, it is approximated that the heat conduction
between particles is not affected by temperature. The conduction part (k c ) can be
obtained from experimental data under normal or elevated temperatures.
In the current radiation model, a packed pebble bed of the HTR-10 without coolant
convection at a very low thermal power was investigated to compute the radiation
contribution (k r ). Here, radiation is considered in the way of effective thermal conduction. Hence, the top and bottom walls were assumed to be adiabatic, and the
temperature at the outer wall was assumed to be constant. In a pseudo-porous media
model, the radial temperature distribution with constant conductivity and the uniform
heat source can be written as
T (x) = T w +
P s
4V b k eff
(R
2
o − x
2
),
(5.187)
where T w is the wall temperature at the outer wall and x is the distance to the center
line. P o , P s , and V b are the outer diameter, total heat source power, and volume of
the packed pebble bed, respectively. k eff is the effective thermal conductivity which
includes conduction and radiation and k r = k eff − k c . There is a linear relationship
between T (u) and x
2 , which is proportional to the effective thermal conductivity.
This trend is also observed in the simulation results shown in Fig. 5.81.
The values of k r (see Fig. 5.82) can be obtained using the least-squares method for
the post-processing of the simulation data. When it is within the super-high temperature region (T > 1, 262
◦ C(Λ > 1)), k r obtained from the SRM model without modification is slightly higher than the others. However, even for T > 1, 262
◦ C, it can
be seen in Fig. 5.82 that k r under different wall temperatures predicted by the SRM+
model is in good agreement with the predicted values obtained using the empirical
correlations, which are described in [5]. Particle radiation increases markedly with
an increase in the temperature. It can be seen in Fig. 5.82 that k r increases from
0.4 W/(m·
◦ C) at 100
◦ C to 10.5 W/(m·
◦ C) at 800
◦ C and 44.6 W/(m·
◦ C) at 1,600
◦ C,
which enhances the ability of the pebble bed to remove the decay heat at high temperatures significantly.
The decay heat power (P d ) with time is shown in Fig. 5.83a [1]. It can be seen
that the decay heat decreases from about 5.6% full power at shut-down to 1% at