370
5 Numerical Models for Pebble-Bed Heat Transfer
is shown in Fig. 5.110a. The inner and outer walls are 0.65 and 2.15 m in radius.
In the simulation of the conduction, the temperature of the inner wall and outer
walls are fixed at 300 K and 800 K, respectively. The discrete particle simulations
are performed in LIGGGHTS with a constant contact area of command “fix cond all
heat/gran/conduction contact_area constant”. The radial temperature distribution in
steady state at half height is shown in Fig. 5.110b. It can be seen that the numerical
results of the macroscopic conduction equation agree with the discrete particle simulation. The computational time is reduced significantly without tracking all particle
motions and the conduction equations.
5.5.3 Efficient Computing of View Factor
Full-range thermal radiation is very difficult to compute accurately and efficiently
since it occurs between the surfaces no matter how far they are. Herein, radiative heat
flux and effective thermal conductivity were derived mathematically by a matrix to
present the thermal radiation between particles in packed beds. A regression model
of the feed-forward neural network was trained by large datasets to compute the
view factor matrix efficiently. A correlation was proposed to predict the radiation
exchange factor under different voidages. Radiative effective thermal conductivity
was found to grow slightly with void fractions. By GPU acceleration, it is feasible
to perform real-time simulation of full-range radiations inside large-scale packed
beds, even for real reactors (e.g., HTR-PM). As a demonstrative application, this
model was applied to simulating thermal radiation of the decay heat removal of a
real reactor, and showed that the highest temperature is still within a design limitation
of the packed bed.
On the one hand, even though the short-range models generally agree with the
experimental data under low-temperature ranges, it is still rather difficult to predict the
radiative flux accurately by the numerical model without considering the long-range
interactions. In particular, this may underestimate the radiation at high temperatures
[5]. On the other hand, because the wavelength of thermal radiation is in the range
of 1–100 m and the spheres are of 60 mm in diameter in a packed bed, the size
parameter is far less than 1, and it is reasonable to apply the geometrical optics in
the radiation simulation [157]. Then, in the conventional integration scheme [43]
or the Monte Carlo method [158], it always takes a rather long time to calculate
the geometrical parameter obstructed view factor from any sphere to all possible
surrounding ones. Therefore, the computational time for a large-scale packed bed
becomes unacceptable. Thus, in conclusion, it is necessary to develop a new model
to predict the full-range parts of radiative heat transfer in packed bed together with
the short-range parts efficiently.
Fortunately, with the fast development of Artificial Intelligence (AI), the nonlinear
regression model provides a universal approximator of a complex function by deep
learning from large data [159]. Moreover, some software is now available for training
the deep neural network, such as the well-known TensorFlow. For thermal radiation
5 Numerical Models for Pebble-Bed Heat Transfer
is shown in Fig. 5.110a. The inner and outer walls are 0.65 and 2.15 m in radius.
In the simulation of the conduction, the temperature of the inner wall and outer
walls are fixed at 300 K and 800 K, respectively. The discrete particle simulations
are performed in LIGGGHTS with a constant contact area of command “fix cond all
heat/gran/conduction contact_area constant”. The radial temperature distribution in
steady state at half height is shown in Fig. 5.110b. It can be seen that the numerical
results of the macroscopic conduction equation agree with the discrete particle simulation. The computational time is reduced significantly without tracking all particle
motions and the conduction equations.
5.5.3 Efficient Computing of View Factor
Full-range thermal radiation is very difficult to compute accurately and efficiently
since it occurs between the surfaces no matter how far they are. Herein, radiative heat
flux and effective thermal conductivity were derived mathematically by a matrix to
present the thermal radiation between particles in packed beds. A regression model
of the feed-forward neural network was trained by large datasets to compute the
view factor matrix efficiently. A correlation was proposed to predict the radiation
exchange factor under different voidages. Radiative effective thermal conductivity
was found to grow slightly with void fractions. By GPU acceleration, it is feasible
to perform real-time simulation of full-range radiations inside large-scale packed
beds, even for real reactors (e.g., HTR-PM). As a demonstrative application, this
model was applied to simulating thermal radiation of the decay heat removal of a
real reactor, and showed that the highest temperature is still within a design limitation
of the packed bed.
On the one hand, even though the short-range models generally agree with the
experimental data under low-temperature ranges, it is still rather difficult to predict the
radiative flux accurately by the numerical model without considering the long-range
interactions. In particular, this may underestimate the radiation at high temperatures
[5]. On the other hand, because the wavelength of thermal radiation is in the range
of 1–100 m and the spheres are of 60 mm in diameter in a packed bed, the size
parameter is far less than 1, and it is reasonable to apply the geometrical optics in
the radiation simulation [157]. Then, in the conventional integration scheme [43]
or the Monte Carlo method [158], it always takes a rather long time to calculate
the geometrical parameter obstructed view factor from any sphere to all possible
surrounding ones. Therefore, the computational time for a large-scale packed bed
becomes unacceptable. Thus, in conclusion, it is necessary to develop a new model
to predict the full-range parts of radiative heat transfer in packed bed together with
the short-range parts efficiently.
Fortunately, with the fast development of Artificial Intelligence (AI), the nonlinear
regression model provides a universal approximator of a complex function by deep
learning from large data [159]. Moreover, some software is now available for training
the deep neural network, such as the well-known TensorFlow. For thermal radiation