266
5 Numerical Models for Pebble-Bed Heat Transfer
5.3.4 Long-Range Radiation Model (LRM)
In the long-range radiation model, it is of vital importance to calculate the view
factors between any two spheres and from sphere to a wall to study the thermal
radiation of a packed pebble bed. Some analytical solutions of view factor between
primitives, such as a sphere, plane, and cylinder are available [40]. The analytical
expression of view factor between two unit spheres without any blockage can be
obtained by the general solution method [41]:
V =
1
π h
π
2
0
2η − sin(2η)
h 2 − 4 cos 2 (η)
sin(2η)dη,
(5.77)
where h is the distance between two unit spheres and h ≥2. For two spheres at contact
(h < 2), the solution is deduced to
V =
4
π h(2 + h)
π
2
arccos
h 2
4
2η − sin(2η)
h 2 − 4 cos 2 (η)
sin(2η)dη +
h
2
16
(h − 2). (5.78)
However, the thermal radiation heat exchange between two spheres in packed
pebble bed is significantly blocked by each other. Adaptive integration [42] and
Tanaka integral [43] are traditional numerical methods to compute the view factor between two spheres or planes subject to any blockage. For complex threedimensional geometries, it is recommended to use the Monte Carlo Method (MCM)
by using ray tracing [30, 44] to calculate the view factor. The MCM can be accelerated by the multithreading and Graphics Processing Unit (GPU) and much faster
than the integral method for computation [45].
Although the view factor between particles of the packed pebble bed is getting small when inter-particle distance h increases (Fig. 5.21), the thermal radiation
exchange still exists between long-distance spheres that are not Voronoï neighboring
particles in the bed. Note that the view factor for all Voronoï neighbors under the
short-range model is only accumulated to about 0.8343 (Fig. 5.21). However, the
averagely cumulative sums of the view factor for two and three peripheral layers of
Voronoï neighbors are about 0.9869 and 0.9991, respectively (e.g., three layers of
particles enclosing the white particle are composed of about 200 nearby particles in
Fig. 5.21a). Thus, it is considered that, under the long-range condition, it is almost
an enclosed space for the while particle to exchange radiative heat with three peripheral layers of Voronoï neighbors. In other words, it is sufficient to compute the view
factors of the white particle within the three layers of particles. It is also similar to
computing the view factors for other particles.
Assuming every particle is an opaque and gray body, its surface radiosity J i is
J i = ε r σ T
4
i + (1 − ε r )G i ,
(5.79)
5 Numerical Models for Pebble-Bed Heat Transfer
5.3.4 Long-Range Radiation Model (LRM)
In the long-range radiation model, it is of vital importance to calculate the view
factors between any two spheres and from sphere to a wall to study the thermal
radiation of a packed pebble bed. Some analytical solutions of view factor between
primitives, such as a sphere, plane, and cylinder are available [40]. The analytical
expression of view factor between two unit spheres without any blockage can be
obtained by the general solution method [41]:
V =
1
π h
π
2
0
2η − sin(2η)
h 2 − 4 cos 2 (η)
sin(2η)dη,
(5.77)
where h is the distance between two unit spheres and h ≥2. For two spheres at contact
(h < 2), the solution is deduced to
V =
4
π h(2 + h)
π
2
arccos
h 2
4
2η − sin(2η)
h 2 − 4 cos 2 (η)
sin(2η)dη +
h
2
16
(h − 2). (5.78)
However, the thermal radiation heat exchange between two spheres in packed
pebble bed is significantly blocked by each other. Adaptive integration [42] and
Tanaka integral [43] are traditional numerical methods to compute the view factor between two spheres or planes subject to any blockage. For complex threedimensional geometries, it is recommended to use the Monte Carlo Method (MCM)
by using ray tracing [30, 44] to calculate the view factor. The MCM can be accelerated by the multithreading and Graphics Processing Unit (GPU) and much faster
than the integral method for computation [45].
Although the view factor between particles of the packed pebble bed is getting small when inter-particle distance h increases (Fig. 5.21), the thermal radiation
exchange still exists between long-distance spheres that are not Voronoï neighboring
particles in the bed. Note that the view factor for all Voronoï neighbors under the
short-range model is only accumulated to about 0.8343 (Fig. 5.21). However, the
averagely cumulative sums of the view factor for two and three peripheral layers of
Voronoï neighbors are about 0.9869 and 0.9991, respectively (e.g., three layers of
particles enclosing the white particle are composed of about 200 nearby particles in
Fig. 5.21a). Thus, it is considered that, under the long-range condition, it is almost
an enclosed space for the while particle to exchange radiative heat with three peripheral layers of Voronoï neighbors. In other words, it is sufficient to compute the view
factors of the white particle within the three layers of particles. It is also similar to
computing the view factors for other particles.
Assuming every particle is an opaque and gray body, its surface radiosity J i is
J i = ε r σ T
4
i + (1 − ε r )G i ,
(5.79)