5.2 Continuum Modeling of Pebble Radiation
247
70,000 graphite spheres packed randomly in an annular pebble bed. The bed is operated in a vacuum condition and the highest pebble temperature is about 1,200
◦ C.
The experimental data shows that thermal radiation contributes to the main part of
total effective thermal conductivity at high temperatures. Furthermore, the radiation exchange factor is known as the non-dimensional radiative effective thermal
conductivity and widely used for estimating particle radiation in engineering. Even
though many empirical correlations were developed which are available for a packed
bed of spheres [10, 25], there is still a lack of a generic physical expression of the
radiation exchange factor based on the packing structure of particles. Furthermore,
there still has no available expression to estimate thermal radiation between complex
non-spherical particles in packed beds based on good understanding of radiation
mechanisms.
In this section, the aim is to develop an Approximation Function Model (AFM)
based on the continuum assumption of densely packed pebble bed, which works as a
reasonable replacement of the existing Radiative Transfer Equation (RTE). Based on
the surface-to-surface radiation mechanism, physical expression is proposed herein
for the radiation exchange factor of a packed bed.
5.2.2.1 Radiation Exchange Factor
In a packed bed with a non-participating medium, the essential characteristics of the
radiative heat transfer are non-contact and long-range interaction [5]. The net flux
between two opaque gray particles through surface-to-surface radiation is given as
Q r,i j = f (ε r )A i X i j σ (T
4
i − T
4
j ),
(5.33)
where T i and T j are the temperatures of particle i and particle j. A i and σ are particle
surface area and the Stefan–Boltzmann constant. ε r is the emissivity, and f (ε r ) is
the independent emissivity term. Basically, it satisfies that f (0) = 0 at ε r = 0 and
f (1) = 1 for black surfaces. It is recommended that f (ε r ) = ε r for dense packed
bed [15, 17]. X i j is the view factor of the two particles, which may be obstructed by
other ones in a bed, and decreases significantly with a distance. In the approximation
function model, it is assumed that, with continuum assumption, the obstructed view
factor X i j for densely packed beds of mono-sized particles is a function of the distance
between particle centers r i j , i.e.,
X i j = X m h i j ,
(5.34)
where X m =
1
2
−
4
3
π is the view factor between two spheres in contact without
overlapping [5]. h i j is defined as the Radiation Interaction Function (RIF). If there
is a particle at x i shown in Fig. 5.6, the radiative flux to the control volume ΔV j at
position x j is given by
ΔQ r,i j = f (ε r )A i X m h(r i j )σ (T
4
i − T
4
j )Δn(r i j ),
(5.35)
247
70,000 graphite spheres packed randomly in an annular pebble bed. The bed is operated in a vacuum condition and the highest pebble temperature is about 1,200
◦ C.
The experimental data shows that thermal radiation contributes to the main part of
total effective thermal conductivity at high temperatures. Furthermore, the radiation exchange factor is known as the non-dimensional radiative effective thermal
conductivity and widely used for estimating particle radiation in engineering. Even
though many empirical correlations were developed which are available for a packed
bed of spheres [10, 25], there is still a lack of a generic physical expression of the
radiation exchange factor based on the packing structure of particles. Furthermore,
there still has no available expression to estimate thermal radiation between complex
non-spherical particles in packed beds based on good understanding of radiation
mechanisms.
In this section, the aim is to develop an Approximation Function Model (AFM)
based on the continuum assumption of densely packed pebble bed, which works as a
reasonable replacement of the existing Radiative Transfer Equation (RTE). Based on
the surface-to-surface radiation mechanism, physical expression is proposed herein
for the radiation exchange factor of a packed bed.
5.2.2.1 Radiation Exchange Factor
In a packed bed with a non-participating medium, the essential characteristics of the
radiative heat transfer are non-contact and long-range interaction [5]. The net flux
between two opaque gray particles through surface-to-surface radiation is given as
Q r,i j = f (ε r )A i X i j σ (T
4
i − T
4
j ),
(5.33)
where T i and T j are the temperatures of particle i and particle j. A i and σ are particle
surface area and the Stefan–Boltzmann constant. ε r is the emissivity, and f (ε r ) is
the independent emissivity term. Basically, it satisfies that f (0) = 0 at ε r = 0 and
f (1) = 1 for black surfaces. It is recommended that f (ε r ) = ε r for dense packed
bed [15, 17]. X i j is the view factor of the two particles, which may be obstructed by
other ones in a bed, and decreases significantly with a distance. In the approximation
function model, it is assumed that, with continuum assumption, the obstructed view
factor X i j for densely packed beds of mono-sized particles is a function of the distance
between particle centers r i j , i.e.,
X i j = X m h i j ,
(5.34)
where X m =
1
2
−
4
3
π is the view factor between two spheres in contact without
overlapping [5]. h i j is defined as the Radiation Interaction Function (RIF). If there
is a particle at x i shown in Fig. 5.6, the radiative flux to the control volume ΔV j at
position x j is given by
ΔQ r,i j = f (ε r )A i X m h(r i j )σ (T
4
i − T
4
j )Δn(r i j ),
(5.35)