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5 Numerical Models for Pebble-Bed Heat Transfer
the conductive Effective Thermal Conductivity (ETC) in a packed bed of spherical
particles, there are still lacks of advanced models which are based on the essences of
physical laws of radiation to compute the ETC and in relation with the particle-scale
pore structure of the packed bed.
A continuum model is proposed here to derive the effective thermal conductivity
of a pebble bed. Based on a concept of continuum, the conductive and radiative heat
transfer between particles in a nuclear pebble bed is described in a uniform form. A
new analytical expression of ETC is derived based on the continuum model to solve
conduction and radiation between pebbles in the packed bed. It is a physics-based
equation determined by the temperature, number density, heat transfer coefficient,
and radial distribution function. The results predicted by this model are in good agreement with existing models and correlations. It will indicate that the local temperature
in the radiation case without internal heat sources is determined by all possible surrounding pebbles weighted by a radiative kernel function. The DEM packing results
are in good agreement with the solution of the continuum model. Both the conductive and radiative continuum models converge to the heat conduction in continuum
mechanics at size factor μ 1.
5.2.1 Uniform Effective Thermal Conductivity (uETC)
In a packed bed filled with mono-sized spheres in stagnant fluid, the conductive or
radiative heat flux between two particles is generally formulated in the continuum
model as the following general expression
Q i j = H (r i j )(T
n
i − T
n
j ),
(5.1)
where r i j , T i and T j are the distance between the two-particle centers, and the temperatures of particle i and particle j, respectively. H (r i j ) is the heat transfer coefficient,
and n is the power:
• For particle–particle conduction at contact, n is 1. H (r i j ) is zero for nonoverlapping pairs. For the pairs at contact, the heat transfer is given by
Q c,i j = H (d)(T i − T j ) =
(T i − T j )
R i j
(5.2)
where d is the particle diameter. R i j is the thermal contact resistance, which is a
function of contact force, surface roughness, and material properties [11, 12].
• For radiative heat transfer, n is 4, and the heat flux between two opaque particles
of a gray surface is written as
Q r,i j = ε r A i X i j (E b,i − E b, j ) = ε r A i X i j (σ T
4
i − σ T
4
j ),
(5.3)
5 Numerical Models for Pebble-Bed Heat Transfer
the conductive Effective Thermal Conductivity (ETC) in a packed bed of spherical
particles, there are still lacks of advanced models which are based on the essences of
physical laws of radiation to compute the ETC and in relation with the particle-scale
pore structure of the packed bed.
A continuum model is proposed here to derive the effective thermal conductivity
of a pebble bed. Based on a concept of continuum, the conductive and radiative heat
transfer between particles in a nuclear pebble bed is described in a uniform form. A
new analytical expression of ETC is derived based on the continuum model to solve
conduction and radiation between pebbles in the packed bed. It is a physics-based
equation determined by the temperature, number density, heat transfer coefficient,
and radial distribution function. The results predicted by this model are in good agreement with existing models and correlations. It will indicate that the local temperature
in the radiation case without internal heat sources is determined by all possible surrounding pebbles weighted by a radiative kernel function. The DEM packing results
are in good agreement with the solution of the continuum model. Both the conductive and radiative continuum models converge to the heat conduction in continuum
mechanics at size factor μ 1.
5.2.1 Uniform Effective Thermal Conductivity (uETC)
In a packed bed filled with mono-sized spheres in stagnant fluid, the conductive or
radiative heat flux between two particles is generally formulated in the continuum
model as the following general expression
Q i j = H (r i j )(T
n
i − T
n
j ),
(5.1)
where r i j , T i and T j are the distance between the two-particle centers, and the temperatures of particle i and particle j, respectively. H (r i j ) is the heat transfer coefficient,
and n is the power:
• For particle–particle conduction at contact, n is 1. H (r i j ) is zero for nonoverlapping pairs. For the pairs at contact, the heat transfer is given by
Q c,i j = H (d)(T i − T j ) =
(T i − T j )
R i j
(5.2)
where d is the particle diameter. R i j is the thermal contact resistance, which is a
function of contact force, surface roughness, and material properties [11, 12].
• For radiative heat transfer, n is 4, and the heat flux between two opaque particles
of a gray surface is written as
Q r,i j = ε r A i X i j (E b,i − E b, j ) = ε r A i X i j (σ T
4
i − σ T
4
j ),
(5.3)