3.3 Effective Thermal Diffusivity and Conductivity
139
void. However, ρ h and c p,h are density and specific heat capacity of helium at different
temperature under the helium condition, respectively. Also, V v = V h = εV , where
V is the total volume of the control volume. ε is the averaged porosity of a local
pebble bed. Since various literature has studied the averaged porosity of random
packing structure in a pebble bed [15], ε is chosen as 0.39 here. In this method, the
pebble zone is considered as the heat transfer continuum by mixing the specific local
porous structure. The heat conduction equation of a porous pebble bed can be derived
from Eq. (3.39), as
1
r
∂
∂r
r λ
∂ T
∂r
=
∂
∂t
(1 − ε)ρ g C p,g T + ερ v C p,v T
, r ∈ [R in , R out ], t ∈ [0, +∞),
(3.40)
where dV is eliminated on both sides of Eq. (3.40). T is the temperature of the
pebble bed between radial coordinates R in and R out . r is the radial coordinate. λ
is the effective thermal conductivity. According to Eq. (3.40), the effective thermal
diffusivities of the pebble bed in vacuum and helium are defined as
α =
λ
(1−ε)ρ g c p,g
, in vacuum,
(3.41)
and
α =
λ
(1−ε)ρ g c p,g +ερ h C p,h
, in helium.
(3.42)
Then, the average density of the pebble bed is ρ pb = (1 − ε)ρ g or ρ pb = (1 −
ε)ρ g + ερ h . In this experiment, the effective thermal diffusivity is first retrieved
by the inverse method and experimental data. Subsequently, the effective thermal
conductivity is calculated through Eqs. (3.41) and (3.42).
The function type of the thermal effective diffusivity of a pebble bed is assumed as
a quadratic polynomial function with respect to temperature in previous researches
[22–24]
α(T ) = p 1 + p 2 T + p 3 T
2
, T ∈ [T min , T max ],
(3.43)
where T min and T max are the minimum and maximum temperatures recorded by
thermocouples. P = ( p 1 , p 2 , p 3 ) is the parameter vector.
The physical properties of graphite and helium should be used to convert α to
λ and to estimate their uncertainties. First, the density and specific heat capacity
of graphite are measured by other specialized methods. Two graphite samples are
the same batch with the graphite balls from the same manufacturer. The average
graphite density of two samples is 1,846 kg/m
3 measured by the third party, and the
volume-expansion thermal coefficient is 4.5 × 10
−6◦ C
−1 , which gives 0.7% density
variation between 30
◦ C and 1,650
◦ C. The error of the graphite density is 50 kg/m
3
given by the manufacturer.
139
void. However, ρ h and c p,h are density and specific heat capacity of helium at different
temperature under the helium condition, respectively. Also, V v = V h = εV , where
V is the total volume of the control volume. ε is the averaged porosity of a local
pebble bed. Since various literature has studied the averaged porosity of random
packing structure in a pebble bed [15], ε is chosen as 0.39 here. In this method, the
pebble zone is considered as the heat transfer continuum by mixing the specific local
porous structure. The heat conduction equation of a porous pebble bed can be derived
from Eq. (3.39), as
1
r
∂
∂r
r λ
∂ T
∂r
=
∂
∂t
(1 − ε)ρ g C p,g T + ερ v C p,v T
, r ∈ [R in , R out ], t ∈ [0, +∞),
(3.40)
where dV is eliminated on both sides of Eq. (3.40). T is the temperature of the
pebble bed between radial coordinates R in and R out . r is the radial coordinate. λ
is the effective thermal conductivity. According to Eq. (3.40), the effective thermal
diffusivities of the pebble bed in vacuum and helium are defined as
α =
λ
(1−ε)ρ g c p,g
, in vacuum,
(3.41)
and
α =
λ
(1−ε)ρ g c p,g +ερ h C p,h
, in helium.
(3.42)
Then, the average density of the pebble bed is ρ pb = (1 − ε)ρ g or ρ pb = (1 −
ε)ρ g + ερ h . In this experiment, the effective thermal diffusivity is first retrieved
by the inverse method and experimental data. Subsequently, the effective thermal
conductivity is calculated through Eqs. (3.41) and (3.42).
The function type of the thermal effective diffusivity of a pebble bed is assumed as
a quadratic polynomial function with respect to temperature in previous researches
[22–24]
α(T ) = p 1 + p 2 T + p 3 T
2
, T ∈ [T min , T max ],
(3.43)
where T min and T max are the minimum and maximum temperatures recorded by
thermocouples. P = ( p 1 , p 2 , p 3 ) is the parameter vector.
The physical properties of graphite and helium should be used to convert α to
λ and to estimate their uncertainties. First, the density and specific heat capacity
of graphite are measured by other specialized methods. Two graphite samples are
the same batch with the graphite balls from the same manufacturer. The average
graphite density of two samples is 1,846 kg/m
3 measured by the third party, and the
volume-expansion thermal coefficient is 4.5 × 10
−6◦ C
−1 , which gives 0.7% density
variation between 30
◦ C and 1,650
◦ C. The error of the graphite density is 50 kg/m
3
given by the manufacturer.