3.2 Experimental Facility and Methodology
125
3.2.2.2 Direct Problem
As for pebble bed, there are plenty of voids between spherical surfaces. However, it
can be regarded as a heat conduction continuum with the effective thermal diffusivity
and conductivity, volume-averaged density and specific heat capacity. For the direct
physical problem of heat conduction in pebble bed, a heat diffusion model is described
as follows with the following assumptions:
• It’s one-dimensional heat conduction at a middle level due to the symmetric configuration of a plant.
• The pebble zone is the heat transfer continuum.
• Effect of variable wall porosity is neglected in the internal pebble bed.
• Boundary heat flow is continuous.
• The derivative of temperature is continuous.
• The thermal properties vary slowly with time and space.
Considering the above assumptions, the mathematical model of heat conduction in
this experiment can be described as
1
r
∂
∂r
r λ(T )
∂ T
∂r
=
∂
∂t
T ¯
ρ(T )c g (T )
, r ∈ [R in , R out ], t ∈ [0, +∞), (3.1)
with initial and boundary conditions
I.C.:
T (r, 0) = I (r )
(3.2)
B.C.: T (R in , t) = f in (t), T (R out , t) = f out (t).
(3.3)
Equation (3.1) is the cylindrical one-dimensional heat conduction equation without interior heat generation. T is the temperature of the pebble-bed zone between
radial coordinates R in and R out ; r is the radial coordinate; ¯
ρ is volume-averaged density; λ is effective thermal conductivity; c g is the specific heat of graphite element.
The R in and R out are the coordinates of inner and outer boundary points, and the t is
time. The f in and f out are the boundary temperature functions of R in and R out , and
I (r ) is the initial temperature distribution.
Considering the temperature dependence of effective thermal diffusivity in pebble
beds at a large temperature range, the mathematical equation of heat conduction is
a representative nonlinear matter which is hard to obtain an analytical solution to
describe the temperature field. However, it’s a simple diffusion equation for numerical
computation.
Noted that the Eq. (3.1)–(3.3), can’t distinguish λ from ¯
ρ and c g independently
and simultaneously since the coupled λ, ¯
ρ and c g are combined as the thermal diffusivity α. Actually, only the heat flow boundary condition can introduce the thermal
conductivity λ independently, and then λ can be solved from heat conduction equations.
125
3.2.2.2 Direct Problem
As for pebble bed, there are plenty of voids between spherical surfaces. However, it
can be regarded as a heat conduction continuum with the effective thermal diffusivity
and conductivity, volume-averaged density and specific heat capacity. For the direct
physical problem of heat conduction in pebble bed, a heat diffusion model is described
as follows with the following assumptions:
• It’s one-dimensional heat conduction at a middle level due to the symmetric configuration of a plant.
• The pebble zone is the heat transfer continuum.
• Effect of variable wall porosity is neglected in the internal pebble bed.
• Boundary heat flow is continuous.
• The derivative of temperature is continuous.
• The thermal properties vary slowly with time and space.
Considering the above assumptions, the mathematical model of heat conduction in
this experiment can be described as
1
r
∂
∂r
r λ(T )
∂ T
∂r
=
∂
∂t
T ¯
ρ(T )c g (T )
, r ∈ [R in , R out ], t ∈ [0, +∞), (3.1)
with initial and boundary conditions
I.C.:
T (r, 0) = I (r )
(3.2)
B.C.: T (R in , t) = f in (t), T (R out , t) = f out (t).
(3.3)
Equation (3.1) is the cylindrical one-dimensional heat conduction equation without interior heat generation. T is the temperature of the pebble-bed zone between
radial coordinates R in and R out ; r is the radial coordinate; ¯
ρ is volume-averaged density; λ is effective thermal conductivity; c g is the specific heat of graphite element.
The R in and R out are the coordinates of inner and outer boundary points, and the t is
time. The f in and f out are the boundary temperature functions of R in and R out , and
I (r ) is the initial temperature distribution.
Considering the temperature dependence of effective thermal diffusivity in pebble
beds at a large temperature range, the mathematical equation of heat conduction is
a representative nonlinear matter which is hard to obtain an analytical solution to
describe the temperature field. However, it’s a simple diffusion equation for numerical
computation.
Noted that the Eq. (3.1)–(3.3), can’t distinguish λ from ¯
ρ and c g independently
and simultaneously since the coupled λ, ¯
ρ and c g are combined as the thermal diffusivity α. Actually, only the heat flow boundary condition can introduce the thermal
conductivity λ independently, and then λ can be solved from heat conduction equations.