360
5 Numerical Models for Pebble-Bed Heat Transfer
5.5.2 Mechanism of Contact Thermal Resistance
In this section, the contact thermal resistance with stagnant fluid is investigated by
modeling the particle–particle and the particle–wall conduction in a discrete particle
simulation. The model predictions of effective thermal conductivity are compared
with various experimental data for validation. The effect of the conductivity ratio
and the void fraction is discussed.
5.5.2.1 Contact Thermal Resistance
The conduction can be divided into two categories: (1) the solid–solid conduction
through contacted particles, including the area contact (PPA) and point contact (PPP);
(2) the conduction through the fluid near the contact point (PFP). In a thermal discrete
element method, the rate of heat conduction between two discrete particles (“i” and
“ j”) at contact is given by
Q
r
i, j =
T p,i − T p, j
R pp
(5.222)
where T p,i and T p, j are the particle temperatures. R pp is the particle–particle contact
thermal resistance. The relationship between the contact resistance and the effective thermal conductivity k e for the packed bed of structured or random packing is
formulated as
k e =
1
π
N (1 − ε r )
1
R pp d
(5.223)
where d, N , and ε r are the particle diameter, average coordination number, and void
fraction of the packed bed, respectively.
Firstly, for the Particle-Particle-Area (PPA) contact, the area contact resistance
when
2k s r c
k f d
1 [113] is written as
R area =
1
2k s r c
(5.224)
where r c is the radius of the contact area. k s and k f are the solid thermal conductivity of particles and fluids conductivity, respectively. Thus, the effective thermal
conductivity of the area contact is given as
k area = N (1 − ε r )k s η,
(5.225)
where η is the dimensionless contact area and it is defined as η =
2r c
πd
.
Secondly, for the PPP conduction, in the packed bed with stagnant fluid (gas or
liquid), the thermal resistance at the contact point at high conductivity ratio (k s k f )
in [142] is written as
5 Numerical Models for Pebble-Bed Heat Transfer
5.5.2 Mechanism of Contact Thermal Resistance
In this section, the contact thermal resistance with stagnant fluid is investigated by
modeling the particle–particle and the particle–wall conduction in a discrete particle
simulation. The model predictions of effective thermal conductivity are compared
with various experimental data for validation. The effect of the conductivity ratio
and the void fraction is discussed.
5.5.2.1 Contact Thermal Resistance
The conduction can be divided into two categories: (1) the solid–solid conduction
through contacted particles, including the area contact (PPA) and point contact (PPP);
(2) the conduction through the fluid near the contact point (PFP). In a thermal discrete
element method, the rate of heat conduction between two discrete particles (“i” and
“ j”) at contact is given by
Q
r
i, j =
T p,i − T p, j
R pp
(5.222)
where T p,i and T p, j are the particle temperatures. R pp is the particle–particle contact
thermal resistance. The relationship between the contact resistance and the effective thermal conductivity k e for the packed bed of structured or random packing is
formulated as
k e =
1
π
N (1 − ε r )
1
R pp d
(5.223)
where d, N , and ε r are the particle diameter, average coordination number, and void
fraction of the packed bed, respectively.
Firstly, for the Particle-Particle-Area (PPA) contact, the area contact resistance
when
2k s r c
k f d
1 [113] is written as
R area =
1
2k s r c
(5.224)
where r c is the radius of the contact area. k s and k f are the solid thermal conductivity of particles and fluids conductivity, respectively. Thus, the effective thermal
conductivity of the area contact is given as
k area = N (1 − ε r )k s η,
(5.225)
where η is the dimensionless contact area and it is defined as η =
2r c
πd
.
Secondly, for the PPP conduction, in the packed bed with stagnant fluid (gas or
liquid), the thermal resistance at the contact point at high conductivity ratio (k s k f )
in [142] is written as