318
5 Numerical Models for Pebble-Bed Heat Transfer
τ = 1 − T
(5.162)
where T is the arithmetic average of non-dimensional temperatures of all particles
in the packed pebble bed. In the simulation, all the particles have the same properties. The Nusselt number is assumed to be constant too, since the fluid properties
are assumed to be constant and there is no acceleration of the flow after the initial
development of the fluid field at the inlet. Thus, the flow-particle heat convection
coefficients for all particles are the same. At the start, all particles have the same
temperature at τ = 0. When all the particles have been cooled to fluid inlet temperature, τ will be 1. The numerical results for different initial temperatures with the
same non-dimensional temperature drop (τ = 0.5) are shown in Fig. 5.64. It can be
seen that there is no notable difference between the non-dimensional particle temperature fields at low temperatures (less than 900 K), for which the particle radiation
flux is not dominant. However, the particle temperature fields tend to be much more
uniform at high temperatures, since the heat transfer processes in packed pebble beds
are enhanced by thermal radiation at high temperatures.
In order to quantify the effect of particle radiation in the heat transfer processes
of packed pebble beds, the range of non-dimensional temperature for all particles is
defined as
θ = T
max − T
min
(5.163)
where T
max and T
min are the maximum and minimum of non-dimensional particle
temperatures, respectively. The standard deviation also represents the quantity of
data dispersion in statistics, which can be calculated as
ϑ =
1
n
n
i=1
(T
i − T ) 2
(5.164)
From the simulation results shown in Fig. 5.65, the values for the range θ as
a function of the temperature drop τ increase from 0 at the start (τ = 0) to its
maximum at approximately τ = 0.5 (the details of θ max and τ = τ (θ max ) are shown
in Table 5.4). Then the values decrease gradually to 0 again at τ = 1, and the shape of
the curve is affected significantly by particle thermal radiation. The trends exhibited
by θ and ϑ for all simulations are similar. It is important to control the highest
temperature in packed pebble beds in engineering applications. Thus, the range of
non-dimensional temperature θ can be used to quantify the degree of uniformity of
the particle temperature field.
The range θ is also affected by heat convection, and its maximum θ max increases
significantly at high Nusselt numbers, as seen in Fig. 5.66. θ max drops from 0.39
(0.382: hereafter the value in parentheses here is obtained by SEM (Sect. 5.3.7),
whereas the value in front of the parentheses is obtained by SRM (Sect. 5.3.2))
at 400 K to 0.31 (0.314) at 1,500 K and 0.26 (0.29) at 1,800 K for N u = 30. For
N u = 60, it reduces to 0.53 (0.544) at 1,800 K from 0.66 (0.648) at 400 K. However,
by visual inspection, the difference between the curves for different Nusselt numbers
5 Numerical Models for Pebble-Bed Heat Transfer
τ = 1 − T
(5.162)
where T is the arithmetic average of non-dimensional temperatures of all particles
in the packed pebble bed. In the simulation, all the particles have the same properties. The Nusselt number is assumed to be constant too, since the fluid properties
are assumed to be constant and there is no acceleration of the flow after the initial
development of the fluid field at the inlet. Thus, the flow-particle heat convection
coefficients for all particles are the same. At the start, all particles have the same
temperature at τ = 0. When all the particles have been cooled to fluid inlet temperature, τ will be 1. The numerical results for different initial temperatures with the
same non-dimensional temperature drop (τ = 0.5) are shown in Fig. 5.64. It can be
seen that there is no notable difference between the non-dimensional particle temperature fields at low temperatures (less than 900 K), for which the particle radiation
flux is not dominant. However, the particle temperature fields tend to be much more
uniform at high temperatures, since the heat transfer processes in packed pebble beds
are enhanced by thermal radiation at high temperatures.
In order to quantify the effect of particle radiation in the heat transfer processes
of packed pebble beds, the range of non-dimensional temperature for all particles is
defined as
θ = T
max − T
min
(5.163)
where T
max and T
min are the maximum and minimum of non-dimensional particle
temperatures, respectively. The standard deviation also represents the quantity of
data dispersion in statistics, which can be calculated as
ϑ =
1
n
n
i=1
(T
i − T ) 2
(5.164)
From the simulation results shown in Fig. 5.65, the values for the range θ as
a function of the temperature drop τ increase from 0 at the start (τ = 0) to its
maximum at approximately τ = 0.5 (the details of θ max and τ = τ (θ max ) are shown
in Table 5.4). Then the values decrease gradually to 0 again at τ = 1, and the shape of
the curve is affected significantly by particle thermal radiation. The trends exhibited
by θ and ϑ for all simulations are similar. It is important to control the highest
temperature in packed pebble beds in engineering applications. Thus, the range of
non-dimensional temperature θ can be used to quantify the degree of uniformity of
the particle temperature field.
The range θ is also affected by heat convection, and its maximum θ max increases
significantly at high Nusselt numbers, as seen in Fig. 5.66. θ max drops from 0.39
(0.382: hereafter the value in parentheses here is obtained by SEM (Sect. 5.3.7),
whereas the value in front of the parentheses is obtained by SRM (Sect. 5.3.2))
at 400 K to 0.31 (0.314) at 1,500 K and 0.26 (0.29) at 1,800 K for N u = 30. For
N u = 60, it reduces to 0.53 (0.544) at 1,800 K from 0.66 (0.648) at 400 K. However,
by visual inspection, the difference between the curves for different Nusselt numbers