This document summarizes a study on heat transfer distribution of randomly packed pebble-bed fuels for a fluoride salt-cooled high temperature reactor. Large-eddy simulation was performed to model the pebble bed and evaluate turbulence models. Results show the pebble heat transfer coefficient is randomly distributed with a Gaussian statistical distribution. A new Nusselt number correlation is developed based on the simulation results to predict heat transfer over the operating ranges of the reactor system.

1. November 13, 2018
Embedded topical meeting on Advanced Thermal Hydraulics –
ANS 208 Winter, Orlando, FL
Seong Gu Kim a, Maolong Liu b, Youho Lee a*, Jeong Ik Lee c
a Dept. of Nuclear Engineering, University of New Mexico (UNM)
b Dept. of Nuclear Science and Engineering, Shanghai Jiao Tong University
c Dept. of Nuclear & Quantum Engineering,
Korea Advanced Institute of Science and Technology (KAIST)
Heat Transfer Distribution of a Randomly packed
Pebble-bed Fuels for Fluoride salt-cooled
High Temperature Reactor (FHR)
1/17

2. <C. Andreades et al., UC Berkeley, Nuclear Technology (2017)>
1. Introduction
<Randomly packed pebble-bed model
for CFD analysis>
 Fluoride salt-cooled high-temperature reactor (FHR) is one of the Gen IV
nuclear systems.
 The pebble bed fuels with a 30mm diameter randomly packed inside the
annulus shape FLiBe reactor’s core.*FLiBe: Molten salt coolant (Li2BeF4)
 FHR’s coolant – High Prandtl number fluid (7.8 – 19 in operating range)
 Pebble beds with 30mm-D are randomly packed inside the core. It
requires a different approach from the conventional water-cooled
reactor.
2/17

3. 1. Introduction – Key results
(i) Large-Eddy Simulation (LES) was
performed to choose proper RANS
model. LES result can be a reference
data for the pebble’s HTC.
<A single Face-Centered Cubic model for
Large-Eddy Simulation>
3/17
(ii) The statistical distribution of the
randomly-packed pebble’s heat
transfer performance was examined.
<Surface temperature of the randomly-
packed pebble beds>
(iii) Based on the CFD results, a new
correlation for the high Pr fluid
coolant (FLiBe) proposed.
<Distribution of heat transfer coefficient in
various Re and Pr ranges>

4. 2. CFD model – Approach
4/17
Random packing
preparation
CFD analysis Post-processing
Create randomly-
packed geometry
Effect of gap size
Grid sensitivity
Number of
pebbles in
single domain Turbulence
model 
Large-Eddy
Simulation
Average pebble
HTC
Statistical
distribution of
HTC
Development of
new correlation
Wall effect

5. 2. CFD model – Create randomly-packed geometry
*Effect of gap size – Aligned geometry
*In-house MATLAB code - Randomly packed-bed
geometry
*Number of pebbles in a single domain
5/17
<Randomly generated packed geometries> Packing
factor: 0.40
Packing
factor: 0.425

6. 2. CFD model – Wall effect
*Wall effect
5/17
 The fluid domain was reduced to
the core region in order to remove
wall effect.
 The diameter and height were
reduced by 2r.
 The models with various space
between the pebble and wall were
tested. In the process, the reduced
domain kept the same.
 The results indicated little
differences as much as 0.15%.
Wall space Pebble HTC
0 mm 3965.8 W/m2-K
1.0 mm 3960.5 W/m2-K
2.5 mm 3963.4 W/m2-K
10 mm 3959.7 W/m2-K

7. 2. CFD model – Grid sensitivity
*Grid sensitivity
Number
of
elements
Average
HTC
[W/m2-K]
Relative
error
3 millions 3,701 3.60%
4 millions 3,619 5.73%
5 millions 3,733 2.76%
6 millions 3,833 0.16%
7 millions 3,839 -
<Vertical plane view of mesh><Transparent view of mesh>
6/17

8. 2. CFD model – Large-Eddy Simulation
<Single FCC model for the Large-Eddy Simulation>
Table. Spatial and temporal discretization for LES
 Grid stretching ratio and time step were
carefully determined and tested to
perform accurate LES.
 WALE sub-grid scale model without wall
function approach.
 A single channel composed of 3.6
millions of cells and 12 millions of nodes.
7/17
Wall y plus Y+ < 1 Time step 4.00e-04 sec
Number of
prism layers
12
Flow-through
time
7.055e-02 sec
Stretching ratio
(Prism layers)
(Flow field)
1.1
2 - 4
Total
simulation
time
4.80e-01 sec
Kolmogorov
time scale
𝑡 𝑛 =
𝜐
𝜀
ൗ1
2
=1.277e-03
sec
Number of
element
3,640,453
CFL number
𝐶 =
𝑈𝛥𝑡
𝛥𝑥
= 0.907
Number of
nodes
12,568,139

9. 2. CFD model – Post-processing
 Boundary conditions determined
based on the conceptual design of
the Mark-I pebble-bed reactor.
 Following expression is used for
calculates pebble heat transfer
coefficient (HTC)
 Average and standard deviation
values of 50 pebble HTC were
obtained.
h : Pebble HTC [W/m2-K]
Qgen : Heat generation from the pebble [W]
AS : Surface area of a fuel [m2]
TS : Average surface temperature of fuel [m2]
Tbulk : Average temperature of cross-sectional area
at the height of fuel center [Co]
<The cross section to obtain bulk temperature> <T* profile versus Z*(Height)> 8/17
Core inlet
temperature
600 oC Coolant
FLiBe
(7Li2BeF4)
Core outlet
temperature
700 oC
Power density of
Active core
22.7 W/cc
Mass flux
317
kg/m2-s
Pressure 1 atm
Diameter of pebble 60 mm Volumetric flow rate 0.52 m3/s
Table. Boundary conditions based on the Mark-I conceptual design
*Pebble HTC defined as:

10.  A time-averaged result of LES
compared to that of the RANS results.
 The average value of differences in
velocity and temperature at each x-
locations considered to the key factor.
 SST model showed the lowest
difference with LES result.
 The randomly-packed models with 50
number of the pebble were solved by
using k-Omega SST turbulence model.
9/17
Average difference
of the velocity at
each x-locations
Average difference
of the temperature
at each x-locations
K-Omega SST
Wall function
3.80% 0.050%
K-Omega SST
Low Re approach
2.72% 0.004%
K-Epsilon two-layer
Low Re approach
6.23% 0.224%
<Line probe>
3. Result -
LES/RANS

11. 3. Result – Statistical Nature of Pebble Heat Transfer
 The HTC result obtained from the
randomly generated 8 cases (400
pebbles) showed a Gaussian
distribution.
 The result shows a significant
variation of the pebble HTC inside
the cylindrical core.
 Pebble HTC is not related to the
radial or axial directions and
randomly distributed inside the
cylindrical domain.
*Statistical distribution of pebble HTC
10/17
Minimum HTC Maximum HTC
95% interval
(1.96𝜎)
3,015 W/m2-K 4,922 W/m2-K
99% interval
(2.33𝜎)
2,835 W/m2-K 5,102 W/m2-K
99.9% interval
(3.09𝜎)
2,465 W/m2-K 5,471 W/m2-K
99% confidence interval
Pebble
Mean=3968
SD = 486.5

12. 3. Result – Effect of Prandtl number on the spread of HTC
11/17
Average, standard
deviation values of
pebble HTC = f(Re, Pr)
1200oC
1000oC
800oC
600oC
16 cases

13. 3. Result – Effect of Prandtl number on the spread of HTC
*Thickness of thermal boundary layer
<The LES result of temperature profiles at the Prandtl number 2.58 and 19.1>
12/17
1,200oC
600oC

14. 3. Result – Pebble HTC’s statistical distribution
13/17
 Because of geometric randomness, the surface temperature
varies many among the pebbles inside the core.
 The safety criteria and material limits for the pebble-bed
reactor need to be reviewed based on this uncertainty.
<Temperature plot of the pebble fuel’s surfaces>
99% interval
2.33𝜎
699oC

15. *Nusselt number correlation for randomly-packed pebbles
with FLiBe coolant
 New Nusselt correlation was proposed based on
the CFD-obtained HTC results.
 Results are used for non-linear curve fitting. A
new correlation has an R-square of 0.989.
 Engineering implication for thermal-hydraulic
design – Average and Deviation of Pebble HTC.
 The ranges of Pr number and Re number cover
most of the operating condition of FLiBe reactor.
Coefficient (with 95%
confidence bounds)
a 0.01238 (±0.00629)
b 0.7479 (±0.0507)
c 0.3444 (±0.0445)
R-square 0.9886
Re Prb c
Nu a
2,024 < Re <17,150
2.58 < Pr < 19.08
𝜙 = 0.40
14/17
3. Result – A new Nusselt number correlation for FLiBe pebble-bed reactor
0.7479 0.3444
0.01238Re PrNu 

16. 3. Result – A new Nusselt number correlation for FLiBe pebble-bed reactor
 The correlations developed for the randomly packed pebble-bed
with other coolants were examined with FLiBe conditions.
 Compared results show that the new correlation is required for
the high Prandtl number and low Reynolds number fluid such
as molten salt. 15/17

17. 3. Result – A new Nusselt number correlation for FLiBe pebble-bed reactor
16/17
<Comparison of Nu vs Re under fixed Pr =19.1> <Comparison of Nu vs Pr under fixed Re =6,000>
Range of mass flow rate: 70% - 200%
Range of temperature: 600 oC – 1200 oC
1200oC
Pr=2.61
600oC
Pr=20
Noramal operating condition
Re = 6,000

18. 4. Conclusions
*Conclusions
(i) Large-Eddy Simulation (LES) was performed by using a single FCC channel. Proper RANS model was selected, and
reference results for design purpose were obtained.
(ii) This study presents a quantification of statistical distribution for randomly-packed pebble’s HTC with molten salt
coolant, FLiBe. Those data could be used for engineering design and safety implications.
(iii) New Nusselt number correlation for the randomly-packed pebble fuels with FLiBe coolant was developed. This
correlation covers most of the FLiBe reactor’s operating & accident conditions of Prandtl number and Reynolds
number. (2.58 < Pr < 19.1, 2025 < Re < 17150)
17/17
havg = f(Re, Pr)
𝜎 = f(Re, Pr)

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