Microsoft Word - PRES22_0064.docx
DOI: 10.3303/CET2294039
Paper Received: 06 May 2022; Revised: 05 June 2022; Accepted: 16 June 2022
Please cite this article as: Wang J., Ma Y., Zeng M., Wang Q., 2022, Buoyancy Impact on Heat Transfer of Lead-Bismuth Eutectic for Nuclear
Applications, Chemical Engineering Transactions, 94, 235-240 DOI:10.3303/CET2294039
A publication of
CHEMICAL ENGINEERING TRANSACTIONS
VOL. 94, 2022 The Italian Association
of Chemical Engineering
Online at www.cetjournal.it
Guest Editors: Petar S. Varbanov, Yee Van Fan, Jiří J. Klemeš, Sandro Nižetić
Copyright © 2022, AIDIC Servizi S.r.l.
ISBN 978-88-95608-93-8; ISSN 2283-9216
Buoyancy Impact on Heat Transfer of Lead-Bismuth Eutectic
for Nuclear Applications
Jinghan Wang, Yangfan Ma, Min Zeng*, Qiuwang Wang
Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, Xi’an Jiaotong University, Xi’an, Shaanxi,
710049, China
zengmin@mail.xjtu.edu.cn
The liquid lead-bismuth eutectic (LBE) is an ideal primary coolant for the fourth-generation advanced nuclear
systems. Due to its special physical properties, especially the low Prandtl number and high density, the natural
convection phenomenon caused by buoyancy is significant. In this study, a circular tube is three-dimensionally
modeled to predict the flow and heat transfer process of LBE. On the base of the model validation, simulations
of the uniformly heated tube with/without buoyancy are performed. The different thermal-hydraulic performance
obtained under different tube arrangements and operating parameters are compared, and the effects of
buoyancy on the convective heat transfer of LBE are discussed. The results show that for LBE flow with smaller
Reynolds numbers, convective heat transfer is significantly enhanced when gravity is opposite to the flow
direction, while the same gravity and flow direction slightly weakens the convective heat transfer capacity. For
the higher Reynolds number LBE flow, the buoyancy impact and the tube arrangement can be ignored. When
the tube is placed horizontally, the heat transfer coefficient of the lower wall is about three times that of the
upper wall. For different engineering applications, an appropriate arrangement is necessary to improve the
efficiency of heat exchanger. This study may contribute to the development and application of LBE-cooled
reactors.
1. Introduction
Nuclear energy has emerged as the largest low-carbon source of electricity for sustainable development and
produces about 18 % of the electricity supply in advanced economies (IAEA, 2020). The problem of
management and disposal of nuclear wastes have risen due to the rapid expansion of nuclear power industrial
scale. Accelerator driven sub-critical system (ADS) has been widely recommended as a promising solution to
proliferate and transmute spent nuclear materials since the 1990s (Chen et al., 2013). Heavy liquid metal (HLM),
e.g. lead-bismuth eutectic (LBE), has been considered as the major coolant for sub-critical reactor like ADS
(Zhang et al., 2020). LBE has many suitable properties for ADS, such as good neutron economy, physical
properties, chemical inertness and strong buoyancy. The strong buoyancy, which provides more advanced
natural circulation ability than sodium-cooled or gas-cooled systems (Wang et al., 2017), has an important
impact on the flow field and temperature field. Mixed convection occurs in an ADS due to fluid gravity
heterogeneity caused by the density variation, and the effects of natural convection on forced convection heat
transfer cannot be ignored.
Many scholars have studied the buoyancy of liquid metals. Jackson (1983) proposed a theoretical model based
on the study of buoyancy of liquid metals in a vertical tube, which provides a criterion for the condition of
significant buoyancy. Niemann et al. (2018) pointed out that flow heat transfer and Reynolds stresses are
strongly influenced by buoyancy, and the turbulent flow of low Pr number fluids at different Reynolds and
Richardson numbers was investigated. Guo et al. (2020) used direct numerical simulations to study the mixed
convection of liquid lead in a horizontal tube, and investigated the effects of different Richardson numbers on
the mixed convection. Marocco et al. (2022) studied turbulent mixed convection of liquid metal flowing vertically
in a uniformly heated tube with Reynolds numbers ranging from 2,650 to 7,500 using large eddy simulations.
In order to improve the thermal-hydraulic performance of LBE, it is necessary to study the related effects of
buoyancy. In this study, a circular tube is three-dimensionally modelled to predict the flow and heat transfer
235
process of LBE. On the base of the model validation, simulations of the uniformly heated tube with/without
buoyancy are performed. The different thermal-hydraulic performance obtained under different tube
arrangements and operating parameters are compared, and the effects of buoyancy on the convective heat
transfer of LBE are discussed.
2. Physical model and numerical methods
2.1 Physical model and boundary conditions
The LBE is heated in a circular tube with an inner diameter of 10 mm and a length of 1.0 m by a constant heat
flux on the wall. The velocity-inlet and outflow conditions are set on the inlet and outlet. The constant heat flux
is applied on the wall surface. Detailed boundary parameters for the simulations are summarized in Table 1.
Table 1: Parameters for CFD simulation
Parameters Values
1,000/10
573.15
0.02 – 2
Tube length/diameter (mm/mm)
Inlet temperature (K)
Inlet velocity (m/s)
Wall heat flux (W/m2) 5×105
Gravitational acceleration (m/s2) 9.81
2.2 Numerical method and mesh independence
The numerical study is executed by the commercial CFD software Fluent. The SIMPLE algorithm is used to deal
with the pressure and velocity coupling. The shear stress transport (SST) k-ω model is selected to study the
flow and heat transfer of LBE, and gravity effect is considered in the simulations. Since the Reynolds analogy
theory is no longer suitable for the LBE due to its special thermal properties, an applicable turbulent Prandtl
number (Prt) model should be constructed to better adapt to the flow and heat transfer characteristics of LBE.
The Prt model for LBE turbulent flows proposed by Cheng and Tak (2006) is used, as shown below:
(1)
where
(2)
The thermal properties of lead-bismuth are estimated based on the Handbook on Lead-bismuth Eutectic of
Nuclear Energy Agency (2015), as shown in Table 2.
Table 2: Thermal properties of LBE
Property Symbol (Unit) Correlation
Density ρ (kg/m3) 11,065-1.293T
Viscosity μ (kg/m·s) 4.94×10-4exp(754.1/T)
Thermal conductivity λ (W/m·K) 3.284+1.617×10-2T-2.305×10-6T2
Specific heat cp (J/kg·K) 164.8-3.94×10-2T+1.25×10-5T2-4.56×105T-2
2.3 Mesh independence and model validation
The computational domain is discretized by the structural mesh generated by ANSYS ICEM software, as shown
in Figure 1. In order to capture the detailed flow and heat transfer in the near-wall region, the mesh refinement
is adopted to ensure that the boundary layer wall function y+ for all the cases is smaller than 1. For the grid
independence verification, five sets of grids are applied in the numerical solution to validate the grid
independence. When the grid number is greater than 1,279,935, the relative deviation of the Nusselt number of
the tube is less than 0.1 %, indicating that the numerical solutions are mesh-independent, as shown in Figure
2. The mesh number would be 1,279,935 considering the computational cost.
1.250.8
4.12
1, 0000.01
1, 000 6, 000
0.018 7.0
t
PePePr
Pe
Pe A
4
4.5 1, 000
= 5.4 9 10 1, 000 2, 000
3.6 2, 000
Pe
A Pe Pe
Pe
236
Figure 1: Mesh configuration
Figure 2: Grid independence test
The heat transfer correlation for LBE given by Cheng and Tak (2006), which agrees well with the experimental
data of Johnson et al. (1953), is shown in Eq.(3). It is used for the verification of the feasibility and accuracy of
present numerical model. It can be seen in Figure 3 that the present numerical model can provide a reasonable
prediction of the flow and heat transfer process of LBE flow.
(3)
Figure 3: Comparison between simulation and heat transfer correlation
3. Results and discussion
The convective heat transfer coefficients of LBE, in vertical upward/downward arrangements of tubes
with/without buoyancy for different inlet velocities, are given in Figure 4. It can be seen that when the gravity is
considered, the LBE heat transfer coefficient of vertical upward pipe is the largest, while that of vertical
downward pipe is the smallest. The convective heat transfer is more sensitive to buoyancy effects at smaller
0.80.018Nu A Pe
237
Reynolds numbers. At an inlet velocity of 0.02 m/s, the hLBE can be increased by approximately 27.6 % overall
by arranging the circular tube vertically upward, while the hLBE can be reduced by 6.7 % when the tube is
vertically downward. This buoyancy impact on convective heat transfer decreases with the increase of inlet
velocity. When the inlet velocity increases to 2 m/s, buoyancy has little effect on the heat transfer. Turbulence
is fully developed and the local convective heat transfer coefficients are high in the entrance region due to the
thin thermal boundary layer. With the continuous development of the thermal boundary layer, the local
convective heat transfer coefficients gradually decrease along the mainstream directions, and then tend to be
stable. I.e., for LBE flow with smaller Reynolds numbers in a tube heated by constant heat flux, convective heat
transfer is significantly enhanced when gravity is opposite to the flow direction, while the same gravity and flow
direction slightly weakens the convective heat transfer capacity. The heat transfer can be improved by using the
vertical upward tube arrangement. For the higher Reynolds number LBE flow, the buoyancy impact and the tube
arrangement can be ignored.
(a)
(b)
(c)
Figure 4: Convective heat transfer coefficients of LBE: (a) vin = 0.02 m/s; (b) vin = 0.05 m/s; (c) vin = 2 m/s
238
Figure 5 illustrates the velocity distribution in the axial section z = 500 mm at different distances from the wall
surface. In the region near the wall (y/R＞0.6), the LBE is accelerated in the upward tube and decelerated in
the downward tube due to the buoyancy force. Accordingly, in the region far from the wall (y/R＜0.6), the LBE
velocity is maximum in the downward tube and minimum in the upward tube to ensure a consistent total mass
flow rate.
Figure 5: Velocity distribution at different distances from the wall (vin = 0.05 m/s, z = 500 mm)
Figure 6 compares the effects of buoyancy on LBE heat transfer horizontal and vertical tubes. Gravity is
perpendicular to the flow direction in the horizontal tube, and the buoyancy force makes a big difference in the
convective heat transfer level between the upper and lower walls. The heat transfer coefficient of the lower wall
is about three times that of the upper wall. This phenomenon of large differences in heat transfer at different
locations on the wall would not occur when gravity is parallel to the flow direction.
Figure 6: Convective heat transfer coefficients of LBE in horizontal and vertical tubes (vin = 0.05 m/s)
4. Conclusions
As an ideal candidate coolant for ADS, LBE has attracted lots of attention in both academia and industrial. In
this paper, a circular tube is three-dimensionally modelled, simulations of the uniformly heated tube with/without
buoyancy are performed. The different thermal-hydraulic performance obtained under different tube
239
arrangements and operating parameters are compared, and the effects of buoyancy on the convective heat
transfer of LBE are discussed. The results show that for LBE flow with smaller Reynolds numbers in a tube
heated by constant heat flux, convective heat transfer is significantly enhanced when gravity is opposite to the
flow direction, while the same gravity and flow direction slightly weakens the convective heat transfer capacity.
At an inlet velocity of 0.02 m/s, the hLBE can be increased by approximately 27.6 % overall by arranging the
circular tube vertically upward. For the higher Reynolds number LBE flow, the buoyancy impact and the tube
arrangement can be ignored. Gravity is perpendicular to the flow direction when the tube is placed horizontally,
and the buoyancy force makes a big difference in the convective heat transfer level between the upper and
lower walls. The heat transfer coefficient of the lower wall is about three times that of the upper wall. This
phenomenon of large differences in heat transfer at different locations on the wall would not occur when gravity
is parallel to the flow direction. For different engineering applications, careful consideration of the impact of
buoyancy and tube arrangements is essential for the overall flow heat transfer performance of lead-bismuth
based heat exchanger.
Nomenclature
cp – specific heat at constant pressure, J/(kg·K)
h – heat transfer coefficient, W/(m2·K)
Nu – Nusselt number, -
Pe – Peclet number, -
Pr – Prandtl number, -
Prt – turbulent Prandtl number, -
R – radius of the tube, m
T – temperature, K
u – velocity, m/s
x, y, z – cartesian coordinates
λ – thermal conductivity, W/(m·K)
μ – dynamic viscosity, Pa/s
ρ – density, kg/m3
Acknowledgments
This study is supported by Innovative Scientific Program of CNNC and the National Natural Science Foundation
of China (Grant No. 52022080).
References
Chen F., Huai X., Cai J., Li X., Meng R., 2013. Investigation on the applicability of turbulent-Prandtl-number
models for liquid lead-bismuth eutectic. Nuclear Engineering and Design, 257, 128-133.
Cheng X., Tak N.-I., 2006. Investigation on turbulent heat transfer to lead–bismuth eutectic flows in circular
tubes for nuclear applications. Nuclear Engineering and Design, 236(4), 385-393.
Guo W., Shams A., Sato Y., Niceno B., 2020. Influence of buoyancy in a mixed convection liquid metal flow for
a horizontal channel configuration. International Journal of Heat and Fluid Flow, 85, 108630.
Nuclear Energy Agency, 2020, Accelerator-driven systems (ADS) and fast reactors (FR) in advanced nuclear
fuel cycles, , accessed 15.05.2022.
Jackson J.D., 1983. Turbulent mixed convection heat transfer to liquid sodium. International Journal of Heat and
Fluid Flow, 4(2), 107-111.
Johnson, H.A., Hartnett, J.P., Clabaugh, W.J., 1953. Heat transfer to molten lead‐bismuth eutectic in turbulent
pipe flow. Journal of Heat Transfer, 75, 1191.
Marocco L., Sala M., Centurelli G., Straub S., Colombo L., 2022. LES simulations and Nusselt number
decomposition of turbulent mixed convection of liquid metals flowing in a vertical pipe. International Journal
of Heat and Mass Transfer, 182, 121977.
Niemann M., Blazquez Navarro R.A., Saini V., Fröhlich J., 2018. Buoyancy impact on secondary flow and heat
transfer in a turbulent liquid metal flow through a vertical square duct. International Journal of Heat and Mass
Transfer, 125, 722-748.
Nuclear Energy Agency, 2015, Handbook on lead-bismuth eutectic alloy and lead properties, materials
compatibility, thermal-hydraulics and technologies. , accessed
15.05.2022.
Wang Y., Xi W., Li X., Huai X., 2017. Influence of buoyancy on turbulent mixed convection of LBE in a uniform
cooled inclined tube. Applied Thermal Engineering, 127, 846-856.
Zhang Y., Wang C., Cai R., Lan Z., Shen Y., Zhang D., Tian W., Su G., Qiu S., 2020. Experimental investigation
on flow and heat transfer characteristics of lead–bismuth eutectic in circular tubes. Applied Thermal
Engineering, 180, 115820.
240