Version of Record: https://www.sciencedirect.com/science/article/pii/S0306454920300736
Manuscript_5df852681d641628371f59f36c7df39e
Numerical Study on Convective Heat Transfer and Friction Characteristics of Molten Salts
in Circular Tubes
Sheng Zhanga, Xiaodong Suna,*, and Elvis E Dominguez-Ontiverosb
a Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, United States
b Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States
ABSTRACT
Molten salts including fluoride, nitrate, chloride, and carbonate salts have been proposed for heattransfer and thermal energy storage applications due to their superior thermal performance at
elevated temperatures. Since it is expensive to perform molten salt heat transfer experiments due
to the high working temperatures, a numerical analysis is carried out to investigate thermal and
hydrodynamic performance of molten salts using a Computational Fluid Dynamics (CFD) tool,
STAR CCM+, and validate the numerical model using existing experimental data and convective
heat transfer correlations, including Dittus-Boelter, Gnielinski, Hausen, and Sieder-Tate
correlations. The analysis shows the hydrodynamic and thermal performance, such as the
hydrodynamic/thermal entrance length, friction factor, and Nusselt number of molten salts in
laminar and turbulent flow regimes can be appropriately modeled. In addition, the widely used
convective heat transfer correlations provide good predictions for molten salt heat transfer.
KEYWORDS
Molten Salts; Convective Heat Transfer; Darcy Friction Factor; Entrance Length; Numerical Study.
*: Corresponding author at: University of Michigan, 2355 Bonisteel Blvd., Ann Arbor, MI 48109-2104, USA.
E-mail address: xdsun@umich.edu
© 2020 published by Elsevier. This manuscript is made available under the Elsevier user license
https://www.elsevier.com/open-access/userlicense/1.0/
1. INTRODUCTION
Molten salts, such as fluoride, nitrate, chloride, and carbonate salts, could be used as heat transfer
fluids (HTFs) and thermal energy storage (TES) fluids. Specifically, fluoride salts including FLiBe
(LiF-BeF2, 66-34 mol%,), FLiNaK (LiF-NaF-KF, 46.5-11.5-42 mol%), and KF-ZrF4 (58-42
mol%) are promising reactor coolants for Fluoride-salt-cooled High-temperature Reactors (FHRs)
due to their good thermophysical and nuclear properties [1]. Nitrate salts, such as KNO3- NaNO2KNO3 (53-40-7 wt%), chloride salts, such as MgCl2-KCl (68-32 mol%), and carbonate salts, such
as Li2CO3-Na2CO3-K2CO3 (32-33-35 wt%), are promising candidates for HTFs and TES media in
Concentrating Solar Power (CSP) plants considering their thermophysical properties, cost,
material compatibility, stability, and flammability [2]. Due to their wide applications as heat
transfer media, it is therefore necessary to investigate thermal and hydrodynamic performance of
these molten salts, especially considering their relatively large Prandtl number.
Figure 1 shows the Prandtl number for six molten salts, with air and saturated water included as
references. The Prandtl number of molten salts as shown in Figure 1 are in the range of 2 to 32 at
their respective potential working temperatures. The high Prandtl number may lead to different
heat transfer characteristics of conventional fluids, such as water and air. Normally, molten salts
work at elevated temperatures to achieve high electric power conversion efficiencies and maintain
large safety margins to avoid potential salt solidification. Due to their high working temperatures,
construction and operation of a high-temperature molten salt facility to investigate molten salt
thermal and hydrodynamic performance is expensive. It is therefore proposed to first numerically
investigate heat transfer and friction characteristics of molten salts, and validate the numerical
results using experimental data from molten salt experiments.
2
35
LiF-BeF (66-34 mol%)
2
LiF-NaF-KF (46.5-11.5-42 mol%)
KF-ZrF (58-42 mol%)
30
4
NaNO3-KNO3 (60-40 wt%)
25
KNO -NaNO -NaNO (53-40-7 wt%)
3
2
3
MgCl -KCl (68-32 mol%)
2
20
Water
Air
15
10
5
0
0
100
200
300
400
500
600
700
800
Temperature (°C)
Figure 1 Prandtl numbers of various molten salts, air, and saturated water
In the literature, experimental studies on molten salt related to the salt thermal hydraulics started
in 1950s. In 1954, Grele and Gedeon [3] investigated the heat transfer characteristics of FLiNaK
in an electrically heated Inconel tube at a temperature range of 540 to 730 °C. The heat transfer
coefficient values estimated from the experimental data were at about 40% of the values given by
the Dittus-Boelter correlation. Formation of a thermal resistance layer due to the intergranular
corrosion to the Inconel tube by the salt was proposed as the reason for the much lower heat transfer
coefficients in the experiments. Later in 1955, Hoffman and Lones [4] experimentally studied the
heat transfer of FLiNaK flowing in three tubes made of Nickel, Inconel, and Stainless Steel (SS)
316. The heat transfer coefficients estimated from the experimental data obtained from the Nickel
and SS 316 tubes agreed well with the Dittus-Boelter correlation results, while the heat transfer
coefficient from the Inconel tube deviated significantly from the Dittus-Boelter correlation results.
This much lower heat transfer coefficient obtained from the experiments was explained due to the
existence of an interfacial thermal resistance in the FLiNaK-Inconel system. However, this
3
conclusion conflicted with Vriesema’s finding in 1979 [5] where the experimental data obtained
from a FLiNaK-Inconel system matched reasonably well the Dittus-Boelter correlation results,
within ±20% uncertainties. The contradictory conclusions may be resulted from different
thermophysical property values used for FLiNaK when the heat transfer coefficient was deduced
[3-5]. Therefore, it is necessary to identify accurate thermophysical properties for FLiNaK when
reducing salt experimental data.
In addition to FLiNaK, other molten salts, including salts with dissolved nuclear fuels, i.e., fuel
salts, were widely investigated as well [6-13]. In 1960, Hoffman and Cohen [6] investigated heat
transfer performance of KNO3-NaNO2-NaNO3 (53-40-7 wt%) in an Inconel tube at a temperature
range of 290 to 442 °C. In recent years, KNO3-NaNO2-NaNO3 (53-40-7 mol%) was studied by
Chen, et al., [7] and Qian, et al., [8-9]. In addition, Wu, et al., [10] and Liu, et al., [11] studied
convective heat transfer of one same molten salt, LiNO3, but proposed different convective heat
transfer correlations. Fuel salts for Molten Salt Reactors (MSRs), such as LiF-BeF2-ThF4-UF4 with
different molar fractions, were studied by Silverman, et al., [12] and Cooke and Cox [13]. In
addition, friction factor of molten salts was experimentally investigated by Vriesema [5], and Jeong
and Bang [14].
It is inconvenient to use various heat transfer coefficient and friction factor correlations for same
molten salts and difficult to acquire local information of interest from the experiments, such as salt
temperature profiles. In addition, it is not cost-effective and perhaps unnecessary to perform extra
experiments for other molten salts of similar ranges of the Prandtl number of the molten salts that
have been widely investigated. The thermal and hydrodynamic characteristics of molten salts, such
4
as the hydrodynamic and thermal entrance lengths, friction factor, and Nusselt number for molten
salts in circular tubes are therefore numerically investigated in this paper.
2. NUMERICAL MODELING
2.1. Numerical Model
A three-dimensional Computational Fluid Dynamics (CFD) tool, STAR CCM+ was used to
investigate the thermal-hydraulic performance of molten salts, including FLiNaK and LiF-BeF2ThF4-UF4 (71.7-16-12-0.3 mol%) in circular tubes. The fluid domain molten salt (FLiNaK and
LiF-BeF2-ThF4-UF4) and solid domain a circular tube made of SS 316 were constructed in STAR
CCM+. The circular tube has an inner diameter of 4 mm and tube thickness of 1 mm as shown in
Figure 2. The tube length is 15 m when the flow in the tube is laminar, while 1 m for turbulent
flow cases. Several mesh models, such as polyhedral mesher, surface remesher, prism layer
mesher, and extruder were enabled for meshing. Boundary conditions include a uniform velocity
inlet at 500 °C, an atmospheric pressure outlet, and constant wall heat fluxes 4 and 100 kW/m2 on
the wall outer surface for laminar and turbulent flows, respectively. In addition, the realizable k-ɛ
two-layer model is used for modeling turbulent flows.
L = 15 m for laminar flows
L = 1 m for turbulent flows
ro = 3 mm
Tube
ri = 2 mm
Molten salt
Figure 2 Dimensions for fluid and solid domains
5
2.2. Grid Independence Study
A grid independence study was first performed to investigate the spatial convergence of numerical
results, which could be characterized by the Grid Convergence Index (GCI) [15] as follows,
=
where Fs, r, p, and
(1)
are the safety factor, grid refinement ratio, order of convergence, and
estimated variable, respectively. The safety factor Fs is empirically set to be 1.25. The grid
refinement ratio for mesh options 1 and 2, r2-1, and order of convergence p [15] are defined as
=
(2)
=
(3)
r3-2 is the grid refinement ratio for mesh options 2 and 3. s and h are respectively the sign function
[15] and grid size [16], which are defined as
= sgn !
$
ℎ = ! &% "
"
(4)
⁄'
(5)
where Vt and N are the grid total volume and quantity, respectively.
A total of six sets of grid size were used for the grid independence study for laminar (Re = 1,800)
and turbulent (Re = 10,000) flows, and results are summarized in Table 1. The GCI values of the
Darcy friction factor and temperature difference between the inlet and outlet for laminar flows are
significantly small, 0.15 and 0.08%, respectively. In addition, the GCI values of the Darcy friction
factor and temperature difference for turbulent flows are 0.62 and 0.48%, respectively. The
simulation results using fine mesh options 1 and 4 are therefore considered to be mesh independent,
6
which are adopted in this study for the thermal and hydrodynamic analyses of molten salts in
laminar and turbulent flow regimes, respectively.
Table 1 Results of grid independence study for FLiNaK
Mesh option
Cell quantity (103)
Grid size (mm)
f
ΔT (°C)
Laminar flow (Re = 1,800)
1 (Fine)
384.16
1.03
0.03338
12.27
2 (Medium)
160.78
1.38
0.03302
12.29
3 (Coarse)
73.93
1.80
0.03020
12.35
GCI (%)
N/A
N/A
0.15
0.08
Turbulent flow (Re = 10,000)
4 (Fine)
139.74
0.59
0.03108
3.87
5 (Medium)
59.86
0.78
0.03101
3.90
6 (Coarse)
24.29
1.05
0.03090
4.01
GCI (%)
N/A
N/A
0.62
0.48
2.3. Data Reduction
The Darcy friction factor used to estimate the frictional pressure loss is defined as
)=
*+,-
./01
Δ
3
(6)
where D, Af, 41, 5, Δpf, and Δz are the inner diameter, cross-sectional area of the tube, mass flow
rate, fluid density, frictional pressure drop, and axial distance, respectively. The average heat
transfer coefficient is defined as
7
ℎ6 =
71
,89 :;689 ;6- <
(7)
whereQ&, Aiw , T iw , and T f are the heat transfer rate, heat transfer area of the tube inner surface,
area-average temperature of the tube inner surface defined by Eq. (8), and arithmetic mean of the
fluid bulk inlet and outlet temperatures defined by Eq. (9), respectively.
=6>? =
=63 =
,-
@ ;89 A,
,89
(8)
B=6>C + =6EFG H
(9)
The fluid bulk inlet (or outlet) temperature is defined as
=6>C =
@ *I $;8J A,
@ *I $A,
(10)
where V and cp are the fluid velocity in the tube axial direction and specific heat capacity,
respectively.
3. REVIEW OF MOLTEN
SALT EXPERIMENTS AND THERMOPHYSICAL
PROPERTIES
3.1. Molten Salt Experiments
Considering the convenience to extract original experimental data for comparison with the CFD
and correlation results, the authors selected eleven salt heat transfer experiments [3-13] and two
hydrodynamic experiments [5,14], which are summarized in Table 2. The working fluids used in
these experiments [3-14] include FLiNaK (LiF-NaF-KF, 46.5-11.5-42 mol%), KNO3-NaNO2NaNO3 (53-40-7 wt% and 53-40-7 mol%), LiNO3, NaBF4-NaF (92-8 mol%), and LiF-BeF2-ThF4UF4 (71.7-16-12-0.3 mol% and 67.5-20-12-0.5 mol%). The salt temperature ranges from 200 to
830 °C.
8
Table 2 Information of the selected molten salt experiments
Molten salt
FLiNaK
(LiF-NaF-KF, 46.511.5-42 mol%)
Heat
transfer
Flow
channel
material
Flow channel
configuration
Inconel X750
Nickel
Inconel
Dimension
(length L and
ID) (mm)
Heating
/cooling
609.6 and 6.23
1182 and 2.98
Circular tube
SS 316
Inconel
600
Salt
temperature
(°C)
Reference
540-730
Grele
(1954) [3]
530-745
Hoffman
(1955) [4]
Heating
1182 and 4.45
1182 and 4.57
2050 and 26
Cooling
575-675
Vriesema
(1979) [5]
Heating
290-442
Hoffman
(1960) [6]
KNO3-NaNO2-NaNO3
(53-40-7 wt%)
Inconel
Circular tube
241.3 and 4.57
KNO3-NaNO2-NaNO3
(53-40-7 mol%)
Inconel
600
Circular tube
1200 and 20
200-300
Chen
(2016) [7]
KNO3-NaNO2-NaNO3
(53-40-7 mol%)
Shell-tube heat
exchanger
(molten salt on
tube side)
350 and 10.45
200-300
Qian (2016)
[8]
KNO3-NaNO2-NaNO3
(53-40-7 mol%)
Shell-tube heat
exchanger
(molten salt on
tube side)
Gas cooled
HX:
350 and 10.45
Molten salt to
salt HX: 500
and 10
258-298
Qian (2017)
[9]
Stainless
steel
Cooling
272-441
LiNO3
Circular tube
1000 and 20
294-441
NaBF4-NaF (92-8
mol%)
LiF-BeF2-ThF4-UF4
(71.7-16-12-0.3 mol%)
Circular tube
Hastelloy
N
LiF-BeF2-ThF4-UF4
(67.5-20-12-0.5 mol%)
Friction
factor
FLiNaK
(LiF-NaF-KF, 46.511.5-42 mol%)
Circular tube
Circular tube
Inconel
600
Circular tube
3500 and
10.52
3500 and
10.52
450-610
Silverman
(1976) [12]
550-765
Heating
622.3 and 4.57
580-830
1200 and
41.25
Adiabatic
575-675
650 and 1.4
Heating
595-620
9
Wu (2009)
[10]
Liu (2009)
[11]
Cooke
(1973) [13]
Vriesema
(1979) [5]
Jeong
(2010) [14]
Three experiments [3-5] investigated FLiNaK heat transfer performance in circular tubes made of
different metals, such as Inconel X-750, Inconel 600, nickel, and SS 316. Although the same type
of FLiNaK salt was investigated in these experiments, inconsistent thermophysical properties were
used during the data reduction for these experiments [3-5] as summarized in Table 3. The
maximum discrepancies in the FLiNaK density, specific heat capacity, thermal conductivity, and
dynamic viscosity used in the data reduction of these experiments [3-5] are 7, 11, 246, and 45%,
respectively. The significant discrepancies in FLiNaK thermophysical properties used in these
experiments, especially the one in the thermal conductivity, result in large differences in the Prandtl
number, Reynolds number, and Nusselt number calculations.
Table 3 Thermophysical properties for FLiNaK used in the experiments [3-5]
5 (kg/m3)
KL (J/kg-K)
M (W/m-K)
N (kg/m-s)
Maximum
Grele (1954) [3]
Hoffman (1955) [4]
Vriesema* (1979) [5]
2555-0.6T(K)
2555-0.6T(K)
2729-0.73T (K)
2093.4
1883
1890
11%
4.5
4.5
1.3
246%
2.5 × 10 U V WXYZ⁄;B[H
2.5 × 10 U V WXYZ⁄;B[H
8.44 × 10
'
3.94 × 10
'
5.50 × 10
2.39 × 10
'
'
discrepancy**
7% for T from 773
to 1073 K
at 773 K
at 873 K
at 973 K
45%
for T from 773 to
1073 K
at 1073 K
*: The kinematic viscosity used by Vriesema is converted to the dynamic viscosity
**: The maximum discrepancy is estimated by |a − c|⁄a × 100%, where a is smaller than b
10
Figure 3 shows the Prandtl number, Reynolds number, and Nusselt number of FLiNaK using
thermophysical properties in the experiments [3-5]. These characteristic numbers, Pr, Re, and Nu
are normalized using their respective maximum value at a temperature range of 500 to 800 °C. The
discrepancies in the normalized Prandtl number, Reynolds number, and Nusselt number are
respectively 100 to 280%, 0 to 35%, and 230 to 235% for T = 500 to 800 °C. Therefore, it is
necessary to identify accurate thermophysical properties for FLiNaK and re-process the original
Normalized Pr
Normalized Re
experimental data to determine the thermal and hydrodynamic characteristics of molten salts.
(a)
(b)
1
0.8
Grele
Hoffman
Vriesema
0.6
0.4
0.2
500
550
600
650
700
750
800
Temperature (°C)
(c)
Figure 3 Comparison of (a) Pr, (b) normalized Re and (c) normalized Nu using FLiNaK
thermophysical properties in the references [3-5]
11
In addition to FLiNaK, the heat transfer performance of KNO3-NaNO2-NaNO3 were widely
investigated as well [6-9]. It should be noted that two different compositions of KNO3-NaNO2NaNO3 were used in the literature [6-9]. Hoffman et al. [6] used a heat transfer salt [17], a eutectic
mixture of KNO3, NaNO2, and NaNO3 based on 53-40-7 weight percent, while Chen, et al. [7] and
Qian, et al. [8-9] used KNO3-NaNO2-NaNO3 based on 53-40-7 molar percent. The eutectic mixture
of KNO3, NaNO2, and NaNO3 at 53-40-7 weight percent is equivalent to the eutectic mixture of
KNO3, NaNO2, and NaNO3 at 44-49-7 molar percent. As these two salts have very similar
chemical compositions, they are expected to thermodynamically behave similarly.
3.2. FLiNaK Properties
Density
Grimes, et al. [18] demonstrated that the rule of additivity of molar volumes was useful to predict
the fluoride salt mixture density:
∑J & f
8
50 = ∑J 8& $8 B;H
8
(11)
8 8
where 50 , h> , i> , and l> B=H are the mixture density, molar fraction of component i, molecular
weight of component i, and molar volume of component i at a temperature of T, respectively. Using
the measured densities of the three fluoride constituent salts LiF, NaF, and KF at two temperatures
of 600 and 800 °C [18], the FLiNaK density can be estimated by the rule of additivity of molar
volumes. The uncertainty in estimating the salt mixture density using Eq. (11) is within 5% [1, 19].
In addition to the model prediction of the FLiNaK density, a number of experimental investigations
were conducted on the FLiNaK density [20-23]. In 2003, Chrenkova, et al., [20] experimentally
measured the FLiNaK density for T = 667 - 897 °C with ±0.4% uncertainties, which was later
12
demonstrated by Cibulkova [21]. Kubikova, et al., [22] and Cheng, et al., [23] in 2013 measured
the FLiNaK density for T = 483 - 609 °C without specifying the associated uncertainty and for T
= 480 - 700 °C with ±0.25% uncertainties. Figure 4 shows a comparison of the FLiNaK density
values used in the references [20-23]. The maximum discrepancy in the FLiNaK density in a
temperature range of 500 - 800 °C is 3.5%. This relatively good agreement demonstrates Eq. (11)
can predict accurately the FLiNaK density. It is recommended to use the Chrenkova’s correlation:
5Bkg⁄m' H = 2408.9 − 0.624=B℃H
(12)
for the FLiNaK density at T = 500 - 800 °C with ±1% uncertainties to cover both Grimes’s and
Cheng’s results.
2200
Model - Grimes; William
Experiment - Chrenkova; Cibulkova
Experiment - Kubikova
Experiment - Cheng
2150
2100
2050
2000
1950
1900
500
550
600
650
700
750
800
Temperature (°C)
Figure 4 The FLiNaK density values used in the references [20-23]
Dynamic Viscosity
In 1963, Powers, et al., [24] measured the dynamic viscosity of FLiNaK for T = 500 - 800 °C with
±20% uncertainties. Chrenkova, et al., [20], Cibulkova, et al., [21] and Kubikova, et al., [25] also
experimentally measured the dynamic viscosity of FLiNaK respectively for T = 500 - 700 °C with
13
±2% uncertainties in 2003, T = 660 - 890 °C with ±2.5% uncertainties in 2006, and T = 529 - 630
°C without specifying uncertainties in 2012 using the torsion pendulum method. In 2014,
Merzlyakov, et al., [26] measured the kinematic viscosity of FLiNak for T = 454 - 871 °C without
specifying the uncertainties. In our current study, the kinematic viscosity of FLiNaK obtained by
Merzlyakov, et al., [26] is converted to the dynamic viscosity using the recommended FLiNaK
density correlation, Eq. (12).
As shown in Figure 5, the FLiNaK dynamic viscosity estimated in the references [18, 20, 24-26]
are within the ±20% uncertainty range of the Powers’ prediction. It is therefore recommended to
use the Powers’ correlation,
N BPa ∙ sH = 4 × 10 U V W
XZ⁄s;B℃H
X'. Ut
for the FLiNaK dynamic viscosity for T = 500 - 800 °C with ±20% uncertainties.
Figure 5 The FLiNaK dynamic viscosity values used in the references [18, 20, 24-26]
Specific Heat Capacity
14
(13)
The specific heat capacity of FLiNaK was given as 1904.8 J⁄Bkg ∙ KH for T = 477 - 557 °C with
±3% uncertainties by Janz, et al., [27] and 1880 J⁄Bkg ∙ KH for T = 500 - 700 °C with ±4%
uncertainties by An, et al., [28]. These values generally agree with Rogers’s prediction [29] as
shown in Figure 6. It is recommended to use
KL BJ⁄Mw ∙ x H = 1880
(14)
for the FLiNaK specific heat capacity for T = 500 - 700 °C with ±5% uncertainties to cover both
Rogers’ and Janz’s results.
Figure 6 The FLiNaK specific heat capacity values used in the references [27-29]
Thermal Conductivity
Rosenthal, et al., [30] suggested the following equation for estimation of the thermal conductivity
of fluoride mixtures.
;
⁄
* ⁄
CHz⁄{
y
M0 = 0.0119 Bf
⁄
(15)
where M0 , =0 , 5, i, and | are the thermal conductivity (W/m·K), melting temperature (K), density
(kg/m3), molecular weight (g/mol), and ion number of fluoride mixtures, respectively. Utilizing
15
the FLiNaK melting temperature 454 °C [31], the recommended Chrenkova correlation for the
FLiNaK density, and ion number n = 2, the FLiNaK thermal conductivity could be estimated by
Eq. (15) as a function of the salt temperature.
Janz, et al., [27], Smirnov et al., [32], and An et al., [28] proposed polynomial equations to estimate
the FLiNaK thermal conductivity for T = 527 - 647 °C with ±25% uncertainties in 1981; T = 517
- 807 °C with ±4% uncertainties in 1987; and T = 500 - 700 °C with ±3.5% uncertainties in 2015,
respectively. Cooper, et al., [33] proposed values of 1.682, 1.508, and 1.45 W⁄Bm ∙ KH for the
FLiNaK thermal conductivity for T = 690, 540, and 620 °C with uncertainties of 13, 8, and 8%,
respectively.
Figure 7 shows a comparison of the FLiNaK thermal conductivity in the references [27-28, 30, 3233]. The maximum discrepancy could be as high as 250% for T = 500 °C and 400% for T = 800
°C. The significant differences may result from the different approaches adopted to measure the
FLiNaK thermal conductivity and impurities in the FLiNaK salt used in these experiments.
Figure 7 The FLiNaK thermal conductivity values used in the references [27-28, 30, 32-33]
16
Janz, et al., [27] and Cooper, et al., [33] used the parallel plate method to measure the FLiNaK
thermal conductivity, which is estimated using the plate temperature difference and heat flux. The
effects of the thermal radiation, natural convection, and axial conduction between the hot and cold
plates result in larger values for the thermal conductivity. In addition, metal impurities, i.e., Fe, Al,
Ni, Pb, Mn, and Mg, etc., presented in FLiNaK salt would lead to overpredicted FLiNaK thermal
conductivity from the experiments.
An, et al., [28] used the laser flash technique developed by Parker, et al., [34] to measure the
FLiNaK thermal conductivity. In their experiments, the thickness of the liquid FLiNaK in their
container was 1.5 mm. Therefore, it is appropriate to assume that the convective heat transfer in
the liquid FLiNaK could be neglected in the experiment. In addition, the FLiNaK salt used in their
experiment was first prepared under an H2/HF environment to reduce the impurity level to be lower
than 0.01 wt%. Therefore, it appears appropriate to assume that the prediction given by An, et al.,
for the FLiNaK thermal conductivity is more accurate.
Smirnov, et al., [32] used the coaxial platinum cylinder method to measure the FLiNaK thermal
conductivity. In their experiments, FLiNaK was purified to reduce the impurity level and therefore
effect of corrosion. In addition, the radiative heat transfer was considered in the experiments.
Therefore, Smirnov’s correlation was suggested by Romatoski and Hu [35] for the FLiNaK
thermal conductivity. Since both Ann’s and Smirnov’s correlations are assumed more accurate
than those by the other references, the following correlation
M sW⁄Bm ∙ KHt = 0.005 + 0.00093=BxH
17
(16)
an arithmetic mean of their results, is recommended for the FLiNaK thermal conductivity for T =
500 - 800 °C with ±10% uncertainties. The thermophysical properties for FLiNaK summarized in
Table 4 are therefore adopted to re-process the original experimental data in the literature [3-5]
and to develop new convective heat transfer correlations for FLiNaK.
Table 4 Adopted thermophysical properties for FLiNaK
Temperature
Uncertainty at 95%
range (K)
confidence level
FLiNaK
Recommendation
Density (kg/m3)
2579.3-0.624 T(K)
±1%
1880
±5%
Specific heat capacity
(J/kg-K)
Thermal conductivity
(W/m-K)
Dynamic viscosity
(kg/m-s)
0.005 + 9.3 × 10
4 × 10 U V W
W
=BKH
773-1073
XZ⁄;B~H
±10%
±20%
The original reported experimental data may change significantly after data reprocessing using
more accurate salt thermophysical property data summarized in Table 4. The three characteristic
numbers, Pr, Re, and Nu originally in Grele’s experiment [3] are compared with their modified
values using recommended thermophysical properties for FLiNaK as shown in Figure 8. It is clear
that these modified values of the characteristic numbers deviate significantly from their original
values, especially for Prandtl and Nusselt numbers. The discrepancies between the original and
modified values of the Prandtl, Reynolds, and Nusselt numbers are 283 to 328%, 17 to 33%, and
331 to 431%, respectively. These large discrepancies are mainly due to the different thermal
conductivity and dynamic viscosity used for FLiNaK in Grele’s experiment [3] and this paper. In
18
addition, the recommended thermophysical properties for FLiNaK result in a significant change
of Nu/Pr1/3 values as shown in Figure 9, which lead to a different heat transfer correlation.
20
104
2.5
2
15
1.5
10
1
5
0.5
0
0
5
10
15
0
20
0
0.5
Original Pr
1
1.5
Original Re
2
2.5
104
(b)
(a)
(c)
Figure 8 Comparison of (a) Pr, (b) Re, and (c) Nu using FLiNaK thermophysical properties
in Grele’s experiment [3] and recommended properties in this paper
19
Figure 9 Comparison of the original heat transfer data in Grele’s experiment [3] and modified
data using the recommended thermophysical properties for FLiNaK
4. RESULTS AND DISCUSSIONS
It is necessary to compare the numerical results with existing molten salt experimental data as well
as friction factor and convective heat transfer correlations summarized in Table 5 to identify their
applicability for the molten salts of interest. In addition, the hydrodynamic and thermal entrance
lengths of molten salts flowing in circular tubes are investigated as well.
20
Table 5 Heat transfer and friction factor correlations for internal flows (circular tube)
Reynolds
Correlation
•⁄€ ≈ 0.05Re
Hydrodynamic
•⁄€ ≈ 4.4Re
Thermal
(fully-developed
flow)
Re ≥ 10W
ڠ
•⁄€ ≈ 0.05RePr
Re ≤ 2,300
) = 64⁄Re
Re ≤ 2,300
•⁄€ ≈ 10
entrance length
factor
Re ≤ 2,300
10 ≤ •⁄€ ≤ 60
entrance length
Darcy friction
number
Re ≥ 10W
Š 10†
‹
) = 0.0055 ˆ1 + ‰2 × 10
+
€ Re
W
) = 0.316⁄Re
⁄'
Œ
⁄W
) = B0.79•|Re − 1.64H
Dittus-Boelter correlation
Nu = 0.023ReZ.• P
4,000 ≤ Re
≤ 5 × 10•
Re ≤ 10U
Convective heat
transfer
--
[36]
---
[38]
--
[39]
0.7 ≤ Pr ≤ 100
Z. W
[36]
--
Re ≥ 10W
Re ≥ 10W
References
[37]
--
C
:N3 ⁄N? <
--
≤ 5 × 10†
Sieder-Tate correlation for turbulent flow:
⁄'
--
3,000 ≤ Re
| = 0.4 for heating and 0.3 for cooling
Nu = 0.027ReZ.• Pr
Prandtl number
[36]
0.7 ≤ Pr
≤ 16,700
Gnielinski correlation:
Nu = 0.012BReZ.•X
− 280HPr
+ B•⁄€ H
Z.W
2300 ≤ Re
™1
⁄'
š:› 3 ⁄› ? <
Z.
Hausen correlation:
Nu = 0.116:Re
⁄'
− 125<Pr
⁄'
≤ 10
†
:N3 ⁄N? <
Z. W
3500 ≤ Re
≤ 1.2 × 10W
Sieder-Tate correlation for laminar flow
Nu = 1.86Re
⁄'
Pr
⁄' B
•⁄€ H
⁄'
:N3 ⁄N? <
Re ≤ 2300
Z. W
21
0.6 ≤ Pr ≤ 10U
[40]
0.7 ≤ Pr ≤ 3
[41]
0.7 ≤ Pr
≤ 16700
[42]
4.1. Hydrodynamic Characteristics
The hydrodynamic characteristics, including the hydrodynamic entrance length, thermal entrance
length, and friction factor of FLiNaK flowing in a horizontal tube are numerically investigated
using the recommended thermophysical properties summarized in Table 4. Reynolds number of
FLiNaK investigated ranges from 100 to 1,800 for laminar flows and 10,000 to 100,000 for
turbulent flows.
4.1.1 Entrance Length
Hydrodynamic Entrance Length
The radial velocity of FLiNaK is set to be uniform at the tube inlet z = 0 to investigate the
hydrodynamic entrance length. Figure 10 shows the computed radial velocity profiles of FLiNaK
for a laminar flow (Re = 400) at different axial locations. The x and y axes are respectively the
nondimensional radial location r/R and velocity l/ B , œH⁄l0 , where R is the tube radius and l0 is
the mean velocity. The radial velocity profile, i.e., the radial distribution of the axial velocity,
changes in the hydrodynamically developing flow region (hydrodynamic entrance region), while
it remains unchanged (both the magnitude and shape) in the hydrodynamically fully-developed
flow region. If the relative change of the local velocity, |sl/ B , œH − l/ B , œ′Ht⁄l/ B , œH|, is less than
1%, the flow is considered to be hydrodynamically fully developed. The hydrodynamic entrance
length for FLiNaK is therefore identified to be 25D at Re = 400.
22
2
1.8
1.6
1.4
z/D = 0
z/D = 2.5
z/D = 5
z/D = 10
z/D = 15
z/D = 20
z/D = 25
z/D = 50
1.2
1
0.8
0.6
0.4
0.2
-1
-0.5
0
0.5
1
r/R
Figure 10 Radial velocity profiles of FLiNaK at different axial locations (Re = 400)
Figure 11 shows the non-dimensional hydrodynamic entrance length Lhyd/D for different Reynolds
numbers in a range of 100 to 1,800, where Lhyd and D are the hydrodynamic entrance length and
tube inner diameter, respectively. The CFD results show that the non-dimensional hydrodynamic
entrance length is linearly dependent on FLiNaK Reynolds number. The relative discrepancy
between the modeling results and a widely used correlation for hydrodynamic entrance length of
laminar flows [36]
•žŸ0>CŸ
,
¡ ⁄€
≈ 0.05¢V
(17)
is within 30% for Re = 100 – 1,800, which was also observed by Srivastava, et al. [43]. Therefore,
the uncertainty of Eq. (17) can be as high as 30% for Re = 100 to 1,800 using the CFD numerical
results as a reference.
23
120
CFD modeling
Lhyd / D = 0.05Re
100
L hyd / D
80
60
40
20
0
0
500
1000
1500
2000
Re
Figure 11 Hydrodynamic entrance length for FLiNaK in laminar flow regime
A similar simulation has also been performed for FLiNaK in turbulent flow regime. Figure 12
shows the non-dimensional hydrodynamic entrance length Lhyd/D for different Reynolds numbers
in a range of 10,000 to 100,000. The modeling results agree well with a widely used correlation
for hydrodynamic entrance length of turbulent flows [36]
10 ≤ •GF
£Fž¤CG,
¡ ⁄€
≤ 60
(18)
However, the modeling results deviate from the correlation [37]
•GF
£Fž¤CG,
¡ ⁄€
≈ 4.4¢V
ڠ
(19)
by 35%, which was also observed by Ferng, et al. [44]. Therefore, the uncertainty of Eq. (19) can
be as high as 35% for Re = 10,000 to 100,000 using the CFD numerical results as a reference.
24
L hyd / D
Figure 12 Hydrodynamic entrance length for FLiNaK in turbulent flow regime
Thermal entrance length
A constant wall heat flux 4 kW/m2 is applied for laminar flows and 100 kW/m2 for turbulent flows
to investigate the thermal entrance length. In addition, the FLiNaK temperature is set to be uniform
at the tube inlet z = 0. Figure 13 shows the computed radial temperature profiles of FLiNaK for a
laminar flow (Re = 400) at different axial locations. The x and y axes are respectively the
nondimensional radial location r/R and temperature = ∗ B , œH, which is defined as
™=3 B , œH − =6? BœHš¦™=3 B0, œH − =6? BœHš, where =3 and =6? are the fluid temperature and azimuthally
averaged wall temperature, respectively. The shape of the FLiNaK radial temperature profile
changes in the thermally developing flow region, while it remains unchanged in the thermally
fully-developed region. If the relative change of the temperature difference over a certain axial
distance §¨™=6? BœH − =3 B , œHš − ™=6? Bœ′H − =3 B , œ′Hš©¦™=6? BœH − =3 B , œHš§, is less than 1%, the
flow is considered to be thermally fully developed. The thermal entrance length is therefore
25
identified to be 500D at Re = 400 and Pr = 22.8 for FLiNaK at 500 °C, which has a 9.6%
discrepancy compared to the results estimated by [36]
•žŸ0>CŸ
,; ⁄€
≈ 0.05¢V›
(20)
A similar simulation has also been performed for FLiNaK in turbulent flow regime. The thermal
entrance length for FLiNaK is identified to be 12.5D at Re = 10,000, which has a 20% discrepancy
compared to [36]
•GF
£Fž¤CG,; ⁄€
≈ 10
(21)
1
0.8
z/D = 15
z/D = 25
z/D = 50
z/D = 125
z/D = 250
z/D = 500
z/D = 750
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
r/R
Figure 13 Radial temperature profiles of FLiNaK at different axial locations (Re = 400)
4.1.2 Friction Factor
The Darcy friction factor in the developing and hydrodynamically fully-developed regions are
numerically investigated for Re = 100 – 1,800 and 10,000 – 100,000. The friction factor decreases
significantly along the flow direction in the developing flow region, while it remains nearly
constant in the fully-developed flow region as shown in Figure 14(a) for laminar flows and Figure
26
14(b) for turbulent flows. The laminar flows shown in Figure 14(a) are hydrodynamically fully
developed at z/D > 115 or earlier, while they are thermally fully developed at z/D > 2,050 or earlier
under the conditions investigated. In addition, the turbulent flows shown in Figure 14(b) are
hydrodynamically fully developed at z/D > 40 or earlier, while become thermally fully developed
at z/D > 20 or earlier under the conditions investigated. These CFD results of FLiNaK in the
hydrodynamically and thermally fully-developed flow regions will be used to estimate
hydrodynamic and thermal performance of molten salts in circular tubes.
0.08
Re = 10 4
Re = 2 x 104
0.07
Re = 4 x 10
0.06
4
Re = 6 x 104
Re = 8 x 104
0.05
Re = 10 5
0.04
0.03
0.02
0.01
0
0
10
10
1
10
2
10
3
z/D
(a)
(b)
Figure 14 Darcy friction factor at different axial locations for FLiNaK in (a) laminar and (b)
turbulent flow regimes
Figure 15 shows comparisons of the Darcy friction factor among our CFD results for FLiNaK in
hydrodynamically and thermally fully-developed flow regions, FLiNaK experimental data, and
widely used friction factor correlations for flows in smooth and rough tubes. No data reprocessing
is needed for the experimental data, such as the Reynolds number and friction factor in the
literature [14] for laminar flows of FLiNaK due to the negligible differences between the
27
thermophysical properties of FLiNaK used in the literature [14] and recommended values in this
paper. As shown in Figure 15(a), the CFD results agree well with both the experimental data and
the correlation for fully-developed laminar flows [36]
) = 64⁄Re
(22)
The relative discrepancies between the CFD results and the experimental data, CFD and
correlation (Eq. (22)) results are within 13.7% and 6.3%, respectively. It is therefore appropriate
to use f = 64/Re for friction factor estimation of FLiNaK in fully-developed laminar flow regime.
0.06
FLiNaK - CFD modeling
FLiNaK - Experiment (Vriesema)
f = 0.316/Re
0.05
1/4
f = (0.79lnRe - 1.64)-2
Darcy friction factor
f = 0.0055[1 + (2 x 10 4 /D + 10 6/Re) 1/3], /D = 10 -3
Darcy friction factor
f = 0.0055[1 + (2 x 10 4 /D + 10 6/Re) 1/3], /D = 3 x 10 -3
0.04
f = 0.0055[1 + (2 x 10 4 /D + 10 6/Re) 1/3], /D = 5 x 10 -3
0.03
0.02
0.01
0
0
2
4
6
8
10
Re
12
10 4
(b)
(a)
Figure 15 Darcy friction factor of FLiNaK in (a) laminar and (b) turbulent flow regimes
The numerical results of the friction factor for hydrodynamically fully-developed FLiNaK
turbulent flows are compared with the experimental data as shown in Figure 15(b). Due to the large
differences between the FLiNaK thermophysical properties used during data reduction in the
literature [5] and recommended values in this paper (Table 4), the original experimental data [5]
has been reprocessed by adopting the recommended properties of FLiNaK. In addition, it needs to
be noted that there is one typo in a figure summarizing the experimental data in Reference [5]: The
28
pressure loss of FLiNaK in a straight-tube test section was misplaced in the column for the pressure
loss of FLiNaK in a Venturi tube. The pressure drop over a Venturi tube was used to calculate the
FLiNaK velocity, which was used to compute the FLiNaK Reynolds number. This typo is
recognized because: (1) The pressure loss in the Venturi tube should be several times larger than
that in the straight-tube test section based on the methodology adopted in Reference [5]. However,
this is the opposite in Figure 3.3-7 of Reference [5]; and (2) The computed friction factor after this
correction (switch the pressure drop experimental data for the straight-tube test section with that
in the Venturi tube) completely agrees with the friction factor plot Figure 3.3-9 in Reference [5].
The Darcy friction factor of FLiNaK for turbulent flows in smooth tubes obtained from our CFD
simulations is compared with two correlations [36,39]:
) = 0.316⁄Re
⁄W
(23)
) = B0.79lnRe − 1.64H
(24)
The relative discrepancy between the CFD and correlation (Eqs. (23) and (24)) results is within
9.7% as shown in Figure 15(b). However, it is in general significantly lower than the experimental
values. This large discrepancy may be due to the increased roughness of the tube inner surface
considering the corrosion effect of the FLiNaK to the test section wall at high temperatures. In
viewing this, we used three relative roughness, ɛ/D = 10-3, 3×10-3, and 5×10-3, to estimate the
friction factor of turbulent flows in a rough tube using the following correlation [38]:
«
) = 0.0055 ª1 + !2 × 10W +
+
Z{
¬-
"
⁄'
®
(25)
The FLiNaK friction factor based on Eq. (25) for ɛ/D = 3×10-3 agrees well (±23.6% uncertainties)
with 95% of the experimental data.
29
4.2. Thermal Characteristics
4.2.1 Laminar Flow Regime
For high-Prandtl number fluids, such as molten salts, the thermal entrance length in a circular tube
can be significantly large for laminar flows. Taking D = 0.01 m, Re = 1,000, and Pr = 10 as an
example, the thermal entrance length L can be as high as 5 m using Eq. (20). It is therefore difficult
to ensure that the flow is thermally developed prior to entering the test section in a limited
laboratory space. The molten salt experimental data presented in References [12, 13] were obtained
in hydrodynamically fully developed, but thermally developing laminar flows in a circular tube.
Therefore, the Nusselt number obtained from these experiments [12, 13] should be larger than
4.36, a theoretical Nu value for thermally fully-developed laminar flows in circular tubes under
constant wall heat flux conditions. As shown in Figure 16(a), the Nusselt number in these two
experiments is about 0.6 – 2.4 times larger than the theoretical value Nu = 4.36, while the CFD
results for thermally fully-developed laminar flows of FLiNaK and LiF-BeF2-ThF4-UF4 agree well
with the theoretical value within 6.9% difference.
16
FLiNaK-CFD modeling
LiF-BeF 2 -ThF 4 -UF4 -CFD modeling
14
LiF-BeF 2 -ThF 4 -UF4 - Exp. (Cooke, et al.)
LiF-BeF 2 -ThF 4 -UF4 - Exp. (Silverman, et al.)
12
Constant heat flux: Nu=4.36
Nu/Pr 1/3
10
8
6
4
2
0
0
500
1000
1500
2000
2500
Re
(b)
(a)
Figure 16 Nusselt numbers of FLiNaK and LiF-BeF2-ThF4-UF4 laminar flows compared with (a)
Nu = 4.36 (constant heat flux) and (b) Sieder-Tate laminar flow correlation
30
The Sieder-Tate correlation (laminar flow) [42]
Nu = 1.86Re
⁄'
⁄' B
• ⁄€ H
Pr
⁄'
(26)
is compared with the experimental data from References [12, 13] as shown in Figure 16(b).
Compared to the theoretical value, Nu = 4.36 for fully-developed laminar flows in circular tubes,
the Sieder-Tate correlation (laminar flow) results are closer to the experimental data by taking the
thermal entrance effect into consideration. However, the maximum discrepancy between the
results of Sieder-Tate correlation and the experimental data is still significant, 26.9%. This large
discrepancy is likely to be resulted from the radiative heat transfer presented in the experiments at
high temperatures. Further investigation of the thermal radiation properties of LiF-BeF2-ThF4-UF4,
such as its absorption coefficient, is therefore suggested. In this study, a curve is fitted based on a
least square regression analysis of the experimental data, which results in
Nu = ReZ.WWU Pr
⁄' B
• ⁄€ H
⁄'
(27)
The experimental data are generally within a ±15% uncertainty range of the above revised
Sieder-Tate correlation, i.e., Eq. (27).
4.2.2 Transitional and Turbulent Flow Regimes
The Nusselt number computed based on the CFD simulations are compared with the experimental
data in molten salt experiments summarized in Table 2 and convective heat transfer correlations
summarized in Table 5 for transitional and turbulent flows, such as the Dittus-Boelter Eq. (28),
Gnielinski Eq. (29), Sieder-Tate (turbulent flow) Eq. (30), and Hausen Eq. (31) correlations.
Nu = 0.023ReZ.• P
C
(n = 0.4 for heating and 0.3 for cooling)
Nu = 0.012BReZ.•X − 280HPr Z.W ™1 + B•⁄€ H
31
⁄'
š:› 3 ⁄› ? <
Z.
(28)
(29)
Nu = 0.027ReZ.• Pr
Nu = 0.116:Re
⁄'
⁄'
− 125<Pr
(30)
⁄'
(31)
Comparison with the Dittus-Boelter Correlation
As shown in Figure 17, the CFD results of FLiNaK and LiF-BeF2-ThF4-UF4 in general agree well
with the experimental data. In addition, both the CFD results and 90% of the total experimental
data points fall within a ±20% uncertainty range of the Dittus-Boelter correlation. Therefore, the
Dittus-Boelter correlation is generally good for modeling salt heat transfer in turbulent flow
regime.
+20%
-20%
Figure 17 Comparison among the CFD results, experimental data, and Dittus-Boelter correlation
for salt heat transfer coefficient
32
Comparison with the Gnielinski Correlation
Similarly, the CFD results of FLiNaK and LiF-BeF2-ThF4-UF4 are within a ±11.4% uncertainty
range of the Gnielinski correlation as shown in Figure 18. In addition, nearly 90% of the
experimental data points fall within a ±20% uncertainty range of the Gnielinski correlation, while
5% of the experimental data points deviate from the Gnielinski correlation results significantly, by
40% and the remaining 5% data points deviate by 150%. Except for the experimental data from
Hoffman [4] who used nickel and SS 316 tubes in the experiments, the Gnielinski correlation in
general predicts accurately the molten salt heat transfer behavior for 2,300 ≤ Re ≤ 50,000.
+20%
-20%
Figure 18 Comparison among the CFD results, experimental data, and Gnielinski correlation for
salt heat transfer coefficient
33
In Hoffman’s experiments, nickel, Inconel, and SS 316 were used as the tube materials. Since the
Nusselt values derived from the FLiNaK-Inconel experimental data are much lower than those
derived from their other two experiments, i.e., FLiNaK-nickel and FLiNaK-SS 316 experiments,
Hoffman concluded that the significant deviation was due to the formation of a corrosion layer on
the inner tube surface during the FLiNaK-Inconel experiment [3]. However, Ambrosek, et al., [45]
and Yoder, et al., [46] disputed the explanation since Inconel and SS 316 should exhibit similar
corrosion resistance to FLiNaK at similar temperatures.
Table 6 summarizes the nominal compositions of Inconel 600, Nickel 200, and SS 316. Chromium
(Cr) contents in Inconel 600 and SS 316 are in very similar ranges. Chromium is more likely to
dissolute in FLiNaK compared with Ni and Fe due to the least formation free energy of CrF2 as
summarized in Table 7 [48]. Therefore, the salt corrosion effect on FLiNaK heat transfer
experiments using Inconel 600 and SS 316 as the tube materials should behave similarly if the
initial purity level in FLiNaK is the same for the three tests and the rest of testing conditions are
very similar, including the salt temperature, salt volume, and test period. A number of publications
[49-51] demonstrated that compared with Inconel 600, SS 316L/316H has a similar or even lower
corrosion rate in FLiNaK salt environment. However, the experimental data from Hoffman’s
FLiNaK-SS 316 experiment are about 100% higher than those from their own FLiNaK-Inconel
experiment. In addition, Nickel 200 is corrosion resistant to FLiNaK, but the experimental data
using the nickel tube show significantly higher values than those using Hastelloy N, another
corrosion resistant alloy to FLiNaK. It is therefore hypothesized that there were most likely some
abnormal conditions un-attended in Hoffman’s FLiNaK-SS 316 and FLiNaK-Nickel experiments.
New heat transfer experiments for FLiNaK in SS 316 and nickel tubes would be necessary.
34
Table 6 Nominal compositions of Inconel 600, nickel 200, and SS 316 (wt%) [47]
Ni
Cr
Fe
Mo
Mn
Si
C
Inconel 600
> 72.0
14.0-17.0
6.0-10.0
--
< 1.0
< 0.5
< 0.15
Nickel 200
> 99.0
--
--
--
< 0.35
< 0.35
< 0.15
SS 316
10.0-14.0
16.0-18.0
Balance
2.0-3.0
< 2.0
< 0.75
< 0.08
Table 7 Formation free energy of fluorides at 1000 K [48]
Compound
CrF2
FeF2
NiF2
MoF6
Free energy (kJ)
-314
-280
-230
-209
Comparison with the Sieder-Tate and Hausen Correlations
Since most of the heat transfer experimental data presented in the literature are in the form of
Nu/Pr1/3, the CFD results are written in the same format and compared with these experimental
data and Sieder-Tate (turbulent) and Hausen correlations as shown in Figure 19. The experimental
data using oil as the working fluid [52] are included as well. The CFD results of FLiNaK and LiFBeF2-ThF4-UF4 agree well with the experimental data, except again for the data derived from
Hoffman’s experiments. In addition, both the CFD results and most of the experimental data are
within a ±20% uncertainty range of the Sieder-Tate correlation (turbulent) for 10,000 ≤ Re ≤
120,000. In addition, Hausen correlation predicts the molten salt heat transfer coefficient with
±20% uncertainties for 5,000 ≤ Re ≤ 10,000, while ±40% uncertainties for 2,300 ≤ Re ≤ 5,000.
35
+20%
-20%
Figure 19 Comparison with the Sieder-Tate and Hausen correlations
It is believed that the larger experimental data compared to the CFD results and Hausen correlation
for Re = 2,300 – 3,000 results from the effect of radiative heat transfer in the experiments.
However, the comparison appears to suggest that the effect of radiative heat transfer in molten salts
could be neglected for flows of large salt Reynolds numbers, such as Re > 3,000 because of
significantly increased convective heat transfer coefficient. A curve is fitted based on a least square
regression analysis of the experimental data located in the transitional flow region as
36
Nu = 0.116BRe
/'
− 115HPr
/'
(32)
The experimental data fall within a ±20% uncertainty range of the above revised Hausen
correlation, i.e., Eq. (32).
5. CONCLUSION
This paper numerically investigated the thermal and hydrodynamic characteristics of molten salts
in circular tubes using a CFD tool, STAR CCM+. The numerical results were validated by
comparing with (1) modified experimental results using more accurate and consistent
thermophysical properties of molten salts and (2) widely used convective heat transfer coefficient
and friction factor models/correlations. Several concluding remarks are summarized as follows:
(1) For the hydrodynamic and thermal entrance lengths of molten salts in laminar and turbulent
flows, the discrepancy between the CFD results and related correlations, Eqs. (17) to (21),
is 9.6% to 35%;
(2) For the friction factor of molten salts in fully-developed laminar flows, the discrepancy
between the CFD results and Eq. (22) is within ±6.3% uncertainties. However, the molten
salt friction factor values in fully-developed turbulent flows estimated by STAR CCM+
and related correlations, Eqs. (23) and (24), are much lower than those from the
experimental data. This large discrepancy most likely results from the increased surface
roughness of the tube due to corrosion by salts in high temperature environments. If a
relative roughness ɛ/D = 3×10-3 is adopted in Eq. (25) to consider the increased surface
roughness due to corrosion, the discrepancy between the correlation results and the
experimental data will decrease to 23.6%;
37
(3) For molten salt heat transfer in laminar and turbulent flow regimes, the Nusselt number
estimated by STAR CCM+ is within ±20% uncertainties of convective heat transfer
correlations, including the Dittus-Boelter, Gnielinski, and Sieder-Tate correlations.
(4) The Dittus-Boelter correlation predicts the molten salt heat transfer coefficient with ±20%
uncertainties for 10,000 ≤ Re ≤ 50,000 and 4 ≤ Pr ≤ 12;
(5) The Gnielinski correlation predicts molten salt heat transfer coefficient with ±20%
uncertainties for 2,300 ≤ Re ≤ 50,000 and 4 ≤ Pr ≤ 19;
(6) The Sieder-Tate correlation for turbulent flows predicts molten salt heat transfer coefficient
with ±20% uncertainties for 10,000 ≤ Re ≤ 120,000 and 4 ≤ Pr ≤ 27, while the Sieder-Tate
correlation for laminar flows underpredicts the molten salt heat transfer coefficient by
26.9% for 400 ≤ Re ≤ 2,300 and 5 ≤ Pr ≤ 10. The larger heat transfer coefficient values
obtained from experimental data in the laminar flow regime result from the effect of
radiative heat transfer in molten salt experiments, while this effect is not considered in our
numerical analyses due to unknown radiation properties of the salts under the conditions
investigated. A revised Sieder-Tate correlation, Nu = ReZ.WWU Pr
/' B
•/€H
/'
, predicts the
molten salt heat transfer coefficient within ±15% uncertainties for 400 ≤ Re ≤ 2,300 and 5
≤ Pr ≤ 10;
(7) The Hausen correlation predicts the molten salt heat transfer coefficient with ±20%
uncertainties for 5,000 ≤ Re ≤ 50,000 and 4 ≤ Pr ≤ 27, while ±40% uncertainties for 2,300
≤ Re ≤ 5,000 and 4 ≤ Pr ≤ 27. A revised Hausen correlation, Nu = 0.116BRe
115HPr
/'
/'
−
, predicts the molten salt heat transfer within ±20% uncertainties for 2,300 ≤ Re
≤ 10,000 and 4 ≤ Pr ≤ 27;
38
ACKNOWLEDGMENTS
This research was performed using funding received from the Department of Energy (DOE) Office
of Nuclear Energy’s Nuclear Energy University Program (NEUP). The authors appreciate the
financial support from the DOE NEUP office and technical support from the technical point of
contact Dr. David Holcomb of the Oak Ridge National Laboratory.
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