ArticlePDF Available

Abstract and Figures

The Small Modular Dual Fluid Reactor (SMDFR) is a novel molten salt reactor based on the dual fluid reactor concept, which employs molten salt as fuel and liquid lead/lead-bismuth eutectic (LBE) as coolant. A unique design of this reactor is the distribution zone, which locates under the core and joins the core region with the inlet pipes of molten salt and coolant. Since the distribution zone has a major influence on the heat removal capacity in the core region, the thermal hydraulics characteristics of the distribution zone have to be investigated. This paper focuses on the thermal hydraulics analysis of the distribution zone, which is conducted by the numerical simulation using COMSOL Multiphysics with the CFD (Computational Fluid Dynamics) module and the Heat Transfer module. The energy loss and heat exchange in the distribution zone are also quantitatively analyzed. The velocity and temperature distributions of both molten salt and coolant at the outlet of the distribution zone, as inlet of the core region, are produced. It can be observed that the outlet velocity profiles are proportional in magnitude to the inlet velocity ones with a similar shape. In addition, the results show that the heat transfer in the center region is enhanced due to the velocity distribution, which could compensate the power peak and flatten the temperature distribution for a higher power density.
Content may be subject to copyright.
Thermal Hydraulics Analysis of the Distribution
Zone in Small Modular Dual Fluid Reactor
Chunyu Liu 1,* , Xiaodong Li 1, Run Luo 1,2 and Rafael Macian-Juan 1
1Chair of Nuclear Technology, Department of Mechanical Engineering, Technical University of Munich
(TUM), Boltzmannstr. 15, 85748 Garching, Germany; (X.L.); or (R.L.); (R.M.-J.)
2School of Resource & Environment and Safety Engineering, University of South China, No. 28,
Changsheng West Road, Hengyang 421001, China
*Correspondence: or
Received: 29 June 2020; Accepted: 4 August 2020; Published: 6 August 2020
The Small Modular Dual Fluid Reactor (SMDFR) is a novel molten salt reactor based on
the dual fluid reactor concept, which employs molten salt as fuel and liquid lead/lead-bismuth
eutectic (LBE) as coolant. A unique design of this reactor is the distribution zone, which locates
under the core and joins the core region with the inlet pipes of molten salt and coolant. Since the
distribution zone has a major influence on the heat removal capacity in the core region, the thermal
hydraulics characteristics of the distribution zone have to be investigated. This paper focuses on the
thermal hydraulics analysis of the distribution zone, which is conducted by the numerical simulation
using COMSOL Multiphysics with the CFD (Computational Fluid Dynamics) module and the Heat
Transfer module. The energy loss and heat exchange in the distribution zone are also quantitatively
analyzed. The velocity and temperature distributions of both molten salt and coolant at the outlet
of the distribution zone, as inlet of the core region, are produced. It can be observed that the outlet
velocity profiles are proportional in magnitude to the inlet velocity ones with a similar shape. In
addition, the results show that the heat transfer in the center region is enhanced due to the velocity
distribution, which could compensate the power peak and flatten the temperature distribution for a
higher power density.
small module dual fluid reactor; molten salt reactor; CFD; thermal-hydraulics;
heat removal by liquid metal
1. Introduction
1.1. The Small Modular Dual Fluid Reactor
The molten salt reactor is one of six Generation IV reactor types, which adopts molten salt as
fuel and has many unique characteristics compared to solid fuel reactors. The study on two-fluid
molten salt reactor can be traced back to 1966, when a two-fluid molten salt breeder reactor (MSBR)
was designed at Oak Ridge National Laboratory (ORNL) [
]. The load-following capability of this
MSBR system was studied for various ramp rates of the power demand [
]. Later a fast breeder reactor
of about 2 GW thermal output using molten chlorides both as fuel and coolant was proposed [3].
A new design of the two-fluid MSBR called the Dual Fluid Reactor (DFR), which employs liquid
lead instead of molten salt for the secondary circuit as coolant, was proposed by the researchers at the
Institute for Solid-State Nuclear Physics (IFK) [
]. Differing from Taube’s design, the high volumetric
heat capacity of the liquid lead enables a significant heat removal capability and, thus, allows operation
with a much higher power density than that of conventional reactors. A series of studies covering
the neutronic characteristics, sensitivity performance, coupled calculations, emergency drain tank
Metals 2020,10, 1065; doi:10.3390/met10081065
Metals 2020,10, 1065 2 of 19
estimation, depletion as well as hydraulic simulations can be found in references [
]. It has to
be noticed that previous studies on the DFR focused mainly on a large type of reactor design with a
thermal power of 3 GW. Yet, a smaller type of less than 0.3 GW would be more flexible and could be
applied to a wider range of industrial processes, such as hydrogen production, water desalination, etc.
In this work a Small Modular Dual Fluid Reactor (SMDFR) is proposed with a nominal thermal
power of 0.1 GW (100 MWt). The scheme is shown in Figure 1. The reactor core of the SMDFR, formed
by hexagonally-arranged fuel tubes and surrounded by the reflector zone, is immersed at the center of
a lead pool inside a cylindrical tank. As shown in Figure 2, the bottom and top of the core zone (from
z = 0.2 m to z = 2.2 m) are joined by a distribution zone (from bottom, z = 0 m, to z = 0.2 m) and a
collection zone (from z = 2.2 m to z = 2.4 m, symmetric with respect to the distribution zone), in order
to distribute and collect the molten salt before and after flowing through the core region. Starting
from the bottom part of the core region (Figure 2), the molten salt (depicted in golden color) goes into
the distribution zone being distributed in the gaps between the hexagonally arranged coolant tubes
(depicted in red) and then flows upwards inside the fuel tubes (depicted in golden color) through a
plate with holes. After reaching the upper core part, it flows into the collection zone and then leaves
the core region towards a chemical processing plant, which is employed for the online processing
of the molten salt. During the operation, the salt coming from the collection zone is continuously
passed to the chemical processing plant, in which the fission products are removed, and nuclear fuel
components can be removed or added to maintain the optimal fuel composition. After being processed,
it is pumped back into the bottom part for the next flow cycle through the core region. The liquid lead
(Figure 2), acting as coolant in the secondary circuit, flows from the bottom center to the top center to
cool down the core, which is heated by the nuclear fission reactions occurring in the molten salt. Upon
leaving the core region, it is pumped to the heat exchanger located in the peripheral region of the
tank and then transfers the heat to the tertiary circuit for utilization. Compared to the original design,
the SMDFR is scaled down to 0.1 GW (100 MW), which makes it particularly suitable for providing
energy for production facilities (e.g., water desalination [
], hydrogen production, or mines) in remote
locations and more flexible, since it does not necessarily need to be hooked into a large power grid,
and is able to be attached to other modules to provide increased power supplies if necessary. In
addition, it also has changed from a pipe-type to a pool-type reactor in order to ensure the capability
of establishing natural circulation during reactor transients or accidents. The hexagonal arrangement
of fuel and coolant tubes realizes a dense lattice in the core and a high power density.
Figure 1. Design schematic of the Small Modular Dual Fluid Reactor.
Metals 2020,10, 1065 3 of 19
Figure 2.
Schematic of the primary circuit (golden) and the secondary ciruit (red) of the Small Modular
Dual Fluid Reactor.
The technical data of the SMDFR are listed in Table 1. In this work a mixture of uranium
tetrachloride and plutonium tetrachloride is chosen as fuel salt and the liquid lead is chosen as coolant.
For the pipe walls silicon carbide is employed. The thermo-physical properties of these materials
are listed in Table 2according to [
]. The formulas in the reference are applied for each 100 K
interval from 800 K to 1600 K (below the boiling point of molten salt: 1950 K [
]), which includes the
operational temperature range. The corresponding interpolated values based on these temperature
points are applied for the calculations.
Table 1. SMDFR technical data.
Parameters Values
Core zone D ×H (m) 0.95 ×2.0
Distribution zone D ×H (m) 0.95 ×0.2
Collection zone D ×H (m) 0.95 ×0.2
Height of core (m) 2.4
Outer reflector diameter (m) 1.25
Tank D ×H (m) 1.65 ×3.4
Number of fuel tubes 1027
Fuel pin pitch (m) 0.025
Outer/interior fuel tube diameter (m) 0.008/0.007
Outer/interior coolant tube diameter (m) 0.005/0.004
Mean linear power density (W/cm) 609
Fuel inlet/outlet temperature (K) 1300/1300
Coolant inlet/outlet temperature (K ) 973/1250
Fuel inlet/in-core velocity (m/s) 3/0.5225
Coolant inlet/in-core velocity (m/s) 5/1.3488
Metals 2020,10, 1065 4 of 19
Table 2. Thermophysical properties of the two fluids and wall material.
Formula Validity Range [K]
Fuel specific heat capacity Cp(g/cm3) 400 [-]
Fuel thermal conductivity k(W/(m·K)) 2 [-]
Fuel density ρ(kg/m3) 1000 ×(5.601 1.5222 ×103×T)[-]
Fuel dynamic viscosity µ(Pa·s) 4.50 ×104[-]
Coolant specific heat capacity Cp(g/cm3)
T21.524 ×106T2
Coolant thermal conductivity k(W/(m·K)) 9.2 +0.011 ×T[600–1300]
Coolant density ρ(kg/m3) 11, 441 1.2795 ×T[600–2000]
Coolant dynamic viscosity µ(Pa·s) 4.55 ×104×e1069/T[600–1473]
Pipe wall heat capacity Cp(g/cm3) 690 [-]
Pipe wall conductivity k(W/(m·K)) 61, 100/ (T115)[300–2300]
Pipe wall density ρ(kg/m3) 3210 [-]
1.2. Simulation Code
COMSOL Multiphysics (version 5.5, COMSOL Inc., Stockholm, Sweden) [
] together with
its CFD (Computational Fluid Dynamics) module [
] and Heat Transfer module [
] is used to
perform the 3D pin-by-pin (for all tubes) thermal-hydraulics calculation. COMSOL Multiphysics is a
multi-physics simulation code package which is used in many industry application, including fluid
flow, and can provide a high accuracy numerical solution by using the Finite Element Method (FEM)
to solve the corresponding set of partial differential equations. The software runs on a standalone PC
with Intel R
CoreTM i7-7700 processor, 64 GB RAM and solid state disk.
1.3. Research Objective
A novel design of the core (including the distribution zone, core zone and collection zone) for
a small modular dual fluid reactor is proposed and the thermal hydraulics characteristics of the
distribution zone are investigated. In order to obtain both systematic and local behaviors of the two
fluids, a 3D computer model is built using COMSOL Multiphysics with the CFD and Heat Transfer
modules and the corresponding numerical simulations are performed. In addition, the sensitivity
analysis is conducted to investigate the system responses of various inlet velocities of both fluids.
2. Modeling and Simulation
The modeling and simulation process presented in this paper is divided into four parts: geometry
extraction and building, setup of governing equations, meshing, selection of the numerical solver and
performing the simulation.
2.1. Geometry
There are many more fuel tubes than those depicted in Figure 3, and a full core calculation
requires very large computational resources. Because of the symmetrical hexagonal configuration of
the core region, a sector of 30
(Figure 4, right) of the lower part of the core with a height of 0.4 m
(from z = 0 m to z = 0.4 m, Figure 4, left) around the central cylindrical axis, only one twelfth of the
complete geometry, was selected to represent the full scale geometry by applying symmetric boundary
conditions (mirror plane) [
]. The number of fuel tubes to be simulated was then decreased from 1027
to 100, among which 72 are complete tubes. A complete tube means that the whole tube was included
in the selected 30
sector and an incomplete tube means only one part of the tube was included.
The tubes at the boundary were incomplete tubes, as shown in Figure 4(right).
In Figure 4(right) three colored domains can be observed: golden, red and grey, which represent
respectively the fuel domain, the coolant domain, and the pipe wall domain. This geometric model
contained the distribution zone with a height of 0.2 m and its upper part (from core zone) with a
Metals 2020,10, 1065 5 of 19
height of 0.2 m. In the fuel domain, the molten salt entered through the inlet pipe from the right side,
flowed into the distribution zone, spread inside the gaps between the hexagonally-arranged coolant
tubes, flowed upwards into the fuel tubes, and then left the core zone through the upper part. In the
coolant domain, the liquid lead flowed into the core zone through the bottom coolant pipes, it then
flowed upwards between the gaps along the fuel tubes, and, finally, it left the core region through the
upper part.
Figure 3.
The arrangement of both fuel and coolant tubes in a real core: left, full height, z = 0 m to z =
2.4 m; upper right, half height, z = 0 m to z = 1.2 m; lower right, z = 0 m to z = 0.4 m.
Figure 4.
A 30
sector of the distribution zone and core zone with a height of 0.4 m, z = 0 m to
z = 0.4 m.
2.2. Governing Equations
Heat transfer between two fluids separated by a solid wall was the main thermal process to be
simulated, with both fluids the molten salt and the liquid lead having high Reynolds numbers (molten
salt: 1.17
, liquid lead: 1.28
). Therefore, the physical COMSOL Multiphysics-interface
“Conjugate Heat Transfer: Turbulent Flow, k-
” is chosen for the calculation, which combines the “Heat
Metals 2020,10, 1065 6 of 19
Transfer in Solids and Fluids” interface and the “Turbulent Flow, k-
” interface and, thus, can be used
to simulate the coupling between heat transfer and fluid flow.
Three compressibility options are available in COMSOL Multiphysics: compressible flow
Ma < 0.3
), weakly compressible flow, and incompressible flow. The option of compressible flow
makes no assumptions for the system and takes into account any dependency that the fluid properties
may have on the variables. The equations of weakly compressible flow look the same as these of
compressible flow except that the density is evaluated at the reference absolute pressure. For the
incompressible flow option, the density is regarded as constant and evaluated using the reference
temperature and pressure. [
] Since the density of molten salt has a strong dependency on its
temperature, the weakly compressible Reynolds-Averaged Navier–Stokes (RANS) equations were
adopted as a compromise between computational cost and accuracy:
ρ(u· 5)u=5 · [pI+K] + F(1)
5 ·(ρu) = 0 (2)
K= (µ+µT)(5u+ (5u)T)2
3(µ+µT)(5 · u)2
is the viscosity term taking into account the interactions between the fluctuating parts of
the velocity field. In order to close the above equations, two additional transport equations with two
dependent variables, the turbulent kinetic energy
and the turbulent dissipation rate
, are introduced:
ρ(u· 5)k=5 · [(µ+µT
)5k] + Pkρe (4)
ρ(u· 5)e=5 · [(µ+µT
)5e] + Ce1
where µTand Pkare the turbulent viscosity and the production term and are given by:
Pk=µT[5u:(5u+ (5u)T)2
3(5 · u)2]2
3ρk5 ·u(7)
The model’s constants used are: Ce1=1.44, Ce2=1.92, Cµ=0.09, σk=1.0, σe=1.3.
For the heat transfer in solids and fluids, the following equations are adopted:
ρCpu· 5T+5 · q=Q(8)
and the heat flux between the fluid with temperature
and the wall with temperature
calculated by:
Since there is no recommended turbulent Prandtl number model for liquid lead flow,
the Kays–Crawford model is adopted:
k(0.3 CpµT
where the Prandtl number at infinity is PrT=0.85.
Metals 2020,10, 1065 7 of 19
2.3. Meshing
The mesh was manually built by defining swept meshes and free tetrahedral meshes of various
sizes for different regions. Since the flows of molten salt and liquid lead vary gradually along
their channels, inside and outside the pipes, the finite elements can be quite stretched in the z-axial
direction. For this reason, swept meshes were adopted in order to reduce the complexity of the
mesh. The remaining parts of the model were relatively irregular and, thus, they were meshed by
free tetrahedral mesh elements. The completely generated mesh (Figure 5) has 599,333 elements with
an average element quality of 0.6259, and its 3D-plot of mesh element quality is shown in Figure 6.
Around 96.9% of the model volume was filled by the mesh elements with qualities greater than
0.1, which is considered accurate enough [
]. From the numerical point of view such a mesh was
acceptable in terms of accuracy and stability of the solution.
Six boundary layers were added for both the molten salt and the liquid lead with a stretching
factor of 1.1. To achieve a high accuracy the value of the wall resolution in viscous units was set to lie
between 11.6 and 100, which corresponded to 4.12
m for the molten salt and to 6.645
for the liquid lead.
Figure 5.
Model with mesh rendering: top, 3D view; middle, top view, xy-plane; bottom, top view
zoomed up.
Metals 2020,10, 1065 8 of 19
Figure 6. Mesh element quality with color indicating its value.
2.4. Numerical Solver
Two fundamental classes of solvers, direct and iterative solvers, were provided by COMSOL
Multiphysics. The direct solvers available within COMSOL Multiphysics are “PARDISO”, “MUMPS”,
and “SPOOLES”, as well as “Dense Matrix” solvers. The Dense Matrix Solver can be only used for
Boundary Element Method models, and therefore not suitable for the model in this paper. Contrary to
direct solvers, iterative solvers approach the solution gradually and consume less memory, rather than
in one large computational step. Three direct solvers, “PARDISO”, “MUMPS”, “SPOOLES”, and one
iterative solver, “GMRES”, were selected for solving the same model, and the convergence criterion
used for all variables was a relative tolerance of 0.005. Their performance comparison is listed in
Table 3.
Table 3. Performance comparison of different solvers.
Solver Solution Time [s] Physical Memory [GB] Virtual Memory [GB]
PARDISO 31,611 51.84 61.23
MUMPS 32,211 41.15 49.17
SPOOLES 103,898 56.43 57.41
GMRES 47,446 24.89 27.13
“PARDISO” was the fastest solver while consuming the highest physical and virtual memories
except for “SPOOLES”. The iterative solver “GMRES” consumed less memory and was slower then the
direct solvers “PARDISO” and “MUMPS”. All these solvers converged to the same results. Since the
sizes of physical and virtual memories were not a bottleneck for the model to be solved, “PARDISO”
was considered as the most efficient solver and was selected for solving all the simulations shown in
this paper.
3. Results and Discussion
The results and discussion section is divided into four parts: verification, hydraulic characteristics,
heat transfer analysis, and sensitivity analysis of inlet velocities.
Metals 2020,10, 1065 9 of 19
3.1. Verification
Since the SMDFR design is a new concept and no experimental data can be used as a reference,
the solution should be verified through the wall resolution in viscous units for the first layer, and also
the consistence of results from different mesh scenarios for the mesh convergence.
The value of wall resolution in viscous units tells how far into the boundary layer the
computational domain starts and should be lower than 500 to ensure that the logarithmic layer
meets the viscous sublayer. As shown in Figure 7, for more than 90% of the fluid domains the value of
the wall resolution in viscous units lay between 11.6 and 100, which means the first boundary layers
approximately corresponded to the beginning of the logarithmic layer and the required accuracy of
the results through the first boundary layers (next to the wall) was achieved. There was no need to
further refine the boundary layer mesh.
Figure 7. Wall lift-off in viscous units [1] at the wall surface.
In order to investigate the mesh convergence, five mesh scenarios (M0 to M4) were selected, M0
was the original mesh with six boundary layers, M1 and M2 had three and eight boundary layers
respectively, and M3 and M4 have coarse and refined meshes separately. The meshing parameters are
shown in Table 4. In order to quantify the simulation results for comparison and analysis, the outlet
surface was divided into 19 channels for molten salt, and 20 channels for liquid lead, along the radial
direction from center to circumference, as shown in Figure 8. The mass flow rate averaged velocity
and temperature of each channel were selected as the quantities to be compared and analyzed.
As shown in Figures 9and 10, all the mesh scenarios delivered consistent results for velocity and
temperature profiles of both molten salt and liquid lead, especially in the peripheral regions. However,
in the center regions (next to the center-line/axis of the core), some divergences were observed for
M1 (black) and M3 (green). In order to ensure a high enough accuracy in the whole domain, M0 (red),
which had consistent results with M2 (magenta) and M4 (blue) but consumed less computational
resources, was verified against the finer meshes and adopted for the final simulations.
Metals 2020,10, 1065 10 of 19
Figure 8. The channel indices of molten salt (fuel) and liquid lead (coolant) at the outlet plane.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 3 4 5 6 7 8 9 10 111213141516171819 20
Figure 9.
Mesh convergence study of outlet velocity distribution: top, velocity profiles of molten salt;
bottom, velocity profiles of liquid lead.
Metals 2020,10, 1065 11 of 19
Figure 10.
Mesh convergence study of outlet temperature distribution: top, temperature profiles of
molten salt; bottom, temperature profiles of liquid lead.
3.2. Hydraulic Characteristics
From Figure 9it can be seen that the molten salt velocity was relatively low in the central regions,
which was caused by the high flowing resistance of the cross-flow through the coolant tubes. In order
to get a better understanding of this phenomenon, a plane of the fuel inlet pipe was obtained by cutting
the model by the blue plane (z =
0.1 m), as shown in Figure 11 (top). A majority of the molten salt
tended to go upwards along the fuel tubes and only a small portion of the molten salt could pass the
peripheral regions and reach the central core regions (Figure 11, bottom). Taking a look into the the
Metals 2020,10, 1065 12 of 19
velocity vector field (Figure 12), one cold and two hot vortices could be observed located at the lower
left corner (cold), top right corner (hot) and lower right corner (hot). Special consideration must be
given to these locations, since the risk of structure material failure was relatively high and vibration
might occur in the regions with vortices.
Figure 11.
Top, plane of the fuel inlet pipe cut by the blue plane; bottom, Velocity vector field on the
plane of the fuel inlet pipe of the molten salt.
Figure 12. Velocity vector field of the molten salt with color indicating its temperature.
Metals 2020,10, 1065 13 of 19
For liquid lead the situation was the contrary, starting from the first two channels it had a high
velocity and after that the velocity became relatively uniform until the last two channels (Figure 9).
Since in the region next to the circumference there was an additional channel surrounding the fuel
channels, a certain portion of liquid lead was distributed to this channel and thus the velocity in
peripheral region was reduced. However, the liquid lead in the central region (first two channels)
accumulated due to the symmetry structure and thus had a relatively high velocity compared to others.
In order to quantify the energy loss during the flowing, Bernoulli’s equation applied for a micro
element of fluid is introduced:
multiplying both sides by a micro flow element
dm =ρ1gu1d A1=ρ2gu2dA2
), and then integrating
both sides by the corresponding cross section areas A1and A2:
is the energy loss in
, which determines the demand of pumping power. Selecting the inlet
and outlet cross section areas of the fluids as
, the energy loss in the distribution zone was
then obtained, which equals
. The
of molten salt in the distribution zone (for this one twelfth core
sector) is 64 W, which was much lower if compared to that of the liquid lead. Since the heat removal
function was mainly accomplished by the liquid lead, the flowing of molten salt is only needed for
chemical processing and its mass flow rate does not have to be large. Due to its high mass flow rate the
of liquid lead (for this one twelfth sector) was quite significant: 14,515 W. Considering the whole
structure and including the collection zone with the assumption of complete symmetry, the
liquid lead was around 348 kW only for the distribution and collection zones. It means that the pump
had to at least overcome this energy loss and it should have provided larger power when taking into
account the energy losses in other parts of the secondary circuit, especially in the heat exchanger.
3.3. Heat Transfer Analysis
From Figure 10 it can be seen that the molten salt temperature was continuously increasing
from the center to the external circumference and the liquid lead had a relative uniform temperature
distribution (Figure 13). This can be easily explained by the characteristics of the cross (lower part) and
co-current flows (upper part). In the center region, there were two factors enhancing the heat transfer:
the long flow path of molten salt crossing the fuel tubes (Figure 14) and the large velocity difference
between molten salt and liquid lead. The heat transfer was weak due to the low velocity difference in
the peripheral region, which resulted in a high molten salt temperature. Since the radial distribution of
power in the core follows a Bessel function of order zero (first period), the heat transfer enhancement in
the center region could compensate the power peak and flatten the temperature distribution to achieve
a higher operational temperature for both molten salt and liquid lead, which means the potential for a
higher power density.
The exchanged heat between molten salt and liquid lead in the distribution zone resulted in
their enthalpies changing and yielding a total thermal power transfer of 1.65 MW for this one twelfth
sector. Considering the whole structure and including the collection zone, the heat power transferred
increased to 39.6 MW, which was around 40% of the total thermal power. It means that the distribution
zone as well as the collection zone not only distributed and collected the fluids, but also had a
significant impact on the total heat removal process of the core, while the remainder thermal energy
was transferred inside the core zone.
Metals 2020,10, 1065 14 of 19
Figure 13. Temperature distribution at the outlet plane.
Figure 14. Temperature distribution at the plane of the fuel inlet pipe.
Metals 2020,10, 1065 15 of 19
3.4. Sensitivity Analysis of Inlet Velocities
In order to investigate the influence of the inlet velocities on the thermal-hydraulic characteristics
of the distribution zone, five inlet scenarios (Table 5: I0 for the original one and I1 to I4 for the changed
inlet velocities) were simulated and compared.
Table 4. Mesh Scenarios.
Parameters M0 M1 M2 M3 M4
Element size Original Original Original Coarse Fine
Number of boundary layers 6 3 8 6 6
Number of elements 599,333 502,525 545,028 515,312 825,333
Average element quality 0.6259 0.6179 0.6222 0.596 0.6758
Minimum element quality 4.447 ×1053.75 ×1063.686 ×1063.467 ×1069.606 ×106
Table 5. Inlet Scenarios.
Parameters I0 I1 I2 I3 I4
Molten salt inlet velocity (m/s) 3.0 2.5 3.5 3.0 3.0
Liquid lead inlet velocity (m/s) 5.0 5.0 5.0 4.0 6.0
As shown in Figure 15, the magnitude of outlet velocity of molten salt was proportional to its
inlet velocity and its profile’s shape was similar. The same conclusion can be made for the liquid lead.
The temperature profile of liquid lead (Figure 16) was not influenced too much by the inlet velocities
thanks to its heat capacity and mass flow rate. However, the thermal behaviour of the molten salt
was influenced by its inlet velocity: a higher molten salt inlet velocity resulted in a higher magnitude
but a similar trend in the temperature profiles and vice-versa. The impact of the liquid lead’s inlet
velocity on the molten salt’s temperature profile was negligible, which means that the heat transfer
between the fluids in the distribution zone occurred mainly in the lower part (cross flow). It has
to be pointed out that the heat transfer in the core zone (co-current flow) was definitely influenced
by the velocity variation of both fluids due to the change of relative velocity between them, and a
strong coupling effect should be considered during transient conditions with rapid velocitiy variation
expected. The thermal hydraulic coupling effects between the molten salt and liquid lead, as well as
the thermal hydraulics-neutronics coupling, will have to be considered for further simulations of the
core, especially in the case of transients and accident situations.
Metals 2020,10, 1065 16 of 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 3 4 5 6 7 8 9 10 11121314151617181920
Figure 15.
Outlet velocity distribution at various inlet scenarios: top, velocity profiles of molten salt;
bottom, velocity profiles of liquid lead.
Metals 2020,10, 1065 17 of 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 3 4 5 6 7 8 9 10 11121314151617181920
Figure 16.
Outlet temperature distribution at various inlet scenarios: top, temperature profiles of
molten salt; bottom, temperature profiles of liquid lead.
4. Conclusions
The computer model built by COMSOL Multiphysics is verified by examining the value of the
wall resolution in viscous units and by performing mesh sensitivity studies. The energy loss and heat
exchange in the distribution zone are quantified, and it is found that the pumping power delivered to
the liquid lead has to be much larger than that required by the molten salt in order to overcome the
large frictional energy loss of the flowing liquid lead. An amount of thermal power corresponding to
about 40% of the total reactor thermal power is transferred in the distribution and collection zones,
which not only affects the hydraulic performance of both fluids but also has a significant influence on
Metals 2020,10, 1065 18 of 19
the heat removal process. Since the radial distribution of power in the core follows a Bessel function
of order zero (first period), the heat transfer enhancement in the center region could compensate the
power peak and flatten the temperature distribution to achieve a higher operational temperature for
both molten salt and liquid lead, which means the potential for a higher power density. The outlet
velocity profiles observed are proportional in magnitude to the inlet velocity ones with a similar shape.
Author Contributions:
Conceptualization, C.L. and R.L.; methodology, C.L.; software, C.L. and X.L.; validation,
C.L.; formal analysis, C.L.; investigation, C.L.; resources, C.L.; data curation, C.L.; writing—original draft
preparation, C.L.; writing—review and editing, C.L. and R.M.-J.; visualization, C.L. and X.L.; supervision, R.M.-J.;
project administration, R.M.-J.; funding acquisition, R.M.-J. All authors have read and agreed to the published
version of the manuscript.
Funding: This research received no external funding.
We would like to thank M. Seidl and X. Wang for their strong support and technical input, to
the research group at IFK for providing basic data and very fruitful discussions on the design characteristics of
the DFR concept.
Conflicts of Interest: The authors declare no conflict of interest.
The following abbreviations are used in this manuscript:
SMDFR Small Modular Dual Fluid Reactor
LBE Lead-Bismuth Eutectic
CFD Computational Fluid Dynamics
MSBR Molten Salt Breeder Reactor
DFR Dual Fluid Reactor
IFK Institue for Solid-State Nuclear Physics
FEM Finite Element Method
RANS Reynolds-Averaged Navier-Stokes
SI International System of Units
All the quantities in this work are expressed according to the International System of Units (SI) and to
the nomenclature listed here below.
uaverage velocity field
Iidentity matrix
Kviscosity term
Fbody force vector
µdynamic viscosity
µTturbulent viscosity
kturbulent kinetic energy
eturbulent dissipation rate
Ce1model constant
Ce2model constant
Cµmodel constant
σkmodel constant
σemodel constant
Cpspecific heat at constant pressure
Tabsolute temperature
qheat flux by conduction
kheat conductivity
nnormal vector
uτfriction velocity
Twwall temperature
Tffluid temperature
T+dimensionless temperature
PrTturbulent Prandtl number
ggravitational acceleration
zaxial coordinate
Across section area
hwenergy loss
Robertson, R.; Briggs, R.; Smith, O.; Bettis, E. Two-Fluid Molten-Salt Breeder Reactor Design Study (Status as of
January 1, 1968); Technical Report; Oak Ridge National Lab.: Oak Ridge, TN, USA, 1970.
Metals 2020,10, 1065 19 of 19
Singh, V.; Lish, M.R.; Wheeler, A.M.; Chvála, O.; Upadhyaya, B.R. Dynamic Modeling and Performance
Analysis of a Two-Fluid Molten-Salt Breeder Reactor System. Nucl. Technol.
,202, 15–38,
doi:10.1080/00295450.2017.1416879. [CrossRef]
Taube, M.; Ligou, J. Molten plutonium chlorides fast breeder reactor cooled by molten uranium chloride.
Ann. Nucl. Sci. Eng. 1974,1, 277–281. [CrossRef]
Huke, A.; Ruprecht, G.; Weißbach, D.; Gottlieb, S.; Hussein, A.; Czerski, K. The Dual Fluid Reactor–A novel
concept for a fast nuclear reactor of high efficiency. Ann. Nucl. Energy 2015,80, 225–235. [CrossRef]
Wang, X.; Macian-Juan, R.; Seidl, M. Preliminary analysis of basic reactor physics of the Dual Fluid
Reactor-15270. In Proceedings of the ICAPP 2015, Nice, France, 3–6 May 2015.
Wang, X. Analysis and evaluation of the dual fluid reactor concept. Ph.D. Thesis, Universitätsbibliothek der
TU München, Munich, Germany, 2017.
Wang, X.; Macian-Juan, R. Comparative Study of Basic Reactor Physics of the DFR Concept Using U-Pu and
TRU Fuel Salts. In Proceedings of the International Conference on Nuclear Engineering. American Society
of Mechanical Engineers, Shanghai, China, July 2-6, 2017; Volume 57830, p. V005T05A001.
Wang, X.; Macian-Juan, R. Comparative Study of Thermal-Hydraulic Behavior of the DFR Using U-Pu and
TRU Salt Fuels. In Proceedings of the International Conference on Nuclear Engineering. American Society
of Mechanical Engineers, Shanghai, China, July 2-6, 2017; Volume 57847, p. V006T08A004.
Wang, X.; Macian-Juan, R. Steady-state reactor physics of the dual fluid reactor concept. Int. J. Energy Res.
2018,42, 4313–4334. [CrossRef]
Wang, X.; Liu, C.; Macian-Juan, R. Preliminary hydraulic analysis of the distribution zone in the Dual Fluid
Reactor concept. Prog. Nucl. Energy 2019,110, 364–373. [CrossRef]
IAEA. Nuclear Desalination. Available online:
nuclear-desalination. (accessed on 30 July 2020).
Desyatnik, V.; Katyshev, S.; Raspopin, S.; Chervinskii, Y.F. Density, surface tension, and viscosity of uranium
trichloride-sodium chloride melts. Sov. At. Energy 1975,39, 649–651. [CrossRef]
Nilsson, O.; Mehling, H.; Horn, R.; Fricke, J.; Hofmann, R.; Mueller, S.; Eckstein, R.; Hofmann, D.
Determination of the thermal diffusivity and conductivity of monocrystalline silicon carbide (300-2300
K). High Temp. High Press. 1997,29, 73. doi:10.1068/htec142. [CrossRef]
Fazio, C.; Sobolev, V.; Aerts, A.; Gavrilov, S.; Lambrinou, K.; Schuurmans, P.; Gessi, A.; Agostini, P.;
Ciampichetti, A.; Martinelli, L.; et al. Handbook on Lead-Bismuth Eutectic Alloy and Lead Properties, Materials
Compatibility, Thermal-Hydraulics and Technologies, 2015 ed.; Technical Report; Organisation for Economic
Co-Operation and Development: Paris, France, 2015.
Huke, A.; Ruprecht, G.; Weißbach, D.; Czerski, K.; Gottlieb, S.; Hussein, A.; Herrmann, F. Dual-fluid reactor.
In Molten Salt Reactors and Thorium Energy; Elsevier: Amsterdam, The Netherlands, 2017; pp. 619–633.
16. Introduction to COMSOL Multiphysics; COMSOL: Stockholm, Schweden, 2019.
17. CFD Module User’s Guide; COMSOL: Stockholm, Schweden, 2019.
18. Heat Transfer Module User’s Guide; COMSOL: Stockholm, Schweden, 2019.
Carasi, B. Compressibility Options and Buoyancy Forces for Flow Simulations. 2016.
Available online:
simulations (accessed on 30 July 2020).
Griesmer, A. Size Parameters for Free Tetrahedral Meshing in COMSOL Multiphysics. 2014.
Available online: comsol-
multiphysics (accessed on 30 July 2020).
2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (
... The characteristics of the steady-state physics of the dual fluid reactor concept were investigated [5,6], and a comparative study of the two fuel options was conducted [7]. The distribution zone of the dual fluid reactor concept has been designed and analyzed from the perspective of thermal hydraulics [8,9]. According to previous studies, the dual fluid reactor concept has a unique feature. ...
... The coolant (liquid lead) flows in the same direction and takes heat generated by the fission reaction taking place in the molten salt. Since this work focuses on the one-dimensional model of the core, a detailed description of the geometry is beyond the scope of this paper, and the reader is referred to the literature [9]. ...
Full-text available
The small modular dual fluid reactor is a novel variant of the Generation IV molten salt reactor and liquid metal fast reactor. In the primary circuit, molten salt or liquid eutectic metal (U-Pu-Cr) is employed as fuel, and liquid lead works as the coolant in the secondary circuit. To design the control system of such an advanced reactor, the uncertainties of the employed computer model and the physicochemical properties of the materials must be considered. In this paper, a one-dimensional model of a core is established based on the equivalent parameters achieved via the coupled three-dimensional model, taking into account delayed neutron precursor drifting, and a power control system is developed. The performance of the designed controllers is assessed, taking into account the model and property uncertainties. The achieved results show that the designed control system is able to maintain the stability of the system and regulate the power as expected. Among the considered uncertain parameters, the reactivity coefficients of fuel temperature have the largest influence on the performance of the control system. The most optimized configuration of the control system is delivered based on the characteristics of uncertainty propagation by using the particle swarm optimization method.
... The reactor core comprises numerous SiC ceramic pipes with small diameters, through which liquid uranium eutectics (U + Cr) flow. These pipes are enveloped by liquid lead, maintained at a lower temperature, thereby facilitating the heat transfer process [15,16]. ...
Full-text available
This study presents the design and computational fluid dynamics (CFD) analysis of a mini demonstrator for a dual fluid reactor (DFR). The DFR is a novel concept currently under investigation. The DFR is characterized by the implementation of two distinct liquid loops dedicated to fuel and coolant. It integrates the principles of molten salt reactors and liquid metal cooled reactors; thus, it operates in a high temperature and fast neutron spectrum, presenting a distinct approach in the field of advanced nuclear reactor design. The mini demonstrator serves as a scaled-down version of the actual reactor, primarily aimed at gaining insights into the CFD analysis intricacies of the reactor while minimizing computational costs. The CFD modeling of the MD intends to add valuable data for the purpose of modeling validation against experiments to be conducted on the MD. These experiments can be used for DFR licensing and design optimization. The coolant and fuel utilized in the mini demonstrator are of low Prandtl number (Pr = 0.01) liquid lead, operating at two distinct inlet temperatures, namely 873 K and 1473 K. The study showed a rapid increase in turbulence due to intense mixing and abrupt changes in flow areas and directions, despite the relatively low inlet velocities. Hot spots characterized by elevated temperatures were identified, analyzed, and justified based on their spatial distribution and flow conditions. Flow swirling within pipes was identified and a remedy approach was suggested. Inconsistent mass flow rates were observed among the fuel pipes, with higher rates observed in the lateral pipes. Although lower fuel temperatures were observed in the lateral pipes, they consistently exhibited higher heat exchange characteristics. The study concludes by giving physical insights into the heat transfer and flow behavior, and proposing design considerations for the dual fluid reactor to enhance structural safety and durability, based on the preliminary analysis conducted.
... The Dual-Fluid Reactor (DFR) concept was presented at the IFK Institute by Huke et al. [17], in which two fluids exist in the core: molten salt as the fuel and liquid lead as the coolant. The neutronic and thermal-hydraulic behaviors of the DFR concept were preliminarily investigated [18,19], and the smaller modular concept (SMDFR) was proposed by Liu et al. [20]. A hybrid MSR system was proposed by Furukawa et al. [21] in Japan, in which the nuclide of 233 U is produced by the accelerator spallation target. ...
Full-text available
A molten-salt reactor (MSR), in which the nuclear waste can be separated and taken away from the liquid fuel through online chemical treatment, has remarkable advantages in safety and economics. To ensure the safe operation of an MSR plant, it is necessary to carry out a controller design and stability analysis of the whole system. In this work, a nonlinear distributed-parameter model of an MSR plant with a graphite-moderated liquid core was established and verified by the reference model. By applying a joint feedforward and feedback control strategy, the power controller and steam temperature controller were designed to adjust the plant load. The Nyquist criterion was used to assess the stability of the control system. The system responses to four different perturbations at the nominal power level were simulated, and the results show that the designed controllers have good reliability. In addition, the control performance was tested in two load change cases: a step load reduction and a linear load change. The dynamic response characteristics of the system parameters show that the developed control system based on the nonlinear distributed-parameter model is capable of fast and accurate regulation.
... Wang et al. (2019b) and Chen et al. (2020) defined an onvehicle smDFR and studied its neutronic feasibility as well as its radiation shielding design. Liu et al. (2020) performed a thermal-hydraulic analysis on a 1/12 model of the distribution zone of the smDFR, but the design parameters were independently determined. ...
The small modular dual fluid reactor (smDFR) adopts a mixture of liquid uranium and plutonium chloride as fuel, with lead as a coolant. With its smaller size, lower weight, and inherent safety, the smDFR can be used as a mobile energy supply for vehicles or ships. Analyses of the smDFR are still in the early stage based on a single or separated physical field, which can hardly capture the impact of the flowing fuel on the neutron field. The coupled neutronic-fluid dynamic calculation is needed for reliable multidimensional, multi-field distribution of the steady-state smDFR core property. In this work, the Monte Carlo code Serpent 2 and the CFD code Open-FOAM are coupled via a Python interface, whereby the neutron field information can be exchanged with the fluid field in a region-average manner. The correctness and reliability of the coupled code are verified with two cases in the pressurized water reactor background, and the results show good agreement. The coupled code is applied to the steady-state analyses of an smDFR, a single-channel case, and a 1/6-core case. The results show that the temperature distribution of the fuel and coolant caused a lower drift of the axial power peak compared to the uncoupled cosine shape power distribution. The 1/6-core results show that the peak temperature of the fuel reaches 980 K at the top-center of the reactor core. Moreover, the influence of fuel thermal properties and inlet velocity are studied with rated power. Through the comparison of mapping schemes, the results can identify the dependence of specific parameters and help to reduce mapping errors. This work shows that through a coupled neutronic-CFD code, the multidimensional and multi-physical fields of the smDFR can be better represented, providing important references for the future design and analyses of smDFR systems.
In this study, building on the author’s previous experience in the simulation of the MSRE system, a similar approach was used to model the dual fluid reactor design. The work was done using a modified version of the system code TRACE capable of simulating molten salt and metal flows and the transport of the neutron precursors with the flowing fuel, and the Monte Carlo based Serpent code to generate the appropriate core neutronic data used in the point kinetics models of TRACE. The authors coupled both codes Serpent and TRACE to calculate the steady‐state power and temperature fields. Based on them, the authors have proposed an optimized geometry and flow rates for the reactor. With the optimized design, the results for several reactivity transient scenarios have been presented: step insertion reactivity and reactor over‐ and undercooled scenarios. The investigated transient scenarios show that the reactor is stable, self‐regulating, and can withstand severe conditions. Radial division of the DFRm core in the Serpent model
Full-text available
A molten salt reactor (MSR) has unique safety and economic advantages due to the liquid fluoride salt adopted as the reactor fuel and heat carrier fluid. The operation scheme and control strategy of the MSR plant are significantly different from those of traditional solid-fuel reactors because of the delayed neutron precursors drift with the liquid-fuel flow. In this paper, a simulation platform of the MSR plant is developed to study the control characteristics under normal and accident conditions. A nonlinear dynamic model of the whole system is built in the platform consisting of a liquid-fuel reactor with a graphite moderator, an intermediate heat exchanger and a steam generator. A new control strategy is presented based on a feed-forward and feedback combined scheme, a power control system and a steam temperature control system are designed to regulate load changes of the plant. Three different types of operation conditions are simulated with the control systems, including transients of normal load-follow operation, a reactivity insertion accident and a loss of flow accident. The simulation results show that the developed control system not only has a fast load-follow capability during normal operation, but also has a good control performance under accident conditions.
Full-text available
The dual fluid reactor (DFR) is a molten salt fast reactor developed by the Institut fuer Festkoerper‐Kernphysik (Berlin) on the basis of the Gen‐IV molten salt reactor and liquid metal–cooled reactor (sodium‐cooled fast reactor and lead‐cooled fast reactor) concepts. In the concept, the fuel is selected with the molten chloride salt and the coolant is the liquid lead. The article establishes the model for the reactor physics analysis and summarizes the preliminary performed calculations and the achieved results so far by the author on the steady‐state reactor physics behavior with the default fuel salt of U‐Pu mixture under zero‐flow‐velocity conditions. They include criticality calculation, neutron spectrum, spatial flux distribution, reaction rates and power distribution, nuclide importance, and temperature coefficient. The conclusion of the article evaluates the reactor physics behavior of the DFR concept and gives comments on the concept itself as well as on the future researches. The article is modified from a part of the PhD thesis of the first author (X. Wang). In this paper, the design of the novel molten salt reactor—dual fluid reactor (DFR)—was validated in the aspect of reactor physics in the steady state by performing a series of calculations including criticality, neutron spectra, temperature feedback coefficient, and sensitivity analysis. The results revealed that this design is feasible in the aspect and shows a sufficient neutronic stability facing possible changes in the fuel salt composition and the temperature fluctuation.
Full-text available
The Dual Fluid Reactor is a molten salt fast reactor developed by IFK in Berlin based on the Gen-IV Molten-Salt Reactor concept and the Liquid-Metal Cooled Reactor. The design aims to combine these two concepts to improve these two concepts. The Dissertation focuses on the concept and performs diverse calculations and estimations on the subjects of neutron physics, depletion and thermal-hydraulic behaviors to validate the new features of the concept. Based on the results it is concluded that this concept is feasible to its desired purpose and with great potential.
Conference Paper
Full-text available
The Dual Fluid Reactor is a novel fast nuclear reactor concept invented by the IFK based on the Generation IV Molten Salt Reactor and the Liquid Metal Cooled Reactor. Since there are attractive features mentioned in this design, the main task of this paper is to verify the model of the whole reactor based on this concept. For this purpose several calculations are presented, including steady state calculations, sensitivity calculations with regard to the nuclide cross sections, the temperature and geometry coefficient of keff as well as the burnup calculation. The Monte Carlo calculation codes MCNP, SERPENT and SCALE are used for the analysis.
This article focuses on the validation of the basic design of the distribution zone of the Dual Fluid Reactor (DFR), which is based on the Generation IV Molten-Salt Reactor concept and the Liquid-Metal Cooled Reactor. The distribution zone located under the core joins the core and input pipes of both fluids. On the top of the core, another distribution zone specified for output is also symmetrically set up; however it will not be mentioned in the paper. This distribution zone partitions the inlet flow of the salt, determines the pressure and velocity distribution of the core inlet and influences the neutronic behaviors in the core. This article uses COMSOL as the tool to develop models and to study the pressure and velocity differences of fuel salt and coolant lead in the “original” and “alternative” configurations/designs of the distribution zone in the DFR concept. COMSOL is also used to determine the initial/boundary conditions of configuration variations of the distribution zone by hydraulic modelling and calculations. The simulation results are analyzed and the optimized design is delivered.
A nonlinear dynamic model for the two-fluid molten-salt breeder reactor (MSBR) system is presented. This work is partly inspired by a preliminary dynamic model of the concept studied at Oak Ridge National Laboratory (ORNL). The core heat transfer model has been revised to accurately reflect the design exemplified in ORNL-4528—the last report on the two-fluid design. A brief description of the reactor system and the effects of reactor poisons and a discussion of temperature feedback mechanisms are presented. This background information is followed by an overview of the modeling approach and a discussion of the revised lumped parametrization, along with detailed descriptions of the modeling methodology and model limitations. All equations and parameters used in the model are presented to aid in model reproduction and adaptation for other molten-salt reactor designs. Model stability is analyzed by observing the uncontrolled response to reactivity perturbations. Simulations illustrate stable behavior at all power levels investigated. Temperature-induced feedback effects lead to stable dynamics for both large and small reactivity transients. Stable and smooth changes in the various nodal temperatures are also observed. The frequency response of the system indicates no dynamics problems at all operating power levels and is consistent with the transient response. Characteristic features in the frequency response plots due to feedback effects are also discussed. Finally, the load-following capability of the MSBR system is studied for various ramp rates of the power demand in the final heat sink. The temperatures in all salt-containing parts of the system are observed to vary about an average during the load-following maneuver. It is observed that the MSBR system exhibits a self-regulating behavior, minimizing the need for external controller action for load-following operations.
The dual-fluid reactor (DFR) is a novel concept of a liquid-fueled heterogeneous fast-spectrum nuclear reactor employing two separate loops, one for fuel and one for the coolant. This design allows for a very high power density, an operating temperature of 1000°C enabling efficient electricity generation, and a very good neutron economy for breeding or transmutation. Both a molten salt version based on undiluted actinide trichlorides and a liquid metal version using an actinide metal alloy melt with an appropriately low solidus temperature are possible. Fuel processing is done on-site employing a pyrochemical processing unit based on fractional distillation. Medical radioisotopes like Mo-99/Tc-99m are byproducts and can be provided immediately. Other possible applications are synthesis of carbon-free fuels at prices competitive with refined oil and radiotomic chemistry. The compactness of the reactor makes it highly economically competitive.
The Dual Fluid Reactor, DFR, is a novel concept of a fast heterogeneous nuclear reactor. Its key feature is the employment of two separate liquid cycles, one for fuel and one for the coolant. As opposed to other liquid-fuel concepts like the Molten-Salt Fast Reactor (MSFR), both cycles in the DFR can be separately optimized for their respective purpose, leading to advantageous consequences: A very high power density resulting in remarkable cost savings, and a highly negative temperature feedback coefficient, enabling a self-regulation without any control rods or mechanical parts in the core.
The thermal properties of monocrystalline silicon carbide of the 6H polytype have been studied. The specimens were n-doped with nitrogen to a carrier concentration of about 5 × 1017 cm-3. The thermal diffusivity, a, was measured with two laser-flash devices, in the range 300-2300 K. The temperature dependence can be described by a/mm2 s-1 = 14600/[(T/K) - 207]. The specific heat capacity, cp, was measured from room temperature up to 1650 K. From the measured thermal difusivities, the density, ρ, and the heat capacity taken partly from measurements and partly from literature, the thermal conductivity, λ, was calculated from λ = aρcp, giving λ(T)/W m-1 K-1 = 61100/[(T/K) - 115]. These data compare well with theoretically calculated data for the high temperature region.