CFD-SIMULATION OF URANIUM HEXAFLUORIDE DURING PHASE
CHANGE
TOMI PAKARINEN
Platom Oy
Jääkärinkatu 33, Mikkeli, Finland
ABSTRACT
A model for simulating the behavior of uranium hexafluoride during melting and
solidification cycles has been developed. First goal was to create a user-defined
material of uranium hexafluoride for commercial computational fluid dynamics
software (FLUENT). The results of the thermo physical properties are presented in
this paper. The material properties were used to create a model that is able to
simulate melting, solidification, evaporation and condensation. The model was
used to obtain knowledge of UF6s behaviour when melting and solidifying the
matter in a two-dimensional cylinder. The results were compared to the results of
an analytical solution. The calculation results are consistent with the simulation.
1. Introduction
Uranium hexafluoride (UF6) is a substance used as a raw material in uranium enrichment. Its
chemical and thermo physical properties make it ideal for the enrichment process because
each phase change is achievable within relatively low pressures and temperatures. UF6 is
stored in an airtight cylinder in which the contents are a mixture solid and gaseous phases.
To feed the material into the enrichment process, the cylinder is first heated and after
reaching the desired temperature, the cylinder is exposed to vacuum. This leads to
sublimation of the solid phase and the gas is fed to the enrichment process. Enriched gas is
then desublimed back to another cylinder. A sample of this enriched UF6 is required to
ensure desired enrichment level. The sample is extracted by heating the cylinder beyond the
melting point (64 °C) after the contents of the cylinder have completely melted and
homogenized. The cylinder is then cooled so that the liquid substance is solidified.
The phenomena related to heating and cooling cycles inside the cylinder are not well known.
This paper introduces a viable method to simulate these using numerical approach with
computational fluid dynamics software. For example, the time required to heat the cylinder to
melt all the contents is usually based on rough estimates and empirical knowledge. Having a
tool that is capable of simulating the behaviour of the substance makes it possible to develop
more efficient processing equipment.
2. CFD-model
2.1 Material properties
Material properties of UF6are not included in any commercial CFD-software. Anyway, Ansys
Fluent software has a capability to add features to the solver code via user-defined functions
(UDF). To add an UF6 feature to Ansys Fluent thermo-physical properties of the media had to
be defined. Anderson et al. (1994) present correlations for UF6 in all three phases in their
paper [1]. These correlations were used to create simple temperature dependent polynomial
functions for viscosities, densities, specific heats and thermal conductivities for UF6. These
properties are presented in Tables 1, 2 and 3.
0,0000013927 T3 – 0,0022709543 T2 +
1,3581332064 T + 128,2597191821
Thermal conductivity (k)
0,0000000006 T3 – 0,0000005324 T2 +
0,0002004505 T – 0,0208911076
Dynamic viscosity (µ)
-0,0000000012 T2 + 0,0000004378 T –
0,0000424922
Heat of vaporization (Hv)
81 589
Density (ρ)
real-gas-redlich-kwong equation of state
Table 1. Thermo-physical properties of gaseous UF6
Specific heat (Cp)
Specific heat (Cp)
Thermal conductivity (k)
421,6603
-0,0000000080 T3 + 0,0000083372 T2 –
0,0032355111 T + 0,6002910871
Dynamic viscosity (µ)
-0,0000000002 T3 + 0,0000002116 T2 –
0,0000941575 T + 0,0149527687
Heat of fusion (Hf)
54743
Density (ρ)
-0,0001953473 T3 + 0,2027000781 T2 74,9930889504 T + 13168,7964459175
Table 2. Thermo-physical properties of liquid UF6
Specific heat (Cp)
492,9424
Thermal conductivity (k)
-3,645e-2 + 1,895e-3 T
Dynamic viscosity (µ)
Heat of fusion (Hf)
54 743
Density (ρ)
4 830
Table 3. Thermo-physical properties of solid UF6
J/kg
W/mK
kg/ms
J/kg
kg/m3
J/kgK
W/mK
kg/ms
J/kg
kg/m3
J/kgK
W/mK
kg/ms
J/kg
kg/m3
Gaseous phase was simulated as real gas using the built-in Redlich Kwong equation of
state. Liquid and solid phases were considered as one single phase. This does not have any
effect on the reliability of the results, because the latent heat content of both phases is taken
into account in the energy equation. Material properties were switched depending on the
liquid fraction of the calculation cell. For specific heats in liquid and solid phases constant
values were used due to the Fluent’s built-in macro used in defining the specific heat. This
macro doesn’t return phase level properties such as liquid fraction of the cell, so specific heat
must be switched based on the temperature only. Solid density was also set to constant
value due to bad convergence of variable solid density.
2.2 Geometry and mesh
The model itself consists of two-dimensional cross-section of the 48G type UF6 cylinder,
which is converted into a mesh that contains 15 500 elements. The mesh is presented in the
figure 1.
Figure 1. Mesh for the model.
Hanging-node adaption is applied to the edges of the geometry since strong natural
convection currents are to be expected.
2.3 Phase change mechanisms
The Volume-of-Fluid multiphase model is chosen to track the interphase between the
phases. Solidification/melting model is enabled to simulate phase change between solid and
liquid UF6. This model is based on melting temperature only and this creates a problem,
because UF6 has a melting point at 64 °C and 152 kPa. When the pressure is not taken into
account, the sublimation/desublimation mechanism is impossible to include into this
particular model without reprogramming the mechanism entirely.
As mentioned earlier, liquid and solid phases are simulated as one single phase.
Solidification is modelled by applying a momentum sink for cells that are below freezing
point. This makes the velocities in the cell zero. Latent heat is taken into account by
modifying the enthalpy in the solid phase [2]. Enthalpy is formulated as
(1)
𝐻 = ℎ + ∆𝐻 ,
where the enthalpy h is defined as
𝑇
ℎ = ℎref + ∫𝑇
ref
𝑐𝑝 𝑑𝑇 .
(2)
Liquid fraction in the domain is defined as
𝛽 = 0, if 𝑇 < 𝑇melt ,
𝛽 = 1, if 𝑇 > 𝑇melt ,
where Tmelt is the melting temperature of the UF6. Latent heat content ∆𝐻 in the equation (1)
is defined as
∆𝐻 = 𝛽𝐿,
(3)
where L is the latent heat. Evaporation and condensation is modelled using a customized
version of Fluent’s own model. This mechanism is based on saturation temperature, which is
a function of pressure. Correlation for vapour pressure is presented in Anderson’s paper [1]
from which the saturation pressure is derived. Saturation temperature is presented in
Figure 2.
350
340
Temperature [K]
330
320
310
300
290
280
270
260
250
0
20000
40000
60000
80000 100000 120000 140000 160000
Pressure [Pa]
Figure 2. Saturation temperature of UF6.
Mass transfer between liquid and gaseous phases is defined through source terms in energy
equation which are defined as
𝑚̇lv = coeff ∙ 𝛼l 𝜌l
𝑇l −𝑇sat
𝑇sat
(4)
,
and
𝑚̇vl = coeff ∙ 𝛼v 𝜌v
𝑇v −𝑇sat
𝑇sat
,
(5)
where 𝛼l is liquid volume fraction, 𝜌l is liquid density, 𝑇l is liquid temperature, 𝑇sat is
saturation temperature, 𝛼v is vapor volume fraction, 𝜌v is vapor density and 𝑇v is vapor
temperature. Coefficients coeff in the equations (4) and (5) are considered as relaxation
1
terms which are left to their default values coeff = 0,1 . Latent energy is taken into account
𝑠
by an additional source term in energy equation and is defined as
𝑆 = 𝑚̇ ∙ ∆𝐻v ,
where 𝑚̇ is mass transfer between phases and ∆𝐻v is heat of vaporization.
(6)
2.4 Simulation parameters
Simulation parameters are presented in the Table 4.
Models:
Solver:
Segregated 2D- 1st order explicit
Multiphase:
Volume-of-Fluid, 2 phases
Energy equation:
Enabled
Viscous:
Laminar
Other models:
Solidification/melting, evaporation/condensation
Solution controls:
Pressure-velocity coupling:
PISO
Discretization of momentum and energy:
2nd order upwind
Materials:
UF6 gas, UF6 liquid, UF6 solid
Phase properties:
UF6-gas
UF6-liquid
Pure solvent melting heat (J/kg) 0
54743
Solidus temperature [K]
0
337
Liquidus temperature [K]
0
337
Critical temperature [K]
503,3
0
4610000
0
0,000727
0
Acentric factor [-]
0,09215
0
Speed of sound [m/s]
real-gas-redlich-kwong
-
condensation
evaporation
melting
solidification
Critical pressure [Pa]
Critical specific volume
[m3/kg]
Phase interactions:
Mass:
Operating conditions:
Op. Pressure [Pa]:
Gravity
Floating (152 kPa initial)
[m/s2]:
-9,81 in y-direction
Op. temperature [K]:
Spec. Op. Density
[kg/m3]
337
0
Boundary conditions
wall'
Cylinder wall
Free stream temperature [K]:
Convection heat transfer coefficient [W/m2K]:
393
32
Wall conductivity [W/mK]:
43
Material:
Carbon Steel
Wall thickness [m]:
0,008
default-interior'
Contents of the cylinder
UDFs
UDF for densities in in each phase, UDF for viscosities in
each phase, UDF for thermal conductivities in each phase,
UDF for specific heats in each phase, UDF for evaporation
and condensation
Table 4. Simulation parameters.
2.5 Simulated case
The melting phase was chosen to be simulated. The cylinder is heated with air that is kept
constant at 120 °C. Airflow is kept turbulent with an electric fan and a constant heat transfer
coefficient outside the cylinder is assumed. The temperature inside the cylinder was set to
63 °C and the gas phase pressure was set to 152 kPa. Melting will begin under these
conditions. Conditions at the beginning of the simulation are presented in the Figure 3.
Figure 3. Situation at beginning of the simulation. Red colour represents solid phase and
blue represents gas phase.
3. Results
Amount of solid matter in the
cylinder [%]
Results clearly show that the model works as intended. Pressure increase inside the cylinder
is caused by sudden drop in density of UF6 during melting which leads to volume increase.
Liquid level rises and the gas phase is compressed to the top of the cylinder. The duration for
all the solid UF6 to melt was approximately 16 hours. The melting process is presented in
Figure 4 and density profiles from data points A, B and C are presented in the Figure 5.
100,00%
A
80,00%
B
60,00%
40,00%
C
20,00%
0,00%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Time [h]
Figure 4. Amount of solid matter inside of the cylinder as the simulation progresses.
Figure 5. Density profile inside the cylinder.
Gas phase pressure is presented in the Figure 6 and mass transfer between phases in the
Figure 7. The increase in pressure in the beginning of the simulation is caused by
evaporating liquid. After the pressure reaches a point where saturation temperature is higher
than gas temperature, the gas begins to condense back to the liquid reducing the pressure.
When adequate amount of solid has melted, the liquid level starts to rise. This reduces the
volume available for the gas and causes the gas to compress in the void space of the
cylinder.
6
Pressure [bar]
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17
Time [h]
Figure 6. Gas phase pressure.
0,300
Mass transfer [g/s]
0,250
0,200
0,150
0,100
0,050
0,000
-0,050 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17
-0,100
-0,150
-0,200
Time [h]
Figure 7. Mass transfer between liquid and gaseous phases. Negative value represents
evaporation and positive value means condensation.
Another interesting phenomenon inside the cylinder is the natural convection. The effect of
natural convection is important when homogenizing the liquid UF6, because it mixes the
liquid evenly inside the cylinder. The sample extracted is reliable representation of the
contents when the liquid is completely homogenized. This phenomenon has never been
visually observed. Currents on the edge of the cylinder are presented in the Figure 8.
Figure 8. Natural convection currents on the edge of the cylinder.
4. Comparison to analytical solution
Simple analytical solution was calculated to compare the simulation results. In the solution it
was assumed that heat transfer area of liquid UF6 varies from 60 % to 90 % of the total
cylinder area (liquid level rise). Heat transfer area from liquid to solid was assumed to vary
from 60 % to 0 % of total cylinder area (melting reduces the area). Total resistance to heat
transfer is defined as
𝑅tot = ℎ
1
a 𝐴cyl
+𝑘
𝐿
cyl 𝐴cyl
+ℎ
1
UF6 𝐴L
+ℎ
1
UF6 𝐴S
,
(7)
where ℎa is convective heat transfer coefficient of air, 𝐴cyl is area of the cylinder, 𝑘cyl
thermal conductivity of the cylinder, ℎUF6 is the convective heat transfer coefficient of UF6, 𝐴L
is the contact area of liquid to the cylinder wall and 𝐴S is the contact area of the solid lump to
the liquid. Convective heat transfer coefficient of UF6 was estimated with correlations
presented by Anderson [3].
The average heating power is defined as
∆𝑇
𝑃=𝑅
tot
(8)
.
Total heat to be applied to the solid UF6 can be calculated from
(9)
𝑄 = 𝑚 ∙ ℎf ,
where 𝑚 is the mass and ℎf is the heat of fusion. Total time for the solid to melt is defined as
𝑄
(10)
𝑡=𝑃.
Results from simulation and analytical solution are presented in the Table 5.
Analytical solution
Numerical solution
Difference
Difference %
Average
Heating
Power
4236 W
3946 W
-290 W
-6,85 %
Melting
Time
765 min
955 min
190 min
24,84 %
Table 5. Comparison between analytical and numerical solutions.
Simulation results are commensurate with the analytical solution. Simulation yields slightly
lower average heating power than the analytical solution. This results from the overestimated
heat transfer areas in analytical solution. Simulation yields 25 % higher duration for the solid
to melt. This can be explained by the assumption in analytical solution that all the heat
applied to the cylinder is consumed by the melting process (i.e., average temperature inside
the cylinder is constant). Average temperature inside the cylinder during the simulation is
presented in the Figure 9. The graph clearly shows that temperature begins to rise implying
that some of the heat is “wasted” as it is consumed by heating the liquid rather than melting
the solid.
Temperature [°C]
66,00
65,50
65,00
64,50
64,00
63,50
0
2
4
6
8
10
12
14
Time [h]
Figure 9. Average temperature inside the cylinder.
5. Summary & discussion
Numerical approach to phase change simulations is a viable option to obtain accurate results
of the phase change simulations. Even though the model could not be yet validated due the
lack of measurement data, the simulation works as expected and the results of the simulation
were commensurate with an analytical solution.
The model still needs some fine tuning and reprogramming. For example sublimation
mechanism is vital for the model to be useful in real life applications. This model could be
used to simulate UF6 behaviour in handling equipment such as transfer pipelines and feeding
stations or it could be used to optimize heat transfer and thus reducing duration of the
heating and cooling cycles.
References
[1] J. C. Anderson, C. P Kerr and W. R. Williams, “Correlation of the thermophysical
properties of uranium hexafluoride over a wide range of temperature and pressure.”, 1994
[2] Ansys Inc., Ansys Fluent Theory Guide, Canonsburg, p. 540, 2012
[3] J. C. Anderson, “Correlation of heat transfer in a cylinder containing uranium hexafluoride
engulfed in fire.”, Oak Ridge National Lab, p. 9-10, 1996