CFD-SIMULATION OF URANIUM HEXAFLUORIDE DURING PHASE CHANGE ABSTRACT

Transcription

1 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 (UF 6) 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. UF 6 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 UF 6 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 UF 6are 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 UF 6 feature to Ansys Fluent thermo-physical properties of the media had to be defined. Anderson et al. (1994) present correlations for UF 6 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 UF 6. These properties are presented in Tables 1, 2 and 3.

2 Specific heat (C p) 0, T 3 0, T 2 + J/kg 1, T + 128, Thermal conductivity (k) 0, T 3 0, T 2 + W/mK 0, T 0, Dynamic viscosity (µ) -0, T 2 + 0, T kg/ms 0, Heat of vaporization (H v) J/kg Density (ρ) real-gas-redlich-kwong equation of state kg/m 3 Table 1. Thermo-physical properties of gaseous UF6 Specific heat (C p) 421,6603 J/kgK Thermal conductivity (k) -0, T3 + 0, T2 W/mK 0, T + 0, Dynamic viscosity (µ) -0, T3 + 0, T2 kg/ms 0, T + 0, Heat of fusion (H f) J/kg Density (ρ) -0, T3 + 0, T2-74, T , kg/m3 Table 2. Thermo-physical properties of liquid UF6 Specific heat (C p) 492,9424 J/kgK Thermal conductivity (k) -3,645e-2 + 1,895e-3 T W/mK Dynamic viscosity (µ) - kg/ms Heat of fusion (H f) J/kg Density (ρ) kg/m3 Table 3. Thermo-physical properties of solid UF6 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 UF 6 cylinder, which is converted into a mesh that contains elements. The mesh is presented in the figure 1.

3 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 UF 6. This model is based on melting temperature only and this creates a problem, because UF 6 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 H = h + H, (1) where the enthalpy h is defined as h = h ref + c p dt. T ref T (2) Liquid fraction in the domain is defined as β = 0, if T < T melt, β = 1, if T > T melt, where T melt is the melting temperature of the UF 6. Latent heat content H in the equation (1) is defined as H = βl, (3)

4 Temperature [K] 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 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 m lv = coeff α l ρ l T l T sat T sat, (4) and m vl = coeff α v ρ v T v T sat T sat, (5) where α l is liquid volume fraction, ρ l is liquid density, T l is liquid temperature, T sat is saturation temperature, α v is vapor volume fraction, ρ v is vapor density and T v is vapor temperature. Coefficients coeff in the equations (4) and (5) are considered as relaxation terms which are left to their default values coeff = 0,1 1. Latent energy is taken into account s by an additional source term in energy equation and is defined as S = m H v, (6) where m is mass transfer between phases and H v is heat of vaporization.

5 2.4 Simulation parameters Simulation parameters are presented in the Table 4. Models: Solver: Multiphase: Energy equation: Viscous: Other models: Segregated 2D- 1st order explicit Volume-of-Fluid, 2 phases Enabled Laminar Solidification/melting, evaporation/condensation Solution controls: Pressure-velocity coupling: PISO Materials: Discretization of momentum and energy: UF6 gas, UF6 liquid, UF6 solid 2nd order upwind Phase properties: UF6-gas UF6-liquid Pure solvent melting heat (J/kg) Solidus temperature [K] Liquidus temperature [K] Critical temperature [K] 503,3 0 Critical pressure [Pa] Critical specific volume [m 3 /kg] 0, Acentric factor [-] 0, Speed of sound [m/s] real-gas-redlich-kwong - Phase interactions: Mass: condensation evaporation melting solidification Operating conditions: Op. Pressure [Pa]: Gravity [m/s 2 ]: Op. temperature [K]: 337 Spec. Op. Density [kg/m 3 ] 0 Boundary conditions wall' Floating (152 kpa initial) -9,81 in y-direction Cylinder wall Free stream temperature [K]: Convection heat transfer coefficient [W/m 2 K]: default-interior' UDFs Wall conductivity [W/mK]: 43 Material: Wall thickness [m]: 0,008 Contents of the cylinder 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 Carbon Steel Table 4. Simulation parameters.

6 Amount of solid matter in the cylinder [%] 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 Results clearly show that the model works as intended. Pressure increase inside the cylinder is caused by sudden drop in density of UF 6 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 UF 6 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 ,00% A 80,00% 60,00% B 40,00% 20,00% C 0,00% Time [h] Figure 4. Amount of solid matter inside of the cylinder as the simulation progresses.

7 Pressure [bar] 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 Time [h] Figure 6. Gas phase pressure.

8 Mass transfer [g/s] 0,300 0,250 0,200 0,150 0,100 0,050 0,000-0,050-0,100-0,150-0, 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 UF 6, 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 UF 6 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 R tot = 1 + L h a A cyl k cyl A cyl h UF6 A L h UF6 A S, (7)

9 where h a is convective heat transfer coefficient of air, A cyl is area of the cylinder, k cyl thermal conductivity of the cylinder, h UF6 is the convective heat transfer coefficient of UF 6, A L is the contact area of liquid to the cylinder wall and A S is the contact area of the solid lump to the liquid. Convective heat transfer coefficient of UF 6 was estimated with correlations presented by Anderson [3]. The average heating power is defined as P = T R tot. (8) Total heat to be applied to the solid UF 6 can be calculated from Q = m h f, (9) where m is the mass and h f is the heat of fusion. Total time for the solid to melt is defined as t = Q P. (10) Results from simulation and analytical solution are presented in the Table 5. Average Heating Power Melting Time Analytical solution Numerical solution Difference Difference % 4236 W 3946 W -290 W -6,85 % 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.

10 Temperature [ C] 66,00 65,50 65,00 64,50 64,00 63, Time [h] 5. Summary & discussion Figure 9. Average temperature inside the cylinder. 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 UF 6 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