Method for predicting uranium dendritic crystal growth in uranium electrolytic refining process based on phase field method

Abstract

The invention discloses a uranium dendritic crystal growth prediction method in a uranium electrolytic refining process based on a phase field method, and belongs to the field of spent fuel aftertreatment. Establishing a phase field model for predicting uranium dendritic crystal growth according to a phase field control equation derived by electrochemical reaction thermodynamics and a free energy function; performing dimensionless processing on the phase field control equation, and inputting the equation into simulation software; determining a phase field simulation domain, performing grid division, and setting an initial crystal nucleus position; setting uranium dendritic crystal phase field model parameters, boundary conditions and solving conditions, determining a specific mathematical relationship between the phase field model parameters and macroscopic measurable parameters through an analytical method, and solving the input control equation by using software; and outputting a result, and carrying out visual treatment to obtain the uranium dendritic crystal growth morphology in the uranium electrolytic refining process. According to the method, uranium dendritic crystal growth prediction under different process parameters such as constant potential, constant current and pulse electrolysis can be achieved, the growth mechanism of uranium dendritic crystals can be comprehensively understood, and time cost and capital investment are saved.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of spent fuel reprocessing, and particularly relates to a method for predicting uranium dendrite growth in a uranium electrolytic refining process based on a phase field method. BACKGROUND

[0002] As a key link in the development of integrated fast reactors and ADS transmutation technology, dry reprocessing technology not only has good irradiation tolerance, but also has a simple process and compact operation. As a core link in the dry reprocessing process of integrated fast reactors, uranium ions are reduced to metal in the solid cathode during the operation of the molten salt electrolytic refining process. The morphology is mostly loose dendritic, which not only reduces the efficiency of uranium separation, but also causes short circuits and other problems. Therefore, in order to ensure the long-term stable and efficient operation of the uranium electrolytic refining system, it is particularly important to study the mechanism of uranium dendrite growth and accurately predict the morphology of uranium dendrites.

[0003] In recent years, researchers have carried out a large number of experimental studies on the growth of uranium dendrites in molten salt electrolytic refining, and have systematically explored the influence of key parameters such as molten salt composition, electrode material, temperature regulation and electrolysis method on the growth of uranium dendrites and the change of its morphology. However, due to the limitations of experimental methods, the morphology of uranium dendrites under specific conditions can only be presented, and it is difficult to deeply reveal the internal mechanism of the growth of uranium dendrites and the change of its morphology. At the same time, due to the constraints of manpower, material resources and time cost, experimental research can not comprehensively cover all operating conditions, and it is difficult to systematically analyze the growth of uranium dendrites under complex and variable process conditions. Therefore, by means of numerical simulation technology, the growth process of uranium dendrites is systematically studied, which realizes the accurate prediction of the growth of uranium dendrites and the change of its morphology under different process parameters, and provides a theoretical basis for optimizing the process parameters of electrolytic refining and regulating the morphology of uranium deposits.

[0004] At present, although the uranium electrolytic refining models developed worldwide play a certain role in engineering parameter optimization, equipment design and other practical applications, they have significant limitations in simulating the growth behavior of uranium dendrites. These models mostly rely on the diffusion control assumption, which seriously ignores the dynamic changes of electrode reaction kinetics in the electrolytic refining process, and cannot restore the true state of key processes such as material migration. In addition, the existing models generally use sharp interface methods, which can only describe the macroscopic geometric characteristics of the interface, and cannot capture the complex fractal structure, competitive growth of dendrite arms and other microphenomena in the dendrite growth process, and restore the dynamic process of dendrite growth. Therefore, they are not suitable for predicting the growth of uranium dendrites in the process of uranium electrolytic refining, and can only be used as an auxiliary means. SUMMARY

[0005] The present application provides a method for predicting the growth of uranium dendrites in the process of uranium electrolytic refining based on the phase field method, in order to solve the technical problems of low calculation accuracy and inefficient simulation of uranium dendrite growth by existing mathematical models.

[0006] This invention provides a method for predicting uranium dendrite growth in uranium electrolytic refining processes based on the phase-field method, comprising the following steps:

[0007] Step 1: Establish a phase-field model for predicting uranium dendrite growth based on the phase-field control equation;

[0008] Step 2: Dimensionlessize the phase field control equations and input them into the simulation software;

[0009] Step 3: Determine the phase-field simulation domain and perform mesh generation, and set the initial nucleus positions;

[0010] Step 4: Set the phase-field model parameters, boundary conditions, and solution conditions, and use simulation software to perform transient solutions on the input control equations;

[0011] Step 5: Output the calculation results and visualize them to obtain the uranium dendrite growth morphology in the uranium electrolytic refining process.

[0012] Furthermore, the phase field control equations include the order parameter evolution equation, the uranium ion diffusion equation, and the potential distribution equation.

[0013] Furthermore, the order parameter evolution equation is:

[0014]

[0015] In the formula: ξ represents the order parameter; t represents the system evolution time; L ξ H represents the interface mobility; H represents the potential barrier; g(ξ) represents the double-well potential function, g(ξ) = Hξ 2 (1-ξ) 2 h(ξ) represents the interpolation function, h(ξ) = ξ 3 (6ξ 2 -15ξ+10); F represents the Faraday constant; η represents the overpotential; α a and α c Let be the charge transfer coefficients for the anodic and cathodic reactions, respectively, satisfying α. a +α c =1; ε ξ Represents the gradient coefficients. The gradient energy coefficient ε ξ The average value of δ represents the anisotropy intensity, ω represents the anisotropy modulus, θ represents the angle between the interface outward normal vector v and the reference axis; R represents the gas constant; T represents the molten salt temperature. This represents the dimensionless concentration of U(III).

[0016] Furthermore, the uranium ion diffusion equation is as follows:

[0017]

[0018] wherein: represents the Laplace operator; D eff represents the effective diffusion coefficient of U(III) in the solid phase, D eff = D s represents the effective diffusion coefficient of U(III) in the liquid phase, D eff = D l represents the effective diffusion coefficient of U(III) at the interface between the solid and liquid phases, D eff (ξ) = D s h(ξ) + D l (1 - h(ξ)), D s and D l respectively represent the diffusion coefficients of U(III) in the solid and liquid phases; c m represents the inverse of the molar volume of uranium metal; c bulk represents the concentration of uranium atoms in the uranium metal.

[0019] Further, the potential distribution equation is:

[0020]

[0021] wherein: σ eff represents the electrical conductivity, σ eff = σ s represents the electrical conductivity in the liquid phase, σ eff = σ1, σ eff (ξ) = σ s h(ξ) + σ l (1 - h(ξ)), σ s and σ l respectively represent the electrical conductivities of the solid and liquid phases; represents the potential gradient, wherein the interface between the solid and liquid phases is the electrode / electrolyte interface; ξ(r, t) represents the order parameter related to position and time.

[0022] Further, in step 2, the dimensionless processing characteristic values include length, time and energy density; the simulation software uses COMSOL software; and the control equation in the dimensionless form is represented as:

[0023]

[0024]

[0025] Further, in step 3, the phase field simulation domain is selected as a square unit representing the morphology properties and characteristics of the real uranium dendrite; the grid type of the grid division is a mapped grid; and the coordinates of the initial crystal nucleus position are (0, y0), and the initial crystal nucleus radius is r0.

[0026] Further, in the step 4, the phase field model parameters include macroscopically measurable parameters and unknown parameters; the macroscopically measurable parameters include uranium physical parameters, molten salt physical parameters, exchange current density, interface anisotropy strength, interface thickness and interface energy; the unknown parameters include gradient coefficient, potential barrier and reaction constant, and the mathematical relationship between the unknown parameters and the macroscopically measurable parameters is determined by an analytical method.

[0027] Further, the gradient coefficient ε ξ is:

[0028]

[0029] In the formula, γ represents the interface energy; W represents the interface width;

[0030] The potential barrier H is:

[0031]

[0032] The reaction constant L η is:

[0033]

[0034] In the formula, κ is a sharp interface kinetic coefficient.

[0035] Further, the boundary conditions are phase field variable boundary conditions, uranium ion concentration boundary conditions and electric potential field boundary conditions.

[0036] The present application has the beneficial effects that:

[0037] 1. The present application adopts a prediction system constructed by a nonlinear continuous order parameter ξ to accurately describe the growth morphology of uranium dendrites under a molten salt system, significantly improves the authenticity and accuracy of uranium dendrite growth prediction under various process conditions, effectively avoids the possible deviation of the traditional description method, and provides a more reliable theoretical basis for the optimization and control of related processes. At the same time, the specific mathematical relationship between the phase field model parameters and the macroscopically measurable parameters is systematically derived and established by an analytical method, solving the problem of lack of clear correlation between the model parameters and the macroscopically measurable parameters in the application of traditional phase field model in the field of electrochemistry, which can improve the accuracy and reliability of the phase field model prediction.

[0038] 2.The application applies the phase field model developed to the prediction of uranium dendrite growth in the uranium electrolytic refining process, breaks through the technical bottleneck of the difficulty in predicting the growth and morphology evolution of dendrites in the traditional method, realizes the prediction of uranium dendrite growth under different process parameters such as constant potential, constant current and pulse electrolysis, and helps to comprehensively understand the growth mechanism of uranium dendrites. The phase field model developed by the application not only greatly improves the accuracy and reliability of the prediction of uranium dendrite growth, but also provides stronger tool support for related industrial production, process improvement and scientific research, and helps to promote the further development and innovation of the technology in this field. The method of the application replaces the traditional trial and error experiment, greatly shortens the process development cycle, and saves time cost and capital investment. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 FIG. 1 is a flowchart of the method for predicting the growth of uranium dendrites in the uranium electrolytic refining process based on the phase field method of the application;

[0040] Figure 2 FIG. 4 is a comparison diagram of the simulation results and experimental results of the growth process of uranium dendrites in the process of constant potential electrodeposition of U(III) in the LiCl-KCl molten salt system in the embodiment of the application;

[0041] Figure 3 FIG. 5 is a comparison diagram of the simulation results and experimental results of the growth process of uranium dendrites in the process of pulse potential electrodeposition of U(III) in the LiCl-KCl molten salt system in the embodiment of the application. DETAILED DESCRIPTION

[0042] In order to more clearly illustrate the purpose, technical scheme and obvious advantages of the application, the application will be described in detail below in combination with the drawings and specific implementation examples. It should be emphasized that the description of these examples is intended to illustrate the application and does not constitute a limitation on the application.

[0043] The application discloses a method for predicting the growth of uranium dendrites in the uranium electrolytic refining process based on the phase field method, as shown in FIG. 1, which comprises the following steps. Figure 1

[0044] Step 1: According to the electrochemical reaction thermodynamics and the change rule of Gibbs free energy, a phase field model for predicting the growth of uranium dendrites is established, which comprises three groups of control equations, i.e., a sequence parameter evolution equation, a uranium ion diffusion equation and a potential distribution equation.

[0045] The sequence parameter evolution equation is described by the following formula:

[0046]

[0047] In the formula, ξ represents the sequence parameter; t represents the evolution time of the system; L represents the length of the sequence parameter; and f represents the function of the sequence parameter. η ​represents the reaction constant; α a and α c are the charge transfer coefficients of the anodic and cathodic reactions, respectively, satisfying α a + α c = 1; δ represents the anisotropy strength; G represents the Gibbs free energy; R represents the gas constant; T represents the molten salt temperature; c m represents the reciprocal of the molar volume of uranium metal; represents the dimensionless concentration of U(III).

[0048] It is assumed that the interface migration velocity is linearly related to the interface free energy reduction, but exponentially related to the overpotential. The order parameter evolution equation is further rewritten as:

[0049]

[0050]

[0051] In the formula, ξ represents the order parameter; t represents the system evolution time; L ξ represents the interface mobility; F grad represents the gradient energy density; g(ξ) represents the double-well potential function, g(ξ) = Hξ 2 (1-ξ) 2 ; h(ξ) represents the interpolation function, h(ξ) = ξ 3 (6ξ 2 -15ξ+10); F represents the Faraday constant; f nos (ξ) represents the interface noise, f nos (ξ) = h'(ξ)χψ, χ represents a random number, and ψ represents the noise fluctuation amplitude; L η represents the reaction constant; η represents the overpotential; α a and α c are the charge transfer coefficients of the anodic and cathodic reactions, respectively, satisfying α a + α c = 1; ε ξ represents the gradient coefficient, represents the average value of the gradient energy coefficient ε ξ ; δ represents the anisotropy strength; ω represents the anisotropy modulus; θ represents the included angle between the interface outward normal vector v and the reference axis; V represents the volume of the integral region; represents the Laplace operator.

[0052] The order parameter evolution equation is further simplified as:

[0053]

[0054] In the formula, ξ represents the order parameter; t represents the system evolution time; Lξ denotes the interfacial mobility; H denotes the potential barrier; g(ξ) denotes the double-well potential function, g(ξ) = Hξ 2 (1 - ξ) 2 ; h(ξ) denotes the interpolation function, h(ξ) = ξ 3 (6ξ 2 - 15ξ + 10); F denotes the Faraday constant; η denotes the overpotential; a a and a c are the charge transfer coefficients for the anodic and cathodic reactions, respectively, satisfying a a + a c = 1; ε ξ denotes the gradient coefficient, denotes the average value of the gradient energy coefficient ε ξ ; δ denotes the anisotropy strength; ω denotes the anisotropy modulus; θ denotes the angle between the outward normal vector v of the interface and the reference axis.

[0055] The uranium ion diffusion equation is described by the following formula:

[0056]

[0057] where: denotes the dimensionless concentration of U(III); t denotes the system evolution time; denotes the Laplace operator; D eff denotes the effective diffusion coefficient of U(III), D eff (ξ) = D s h(ξ) + D l (1 - h(ξ)); D s and D l denote the diffusion coefficients of U(III) in the solid and liquid phases, respectively, D eff = D s in the solid phase and D eff = D l in the liquid phase, at the interface between the solid and liquid phases, D eff is the interpolation of both; F denotes the Faraday constant; R denotes the gas constant; T denotes the temperature of the molten salt; φ denotes the electric potential; R U(III) denotes the reaction source term, where the interface between the solid and liquid phases is the electrode / electrolyte interface.

[0058] Further, the mass transfer of U(III) is simplified as:

[0059]

[0060] where: denotes the dimensionless concentration of U(III); t denotes the system evolution time; represents Laplace operator; D eff represents effective diffusion coefficient of U(III) in solid phase, D eff = D s represents effective diffusion coefficient of U(III) in liquid phase, D eff = D l represents effective diffusion coefficient of U(III) at interface between solid phase and liquid phase, D eff (ξ) = D s h(ξ) + D l (1-h(ξ)), D s and D l respectively represent diffusion coefficient of U(III) in solid phase and liquid phase; c m represents the reciprocal of molar volume of uranium metal; c bulk represents uranium atom concentration in uranium metal; ξ represents order parameter.

[0061] The potential distribution equation is described by the following formula:

[0062]

[0063] In the formula: σ eff represents conductivity, σ eff = σ s represents conductivity in liquid phase, σ eff = σ1, σ eff (ξ) = σ s h(ξ) + σ l (1-h(ξ)), σ s and σ l respectively represent conductivity of solid phase and liquid phase; represents potential gradient, wherein the interface between solid phase and liquid phase is electrode / electrolyte interface; F represents Faraday constant; c m represents the reciprocal of molar volume of uranium metal; t represents system evolution time; ξ represents order parameter; ξ(r,t) represents order parameter related to position and time.

[0064] Step 2: The obtained control equation is dimensionless, and then the dimensionless control equation is input into COMSOL software, and the control equation is solved by the mathematical module of COMSOL software;

[0065] The dimensionless form of the control equation is:

[0066]

[0067] Step 3: According to the uranium dendrite morphology properties and characteristics obtained from the experimental results, the simulation domain is set to 1*1mm 2A square, where the left and right boundaries represent the cathode surface and the electrolyte surface, respectively; at the same time, one or more semi-circular initial crystal nuclei are set on the left boundary of the simulation domain, which will evolve into dendrites during the electrodeposition process;

[0068] Step 4: Solve for the required parameters based on the governing equations. The phase-field model parameters include uranium physical property parameters, molten salt physical property parameters, exchange current density, interfacial anisotropy intensity, interfacial thickness, and interfacial energy. The gradient coefficient, potential barrier, and reaction constant in the process of solving the governing equations are obtained analytically.

[0069] The gradient coefficient is calculated using the following formula:

[0070]

[0071] Where: ε ξ γ represents the gradient coefficient; W represents the interface energy; and W represents the interface width.

[0072] The potential barrier is calculated using the following formula:

[0073]

[0074] In the formula: H represents the potential barrier; γ represents the interface energy; W represents the interface width.

[0075] The reaction constant L η The calculation is performed using the following formula:

[0076]

[0077] In the formula: κ is the sharp interface dynamic coefficient; γ represents the interface energy.

[0078] The boundary conditions are set as phase field variable boundary conditions, uranium ion concentration boundary conditions, and electric potential field boundary conditions; the boundary conditions are set as shown in Table 1.

[0079] Table 1 Boundary conditions for the uranium dendrite phase field model

[0080]

[0081] The solution conditions are set using a fully coupled computational method based on Newton-Raphson iteration.

[0082] Step 5: Perform simulation calculations to obtain the calculation results, and then perform visualization processing to obtain the growth morphology of uranium dendrites.

[0083] Numerical simulations were performed on the growth process of uranium dendrites during constant potential electrodeposition of U(III) and pulsed potential electrodeposition of U(III) in the LiCl-KCl molten salt system, enabling the prediction of uranium dendrite growth under different process parameters.

[0084] Experimental results and phase field simulation results after constant potential electrodeposition for 560 s at different cathode potentials, such as Figure 2 As shown. When At that time, both experimental and simulated dendrites exhibited a needle-like structure; when At this point, a small number of secondary dendrites appear on the dendrite surface. When the cathode potential increases to -0.10V, the number of secondary dendrites increases significantly. As the length of the secondary dendrites increases, tertiary dendrites appear, forming a typical tree-like structure.

[0085] Experimental results and phase-field simulation results after pulse potential electrodeposition for 560 s under different process conditions, such as Figure 3 As shown, the dendrite types are unbranched dendrites, regularly branched dendrites, semi-unbranched dendrites, and locally networked dendrites.

[0086] It is evident that the phase-field model proposed in this application can correctly couple the growth of uranium dendrites with the concentration and potential of U(III), thereby enabling the prediction of uranium dendrite growth during uranium electrolytic refining under different process parameters.

[0087] In summary, this paper elucidates a method for predicting uranium dendrite growth in the uranium electrolytic refining process based on the phase-field method, using a specific case study. The accuracy of the model is verified by comparing simulation and experimental results, demonstrating the reliability of the method. This invention uses a phase-field model to predict uranium dendrite growth during uranium electrolytic refining under different process parameters. Employing this method contributes to a comprehensive understanding of the uranium dendrite growth mechanism, saving time and financial investment.

[0088] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for predicting dendrite growth of uranium in a phase-field method based process for the electrorefining of uranium, characterized in that: The method comprises the following steps: Step 1: establishing a phase field model for predicting growth of uranium dendrites according to a phase field control equation; Step 2: performing dimensionless processing on the phase field control equation and inputting the equation into simulation software; Step 3: determining a phase field simulation domain and performing grid division, and setting an initial crystal nucleus position; Step 4: setting phase field model parameters, boundary conditions and solution conditions, and performing transient solution on the input control equation by using the simulation software; Step 5: outputting calculation results, and obtaining a uranium dendrite growth morphology in a uranium electrolytic refining process through visual processing.

2. The method of claim 1, wherein the method is used in a process for the prediction of uranium dendrite growth in the electrolytic refining of uranium based on the phase field method, characterized in that: The phase field control equation comprises an order parameter evolution equation, a uranium ion diffusion equation and a potential distribution equation.

3. The method of claim 2, wherein the method is characterized by: The order parameter evolution equation is: where: ξ represents the order parameter; t represents the system evolution time; L ξ represents the interfacial mobility; H represents the potential barrier; g(ξ) represents the double-well potential function, g(ξ) = Hξ 2 (1 - ξ) 2 ; h(ξ) represents the interpolation function, h(ξ) = ξ 3 (6ξ 2 - 15ξ + 10); F represents the Faraday constant; η represents overpotential; α a and α c are charge transfer coefficients of anode reaction and cathode reaction, respectively, satisfying α a + α c = 1; ε ξ denotes the gradient coefficient, denotes the gradient energy coefficient ε ξ denotes the average value, δ denotes the anisotropy strength, ω denotes the anisotropy modulus, θ denotes the angle between the interface outward normal vector v and the reference axis; R denotes the gas constant; T denotes the molten salt temperature; denotes the dimensionless concentration of U(III).

4. The method of claim 2, wherein the method is characterized by: The uranium ion diffusion equation is: where: represents the Laplace operator; D eff represents the effective diffusion coefficient of U(III) in the solid phase, D eff = D s represents the effective diffusion coefficient of U(III) in the liquid phase, D eff = D l represents the effective diffusion coefficient of U(III) at the interface between the solid and liquid phases, D eff (ξ) = D s h(ξ) + D l (1 - h(ξ)), D s and D l represent the diffusion coefficients of U(III) in the solid and liquid phases, respectively; c m represents the inverse of the molar volume of the uranium metal; c bulk represents the concentration of uranium atoms in the uranium metal.

5. The method of claim 2, wherein the method is characterized by: The potential distribution equation is: where σ eff represents the electrical conductivity, σ eff = σ s represents the electrical conductivity in the liquid phase, σ eff = σ1represents the electrical conductivity at the interface between the solid and liquid phases, σ eff (ξ) = σ s h(ξ) + σ l (1 - h(ξ)), σ s and σ l represent the electrical conductivities of the solid and liquid phases, respectively; represents the potential gradient, where the interface between the solid and liquid phases is the electrode / electrolyte interface; and ξ(r, t) represents a order parameter dependent on position and time.

6. The method of claim 1, wherein the method is characterized by: In the step 2, characteristic values of the dimensionless processing include length, time and energy density; the simulation software adopts COMSOL software; and the dimensionless form of the control equation is represented as:

7. The method of claim 1, wherein the method is characterized by: In the step 3, the phase field simulation domain is selected as a square unit representing properties and characteristics of a real uranium dendrite morphology; a grid type of the grid division is a mapping grid; and coordinates of the initial crystal nucleus position are (0, y0), and an initial crystal nucleus radius is r0.

8. The method for predicting uranium dendrite growth in uranium electrolytic refining process based on phase-field method according to claim 1, characterized in that: In the step 4, the phase field model parameters include macroscopically measurable parameters and unknown parameters; the macroscopically measurable parameters include uranium physical parameters, molten salt physical parameters, exchange current density, interface anisotropy strength, interface thickness and interface energy; the unknown parameters include gradient coefficients, potential barriers and reaction constants, and a mathematical relationship between the unknown parameters and the macroscopically measurable parameters is determined through an analytical method.

9. The method of claim 8, wherein the method is characterized by: The gradient coefficient ε ξ is: In the formula, γ represents interface energy; and W represents interface width. The potential barrier H is: The reaction constant L η is: In the formula, κ is a sharp interface kinetic coefficient.

10. The method of claim 1, wherein the method of predicting dendrite growth in a phase- field method based electrorefining process of uranium is characterized by: The boundary conditions are set as phase field variable boundary conditions, uranium ion concentration boundary conditions and electric potential field boundary conditions.

Images