A method for calculating the permeability of the mushy zone during alloy solidification

Abstract

The present invention discloses a method for calculating the permeability of the mushy zone during alloy solidification, comprising the following steps: 1) obtaining the physical property parameters of the actual alloy material first; 2) establishing a phase field model for the directional growth process of multiple dendrites during the alloy solidification process based on the Ginzburg-Landau theory; 3) simulating the microstructure evolution process under different undercooling conditions by using the established multiple-dendrite phase field model; 4) calculating the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained by simulation by using the fractal theory; 5) calculating the permeability K according to Darcy's law by using the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained. The method of the present invention is based on a multi-phase field numerical model combined with the fractal theory, effectively realizing the prediction of the permeability of the alloy solidification mushy zone, avoiding the deviation introduced by using empirical formulas, and laying an important foundation for the accurate description of the liquid phase flow between dendrites during the alloy solidification process.

Description

Technical Field

[0001] The present invention belongs to the technical field of alloy casting, and particularly relates to a method for calculating the permeability of the mushy zone during alloy solidification. Background Art

[0002] Solidification is the basic link in metal forming and has an important impact on the microstructure and properties of subsequent material processing and products. Among them, nucleation and dendrite growth are the sources of the solidification process and determine the formation and evolution of the as-cast microstructure. Due to the complex physical processes involved, the development of quantitative research on the as-cast microstructure during the solidification process has always been a frontier issue in materials science. Most metal materials are alloys composed of different components. Due to the change in solubility during the solidification process, microsegregation of alloy components in the matrix is inevitable, which is also the reason for the uneven composition and even tissue deviation of as-cast products. The redistribution of solute elements at the solid-liquid interface causes their uneven distribution inside and at the grain boundaries of the grains, which in turn leads to microsegregation and dendritic growth.

[0003] As a typical solidification structure of alloys, dendrites are the basis for the formation of solidification defects. On the one hand, the dendritic network hinders the replenishment of the solidification shrinkage between dendrites by the external molten steel, and on the other hand, it intensifies the suction of solute-rich intergranular matter at the end of solidification, thus promoting the formation and development of intergranular cracks and central segregation. Therefore, the seepage characteristics of the dendritic network are the key link coupling macroscopic transport phenomena and microscopic tissue evolution and revealing the formation mechanism of intergranular cracks and central segregation.

[0004] At present, the permeability of the mushy zone of alloys such as superalloys and steels is mostly determined by the empirical formula of non-ferrous alloys, and its direct determination method has not been reported. In view of the important role of the mushy zone permeability in describing the segregation defects of alloy solidification structures, etc., it is urgent to find a feasible determination method.

[0005] In view of this, the inventor of the present invention proposes a method for calculating the permeability of the mushy zone during alloy solidification to solve the above practical problems. Summary of the Invention

[0006] The purpose of the present invention is to overcome the above-mentioned disadvantages of the prior art and provide a method for calculating the permeability of the mushy zone during alloy solidification. Based on the multi-phase field numerical model combined with the fractal theory, and then according to Darcy's law, the permeability K of the mushy zone during alloy solidification is calculated, and the corresponding relationship with the solidification conditions is established, avoiding the deviation introduced by using the permeability empirical formula of non-ferrous alloys, and laying an important foundation for the accurate description of the liquid phase flow between dendrites during alloy solidification.

[0007] The purpose of the present invention is solved by the following technical solutions:

[0008] A method for calculating the permeability of the mushy zone during alloy solidification, comprising the following steps:

[0009] Step 1: First, obtain the physical property parameters of the actual alloy material;

[0010] Step 2: Then, based on the Ginzburg-Landau theory, establish a phase-field model for the dendritic directional growth process during alloy solidification;

[0011] Step 3: Use the established dendritic phase-field model to simulate the microstructure evolution process under different undercooling conditions;

[0012] Step 4: Use fractal theory to calculate the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained by simulation;

[0013] Step 5: According to Darcy's law, use the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained to calculate the permeability K.

[0014] Furthermore, the physical property parameters of the alloy material in Step 1 include liquidus slope, partition coefficient, solid solute diffusivity, liquid solute diffusivity, anisotropy strength, melting point, interface energy, and molar volume.

[0015] Furthermore, the specific process of establishing the phase-field model in Step 2 is as follows:

[0016] Introduce a continuously varying order parameter - the phase-field variable φ (when φ = 1, it represents the solid phase; when φ = -1 or 0, it represents the liquid phase), and in the solid-liquid two-phase region, φ is a value between 0 and 1;

[0017] The phase-field and concentration control equations including the phase-field parameter φ are as follows:

[0018]

[0019]

[0020] Among them, in the formula, φ is the phase-field variable, c is the concentration, M and ε are phase-field parameters related to the interface characteristics, and D(φ) is the solute diffusion coefficient.

[0021] Furthermore, in Step 4, the box-counting method is used to calculate the fractal dimension D during dendritic growth under different undercooling conditions.

[0022] Furthermore, the specific calculation process is as follows:

[0023] First, divide the dendritic area simulated by the phase-field into boxes with size r, calculate the number of all boxes containing the S / L interface, denoted as N(r);

[0024] Then change the size r of the box and repeat the same steps to obtain different N(r), and get the following relationship for the fractal dimension:

[0025] N(t) = r D (3)

[0026] Wherein, in the formula, D is the fractal dimension, and the value of the fractal dimension D is the ratio of logN(r) to logr.

[0027] Furthermore, in step five, according to Darcy's law, the mushy zone during alloy solidification is regarded as a medium with multiple flow channels, and the expression of the permeability K is as follows:

[0028]

[0029] Wherein, in the formula, g L is the liquid fraction, n is the number of flow channels per unit area, τ is the tortuosity factor. The tortuosity factor τ is introduced to solve the situation where the flow channels are not straight and asymmetric, and it is assumed that the number of channels is equal to the number of regions between dendrite arms. The spacing between channels is equal to the dendrite arm spacing d1, and the expression of the permeability K is as follows:

[0030]

[0031] K = g L 2 d1 2 / 8πτ 3 (6)

[0032] Assume that the tortuosity factor τ corresponds to the complexity of the dendrite morphology, and the fractal dimension D is equal to the tortuosity factor τ. Using the dendrite arm spacing value d1 in the alloy, the permeability K during the microstructure evolution under different undercooling conditions is calculated.

[0033] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0034] The method for calculating the permeability of the mushy zone during alloy solidification according to the present invention first establishes an alloy multi-dendrite growth model based on the multi-phase field theory, dynamically reproduces the dendrite network structure in the mushy zone, solves the problem that it is difficult to quantitatively characterize the mushy zone through experiments due to high temperature and opacity, and describes the complexity of the dendrite network structure through the fractal theory. Combining with Darcy's law (Darcy), the variation law of the permeability of the mushy zone with the liquid fraction is calculated. Based on the multi-phase field numerical model combined with the fractal theory, the present invention effectively realizes the prediction of the permeability of the mushy zone during alloy solidification, establishes the corresponding relationship with the solidification conditions, avoids the deviation introduced by using the empirical formula of the permeability of colored alloys, and lays an important foundation for the accurate description of the liquid flow between dendrites during alloy solidification. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The drawings herein are incorporated into the specification and form a part of the specification, and are used together with the specification to explain the principles of the present invention.

[0036] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required in the description of the embodiments or the prior art. Obviously, for those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0037] Figure 1 It is a flowchart of a method for calculating the permeability of the solidification mushy zone of an alloy according to the present invention;

[0038] Figure 2 It is a schematic diagram of the simulation results of dendritic directional growth under different undercooling conditions in an embodiment of the present invention;

[0039] Figure 3 It is a curve graph showing the change of the permeability of the alloy directional growth network structure with the liquid fraction in the dendritic growth calculation domain under different undercooling conditions in an embodiment of the present invention. Detailed implementation manners

[0040] Here, the exemplary embodiments will be described in detail, and the examples are shown in the drawings. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The implementation manners described in the following exemplary embodiments do not represent all implementation manners consistent with the present invention. On the contrary, they are only examples of devices consistent with some aspects of the present invention detailed in the appended claims.

[0041] To enable those skilled in the art to better understand the technical solutions of the present invention, the present invention will be further described in detail below with reference to the drawings and embodiments.

[0042] See Figure 1 As shown, the present invention provides a method for calculating the permeability of the solidification mushy zone of an alloy, and the method includes the following specific steps:

[0043] Step 1: First, obtain the physical property parameters of the actual alloy material;

[0044] Among them, the physical property parameters of the alloy material include the liquidus slope, partition coefficient, solid solute diffusivity, liquid solute diffusivity, anisotropy strength, melting point, interfacial energy, and molar volume, etc.

[0045] Step 2: Then, based on the Ginzburg-Landau theory, establish a phase field model for the multi-dendritic directional growth process during alloy solidification;

[0046] The specific approach is to introduce a continuously varying order parameter - the phase field variable φ (when φ = 1, it represents the solid phase, and when φ = -1 or 0, it represents the liquid phase), and in the solid-liquid two-phase region, φ is a value between 0 and 1;

[0047] The phase field and concentration control equations containing the phase field parameter φ are as follows:

[0048]

[0049]

[0050] Among them, in the formula, φ is the phase field variable, c is the concentration, M and ε are phase field parameters related to the interface characteristics, and D(φ) is the solute diffusion coefficient.

[0051] Step 3: Use the established multi-dendrite phase field model to simulate the microstructure evolution process under different undercooling conditions;

[0052] Step 4: Use the fractal theory to calculate the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained by simulation;

[0053] Among them, in Step 4, the box-counting method is used to calculate the fractal dimension D in the dendrite growth process under different undercooling conditions.

[0054] The specific calculation process is as follows:

[0055] First, divide the dendrite area simulated by the phase field into boxes with a size of r, and calculate the number of all boxes containing the S / L interface, denoted as N(r);

[0056] Then change the size r of the box and repeat the same steps to obtain different N(r), and obtain the following relational expression for the fractal dimension:

[0057] N(r) = r D (3)

[0058] Among them, in the formula, D is the fractal dimension, and the value of the fractal dimension D is the ratio of logN(r) to logr.

[0059] Step 5: According to Darcy's law, use the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained to calculate the permeability K;

[0060] Specifically, according to Darcy's law, the mushy zone during the alloy solidification process is regarded as a medium with multiple flow channels, and the expression of the permeability K is as follows:

[0061]

[0062] Among them, in the formula, g L is the liquid fraction, n is the number of flow channels per unit area, τ is the tortuosity factor. The tortuosity factor τ is introduced to solve the situation where the flow channels are not straight and asymmetric, and it is assumed that the number of channels is equal to the number of regions between dendrite arms, the spacing between channels is equal to the dendrite arm spacing d1, and the expression of the permeability K is as follows:

[0063]

[0064] K = g L 2 d1 2 / 8πτ 3 (6)

[0065] Assume that the tortuosity factor τ corresponds to the complexity of the dendrite morphology, and the fractal dimension D is equal to the tortuosity factor τ. Using the dendrite arm spacing value d1 in the alloy, the permeability K during the microstructure evolution under different undercooling conditions is calculated.

[0066] To further verify the method of the present invention, the inventors made the following specific embodiments, which take the dendritic directional evolution process of an Al-2%Si binary alloy as an example to illustrate the implementation and effect of the method for calculating the permeability between dendrites in the solidification mushy zone of the alloy.

[0067] This embodiment specifically includes the following steps:

[0068] 1) First, obtain the physical property parameters of the actual alloy material;

[0069] The physical property parameters and simulation parameters of the Al-2%Si binary alloy used in the microstructure model are shown in Table 1. Discretize the time, space, and state of the computational simulation, mainly referring to defining the time step of the simulation, the size of each grid, the number of grids divided, and the state of each grid. Before the simulation calculation, assign an initial value to each grid in the calculation area.

[0070] Table 1 shows the physical property parameters and simulation parameters of the Al-2%Si binary alloy

[0071]

[0072] 2) Then, based on the Ginzburg-Landau theory, establish a phase-field model for the multi-dendritic directional growth process during alloy solidification;

[0073] During the process of constructing the model, introduce a continuously varying order parameter - the phase-field variable φ (when φ = 1, it represents the solid phase; when φ = -1 or 0, it represents the liquid phase). In the solid-liquid two-phase region, φ is a value between 0 and 1;

[0074] The phase-field and concentration control equations including the phase-field parameter φ are as follows:

[0075]

[0076]

[0077] Among them, in the formula, φ is the phase-field variable, c is the concentration, M and ε are phase-field parameters related to the interface characteristics, and D(φ) is the solute diffusion coefficient.

[0078] 3) The established multi-dendrite phase field model is used to simulate the microstructure evolution process under different undercooling conditions;

[0079] The undercooling degrees of ΔT = 23 and 30 K are selected to simulate the grain orientation evolution process; the number of grids in the simulation area is set to 1500×1500, and the grid size Δx = Δy = 1.0×10 -8 m, and three grain orientation growths are set on the right side of the calculation domain.

[0080] Based on the above phase field model, a computer program is written, and the dendrite growth and evolution results can be obtained and exported, that is, the simulation results of dendrite orientation growth under different undercooling conditions as shown in Figure 2 are obtained.

[0081] 4) The fractal dimension D in the microstructure evolution process under different undercooling conditions obtained by simulation is calculated using the fractal theory;

[0082] The box-counting method is used to calculate the fractal dimension D of the dendrite network structure in the results of the dendrite growth process simulated in step 3) under different undercooling conditions. Table 2 shows the fractal dimension and liquid fraction of the dendrite network structure of Al-Si under different undercooling conditions. The specific basic process is as follows:

[0083] First, the dendrite area simulated by the phase field is divided into boxes with a size of r, and the number of all boxes containing the S / L interface is calculated, denoted as N(r); then the box size r is changed, and the same steps are repeated to obtain different N(r), and the following relational expression for the fractal dimension is obtained:

[0084] N(r) = r D (3)

[0085] where D in the formula is the fractal dimension, and its value is the slope of the plotted curve obtained from the ratio of logN(r) to logr.

[0086] Table 2 shows the fractal dimension and liquid fraction of the dendrite network structure of Al-2%Si under different undercooling conditions

[0087]

[0088] 5) According to Darcy's law, the permeability K is calculated using the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained.

[0089] According to Darcy's law, the mushy zone during the alloy solidification process is regarded as a medium with multiple flow channels, and the expression of the permeability K is as follows:

[0090]

[0091] where g in the formulaL where \(f\) is the liquid fraction, \(n\) is the number of flow channels per unit area, and \(\tau\) is the tortuosity factor. The tortuosity factor \(\tau\) is introduced to account for non - straight and asymmetric flow channels. It is assumed that the number of channels is equal to the number of regions between dendrite arms, and the spacing between channels is equal to the dendrite arm spacing \(d_1\). The permeability \(K\) is expressed as follows:

[0092]

[0093] \(K = g\) L 2 \(d_1\) 2 / \(8\pi\tau\) 3 (6)

[0094] Assume that the tortuosity factor \(\tau\) corresponds to the complexity of the dendrite morphology, and the fractal dimension \(D\) is equal to the tortuosity factor \(\tau\).

[0095] Using the dendrite arm spacing value \(d_1 = 100\ \mu m\) in the Al - 2% Si alloy, the permeability \(K\) during the microstructure evolution under different undercooling conditions is calculated. Table 3 shows the calculated permeability values of the dendritic network structure of Al - 2% Si under different undercooling conditions, that is, as shown in Figure 3 the curve of the permeability of the alloy's directional growth network structure varying with the liquid fraction in the dendrite growth calculation domain under different undercooling conditions.

[0096]

[0097] The above - mentioned are only specific embodiments of the present invention, enabling those skilled in the art to understand or implement the present invention. Various modifications to these embodiments will be obvious to those skilled in the art. The general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention.

[0098] It should be understood that the present invention is not limited to the above - described content and can be modified and changed without departing from its scope. The scope of the present invention is only limited by the appended claims.

Claims

1. A method for calculating the permeability of the mushy zone during alloy solidification, characterized in that It includes the following steps: Step 1: First, obtain the physical property parameters of the actual alloy material; Step 2: Then, based on the Ginzburg-Landau theory, establish a phase-field model for the directional growth of multi-branched dendrites during the alloy solidification process; Step 3: Use the established multi-branched dendrite phase-field model to simulate the microstructure evolution process under different undercooling conditions; Step 4: Use the fractal theory to calculate the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained by simulation; Step 5: According to Darcy's law, use the fractal dimension D in the microstructure evolution process under different undercooling conditions obtained to calculate the permeability K; In Step 4, the box-counting method is used to calculate the fractal dimension D during the dendrite growth process under different undercooling conditions. The calculation process is as follows: First, divide the dendrite area simulated by the phase field into boxes with a size of r, and calculate the number of all boxes containing the S / L interface, denoted as N(r); Then, change the size r of the box and repeat the same steps to obtain different N(r), and obtain the following relationship for the fractal dimension: N(r) = r D (3) Wherein, in the formula, D is the fractal dimension, and the value of the fractal dimension D is the ratio of logN(r) to logr; In Step 5, according to Darcy's law, the mushy zone during the alloy solidification process is regarded as a medium with multiple flow channels, and the expression of the permeability K is as follows: where g in the formula L is the liquid fraction, n is the number of flow channels per unit area, τ is the tortuosity factor. The tortuosity factor τ is introduced to address the non-straight and asymmetric situation of the flow channels. It is assumed that the number of channels is equal to the number of regions between dendrite arms, and the spacing between channels is equal to the dendrite arm spacing d1. The expression for the permeability K is as follows: K = g L 2 d1 2 / 8πτ 3 (6) Assume that the tortuosity factor τ corresponds to the complexity of the dendrite morphology, and the fractal dimension D is equal to the tortuosity factor τ. Using the dendrite arm spacing value d1 in the alloy, calculate the permeability K in the microstructure evolution process under different undercooling conditions.

2. The method for calculating the permeability of the solidification mushy zone of an alloy according to claim 1, wherein The physical property parameters of the alloy material in Step 1 include the liquidus slope, partition coefficient, solid solute diffusivity, liquid solute diffusivity, anisotropy strength, melting point, interface energy, and molar volume.

3. A method for calculating the permeability of the solidification mushy zone of an alloy according to claim 1, characterized in that, The specific process of establishing the phase-field model in Step 2 is as follows: Introduce a continuously varying order parameter - the phase-field variable φ; when φ = 1, it represents the solid phase, when φ = 0, it represents the liquid phase, and when φ is greater than 0 and less than 1, it represents the solid-liquid two-phase; The phase-field and concentration control equations including the phase-field parameter φ are as follows: Among them, in the formula, φ is the phase-field variable, c is the concentration, M and ε are phase-field parameters related to the interface characteristics, and D(φ) is the solute diffusion coefficient.

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