CN110321604B - Numerical simulation method for growth of single dendrite during solidification of ternary alloy - Google Patents
Numerical simulation method for growth of single dendrite during solidification of ternary alloy Download PDFInfo
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Abstract
The invention discloses a numerical simulation method for growth of single dendrite during solidification of ternary alloy. The method comprises the following specific steps: simplifying the solidification condition, establishing a dendritic crystal growth model and a solute redistribution and diffusion model, compiling a computer program based on the established model, importing the computer program into simulation software for calculation, and finally obtaining a simulation result of dendritic crystal growth in the solidification process. The invention can simulate the growth morphology of dendritic crystals and the distribution state of solute components in the solidification process of ternary alloy, and can simulate the influence of factors such as supercooling degree, disturbance amplitude, anisotropic strength and the like on the solidification process, thereby playing a guiding role in practical engineering application.
Description
Technical Field
The invention belongs to the technical field of numerical simulation of metal material welding processes, and particularly relates to a numerical simulation method for growth of a single dendrite during solidification of a ternary alloy.
Background
The welding pool has the characteristics of high temperature, instantaneity, dynamic property and the like, so that the traditional experimental method is difficult to research the solidification process of the welding pool with transient change. With the rapid development of computer technology, numerical simulation is used as a new technology for researching the solidification process of a welding molten pool, various phenomena and evolution rules in the solidification process of alloy can be accurately quantified, and the defects of the traditional experimental method are overcome.
The numerical simulation method of the solidification process of the welding molten pool comprises a deterministic method, a Monte Carlo Method (MC), a phase field method (PF), a cellular automaton method (CA) and the like. The cellular automata method is based on a probabilistic random thought and is based on a grain nucleation and growth physical mechanism, so that the physical basis is clear, and the cellular automata method has high flexibility and capability of representing complex actual conditions.
At present, a CA model of a solidification structure is mainly suitable for binary alloys, but most of the CA model is ternary alloys in actual engineering and application, and the numerical simulation of the solidification structure of the ternary alloys is rarely reported. Therefore, establishing a ternary alloy dendrite growth CA model is particularly important.
Disclosure of Invention
The invention aims to provide a numerical simulation method for growth of a single dendrite during solidification of a ternary alloy, which solves the problem that a CA model applicable to a solidification structure of the ternary alloy is lacked in the prior art.
The invention adopts the technical scheme that a numerical simulation method for the growth of a single dendrite during the solidification process of ternary alloy is implemented according to the following steps:
step 1: simplifying the model conditions;
and 2, step: capturing a rule definition;
and 3, step 3: establishing a growth model;
and 4, step 4: solute redistribution and diffusion model establishment;
and 5: and calculating and exporting results.
The invention is also characterized in that:
the whole solidification process is divided into three states of liquid phase, solid phase and interface (solid-liquid coexistence);
dynamic supercooling is omitted in the model, and only temperature supercooling, component supercooling and curvature supercooling are considered;
setting solute components B and C in the solidification process of the ternary alloy, neglecting mutual diffusion between solutes and only considering the self-diffusion of the solutes;
in order to make the simulation result more accurate, eight neighborhoods (Moore neighborhoods) are adopted in the cell neighborhood relationship.
The step 2 is implemented according to the following steps:
step 2.1, dividing the simulation area into square grids, wherein each grid is a cell;
step 2.2, defining a solid-phase cellular in the center of a molten pool in advance, wherein 8 neighboring cellular around the cellular are interface cellular, and the rest cellular are in a liquid-phase state;
step 2.3, selecting the solid-phase unit cell obtained in the step 2.2, and when the solidification starts, performing solid phase fraction solving and judgment on neighbor unit cells around the solid-phase unit cell, wherein if the solid phase fraction of the neighbor unit cell is more than 1, the unit cell is converted into a solid-phase unit cell, and the liquid-phase unit cells around the unit cell are captured to form new interface unit cells;
and 2.4, solving and judging the solid phase fraction of the interface cells around the new solid-phase cells obtained in the step 2.3, wherein if the solid phase fraction of the neighbor cells is more than 1, the cells are changed into the solid-phase cells, the liquid-phase cells around the cells are captured to be the new interface cells, and so on until the whole molten pool is completely solidified.
step 3.1, calculation of Total supercooling degree, t n The total subcooling at a time can be given by:
ΔT(t n )＝T l -T(t n )+ml 1 ×(Cl 1 (t n )-C1)+ml 2 ×(Cl 2 (t n )-C2)-Γ(θ)×k(t n )
in the formula: t is l Is the liquidus temperature; t (T) n ) Is t n The temperature of the liquid metal at the moment; ml of 1 、ml 2 The liquid-phase line slopes of solute components B and C; cl 1 (t n )、Cl 2 (t n ) Solute components B and C are respectively at t n Temporal liquid phase solute concentration; c1 and C2 are initial solute concentrations of solute components B and C respectively; Γ (θ) is a Gibbs-Thompson coefficient; k (t) n ) Is t n The interface curvature at that time;
step 3.2, based on t obtained in step 3.1 n Total supercooling degree at time v (t) n ) The dendrite tip growth rate was calculated as shown in the following equation:
v(t n )＝μ k (θ)×ΔT(t n )
in the formula: mu.s k (θ) is the interfacial kinetic coefficient;
step 3.3 based on v (t) obtained in step 3.2 n ) For Δ f s The increase in interfacial cell solid fraction was calculated as shown in the following formula:
in the formula: g is an adjacent position grid state parameter; Δ t is the time step; a is a disturbance factor; rand () can generate a function of random number at [0,1 ];
step 3.4, based on Δ f obtained in step 3.3 s To f for s n The solid phase fraction of the interfacial cells was calculated as shown in the following formula:
f s n+1 ＝f s n +Δf s
in the formula: f. of s n+1 Is the solid phase fraction of the interface cells at the next moment; f. of s n And the solid phase fraction of the interface cells at the current moment.
step 4.1, based on f obtained in step 3.3 s n For Δ C i The excess solute discharged from the interface cells converted into solid phase cells is calculated as follows:
ΔC i ＝Cl i ×(1-k i )×Δf s
in the formula: cl i Represents the solute concentration of the liquid phase of the component i (B or C); k is a radical of i Represents the solute equilibrium distribution coefficient of the i component;
step 4.2, based on Δ f obtained in step 3.3 s And f from step 3.4 s n The solid phase components that have solidified are calculated as follows:
in the formula: cs i Represents the solid phase solute concentration of the i component;
step 4.3, based on Δ C obtained in step 4.1 i And Cs from step 4.2 i The diffusion of solute components was calculated as shown in the following formula:
in the formula: dl i 、Ds i Respectively represent the liquid phase diffusion coefficient and the solid phase diffusion coefficient of the i component.
step 5.1, compiling a computer program based on the models established in the step 3 and the step 4;
and 5.2, calculating according to the computer program obtained in the step 5.1 and deriving a result.
The invention has the beneficial effects that: on the basis of a binary alloy CA model, the invention provides a numerical simulation method for single dendritic crystal growth in a ternary alloy solidification process, which can simulate the growth morphology of dendritic crystals and the distribution state of solute components in the ternary alloy solidification process, and can simulate the influence of factors such as supercooling degree, disturbance amplitude, anisotropic strength and the like on the solidification process, thereby playing a guiding role in practical engineering application.
Drawings
FIG. 1 is a solidification process microstructure simulation flow chart of a simulation model of a numerical simulation method for growth of single dendrite during solidification of ternary alloy;
FIG. 2 is a schematic diagram of a cell after the space and state of a simulation model of the numerical simulation method for the growth of single dendrite during the solidification process of the ternary alloy are dispersed;
FIG. 3 is a diagram showing the distribution state of solute in the liquid phase of Al component and the morphology of dendrite growth when the preferential growth direction of Ti-6Al-4V alloy simulated in example 1 of the present invention is 0 ℃;
FIG. 4 is a graph showing the solute concentration distribution and dendrite growth morphology of the V component liquid phase when the preferred growth direction of the Ti-6Al-4V alloy simulated in example 1 of the present invention is 0 °;
FIG. 5 is a graph showing the distribution state of solute in Al component liquid phase and the growth morphology of dendrites when the preferential growth direction of Ti-6Al-4V alloy simulated in example 1 of the present invention is 30 °;
FIG. 6 is a diagram showing the distribution state of solute in liquid phase of V component and the growth morphology of dendrite crystal when the preferential growth direction of Ti-6Al-4V alloy simulated in example 1 of the present invention is 30 ℃;
FIG. 7 is a graph of the Al component liquid phase solute concentration distribution and dendrite growth morphology when the anisotropic strength of the Si alloy is 0 according to the simulation of Fe-0.8-C-0.3% in example 2 of the present invention;
FIG. 8 is a graph of the V component liquid phase solute concentration distribution and dendrite growth morphology when the anisotropic strength of the Si alloy is 0 according to the simulation of Fe-0.8-C-0.3% in example 2 of the present invention;
FIG. 9 is a graph showing the liquid phase solute concentration distribution state of Al component and dendrite growth morphology when the anisotropic strength of Si alloy is 0.2% according to the simulation of Fe-0.8-C-0.3% according to example 2 of the present invention;
FIG. 10 is a graph of V-component liquid phase solute concentration distribution state and dendrite growth morphology when the anisotropic strength of Si alloy was 0.2% by Fe-0.8-C-0.3% as simulated in example 2 of the present invention;
FIG. 11 is a graph of the Al component liquid phase solute concentration distribution and dendrite growth morphology for a 5K supercooling degree Si alloy simulated by example 3 according to the present invention as Fe-0.6-C-0.4%;
FIG. 12 is a graph of the V component liquid phase solute concentration distribution state and dendrite growth morphology when the supercooling degree of the Si alloy is 5K, simulated Fe-0.6-C-0.4% according to example 3 of the present invention;
FIG. 13 is a graph of Al component liquid phase solute concentration distribution and dendrite growth morphology for a 10K supercooling degree Si alloy simulated by example 3 according to the present invention as Fe-0.6-C-0.4%;
FIG. 14 is a graph of the V component liquid phase solute concentration distribution and dendrite growth morphology when the supercooling degree of the Si alloy is 10K according to the invention in example 3, which was simulated by Fe-0.6-C-0.4%.
Detailed Description
The invention is described in detail below with reference to the drawings and the detailed description.
The invention relates to a numerical simulation method for single dendrite growth in the solidification process of ternary alloy, which comprises the following specific steps as shown in figure 1,
step 1: simplifying the model conditions;
the whole solidification process is divided into three states of liquid phase, solid phase and interface (solid-liquid coexistence);
dynamic supercooling is omitted in the model, and only temperature supercooling, component supercooling and curvature supercooling are considered;
setting solute components B and C in the solidification process of the ternary alloy, neglecting mutual diffusion between solutes and only considering the self-diffusion of the solutes;
in order to make the simulation result more accurate, eight neighborhoods (Moore neighborhoods) are adopted for the cell neighborhood relationship.
And 2, step: the capture rule definition is specifically implemented according to the following steps:
step 2.1, dividing the simulation area into square grids, wherein each grid is a cell;
step 2.2, a solid-phase cellular is defined in the center of the molten pool in advance, 8 neighboring cellular around the cellular are interface cellular, and the rest cellular are in a liquid phase state, as shown in fig. 2;
step 2.3, selecting the solid-phase unit cell obtained in the step 2.2, and when the solidification starts, performing solid-phase fraction solving and judgment on neighbor unit cells around the solid-phase unit cell, wherein if the solid-phase fraction of the neighbor unit cell is more than 1, the unit cell is converted into a solid-phase unit cell, and the liquid-phase unit cells around the unit cell are captured to form new interface unit cells;
and 2.4, solving and judging the solid phase fraction of the interface cells around the new solid-phase cells obtained in the step 2.3, wherein if the solid phase fraction of the neighbor cells is more than 1, the cells are changed into the solid-phase cells, the liquid-phase cells around the cells are captured to be the new interface cells, and so on until the whole molten pool is completely solidified.
And step 3: establishing a growth model, specifically comprising the following steps:
step 3.1, calculation of Total supercooling degree, t n The total subcooling at a time can be given by:
ΔT(t n )＝T l -T(t n )+ml 1 ×(Cl 1 (t n )-C1)+ml 2 ×(Cl 2 (t n )-C2)-Γ(θ)×k(t n )
in the formula: t is l Is the liquidus temperature; t (T) n ) Is t n The temperature of the liquid metal at that time; ml of 1 、ml 2 The liquid line slopes of solute components B and C; cl 1 (t n )、Cl 2 (t n ) Solute components B and C at t n The liquid phase solute concentration at that moment; c1 and C2 are initial solute concentrations of solute components B and C respectively; gamma (theta) is a Gibbs-Thompson coefficient; k (t) n ) Is t n The interface curvature at that time;
step 3.2, based on t obtained in step 3.1 n Total supercooling degree at time v (t) n ) The dendrite tip growth rate was calculated as follows:
v(t n )＝μ k (θ)×ΔT(t n )
in the formula: mu.s k (θ) is the interfacial kinetic coefficient;
step 3.3 based on v (t) obtained in step 3.2 n ) For Δ f s The increase in interfacial cell solid fraction was calculated as shown in the following formula:
in the formula: g is an adjacent position grid state parameter; Δ t is the time step; a is a disturbance factor; rand () can generate a function of random number at [0,1 ];
step 3.4,. DELTA.f based on step 3.3 s To f for s n The solid phase fraction of the interfacial cells was calculated as shown in the following formula:
f s n+1 ＝f s n +Δf s
in the formula: f. of s n+1 Is the solid phase fraction of the interface cells at the next moment; f. of s n Is the solid phase fraction of the interface cells at the current moment.
And 4, step 4: solute redistribution and diffusion model establishment are specifically implemented according to the following steps:
step 4.1, based on f obtained in step 3.3 s n For Δ C i The excess solute discharged from the interface cells converted into solid phase cells is calculated as follows:
ΔC i ＝Cl i ×(1-k i )×Δf s
in the formula: cl i Represents the solute concentration of the liquid phase of the i (B or C) component; k is a radical of formula i Represents the solute equilibrium distribution coefficient of the i component;
step 4.2, based on Δ f obtained in step 3.3 s And f from step 3.4 s n The solid phase components that have solidified are calculated as follows:
in the formula: cs i Represents the solid phase solute concentration of the i component;
step 4.3, based on Δ C obtained in step 4.1 i And Cs from step 4.2 i The diffusion of solute components was calculated as shown in the following formula:
in the formula: dl i 、Ds i Respectively represents the liquid phase diffusion coefficient and the solid phase diffusion coefficient of the i component.
And 5: the calculation and result derivation are specifically implemented according to the following steps:
step 5.1, compiling a computer program based on the models established in the step 3 and the step 4;
and 5.2, calculating according to the computer program obtained in the step 5.1 and deriving a result.
Example 1
The method of the present invention is simulated by taking Ti-6Al-4V ternary alloy as an example, and the thermophysical property parameters of the alloy used in the simulation are shown in Table 1:
TABLE 1
Table 1 shows the thermophysical parameters used in the calculation of the Ti-6Al-4V alloy simulation.
As shown in fig. 3, 4, 5, and 6, it can be seen that the secondary dendrites on the primary dendrite arms are not developed and only a small amount of fine secondary dendrites are formed when the preferred growth direction is 0 °, and coarse secondary dendrites are formed on the primary dendrite arms when the preferred growth direction is 30 °. And the diffusion layer of the Al element on the interface is larger than the diffusion layer of the V element on the interface.
Example 2
The method of the present invention was simulated using a ternary Si alloy as an example of Fe-0.8% by C-0.3%, and the thermal property parameters of the alloy used in the simulation are shown in Table 2:
TABLE 2
Table 2 represents the thermal property parameters used in the calculation of Fe-0.8% C-0.3% of Si alloy in the simulation.
As shown in fig. 7, 8, 9 and 10, when the anisotropy is 0, the equiaxed crystal grows uniformly from the original crystal grain to the periphery with the primary dendrite, and the dendrite arms have many secondary dendrites, and the overall dendrite morphology is snowflake-shaped. When the anisotropy is 0.2, primary dendrites do not grow dispersedly around but follow a crystal structure, and developed secondary and tertiary dendrites are observed on primary dendrite arms.
Example 3
The method of the present invention was simulated using a ternary Si alloy containing Fe-0.6% by C-0.4% by weight, and the thermal property parameters of the alloy used in the simulation are shown in Table 3:
TABLE 3
Table 3 shows the thermal properties parameters used in the calculation of the% by weight of Fe-0.6-0.4% by weight of Si alloy in the simulation.
As shown in fig. 11, 12, 13 and 14, it can be seen that when the supercooling degree is 5K, the primary dendrite arms are short, and the secondary dendrites on the primary dendrite arms are less in number and are not developed. When the supercooling degree is 8K, the primary dendrite is obviously increased, and developed secondary dendrite exists on the primary dendrite arm.
According to the three embodiments, the influence of the growth morphology of the dendrite, the solute distribution state, the preferred crystal orientation, the anisotropic strength and the supercooling degree on the growth morphology of the dendrite in the solidification process of the ternary alloy can be successfully simulated.
Claims (3)
1. A numerical simulation method for single dendrite growth in a ternary alloy solidification process is characterized by comprising the following steps:
step 1: simplifying the model conditions;
step 2: capturing a rule definition;
the step 2 is specifically implemented according to the following steps:
step 2.1, dividing the simulation area into square grids, wherein each grid is a cell;
2.2, defining a solid-phase cellular in the center of the molten pool in advance, wherein 8 neighboring cellular around the cellular are interface cellular, and the rest cellular are in a liquid phase state;
step 2.3, selecting the solid-phase unit cell obtained in the step 2.2, and when the solidification starts, performing solid-phase fraction solving and judgment on neighbor unit cells around the solid-phase unit cell, wherein if the solid-phase fraction of the neighbor unit cell is more than 1, the unit cell is converted into a solid-phase unit cell, and the liquid-phase unit cells around the unit cell are captured to form a new interface unit cell;
step 2.4, solving and judging the solid phase fraction of the interface cells around the new solid phase cell obtained in the step 2.3, wherein if the solid phase fraction of the neighbor cell is more than 1, the cell is converted into a solid phase cell, and the liquid phase cells around the cell are captured to be the new interface cell, and so on, until the whole molten pool is completely solidified;
and 3, step 3: establishing a growth model;
the step 3 is specifically implemented according to the following steps:
step 3.1, calculation of Total supercooling degree, t n The total subcooling at a time can be given by:
ΔT(t n )＝T l -T(t n )+ml 1 ×(Cl 1 (t n )-C1)+ml 2 ×(Cl 2 (t n )-C2)-Γ(θ)×k(t n )
in the formula: t is l Is the liquidus temperature; t (T) n ) Is t n The temperature of the liquid metal at that time; ml of 1 、ml 2 The liquid line slopes of solute components B and C; cl 1 (t n )、Cl 2 (t n ) Solute components B and C at t n The liquid phase solute concentration at that moment; c1 and C2 are initial solute concentrations of solute components B and C respectively; gamma (theta) is a Gibbs-Thompson coefficient; k (t) n ) Is t n The interface curvature at that time;
step 3.2, based on t obtained in step 3.1 n Total supercooling degree at time v (t) n ) The dendrite tip growth rate was calculated as shown in the following equation:
v(t n )＝μ k (θ)×ΔT(t n )
in the formula: mu.s k (θ) is the interfacial kinetic coefficient;
step 3.3 based on v (t) obtained in step 3.2 n ) For Δ f s The increase in interfacial cell solid fraction was calculated as shown in the following formula:
in the formula: g is an adjacent position grid state parameter; Δ t is the time step; a is a disturbance factor; rand () can generate a function of random number at [0,1 ];
step (ii) of3.4 Δ f based on step 3.3 s To f for s The solid phase fraction of n interfacial cells was calculated as shown in the following formula:
in the formula:
is the solid phase fraction of the interface cells at the next moment; f. of s n is the solid phase fraction of the interface cells at the current moment;
and 4, step 4: solute redistribution and diffusion model establishment;
the step 4 is specifically implemented according to the following steps:
step 4.1, based on f obtained in step 3.3 s n To Δ C i The excess solute discharged from the interface cells converted into solid phase cells is calculated as shown in the following formula:
ΔC i ＝Cl i ×(1-k i )×Δf s
in the formula: cl i Represents the solute concentration of the liquid phase of the component i (B or C); k is a radical of formula i The solute equilibrium distribution coefficient of the i component is expressed;
step 4.2, based on Δ f obtained in step 3.3 s And f from step 3.4 s n The solid phase components that have solidified are calculated as follows:
in the formula: cs i Represents the solid phase solute concentration of the i component;
step 4.3, based on Δ C obtained in step 4.1 i And Cs obtained in step 4.2 i The diffusion of solute components is calculated as follows:
in the formula: dl i 、Ds i Respectively representing the liquid phase diffusion coefficient and the solid phase diffusion coefficient of the i component;
and 5: and (4) calculating and exporting a result.
2. The method for numerical simulation of single dendrite growth during solidification of ternary alloy according to claim 1, wherein the step 1 simplifying model conditions comprises:
the whole solidification process is divided into three states of liquid phase, solid phase and interface;
dynamic supercooling is omitted in the model, and only temperature supercooling, component supercooling and curvature supercooling are considered;
setting solute components B and C in the solidification process of the ternary alloy, neglecting mutual diffusion between solutes and only considering self-diffusion of the solutes;
in order to make the simulation result more accurate, eight neighborhoods are adopted in the cell neighborhood relation.
3. The numerical simulation method for single dendrite growth in the ternary alloy solidification process according to claim 1, wherein the step 5 is specifically implemented according to the following steps:
step 5.1, compiling a computer program based on the models established in the step 3 and the step 4;
and 5.2, calculating according to the computer program obtained in the step 5.1 and deriving a result.
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