Calculation force of thermal expansion and cold contraction
Publish: 2021-04-16 05:35:40
1. Taking the establishment of the calculation model for the instantaneous linear thermal expansion coefficient of carbon steel as an example:
when the temperature of the material changes from Tref (reference temperature of the benchmark) to t, the relative change of the material length L is:
(1)
according to the density ρ It is inversely proportional to L3 ε Th and ρ There is the following relationship between the two factors:
(2)
(3)
then the instantaneous linear thermal expansion coefficient is defined as:
(3)
. Therefore, the key to obtain the instantaneous linear thermal expansion coefficient is to determine the density of carbon steel at different temperatures< According to the characteristics of solidification structure ring cooling (see Fig. 1), carbon steels with 〔 C 〕≤ 0.8% were divided into the following four groups according to carbon content:
I. 〔 C 〕 < 0.09%:
L → L+ δ → δ → δ+γ → γ → α+γ → α+ Fe3C
Ⅱ.〔C〕=0.09 %～0.16 %:
L→L+ δ → δ+γ → γ → α+γ → α+ Fe3C
Ⅲ.〔C〕=0.16 %～0.51 %:
L→L+ δ →L+ γ → γ → α+γ → α+ Fe3C
Ⅳ.〔C〕=0.51 %～0.80 %:
L→L+ γ → γ → α+γ → α+ The solidification structure of Fe3C
carbon steel is a multiphase mixed system, and its density is determined according to formula (4) and formula (5), namely:
(4)
F1 + F2 +... + fi = 1 (5)
where FI is the mass fraction of component I in the system, which can be determined by program calculation according to the lever rule and phase diagram. Component I (I is L δ、γ、α Or Fe3C) as a function of temperature and carbon content ρ〔 T,(i)〕= ρ I (T, c), whose value is from reference [6]
when calculating the coefficient of linear thermal expansion, the solis temperature is selected as the reference temperature. The coefficient of thermal expansion decreases linearly from the value at the solis to zero at the zero strength temperature (that is, the temperature corresponding to the solid fraction FS = 0.8). Above the zero strength temperature, the coefficient of thermal expansion remains zero. In this way, the thermal stress in the liquid region can be avoided< 1 Fe-C phase diagram
1.2 brief introction of slab thermal elastic plastic stress model
1.2.1 calculation of slab temperature field
the heat transfer in drawing direction is ignored, and according to the symmetry, 1 / 4 section slab is taken, and its quadrilateral 4-node isoparametric element grid is shown in Figure 2. The governing equations of unsteady two-dimensional heat transfer are as follows:
Fig.2 calculation domain and FEM meshed for analysis
(6)
the initial temperature is casting temperature, the measured value of heat flux on slab surface is q = 2 688-420 T1 / 2 kW / m2, and the central symmetry line is adiabatic boundary. The thermophysical parameters used in the model vary with temperature, and the equivalent specific heat capacity C is used to consider the effect of latent heat. In addition, the convection effect in the liquid region is achieved by appropriately enlarging the thermal conctivity of the liquid region
1.2.2 calculation of slab stress field
in order to make use of the calculation results of temperature field, the slab grid generation method consistent with the temperature field is adopted. In the system, the mold copper plate is a rigid contact boundary, and the mold taper is characterized by controlling its motion trajectory (including motion direction and velocity). If the distance between a node on the slab surface and the copper plate is less than the specified contact criterion, it is considered that the contact occurs here, and the contact constraint is imposed on the node (to avoid the node crossing the copper plate surface), otherwise it is treated as a free boundary
in the calculation, the liquid and solid regions are regarded as a whole, and the mechanical parameters of the material higher than the liquis temperature are specially treated, so that the stress state of the liquid region can maintain a uniform static pressure state, and the static pressure of the liquid steel applied on the outside can be basically transferred to the inside of the solid shell. According to the symmetry, a fixed displacement constraint in the vertical direction should be applied on the central symmetry line, but because only the displacement field of the shell is concerned, and the thickness of the shell is generally not more than 15 mm, the constraint is only applied within 15 mm from the surface. The range beyond 15 mm is basically liquid phase region, and hydrostatic pressure is applied on its outer edge (at the symmetrical line) (the pressure value is proportional to the distance from the meniscus)< The force balance equation of the above system is:
(7)
where [k] is the total stiffness matrix of the system{ δ i} Is the node displacement array{ Rexter} is the equivalent nodal load array caused by external forces (static pressure of molten steel and contact reaction force of copper wall){ R ε 0} is the equivalent nodal load array caused by thermal strain. Considering the effect of peritectic phase transition, the calculation of {r} ε The linear thermal expansion coefficient curve of carbon steel calculated above is used when the temperature is less than 0}
the thermal elastic plastic model is used in the calculation. It is assumed that the slab section is in the generalized plane strain state, obeys Mises yield criterion and isotropic strengthening law, and the hardening curve is piecewise linear
2 calculation results and discussion
three kinds of carbon steel with carbon content of 0.045%, 0.100% and 0.200% were taken as calculation objects, and the same calculation conditions were adopted, that is, the section size of slab was 150 mm × The casting temperature is 1 550 ℃, the length of mould is 700 mm, the taper is 0.8%, and the distance between meniscus and top of mould is 100 mm
2.1 instantaneous thermal expansion coefficient of three kinds of carbon steel
Fig. 3 shows the calculated instantaneous linear thermal expansion coefficient curve of carbon steel. It can be seen that when [C] = 0.045%, the coefficient of thermal expansion changes suddenly below the solis temperature. This is e to the primary corrosion of molten steel after solidification δ Phase → γ The transformation of phase and the change of specific volume make the coefficient of thermal expansion rise sharply; When [C] = 0.100%, the coefficient of thermal expansion changes abruptly from the two-phase region. This is because when the liquid steel solidifies, the liquid phase and δ The peritectic reaction takes place and the phase is transformed into γ Phase, remaining δ One after another γ Phase transition. The change of specific volume in the transformation process also leads to the sharp rise of thermal expansion coefficient
Fig. 3 instantaneous linear thermal expansion coefficient curve of carbon steel
in the three curves, the starting point of non-zero value is the corresponding point of zero strength temperature
A, B and C are the corresponding points of solis temperature
Fig.3 instant linear thermal expansion
coefficient of carbon steel
in addition, [C] = 0.045% δ Phase → γ The phase transition temperature range is narrow and the transition is fast (see Fig. 1), so the abrupt change value of linear thermal expansion coefficient is large. In contrast, the abrupt change of coefficient of thermal expansion with [C] = 0.100% is smaller. However, because of the wide temperature range of the latter, the temperature range of abrupt change of the coefficient of thermal expansion is also wide. It can be inferred that the effect of peritectic phase transformation on the solidification shrinkage of primary shell at [C] = 0.100% is greater than that at [C] = 0.045% δ Phase → γ Phase transition
[C] = 0.200% steel has no abrupt change in coefficient of thermal expansion. The reason is that although peritectic transformation also occurs, it only occurs at a certain temperature level (about 1 495 ℃), so it has little effect on the coefficient of thermal expansion< The results show that the surface shrinkage of 〔 C 〕 = 0.045%, 0.100% and 0.200% of the three steels varies along the drawing direction and cross section direction (the space inclined plane at the bottom is the mold copper plate
Fig. 4 〔 C 〕 = 0.045%) b) 〔C〕=0.100 % ( c) [C] = 0.200%
Fig.4 surface shrinkage of billet
inner wall). It can be seen from the figure that the corner of the slab shrinks and breaks away from the copper plate of the mould at the early stage of solidification, and almost always contacts the copper plate near the middle (only the steel with [C] = 0.100% keeps separation near the outlet). The closer to the corner, the earlier the contraction and detachment, and the larger the contraction
under the static pressure of molten steel, the shrinking shell will be pressed back to the mold copper plate, so that the shell shrinkage fluctuates [the surface of shrinkage surface is dog tooth shape (see Figure 4)]. The shell near the meniscus is thinner and the fluctuation is obvious. In addition, the closer to the corner, the more obvious the fluctuation is. The shrinkage fluctuation of primary shell will lead to stress concentration, which is easy to ince surface defects such as cracks
comparing the surface shrinkage of three kinds of carbon steel billets, it can be seen that: [C] = 0.100% steel has the most significant shrinkage, the largest shrinkage fluctuation (meniscus area), and the most extensive expansion along the cross section direction The shrinkage of 0.200% steel is the smallest
2.3 shrinkage of primary shell at the corner of meniscus region
Fig. 5 shows the shrinkage of corner of three kinds of carbon steel near meniscus region. It can be seen that: within 20 mm from the meniscus, the corner of the slab is separated from the mold copper plate, and the steel with 〔 C 〕 = 0.045% is the earliest to separate from the mold copper plate, because the solis temperature of the steel is the highest, and it is the earliest to solidify and form the shell C] = 0.100% steel shrinks strongly after forming primary shell, but it is pressed back by the increased hydrostatic pressure 50 mm away from the meniscus, and then continues to shrink. The shrinkage of the primary shell of this steel is the most significant, and the shrinkage fluctuation is also the largest, so it is most likely to ince the surface defects of the slab The results show that the shrinkage and shrinkage fluctuation of primary shell of 0.045% steel decrease obviously C] = 0.200% steel has the smallest shrinkage and shrinkage fluctuation< 5 shrinkage of initial shell of pellet corner at Meniscus
3 conclusion
(1) for peritectic steel with carbon content around 0.1%, the shrinkage of primary shell at the top of mold and near the corner is very irregular, which is easy to ince surface defects
(2) the irregular shrinkage of the shell is mainly concentrated in the range of 100 mm below the meniscus. Therefore, the taper of the upper part of the mold is not suitable for shell shrinkage. Therefore, the casting speed should be improved by optimizing the mold taper. An important guiding principle is to use a larger taper in the upper part of the mold to make the shell in good contact with the copper plate.
when the temperature of the material changes from Tref (reference temperature of the benchmark) to t, the relative change of the material length L is:
(1)
according to the density ρ It is inversely proportional to L3 ε Th and ρ There is the following relationship between the two factors:
(2)
(3)
then the instantaneous linear thermal expansion coefficient is defined as:
(3)
. Therefore, the key to obtain the instantaneous linear thermal expansion coefficient is to determine the density of carbon steel at different temperatures< According to the characteristics of solidification structure ring cooling (see Fig. 1), carbon steels with 〔 C 〕≤ 0.8% were divided into the following four groups according to carbon content:
I. 〔 C 〕 < 0.09%:
L → L+ δ → δ → δ+γ → γ → α+γ → α+ Fe3C
Ⅱ.〔C〕=0.09 %～0.16 %:
L→L+ δ → δ+γ → γ → α+γ → α+ Fe3C
Ⅲ.〔C〕=0.16 %～0.51 %:
L→L+ δ →L+ γ → γ → α+γ → α+ Fe3C
Ⅳ.〔C〕=0.51 %～0.80 %:
L→L+ γ → γ → α+γ → α+ The solidification structure of Fe3C
carbon steel is a multiphase mixed system, and its density is determined according to formula (4) and formula (5), namely:
(4)
F1 + F2 +... + fi = 1 (5)
where FI is the mass fraction of component I in the system, which can be determined by program calculation according to the lever rule and phase diagram. Component I (I is L δ、γ、α Or Fe3C) as a function of temperature and carbon content ρ〔 T,(i)〕= ρ I (T, c), whose value is from reference [6]
when calculating the coefficient of linear thermal expansion, the solis temperature is selected as the reference temperature. The coefficient of thermal expansion decreases linearly from the value at the solis to zero at the zero strength temperature (that is, the temperature corresponding to the solid fraction FS = 0.8). Above the zero strength temperature, the coefficient of thermal expansion remains zero. In this way, the thermal stress in the liquid region can be avoided< 1 Fe-C phase diagram
1.2 brief introction of slab thermal elastic plastic stress model
1.2.1 calculation of slab temperature field
the heat transfer in drawing direction is ignored, and according to the symmetry, 1 / 4 section slab is taken, and its quadrilateral 4-node isoparametric element grid is shown in Figure 2. The governing equations of unsteady two-dimensional heat transfer are as follows:
Fig.2 calculation domain and FEM meshed for analysis
(6)
the initial temperature is casting temperature, the measured value of heat flux on slab surface is q = 2 688-420 T1 / 2 kW / m2, and the central symmetry line is adiabatic boundary. The thermophysical parameters used in the model vary with temperature, and the equivalent specific heat capacity C is used to consider the effect of latent heat. In addition, the convection effect in the liquid region is achieved by appropriately enlarging the thermal conctivity of the liquid region
1.2.2 calculation of slab stress field
in order to make use of the calculation results of temperature field, the slab grid generation method consistent with the temperature field is adopted. In the system, the mold copper plate is a rigid contact boundary, and the mold taper is characterized by controlling its motion trajectory (including motion direction and velocity). If the distance between a node on the slab surface and the copper plate is less than the specified contact criterion, it is considered that the contact occurs here, and the contact constraint is imposed on the node (to avoid the node crossing the copper plate surface), otherwise it is treated as a free boundary
in the calculation, the liquid and solid regions are regarded as a whole, and the mechanical parameters of the material higher than the liquis temperature are specially treated, so that the stress state of the liquid region can maintain a uniform static pressure state, and the static pressure of the liquid steel applied on the outside can be basically transferred to the inside of the solid shell. According to the symmetry, a fixed displacement constraint in the vertical direction should be applied on the central symmetry line, but because only the displacement field of the shell is concerned, and the thickness of the shell is generally not more than 15 mm, the constraint is only applied within 15 mm from the surface. The range beyond 15 mm is basically liquid phase region, and hydrostatic pressure is applied on its outer edge (at the symmetrical line) (the pressure value is proportional to the distance from the meniscus)< The force balance equation of the above system is:
(7)
where [k] is the total stiffness matrix of the system{ δ i} Is the node displacement array{ Rexter} is the equivalent nodal load array caused by external forces (static pressure of molten steel and contact reaction force of copper wall){ R ε 0} is the equivalent nodal load array caused by thermal strain. Considering the effect of peritectic phase transition, the calculation of {r} ε The linear thermal expansion coefficient curve of carbon steel calculated above is used when the temperature is less than 0}
the thermal elastic plastic model is used in the calculation. It is assumed that the slab section is in the generalized plane strain state, obeys Mises yield criterion and isotropic strengthening law, and the hardening curve is piecewise linear
2 calculation results and discussion
three kinds of carbon steel with carbon content of 0.045%, 0.100% and 0.200% were taken as calculation objects, and the same calculation conditions were adopted, that is, the section size of slab was 150 mm × The casting temperature is 1 550 ℃, the length of mould is 700 mm, the taper is 0.8%, and the distance between meniscus and top of mould is 100 mm
2.1 instantaneous thermal expansion coefficient of three kinds of carbon steel
Fig. 3 shows the calculated instantaneous linear thermal expansion coefficient curve of carbon steel. It can be seen that when [C] = 0.045%, the coefficient of thermal expansion changes suddenly below the solis temperature. This is e to the primary corrosion of molten steel after solidification δ Phase → γ The transformation of phase and the change of specific volume make the coefficient of thermal expansion rise sharply; When [C] = 0.100%, the coefficient of thermal expansion changes abruptly from the two-phase region. This is because when the liquid steel solidifies, the liquid phase and δ The peritectic reaction takes place and the phase is transformed into γ Phase, remaining δ One after another γ Phase transition. The change of specific volume in the transformation process also leads to the sharp rise of thermal expansion coefficient
Fig. 3 instantaneous linear thermal expansion coefficient curve of carbon steel
in the three curves, the starting point of non-zero value is the corresponding point of zero strength temperature
A, B and C are the corresponding points of solis temperature
Fig.3 instant linear thermal expansion
coefficient of carbon steel
in addition, [C] = 0.045% δ Phase → γ The phase transition temperature range is narrow and the transition is fast (see Fig. 1), so the abrupt change value of linear thermal expansion coefficient is large. In contrast, the abrupt change of coefficient of thermal expansion with [C] = 0.100% is smaller. However, because of the wide temperature range of the latter, the temperature range of abrupt change of the coefficient of thermal expansion is also wide. It can be inferred that the effect of peritectic phase transformation on the solidification shrinkage of primary shell at [C] = 0.100% is greater than that at [C] = 0.045% δ Phase → γ Phase transition
[C] = 0.200% steel has no abrupt change in coefficient of thermal expansion. The reason is that although peritectic transformation also occurs, it only occurs at a certain temperature level (about 1 495 ℃), so it has little effect on the coefficient of thermal expansion< The results show that the surface shrinkage of 〔 C 〕 = 0.045%, 0.100% and 0.200% of the three steels varies along the drawing direction and cross section direction (the space inclined plane at the bottom is the mold copper plate
Fig. 4 〔 C 〕 = 0.045%) b) 〔C〕=0.100 % ( c) [C] = 0.200%
Fig.4 surface shrinkage of billet
inner wall). It can be seen from the figure that the corner of the slab shrinks and breaks away from the copper plate of the mould at the early stage of solidification, and almost always contacts the copper plate near the middle (only the steel with [C] = 0.100% keeps separation near the outlet). The closer to the corner, the earlier the contraction and detachment, and the larger the contraction
under the static pressure of molten steel, the shrinking shell will be pressed back to the mold copper plate, so that the shell shrinkage fluctuates [the surface of shrinkage surface is dog tooth shape (see Figure 4)]. The shell near the meniscus is thinner and the fluctuation is obvious. In addition, the closer to the corner, the more obvious the fluctuation is. The shrinkage fluctuation of primary shell will lead to stress concentration, which is easy to ince surface defects such as cracks
comparing the surface shrinkage of three kinds of carbon steel billets, it can be seen that: [C] = 0.100% steel has the most significant shrinkage, the largest shrinkage fluctuation (meniscus area), and the most extensive expansion along the cross section direction The shrinkage of 0.200% steel is the smallest
2.3 shrinkage of primary shell at the corner of meniscus region
Fig. 5 shows the shrinkage of corner of three kinds of carbon steel near meniscus region. It can be seen that: within 20 mm from the meniscus, the corner of the slab is separated from the mold copper plate, and the steel with 〔 C 〕 = 0.045% is the earliest to separate from the mold copper plate, because the solis temperature of the steel is the highest, and it is the earliest to solidify and form the shell C] = 0.100% steel shrinks strongly after forming primary shell, but it is pressed back by the increased hydrostatic pressure 50 mm away from the meniscus, and then continues to shrink. The shrinkage of the primary shell of this steel is the most significant, and the shrinkage fluctuation is also the largest, so it is most likely to ince the surface defects of the slab The results show that the shrinkage and shrinkage fluctuation of primary shell of 0.045% steel decrease obviously C] = 0.200% steel has the smallest shrinkage and shrinkage fluctuation< 5 shrinkage of initial shell of pellet corner at Meniscus
3 conclusion
(1) for peritectic steel with carbon content around 0.1%, the shrinkage of primary shell at the top of mold and near the corner is very irregular, which is easy to ince surface defects
(2) the irregular shrinkage of the shell is mainly concentrated in the range of 100 mm below the meniscus. Therefore, the taper of the upper part of the mold is not suitable for shell shrinkage. Therefore, the casting speed should be improved by optimizing the mold taper. An important guiding principle is to use a larger taper in the upper part of the mold to make the shell in good contact with the copper plate.
2. My understanding is that compared with gas, the thermal expansion rate of liquid is very small, but the pressure generated by the expansion of liquid is very high. In other words, because there is no compression space for the molecular spacing, it is difficult to be compressed. Therefore, if only the liquid in the container is heated and expanded, and the container is not heated without thermal deformation, the pressure of liquid expansion in the container will cause the deformation (expansion) of the container, so that the pressure generated by thermal expansion in the tank will be released
as for whether there is a relevant thermodynamic calculation method, I am not sure. Is this accurate calculation really meaningful?
as for whether there is a relevant thermodynamic calculation method, I am not sure. Is this accurate calculation really meaningful?
3. When the temperature of the material changes from Tref to t, the relative change of the material length L is:
(1)
according to the density ρ It is inversely proportional to L3 ε Th and ρ There is the following relationship between the two:
(2)
(3)
(3)
. Therefore, the key to obtain the instantaneous linear thermal expansion coefficient is to determine the density of carbon steel at different temperatures.
(1)
according to the density ρ It is inversely proportional to L3 ε Th and ρ There is the following relationship between the two:
(2)
(3)
(3)
. Therefore, the key to obtain the instantaneous linear thermal expansion coefficient is to determine the density of carbon steel at different temperatures.
4. The generation of internal stress is e to the different thermal expansion coefficient between two different materials. Plastic deformation does not proce internal stress, only elastic deformation can proce internal stress. That is to say, there is no internal stress when the metal is molten. When it is cooled to solid state, the deformation mode of each part is different, that is to say, stress is proced. This kind of stress can not be completely removed, it can only be reced by changing the welding method or material, and the post welding treatment can also eliminate part of the internal stress. External stress? It is not clear about this statement, so it is recommended to look for information on stress.
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