# Verification of the adequacy of the mathematical model of the solidification process of titanium ingots produced in electron beam cold hearth furnaces.

The solidification of metal is accompanied by complicated and
high-rate physical processes of heat transfer, hydrodynamic flow, and
radiation. In practice, it is often not possible to measure the values
of these parameters of the processes with the sufficient accuracy.

In addition, full-scale experiments in metallurgy are associated with considerable material consumption due to the size and cost of ingots.

Of special importance are the numerical experiments using the method of mathematical simulation and computer calculations which make it possible, with a relatively small consumption and the minimum number of experimental data, to compile the quantitative and quantitative patterns of the phenomena taking place in the metallurgical processes.

The advantage of the numerical methods is that they take into account not only the mean parameters of the physical quantities and but also can be used to determine the distribution of these parameters in the space and time in the conditions very similar to those in the actual processes.

The aim of modelling is to avoid carrying out expensive full-size experiments by calculating the distribution of the required parameter at any moment of time with the required accuracy.

In practice, the number of experiments is usually reduced by calculating the nature of distribution and the tendency of the variation of the given parameter. However, the experimental verification of these mathematical models is still necessary because it provides the actual pattern of the processes in the ingot.

The main principles of modelling the thermophysical processes in the ingots, produced by the methods of special electrometallurgy, have been formulated in [1, 2]. Recently, several mathematical models have been developed, describing the individual stages of the process of electron beam melting in an intermediate container (cold hearth): melting of the initial charge, mixing and evaporation of the metal and impurities in an intermediate container, formation of the ingot in the solidification mould (including the displacement of the shrinkage cavity), cooling of the ingot (3-6).

The E.O. Paton Electric Welding Institute has developed and subsequently improved the mathematical model of solidification of cylindrical ingots in electron beam melting in an intermediate container (EBMIC) which makes it possible to determine the distribution of temperatures in the cylindrical ingot at any moment of time and, consequently, the configuration of the liquid pool and of the zone of the solid-liquid state of the metal in relation to the technological parameters of the electron beam melting (process productivity, periodicity of pouring in the melt into the solidification mould and the power of electron beam heating) (3).

However, until recently, the adequacy of the model has been confirmed only indirectly. For example, in (7) the authors describe the effect of the depth of the liquid pool, which depends on the power of electron beam heating and melting productivity, on the structure of the titanium ingot.

The aim of the present work is the experimental verification of the adequacy of the model of thermophysical processes in the solidification mould.

In this model, the liquid metal was poured into the solidification mould in portions, and the ingot was periodically withdrawn. The surface of the ingot was heated with two electron beams, and the power of one of these beams w1 was uniformly distributed in the central zone, and the power of the other beam [w.sub.2] was concentrated in the peripheral zone (Figure 1).

[FIGURE 1 OMITTED]

The controlled technological parameters in the mathematical model were the power of the beams [w.sub.1], [w.sub.2], the periodicity of pouring t, the height of the portion poured into the solidification mould h, the displacement of the peripheral beam from the centre to the wall of the solidification mould d.

The formal criterion of the depth of the liquid pool was the radius of the ingot R, generally accepted in the evaluation of the solidification conditions (3). According to the data in the graph (Figure 2), it may be seen that the depth of the liquid pool does not exceed the ingot the radius (for the in-got diameter of 600 mm) and the melting productivity of 350 kg/h (8). Therefore, the melting rate of 300 kg/h was selected for the experiments.

[FIGURE 2 OMITTED]

The mathematical model (3) was used to simulate three melting regimes of the ingots with a diameter of 600 mm made of VT1-0 titanium alloy. The problem was sold numerically using the finite difference method. The constant technological parameters were: t = 180 s; h = 0.012 m; d = 0.006 m. The three melting regimes were simulated by changing the energy transferred by the electron beams to the free surface of the ingot (capital table 1).

The calculations yielded the temperature field in the ingot for different conditions of electron beam heating the surface of the metal in the solidification mould in EBMIC (Figure 3).

[FIGURE 3 OMITTED]

The processes of formation of the liquid pool in the titanium ingots in melting in the electron beam furnaces with the intermediate container were also investigated by experiments.

The calculated regimes of electron beam heating of the cylindrical ingot in the solidification mould in UE-5812 electron beam equipment (9) were used for the experimental melts of ingots of VT1-0 titanium with a diameter of 600 mm.

The ingots were melted by the EBMIC technology with the horizontal feed of the consumable billet, with the buildup of the metal in the intermediate container and its pouring into the solidification mould with the formation of the ingot.

The upper end of the ingot was heated with two axial electron beam guns in which the energy distribution was in the form of concentric circles.

The power the first beam was uniformly distributed in the central region, and the power of the second beam in the periphery with the displacement to the wall of the copper water-cooled solidification mould.

Both the total power [w.sub.1] + [w.sub.2] and the power distribution between the electron beam guns (Table 1) were varied. The metal was supplied in portions into the solidification mould.

The experimental results show that the depth and volume of the liquid metal pool continuously increase at the start of melting into a constant level indicating the establishment of the quasi-stationary regime in the upper part of the ingot which is not influenced by the variation of the thermal regime (3).

After the establishment of the quasi-stationary regime for the given melting rate and producing the ingots with the height of 1000 mm, the liquid metal was poured from the entire volume of the metal pool by melting the metal crust at the periphery of the ingot. The produced ingots were sectioned into two halves along the axis of symmetry (Figure 3).

The formation of the billet shows that they repeat with sufficient accuracy the shape of the calculated temperature field. The depth of measured from the outer edge of the head part of the ingot to the lowest point of the resultant crater (Figure 4). The results of the measurements are presented in Table 1.

[FIGURE 4 OMITTED]

The deviation of the calculated data from the experimental ones was determined from the following equation:

D = ([H.sub.e] - [H.sub.p]).100/[H.sub.p], where D is the deviation of the calculated values from the experimental data, %; [H.sub.e] is the experimental value of the depth of the liquid pool, mm; [H.sub.e] is the calculated depth of the liquid pool, mm.

Comparison of the experimental results with the results of mathematical modelling shows that, on the whole, the model reflects accurately the physical phenomenon of the formation of the liquid pool in the ingot. The calculated values of the depth of the liquid pools are in agreement with the experimentally measured values in the range 5-12%. The experiments shows that on the whole the model reflects with sufficient accuracy the variation of the depth of formation of the liquid pool with the variation of the power in heating the ingot.

References

(1) Bello G.P., et al., Probl. Spets. Elektrometalurgii, 1996, No. 4, 27--7.

(2.) Paton B.E., et al., Electron beam welding, Naukova dumka, Kiev, 1997.

(3.) Paton B.E., et al., Electron beam melting of titanium, Naukova Dumka, Kiev, 2006.

(4.) Lesnoi A.B., et al., Probl. Spets. Elektrometalurgii, 2001, No. 2, 17--1.

(5.) Kalinyuk A.N., et al., Probl. Spets. Elektrometalurgii, 2002, No. 1, 20--5.

(6.) Zhuk G.V., Probl. Spets. Elektrometalurgii, 1998, No. 2, 21--5.

(7.) Zhuk G.V., et al., Protsessy Lit'ya, 2003, No. 4, 79--2

(8.) Zhuk G.V., Sovremen. Elektrometallurgiya, 2008, No. 2, 17--0.

(9.) Trigub N.P., et al., Sovremen. Elektrometallurgiya,, 2007, No. 1, 11--4.

V.A. Berezos

E.O. Paton Electric Welding Institute, Kiev

In addition, full-scale experiments in metallurgy are associated with considerable material consumption due to the size and cost of ingots.

Of special importance are the numerical experiments using the method of mathematical simulation and computer calculations which make it possible, with a relatively small consumption and the minimum number of experimental data, to compile the quantitative and quantitative patterns of the phenomena taking place in the metallurgical processes.

The advantage of the numerical methods is that they take into account not only the mean parameters of the physical quantities and but also can be used to determine the distribution of these parameters in the space and time in the conditions very similar to those in the actual processes.

The aim of modelling is to avoid carrying out expensive full-size experiments by calculating the distribution of the required parameter at any moment of time with the required accuracy.

In practice, the number of experiments is usually reduced by calculating the nature of distribution and the tendency of the variation of the given parameter. However, the experimental verification of these mathematical models is still necessary because it provides the actual pattern of the processes in the ingot.

The main principles of modelling the thermophysical processes in the ingots, produced by the methods of special electrometallurgy, have been formulated in [1, 2]. Recently, several mathematical models have been developed, describing the individual stages of the process of electron beam melting in an intermediate container (cold hearth): melting of the initial charge, mixing and evaporation of the metal and impurities in an intermediate container, formation of the ingot in the solidification mould (including the displacement of the shrinkage cavity), cooling of the ingot (3-6).

The E.O. Paton Electric Welding Institute has developed and subsequently improved the mathematical model of solidification of cylindrical ingots in electron beam melting in an intermediate container (EBMIC) which makes it possible to determine the distribution of temperatures in the cylindrical ingot at any moment of time and, consequently, the configuration of the liquid pool and of the zone of the solid-liquid state of the metal in relation to the technological parameters of the electron beam melting (process productivity, periodicity of pouring in the melt into the solidification mould and the power of electron beam heating) (3).

However, until recently, the adequacy of the model has been confirmed only indirectly. For example, in (7) the authors describe the effect of the depth of the liquid pool, which depends on the power of electron beam heating and melting productivity, on the structure of the titanium ingot.

The aim of the present work is the experimental verification of the adequacy of the model of thermophysical processes in the solidification mould.

In this model, the liquid metal was poured into the solidification mould in portions, and the ingot was periodically withdrawn. The surface of the ingot was heated with two electron beams, and the power of one of these beams w1 was uniformly distributed in the central zone, and the power of the other beam [w.sub.2] was concentrated in the peripheral zone (Figure 1).

[FIGURE 1 OMITTED]

The controlled technological parameters in the mathematical model were the power of the beams [w.sub.1], [w.sub.2], the periodicity of pouring t, the height of the portion poured into the solidification mould h, the displacement of the peripheral beam from the centre to the wall of the solidification mould d.

The formal criterion of the depth of the liquid pool was the radius of the ingot R, generally accepted in the evaluation of the solidification conditions (3). According to the data in the graph (Figure 2), it may be seen that the depth of the liquid pool does not exceed the ingot the radius (for the in-got diameter of 600 mm) and the melting productivity of 350 kg/h (8). Therefore, the melting rate of 300 kg/h was selected for the experiments.

[FIGURE 2 OMITTED]

The mathematical model (3) was used to simulate three melting regimes of the ingots with a diameter of 600 mm made of VT1-0 titanium alloy. The problem was sold numerically using the finite difference method. The constant technological parameters were: t = 180 s; h = 0.012 m; d = 0.006 m. The three melting regimes were simulated by changing the energy transferred by the electron beams to the free surface of the ingot (capital table 1).

The calculations yielded the temperature field in the ingot for different conditions of electron beam heating the surface of the metal in the solidification mould in EBMIC (Figure 3).

[FIGURE 3 OMITTED]

The processes of formation of the liquid pool in the titanium ingots in melting in the electron beam furnaces with the intermediate container were also investigated by experiments.

The calculated regimes of electron beam heating of the cylindrical ingot in the solidification mould in UE-5812 electron beam equipment (9) were used for the experimental melts of ingots of VT1-0 titanium with a diameter of 600 mm.

The ingots were melted by the EBMIC technology with the horizontal feed of the consumable billet, with the buildup of the metal in the intermediate container and its pouring into the solidification mould with the formation of the ingot.

The upper end of the ingot was heated with two axial electron beam guns in which the energy distribution was in the form of concentric circles.

The power the first beam was uniformly distributed in the central region, and the power of the second beam in the periphery with the displacement to the wall of the copper water-cooled solidification mould.

Both the total power [w.sub.1] + [w.sub.2] and the power distribution between the electron beam guns (Table 1) were varied. The metal was supplied in portions into the solidification mould.

Table 1. Conditions of electron beam heating of the surface of VT 1-0 ingots with a diameter of 600 mm in the solidification mould Power, kW Liquid bath depth, mm Deviation of calculated from experimental values, % Regime Centre Periphery Calculated Experiments No [H.sub.c] [H.sub.e] 1 200 120 142 150 5.3 2 160 80 131 139 5.7 3 100 110 89 80 11.2

The experimental results show that the depth and volume of the liquid metal pool continuously increase at the start of melting into a constant level indicating the establishment of the quasi-stationary regime in the upper part of the ingot which is not influenced by the variation of the thermal regime (3).

After the establishment of the quasi-stationary regime for the given melting rate and producing the ingots with the height of 1000 mm, the liquid metal was poured from the entire volume of the metal pool by melting the metal crust at the periphery of the ingot. The produced ingots were sectioned into two halves along the axis of symmetry (Figure 3).

The formation of the billet shows that they repeat with sufficient accuracy the shape of the calculated temperature field. The depth of measured from the outer edge of the head part of the ingot to the lowest point of the resultant crater (Figure 4). The results of the measurements are presented in Table 1.

[FIGURE 4 OMITTED]

The deviation of the calculated data from the experimental ones was determined from the following equation:

D = ([H.sub.e] - [H.sub.p]).100/[H.sub.p], where D is the deviation of the calculated values from the experimental data, %; [H.sub.e] is the experimental value of the depth of the liquid pool, mm; [H.sub.e] is the calculated depth of the liquid pool, mm.

Comparison of the experimental results with the results of mathematical modelling shows that, on the whole, the model reflects accurately the physical phenomenon of the formation of the liquid pool in the ingot. The calculated values of the depth of the liquid pools are in agreement with the experimentally measured values in the range 5-12%. The experiments shows that on the whole the model reflects with sufficient accuracy the variation of the depth of formation of the liquid pool with the variation of the power in heating the ingot.

References

(1) Bello G.P., et al., Probl. Spets. Elektrometalurgii, 1996, No. 4, 27--7.

(2.) Paton B.E., et al., Electron beam welding, Naukova dumka, Kiev, 1997.

(3.) Paton B.E., et al., Electron beam melting of titanium, Naukova Dumka, Kiev, 2006.

(4.) Lesnoi A.B., et al., Probl. Spets. Elektrometalurgii, 2001, No. 2, 17--1.

(5.) Kalinyuk A.N., et al., Probl. Spets. Elektrometalurgii, 2002, No. 1, 20--5.

(6.) Zhuk G.V., Probl. Spets. Elektrometalurgii, 1998, No. 2, 21--5.

(7.) Zhuk G.V., et al., Protsessy Lit'ya, 2003, No. 4, 79--2

(8.) Zhuk G.V., Sovremen. Elektrometallurgiya, 2008, No. 2, 17--0.

(9.) Trigub N.P., et al., Sovremen. Elektrometallurgiya,, 2007, No. 1, 11--4.

V.A. Berezos

E.O. Paton Electric Welding Institute, Kiev

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Author: | Berezos, V.A. |
---|---|

Publication: | Advances in Electrometallurgy |

Date: | Jul 1, 2010 |

Words: | 1519 |

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