- Email: [email protected]
Computer simulation of the heat transfer during electron beam melting and refining
Vacuum 53 (1999) 87 — 91
Computer simulation of the heat transfer during electron beam melting and refining K. Vutova*, G. Mladenov Bulgarian Academy of Sciences, Institute of Electronics, Blvd. Tzarigradsko shosse 72, 1784 Sofia, Bulgaria
Abstract A computer model and software for simulation of electron beam melting are developed. A two-dimensional modeling is done, for an ingot, casted in a cylindrical copper water-cooled crucible. Melting of copper, titanium and aluminum were simulated. The pouring of the molten material increases the energy input and the depth of the molten pool. In the case of titanium, due to the molten metal stirring an assumption for limitation of upper surface ingot temperatures was done. The important role of the temperature distribution, near to cooled crucible wall contact zone, in the process of thermal balance is discussed. Some qualitative conclusions for non-steady thermal transfer in this zone are made. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Electron beam melting and refining; Heat transfer; Heat flow; Molten pool
1. Introduction The optimization of vacuum metallurgy processes is a way to achieve production of high-quality metals or coated materials. The results of electron beam (EB) melting and refining of the metals and alloys are in close connection with the: temperature distribution in the treated ingot, melting pool shape, volume and convection of the liquid metal. The heating energy input and speed of melting, are main controllable parameters for obtaining alloys with tight composition specification. The liquid metal pool’s geometry depends on EB power and energy allocation, heat flows distribution, thermal and physical material properties. The ratio of the pool’s volume and the surface area, and the stirring in the liquid metal pool, define to a high degree the process of impurities transport and metal refining. The shape of the liquid/crystallized material interface, together with the temperature gradients in adjacent zones determine the parameters of the casted ingot structure. To minimize macrosegregation process parameters we must provide thin mushy-zone and small curvature of the crystallization front. In our previous papers [1, 2] a steady-state model for computer simulation of a metal ingot casting in a cylin-
* Corresponding author. Tel.: 00359 2 7144680; fax: 00359 2 9753201; e-mail: [email protected]
drical cooper water-cooled crucible, with a movable water cooled bottom was built. The case of EB melting without adding of material was studied. In this paper, this model is extended for the case of pouring of material into the crucible, by the melted droplets, from a primary block or by liquid metal flow, over the weir from the cold heart (thermal energy of the added heated liquid metal is taken into account). The thermal distributions in the heated ingots for copper, aluminum and titanium at beam power ranging between 7.5 and 100 kW were calculated. It is shown that the heat contact in the ingot/cooled crucible and ingot/cooled bottom interfaces determines the liquid metal temperatures. In such a way they have definitive effect on the ingot crystallization and refining as well as for the energy efficiency of the technology process.
2. Two-dimensional heat model The steady-state axisymmetrical thermal flow in a cylindrical ingot, confined in a water-cooled crucible can be calculated as follows. Let the electron beam be distributed uniformly on the central part of the top surface of the ingot. The used EB energy input is reduced by the energy of the reflected electrons. The temperature distribution in the ingot cross-section can be
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 9 8 - 4
88
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
described by 1/r*/*r(r*¹/*r)#*¹/*z#»/a *¹/*z"0,
(1)
where r and z are the cylindrical coordinates, » is the casting velocity of the ingot, a is the temperature diffusivity determined as a"j/C o where C is the mean specific heat for the temperature range between the room and fusion temperatures, o is the material density, j is the material thermal conductivity. The last term in Eq. (1) takes into account the transfer of thermal energy from the material, moving with velocity », coincident to the z-axis. The value of the liquid thermal conductivity multiplication factor, used to simulate stirring and mixing in the melting pool, is between 1 and 2 [3]. In this case we consider the value of the ingot material’s thermal conductivity to be equal to the value of the solid’s thermal conductivity at an intermediate (between room and melting point) temperature and to be independent from the ingot’s thermal distribution. The volume heat drains at the crystallization front, due to the latent heat of melting, are assumed to be negligible. The boundary conditions can be formulated on the base of the assumptions of three heat transfer mechanisms. An ideal heat contact and heat transfer by thermal conductivity exist only between the liquid metal and cooled crucible at a narrow upper part of the interface, between the ingot’s side wall and the inner crucible wall. A Newton heat flux, which is proportional to the temperature difference ¹ !¹ (where ¹ is the wall temper ature and ¹ is the surrounding temperature), occurs at the solid/vacuum/solid interfaces including the narrow gaps between crystallized ingot surface and a nearly situated other surface. This kind of heat transfer occurs in areas of the nonideal heat contact, between the solid ingot side wall and the cooled crucible inner wall, as well as at the ingot’s bottom. The absence of ideal contact is a result of the solidified ingot shrinkage. The third mechanism — heat transfer through a radiation flux, can be determined by Stefan—Boltzman equation. Due to the higher temperatures, the radiative flux is more considerable at the top surface of the ingot and is observable at the side wall. Let us denote the areas of the ingot boundaries in which different heat transfer mechanisms can be assumed as follows: (i) S — ideal thermal contact area in which thermal conductivity take place, (ii) S — the area where Newton heat interchange occurs between surfaces at close distance through gas molecules, (iii) S — heat radiation area. Then the boundary condition equation can be written as j *¹ /*r(S #S #S ) "j *¹ /*r S #a(¹ !¹ )S #ep(¹!¹)S , (2)
where ¹ and ¹ are the temperatures from both sides of the boundary, respectively, j and j are the thermal conductivity of the molten material and of the copper crucible respectively, a is the heat transfer coefficient, e is the emissivity, and p is the Stefan—Boltzman constant.
3. Results of simulation with 2D heat model Fig. 1 shows the typical temperature distributions for an aluminum ingot with diameter 60 mm, heated by electron beam with a power P"7.5 kW (after the correction for the reflected electron energy losses). The heated area is a circle of diameter 20 mm on the top surface and the other computation parameters are given in Table 1. The heat fluxes for the EB melting and refining of Al at three casting velocities are summarized in Table 2, together with the assumed heat contact conditions. Note that Q means the heat, added by the pouring metal; P is the total heating power at the top ingot surface, taking in account the radiation losses; P is the thermal losses at the pool/crucible interface. The first and second terms of Eq. (2) determine the heat losses through the areas S and S or through the corresponding contact rings
Fig. 1. Temperature contours and the liquid/solid boundaries, calculated at a beam power P"7.5 kW for Al ingot. ¸ is the ingot length, 2R is the ingot diameter, 2R is the molten pool diameter, h is the width of the solidified skin ring, h is the depth of the molten pool, measured at the ingot axis.
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
89
Table 1 Material characteristics, used in calculations [4] No.
Parameters
1 2 3 4 5
Values
Thermal conductivity Melting temperature, ¹
Heat capacity, C Thermal diffusivity, a Heat content, C o¹
Dimensions
Cu
Ti
Al
318.1 (at 1280 K) 1356 0.38 1.13;10\ 4612
13 (at 973 K) 1938 0.58 0.534;10\ 6120
184.5 (at 920 K) 823 1.087 0.63;10\ 2415.5
(W/m K) (K) (J/g K) (m/s) (J/cm)
Table 2 Calculated fluxes and solidified scin dimensions for EB melting of Al with P"7.5 kW C & crucible
h* (mm)
Puller C &
161
0.88
3
0.55
7555
6
946
1.0
3
0.55
8300
9
2710
1.0
3
0.6
No.
» (mm/min)
1
3
2 3
Q (W)
P (W)
10 000
P (W) 5470 71.4% 6050 71.2% 6250 61.2%
P (W) 60 80 120
*h (mm)
P (W)
R 1 (mm)
h (mm)
h 1 (mm)
2075 27.1% 2000 27.5% 2230 21.9%
23
28
14
1
24
29
12.5
6
25
30
10
12
h 1 (mm)
*h (mm)
Table 3 Calculated fluxes and solidified skin dimensions for EB melting of Ti with P"10 kW C & crucible
h* (mm)
Puller C &
P (W)
P (W)
P (W)
P (W)
R 1 (mm)
h (mm)
486
1.0
5
1.7
10 400
1490
30
9
6
973
1.0
7
1.7
10 900
25.5
32
13
7
9
1460
1.0
8
1.7
11 400
1500 14.3% 1550 14.1% 1640 14.3%
25
6
5800 55.3% 6300 57.4% 8400 73.3%
26
34
12
10
No.
» (mm/min)
1
3
2 3
Q (W)
with heights h and h , respectively. These heat fluxes are a main part of the total heat flux through the crucible water. It is assumed that their values are interconnected and cannot be determined independently. Taking into account the nonideal contact, we can reduce the choice of parameters: S , S , corresponding to h , h , to the choice of a common h* (or S*). A heat effectiveness coefficient C can be defined as a ratio between the heat flows at & a real contact, taking into account the non-steady nature of this flow, and an ideal and Newton heat contacts at the corresponding boundary. P is the radiative flux from the rest side ingot’s wall and P is the flux at the ingot’s bottom/puller interface. The heat fluxes for the EB melting and refining of Ti, at the same three casting velocities as in the case of Al ingot, are summarized in Table 3, together with the assumed
1520 1760
heat contact conditions. In this case, according to the experimental data we had to eliminate the radiation energy losses from the ingot’s top surface. In the real process, the convection flow of the liquid metal considerably diminuates the surface temperatures. The calculations for EB melting of Cu at the beam power 100 kW are given in Table 4. It can be noted an unexpected break in the solidified skin around the molten pool, observed in all cases of computer simulation of the ingot/crucible interface. This molten part of the side ingot wall (with width of *h) is bellow the top ring, where it is assumed an ideal heat contact, causing its solidification. The change of the ideal contact ring’s width do not modify this situation. If we extend the size of this contact area, we will intensify the energy losses up to unacceptable values (in comparison
90
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
Table 4 Calculated fluxes and solidified skin dimensions for EB melting of copper with P"100 kW No.
» (mm/min)
Q (W)
C & crucible
0.9 0.7
h* (mm)
Puller C &
P (W)
P (W)
P (W)
P (W)
20
0
83 500
66 600 66.6% 67 300 66.3% 69 200 65.4%
300
16 700 23 16.7% 17 000 23 16.75% 17 700 23 16.73%
1
1
2
6
1500
0.7
20
0.9
84 300
3
20
5800
0.7
20
0.9
86 400
with the experimentally obtained losses). To overleap this paradox in 2D calculations, one can divide the cooling ring area into two different rings with approximately ideal heat contact — one at the top ingot surface and the second — in the middle of the observed molten part of the side ingot wall. Such second contact ring below the top liquid metal/crucible contact ring was observed experimentally in [5] for the vacuum arc melting of big ingots (were it is easy to be observed).
4. Comments on situation at the ingot/crucible contact boundary The boundary contact between the casting ingot and crucible can be an ideal heat contact, if the temperatures at all contact points are the same. In the case of nonideal contact, the temperatures at the interface between two materials changes abruptly. In the case of 3D non-steady approach, we can assume that there are areas with an ideal thermal contact. The direct contact between the liquid and solid material causes the existence of an ideal thermal contact. It can be assumed only for a short time and for small segment surfaces from the ingot’s side surface upper part. After that, these surfaces transfer the heat to the liquid metal interface and solidify. The molten pool in this segment does not reach more the cooled wall. In this time we can observe temporally breaks of the solidified skull of the molten pool in the neighboring areas. These breaks are caused by the increased heating from the melted metal to the surrounding area. It is a result of the local heat flow diminuation through the shrank solidified segment. The roughness of the ingot side wall is connected with this shrinkage and breakage of neighboring spots of solidified skull. Experimentally non-steady heat flows and temperatures can be measured in the contact region. In [6] data for a in the case of liquid metal in good contact with a metal wall (of order of 10 W/m K) and for frozen metal in good contact with metal mold wall (10 W/m K) are given. Our calculated values of the heat flow density are in the range of 5000—20 000 kW/m. If we assume a temperature drop of 1000 K, the corres-
300 400
R 1 (mm)
h (mm)
h 1 (mm)
*h (mm)
52
29
22
52
27
23
55
28
25
ponding values of heat transfer coefficient a are it the same order with the mentioned data for the liquid/solid contact case. It can be noted, that the cristalization process of the casted block scin is more probable, in the case of higher heat flow value, due to the mentioned non-steady nature of the contact processes. This means, that the above cases are an abstract names only — in the first case, the solid/solid contact take place at longer time, than the liquid/solid contact. An exact heat model must take in account the nonsteady character of the heat transfer in the contact area. In the case of liquid metal fluxes simulation, this phenomenon will give additionally a reason for 3D turbulence in crucible/ingot interface proximity region.
5. Conclusions The proposed model and computer programs can be used for the development of a mathematical model, including liquid metal fluxes simulation. The simulation results are very useful when optimizing particular technological processes in EB melting and refining. As an extension to the real 3D non-steady cases, where operator moves the liquid metal on top of the melted ingot surface, by moving the heated spot of the deflecting electron beam, some parts of contacted area’s segments, are assumed to be temporally in ideal thermal contact, after which they quickly solidify and shrink, making non-ideal heat contacts with crucible. Sometimes, neighboring ingot wall spots melt, make an ideal thermal contact and transfer heat by the thermal conductivity. Then these areas are cooled and decrease their heat transfer. Some qualitative conclusions for non-steady thermal transfer in this zone are made.
Acknowledgements The authors gratefully acknowledge the financial support of the Bulgarian National Fund of Scientific Investigations at the Ministry of Education, Science and Technologies, supporting this work (MM-518).
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
References [1] Vutova K, Vassileva V, Mladenov G. Simulation of the heat transfer process through treated metal, melted in a water-cooled crucible by an electron beam. Vacuum 1997;48(2):143—8. [2] Vutova K, Mladenov G, Vassileva V. Computer simulation of the heat processes at electron beam melting of copper. Proc Nat Conf ‘‘Electronica’96’’, Botevgrad, 10—11 October 1996: 173—7 (in Bulgarian). [3] Tripp D, Mitchell A. Thermal regime in an EB hearth. Proc Int Conf on Electron Beam Melting and Refining — State-of-the-Art 1985, Reno, Nevada, USA, 1985, (II): 14—9.
91
[4] Samsonov GV. Chemo-physical properties of elements. Kiev: Naukova Dumka Publ House, 1965 (in Russian). [5] Shalimov AG, Gotin WN, Toulin NA. Intensification of the processes of the special metallurgy. Moscow: Metallurgia Publ House, 1988:273—80 (in Russian). [6] Ransing RS, Zheng Y, Lewis RW. Potential applications of intelligent preprocessing in the numerical simulation of casting. In: Lewis RW, editor. Numerical methods in thermal problems, vol. VIII, Pt. 2. Swansea: Pineridge Press, 1993:361—75.
Computer simulation of the heat transfer during electron beam melting and refining K. Vutova*, G. Mladenov Bulgarian Academy of Sciences, Institute of Electronics, Blvd. Tzarigradsko shosse 72, 1784 Sofia, Bulgaria
Abstract A computer model and software for simulation of electron beam melting are developed. A two-dimensional modeling is done, for an ingot, casted in a cylindrical copper water-cooled crucible. Melting of copper, titanium and aluminum were simulated. The pouring of the molten material increases the energy input and the depth of the molten pool. In the case of titanium, due to the molten metal stirring an assumption for limitation of upper surface ingot temperatures was done. The important role of the temperature distribution, near to cooled crucible wall contact zone, in the process of thermal balance is discussed. Some qualitative conclusions for non-steady thermal transfer in this zone are made. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Electron beam melting and refining; Heat transfer; Heat flow; Molten pool
1. Introduction The optimization of vacuum metallurgy processes is a way to achieve production of high-quality metals or coated materials. The results of electron beam (EB) melting and refining of the metals and alloys are in close connection with the: temperature distribution in the treated ingot, melting pool shape, volume and convection of the liquid metal. The heating energy input and speed of melting, are main controllable parameters for obtaining alloys with tight composition specification. The liquid metal pool’s geometry depends on EB power and energy allocation, heat flows distribution, thermal and physical material properties. The ratio of the pool’s volume and the surface area, and the stirring in the liquid metal pool, define to a high degree the process of impurities transport and metal refining. The shape of the liquid/crystallized material interface, together with the temperature gradients in adjacent zones determine the parameters of the casted ingot structure. To minimize macrosegregation process parameters we must provide thin mushy-zone and small curvature of the crystallization front. In our previous papers [1, 2] a steady-state model for computer simulation of a metal ingot casting in a cylin-
* Corresponding author. Tel.: 00359 2 7144680; fax: 00359 2 9753201; e-mail: [email protected]
drical cooper water-cooled crucible, with a movable water cooled bottom was built. The case of EB melting without adding of material was studied. In this paper, this model is extended for the case of pouring of material into the crucible, by the melted droplets, from a primary block or by liquid metal flow, over the weir from the cold heart (thermal energy of the added heated liquid metal is taken into account). The thermal distributions in the heated ingots for copper, aluminum and titanium at beam power ranging between 7.5 and 100 kW were calculated. It is shown that the heat contact in the ingot/cooled crucible and ingot/cooled bottom interfaces determines the liquid metal temperatures. In such a way they have definitive effect on the ingot crystallization and refining as well as for the energy efficiency of the technology process.
2. Two-dimensional heat model The steady-state axisymmetrical thermal flow in a cylindrical ingot, confined in a water-cooled crucible can be calculated as follows. Let the electron beam be distributed uniformly on the central part of the top surface of the ingot. The used EB energy input is reduced by the energy of the reflected electrons. The temperature distribution in the ingot cross-section can be
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 9 8 - 4
88
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
described by 1/r*/*r(r*¹/*r)#*¹/*z#»/a *¹/*z"0,
(1)
where r and z are the cylindrical coordinates, » is the casting velocity of the ingot, a is the temperature diffusivity determined as a"j/C o where C is the mean specific heat for the temperature range between the room and fusion temperatures, o is the material density, j is the material thermal conductivity. The last term in Eq. (1) takes into account the transfer of thermal energy from the material, moving with velocity », coincident to the z-axis. The value of the liquid thermal conductivity multiplication factor, used to simulate stirring and mixing in the melting pool, is between 1 and 2 [3]. In this case we consider the value of the ingot material’s thermal conductivity to be equal to the value of the solid’s thermal conductivity at an intermediate (between room and melting point) temperature and to be independent from the ingot’s thermal distribution. The volume heat drains at the crystallization front, due to the latent heat of melting, are assumed to be negligible. The boundary conditions can be formulated on the base of the assumptions of three heat transfer mechanisms. An ideal heat contact and heat transfer by thermal conductivity exist only between the liquid metal and cooled crucible at a narrow upper part of the interface, between the ingot’s side wall and the inner crucible wall. A Newton heat flux, which is proportional to the temperature difference ¹ !¹ (where ¹ is the wall temper ature and ¹ is the surrounding temperature), occurs at the solid/vacuum/solid interfaces including the narrow gaps between crystallized ingot surface and a nearly situated other surface. This kind of heat transfer occurs in areas of the nonideal heat contact, between the solid ingot side wall and the cooled crucible inner wall, as well as at the ingot’s bottom. The absence of ideal contact is a result of the solidified ingot shrinkage. The third mechanism — heat transfer through a radiation flux, can be determined by Stefan—Boltzman equation. Due to the higher temperatures, the radiative flux is more considerable at the top surface of the ingot and is observable at the side wall. Let us denote the areas of the ingot boundaries in which different heat transfer mechanisms can be assumed as follows: (i) S — ideal thermal contact area in which thermal conductivity take place, (ii) S — the area where Newton heat interchange occurs between surfaces at close distance through gas molecules, (iii) S — heat radiation area. Then the boundary condition equation can be written as j *¹ /*r(S #S #S ) "j *¹ /*r S #a(¹ !¹ )S #ep(¹!¹)S , (2)
where ¹ and ¹ are the temperatures from both sides of the boundary, respectively, j and j are the thermal conductivity of the molten material and of the copper crucible respectively, a is the heat transfer coefficient, e is the emissivity, and p is the Stefan—Boltzman constant.
3. Results of simulation with 2D heat model Fig. 1 shows the typical temperature distributions for an aluminum ingot with diameter 60 mm, heated by electron beam with a power P"7.5 kW (after the correction for the reflected electron energy losses). The heated area is a circle of diameter 20 mm on the top surface and the other computation parameters are given in Table 1. The heat fluxes for the EB melting and refining of Al at three casting velocities are summarized in Table 2, together with the assumed heat contact conditions. Note that Q means the heat, added by the pouring metal; P is the total heating power at the top ingot surface, taking in account the radiation losses; P is the thermal losses at the pool/crucible interface. The first and second terms of Eq. (2) determine the heat losses through the areas S and S or through the corresponding contact rings
Fig. 1. Temperature contours and the liquid/solid boundaries, calculated at a beam power P"7.5 kW for Al ingot. ¸ is the ingot length, 2R is the ingot diameter, 2R is the molten pool diameter, h is the width of the solidified skin ring, h is the depth of the molten pool, measured at the ingot axis.
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
89
Table 1 Material characteristics, used in calculations [4] No.
Parameters
1 2 3 4 5
Values
Thermal conductivity Melting temperature, ¹
Heat capacity, C Thermal diffusivity, a Heat content, C o¹
Dimensions
Cu
Ti
Al
318.1 (at 1280 K) 1356 0.38 1.13;10\ 4612
13 (at 973 K) 1938 0.58 0.534;10\ 6120
184.5 (at 920 K) 823 1.087 0.63;10\ 2415.5
(W/m K) (K) (J/g K) (m/s) (J/cm)
Table 2 Calculated fluxes and solidified scin dimensions for EB melting of Al with P"7.5 kW C & crucible
h* (mm)
Puller C &
161
0.88
3
0.55
7555
6
946
1.0
3
0.55
8300
9
2710
1.0
3
0.6
No.
» (mm/min)
1
3
2 3
Q (W)
P (W)
10 000
P (W) 5470 71.4% 6050 71.2% 6250 61.2%
P (W) 60 80 120
*h (mm)
P (W)
R 1 (mm)
h (mm)
h 1 (mm)
2075 27.1% 2000 27.5% 2230 21.9%
23
28
14
1
24
29
12.5
6
25
30
10
12
h 1 (mm)
*h (mm)
Table 3 Calculated fluxes and solidified skin dimensions for EB melting of Ti with P"10 kW C & crucible
h* (mm)
Puller C &
P (W)
P (W)
P (W)
P (W)
R 1 (mm)
h (mm)
486
1.0
5
1.7
10 400
1490
30
9
6
973
1.0
7
1.7
10 900
25.5
32
13
7
9
1460
1.0
8
1.7
11 400
1500 14.3% 1550 14.1% 1640 14.3%
25
6
5800 55.3% 6300 57.4% 8400 73.3%
26
34
12
10
No.
» (mm/min)
1
3
2 3
Q (W)
with heights h and h , respectively. These heat fluxes are a main part of the total heat flux through the crucible water. It is assumed that their values are interconnected and cannot be determined independently. Taking into account the nonideal contact, we can reduce the choice of parameters: S , S , corresponding to h , h , to the choice of a common h* (or S*). A heat effectiveness coefficient C can be defined as a ratio between the heat flows at & a real contact, taking into account the non-steady nature of this flow, and an ideal and Newton heat contacts at the corresponding boundary. P is the radiative flux from the rest side ingot’s wall and P is the flux at the ingot’s bottom/puller interface. The heat fluxes for the EB melting and refining of Ti, at the same three casting velocities as in the case of Al ingot, are summarized in Table 3, together with the assumed
1520 1760
heat contact conditions. In this case, according to the experimental data we had to eliminate the radiation energy losses from the ingot’s top surface. In the real process, the convection flow of the liquid metal considerably diminuates the surface temperatures. The calculations for EB melting of Cu at the beam power 100 kW are given in Table 4. It can be noted an unexpected break in the solidified skin around the molten pool, observed in all cases of computer simulation of the ingot/crucible interface. This molten part of the side ingot wall (with width of *h) is bellow the top ring, where it is assumed an ideal heat contact, causing its solidification. The change of the ideal contact ring’s width do not modify this situation. If we extend the size of this contact area, we will intensify the energy losses up to unacceptable values (in comparison
90
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
Table 4 Calculated fluxes and solidified skin dimensions for EB melting of copper with P"100 kW No.
» (mm/min)
Q (W)
C & crucible
0.9 0.7
h* (mm)
Puller C &
P (W)
P (W)
P (W)
P (W)
20
0
83 500
66 600 66.6% 67 300 66.3% 69 200 65.4%
300
16 700 23 16.7% 17 000 23 16.75% 17 700 23 16.73%
1
1
2
6
1500
0.7
20
0.9
84 300
3
20
5800
0.7
20
0.9
86 400
with the experimentally obtained losses). To overleap this paradox in 2D calculations, one can divide the cooling ring area into two different rings with approximately ideal heat contact — one at the top ingot surface and the second — in the middle of the observed molten part of the side ingot wall. Such second contact ring below the top liquid metal/crucible contact ring was observed experimentally in [5] for the vacuum arc melting of big ingots (were it is easy to be observed).
4. Comments on situation at the ingot/crucible contact boundary The boundary contact between the casting ingot and crucible can be an ideal heat contact, if the temperatures at all contact points are the same. In the case of nonideal contact, the temperatures at the interface between two materials changes abruptly. In the case of 3D non-steady approach, we can assume that there are areas with an ideal thermal contact. The direct contact between the liquid and solid material causes the existence of an ideal thermal contact. It can be assumed only for a short time and for small segment surfaces from the ingot’s side surface upper part. After that, these surfaces transfer the heat to the liquid metal interface and solidify. The molten pool in this segment does not reach more the cooled wall. In this time we can observe temporally breaks of the solidified skull of the molten pool in the neighboring areas. These breaks are caused by the increased heating from the melted metal to the surrounding area. It is a result of the local heat flow diminuation through the shrank solidified segment. The roughness of the ingot side wall is connected with this shrinkage and breakage of neighboring spots of solidified skull. Experimentally non-steady heat flows and temperatures can be measured in the contact region. In [6] data for a in the case of liquid metal in good contact with a metal wall (of order of 10 W/m K) and for frozen metal in good contact with metal mold wall (10 W/m K) are given. Our calculated values of the heat flow density are in the range of 5000—20 000 kW/m. If we assume a temperature drop of 1000 K, the corres-
300 400
R 1 (mm)
h (mm)
h 1 (mm)
*h (mm)
52
29
22
52
27
23
55
28
25
ponding values of heat transfer coefficient a are it the same order with the mentioned data for the liquid/solid contact case. It can be noted, that the cristalization process of the casted block scin is more probable, in the case of higher heat flow value, due to the mentioned non-steady nature of the contact processes. This means, that the above cases are an abstract names only — in the first case, the solid/solid contact take place at longer time, than the liquid/solid contact. An exact heat model must take in account the nonsteady character of the heat transfer in the contact area. In the case of liquid metal fluxes simulation, this phenomenon will give additionally a reason for 3D turbulence in crucible/ingot interface proximity region.
5. Conclusions The proposed model and computer programs can be used for the development of a mathematical model, including liquid metal fluxes simulation. The simulation results are very useful when optimizing particular technological processes in EB melting and refining. As an extension to the real 3D non-steady cases, where operator moves the liquid metal on top of the melted ingot surface, by moving the heated spot of the deflecting electron beam, some parts of contacted area’s segments, are assumed to be temporally in ideal thermal contact, after which they quickly solidify and shrink, making non-ideal heat contacts with crucible. Sometimes, neighboring ingot wall spots melt, make an ideal thermal contact and transfer heat by the thermal conductivity. Then these areas are cooled and decrease their heat transfer. Some qualitative conclusions for non-steady thermal transfer in this zone are made.
Acknowledgements The authors gratefully acknowledge the financial support of the Bulgarian National Fund of Scientific Investigations at the Ministry of Education, Science and Technologies, supporting this work (MM-518).
K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91
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