simulation of the heat transfer process through treated metal, melted in a water-cooled crucible by...

Pergamon

Vacuum/volume 48/number Z/pages 143 to 14811997 0 1997 Elsevier Science Ltd

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Simulation of the heat transfer process through treated metal, melted in a water-cooled crucible by an electron beam K Vutova, V Vassileva and G Mladenov, Laboratory of Electron Beam Technologies, Institute of Electronics, Bulgarian Academy of Sciences, blvd. Tzarigradsko shosse 72, 1784 Sofia, Bulgaria

received 74 June 7996

A two-dimensional axisymmetrical steady-state base thermal model is developed for electron beam (EB) melting and refining. A computer simulation is carried out, according to the proposed model, for various powers of EB heat source. Temperature distributions and heat flows in a cylindrical copper ingot, confined in a copper water-cooled crucible are calculated. The calculated and experimental observed shapes and/or depths of the metal pools are compared and a reasonable agreement is observed. 0 1997 Elsevier Science Ltd. All rights reserved

Introduction Model

In despite of more than 30 years of development and substantial industrial growth in the field of electron beam (EB) melting and refining, there still exist a number of problems incompletely solved. Some of them are: real time assessment and control of the liquid metal pool’s depth and shape. The volume and the geometry characteristics of the metal pool during EB melting and refining determine to a high degree the impurities transport and the crystallization properties of the produced metal ingot. They are a function of many parameters such as the EB power and energy distribution, thermal and physical properties of the liquid and solid metal, the quantity and the heat content of the added (by pouring) metal to the ingot, the heat contact characteristics at the interfaces ingot-crucible and ingot-puller.

We consider steady-state, axisymmetric thermal flow in a cylindrical ingot, shown in Figure 1. The treated ingot is heated by an EB, distributed on a central part of the top surface of the ingot. The initial EB energy is reduced by the energy of the reflected electrons.

The temperature distribution in the limited region G, is described by Poisson equation:

$ x g x ( 7 xa x !& + a’T + p x g =.f’(x,y,z). dY2

(1)

In this paper we investigate the special case when only the top of the ingot is cast (no material is added to the ingot by pouring from the melting primary block or from the cold hearth). A steady-state thermal model is proposed for the process of EB melting and refining, under some special conditions. Calculations are done using this model and the obtained results are compared with corresponding experimental data. Useful conclusions are made for the specific application of the developed model. The so called ‘method of the disks” is an example of the practical use of the above case. Other examples of the studied processes are: remelting with small melting rates and long metal refining time (the overall processing time), the reversive process of the ingot evaporation for a moved strip coating, or the production of bulks by condensation from the generated EB’s vapours.

The value of the liquid thermal conductivity multiplication factor, used to simulate stirring or mixing processes in the melting, described in Ref. 2 is between 1 and 2. The molecular thermal

conductivity of the liquid phase is 1-2 times lower then the thermal conductivity of the solid phase.i Taking into account that our goal is to clarify the basic phenomena of the heating, using an EB source, the above facts make it possible to assume that the thermal conductivity is temperature independent.

Equation (1) must be solved under appropriate boundary conditions. We use cylindrical coordinates. Let us denote with G,, G,, G3 and G, (see Figure 1) the ingot surfaces (i.e. boundaries of the region G) as follows:

l G, is the ingot top surface; l G, is the molten pool side surface, which is in contact with the

inner copper, water cooled crucible’s surface;

143

K Vufova et a/: Simulation of the heat transfer process through treated metal

Centre spot

Figure 1. The boundary denotation of the ingot.

l G3 is the part of the ingot side surface, which is the interface between the ingot metal and the vacuum;

l G, is the bottom of the ingot.

Then the corresponding boundary conditions are given by the following equations:- at G,:

1,x $& = -P,+cXxox(T4-_T,4), (2)

- at G2:

-at G,:

aT 1, x +R = -axax(T4-T.;,),

- at G4:

(4)

where r and z are the coordinates, a is the emissivity, ~7 is the Stefan-Boltzman constant, 3, is the mean ingot thermal conductivity value, T is the temperature, T,, is the ingot surface temperature.

In the axisymmetrical case eqn (1) can be written as:

i a a a2T _ x-x TX- +--0. r ar ( ‘? f3r a22

We use a square grid of points with a step h(r, = ih, z, = jh, where i,j= 0, 1,2, ). The approximation of eqn (6) is:

- for r,, # 0:

WY + uT)ij = 0 2 (7)

- for r,, = 0:

A; = 2A,. (8)

Then we obtain:

(PA, + A,> T), = 0,

where:

144

(9)

1 UWi= 1 x(u,,m,-2uo+u,;+,).

h- t (11)

Results of the computer simulation

Figure 2 shows the typical temperature distribution in a copper ingot (the length is lOOmm), heated by an electron beam, irra- diating the central part of the top surface. The ingot’s diameter is 60mm and the diameter of the regularly heated spot, by the oscillating electron beam with a power of 10.03 kW, is 20mm. Figure 2 presents half of the ingot cross-section. The equi-temperature curve at 1300 K is the boundary between the liquid pool and the solid crystallized metal. The beam power in this case is chosen under the condition that the liquid metal pool is extended to cover the whole top surface of the ingot. The material properties, used in our computer simulations are listed in Table 1. Our assumptions during the calculations (Figures 2 and 3) are: (i) the two-phase or ‘mushy’ zone in the region between the solid and the liquid metal phases is negligible (with a small length, compared to that of the ingot size), (ii) the direct heat contact between the heated ingot and the inner wall of the copper crucible (G,) is negligible and the ingot side surface is cooled only by radiation losses, (iii) the beam power value is reduced by the energy of the back scattered and secondary electrons, (iv) the time variation of

---- 1500 K ..‘... 1160 K -.-.

20

1400 K - - 1000 K -- I- 1360K -8OOK ---- 1320 K - 385 K

- 1260K

A 0 10 20 30

Distance from the ingor centre (mm)

Figure 2. Temperature contours of the electron beam melted ingot

Table 1. Material characteristics, used in our calculations 3

Parameters Value

Thermal conductivity A, 318.1 (W/m x K) Melting temperature T 1356 (K)

K Vutova et al: Simulation of the heat transfer process through treated metal

I 00.00

c I 2 7 c

80.00

20.00 / -

0J.K ,- I

0.00 20.00 30.00

Distance from the. ingot centte (mm)

Figure 3. The liquid/solid boundaries, calculated at beam powers 7.5 kW (curve 1) and IO kW (curve 2). The energy losses at Gzand G3 are only radiative.

the energy flow surface distribution, due to the movement and the instability of the beam, as well as the changes of the pool surface smoothness and shape, are negligible, (v) the heat contact between the ingot bottom and the puller surface (G4) is ideal, (vi) the flows and stirring of the liquid metal in the pool are negligible, (i.e. the ingot has isotropic thermal conductivity) and (vii) the thermal conductivity is chosen to be constant for the liquid pool and solid metal ingot.

In Figure 3 shows the liquid pool shape as a function of the beam power density at the melt surface.

From Figures 4 and 5 one can see that the boundary (contact) conditions and the mean thermal conductivity value Iz of the ingot material, influence the shape of the liquid metal pool as well as the solidified metal shell thickness and shape. In the calculations, concerning Figure 4, we assume ideal heat conditions at the interfaces G, and G,. The results shown in Figures 5-8 are obtained, when we assume a non-ideal heat contact and a different initial value of I for the ingot. Non-ideal heat contact means that the temperature at the interface between the two materials changes abruptly; the interface surface size is bigger than the ideal contact area, but the heat flows from both sides of the contact boundary remain equal.

The surface geometry, the temperature drop and the heat contact parameters (i.e. resistance) of the interfaces between the melting ingot and water cooled surfaces of the crucible, and between the movable puller and the bottom of the ingot (G,, G,) are not well defined. In our calculations (Figures 5-8, Table 2) we assume, that the real heat flows Q, through the interface areas S, are less than the heat flows and the corresponding area surface

I00.00

X0.0(1

s 60.00

H ‘j 4 40.00

2o.w

0s~:

2 I T

I

0.00 20.00 30.00

Distance from the ingot centre (mm)

Figure 4. The calculated liquid/solid boundaries at different widths of the ideal heat contact: curve 1 at width 1 mm; and curve 2 at width 2mm. The beam power is 15 kW.

value (Qi, SJ in the case of ideal heat contact. A heat effectiveness coefficient C, I 1 can be defined as:

c~xQ,=Qr, (12) where:

Six ATi’ (13)

The heat flows distributions are given in Table 2 and partially in Figures 7 and 8. P,, P2, P, and P4 are the energy fluxes through the boundary surfaces G,, Gz, Gj and G+ Figures 7 and 8 represent the dependence of the heat flows Qb and Q, at the ingot bottom surface and at the interface ingot/crucible wall, respectively. These characteristics are calculated using 1 = 0.81 x 1, for the ingot. The results in Figure 7 are obtained at CH = 0.78 for the interface ingot/crucible. The value of the heat effectiveness coefficient C,., is 0.78 for the ingot bottom/puller in Figure 8.

The calculated depths of the liquid metal pool, measured at r = Omm are given in Figure 6.

Experimental results and discussion

The melting experiments with copper were done in an EB plant with a maximum beam power of 60 kW. The experimental conditions were the same as in the ‘disks melting method’.’ The scheme of the experiment is seen in Figure 9. The copper samples (4) (Figure 9) with a deformed crystallization structure (their diameters are 58 mm and their lengths are 60 mm) are situated in

145


I

0 20 30

I>istnnce from the ingot centre (mm)

Figure 5. The liquid/solid boundaries, calculated at a beam power 15 kW and a heat contact width 1 mm: curve 1 - ideal thermal contact; curve 2 - C, = 0.78 for Gz and Gq, 1 = 0.84 x 10; curve 3 - C, = 0.78 for G2 and Gq, i. = 0.83 x j.0.

01 I I I I I

5 I 9 II 13 15

Beam power (kW)

Figure 6. Liquid metal pool depth dependence on the heating electron beam power. The calculations is done: for power of 7 5 kW and 10 kW, assuming ideal heat contact at Gz and Gq; for 1.5 kW at the same conditions as in Figure 5, curve 2.

the cooled copper crucible (3) (Figure 9) and on the cooled surface of the puller (5) (Figure 9). During the first 2-3 min the upper copper sample is preheated, using a low power defocused beam. After that a beam of chosen power (the beam diameter is 20 mm) is directed to the central part of the upper surface of the highest

146

I-

,_

CH

Figure 7. Variation of the heat flow Qb through the ingot bottom G, vs the heat effectiveness coefficient C, for this interface. I = 0.81 x I,, CH = 0.78 for GZ. The beam power is 15 kW, the heat contact width is 1 mm.

CH Figure 8. Variation of the heat flow Qc through the interface ingot/crucible wall G, vs the heat effectiveness coefficient C, for this interface. 1 = 0.81 x &,, C, = 0.78 for G,. The beam power is IS kW, the heat contact width is 1 mm.

situated sample. The length of the EB treatment is chosen to be in the range of 10-30 min.

In Figure 10 are given the typical metallographic pictures of the cross sections of the copper samples after three different thermal treatments. In Table 3 are shown the obtained data for the melted surface areas, the weight of the evaporated metal G and the depth of the liquid metal pool.

From Table 2 one can see that the heat contact at the two interfaces, ingot-crucible and ingot-puller, is a very important parameter. The copper crucible wall and the puller surface can remove large parts of the input energy. This contact is nonideal, due to: the rough ingot surface and the fact that the protecting solid shell diameter is mechanically shortened by the solidified liquid metal, being in contact with the water cooled walls. During the real process of refining (when the ingot lengthens, the puller is withdrawn to keep the upper pool surface at a constant elev- ation) the mechanic deformations of this she11 decrease the heat contact at the interface ingot/crucible.

The comparison between calculated and experimental depth’s values of the liquid pool (Figure 6 and Table 3) makes it possible to conclude that the mode1 used and the assumed approximations are rather good in the case, when the value of the beam power is 10-15 kW. For the lower beam power the energy losses through the boundary Gz represent a considerable part of the input heat energy. Thus, the exact value choice of the parameters in the region of the non-idea1 contact is of great significance.

The values in Table 3 and the results, obtained from our metallographic investigations are a verification of the ‘method of

K V&ova et a/: Simulation of the heat transfer process through treated metal

Table 2. Heat flows at different EB powers and heat contact conditions

N EB power I

(kW)

Heat contact at G2 Conditions at Pi (W) p, (W) p3 (W) p4 (W)

G,

1 1.5 J .0 not (only radiation) ideal 7340 - 260 7090

‘) 2 10.00 & -II- fFzal= 9650 - 510 8850

3 10.03 Al -II- ideal 9676 - 510 8865

4 15.00 I, ideal, hi = 1 mm ideal 14580 5520 430 8430

5 15.00 0.84& Cn = 0.78, h = 1 mm Cu = 0.78 14370 5700 600 7650

6 15.00 0.831” C, = 0.78, h = 1 mm Cn = 0.78 14350 5800 600 7700

7 15.00 & ideal, h, = 2 mm ideal 14753 7700 200 6400

1 -Vacuum chamber 2 -Observation window 3 -Water cooled crucible 4 - Copper samples 5 -Puller 6 -Electron gun 7 -Electron beam 8 -Water circuit

Figure 9. The scheme for the electron beam melting, using the ‘disks melting method’.

the disks’.’ We can conclude, that (i) the upper ingot area must not be totally melted for correct data observation and (ii) the use of the shorter experimental time is preferable. In the opposite case the growth of the dimensions of the solid grains of the base metallographic structure, leads to a similarity with the melted grains dimensions of the ingot metal.

Conclusions

A steady-state base thermal model of the EB melting and refining of cylindrical ingots for various power densities is developed. The distributions of the temperatures, heat flow values, some shapes and depths of the liquid metal pool are calculated, using different process parameters.

Some experiments with short ingots are produced to check the main assumptions in the developed model. The accordance in the liquid pool depth is satisfactory when the beam power exceeds 10 kW. In all the cases the non-ideal heat contact parameters are important, but under 10 kW limit the uncertainty of these

Figure 10. Metallographic pictures of the upper part of copper ingots. (a) at a beam power 7.5 kW, (b) at a beam power 10 kW and (c) at a beam power 15 kW. The beam heating time is lOmin, the beam spot is 20mm.

parameters is of a considerable importance for the calculated and experimental data declinations.

A verification and some conclusions for the ‘method of the disks’ melting were done. The factors that can be taken into

147


Table 3. Experimental data for S and h of the pool, and the evaporated ingot weight G

f (min) 10 20 30 P (kW

7.5 S (cm*) 25.44 28.27 28.27 G (g) 0.60 0.99 1.95 h (cm) 2.46 1.51 2.52

10.0 S (cm*) 26.38 28.27 28.21 G (g) 3.89 7.83 9.01 h (cm) 2.30 1.75 2.34

15.0 S (cm*) 28.27 28.27 28.27 G (g) 14.52 20.80 21.90 h (cm) 1.96 2.19 2.52

account in the further improvements of the described base model are: (i) the addition of the liquid metal in the crucible, coming

from the melted starting block or from the water cooled hearth (metal pouring); (ii) the liquid metal fluxes in the pool (liquid metal stirring or convection) and (iii) the influence of the mushy or two-phase zone (between the liquid and the solid phases) on the crystallization processes.

Acknowledgements

The authors gratefully acknowledge the financial support of the Bulgarian National Fund of Scientific Investigations at the Min- istry of Education, Science and Technologies.

References

1. Kurapov, Vu. A., Physics and chemistry of material treatment , 1969, 1, 58-63 (in Russian). 2. Tripp, D. and Mitchell, A., Thermal Regime in an EB Hearth. Proc. Int. Conf. Electron Beam Melting and Refining-State of the Art 1985, Reno, Nevada, USA, 1985 14(B). 3. G. V. Samsonov, Chemo-physical Properties of Elements. Naukova dumka Publ. House, Kiev, 1965 (in Russian).

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