Vacuum 53 (1999) 87—91

Computer simulation of the heat transfer duringelectron 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 aningot, casted in a cylindrical copper water-cooled crucible. Melting of copper, titanium and aluminum were simulated. The pouring ofthe molten material increases the energy input and the depth of the molten pool. In the case of titanium, due to the molten metalstirring an assumption for limitation of upper surface ingot temperatures was done. The important role of the temperaturedistribution, near to cooled crucible wall contact zone, in the process of thermal balance is discussed. Some qualitative conclusions fornon-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 isa way to achieve production of high-quality metals orcoated materials. The results of electron beam (EB)melting and refining of the metals and alloys are in closeconnection with the: temperature distribution in thetreated ingot, melting pool shape, volume and convectionof the liquid metal. The heating energy input and speed ofmelting, are main controllable parameters for obtainingalloys with tight composition specification. The liquidmetal pool’s geometry depends on EB power and energyallocation, heat flows distribution, thermal and physicalmaterial properties. The ratio of the pool’s volume andthe surface area, and the stirring in the liquid metal pool,define to a high degree the process of impurities transportand metal refining. The shape of the liquid/crystallizedmaterial interface, together with the temperature gradi-ents in adjacent zones determine the parameters of thecasted ingot structure. To minimize macrosegregationprocess parameters we must provide thin mushy-zoneand small curvature of the crystallization front.

In our previous papers [1, 2] a steady-state model forcomputer simulation of a metal ingot casting in a cylin-

*Corresponding author. Tel.: 00359 2 7144680; fax: 00359 2 9753201;e-mail: vutova@iegate.ie.bas.bg

drical cooper water-cooled crucible, with a movablewater cooled bottom was built. The case of EB meltingwithout adding of material was studied.

In this paper, this model is extended for the case ofpouring of material into the crucible, by the melteddroplets, from a primary block or by liquid metalflow, over the weir from the cold heart (thermal energyof the added heated liquid metal is taken into account).The thermal distributions in the heated ingots forcopper, aluminum and titanium at beam power rang-ing between 7.5 and 100 kW were calculated. It isshown that the heat contact in the ingot/cooledcrucible and ingot/cooled bottom interfaces determinesthe liquid metal temperatures. In such a way they havedefinitive effect on the ingot crystallization and refiningas well as for the energy efficiency of the technologyprocess.

2. Two-dimensional heat model

The steady-state axisymmetrical thermal flow in acylindrical ingot, confined in a water-cooled cruciblecan be calculated as follows. Let the electron beam bedistributed uniformly on the central part of the topsurface of the ingot. The used EB energy input isreduced by the energy of the reflected electrons. Thetemperature 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

described by

1/rL/Lr(rL¹/Lr)#L2¹/Lz2#»/a L¹/Lz"0, (1)

where r and z are the cylindrical coordinates, » is thecasting velocity of the ingot, a is the temperature diffus-ivity determined as a"j/C

1o where C

1is the mean

specific heat for the temperature range between the roomand fusion temperatures, o is the material density, j is thematerial thermal conductivity.

The last term in Eq. (1) takes into account the transferof thermal energy from the material, moving with velo-city », coincident to the z-axis. The value of the liquidthermal conductivity multiplication factor, used to simu-late stirring and mixing in the melting pool, is between1 and 2 [3]. In this case we consider the value of the ingotmaterial’s thermal conductivity to be equal to the valueof the solid’s thermal conductivity at an intermediate(between room and melting point) temperature and to beindependent from the ingot’s thermal distribution. Thevolume heat drains at the crystallization front, due to thelatent heat of melting, are assumed to be negligible.

The boundary conditions can be formulated on thebase of the assumptions of three heat transfer mecha-nisms. An ideal heat contact and heat transfer by thermalconductivity exist only between the liquid metal andcooled 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 temper-ature difference ¹

8!¹

0(where ¹

8is the wall temper-

ature and ¹0

is the surrounding temperature), occurs atthe solid/vacuum/solid interfaces including the narrowgaps between crystallized ingot surface and a nearlysituated other surface. This kind of heat transfer occurs inareas of the nonideal heat contact, between the solidingot side wall and the cooled crucible inner wall, as wellas at the ingot’s bottom. The absence of ideal contact isa result of the solidified ingot shrinkage. The third mech-anism — heat transfer through a radiation flux, can bedetermined by Stefan—Boltzman equation. Due to thehigher temperatures, the radiative flux is more consider-able at the top surface of the ingot and is observable atthe side wall.

Let us denote the areas of the ingot boundaries inwhich different heat transfer mechanisms can be assumedas follows:(i) S

1— ideal thermal contact area in which thermal

conductivity take place,(ii) S

2— the area where Newton heat interchange occurs

between surfaces at close distance through gas mol-ecules,

(iii) S3

— heat radiation area.Then the boundary condition equation can be written

as

j1L¹@/Lr(S

1#S

2#S

3)

"j2L¹A/Lr S

1#a(¹

8!¹

0)S

2#ep (¹4

8!¹4

0)S

3, (2)

where ¹@ and ¹A are the temperatures from both sidesof the boundary, respectively, j

1and j

2are the

thermal conductivity of the molten material and of thecopper crucible respectively, a is the heat transfer coeffic-ient, e is the emissivity, and p is the Stefan—Boltzmanconstant.

3. Results of simulation with 2D heat model

Fig. 1 shows the typical temperature distributions foran aluminum ingot with diameter 60 mm, heated byelectron beam with a power P"7.5 kW (after the correc-tion for the reflected electron energy losses). The heatedarea is a circle of diameter 20 mm on the top surface andthe other computation parameters are given in Table 1.The heat fluxes for the EB melting and refining of Al atthree casting velocities are summarized in Table 2, to-gether with the assumed heat contact conditions. Notethat Q means the heat, added by the pouring metal; P

1is

the total heating power at the top ingot surface, taking inaccount the radiation losses; P

2is the thermal losses at

the pool/crucible interface. The first and second termsof Eq. (2) determine the heat losses through the areasS1

and S2

or through the corresponding contact rings

Fig. 1. Temperature contours and the liquid/solid boundaries, cal-culated at a beam power P"7.5 kW for Al ingot. ¸ is the ingot length,2R is the ingot diameter, 2R

4is the molten pool diameter, h

4is the width

of the solidified skin ring, h0

is the depth of the molten pool, measuredat the ingot axis.

88 K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91

Table 1Material characteristics, used in calculations [4]

No. Parameters Values Dimensions

Cu Ti Al

1 Thermal conductivity 318.1 (at 1280 K) 13 (at 973 K) 184.5 (at 920 K) (W/mK)2 Melting temperature, ¹

.1356 1938 823 (K)

3 Heat capacity, C1

0.38 0.58 1.087 (J/gK)4 Thermal diffusivity, a 1.13]10~4 0.534]10~5 0.63]10~4 (m2/s)5 Heat content, C

1o¹

.4612 6120 2415.5 (J/cm3)

Table 2Calculated fluxes and solidified scin dimensions for EB melting of Al with P"7.5 kW

No. »

(mm/min)Q(W)

CH

crucibleh*(mm)

PullerC

H

P1

(W)P2

(W)P3

(W)P4

(W)R

S(mm)

h0

(mm)hS

(mm)*h(mm)

1 3 161 0.88 3 0.55 7555 5470 60 2075 23 28 14 171.4% 27.1%

2 6 946 1.0 3 0.55 8300 6050 80 2000 24 29 12.5 671.2% 27.5%

3 9 2710 1.0 3 0.6 10 000 6250 120 2230 25 30 10 1261.2% 21.9%

Table 3Calculated fluxes and solidified skin dimensions for EB melting of Ti with P"10 kW

No. »

(mm/min)Q(W)

CH

crucibleh*(mm)

PullerC

H

P1

(W)P2

(W)P3

(W)P4

(W)R

S(mm)

h0

(mm)hS

(mm)*h(mm)

1 3 486 1.0 5 1.7 10 400 5800 1490 1500 25 30 9 655.3% 14.3%

2 6 973 1.0 7 1.7 10 900 6300 1520 1550 25.5 32 13 757.4% 14.1%

3 9 1460 1.0 8 1.7 11 400 8400 1760 1640 26 34 12 1073.3% 14.3%

with heights h1

and h2, respectively. These heat fluxes are

a main part of the total heat flux through the cruciblewater. It is assumed that their values are interconnectedand cannot be determined independently. Taking intoaccount the nonideal contact, we can reduce the choice ofparameters: S

1, S

2, corresponding to h

1, h

2, to the choice

of a common h* (or S*). A heat effectiveness coefficientC

Hcan be defined as a ratio between the heat flows at

a real contact, taking into account the non-steady natureof this flow, and an ideal and Newton heat contacts at thecorresponding boundary. P

3is the radiative flux from the

rest side ingot’s wall and P4

is the flux at the ingot’sbottom/puller interface.

The heat fluxes for the EB melting and refining of Ti, atthe same three casting velocities as in the case of Al ingot,are summarized in Table 3, together with the assumed

heat contact conditions. In this case, according to theexperimental data we had to eliminate the radiationenergy losses from the ingot’s top surface. In the realprocess, the convection flow of the liquid metal consider-ably diminuates the surface temperatures.

The calculations for EB melting of Cu at the beampower 100 kW are given in Table 4.

It can be noted an unexpected break in the solidifiedskin around the molten pool, observed in all cases ofcomputer simulation of the ingot/crucible interface. Thismolten part of the side ingot wall (with width of *h) isbellow the top ring, where it is assumed an ideal heatcontact, causing its solidification. The change of the idealcontact ring’s width do not modify this situation. If weextend the size of this contact area, we will intensify theenergy losses up to unacceptable values (in comparison

K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91 89

Table 4Calculated fluxes and solidified skin dimensions for EB melting of copper with P"100 kW

No. »

(mm/min)Q(W)

CH

crucibleh*(mm)

PullerC

H

P1

(W)P2

(W)P3

(W)P4

(W)R

S(mm)

h0

(mm)hS

(mm)*h(mm)

1 1 0.9 0.7 20 0 83 500 66 600 300 16700 23 52 29 2266.6% 16.7%

2 6 1500 0.7 20 0.9 84 300 67 300 300 17000 23 52 27 2366.3% 16.75%

3 20 5800 0.7 20 0.9 86 400 69 200 400 17700 23 55 28 2565.4% 16.73%

with the experimentally obtained losses). To overleap thisparadox in 2D calculations, one can divide the coolingring area into two different rings with approximatelyideal heat contact — one at the top ingot surface and thesecond — in the middle of the observed molten part of theside ingot wall. Such second contact ring below the topliquid metal/crucible contact ring was observed experi-mentally in [5] for the vacuum arc melting of big ingots(were it is easy to be observed).

4. Comments on situation at the ingot/cruciblecontact boundary

The boundary contact between the casting ingot andcrucible can be an ideal heat contact, if the temperaturesat all contact points are the same. In the case of nonidealcontact, the temperatures at the interface between twomaterials changes abruptly. In the case of 3D non-steadyapproach, we can assume that there are areas with anideal thermal contact. The direct contact between theliquid and solid material causes the existence of an idealthermal contact. It can be assumed only for a short timeand for small segment surfaces from the ingot’s sidesurface upper part. After that, these surfaces transfer theheat to the liquid metal interface and solidify. The moltenpool in this segment does not reach more the cooled wall.In this time we can observe temporally breaks of thesolidified skull of the molten pool in the neighboringareas. These breaks are caused by the increased heatingfrom the melted metal to the surrounding area. It isa result of the local heat flow diminuation through theshrank solidified segment. The roughness of the ingotside wall is connected with this shrinkage and breakageof neighboring spots of solidified skull. Experimentallynon-steady heat flows and temperatures can be measuredin the contact region.

In [6] data for a in the case of liquid metal in goodcontact with a metal wall (of order of 104 W/m2K) andfor frozen metal in good contact with metal mold wall(103 W/m2K) are given. Our calculated values of the heatflow density are in the range of 5000—20 000 kW/m2. Ifwe assume a temperature drop of 1000 K, the corres-

ponding values of heat transfer coefficient a are it thesame order with the mentioned data for the liquid/solidcontact case. It can be noted, that the cristalizationprocess of the casted block scin is more probable, in thecase of higher heat flow value, due to the mentionednon-steady nature of the contact processes. This means,that the above cases are an abstract names only — in thefirst case, the solid/solid contact take place at longertime, than the liquid/solid contact.

An exact heat model must take in account the non-steady character of the heat transfer in the contact area.In the case of liquid metal fluxes simulation, this phe-nomenon will give additionally a reason for 3D turbu-lence in crucible/ingot interface proximity region.

5. Conclusions

The proposed model and computer programs can beused for the development of a mathematical model, in-cluding liquid metal fluxes simulation. The simulationresults are very useful when optimizing particular tech-nological processes in EB melting and refining.

As an extension to the real 3D non-steady cases, whereoperator moves the liquid metal on top of the meltedingot surface, by moving the heated spot of the deflectingelectron beam, some parts of contacted area’s segments,are assumed to be temporally in ideal thermal contact,after which they quickly solidify and shrink, makingnon-ideal heat contacts with crucible. Sometimes, neigh-boring ingot wall spots melt, make an ideal thermalcontact and transfer heat by the thermal conductivity.Then these areas are cooled and decrease their heattransfer. Some qualitative conclusions for non-steadythermal transfer in this zone are made.

Acknowledgements

The authors gratefully acknowledge the financial sup-port of the Bulgarian National Fund of Scientific Invest-igations at the Ministry of Education, Science and Tech-nologies, supporting this work (MM-518).

90 K. Vutova, G. Mladenov / Vacuum 53 (1999) 87—91

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