q5uq//f- - 2 UCRLJC-118185 PREPRINT

Process Simulator for Time-Dependent Material and Energy Flow in a Continuous Casting System

K. W. Westerberg M. A. McClelland

This paper was prepared for submittal to the Modelling of Casting, Welding, and Advanced Solidification Processes MI

London, England September 10-15,1995

I February 6,1995

This isa preprint ofa paper intended for publication ina joumalorpmceedings. Since changes may be made before publication, this preprint is made available with the understanding that it will not be cited or reproduced without the permission of the author.

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Process Simulator for Time-Dependent Material and Energy Flow in a Continuous Casting System*

K. W. Westerberg Aspen Technology, Inc.

Ten Canal Park Cambridge, MA 02141 U.S.A.

M. A. McClellandt Lawrence Livermore National Laboratory

Livermore, CA 94550 U.S.A. P.O. BOX 808, L-460

Abstract A process simulator is developed for the material and energy flow in a continuous

casting system which utilizes an electron-beam energy source. A time-dependent, one-dimensional model is used which accounts for energy transport within the in- got and transport to the surroundings by conduction, thermal radiation, and the formation of secondary electrons. Also included are the mass and energy additions associated with the poured metal. A modified finite element method is used to solve the energy equation while tracking boundaries at the pool surface and solid- ification zone. Model parameters are determined using results from a steady-state experiment and a more detailed twc+dimensional model for fluid flow and energy transport. For a transient experiment with pouring, a comparison is made between predicted and measured heat flows.

keywords: continuous casting, electron beam, finite element method, free sur- faces.

Introduction The continuous casting of ingots is important in the processing of metal alloys. Electron- beam systems are often employed in applications involving tight specifications for com- position. A casting process to recycle scrap uranium with 6 wt% niobium (U-6 w%Nb) is shown in Figure 1. In this system, solid U-Nb is fed to a water-cooled hearth where it is melted and mixed. Liquid flows over a weir into a cooled crucible where it solidi- fies and is continuously drawn as a cylindrical ingot. The e-beam system provides the energy to melt the solid feed and maintain liquid pools.

For this U-Nb casting system, there are several factors which make it a challenge to meet composition specifications. One is macrosegregation, the transport of species by fluid flow in the "mushy" or two-phase zone [l]. In addition, there are transient phenomena such as the lengthening of the ingot and pour irregularities.

For the processing of U-6 w%Nb, multidimensional models have been developed for fluid flow along with energy and species transport. Bertram and Zanner ([2], [3]) investigated the vacuum-arc-remelt process, and obtained a favorable comparison be- tween calculated and measured pool volumes. Schunk et aI. [4] performed two and

b 'Work performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore

fAuthor to whom correspondence should be addressed. National Laboratory under Contract W-7405-ENG-48.

1 M ER

Pour stream

scrap 0

Electron beam /

\ n t t f t h beam I

Solid

-

Model equations

Figure 1: An electron-beam system for continuously casting U-6 w%Nb ingots.

three-dimensional calculations for an dectron-beam system. For the system of Fig- ure 1, Westerberg and McClelland [5] developed a two-dimensional finite element model for fluid and energy transport in which a deforming mesh was used to track interface locations. This model provided a good representation of heat flows measured in a steady-state experiment involving a crucible ingot.

Since a large computational effort is generally required to resolve intense pool flows, the application of multidimensional models is often limited to steady-state cases or transient cases of short duration. For transient phenomena of a longer time scale, it is often beneficial to employ lumped-parameter or one-dimensional models.

material and energy flow in the cast ingot of Figure 1. A modified Galerkin finite ele- ment method is used to solve the energy equation and track the boundaries of the pool. Model parameters are obtained from previous steady-state measurements and results from the two-dimensional model [5]. For a case with pouring, model predictions for heat flows are compared with experimental values.

In this study, a process simulator is developed for the time-dependent, one-dimensional

A time-dependent, one-dimensional model is developed which incorporates the pool, two-phase, and solid regions of the cylindrical crucible ingot (see Figures 1 and 2). The effects of thermal gradients in the axial direction are included, but those in the radial direction are taken to be small. The model accounts for the material and energy

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Secondary Thermal

by contact tR4 Figure 2: Domain for transient, one-dimensional model of fluid flow and energy trans- port in a cylindrical crucible ingot.

additions of the pour stream and thermal interactions with the electron beam and water-cooled surfaces.

A mass balance gives the one-dimensional growth rate of the ingot resulting from the metal pour:

Here mpovr is the mass flow rate of the pour stream. The solid density, ps, and the radius of the ingot, R, are taken to be constant and uniform.

The time-dependent, one-dimensional energy equation includes accumulation and conduction contributions along with a term to account for radial heat losses to the surroundings:

A convection term is not included, since the coordinate system moves with the ingot and the effects of thermal expansion are neglected (see Figure 2). Here k is the thermal conductivity, and the parameter c is a dimensionless coefficient that can be used to increase the conductive transport t o approximate convective effects that are otherwise not included in Eqn. (2). In the mushy zone, C, includes the effects of latent as well as sensible heat.

For the case of a well-stirred pool and a pour stream of constant composition, the temperature at the lower boundary of the pool is taken to be constant. This temper-

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ature of immobilization, Tim, has a value between the liquidus temperature, Tlip, and the solidus temperature, Tsol. We use T;, < Ti;, since some solidification is required before the formation of a pool boundary with mechanical integrity. For nonequilibrium solidification of U-Nb, Tsol is the melting-point temperature for pure uranium.

At the top surface of the pool, the net energy flux, qL, includes contributions from the electron beam, the sensible heat of the poured liquid, secondary electrons, and thermal radiation:

Here Qb is the total incident e-beam power and 7 is the fraction of energy absorbed by the pool. The emissivity, &(T) , is a function of temperature.

At the crucible wall, the heat flux includes contributions from contact and thermal radiation. In the pool region, heat transport is assumed to occur through conductive contact alone:

QR = hliq,L(T - Tco) for T 2 Tim (4) Here the subscript “1” denotes a result for the one-dimensional model. For the mushy- zone and solid regions, the contact is treated as negligible and energy transfer to the surroundings occurs exclusively by thermal radiation:

For the puller, the effects of thermal radiation and conduction are lumped together in a single heat transfer coefficient hpull:

Finite Element Method A finite element method is applied in which the mesh deforms to track the upper and lower boundaries of the pool [6]. Specified nodes remain on these boundaries, while interior nodes maintain their relatively spacing within each material phase.

The mass and energy balances (Eqns. (1)-(2)) are discretized using the Galerkin finite element method [7] in which the temperature is represented by one-dimensional, quadratic basis functions. In the discretized form, a convection term appears in Eqn. (2) to account for the motion of the deforming mesh [8]. The heat flux conditions (Eqns. (3) and (6)) for the upper and lower ingot surfaces are incorporated as natural boundary conditions, while the radial conditions (Eqns. (4)-(5)) are included in Eqn. (2). In the assembly of boundary conditions, the mass balance Eqn. (1) and the isotherm condition T = T,, are applied as “distinguished” conditions for determining the upper and lower boundary locations of the pool.

The trapezoid rule with error control is used to perform the time integration [9]. At each time step the Newton-Raphson method is applied to solve the resulting algebraic equations for the temperature field and two pool boundary locations. In the computer code “MELT“ these expressions are arranged in an “arrow” matrix and solved using a method given by Westerberg et ai. [9]. In a typical transient calculation, 60 elements were employed, requiring about 1 minute of computation time on a DEC Alpha 3000 (OSF/ 1).

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Experimental Arrangement An electron-beam casting system was employed to measure energy flows from the cru- cible ingot in experiments with and without pouring (see Figure 1). These measurements are used for model verification and the determination of model parameters as described in the next section.

A 35 [kV] beam from a 250 [kW] commercial electron gun is swept over the surfaces of U-6 w%Nb crucible and hearth ingots. The beam pattern on the crucible ingot consisted of two concentric rings. The copper crucible (D = 14 [cm], L = 20 [cm]) and puller are cooled with water from two circuits. Heat flows are calculated from measurements of water flow rates and differences between thermocouple measurements (A 0.1 [“CJ) of the inlet and outlet water temperatures.

In these experiments, the pouring of liquid into the crucible is initiated by feeding uranium bars with niobium cores (6w%) to the hearth, resulting in flow over the weir. The length of the crucible ingot is determined from measurements of the puller position and camera measurements of the pool level.

During both steady-state and transient experiments, the pool level was kept at a p proximately 1 [cm] below the crucible lip. Consequently, a significant fraction of the skip energy was absorbed by the crucible-wall cooling circuit. Since this circuit also includes thermal contributions from the side of the ingot, i t is desirable to separate these components for model comparison. A simple model involving analytical expres- sions for exchange factors was developed for the twcwing beam profile (Westerberg and McClelland [5 ] ) . For this study, approximately 11% of the skip energy was absorbed by the crucible circuit.

Comparison of Results In this section, parameters for the process simulator are obtained using steady-state heat flow measurements along with calculations from a two-dimensional finite element model for liquid flow and heat transport. For a transient casting experiment involving a pour, heat flow measurements are compared with predictions from the process simulator.

Material properties The material properties for U-6 w%Nb are listed in Table 1. The immobilization tem- perature is provided by Bertram and Zanner ([2], [3]), and the solidification temperature is the melting-point temperature for uranium, consistent with nonequilibrium solidifica- tion theory [l]. Although the expression for the emissivity applies for the liquid phase, it is also used for radiative heat transfer from the solid ingot to the crucible wall. For the mushy zone (Tsol < T < Tljq), the effective heat capacity includes both sensible and latent heat effects:

‘ C P S ,

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Figure 3: Steady-state results for two-dimensional model of fluid flow and energy transport in the crucible ingot of Run No. 1 [5]. &a = 32.4 [kW], L = 12 [cm], R = 7 [cm]. (a) Deformed mesh, (b) Temperature contours, (c) Stream function contours.

The other liquid and solid properties are evaluated at Ti, and Tsol, respectively. These properties are taken to be uniform in a given phase with the mushy-zone and solid regions treated as a single phase.

Table 1: Material properties for U-6 w%Nb.

PI = 16,010 kg/m" Tsol = 1132 "C P s = 16,600 kg/m3 Y = 0.6 Cpl = 210 J/kg-K &(T) = U O + U ~ T + U ~ T ~ Cps = 171 J/kg-K uo = 0.09136 Mi = 6.53 x lo4 J/kg a1 = 7.252 x lov5 K-'

Ti, = 1277 "C = 1332 "C a2 = -6.150 x lo-' K-2

Steady-state results and determination of heat transfer coefficients

In an earlier study [5], a steady-state experiment (Run No. 1) was performed in which an electron beam with 32.4 [kW] of power was applied to a short ingot ( L = 12 [cm]) in the absence of pouring. The measured heat flows from this experiment were used to determine heat transfer coefficients in a two-dimensional finite element model incor-

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porating fluid flow and energy transport. The values of hliq,2 = 2500 [W/m2-K] and hpull,2 = 100 [W/m2-K] yielded heat flows which agree well with the experimental values (see Table 2). (The subscript “2” denotes a result from the two-dimensional model.) This study also revealed tha t the predicted puller and crucible heat flows are sensitive to variations in hpull,2, but not hliq,2 which leaves greater uncertainty in the latter quantity.

Table 2: Comparison of steady-state measurements from experiment (Run No. 1) with calculations from the one- and two-dimensional models.

experiment onedimensional two-dimensional model model

pool depth [cm]

temperature [“C] (centerline, top)

2.20

1523

2.23

1490

heat flows [kW] crucible” 16.0 16.1 16.4 ingot top 15.0 14.8 14.6 puller 1.4 1.5 1.5

total 32.4 32.4 32.5 - - -

* The total energy removed by the furnace was 97% of the incident e-beam power for the hearth and crucible. The crucible heat flow does not include the skip contribution from the t6p of the ingot.

The mesh, streamlines, and temperature contours are shown in Figure 3 for the two- dimensional model. It is observed that the pool is shallow and there is a large two-phase region (1132 < T < 1332 [“C]) in the ingot. These results, the heat transfer coefficients given above, and the heat flows of Table 2 all indicate poor contact between the ingot and surrounding water-cooled surfaces, except where the pool meets the wall.

The measured heat flows and twedimensional model results are used to obtain three process simulator parameters: hl;,,l, h,n,I and c. It is observed tha t the temperature field of Figure 3 is approximately one-dimensional near the puller. Since the two models exhibit similar behavior in this region, we use hli,,l = h1;,,2 = 100 [W/m2-K]. With the additional choices of c = 1 and hl;,,l = 1100 [W/m2-K] the process simulator yields a good representation of the two measured heat flows along with the center-line pool depth and top surface temperature calculated with the two-dimensional model (see Table 2).

This value of unity for c suggests no increase in conductive transport in the axial direction to account for thermal convection. However, the effects of thermal convection are present with the use of the one-dimensional approximation which entails perfect en- ergy transport in the radial direction. The value hliq,l = 1100 [W/m2-K] is considerably smaller than values of O(10,OOO [W/m2-K]) for liquid-crucible contact [lo]. However, the actual liquid-crucible contact area is much smaller than the value given by the pro- cess simulator (see Figures 2 and 3), and it is better to regard hljq,l as a composite value representing liquid-crucible and soiid-crucible contact in the vicinity of the pool.

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i: & c

3 (d 0) IG

25

20

15

I I I I I I I I I I I I I I (b) - exDeriment n

I I I I I I I I I I I I I

-5 0 5

Figure 4: Comparison of transient pour results from Run No. 2 and predictions of one- dimensional model (material properties listed in Table 1). &b = 25 [kW], R = 7 [cm]. (a) Model ingot length. (b) Crucible energy flow. The crucible heat flow does not include the skip contribution from the tbp of the ingot.

Transient results with pouring

In Run No. 2, an experiment was performed in which 25 [kW] of e-beam power was a p plied to a U-6 w%Nb ingot of initial length 21.9 [cm] (see Figure 4a). After steady-state conditions were achieved a pour was initiated in which the ingot length was increased to 30.8 [cm] in 4.5 [min] (inpour = 300 [kg/h]). The experiment concluded 5 [min] later. The measured crucible heat flow is initially 13.5 [kW], increases to 22.2 [kW] during the pour, and decreases to 14 [kW] after the pour (see Figure 4b). Many of the high frequency fluctuations of amplitude ~ 1 . 5 [kW] are associated with incremental changes of 0.1 [“C] in measurements of the inlet and outlet cooling water temperatures.

Predictions by the process simulator were obtained using parameters from the steady- state results of Run No. 1. In addition, the length of the ingot was taken to increase linearly during the pour (see Figure 4a). For the periods before and after the pour the simulator predictions and measurements of the crucible heat flow agree to within 10% (see Figure 4b). This agreement is quite good, considering the approximations in the simulator model and the differences between conditions in Run Nos. 1 and 2. However, differences between predicted and measured values are as large as 25% during the pour. The total energy transferred to the crucible during the pour interval can be determined by calculating the area under a heat flow curve. The respective model and experimen- tal values of 5.00 and 4.84 [MJ] are in good agreement. However, the shapes of the

\

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heat-flow curves are somewhat different. One explanation is that the simulator model does not completely describe the response characteristics of the casting system. It is also probable that the pour rate is not constant as assumed in the model. Nonetheless, the simulator captures the key features of this continuous casting process.

Conclusions A theoretical and experimental analysis is made of an e-beam continuous casting sys- tem for U-6 w%Nb. A process simulator is developed for the time-dependent, one- dimensional material and energy transport in the casting of a cylindrical ingot. Tem- perature fields, heat flows, and pool boundary locations are calculated with a modified Galerkin finite element method which employs meshes that stretch with the growing ingot. Parameters for the process simulator are obtained from previous steady-state measurements and results from a twc+dimensional finite element model incorporating fluid flow and energy transport. For a transient case involving the pouring of U-Nb metal, the simulator provides a satisfactory prediction for the measured heat flows.

Acknowledgement Robert H. McKoon and the Mars Melt Furnace staff are acknowledged for their support in operating the casting system and providing the experimental measurements.

References [l] M. C. Flemings. Solidification Processing. McGraw-Hill, New York, 1974.

[2] F. J. Zanner and L. A. Bertram. Computational and experimental analysis of a U-6 w%Nb vacuum consumable arc remelted ingot - a progress report for the Sandia macrosegregation study. Technical Report SANDSO-1 156, Sandia National Laboratories, 1980.

[3] L. A. Bertram and F. J. Zanner. Interaction between computational modeling and experiments for vacuum consumable arc remelting. In H. D. Brody and D. Apelian, editors, Modeling of casting and welding processes, pages 333-349. The Metallurgi- cal Society of AIME, Warrendale, PA, 1981.

[4] P. R. Schunk, R. W. Fisher, and F. J. Zanner. Two and three dimensional flow simulations of ingot growth in an e-beam furnace. In Proceedings of the 1991 Vacuum Metallurgy Conference on Specialty Materials, pages 41-43. Pittsburgh, Pennsylvania, September 13-15 1991.

[5] K. W. Westerberg and M. A. McClelland. Modeling of material and energy flow in an EBCHR casting system. In R. Bakish, editor, Electron Beam Melting and Refining State of the Art 1994. Bakish Materials Corp., Englewood, NJ , 1994.

[SI K. W. Westerberg, M. A. McClelland, and B. A. Finlayson. The interaction of flow, heat transfer, and free interfaces in an electron-beam vaporization system for metals. Paper #124e, 1994 AIChE Annual Meeting, San Francisco, CA, 1994.

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0. C. Zienkiewicz. The Finite Element Method. McGraw-Hill, London, third edi- tion, 1977.

D. R. Lynch. Unified approach to simulation on deforming elements with applica- tions to phase change problems. J. Comp. Phys., vol. 47, pp. 387-411, 1982.

K. W. Westerberg, C. Wiklof, and B. A. Finlayson. Time-dependent finite-element models of phase-change problems with moving heat sources. Numer. Heat Transfer, Part B, vol. 25, pp. 119-143, 1994.

R. S. Ransing, Y. Zheng, and R. W. Lewis. Potential applications of intelligent preprocessing in the numerical simulation of castings. In R. W. Lewis, editor, Nu- merical Methods in Thermal Problems, volume VIII, Pt. 2, pages 361-375. Pineridge Press, Swansea, 1993.

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