aberg paper

8/13/2019 Aberg Paper

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Jonas Aberg, Michael Vynnycky, Hasse Fredriksson

Modelling of thermal stresses in industrial continuous castingprocesses

Abstract This paper reports on progress in the imple-mentation of COMSOL Multiphysics 3.2 to model ther-mal stresses in a three-dimensional solidifying shell, asoccurs typically in the industrial continuous casting ofcopper, copper alloys and steel. Computer memory re-

quirements prohibit a direct 3D numerical simulation ofthe temperature and the stresses. Instead, we use the factthat casting geometries are usually slender to divide thecalculation into three steps that are each less memory-intensive: (i) heat transfer and solidification is simulatedusing an arbitrary Lagrangian-Eulerian model; (ii) thetemperature solution is used to solve the force equilib-rium equations in generalised plane strain mode appro-priate for a moving body; (iii) this solution is used tocompute the accumulated stress and strain in the body.Comparison between an analytical solution and a nu-merical solution, that requires the simultaneous use ofseveral of Comsol Multiphysics peripheral features, is

presented.

1 Introduction

In industrial continuous casting processes, molten metal,typically copper, aluminium and steel alloys, passes ver-tically downwards through a cooled mold; subsequently,a solidified shell forms adjacent to the mold walls and iswithdrawn at uniform speed. An idealized schematic ofsuch a process is given in Fig. 1, which shows a moltenregion (within the inverted pyramid) surrounded by a so-

lidified metal shell that is pulled vertically downwardswith casting speed,Vcast; in general, the mold walls whichsurround the solidified shell(not shown here) are not inthe form of a square, neither are they necessarily vertical[1].

The industrial importance of the process demands adetailed understanding of the factors that influence prod-uct quality. Of key interest is the heat transfer that occursduring solidification, since this is thought to be respon-sible for the appearance of surface and half-way crackswithin the solidified shell [28]. As a result, a number of

J. Aberg , KTH/Metallernas GjutningBrinellv. 23, 10044 Stockholm, Sweden

Tel.: +46-8-7906151Fax: +46-8-216557E-mail: [email protected]

Fig. 1 : Schematic of a casting process with a liquid interior (re-gion within the inverted pyramid), a solid shell (region below the

pyramid) that is withdrawn vertically downwards with speed Vcast

attempts have been to compute the levels of thermally-induced stresses and strains that develop in the solidify-ing shell [914]. These works either simplify the prob-lem to one dimension [911] or develop specially-designedgeometry-dependent code to solve the structural equilib-rium equations to obtain the thermally-induced stressesand strains in the shell [1214].

Here, we report on progress to employ Comsol Mul-tiphysics 3.2 [15], to tackle this problem. The idea is tocouple the heat transfer and structural mechanics mod-

ules that exist by default in COMSOL Multiphysics, soas to develop a numerical model that can be used for anycontinuous casting process, irrespective of cross-section

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Stockholm


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geometry. Computer memory requirements prohibit themost obvious approach: a direct 3D numerical simula-tion of the temperature and the stresses. Instead, we usethe fact that casting geometries are usually slender to di-vide the calculation into three steps that are each lessmemory-intensive:

heat transfer and solidification is simulated using anarbitrary Lagrangian-Eulerian model;

the temperature solution is used to solve the forceequilibrium equations in generalized plane strain, al-beit modified to take account of the translational mo-tion of the solidified shell;

this solution is used to compute the accumulated stressand strain in the shell.

A benchmark comparison between an analytical so-lution and a numerical solution, that requires the simulta-neous use of several of Comsol Multiphysics peripheralfeatures, is presented.

2 Problem formulation

We assume a steady state problem. Most generally, thetemperature and stress components would have to be com-puted simultaneously; for the benchmarking we will per-form here, however, we will assume that the thermal andstructure mechanical problems decouple, so that the tem-perature distribution can computed first, and is then usedas input to the stress calculations. We consider here thesolidification of a pure metal, and assume that the moltenmetal streams uniformly downwards with the same speed

as the solidifying shell, i.e. Vcast.

Fig. 2 : Cross-section of a casting geometry

A detailed derivation of the relevant heat transfer andforce equilibrium (Navier) equations are given by Jablonka[12] and Schwerdtfeger et al. [13]; it is their mathemat-ical formulation in generalized plane strain mode for asteady-state continuous casting process that we imple-

ment, although without their restriction to axisymmetricgeometries. Mathematically, the most general situationis equivalent to that shown in Figure 2, which shows the

solidification front advancing as a function oft, wheret=z/Vcast.

For heat transfer, we have

CpsTst

=ks

2Tsx2

+2Ts

y2

in the solid,

CplTlz

=kl

2Tlx2

+2Tl

y2

in the liquid,

where ks and kl are the thermal conductivities of solidand liquid metal, respectively,cpsand cplare the specificheat capacities of solid and liquid, respectively, is thedensity,andHfis the latent heat of fusion; we give theboundary condition associated with the release of latentheat at the solidification front later.

Stresses will occur only within the solidified shell.There, the force equilibrium equations are given, in gen-eralized plane strain mode, for the stress componentsx,y,xy by

xx

+xy

y=0, (1)

xy

x+

y

y=0, (2)

where the dots here denote differentiation with respect tot.Further,x,y,zare related to the strain components,x,yand z,and the temperatureT by

xyz= E

(1 + ) (12) 1

1

1xyz

ET

(12)

1 0 00 1 0

0 0 1

,

where E is the Youngs modulus, is the Poisson ra-tio and is the thermal expansion coefficient, and thestrain components are in turn related to the displace-ments,u,v,w,by

x=u

x, y=

v

y, z=

w

z.

Furthermore, compatibility conditions would give that, ifthe cross-section of the casting geometry does not pos-sess symmetry in either x or y, then z must be of theform

z=a (t) + b (t)x + c (t)y, (3)

where a(t),b (t) ,c (t) are functions to be determined; thisis consistent with the situation in [13] for radially sym-metric casting, where (3) reduces simply to z=a (t) .

For the purposes of benchmarking, we consider, in-stead of the geometry in Figure 2, a problem that is known

to have an analytical solution for which z = a (t); thegeometry and relevant boundary conditions are summa-rized in Figures 3 and 4. In particular, Figure 4 contains



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Fig. 3 : Boundary conditions for heat transfer

Fig. 4 : Boundary conditions for structure mechanics

two conditions that have genuine physical relevance incontinuous casting: v=0 at y=0 expresses the fact thatthere is direct contact between the shell and the mold,andtherefore that there is no normal displacement of the shellsurface, whilst y= 0 at the solidification front comesfrom the fact that the metallostatic pressure will be muchsmaller than the induced thermal stresses; and six othersthat are necessary for an analytical solution.

These boundary conditions will not suffice, however,to determine a(t), and therefore additional considera-tions are required. A further condition results from thefact that z has to be compensated by the total externalforce,Fz,acting on the cross-sectional area, i.e.

Fz=

yc(t)0

zdy.

Different assumptions can be made to obtain Fz; this isdiscussed briefly for the continuous casting of steel bySchwerdtfeger et al. [13]. In the mold region, Fz willequal the weight of the steel column above the cross sec-tional area of the shell if the friction within the mold isnegligible and the strand is held by the guiding systembelow the mold, i.e. the slice of the shell is subjected toan overall compression. In principle, there can also beoverall longitudinal tension in the shell with positive Fz,exerted by the withdrawal machine, if the friction in themold is large. If friction and weight force compensateeach other,Fz=0,so that

yc(t)0

zdy=0, (4)

which then becomes yc(t)0

zdy=0. (5)

3 Numerical modelling in COMSOL Multiphysics

A first approach was to use a common domain for bothsolid and liquid phases, with a source term for the heatequation to describe the release of latent heat at the so-lidification front [16], and a further source term for thet-derived Naviers equation in order to numerically dampout all stresses in the liquid phase. This method suffersfrom several drawbacks:

a fine mesh is required throughout the solution do-main in order to resolve the release of latent heat atthe melting temperature;

whilst a fine mesh gave good agreement with the an-alytical solution for the temperature,T,it proved ex-ceedingly difficult to obtain an oscillation-free pro-file for T, which then led to oscillations in the pro-files for the stress components.

Consequently, we opted to implement the arbitraryLagrangian-Eulerian (ALE) method as described in theCOMSOL Multiphysics 3.2 documentation. With refer-ence to Figure 3, this method tracks the solid-liquid in-terface; all computations for heat transfer are performedon this domain and additional PDEs are solved for thedeformed mesh. Furthermore, the structure mechanicalcalculations can be switched off in the liquid region, sothat there is no need for ad hoc damping terms there.

Consequently, the problem can be solved accurately andwithout numerical oscillations using several hundred el-ements in a few minutes; this compares with the tens ofthousands of elements and several hours that are requiredwhen using the approach with sources.

On top of the ALE framework, the coupling betweenheat transfer and structure mechanics that exists withinComsol Multiphysics still has to be extended in severalways in order to solve continuous casting problems:

(a) The Navier equations are normally for the stress com-ponents in a stationary body; in continuous casting,the solid body is translating with a casting speed,and so the Navier equations in this case are for the

derivatives, with respect to the direction of casting,of the stress components. Once these are determinedby COMSOL Multiphysics, they must be integrated,with respect to that direction, back to the initial pointof solidification to determine the actual stress com-ponents at a particular point in space; this is illus-trated in Fig. 1.

(b) By default, COMSOL Multiphysics has only an in-built plane strain approximation mode, and furtherprogramming is required to extend this to the gen-eralized plane strain case. With this approximation,a 3D steady-state casting problem can be reformu-lated as a 2D time-dependent problem, an approach

that alleviates computer memory requirements, sincethe problem can be solved numerically z-plane byz-plane, rather than for the whole 3D geometry at once.



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(c) The ODE settings option is used to computez.

The solution algorithm therefore consists of the fol-lowing steps:

1. Solve for the temperature,T.2. Solve for

x,

y,

xyunder the generalized plane strain

approximation and ODE settings in each consecutivez-plane.

3. Find the actual stress components, i.e.x,y,z,xyat a particular point (x,y,z) in space, by integratingx, y, z and xy vertically back to the initial pointof solidification, as shown in Fig. 1, i.e.

(x,y,z) =

zzc(y)

x,y,z

dz, (6)

or alternatively

(x,y,t) = t

tc(y)

x,y,tdt (7)where = x,y,z,xy.

4 Results

The analytical solution for the temperature is that for aclassical 1D time-dependent Stefan problem involvinga mold wall at constant cooling temperature, Tmold, andsemi-infinite pure melt that is initially at a temperature,Tcast,higher than the melting temperature,Tmelt. The so-lution for the temperature is then given by

T(y,t) =

Tmold+ (TmeltTmold)erf

y2

st

erf()

if 0 0. Forthe results we present here, we have taken tinit=0.1 s;solutions were also obtained for much smaller values oftinit.A comparison of analytical and numerical solutions

at t= 1.5 s and 3 s is given in Figure 5; here, we haveused Tmold= 1000 K, Tmelt= 1356 K and Tcast=1393K, and the values for the physical parameters are those

Fig. 5 :Tas a function ofy

Fig. 6 : Tas a function ofy

of pure copper. Furthermore, Figure 6 compares T; ev-

ident here is how the numerical solution is able to re-capture the jump discontinuity in Tat the solidificationfront. Note incidentally that, in both figures, the ana-lytical solution, which is valid for solidification into ansemi-infinite medium, begins to differ from the numeri-cal solution att= 3 s.

The stress field for this case, when a linear elasticmodel is used, appears not to be given in literature (noteven by Weiner and Boley [9], whose solution is for anelastic-plastic model); thus, we have had to derive it fromscratch, and for future use we note here that

x= z= E

(1) 1

yc(t)

yc(t)

0

T dy

T ,

y 0,

and

z= x=

yc(t)

yc(t)0

T dy,

y=

1

(1 + )T 2

yc(t)

yc(t)0

T dy

.

Implementing tc(y) =y2/42s in equation (6), we fi-

nally arrive at

z(x,y,z) = E

1

(TmeltTmold)

4

erf

1exp

2



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log

y

2

sz/Vcast

T

y,

z

Vcast

Tmelt

.

Fig. 7 :z as a function ofy

The development ofzwithtis given in Fig. 7. Fur-thermore, Fig. 8 gives the z-profile as a function of t,which, for this simplified geometry, will not depend on

xand y,and is given by

z=(TmeltTmold)

2

terf

1 exp

2

. (8)

In all cases, the agreement between the analytical andnumerical solutions is very good.

Fig. 8 : z as a function oft

5 Conclusions

We have reported on progress in the implementation ofComsol Multiphysics 3.2 to compute thermal cracks inindustrial continuous casting. The most practical way tosolve the benchmark example chosen was found to be to

use an ALE model, combined with the generalized planestrain mode and the ODE Settings option, in order tosatisfy an integral constraint that the solution must fulfil.

Immediate and obvious extensions are: (i) non-planarsolidification fronts, as occur in actual continuous cast-ing processes; (ii) metal alloys, where the ALE-approachshould be applied twice: once at the solidus isotherm,and once at the liquidus isotherm; (iii) more realisticstress-strain constitutive relations. Work on these is in

progress.

Acknowledgements The authors would like to thank Elektrokop-par, Outokumpu, Jernkontoret and VINNOVA for financial sup-port, and Comsol AB for assistance with certain aspects of the nu-merics. This work has been performed, in part, within the frame-work of the Faxen Laboratory.

References

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