J Eng Math (2010) 68:129–152 DOI 10.1007/s10665-009-9341-8
Air gaps in vertical continuous casting in round moulds M. Vynnycky
Received: 12 April 2009 / Accepted: 28 September 2009 / Published online: 15 October 2009 © Springer Science+Business Media B.V. 2009
Abstract A recent asymptotic thermomechanical model for the formation and evolution of air gaps in vertical continuous casting is extended from an idealized geometry to a cylindrical one that is of actual industrial relevance. The differences between the models, in particular as regards the criterion for the onset of air-gap formation for the two geometrical configurations, are noted. Parameter regimes for which the thermal and mechanical problems decouple are discussed. In such cases, corresponding to thermal stresses dominating viscosplastic ones, asymptotic analysis helps to reduce the model to a moving-boundary problem for the temperature, along with a boundary condition in integro–differential form that describes the evolution of the air gap. Sample computations are carried out using parameters for the continuous casting of copper, and the value of the model results as a new and useful benchmark for verifying 3D numerical codes describing the thermomechanics in continuous casting models is highlighted. Keywords
Air-gap formation · Asymptotics · Continuous casting
1 Introduction Air-gap formation in the industrial continuous casting of metals and metal alloys has long been recognized as having an adverse effect on process efficiency. A schematic of the situation is given in Fig. 1, which shows molten metal, typically copper, aluminium or steel alloys, passing vertically downwards through a cooled mould, solidifying and being withdrawn at casting speed, Vcast . At a mould wall, there is typically first a region where liquid metal is in contact with the mould wall, followed by a region where the solidified shell is in contact; after this, at z = z gap , an air gap begins to form between the solidified shell and the mould wall. Eventually, at some location z = z mid , complete solidification occurs at the centreline. In particular, the formation of the air gap prohibits effective heat transfer between the mould and shell, leading to longer solidification lengths and requiring supplementary process-design considerations, such as mould tapering. In view of the detrimental effect that the air gap has on process efficiency, mathematical models of varying degrees of complexity have been derived to describe the phenomenon. Early models for predicting the onset of airgap formation were analytical [1–4]; most subsequent models [5–11] have been solely numerical. However, whilst M. Vynnycky (B) MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail:
[email protected]
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M. Vynnycky r=W
r=0
Fig. 1 Schematic of vertical continuous casting with air-gap formation
r=WM
meniscus
z=0
M E LT
zgap
r=rm (z)
M O U L D
M O U L D A I R
S O L I D
S O L I D
T=T o(z)
A I R
r
zmid r=ra (z) Vcast
z z=L
able to capture the thermomechanical interaction of gap formation and evolution, such models are computationally expensive and unwieldy: for example, they do not give a qualitative understanding of the air-gap’s dependence on different operating parameters, or indeed whether it is possible to avoid air-gap formation completely. An exception to all of the above is a recent model [12] that uses asymptotic methods; that model was, however, derived for an idealized two-dimensional configuration and the purpose of this paper is to extend that model to a geometry of actual industrial interest. Although most continuous casting configurations are fully three-dimensional, axisymmetric cylindrical casting in round moulds, as used for the manufacture of bars, is also of wide interest [6,13–23]. Furthermore, this configuration presents an ideal case for extending the asymptotic air-gap model in [12], since substantial analytical progress can be made and it proves possible to test alternative hypotheses regarding the appropriate boundary conditions for the onset of air-gap formation and gap evolution; in addition, the model solutions obtained can be later used as benchmarks for testing 3D codes. The layout of the paper is as follows. In Sect. 2, we formulate the appropriate thermomechanical problem for a cylindrical geometry in which the generalized plane-strain approximation holds. In Sect. 3, we nondimensionalise the governing equations, perform asymptotic reduction and derive a criterion for air-gap formation in terms of the operating parameters. From the asymptotics, we see that the thermal and mechanical problems decouple if thermal stresses dominate viscoplastic stresses. In Sect. 4, a finite-element method is used to solve the governing equations; in particular, two different models for describing the evolution of the air gap are compared, and the effect of the casting speed on the results for these is explored in detail. Conclusions are drawn in Sect. 5.
2 Mathematical formulation We consider a steady-state problem with cylindrical symmetry, as shown in Fig. 1, in which pure liquid metal at its melting temperature, Tmelt , enters a mould region at z = 0, solidifies and is withdrawn at a casting speed Vcast ; the extension to the case where the liquid metal is at a temperature greater than its melting temperature will be considered elsewhere, but the working assumption that we use allows us to take rm (0) = W and thereby to avoid extraneous details that are not essential here. Subsequently, an air gap starts to form at the inner mould surface at z = z gap . For 0 < z < z gap , solidification occurs in the region rm (z) < r < W , whereas for z > z gap , air occupies
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Air gaps in vertical continuous casting in round moulds
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ra (z) < r < W and solid occupies rm (z) < r < ra (z). Eventually, after complete solidification has occurred at z = z mid , the solid region occupies 0 < r < ra (z). 2.1 Heat transfer The heat-transfer aspects are as follows. For 0 < z < z gap and rm (z) < r < W , and then z > z gap and rm (z) < y < ra (z), we have ks ∂ ∂ Ts ∂ Ts = r , (2.1) ρcps Vcast ∂z r ∂r ∂r where ks is the thermal conductivity of the solid metal, cps is its specific heat capacity and ρ its density. In Eq. 2.1, we use the fact that casting geometries are often slender, which motivates us to assume that ∂ 2 /∂z 2 ∂ 2 /∂r 2 . For boundary conditions at r = rm (z), we have Ts = Tmelt ,
(2.2)
and the Stefan condition, ∂ Ts drm = ρH f Vcast , (2.3) ks ∂r dz where H f is the latent heat of fusion; this form for (2.3) also makes use of the fact that the geometry is slender and that there is no heat flux in the liquid phase. However, once solidification is complete, at z = z mid , we treat r = 0 as a symmetry axis, so that (2.2) and (2.3) are replaced by just ∂ Ts = 0. (2.4) ∂r Separate considerations are necessary for 0 < z < z gap and z > z gap . For 0 < z < z gap , where the solid shell is in contact with the mould, it is reasonable to assume continuity of temperature and heat flux; i.e., at r = W , ∂ Ts ∂ TM = kM , (2.5) Ts = TM , ks ∂r ∂r where k M is the thermal conductivity of the mould. In addition, assuming only heat conduction in the slender mould, we have ∂ TM 1 ∂ r = 0 for W ≤ r ≤ W M , (2.6) r ∂r ∂r subject to TM = To (z) at r = W M ,
(2.7)
where To (z) is the experimentally measurable temperature at the outer surface of the mould, although in practice, measurements may also be made by means of thermocouples located within the mould itself. So, r , (2.8) TM (r, z) = To (z) + A M (z) log WM where A M (z) is to be determined; substituting in (2.5), we arrive at kM ∂ Ts = (2.9) ks (Ts − To (z)) at r = W. ∂r W log (W/W M ) For z > z gap , on the other hand, we assume heat conduction only in the air gap and continuity of temperature and heat flux at the mould/air and air/shell interfaces, giving just 1 ∂ ∂ Ta r = 0 for ra (z) ≤ r ≤ W, (2.10) r ∂r ∂r with additional interface conditions
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∂ Ta ∂ TM = kM at r = W, ∂r ∂r ∂ Ta ∂ Ts Ta = Ts , ka = ks at r = ra (z), ∂r ∂r
Ta = TM , ka
(2.11) (2.12)
where ka is the thermal conductivity of air; whether conduction is the dominating mechanism for heat transfer has been debated previously [24,25], but it is convenient to assume so here, in order to make analytical headway. Now, using (2.8) and Ta (r, z) = Aa (z) log
r + (Ts )r =ra (z) ra (z)
(2.13)
as the general form of the solution to (2.10), boundary conditions (2.11) and (2.12) lead to W W + (Ts )r =ra (z) , ka Aa (z) = k M A M (z), To (z) + A M (z) log = Aa (z) log WM ra (z) giving ka (Ts )r =ra (z) − To (z) A M (z) = , ka log WWM − k M log raW(z)
k M (Ts )r =ra (z) − To (z) Aa (z) = . ka log WWM − k M log raW(z)
Following straightforward manipulation to eliminate Ta , we obtain ks
1 ∂ Ts ka k M {Ts − To (z)} = ∂r ra (z) k log W − k log W a M WM ra (z)
at r = ra (z).
(2.14)
Of use to process engineers is the heat-transfer coefficient, h, between the solidified shell and the mould, defined in dimensional variables from the equation ∂ Ts = −h (Ts )r =ra − TM (W, z) ; (2.15) ks ∂r r =ra retracing the manipulation that led to (2.14), we arrive at h=
ka ra (z) log raW(z)
.
(2.16)
Since governing equation (2.1) is a parabolic PDE, an initial condition for Ts is necessary at z = 0. As this is where solidification starts, we set Ts = Tmelt .
(2.17)
Note that although mathematical analysis of Stefan problems such as this one indicates that the correct initial data ought to be for the heat content (enthalpy) of the liquid at z = 0, one should bear in mind that this model is actually derived from a steady-state problem in which it makes more physical sense to prescribe the temperature of the molten metal, as it comes from an overlying tundish; hence, it is reasonable to set the temperature at z = 0. In addition, we must have rm (0) = W,
(2.18)
i.e., the solid phase is initially of zero thickness. Consequently, the problem is initially degenerate; we demonstrate later how to deal with this potential difficulty.
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2.2 Structure mechanics In (r, θ, z) coordinates, the equilibrium of forces is given by ∂σr 1 ∂τr θ ∂τr z σr − σθ + + + = 0, ∂r r ∂θ ∂z r 1 ∂σθ ∂τθ z τr θ + τθr ∂τθr + + + = 0, ∂r r ∂θ ∂z r τzr 1 ∂τzθ ∂σz ∂τzr + + + = 0, ∂r r r ∂θ ∂z where σr , σθ , σz , τr θ , τr z , τθ z are the stress components, with
(2.19) (2.20) (2.21)
τr θ = τθr , τr z = τzr , τθ z = τzθ . Here, as in other continuous-casting models [12,19], we invoke the generalized plane strain approximation, by which we mean that the length scale in the z-direction is much greater than that in the r -direction; this enables us to reduce (2.19)–(2.21) to 1 ∂τr θ σr − σθ ∂σr + + = 0, ∂r r ∂θ r ∂τθr 1 ∂σθ τr θ + τθr + + = 0. ∂r r ∂θ r In addition, if we exclude bending, we can set τr θ = τθr = 0, as in [19], leaving just
(2.22) (2.23)
∂σr σr − σθ + = 0, (2.24) ∂r r ∂σθ = 0; (2.25) ∂θ there is, however, axial stress, i.e., σz = 0. However, as discussed in detail by [19,26], a more convenient form for these when considering a body translating in the z-direction is σ˙ r − σ˙ θ ∂ σ˙ r + = 0, ∂r r ∂ σ˙ θ = 0. ∂θ where the dots denote differentiation with respect to z. The stress components are then related to the elastic strain rates—˙εrel , ε˙ θel , ε˙ zel —through ⎡ ⎤ ⎡ ⎤ ⎡ el ⎤ σ˙ r 1−ν ν ν ε˙r E ⎣ σ˙ θ ⎦ = ⎣ ν 1 − ν ν ⎦ ⎣ ε˙ el ⎦ ; θ (1 + ν)(1 − 2ν) σ˙ z ε˙ zel ν ν 1−ν
(2.26) (2.27)
(2.28)
here E is the Young modulus and ν is the Poisson ratio; as in [12], we take both of these to be constant. We will assume that thermal εrth , εθth , εzth , elastic and inelastic strains εri , εθi , εzi occur simultaneously and that they are additive, so that ε j = εelj + εthj + εij ,
j = r, θ, z,
(2.29)
where εr , εθ and εz are the strains. Assuming the material to be isotropic with respect to thermoelasticity, we have ε˙rth = ε˙ θth = ε˙ zth = α T˙s , so that ⎫ ⎧ ⎡ ⎤ ⎤ ⎡ ⎤⎡ ε˙r − ε˙ri σ˙ r 1−ν ν ν ⎬ ⎨ 1 E ⎣ σ˙ θ ⎦ = ⎣ ν 1 − ν ν ⎦ ⎣ ε˙ θ − ε˙ i ⎦ − α T˙s I , (2.30) θ ⎭ (1 − 2ν) ⎩ (1 + ν) σ˙ z ν ν 1−ν ε˙ z − ε˙ zi
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where α is the thermal expansion coefficient, I is the identity matrix and the strain components are in turn related to the radial and axial displacements, u and v respectively, by ∂ u˙ u˙ ∂ v˙ , ε˙ θ = , ε˙ z = . (2.31) ∂r r ∂z Also, as in [19], a specific constraint is exerted on the longitudinal strain, εz . Since the strand geometry is close to a long tube, the circular slices of the strand travelling downward keep their horizontal cross-sectional faces parallel. Consequently, the change of z strain in the slices is independent of the radial coordinate r (state of generalized plane strain), but is a function of z; hence, we have ε˙ z = ε˙ z (z). Although it is beyond the scope of this paper to consider a detailed model for the viscoplastic strains, we will nevertheless go as far establishing their orders of magnitude. To this end, we introduce the empirical strain hardening equation suggested by [19],
ε˙r =
dεri = dτ dεθi = dτ dεzi = dτ where
k (2σr − σθ − σz ) , 2 k (2σθ − σz − σr ) , 2 k (2σz − σr − σθ ) , 2
n Q i m−1 , k = A0 εe σe exp − RTs
(2.32) (2.33) (2.34)
(2.35)
and τ denotes time; in the present context, d/dτ in (2.32)–(2.34) is interpreted as Vcast d/dz. In (2.35), R is the universal gas constant, Q is a model constant, σe is the effective stress, given by 1 σe = √ (σr − σz )2 + (σθ − σz )2 + (σr − σθ )2 , 2 and the effective strain, εei , can be obtained by integrating the effective strain rate, which is expressed by √ 2 2 2 2 i i ε˙ e = ε˙r − ε˙ zi + ε˙ri − ε˙ θi + ε˙ θi − ε˙ zi . 3 The structure-mechanical problem is then fully specified by imposing three conditions. Normally, these consist of boundary conditions at r = W and r = rm (z), as well as a global constraint [19]. Considering first r = rm (z), we will have σr = σθ = σz = − p(z),
(2.36)
where p(z) is the hydrostatic pressure; we write this as p = p0 + ρgz, with p0 as the pressure at z = 0 and ρ as the liquid metal density, which we take in this paper to be the same as the solid metal density. Then ∂σr = − p(z), ˙ ∂r whereupon, using (2.24) and (2.36), we obtain σ˙ r + r˙m
(2.37)
σ˙ r = − p(z) ˙ at r = rm (z) .
(2.38)
For z > z mid , we set symmetry conditions at r = 0, ∂v = 0, ∂r which lead to ∂ v˙ = 0. u˙ = 0, ∂r u = 0,
123
(2.39)
(2.40)
Air gaps in vertical continuous casting in round moulds
135
The other boundary lies at r = W for z ≤ z gap , where we set zero normal displacement, u = 0,
(2.41)
which then gives u˙ = 0.
(2.42)
Also, because or the slenderness of the geometry, we will have zero shear stress in the z-direction at leading order, so that τr z = 0,
(2.43)
leading to τ˙r z = 0;
(2.44)
this condition holds for z > z gap also. Less straightforward is which boundary condition to set once the air gap has formed, i.e., at r = ra (z) for z > z gap . Schwerdtfeger et al. [19] associate the air gap with the condition σn = − pa ,
(2.45)
where n(= r ) is the coordinate normal to the mould and pa is the pressure of the surrounding atmosphere. On the other hand, Richmond and Tien [2] associate the onset of the air gap with σt = 0,
(2.46)
where t is the coordinate tangential to the mould in the horizontal plane, i.e., that the lateral stress should vanish when the air gap forms. Furthermore, from the analysis in [12], it became clear that, for that geometry (where t = x, n = y), σn = − pa would be inconsistent with a solution structure that assumes that the shear stresses are zero: in particular, the solution must be of the form σ y ≡ − p(z) (= 0) if τx y = 0. In order not to lose analytical tractability, Vynnycky [12] retained (2.46), even after the gap had formed. However, one benefit of the cylindrical geometry is that we can make analytical headway for both cases and can therefore explore the consequences of both assumptions. Denoting as model I the case where boundary condition (2.45) is used for z > z gap , and model II as the case where (2.46) is used instead, we describe the development for model II in the main text and relegate the description of model I to the appendix. However, additional considerations are required in order to determine ε˙ z (z). A further condition results from the fact that σz has to be compensated by the total external force, Fz , acting on the cross-sectional area, i.e. ⎧ W ⎪ ⎪ ⎪ ⎪ ⎪ 2π r σz dr if z ≤ z gap , ⎪ ⎪ ⎪ ⎪ ⎪ rm (z) ⎪ ⎪ ⎪ ra (z) ⎪ ⎨ Fz = 2π (2.47) r σz dr if z gap < z < z mid , ⎪ ⎪ ⎪ ⎪ rm (z) ⎪ ⎪ ⎪ ra (z) ⎪ ⎪ ⎪ ⎪ ⎪ 2π r σz dr if z mid < z < W. ⎪ ⎪ ⎩ 0
Different assumptions can be made to obtain Fz ; this is discussed briefly for the continuous casting of steel by Schwerdtfeger et al. [19]. In the mould region, Fz will equal the weight of the liquid metal column above the cross-sectional area of the shell if the friction within the mould is negligible and the strand is held by the guiding system below the mould, i.e., the slice of the shell is subjected to an overall compression. Hence, we set
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⎧ ⎪ −π p0 W 2 − rm2 (z) ⎪ ⎪ ⎪ ⎪ z ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ −πρg dz r (z ) − r if z ≤ z gap , (z) ⎪ m m ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ 2 2 ⎪ r + ρgz (z))) (z) − r (z) −π p ( (r ⎪ 0 m a a m ⎪ ⎪ ⎪ z ⎪ ⎪ ⎪ ⎨ −πρg rm2 (z ) − rm2 (z) dz if z gap < z < z mid , Fz = ⎪ z m (ra (z)) ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ 2 ⎪ ⎪ −π p + ρgz (z) − πρg W 2 (z − z mid ) r ( ) 0 mid a ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ ⎪ ⎪ z ⎪ ⎬ ⎪ ⎪ 2 2 ⎪ ⎪ if z mid < z < W, r (z ) dz − − r (z) ⎪ a m ⎪ ⎪ ⎩ ⎭
(2.48)
z m (ra (z))
where z m (r ) is the inverse function of rm (z); in each of the cases in (2.48), the first term containing p0 is due to the ambient pressure at the meniscus level, whereas the remainder is due to the weight of the liquid metal column. Differentiating (2.47) and (2.48) with respect to z and equating then gives ⎧ ⎪ W ⎪ ⎪ ⎪ ⎪ r σ˙ z dr = 0 if z ≤ z gap , ⎪ ⎪ ⎪ ⎪ ⎪ rm (z) ⎪ ⎪ ⎪ ⎪ ⎨ ra (z) (2.49) r σ˙ z dr = F (z) if z gap < z < z mid , ⎪ ⎪ ⎪ ⎪ rm (z) ⎪ ⎪ ⎪ ra (z) ⎪ ⎪ ⎪ ⎪ ⎪ r σ˙ z dr = F (z) if z mid < z < L , ⎪ ⎪ ⎩ 0
where F (z) = ( p0 + ρgz m (ra (z))) ra (z)˙ra (z). Note that in deriving (2.49), use has been made of condition (2.36). 3 Analysis 3.1 Nondimensionalisation To nondimensionalise, we write r z p rm ra R= , Rm = , Ra = , Z= , P= , W W W L EαT u Tmelt − To (z) v Tmelt − Ts U= , V = , = , o = , W αT LαT T T σr σθ σz (3.1) , θ = , Z = ,
R = EαT EαT EαT εi εi εi εr εθ εz , θ = , Z = , iR = Ri , θi = θi , iZ = zi , R = αT αT αT ε ε ε i where T and ε are temperature and inelastic strain scales, respectively, to be determined. For later use, we also write Ra (Z ) = 1 − δa (Z ) ,
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Air gaps in vertical continuous casting in round moulds
137
where a (Z ) is an O(1) function and δ is a dimensionless parameter which will be determined shortly. Equation 2.1 then becomes 1 ∂ ∂ ∂ = R , (3.2) Pe ∂Z R ∂R ∂R is the reduced Péclet number, given by where Pe ρc V L W 2 = ps cast . Pe ks L
(3.3)
The boundary conditions for are then = 0,
dRm Pe ∂ =− at R = Rm (Z ), ∂R St dZ
∂ = −κ ( − o (Z )) at R = 1, for Z < Z gap , ∂R and { − o (Z )} kM ∂ = ∂R ks (1 − δ (Z )) log W + k M log 1 − δ (Z ) a a WM ka
(3.4) (3.5)
(3.6)
at R = 1 − δa (Z ) for Z > Z gap , where Z gap = z gap /L, κ=
kM , ks log WWM
(3.7)
and St is the Stefan number, given by St =
cps T ; H f
(3.8)
a further dimensionless parameter that will turn up later is γ = δ/αT . As in [12], the air gap makes a leading-order contribution to heat transfer if kM W ∼ log log 1 − δa (Z ) , WM ka and we expect by analogy that δ 1, in which case it is appropriate to set δ = −ka /k M log (W/W M ); then (3.6) simplifies to −κ ( − o (Z )) ∂ = . ∂R (1 − δa (Z )) (1 + a (Z ))
(3.9)
Lastly, the initial conditions (2.17) and (2.18) are, respectively, = 0 at Z = 0,
(3.10)
Rm (0) = 1.
(3.11)
For the structural-mechanics part of the problem, we have, from (2.30), ⎫ ⎧ ⎤ ⎤ ⎡ ⎤⎡ ˙R ˙ R − χ ˙ iR
1−ν ν ν ⎬ ⎨ 1 1 ⎣ ⎣ ν 1 − ν ν ⎦ ⎣ ˙θ − χ ˙ i ⎦ + I ˙θ ⎦ = ˙ , θ ⎭ (1 − 2ν) ⎩ (1 + ν) ˙Z
ν ν 1−ν ˙ Z − χ ˙ iZ where χ = εi /αT . Writing P(Z ) = P0 + P1 Z , with ⎡
P0 =
p0 , EαT
P1 =
ρgL , EαT
(3.12)
(3.13)
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we have, from (2.36),
R = θ = Z = −P(Z ) at R = Rm (Z ),
(3.14)
for Z ≤ Z mid ; for Z > Z mid , we will have symmetry conditions at R = 0, i.e., ∂ V˙ = 0, ∂R where dots now denote differentiation with respect to Z . For the boundary facing the mould, we have
U˙ = 0,
(3.15)
U˙ = 0 at R = 1 for Z ≤ Z gap , ˙ θ = 0 at R = 1 − δa (Z ) for Z > Z gap .
(3.16)
Finally, Eq. 2.49 becomes ⎧ 1 ⎪ ⎪ ⎪ ⎪ ˙ Z dr = 0 ⎪ r if Z ≤ Z gap , ⎪ ⎪ ⎪ ⎪ ⎪ R (Z ) m ⎪ ⎪ 1−δ ⎪ ⎪ ⎨ a (Z ) ˙ Z dr = G(Z ) if Z gap < Z ≤ Z mid , r ⎪ ⎪ ⎪ ⎪ Rm (Z ) ⎪ ⎪ ⎪ a (Z ) ⎪ 1−δ ⎪ ⎪ ⎪ ˙ Z dr = G(Z ) if Z mid < Z ≤ 1, ⎪ r ⎪ ⎪ ⎩
(3.17)
(3.18)
0
where G(Z ) = −δ (P0 + P1 Z m (1 − δa (Z ))) (1 − δa (Z )) ˙ a (Z ); here, Z m (R) is the inverse function of Rm (Z ). To make further analytical progress, we need to consider the orders of magnitude of the various dimensionless parameters that are present: St, κ, δ, γ , P0 , P1 , χ . Pe, Using the data in Table 1, we obtain ∼ 0.17; κ ∼ 2, δ ∼ 10−4 , Pe
(3.19)
for the other five parameters, we need to determine T . This quantity ought to characterize the temperature difference across the solidified shell; furthermore, it should be chosen so that ∼ O(1). However, it is not straightforward to know the appropriate scale a priori, because the only temperature associated with the boundary condition at the outer surface of the shell is actually that associated with the mould, To (z), although this temperature is likely to be much lower than that at the outer surface of the shell. Nevertheless, using this value does set an upper bound on T ; hence, we take T = Tmelt − Tomin , where Tomin = min (To (z)| z ≥ 0). This gives St ∼ 2, γ ∼ 5 × 10−3 ,
P0 ∼ 10−4 ,
P1 ∼ 10−5 ;
whilst these orders of magnitude depend on the fact that we have taken T ∼ 103 K, we note that even if we had take T ∼ 102 K, leading to St ∼ 0.2, γ ∼ 5 × 10−2 ,
P0 ∼ 10−3 ,
P1 ∼ 10−4 ,
there will be no quantitative differences in what follows. Much less straightforward is how to establish an estimate for χ . Equations (2.32)–(2.34) suggest that n−1 Vcast exp RTQmelt εi ∼ . A0 L (EαT )m
123
(3.20)
Air gaps in vertical continuous casting in round moulds Table 1 Parameters for computations
139
Value
Unit
cps
485
J kg−1 K−1
E
3 × 1010
N m−2
g
9.81
ms−2
ka
0.1
W m−1 K−1
kM
160
W m−1 K−1
ks
335
W m−1 K−1
L
0.38
m
p0
105
N m−2
Tmelt
1356
K
Tomin
323
K
Vcast
0.03
m s−1
W
0.0135
m
WM
0.0165
m
α
2.25 × 10−5
K−1
ρ
8000
kg m−3
ν
0.35
–
H f
205000
J kg−1
However, earlier work specifically for copper by Freed [27], for which model constants, i.e. A0 , m, n, Q, are available, suggests that L A0 EαT m Q , (3.21) exp − εi ∼ Vcast C RTmelt where C is a further model constant; thus, we combine the two approaches to give, on setting n = 0, L A0 (αT )m−1 E m Q χ∼ exp − , Vcast C RTmelt
(3.22)
for which we have used [27] A0 = 5 × 107 s−1 , C = 1.43 × 107 Pa, m = 5, Q = 2 × 105 Jmol−1 , as well as R = 8.314 Jmol−1 K−1 . At this stage, the approach can considered consistent if we arrive at χ 1; be by this, we mean that the scalings adopted for σ j j=r,θ,z and ε j j=r,θ,z in (3.1), which were based on thermal rather than viscoplastic stresses, are indeed the correct ones. However, the high value of m in (3.22) means that the nominal value for χ is very sensitive to the exact values for T and E that are used; furthermore, Tmelt is most likely an overestimate for the temperature scale, and this will in turn lead to an overestimate in χ . Whilst the issue can presumably be resolved by numerical computations for the full thermoviscoplastic problem, which we try avoid here in favour of a more qualitative analytical approach, we proceed by assuming that χ 1 and thereby neglecting the viscoplastic strains; in support of this, we note that the computations of Schwerdtfeger et al. [19] for the continuous casting of steel indicated nothing qualitatively new if viscoplasticity is included, although there are of course quantitative differences. A further motivation for assuming that χ 1 is that it enables us to make a direct comparison with the air-gap model in [12], for which it was shown analytically that viscoplasticity played no role in determining the width of the air gap. No less significant is the fact that δ 1, which permits still further simplification.
123
140
M. Vynnycky
3.2 δ 1 For the thermal model, we now have Eq. 3.2, subject to initial conditions (3.10) and (3.11) and the following boundary conditions: • for 0 < Z ≤ Z gap , boundary conditions (3.4) and (3.5); • for Z gap < Z ≤ Z mid , (3.5) and ∂ −κ = (3.23) ( − o (Z )) at R = 1; ∂R (1 + a (Z )) • for Z mid < Z ≤ 1, (3.23) and ∂ = 0 at R = 0. (3.24) ∂R To determine Ra (Z ) and Z gap , we return to the structure mechanics. Whereas (3.16) is retained in its existing form, (3.17) becomes ˙ θ = 0 at R = 1 for Z > Z gap .
Also, Eq. 3.18 now becomes, at leading order in δ, ⎧ 1 ⎪ ⎪ ⎪ ˙ Z dr = 0 if Z ≤ Z gap , ⎪ r ⎪ ⎪ ⎪ ⎪ ⎪ Rm (Z ) ⎪ ⎪ ⎪ 1 ⎪ ⎨ ˙ Z dr = 0 if Z gap < Z ≤ Z mid , r ⎪ ⎪ ⎪ ⎪ Rm (Z ) ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ˙ Z dr = 0 ⎪ r if Z mid < Z ≤ 1. ⎪ ⎩
(3.25)
(3.26)
0
Using (3.12) and the nondimensional version of (2.26), ˙R ˙R − ˙θ ∂
+ = 0, ∂R R we have (1 − 2ν) ∂ ˙ + (1 − ν) ˙ R + ν ˙θ + ν ˙ Z + (1 + ν) (˙ R − ˙θ ) = 0, ∂R R which can be written, on recalling that ˙ Z = ˙ Z (Z ), as ˙ 1 ∂ ˙ ∂ ∂ R U = −(1 + ν) . (1 − ν) ∂R R ∂R ∂R Using (3.2) and integrating twice with respect to R gives 1 + ν ∂ 1 f 2 (Z ) ˙ + f 1 (Z )R + , (1 − ν) U = − ∂R 2 R Pe
(3.27)
(3.28)
(3.29)
(3.30)
where f 1 (Z ) and f 2 (Z ), as well as ˙ Z (Z ), are functions to be determined from the boundary conditions. For later use, ˙ R and ˙θ can be written as 1 + ν ∂ 2 1 1 f 2 (Z ) , (3.31) − f ˙ R = + (Z ) − 1 1−ν ∂ R2 2 R2 Pe 1 + ν 1 ∂ 1 1 f 2 (Z ) , (3.32) − + f 1 (Z ) + ˙θ = 1−ν R ∂R 2 R2 Pe and hence
123
Air gaps in vertical continuous casting in round moulds
141
⎡ ⎤ ⎤⎡ ⎤ ⎡ 1 ∂ ˙R 1/2 −(1 − 2ν) ν (1 − ν) f1
R ∂R 1 1 ⎢ ∂2 ⎣ 1/2 (1 − 2ν) ν (1 − ν) ⎦ ⎣ f 2 /R 2 ⎦ + ˙θ ⎦ = (1 − ν) ⎣ ⎣ 2 (1 + ν)(1 − 2ν) Pe ∂ 2 ∂ R 1 ˙Z
˙ Z ν 0 (1 − ν)2 + 2 ⎡
∂R
∂ R ∂R
⎤ ⎥ ⎦
(3.33)
For Z < Z gap , we have, from (3.16) and (3.30), 1 (1 + ν)ϑ1 , f1 + f2 = 2 Pe where ϑ1 = (∂ /∂ R) R=1 . Also, using ˙ ) at R = Rm (Z ) , ˙ R = − P(Z
(3.34)
(3.35)
we apply (3.4) to end up with
˙ 1 f2 −1 Rm ˙ − (1 − ν) P . f 1 − (1 − 2ν) 2 + ν (1 − ν) ˙ Z = (1 + ν)(1 − 2ν) St 2 Rm Rm
Turning to (3.26), we have 2(1 + ν)(1 − 2ν) −1 Rm R˙ m + ϑ1 . ν f 1 + (1 − ν)2 ˙ Z = − PeSt 2 1 − Rm Pe
(3.36)
(3.37)
Hence,
3ν − 1 ϑ1 Rm R˙ m 2 2 2 + 1+ Rm − (1 − ν)Rm 1 − Rm P˙ , (3.38) 1−ν St Pe ˙ Rm 1+ν (1 − ν 2 )Rm Rm ϑ1 3ν − 1 2 2 2 ˙ f2 = (1 − ν)R R R , 1 − R P − − 1 + 1 + m m m m 2 1−ν 1−ν St 1 − Rm Rm Pe (3.39) 2 ϑ1 2(1 + ν) 1 + Rm Rm R˙ m 2 ˙ ν Rm , (3.40) ˙ Z = P− + 2 Rm 1 − Rm St Pe 2(1 − ν 2 ) f1 = 2 1 − Rm Rm
where 2 Rm = (1 + ν)Rm + 1 − ν.
For Z gap < Z ≤ Z mid , we retain (3.36) and (3.37), but replace (3.34) by using (3.25); it is, however, more convenient to differentiate it with respect to Z and use it in the form ˙ θ = 0 at R = 1.
(3.41)
So,
˙ − ϑ1 (1 + ν)(1 − 2ν) Pe 1 , f 1 + (1 − 2ν) f 2 + ν (1 − ν) ˙ Z = − 2 Pe and hence, 2 + (1 + ν) 2 P˙ ˙ + (1 − ν) Rm 2 (1 − ν) 2 (3ν − 1) Rm PeRm R˙ m f1 = − + ϑ , 1 2 4 1 − Rm St 1 + Rm Pe 2 R˙ (1 + ν)Rm ϑ ˙ + 1− m , (1 − ν) P˙ − f2 = 2 Rm St 1 + Rm Pe 2 (1 − ν) R 2 + 1 + ν ϑ1 + Pe Rm R˙ m ˙ m 2ν St 2 ˙ Rm − . ˙ Z = P+ 2 4 (1 − ν) 1 − Rm 1 + Rm 1−ν Pe For Z mid < Z ≤ 1, we retain (3.37), which simplifies to 2(1 + ν)(1 − 2ν)ϑ1 , ν f 1 + (1 − ν)2 ˙ Z = − Pe
(3.42)
(3.43) (3.44)
(3.45)
(3.46)
123
142
M. Vynnycky
and (3.42). Using (3.15), we have f 2 (Z ) = 0, which also avoids the appearance of infinite stresses and strains at R = 0. Hence, 2 ˙ − (1 + ν)ϑ1 , (3.47) f1 = − (1 − ν) Pe Pe (1 + ν)ϑ1 2 ˙ − ν ˙ Z = . (3.48) 1−ν Pe At this point, we note that for Z ≥ Z gap , (U ) R=1 = −γ a (Z ). Now, since U˙ R=1 = (R ˙θ ) R=1 , we can use (3.4) to write ⎧ 2 m R˙ m /St ⎪ ⎪ 4ν Rm ϑ1 + PeR ⎪ ⎪ 4 ⎪ (1 − ν) 1 − Rm Pe ⎨ for Z gap < Z ≤ Z mid , 2 +1−ν 2 P˙ ˙ (3.49) U˙ R=1 = (1 + ν)Rm 2ν Rm ⎪ + − ⎪ 2 2 ⎪ 1 + R − ν) 1 + R (1 ⎪ m m ⎪ ⎩ ˙ − for Z mid < Z ≤ 1. Returning to (3.23), we see that this can be written as ∂ −κ ( − o ) , = ∂R 1 − γ −1 (U ) R=1
(3.50)
where Z (U ) R=1 =
U˙
R=1
dZ .
(3.51)
Z gap
Hence, boundary condition (3.50) can be written in integro–differential form solely in terms of , and the problem as a whole for has now been rewritten in a form that does not contain any structure-mechanical quantities, i.e., we have decoupled the thermal and mechanical problems. Note also that the problem for V completely decouples from the equations for U , and can be solved for once U is determined; we omit this here, since the problem of most interest is that for U and . 3.3 Onset of air-gap formation (analysis for Z 1) Since the air gap is thought to start to form just a short distance below the meniscus, it is instructive to consider the analysis for Z 1, where regular series expansions for and Rm in terms of Z ought still to be valid. Setting = (1 − Rm (Z )) F (Z , η) , η =
1− R , 1 − Rm (Z )
we may write (3.2) as
(1 − Rm ) − R˙ m F + (1 − Rm ) FZ + η R˙ m Fη = Pe with boundary conditions (3.4) and (3.5) becoming −1 R˙ m at η = 1, F = 0, Fη η=1 = PeSt Fη = κ ((1 − Rm ) F − o (Z )) at η = 0,
(3.52)
[1 − η (1 − Rm )] Fη 1 − η (1 − Rm )
η
,
(3.53)
(3.54) (3.55)
respectively. As Z → 0, a self-consistent boundary-value problem is obtained if Rm (Z ) = 1 − λZ + o(Z ), where λ is a strictly positive constant whose value is to be determined. However, it turns out to be more convenient to write F = F0 (η) + Z F1 (η) + · · ·, Rm (Z ) = 1 − λ1 Z − λ2 Z 2 + . . .
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Air gaps in vertical continuous casting in round moulds
143
At Z 0 , Eq. 3.53 reduces to F0ηη = 0,
(3.56)
subject to F0η = −κ o (0) at η = 0, −1 λ1 at η = 1, F0 = 0, F0η = −PeSt
(3.57) (3.58)
respectively, giving λ1 = κ Pe
−1
St o (0),
F0 (η) = κ o (0) (1 − η) .
˙θ As regards air-gap formation, we need to consider lim Z →0 , where R=1 ˙ 2 Rm Rm + ϑ1 4ν Rm St Pe ˙θ
=− R=1 2 2 (1 − ν) ((1 + ν)Rm + 1 − ν) 1 − Rm 2 P˙ ˙ 2ν Rm + . − 2 + 1 − ν) ((1 + ν)Rm 1−ν At leading order in Z , which is Z 0 , we obtain −1 + F1η ν 2λ2 PeSt λ1 F0 (0) η=0 ˙θ ;
= − ν P1 + R=1 (1 − ν) λ1 1−ν Pe hence, we need to consider (3.53) at Z 1 , since F1 in (3.62) has not yet been determined. Equation (3.53) at Z 1 gives 1 F0 + 1 − Peλ 1 η F0η + ηF0ηη = F1ηη , λ1 Peλ subject to ˙ o (0) at η = 0, F1η = κ λ1 F0 − −1 at η = 1. F1 = 0, F1η = −2λ2 PeSt
(3.59) (3.60)
(3.61)
(3.62)
(3.63)
(3.64) (3.65)
Using (3.60), we have κ 2 St 2o (0) (κSt o (0) − 1) = F1ηη , Pe and so κ 2 St 2o (0) F1 = (κSt o (0) − 1) η2 + A1 η + B1 , 2Pe where A1 and B1 are constants of integration. Applying the boundary conditions, we obtain ˙ o (0)Pe κ −κ 2 St 2o (0) + , A1 = − Pe ˙ o (0) − κ 2 St2 3o (0) κ κ (1 − 2κ) St 2o (0) + 2Pe B1 = , 2 Pe ˙ o (0) − κ 2 St2 3o (0) κSt κ (1 − κ) St 2o (0) + 2Pe λ2 = , 2 2Pe and hence we arrive at νκ o (0) κ 2 St 2o (0) ˙θ + − ν P1 . = lim R=1 (1 − ν) Z →0 Pe Pe
(3.66)
(3.67)
(3.68) (3.69) (3.70)
(3.71)
123
144
M. Vynnycky
˙θ By analogy with [12], an air gap is more likely to form if lim Z →0 > 0, i.e., R=1 κ 2 St 2o (0) νκ o (0) + − ν P1 > 0. Pe Pe (1 − ν) Taking the relevant square root, we have ! 2 1 ν ν PeP ν o (0) > + , − 2 κ St 2κSt (1 − ν) 2κSt (1 − ν) or, in dimensional terms, ⎧! ⎫ 2 ⎨ 2 νH f ⎬ νgH f ρcps Vcast W νH f ks log (W M /W ) + To (0) < Tmelt − − , ⎩ cps k M Eα 2 (1 − ν) 2 (1 − ν) ⎭
(3.72)
(3.73)
which is substantially different to the criterion obtained by [12]; the reason for this is discussed later. Also, we now have, as Z → 0, 2 νκ o (0) κ St 2o (0) + − ν P1 Z ,
θ ∼ −P0 + (1 − ν) Pe Pe indicating that the gap should start at −1 2 νκ o (0) κ St 2o (0) Z gap ≈ + − ν P1 P0 . (3.74) (1 − ν) Pe Pe This equation highlights rather well the different techniques that could be applied in order to delay the onset of by increasing air-gap formation, i.e., to increase Z gap : (i) to lower o (0), by decreasing To (0); (ii) to increase Pe, Vcast ; (iii) to decrease κ, by increasing W M . 3.4 Summary of reduced model equations Following reduction for δ 1 and the use of variable transformation (3.52), the remaining task at hand is the solution of [1 − η (1 − Rm )] Fη η (1 − Rm ) − R˙ m F + (1 − Rm ) FZ + η R˙ m Fη = Pe , (3.75) 1 − η (1 − Rm ) subject to initial conditions F = κ o (0) (1 − η) at Z = 0, (3.76) Rm (0) = 1, and boundary conditions ⎧ for 0 < Z ≤ Z gap , ⎨ κ ((1 − Rm ) F − o (Z )) − Rm ) F − o (Z )) Fη = −κ ((1 for Z > Z gap , ⎩ 1 − γ −1 (U )η=0 at η = 0, where (U )η=0 = (U ) R=1 , as given by (3.51); and " −1 R˙ m for 0 < Z ≤ Z mid , F = 0, Fη η=1 = PeSt for Z mid < Z ≤ 1, Fη η=1 = 0
at η = 1. Furthermore, Z gap will be given by the solution to θ Z gap = 0, i.e., $ Z gap# 2 Z −PeSt ˙ )/2 −1 Rm Z R˙ m Z − ϑ1 − P(Z ˙ 4ν Rm + P0 = dZ , Rm (Z ) 1−ν
(3.77)
(3.78)
(3.79)
(3.80)
0
whereas Z mid is such that Z mid = {Z |Rm (Z ) = 0} . As in [12], the model equations were then solved using the finite-element software Comsol Multiphysics [28].
123
Air gaps in vertical continuous casting in round moulds
145
4 Model results Although nondimensional variables were useful for the analysis in Sect. 3 for easily identifying orders of magnitudes of various terms, as well as formulating the leading-order problem, it turns out to be more useful to return to dimensional variables in order to consider actual model predictions. As a basis for computations, we use the geometry and operating conditions given in Table 1, and choose To (z) of the same form as used in [12], i.e., To (z) = T0 + (To (0) − T0 ) exp(−z/z c ),
(4.1)
where z c = − (To (0) − T0 ) /T˙o (0); also, as earlier, we take T0 = 323 K. Consequently, we are able to compare directly the effect of the change in geometry.
4.1 Air-gap formation Figure 2 shows the curve
⎧! ⎫ 2 ⎨ 2 νH f ⎬ νgH f ρcps Vcast W νH f ks log (W M /W ) To (0) = Tmelt − + − , ⎩ cps k M Eα 2 (1 − ν) 2 (1 − ν) ⎭
(4.2)
and the corresponding curves from the earlier model in [12], given by (Tmelt − To (0))2 = −
2 T˙ (0) H f ρVcast ks HM o
k 2M
,
(4.3)
where all notation is the same as for the cylindrical model, except with HM in the earlier model corresponding to W M − W here; we plot To (0) as a function of Vcast for two different values of T˙o (0), −3 × 103 and −3 × 104 Km−1 . With reference to Eq. 3.73, an air gap is likely to form for combinations of T0 (0) and Vcast that lie below the curve given by (4.2), whereas an air gap is likely to form for combinations of T0 (0) and Vcast that lie above these curves given by (4.3). There is clearly a great difference between the two. Recalling the notation for n and t introduced in conjunction with (2.45) and (2.46), the principal reason for this is the fact that limt→0 (σ˙ t )n=0 = 0 for the present case, whereas limt→0 (σ˙ t )n=0 = 0 in [12]; the latter case then leads to T˙o (0) also appearing in the air-gap formation criterion, although it does not appear in the former case. Furthermore, we find that, for the values of Vcast of interest, Eq. 4.2 gives a horizontal line, indicating that more or less all reasonable combinations of To (0) and Vcast will result in an air gap. Fig. 2 T0 (0) as a function of Vcast using (4.2) and (4.3)
1400 1300 .
To(0)=−3x103 Km−1
1200
To (0) [K]
1100 1000 900 .
To(0)=−3x104 Km−1
800 700 Eqn. (4.2) Eqn. (4.3) Eqn. (4.3)
600 500
0
0.02
0.04
V
cast
0.06
0.08
0.1
−1
[ms ]
123
146
M. Vynnycky
1.4
0.014 no air−gap model model I model II
0.01
1
0.008
0.8
0.006
0.6
0.004
0.4
0.002
0.2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
z [m]
Fig. 3 Comparison of rm vs. z, obtained with air-gap models I and II and no air-gap model for To (0) = 1,300 K and Vcast = 0.03 ms−1
model I model II
1.2
W–r a [m]
rm [m]
0.012
x 10−4
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m] Fig. 4 Comparison of W −ra vs. z, obtained with air-gap models I and II for To (0) = 1,300 K and Vcast = 0.03 ms−1
4.2 Model comparison Now, we fix To (0) = 1,300 K, T˙o (0) = −3 × 104 Km−1 and Vcast = 0.03 ms−1 and compare results for models I and II, as well as the case when the air-gap model is omitted completely. Figure 3 compares the profiles for rm . As is evident, models I and II predict a far greater solidification depth than the model without the air gap, and this is analogous to the result in [12]; for this value of Vcast , the difference in predictions between models I and II is not particularly great. Figure 4 compares the profiles for W − ra , i.e., the air-gap thickness. For both of these, the air-gap thickness is of the order of magnitude computed in models and inferred from experiment by other authors, albeit for the continuous casting of steel [6,8]; in addition, the profiles are quantitatively similar to those computed in [23]. Several features are noteworthy in Fig. 4: model I predicts a wider air gap than model II; once complete solidification has occurred, the air gap actually widens still further, unlike the corresponding result in [12]. Furthermore, for these values of T0 (0) and Vcast , we compute that Z gap ∼ 10−4 ; hence, the air gap starts almost instantaneously, which is why its point of inception is indistinguishable from the origin on this plot. Note also the kinks in the profiles for W − ra , which correspond to the location of z mid ; here, we have omitted the corresponding curve for the no air-gap model, since it implicitly assumes that ra ≡ W . Figure 5 shows the impact of the air-gap thickness on the temperature at the outer surface of the solidified shell, i.e., (Ts )r =ra : the wider air gap present in model I results in a slower decrease in temperature with z. In both cases, the temperature increases slightly at the value of z corresponding to the location of z mid ; once again, we omit the corresponding curve for the no air-gap model, since (Ts )r =ra = (TM )r =W for this model. Figure 6 shows the temperature at the inner mould surface, i.e., (TM )r =W , for all three models. In spite of the difference in the values of (Ts )r =ra for models I and II, there is scarcely any difference in the profiles for (TM )r =W for the two models. Note, however, that the model without the air gap predicts a much higher temperature at this surface, particularly after complete solidification has occurred. Figure 7 shows the heat flux, Q, given by ∂ TM ; Q = kM ∂r r =W once again, the results of models I and II resemble each other, but differ greatly from those of the model without air gap. It is only after z ≈ 0.15 m that the profiles for all three models begin to agree.
123
Air gaps in vertical continuous casting in round moulds
147
1400
1400 model I model II
1300
no air−gap model model I model II
1200
[K]
1000
M r=W
1100
800
(T )
(T )
s r=ra
[K]
1200
1000
600
900
400
800 700
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
200
0.4
0
0.05
0.1
0.15
Fig. 5 Comparison of (Ts )r =ra vs. z, obtained with air-gap models I and II for To (0) = 1,300 K and Vcast = 0.03 ms−1
0.25
0.3
0.35
0.4
Fig. 6 Comparison of (TM )r =W vs. z, obtained with air-gap models I and II and no air-gap model for To (0) = 1,300 K and Vcast = 0.03 ms−1
x 106
10000
−2
9000
−4
8000
−6
7000
−8
−2 −1 h [Wm K ]
−2 Q [Wm ]
0
0.2
z [m]
z [m]
no air−gap model model I model II
−10 −12
model I model II
6000 5000 4000 3000
−14
2000
−16
1000
−18 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m]
Fig. 7 Comparison of Q vs. z, obtained with air-gap models I and II and without for air-gap model for To (0) = 1,300 K and Vcast = 0.03 ms−1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m]
Fig. 8 Comparison of h vs. z, obtained with air-gap models I and II for To (0) = 1,300 K and Vcast = 0.03 ms−1
Finally, Fig. 8 shows the heat-transfer coefficient, h, obtained with models I and II. These resemble each other qualitatively, even in so far as the customary kink at z = z mid . It is, however, model II, with its slightly narrower air gap, that has marginally larger values for h; here also, we omit the curve for the no air-gap model, since it would predict an infinite heat-transfer coefficient. Although model I appears to be the more generally accepted one for describing conditions at the surface of the solidified shell [13,17,19], it is nevertheless instructive to consider the effect of changes in Vcast on both models; we proceed to this next.
4.3 Effect of Vcast on models I and II Figures 9a and b compare the profiles for rm for models I and II, respectively, for three different casting speeds: Vcast = 0.01, 0.03 and 0.1 ms−1 ; these are the same speeds as in [12]. Whereas complete solidification occurred there for only the two lowest speeds, it is clear that is has occurred for all three speeds here. Hence, in general,
123
148
M. Vynnycky
(a) 0.014
(b) 0.014
0.012
V
0.012
cast =0.01
0.01
V
cast =0.01
V
[m] m
0.006
r
0.006
r
[m]
cast
0.008
0.004
0.004
0.002
0.002 0
0.05
0.1
0.15
0.2
0.25
0.3
cast =0.03
=0.1 ms−1
0.008
0
V
0.01
m
cast
V
ms−1
−1 cast =0.03 ms
V
0.35
0
0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
ms−1 ms−1
=0.1 ms−1
0.35
0.4
0.35
0.4
z [m]
z [m]
Fig. 9 rm vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II x 10−4
(a)
−1
=0.01 ms
V
=0.01 ms
V
=0.03 ms−1
V
=0.03 ms−1
V
=0.1 ms−1
V
=0.1 ms−1
cast cast cast
−1
cast cast cast
W–r [m]
2
2
a
a
W–r [m]
x 10−4
(b) V
1
0
1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0
0.05
z [m]
0.1
0.15
0.2
0.25
0.3
z [m]
Fig. 10 W − ra vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II
solidification depths corresponding to a particular value of Vcast are much shorter for the axisymmetric configuration, as might be expected. Figures 10a and b show W − ra (z) for models I and II, respectively, for these three casting speeds. As in [12], the air-gap thickness is much greater at lower casting speeds; however, a significant difference is that the width of the air gap continues to increase for z > z mid , and furthermore at a rate greater than that for z < z mid . Consequently, we do not have, as occurred in [12], the possibility that the air gap can vanish before the end of the mould; hence, the present result is more in line with those of other authors, albeit obtained for the continuous casting of steel [6,8]. Figures 11a and b show the temperature at the outer surface of the solidified shell, whereas Fig. 12a and b show the temperature at the mould wall at r = W ; for the latter plots, the profile for To (z) has been included for reference. From these, it is evident that the kinks present in the temperature profiles in Fig. 11a and b at z = z mid have been smeared away in Fig. 12a and b; also, whilst there are substantial quantitative differences in the profiles for different values of Vcast in Fig. 11a and b, these are almost negligible in Fig. 12a and b. Figures 13a and b show the heat flux, Q, which turns out to be the same order of magnitude as in [12]. Figures 14a and b show the heat-transfer coefficient; although qualitatively similar to the profiles in [12] near the start of solidification, it continues to decrease, rather than increase, as the air gap widens after z = z mid .
5 Conclusions In this paper, we have developed an asymptotic thermomechanical model for air-gap formation in vertical continuous casting in round moulds. Although the model is based on an earlier counterpart in an idealized 2D geometry [12], there are significant differences: the thermal and mechanical models cannot be decoupled, unlike in the earlier
123
Air gaps in vertical continuous casting in round moulds 1400
(b)
1300
1200
1200
(T ) r=r [K]
a
1100
s
s
1400
1300
a
(T ) r=r [K]
(a)
149
1000 V
900
V V
800
0
cast
=0.01 ms−1
cast
=0.03 ms−1
cast
=0.1 ms−1
0.05
0.1
1100 1000 V
900
V V
0.15
0.2
0.25
0.3
0.35
800
0.4
0
cast
=0.01 ms−1
cast
=0.03 ms−1
cast
=0.1 ms−1
0.05
0.1
0.15
z [m]
0.2
0.25
0.3
0.35
0.4
z [m]
Fig. 11 (Ts )r =ra vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II
(a)
(b)
1400 1200
V
cast =0.01
1200
−1
ms
−1 cast =0.03 ms
[K]
cast =0.1
ms−1
M r=W
V
800
(T )
(T )
M r=W
[K]
V
1000
1400 V
cast =0.01
1000
V
cast =0.1
ms−1
800 600
600
400
400 T
T
o
200
−1
ms
−1 cast =0.03 ms
V
0
0.05
o
0.1
0.15
0.2
0.25
0.3
0.35
200
0.4
0
0.05
0.1
0.15
z [m]
0.2
0.25
0.3
0.35
0.4
z [m]
Fig. 12 (TM )r =W vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II 0
x 10
6
(b)
6
0
−1
−1
−2
−2 −2 Q [Wm ]
−2 Q [Wm ]
(a)
−3 −4 −5
−1 cast =0.01 ms
V V
cast =0.03
−6
V
cast =0.1
ms−1
−3 −4 −5
V
cast =0.01
V
cast =0.03
−6
ms−1
V
cast =0.1
−7 −8
x 10
ms−1 ms−1
ms−1
−7 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
−8
0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m]
z [m]
Fig. 13 Q vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II
(b) 10000
(a) 10000
9000
9000
8000
V
Vcast =0.03 ms−1
7000
Vcast =0.03 ms−1
cast =0.01
7000
V
cast =0.1
6000
−2 −1 h [Wm K ]
−2 −1 h [Wm K ]
ms−1
V
8000
ms−1
5000 4000 3000
V
cast =0.1
6000
ms−1
ms−1
5000 4000 3000 2000
2000
1000
1000 0
cast =0.01
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m]
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [m]
Fig. 14 h vs. z for Vcast = 0.01, 0.03, 0.1 ms−1 for: a model I, b model II
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model, and the thickness of the air gap will therefore depend, in general, on the thermoviscoplastic constitutive relation that is used to describe the cooling behaviour of the solidified metal shell. However, such decoupling is possible in the thermoelastic limit, which is the situation that was motivated and focused on here. Even so, results obtained using parameters for the continuous casting of copper differed significantly from those in [12], in several ways: through the criterion for the onset of air-gap formation just after solidification starts; through the fact that the air gap was found to widen still further after complete solidification had occurred. The analytical framework presented here can now be extended in many ways: for example, to include the effect of mould taper, which can also be treated asymptotically, mould contraction, which would require solving for the stresses and displacements in the mould itself, and the effect of superheat, i.e., with the temperature of the incoming molten metal being greater than the melting temperature. Furthermore, the work is a precursor to considering a more complete model that includes viscoplasticity, as well continuous casting processes other than downward vertical, e.g. [16]. On another tack, however, the model results provide a new and useful benchmark for verifying 3D numerical codes describing the thermomechanics in continuous casting models, which is normally done using the classical solution in [29]; such verification would, in itself, be a prerequisite for development of models for processes with arbitrary casting cross-section [30]. Acknowledgements The author would like to acknowledge the constructive comments made by the referees during the preparation of this paper, as well as the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative Grant 06/MI/005.
Appendix A: Analytical forms for model I For model I, we replace (3.17) by
R = −Pa at R = 1 − δa (Z ) for Z > Z gap ,
(A1)
where Pa = pa /EαT ; Eq. A1 subsequently gives, at leading order in δ, ˙ R = 0 at R = 1,
(A2)
instead of (3.41); hence, other than for determining R (R, Z ), the model results will not depend on the value chosen for Pa . Then (3.42) is replaced by 1 (1 + ν)(1 − 2ν)ϑ1 , (A3) f 1 − (1 − 2ν) f 2 + ν (1 − ν) ˙ Z = − 2 Pe and f 1 , f 2 and ˙ Z for Z gap < Z < Z mid are now given by Rm R˙ m 2 ϑ1 2 2 ˙ + (1 − ν) , (A4) f1 = R P − (1 − 3ν) m 2) (1 − Rm St Pe Rm ϑ1 (1 + ν)Rm R˙ m ˙ , (A5) f2 = (1 − ν)Rm P − − 2) (1 − Rm St Pe 2 Rm R˙ m ϑ1 2 ˙ ˙ Z = − , (A6) ν Rm P+ + 2 (1 − Rm ) St Pe instead of (3.43)–(3.45). For Z mid < Z < 1, we end up with 2(3ν − 1)ϑ1 , f1 = Pe 2ϑ1 ˙ Z = − , Pe instead of (3.47) and (3.48).
123
(A7) (A8)
Air gaps in vertical continuous casting in round moulds
Finally, U˙ R=1 is now given by ⎧ m St−1 R˙ m − P˙ Rm 2 ϑ1 + PeR ⎪ ⎪ ⎨− 2) − Rm Pe(1 U˙ R=1 = ⎪ 2ϑ ⎪ ⎩− 1 Pe
151
for Z gap < Z ≤ Z mid ,
(A9)
for Z mid < Z ≤ 1,
instead of (3.49).
References 1. Savage J (1962) A theory of heat transfer and air gap formation in continuous casting molds. J Iron Steel Inst 198:41–47 2. Richmond O, Tien RH (1971) Theory of thermal stresses and air-gap formation during the early stages of solidification in a rectangular mold. J Mech Phys Solids 19:273–284 3. Kristiansson J-O (1982) Thermal stresses in the early stage of the solidification of steel. J Therm Stresses 5:315–330 4. Tien RH, Richmond O (1982) Theory of maximum tensile stresses in the solidifying shell of a constrained regular casting. J Appl Mech 49:481–486 5. Grill AK, Sorimachi K, Brimacombe JK (1976) Heat flow, gap formation and break-outs in the continuous casting of steel slabs. Metall Mater Trans B 7B:177–189 6. Kelly JE, Michalek KP, O’Connor TG et al (1988) Initial development of thermal and stress fields in continuously cast steel billets. Metall Mater Trans A 19A:2589–2602 7. Bellet M, Decultieux F, Menai M et al (1996) Thermomechanics of the cooling stage in casting processes: three-dimensional finite element analysis and experimental validation. Metall Mater Trans B 27B:81–99 8. Huespe AE, Cardona A, Fachinotti V (2000) Thermomechanical model of a continuous casting process. Comput Methods Appl Mech Eng 182:439–455 9. Kim KY (2003) Analysis of gap formation at mold-shell interface during solidification of aluminium alloy plate. ISIJ Int 43:647–652 10. Li C, Thomas BG (2004) Thermomechanical finite-element model of shell behavior in continuous casting of steel. Metall Mater Trans B 35B:1151–1172 11. Sun D, Annapragada SR, Garimella SV et al (2007) Analysis of gap formation in the casting of energetic materials. Numer Heat Transf Part A 51:415–444 12. Vynnycky M (2009) An asymptotic model for the formation and evolution of air gaps in vertical continuous casting. Proc R Soc A 465:1617–1644 13. Cherukuri HP, Johnson RE (2001) Modelling vertical continuous casting with temperature-dependent material properties. Int J Mech Sci 43:1243–1257 14. Das SK, Sarkar A (1996) Computational modelling of thermal transport phenomena in continuous casting process based on nonorthogonal control volume approach. Commun Numer Methods Eng 12:657–671 15. Guo LL, Wang XD, Zhan HY et al (2007) Mould heat transfer in the continuous casting of round billet. ISIJ Int 47:1108–1116 16. Härkki M, Miettinen J (1999) Mathematical modeling of copper and brass upcasting. Metall Mater Trans B 30:75–98 17. Johnson RE, Cherukuri HP (1999) Vertical continuous casting of bars. Proc R Soc A 455:227–244 18. Lee JE, Han HN, Oh KH et al (1999) A fully coupled analysis of fluid flow, heat transfer and stress in continuous round billet casting. ISIJ Int 39:435–444 19. Schwerdtfeger K, Sato M, Tacke K-H (1998) Stress formation in solidifying bodies. Solidification in a round continuous casting mold. Metall Mater Trans B 29:1057–1068 20. Yao M, Yin HB, Fang DC (2004) Real-time analysis on non-uniform heat transfer and solidification in mould of continuous casting round billets. ISIJ Int 44:1696–1704 21. Yao M, Yin HB, Wang JC et al (2005) Monitoring and analysis of local mould thermal behaviour in continuous casting of round billets. Ironmaking Steelmaking 32:359–368 22. Yin HB, Yao M, Zhan HY et al (2006) 3D stress model with friction in and of mould for round billet continuous casting. ISIJ Int 46:546–552 23. Yin HB, Yao M (2007) Inverse problem-based analysis on non-uniform profiles of thermal resistance between strand and mould for continuous round billets casting. J Mater Process Technol 183:49–56 24. Ho K, Pehlke RD (1984) Mechanics of heat transfer at a metal-mold interface. AFS Trans 61:587–598 25. Nishida Y, Droste W, Engler S (1986) The air gap formation process at the casting mold interface and the heat transfer mechanism through the gap. Metall Mater Trans B 17:833–844 26. Jablonka A (1995) Spannungen und Deformationen in Elektro-Schlacke-Umschmelzblöcken während des Umschmelzens, der nachfolgenden Abkühlung und der sich anschliessenden Wärmebehandlung. Dissertation, Technische Universität Clausthal, Clausthal-Zellerfeld, Germany 27. Freed AD (1988) Thermoviscoplastic model with application to copper. NASA Technical Paper 2845
123
152
M. Vynnycky
28. Comsol Multiphysics 3.5a. http://www.comsol.com 29. Weiner JH, Boley BA (1963) Elasto-plastic thermal stresses in a solidifying body. J Mech Phys Solids 11:145–154 30. Åberg J, Vynnycky M, Fredriksson H et al (2005) An on-site experimental heat flux study and its interpretation in a FEMLAB finite element simulation of continuous casting of copper in the South-Wire process. Trans Indian Inst Met 58:509–515
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