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20.- Modeling of Continuous Casting of Steel in Dog-Bone shaped Molds

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20.- Modeling of Continuous Casting of Steel in Dog-Bone shaped Molds
Case 20
Modeling of Continuous Casting of Steel in Dog-Bone
shaped Molds
An investigation was undertook progress by a steel producer to continuously cast steel
slabs using a mold with dog-bone cross section. While plant trials were being undertaken,
the produced decided to gain additional insight in to the behavior of the system by means
of mathematical modeling. A study must commissioned to develop a mathematical model
of the peocess of solidification of steel inside the dog-bone shaped mold.
The model consisted of a two dimensional slice of steel on the x − y plane that moved
downwards at the casting velocity vc though the mold. The mathematical model requires
statement of the energy balance equation
ρCp
∂ ∂T
∂ ∂T
∂T
=
(k
)+
(k
)
∂t
∂x ∂x
∂y ∂y
By incorporating the latent heat into the specific heat thus yielding an effective specific heat
formulation, the solidification process is readily taken into account.
The above must be solved subject to the statement of boundary conditions specifying the
inlet temperature of steel in the mold as well as the heat extraction rate though the mold
wall.
The main complication here is the fact that one of the mold boundaries is not aligned
with the corresponding coordinate direction. This creates difficulties in implementing the
boundary condition.
Non-orthogonal curvilinear coordinate transformations were used to transform the region
with irregular boundary into a simple geometrical shape. The irregular boundary is mapped
onto a simple boundary which coincides with a coordinate line in the transformed system
of coordinates. Since excellent numerical solvers for the heat equation are already available,
the objective of the research was to use the transformation to produce a set of equations
that could then be solved by the standard software. The specific software requested to be
used was the finite volume method embedded in the fortran code 2-EFIX.
In standard finite volume codes such as 2-EFIX, TEACH, and CONDUCT the discretized
energy equation for a given finite volume has the form
aP TP = aE TE + aW TW + aN TN + aS TS + b
1
(1)
where
∆y
∆x
∆y
aW = αw
∆x
∆x
aN = αn
∆y
∆x
aS = αs
∆y
∆x∆y
a0P =
∆t
A
b=
∆x∆y + a0P TP0
ρCp
(2)
aE = αe
and
aP = aE + aW + aN + aS + a0P −
(3)
(4)
(5)
(6)
(7)
B
∆x∆y
ρCp
(8)
Since there is one equation like the above for each volume element we have a system of
algebraic equations. The system can be solved by line by line iteration with relaxation
to yield the values of TP for all P. The goal here is to produce appropriate values of the
coefficients so as to be able to use the code to solve the solidification problem in the irregular
domain.
The key to the transformation is to consider the irregular boundary to be describable
as a function of x, δ(x). Discretization proceeds by subdividing the region of interest into
a collection of finite volumes. Consider a single irregular volume element with boundaries
l1 , l2 , l3 and l4 . The coordinate transformation must be such such that the transformed
volume becomes rectangular.
In the transformed domain the coordinates are ξ and η. The transforming equations are
ξ=x
and
η=
y
δ(x)
Further, defining a new quantity β as
β=η
dδ
dδ
=η
dx
dξ
Therefore, applying the chain rule gives
∂ ∂ξ
∂ ∂η
∂
β ∂
∂
=
+
=
−
∂x
∂ξ ∂x ∂η ∂x
∂ξ
δ ∂η
2
and
∂ ∂ξ
∂ ∂η
1 ∂
∂
=
+
=( )
∂y
∂ξ ∂y ∂η ∂y
δ ∂η
The nabla operator then becomes
∇=[
β ∂
1 ∂
∂
−
]ex + ( ) ey
∂ξ
δ ∂η
δ ∂η
The normal vectors at the four element surfaces are, for l1 ,
n = −ex
For l3 ,
n = ex
For l2
n=
−βex + ey
∇η
=
|∇η|
γ 1/2
And for l4
βex − ey
γ 1/2
n=
where γ = 1 + β 2 .
To see how the required equations are obtained consider first the steady state case. An
energy balance on a finite volume is expressed as
4 Z
X
i=1 li
n · k∇T dl = 0
The four integrals above are
Z
I1 =
l1
Z
I3 =
Z
I2 =
l2
l3
kn · ∇T =
kn · ∇T =
kn · ∇T =
Z
l1
∂T
β ∂T
−
]dl1
δ ∂η
∂ξ
Z
l3
Z
l2
k[
k[
k[−
∂T
β ∂T
+
]dl2
δ ∂η
∂ξ
β ∂T
γ 1/2 ∂T
− 1/2
]dl3
δ ∂η
γ ∂ξ
3
and
Z
I4 =
l4
kn · ∇T =
Z
l4
k[−
β ∂T
γ 1/2 ∂T
+ 1/2
]dl4
δ ∂η
γ ∂ξ
Now, note that
dl1 = δdη
dl3 = δdη
dl2 = γ 1/2 dξ
dl4 = γ 1/2 dξ
Next, introduce the new variables
Ω=−
γ ∂T
δ ∂η
Γ = −δ
∂T
∂ξ
Λ=β
∂T
∂ξ
Ψ=β
∂T
∂η
and
With the above, the expressions the Ii ’s become
Z
I1 =
Z
I3 =
l3
Z
I2 =
l2
l1
k[Ψ + Γ]dη = k[Ψ + Γ]∆η
k[−Γ − Ψ]dη = −k[Ψ + Γ]∆η
k[−Ω − Λ]dξ = −k[Ω + Λ]∆ξ
4
and
Z
I4 =
l4
k[Ω + Λ]dξ = k[Ω + Λ]∆ξ
Using piecewise approximations for the derivatives yields
Z
L
kn · ∇T dl = [k(Ψ + Γ]|1 ∆η − [k(Ψ + Γ]|3 ∆η + [k(Ω + Λ]|4 ∆ξ − [k(Ω + Λ]|2 ∆ξ = 0
Define now the fluxes Ji , i = 1, 2, 3, 4 as
J1 = aW (TW − TP ) = k1
J2 = aN (TP − TN ) = k2
γ∆ξ/δ
(TP − TN )
∆η
J3 = aE (TP − TE ) = k3
J4 = aS (TS − TP ) = k4
δ∆η
(TW − TP )
∆ξ
δ∆η
(TP − TE )
∆ξ
γ∆ξ/δ
(TS − TP )
∆η
and the ”coordinate source” function S
S = ([kΨ∆η]3 − [kΨ∆η]1 ) − ([kΛ∆ξ]4 − [kΛ∆ξ]2 )
The heat equation thus becomes
J 1 − J3 + J4 − J2 = S
All is left now is to discretize S. This is done by using weighted piecewise approximations
for the derivatives, i.e.
[kΨ∆η]3 = [kβ
[kΨ∆η]1 = [kβ
∂T
TN E − TSE
TN − TS
∆η]3 = [kβ]3 [(
)fe + (
)(1 − fe )]∆η
∂η
ηN E − ηSE
ηN − ηS
∂T
TN W − TSW
TN − TS
∆η]1 = [kβ]1 [(
)fw + (
)(1 − fw )]∆η
∂η
ηN W − ηSW
ηN − ηS
[kΛ∆ξ]4 = [kβ
∂T
TSE − TSW
TE − TW
∆ξ]4 = [kβ]4 [(
)gs + (
)(1 − gs )]∆ξ
∂ξ
ξSE − ξSW
ξE − ξW
5
and
[kΛ∆ξ]2 = [kβ
∂T
TN E − TN W
TE − TW
∆ξ]2 = [kβ]2 [(
)gn + (
)(1 − gn )]∆ξ
∂ξ
ξN E − ξN W
ξE − ξW
and where
fe = fw = gs = gn =
1
2
if a uniform mesh is used and are adjusted accordingly if variable meshes are employed.
The transient term is easily incoporated into the above procedur and the final result is
the desired system of equations in the transformed domain.
A selected sample of results are shown in the attached figures.
6
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