Numerical Study of Impingement Location of Liquid
Jet Poured from a Tilting Ladle with Lip Spout
R. CASTILLA, P.J. GAMEZ- MONTERO, G. RAUSH, M. KHAMASHTA,
and E. CODINA
A new approach for simulating liquid poured from a tilting lip spout is presented, using neither
a dynamic mesh nor the moving solid solution method. In this case only the tilting ladle is
moving, so we propose to rotate the gravitational acceleration at an angular velocity prescribed
by a geometrical and dynamical calculation to keep the poured flow rate constant. This angular
velocity is applied to modify the orientation of the gravity vector in computational fluid
dynamics (CFD) simulations using the OpenFOAM
Ò
toolbox. Also, fictitious forces are
considered. The modified solver is used to calculate the impingement location for six spout
geometries and compare the jet dispersion there. This method could offer an inexpensive tool to
calculate optimal spout geometries to reduce sprue size in the metal casting industry.
DOI: 10.1007/s11663-017-0920-1
Ó The Author(s) 2017. This article is published with open access at Springerlink.com
I. INTRODUCTION
THE cast-iron foundry industry produces cast parts
for automotive, agriculture, transportation, energy,
aerospace, manufacturing industry, etc., with 90 pct of
manufactured products containing cast-iron parts. All
foundry processes generate a certain level of rejected
parts, which are irrecoverably lost, being closely related
to the type of casti ng and processes used and the
equipment available. As quality demands from end-
users of castings increase, it is essential that cast-iron
technology move forward, together with green manu-
facturing as a first step towards sustainability.
Extensive work has been carried out by the foundry
community to minimize casting defects such as porosity,
slag, and clogging, but few literature reports consider
casting metal using more versatile melt control
technology.
The flow control system in a tundish has a significant
impact on the quality and level of rejection when
pouring casts. If the metal does not flow in a consistent
stream, casting defects may result from oxidation, air
entrapment, and erosion of the pouring mold, among
other causes.
In the cast-iron industry, it is usual for molten metal
to be transfered from a ladle with a lip-axis pour
design.
[1]
This method is fast and reliable, but inaccurate
unless a sophisticated control system is used.
[2–4]
In
recent years, our group has been working on technolo-
gies that aim to transform the conventional
(batch-by-batch) foundry process into a flexible (mold-
by-mold) process. This requires that the ladle become a
furnace as well, that the tundish be eliminated from the
process, and that the molten metal be poured directly
into the mold. Moreover, it is convenient for the pouring
flow rate to be constant to obtain uniform refilling of the
mold. Such a flexible mold-by-mold process is charac-
terized by, firstly, combination of the melting, treat-
ment, and pouring processes into a single cast-iron
production cell. Also, this requires integration of an
artificial-intelligence-based control system to monitor
local structures, phases, and mechanical properties to
guarantee high-quality casting in the foundry. Finally, a
robot cell will be in charge of the metal finishing process.
This also reduces the melt temperature and transport
while improving validation of the cast pieces.
In recent decades, extensive effort has been invested in
development of CFD simulation methods for applica-
tion in the casting industry, especially for multiphase
flows.
[5–7]
In the particular case of sim ulations of pour
tilt casti ng, Prakash et al.
[8]
used the smoothed particle
hydrodynamics method to sim ulate the oxidation pro-
cess during furnace emptying, and Kuriyama et al.
[9]
used the software Flow 3D to optimize the pouring
velocity in aluminum tilt casting. Pauty et al.
[10]
carried
out two-dimensional (2D) numerical simulations of the
liquid transport and thermal convective velocity in a
tilting furnace, finding good agreement with water
experiments, while admitting that the extension of the
method to three-dimensional (3D) flow simulations of
molten metal would require a dramatic increase of
computational resources and time. Fortunately,
advances in computer technologies in recent decades
have made this kind of complex 3D fluid flow simulation
more affordable; For instance, Davila et al.
[11]
com-
puted the 3D flow during drainage of a ladle to
understand vortex mechanisms in this process.
R. CASTILLA, P. J. GAMEZ-MONTERO, G. RAUSH, and E.
CODINA are with the LABSON, Department of Fluid Mechanics,
Universitat Politecnica de Catalunya, 08222, Terrassa, Spain. Contact
e-mail: castilla@mf.upc.edu M. KHAMASHTA is with the LABSON,
Department of Mechanical Engineering, Universitat Politecnica de
Catalunya, 08222, Terrassa, Spain.
Manuscript submitted July 26, 2016.
Article published online February 7, 2017.
1390—VOLUME 48B, APRIL 2017 METALLURGICAL AND MATERIALS TRANSACTIONS B
We present herein a study of the impingement
location of liquid poured from a tilting ladle having a
lip spout with different curvatures. It has been reported
that a curved lip provides better jet control and reduces
air entrainment.
[12]
The purpose of this work is to
present a numerical method with low computer power
requirements to calculate the liquid trajector y in such a
pour tilt casting process. The results could be used to
reduce jet dispersion to minimize liquid spilling and
reduce sprue size.
Section
II develops the theoretical background to
calculate the angular velocity required to achieve
constant flow rate. The resul ts are the angular velocity
and tilting angle vs time. In the next stage, these data
are used in CFD simulations. The numerical approach
is described in Section III. The simulations were
performed using the OpenFOAM
Ò
toolbox, which is
based on the finite-volume method (FVM). As this is a
dynamic process with moving boundaries, the habitual
strategy is to use a dynamic mesh. Another option is
the FAVOR
Ò
technique.
[13,14]
These two methods are
very flexible and allow independent movement of
several objects in the domain. However, we propose
an alternative method where neither the mesh nor
object move, but the gravitational acceleration. This
method is simpler and faster than the dynamic mesh or
FAVOR
Ò
approach, but has the disadvantage that
only one moving object is allowed (in this case, the
tilting ladle) and the postprocessing is more compli-
cated. The results are presented in Section
IV; finally,
in Section
V, these results are discussed and conclu-
sions drawn.
II. THEORETICAL BAC KGROUND
The theoretical calculation of the flow rate poured
from the furnace is based on conservation of mass of
liquid. The flow rate of liquid is equal to the time
derivative of the mass inside the ladle, thus
@
@t
Z
VC
qdV ¼qq;
½1
where q is the flow rate through the spout. Since the
density q is constant , it can be eliminated from Eq. [
1].
Assuming that the free surface of the liquid always
remains horizontal, the volume of liquid can be
divided into two regions: the volume above a horizon-
tal plane passing through the lower point of the liquid
exit surface, V
t
, and the volume below this plane, V
b
.
The height of V
t
is h, so that
V
t
¼ hA
FS
; ½2
where A
FS
is the area of the free surface. Figure
1
depicts a scheme of the pouring liquid with arbitrary
tilting angle h.
Thus, Eq. [
1] yields
dV
t
dt
¼q
dV
b
dt
;
½3
which can be expressed in terms of the angle h instead
of time, thus
x
dV
t
dh
¼q þ x
dV
b
dh
;
½4
where x ¼dhdt is the angular tilting veloci ty of the
ladle. Using Eq. [
2]in[4] yields
A
FS
dh
dh
þ h
dA
FS
dh
¼
q
x
dV
b
dh
:
½5
The lower volume, V
b
, is known to be a cylindrical
wedge and can be calculated based on geometric
arguments if the spout lip is neglected (i.e., assuming
that the ladle is a perfect cylinder). If the free surface
does not cut the cylinder (ladle) bottom, the volume can
be easily calculated as
[15]
V
b
¼
1
2
pR
2
H þ H
s
ðÞ¼pR
2
H 1
R
H tan h
: ½6
If, otherwise, the plane cuts the cylinder bottom, the
calculation is more complex . In the particular case in
which the plane cuts the bottom circle across its
diameter, known then as a cylindrical hoof, the volume
θ
V
b
V
t
A
FS
h
q
H
H
s
Fig. 1—Scheme for pouring liquid for arbitrary tilting angle h (not
to scale).
θ
b
0
b
H
Fig. 2—Scheme of partially full inclined cylinder.
METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 48B, APRIL 2017—1391
is one-sixth the volume of the square prism in which the
cylinder is inscribed, as shown by Archimedes
[16]
V
b
¼
2
3
R
2
H:
½7
For the general case, the volume has to be calculated
by integration, giving
V
b
¼
R
3
tanh
pg
H
g
0
ðÞg
H
arccosg
H
þ
1
3
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1g
H
p
2þg
2
H
þg
0
arccosg
0
1
3
ffiffiffiffiffiffiffiffiffiffiffiffi
1g
0
p
2þg
2
0
Þ
;
½8
where
g
0
¼
b
0
R
1; ½9
g
H
¼g
0
þ
Htanh
R
;
½10
where b
0
is the height of the liquid on the opposite
side of the cylinder (see Figure
2). For our case of
pouring liquid, b
0
¼0 and Eq. [
8] reduces to
V
b
¼pHR
2
R
3
tanh
pþgarccosg
1
3
ffiffiffiffiffiffiffiffiffiffiffiffi
1g
2
p
2þg
2
;
½11
where g ¼HtanhR1. Alternatively, one can use the
following expression from Reference
15:
V
b
¼
1
3
HR
2
3sin/3/cos/sin
3
/
1cos/
;
½12
where
/¼
p
2
þarctan g
ffiffiffiffiffiffiffiffiffiffiffiffi
1g
2
p
:
:
Both equalities, i.e., Eqs. [
11] and [12], reduce to the
Archimedes theorem, Eq. [
7], in the case of the cylin-
drical hoof (i.e., g ¼0 in Eq. [
11]).
The area of the free surface is trivial when the plane
does not cut the bottom of the cylinder, being an ellipse
with area
Fig. 3—Theoretical estimation of dimensionless angular velocity vs
dimensionless time for constant flow rate.
Fig. 4—Theoretical estimation of rotation angle vs dimensionless time for constant flow rate.
Fig. 5—Original spout geometry (lengths in mm).
1392—VOLUME 48B, APRIL 2017 METALLURGICAL AND MATERIALS TRANSACTIONS B
A
FS
¼
pR
2
sin h
;
½13
but is more complicated when the plane cuts the bot-
tom. In this case, it is a segment of an ellipse, and,
according to Hugues et al.,
[17]
can be calculated as
A
FS
¼
R
2
sin h
w sin wðÞ;
½14
where
w ¼ 2 arccos gðÞ:
The flow rate in Eq. [
5] can be estimated using the
expression for the flow rate over a circular weir
[18]
q ¼ 0:3926C
d
ffiffiffiffiffi
2g
p
h
3=2
Db
1=2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 0:2200b
p
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 0:7730b
p
;
½15
Fig. 6—Geometries of simulated spouts (lengths in mm) with spout angle of (a) 0 deg, (b) 10 deg, (c) 20 deg, (d) 30 deg, and (e) 40 deg.
METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 48B, APRIL 2017—1393
where b ¼ h=D and C
d
is the discharge coefficient,
which is also a function of b. Nevertheless, a simplified
expression
[19]
is usually applied:
q ¼ KD
n
1
h
n
2
; ½16
where n
1
¼ 0:693, n
2
¼ 1:807, and K ¼ 1:598 when S.I.
units are used.
Given the flow rate poured, which is kept constant,
Eq. [
16] leads to the height of the free surface as
h ¼
q
KD
n
1
1
n
2
:
½17
Thus, the angular velocity as a function of angle can
be calculated from Eq. [
5], provided that the flow rate q
and the free surface height h are constant,
xhðÞ¼
q
h
dA
FS
dh
þ
dV
b
dh
;
½18
where the derivative of the free surface area is calcu-
lated from Eqs. [
13] and [14], and the derivative of the
bottom volume is obtained from Eqs. [
6]and[11].
Fig. 7—Slice of mesh for original geometry.
Fig. 8—Simulated pouring for t
¼ 0:3 and t
¼ 0:6 for original case (a, b) and simulation 1 with spout angle of 0 deg (c, d)).
1394—VOLUME 48B, APRIL 2017 METALLURGICAL AND MATERIALS TRANSACTIONS B